Rotodynamic Pumps

NEW AGE

ROTODYNAMIC PUMPS (Centrifugal and Axial)

Non-met allic Containment Gas

o

K.M. Srinivasan

(f.D

NEW AGE INTERNATIONAL PUBLISHERS


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K.M. Srinivasan B.E.(Hons), PhD.(USSR)

Dean (R&D) Mechanical Sciences Department of Mechanical Engineering Kumaraguru College of Technology Coimbatore, Tamil Nadu

PUBLISHING FOR ONE WORLD

NEW AGE INTERNATIONAL (P) LIMITED, PUBLISHERS New Delhi· Hangalorc • Chennai • Cochin • Guwahati • Hydcrabad .Ialandhar· Kolka!a • Lucknow • Mumbai • Ranch; Visit us at www.newagepublishers.corn

Copyright © 2008, New Age International (P) Ltd., Publishers Published by New Age International (P) Ltd., Publishers All rights reserved. No part of this ebook may be reproduced in any form, by photostat, microfilm, xerography, or any other means, or incorporated into any information retrieval system, electronic or mechanical, without the written permission of the publisher. All inquiries should be emailed to [email protected]

ISBN (13) : 978-81-224-2976-3

PUBLISHING FOR ONE WORLD

NEW AGE INTERNATIONAL (P) LIMITED, PUBLISHERS 4835/24, Ansari Road, Daryaganj, New Delhi - 110002 Visit us at www.newagepublishers.com

THIS BOOK is dedicated to My Parents Sri. K. MUTHUSAMY PILLAI And Smt. K.T. SAMBOORNAM As well as To my Professor and guide Dr Dr.. A.A. LOMAKIN And Dr APIR Dr.. A.N. P PAPIR Leningrad P olytechnic, Leningrad, K-21, USSR (at present called as St. Petersburg Polytechnic, Polytechnic, St. Petersburg, Russia) Who brought me to this level

Comp-1/Newage/Pump-co.pm6.5—29.12.07

3.1.08


PREFACE It was my very long felt ambition to provide a detailed and full information about the theory, design, testing, analysis and operation of different types of rotodynamic pumps namely Centrifugal, Radial, Diagonal and Axial flow types. I have learned a lot during the period 1959–62 about pumps at PSG College of Technology, Coimbatore, while working as Senior Research Assistant for CSIR Scheme on Pumps, Turbo chargers and flow meters. At the same time, I was undergoing training in foundry, pattern making, moulding, production, testing and design for different pumps at PSG Industrial Institute, Coimbatore and also during the period 1967 and 1975. I cannot forget my study at Leningrad Polytechnic, Leningrad K-21, USSR (now St. Petersburg Polytechnic, St. Petersburg, Russia), for my doctorate degree in pumps. Dr. A.A Lomakin, Dr. A.N. Papir, Dr. Gurioff, Dr. N.N. Kovaloff, Dr. A.N. Smirnoff, Dr. Staritski, Dr. Gorgidjanyan, Dr. Gutovski are the key professors who made me to know more about pumps from fundamentals to updated technology. I am very much grateful to Dr. A.A Lomakin and Dr. A.N. Papir, who were my professors and guides for my doctorate degree in pumps. As a consultant, for different pump industries in India and abroad, I could understand the field problems. My experience, since 1959 till date, has been put up in this book to enable the readers in industries, and in academic area, to design, to analyze and to regulate the pumps. Complete design process for pumps, losses and efficiency calculation, based on boundary layer theory for axial flow pumps are also given. Computer programmes for the design of pump and for profile loss estimation for axial flow pumps are also given. All the design examples in the last chapter are real working models. The results are also given with pump drawings. I do hope that the reader will be in a position to understand, design, test and analyze pumps, after going through this book. I shall be very much honoured if my book is useful in attaining this. I am grateful to my wife Smt. S. Nalini, my sons Sri S. Muthuraman and Sri S. Jaganmohan and my daughter Smt. S. Nithyakala, who were very helpful in preparing the manuscript and drawings. Last but not the least I am grateful to the editorial department of M/s New Age International (P) Ltd. Publishers for their untiring effort to publish the book in a neat and elegant form, in spite of so many problems they come across while formulating this book from the manuscript level to this level. Constructive criticisms and suggestions are highly appreciated for further improvement of the book.

K.M. SRINIVASAN


CONTENTS PREFACE

(vii)

1 INTRODUCTION

1–5

1.1 Principle and Classification of Pumps 1 1.1.1 Principle 1 1.1.2 Classification of Pumps 1

2 PUMP PARAMETERS

6–33

2.1 Basic Parameters of Pump 6 2.1.1 Quantity of Flow or Discharge (Q) of a Pump 6 2.1.2 Total Head or Head of a Pump (H) 6 2.1.3 Total Head of a Pump in a System 7 2.1.4 Power (N) 11 2.1.5 Efficiency (η) 11 2.2 Pump Construction 12 2.3 Losses in Pumps and Efficiency 15 2.3.1 Hydraulic Loss and Hydraulic Efficiency (ηh) 15 2.3.2 Volumetric Loss and Volumetric Efficiency (ηv) 15 2.3.3 Mechanical Loss and Mechanical Efficiency (ηm) 16 2.3.4 Total Losses and Overall Efficiency (h) 16 2.4 Suction Conditions 16 2.5 Similarity Laws in Pumps 19 2.5.1 Similarity Laws 19 2.5.2 Specific Speed (ns) 22 2.5.3 Unit Specific Speed (nsq) 23 2.6 Classification of Impeller Types According to Specific Speed (ns) 24 2.7 Pumping Liquids Other than Water 26 2.7.1 Total Head, Flow Rate, Efficiency and Power Determination for Pumps 26 2.7.2 Effect of Temperature 27 2.7.3 Density Correction (ρ or γ) 27 2.7.4 Viscosity Correction 28 2.7.5 Effect of Consistency on Pump Performance 32 2.7.6 Special Consideration in Pump Selection 33 (ix)

(x)

CONTENTS

3 THEORY OF ROTODYNAMIC PUMPS

34–64

3.1 Energy Equation using Moment of Momentum Equation for Fluid Flow through Impeller 34 3.2 Bernoulli’s Equation for the Flow through Impeller 35 3.3 Absolute Flow of Ideal Fluid Past the Flow Passages of Pump 38 3.4 Relative Flow of Ideal Fluid Past Impeller Blades 40 3.5 Flow Over an Airfoil 43 3.6 Two Dimensional Ideal Flow 45 3.6.1 Velocity Potential 45 3.6.2 Rotational and Irrotational Flow 45 3.6.3 Circulation and Vorticity 47 3.7 Axisymmetric Flow and Circulation in Impeller 48 3.7.1 Circulation in Impellers of Pump 49 3.7.2 Vorticity and Circulation Around Impeller Blades 49 3.8 Real Fluid Flow after Impeller Blade Outlet Edge 50 3.9 Secondary Flow between Blades 51 3.10 Flow of a Profile in a Cascade System—Theoretical Flow 52 3.11 Fundamental Theory of Flow Over Isolated Profile 53 3.12 Profile Construction as per N.E. Jowkovski and S.A. Chapligin 55 3.13 Development of Thin Plate by Conformal Transformation 58 3.14 Development of Profile with Thickness by Conformal Transformation 58 3.15 Chapligin’s Profile of Finite Thickness at Outlet Edge of the Profile 59 3.16 Velocity Distribution in Space between Volute Casing and Impeller Shroud 61 3.17 Pressure Distribution in the Space between Stationary Casing and Moving Impeller Shroud of Fluid Machine 63

4 THEORY AND CALCULATION OF BLADE SYSTEMS IN CENTRIFUGAL PUMP 4.1 4.2 4.3 4.4

4.5 4.6

Introduction 65 One Dimensional Theory 65 Velocity Triangles 66 Impeller Eye and Blade Inlet Edge Conditions 69 4.4.1 Inlet Velocity Triangle 70 4.4.2 Normal or Radial or Axial Entry of Fluid at Impeller Inlet 72 Outlet Velocity Triangle: Effect due to Blade Thickness 73 4.5.1 Outlet Velocity Triangle: Effect of Finite Number of Blades 74 Slip Factor as per Stodola and Meizel 75

65–129

(xi)

CONTENTS

4.7 4.8 4.9 4.10 4.11 4.12 4.13 4.14 4.15 4.16 4.17 4.18 4.19

4.6.1 Slip Factor as Defined by Karl Pfliderer 77 4.6.2 Slip Factor as per Proscura 79 Coefficient of Reaction (ρ) 81 Selection of Outlet Blade Angle (β2) and its Effect 83 Effect of Number of Vanes 86 Selection of Eye Diameter D0, Eye Velocity C0, Inlet Diameter of Impeller D1 and Inlet Meridional Velocity Cm1 89 Selection of Outlet Diameter of Impeller (D2) 92 Effect of Blade Breadth (B2) 92 Impeller Design 103 Determination of Shaft Diameter and Hub Diameter 106 Determination of Inlet Dimensions for Impeller 107 Determination of Outlet Dimensions of Impeller 108 Development of Flow Passage in Meridional Plane 109 Development of Single Curvature Blade—Radial Blades 111 Development of Double Curvature Blade System 113 4.19.1 Importance of Diagonal Impellers 113 4.19.2 A General Solution for the Flow through the Vane System 114 4.19.3 Axisymmetric Flow of Fluid 115 4.19.4 Flow Line and Vortex Line in Axisymmetric Flow 116 4.19.5 Differential Equation for the Cross-section of Vane with the Flow Surface 118 4.19.6 Construction of Vane Surface when Wu = 0 118 4.19.7 Construction of Vane Under Equal Velocity Construction 120 4.19.8 Construction of Vane Surface Under Equal Velocity Flow for the Given w(s) 121 4.19.9 Conformal Transformation of Vane Surface 125 4.19.10 The Method of Error Triangles 126

5 SPIRAL CASINGS (VOLUTE CASINGS) 5.1 5.2 5.3 5.4 5.5 5.6 5.7

130–146

Importance of Spiral Casings 130 Volute Casing at the Outlet of the Impeller 131 Method of Calculation for Spiral Casing 132 Design of Spiral Casing with Cur = Constant and Trapezoidal Cross-section 134 Calculation of Trapezoidal Volute Cross-section Under Constant Velocity of Flow CV = Constant (Constant Velocity Design) 135 Calculation of Circular Volute Section with Cur = Constant 137 Design of Circular Volute Cross-section with Constant Velocity (CV) 138

(xii)

CONTENTS

5.8 5.9 5.9 5.9 5.10 5.11 5.12 5.13 5.14

Calculation of Diffuser Section of Volute Casing 139 (A) Design of Diffuser 140 (B) Calculation of Spiral Part of Diffuser Passage 141 (C) Calculation of Diverging Cone Part of the Diffuser 142 Return Guide Vanes 143 Design of Suction Casing at Inlet of the Impeller 144 Straight Convergent Cone 144 Spiral Type Approach Ring 144 Effect due to Volute 146

6 LOSSES IN PUMPS 6.1 6.2 6.2 6.2 6.2 6.3 6.3 6.4

Introduction 147 (A) Mechanical Losses 147 (B) Losses due to Disc Friction (∆Nd ) 147 (C) Losses Stuffing Box (∆NS) 149 (D) Bearing Losses (∆NB) 154 (A) Leakage Flow through the Clearance between Stationary and Rotatory Wearing Rings 154 (B) Leakage Flow through the Clearance between Two Stages of a Multistage Pump 159 Hydraulic Losses 161

7 AXIAL AND RADIAL THRUSTS 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 7.10 7.11 7.12

164–181

Introduction 164 Axial Force Acting on the Impeller 165 Axial Thrust in Semi-open Impellers 167 Axial Thrust due to Direction Change in Bend at Inlet 168 Balancing of Axial Thrust 169 Axial Thrust taken by Bearings 170 Radial Vanes at Rear Shroud of the Impeller 170 Axial Thrust Balancing by Balancing Holes 171 Axial Thrust Balancing by Balance Drum and Disc 172 Radial Forces Acting on Volute Casing 177 Determination of Radial Forces 177 Methods to Balance the Radial Thrust 180

8 MODEL ANALYSIS 8.1

147–163

Introduction 182 8.1.1 Real Fluid Flow Pattern in Pumps 187

182–194

(xiii)

CONTENTS

8.2 8.3 8.4

Similarity of Hydraulic Efficiency 191 Similarity of Volumetric Efficiency 192 Similarity of Mechanical Efficiency 193

9 CAVITATION IN PUMPS 9.1 9.2 9.3 9.4 9.5 9.6

Suction Lift and Net Positive Suction Head (NPSH) 195 Cavitation Coefficient (s) Thoma’s Constant 200 Cavitation Specific Speed (C) 201 Cavitation Development 201 Cavitation Test on Pumps 203 Methods Adopted to Reduce Cavitation 211

10 AXIAL FLOW PUMP 10.1 10.2 10.3 10.4 10.5 10.6 10.7 10.8 10.9 10.9 10.10 10.11 10.12 10.13 10.14

195–215

216–292

Operating Principles and Construction 216 Flow Characteristics of Axial Flow Pump 218 Kutta-Jowkovski Theorem 218 Real Fluid Flow over a Blade 222 Interaction between Profiles in a Cascade System 223 Curved Plates in a Cascade System 224 Effect of Blade Thickness on Flow Over a Cascade System 233 Method of Calculation of Profile with Thickness in a Cascade System 234 (A) Pump Design by Direct Method (Jowkovski’s Method, Lift Method) 243 (B) Design of Axial Flow Pump as per Jowkovski’s Lift Method— Another Method 247 Flow with Angle of Attack 255 Correction in Profile Curvature due to the Change from Thin to Thick Profile 256 Effect of Viscosity 259 Selection of Impeller Diameter and Speed 260 Selection of Hub Ratio 261  l

10.15 Selection of   — Aspect Ratio at Periphery 263 t peri 10.16 Calculation of Hydraulic Losses and Hydraulic Efficiency 268 10.17 Calculation of Profile Losses using Boundary Layer Thickness (δ**) 271 10.17.1 Notations and Abbreviations 271 10.17.2 Determination of Profile Losses and Hydraulic Efficiency 274 10.17.3 Determination of Momentum Boundary Layer Thickness (δ**) 277 10.17.4 Computer Programme 283 10.18 Cavitation in Axial Flow Pumps 283

(xiv)

CONTENTS

10.19 Radial Clearance between Impeller and Impeller Casing 288 10.20 Calculation for Axial Flow Diffusers 289 10.21 Axial Thrust 291

11 TESTING, PERFORMANCE EVALUATION AND REGULATION OF PUMPS 293–338 11.1 Introduction 293 11.2 Pump Performance—Relation between Total Head and Quantity of Flow 293 11.3 Pump Testing 301 11.4 Systems and Arrangements 306 11.5 Combined Operation of Pumps and Systems 310 11.6 Stable and Unstable Operation in a System 312 11.7 Reverse Flow in Pump 315 11.8 Effect of Viscosity on Performance 317 11.9 Pump Regulation 232 11.10 Effect of the Pump Performance when Small Changes are made in Pump Parts 336

12 PUMP CONSTRUCTION AND APPLICATION 12.1 12.2 12.3 12.4 12.5 12.6 12.7 12.8 12.9

339–374

Classification 339 Pumps for Clear Cold Water and for Non-Corrosive Liquids 339 Other Pumps 346 Axial Flow Pumps 354 Condensate Pumps 357 Feed Water Pumps 361 Circulating Pumps 363 Booster Pumps 365 Pump for Viscous and Abrasive Liquids 370

13 DESIGN OF PUMP COMPONENTS Design No. D1-A : Design of a Single Stage Centrifugal Pump 375 Design No. D1-A1 : Computer Programming in C++ for Radial Type Centrifugal Pump Impeller and Volute 381 Design No. D1-B : Design of a Multistage Centrifugal Pump 395 Design No. D2 : Spiral Casing Design 409 D2-A : Spiral Casing Design Under Cur = Constant and Trapezoidal Cross-Section 411 D2-B : Spiral Casing Design with CV = Constant and Trapezoidal Cross-section 414 D2-C : Design of Suction Volute 417

375–486

(xv)

CONTENTS

Design No. D3 Design No. D4 Design No. D5 Design No. D6 Design No. D7 Design No. D8

: Design of Axial Flow Pump 418 : Correction for Profile Thickness by Increasing Blade Curvature (β) 427 : Calculation of Correction for Blade Thickness using Thickness Coefficient (χ) 429 : Design of Axial Flow Pump 431 : Profile Losses Calculation 473 : Design of Axial Flow Pump—as per method Suggested by Prof. N.E. Jowkovski 482

APPENDICES y Appendix I : Equations Relating Cy, max , δ° for Different Profiles l

Appendix II : ISI Standards Appendix III : Units of Measurement—Conversion Factors

487–508 487 495 502

LITERATURE—REFERENCES

509–518

INDEX

519–520


1 INTRODUCTION 1.1 PRINCIPLE AND CLASSIFICATION OF PUMPS 1.1.1 Principle Newton’s First law states that “Energy can neither be created nor be destroyed, but can be transformed from one form of energy to another form.” Different forms of energy exists namely, electrical, mechanical, fluid, hydraulic and pneumatic, pressure, potential, dynamic, wave, wind, geothermal, solar, chemical, etc. A machine is a contrivance, that converts one form of energy to another form. An electric motor converts electrical energy to mechanical energy. An internal combustion engine converts chemical energy to mechanical energy, etc. A pump is a machine which converts mechanical energy to fluid energy, the fluid being incompressible. This action is opposite to that in hydraulic turbines. Most predominant part of fluid energy in fluid machines are pressure, potential and kinetic energy. In order to do work, the pressure energy and potential energy must be converted to kinetic energy. In steam and gas turbines, the pressure energy of steam or gas is converted to kinetic energy in nozzle. In hydraulic turbine, the potential energy is converted to kinetic energy in nozzle. High velocity stream of fluid from turbine nozzle strikes a set of blades and makes the blades to move, thereby fluid energy is converted into mechanical energy. In pumps, however, this process is reversed, the movement of blade system moves the fluid, which is always in contact with blade thereby converting mechanical energy of blade system to kinetic energy. For perfect conversion, the moving blade should be in contact with the fluid at all places. In other words, the moving blade system should be completely immersed in fluid.

1.1.2 Classification of Pumps 1.1.2.1 Classification According to Operating Principle Pumps are classified in different ways. One classification is according to the type as positive displacement pumps and rotodynamic pumps. This classification is illustrated in Fig. 1.1. In positive placement pumps, fluid is pushed whenever pump runs. The fluid movement cannot be stopped, otherwise the unit will burst due to instantaneous pressure rise theoretically to infinity, practically exceeding the ultimate strength of the material of the pump, subsequently breaking the material. The motion may be rotary or reciprocating or combination of both.

2

ROTODYNAMIC PUMPS (CENTRIFUGAL AND AXIAL)

PU M PS PUMPS

PositiveDisplacement D isplacem ent PPumps um ps Positive

Reciprocating Type

Piston plunger

Rotary Type

O ther Pumps P um ps Other

R otodynam ic PPumps um ps Rotodynamic

Jet Pump Hydraulic Ram

Centrifugal, Mixed and Axial Flow Regenerative

Vane, Lobe Screw, Gear Perialistic, Metering, Diaphram, Radial piston, Axial piston

Fig. 1.1. Pump classfication

The principle of action, in all positive displacement pumps, is purely static. These pumps are also called as ‘static pumps’. The pumps, operated under this principle, are reciprocating, screw, ram, plunger, gear, lobe, perialistic, diaphram, radial piston, axial piston etc. In rotodynamic pumps, however, the energy is transferred by rotary motion and by dynamic action. The rotating blade system imparts a force on the fluid, which is in contact with the blade system at all points, thereby making the fluid to move i.e., transferring mechanical energy of the blade system to kinetic energy of the fluid. Unlike turbine, where pure pressure or potential energy is converted to kinetic energy, in pumps, the kinetic energy of the fluid is converted into either, pressure energy or potential energy or kinetic energy or the combination of any two or all the three forms depending upon the end use in spiral or volute casing, which follows the impeller. In domestic, circulating and in agricultural pumps, the end use is in the form of potential energy i.e., lifting water from low level to high level. In process pumps, used for chemical industries, the fluid is pumped from one chamber under pressure to another chamber under pressure. These chambers may be at the same level (only pressure energy conversion) or may be at different levels (pressure and potential energy conversion). Pumps used for fire fighting, for spraying pesticides, must deliver the liquid at very high velocity i.e., at very high kinetic energy. These pumps convert all available energy at the outlet of the impeller into very high kinetic energy. In turbines, the fluid is water or steam or chemical gas-air mixture at constant pressure and temperature, whereas, pumps deal with fluid at different temperatures and viscosities such as water, acids, alkaline, milk, distilled water, and also cryogenic fluids, like liquid hydrogen, liquid oxygen, liquid nitrogen, liquid ammonia, which are in gaseous form under normal temperatures. Pumps are also used to pump solid-liquid, liquid-gas or solid-liquid-gas mixtures, with different percentage of concentration called ‘consistency’. Hence pumps are applied in diversified field, the pumping fluid possessing different property, namely, viscosity, density, temperature, consistency, etc.

3

INTRODUCTION

A third category of pump, called jet pump, wherein, the fluid energy input i.e., high head low discharge of fluid is converted into another form of fluid energy i.e., low head and high discharge. These pumps are used either independently or along with centrifugal pumps. The reverse of Jet pump is ‘Hydraulic Ram’ wherein low head and high discharge of water is converted into high head and low discharge. Hydraulic Rams are installed at hills near a stream or river. The natural hill slope is the low head input energy. Large quantity of water at low head is taken from the river. A portion of water is pumped at high pressure and is supplied to a nearby village as drinking water. Remaining water is sent back to the river. This system does not need any prime mover like diesel or petrol engine or electric motor. Repair and maintenance is easy, in hydraulic ram since moving part is only the ram.

1.1.2.2 Classification According to Head and Discharge Another classification of pump is according to the head and discharge or quantity of flow to be pumped. Any customer, who is in need of a pump specifies only these two parameters. A quick selection of the pump is made referring standard charts for selecting the pump. Fig.1.2 gives the selection of pump according to head and discharge. 10000 H.m PISTON 1000

CENTRIFUGAL 100

10 AXIAL 1

10

100

1000

10000

100000 3 Q.m /hr

Fig. 1.2. Pump selection as per head and discharge

1.1.2.3 Classification According to Specific Speed Most accurate method of pump selection is based on the non-dimensional parameter called ‘specific speed’ which takes into account speed of the pump along with head and discharge. Specific speed,

ns = 3.65

n Q H 3/ 4

...(1.1)

where ns–specific speed, n–speed in rpm, Q–discharge in m3/sec, H–head in m. If pressure rise is known instead of total head then p = γH, where p–pressure rise of pumping fluid in N/m2 and γ–specific weight of the fluid at the given temperature in N/m3. It is essential that all parameters must be

4


converted to equivalent water parameters before substituting them in equation 1.1. Fig.1.3, illustrates the pump selection according to the specific speed of the pump. Centrifugal (radial flow) Low

Medium

High

Diagonal and mixed flow

Propeller and axial flow

ns = 50 ÷ 80

n s = 80 ÷ 150

ns = 150 ÷ 300

ns = 300 ÷ 500

n s = 500 ÷ 1000

b2

D2 = 1,8 to 1,4 D0

H–Q

H–Q

D2

D0

D2 = 2 to 1,8 D0

D0

D2

D0

D0

D0 D2 = 2,5 to 1,8 D0

b2

D2

D2

b2

D2

b2

D2 = 1,4 to 1,2 D0

D2 = 0,8 D0

H–Q

N– h

Q

–Q

Q N– h–

Q

N–

H–Q

Q

Q h–

h

Q N– –Q

H–Q N–Q Q h–

Fig. 1.3. Classification according to specifc speed

From Fig.1.3, it is evident that, at low specific speeds, centrifugal pumps; at medium specific speeds, mixed flow pumps and at high specific speeds, axial flow pumps are used. All of them are classified as rotodynamic pumps. At very low specific speeds, however, positive displacement pumps are used. Referring to the equation (1.1), it is seen that positive displacement pumps are used for very high head-very low discharge conditions. Ship propellers and aircraft propellers are of very high specific speed units beyond 1200 i.e., used for very low head-very high discharge conditions.

1.1.2.4 Classification According to Direction of Flow in Impeller Another classification of pumps is according to the direction of flow of fluid in impeller of the pump such as radial or centrifugal flow, mixed or diagonal flow and axial flow. Fig.1.4, illustrates the position of blade system in the impeller passage of a pump. Considering the flow of fluid in impeller, (Fig.1.4) if the flow direction is radial (2-1) and (3-1) i.e., perpendicular to the axis of rotation, the pump is called radial flow centrifugal pump. If the flow is axial (6-5) i.e., parallel to the axis of rotation, the pump is called axial flow pump. If the flow is partly axial and partly radial (4-2) and (4-3) i.e., diagonal, it is called mixed flow pump or diagonal flow pump. It is evident, from the Fig.1.4, that all these pumps are rotodynamic pumps i.e., rotary blade passage and dynamic action of blade system in the fluid passage.

5

INTRODUCTION

b 2′′ b2

Outlet, Delivery of water

a2

a2 III

D 2′

IV 6

(b) Mixed

1 2 3

Ds

a1

D1

(a) Radial

Inlet, entry of water

Ds

II

D3

D ′3′

a1

D2

I

(c) Axial

5 4 Shaft

90° axis

(d) Relative location

Fig. 1.4. Position of blade system in different types of impellers

2–1 Centrifugal — Radial flow — very high head and very low flow. 3–1 Centrifugal — Radial flow — high head and low flow. 4–2 Mixed flow — Medium head and medium flow — low range. 4–3 Diagonal flow — Medium head and medium flow — higher range. 6–5 Axial flow, propeller — low head and high flow. Radial type centrifugal pumps have higher impeller diameter ratio (outlet to inlet diameter) and the blade is longer. Mixed flow pumps have medium diameter ratio and axial flow pumps have equal inlet and outlet diameters. This indicates that radial flow pumps work mostly by centrifugal force and partly by dynamic force, whereas, in axial flow pumps, the pressure rise is purely by hydrodynamic action. In mixed and diagonal flow pumps, however, the pressure rise is partly by centrifugal force and partly by hydrodynamic force.

2 PUMP PARAMETERS 2.1 BASIC PARAMETERS OF PUMP A pump is characterised by three parameters i.e., 1. Total head (H), 2. Discharge or quantity of flow (Q), and 3. Power (N).

2.1.1 Quantity of Flow or Discharge (Q) of a Pump Quantity of flow or rate of flow or discharge (Q) of a pump is the flow of fluid passing through the pump in unit time. The rate of flow or discharge in volumetric system is expressed as

unit weight flow unit volume flow i.e., m3/sec, m3/hr, lit/sec etc., and in gravimetric system as i.e., unit time unit time tons/day, kg/hr, kg/sec etc. The relation between gravimetric or weight (W) and volumetric (Q) flow rate is given by W = γQ where γ is specific weight of the fluid.

2.1.2 Total Head or Head of a Pump (H) Total head of a pump (H) is defined as the increase in fluid energy received by every kilogram of the fluid passing through the pump. In other words, it is the energy difference per unit weight of the fluid between inlet and outlet of the pump. Referring to Fig. 2.1, the energy difference per unit weight of the fluid (E) between inlet (E1) and outlet (E2) will be

H

H =Z2 – Z1 Hd

Z2 p2 = pd Z2

Z1

V

G

– Hs

X2

+ Hs X1

Z1

p1 = ps

Fig. 2.1. Head measurement in pumps 6

7

PUMP PARAMETERS

Einlet

p1 C12 Z + + = E1 = 1 γ 2g

Eoutlet

p2 C22 Z + + = E2 = 2 γ 2g

    

...(2.1)

p — the pressure in N/m2 (Pascal–Pa) Z — the level or position above or below reference level in ‘m’ C — the flow velocity of the fluid in m/sec γ — specific weight of the fluid in kg/m3 (or) N/m3 g — acceleration due to gravity in m/sec2 Suffix 1 — indicates inlet condition of the pump 2 — indicates outlet condition of the pump Total head H will be

where

H = (E2 – E1) = and is expressed as

(C22 − C12 ) ( p2 − p1 ) + (Z2 – Z1) + γ 2g

...(2.2)

kgf.m N.m or = m. kgf N

2.1.3 Total Head of a Pump in a System A pump installation consists of pump and system. Pumps are selected to match the given condition of the system, which depends upon the system head (Hsy), quantity of flow (Q), density (ρ), the viscosity (µ), consistency (C), temperature (T), and corrosiveness of the pumping liquid. If the pumping liquid is other than water at different temperatures and pressures such as milk, distilled water, acid, alkaline solutions, as well as liquid ammonia, liquid oxygen, liquid hydrogen, liquid nitrogen or any other chemical solutions under higher temperatures and pressures, solid-liquid solution, liquid-gas solutions etc., the pump parameters in liquid must be changed into equivalent water parameters. The quantity (Q) and the total head (H) of the pump must coincide with the conditions of external system such as pressure, and location of the system. Normally the pump is selected with 2 to 4% higher value in total head than the normal value of system head. A system consists of pipelines with fittings such as gate valve or butterfly valve or non-return valve or any other valve along with bends, tee joints, reducers etc., at the delivery line of the pump as well as foot valve, strainer, bend, etc., at the suction line of the pump. The system is an already available pipeline in the field or at the working area, to suit the prevailing conditions in the field or working area. It is a fixed system for that particular place. System varies from place to place. Referring to the Fig. 2.2, the pipe 2-d refers to the delivery side and s–1 refers to the suction side of the system. For all calculations in a pumping system, the axis of the shaft of the horizontal pump is referred as reference line. For vertical pumps, the inlet edge of the blade of the impeller will be the reference line. Since the difference between the inlet edge of the blade and the centre line of the outlet edge of the blade is usually small, it is neglected and the centre line of the outlet edge of the blade is taken as reference line. Anything above or after the reference line is called delivery side (marked with suffix ‘d’) and anything below or before the reference line is called suction side (marked with suffix ‘s’) of a pump.

8


Referring to Fig. 2.2, the equation for suction and delivery pipelines of the system can be written as follows. Since no energy is added or subtracted in these lines during the flow through the system, For (2 – d) delivery line E2 = Ed + hf (2 – d) i.e.,

pd = p2 pd d

p C2 p2 C + Z 2 + 2 = d + Z d + d + h f (2 − d )  γ 2g γ 2g

For (s–1) suction line Es = E1 + hf (s–1)

p C2 i.e., s + Z s + s γ 2g

p1 C12 + + + h f ( s −1) Z = 1 γ 2g

hd

  ...(2.3)   H 

C2

2

X 1

 p2 C2   p C2  + Z 2 + 2  –  1 + Z1 + 1  Hp = E2 – E1 =  2g   γ 2g   γ

G Reference line

The values hf (2 – d) and hf (s – 1) include major frictional losses and all minor losses. The total head of the pump as per equation 2.2 is hs

C1 V

h fs

ps=p 1 S

= Ed +hf (2 –d) – Es + hf (s – 1)

 pd  p Cd2 C2 + + + h f (2 − d )  –  s + Z s + s − h f ( s −1) Z Hp =  d   γ 2g 2g  γ   =

hfd

  

Fig. 2.2. Pump in a closed system

 C 2 − Cs2  pd − ps + (Zd – Zs) +  d  + hf (2 – d) + hf (s – 1) γ  2g 

pd − ps = + hs + hd + hf (d) + hf (s) + γ

 Cd2 − Cs2    = H sy  2g 

...(2.4)

H H syst = f(Q) H

pd – p s γ

O

Operating point (H sy = H p)

H p = f(Q) + h s + hd Q

Fig 2.3. Head of pump and system

9

PUMP PARAMETERS

Equation 2.4 shows that, if a pump is connected to a system, the pump and the system will operate only at a point where Hp = Hsy. Fig. 2.3 shows graphically this condition.

C2 = KQ2,where K is the For both major and minor losses combined together hf = constant × 2g pd − ps sum of all constants (major and minor). The system head Hsy= + hs + hd + (Kd + Ks) Q2. If a γ curve Hsyst= f (Q) is drawn, it will be a parabola moving upwards, i.e., increase of head when the flow Q increases. (Fig. 2.3). If this curve is superimposed with H–Q curve of the pump, the meeting point will be (Hp = Hsyst) the operating point of the pump for that system. Different Hsy curves can be drawn by changing hs or hd or pd or ps as well as by changing pipe size Dp, pipe length lp, in suction and delivery, or by adding or removing or changing bends. Tee, crossjoints or by changing the valves in the system. Change of every individual parts mentioned above changes the Hsyst–Q curve. If these curves are superimposed on pump H–Q curve, the operating point for each system can be determined (Fig. 2.4).

H st = hs + h d +

pd – p s γ

Head m.

H

P1, P2, P3,P4 Operating points Hsyst 4 – Q H syst 3 – Q

P4 P3

H syst 2 – Q

P2 P1

Q4 Q3 Q2 Q1

H syst 1 – Q

(H p – Q) Q

Quantity m3/sec, Lt/sec.

Fig. 2.4. Different systems operating on one pump

Referring to equation 2.4, if suction and delivery chamber pressures are very high, when compared to the potential and kinetic energies, then the pump is called process pump. If the suction and delivery chambers are open type, then pd = ps = patm and if hd, hs are very high, then these pumps are called domestic or agricultural or circulating pumps. If velocity C2 is very large, when compared to other parameters and pd = ps = patm and hs and hd may be positive or zero, then these pumps are called fire fighting pumps, sprayer pumps. Rearranging equation 2.2

pd p2 Cd2 − C22 = + (Z – Z ) + + hf (2 – d) d 2 γ γ 2g

10


pd Cd2 − C22 + hd + hfd + ...(2.5) γ 2g If a pressure gauge is connected very close to the delivery side of the pump at point 2, it will read =

p  the delivery chamber pressure  d  , static delivery height (hd) delivery line frictional losses (hf) (both  γ  major and minor losses) and the difference between the velocity head or kinetic energy at delivery  C2  chamber  d  and immediately after the delivery of liquid from pump i.e., at the outlet of the volute  2g 

 C2  p casing  2  . If the delivery chamber is a closed one, then d will be real and normally above γ  2g  atmosphere

Cd2 will be equal to zero. The pressure gauge P2 will read 2g pd p2 C2 = + hd + hfd – 2 γ γ 2g

...(2.6)

C22 where C2 is the velocity at the delivery pipe, and will be the kinetic energy at the delivery pipe. 2g patm pd Cd2 In case the delivery chamber is open to atmosphere then γ = γ and will be real. The 2g velocity Cd = C2 and the velocity head at the delivery pipe is

Cd2 − C22 = 0. The pressure gauge (P2) 2g

will read

p2 = hd + hfd (gauge pressure) γ p = atm + hd + hfd (absolute pressure) ...(2.7) γ If a pressure gauge is connected at the end of suction pipe and very near to the pump inlet at point 1, it will read p1 ps = + (Zs – Z1) + γ γ

 Cs2 − C12   2 g  – hf (s – 1)  

Cs2 − C12 ps – h – h + = s fs 2g γ

...(2.8)

Cs2 ps If the suction chamber is closed, γ will be read and Cs = 0, = 0. Then 2g  p1 C12  ps h + h + = –  s  fs γ 2 g  γ  where C1 is the fluid velocity at suction pipe.

...(2.9)

11

PUMP PARAMETERS

patm ps p = . The pressure 1 will be γ γ γ negative i.e., under vacuum. A vacuum gauge (V) instead of pressure gauge P1 must be connected at point 1. The velocity Cs = 0 and so If the suction chamber is open to atmosphere then

 C12  p1 pat h + h +  s  fs 2 g  absolute γ = γ –   C12  h + h +   s fs =  2 g  vacuum 

or

...(2.10)

ps , the suction chamber γ pressure is not sufficiently higher than the vacuum in the suction side of the pump. In this case also only vacuum gauge must be connected at point 1. That’s why if the suction chamber is closed, a pressure cum vacuum gauge and if suction chamber is open to atmosphere a vacuum gauge is connected at point 1 i.e., at the end of suction pipe or immediately before the inlet of the pump. Since total head of the pump (Hp) = Total head of the system (Hsyst) Vacuum gauge will read only vacuum. The same condition will exist if

 Cd2

Hp = Hsyst = P2 + V + X + 

−

Cs2   2 g  for open system

 2g = P2 – P1 + X for closed system

...(2.11) where X is the difference in height between delivery pressure gauge (P2) and suction gauge (P1 or V). If P2 is at a higher level than P1, X is positive. If P2 is at a lower level than P1 then X1 is negative. If P2 and P1 are at the same level X = 0.

2.1.4 Power (N) Power is defined as the amount of energy spent to increase the energy of the fluid passing kgf.m N.m or or watts or sec sec kilowatts. If ‘W’ is the weight of fluid passing through the pump and the energy increase per unit weight of the fluid between inlet and outlet of the pump is ‘H’, power N will be

through the pump from inlet to outlet of the pump and is expressed in

WH γ QH = in kW or watts. constant constant where W = γ Q, if W is expressed in kgf, the constant will be 102, and if expressed in Newton the constant will be 1000 in order to get the power in kW.

N0 =

η) 2.1.5 Efficiency (η The power supplied to the pump will be higher than the energy spent in converting mechanical energy to fluid energy due to various losses, namely, hydraulic, volumetric and mechanical losses. The ratio of actual power utilized to the power supplied is called efficiency (η).

12


 γQH  power spent  N 0 =  γ QH const (C )   η = = C .N power supplied( Nth ) th or

Nth =

N0 γ QH = η C.η

...(2.12)

2.2 PUMP CONSTRUCTION Any pump consists of an impeller having specified number of curved blades called vanes, kept in between two shrouds. The impeller is the rotating element responsible for the conversion of mechanical energy into fluid energy. This impeller is connected, through a shaft and coupled, to the prime mover for rotation. The connection may be a direct drive or indirect drive, through belt or gear system. The shaft is supported by one or two fixed bearing supports depending upon the pump duty and one floating sleeve bearing support along with either mechanical seal or asbestos packed stuffing box. This floating support is arranged to take care of liner thermal expansion of shaft, towards the impeller side but not at the prime mover side and at the same time acting as load bearing unit. The mechanical seal material or the packing material is selected according to the type of pumping liquid such as acidic, alkaline, neutral, milk, distilled water, cryogenic liquids like ammonia, hydrogen, oxygen, nitrogen, two phase fluids such as solid-liquid, liquid-gas etc. A gland provided in the stuffing box keeps the packing material or seal in position. The impeller is rotated inside a sealed spiral casing or volute casing. Suction and delivery pipes are connected to the suction side and delivery side of the spiral casing through respective flanges. Since volute casing is a non rotating part and impeller is a rotating element, sufficient clearance should be provided between them. The fluid enters the suction side of the impeller, called eye of the impeller with low energy. Due to conversion of mechanical to fluid energy, the fluid leaving the impeller will be with higher energy, mostly with more kinetic energy. Due to the energy difference between inlet and outlet of impeller and due to the clearance between volute casing and impeller, a part of fluid flows from impeller outlet to the eye of the impeller at the suction side and towards the stuffing box side at the back. In order to control this leakage flow, wearing rings, at the casing and at the impeller at front and back side are provided. The amount of clearance and different forms of wearing rings used depends upon the pumping fluid (temperature, consistency etc.). The mechanical seal and the packing in stuffing box reduces this leakage still further at the rear side. The volute casing and the impeller with shaft are fitted to the bracket which has the bearings to support the shaft. This bracket base is mounted in a common base plate, which has the provision to mount the prime mover. The pump and prime mover will be kept on a common base plate. In Figs. 2.5, 2.6 and 2.7, three types of pump assemblies are given for single suction pumps. However, the construction differs for double suction pumps and multi stage pumps.

13

PUMP PARAMETERS

2 11

12 7 16 6

10 14 9

4

15

15 3 5 1

8

13

1. 2. 3. 4.

5. 6. 7. 8.

Suction flange Delivery flange Impeller Volute casings

13. 14. 15. 16.

9. Flexible coupling (pump side) 10. Flexible coupling (motor tside) 11. Gland 12. Bearing cap

Bearing bed Shaft Deep groove ball bearing Bush

Impeller nut Coupling nut Air cock Grease cup

Fig. 2.5. Single bearing supported pump with split type volute casing 2 1

2

18

3

15

14 24 10

21

6 22

20

29

36 40

27

8 28 32

33 7

26

11

5 44 19

42 17

12

43

31 26

1. Spiral casing 2. Intermediate casing 3. Cooling room cover 4. Supporting foot 5. Pump shaft 6. Left-hand impeller 7. Radial ball bearing 8. Radial roller bearing (only for bearing bracket)

39

30 9 38 13 18 37 34 41 40 25

16 35 9. Bearing bracket 10. Bearing bracket intermediate 11. Bearing cover 12. Flat seal 13. Flat seal 14. Flat seal 15. Flat seal 16. Flat seal

17. Flat seal 18. Seal ring 19. Radial seal ring 20. Gland 21. Stuffing box ring 22. Bottom ring 23. Block ring 24. Stuffing box 25. Splash ring

4

26. Wearing ring 27. Shaft sleeve 28. Disk 29. Pin 30. Oil level regular 31. Hexagon screw 32. Hexagon screw 33. Stud bolt 34. Stud bolt

35. Stud bolt 36. Stud bolt 37. Locking screw 38. Threaded pin 39. Inner hexagon screw 40. Nut 41. Nut 42. Impeller nut 43. Fitting key 44. Fitting key

Fig. 2.6. Back pullout-double bearing type pump with combine volute casing

14


6

1 7 2 8 3 5

10

4

9

Fig. 2.7. Heavy duty pump

Basically pump construction consists of three sub-assemblies namely (1) shaft assembly (2) casing assembly and (3) base assembly or bracket assembly. Shaft assembly, consists of impeller, impeller key, impeller nut, shaft, bushes at stuffing box, bearing inner races, pump coupling, key, and coupling nut, all mounted on a common shaft. The shaft is connected to the prime mover either through belt drive, or direct. This assembly is the only rotating assembly and hence this assembly must be perfectly balanced. But, all components in this assembly are machined components except impeller, viz., inside surface of shrouds and the blade surfaces. These surfaces are normally rough cast surfaces and could not be machined. Hence impeller only is balanced and assembled on the shaft. Casing assembly consists of suction side or front side bracket, rear side or coupling side bracket of the volute casing. However, volute casing construction changes depending upon the pumping fluids. For pumping high consistency liquid, two phase fluids, suction side bracket, coupling side bracket and volute casing are made up of three separate pieces (Fig. 2.7). For ordinary pumping liquids like water, milk, etc. suction side bracket and volute casing are single unit (Fig. 2.6). In agricultural pumps, casing is made into two halves (Fig.2.5). Suction side bracket and one half of the casing become one part. Coupling side bracket and other half the casing become another part. Coupling side bracket will also have stuffing box or mechanical seal chamber. For higher capacity pumps, the base assembly or bracket assembly consists of a bracket with provisions for assembling front and rear bearings, and bearing caps. In agricultural pumps (Fig. 2.5), however, the stuffing box and gland at the front side of the bracket and bearing chamber and bearing cap on the other side of the bracket will be the normal construction. In low capacity pumps, the bracket is fitted on a base plate along with the prime mover. The casing will be connected to the bracket. In such pumps, the entire weight of delivery pipe with fluid, the suction pipe with fluid and all minor fitting like valve, bend etc. will be connected to the casing delivery side and suction side respectively as a overhung unit. In higher and medium capacity pumps, pumps with heavy liquids, two phase fluids will have the base at the casing which is connected to the common base plate. Such assemblies are called ‘back pull out’ assembly (Fig 2.6). This assembly is a convenient assembly, where in all parts, except casing can be removed by pulling the entire assembly backwards for any repair and maintenance. The pipe system need not be disturbed. However, the prime mover has to be removed from base plate, in order to remove the pump assembly parts.

15

PUMP PARAMETERS

2.3 LOSSES IN PUMPS AND EFFICIENCY Theoretically, all the energy supplied to the pump by the prime mover, in the form of mechanical energy, should be converted into fluid energy. Owing to manufacturing inaccuracies and entirely different flow conditions prevailing in pump, entire energy input (mechanical energy) is not converted into fluid energy. Referring to Figs. 2.5, 2.6 and 2.7, 100% mechanical energy supplied at the coupling side of the pump by the prime mover is reduced, due to energy absorption in bearings, stuffing box, disc friction. Hence, the energy input at the impeller will be less than the energy input at the pump coupling. Due to surface roughness inside impeller and due to the leakage flow through clearance, there will be further reduction in the energy input to the impeller. Hence, the energy output from the pump is less than the energy input to the pump. The difference between energy input and energy output of the pump is called losses in pump. The ratio of energy usefully utilized for work to the energy supplied is called efficiency. In other words, efficiency is the ratio of output energy to the input energy of the machine in doing work. Three kinds of losses prevail in fluid machines namely, (1) Hydraulic loss (2) Volumetric loss and (3) Mechanical loss. The sum of all losses will be the total loss. Overall efficiency is the product of hydraulic efficiency, volumetric efficiency and mechanical efficiency.

η h) 2.3.1 Hydraulic Loss and Hydraulic Efficiency (η Due to surface roughness at the inner side of the impeller, through which the fluid passes, losses due to friction and losses due to secondary flow, take place, as a result of which energy loss take place. Actual head developed (Ha) will be less than the theoretical head (Hth) by the amount ∆H = Hth – Ha. ∆H is called the hydraulic loss. Hydraulic efficiency (ηh) is the ratio between, actual head to the theoretical head. Hydraulic loss, ∆H = Hth – Ha  Ha Hth − ∆H Ha ∆H  Hydraulic efficiency, ηh = = = = 1– ...(2.13) Hth Hth Ha + Hth Hth    ∆H = (1–ηh) Hth

η v) 2.3.2 Volumetric Loss and Volumetric Efficiency (η In order that the impeller can rotate inside the stationary casing, proper clearance is provided at the front and rear side of the impeller at wearing rings. Due to pressure difference between impeller outlet and impeller inlet at the front side of the impeller as well as the pressure difference between impeller outlet and slightly higher than atmospheric pressure at the stuffing box, part of fluid coming out of the impeller leaks through the clearances on both sides of the impeller. As a result the quantity coming out of the pump, the actual quantity (Qa) will be less than the quantity passing through the impeller, i.e., theoretical quantity (Qth) by the amount of leakage quantity passing through the clearances (∆Q), i.e., ∆Q = Qth – Qa. Volumetric efficiency (ηv) is the ratio between actual quantity and theoretical quantity ∆Q = Qth – Qa

 Q Qa Qth − ∆Q ∆Q  a ηv = = = =1–  Qa + ∆Q Qth Qth Qth   ∆Q = (1–ηv) Qth

...(2.14)

16


η m) 2.3.3 Mechanical Loss and Mechanical Efficiency (η Energy loss in ball, roller or thrust bearings (∆NB), in bush bearings at stuffing box or in mechanical seal portion (∆Ns), and the disc friction losses (∆ND ) due to the impeller rotation inside the volute casing, which is filled with fluid are classified as mechanical losses (∆N ). The energy received at the impeller side of the shaft, i.e., actual power (Ni) for energy conversion into fluid energy will be less than the energy supplied at the coupling side by the prime mover, i.e., theoretical power (Nth), i.e., ∆N = Nth – Ni. The ratio between actual power (Ni) and the theoretical power (Nth) is the mechanical efficiency (ηm) ∆N = ∆ND + ∆NB + ∆Ns ∆N = Nth – Ni

i.e.,

ηm =

N th – ∆N Ni Ni ∆N = = =1– N th N th N i + ∆N N th

∆N = (1 – ηm) . Nth

        

...(2.15)

2.3.4 Total Losses and Overall Efficiency (h) Total losses = Hydraulic loss + Volumetric loss + Mechanical loss = ∆H + ∆Q + ∆N. Since

Qa ηv = Q , output energy (N0) = γ Qa.Ha = γQth.ηv . Hth .ηh th

Taking Ni = γ Qth Hth where Ni = power available at the impeller end of the shaft, Ni = Nth – ∆N. Therefore,

N0 = Ni ηv ηh = Nth ηm . ηv . ηh. Since ηm =

Overall efficiency, η =

N0 = ηm . ηv . ηh N th

Ni N th ...(2.16)

2.4 SUCTION CONDITIONS Normal and dependable operation of a pump depends mostly on suction conditions of the pump i.e., pressure at the inlet edge of the impeller blade (Fig. 2.8). Referring to the equations (2.8) and (2.9), the pressure p1 at the impeller inlet is less than the pressure at the suction chamber ps. If the suction chamber pressure ps is low or if the suction chamber is open to atmosphere i.e., ps = patm, the pressure at point 1, the inlet edge of the blade of the impeller will be under vacuum (Equation 2.10). If this pressure, p1 is lower than the local vapour pressure of the pumping fluid, corresponding to the temperature of the liquid at impeller eye (pvp), then the liquid at this point will be boiling. In other words, liquid will not be in liquid form, instead it will be in gaseous form and pumping cannot be done. Hence, the pressure at the inlet of the impeller, i.e., at the eye of the impeller, must be above vapour pressure of the flowing fluid corresponding the temperature of the fluid.

17

PUMP PARAMETERS

∆h 2 Cs 2 = C 0 2g 2g Xs D1 2

D0 2

C0

B

Radial flow

Axial flow

Hs

A

Fig. 2.8. Suction conditions in a pump

or If

pvp  p1 Cs2  ps  > = –  hs + h fs + γ γ 2g  γ   ps pvp    p1 pvp  Cs2  ...(2.17)  γ − γ  =  γ − γ  –  hs + h fs + 2 g  > 0   ps patm = i.e., if the suction chamber is open to atmosphere, then γ γ

  p1 pvp   patm pvp  Cs2  − − + + h h > 0 =  –  s fs  γ γ  γ  2 g   γ  must be greater than zero or in other words, always it should be positive i.e.,   patm pvp  Cs2   γ − γ  >  hs + h fs + 2 g     patm pvp   p1 − p vp  − is termed as H and is called Net Positive Suction Head (NPSH).   is called  sv γ  γ    γ NPSH available. The two terms patm and pvp cannot be altered, since these values patm, the atmospheric pressure at the place where pump is running and pvp is the vapour pressure, which depends upon the

 C2  temperature of the pumping liquid, are fixed values. The term  hs + h fs + s  is called NPSH required 2g   which is depending upon, the pump, viz., flow rate, pipe length and size, and the level of suction chamber with respect to the reference line of the pump. All these can be altered during pump erection at site.

18


Hence NPSH (Net) (Hsv) = NPSH (available) – NPSH (required)  patm pvp   C2  − Hsv =  ...(2.18)  –  hs + h fs + s  γ   2g   γ  C2  ( H atm − H vp ) −  hs + h fs + s  2g  H sv  σ = = ...(2.19) H H where σ is called Thoma’s constant. All pump manufactures give this value i.e., Hsv or σ by conducting test on water in the laboratory. Depending upon the site conditions, pump erection is carried out so that pump can work without cavitation. In order to have a safe operation, a reserve in the NPSH is introduced and suction lift or suction head is calculated accordingly.

 C2  KHsv = (Hatm – Hvp) –  hs + h fs + s  2g   Normal values of K will be 1.15 to 1.40. Therefore, hs will be  C2  hs = (Hatm – Hvp) –  h fs + s  – KHsv 2g   In case the pumping liquid is other than water Hsv (L) =

H sv ( w )

=

...(2.20)

...(2.21)

γ w H svw γL

...(2.22)

SL where SL is the specific gravity of the liquid γL and γw are the specific weights of liquid and water respectively. 11 10 9

8

8

Additional Suction Head in Metres

Vapour Pressure ion Metres of Water Column

12

7 6 5 4 3 2 1 0 10

20 30

40

50

60 70 80

Water Temperature °C (a)

90 100 110

7 6 5 4 3 2 1 0 100 125 150 175 200 Water Temperature °C

(b)

Fig. 2.9. Vapour pressure of water at different temperatures

225

19

PUMP PARAMETERS

2.5 SIMILARITY LAWS IN PUMPS 2.5.1 Similarity Laws A complete study of fluid flow and the flow pattern in impeller, in casing and in various other elements of pump by theoretical means could not be achieved. Thats why, experimental coefficients are used along with the theoretical equations to solve the problems in pumps. These experimental coefficients are obtained by conducting experiments on different pumps and obtaining results with the help of similarity laws and dimensional analysis. Similarity and dimensional analysis is a process of obtaining the property and characteristics of another similar pump from the available property and characteristics of a pump on which experiment was carried out and the results known. A functional relationship between different parameters of the pump tested and the pump for which the calculations are needed is established by this law. Using dimensional analysis under geometrical similarity, different expressions, connecting pump head (H), quantity (Q), power (N) and speed (n) with the impeller diameter (D), which is the standard reference linear dimension for a pump, and the properties of fluid, such as density (ρ), viscosity (µ) and gravitational acceleration (g) can be established. The following Table 2.1 gives the dimensions and units of different parameters used for non-dimensional analysis. TABLE 2.1: Units and dimensions Parameter

Dimensions

Symbol

1.

Head

H

metre (m)

L

2.

Quantity

Q

m3/second (sec)

L3/t

Newton . m sec

ML2

3.

Power

N

4.

Speed

n

5. 6.

Diameter Gravitational acceleration

D

7.

Density

g ρ

8.

Viscosity

µ

1 sec m

t2 1 t L

m/sec2

L/t2

kg/m3

M/L3

kg m sec

M Lt

As per the laws of dimensional analysis, there are 8 parameters with 3 dimensions. Hence, (8 – 3) = 5 non-dimensional parameters can be evolved. After solving, we get the following nondimensional parameters. (1)

µ  ρVL  which is Reynold’s number  Re =  ρ n D2 µ  

(2)

Q V   which is Struhaul’s number  Sh =  called unit discharge KQ in fluid machines nL nD 3  

20


(3)

N called unit power (KN) ρn3 D 5

 V2  F = which is Froude number   r  gl  n2 D  H (5) D

(4)

g

Multiplying non-dimensional parameters (4) and (5), we get another non-dimensional number H gH . However, since g is a constant, 2 2 is used, in practice which is called unit head (KH) in fluid 2 2 n D n D machines. Based on the above non-dimensional parameters, a functional relationship between unit power (KN) and the unit discharge (KQ) i.e., KN = f (KQ) as well as unit head (KH) and unit discharge (KQ). viz., KH = f (KQ) can be established. Ni 3

ρn D

5

 µ  Q  Q  , R ,  = f 2 3  = f  e  n D3   ρnD nD  

...(2.23)

 Q   ...(2.24) Ni = ρn3D5 f  Re , n D3   where, Ni (internal power) or the power input at the impeller unit i.e., the power input at the coupling side minus mechanical losses in bearings, stuffing box, and disc friction. Also

gH = f n2 D2

F µ GH ρ n D

2

,

Q nD 3

I JK

=f

FG R , Q IJ H nD K e

3

...(2.25)

Q  n2 D2  f  Re , or H = ...(2.26)  nD3  g  Equations (2.24) and (2.26) give the relation between the internal power (Ni) and head (H) with Reynold’s number and unit discharge (KQ). The effect of Reynold’s number is not considered, since the tests are conducted in auto model region i.e., at high Reynold’s number (Re > 105), where the coefficient of friction ‘f ’ remains constant and is independent Reynold’s number (Re). This value H will be approximate, since effect due to frictional losses is not considered. Considering two identical pumps viz., prototype (suffix p) and model (suffix m) i.e., pumps of the same series which are geometrically similar, i.e., linear dimensions are proportional and kinematically similar, i.e., flow directions are same within the impeller and in casing, i.e., blade angles are same, velocity triangles are identical.

For Head or

gH p n 2p

D p2

Hp n 2p

D p2

Hp Hm

= =

=

gH m nm2 Dm2 Hm nm2

...(2.27)

Dm2

gn 2p D p2 gnm2

Dm2

=

K2

FG n IJ Hn K p

m

2

where

 Dp  K=    Dm 

...(2.28)

21

PUMP PARAMETERS

Qp

For Quantity

n p D 3p Qp

or

Qm Np

For Power

ρ p n3p D5p Np

or

Nm

=

=

=

Qm nm Dm3 n p D3p

= K3

nm Dm3

FG n IJ Hn K p

...(2.29)

m

Nm

ρm nm3 Dm5 ρ p n3p D5p

=

ρm nm3 Dm5

=

K5

FG n Hn

IJ FG ρ IJ K Hρ K 3

p

m

p

...(2.30)

m

If the pumping liquid is same for both prototype and for model ρp = ρm, then

Np

K5

FG n IJ Hn K

3

p

= ...(2.31) Nm m Equations (2.28), (2.29) and (2.30) are called similarity equations for pumps, and include the scale µ and relative Roughness effect effect, i.e., include change in the effect of Reynold’s number Re = ρnD 2 ε   . D However, exact values, which include the change in the corresponding efficiencies between prototype and model, are given below :

Qp Qm Hp Hm N ip

The value

FG η IJ Hη K vp

N im

FG n IJ FG η IJ Hn K Hη K F n I  η  =K G J  Hn K η  Fn I Fρ I  η =K G J G J  Hn K Hρ K η = K3

p

vp

m

vm

2

2

p

hp

m

hm

3

5

p

p

mp

m

m

mm

  

U| || | V| || || W

...(2.32)

takes into account the change in volumetric efficiency connected with the

vm

change in the relative values of wearing clearances, balancing holes and usually connected with the  ηhp  change in scale K. The value  is the change in hydraulic efficiency which is a function of  ηhm  ηmp Reynold’s number and scale K. The value is the change in the relative values of mechanical ηmm losses in bearings, stuffing box and for disc friction. The equations developed under similarity laws for pumps are most important for test result analysis and widely used in pump industries, to analyse the

FG H

IJ K

22


performance of model tested in the laboratory, with the test results obtained from the prototype, tested in industries such as test at different speeds, test at different diameters, tests on liquids other than water etc., and also to develop new pumps.

2.5.2 Specific Speed (ns) Specific speed (ns) is defined as the speed of a geometrically similar pump which consumes 1 (metric) hp and develops 1 m of total head, the pumping liquid being water under normal temperature of 4°C and at atmospheric pressure of 1.0336 kgf/cm2, and γ = 1000 kgf/m3 viscosity µ = 1 centipoise or ν = 1 centistoke i.e., n = ns, when N = 1 hp and H = 1 m. Since,

γQH . 75 γ = 1000 kgf/m3

N (hp) =

Substituting the values

N (hp) = 1 hp, H = 1 m

1 × 75 = 0.075 m3/sec. 1000 × 1 Referring to equation for unit power, KN and substituting the values. N 1 = ρn 3 D 5 ρns3 Ds5 Q =

N =

ρn3 D 5 ρns3 Ds5

=

FG n IJ Hn K

K5

3

...(2.33)

s

gH g .1 2 2 = n D ns2 Ds2

F nI F DI H = G J .G J Hn K HD K 2

or

s

2

=

s

K2

FG n IJ Hn K

2

...(2.34)

s

Combining equations 2.33 and 2.34

FG n IJ and H = K FG n IJ = Hn K Hn K F n I or n = n N = G J Hn K H 6

2

N

K10

5

s

H5 N2

ns = Since

4

4 s

2

5

n N

...(2.35)

H5 4 γ QH N = 75

 ns =   Since

s

4

s

λ = 75

γ = 1000 kgf/m3

10

10

n Q 1000   ⋅ 3.65 3 / 4 75  H

23

PUMP PARAMETERS

Hence

n

ns =

N 5/ 4

= 3.65

n Q

H H 3/ 4 Equation (2.35) is used for turbines and equation (2.36) is adopted for pumps.

...(2.36)

2.5.3 Unit Specific Speed (nsq) Unit Specific Speed (nsq) is defined as the speed of a geometrically similar pump delivering 1 m3/sec of discharge and develops 1 m head i.e., n = nsq where Q = 1 m3/s and H = 1 m, i.e., ns =

n Q H 3/ 4

.

Combining

gH Q and into one by removing ‘D’ 2 2 n D nD 3

Q ∝ nD3 gH ∝ n2D2

or

D3 ∝

Q n

or

D6 =

Q2 n2

or

D2 ∝

gH n2

or

D6 =

g3H 3 n6

Therefore,

or

Q2 g3H 3 2 ∝ n n6

or

n 4 Q2 = Constant g3H 3

or

n 6Q2 = Constant n2 g 3 H 3 n Q

= Constant (nsn) ...(2.37) ( gH ) 3/ 4 Equation (2.37) is called non-dimensional specific speed (nsn). Since g is a constant, it can be taken to the right hand side. Unit specific speed, nsq = Similarly, combining

n Q H 3/ 4

.

gH N and into one and by removing ‘D’ in both expressions n2 D2 ρn 3 D 5

gH ∝ n2 D2 or D2 ∝ N∝

So

or

ρn3D5

or

D5

∝

gH n2

D10 ∝

or

D10

or

n10 N 2 = Constant g 5 H 5ρ 2 n 6

N ρn3

N2 g5 H 5 ∝ ρ2 n 6 n10 n10 N 2 = Constant ρ2 g 5 H 5

g5H 5

or

n10

N2 ∝ 2 6 ρ n

24


n

or

N

ρ g 5/ 4 H 5/ 4

= Constant = nsn

...(2.38)

where nsn is the non-dimensional specific speed. Since N = γ QH = ρg QH, substituting this value in the above equation

n

ρ

g

ρg

5/ 4

Q H

H

5/ 4

= Constant

n Q

= Constant = nsn ( gH )3/4 which is the same nsn as defined earlier. While calculating the specific speed, all efficiencies i.e., volumetric, hydraulic, mechanical and overall efficiencies are assumed to remain same for one value of ns i.e., for one series, independent of size, capacity, head of the pump, of same ns. This is not correct since larger size and capacity pumps will have higher efficiency than smaller capacity units of same ns. This is the only drawback in the calculation of specific speed. Referring to the specific speed equation, it can be said that each value of specific speed, ns refers to one particular series of geometrically similar pumps i.e., a number of pumps with different H, Q, n can be developed, all having same (ns) specific speed. From the above it can be concluded that each value of ns refers one particular series of geometrically and kinematically similar pump, each pump in this series will be identical to the other. It can also be said that for the same value of head and discharge (H – Q) different types of pumps in different series can be obtained with different specific speed, by changing the speed n. Each pump will be different in type and construction. But due to limited suction conditions and due to cavitation and subsequent vibration, noise and damage of pump parts at higher speeds, high speeds are not recommended unless otherwise needed. Moreover, maximum efficiency can be obtained only at a particular speed for the given head (H) and discharge (Q) i.e., for given ns only at one particular speed. In fact, the specific speed, ns is calculated at the maximum efficiency point only. Normally pumps are driven by electric motor (speed will be 720, 960, 1450, 2990 rpm) or by I.C. Engines (750 or 1000 rpm) or by Turbines (25000 to 50000 rpm). Hence, pumps are always selected or developed to give maximum efficiency at these speeds. The value of specific speed, the type of pump will be always selected for the given H – Q of pumps and from the speed, n of the prime mover coupled to the pump. or

2.6 CLASSIFICATION OF IMPELLER TYPES ACCORDING TO SPECIFIC SPEED (nS) The shape and type of impeller depends upon the specific speed ns. For the same head and discharge, the specific speed (ns) is directly proportional to the speed (n). ns increases when the speed is increased. When the speed increases, the shape and type of impeller change. In first approximation the pump head (H) is directly proportional to the peripheral velocity or blade velocity (u). This is evident from the non-dimensional equation H ∝ n2 D2 ∝ u2. When speed (n) decreases the diameter (D) increases.

25

PUMP PARAMETERS

Outer diameter (D2) of the impeller is the characteristic linear dimension or the reference diameter D. So increase in speed n decreases the diameter D2 and correspondingly the size and weight of the pump is reduced which is naturally most advantageous, provided suction conditions do not have any limitations. The eye diameter (D0) or the inlet diameter (D1) is determined from the quantity of flow (Q). D0 or D1 D D and slightly reduces when speed is increased. So the ratio 2 or 2 reduces with the increase of ns. D0 D1 b Also for the given quantity, the diameter D2 reduces, the breadth b2 increases. So 2 increases with the D2 increase of ns. When ns the specific speed increases, the flow rate (Q) increases and total head (H) decreases. High head-low discharge pumps have low specific speed. The pumps have higher value of (D2/D1) and low value of (b2/D2). Impeller blades are in radial direction and of single curvature design. These pumps are called radial flow centrifugal pumps. Medium head-medium discharge pumps have medium specific speed. These pumps have medium D b value of 2 and 2 . At lower range of medium specific speed, the impeller blades have double D1 D2 curvature at inlet and single curvature at outlet. The outlet edge of the blade is parallel to the axis. The inlet edge of the blade extends towards the eye of the impeller in order to reduce blade loading since outer diameter D2 is reduced. When the specific speed increases further the inlet and outlet edges are inclained i.e., neither radial nor axial. The blades have double curvature design. Flow through the impeller is neither radial nor axial, but is in mixed or diagonal direction. These pumps are called mixed flow pumps or diagonal flow pumps. Low head-high discharge pumps have high specific speed. Inlet and outlet edges of impeller blade are almost perpendicular to the flow direction. The blades are of double curvature design. These pumps are called axial flow pumps. Very low head and very high discharge condition gives very high specific speed. The fluid flow direction in impeller is axial. Ship propellers belong to this category. In general, pumps are classified as radial, mixed, diagonal or axial, depending upon the fluid flow through the impeller passage. All positive displacement pumps have very low discharge and very high head and hence very low specific speed. Theoretically, specific speed changes from 0 to ∞ i.e., from zero discharge to zero head as well as change in speed. Practically very low speed and very high speeds could not be attained, so also very low head and very high discharge are limited and hence the specific speed. D D

D1

C

D2

B 80

350

450

A ns

800

Fig. 2.10. Form and shape of impeller for

D2 D1

26


Figs. (1.3) and (2.10) give different forms or shapes of impellers and their range of specific speeds as well as the range of diameter ratio (D2/D1). TABLE 2.2: Specific speed of pumps Positive displacement pumps

Type of impeller

ns=

3.65 n

n H

Diagonal

Propeller

Propeller

Q

H 3/ 4

D2 D1

nsq =

Centrifugal Radial Mixed Low Normal Higher discharge discharge discharge

Q 3/ 4

( gH ) 3/ 4

Axial Ship propellers

8–35

40–80

80–150

150–300

300–400

400–600

600–1200

1200–1800 and above

–

≈ 2.5

≈2

1.8–1.4

1.3–1.15

1.15–1.1

0.8–0.6

0.6–0.55

10–22

22–41

41–82

82–110

110–165

165–330

330–495

1.8–4.0

4.0–7.4

7.4–14.8

14.8–19.8

19.8–29.8

29.8–59.5

59.5–89.3

2–10

n Q nsq =

Mixed

0.36–1.8

2.7 PUMPING LIQUIDS OTHER THAN WATER 2.7.1 Total Head, Flow Rate, Efficiency and Power Determination for Pumps when Pumping, Liquids other than Water Unlike turbines; pumps are used not only for pumping clear cold water at normal temperatures, but also for pumping liquids with different properties such as different densities, different viscosities and different consistencies, pumping not only at normal temperatures, but also at cold or hot temperatures. Liquids may be corrosive or non-corrosive, two phase fluids such as gas-liquid or solid-liquid mixtures, milk, distilled water, acids, alkaline solutions, cryogenic liquids like liquid hydrogen, liquid oxygen, liquid nitrogen, liquid ammonia, molasses, tar, petrol, diesel, crude-oil etc. It is not possible to design each pump for each liquid and test them in the laboratory with the pumping liquid at the actual field working conditions. Pump design is always carried out for clear water at normal temperature. Water is considered as reference liquid for all the liquids mentioned above. For pumping liquids with viscosity and consistency, correction coefficients KH , KQ and Kη (or Ke) are used for converting the liquid parameters to equivalent water parameters. These coefficients are taken from standard recommended graphs and tables. These values are the consolidated results from a number of experiments by many authors and recommended by International Hydraulic Institute and Bureau of Indian Standards | 46 |. Suitable pump is then selected from the commercially available water pumps for which performance characteristics are known.

27

PUMP PARAMETERS

2.7.2 Effect of Temperature Increase in the temperature of the liquid decreases the density, viscosity and consistency and increases vapor pressure of the liquid. Due to high temperature of pumping liquid, the dimensions of pump parts change at running condition, due to thermal expansion of the material of the pump parts. Extra dimensional allowances in clearances are given depending upon the temperature of the pumping liquid and coefficient of thermal expansion of the material of the pump parts. These pumps are brought to the running temperature by filling with the pumping liquid or by external heating, before starting of the pump for smooth and vibration free operation. These pumps will not be started at normal temperatures and also should not be used for liquids at other than the recommended temperature. Increase in vapor pressure due to increase in temperature of the pumping liquid changes the net NPSH value and also reduction in suction lift. The system at suction side of the pump must be suitably altered for cavitation free operation of the pump. Recommended changes are given in chapter 9 of this book.

ρ or γ ) 2.7.3 Density Correction (ρ Pumping pressure ‘p’ and the total head (H) are related by the hydrostatic equation p = γH = ρ g H where ‘γ’ is the specific weight and ‘ρ’ is the density of the pumping liquid and ‘g’ is the gravitational acceleration. For the same pumping pressure, total head of the pump changes according to the specific weight (γv) or the density (ρv) or the specific gravity (Sv) of the pumping liquid i.e., p = γw Hw = γv Hv = Sv γw Hv Since rv = Sv γw. Suffix ‘w’ is for water and suffix ‘v’ is for the viscous liquid. ∴

Hw =

γv H v = Sv Hv γw

Although theoretically density has no influence on flow rate i.e., Qw = Qv, practically Qv changes by 2 to 3% Qw and even up to 5% at higher density of pumping liquid due to the influence of surface tension. For high temperature liquid pumping at t°C, the density of pumping liquid (ρt°C) is calculated as (equation 2.39).

ρ15 C

ρt°C =

...(2.39)

1 + β t °C (t C − 15C ) where (βt°C) is the coefficient and (ρ15°C) is the density at t = 15°C. Table 2.3 gives the values of (βt°C) for different values of (ρ15°C). TABLE 2.3: Density correction coefficients ρ15°C

0.7

0.8

0.85

0.9

0.95

βt°C

82 × 10–10

77 × 10–5

72 × 10–5

64 × 10–5

60 × 10–5

28


2.7.4 Viscosity Correction Performance of centrifugal pump changes when the viscosity of the pumping liquid changes. For higher viscous liquids, total head (Hv), flow rate (Qv) and efficiency (ηv) reduce considerably. Correspondingly, power consumption (Niv) increases. Head-discharge graph droops down more. Overall efficiency reduces. Optimum efficiency shifts to lower flow rate condition. Power consumption increases considerably especially at high viscous liquid pumping due to higher reduction in efficiency. However, shut off head of viscous liquid remains same as that of water. Fig. 12.27 shows the change in pump parameters when viscosity of the pumping liquid changes. However, up to liquid viscosity 20 C.S., pump performance for viscous liquid pumping does not change with respect to the pump performance pumping with water. Correction is applied only if the pumping liquid viscosity is more than 20 C.S. Figure 2.12 gives the values of coefficient for flow rate (KQ), coefficient for total head KH and coefficient for efficiency (Kn or Ke) for different values of Qv, Hv and νv, where νv is the viscosity of liquid in (S or SSU). If the temperature of pumping liquid is higher, viscosity (νt°C) at the temperature (t°C) is calculated as 0.01775 ...(2.40) ν t°C(C.S.) = 1 + 0.0337 t ° + 0.00023 t ° 2 νt°C must be taken while referring the Fig. 2.12. However, this graph can be referred only for : (a) Pumps of radial type centrifugal pumps under the normal operating range, having open or closed impellers. It cannot be used for mixed and axial flow pumps or for pumps of special design of impellers such as s-type impellers, single blade or two blade impellers or for nonuniform liquids like, slurries paperstocks etc., since it may produce widely varying results, depending upon the particular characteristics of the liquids. (b) Sufficient NPSH should be available in water parameters in order to avoid cavitation. Relation between viscous and water parameters is expressed as Qv =

KQ . Q W

Hv =

KH . Hw

ηv =

Kη . η w

Niv =

( Sv × γ w × Qv × H v ) (kW) 1000 ηv

...(2.41)

2.7.4.1 Determination of Water Parameters for the Given Head, Quantity and Viscosity of the Pumping Liquid For the given total head (Hv), quantity (Qv), efficiency (ηv) and specific gravity (Sv) at the pumping temperature (t°C) of the viscous liquid to be pumped, equivalent water parameters (Hw, Qw, ηw, Niw) can be determined referring the graph (Figures 2.11 and 2.12). The procedure is as follows: From the point of given viscous quantity (Qv) (Point A) in X-axis, a vertical line is drawn to meet the given viscous head (Hv) line (Point B). From this meeting point of Hv and Qv (Point B) a horizontal line, either left or right, is drawn to intersect the given viscosity (νv) line (Point C). From the point C, a

29

PUMP PARAMETERS 100

Water pump peak efficiency %

Head

90 80 70 90 80 70

60 100

20

50 60

40

30

Capacity

90 80 70 Water pump peak efficiency %

60 50 100

20

30

40 50 60 70 80

90

90 80

Efficiency

70 60 50 40 Water pump peak efficiency % 30 20

90 20

30

40

50

70

60

80 300 200 150 100 75 50 40 30 20 15 10

00 15,0 0 ,00 1 0 00 50 0 0 80 00 40 00 30 0 0 20 0 0 15 00 10 0 90 0 80 0 40 0 30 0 20 0 15 0 10 90 80

40

Head per stage in m at peak efficiency for water at actual operating r.p.m.

10

5m

V is co s ity

300 200 150 100 75 50 40 30 20 15

30

40

30

50 60 70 80 100

40 50 60

150

200

80 100

300

150 200

300

400 500 600 700 800 1000

400

600 800 2000

1400

20

1800 2200

15

1000

3

10

m /hr

3000 4000 5000 6000 8000 imp gpm

Fig. 2.11. A viscosity correction nomogram based on that quoted by (from Davidson (3), 1993, Process Pump Selection—A System Approach, Second Edition IMechE, London)

30

ROTODYNAMIC PUMPS (CENTRIFUGAL AND AXIAL) 1.0 0.9

KH

Correction Factors

0.8

0.60 0.80 1.00 1.20

0.7 0.6 1.0 0.9 0.8

KQ

0.7 0.6 0.5

Kη

0.4

1670 2280 3190

33.4 45.2 60.5 75 114 132 190 223 304 350 436 610 760 915 1217

11.8 16.5 21.2

6.25

0.3 0.2

Centistokes

2000

160

220

1500

100 120

80

1000

800

20 25 30 40 50 60

500

600

8

10 15

400

4.5

6

300

200

2.5 3

1.5

2

130

10

100

60

50

30

20

Engler

200 150 100 80 60 40 30 25 20 15 10 8 6 4

300 430

Hm

Q imp gpm

Fig. 2.12. Performance correction chart for viscous liquids

vertical line is drawn to meet the correction curves Kη, KQ and KH at peak water efficiency points D, E, F respectively. The values Kη, KQ and KH are the correction coefficients. By using the equation (2.41), equivalent water parameters QW, HW, ηW can be calculated. For multistage pumps, the total head (Hv) must be the total head per stage only i.e., Hv = [(HV) multistage/number of stages]. Based on the water parameters (HV and QV), suitable pump can be selected from the commercially available pumps.

2.7.4.2 Determination of Viscous Parameters When Water Parameters are Known For the given Hw, Qw, ηw values of water pump, equivalent viscous parameters Hv, Qv, and ηv can be determined, referring the graph (Figures 2.11 and 2.12). From the performance characteristics of the available water pump, namely Hw = f (Qw), ηw= f (Qw) and Niw = f (Qw), where Qw is the quantity at the

31

PUMP PARAMETERS

maximum efficiency condition and Hw, ηw, Nw are the corresponding values at Qw, the values of Hw, ηw, Nw for 0.6 Qw, 0.8 Qw, 1.0 Qw and 1.2 Qw are determined. As first approximation, all the above determined water parameters are assumed as viscous liquid parameters, so that graph (Figs. 2.11 and 2.12) can be referred to find KH, KQ, and Kη for all four capacities, following the same procedure as mentioned. Using the equation (2.41), equivalent values of HV, ηV, and QV can be calculated for all four Qw capacities. Two graphs Hw, ηV, NW = f (QW) and Hv, ηV, NV = f (QV) are drawn taking shut off head is same for water and for viscous liquid pumping. From this curve QV, can be found out for the given value of Qw, and other values. One such graph is given in Figure 2.13. Q 0, 6

H N

H

0, 8 Q

1, 0 Q

η

1, 2 Q N

Water parameters Viscous liquid parameters

η Q

Fig. 2.13. Determination of viscous parameters from water parameters of pump

Example: A water pump has the following details as per the performance graph: Optimum efficiency condition ηW (max)= 80% is at QW = 150 m3/hr. Corresponding Hw = 40 m, Nw= 28 kW. Pumping liquid viscosity is 57 CS. Referring to the performance characteristic of water pump, the values of HW, ηW, NW, for 0.6 Qw = 90 m3/hr, 0.8 Qw = 120 m3/hr and for 1.2 Qw = 180 m3/hr are found out. Referring the conversion graphs (Figs. 2.11 and 2.12), the values of Kη,KH, and KQ for all four capacities are determined. Using equation (2.41), HV ,QV, ηV, and the power required for viscous fluid pumping NV, are calculated. All these values are given in Table 2.4. TABLE 2.4: Viscous parameter determination from water parameters % QW values Parameters

0.6

0.8

1.0

1.2

Flow rate m3/hr

Qw

90

120

150

180

QV = KQ.QW

QV

88.2

117.5

147

176.5

Total head m

HW

44

42

40

36

HV = KH . HW

HV

43.2

40.8

38

33.5

Efficiency %

ηW

70

78

80

77

ηV = KηηW

ηV

49

54.5

56

54

Input power kW

NW

15.7

17.9

20.9

23.1

NV

21.6

24.6

27.6

29.8

32


Based on the results tabulated in above table (2.4), HV, ηV, NV = f (QV) are drawn in the same scale and in the same available performance characteristics of water pump, taking shut off head same for both liquids. From this graph (Fig. 2.13), for any value of QW, HW, ηW, corresponding values of QV, HV, ηV and NV can be determined.

2.7.5 Effect of Consistency on Pump Performance Pumps in chemical and process industries, handle two phase fluids i.e., liquid with another nonmixing liquid, liquids with solids in suspension, gas particles in liquids. Apparently average specific gravity of such mixtures is different from specific gravity of liquid alone. The problem becomes more difficult, if the liquid is other than water, which is very common in chemical industries. As a result, the net pumping head, flow rate, power, NPSH of the mixture change. So the pump parameters of the mixture is converted into equivalent water parameters by using experimental coefficients called ‘consistency factor’. ‘Consistency’ is defined as the percentage by volume or by weight (or specific gravity) of the solid content or gas content or other liquid present in suspension in the whole pumping mixture. It is the property of material by which, a permanent change of shape is resisted and is also defined by the complete force-flow relationships. As done for viscous fluids, the experimentally determined conversion factors are used to determine the liquid parameters. The following equations are used for such conversion:  Pulp (or) stock rating for Q or H ( Qs or H s ) Water Rating (Qw or HW) =  Conversion factor for Q or H ( Eq or EH )  ... (2.42)   HS = EH , HW , QS = EQ . QW Water efficiency (ηW) × Conversion factor (Eη) = Pulp or stock efficiency (ηs) ηW × Eη = ηs

Table 2.5 gives the conversion factor for pulp or stock pumping at different consistency conditions | 5 |. TABLE 2.5: Consistency conversion coefficient Pulp or stock consistency %

EQ

EH

Eη

1.0 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0

0.99 0.99 0.98 0.97 0.96 0.92 0.87 0.80 0.72 0.62 0.52 0.42

1.00 1.00 1.00 1.00 0.99 0.98 0.97 0.95 0.93 0.90 0.87 0.83

0.99 0.99 0.98 0.97 0.95 0.90 0.85 0.76 0.67 0.56 0.45 0.35

PUMP PARAMETERS

33

Such conversion factors are available for different liquid mixtures from the manufactures such as KSB pumps, pump manual or International Hydraulic Institute Standards. Rotodynamic pumps can be used only up to 7% consistency. For consistencies above 7%, positive displacement pumps must be used. Correct design, construction and material of pump parts must be followed especially for impeller blade shape, casing shape and location, sealing arrangement, and cooling arrangements such as external cooling or mother liquid circulation for cooling and sealing to suit the pumping fluid and operating conditions. In this book, water handling pumps and their constructions are only dealt with and discussed in chapter 13. For special pumps, however, handling hydrocarbons and other high consistency liquids, specific manufacturer’s recommendation must be referred.

2.7.6 Special Consideration in Pump Selection Normally pumps are manufactured as per the manufacturer’s standard of production range. Any customer selects pump for his requirement from the available standard ranges. Sometimes, pumps are selected according to space availability in the field such as in ships, rigs, railways, in general for transport systems and sometimes to replace the existing pump with the new pump especially in mechanical and process industries. In such cases, efficiency is not considered as a major factor, instead functional applications such as fitting the pump in the space available, non-stop or continuous operation even at emergency conditions are considered as important. Such conditions change from field to field and installation to installation. Pumps must be designed and constructed and must work as per the requirement of prevailing conditions at the fluid.

3 THEORY OF ROTODYNAMIC PUMPS

3.1 ENERGY EQUATION USING MOMENT OF MOMENTUM EQUATION FOR FLUID FLOW THROUGH IMPELLER Energy transfer from the impeller blade to the fluid, per unit mass (or weight) of fluid flow, when fluid passes through the impeller, can be developed by using momentum equation between point ‘O’, just before the impeller blade and point ‘3’ just after the blade. The cylindrical contour surface passing through point O and point 3 are shown in figure (3.1). The contour circles drawn with radius ‘r1’ passing through the point O and with radius ‘r2’ passing through point 3 are connected to the front and rear shrouds (Fig. 3.1). Pressure and velocity forces, on both sides of the shrouds, are equal and opposite and hence get cancelled. Only two forces, due to absolute velocities, one acting on the outer cylindrical surface 3 and another on the inner cylindrical surface ‘O’ are responsible for energy transfer. Taking moment of this momentum at inlet and at outlet i.e., moment of tangential component of these forces with respect to the centre of the circle and since l0= r0 cos α0, r0= r1, C0 cos α 0 = Cu0= Cu1 and l3 = r3 cos α3, r3= r2, C3 cos α3 = Cu3 = Cu2, the reactive moment due to the tangential forces acting on the cylindrical surfaces 3 and 0 will be C3

α3 C u3

3

Contour C0

3 Contour line

r3=r2

α0

r0=r1

l3

C u0 l0 α0 r 1

0

r2 0

II

I

Fig. 3.1. Moment of momentum equation as applied to impeller

Moment M 0 = C0 l0 = C0 r0 cos α 0 = Cu1r0 = Cu1r1 Moment M 3 = C3l3 = C3 r3 cos α 3 = Cu 3 r3 = Cu 2 r2

...(3.1)

Taking into account, moment Mf due to friction, created due to the fluid passing through blade passages, total moment M will be M = M3 + M 0 + M f 34

35

THEORY OF ROTODYNAMIC PUMPS

rQ (Cu 2 r2 − Cu1 r1 ) + M f ...(3.2) g For ideal fluid flow, Mf = 0. Energy transfer per unit weight of fluid flow through the impeller of a pump i.e., the theoretical head developed under infinite number of blades, with infinitesimally smaller vane thickness, will be =

Hth ∞ =

M ω  Cu 2 − Cu1  =   γQ g  

...(3.3)

where Mω = N, γQ = W and u = ω r. Equation 3.3 is the Eular’s equation for the head developed by a pump.

3.2 BERNOULLI’S EQUATION FOR THE FLOW THROUGH IMPELLER Eular’s equation for an elementary flow along a streamline (S) is given by

1 ∂p dC ∂C ∂C ∂s ∂C ∂C ∂C ∂  C 2  . = = + = + C = + Fs – ρ ∂ ∂t ∂s ∂t s dt ∂t ∂s ∂t ∂s  2  where, Fs = Resolved component unit of mass along the direction of the streamline S p = pressure C = velocity (absolute) ρ = density For an elementary length ‘ds’ on the streamline the equation (3.4) can be written as Fs ds – For steady flow condition

∂  C2  ∂C 1 ∂p ds –   ds = ds ρ ∂s ∂s  2  ∂t

...(3.4)

...(3.5)

∂C = 0. ∂t ∂  C2 1 ∂p ds –  ρ ∂s ∂s  2

 ...(3.6)  ds = 0.   mg  The force due to unit mass is the gravitational force ‘g’  =  which is directed downwards.  m  Fg = – g. Therefore,

Fs ds –

Taking vertically upward direction of Z-axis as +ve direction and

dZ ds Substituting this value of Fs in equation 3.6 and changing the sign

+g

or

Fs – Fg ( cos Z , ds ) = – g

...(3.7)

dZ 1 ∂p ∂  C2  ds + ds +   ds = 0 ds ρ ∂s ∂s  2 

...(3.8)

gdZ +

dp C2 +d = 0 ρ 2

36


For compressible flow, density ‘ρ’ is a function of the pressure p i.e., ρ = f ( p). Integrating the equation (3.8) with respect to ds

dp C 2 + = Constant ...(3.9) gZ + ∫ ρ 2 For incompressible fluid, the density ‘ρ’ is constant. The specific weight γ = ρg. Hence, the equation (3.9) can be written for unit weight of fluid as, p C2 +Z + = Constant γ 2g

...(3.10)

Equations (3.8), (3.9) and (3.10) are called Bernoulli’s equation derived from fundamental Eular’s equation of motion under steady absolute flow condition along a streamline. It is evident that, this equation cannot be applied for the change of energy of ideal fluid under unsteady absolute motion of fluid in impellers. Perhaps this equation can be applied for other elements like approach pipe with or without inlet blades, volute casing, diffuser, return passage of multistage pumps, which are non-moving or stationary elements, where steady flow prevails under optimum conditions. For impellers, however, steady flow condition can be applied for relative velocity of flow of fluid since this velocity is actual velocity flowing past the blades. Referring the equation (3.7) the force Fs in impeller blades consists of the gravitational force Fg and inertia force (since blade is moving) namely centrifugal force FCF and Coriolis force Fc . Fs = Fg + FCF + Fc ...(3.11) dZ and is directed ds = ω2 r, where ‘ω’ is the angular velocity and

For unit mass flow along the streamline ‘S’, the gravitational force Fg = – g towards downward direction. The centrifugal force FCF

‘r ’ is the radius, and is directed towards radial direction. Coriolis force, Fc = ω w sin (ω w) , is directed normal to the direction of relative velocity, vector ‘w’ and angular velocity ‘ω’. Since ds = w dt along the streamline, the resolved component of the total mass force Fs will be Fs = fg cos (Fg.ds) + FCF cos (r.ds) + Fc cos (Fc.ds) ω

Wu W. sin (ω,w)

Fc F cu Fcr acu

W

Wr

a cr

u

Wz

Fig. 3.2. Vector diagram for Coriolis component Fa determination of Mz

37


Taking axis of rotation vertically upwards as +ve direction the resolved component of the mass force in relative motion along a streamline will be Fs = – g

dz dr + ω2 ds ds

...(3.12)

substituting the value of Fc in equation (3.6) and since, Fcs = Fc cos (Fc.ds) = 0, because of the direction of Fc normal to the direction of w on the elemental strip ‘ds’ where the relative velocity ‘w’ is tangential to the streamline –g

Simplifying

dr ∂  w2  1 ∂p dZ ds – ds –   ds = 0 ds + ω 2 r ds ρ ∂s ∂s  2  ds

...(3.13)

2  w2  dp 2 r  +d gdZ – ω d   +  =0  2 ρ  2 

...(3.14)

Integrating the above equation (3.14) and since u = ωr

dp w2 – u 2 + = Constant ...(3.15) ρ 2 For an incompressible fluid flow, the density ‘ρ’ is constant and independent of pressure ‘p’. Hence, the above equation can be written as gZ + ∫

p w2 − u 2 + = Constant ρ 2

...(3.16)

ρ w2 − u 2 +Z + = Constant γ 2g

...(3.17)

gZ +

for unit mass flow for unit weight flow

The equation (3.16) and (3.17) represent Bernoulli’s equation for a relative flow past impeller blades and is used for an indepth study of flow analysis i.e., interaction between blades and flow past the impeller blade. This equation is similar to the Bernoulli’s equation derived for an absolute flow used for analysis of all non-moving flow passages of the pump. Applying equation (3.17) between point 0 and point 1 (which lies on the inlet edge of the blade), where steady flow exists in absolute and in relative flows.

p1 w2 − u12 p0 w2 − u02 + Z1 + 1 + Z0 + 0 = γ 2g γ 2g

...(3.18)

Relative velocity ‘w’ can be expressed in terms of absolute velocity ‘C’ and blade velocity ‘u’. Referring to the velocity triangle (Fig. 4.1)

w2 = C 2 + u 2 − 2u Cu So,

...(3.19)

w2 − u 2 C 2 Cu1 u = − 2g 2g g

Equation 3.27 can be written as 2 p1 C C u + Z1 + 1 − u1 1 = p0 + Z 0 + C0 − Cu 0 u0 γ g 2g g g 2g

...(3.20)

38


Since total energy E =

p C2 +Z+ , the above equation (3.20) can be written as γ 2g

Cu1 u1 C u = E0 – u 0 0 g g C u − Cu 0 u0 E1 − E0 = u1 1 or ...(3.21) g Correspondingly the energy difference between point 3 and 0 which is the total head Hm developed by the pump, is C u − Cu 0 u0 Hm = E3 – E0 = u 3 3 g C u − Cu1u1 and Hth ∞ = u 2 2 ...(3.22) g E1 –

Also

u3 = u2 , Cu 3 = Cu 2, Cu 0 = Cu1 and u0 = u1

This equation (3.22) is the fundamental Eular’s equation for rotodynamic pumps. Hm is the monometric head applied for finite number of blades with finite thickness and Hth∞ is the theoretical head applied for infinite number of blades with infinitesimally smaller thickness.

3.3 ABSOLUTE FLOW OF IDEAL FLUID PAST THE FLOW PASSAGES OF PUMP The integral form of equation for the ideal fluid flow as per Gromeko-Lamb |67| is

→

∂C 1 grad p = + grad F– ∂t ρ

 C2  → →   + rot C × C 2  

where C is the velocity vector.

...(3.23) →

Taking Z-axis in vertically upward direction as +ve direction , mass force F under absolute flow through the passages of pump parts, is →

Mass force F = – grad ∏ = – gz

...(3.24)

where g = acceleration due to gravity, which is acting vertically downwards. The density ‘ρ’ is a function of pressure ‘p’. ρ = f (p) ...(3.25) For compressible flow it depends upon the process. Under baratropic conditions, for isentropic process ρ = (p)1/γ. For adiabatic process ρ = ( p)1/k. For isothermal process ρ = p–1 = p. For incompressible flow, ρ = Constant (p0). In general, under baratropic condition. p

P (p) =

dρ

∫ ρ ( p)

p0

...(3.26)

39


Gradient of the above function will be grad P =

1 dp grad p = grad p ρ dρ

...(3.27)

For the condition (equations 3.23 and 3.24) as per Gromeko-Lamb equation, (3.23) can be written as:

→

 C2  → → ∂C + P + ∏ + rot C × C = 0 + grad  ∂t  2 

...(3.28)

→

→ ∂C For absolute flow + grad E + Ω × C = 0 ...(3.29) ∂t → C2 + P + ∏ and rot C = Ω. where E = 2 Equation (3.29) represents the ideal fluid flow under baratropic condition (for liquids and gases) and under potential field of mass force.

→

∂C Under steady flow condition = 0. For absolute flow stationary conditions prevailing in diffusers ∂t volute casings etc., (non-moving parts) equation (3.29) can be written as →

→

grad E + Ω × C = 0

...(3.30)

→

→

For axisymmetric flow Ω = 0. Such conditions prevail in approach pipe at pump inlet. So

→

→

→

Ω × C = 0, where the vortex vector Ω is parallel to the velocity C . Such things exist in propeller and screw type units, because, the interaction between fundamental vortex and the flow becomes zero. The vortex motion developed in impellers continues up to outlet also. Integrating equation (3.30) E =

C2 C2 + +P+∏= 2 2

p

∫

p0

dp + gZ = Constant ρ

...(3.31)

which is the Bernoulli’s theorem for the entire stream tube which is called as Lagranze’s equation, |67|. For incompressible fluid flow

p C2 +Z+ = Constant γ 2

...(3.32)

Equations (3.31) and (3.32) can be and widely used for compressible and for incompressible flow in fluid machines. For unsteady absolute flow, which is normally existing in impeller blades and in places where flow is changing from stationary to moving. Integrating the equation (3.29) under axisymmetric conditions. →

→

−

→

Ω = rot C = 0 and for potential function φ of velocity C , grad φ = C and hence

∂C ∂φ . = grad ∂t ∂t

40


Equation (3.29) under axisymmetric flow condition will be  ∂φ  grad  + E  = 0  ∂t  ∂φ ∂φ C 2 +E = + + P + Π = f (t) ...(3.33) ∂t ∂t 2 This equation is called Lagrange-Koshi’s equation | 67|. The function f (t) is a function of time and is determined from the boundary conditions. Equation (3.33) i.e., equation of Langrange-Koshi plays a very important role in unsteady flow as similar to Bernoulli’s equation in steady flow. Equation (3.33) for incompressible flow will be

Since

1 ∂φ C 2 p + + = f (t) g ∂t 2 g γ

...(3.34)

3.4 RELATIVE FLOW OF IDEAL FLUID PAST IMPELLER BLADES Considering the integral form of Gromeko-Lamb equation (equation 3.23) in vector form for a →

relative flow of fluid past impeller blades, absolute velocity of fluid C before entering the impeller → blade at inlet is changed to relative velocity w , while flowing through the impeller due to rotation of →

impeller with a peripheral velocity u . So also at outlet. The relation between these velocities can be written as : →

→

→

→

→

→

C1 = w1 + u 1 and C 2 = w2 + u 2

The force due to mass ‘F’ under relative flow consists of three elements namely →

→

→

→

F = Fg + FCF + Fc where

...(3.35)

→

Fg = gravitational force acting vertically downwards →

FCF = centrifugal force and →

FC = Coriolis component of force acting normal to the flow at any given point on the streamline. The gravitational force Fg is a potential function Π →

F g = grad ∏ = – gZ since Z-axis is the reference axis and vertically upward direction is taken as +ve. The centrifugal force is also a potential function and expressed as

...(3.36)

ω2 u2 = grad r 2 = ω2r ...(3.37) 2 2 where, u = ωr, ω = angular velocity, r = the selected radius on the stream referred with respect to Z-axis, the axis of rotation. The Coriolis component can be written in general form as →

F CF = grad

→

→

→

FK = – 2 ( ω × w )

...(3.38)

41

THEORY OF ROTODYNAMIC PUMPS →

→

→

→

→

The vectorial form of the relative velocity w will be rot w . Since C = w + u →

→

→

→

→

→

→

→

→

w × rot w = w × rot ( C – u ) = w × rot C – w × rot u →

→

... (3.39)

→

rot u = rot ( ω × r ) = 2ω Same thing can be obtained by selecting or elementary contour of radius ‘r’ in the plane, normal to the Z-axis of rotation. The resolved component of these vortex along the direction Z is →

rotz u = lim

r →0

→

→

2π r 2 ω 2π r u lim = = 2ω r →0 π r2 π r2

→

→

→

→

w × rot w = w × rot C + ( ω × w ) Substituting all the values in equation (3.23) of Gromeko-Lamb 1 → → u2 w2 → ω w – grad Π + grad – 2 ( × ) – ρ grad p – grad + w × rot C 2 2 So

→

→

+ 2 (ω × w ) –

...(3.40)

dw =0 dt

...(3.41)

Simplifying →  w2 – u 2  dw → = w rot C grad  ∏ + P +  + dt 2  

...(3.42) →

Under steady flow conditions of relative velocity w ,

dw =0 dt

Integrating equation under two conditions: →

(1) When axisymmetric potential flow of absolute velocity exists i.e., rot C = 0. (2) When the vector of absolute flow vortices are parallel to the relative velocity under vortex →

→

flow of absolute flow, i.e., w × rot C = 0 So,

 w2 – u 2   = 0 grad  ∏ + P + 2  

...(3.43) p

Integrating the above equation (3.43) and since Π = g Z and P =

∫

p0

dp ρ

dp w2 – u 2 + gZ + = Constant ...(3.44) ρ 2 Equation 3.44 is the equation for the relative flow under steady potential flow conditions. For compressible fluids the density ‘ρ’ is a function of pressure ‘p’ and depending upon the process. For incompressible flow ρ = Constant. Changing the equation from unit mass to unit weight. γ = ρg

∫

p w2 – u 2 = Constant +Z + γ 2g

...(3.45)

42


Equation (3.45) holds good for the entire flow in the channel for a steady and potential (axisymmetric) flow. Applying Bernoulli’s equation between points x and y (Refer Fig. 3.3 and 3.13) located on both sides of the impeller blade at the same radius ‘r’.

x

r r y

Impeller blade passage at r ′ f

1 2

1 2

2

py

px

r

r p1

x p2

y

(a )

(b)

Fig. 3.3. (a) Pressure variation in radial flow impeller passage (b) Pressure variation across impeller passage of axial flow pump ω

ω

y x r

r (a) Theoretical

(b) Actual

Fig. 3.4. Velocity distribution between blades

py px wy2 − u y2 w 2 − ux2 + Zx + x = + Zy + γ 2g 2g γ ux = uy and Zx = Zy since point x and y are located on the same radius and the difference in level is negligibly small, the above equation can be written as

43


p y wy px wx + + = γ 2g γ 2g Useful work done by the impeller blades under finite number of blades with finite thickness is due to the interaction between the blade and the flow of fluid and due to the local pressure difference between leading and trailing side of the impeller blade. The pressure at the point x, located on the leading side of the blade on the radius, is higher than the pressure at point y, which is located at the trailing side of the blade at the same radius. In the same manner the pressure at point x′ is same as at x since both are at leading surfaces but at adjacent blade. This means the pressure across the channel between two successive blades (x′ – y) located at the same radius are not equal. px = ( px′) > py. Across the channel, pressure changes uniformly. Correspondingly, wx= wx′ < wy the relative velocity across the channel gradually decreases for x′ to y. (Figs. 3.3 and 3.4).

3.5 FLOW OVER AN AIRFOIL The flow of fluid over a blade kept in space is of three types: (1) Plain flow over the blade [Fig. 3.5 (a)], (2) Circulatory flow [Fig. 3.5 (b)] and (3) Combination of these two flows [Fig. 3.5 (c)]. In pure plain flow, fluid flows over blade without any circulation or vortex. It is a uniform, steady, potential (axisymmetric) flow. In pure circulatory flow, the fluid flows around the blades, encircling the blade. There is no plain flow. The line integral of the flow velocity around their closed contour gives circulation, sum of which is zero in an endless flow (i.e., in space where flow starts from infinity and ends in infinity).

A

(a) Plain flow

B

(b) Circulatory flow

(c) Combined flow

(d) Flow after the airfoil

Fig. 3.5. Fow over an airfoil

Pure plain flow or pure circulatory flow acting individually on the blade does not produce any force on blade. The pressures on both sides of the blade are equal in both type of flows. When these two flows are combined together, a pressure difference is developed due to the difference in flow velocities between two sides of the blade, by which energy transfer between fluid and the blade takes place. In rotodynamic machines (pumps, fans, blowers, compressors and turbines) instead of one blade, a number of blades, i.e., a cascade system is adopted for energy transfer between blades and fluid. This pressure difference in axial flow pump is due to the flow over the cascade system with angle of attack. The flow velocity is determined from the flow rate. But in radial flow pump this pressure difference is produced

44


not only from the main flow (plain flow combined with circulatory flow), but also due to the rotation of the impeller i.e., Coriolis component. The Coriolis component does not give any flow, but increases the velocity difference further as a result of which circulation is increased. The relative velocity at the trailing side of the impeller blades is higher, whereas the relative velocity at the leading side of the blade is lower (Fig. 3.7 and 3.8). Due to this, relative velocity across the channel from trailing side of the blade to the leading side of the sucessive blade is not constant, instead changing. Absolute velocity of flow C is axial and potential, without any rotation or vortex, but the relative velocity is not axisymmetric i.e., with vortex, induced due to peripheral velocity u. This can be shown by the following. Since the relative velocity →

→

→

→

→

→

→

→

w = C – u ; rot w = rot ( C – u ) = rot C – rot u →

Since absolute velocity of flow is axisymmetric and potential rot C = 0. →

→

→

→

→

So, rot w = rot u = rot (ω − r ) = 2 ω This means the relative velocity is with vorticity or circulation. Consider the flow of ideal fluid, in a completely closed cylindrical container and the container moves in a circular path (Fig. 3.6) with the centre of rotation ‘O’. The fluid body AB in the closed channel keeps its position same without any rotation at all positions when the channel is moved in a circular path. i.e., the pointer is always pointing the same upward direction. In other words, the absolute velocity direction is without any circulation or vortex, even when the channel moves in a circular I B path with centre ‘O’. But the same line AB rotates, if it is considered with respect to the circle. When A the circle is rotated in anticlockwise direction, at II IV the section I the arrow AB is perpendicular to the circle. At section II, it is parallel to the circle. This B B is repeated to section III and IV but rotated by O A A 180° with respect to section I and II. The fluid in the container rotates in opposite direction i.e., III clockwise direction, with respect to the circle with B an angular velocity ‘ω’. This indicates that the A relative velocity is with circulation or vortex. The same situation prevails in impeller channel Fig. 3.6. Relative rotary motion of fluid in a closed (Fig. 3.7) when the channel is closed at inlet and channel when moving in a circular path at outlet.

Fig. 3.7. Circulatory motion of fluid in impeller blade passage (channel vortex)

45


Adding the plain flow i.e., the potential absolute flow moving with constant velocity over both sides of the blade to this relative circulating flow i.e., when the channel is no longer a closed one, the resultant flow gives velocity difference between trailing side and leading side of the impeller, and hence the energy transfer from blade to fluid. The velocity triangles at inlet and at outlet are shown in Fig. 3.8. C2 u2 α2 α′2

ω

E

C2′ w1

w2 β2

ω1

R

S

w1

K

ω1

B2

β′2

w2′

c1

B1

B

A

P

M

D

r1

r2

c1′ α′1 u1

N L 0

Fig. 3.8. Relative velocity of flow in impeller passage at normal conditions

As mentioned earlier, different velocities at the outlet of the impeller w2 equalises after some distance. Constant and uniform velocity ‘w3’ exist after the impeller, which again shows that axisymmetric absolute flow prevails after the impeller.

3.6 TWO DIMENSIONAL IDEAL FLOW 3.6.1 Velocity Potential Velocity potential φ is defined as φ = ∫ Cs ds between any two points in the potential field and Cs is the velocity tangential to the elementary path connecting these two points, independent of the path taken between two points. Such a flow is called potential flow. Circulation is equal to zero for a potential flow. Presence of isolated vorticity does not change the potential flow. Circulation along a closed contour not enclosing the vorticity is also equal to zero and remains constant at all times along and contour.

3.6.2 Rotational and Irrotational Flow Two basic motions of fluid, namely translation and rotation can act either independently or collectively. Fig. 3.9 shows such motions.

46


dθ

dy dx (a)

(b)

Fig. 3.9. Translation and rotation in fluid motion

So also a deformation of a fluid element, represented by a square in Fig. 3.10, can be either linear or angular.

(a)

(b)

Fig. 3.10. Linear and angular deformation in fluid motion

All the above mentioned fluid motion can take place either individually of collectively. Consider a fluid motion with rotation and translation as shown in Fig. 3.11. During the time interval ‘dt’ point ‘A’ in the fluid element aAb moves to point A′ and takes new position a′, A′, b′. When deformation also takes place, the angles of rotation α and β are not equal. Average rate of rotation ‘ω’ in ω Aa + ω Ab α +β = time ‘dt ’ will be ω = 2 2dt

y

∂vy

v x dt ∂ y

dy dt a′

β

a A′

dy

A

dx

b

α b′

Arc = Radius ∂u dydt . β = ∂y

So

ωz =

dxdt

v ydt

Fig. 3.11. Rotation, translation and deformation in fluid element

taking α and β as small values ( tan θ = θ )

and

∂x

...(3.46)

Taking anticlockwise direction as positive and

α =

∂vy

∂v ∂v 1 dxdt . = dt ∂x ∂x dx 1 ∂u =– ∂y dt dy

α +β 1  ∂v ∂u  1 1  ∂v ∂u  = dt  . =  dt –  –  2dt 2  ∂x ∂y  dt 2  ∂x ∂y 

x

47


3.6.3 Circulation and Vorticity

z

→

y

vx +

∂v x ∂y

dy

B

dy

Circulation ‘ Γ ’ is the line integral of velocity around an element. A study of circulatory motion can be understood by studying vortex motion under potential flow condition. The velocity components on all four sides of the fluid element ABCD are shown in Fig. 3.12(a). The fluid element is rotating in anticlockwise direction with an angular velocity ‘ω’. Since the centre of rotation is not known it is convenient to relate the sum of the products of velocity and distance around the contour of the element which is the sum of the line integral of velocity around the element. This is called ‘circulation, Γ ’. Since area of the element dA = dxdy.

C Direction of integration

vy A

vx

vy +

∂x

dx

D

dx x

Fig. 3.12. (a) Circulation in fluid element

→

Γ = C ⋅ ds Circulation Taking anticlockwise direction as positive direction for integration. ∂v   Γ ABCD = udx +  v + ∂x dx  dy –  

=

∂v y

...(3.47)

 ∂u   u + dy  dx – vdy = 0 ∂y  

∂v ∂u dxdy – dxdy ∂x ∂x

 ∂v ∂u  =  –  dxdy = 2ωdA ...(3.48)  ∂x ∂y  It is evident from the equation (3.48) that the circulation around a contour of an element is equal to the sum of vortices within the area of the contour.

v θ dA C

Fig. 3.12. (b) Circulation and vorticity

This is known as Stokes theorem. Mathematically, it is represented as

z

→

→

z

z

ΓC = C ⋅ ds = C cos θ ds = 2ωdA = 2ωA Vorticity

Ω=

→ Γ = rot C = 2ω Area

s

...(3.49)

48


is twice the angular velocity of fluid rotating as a solid body. Taking anticlockwise direction as +ve direction, the component of vorticity in polar (r, θ, z) coordinates will be ∂C   ∂C Ωu = 2ωθ = 2ωu  r – z  ∂ ∂r  z  ∂ (Cu r )  1  ∂C Ω π = 2ωr =  z – ∂z  r  ∂θ

1  ∂(Cu r ) ∂Cr  Ω z = 2ωz =  –  ∂θ  r  ∂r For an irrotational flow ωu = 0. For a potential and incompressible flow Cr= 0. Circulation along a closed contour is constant and is equal to the intensity of vorticity. i.e., ∂Cz = 0 or Cz = Constant. ∂r In axial flow pump, the existence of potential flow gives equal flow velocity at all radii. Under axisymmetric, potential flow condition. ∂C z ∂Cr = = 0. ∂θ ∂θ ∂ (Cu r ) ∂ (Cu r ) = = 0 or Cu r = Constant ∂z ∂r at all radii of the impeller inlet and outlet. However, under potential flow in meridional sections only

Also

ωu = 0. Whereas ωr and ω z ≠ 0 and henc Cu r ≠ constant in these direction.

3.7 AXISYMMETRIC FLOW AND CIRCULATION IN IMPELLER Ideal fluid flow through the elements of fluid machines is an axisymmetric flow. Kelvin’s theorem and Lograngan’s theorem are the most important expressions for the study of dynamics of ideal fluid flow. Circulation ‘ Γ ’ is defined as the line integral of the velocity along a closed contour.

Γ= →

→

z

→→

q dl

c

→

→

where q is the velocity vector dl is the differential of arc length of the closed curve q . dl is the scalar product of these two vectors. Kelvin’s theorem is the time rate change of circulation for a closed fluid curve. i.e., DΓ = 0. Dt Kelvin’s theorem states that for a barotropic ideal fluid acted on by gravitational force with potential, the circulation along a closed fluid contour, remains constant with respect to time. When applied to radial flow fluid machines, the Kelvin‘s equation is written as

49


d dt →

→ →

∫ C dr = 0

→

where C = q and dr is dl. This equation confirms the application of Eular’s equation for fluid machine design. Kelvin’s theorem can be used only for absolute flow for both non-stationary and stationary elements, where the gravitational force is under potential.

3.7.1 Circulation in Impellers of Pump As per Kelvin’s theorem, if axisymmetric or potential flow exists in ideal fluid flow before entering the impeller, then same potential flow prevails, when fluid flows through the impeller also. Circulation along the closed contour of the fluid flow must also be equal to zero. But this statement is correct only if the fluid flow is an unified flow, i.e., only for a fluid flow through impeller. If the closed contour encloses, a solid body apart from the fluid, for example, impeller blades, then fluid flow cannot be taken as unified flow and correspondingly circulation along the closed contour under potential flow cannot be zero. This is evident from the fact that pressure at the leading side of impeller blade is higher than that at trailing side. Correspondingly, flow velocities at the leading side is lower than that at leading side of the blade, due to the interchange of momentum from blade to fluid (effect due to finite number of blades with finite thickness).

3.7.2 Vorticity and Circulation around Impeller Blades Consider a closed contour enclosing one impeller blade 11′2′2 (Fig. 3.13) of a radial flow pump. Lines 12 and 1′2′ are two identical stream lines kept at a distance of 2πr   2πr1 and ‘t’ the pitch  t =  at inlet and at outlet. t1 = Z Z   2πr2 t2 = , where r1 = inlet radius, r2 = outlet radius and Z Z = number of impeller blades. Lines 11′ and 22′ are the arc of circles at inlet radius r1 and at outlet radius r2 respectively, connecting the two streamlines.

t2 = 2 πr 2z 2

2′ 7

6

4

1

5 ω

db 3

x′

2

ds r2

r1 1′ y

Consider the absolute flow of the fluid along the x contour 11′2′2′ (Fig. 3.13). The streamlines 12 and 1′2′ are identical, but located one on each side of the blade, symmetrically to the blade. Circulation along the streamlines Fig. 3.13. Vorticity and circulation along a are equal in magnitude but apposite in sign due to the change moving impellerl blade in the direction of movement along the contour. On integration, we get

Γ cb =

=

z

C ds cos (C, ds) =

b122′1′ 1g

2π (Cu2 r2 – Cu1 r1) z

∫ Cu

22 ′

2

ds −

∫ Cu ds

11′

1

...(3.50)

50


where Cu 2 and Cu1 are the average values of the tangential component of absolute velocities at radii r2 and r1 respectively and –ve sign for Cu1 is due to the direction of Cu1 opposite to Cu 2 in the closed contour. In order to find the circulation ‘ Γ cb’ along the blade, the contour 47654, enclosing the blade is connected to the previous contour 11′2′21 through the line 34. Consider the circulation along the contour 1345674322′1′1. Since this contour does not include the impeller blade, the flow can be considered as axisymmetric or potential. As per Kelvin’s theorem circulation along the contour must be equal to zero. Integrating (Fig. 3.13) = + + Γ Γ Γ + Γ Γ =0 (1345674322′1′1) (122′1′1) (34) (45674) (43) Since Γ = – Γ = – Γ cb and Γ + Γ = 0 (45674) (47654) (34) (43) Γ 2π (122′1′1) = (Cu2 r2 – Cu1 r1 ) as per equation (3.50)

and

z

Γ cb

=

2π (Cu2 r2 – Cu1 r1 ) z

...(3.51)

i.e., under axisymmetric or potential absolute flow, circulation along any contour enclosing the blade, including the blade contour also will be constant. Since same value of Γcb exist on all other impeller blades also, circulation for the impeller, possessing Z number of blades will be

Γimp = Σ Γ cb = z Γ cb = 2π ( Cu 2 r2– Cu1 r1 )

...(3.52)

3.8 REAL FLUID FLOW AFTER IMPELLER BLADE OUTLET EDGE Let us consider the flow of fluid before and after the outlet edge of an impeller blade in a cascade system under finite number of blades with finite thickness (Fig. 3.14). S

w1 δ

A w 3u w3

β3

b

a

β2

w 2m w3m

w 2u = w3u t-S

w 2u

t

β3

β0

w3 w2

w2m

d C w3m

t

Fig. 3.14. Real fluid flow after impeller blade outlet edge

The geometric and kinematic parameters of the blade system are: outlet blade angle β2 , blade thickness ‘δ2’, pitch ‘t2’ outlet flow velocity on the blade ‘w2’ and after the blade ‘w3’. The tangential and normal components, of these velocities, when resolving with respect to the blade movement are wu2 = w2 cos β2 , wm 2 = w2 sin β2 , wu 3 = w3 cos β3 and wm3 = w3 sin β3.


51

Fluid uniformly flows over the blades at outlet tangentially without shock. The flow area of the passage between two successive blades before leaving the outlet edge will be (t2 – S2), where

S2 =

δ2 since the flow area is reduced due to the vane thickness. sin β2

After some distance, fluid stream coming out from both sides of a blade converges into one stream. A no flow area prevails (marked as A in Fig. 3.14) after the blade thickness area at outlet. The flow area is increased from (t – S) to t. Correspondingly, the flow velocity is reduced to t − S2 . w3m = w2 m 2 t2 In order to find the relation between w2u and w3u, a controlled surface abcd enclosing the no flow area A as well as covering the outlet edge of the blade is taken for analysis. The lines ‘ad’ and ‘bc’ are two identical streamlines kept at a distance of pitch ‘t’ between them. Other two lines ‘ab’ and ‘cd’ are parallel to the direction of movement of the cascade system. Considering the force on the surface ‘ad’ and ‘bc’, the forces are equal and opposite at each and every point considered along the streamline ‘ad’ and ‘bc’ respectively and hence they cancel each other (a couple produced by these two forces are neglected).

γ Considering the surfaces ‘ab’ and ‘cd ’, the forces on surface ab will be FD 2 = g ∆Qw2 inclined at γ an angle of β2 to the blade movement and on the surface ‘cd ’ will be FD3 = g ∆Qw3 inclined at an angle of β3 to the blade movement. Resolved components of these two forces are equal and opposite, hence γ γ they cancel each other, i.e., ∆Qw2 cos β2 = ∆Qw3 cos β3. g g Since w2 cos β 2 = wu2 and w3 cos β2= wu3, wu2 = wu3, i.e., the tangential components of the velocities before and after the outlet blade edges are equal. So the change in the relative velocities w2 and w3 is due to the flow area change from (t – S) to ‘t ’ before and after the outlet blade edge and is only due the change in meridional flow velocities wm2 and wm3 and correspondingly the blade angle from β2 to β3. The corresponding velocity triangles are shown in Fig. 3.14. This is effective due to the application of finite number of blades in cascade system. The same rule is applicable to the inlet edge also.

3.9 SECONDARY FLOW BETWEEN BLADES (Fig. 3.15) For energy transfer in pump, pressure at the trailing side of the blade should be greater than the pressure at the leading side, correspondingly pressure and velocity difference prevails at the passage between two blades. This pressure difference induces a circulatory secondary flow along with the main flow in the flow passage in axial direction from inlet to outlet at the same radius. At the same time from hub to periphery in the radial direction another circulation secondary flow exists due to the boundary layer in real fluid flow. The circulation in radial directions increases. However, the real fluid flow more or less concides with ideal fluid flow. Hence, the secondary flow effect is neglected normally in axial flow pumps.

52


If inlet entry is normal Cu = 0 and Cr = 0. Circulation and hence the total head remains constant at all radius. In practice, however, it is found that circulation slightly increases near hub and considerably at periphery due to finite number of blades and subsequent secondary flow. However, it is very small and hence neglected. Pumps designed with Cu r = constant give very high efficiency (93 to 98%) in spite of complicated three dimensional flow pattern actually existing at all radii of the impeller inlet and outlet.

h

Fig. 3.15. Secondary flow in axial flow pump

3.10 FLOW OF A PROFILE IN A CASCADE SYSTEM—THEORETICAL FLOW Flow over an impeller blade of a pump or airfoil of axial flow pump is with a relative velocity ‘w’, → → → which is resultant of absolute and peripheral velocities. (C and u) w = C – u . This flow can be considered as the flow with many vortices and circulation is due to the action of these vortices. Intensity of such vortices acting an elementary blade length ‘ds’ in the form of circulation ‘ Γwb ’ will be (Fig. 3.4) Γwb =

z

→

→

w ⋅ ds →

→

→

Since circulation Γ is the line integral of velocity around the element. Since w = C – u Γwb =

z z z

→

z FGH C − uIJK .ds →

→

C ⋅ ds = →

→

Γ cb =

z

→

z

→

→

→

C ⋅ ds − u ⋅ ds = Γcb − u ⋅ ds

s

Since,

→

s

s

=

→

s

→

...(3.53)

s

→

C ⋅ ds

s

Applying Stokes theorem, which states that circulation around a contour is equal to the sum of the vortices within the area of the contour, to the above equation

z s

Hence,

→

→

u ⋅ ds =

z

z

→

z

u cos θ . ds = (rot u) n dA = 2ωdA = 2ωA

s

Γwb = Γcb + 2ωA

s

s

...(3.54)

53


If the impeller blade or airfoil is very thin, area A will be small and hence 2ωA being very small, when compared to ‘ Γcb’ it is neglected. Hence 2π . (Cu2 r2 – Cu1r1) ...(3.55) z In axial flow pumps, impeller blades are airfoils. For analysis, a cylindrical section of cascade of impeller blades is considered as equal to blades with finite thickness, displaced at a distance of pitch ‘t’ between two successive blades, and spread over from (∞) to (∞). The Z-axis of the coordinates coincides with the axis of rotation of the cylindrical section. Projection of this cylindrical section perpendicular to Z-axis will be zero. Hence the equation (3.54) can be considered for the cylindrical section or cascade.

Γwb = Γcb =

The boundary layer thickness ‘δ’ in real fluid flow over blades, is very small, about 1% of blade chord length ‘l’. The relative velocity on the blade is zero. Hence, circulation will be zero. The flow velocity beyond the boundary layer thickness can be considered as ideal fluid flow. Hence, neglecting the circulation in the boundary layer thickness, since it is very small, the equation (3.55) can be applied to the relative velocity of flow over impeller blades of axial flow pump also. Head developed by the impeller blade as per Eular’s equation is

Cu 2 u2 – Cu1 u1 ω = (Cu2 r2 – Cu1 r1) g g

H∞ =

ωΓimp

=

2πg 2πgH m

Γimp = z Γ =

or

...(3.56)

ω t

p 2 , z2

2′

2

u

p 1, z 1

1′

1 t

Jowkovski’s theorem

Fig. 3.16. Flow over a profile of a cascade system

3.11 FUNDAMENTAL THEORY OF FLOW OVER ISOLATED PROFILE Theoretical flow of fluid over a cylinder with plain flow combined with source, sink and vorticity (circulation) is usually considered for the study of flow over isolated airfoils or hydrofoils used in axial flow machines.

54


y

y C

φ=0

–a

x A

0

B

=

=

0 0

a α0

x B

D

φ

φ

0

φ

=

0

φ=

φ=0 A

C

D (a)

(b)

v0

(c)

Fig. 3.17. (a) Plane flow with doublet (b) Plane flow with doublet and circulation (c) Actual flow

Plain straight flow with infinite velocity (C∞) combined with a source and sink of same intensity (q) kept at origin i.e., a doublet located at point ‘O’ [Fig. 3.17 (a)] leads to a flow over a cylinder. The velocity | C | at any point on the cylinder surface, located at an angle ‘θ’ from X-axis will be | C |= 2C∞ sin θ. The direction of velocity | C | will be tangent to the cylinder surface at the point considered. At point A and B, instantaneous velocity | C | = 0. Since θ = 0 and 180°. At points C and D velocity | C | = | Cmax | = 2C∞ since θ = 90° and 270°. Applying Bernoulli’s equation, pressure at any point in the cylinder surface can be determined from the known velocity. It is evident that velocities and pressures are equal at symmetrical points. The vectorial sum of all pressure is zero. There is no flow separation under theoretical flow. In practice, however, due to real viscous fluid flow, flow separation takes place at the outer half of the cylinder, which is called Dalambir’s paradox [Fig 3.17(c)]. Due to addition of circulation (vorticity Γ ) to the above mentioned flow, i.e., plain flow with doublet and with vorticity located at the centre of the cylinder i.e., at the origin of the axis, the resultant velocity at any point on the cylinder changes although symmetrical with respect to Y-axis. Due to the introduction of vorticity, points A and B are shifted with downward direction towards point D [Fig. 3.17(b)]. The velocity at point C is greater than the velocity at point D. Correspondingly, pressures at point D is higher than pressure at point C, as per Bernoulli’s equation. This results in net upward force called lift force on the cylinder. The cylinder is made to raise upward. This effect is called Magnus effect. The higher the lift force will be, if the intensity of vorticity is higher. This lift force will be Y = ρC∞Γ ...(3.57) where ρ is the density of the fluid, C∞ is the infinite velocity of approach to the profile and Γ is the vorticity induced. This is called the theorem of Kutta-Jowkovski about lift force on a profile of any form. The angle of shift of point A from X-axis ‘α0’ is given by sin α0 =

Γ 4πaC ∞

...(3.58)

55


where ‘a’ is the radius of the cylinder. Same condition prevails but reversed, if the rotation of vorticity in the direction of circulation is reversed. The lift force will be towards the downward direction and the points A and B shift towards the point C. In that case CD > CC and PC > PD. In general, the lift force Y is determined as Γ = 4πaC∞ sin α 0 and Y = 4πaρC

2 ∞

  

sin α 0

...(3.59)

Actually, in fluid machines flow of fluid over a profile takes place at an angle θ∞ to X-axis, i.e., with an angle of attack α∞. The infinite velocity C∞, is directed at an angle of α∞ from X-axis, when passing over the profile [Fig. 3.18(a)]. y

η

y B

x

C*

z plane C

A C∞

ζ

x C ∞∗

θ∞

(a)

ζ plane

θ∞

(b)

Fig. 3.18. (a) Flow with angle of attack, (b) Conformal transformation of airfoil from cylinder

3.12 PROFILE CONSTRUCTION AS PER N.E. JOWKOVSKI AND S.A. CHAPLIGIN Theoretical fluid flow study on airfoils, used in fluid machines, is done from the known flow study on cylinder using conformal transformation as suggested by Prof. Jowkovski and Prof. Chapligin. Cylinder in Z = x + iy plane is transformed into a plate or a profile which is in ζ = ξxi + ζ plane, based on a mathematical relation z = f (ζ). [Fig. 3.18(b)] While doing so, the magnitude and direction of the infinite velocity C∞ of the fluid approaching the blade, the circulation along the contour of the blade and the forces acting on the flow by the blade on both blades remain same. Necessary conditions are | f ′ (z)z →∞ | = 1 and arg f ′ (z) = 0  or

dζ dz z →∞

...(3.60)

  

= 1

The transformation function, to meet the above condition, is given by ζ = Z+

a2 = Z + Z′ , Z

where

Z′ =

a2 Z

...(3.61)

56

ROTODYNAMIC PUMPS (CENTRIFUGAL AND AXIAL) l = 4a

Z and Z′ are real and imaginary planes containing circles K and K′ respectively (Fig. 3.19) and ‘a’ is the radius of the cylinder. The relation between the plane Z and the radius ‘a’ is given by Z = a eiθ = a (cos θ − i sin θ) = (x + iy) ...(3.62) Combining equation (3.61) and (3.62)

α

ζ = a (cos θ + i sin θ) + a (cos θ − i sin θ)

Q A1 A2 z 0′ z0 v v O A1′

a2 as per equation (3.63) Z Referring to Fig. 3.20. Point ‘A′ ’ of the Z′ plane, located at the circle K′, is the inverse of point A1 of the Z plane, located at the circle K, with respect to the circle Q by the relation. r1eiθ1

+ 2a

K

of two vectors Z and

a2

+a

v∞

Fig. 3.19. Transformation of circle to plate

= 2a cos θ ...(3.63) It is evident from the equation (3.63) that cylinder of radius ‘a’ is equivalent to a plate having a distance of –2a and +2a from the origin. Plate length is l = 4a. The vector ζ is equal to the geometrical sum

Z′ = r1′ eiθ1 =

O

–a

– 2a

K′

Fig. 3.20. Construction of inverse of circle

=

a 2 − iθ e 1 r1

...(3.64)

a2 argument θ1′ = – θ1 for the complex point Z′. Points A1 and A2 are inversely r1 located with respect to the circle of radius ‘a’ in such a way that r1 r2 = a2. Module r1′, inverse of point A1 is determined as module of point A2, i.e., r1′ = r2. The mirror image of A2 is A′1 with an argument θ1′ = θ1 and modules |r1′ |. The relation between Z1 and its inverse Z′1 is given as Module | r1′ | =

x′+ iy′ =

a 2 ( x − iy ) a2 = 2 x + iy x + y2

...(3.65)

Τhe real and imaginary parts are x′ =

a2 x a2 y and y′ = – x2 + y 2 x2 + y 2

a 2 x′ a2 y′ In the same manner x = 2 and y = 2 ...(3.66) x ′ + y ′2 x ′ + y ′2 The radius R of the real circle K with centre at z0 = x0 + iy0 relative to the circle Q can be written as (x – x0)2 + (y – y0)2 = R2

...(3.67)

57


From the above equations (3.66) and (3.67), the parameters for inverse circle K′ i.e., R′ x′0 and y0′ can be determined. a 4 ( x ′2 + y ′2 )

(x

2

+y

– 2x0

)

2 2

a 2 x′

+ 2y0

( x′2 + y ′2 )

a 2 y′ ( x ′2 + y ′2 )

– (R2 – x02 – y20 ) = 0

or

a4 – 2a2x0x′ + 2a2y0 y′ – (R2 – x20 – y20) (x′2 + y′2) = 0

or

x′2

+

y′2

+

2a 2 x0 x ′ ( R 2 − x02 − y02 )

–

2a 2 y0 y ′ ( R 2 − x02 − y02 )

–

a4 ( R 2 x 20 − y 02 )

=0

This equation can be modified and rewritten in the following form:

 a 2 x0  x′ + 2 R − x 02 − y 20  =

=

2

  a 2 y0  +  y ′ + 2 R − x 20 − y 20   a4

(R 2 − x02 − y02 )

  

2

 x02 + y02  1 + 2 2 2    R − x0 − y0 

a4 R2

...(3.68)

( R 2 − x02 − y02 ) 2

From the above equation (3.68), equation for inverse circle K′ relative to circle Q is R′ = R

and

y0′ = y0

a2 ( R 2 − x02 − y02 )

;

x′0 = – x0

a2 ( R 2 − x02 − y02 )

a2

...(3.69)

( R − x02 − y02 ) 2

by

Rb

B

z

A yb – 2a

ξ

a a

–a

x + 2a

z′ a

a

b = 4a

Fig. 3.21. Transformation of circle into thin curved plate of an arc of a circle

58


3.13 DEVELOPMENT OF THIN PLATE BY CONFORMAL TRANSFORMATION For transforming cylinder circle ‘a’ into an arc i.e., a camber thin plate, the centre of the real circle ‘K ’ with radius Rb is located vertically above the centre of the cylinder circle ‘a’ on the imaginary axis in Z-plane Z-axis (Fig. 3.21) such that x0= 0, y0 = yb and R2b – y2b = a2. Substituting this value in equation (3.38) (3.70) R′b = Rb . x′0 = 0 and y′0 = yb which indicates that inverse circle K′ with radius R′b relative to the main circle ‘a’ exactly coincides with the real circle ‘K’ with radius Rb. Centre z0 and z′0 of the circles K and K′ concides each other. Transformation of real and imaginary circles K and K′ in Z-plane to ζ-plane is done by the geometrical summation of vectors Z (real) and Z′ (imaginary) (Figs. 3.20, 3.21). Since in Z-plane real and imaginary circles K and K′ coincide each other, z and Z′ vectors drawn from the origin ‘O’ meet the same circle K and K′. Thus conversion of entire circle with radius Rb = R′b represents in ζ-plane an arc i.e., cambered thin plate, with a chord length of l = 4a.

3.14 DEVELOPMENT OF PROFILE WITH THICKNESS BY CONFORMAL TRANSFORMATION Prof. Jowkovski developed a cambered profile with thickness i.e., airfoil by shifting the centre z0 of the real circle K (Fig. 3.22) along the line ab towards the negative x direction from the imaginary axis. The radius of the circle K is equal to ‘az0’ and the centre of this circle is z0. K R R′ z ζ

z0 v0

II

b

v0′

z 0′ a

II a

z′

4a

Fig. 3.22. Transformation of circle into thick profile—Jowkovshi’s profile

From equation (3.69), we can write

tan θ′0 =

y0′ = x0′

 a2 y0   R 2 − x 20 − y 20 

   

 a2 − x0   R 2 − x 2 − y 20  0

   

= −

y0 x0

= − tan θ

...(3.70)

59


 y0 or θ′0 = θ0 . Inclination of the line az0 = arc tg   a + x0 From equation (3.70), we can write

  and inclination of the line az′0 = arc tg 

y0′ y0 a 2 = a − x0′ a( R 2 − x02 − y02 ) + a 2 x0

 y0′   a – x0′

 . 

...(3.71)

Referring Fig. 3.22, R2 = y20 + (x0 + a)2 So,

y0′ (a − x0 )

=

y0 a 2 (a + x0 a ) 3

2

=

y0 ( a + x0 )

...(3.72)

i.e., points z0 and z′0 lie on the same straight line ‘ab’. z0 lies at an angle of θ0 in the negative direction, whereas z′0 lies at an angle θ′0 (= θ0) in the positive direction of X-axis. The inverse circle K′ passes through the point +a′ with radius (az′0 ). The geometrical summation of lines drawn from origin ‘O’ at the same angle with respect to X-axis meet the circle K and K′ in Z plane, gives a point on the thick profile in ζ-plane. The drawback in this process is that the thickness of the profile at outlet is zero, which is not practically possible. Profile shape developed by Prof. Jowkovski for the given arc is determined by the single parameter, namely the distance of z0 from the imaginary axis. The magnitude of this determines profile thickness. Profile thickness at the middle section is given by 2 (R – R′ ) = 2 (z0 – z′0).

3.15 CHAPLIGIN’S PROFILE OF FINITE THICKNESS AT OUTLET EDGE OF THE PROFILE In order to avoid zero profile thickness at the outlet edge, Prof. Chapligin suggested the formation of thick profile from the inverse of ellipse. He suggested that the main circle K is selected such that it touches the real axis at point x1 > a (Fig. 3.23). Then inverse circle K ′ will meet real axis at point

x′ =

a2 . x1

The distance x1 – x1′ = δ can be taken as the parameter for thickness of outlet edge. Referring the Fig. 3.23 and the equation relating the radius and coordinates of centre of the circle K and its inverse K′ a relation can be obtained. i.e., R x y = 0 = 0 ′ R x0′ y0′

from which ∠θ0′ = ∠ θ0 and ∠ 0x1 z0 = ∠ 0x1′ z′0 = i.e.,

x1 z0 || x1′ z′0

...(3.73) θ 2

60


VI

R R1

V 5

6 6′

5′

IV

7

R ′1

VII

7′ VIII

4

8

8′ III II

1 1′

I

9′

3′ 2′

2

IX 9

4′

3

z ′0

z0

δ

10′

1 1′ 0

3 3′

III

10

2

2′

II

4

8 5

IV

6

5′

6′

x1 11′

XI

10′

X

9

l

4′

X

7

8′

9′

7′

V

IX

VIII

VII VI

Fig. 3.23. Profile formation with finite outlet edge thickness—Chapligin’s method

where θ is the angle of curvature of the centre line of the profile. For the given value of length ‘l’ thickness at the centre ‘∆’ and angle of curvature of the centre line θ, referring Fig. 3.23 and the equation (3.73) the radius Rb of the main circle K and the coordinates of its centre x 0 and y 0 will be R b= x0 =

1  θ ∆  sec +  2 2 l Rb θ θ  cos  ∆ cos − δ  l 2 2 

      

...(3.74)

θ 2 The outlet edge thickness δ is selected as = 0.5 to 0.6 ∆. As per equation (3.68) the main circle K is shifted from point x1 toward the beginning of the coordinate of the parameter δ to get point x′1. Since

y0 = Rb sin

θ′0 = θ0 , a line parallel of x1 z0 is drawn from x′1 to meet the line O z′0. This line is the mirror image with respect to Y-axis of the line O z0. This process gives the centre of inverse circle z′0. The radius Rb of the inverse circle K′ is determined by the line z′0 x1′ and is equal to R′b = Rb – ∆. Here also the point on ζ-plane − is determined from the Z-plane as the vector summation ζ = Z + Z ′ .

61


For ease in construction of this profile, an additional circle R is drawn from the origin, with a suitable radius which encloses the main circle K and inverse circle K′. The circle R is divided into a number of divisions, about 20 to 26, so that, from each point of the upper half, corresponding symmetrical point at the lower half of the circle with respect to X-axis is determined. Symmetrical points are identified with the same designation such as I, II etc. Very near to the inlet and outlet edges more number of points are selected. Above X-axis, the meeting of radius of supplement circle I–0, II–0 etc., with main circle K are designated as 1, 2, 3 etc., and with inverse circle K′ with 1′, 2′, 3′ etc. Below X-axis the meeting of radius of supplementary circle with main circle is designated as 1′, 2′, 3′ etc. and with inverse circle as 1, 2, 3 etc. Middle point of the lines joining identical points 1–1, 1′–1′, 2–2, 2′–2′ etc. gives the profile of Chapligin. If profiles are to be drawn for actual flow condition, then the axis of the coordinates should be rotated to an angle corresponding to, angle of attack of the profile in cascade. The coordinates of the circle, x0 must be shifted along X-axis towards inlet edge of the profile and co-ordinate y0 must be shifted towards the outlet edge.

3.16 VELOCITY DISTRIBUTION IN SPACE BETWEEN VOLUTE CASING AND IMPELLER SHROUD Flow through the space between volute casing and impeller shroud has the following characters: (1) Very near to the impeller shroud, the fluid flow velocity will be equal to the shroud velocity at the point considered. (2) Due to stationary condition, there is no flow near casing. Fluid velocity changes from casing wall to shroud wall i.e., changes from ‘0’ to ‘u’. Apart from that due to rotation of impeller, the fluid will be thrown out towards periphery near the impeller, but returns back towards inward direction from periphery near the casing wall, as a result of which a vortex, circulatory flow exists along with the mainflow QL1, passing through the wearing ring clearance. Considering an elementary radial height ‘dr’ and axial length ‘l’ with inner radius ‘ra’ and outer radius ‘rb’ (Fig. 3.24) and applying moment of momentum equation relative to axis of the pump. M = ∫ ρCn dA. Cu r

...(3.75)

r

p2

p2 ∆r

l

b

Q sz

2

2

u 2 u–′ 2 2g

p

Q s3

Q′

r

Di

ra rb

p1

γ

pi

l

p1 p′ i pi

(a)

(b)

Fig. 3.24. (a) Flow through the space between casing and impeller shroud (b) Pressure distribution

62


where

ρ — the fluid density Cn — normal component of absolute velocity C Cu — tangential component of absolute velocity C M — moment of external forces acting of the surface area ‘dA’ dA — elementary surface area.

Elementary area dA is the sum total of impeller shroud, area at casing surface and outer and inner cylindrical surfaces at rb and ra respectively. The moment of normal components of the absolute velocity on impeller shroud surface and casing surface are equal to zero. So,

M=

∫ ρCr dA . Cu rb

Ab

–

∫ ρCr dACu ra

...(3.76)

Ar

Under turbulent flow conditions, the flow velocity in the space will be constant but increases from zero to this velocity near the casing boundary layer and from this velocity to the impeller shroud very near to impeller shroud, which is rotating with a velocity ‘u’. Velocity u = ωr (Fig. 3.15). Taking average value of the resolved component of absolute velocity Cu, equation (3.76) can be written as M = (Cu r )b ρ

∫ Cr dA − (Cu r )a ρ ∫ Cr dA

Ab

But

∫ Cr dA = ∫ Cr dA

Ab

∴

...(3.77)

Aa

= QL1, the leakage flow

...(3.78)

Aa

M = ρ QL1, [ (Cu r )b − (Cu r ) a ] = ρ QL1. ∆ (Cu r ) a − b

Moment of external forces M = MI – Mv + Mfa+ Mfb where MI — Inducing moment of friction at impeller surface

...(3.79) ...(3.80)

Mv — Breaking moment of friction at casing surface Mfa and Mfb are the moment of friction of the control section Aa and Ab. But Mfa and Mfb , the moment of friction of the control sections are negligibly small by magnitude as well as when compared with the magnitude of MI and Mv the moment of friction at impeller and casing surfaces. Hence M = MI – Mv = ρ QL1. ∆ (Cu r ) a − b ...(3.81) From non-dimensional analysis, moment M can be expressed as

 C2  M = µ  ρ A r 2   where µ — coefficient of friction C — velocity relative to the surface A — area of the surface Combining equations (3.81) and (3.82) M = MI – Mv = µ.ρ.2πr dr = c

(u − Cu )2 C 2 . r – µ. ρ .2πr dr u .r 2 2

...(3.82)

...(3.83)

63


Comparing equations (3.79) and (3.83) µ. πr 2 [(u – Cu)2 – Cu2 ] dr = QL1.∆ (Cu r)a–b

...(3.84)

Since ∆ (Cu r ) a − b and QL1 are very small, their product is negligibly small and hence can be assumed as zero. Simplifying (u – Cu)2 – Cu2 = 0 u ...(3.85) 2 Hence, under normal conditions of wearing ring, the flow velocity in the space between casing and impeller will be half the peripheral velocity of the impeller at the point considered. When wearing ring clearance is fully damaged due to wear of the ring, QL1 considerably increases and since ∆ (Cur) is too small, the product QL1. ∆ (Cu r)a-b can be taken as zero. Equation (3.85) can be taken for calculations.

or

Cu =

3.17 PRESSURE DISTRIBUTION IN THE SPACE BETWEEN STATIONARY CASING AND MOVING IMPELLER SHROUD OF FLUID MACHINE When fluid moves in the space between stationary volute casing and impeller shroud, which is rotating with a velocity u, the fluid can be considered as the fluid moving as a solid body. Based on this, pressure distribution can also be determined. Taking r, u, z coordinates, for an elementary fluid section dr, dz, rdθ the basic hydrodynamic equations in the space can be written as Fr – –

1 ∂p = 0 ρ dr

1 ∂p = 0 ρ rd θ –

1 ∂p = 0 ρ dz

dCr dCu dC z , , are is equal to zero. The mass force is the dt rdθ dz centrifugal force ‘FCF’ and is directed in the radially outward direction. Hence

Since the total differential of velocity

where

dp = FCF = ρω2f r dr ωf — is the angular velocity of the moving fluid ρ — density of the fluid

FCF — centrifugal force of the unit mass considered r — radius of the elementary mass.

...(3.86)

64


Taking p2 and p1 as the pressures at outlet and at inlet of the impeller passage respectively, at radius r2 and r1 respectively the pressure p at any radius ‘r’ can be written as r2

p2 – p =

∫ ρ ωf2r dr = ρω f

2

r

r22 − r12 2

...(3.87)

Under normal flow conditions through such flow passage referring equation (3.85), the angular velocity of flowing fluid, ‘ωf’ is ωf = =

2πn ω , where ω is the angular velocity of the impeller shroud ω = 60 2

u , where u peripheral velocity of the impeller at radius ‘r’. Equation (3.87) can now be written as r u22 − u12 p2 − p ω2 r22 − r 2 = . = 8g γ 4 2g

  u22 1  r   u22 − u 2 −   p = p2 – γ = p2 – γ 8g 8 g   r2   2

or

...(3.88)

Equation (3.88) shows a parabolic pressure distribution along the radial direction [Fig. 3.24 (b)]. Equation (3.88) is used to determine axial thrust at front and rear side of the impeller. In case the clearance between the stationary and rotary members is damaged, the above formula (equation 3.88) cannot be applied. The flow follows the Bernoulli’s law. Neglecting the effect at the surface roughness, applying Bernoulli’s equation.

p C2 p2 C22 + + = γ 2g γ 2g But C2 = C m2 + C2u and assuming Cm remains constant throughout the passage 2 p2 Cu′2 + γ 2g

=

Cu2 p + γ 2g

p = p2 – γ

or

Cu2 − Cu′ 2 2g

Since energy is constant throughout the passage Cu′ 2 r2 = Cu r

C′ 2 p = p2 – γ u 2 2g

and Taking Cu′ 2 =

  r 2    2  − 1   r1    

u2 , which is prevailing mostly at this space and rearranging 2 2 2  u22  r2  1 −  r     p = p2 – γ   8 g  r    r2  

...(3.89)

Hence, in case of damaged wearing rings, the pressure drop across the passage increases at a faster 2

 r2  rate by   times than for normal wearing ring. r

4 THEORY AND CALCULATION OF BLADE SYSTEMS IN CENTRIFUGAL PUMP 4.1 INTRODUCTION The impeller is the main element in a centrifugal pump. Entire construction of a pump depends upon the impeller. Impeller design forms the most important part in pump design. The fundamental equation of impeller, determines the head developed by the impeller with respect to the increase in the moment of momentum of the fluid flowing through the impeller i.e., to get a relation between dynamic and kinematic parameters of impeller. But this fundamental equation does not give any relation between the form and shape (dimensions) of the blade system with the change in the moment of momentum of the fluid in impeller. A kinematic study of the ideal fluid flow through the impeller based on hydrodynamic action in general is yet to be determined and found to be an unsolved problem till today. The real fluid flow conditions are still determined from the ideal fluid flow confition only. The study of fluid flow in impeller is done by the use of theoretical equations along with the correction factors which are determined from experiment. For the calculation of blade system in impeller, wherein, the length of the flow passage between two blades is much longer than the width of the passage, elementary one dimensional theory can be used successfully. In case, the blades are kept at a distance apart i.e., the width is longer, the interaction between two successive blades can be neglected, and the blade can be considered as an isolated blade. Hence, two elemertary theories are existing for the impeller calculation. Application of the correct procedure is based on the correction factors, which are determined by experiments and also based on its boundary conditions. If the theoretical means of approach for the impeller design, coincides with real fluid flow, the design is considered as most satisfactory design.

4.2 ONE DIMENSIONAL THEORY Elementry one dimensional theory for the centrifugal pump design is given by the mathematician Leonard Eular (1707–1783), member of the St. Petersburg Academy of Sciences. His one dimensional theory is still considered as the fundamental theory for the centrifugal pump design. In early days, Eular’s one dimensional equation agreed perfectly with practical result due to the fact that each impeller passage was constructed as individual tubes, and the speed of the pump was very low. Impeller flow passage were too longer than normal. The length to breadth ratio for the impeller passages was higher. Due to the introduction of electric motors, I.C. Engines and high speed turbines, as pump prime movers, pump speed was increased. Correspondingly, the outer diameter D2 was decreased. This in turn decreased the flow passage length, the passage width remaining same. Existing one dimensional theory did not agree with the practical results, when length to breadth ratio of impeller passage is reduced. The flow pattern in impeller passage is completely changed. 65

66


As per the existing one dimesional theory, impeller blades are considered to contain infinite number of blades kept at equal distant apart with infinitesimally smaller blade thickness. Practically finite number of blades with finite thickness are adopted in impellers, diffuser etc. In order to have a complete agreement between theoretical design and practical results, corrections are introduced in the actual design of finite number of blades with finite thickness. Under theoretical flow, a study through impeller passage, i.e., for conditions of infinite number of blades, with infinitesimally smaller blade thickness, the flow is axisymmetric. At any radius, the average flow velocity is constant in the impeller passage and is calculated from continuity equation. The direction of fluid flow on the blade is the tangent to the blade drawn at the point, at the given radius ‘r’, where the velocity is determined. In real fluid flow i.e., for finite number of blades with finite thickness, Eular’s one dimensional theory is applied for impeller design with corrections, agrees perfectly with practical results. In actual flow with finite number of blades with finite blade thickness, the velocity at any radius across the flow passage width between two successive blades of impeller passage is not constant (Fig. 3.4). The flow is not axisymmetric due to the interaction between the blade and fluid. The fluid is pushed by the blade. This is the main reason, that all the flow passages must be completely immersed within the flowing fluid or must be filled with flowing fluid completely in all rotodynamic machines. The theoretical head (H∞), determined, as per the Eular’s one dimensional theory for infinite number of blades will not be equal to the actual head (Hm) determined as per finite number of blades condition. (H∞) and Hm are related as ...(4.1) H∞ = (1 + p) Hm where, p is the correction coefficient for finite number of blades application. Different authors developed different values of correction coefficient ‘p’ in different form. Application of this coefficient in the equation 4.1 to determine the total head developed gives a very good result.

4.3 VELOCITY TRIANGLES C2

u2

B2 C2

C m2 α2

w2

α2

C u2 Ci

β2

B

β2

w2 D2

A2

αi

w1

C m2

C u2 u2

C ui

β1

(b)

r2

A

α1

C1 B1

C u1 0 E

C1 u1

K

α1

C m1

C 1u u1 (a)

(c)

Fig. 4.1. Velocity triangles with symbols

w1 β1

D1

67

THEORY AND CALCULATION OF BLADE SYSTEMS IN CENTRIFUGAL PUMP

The following symbols are used in drawing velocity triangles : (Fig. 4.2) u — Vane or blade velocity =

πDn (m/sec). 60

C —Absolute velocity of flow of fluid i.e., velocity of the fluid with reference to the earth or any non-moving object. w —Relative velocity of the fluid in the blade passage, i.e., the velocity of the fluid with reference to the blade or impeller, in other words, the velocity of the fluid inside the blade passage, when the blade velocity is brought to zero. →

→

→

→

→

→

w = C – u or C = w + u α —Absolute angle, the angle between the absolute velocity ‘C ’ and blade velocity ‘u’

β — Vane angle or blade angle—the angle between the relative velocity ‘w’ and vane or blade velocity ‘u’. C m2 C m3

δ2

β2

2

C m0

B2

C0

t2 C m1 r2

B1 r1

r0

β1 δ

t1 δ1

S2

S1

r1

r2

(a)

(b)

Fig. 4.2. Symbols and suffices used in impeller

Suffix 0—indicates the conditions before the impeller blade entrance edge and at impeller eye. 1—indicates the conditions on the impeller blade entrance edge. 2—indicates the conditions on the impeller blade outlet edge. 3—indicates the conditions after the impeller blade outlet edge. The relative velocity of fluid in the impeller passage ‘w’ is equal to the vectorial subtraction of →

→

→

→

→

→

absolute velocity ‘C ’ and the blade velocity ‘u’, w = C – u or C = w + u (Fig. 4.1). The direction of blade velocity ‘u’ is always tangential to the circle of radius ‘r’, whereas the direction of the relative velocity of the fluid ‘w’ at any point on the blade will be the tangential to the blade curve at the given

68


radius ‘r’. These two velocities i.e., relative velocity ‘w’ is inclined at an angle ‘β’ with respect to the blade velocity ‘u’. A parallelogram is drawn, with the relative velocity vector ‘w’ and blade velocity vector ‘u’ at the radius ‘r’ at the point on the blade in the impeller passage. The diagonal of parallelogram will be the absolute velocity, C, both in magnitude and in direction. If all these three velocity vectors are drawn in position, we get a triangle called ‘velocity triangle’ (Fig. 4.1). The subtended angle between absolute velocity ‘C’ and blade velocity ‘u’ is called absolute angle (α) and the angle subtended between relative velocity ‘w’ and blade velocity ‘u’ is called the blade or vane angle (β). → → →

The velocities C , u , w at any radius ‘r’ between inlet and outlet of impeller blade passage can be obtained by constructing velocity triangle at the point on the blade at radius ‘r’ (Fig. 4.1). By constructing such triangles at different radii ‘r’ between inlet radius ‘r1’ and outlet radius ‘r2’, we can find the velocity distribution in impeller blade system. One such velocity distribution in impeller blade passage is given in Fig. 4.3.

θ° 100 25

φ190

φ150

9 80 20 8

φ122

φ136

φ 178 φ 164

γ

11

φ 50

φ110

φ 105

along the streamline w

Cm

7 60 15 6 5 40 10 4 20

3 5 2 1 r 1= 55 61 68

Cm θ

δ = 2 mm

w 30

5 mm

8

Meridional section

δ

75

82 89 r2= 95

Fig. 4.3. Velocity and angle variation in impeller passage

If width of the passage is very small, as per one dimensional theory of flow, fluid enters the inlet edge tangentially and hence there is no shock loss at entry. Fluid leaves the outlet edge tangentially and hence there is no loss, at exit. From inlet to outlet. Fluid moves tangentially over the blade. The direction of the fluid at any point in the impeller passage will be the direction of the blade at that point i.e., will be the blade angle ‘β’ at that point, which is the angle between relative velocity ‘w’ and blade velocity, ‘u’. The velocity triangles are shown in Fig. 4.4. Flow velocities wm and Cm in the impeller passage are equal and determined as wm0 = w0 sin β0 and Cm0 = C0 sin α0 Referring to (Fig. 4.4), the relative velocity of the fluid will be tangential to the blade at all points, on the line AB. Actually, the fluid moves along the direction of absolute velocity (C1 to C2) from α1 to α2 line AB′ . The fluid at point x on the blade will be actually at x′ i.e., on the absolute velocity line. If the time taken for the fluid to travel from point A to x on the blade is ‘t’ and if the angular velocity of the blade is constant and is equal to ‘ω’ then the included angle θ will be θ = ωt.

69


The real fluid flow differs from the above mentioned theoretical one dimensional flow. u2

C2

α2

C′

α

u

w

w2 B

β

w 1 C′

1

β1

β2

D2

ϕ

α1

x′

A B′ D1

Fig. 4.4. Graphical representation of velocity triangle

4.4 IMPELLER EYE AND BLADE INLET EDGE CONDITIONS Fluid enters the impeller eye, with a velocity C0 and is maintained constant until it reaches the point before inlet edge. The flow velocity at the impeller eye is calculated by using continuity equation as : (Refer Fig. 4.4). If there is no impeller hub or shaft Qth = C0 . π D02 4 Example: End suction agricultural pumps, domestic pumps. If the hub or shaft is protruding into the impeller eye Qth = C0 .

π D02 − d h2 4

(

)

      

...(4.2)

where dh is the hub diameter. Example: Double suction pump, multistage pump. When the specific speed ‘ns’ is very low (ns = 40 to 50) the inlet edge of the impeller blade will be parallel to axis of the pump after bend portion of the impeller passage. When the specific speed ‘ns’ is 60 to 150, the impeller blade is extended towards the bend portion at inlet of the impeller passage to reduce the blade loading. Blade edge at inlet will be neither parallel nor perpendicular to the pump axis, instead it is inclined. (Fig. 3.2). Since quantity ‘Q’ is low and the total head ‘H’ is higher for lower specific speed pumps, the breadth ‘B2’ will be smaller and diameter ‘D2’ will be larger. The meridional velocity change along the transverse section of the inlet edge i.e., from shroud to shroud will be negligible and hence is assumed to be constant throughout the passage. For infinite

70


number of blades with infinitesimally smaller blade thickness, the streamlines in the impeller passage are congruent and the flow is considered as one dimensional. The flow velocity Cm0 is determined from the velocity (C0). The velocity C0 is determined from the equation (4.2). Usually the meridional flow velocity Cm0 is assumed to be equal to C0. Sometimes, it is increased, Cm0 = 1.03 to 1.05 C0, in order to get smooth, uniform flow at the bend portion of the impeller passage. In real fluid flow, however, finite number of blades, with finite thickness are used. Referring to Fig 4.3, the blade thickness ‘δ’ will be occupying a circumferential distance of ‘s’ due to the blade angle ‘β’ and is determined as

s=

δ sin β

...(4.3)

If there are ‘Z’ number of impeller blades, the actual circumferential length available for the flow of fluid is (πD − Zs ) instead of πD.

4.4.1 Inlet Velocity Triangle Due to finite vane thickness the inlet area, blocked by one vane will be s1 =

blocked by Z number of blades is Zs =

δ1 . Total length sin β1

Z δ1 sin β1

. Total available area due to this will be (πD1 – Zs1)B. Due

to this the flow velocity before the inlet edge, Cm0 is increased to Cm1 on the blade, i.e., the moment fluid touches the inlet edge. The meridional flow velocity Cm, the resolved component of absolute velocity C, in radial direction at points 0 and 1 are equal to corresponding value of wm the resolved component of relative velocity w in the same direction. Cm0 = C0 sin α0 = wm0 = w0 sin β0 and

...(4.4)

Cm1 = C1 sin α1 = wm1 = w1sin β1

Since quantity of flow Q is same, at inlet Qth = πD.B.Cm0 before the inlet edge and Qth is equal to = (πD1 – zs1) B1 Cm1 on the blade  πD1  Cm1 =   Cm0 = Cm0 .K1  πD1 − zs1 

where,

K1 =

πD1 1  =  zs πD1 − zs1  1– 1  πD1

 =  

...(4.5)

1   Ζδ1  1− π D1 sin β1 

   

...(4.6)

71


φ-6.10

94.9 76

φ

φ

40

φ

14R 150R 14

5.5 12° 38

62 35

20R 3R

28 14 22

25 Hy 45 200

14.8

16.4

24

18.4

25

13

21.1

35

φ

φ

0.000 φ ++ 0.020

Fig. 4.5. (a) Impeller eye without hub

18 17.5 17 16.5 15 14 13 12

φ φ

φ

8.5 φ 9 φ 9.5 φ 10 φ 10.5 φ 11 φ 11.5

φ φ φ φ

12

φ φ

11.5

Fig. 4.5. (b) Impeller eye with hub

φ

φ

72


Since the inlet blade velocity ‘u1’ remains same at points 0 and 1 inlet velocity triangle A0, B1, C1 at point O, before the inlet edge changes to A1B1C1 at point 1 on the inlet edge of the blade (Fig. 4.6) due to the increase in ‘Cm’ from ‘Cm0’ to ‘Cm1’. A1

1

w1

δ1

B1

C m1

α0

C0 α1

C m0

C

C m0

C m1

A0

w0

β0

t1

β1

β1

s1

u0 = u1

C u0 = C u1 (a)

(b) β 2 B 2c 1,0= c m1

w1 w 1,0

c ′1= c ′m1 R2

B 1 β1

C1

β1

C u1 R1

β1

β10

w1′ β′1

u1

s1 (c)

Fig. 4.6. (a), (b), (c) Inlet velocity triangle—effect of blade thickness and normal entry

Practically the inlet edge is rounded off in order to make the fluid to enter the blade tangentially without any shock. Because of this, entry losses are reduced and the hydraulic efficiency is increased. However, in practice, the inlet blade angle B10 is increased by an angle ‘δ’ in order to reduce shock losses at entry and also to improve cavitational characteristics. The inlet angle B10 is increased by δ = 3° to 10° and in special cases up to 15°. Actual blade angle at inlet β1 will be Fig. 4.6 (c). ...(4.7) β1= β1, 0 + δ = β1, 0+ (3° to 10°) At optimum conditions δ = 0 because, fluid flows tangentially to the blade. For the optimum cavitational conditions, it is recommended to have β1, 0= 16° to 20° and after correction for vane thickness and adding angle of attack δ, final value of β1= 18° to 25°.

4.4.2 Normal or Radial or Axial Entry of Fluid at Impeller Inlet The direction of flow of fluid in approach pipe or in suction pipe before entering the impeller is normal to the area of cross-section. The flow rate is calculated as per the continuity equation (4.2). 2

πD 0

π( D02 − d h 2 ) C0 depending upon whether the impeller hub, on pump shaft 4 4 is protruding or not. D0 is the eye diameter or inlet diameter of pipe, dh is the impeller hub diameter and ds is the pump shaft diameter, C0 is the flow velocity normal to the area of cross-section, eye of the impeller. The velocity C0 will be in axial direction. At inlet, before entering the impeller inlet edge, the

Qth =

. C0 (or)

73


flow direction will be radial in centrifugal pump, diagonal in mixed flow pump and axial in axial flow pump. The flow rate is calculated as Q = πD0 B0 . Cm0 = πD1B1 Cm1

...(4.8)

This condition of flow is called normal entry (i.e., Axial entry in axial flow pump and radial entry in centrifugal pump). Inlet velocity triangle, under such condition, is as shown in Fig. 4.6 (c). From the figure it is evident that C0 = Cm0, Cu0 = 0, α0 = 90°. Correspondingly C1 = Cm1, Cu1 = 0, α1= 90°. So, the Head developed Hm =

Cu 2 u2 − Cu1 u1 g

=

Cu 2u2 g

...(4.9)

For normal entry Cu0 u0 = Cu1 u1 = 0. This condition is adopted in all pump designs by which blade loading and pressure intensity on the blade will be at a reduced level. In order to get durability and dependability in operation of pump and for stable operation of pump sometimes α1 is taken between 85° and 90°. This reduces slightly the inlet pressure before the impeller entry, due to the reduced work load on impeller blades as per the equation (4.9). In practice, however, due to the rotation of impeller, the fluid also gets rotated before the blade inlet slightly. As a result a forced vortex is developed. Initial conditions of flow at inlet is determined, mostly by the prewhirl developed, due to impeller rotation near impeller eye. That is why, this action is considered as the change in the moment given by the inlet guide blade or by the influence of suction pipe, which does not lie in the same plane. The effect of this action is the reduction in input energy due to pressure reduction. This pressure reduction is not due to frictional losses occurring due to friction, taking place in the rotating impeller passages. Impeller friction losses, are separately given as hf (1 –2) in the moment of momentum equation for pumps as Cu 2 u2 − Cu1 u1 Hm = + hf (1 – 2) ...(4.10) g In order to accommodate for shock losses reduction, the inlet blade angle β1 is slightly increased over and above the angle necessary to meet the reversed direction of flow at inlet. The flow rate is also slightly increased over and above the loss of flow in clearance, and reduction in the area of cross-section at inlet as well as for induced prewhirl. That’s why normal entry is assumed even in the absence of the inlet guide blades. If inlet guide blades are used, the inlet pressure is reduced before the entry into the impeller and the cavitational characteristics is also reduced. It is found, that higher efficiency is attained when α1is slightly lower than 90° i.e., 85° < α < 90°.

4.5 OUTLET VELOCITY TRIANGLE : EFFECT DUE TO BLADE THICKNESS Due to vane thickness, effective area at outlet (A2) is decreased. A2 = (πD2 − Zs2 ) B2 where

s2 =

δ2 sin β2∞

The flow velocity Cm2 on the outlet edge of the blade is reduced to Cm3 immediately after the blade. Since blade velocity u2 = u3 and total energy remains constant at outlet Cu2∞ the whirl velocity at outlet remains same Cu2∞ = Cu3∞  Zδ 2  Q = Cm2 .  πD2 −  B = Cm3 ( πD2 B2 ) sin β2∞  2 

...(4.11)

74


Cm2 = Cm3

= Cm3

K2 =

where,

πD2 Zδ 2 (πD2 − ) sin β2∞ 1 = Cm3 . K2   Zδ 2 1 −   πD2 sin β2∞ 

...(4.12)

1   Zδ 2 1 −   πD2 sin β2∞ 

...(4.13)

Cm2 = C2∞ sin α 2∞ = Wm2= Wm2∞ sin β2∞

and

Cm3 = C3∞ sin α 3∞ = Wm3 = W3∞ sin β3∞

...(4.14)

The outlet velocity triangle before and after the outlet edge of the blade is given in Fig. 4.7. A2 C2 B2

α3

α2 C3

C u2 = C u3 (a)

C m3

t2

w2

β3 β2

s2

C m2

A3 w3

β2 δ2

C2 u2 (b)

Fig. 4.7. Outlet velocity triangle—effect of blade thickness

In order to get higher head and efficiency, the outlet edge of the blade is made as sharp edged as shown in continuous lines (Fig. 4.7). This reduces the area blocked by blade at outlet and the flow resembles like flow with infinite number of blades with infinitesimally smaller thickness. However, angle of sharpness must be properly selected, so that there should not be any flow separation. The outlet velocity triangle A2 B2C2 due to area increase and subsequent reduction in flow velocity Cm2 to Cm3, will change into A3 B2 C2. Correspondingly, the direction and magnitude of absolute and relative velocities change (Fig. 4.7).

4.5.1 Outlet Velocity Triangle: Effect of Finite Number of Blades The direction of the flow of fluid at outlet of the impeller, under elementary theory of blade system, must be tangential to blade position at outlet. In other words, the fluid angle will be same as blade angle at outlet. Also under infinite number of blades with infinitesimally smaller vane thickness, the flow velocity distribution i.e., the relative velocity w and the meridional velocity Cm at any radius, across the channel should be equal i.e., from the trailing side or suction side of the blade to the leading side or pressure side of the next blade (Refer Fig. 3.3). Correspondingly, the velocity has the same value at leading and trailing sides of the impeller blade. Considering any blade in such a system, as per Bernouli’s equation the pressures between the leading side and the trailing side of the blade are same due to equal velocity on both sides. Under this condition, there cannot be energy transfer from mechanical to fluid by the blade system. In other words,

75


pumping will not exist. In order to have pumping or to change mechanical energy to fluid energy, the pressure at the leading side of the blade must be higher than the pressure at the trailing side of the blade. Correspondingly, the velocity (w and Cm) at the trailing side will be higher than the velocity at the leading side of the blade. When impeller rotates, the leading side of the blade exerts a force on the fluid in contact and makes the fluid to move. This unequal velocity distribution within the impellers passage can be considered as consisting of two types of flow: (1) Constant velocity of flow across the entire impeller passage combined with, and (2) A circulating velocity moving from trailing edge to the leading edge and then back to the trailing edge (Fig. 3.6). Due to this circulatory motion, a tangential velocity is created at the outlet edge of the blade, which is opposite to the direction of motion of blade and is in the same direction of blade motion at the inlet edge of the blade. Considering the outlet, the tangential velocity (∆Cu) created in the opposite direction reduces the original tangential velocity Cu2∞ to Cu2 correspondingly the total head is reduced from H∞ to Hm. Both these total heads are connected by the equation H∞ = (1 + p)Hm, where ‘p’ is the correction coefficient. Various authors derived different methods to determine the value of the coefficient ‘p’.

4.6 SLIP FACTOR AS PER STODOLA AND MEIZEL |109| Due to the flow change from theoretical to actual, in the impeller passage, outlet blade angle β2 reduces and the relative velocity w2 increases (Fig. 4.10). ω ω

90°

r2

0° β2 < 9

r2

ra

ra < r2

ra > r2

Fig. 4.8. (a) Determination of effective radius π – β2 2 C

∆ W max

Relative circulatory flow

t2 A β2 2

90° B

ω

Main flow

Fig. 4.8. (b) Flow in impeller passage

76


Stodola and Meizel suggested that ∆w2 is proportional to u2. The blade velocity at outlet ∆w2 = xu2. In order to determine the value of x, Meizel considered the flow in impeller passages consists of (1) flow with constant velocity in impeller passage along with (2) a circulatory flow with an angular velocity ω, rotating opposite to the blade rotation. He assumed that maximum value of relative circulation velocity ∆w2max occurs at the middle of the passage. The plain flow with equal velocity is along the streamline, whereas the velocity vector of the circulatory flow is perpendicular to this plain flow direction, with the result, combined velocity w2 is changed from one end to another end in impeller passages. Applying Stokes theorem, and referring to Fig. 4.8 (b) the circulation along the contour ABC will be

Γ = 2ωA = ΓAB + ΓBC + ΓCA where A is the area ABC. Since contour AB and BC are perpendicular to the streamline, circulation Γ AB = 0 and Γ BC = 0 and ΓAC = ∆w2t = ∆w2

2πr2 2πr2 , since t = Z Z 2πr

2 Γ = ΓAC = ∆w2 Z = 2ωA.

∴

The Area

ABC =

β β 1 t2 t cos .t sin = sin β2 . 2 2 2 4

∴

∆w2 =

ωπr2 sin β2 ωt sin β2 2ω t 2 sin β2 = = . Z 2 t 4

= Since ∆w2= xu2,

x =

π sin β2 . u2 Z π sin β2 Z

If β2 is increased the value x is also increased Cu2 = Cu2∞ – ∆w2 = u2 –

Cm 2 π sin β2 – u2. Z tgβ2∞

C  π sin β2  = u2  1 − – m2  Z  tgβ2∞ 

...(4.15)

The following assumptions were made by Meizel in deriving the above equation : 1. The circulatory velocity vector is perpendicular to the main flow streamline, which is not always correct. 2. The circulatory vortex moves in a closed contour which is not correct since inlet and outlet passages are open for flow. Only two sides of the blades act as closed contour.


77

3. The relative velocity w2∞ is tangential to the blade at outlet i.e., β2 of flow = β2 of the blade and flow is parallel to each other at all points of impeller outlet passage. This is correct only for more number of blades ( z ≥ 10) . For smaller number of blades, the correction factor called slip factor does not agree. Also it is assumed that inlet flow conditions, will not affect outlet flow confitions, which is also not true. In general, the slip factor equation given by Stodola and Meizel agrees with the experimental results for higher number of impeller blades.

4.6.1 Slip Factor as defined by Karl Pfliderer |97| Karl Pfliderer established a relationship for slip factor based on the blade loading (Fig. 4.9) which is based on the following assumptions : 1. Pressure drop across the unit length of middle streamline is constant in meridional section. 2. Unequal pressure and relative velocity distribution S2 exists in impeller passage before the outlet edge of the impeller blade i.e., high relative velocity and ds low pressure at the trailing side of the impeller blade dr and low relative velocity and high pressure at the leading side of the blade. High relative velocity at S1 the trailing face remains same whereas the low r velocity at the leading side gradually increases and becomes equal to the high velocity at the outlet edge. Hence for the normal entry condition (Cu1= 0), Karl Pfliderer defined a relation between H∞ and Hm Fig. 4.9. (a) Slip factor as per Pfliderer (equation 4.10) with a slip coefficient ‘p’ as H∞ = (1 + p) Hm where,

p =

ψ r 22 Z S

Z — No. of impeller blades. ψ — Coefficient depending upon the blade configuration. r2

S — Static moment of the central streamline =

∫ r ds.

r1

If the blades are radial or nearly radial ds = dr r2

S =

∫ rdr

r1

and

p =

ψ Z

=

r 22 − r12 2

1 r  1−  1   r2 

2

...(4.16)

78


yH Z 12

0

30

10 8 6 5 4 3 2 βepad

60

z

16 12 0,8 10 8 6 5 4 0,6 3 2 0,4 0

0,2

0,4

0,6

0,8 r1/r2

K 1,0 0,8 0,7 0,6 0,5

β

0,4 0,35 0,3 0,25

r1 r2

z

0,2 0,16 0,14 0,12 0,10

(a) y H = f(β) when

(b) y H = f

0,08 0,07 0,06 0,05

90°

45

45 25

20

15 10

5

0

(c) k = f

r1 =0 r2

r1 when β = 90° r2 r1 , z, β r2

0,04 0,03 0,02

0,01 0,01

0,02 0,03 0,04 0,05

0,1

0,2

0,3 0,4 0,5 0,6 0,8(r1/r2)

Fig. 4.9. (b) Correction coefficient for finite number of vanes as per S.S. Rudinoff |104|

Karl Pfliderer recommended the value of coefficient as ψ = (0.55 to 0.68) + 0.6 sin β2

...(4.17)

The value of ψ, calculated as per the above equation, coincides with practical results, only for radial type pumps, having

r1 < 0.5 and with backward curved blades. For radial blades β2 = 90° r2

ψ ≈ 1.8 i.e., nearly 50% more than normal value. For forward curved blades, it increases further. The corrected value of ψ as recommended by Pfliderer is ψ = (0.6 to 0.65) (1+ sin β2 )

...(4.18)


79

This equation is applicable for diffuser type pumps, where the inlet edge of the diffuser is kept very near to the impeller outlet edge. ψ increases if this distance increases. For volute pumps and for vaneless diffuser pumps, ψ values becomes higher. The approximate values are given below: ψ = 0.65 to 0.85 for volute ψ = 0.68 for vaned diffuser ψ = 0.85 to 1.0 for vaneless diffuser Also when α 1 ≤ 10°, ψ increases approximately by 30%. A normal value of α1 ≈ 20° is recommended for pumps for which ψ is minimum. When ψ is minimum,the power consumption is also reduced. Pfliderer’s slip factor gives a good result for pumps n ≤ 150 with back ward curved vanes. Slip factor ‘p’ increases with the increase of nS and it depends upon the surface roughness of the flow passage also. Extending the inlet edge towards the eye side as well as change in the static moment of the middle streamline ‘S ’ does not change the slip factor and hence Hm does not increase. In general Hm calculated as per Stodota-Meizel formula is found to be nearer to the experimentally determined value of Hm than Hm calculated as per Pfliderer.

4.6.2 Slip Factor as per Proscura |93| Professor Proscura mentioned that the flow of fluid in rotating curved blades of impeller is the combination of two flows: (1) plain flow with uniform and constant relative velocity across the entire flow passage width from leading side of one blade to the trailing side of the next blade of a stationary curved blade cascade system, determined by using conformal mapping from the stationary straight blade cascade system and (2) axial vortex flow. 2. Considering the flow due to axial vortex (2ω) developed within the impeller flow passage, a gave the relation between Hm and H∞ is given 2    r1   sin β2 +   sin β1   π   r2  Hm = 1 − .  Hm 2  r1   Z  1−     r  2  

≈

1 2

r  sin β2 +  1  sin β1 π  r2  1+ 2 Z r  1−  1   r2 

Considering equation (4.1), H∞ = (1 + p) Hm the value ‘p’ is 2

r  sin β2 +  1  sin β1 π 2ψ 1  r2  = p= 2 2 Z Z  r1   r1  1−   1−    r2   r2 

...(4.19)

80


where,

π ψ= 2

2

sin β +  r1  sin β    2 1   r2    2

 r1    ≈ 1.6  sin β2 +  r  sin β1     2

...(4.20)

which is similar to the equation (4.13) determined by Pfliderer. w2

Equation (4.15) is determined only for ideal fluid flow with finite number of blades and not for real fluid having friction losses due to viscosity. Before leaving the outlet edge, due to the slip factor, the relative velocity at outlet w2∞ deviates from the original direction (Fig. 4.10). Since flow rate is same Cm2∞ = Cm2 and the outlet blade angle reduces from β2∞ to β2. With the result w2 > w2 ∞ , C2 α2∞ β2< β2∞, wu2> wu2∞,Cu2< Cu2∞

w 2∞

C m2 C 2∞

C2 α

2∞

α2

C u2 C u2∞ = (1 + p) C u2

β2 β 2∞

u2

Fig. 4.10. Outlet velocity triangle–effect of finite number of blades

Cu 2 u2 Cu 2 Hm Hm 1 = = = = Cu 2∞ u2 Cu 2∞ H∞ H m (1 + p) (1 + p )

When the fluid comes out of the outlet edge, the flow velocity Cm2 reduces to Cm3 owing to the  Z δ2  sudden increase in area from  πD2 − B to πD2B2, due to the absence of vane thickness. Cu2 sin β2  2 

remains same, as defined earlier, i.e.,Cu2 = Cu3 with the result, the outlet velocity triangle changes. Since u2= u3, Cu2 = Cu3, wu2 = wu3, Cm3< Cm2, α 3 < α 2, β3 < β2.

whereas,

Hm =

C u − Cu 0u0 H∞ H = = u3 3 g ηh (1 + p )

H∞ =

Cu 2u2 − Cu1u1 g

...(4.21)

Figure 4.11 gives a comparison of H–Q curve with correction for the effect due to finite number of blades by different authors. It is suggested that the correction coefficient for the finite number of blades can be carried out as per Stodola-Meizel, if active radius ‘ra’ is considered or otherwise method suggested by Karl Pfliderer can be applied.


H1

H 1∞

u 22 π sin β2 1– g Z

u 22 g

u 22 g 1 1+p

H m as per Pfliderer H m as per Stodola-Meizel

Qp

Fig. 4.11. (a) Comparison of H–Q curve with different correction coefficients 1.0 d c

0.9

b a

0.8 a. G.F. Proskura b. Pfliderer c. Stodola-Meizel d. Rudinoff

0.7

0

10

20

30

40

50

60

70

80

90 100

Fig. 4.11. (b) Correction coefficient for finite number of vanes

4.7 COEFFICIENT OF REACTION (ρ) Total head

H m = E2 – E1 =

C22 − C12 p2 − p1 + Z2 – Z1 + 2g γ

= Hp+ Hdy where Hp = and

Hdy = Also,

p2 − p1 + (Z2 –Z1) γ C22 − C12 2g Hm =

Cu 3u2 − Cu 0u1 g

Referring to velocity triangle w22 = C22 + u22 – 2u2 Cu 2

81

82


w12 = C12 + u21 – 2u1Cu1 and Cu0 = Cu1, u0 = u1

and Hence,

u2 Cu 2 − u1 Cu 0 g

=

Hp∞ =

C22 − C12 u 2 − u12 w2 − w22 + 2 + 1 2g 2g 2g

...(4.22)

C 2 − C12 u22 − u12 w2 − w22 and Hdy = 2 + 1 2g 2g 2g

...(4.23)

Pressure head Hp∞ indicates the difference in pressure and potential energy between inlet and outlet of the pump, which is the sum of the pressure energy due to centrifugal force (Coriolis component) u 2 − u12 and due to the flow over the blade system (due to relative velocity). 2 is the increase in pressure 2g energy of the fluid within the impeller due to the rotation of the impeller, under no flow conditions i.e.,

w12 − w22 is the increase in the pressure energy of the flowing 2g fluid over the impeller blade system due to velocity reduction from inlet to outlet, when the impeller is stationary. If both these flows are combined, a circulatory flow is developed, which gives a tangential momentum at outlet and at inlet to develop total energy. For real fluid flow with friction

purely by the centrifugal force, whereas

p2 − p1 u22 − u12 w2 − w22 + 1 = + (Z2 – Z1) + hfim + hfv γ 2g 2g where ‘hfim’ is the hydraulic loss in impeller and ‘hfv’ is the hydraulic loss in volute or in diffuser. The coefficient of reaction ‘ ρ ’ is the ratio of pressure head developed to the total head

ρ =

Hdy =

H p∞ H∞

=

H ∞ − H dy H∞

 H dy  =1–    H∞ 

C2u1 + Cm22 − Cu22 − Cm21 C22 − C12 = 2g 2g

...(4.24)

=

Cu22 − Cu21 2g

Since Cm2 ≈ Cm1 and for normal entry Cu1 = 0. Hence Hdy

Hence,

gH m2 C 2u 2 = = 2g 2u 22

ρ = 1–

H dy Hm

= 1–

...(4.25)

gH m Cu22 =1– 2 gH m 2u22

...(4.26)


H p = Hm – Hdy = Hm – gH m2

= Hm –

2u22

83

C 2u 2 2g

 gH m  = Hm 1 –  2u22  

...(4.27)

β 2) AND ITS EFFECT 4.8 SELECTION OF OUTLET BLADE ANGLE (β Under normal entry, Total head (H) depends upon the oulet dimensions of impeller, D2, B2, β2, δ2, Z and speed n. Hm = where,

Cu 2∞ =

Cu 2∞u22 1 . 1 Cu 2∞u2 ⋅ ⋅ = 1+ p 1+ p g g Cu 2∞ u2

From outlet velocity triangle Cu2∞ = u2 – wu2∞ = u2 – = u2 –

Hm =

Cm 2 ∞ = u2 – Cm2 cot β2∞ tan β2∞ Q

Zδ2   πD2 − sin β2∞ 

  B2 .tan β2∞ 

Q 1  u2   u2 −    Zδ2  g  1+ p    .tan D B π − β 2∞   2 sin β  2      2∞ 

...(4.28)

...[4.28 (a)]

...(4.29)

For the given dimensions of impeller, Hm is a function of C u 2∞ and C u 2∞ is function of β2. Blade shape changes, when β2 change from < 90° to > 90° for the same direction of rotation of impeller. At outlet blades are curved backwards when β2< 90°, radial when β2 = 90° and curved forward when β2 > 90°. Correspondingly the flow passage between blades of the impeller also changes. Fig. 4.12 illustrates the blade shape, and the shape of the passage when β2 changes from < 90° to > 90°. For β2 < 90°, blade passage is longer, the angle of divergence is smaller. Flow can be smooth, without any flow separation. For β2 = 90° and β2 > 90°, the passage length is reduced and angle of divergence is increased, which induces flow separation and subsequent hydraulic losses due to secondary flow. β2 < 90° is commonly adopted for pumps, to get higher efficiency since the flow passage is divergent. β2 = 90° is used in turbines, and in return guide values where flow is through a convergent passage. The longer length of the flow passage induces more hydraulic losses. Thus outlet blade angle β2 influences considerably on the performance of the pump. Converting all the values of head, H∞, Hp∞, Hdy∞, ρ∞, into a non-dimensional unit, as percentage of total value of (u22/2g).

84

ROTODYNAMIC PUMPS (CENTRIFUGAL AND AXIAL) β 2>90° ω

ω

ω

β2

=9 0°

β 2<90°

(a)

(b)

(c)

Fig. 4.12. Blade shapes for different β 2 (a) β 2 < 90°° (b) β 2 = 90°° (c) β 2 > 90°°

H dy∞ =

=

H∞ =

H dy∞ (u22

/ 2g )

2 gCu 2∞ 2 gu22 H∞ (u22 / 2 g )

=

2 gH dy u22 2

 Cu 2∞  2 =   = (Cu 2 ∞ )  u2  =

Cu 2∞ . u2 g (u22 / 2 g )

=

Cu 2∞ u22 g (u22 / 2 g )

...(4.30)

= 2 Cu 2∞

2  Hp∞ = H∞ – Hdy∞ =  2 Cu 2∞ – (Cu 2∞ )  = Cu 2∞ ( 2 − Cu 2∞ )  

ρ∞ =

H p∞ H∞

=

 Cu 2∞  (2 − Cu 2∞ )Cu 2∞ = 1 −  2Cu 2∞ 2  

...(4.31) ...(4.32)

...(4.33)

The following table (4.1) gives the variation of H∞ , Hdy∞ , Hp∞ , ρ∞ for different values of Cu 2∞, calculated as per the equations 4.30, 4.31, 4.32 and 4.33, when Cu 2∞= 0, β2∞= 0, when Cu 2∞ = 2, β2∞= 180° when Cu 2∞ = 1β2∞= 90°. β2∞ can also be calculated from the equation 4.34. β 2∞ as a function of Cu 2∞ TABLE 4.1: Variation of H∞ , Hdy∞ ∞ , Hp∞ ∞ , ρ ∞ ,β

Cu2∞

H∞

Hp∞ ∞

Hdy∞ ∞

ρ∞

0 0.5 0.75 1.0 1.5 2.0

0 1.0 1.5 2.00 3.00 4.00

0 0.75 0.9375 1.00 0.75 0

0 0.25 0.5625 1.000 2.25 4.00

1.0 0.75 0.625 0.50 0.25 0

85


C u 2∞ =

Cm 2∞ Cu 2∞ u2 − Cm 2∞ cot β 2∞ = = 1 – Cm 2∞ cot β2 or tan β2 = 1 − Cu 2∞ u2 u2

...(4.34)

A graph is also drawn referring the Table 4.1.

H T∞

ρ

ρ

0,75

3,0

H T∞, H d∞, H p∞

H d∞ 2,0

0,5

1,0

0,25 H p∞

0 0.5

1,0 —

—

1,5 —

C 2u∞ —

Fig. 4.13. Graph H∞ , Hdy ∞ , Hp ∞ , ρ∞ = f (c2u ∞ )

The velocity triangles are shown in Fig. 4.14 for three condition namely β2 < 90°, β2 = 90° and β2 > 90°.

2a

C

C 2m

C 2c C 2b

β2b

β2c

β2a

C 2ua

C 2ub C 2uc

Fig. 4.14. Velocity triangles for β 2 < 90°° , β 2 = 90°° and β 2 > 90°° (a) β 2a < 90°° (b) β 2b = 90°° (c) β 2c > 90°°

Corresponding impeller blade shapes are also indicated in the Fig. 4.12. Total head (H∞ ) and coefficient of reaction ‘ρ’ are directely proportional to Cu 2∞. Dynamic head, Hdy is proportional to ( Cu 2∞ )2 and pressure Head (Hp ∞ ) changes inversely to ( Cu 2∞ )2.

86


When β2∞ < 90°, the blades are backward curved, the angle of divergence of flow passage in impeller is narrow and, hence secondary flow losses are less. Hydraulic efficiency is higher. Also Hp > Hdy i.e., a greater part of the outlet energy is in the form of pressure energy. Only a smaller portion of total head i.e., Hdy is converted to pressure energy, which is the basic requirement for pumps. That why pumps and blowers are designed with impeller blades as backward curved blades. Normally β2, the outlet blade angle lies between 20° to 35° for nS ranging from 50 to 400. Recommended values are: β2 ≤ 30° for nS < 100 β2 ≤ 25° for 100 nS < 200 β2 ≤ 20° for 250 nS < 400 Total Head increases with the increase of β2. The performance of pump H–Q curve gradually rises and becomes more and more a straight line when β2 increases. Smaller values of β2 is selected for more steep H–Q curves and larger values of β2 are for more flat H–Q curves. β2 is always selected for maximum efficiency condition of operation. As per stepanoff | 112 | average static condition of optimum blade angle of outlet β2 lies, between 22°–23°. For increase in head, angle β2 may be increased to 28° to 30° without any sacrifice in efficiency. Minimum blade angle β2 can be 15° to 17° and never less than this value for pumps of any specific speed (nS) or any size.

4.9 EFFECT OF NUMBER OF VANES Selection of number of blades (Z) in impellers and in diffusers (Zd) is very important. It influences on the H–Q characteristics of the pump, pump efficiency as well as suction characteristics of the pumps. Selection of less number of blades as well as shorter length of blades, give higher angle of divergence in impeller passages, which increases secondary losses in impeller namely circulatory losses between two blades due to large pressure differences between leading and trailing sides of the blade as well as losses at exit due to flow separation. Cavitational characteristics of the pump also reduces. H–Q curve will be lower than the normal curve. Blade loading will be higher. This increases the blade thickness which again reduces the blade passage and corresponding increase in w and Cm which increases frictional losses. Too long the impeller blades and more number of blades increase the frictional losses, although blade loading and secondary losses are reduced due to less angle of divergence. Hence correct selection of number of blades is absolutely necessary. In Fig. 4.15, the effect of number of vanes on pump performance is given. It is evident from the graph that, number of vanes influences in pump performance. Based on the channel width in plan of the impeller and the blade length which ensures proper angle of divergence, Karl Pfliderer | 97 | has established an expression for the determination of number of blades (Z).

1

Z

=

3

Z=

H Vs Q

8

Z=4

Z=5

2

10

Z = 10

Specification: Head : 12.8 m Flow rate: 6 lps Speed : 1440 rpm n s = 60 Size = 50 mm × 40 mm

5

6

Z=8

Z=6

Z = 10

Fig. 4.15. (a) Effect of no. of vanes on pump performance

(Q) Quantity of Flow, lps

4


20 15 10 100% 50% 25% 0

Total head, (H) m Efficiency ( η)

7

87

Z=5

Z=6

Z=4

Z=5 Z=8 Z=4

8

88

ROTODYNAMIC PUMPS (CENTRIFUGAL AND AXIAL) β2

90° 71° 44°

0

2

4

6

8

10

12

30° 21° 16° 14° 14

Z

β 2) as per Pfliderer Fig. 4.15. (b) Selection of number of vanes, z = f(β

Z = 6.5

r2 + r1 r2 – r1

. sin

β1 + β2 2

...(4.35)

where, 6.5 in the constant derived from experimental results. This expression gives nearly correct value for normal backward curved blades. If the blades are too long and too much bent in backward direction, i.e.,for smaller values of β1 and β2, this expression gives more number of blades than normally used in practice. Karl Pfliderer, based on the results from Hanson has given a graph for selection of number of blades as a function of outlet blade angle β2 for diffuser pumps. [Fig. 4.15 (b)]. In Fig. 4.15 (a) the effect of number of blades on pump performance is given. From the graph it is seen that for best performance number of vanes are found to be between 5 and 6 for radial flow pump. Most of the pumps of different specific speeds have number of blades between 5–8 in impeller and 1 or 2 blades, more than impeller blades, in diffusers. For nS ≤ 150, Z will be 6 to 8, for nS ≥ 150 and D2 ≤ 1000 mm, Z = 6 to 7, for D2< 120 mm, Z = 6 to 5. Most of the high efficiency pumps have an included angle θ between inlet and outlet edges of blades in plan (Fig. 4.16) between 80° to 150°, optimum being 110° to 120° and θ/tang between 1.2 to 2.2 360 where tang = for pumps of specific speed nS ranging from 130 to 400 and D2 ranging from 100 mm Z to 300 mm. n%

Efficienc y

90 3 80

2

1 4

70

5

For ns = 300 to 400 (1) D 2 = 300 mm (2) D 2 = 116 mm For ns = 180 – 250 (3) D 2 = 116 mm For ns = 130 – 180 (4) D 2 = 116 mm (5) for ns = 60 to 100 D2 = 300 mm θav / tang

Angular displacement θav / t ang

Fig. 4.16. Selection of number of vanes from the graph, η = f

Fθ GH t

a

ang

ns , D 2

I JK


89

4.10 SELECTION OF EYE DIAMETER (D 0), EYE VELOCITY (C 0), INLET DIAMETER OF IMPELLER (D1) AND INLET MERIDIONAL VELOCITY (Cm1) Impeller eye diameter D0 and impeller inlet diameter D1 are selected for best hydraulic efficiency and for best cavitational characteristics of the pump. Impeller eye velocity C0 for pumps with protruding shafts and hubs such as multistage pumps, double suction pumps is given by Qth =

π Q = (D02 – d 2h ). C0 4 ηv

...(4.2)

where dh is the hub diameter. ∴

4Qth

C0 =

πD02

 d h2   1 – 2  D0  

...[4.2 (a)]

For pumps without protruding shaft and hub i.e., dh = 0 such as end suction, single stage, single entry pumps C0 =

4Qth

...[4.2 (b)]

πD02

From similarity laws unit discharge (KQ) is defined as KQ =

Q nD 3 3

or

D0 = 3

where,

K =

3

3 Qth 1 Qth ⋅ K ⋅ = KQ n n

...(4.36)

1 KQ

Substituting the value of D0 from the equation (4.36) into equation [4.2 (a)]

4 KQ2 / 3  Qn 2 / 3  Qth 4  C0 =  =  2 π 3   d  2   Q 2 / 3  d h2     Qth   h π 1 − 1 −      K n   D 2   D Q  0    = constant (K1) . 3 Q. n 2

where,

K1 =

4 KQ2 / 3  d2  π 1 − h2   D   

...(4.37)

90


For better cavitational characteristics and hydraulic efficiency, C0= 2 to 4 mps and the meridional velocity (Cm1) before the blade inlet edge Cm1= C0 or 1.05 to 1.10 C0, since increase in velocity Cm reduces the efficiency and cavitation effect. From inlet to outlet in the impeller passage, the relative velocity w, meridional velocity Cm and hence the absolute velocity C gradually reduces. This means the blade passage is a divergent passage. For better results, angle of divergence should be within 10°. This is achieved when

w1 = 1.05 to 1.15 w2

and Cm2 = 0.85 to 0.9 Cm1. From inlet velocity triangle for normal entry 2

w21

 4Q  2 + u2 = = Cm1  2 + 1  πD0 

π π D2 = 4 0 4

Taking, Cm1= C0,

FG πD n IJ H 60 K

2

1

(D12 – dh2 )

D12 = D02 + dh2

or

2

2  4Q  π n 2   w1 =   (D20 + dh2 ) 2  +   60   πD0 

∴

...(4.38)

2

w1 M sec

2

2

w1

2

π 60

2 2 2 n D1

= u1

4 π

2

Q D 21

2

D 0 opt

2

2

2

2

– dh

= C m1

2

D0(m )

2 , u2 = f (D2) Fig. 4.17. Determination of optimum eye diameter graph, w21, Cm 1 0

For good cavitational characteristics w1 must be minimum. Differentiating equation (4.38) with respect to D0 i.e.,

dw1 = 0 and simplifying dD0 2

2

2  π n  4 Q − 2 =0    π  60  D06

which is same as equation (4.36).

3

or

D0 = K

Q n

...(4.39)

91


From experimental results, the constant K is ranging from 3.6 to 5.0. For single stage end suction pumps of D1 > 70 mm, K = 4.0 to 4.5 and for same pumps if D1< 70 mm, K = 4.5 to 5.00. For all multistage pumps except for Ist stage K = 3.6 – 3.9 | 76 |. As per Karl Pfliderer | 97 |, Lomakin | 69 |, Stepanoff | 112 |, and Karrasik | 54 |, the best inlet blade angle without including the angle of attack for better cavitational characteristics and for better hydraulic efficiency is β10 = 15° to 20°. Taking this into consideration and from the inlet velocity triangle (Fig. 4.1), under normal entry condition,

tan β10

4Q 60 × 4 × Q C0 Cm1 πD02 = = = = u1 π2 D03 n u1 πD0 n 60 =

or

D03 =

240 π

2

D03

240 π

2

⋅

×

Q n = tan 15° to tan 20° 1 Q ⋅ tan (15 to 20) n

 240  1 D0 =  2 .   π tan (15° to 20°) 

1/ 3 3

Q = 4.5 to 4.0 n

3

Q n

Correspondingly finding the value of C0 from the above equation, Qth

2 π 2   dh   4Qth = D0 1 −    C0 or D0 = 4   D0     d 2    π  1 −  h   C0   D0    

tan β10 =

tan (15° to 20°) =

60 C0 Cm1 60 C0 = = πn π D0 n u1

  d 2  π 1 −  h   C0   D0     4 Qth

2

d  C 3/ 2 1−  h  ⋅ 0  D0  π n Qth

30

 π tan (15° to 20°)  2/3   n Qth  2 C0 =  30   dh   1–       D0   

...(4.40)

92


Substituting the values for tan β10 = tan (15° to 20°) = 0.268 to 0.364 C0 = 0.063 to 0.077

3

Qth n 2

= 0.06 to 0.08 3 Qn 2

...(4.41)

4.11 SELECTION OF OUTLET DIAMETER OF IMPELLER (D2) As a first approximation D2, the outer diameter of impeller is determined from fundamental Eular’s equation. For normal entry Hm =

H C u = u2 2 ηh g

Cu2 can be selected from (0.8 to 0.5) u2 for nS ranging from 150 to 250.

u22 Cu 2u2 Hm = = (0.8 to 0.5) g g

Hence,

Having known the speed n, D2 =

or u2 =

gH m (0.8 to 0.5)

60 u2 πn

This diameter D2 is used to determine β2, Z, p. From the value p H∞ = (1 + p) Hm is determined. From outlet velocity triangle, for normal entry condition H∞ =

  C Cu 2∞ u2 = u2  u2 − m 2  tan β2  g 

 Cm 2  u2 =   +  2 tan β2 

from which

2

 Cm 2    + g H∞  2 tan β2 

...(4.42)

The diameter D2 determined under second approximation by equation (4.42) is corrected again after finalizing the correct value of area reduction coefficient ‘Κ2’ and outlet blade angle ‘β2’.

4.12 EFFECT OF BLADE BREADTH (B2) For the given value of Q,H and the determined optimum value of β2, and Cm2, maximum diameter at outlet of the impleller D2 is determined. Minimum value of Β2, the breadth at outlet can be determined from Q, D2 and Cm2. This is achieved when the coeficient of reaction ‘ρ’ is maximum by which Cm2 is maximum. Since β2 is independent of nS, β2 is selected mostly from the practical results. The total head H ∞ will be H ∞ = (1 + p) Hm = (1 + p).

H ηh

93


where

Hm =

=

H Cu 2u2 = for normal entry at inlet ηh g

(1 + p ) ⋅ Cu 2u2 ηh

g

Referring to the outlet velocity triangles, the total head (H) is given as Hm =

=

where

 Cm 2  u2 Cu 2u2 u = 2 (u2 – wu2) = =  u2 −  tan β2  g g g  u2 g

 K 2Qth  u2 – πD2 B2 tan β2 

B2 =

 K 2 Qth  u2  u 2 –  = 2 g  πD2 B2 tan β2  

...(4.43)

B2 D2

Introducing non-dimensional coefficients unit discharge

Q    KQ =  and Unit Head nD3  

H    K H = 2 2  into the equation (4.40), we get n D  

 K 2 KQ nD23 πD2 n  K H n2 D 2 π D n −   = 2 2 g  ηh ηv .π.D2 B2 tan β2 

Introducing specific speed ns =

3.65n Q H(

3 / 4)

 219 K Q    = or K = H   ns ( K H )(3 / 4)

219 KQ

4/3

into the above

equation and simplifying

B2 =

K 2 KQ

( )

2/3  g KQ (219)4 / 3  2   ηv tan β2 π – 4/3   η1 ( ns )  

...(4.44)

For radial type centrifugal pumps ns ranges from 40 to 300, KQ ranges from 0.02 to 0.22, K2 ranges from 1.05 to 1.25, β2 ranges from 20° to 30°.

94

70

80


B = 12

60

B=9 B=6

50 40 30 20 10

Efficiency (η)

B = 16

0

0.01

0.02 (KQ ) Unit Discharge

0.03

Fig. 4.18. (a) Efficiency—unit discharge characteristics η – KQ ) (η

0.04

95

0.6

0.7


0.4

B = 12

0.3

B=9

0.2

B=6

0.1

(K H) Unit Head

0.5

B = 16

0

0.01

0.02

0.03

(KQ ) Unit discharge

Fig. 4.18. (b) Unit head—unit discharge characteristics (KH – KQ)

0.04

96


0.3

0.25

0.2

KQ 0.15

0.1

0.05

0 40

45

50

55

60

70

80

90

100 ηs

115

130

150

175

200

250

300

Fig. 4.18. (c) Allowable range of KQ for different specific speeds (Data collected from different pumps working η ranges from 65% to 81%) 40

45

50

60

70

0.8000 55

80

90

115

100

130

KH

150

175 200

0.3000 0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.20

0.22

KQ

Fig. 4.18. (d) Allowable range of KQ and KH for different specific speeds of Radial type centrifugal pumps (Data collected from pumps working η ranges from 65% to 81%)

97


0.1600 0.1400 0.1200 0.1000 B

0.0800 0.0600 0.0400 0.0200 0.0000 40

45

50

55

60

70

80

90

100 115 130 150 175 200 250 300 ns

—

Fig. 4.18. (e) B vs ns for K2 = 1.05, η h = η v = 0.84, β 2 = 22°°

0.1200

0.1000

0.0800

B

0.0600

0.0400

0.0200

0.0000 40

50

60

80

100

130

175

ns

—

Fig. 4.18. (f ) B vs ns for K2 = 1.05, η h = η v = 0.84, β 2 = 30°°

250

B

IJ K

98

FG H

2 TABLE 4.2: Selection of impeller blade breadth at outlets B 2 = D 2

K2 = 1.05 η h = η v = 0.84

η h = η v = 0.90 β 2 = 30

β 2 = 22

β 2 = 22

β 2 = 30

KQ

B

KQ

B

KQ

B

KQ

B

KQ

B

KQ

B

KQ

B

KQ

B

40

0.02

0.0395

0.013

0.0111

0.02

0.0276

0.013

0.0077

0.02

0.0272

0.013

0.0093

0.02

0.0191

0.013

0.0065

45

0.025

0.0473

0.015

0.0116

0.025

0.0331

0.015

0.0081

0.025

0.0331

0.015

0.0098

0.025

0.0231

0.015

0.0069

50

0.031

0.0596

0.02

0.0167

0.031

0.0417

0.02

0.0117

0.031

0.0414

0.02

0.0140

0.031

0.0290

0.02

0.0098

55

0.037

0.0679

0.023

0.0182

0.037

0.0475

0.023

0.0127

0.037

0.0479

0.023

0.0154

0.037

0.0335

0.023

0.0108

60

0.042

0.0670

0.027

0.0211

0.042

0.0469

0.027

0.0148

0.042

0.0491

0.027

0.0179

0.042

0.0344

0.027

0.0125

70

0.053

0.0704

0.034

0.0247

0.053

0.0493

0.034

0.0173

0.053

0.0540

0.034

0.0212

0.053

0.0378

0.034

0.0148

80

0.067

0.0832

0.043

0.0304

0.067

0.0582

0.043

0.0213

0.067

0.0648

0.043

0.0262

0.067

0.0454

0.043

0.0183

90

0.081

0.0923

0.053

0.0366

0.081

0.0646

0.053

0.0256

0.081

0.0733

0.053

0.0316

0.081

0.0513

0.053

0.0221

100

0.097

0.1050

0.062

0.0411

0.097

0.0735

0.062

0.0288

0.097

0.0842

0.062

0.0357

0.097

0.0589

0.062

0.0250

115

0.118

0.1128

0.077

0.0488

0.118

0.0789

0.077

0.0342

0.118

0.0926

0.077

0.0427

0.118

0.0648

0.077

0.0298

130

0.134

0.1109

0.092

0.0558

0.134

0.0776

0.092

0.0391

0.134

0.0933

0.092

0.0490

0.134

0.0653

0.092

0.0343

150

0.164

0.1245

0.112

0.0645

0.164

0.0872

0.112

0.0451

0.164

0.1062

0.112

0.0570

0.164

0.0743

0.112

0.0399

175

0.193

0.1295

0.14

0.0770

0.193

0.0906

0.14

0.0539

0.193

0.1123

0.14

0.0684

0.193

0.0786

0.14

0.0479

200

0.222

0.1364

0.158

0.0812

0.222

0.0954

0.158

0.0568

0.222

0.1196

0.158

0.0727

0.222

0.0837

0.158

0.0479

250

0.257

0.1345

0.193

0.0905

0.257

0.0941

0.193

0.0633

0.257

0.1202

0.193

0.0817

0.257

0.0841

0.193

0.0572

300

0.284

0.1341

0.225

0.0990

0.284

0.0938

0.225

0.0693

0.284

0.1211

0.225

0.0900

0.284

0.0847

0.225

0.0630


ηs

B

IJ K

K2 = 1.25 η h = η v = 0.84

η h = η v = 0.90 β 2 = 30

β 2 = 22

β 2 = 22

β 2 = 30

KQ

B

KQ

B

KQ

B

KQ

B

KQ

B

KQ

B

KQ

B

KQ

B

40

0.02

0.0470

0.013

0.0132

0.02

0.0329

0.013

0.0092

0.02

0.0324

0.013

0.0110

0.02

0.0227

0.013

0.0077

45

0.025

0.0563

0.015

0.0138

0.025

0.0394

0.015

0.0096

0.025

0.0394

0.015

0.0117

0.025

0.0275

0.015

0.0082

50

0.031

0.0709

0.02

0.0199

0.031

0.0496

0.02

0.0139

0.031

0.0493

0.02

0.0167

0.031

0.0345

0.02

0.0117

55

0.037

0.0808

0.023

0.0217

0.037

0.0566

0.023

0.0152

0.037

0.0570

0.023

0.0184

0.037

0.0399

0.023

0.0128

60

0.042

0.0798

0.027

0.0251

0.042

0.0558

0.027

0.0176

0.042

0.0585

0.027

0.0213

0.042

0.0409

0.027

0.0149

70

0.053

0.0838

0.034

0.0294

0.053

0.0586

0.034

0.0206

0.053

0.0643

0.034

0.0252

0.053

0.0450

0.034

0.0176

80

0.067

0.0990

0.043

0.0362

0.067

0.0693

0.043

0.0253

0.067

0.0772

0.043

0.0311

0.067

0.0540

0.043

0.0218

90

0.081

0.1099

0.053

0.0436

0.081

0.0769

0.053

0.0305

0.081

0.0873

0.053

0.0377

0.081

0.0611

0.053

0.0264

100

0.097

0.1250

0.062

0.0489

0.097

0.0875

0.062

0.0342

0.097

0.1003

0.062

0.0425

0.097

0.0702

0.062

0.0297

115

0.118

0.1342

0.077

0.0581

0.118

0.0939

0.077

0.0407

0.118

0.1102

0.077

0.0508

0.118

0.0771

0.077

0.0355

130

0.134

0.1320

0.092

0.0665

0.134

0.0924

0.092

0.0465

0.134

0.1110

0.092

0.0584

0.134

0.0777

0.092

0.0409

150

0.164

0.1483

0.112

0.0767

0.164

0.1038

0.112

0.0537

0.164

0.1264

0.112

0.0678

0.164

0.0884

0.112

0.0475

175

0.193

0.1542

0.14

0.0917

0.193

0.1079

0.14

0.0642

0.193

0.1337

0.14

0.0815

0.193

0.0936

0.14

0.0570

200

0.222

0.1623

0.158

0.0967

0.222

0.1136

0.158

0.0677

0.222

0.1424

0.158

0.0866

0.222

0.0997

0.158

0.0606

250

0.257

0.1601

0.193

0.1077

0.257

0.1120

0.193

0.0754

0.257

0.1430

0.193

0.0973

0.257

0.1001

0.193

0.0681

300

0.284

0.1596

0.225

0.1179

0.284

0.1117

0.225

0.0825

0.284

0.1441

0.225

0.1071

0.284

0.1009

0.225

0.0750

99

ηs


FG H

2 TABLE 4.3: Selection of impeller blade breadth at outlets B 2 = D 2

100


0.1400 0.1200 0.1000 B

0.0800 0.0600 0.0400 0.0200 0.0000 40

45

50

55

60

70

80

90

100 115 130 150 175 200 250 300 ns

—

Fig. 4.18. (g) B vs ns for K2 = 1.05, η h = η v = 0.90, β 2 = 22°°

0.0900 0.0800 0.0700 0.0600 0.0500 B 0.0400 0.0300 0.0200 0.0100 0.0000 40

45

50

55

60

70

80

90

100 115 130 150 175 200 250 300 ns

—

Fig. 4.18. (h) B vs ns for K2 = 1.05, η h = η v = 0.90, β 2 = 30°°

101


0.1800 0.1600 0.1400 0.1200 0.1000 B 0.0800 0.0600 0.0400 0.0200 0.0000 40

45

50

55

60

70

80

90

100 115 130 150 175 200 250 300 ns

—

Fig. 4.18. (i) B vs ns for K2 = 1.25, η h = η v = 0.84, β 2 = 22°°

0.1200

0.1000

0.0800

B

0.0600

0.0400

0.0200

0.0000 40

50

60

80

100

130

175

ns

—

Fig. 4.18. (j) B vs ns for K2 = 1.25, η h = η v = 0.84, β 2 = 30°°

250

102


0.1600 0.1400 0.1200 0.1000 B

0.0800 0.0600 0.0400 0.0200 0.0000 40

45

50

55

60

70

80

90

100 115 130 150 175 200 250 300 ns

—

Fig. 4.18. (k) B vs ns for K2 = 1.25, η h = η v = 0.90, β 2 = 22°° 0.1200

0.1000

0.0800

0.0600 B 0.0400

0.0200

0.0000 40

45

50

55

60

70

80

90

100 115 130 150 175 200 250 300 ns

—

Fig. 4.18. (l) B vs ns for K2 = 1.25, η h = η v = 0.90, β 2 = 30°°

103


Overall efficiency of pumps ranges form η = 65% to 81%, ηv is assumed as equal to ηh and is taken as = η . Using excel programme, the values of B2 for the above mentioned variations were calculated and graph B2 = f (ns) were drawn (Figs. 4.18 c, d, e, f, g, h, i, j, k, l), for different values of KQ, K2, ηh, ηv and β2. These graphs can be referred for the selection of outlet blade breadth for radial flow impellers. An experiment on a radial type centrifugal pump was also conducted. The blade breadths (B2 and B1) at inlet and at outlet were changed keeping the inclination of the shrouds with vertical same at all time. The results are given in Figs. 4.18 (a), (b) in KH – KQ and η – KQ graphs. Experimental results agree with the theoretically determined values. Normally B2will be smaller for diffuser pumps, when compared to volute casing pumps.

4.13 IMPELLER DESIGN Three fundamental parameters namely (1) Total Head ‘H’, (2) quantity of flow ‘Q’ and (3) either speed of the pump ‘n’ or suction head ‘HS’ are necessary for impeller design. The speed of rotation ‘n’ however, is related to the size of the pump, and cavitational characteristics of the pump. If suction head ‘HS’ is known, the speed can be determined from suction specific speed (C). From the known value of H, Q , n, specific speed ‘ns’ for the pump is calculated by which the type of pump can be determined. When speed ‘n’ is increased for the given value of Q and H. specific speed ‘ns’ increases. The type of pump changes such as radial, or diagonal or mixed or axial flow. Also the overall size of the pump is reduced. It is found that maximum hydraulic as well as overall efficiencies are attained between ns = 150 to 200 for radial type centrifugal pumps. Fig. 4.19 gives the relation between efficiency (η) as a function of specific speed ‘ns’ for different eye diameters of the impeller. 12% 100 a c 50 0

85

200 250

400 420

600

800

b

875 n s

Fig. 4.19. Graph h = f (ns, DO) (a) DO > 200 mm, (b) DO = 50 to 200 mm, (c) DO = 10 to 50 mm

Suction head ‘Hs’ is reduced, when speed is increased. Cavitational specific speed ‘C’ can be taken as C = 800 to 1000 from which suction head ‘Hs’ can be determined under first approximation using the formula.

 n Qp  pvp p  –  Hs = atm –  C  γ γ  

4/3

...(4.45)

104


where, patm— atmospheric pressure pvp — vapour pressure and γ

— specific weight of liquid

Detailed analysis on cavitation is dealt with in chapter 9. Correct suction head ‘Hs’ or speed ‘n’ can be established by applying cavitation conditions. In pump industries, pump is selected mostly from among the available models manufactured in the industry. For the available data of H, Q, pump model, so selected, must be capable of meeting the hydraulic and constructional requirements of the field conditions. For example, impellers of multistage pump having hub extended into the impeller eye should not be selected for a single stage end suction pump, since the entry in multistage pump impeller is different from entry of liquid in single stage end suction impellers. In single stage end suction pumps entry is radial, whereas in multistage pump entry at suction need not be radial. By applying model analysis, the available models are selected to suit the new requirements. If pumps are not available from the existing models, new designs are made using systematic design procedure. Total head of single stage pumps with standard speed of rotation 1440 rpm (ns ≈ 40) will be H ≤ 30 m, in order to keep the impeller size and weight of the pump within the limit. If head for the single stage pump is more than 30 m, then the impeller size and corresponding the total weight of the pump considerably increases. Hence, head and quantity for a single stage pump should be selected up to miximum of 30 m for n = 1440 rpm. If pumps are in series, then head per stage will be H =

HT where i

H — Head of single stage pump, HT — Total head of the multistage pump, i is the number of stages. If the pumps are in parallel then quantity of flow per pump will be Qp=

QT , where Qp—quantity of flow i

for one pump, QT—Total quantity required and i—Number of pumps to be kept in parallel. If a double suction pump is used then Q =

QT . In case of single stage pump, excess quantity is required to take care 2

of axial thrust, leakage through wearing rings, stuffing box cooling, etc. Actual quantity must be increased by an extra of 3 to 10% i.e., Qp = Qact = (1.03 to 1.1) Q when selecting pumps for usage in field. Multistage pumps are used for boiler feed, mines etc. When high suction characteristics are essential such as condensate or for gas-liquid pumping, speed of rotation must be selected a little lower than normal. A double suction pump is preferred. If a multistage pump is used for such conditions, the first stage impeller must be specially designed.The suction head Hs is determined as per the equation 4.45. For the calculated specific speed, approximate overall efficiency (η) can be obtained by referring the graph (Fig. 4.19). The power of the prime mover will be Ni =

γQH . η

105


N% 1

100

h N % 5

2 95

5

5a

3 90

4

10 5b

6

85

15

80

20

75

25

0

70

140

210

240

350

30 490 ns

420

Fig. 4.20. Energy balance for pumps of different ns (1) Mechanical loss (2) Impeller loss (3) Discfriction loss (4) Volumetric loss (5) Hydraulic loss 5a. Volute loss 5b. approach pipe (6) useful output 100

0 1 5

95 2

90

3

10

4 15

85 5 80

20

75

25

70

30

65 60 55 50

Fig. 4.21.

0

20

40

60

80

100 120

140

Q % Energy balance for a pump (1) Mechanical loss (2) Volumetric loss Qnor (3) Hydraulic loss (4) Useful power (5) Recirculation loss.

Fig. 4.20 gives an energy balance for different specific speeds of pump and Fig. 4.21 for one pump. Referring to these figures the mechanical losses, volumetric losses and hydraulic losses, and

106


corresponding efficiency can be determined. Overall efficiency η = ηv× ηm× ηh, where ηv, ηm, ηh are the volumetric, mechanical and hydraulic efficiencies respectively. It is necessary to reduce the volumetric eifficiency by 1 to 2% depending upon the conditions, in case excess volume is used for axial thrust balancing and stuffing box cooling. Prof. A.A. Lomakin (69) has suggested that volumetric and hydraulic efficiencies can be determined as per the equation given below. Mechanical efficiency can be assumed as 1% for larger pumps and 1.5% for smaller pumps. Volumetric efficiency ‘ηv’ is given by 1 = 1 + 0.68 (nS)–2/3 ηv

...(4.46)

Qact where Qact is the quantity of flow for one pump. ηv the volumetric efficiency lies between ηv 85% and 95% for pumps. Hydraulic efficiency (ηh) is given by

and Qth =

ηh = 1 –

0.42 (log D0( Nom ) – 0.172)2 3

where,

and Hth =

D0(nom) = (4.5 to 4.0)

...(4.47)

Q (metres) n

H act where Hact – total head for one pump. ηh the Hydraulic efficiency lies between 75% to ηh

95% and depends upon the shape of the vane passages, surface roughness of the passages and size of the impeller. Mechanical efficiency ηm lies 1% for larger pumps and 1.5% for smaller pumps. Based on the head, quantity of flow, the power required to drive the pump, Ni =

γQH , where γ is η

the specific weight of the pumping liquid. The above equation can be written in different forms such as

PQ WH since γQ = W, weight of the pumping liquid flowing per unit time. Also, Ni = , since η η γH = p, the total pressure required for the pumping liquid. Ni is the power input to the pump at coupling and is equal to the power output from the prime mover (Nop). If the efficiency of the prime mover (ηpr)

Ni =

is known, the power input to the prime mover Nipr =

N op – Ni η pr

.

4.14 DETERMINATION OF SHAFT DIAMETER AND HUB DIAMETER Having known the total head, quantity of flow and power, the shaft diameter ‘dS’ can be determined based on the material selected for shaft, its yield strength for bending and the torque to be transmitted. A factor of safety of 2 to 6 is used depending upon the type of operation of the pump. In order to take care of the operation of pumps under overloading condition, a 10% to 15% extra power, over and above normal rated power is taken for shaft diameter design.


107

Power ‘N’ required is (1.1 to 1.15) Ni = Tω where T is the torque transmitted in N.m and ω is the 2π n , where n is the speed rpm. If ‘fu’ is the ultimate strength of the 60 shaft material selected, the yield strength ‘fs’ for bending, fatique and shear operating condition,

angular velocity of the shaft, ω =

fs =

fu , where FS is the factor of safety (2 to 6). Shaft diameter ds is determined from the formula FS

π d s3 f = T. 16 s dh = 1.2 to 1.3 ds ...(4.48) Hub diameter ‘dh’ will be depending upon the pump capacity. It is necessary to select the hub diameter to accommodate impeller key with sufficient space especially for smaller pumps.

4.15 DETERMINATION OF INLET DIMENSIONS FOR IMPELLER Normally the eye velocity C0 will be 3 to 5 mps. However, it can be determined as, C0 = 0.06 to 0.08

3

Qn 2

...(4.41)

Eye diameter ‘D0’ is determined as ...(4.2) Qth = C0 . π (D20 – dh2 ) or = C0 π D20 4 4 depending upon type of construction of the pump such as multistage pumps or double suction pumps or single stage end suction pumps. Eye diameter D0, is rounded off to the nearest standard pipe size and then correct value of C0 is again determined from the continuity equation (4.2). The position of the inlet edge of the blade in impeller must be selected based on the required cavitational characteristics. Radial type, low specific speed, centrifugal pumps will have the inlet edge of the impeller parallel to thie axis. At higher ranges of specific speeds (ns = 150 to 250) the inlet edge of the impeller is extended into impeller eye, in order to provide better cavitational characteristics. The inlet edge of the impeller blade will be inclined (diagonal) instead of purely parallel to shaft axis. In other words, the inlet edge of the impeller blade gradually extends from purely axial to diagonal when specific speed of radial type centrifugal pumps increases, in order improve the cavitation characteristics of the pump. Diameter D1 is selected as D1 = 0.70 to 1.1 D0 , when specific speed ranges from 300 to 70. Taking, Cm0 the meridional velocity before the blade inlet as Cm0 = C0 or 1.05 to 1.1 C0 the breadth ‘B1’ at inlet is calculated from the continuity equation. Qth = π D1 B1 Cm0 or

B1 =

Qth πD1Cm 0

...(4.49)

The inlet blade angle β1 is determined as tan β10 =

Cm1 KC = 1 m0 u1 u1

...(4.50)

108


and

Cm1 = K1Cm0 =

where,

t=

πD 1Cm 0 1.Cm0 t = = Zδ Zδ δ πD − 1− t− πD sin β10 sin β10 sin β10

...(4.51)

π D1 . Z

Selection of number of blades may be carried out referring Figs. 4.15 and 4.16. Normally number of vanes is selected as Zi = 6 to 8 depending upon the specific speed. The pitch or blade spacing (t) can πD1 . Vane thickness can be selected for strength and at the same time as minimum Z thickness as possible to get more flow passage area between any two blades and also to get proper vane shape while casting in foundary. As first approximation β10 is determined from equation (4.50). This value is substituted in equation (4.51) and the coefficient K1 is calculated.

be calculated as t1 =

This value is now substituted in equation (4.48) to get new value of β10 .This value is substituted in equation (4.49) to get new value of K1. This value of K1 is now substituted in equation (4.48) to get the second value of β10. This process is repeated until two successive values of β10 and K1 are same. The blade angle β1 is determined by adding the angle of attack ‘δ ’ i.e., β1= β10 + δ as mentioned earlier. Final value of Cm1 is determined from Cm1= K1Cm0. Thus, all parameters for impeller blade inlet D1, B1, Cm1, K1, u1, β1 and β10, Z are available for further the calculation to determine the parameters at impeller blade outlet.

4.16 DETERMINATION OF OUTLET DIMENSIONS OF IMPELLER The relative velocity ‘w1’ at inlet will be w1 =

Cm1 ( = K1Cm 0 ) sin β1

...(4.52)

The meridional velocity at outlet Cm2 is selected as Cm2 = 0.8 to 0.9 Cm1 and the relative velocity at outlet w2 is determined as

w1 = 1.1 to 1.15, since the blade passage is a divergent passage. It is also w2

necessary to take uniform change of w and Cm between inlet and outlet of impeller passage, in order determine the blade angle β at different radii between inlet and outlet of the impeller blade passage. Also the blade shape and flow passage between blades form, more uniformly, by which impeller secondary losses will be less and hydraulic efficiency will be higher. Outlet parameters are determined by approximate method, and then corrected, since the coefficient ψ and p to determine the total head reduction due to finite number of blades, determination of number of blade, are all function of outlet blade angle and outlet diameter. As first approximation Cu 2 is selected as Cu 2 = 0.8 to 0.5 for specific speeds 75 to 250.


Manometric Head

Hm =

u2 =

or

109

Cu 2 u22 H uC = 2 u2 = for normal entry at inlet g ηh g gH m Cu 2

and D2 =

60 u2 πn

D2 determined from first approximation, is used to determine outlet blade angle β2, number of blades Z and the head correction coefficient ψ and p. From velocity triangles at inlet and at outlet, w1 =

Cm1 C and w2 = m 2 from which sin β1 sin β 2

Cm 2 sin β1 K 2 Cm3 sin β1 Cm 2 sin β1 w2 ⋅ ⋅ ⋅ ⋅ = = Cm1 sin β 2 K1 Cm 0 sin β 2 sin β 2 Cm1 w1

sin β2 =

or

Since β1 is known,

Cm 2 w2 K 2Cm 3 w2 ⋅ ⋅ sin β1 = sin β1 Cm1 w1 K1Cm 0 w1

...(4.53)

Cm 2 w2 , , value β2 can be determined. Values ψ, p Z are determined from Cm1 w1

equations 4.18 and 4.19. The value H∞ = ((1 + p) Hm is determined. The outlet vane velocity u2 is determined from equation (4.39) and then D2 =

K2 =

Relative velocity at outlet w2 =

t2 t2 −

δ2 sin β 2

60 u2 . Outlet breadth, B2 is determined as πn and Cm3 =

Cm 2 K 2 Qth = π D2 B2 Cm3

Cm 2 . sin β 2

If D2 value determined by I and II and approximation vary too much, then D2 determined from IInd approximation should be substituted in all equations to determine the outlet dimensions and the process should be repeated until successive values of D2 are same.

4.17 DEVELOPMENT OF FLOW PASSAGE IN MERIDIONAL PLANE After determining inlet and outlet parameters of impeller blade, the development of flow passage in meridional plane (elevation) should be determined before developing the blade shape in plan. Selection and formation of flow passage depend upon the specific speed of the pump. The radius of curvature at the bend portion must be as large as possible in order to provide a smooth change over from axial to radial direction. The criteria for construction of such flow passage is to provide an uniform

110


change in area from eye to outlet of impeller and at the same time providing velocity C0 at eye, Cm0 at inlet Cm3 at outlet. From the established dimensions at inlet and at outlet for the impeller, a graph indicating the variation of Cm, w, β, δ, B, from inlet to outlet as a function of diameter D should be prepared. The uniform change in Cm and w is suitably assumed between inlet and outlet and the graph is drawn. (Fig. 4.3). Referring this graph Cm and w for any diameter can be found out. The blade angle β will be β = sin–1 Cm . Similarly, the blade thickness ‘δ’ can be assumed. Blade thickness is always determined w based in the blade loading and the facility available at foundry to cast as minimum thickness as possible, which provides more flow passage area. Normally blade thickness is gradually increased from inlet to some distance approximately up to 1/3 to 2/5 of the blade length and then decreases up to outlet. Usually 4 mm to 6 mm for smaller pumps and 10 mm to 12 mm for larger pumps are selected. A graph δ = f (D) is drawn. The breadth of the blade at any diameter can be determined from the equation.

 Zδ  Qth =  πD −  B.Cm. sin β   The value β, δ, Cm are taken from the graph for the selected diameter ‘D’ A graph B = f (D) is drawn in the same graph. From impeller eye to blade inlet edge, the graph can be extended to get complete the flow passage. The continuity equation at impeller eye portion will be Qth = This can be changed as Qth = the equivalent breadth B = Similarly, Qth =

π D02 .C0 for end suction pumps. 4

D0 π D02 C0 = πDm B C0, where Dm is mean diameter = and B is 2 4

D0 . 2

π 2 2 (D – d ).C for double suction and multistage pumps. This can be modified as 4 0 h 0

Qth =

π (D20 – dh2 ).C0 = πDm BC0 4

D0 + d h D0 – d h and B = 2 2 Depending upon the specific speed, the shape of the middle stream line (Dm from eye to inlet and D from inlet to outlet) is drawn. It should be remembered, that the radius of cruvature at the bend, where the flow direction changes from axial to radial must be as large as possible at inner and at outer shrould for better performance. On this streamline, a number of circles are drawn, at frequent intervals, selecting different diameters of impeller passage. The diameter of these circle drawn on the streamlines is equal to the breadth ‘B’. For the selected diameter (D) this value of B can be obtained from the graph. Lines are drawn at both ends of the circle such that the line drawn must be tangent to all cirlces (Fig. 4.22). These two lines form inner and outer shrouds of the impeller.

where,

Dm =

111


D0

r1

r

r2

dr

ds

If an arc is drawn connecting the meeting tangent points on S2 shrouds and the centre of circle the angle between the arc and tangent should be 90° i.e., normal (Fig. 4.22). A graph can be drawn between area ‘A’, A = f (S). Which must have the shape as shown in the Figs. 4.5 (b) and 4.22. For better cavitational characteristics the rate of area increase at the bend portion, where the flow changes from axial to radial direction must be at a larger rate than the area increase at the radial portion. By providing considerable increase in area at the inlet section, the rate of increase in area at the radial direction will be at a lower rate. Moreover, significant increase in area at inlet compensates the area reduction due to vane thickness at inlet. Radius of curvature at the bend portion of the meridional passage at the outer side must be as large as possible since smaller radius of curvature at this point yields high velocity of flow as well as flow separation after the bend, which drastically reduces cavitational property and hydraulic efficiency. Flow separation at this point will create very poor flow in the following radial portion as well as at inlet Fig. 4.22. Vane development in meridional section of the impeller, all will reduce the hydraulic efficiency. In general, meridional flow passage development must possess, 1. Smooth, streamlined and uniform area change from eye to the outlet must be ensured. 2. The radius of curvature at the outer side of the bend portion must be as large as possible. 3. Contour of flow passage must be in the same pattern as that recommended for that specific speed. D2 D2 The diameter reduces, when specific speed nS increases. When < 1.6, the surface area of D1 D1 the vane significantly reduces if the inlet edge of the blade lies in the radial portion of the passage. The blade loading will be higher, which inturn, reduces the cavitational characteristics. Hydraulic losses are increased. To overcome this, the blade inlet is extended into the bend portion. The inlet edge of the blade, instead of being parallel to axis, will be inclined. The blade passage changes from diagonal at inlet to radial at outlet. This inturn reduces the blade velocity and relative velocity at inlet. This reduces hydraulic losses and improves cavitational characteristics and reduces blade loading. Due to the inclined location of inlet edge, the radius from hub to outer changes. Since meridional velocity Cm is constant throughout the inlet cross-section, blade angle β1 reduces from hub to periphery. Blade curvature changes, from single curvature to double curvature. The inlet edge will be diagonal and outlet edge will be parallel to axis for specific ns = 200 to 300. When specific speed increases still further i.e., for ns = 350 to 500 the outlet edge also becomes inclined and the pumps will be mixed or diagnal type in stead of radial.

4.18 DEVELOPMENT OF SINGLE CURVATURE BLADE—RADIAL BLADES Single curvature blade or plane vane development is adopted for pure radial blades, where the inlet and outlet edges lie parallel to axis. The specific speed ‘ns’ of such pump will be less than 100 i.e., ns < 100 and normally the diameter D2 < 70 to 100 mm.

112


Vane development, either by single or by double arc method or by step by step method called as point by point method, must provide uniform variation in relative velocity ‘w’, meridional velocity Cm and angle of divergence from inlet to outlet along the flow passage i.e., from S1 to S2 (Fig. 4.23).

G

S2

E

β

β2

S2 B

D β1

θ

r

dr

dθ

θk

ri

rk

β

S1

δ

r0

A dθ ri θ

H

Fig. 4.23. (a) Single curvature plane vane development

Fig. 4.23. (b) Vane development by point by point method

Blade thickness ‘δ’ is selected either constant or changing from inlet to outlet, smaller thickness at inlet and at outlet end and higher thickness at the middle. However, blade thickness is determined based in the blade loading and the type of casting adopted in foundry for casting the impeller. The vane thickness will be a little higher at inlet than that at outlet and will be rounded off at inlet for shockness entry. For smaller pumps the blade thickness will be 3 mm at inlet, 5 mm to 6 mm at the middle and 1 mm to 2 mm at the outlet. For larger pumps blade thickness is increased up to 10 to12 mm. Selection of minimum thickness provides a larger flow passage between blades. The velocities Cm and w in the flow passage is reduced, which yields to higher hydraulic efficiency. Flow is also without separation for a wide range of flow rate. Now-a-days airfoils are used, for maximum economy and for better anticavitating property. These profiles are positioned on the stream line ‘S1 to S2’ determined by point by point method. Referring to Fig. 4.23 the differiential equation at any point between ‘S1 to S2’ for the central steam line in plan can be written as tan β =

dr dr or dθ = r tan β rdθ

Taking θ = 0 when r = r1 and ‘β’ from the graph (Fig. 4.3) β = f (D) and integrating. θ=

θ

r2

0

1

dr

∫ dθ = r∫ r tan β

...(4.54)

113


Integration is carried out by step by step summation of dθ

1 = B (r) r tan β

Taking,

∆θ =

Bi + Bi + 1 ∆ri 2

where ∆θ and ∆r are the increment in central angle and radius Bi and Bi+1 are the integrals at the beginning and at the end of selected radius. Total value of θ will be i =i

θi =

∑ i =1

Bi + Bi + 1 ∆ri 2

...(4.55)

All calculations are carried out in Tabular form.

θ deg

θi =Σ∆θ rad

∆θ = B × ∆s

Bi + Bi +1 2

B=

∆S or ∆r

1 r tan β Bi =

tan β

β

 Cm δ  +  sin β =  t  w

2πr Z t=

δ

w

Cm

b

r

S

S.No.

TABLE 4.4: Plane vane development

The values of S or r can be arbitrarily selected for which Cm, W, β can be taken from the velocity distribution graph (Figs. 4.23 and 4.3). The middle streamline is constructed from the table where θ and the corresponding r are known. Blade thickness is added on the streamline, to get the blade in complete shape.

4.19 DEVELOPMENT OF DOUBLE CURVATURE BLADE SYSTEM 4.19.1 Importance of Diagonal Impellers Increase in speed of the impeller reduces the overall dimensions, total weight and the cost of the pump. The specific speed of the pump ns increases. Diameter ratio D2/D1 reduces. If radial vanes are provided when D2/D1< 1.6 and specific speed is 150 ≤ ns ≤ 250, the specific, load on the vane increases, due to the reduction in the effective vane area. Cavitational property of the pump also reduces. In order to overcome this, the vane is extended into the impeller eye i.e., vane will be diagonal at the inlet instead of radial.

114


If the increase in specific speed is still further, 300 ≥ ns ≥ 600 the outlet edge of the vane also becomes diagonal. Each streamline of the vane will have its own configuration, i.e., the vane angles β1 and β2 are different from hub to periphery. The vane will be in the twisted form i.e., double curvature. Due to the change in direction of flow for axial, to diagonal, uniform steady flow no longer exists. The velocity field considerably changes at the inlet and at outlet. This complicates the pattern of flow. Existing elemental theory of pumps with average velocity assumption along the circumferential and along the radial directions cannot be assumed. A simple but considerably accurate scheme has to be developed. Axisymmetric flow, i.e., flow with infinite number of vanes is commonly adopted for this type of flow. Theoretical investigation under axisymmetric flow with infinite number of vanes in meridional section of flow will be equal velocity construction. This has been suggested by so many authors. One of the methods of construction for diagonal type of impellers is the assumption of constant head along all surfaces of revolution where the flow line lies. By applying Kelvin’s theorem, a vortex free flow i.e., potential flow ωu = 0, suggested by Bowersfield is attained in the vane system as a result of which the circulation along any contour is constant.

4.19.2 A General Solution for the Flow Through the Vane System Considering general flow conditions, due to the perpendicularity of the normal nf to the surface f → and the relative velocity vector w , the flow on vane surface ‘f ’ in a relative form can be written in the form, cos (n f , w) = 0 By applying cosine law between two crossing lines in the cylindrical coordinates (r, θ, z). _ cos (n f , w) = cos (n f , r ) cos ( w, r ) + cos ( n f , u ) . cos ( w, u ) + cos (n f , z ) . cos ( w, z ) ...(4.56) Equation (4.56) can be written as

∂f ∂f ∂f wr + wu + wz = 0 ∂z ∂r r ∂θ

...(4.57)

Cosine angle of the normal nf to the coordinates is proportional to the partial differential of the function (r, θ, z) along the corresponding coordinates and cosine angle of the vector w with the coordinate is proportional to the corresponding components of the velocity. Taking into account the relation between absolute and relative velocities. ∂f ∂f ∂f Cr + (u – Cu) + C = 0 ∂r r ∂θ ∂z z

...(4.58)

The vector Ω of the vorticity is also perpendicular to n f , since the surface f is a vortex surface. Similarly, the condition of perpendicularity n f , and Ω can be written as ∂f ∂f ∂f Ω + Ω + Ω = 0 ∂r r r ∂θ u ∂z z

...(4.59)


115

__ Considering an element dS on the surface ‘f ’ which lies tangential to the surface and perpendicular __ to nf and designating dS components as dr, rdθ and dz, we get ∂f ∂f ∂f dr + rd θ + dz = 0 ∂r r ∂θ ∂z

...(4.60)

From the relativity theory equations 4.56, 4.57, 4.58 can be written in the form dr rd θ dz Cr u − Cu C z = 0 Ωr Ωu Ω z

...(4.61)

This differential equation expresses the condition for the flow of vanes under vortex free absolute flow of fluid without any boundary limitations. However, the components of velocities can be determined only under axisymmetrical relative flow.

4.19.3 Axisymmetric Flow of Fluid Axisymmetric flow can be conveniently determined in cylindrical coordinates. Continuity equation in cylindrical coordinates can be obtained, considering the flow through the surface of elementary volume with dr, rdθ and dz as boundaries. [Fig 4.24 (b)]. If q is the flow through one side: qabcd = ρCr rdθ, dz, dt ∂ (ρCr )   dr  dθ, dz, dt qefgh = ρCr + ∂r  

Correspondingly: qadhe = ρCu drdzdt ∂ (ρCu )   qbegf =  ρCu + d θ  dru , dz, dt ∂θ  

qaefb = ρCZ rdθ dz dt

and

 ∂ (ρCZ )  qehgc = ρCZ +  rd θ dr dt ∂Z  

The total flow through all sides of the elementary volume will be dt

∫ ρCn f

∂ (ρC z )   ∂ (ρ Cr r ∂ (ρ Cu ) + +r df =  dr dθ dz dt ∂θ ∂z   ∂r

where Cr, Cu, Cz — the component velocities on the cylindrical coordinates. Cn — Projection of velocity along the direction of the normal to the elemental surface df.

116


∂ρ dr, rdθ dz, dt. This equation of balance ∂t is divided on both sides by dr, rdθ dz, dt. Continuity equation in cylindrical coordinates is obtained.

Increase in the mass flow inside the given volume is

∂ρ 1 ∂ (ρr Cr ) 1 ∂ (ρCu ) ∂ (ρ C z ) + r +r + =0 ∂r ∂Z ∂t ∂θ

...(4.62)

For incompressible flow where ρ = constant ∂ ( r Cu ) ∂ ( r Cr ) ∂ (r C z ) + + = 0 ∂r ∂Z r ∂θ

...(4.63)

∂ (r Cu ) = 0 r ∂θ

For axisymmetric flow

∂ ( r Cr ) ∂ ( rC z ) + = 0 ∂r ∂Z

...(4.64)

II II II II

Hence,

1 2 3 4 5 dσ 1 2 3 Si

σk

4 5

dz

r

a

Fig. 4.24. (a) Flow pattern in mixed flow pumps

dr

d b

g

h

rd θ

rh

r0 rm

c

dθ

f j θ )d dr + k (r

Fig. 4.24. (b) Elementary section for the determination of continuity equation and vorticity components in cylindrical co-ordinates system

ds

hf

t

dr

s2

r

s1 dθ

dr

θ

r

R

R dR dl

Fig. 4.24. (c) Construction of vane section on the flow surface

4.19.4 Flow Line and Vortex Line in Axisymmetric Flow The flow function ψ in axisymmetric flow will be rCr =

∂ψ ∂ψ ; – r Cz = ∂Z ∂r

...(4.65)


117

From equations (4.64) and (4.65), we get

∂2ψ ∂2ψ = ∂r ∂Z ∂Z ∂r The streamline equation for a two dimensional flow will be dr dz = or Cr dz – Cz dr = 0 Cr Cz from which

rCr dZ – r CZ dr =

...(4.66)

∂ψ ∂ψ + dr = dψ = 0 ∂zdz ∂r

This means that ψ function is constant along flow line. The streamline in meridional section corresponds to the flow surface in space. The components of vorticity Ω of absolute velocity in cylindrical coordinate can be determined from the circulation along the contour of the elemental volume considered above. Ωr = rotr C =

∂C z ∂Cu − r ∂θ ∂z

Ω u = rotu C =

∂Cr ∂C z − ∂z ∂r

Ωz = rotz C =

∂ (rCu ) ∂Cr − r ∂r r ∂θ

...(4.67)

For axisymmetric flow Ωr = –

∂ Cu ∂ (Cu r ) =− ∂z r ∂z

Ωu =

∂ Cr ∂ Cz =− ∂z ∂r

Ωz =

∂ Cu r r ∂r

...(4.68)

The meridional component of the vector Ωm = Ωr + Ω z

...(4.69)

The equation of vortex line Ωm is

dr dz = or Ω z dr – Ωr dz = 0 Ωr Ωz From equations 4.68 and 4.70, we get Ωzdr – Ωr dz =

∂ (r Cu ) ∂ (r Cu ) 1 dr = dz = d ( rCu) = 0 r r ∂r r ∂z

From which we get that vortex line coincides with the line r Cu = constant

...(4.70)

118


4.19.5 Differential Equation for the Cross-section of Vane with the Flow Surface The vane surface f is determined by crossing points of line 1 with the surface S. The relative velocity is tangential to both vane surface f and flow surface S. Hence, it is directed along the line of crossing of those two lines which represents flow line. [Fig. 4.24 (c)]. Let us construct a conical surface tangential to the flow surface S, such that it contains an element ds. The differential equation of flow line 1 on the surface S [Fig. 4.24 (c)] can be represented in the form of Rd θ ds = w u wm

...(4.71)

where R and θ are the corresponding radius and angle along the spreaded cone surface. Transferring from relative velocity to absolute velocity, we get Cm wm ds = w = u − Cu Rdθ u

...(4.72)

From the figure 4.24 (c), we get Rdθ = rdθ Since the elemental circumferential line in conical surface and in plan are equal. Equation 4.71 can now be written as tan β =

Cn ds ds = = u − Cu Rdθ rdθ

...(4.73)

which is the differential equation of line crossing the vane on the flow surface S. The above function can be rewritten as ds =

r 2 Cn ωr 2 − Cu r

= dθ

...(4.74)

4.19.6 Construction of Vane Surface when Ω u = 0 The simplest construction of vane surface is obtained when we consider Ωu = 0 which means the increase in energy of flow in impeller is proportional to change in the moment Cu r , and Ωu is independent. Ωu =

∂ Cr ∂ C z – =0 ∂z ∂r

...(4.75)

Potential function Φ of the meridional velocity Cm Cz =

∂Φ ∂Φ and Cr = ∂r ∂Z

Considering both, we get

∂2Φ ∂2Φ = ∂r ∂Z ∂Z ∂r

...(4.76)

119


The equipotential lines (Φ = constant) are determined as dΦ =

∂Φ ∂Φ dr + dz = Cr dr + Cz dz = 0 ∂r ∂Z

dz dr = Cr Cz

or

...(4.77) w(s)

C ′m w 2 m/sec

a e

12,0 10,0 8,0

5.0 .0 4 4. 5

3. 5 3 2 . .0 5 2 1 . .0 5

a b c d

c

d

v 1m

6,0

b

a c

4,0

b

d

e

2,0

e

0

40

80 120 160 160 200 240 280 320360 400

(a)

s mm

(b)

Fig. 4.25. Potential flow pattern in vane passage and the velocity distribution along the streamline

45 6 a01 2 3 b c d

7

9 8 3

4

2

1 0

5 6 7

e

8 9

Fig. 4.26. Vane construction under Ω u = 0 F l

II II

a b c d n

δ(σ)

II II b

c

d

a ∆a σ* ∆b σ* ∆c σ* ∆d σ*

(a)

e

12

3

4

7 6 5

5 6

e

(b)

σ ) (b) Construction of ‘s’ lines and s lines for equal velocity flow Fig. 4.27. (a) Graph f (σ

120


Ωu = 0 construction is theoretically based. Practical results coincide, if the meridional flow pattern is uniform. If flow separation prevails, this condition cannot be obtained. Uniform flow can be obtained if radius of curvature is large in meridional section, which at the same time increases the size of impeller in axial direction.

4

5

6

10

11

σ*

9

δ(∆σ) 100 ∆σ

8

δ (∆σ) = ∆σ – ∆σ*

7

i

∆σ*

3

σ = Σ ∆σ

2

F = Σ ∆F 1

1

∆σ

r

∆F = rav ∆σ

k

rav =

i

ri + ∆σ 2

TABLE 4.5: For the calculation of flow line under equal velocity flow

12

1 2 a

3 4 5

4.19.7 Construction of Vane Under Equal Velocity Construction Since potential flow does not agree, especially for pumps, another method, what is called, equal velocity distribution is adopted. This construction for pump is an extension of the method used for radial impellers. This is based on the principle that the calculation for each streamline is based on the equal meridional velocity Cm in the impeller passage, under the known value of H∞. Due to inadequate theoretical background it is more or less done based on experimental results-obtained from best impellers. Using trial and error method, for the given Cm, the flow line S is determined as first approximation. For this the entrance area to exit area f is divided into a number of equal areas. By eye judgement the position of S and equipotential line σx at the intermediate places are located [Fig. 4.27 (b)]. The flow through σx is equal to Q1 =

n

∫ 1 Cmx

2 π rdσ = 2 πCmx

n

∫ 1 rdσ

...(4.78)

Since Cm is constant along σx the velocity in the section σx is equal to Cmx =

where,

r =

Q1 2π r02 ∫

n 1

rd σ

R dσ and dσ = r0 r0

...(4.79)

121


The ratio Cmx to C0 is given by —

Cmx

=

2 1 − r h2 Cmx 1 − rhub = = n C0 2F 2 ∫ rd σ

...(4.80)

1

n

The values F = ∫ rdσ is determined by integration (Table 4.5). 1

As per the table, F (σ) is constructed [Fig.4.27 (a)] and then this value F is divided into (n – 1) equal parts, which is equal to n flow lines. The value ∆σ* is determined by second approximation and on it the new position of flow line in section σ is established along which the first approximation flow line S for the entire surface is constructed [Fig. 4.27 (a)]. The second approximation of flow line S is determined by constructing all σ lines. For that the flow line k = 1 is divided into m – 1 equal parts corresponding to the assumed m σ lines [Fig. 4.27 (b)]. Then σ lines are constructed and the table is developed. The positions of flow lines ‘S’ and fr lines to S are now corrected. The same is continued for third time. The lines fr to σ are equal velocity lines [Fig. 4.27 (b)].

4.19.8 Construction of Vane Surface Under Equal Velocity Flow for the Given w(s) From the calculation of vane surface under equal velocity constructions, the change in w = f (S) is determined. Each vane section (flow lines) is calculated separately. These are common for all these lines and is the head H∞. The vane surface at the entrance is formed as twisted in order to provide shockless entry for some discharge Qδ = 0 = m Q′ at all entrance edges. The coefficient m is determined from the angle δ; selected along one flow line, usually the leading edge. The shockless entry for the leading edge (marked K =1) is determined as tan β110 =

K11Cm11 u11

...(4.81)

K1 — correction coefficient for vane thickness u1 — peripheral velocity Taking angle of attack δ1, for the first line, we get β11 = β110 + δ1 tan β11 =

and

K11 (Cm′ 1 )δ = 0 u11

where (C′m1)δ = 0 is the meridional velocity corresponding to shockless entry at the inlet edge. The ratio

(δ′) δ=0 (Cm′ 1 )δ =0 tan β11 = = = m0 Q′ Cm′ 1 tan β110

...(4.82)

122


which is constant for twisted surface at entrance. The entrance angle β1K for all entrance edges lying in a flow sk is determined by tan β1K = m tan β1K, 0. ...(4.83) Cm , K

Cm′ ,k

=

u0

u1k ωr1k = = r1k u0 ωr0

, u1k =

tan β1h = m K1k

Cm1k r1k

...(4.84)

Cm1k is determined from the equal velocity construction. If entrance edge lies on the orthogonal line then it is constant along the entrance edge. P1 = 2 πr1k tan β1k ...(4.85) The entrance edge location is based on specific speed and experience and then subsequently corrected. The calculation β1K + w1n along flow lines are carried is a tabular form (Table 4.6). The value K1 is assumed, corrected and then w1K is determined. The exit edge is usually parallel to axis. If inclined, the same procedure is followed for outlet edge also.

—

W 1n =

w1n =

Cm1n ωin = K1k sin β1k u0

Cm1n w1n = sin β u0 1k

w2k = K2k

Cm 2 K sin β2 K

15

16

17

18

w1k = K1 k v m1 K sin β 1k

14

K*1k

13

19

20

tk–

12

∆ sk sin/pk

11

∆s / sin β1k

10

2prk tk = z

9

∆k ∆ξk = cos γk

8

cos γk

7

sin β1k

m

6

β1k

tan β10,k

5

tan β1,k = m tan β 10k

tan β 1,0,k = K1,k v*m,k

4

K1 ,k

3

v*m1

r1

2

γ1

S

1

∆1

K

TABLE 4.6: Method of calculation for vane entrance

...(4.86)

123


ω2 K ∞ is usually selected as 1 to 0.75 ω1k

sin β2k =

H∞ =

C mγ w1k K , 2k , sin β1k w2 k ∞ K1k C m1

...(4.87)

2gH ∞

...(4.88)

u02

Cm 2 1 u2 ⋅ + = u 2 tan β2 u0 0

from which

2

 Cm 2 gH ∞ 1   u 2 tan β  + u 2 0 2 0 2

 K 2Cmγ  H K 2 Cm 2  + ∞ +  r2 = 2 2 tan β2  2 tan β2 

from which

...(4.89)

Similarly, for other sections; for constant values of H∞ and for the assumed value of r2 the values

K 2 Cm 2 from equations 4.89 is found out from which β2K is calculated. tan β2 tan β2K = tan β2

K 2 k Cm 2 k K 2 Cm 2

...(4.90)

Calculation is done in a tabular form (Table 4.7). The coefficient K2k is assumed suitably. The construction of vane is carried out as per the pattern of change of w(S) along the flow line S. The procedure of w(S) is obtained as per the relative velocity at entrance (Table 4.6) and at exit (Table 4.7) such that change of w along (S) is uniform. The presence of maximum and minimum velocity at intermediate points indicate the losses due to conversion of kinetic energy to pressure.

14

15

w 2k = K*2k vm2k sin β2

13

K*2k

12

∆S sin β2k

11

tk –

10

∆S sin β2k

2 π rk z

9

tk =

∆S

8

cos γ

7

sin β2

6

cos β

tan β2

5

m*2

4

tan K2

3

γ2

2

∆2

S2

1

r2

K

TABLE 4.7: Method of calculation for vane exit

16

17

124

15

16

17

18

19

20

21

lu = Σ∆ lu

14

∆lu = rav ∆θ

13

r 1 + r θ1 2

12

rav =

11

∆ = Σ∆θ

10

∆θ = 18 17

9

B av

8

∆s

1 r tan β

7

B=

v m1 w ∞1

tan β

r1

β

S1

sin β 6 + 12

6

∆s / t

vm* w ∞

5

∆s

w∞

4

cos γ

vm *

3

γ

r

2

D

S

1

t

l

TABLE 4.8: Method of vane surface for the given value of w(s)

22

23

m

m-1

l

Note : 1. Circles are indicated the N 2 of tabular column. While calculating, put the exact values. 2. (21) to (22) are for conformal transformation.


m-2

125


The calculation of vane as per w(s) is carried out in the tabular form normally adopted for radial vanes, corresponding to the methods explained above, or in the form Table 4.8 (in non-dimensional form). The calculation is carried out from arbitrarily selected exist edge, to the inlet edge, which is obtained from the calculation. The pattern of change of w(s) and the thickness ∆(s) is obtained such that the position of entrance edge of the vane lies at the desired level and also not so much deviated in the value of central angle θlk (which is shown in the plan). The change of θlk from the leading edge to the trailing edge must also be uniform, to ensure the uniform surface. The change of vane thickness ∆ (s) for all streamlines must be agreed upon correspondingly so as to get uniform change at the meridional section.

4.19.9 Conformal Transformation of Vane Surface The obtained form of vane with streamlines and thickness is constructed in plane surface by conformal transformation (Fig. 4.28). In conformal transformation the angle of inclination of vane β is kept constant and is used for construction.

ds ds = dlu = 0 rd θ where dlu is the projection of elemental length dl along the direction tangential to ‘u’ i.e., dlu = rdθ The relation between r and θ is obtained from Table 4.8. Taking step integration method, we can write tan β =

Since

i=m

lu10 =

∑ i =1

ri + ri +1 δθ. 2 2π z

7 6 45 3 12 s

∆s

∆

rdθ

r1 h

6

7

8

16 15 14 13 12 11 10 9

a1

12 3 4

56 7 a2

∆θ

r

γ

∆θ = 2π kz

87

3 2 1 5 4

p1

9

6

s

1 2 3 45

14 12 11 1 0

123 4 5 6 7 ∆s . r d θ

δ

Fig. 4.28. Conformal transformation of vane surface

Which is carried out in a tabular form (Table 4.8), column 21, 22, 23. As per the values S , ∆u the conformal transformation in plan is constructed.

126


The interdisplacement of streamlines is carried out based on the experience. The most advantageous are: (1) The entrance edge at plan must be radial or slightly inclined (point of entrance edge of leading edge) by an angle of 10–15° toward the opposite side of rotation of impeller. (2) The exit edge is parallel to axis or the lag. The edge point of the impeller may be inclined towards the direction of rotation. Two methods are adopted for mixed flow pump design. The first rather old method is one in which the vane inlet and outlet edges are developed on a cone as a plane cylindrical vane and then transferred to the plan view from which patterns are made. In the second developed by Kaplan and called as ‘error triangles method’ the vane is developed with true angularity, length and thickness. The flow lines are then replotted in plan view. The second method will be discussed in detail here. In order to layout the vane in plan and in meridional section it is necessary to have the following quantities: (a) Meridional velocities at inlet and at outlet i.e., Cm1 and Cm2 respectively. (b) The impeller outer diameter D2 or peripheral velocity u2. (c) Vane angles at inlet and at outlet i.e., β1 and β2 respectively. The following points are most useful: 1. The vane can be extended with the impeller eye which will (a), improve the overall efficiency of the pump since overlap is more (b), reduces the outer diameter and (c), less shock and disc friction losses. 2. The profile in meridional section will be drawn for uniform change over from Cm1 to Cm2. 3. The flow lines a1, a2 represent the true radial sections of the flow lines, which should be gradual to avoid sharp corners. 4. Number of flow lines selection depend purely on experience. These are selected based on equal area construction. Further, it is assumed that the meridional velocity is constant along the normal and is equal to the average velocity. Naturally the velocities at inlet and at outlet edges are same provided they lie along the normal. Usually normals are draw first by eye and then these are divided into parts based on the law. 2πr1b1 = 2πr2b2.

4.19.10 The Method of Error Triangles Any flow line shown in a perspective view can be conveniently divided into number of parallel planes. The curve C1, C2 with the parallel circles, a number of meridional planes can be divided into a number of sections f1 , f2....The intersection of the planes with the surface of the shroud will form a number of parallel circles. Through the points of intersection of the curve C1, C2 be drawn which will section the shroud surface along the curved lines at g1, g2 .... These lines together with the section of parallel circles h1, h2.... and the curve C1, C2 form a number of curved triangle. The accuracy of the line C1, C2 will be more if greater number of sections are taken. On elevation the lines C1, C2 will appear as shown in Fig. 4.29 (b), whereas in plan the same line will appear as shown in Fig. 4.29 (d). The true representation of the lines C1, C2 is given by Fig. 4.29 (c). Where in all the curved triangles are transferred to a plane such that h1, h2 .... of the parallel circles form horizontal parallel lines and the sections g1, g2 ..... of the curved vertical lines will become flat vertical lines. The following procedure is adopted for the impeller vane lay out by error triangle method. 1. The meridional section (elevation) of the impeller passage is drawn [Fig. 4.30 (b)]. 2. Different streamlines are constructed based upon the methods indicated.


127

3. The vane development on the plane [Fig. 4.30 (b)] is drawn which corresponds to the exact vane angle at inlet and at outlet and vane length. Vane thickness is also added. The selection of vane thickness depends upon the moulding facility available. For normal conditions it is assumed as 5 to 6 mm. In order to draw vane development, flow line (a1, a2 or b1, b2 or c1, c2) is divided into a preferably a number of equal parts such as 1a , 2a, 3a. The parallel lines g1, g2 ..... are drawn the distances between them corresponds to the distance between 1a , 2a , .... 1b , 2b ...,1c 2c .... It is always better to draw vane sections of several flow lines in this relative positions. This will enable us to determine the inlet edge of the vane. 4. The vane sections are now transferred from the vane development to the plan view. [Fig. 4.30 (c)]. In the plan view an arbitrary point is selected. The arcs of the parallel circles are drawn with the radii taken from the elevation 1a , 2a ... The displacement of one point with respect to the other is taken from the vane development (h1, h2 ...). By joining the points with a smooth curved line the plan projection of the flow line is obtained. Usually leading edge is taken for the vane development. In order to get the trailing edge, vane thickness is measured in the plane vane development along the parallel circles at the required point and laid out at the corresponding points along the circumference. The line joining all these points will give us the trailing edge. Same procedure is to be followed for other streamlines also. 5. The next procedure is to draw pattern section for the construction of which the flow lines on the elevation and plan view are taken as preliminary guidelines. Since these lines are not sufficient enough to prepare vane pattern sections a second set of construction lines are constructed. A number of equispaced lines are drawn in the elevation view as indicated in Fig. 4.30 (d) i.e., A, B, C, D... The intersection of these lines with meridional streamlines are transferred to the plan view. Line joining these transferred point must be smooth, on the plan view. If these lines do not form smooth, and uniformly spaced, it is an indication that the change in the angularity on the vane development was too abrupt in any of the flow lines. The second set of construction lines can also be drawn starting from the plan view. This is done by drawing different radial lines I, II ... [Fig. 4.30 (e)] and then transferring them to elevation. Here also the line joining the intersecting points must be smooth. However, the first method is more advantageous since the second set of construction lines can be taken for pattern making. 6. The next procedure is to get vane pattern sections for which the contour lines on the plan view are taken. Wooden boards of thickness A, B, C, D .... are cut to the shape along the corresponding contour lines and then stacked one over the other in the proper order. We get the resulting vane pattern section in steps [Fig. 4.30 ( f )]. These steps are now filled with wax to get a smooth surface leading edges. Similarly, the same procedure is adopted for trailing edge also. 7. The best form of impeller channel will be one where in the vane makes 90° with the channel. This channel form can be improved by moving the flow lines on the plan view [Fig. 4.30 (e)] through a certain angle.

128


g6 g5 g4 g3

r6 r7 r5 r1

(a) f5 f4 f3 f2 f1

c1 h1

h2

h3

h6

c2

f6

g1 g2 g 3 g 4 g5 g6

c1

g3 g2 h2 g1 h1

g2

r4

C1

f2 f1

g1

r3

(b)

s2

r2

c2 h f 5 g6 6 f4 g 5 h 5 f3 g h 4 4 f6

h5 h4

r7 r6

h3 h2

r4

h4 h 5 h 6

h1 r1

(c)

(d)

Fig. 4.29. Impeller flow line development on a plane b2

c2 g1 1 b 1c g 2 2 c r1a 2 g3 b r2a 3 b 3c r3a 4c 5 6a a r4a 54b 5 c 7 b r 5a 8a a 6c 6b a1 7c 7b 9b 8b 8c b1 10 9c b h8 10c a1 11c c1 a2 1a 2a 3a 4a

a2b2c2

h5

h4

h3

g3

g2

h2 g1

h1

su 1

su1 2

s u2

3 4

h6

5 6 7

h7

8 9 10 11

b1 c1

(a)

(b) H G F E D C BA b c I a2 2 2

su

h1

g 1 h g h3 2 2 g3

III IV V

VIII

s u2 D2

VII VI a1

r1a

γ ′3a γ 3a γ′4a γ 4a

II III IV V

r2a

VI VII

b1

a1

C1

b1 c1 (c)

I

II

(d)

129


F–E G–H B–C

II V

IV VII

IV III

BC A

F–G

E– D

III VI

V VIII

DI E

VI

II

G

II

I c1

a1

VIII C.D.

VIII

B G

a2

CD

(e)

E

F

VI VII

III

VII

F

V

IV

I (f)

Fig. 4.30. (a) to (f) Mixed flow impeller profile and plane vane development

5 SPIRAL CASINGS (VOLUTE CASINGS)

5.1 IMPORTANCE OF SPIRAL CASINGS Spiral or volute casing is an approach or suction channel kept before the impeller inlet as well as a delivery channel kept after impeller outlet. The channel passage may be in the form of vaneless spiral casing or in the form of vaned or vaneless divergent passage called diffuser or return guide passage. Design of casing or diffuser must ensure the following: 1. Axisymmetric and equal velocity distribution of flow must always be ensured, since at optimum conditions, the flow in impeller is axisymmetric. Hydraulic efficiency is also higher. 2. Must uniformly and efficiently convert kinetic energy coming out from the impeller outlet into useful pressure energy. 3. Momentum at the outlet of impeller must be completely converted in volute casing and momentum at the casing outlet should be zero. Fig. 5.1 depicts different forms of casings adopted in pumps. Normally about 25% of kinetic energy is converted into pressure energy in casings.

Volute or spiral casing

Concentric passage with diffuser blades

Return guide vanes

Diffuser with blades

Discharge cone (a) Volute

(b) Diffuser

(c) Concentric with diffuser blades

Fig. 5.1. Different forms of casings 130

131

SPIRAL CASINGS (VOLUTE CASINGS)

Casing Diffuser

Impeller

Fig. 5.2. Diffuser pump

5.2 VOLUTE CASING AT THE OUTLET OF THE IMPELLER Dynamic head available at the outlet of the impeller can be expressed as Hdy = (1 – ρ) Hm =

gH m

...(5.1)

2u22

If the coefficient of reaction ρ is high, the kinetic energy will be higher at outlet of the impeller. The casing must be properly designed to convert this high kinetic energy into useful pressure energy. Volute casings at outlet are of two types: (1) Spiral shaped, vaneless form of casing and (2) Diffuser type vaned system of casing. ρ

ρ ρ

b3

b3 α

α

R3

b3

α = 180°

(a) Circular

α α = 35° to 45°

(b) Circular

(c) Trapezoidal

Fig. 5.3. Shapes of volute cross-sections

Technical and constructional features are different for each type of casing. Spiral casing at outlet consists of spiral shaped channel 02′4′68 followed by a diffuser passage 8– 9 (Fig. 5.4). The spiral portion connects the impeller outlet to the diffuser 8–9 under axisymmetric flow. It not only collects the fluid coming out around the circumference of the impeller but also converts about 75% of kinetic energy into pressure energy. The remaining 25% of kinetic energy is converted at the diffuser. Casing plays a major role in improving hydraulic and overall efficiencies. Lateral cross-section of the casing passage will be either trapezoidal or circular (Fig. 5.3) connected by two tangents, with an angle of divergence at the inlet α approximately = 35° to 45°. Spiral form of volute cross-section gradually increases along the flow direction due to gradual increase in flow.

132


Experimental investigation shows that trapezoidal cross-section gives higher hydraulic efficiency than circular cross-section at higher specific speeds and vice versa at lower specific speeds. Experiments indicate that spiral casing gives higher efficiency than diffusers at all partial flow conditions either lower or higher than optimum conditions. But, at optimum condition, diffuser gives higher efficiency than spiral casing. Spiral casing design is adopted for variable flow operations, whereas diffuser design is adopted for fixed optimum flow conditions of operation. Spiral casings are used for single impeller design whereas diffusers are used for multistage pumps to reduce pump weight.

5.3 METHOD OF CALCULATION FOR SPIRAL CASING Total energy remains constant after the impeller and also when flow passes through casing or diffuser. Two methods are adopted for volute design i.e., (1) the velocity of flow Cu changes according to free vortex pattern when passing through casing i.e., Cur = constant. The flow is assumed to be axisymmetric and ideal and (2) constant velocity, Cu = constant, in the spiral casing. In practice it is found that constant velocity design gives higher efficiency than free vortex design for pumps and vice versa for hydraulic turbines, due to increase in area of flow. Considering an element 1234 of the fluid, with a mass ∆m moving in the spiral passage (Fig. 5.4), external forces acting on the mass are : (a) tangential force ‘PAu’ and (b) normal force ‘PAn’ developed due to hydrodynamic pressure. In ideal fluid, the tangential force inside the fluid friction is equal to zero i.e., ± PAu = 0. The normal force ‘PAn’ acting on surfaces 1–3 and 2–4, due to symmetry in pressure under axisymmetric flow condition is PAn 1–3 and PAn 2–4. Moment of these forces, with respect to point ‘O’ are equal and opposite. The direction of the forces acting in the surfaces 1–2 and 3–4 pass through the point O and hence moment of these forces about point ‘O’ is zero. Thus the moment of all surface forces acting on the elementary volume of fluid in the spiral casing is zero i.e., ± ∆Mz = 0. Thus the moment of momentum remains constant in the elementary fluid i.e., d (∆mrCu ) =0 dt ∆mr Cu = Constant CuR = Cu3R3 = Cu2R2 = Constant

∆Mz =

or or

For normal entry at pump inlet, Cu1 = 0 and Hm =

...(5.2)

Cu u C u = u2 2 g g

gH m ΓB = 2π ω where ΓB = 2πR2Cu2 and is constant throughout spiral passage of the casing. With the increase in radius ‘R’ in spiral passage the tangential velocity decreases, correspondingly the pressure energy increases. The flow rate gradually increases and proportional to the volute angle θ (Fig.5.4). Since there is no flow perpendicular to the spiral section under steady flow condition, applying continuity equation, total mass flow at any section will be

CuR = Cu2R2 =

W = g

∫ ρCn dA S

=

∫ ρ Cn dA f1

+

∫ ρ Cn dA

S1

+ ... +

∫ ρ Cn dA = 0

fK

133


Mass flow through the section between volute angle θ1 and θ2 will be –

W1 = g

W2 = g

∫ ρ Cn dA and

A1

∫ ρ Cn dA

A2

Under axisymmetric flow the velocity Cn = Cr3 at the surface ‘A3’ the flow will be

∫ ρCn dA

= – ρ Cr3 b3R3 (θ2 – θ1)

A3

W2π (θ2 – θ1) 2π where W2π= ρg Cr3 b3 2πR3 which is the total flow of the pump at entrance to the spiral channel. Taking θ = 0 at the tongue of the volute section.

Wθ2 – Wθ1 = ρg Cr3 b3 R3 (θ2 – θ1) =

dr

b

9

8 ∆R

b3

0

p cu

dθ

R3 R2

θ2 0 θ1

2′

C u2 2

p2

R R3

dr 6

1 2

3

α

r

R3

R

4

∆R

and

A1

b3

A2

fk

4′ Fig. 5.4. Scheme for spiral casing at outlet

θ1 = 0 and Wθ1 = 0 θ θ° or Qθ = Q2π × ...(5.3) 2π 360° Flow, thus, increases along the spiral passage in proportion to the angle of the volute ‘θ’, referred with respect to the initial tongue. Flow across the elementary area ‘dA = bdR’ will be Wθ = W2π

dθ = bdR Cu = bdR

ΓB 2πR

...(5.4)

Integrating within the limits R3– r Q =

ΓB 2π

r

b

∫R

dr

...(5.5)

R3

Integration is carried out by step by step method in tabulation form (Table 5.1). The function Bi =

bi . Elementary flow rate through the area dA = b∆r is determined as ri

134


Γ B Bi + B(i +1) ∆Ri 2π 2 and total quantity of flow Q is determined as

∆ Qi =

Q =

5.4

ΓB 2π

i=n

Bi + B(i +1)

i =1

2

∑

∆Ri

...(5.6)

DESIGN OF SPIRAL CASING WITH Cur = CONSTANT AND TRAPEZOIDAL CROSS-SECTION

Flow from the impeller enters the circumferencial section of volute having a radius R3, which is determined as ...(5.7) R3 = (1.03 to 1.05) R2 in order to get uniform flow across the section at R3, since, uneven flow velocity and quantity exists at radius R2 i.e., at impeller exit, due to finite number of impeller blades. Breadth b3 at the entrance of the spiral section at the radius R3 is taken as ...(5.8) b3 = b2 + (0.04 to 0.05) D2 in order to take care of disc friction losses and trouble free running of pump. Fluid leaves the impeller at radius R2 with an absolute velocity C3 and at an angle α3. Corresponding velocity at the entry into spiral casing at radius R3 is determined as C3 R 2 = C 4 R 3 Taking volute cross section divergent angle ‘αV’ as 35° to 45° and breadth ‘b3’ as per the equation 5.8 and since these values are constant at all circumferencial portions, (Fig. 5.5) construction of trapezoidal cross-section under C3R = constant and the quantity of flow Q at any section is determined from equation (5.6). Calculations are carried out in tabular form (Table 5.1). b360

Ab=A a

∆r

bθ

bi

h ′θ

∆R

b3 = b4

Q 45° ∆Q

R 135°

b3 = b4 R3

α

4

Q135°

Q 225° Q 315°

Q

Qi

ri

3

Q 360° (old) R4

Tongue R 4 = R 3 + ∆ R

α

Q 360°(new)

Fig. 5.5. Volute construction for trapezoidal cross-section and free vortex design, Cur constant and also for Cv = constant

R3

135


TABLE 5.1: Calculation of volute section—Trapezoidal cross-section and CuR = constant (Free vortex design) S. No.

R

b

B=

1

2

3

4

b R

∆R

Bi + Bi+ 1 2

5

6

∆ Qi =

B × (6) × (5) 2π 7

1

Qi = ∑∆Q 8 0

2

Referring the table 5.1, A graph r = f (Q) (Fig. 5.5) is drawn. X-axis i.e. the value Q is now divided into definite number of equal parts normally 8 equal sections such as Q45°, Q90°, Q135°, Q180°, Q225°, Q270°, Q315°, Q360°. Projecting these points upwards to meet the R = f (Q) curve and then drawing horizontal lines from these meeting point to y-axis i.e., ‘R’ axis, the radius at which corresponding quantity of flow through the trapezoidal cross-section can be determined. By projecting these horizontal lines futher to meet the trapezoidal cross-section gives the corresponding areas for the corresponding quantity Qθ. Final section for Q 360°. Starts at R4 = R3 + ∆t, where ∆t is the volute tongue thickness. Normally ∆t= 2 to 3 mm. Due to this, the quantity of flow at the last section Q360 will be higher than the normal Q360 taken as per the graph r = f (Q). This is represented in the Fig. 5.5 Q360 (new) = Q360 (old) + ∆Q at tongue. In some of the volute designs the tongue starts after some angle θ from initial position of θ = 0. This is due to the fact that huge noise and subsequent vibration takes place due to the fluid passing through the gap between the impeller outlet diameter and volute starting point at θ = 0 at high velocity since this gap acts as nozzle. To avoid this, volute tongue starts a little away from the point θ = 0. Normally this value will be θt = 17° to 21°. The contour of the trapezoidal cross-section obtained will have sharp corners which increases the hydraulic losses. Also, flow does not exists at the corner point. In order to reduce hydraulic losses and ensure flow through entire area of trapezoidal cross-section, the sharp edged corners are rounded off in such a way that the area added at the middle ‘Ab’ will be equal to the area reduced due to rounding off at the corners ‘Aa’ Aa Cua = Ab C ub, Aa hence,

Γ Γb = Ab b 2πrb 2πra

r Aa = a where Ra and Rb are the radius of centre of gravity of areas Aa and Ab . rb Ab

5.5 CALCULATION OF TRAPEZOIDAL VOLUTE CROSS-SECTION UNDER CONSTANT VELOCITY OF FLOW C V = CONSTANT (CONSTANT VELOCITY DESIGN) Constant velocity of flow through all volute sections CV is determined as CV R = Cu2R2, where R is the radius of the centre of gravity of the last volute cross-section.

136


Area of the last volute section will be A360° =

Q CV

Area of any volute cross-section at angle θ from the tongue will be θ° 360° Construction of volute section will be same as that mentioned in the previous section.

Aθ = A360°

The velocity CV can be determined by the law of similarity as CV = ΦV 2gH . The value ΦV can be determined from the graph, ΦV = f (nS), (Figs. 5.6, 5.7, 5.8). These values are determined based on the experimental results by different authors. |4|, |12| Considering the trepezoidal cross-section (Fig. 5.4) 1 h (b + b3). Angle α can be selected as α = 35° to 45° in order to avoid flow Area Aθ = 2 θ θ separation due to divergence. Selecting the value α.

(bθ − b3 ) 2 hθ

= tan

Aθ = =

FG H

α α or bθ = 2 hθ tan + b3 2 2

FG H

IJ K

1 1 α h (b + b3) = h 2 hθ tan + b3 + b3 2 θ θ 2 θ 2

FG h H

IJ K

FG H

IJ K

IJ K

α α + hθ b3 = hθ hθ tan + b3 . 2 2

2 θ tan

Qθ = CV × Aθ = hθ

FGh tan α + b IJ × φ H 2 K θ

3

...(5.9)

2gH

By selecting hθ from minimum to maximum up to the value Q360, a graph hθ = f (Qθ ) can be drawn similar to Fig. 5.5. From this graph Qθ for Q = 45°, ....... 360° can be determined for different values of hθ. As a check the values aq = f (Qθ ) can also be drawn and checked with the constructed values. The construction of trapezoidal cross-section for CV = Constant is same as for Cur = constant. (Same as Fig. 5.5). Entire calculations can be brought out in a tabular form (Table 5.2) TABLE 5.2: Calculation for area and flow rate under CV = constant and trapezoidal cross-section CV = φ V S.No.

α=°

2gH =

hθ mm 5 10 15 up to a value until Qθ > Q360 = Q total

A θ = hθ

b3 = mm.

FG h tan α + b IJ H 2 K θ

3

Qθ = Aθ × CV

aθ = 2hθ tan

α + b3 2

For check up

137


TABLE 5.3: Calculation of flow rate at different θ values (CV = const., trapezoidal cross-section) θ

S.No.

1 2 3

Qθ

0

0

45° to 360° at constant

Q = Q360

interval

The height of the last trapezoidal section is determined as h360 = (Kp× r2) – r3. where Kp is an experimental coefficient given in Fig. 5.5. The value of Kp given are for double suction pumps. For single suction pumps the value Kp will be less by 10 to 15%.

5.6 CALCULATION OF CIRCULAR VOLUTE SECTION WITH Cur = CONSTANT Applying equation 5.5 for a circular crosssection (Fig. 5.6) volute design with Cu r =constant.

1 ΓB = π

r3

b( r ) dr r

a1 +ρi

ρ21

∫

 = ΓB  ai – 

r

a 2i

Qθ =

fv

− ρ 2i

2

dr

r3

dr 5

  

3 4

Since, b (r) = 2 ρ2i − (r − ai )2 Since

1

v 6

− ( r − ai )

ai − ρi

0

a

∫

R

7 r

1 Γ 2π B

b

r3

Qθ =

p

R

Quantity Qθ at an angle θ from initial position will be

8

Fig. 5.6. Volute design with circular crosssection and free vortex (Cur = const.)

θ . Q360. 360

Substituting this value in the above equation θ° = where K =

360Γ B Q360

FG a – H

a 2 − ρ2

IJ = K FGa – K H

a 2 − ρ2

IJ K

360Γ B 720πg H m = . Q360 Q360 ω

Since a = r3 + ρ, substituting this value and after simplification ρ =

θ° + K

2

θ° r3 . K

...(5.10)

138


Calculations are made in Tabular form (Table 5.4). θ° values are normally selected as 45°, 90°, 135°, 180°, 225°, 270°, 315° and 360°. TABLE 5.4 : Calculation of circular volute with Cu r = constant K=

S. No.

θ°

θ° K

1

2

3

Hm 720πg ⋅ ω Q360 2

θ° r K 3 4

r3 =

4

ρ = (3) + (5)

5

6

θ° – selected uniformly at 45° interval

As mentioned earlier, final area at spiral outlet before entering the diffuser will be the sum of calculated area and the tongue area.

5.7 DESIGN OF CIRCULAR VOLUTE CROSS-SECTION WITH CONSTANT VELOCITY (CV) Flow velocity in spiral casing is taken as CV = φV 2gH . The value φV is taken from the Nomogram (Figs. 5.6, 5.7, 5.8). Area at final section and corresponding radius of volute section will be QT f = πρ 2 = CV

ρ=

and

QT CV π

Quantity at any section will be proportional and will be Qθ =

QT .θ° 360

From the quantity Qθ , ρθ at any section is calculated. D3 D2 1,3

0,5

1,2

0,4

1,1

0,2

1

b3 b2

φ v, φ d, φ vi φv φd

0 φvi 50

D 3/D 2

b 3/b 2

2,0 1,5 1

100

150

200 ns

Fig. 5.7. Design constants as suggested by Artisikoff |4|

139


D3 D2 1,5

5

φv 0,5

4

0,4

φv

1,4

3 K p,K rs 2

0,3

Krs

1,3

1

0,2 01

1,2 K1p

1,1

D /D 0 3 2 40

0

100

1,0 300 400 n s

150 200

Fig. 5.8. |4| 50

40 D3 – D2 × 100 D2

Volute velocity coefficient k 3

0.50

αv

k3

0.40

30

20 Volute angel αv degrees 10

0.30

8 6 2

k 3 × 100 0.20

4 3

0.5 42

57

71

106 142

212

263 354 424 495 565 595 706 1060

(ns) specific speed

Fig. 5.9. Volute constants Cv = C3 = k3 =

2gH as per A.J. Slepanoff |112|

5.8 CALCULATION OF DIFFUSER SECTION OF VOLUTE CASING Diffuser connects spiral casing outlet with the delivery pipe of the pump. Velocity of flow Cd gradually reduces from CV at final section of spiral section of volute to Cd the velocity at delivery pipe (Cd = 3 to 5 m/sec.). In order to ensure uniform flow without any separation the angle of divergence is normally selected as ε° = 8 to 10°. If the shape of spiral section at the last stage is not circular, the shape

140


is determined by assuming equal area of circular cross-section for the inlet of the diffuser and calculations are made accordingly such that the axis is in a straight line. However, the final length of the diffuser should be decided taking into account the construction feasibility of pump delivery flange with pipe flange.

 Rp − R f  ε°  =  L 2   where RP and Rf are the radial length of the pipe and final spiral section of the volute casing, with respect to the axis of the diffuser. Velocity at the diffuser outlet can be determined by the equation Cd = φd

2gH . Value φd can be obtained from Figs. 5.6, 5.7, 5.8.

5.9 (A) DESIGN OF DIFFUSER As similar to spiral casing the diffuser is also receiving the fluid from the impeller outlet and converting the available kinetic energy of fluid into useful pressure energy (Fig. 5.10). It consists of a number of diffuser passages kept at equal space around the circumference from inlet to outlet of the diffuser. Each channel consists of a spiral section abc and diffuser section bcde. Spiral section of the diffuser is constructed with constant width b3 = b2 + (0.04 to 0.05) D2 . The diffuser part forms a straight channel with angle of divergence either in one plane or in both planes perpendicular to each other with either straight or curved axis. d

c

(a)

ε

dr dθ

α3

r θc θt

b3

A I

a3

I

0

a

III

III

e

b

θ ∆3

A–B

II

II ε

II III b III (b)

II

Passage 1– 2 – 3 – 6

I

b3

I

6–3–4–5

6 2

3 4

1 5

Fig. 5.10. Diffuser and return guide vanes for radial type centrifugal pumps (a) Diagonal (b) Radial

141


The diffuser passage is formed between two walls of the stage and is connected to the return passage. The return passage guides the outcoming fluid from the previous stage diffuser to the next stage impeller suction. The diffuser vanes and return guide vanes are often cast as a single unit. The diffuser is of two types: diagonal and radial. In radial type diffuser the return passage is connected by vaneless U-shaped passage, where in the fluid turns through 180°. In diagonal type of diffuser, the diffuser and return blades are kept one after another with no space in between them. Referring the Fig. 5.10, the section II–II, the channel, deviates in axial direction and connects the return passage [Fig. 5.10 (a)]. These type of diffusers possess smaller dimensions in radial direction and also gives higher hydraulic efficiency when compared to radial type diffusers. Diffusers with return guide vanes are used in multistage pumps by which overall dimensions and weight of the pump reduces considerably. At the same time, axial thrust at the impeller is balanced at all regions of operation of pump.

5.9 (B) CALCULATION OF SPIRAL PART OF DIFFUSER PASSAGE Referring to Fig. 5.10 under axisymmetric plane, ideal fluid flow conditions, the wall ‘a – c’ of the spiral passage consists of cylindrical surface formed parallel to z-axis of the diffuser. Contour ‘a – c’ in its own plane perpendicular to z-axis, is the streamline of plane axisymmentric flow. Differential equation of the streamline ‘a – c’is determined from the condition, that the flow of the fluid dr and rdθ as per the coordinates is proportional to the corresponding resolved components of the absolute velocity Cr and Cu. Cr dr = Cu rdθ

In axisymmetric flow the tangential component of absolute velocity is determined Cu r = Cu2 R2 or R Cu = Cu2 2 . From continuity equation, the radial component of absolute velocity Cr is determined as r Q bR 2π R2b2Cm 2 Cr = k = k3 = k3 Cm′ 2 2 2 2π r b3 3 b3r 2π r b3 Therefore,

b2 Cm′ 2 dr = k3 = tan α3, is a constant b3 Cu 2 rdθ

tan α3 = k3

b2 Cm′ 2 = Constant b3 Cu 2

dr = tan α3dθ r Integrating between limits θ1 = 0 when R3 = r and R3 = θ when r = r, the equation for the streamline a – c is r = R3 e θ tan α3 ...(5.11)

also

which indicates that the streamline is logarithmic spiral. Thus, the spiral part of the diffuser is designed as logarithmic spiral.

142


5.9 (C) CALCULATION OF DIVERGING CONE PART OF THE DIFFUSER The diverging cone part of the diffuser bcde follows immediately after the logarithmic part of the diffuser. In this cone remaining kinetic energy is converted to pressure energy. The lateral cross-section of the cone is designed as straight cone. The height of the cone at inlet a3 is determined by the triangle bch [Fig. 5.10 (a)] as a3 =

RC – R3 – ∆3 cos α3

where RC is the radius at point C of the spiral canal, ∆3 blade thickness of inlet edge. The inlet edge is always rounded off for shockness entry. RC will be RC = R3eθC tan α3 2π – ∆θ—angle at the centre of spiral canal. Angle ∆θ is determined approximately from Zd the triangle ‘bch’ and the triangle abc. __ 2πR3 R3 ∆θ = bc sin α3 = sin2 α3 Zd

where θC =

∆θ =

or

So,

2π sin2 α3 Zd

π R3  zd sin 2α 3  − 1 – ∆ e a3 = 3 cos α3  

...(5.12)

Number of Diffuser vanes Zd will be between 3 to 8. It will be always 1 or 2 blades more than impeller blades. The best efficiency of diffuser can be attained when the entry section of the diverging cone is in the form of a square i.e., a3 = b3. The reduction in length due to blade thickness at entry is a3 + ∆ 3 ≈ 1.1 to 1.15. a3

The angle to divergence ε = 10° to 12° at one plane, if the other plane is straight. If both the planes diverge, then ε = 6 to 8°. Curved form of divergence cone axis enables to reduce the outer diameter of the diffuser, which inturn reduces the weight of the pump. Hydraulic efficiency, however, reduces to a certain extent. E.V. Dondoff [4], assuming that Cu r = constant at inlet divergent cone, suggests that height of entry section of divergent cone can be determined by the equation.

  2k a3 = (R3 + ∆3) 1 − 1 − A2    b3   where

A2 =

ωQ 2πQ = Γb Z d Zd H m g

and K = Correction coefficient determined as a function of the specific speed ‘ns ’ (Fig. 5.12)

...(5.13)

143


section AA–CC section CC–EE

c

α8

α5

7 8 9 10 11 12 13 14 15 16 17

D4 = 315 D 5 = 375

16

ε

11 Blades α8

90 +ε+

°

r

B

Return guide passage 9 vanes

8 10 11 12 13 14 15 16 17

α8

α4

(b) A

7

18

D 2 = 308

6 5 4 3 1 2

Blades constructed by point by point method

ϕ

(a )

s

Blade construction and graph – – as per circular arc method – point by point method

(b)

α

Blades constructed by circular arc method

c 5 12 3 4 5 6

Diffuser passage

7 8 9 10 11 12 13 141516 17

l Vaneless (c)

Return guide vane

Fig. 5.11. Diffuser, vaneless passage, return guide vanes design and construction (a) Vaneless ring between diffuser and return guide vanes (b) Another construction of vaneless ring (c) Velocity, angle of flow in passages. K 1,4 1,3 1,2 1,1 1,0 0,9 0,8 0,7 70

75

80

85

90

95

100 ns 105

Fig. 5.12. Correction coefficient for the divergent passage of the diffuser

5.10 RETURN GUIDE VANES Return guide vanes are used in multi-stage pumps. It is located immediately after the U-shaped bend in the diffuser. Normally spiral shaped portion and the divergent portion of the diffuser are kept on one side of a plate and the return guide vanes on the other side of a plate and are cast in manufacture as one piece. This facilitates easy casting in foundry as well as easy machining. A circular plate forms as

144


disc between rear side of the impeller and spiral shaped diffuser passage as well as another similar circular plate forms as disc between front side of the impeller and return guide vanes. These two plates form the cover between impeller and diffuser. All other designs are used for single stage pumps. (Figs 5.10 and 5.11)

5.11 DESIGN OF SUCTION CASING AT INLET OF THE IMPELLER Suction or approach casing for a pump (Fig. 5.11) consists of (1) a straight convergent cone (2) bend type curved convergent pipe (3) ring type chamber (4) spiral shaped chamber or casing and (5) return guide vanes. (Fig. 5.13)

5.12 STRAIGHT CONVERGENT CONE Straight convergent cone with angle of convergence ε = 17° to 21° are fitted infront of the suction. Sometimes radial ribs 4 to 6 Nos, called ‘baffles’ are fitted to ensure uniform, axisymmetric normal flow at suction (Cu1 = 0). An increase in velocity by 15–20% between inlet and outlet of the convergent section is normally recommended. If the space before the pump suction is limited, bend type convergent pipe is used with or without radial ribs at the exit of the bend. Ring type casing consists of a circular chamber of constant area followed by a converging type annular ring kept before the impeller eye. In these types, shaft is extended from the impeller to the suction side bearing through the ring casing. These pumps have bearings on both sides of the impeller. However, flow velocity decreases due to gradual reduction in quantity, the area of flow remaining constant along the ring. Due to uneven velocity distribution around the ring, fluid approaches impeller inlet edge with non uniform, unequal velocity and also with vortex motion Cu1 ≠ 0. Due to centrifugal force, at the bend Cu1 is not constant from shaft to inlet periphery. These type of casings are used mostly in multistage pumps.

5.13 SPIRAL TYPE APPROACH RING Spiral type approach ring followed by annular convergent ring at suction are used to overcome the drawbacks in ring type constant area suction casings. Constant velocity at all point of the flow passage in the spiral channel is maintained. Spiral shaped suction casing or approach channel consists of an entry tube ‘10–9’ followed by ‘864 20’ spiral channel and a convergent annular rings ‘ab’ (Fig. 5.13). Spiral shape ends with a radial rib, which avoids the fluid to enter back to the inlet approach tube. Total quantity ‘Q’ enters the chamber at ‘10’ and passes into ‘point 9’. After point 9, almost half the quantity enters directly into the impeller through convergent cone. Remaining quantity passes through the spiral passage of the casing, where quantity gradually reduces due to uniform entry into impeller along the spiral passage. Flow velocity ‘C ’ gradually increases in the passage 10 to 9 and remains constant in the spiral passage i.e., from 9 to 0. Flow velocity ‘CC’ increases to impeller eye velocity ‘C0’, when fluid passes through convergent annular bend ring ‘ab’. Normally the velocity before the annular ring CC = (0.7 to 0.85) C0 . Impeller will be suspended between two bearings. The shaft passes through the suction casing to the suction side bearing. Uniform velocity is ensured at the impeller eye in this type of design. It is recommended that

145


(a) a

a

(b) 4

3 2 1 0 1

b

b

2

3

4

(c) a b

2

4

0 9

6

10 a b

R8

8

Rg

9 10

(d)

Fig. 5.13. Different forms of suction casing for centrifugal pump (a) Approach bend with converging passage (b) Concentric ring with constant area of cross-section (c) Symmetrical half spiral casing (d) Single spiral casing 3 4 1 f 4 1

5

2 1f 8 1

1 0

db Dφ

D1

f1 = 6

2 π 2 (D – d b ) 4 1

10

3f 8 1

D4 D0

f4

1 f 2 1 8

α

D5

0 1 2 3 4 5 6 7 8

b

7

9

Fig. 5.14. Suction volute casing

146


D0 , R0 = (0.5 to 0.6) R8 2 R4 = 0.75 R8 , R9 = 1.5 R8 Coefficient krs is a function of ‘nS ’, the specific speed. The value krs can be taken from the graph [Figs. 5.7, 5.8 (b)].

R8 = (krs – 1)

The velocity Cvi is calculated as CVi = ΦVi

2gH , where φVi is taken from the graph (Fig. 5.7).

0,60 84,0

84,5 84,5

0,55

100 80

0,50 0,45

60

0,40

40

0,35

20

Efficiency % η/ηmax

Head coefficient ψ =

u 22

ψ

0

0 0,02 0,04 0,06 0,08 0,10 0,12 0,14 0,16 ϕ

Flow coefficient φ =

C m2 u2

Fig. 5.15. Performance variation due to three different volutes with same impeller

5.14 EFFECT DUE TO VOLUTE It is found that the increase in volute cross-section area is attained when volute is designed by constant velocity method than by Cu r = constant method. Increase in area of volute cross-section by about 5 to 7% may increase the overall efficiency by 2 to 5%. Optimum efficiency will shift to higher flow rate point. Decrease in volute area decrease the maximum efficiency and shifts optimum efficiency point towards lower flow rate. Shut off head slightly increases at higher specific speeds. Fig. 5.15 shows the test results of the same impeller tested with three volutes.

6 LOSSES IN PUMPS

6.1 INTRODUCTION Losses in pumps can be classified as: 1. Hydraulic, 2. Volumetric, and 3. Mechanical. Determination of hydraulic losses by theoretical means is still not possible. Intensive research is still going on. Since volumetric losses and mechanical losses can be determined accurately by theoretical means, hydraulic efficiency is determined from volumetric, mechanical and overall efficiencies. However, empirical formula for the determination of hydraulic losses is available by which hydraulic efficiency can be approximately determined.

6.2 (A) MECHANICAL LOSSES Power input ‘NI’ available from the prime mover output i.e., at the coupling side of pump shaft, gets reduced by an amount of ‘∆N’ due to losses in bearings ‘∆NB’, due to losses in stuffing box, ‘∆NS’ and losses due to disc friction, ‘∆Nd’ i.e., ∆N = ∆NB + ∆NS + ∆Nd . Correspondingly power available at the impeller side of the pumpshaft, Ni = NI – ∆N. Mechanical efficiency, ηm =

N

i

NI

=

Ni N I − ∆N = N NI i + ∆N

...(2.15)

∆ Nd ) 6.2 (B) LOSSES DUE TO DISC FRICTION (∆ Losses created due to the rotation of a solid body, inside a closed and fluid filled chamber, is called “Disc friction losses”. Pump impeller rotates as a solid body inside spiral casing chamber, which is filled with fluid, possesses the same phenomena and hence losses created by the rotating impeller inside a water filled spiral casing is the disc friction losses of the pump, ∆Nd. Fluid inside the space between impeller shroud and casing wall rotates at half the velocity of the impeller velocity. The induced moment of friction of fluid on the disc is equalised by the frictional moment of the fluid on casing wall. Due to centrifugal force, fluid near impeller shroud is thrown towards pheripery. This fluid returns back near the casing wall. As a result, a circulation i.e., a secondary vortex flow prevails in this space. 147

148


From dimensional analysis, the moment of friction of disc, on one side, can be written as M = Cf ρ ω2 r25 where,

Cf — coefficient of friction. ρ — density of the fluid. ω — angular velocity of the disc (impeller). r2 — outer diameter of the disc (impeller). ωr2 ν

2

Coefficient of friction, Cf depends upon the type of flow and hence Reynold’s number ‘R0’ = for the disc. (a) For laminar flow (Re < 2 × 104) | 67 | π r2 Cf = . + Re Re S

3  S  0.0146 +  S  0.1256         r2    r2  

...(6.1)

where, S — the distance of the casing hall from the wall of the rotating disc. (b) For transition flow Re = 2.104 to 105 | 67 | Cf =

1.334 Re

Cf =

0.0465 5 Re

...(6.2)

(c) For turbulant flow

A graph, Cf = f (Re ), is drawn in Fig. 6.1 for all the three regions of operation. 0,3 – 1 0,1 – 1 r2

0,9 – 2 0,7 – 2 0,5 – 2 lg C f 0,3 – 2

0,1 – 2 0,9 – 3 0,7 – 3 0,5 – 3 0,3 – 3 0,1 – 3 3

4

5 lg R e

Fig. 6.1. Cf = f (R)e

6

7

149

LOSSES IN PUMPS

Power lost in disc friction losses Nd =

2 Mω = 2 ρCf ω3r5 constant

...(6.3)

∆ NS) 6.2 (C) LOSSES STUFFING BOX (∆ 2

3

1

4 s

R r

σr

x

σx

p0

dx

p0

σx

pH

Stuffing box consists of a chamber ‘4’ containing a flexible asbestos packing ‘1’. Packing is kept in position by the gland ‘2’. By tightening the gland bolts, the gland is axially moved towards the chamber and compresses the packing. This packing has direct contacts with the shaft or shaft sleeve,‘3’. Since shaft or shaft sleeve is a rotating element a small clearance will be existing between packing and shaft sleeve or shaft, through which fluid passes from the impeller outlet through the space between casing and rear impeller shroud. This leakage flow can be adjusted by compressing the packings, with the help of gland since packing is stuffed inside a chamber. This chamber is called stuffing box. Although theoretically no flow through stuffing box can be made, practically small quantity of water as droplets must come out through stuffing box in order to avoid (1) air entering into impeller through stuffings box, (2) to cool the packings. Since this leakage quantity is very small, it is usually neglected.

x

Fig. 6.2. Stuffing box

Due to flexibile nature of packings the axial force σx created due to tightening the gland is changed into radial for σr acting on shaft in radial direction. σx = Kσr

...(6.4)

where K is the coefficient depends upon the packing property and is always > 1. In order to maintain leak proof σx > p0, the pressure at the inner side of the casing and impeller shroud and very near to the shaft. Pressure σx must gradually reduce from the gland to the impeller side. Considering an elementary thickness ‘dx’ (Fig. 6.2) of the packing, equilibrium is maintained. When,

2π (R +r) µ σrdx = – π (R2 – r2) dσx

...(6.5)

where µ1 is the frictional coefficient of packing. Combining equations 6.4 and 6.5 and rearranging. When,

dσ x 2µ1 = – dx. σx (R − r )K

Taking σx = P0 and integrating ‘x’ up to length ‘l’. P0 = – log σx

l

2µ1

∫ K (R − r) x

dx = –

2µ1 (l − x) K (R − r )

150


σx = ( P0 )

or

 2µ(l − x )   R − r 

...(6.6)

µ1 = µ2 K It is evident, that the pressure p, gradually increases and is maximum when x = l. It will be

where

pmax = σx = 0 = p0 e

2µ2

l (R − r )

= p0 e

2µ 2

l S

= p0 e

2µ 2 z

...(6.7)

Where, S is the thickness = (R – r) and Z is the number of packing rolls inside stuffing box =

l . S

Elementary friction force, dT = 2π r dx µ1 σx eµ l − x 2

T = 2π r µ1 p0 ∫ e

s

0



− 2µ

l

2µ 2 µ1 l  − 2µ 2 p0 e S  s  = π rS 1− e µ2   l

...(6.8)



l 2 a 2µ 2 1− e π r 2 s µ1  watts. Coefficient µ, is 0.02 to 0.1. About p0 e s  . Power NS = Tωr =  const µ 2 5 to 7 (= Z) packing rolls are used for normal pumps. Practically frictional coefficient, µ considerably reduces due to the introduction of cooling water as mentioned earlier.

2

3

1

l

s

ln

7

d

D

p

p0

p0

4

dn

(d) With cooling

(a) Normal

B 5

A E p0

D0

(b) With lantern ring at the middle

(e) External cooling δ

6

g

6

p0

p0 lu4

(c) With cooling circulation

10 (f) External and internal cooling

Fig. 6.3. Different types of stuffing box arrangements and with cooling systems

151

LOSSES IN PUMPS

Fig. 6.4. Stuffing box with the classic cooling water jacket cooling the outer diameter of the gland

Fig. 6.5. Stuffing box with unclear lantern ring for sealing water supply

Fig. 6.6. Stuffing box with externally cooled circuit to reduce the temperature of the pumped medium in the gland area

152


Fig. 6.7. Stuffing box with lantern type end ring for cold water injection

Fig. 6.8. Stuffing box of special design with hollow shaft sleeve to cool the inner diameter of the gland

Fig. 6.9. Stuffing box with double cooling effect and duplicate cooling feed cooling inner and diameter of gland

153

LOSSES IN PUMPS

Fig. 6.10. Stuffing box with double cooling effect and single cooling feed cooling inner and outer diameter of gland

Cu 2 , Cu1 , r1

Fig. 6.11. Stuffing box with double cooling effect and single cooling feed and also introducing cooling liquid upstream of the packing end ring for cooling inner and outer diamter of gland

Fig. 6.12. Gland area of feed pump with injection type shaft, intensive cooling and differential type balancing device absorbing pressure fluctuations of feed pump suction pressure sealing water pressure

154


6.2 (D) BEARING LOSSES ( ∆ NB) Bearing losses depend upon the type of bearing used such as, ball, roller, angular contact, thrust bearing. Based on the hydrodynamic theory of lubrication in bearings, power loss in bearings can be calculated. One such formula is given below. Power loss ∆NB in bearing will be ∆NB =

ω r.T 2π.η r = (ωr)2 . . l constant constant δ

T = η

where torque

...(6.9)

u 2πrl δ

η — Coefficient of viscosity of the lubricating oil used. u = ωr — Velocity of ball or roller centre. r and l — Radius and length of the ball or roller. δ — Radial clearance in the bearing.

6.3 (A) LEAKAGE FLOW THROUGH THE CLEARANCE BETWEEN STATIONARY AND ROTATORY WEARING RINGS Leakage flow is controlled by the clearances ‘b’. b = 0.003r, for smaller pumps and b = 0.2 + (D1 – 100) 0.001 in mm for larger pumps. ‘b’ normally lies between 0.15 and 0.25 mm. Larger clearance leads to higher volumetric losses and corresponding lower volumetric as well as overall efficiencies. Figs. 6.13 and 6.14 indicate the change in the performance due to increased clearance.

600

120

H, η in % of (H, η) norm

H 500

100 η

80

400

60

300 200

40 1

Axial force in % of force under Q norm

2

100

20

0

20

40

60

80

100

120

140

Q in % of Q norm

Fig. 6.13. Effect of clearance at shaft between 2 stages H, η and axial thrust (1) Axial thrust under normal clearance 0.2 mm (2) Under increased clearance 1.5 mm

155

LOSSES IN PUMPS

0.45 MM 0.575 MM

20

Without balancing holes

18

0.64 MM

0.74 MM

12

With balancing holes

0.64 MM 0.575 MM

8

10

0.45 MM

4

6

Average flow is 6.6 gpm/hole 26.4 gpm for all 4 holes

2

Total head in metres

14

16

0.74 MM

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17 18

Quantity of flow in Ips.

Fig. 6.14. Effect of wearing ring clearance and balancing holes

19

20

156


Referring of Fig. 6.15 the flow through the clearance QL1 can be determined from one dimensional theory Q = K. AV = K. A . 2gH .

(a)

(b) Y

(c)

(d)

X

(e)

(f)

(g)

Fig. 6.15. Different types of wearing rings

Applying this principle to the flow through the clearance ‘QL1’ or called as leakage flow, will be QL1 = µ Ai where,

Ai Di b pi p1

— — — — —

2g

pi − pI = µπ D b 2 gH pi i r

...(6.10)

Area of the clearance Clearance diameter Clearance width Pressure before clearance Pressure after clearance, at suction side of the impeller

pi − p1 head loss in clearance γ µ — Flow coefficient. Normally b = 0.003 r and should never be less than 0.15 mm for any type of wearing ring construction. The pressure drop across the wearing ring [Fig. 6.15 (a)] between any point inside wearing ring and inlet Hpi —

Hpi = Substituting the value for

pi − p1 p − pi p − pi p − p1 = 2 – 2 = Hp – 2 γ γ γ γ

...(6.11)

p2 − pi from the equation (3.89) γ

2  γu22   r   1 −   for normal wearing ring Hpi = Hp – 8 g   r2  

...(6.12)

2 2  γu22  r2  1 −  r     for damaged wearing ring ...(6.13) = Hp –   8 g  r    r2   Referring to the figure [6.15 (a)], the losses through the wearing ring consists of loss at entry, loss in the passage and loss at exit.

157

LOSSES IN PUMPS

VL2 hc = Loss at entry due to sudden entry = 0.5 2g he = Loss at exit, due to sudden exit = 1.00

VL2 2g

Loss in the passage having length ‘l’ and clearance ‘b’ and diameter ‘D’.

lVL2 hf = λ 8 gR where, VL is the velocity in the clearance =

QL πDi bi

R = the Hydraulic Radius =

πDi bi Area b = = i Perimeter 2πDi 2

So,

hf =

λlVL 2 4 gbI

Total loss

hL = he + hf + hc = Hpi

∴

 λli  VL 2 0.5 1.00 + + Hpi =  2bi  2 g  2

 λli   Q  1 + 1.5 ⋅  L1  =   2bi   πDi bi  2 g QL1 =

1 λli + 1.5 2bi

. πDi . bi . 2 gH pi

...(6.14)

where Hpi is calculated as per the equation (6.12) or (6.13). Comparing equations (6.14) and (6.11) µ will be µ=

1 λli + 1.5 2bi

For high pressure pumps Hpi will be higher due higher delivery pressure. The clearance cannot be altered since efficiency has to be maintained at high level as well as for ease in manufacture. So the leakage flow QL1 will be higher. Correspondingly, the volumetric efficiency and overall efficiency reduce. To maintain efficiency at higher level, QL has to be reduced. This is achieved by increasing the length of leakage path. Correspondingly, for the fixed value of area, µ value is changed. Different wearing ring forms are shown in Fig. 6.15. Referring to Fig. 6.15.

158


QL12 pI − p x 1 = 2 g (µ1. A1 ) 2 γ px − p y

=

γ ( p y − p2 ) γ ∴

Hpi

=

QL12

1

2 g (µ2 . A2 ) 2

QL12

1

2 g (µ3 . A3 ) 2

1 1 QL12  1 pi − p2  = = + + 2 2 2   (µ A ) (µ A ) (µ A ) 2g  1 1 γ 2 2 3 3 

=

µ =

QL12 2g

2 2 1  λ1l1  A1    λ 2 .l2  A1   λ3 .l3  1.5 1.5 1.5 + + + + +        A12  2b1 2b2 A2   2b3    A3   

1 2 2   A  2  A 2  λ3l3  A1  λ1l1 λ 2l2  A1  1 +  1  +  1   + + + 1.5     2b1 2b2  A2  2b3  A3    A2   A3  

...(6.15)

In the similar manner, µ can be calculated for other configurations. The value for λ is calculated similar to the procedure followed for pipes. Equivalent pipe diameter ‘d’ for the clearance b will be d = 4R = 2b Reynold’s number for the clearance b is determined as u  2b . ν 2s +  1  ud 2 Re = = ν ν

2

...(6.16)

u . 2 Normal value of λ will be 0.04 to 0.08. For low viscous fluids, λ = 0.4.

since the velocity of the fluid, ui =

For pumps of Di > 100 mm li the length of clearance passage

l = 0.12 to 0.15 and µ = 0.5 to 0.6. DI

l = 0.2 to 0.25. Model analysis does not carried out for clearances. For protoDi types, keeping clearance width ‘bi’ same, the length ‘li’ is increased. Increase in length li increases the l losses and reduces the leakage QL1. When 1 > 0.25, µ reduces only to a smaller extent, but li increases Di considerably. The type of wearing ring construction used depends upon pump construction. li should always be ≥ 20 mm and µ ≤ 0.65 considering techno-economical condition.

When Di < 100 mm

159

LOSSES IN PUMPS

Prof. A.A. Lomakin |69| recommends that volumetric efficiency, ηvol can be calculated as  1  –2/3  η  = 1 + 0.68 nS .  V

6.3 (B) LEAKAGE FLOW THROUGH THE CLEARANCE BETWEEN TWO STAGES OF A MULTISTAGE PUMP p′ − p′ Hp3 =

1

i

γ

where p1′ = P1 + γHi, H1 = total head, and p′1 = pressure at the hub of the impeller. Hp3

u22 − u 2h p2 − p1 = H1 − + γ 8g = H1 − H p +

= Since,

gH m 2u22

2 u22   rh   1 −    8 g   r2    

  r 2  u22 + − 1 −  h   8g   r2  

H1 – Hp = Hdy QL3 = µ . πdh b . 2 gHp3

...(6.17)

Vu

R

d

Fig. 6.16. Vortex formation at bend

Fig. 6.18. Flow in divergent passage

Fig. 6.17. Vortex formation due to sudden contraction

Fig. 6.19. Flow separation at imepller outlet due to shroud

160


Cu Cr B

A

1

Fig. 6.20. Velocity distribution at outlet of impeller

Fig. 6.21. Flow separation and return flow at the outlet edge of impeller due to break effect

q = 0,15

q=0

Fig. 6.22. Secondary flow at q =

y

x

2

Q <1 Q norm

A

r

A A–A x

y

Fig. 6.23. Secondary flow between shrouds

Fig. 6.24. Flow separation at impeller outlet

161

LOSSES IN PUMPS

Diffuser blade leading edge Impeller blade trailing edge Gap "B" Leakage flow Inlet guide vane

Gap "A"

Casing

Impe lle r Impeller blade leading edge

Fig. 6.25. Secondary flow in pump impeller at partical flow conditions

Fig. 6.26. Vortex on finite blade length

6.4 HYDRAULIC LOSSES Hydraulic losses are the losses due to: friction in the main flow when fluid passes through: (a) suction or approach pipe, (b) impeller vane passages, (c) diffuser or volute casing flow passages and (d) outlet passage or discharge cone of the spiral casing of a pump. Friction arises due to (1) fluid flow over the surface roughness of the above mentioned pump parts. Heat developed by friction is carried away by water and (2) secondary flow prevailing at places of unequal distribution of pressure in the flow passage. (3) vortex λ formation along the inner side of blade configuration, (4) wake formation after the blades. Shock entry are a few examples for the secondary flow. Exact study on hydraulic losses has not so far being done, as it is extremely difficult. Theoretically developed equations along with the experimentally determined coefficients are used to determine the hydraulic losses. In general, the shape of flow passage of each element is designed not only as per dimensions but also to provide stream lined flow passages at entry as well as at exit of each elements of the pump in order to ensure minimum hydraulic losses. For example, uniform flow at casing can be ensured only if uniform flow exists at the outlet of the impeller. Similarily flow must be stream lined, steady and uniform at the exit of the approach pipe, so that same type of flow can exist at the inlet as well as at the impeller passages. Hydraulic losses can be classified as follows: (1) Losses under steady flow conditions (a) relative flow through impeller (b) absolute flow in casing, approach and exit passages (2) Losses due to unsteady flow and (3) Losses of hydraulic nature but attributed to mechanical losses. Surface roughness of the flow passage in pump parts, such as approach pipe, impeller, spiral casing and exit pipe is the main cause for hydraulic losses. Smooth surfaces give lower hydraulic losses.

162


Impeller flow channel losses can be calculated taking relative velocity ‘w’. Average value of relative velocity should be taken, since relative velocity in impeller across the circumferencial length is not constant due to circular secondary flow prevailing due to finite vanes (Figs. 6.21, 6.22, 6.23, 6.24 and 6.25). Flow separation from the outlet edge of impeller due to insufficient kinetic energy in the boundary, larger at all regions of operation except at optimum flow rate condition and subsequent increase in losses in casing due to improper flow from impeller, are the causes for the development of secondary flow. Spannhake | 110 | has given a detailed analysis about the secondary flow in impeller passages. Experimental investigation in spiral casing has clearly indicated, the presence of unequal pressure and velocity distribution not only due to non symmetrical flow from impeller, but also due to curvature of the casing. Experiments conducted indicate about 2% velocity variation along the radial direction at optimum flow region, and about 40% at Q = 0.4 (optimum region). So also under unsteady flow a higher percentage of deviation in pressure and velocity exists at the return passage due to the presence of bend kept after the diffuser of multistage pumps, which creates non uniform flow at the suction of the next impeller. At partial flow regions of operation, the pressure variation across the radial direction changes to a larger extent at inlet of the pump impeller. Flow reverse takes place from hub to periphery. Due to the presence of bend in the impeller, flow could not follow the passage, instead it again deviates from the main stream, which again creates reverse flow at the outlet of the impeller (Fig. 6.25). Flow velocity through the channel increases due to the restricted flow area. Apart from that due to larger difference in pressure and velocity between leading and trailing surfaces of the impeller channel at partial flow regions, the flow reverse takes place at the outlet of the impeller. It gradually extends in both direction along the flow passage of the impeller, then to the casing on one side, extending to suction side of the impeller on the other side. Due to this effect, the steepness of head discharge curve increases at higher specific speeds. Rate of increase in steepness is higher for higher specific speed pumps. ηh% 100

ηh

as per A.A. Lomakin

ηh

As per A.J. Stepanoff

75 as per Stodila-Meizen

50 0

100

200

300

D q MM

Fig. 6.27. Comparison of hydraulic efficiency as recommended by different authors

163

LOSSES IN PUMPS

12 10

1

%

N NT

8

2

6 4 2

3

0 70 140 210 280 350 420 ns (1) Disc friction (2) Volumetric (3) Bearing and stuffing box

Fig. 6.28. Power loss = f (ns )

All flow passages in pumps are diffuser type passages, where in flow area gradually increases. The velocity is changed into pressure. Angle of divergence of the diffuser passage must be between 5° to 10°. Radius of curvature of bend portion R should be more than 7 to 8 times the width of the passage R = 7 to 8 (Fig. 6.16). Losses ‘hf ’ due to sudden contraction and due to sudden expansion can be calculated by using the standard formula, For sudden contraction hf (contraction) = 0.5

V2 V2 and = 0.15 to 0.35 for gradual contraction. 2g 2g

For best results i.e., for minimum hydraulic losses, angle of convergence can be 17° to 21°. For sudden expansion hf (exp) = φ

(Vent − Vexi )2 2g

...(6.18)

where φ the coefficient is ‘1’ for sudden expansion and 0.4 to 0.5 for gradual expansion if the angle of divergence 8 to 10°. Since hydraulic losses are complicated, in nature, it is always calculated from the experimental results. Approximately hydraulic efficiency is assumed as ηh =

η , where ηh = hydraulic efficiency and

η = overall efficiency, A.A. Lomakin suggests that ηh = 1 –

0.42 (log Dn0 − 0.172) 2

...(6.19)

and prototype efficiency is calculated as   ηhp = 1 – ( 1– ηhm)  log D1nom − 0.172   log D1nop − 0.172   

2

...(6.20)

7 AXIAL AND RADIAL THRUSTS

7.1 INTRODUCTION The impeller of the pump, when running inside casing, which is filled with the fluid, is acted on by a number of forces. Sometimes these forces are significant in pumps either at all regions of operation of pumps or when operated at other than optimum regions of operation. These forces must be properly taken care of by proper construction of the pump. These forces are divided into two types : (1) mass force i.e., weight of pump parts and weight of the fluid filled inside the pump space and inertia force i.e., the centrifugal force and forces due to coriolis acceleration and (2) surface force, acting on the pumps due to the fluid pressure and its reaction. Total weight of the pump and the weight of the fluid inside casing, will be acting at its centre of gravity in the downward direction, whereas hydrostatic pressure will be acting in upward direction as buoyant force. Inertia force is the centrifugal force of the rotating parts. Fc =

W 2 ωr g

where, W — weight of moving parts. ω — angular velocity of the moving parts. r — the eccentricity between centre of gravity of moving part and the centre of the axis of the shaft. If the centre gravity of impeller falls exactly on the shaft axis then r = 0 and the centrifugal force is zero. This is achieved by dynamically balancing all rotating parts. Since inside surface of the impeller is the only unmachined part among all rotating parts, only impeller is dynamically balanced, before assembly. The surface force ‘F ’ by the flowing fluid on impeller and casing consists of ‘FZ’, the axial force and ‘Fr’ radial force. Axial force acting on the impeller is considerable whereas axial force on casing equalises each other and hence zero. Correspondingly radial force acting on impeller, due to uniform and constant flow at each and every point of impeller circumference, is zero, but radial force acting on the casing is not zero as this force changes in magnitude and direction from tongue to delivery mouth of the casing due to gradual increase in flow from impeller to casing. 164

165

AXIAL AND RADIAL THRUSTS

7.2 AXIAL FORCE ACTING ON THE IMPELLER Referring to Fig. 7.1, the flow at outlet of the impeller consists of (1) main flow ‘Qa’ into the casing, (2) leakage flow at the front end of the impeller, passing through the clearance between casing and impeller front shroud, then through the wearing ring clearance and finally reaching the suction side at impeller eye and (3) leakage flow at the rear end of the impeller passing through the space between rear shroud of the impeller and casing wall, then through rear wearing ring clearance and finally either passing through the stuffing box to atmosphere or passing through the balancing hole back to impeller eye. r

p2 Contour

A1

pL

p2 pR

p R – pL

r

Shaft axis

r0

rh

r

r2

A2

z-axis

p1

Fig. 7.1. For axial thrust calculation

This fluid pressure acts on both shrouds and on both sides of the shroud i.e., at inner and at outer side of the impeller shrouds. If FZ0 is the force due to fluid pressure at the outer side of the impeller shroud and FZi is the force due to fluid pressure at the inner side of the impeller shrouds and since these pressures are acting parallel to Z-axis, i.e., shaft axis, the total force acting in the impeller will be FZ = FZ0 + FZi . FZi the internal force is calculated by applying impulse momentum equation to the unit mass passing through the impeller. If AT is the surface area of the two sides of shrouds and blades of the impeller, and A1 and A2 are the area at the suction and delivery surfaces of the impeller, neglecting the weight of the of the impeller and the weight of the fluid enclosed at outside and inside the impeller passage the resolved component of net hydrodynamic force, applying Eular’s law, is written as, FZ A 1+ FZ A 2 – FZi = –

W (CZ2 – CZ1) g

...(7.1)

where FZ A1 and FZA 2 are the axial component of the pressure force on surface A1 and A2 of the impeller respectively W = γQ – Weight flow rate FZ A 1 and FZA2 are the axial component of the absolute velocity at entrance and at exist respectively. Axial force at the inner surface ‘FZ1’ cannot be determined, since pressure distribution at the inner surface is not known due to the presence of the impeller vane. It can be determined, however, with the help of Eular’s equation. In Eular’s equation, the blade induces force on fluid. The same force prevails with negative sign if the fluid induces force in the surface, hence, the negative sign on the equation (7.1). The direction of the force ‘FZ A 2’ at the surface A2 at the outlet edge of the impeller is perpendicular to axis since CZ2 is perpendicular to axis. Hence, it is zero. The force ‘FZ A 1’ will be

166

ROTODYNAMIC PUMPS (CENTRIFUGAL AND AXIAL) Rc

FZA1 =

∫ pdA = R∫ pi 2πrdr=p1 . π (R02−R1h2)

A1

...(7.2)

h

where, p1 is the pressure at impeller inlet and is equal to hS the suction pressure. R1C and R1h are the radius at periphery and at hub of the impeller eye. For normal entry the fluid velocity CZ1 = C0. Pressure p1 at the impeller eye is the suction pressure which is negative, hence the direction of FZ1 will be opposite i.e., opposite to the fluid flow direction at impeller eye and is parallel to Z-axis (shaft axis). Equation (7.1) can be written as

w γQ p1π (R210– R21h ) – FZ1 = – g . C0 = – C g 0 or

FZi =

γQ C + p1π (R21c – R21h) g 0

...(7.3)

Axial force ‘ FZ0’ acting on the outer surface will be integral of elementary hydrodynamic force pdA cos ( p z ) on the surface i.e., FZ0 =

∫ pdA cos ( pz )

A0

The value dA cos ( pz ) is the projection of the external surface dA along the direction Z, dAZ = rdθ dr .Therefore 2π

R2

FZ0 =

∫

R0

rdr

∫ 0

pF d π –

2π

R2

∫

Rh

rdr

∫ pR d θ

...(7.4)

0

where, pF and pR are the pressure at the frontside and backside of the impeller outer surface. Since the space is a free surface at any radius r the angular velocity of fluid ‘ωf’ will be half of the ω impeller angular velocity ‘ω’ i.e., ‘ωf’ = 2 Referring to the equation (3.69), pressure at the space at any radius will be 2 u22   r   1 −    ...(3.69) p = p2 – γ 8 g   r2     Therefore, the pressures at front and rear side space between impeller and casing, from outer radius R2 to the eye radius R0 are equal in magnitude and opposite in direction and hence cancel each other. But the pressures at the rear end of the impeller from R0 and Rh changes as per equation (3.69), whereas the pressure at the front and of the impeller is the suction pressure pS = p1. Equation (7.4) can be written as 2  u 22  r     = ∫  p2 − γ 1 −   2πr dr  g R 8   2   RH    R0

FZ0

167


 p2 u 2  R02 + Rh2   2 2 = γπ R0 − Rh  − 1 −  2R22    γ 8 g  FZ = FZ0 + FZi j

(

∴ Net force

)

...(7.5)

 p2 – p1 u 2  R02 + Rh2   γQ 2 2 − − C0 1− FZ = γπ R0 − Rh   2  g 8g  2R2    γ

(

)

L F I OP rQ iMM G JK PQ − g ⋅ C H N

R02 + Rh2 u2 2 2 = γπ R0 − Rh H p − 8 g 1 − 2 R 2 2

d

0

A

...(7.6)

B

The value marked ‘A’ is normally very large when compared to the value ‘B’ in the equation (7.6). Hence, the value B is usually neglected. The direction of this axial force will be opposite to the direction of flow at suction i.e., the tendency of this force FZ is to remove the impeller from the shaft. This force is carried over to rear bearing of the pump through the impeller nut and pump shaft. When the wearing ring clearance is in fully damaged condition. Equation (3.70) should be considered instead of equation (3.69). p2 = p2 − γ

u22  R22    8 g  r2 

 r22  1 −   R22  

...(3.70)

R2

There,

FZ02 =

∫ ( p2 − p f ) 2π rdr

R0

2   u2  R   u2 r2   = 2π ∫   p2 − γ 2  2  1 − 2   − 2π  p2 − γ 2 8 g  r   R2   8g  R0    R2

 − 2 

...(7.7)

Net Axial force FZ = FZ02 (as per equation) (7.6) + FZ (as per equation) (7.7)

...(7.8)

(

2 2 = γπ R2 − Rh



 r 2   1 −     rdr  R22   

) 8ug  R R− R 2 2

2 2

2 2

2 h

ln

R22 Rh2

+

R22 + Rh2 2 R22

Due to continuous operation of pump, erosion at the inlet and at the outlet of wearing ring passage gradually removes the material of the wearing ring, as a result of this, wearing ring clearance increases. RC This wear is more in high pressure units for low specific speed pumps, nS = 50 to 80 and < 0.5. R2

7.3 AXIAL THRUST IN SEMI-OPEN IMPELLERS Axial thrust in semi-open impellers will be more than that in closed impellers, due to the fact that the axial thrust on the rear shroud is partially balanced from (Point 2 to point 1) outlet edge to inlet edge of the impeller due to equal pressure variation at both ends of rear shroud (Fig. 7.2). Net pressure at the rear shroud will be

168


H2

r1

r2

H2

rh

 1 u 22 – u 2S   ...(7.9) FZR = γ (A2 – As)  H 2 − 8 2g   where A2 and AS are the surface area at radii R2 and RS . At the front side i.e., at the suction side of the impeller since there is no shroud, it is assumed that the pressure between outlet and inlet edge of the impeller i.e., from R2 to R1 varies in a straight line form. Axial thrust will be γH 2 FZ f = (A2 – A1) 2 The resultant axial thrust will be

Fig. 7.2. Axial thrust in semi-open impellers

...(7.2)

FZi = FZR – FZf = γ (A2 – As)

 H2  1 u 22 − uS2  H2 −  – γ  (A2 – AS ) 2  8 2g   

...(7.10)

7.4 AXIAL THRUST DUE TO DIRECTION CHANGE IN BEND AT INLET Actual axial thrust will be lower than the value calculated as per the equation (7.10) due to the centrifugal force created at inlet due to the direction change from axial to radial in centrifugal pumps. Since, there is no blade in this space only absolute velocity should be considered. Centrifugal force can be calculated as γ ...(7.11) Pt = k QC0 g where k is the experimental coefficient 0.85 to 0.45 for nS = 50 to 250. This centrifugal force acts opposite to the main axial thrust. Net axial thrust P will be PNet = PZi – Pt ...(7.12) For higher nS pumps Ft is small and hence Kt neglected. For single stage radial type centrifugal pump, axial force can be determined as 1,0

D

W

Dcp π 2 – d2h ) H3γ ...(7.13) P = (Dwi dh 4 0,5 where Dwi is the diameter of impeller wearing 50 100 150 200 250 ns ring, dh hub diameter and H3 is the average static head H3 = 0.6 H for nS < 60 and 0.8 H for nS 60 Fig. 7.3. Coefficient to calculate axial thrust due to direction change at inlet of impeller (at bend) to 200. Experiments were conducted by the Institute of Hydraulic Machines Moscow |131| based on the above equation. The above equation in final form is written as π ...(7.14) P = Kt γH (Dw2 – d2h ) 4 The experimental coefficient Kt is given in Fig. 7.3.

169


7.5 BALANCING OF AXIAL THRUST Balancing of axial thrust is carried out by the following methods: (a) By proper use of rear bearing i.e., bearing near coupling side of the pump, such as ball bearing, angular contact bearing, tapered roller bearing, or thrust bearing. (b) By providing radial vanes at the outer side of rear shroud i.e., at the space between impeller rear shroud and the casing wall [Fig. 7.5 (d)]. (c) By providing balancing holes, below the inlet edge of the impeller at the rear shroud. [Fig. 7.6 (c)] (d) By providing balancing disc or balancing drum. [Fig. 7.6 (b)] (e) By arranging opposed impellers either suction end facing each other i.e., single suction at the middle and double delivery at both ends or suction end opposing each other i.e., double suction and single delivery. This may be embedded into one unit as in double suction pumps or individual impellers. [Figs. 7.4, 7.5 and 7.6 (a)]

FZ

FZ

Fig. 7.4. Balancing by individual two suction facing impellers

Fig. 7.5. Balanced individual opposed impellers

Suction connection tube Q

Q

(a) Double suction

D

(b) Balancing disc

C

(c) Balancing hole (d) Radial blades

Fig. 7.6. Different methods of balancing axial thrust

Arrangement c, d are provided for single stage pumps whereas arrangements a and b are provided for multistage pumps. Double suction pump is a single stage opposed impeller pump [Fig. 7.6 (a)].

170


Balancing by (a) Balancing disc or drum (a) 1

2

3

4

5

6

(b), (c), (d), (e), self balancing by impeller positioning (b)

1

3

5

(c)

1

(d)

1

2

3

(e)

1

2

3

3

6

4

4

5

6

4

5

6

5

2

2

6

4

Fig. 7.7. Different balancing methods by impeller position in multistage balancing.

7.6 AXIAL THRUST TAKEN BY BEARINGS Coupling side bearing of the pump is always designed to take care of the axial thrust. For smaller axial loads deep groove ball bearings are used. For higher axial thrusts, deep groove ball bearings are replaced by angular contact ball bearings. For still higher axial thrusts the bearing is replaced by tapered roller bearing or a pure thrust bearing along with ball bearing. Use of suitable bearing, keeps the pump bearing housing compact, thereby pump weight is considerably reduced and also the cost.

7.7 RADIAL VANES AT REAR SHROUD OF THE IMPELLER Location of radial blades at the rear shroud is shown in Fig. 7.8. The magnitude of axial thrust reduction depends purely upon the size of the radial blades as well as the clearance between casing wall and the radial blade. Extra power consumed due to this additional radial vanes does not exceed the power spent for the additional clearance provided at the rear shroud to control leakage. This power always remains constant. The blade width will be normally 3 to 4 mm. But it is difficult to maintain the clearance between casing wall and the radial blade, since casing wall is an unmachined surface.

171


p2

p2 a b r0

C

r8m

∆p

Fig. 7.8. Axial thrust balancing by radial blades a = p2 – γ

2

2

u 2 – u 2p ; 8g

b=γ

2

u p – u2 u 2 – u2 ; c=γ . 2g 8g

Due to the presence of rear radial blades, which is rotating at a blade velocity ‘u’ the fluid also 1 rotates at a velocity of ‘u’ instead of u as in the case of without radial blades condition. This reduces 2 the pressure at the rear space. At any radius ‘rp’ the pressure drop ∆p will be ∆p =

3 γ 2 (u p – u2) 8 g

...(7.15)

when radial blades are present the radial thrust reduces to ∆Fzp

 u 2p − uh2  3 γ 2 2 ⋅ π rp − rh  =  8 g  2 

(

)

...(7.16)

In order to have perfect axial thrust balancing FZi = ∆Fzp. ...(7.17) The diameter of the radial blades can be determined from equation (7.16) and (7.17). In some of the pump constructions, the radial blades are provided in casing wall instead of at 1 impeller rear shroud. Since blade is stationary, the fluid velocity will be u instead of ‘u’. In practice, 2 1 it is found that fluid velocity is slightly more than ‘ u’. 2

7.8 AXIAL THRUST BALANCING BY BALANCING HOLES This method of balancing is done by providing an orifice hole below the inlet edge of impeller blade, connecting the chamber kept between the impeller rear shroud and casing wall [Figs. 7.6 (c), 7.9]. Referring to the pressure distribution prevailing on both sides of impeller, non balancing of axial thrust exists only at impeller eye area. The pressure is suction pressure (negative pressure) at the front side of the impeller shroud below the inlet edge of the impeller, whereas, pressure at the rear side of the impeller shroud is +ve (space C). Perfect balancing can be attained if these two pressures are made equal by providing an orifice type hole (D) connecting these two spaces. Although theoretically the axial thrust is balanced, practically a slight unbalanced force exists either +ve or –ve. This is due to the fact that, the pressure drop across the front and rear wearing rings A and B are not equal owing to the difference in

172


wearing clearance at A and at B due to gradual wear in running. The flow qL1 and qL3 are also not equal. The pressure at chamber D is always slightly greater than the suction pressure at impeller eye, due to hydraulic losses in orifice. But magnitude of this axial thrust is very small, and it is taken care of by bearing. This facility enables to provide smaller capacity bearings. C D The diameters of wearing ring ‘Di’ at both ends of the impeller are selected as equal. The sum total of the area of orifices ΣAOri must be more than the wearing ring area ‘Aw’, in order to have perfect balancing. Fig. 7.9. Axial thrust balancing by Normally ΣAOri ≥ 4 to 5 A w for better cavitational balancing holes property. Although this process is simple and easy for manufacture, the flow quantity ‘qC3’ through the orifice is larger than the normal ‘qL3’ through stuffing box, as a result of which the main flow ‘Qact’ is smaller and the flow through the impeller Qth is larger. Volumetric efficiency ηv, overall efficiency η and the total head ‘H’considerably reduce. Vacuum before the inlet edge of the impeller is also reduced. Correspondingly, the cavitational characteristics also reduce, due to reduction in allowable HVac for a wide range of operation of pump. Due to this drawback, this method is not recommended and not followed for pump having low quantity and low head.

7.9 AXIAL THRUST BALANCING BY BALANCE DRUM AND DISC Axial thrust balancing by balancing drum and by balancing disc provides perfect balancing. This method is adopted in multistage pumps and is fitted after the last stage of the pump. The main advantage of the arrangement is that, perfect balancing is achieved at all points of operation automatically and also it can be used for any number of stages in pump. For trouble free running, provision must be given to allow the rotating system to move slightly in axial direction in both directions (≈ 0.5 mm). Balancing drum [Fig. 7.6 (a)] consists of a chamber, where the drum is located. This chamber is connected by a Tube T q3

ϕ∆p d

p5

K2

Re

ps Q Σ Fzi

Fd p3

p5

Ra

K1

∆p d

ld2

b2

R sh

b p4

Fig. 7.10. Axial thrust balancing by balancing disc

Rb

p2

b

Q

p4

p5

173


pipe to the impeller eye of the I stage impeller. The size of the drum is selected such that ΣFZ, the total axial thrust of all the impellers, of the multistage pump is equalised by the axial force (FD) developed by balancing drum in the opposite direction. Practically, there may be a very small amount of unbalanced axial thrust, which is developed due to fluctuation in the leakage flow q3. Also, there will be unequal balancing of axial thrust, when the clearance between rotary and stationary number of the drum wears out, due to aging. Such unbalanced forces are taken care of by bearings. Balancing disc works under the same principle as that of balancing drum (Figs. 7.11 and 7.12). The leakage flow q3 of the fluid from the last stage of the impeller enters the chamber (K1) located between balancing disc (1) and stationary ring (2) through the axial clearances b. The chamber ( K2) is connected to the impeller eye area of the I stage with the help of pipe. When pump is running, the leakage flow (q3) from the main flow (Qth) through the impeller, under the pressure p3 flows through the axial clearance (b) to the disc chamber K1 where the pressure is p4 and then through the clearance (b1), with certain losses. Flow then enters the chamber outlet K2, from which flow passes through the tube T, enter the eye of the impeller. The pressure drops from p5 to p1 which is greater than the suction pressure PS (i.e., P5 > p1> pS ). Suciton

p1

Q

p0

q3

F zI

d y∂

d∂

Q

Fig. 7.11. Balancing drum (another form)

FzD

Fig. 7.12. Axial thrust balancing by balancing drum

Pressure (p4) in the chamber (K1) induces a force at the bottom of the disc clearance passage. If this pressure (p4) is larger than total axial thrust Σ FZ, the moving disc moves away from the stationary ring. The disc clearance (b1) now increases. This in turn increases the leakage flow (q3) and also the losses in the clearance. As a result of this, the pressure (p4) drops down and the disc moves towards the stationary ring which in turn reduces the clearance (b1) and losses in the disc. This process repeats and the clearance (b1) goes on changing, until the pressure (p4) equalises the axial thrust ΣFZ.. At this stage the clearance b1 remains constant. The leakage quantity (q3) flows through the tube to the impeller eye of the first stage of the impeller. The pressure drop (p4 – p5) at the disc clearance, the leakage flow (q3), the dimensions of the clearance, the connecting pipe dimensions to carry the leakage quantity q3 back to the inlet of the Ist stage impeller are to be determined as follows: The pressure drop, ∆ p (p4 – p5) across the disc clearance, to get complete balancing of axial thrust will be ∆P =

ΣFZ ψ π ( Ra2 − R 2h )

...(7.18)

174


where, Ψ is the coefficient depending upon the pressure distribution across the disc Ψ < 1, Ra is outer diameter of disc and Rh is the outer diameter of the shaft sleeve. Taking an uniform change of pressure across the clearance ‘b1’, the coefficient Ψ depends upon the dimensions of the disc only 2

ψ =

 r  r  r  (1 − φ) 1 + e  + (1 + 2 φ)  e  − 3  b   Ra   Ra   Ra   r  3 1 − b   Ra 

2

2

...(7.19)

where, φ the coefficient depends upon the pressure drop at the entry to and exit from disc clearance and the losses in the clearance and is taken as φ = 0.18 to 0.25. The leakage quantity (q3 ) will be q3 = µ2 π reb1 2 g

∆p γ

...(7.20)

The flow coefficient µ will be µ =

1 r2 ( R − r )r λ a e e + e2 + 0.5 2 Ra b1 Ra

...(7.21)

The pressure p4 before the disc can be determined from the pressure drop across the axial clearances b. i.e.,

  ω2 2 γ H Z − + H − ( 1) ( R2 − rS2 )  – ∆P – (p5 – ps) (p3 – p4 ) =  p 8g   where,

H Z ps p5

...(7.22)

— — — —

Head developed per stage. Number of stages. Suction pressure at impeller eye of 1st stage. The pressure in the balancing chamber outlet (not more than 5 to 8 kg/cm2 so that the stuffing box can work without any trouble). The pressure drop (p3 – p4) across the axial clearance (b) will be q3 = µs As 2 g

p3 − p4 γ

...(7.23)

Knowing q3 from equation (7.20) and the pressure drop from equation (7.19) the area µs As =

q3 p − p4 2g 3 γ

...(7.24)

175


For better operation the clearance b1= (0.0010 to 0.0012) Ra and will be 0.6 to 0.8 mm. Now, hence the length ‘L’ for the clearance can be determined. The radius Ra of the disc is selected slightly less than the outer radius of the impeller. The diameter Rb = (1.2 to 1.5) Rsh where Rsh = the shaft radius. The inner radius Re is fixed, based on the sufficient length (ld ) of the disc. The pressure drop ‘Ψ∆p’ is taken as constant. The coefficient ψ is determined from the condition that the force Fd determined from the actual pressure distribution is equal to the pressure distribution on the complete surface of the disc i.e., ∑Fzi = Fd = ψ ∆pd π

(

−

Ra2

Re

Ra

Rb

Re

= ∫ ∆pd 2π rdr +

Rb2

)=

Ra

∫ ∆p 2π rdr

Rb

∫ ∆pd 2π rdr

...(7.25)

The pressure distribution on both sides of the disc and the pressure drop ∆p change according to radius. Pressure on the right side of the disc p5 is constant and approximately 4 to 8 kg/cm2, for trouble free operation of stuffing box. The pressure ‘p4’ at the left side of the disc is also constant. The pressure drops from p4 to p5 due to losses in the balancing disc clearance φ∆p where the coefficient φ will be φ =

1.5

...(7.26)

2

λld Ra  Re  . + + 0.5 2bd Re  Ra 

where ld = Ra – Re and λ, the coefficient of friction, depends upon the Reynold’s number of the flow ‘Re’ 2

u  2b2 C a2 +  a   2  Re = ...(7.27) ν where Ca is the flow velocity at entrance and ν the kinematic viscosity of the fluid. Normally λ = 0.4 to 0.8 and φ will be 0.15 to 0.25. The pressure drop (∆p) in the disc clearance can be taken as proportional to radius of the disc. It can be expressed

∆p = ∆pd (1 – φ)

Ra − r Ra − Re

...(7.28)

Substituting this value of ∆p in equation (7.21) Re

Fd =

∫

Rb

=

Ra

∆ Pa 2π rdr +

∫ ∆ Pa (1 – φ)

Re

Ra − r 2π rdr Ra − Re

(

)

2 2  2 2  (1 − φ) ( Ra + Ra R1 ) + (1 + 2φ) Re − 3R b  π R a − Re ∆pd

(

3 Ra2 − Rb2

)

176


from which 2

 R  R  R  (1 − φ)  1 + e  + (1 + 2φ)  e  − 3  b  Ra    Ra   Ra 

ψ =

where, µd =

  R 2  3 1 −  b     Ra     ∆pd = µ2π Reb2 γ

q3 = µd A3 2 g

The flow

1

λ d Re R 2e . + + 0.5 2b Ra R 2a

2

2g

∆pd γ

...(7.29)

...(7.30)

.

The length of the clearance ‘ly’ at the shaft sleeve before the disc will be q3 = µy Ay 2 g

p3 − p4 = µy 2π Rh by γ

2g

p3 − p4 γ

...(7.31)

where, by is the radial width of the clearance and µy the coefficient is µy =

1 λl y

+ 1.5

2by

The pressure drop (p3 – p4) = (p2 – p5) – (p2 – p3) – ∆ pd p2 = p1 + γ [H (Z – 1) + Hp] H — Total head of the pump Z — Number of stages

u 22 (p2 – p3) = γ 8g

 Rb2 1 − 2  R2

  

...(7.32)

...(7.33)

The length of the tube ‘lt’, connecting the suction side of the Ist stage and outlet chamber of the disc is approximately the length of the pump assembly. The tube diameter dt is determined from the equation (7.34) q3 = µt A –

2g

p5 − p1 γ

πdt 2

=

4

λlt + 1.5 dt

2g

p5 − p1 γ

...(7.34)

177


7.10 RADIAL FORCES ACTING ON VOLUTE CASING Radial forces in spiral casing, occur only where axisymmetry is not maintained in flow at the impeller outlet. Theoretically, axisymmetry can prevail only at the optimum efficiency conditions. Practically, axisymmetry cannot be maintained due to uneven flow velocity across the breadth at the outlet of the impeller. At partial flow conditions due to less flow, the flow is in the form of diffuser, due to larger area of casing, whereas at higher flow the flow passage becomes smaller and forms a convergent passage. In all these cases, flow cannot be axisymmetric. In Fig. 7.13, the pressure distribution at the inlet of the spiral casing and the location of minimum (F1) and maximum (F2) forces acting on the casing are illustrated. The net radial force acting on the rotating shaft creates fatigue and deflection. The clearance provided between wearing rings and between shaft sleeve and shaft must take care of this deflection while the pump runs from minimum to maximum flow. The impeller side bearing or the front bearing must be designed to take this radial load. 90

90

p2

180

360

F1

180

0

360

270 F2

270 (a)

(b)

Fig. 7.13. Pressure distribution across volute

7.11 DETERMINATION OF RADIAL FORCES Forces acting on the external (Fe) and internal (Fi) surfaces in x and y coordinates can be determined as, Fxe =

∫ p dA cos (nx)

Ae

Fye =

for external surface

...(7.35)

∫ p dA cos (ny)

Ae

where n is the normal to elementary surface considered p is the pressure, taken from the known pressure distribution. Applying moment of momentum equation between inlet and outlet, in order to determine the internal forces.

178


– Fxi =

∫ ρ CndA C cos (C x) – ∫ ρ Cn dA C cos (C x)

Ae

– Fyi =

A1

...(7.36)

∫ ρ CndA C cos (C y) – ∫ ρ CndA C cos (C y)

A2

A3

where, A1 and A2 are the inlet and outlet area of the impeller C and Cn are the absolute velocity and its components, resolved normal to the surface dA. ρ is the fluid density. Since the fluid exerts a force on impeller, the negative sign is given for Fxi and Fyi. For normal entry at inlet C = C0 and is parallel to axis. Hence, Cn = 0. The integral for the area ‘A’ becomes zero. The equations are reduced to – Fxi =

and

– Fyi = Total forces,

and

∫ ρ CndA C cos (C x )

A2

∫ ρ CndAC cos (C y)

...(7.37)

A2

Fx = Fxe + Fxi and Fy = Fye + Fyi

F x2 + F y2

F =

and angle θ = Arc tan

Fy Fx

...(7.38)

Experimental investigation on different pumps shows that the pressure is almost constant around the circumference of the impeller outlet (or at inlet to spiral casing) at optimum efficiency condition i.e., flow is axisymmetric. At high flow rate, (above normal flow) the pressure falls from tongue to outlet of spiral casing. At below normal flow conditions the pressure increases from tongue to outlet end of spiral casing. At very small flow rate, the flow reversal takes place near the tongue due to high pressure. For calculation purpose, it can be assumed that pressure variation is uniform i.e., in straight line, from tongue to outlet of spiral casing at below normal and above normal flow rates (Fig. 7.13). Total head is the sum of pressure head (Hp) and dynamic head (Hdy). Considering a uniform straight line variation around the circumference of the impeller, the pressure at any angle θ of the volute can be written as θ   p = γHm = γ  Hp+ Hdy  2π  

...(7.39)

and dA = b2R2d θ and cos (nx) = cos θ in equation (7.31). The radial thrust on the external side will be (Fxe) Fxe =

∫

2π

p dA cos (n x) = –

Ae

0

2π

=–



θ

∫ γ 2π 0

θ



∫ γ  H p + 2π H dy  b2 r2 dθ cosθ

Hdy b2 r2 dθ cos θ

179


γH dy

=–

2π

2π

∫ θ cos θ dθ –

b2 r2

0

γHdy b2 r2 [ θ sin θ + cos θ]02π = 0 2π ...(7.40)

2π

∫ cos θ dθ

Since,

=0

0

The external force in y direction Fye will be 2π

Fye = –

∫ 0

= − =–

2π θ   θ H dy  b r dθ sinθ = ∫ γ γ Hp + H b γ d θ sinθ 2 2 2π 2π dy 2 2   0

γH dy 2π

2π

∫ θ sin θ d θ

b2 r2

= −

0

γH dy

b2 r2 [− θ cos θ + sin θ] 2 π 0

2π

γH dy

b r (–2π ) = r Hdy b2 r2 ...(7.41) 2π 2 2 The forces Fxe and Fye calculated as per the equation (7.35) are directed in radial direction towards axis, near the tongue i.e., towards the smaller sections of spiral casing. The radial forces at the internal surface Fxi and Fyi can be determined from the experimental results. From the analysis, it is found that the tangential component Cu2 at the outlet of the impeller is constant at all point around the circumference →

Since

→

→

C2 = Cr 2 + Cu 2 – Fxi =

∫ ρ Cr2 dA C′2 (C2 x) = ∫ ρ C2r2 dA cos (Cr2 x)

A2

A2

+

∫ ρ Cr2dA C′u2 cos (C′u2 x) and

A2

– Fyi =

∫ρ

A2

...(7.42)

∫ ρ C r2 dA cos (Cr2 y)

Cr2 dA C′2 cos (Cu2 y) =

2

A2

+

∫ ρ Cr2

dA cos (Cr2 y)

A2

Q , radial velocity at any πD2 b2

Taking the radial velocity at the outlet of the spiral casing as Cr20 = angle θ of the spiral casing will be θ   Cr2 = 1 −  Cr20 2π   2π

So,

θ   Fxi = ∫ ρ Cr20 1 −  b r d θ cos θ –  2π  2 2 0

= ρ C2r20 r2b2

2π

2π



θ 

∫ ρ Cr20 × 1 − 2π  × Cu2′ b2 r2 d θ sin θ 0

θ   ∫ 1 − 2π  cos θ d θ – ρ Cr20 b2 r2 0

2π



θ 

∫ 1 − 2π  0

sin θ dθ

180


= 2b2 r2γ

C2 C 2r 20  Cu′ 2 1  −  b D γ r 20  2 g  Cr 20 π  2 2 2 g

2π

– Fyi =

2  ∫ ρ Cr20 1 −

0



2

θ  b r d θ sin θ +  2 2 2π 

 1 1 −    tanα ′ π    2

2π

∫ ρ Cr20 0

2

×  1 − θ  Cu′ 2 b2 d θ cos θ = ρC2r 2 0 b2 r2  2π  + ρ Cr20 C′u2 b2 r2

2π



...(7.43)

2π

2

θ   ∫ 1 − 2π  sin θd θ 0

θ 

∫ 1 − 2π  cos θd θ 0

2

= – 2b2r2γ where, C′u 2 =

C r 20 C r220 = – b2D2γ 2g 2g

...(7.44)

gH m , Cr20 = radial velocity at impeller outlet at optimum efficiency condition, α′2 is the u2

absolute angle at outlet of the impeller. tan α′2 =

Cr 20 . Cu′ 2

While calculating the forces acting on outside surfaces of the impeller, only the cylindrical surface of the impeller outlet is considered. The calculated value will be slightly lower than the actual. The direction of the resultant will deviate about 20° with respect to vertical towards the volute tongue. The derived equations can also be applied for other regions of operation either for part loaded or for overloaded conditions. A.J. Stepanoff |112| recommends following empirical rule for the radial thrust in pumps

  Q Fr = 360 1 −    Qopt 

   

2

 Hb′ D 2 2  

...(7.45)

where H — Total head, D2 — outer diameter of the impeller b2′ = b2 + 2t b2 — Outlet breadth of impeller and t — thickness of shroud at outlet.

7.12 METHODS TO BALANCE THE RADIAL THRUST Hydraulically balancing the radial thrust at all regions of operation is achieved by providing flow symmetry in casing design. Dividing the volute flow passage into two equally, symmetrical channels, each channel covering 180° of flow and kept opposite to each other provides complete symmetry at all regions of operations. Radial forces developed in each channel are equal in magnitude, but acting opposite to each other at any point around the circumference, at all regions of operation. In the same manner, number of channel can be increased. Provision of a number of vaned channels more than two called, diffuser, provides perfect symmetry and equalises the radial thrust.

181


In some of the volute designs for single stage pumps two volutes, each covering 180° of total flow angle, are provided, with two outlet mouths (Fig. 7.14). In some other designs, two half volutes are provided each covering 180° of flow area from impeller outlet (Fig. 7.14). The total flow enters a single outlet mouth of volute. In both cases, the radial thrust created at any point equalises between two half volute thereby net radial force is zero. In multistage pumps, the outlet flow from the impeller enters two spiral passage, which are kept 180° apart, which equalises the radial thrust. diffuser section

(a)

Fig. 7.14. Volute designs to balance radial thrust

(b)

8 MODEL ANALYSIS

8.1 INTRODUCTION Actual pump parameters differ from the theoretical values, due to the presence of viscosity in real fluid and complicated flow passages in pumps. Two identical pumps differ in quality due to the presence of different dimensions of surface roughness in flow passages. Model analysis and model testing of pumps give an option to overcome all the above mentioned difficulties and also gives all necessary information to design new pumps, so that, these pumps can be operated in a wide range of operation, with quality. Two pumps, model and prototype units, can be identical, if these pumps are similar geometrically, kinematically and dynamically. Geometrical similarity indicates linear proportionality of all dimensions including surface roughness of pump parts between model and prototype units. Kinematic similarity indicates that fluid flow direction in all elements of model and prototype, at identical points remain same. Combining these two similarities, we get that the absolute, relative and blade velocities between model and prototype are proportional but in the same direction. Dynamic similarity indicates the proportionality of the forces acting at the identical points of model and phototype units. Referring the Navier-Stokes equation for a three dimensional incompressible fluid flow, geometrical similarity and kinematic similarity are included, if dynamic similarity is considered. Most important non-dimensional parameters such as Reynold’s number (Re), Froude number (Fr), Struhaul’s number (Sh) and Eular’s number (Eu) are considered for dynamic similarity for incompressible viscous flow through pumps. These numbers must be same for model and for prototype. CA

wA w A′

αA′ = αA

βA′

C A′

A A′ B′

B

βA = β A′

uA′ Model

CA′ CA

Prototype

Fig. 8.1. Geometrical and kinematic similarity 182

=

uA w A′ wA

183

MODEL ANALYSIS

Inertia force Vl = Viscous force ν

Reynold’s number

Re =

Froude number

V2 Inertia force Fr = = gl Gravitational force

Struhaul’s number

Sh =

...(8.1)

Inertia force V = nl Unsteady, periodical forces

where V is the velocity, l is the linear dimension, ν is the kinematic viscosity, and n is the speed. During model test, all the above three non-dimensional numbers cannot be studied simultaneously. Since these three numbers do not depend on each other, they are studied individually. Reynold’s number is studied for a pressure flow, closed conduit flow of viscous fluid, such as flow in fully submerged condition, flow in pipes and flow of fluid through pumps under completely filled condition. Froude number is studied for a free flow such as open channel flow, flow of ship in water. In pumps, this number is studied under fully developed cavitation condition, where flow separation exists. Struhauls number is used for unsteady, periodical flow, in pumps, impeller as a whole, propulsion of ships. When volumetric forces are not considered, Reynold’s number and Froude number can be studied together by another number called Eular’s number (Eu). Eular’s Number

Eu =

p ρV

2

=

Pressure force Inertia force

where p is the pressure drop. Since geometric similarity is the proportionality of linear dimensions of identical parts of model and prototype, using suffices ‘p’ for prototype and ‘m’ for model for all equation hereafter,

lp

Dp Bp lm = λl Dm = Bm where ‘l’ is the linear dimension, D is the diameter, B is the breadth and ‘λl’ is the proportionality coefficient. Kinematic similarity in pumps indicate that flow directions are same for model and for prototype, i.e., flow angles namely absolute angle ‘α’ and blade angle ‘β’ remain same in model and in prototype i.e., αm = αp and βm= βp . Since linear dimensional are already proportional between model and prototype, velocity triangles are similar i.e.,

Cp Cm up um Theoretical Flow rate,

Qthp Qthm

=

=

=

wp wm

=

up um

πD p n p πDm nm

=

Cmp Cmm

= λl

πD p B p Cmp πDm Bm Cmm

=

Cup Cum

np

...(8.2)

nm = λ3l

np nm

...(8.3)

184


For geometrical similarity,

Dp Dm

=

Bm D p3 n p

Qp

Hence,

Bp

Qm = Dm3 nm

...(8.4)

Qa = ηv .Qth,

Actual flow rate,

Qap Qam

=

H thp

Theoretical total head,

λ3l

ηvp .n

ην p D3p n p

p

= η . = ηνm Dm3 nm vm nm

u 2p

Cup u p

H thm = Cum um = u 2 m

2 = λl

n 2p nm2

=

...(8.5)

D 2p n2p Dm2 nm2

...(8.6)

Actual total head, Ha = ηh Hth.

H ap

Hence,

H am

=

λ2l

ηhp n2p ηhm n 2m

=

ηhp D 2p n 2p

...(8.7)

ηhm D 2m n 2m

Theoretical power, Nth = γQH and actual power, Na = ηNth

N ap N am

=

η p Nthp η p γ pQ p H p ⋅ = ηm Nthm ηm γ m Qm H m

=

η p γ p n3p D5p η p γ p D3p n p D 2p n2p ⋅ ⋅ ⋅ 3 ⋅ 2 2 = 3 ηm γ m nm ηm γ m D m nm D m n m D5m

=

η p γ p 5 n3p ⋅ ⋅ λl ⋅ 3 ηm γ m nm

...(8.8)

If pumping fluid is same in prototype and in model γp = γm. In order to compare the performance of different pumps, and also to get complete characteristics of one series of pumps a term specific spread (ns) is used. It is defined as the speed of a pump which is geometrically similar for one pump series and consumes 1 hp of power under 1 m of total head. The efficiencies of these pump series remain constant independent of its sizes i.e., n = ns when N = 1 hp and H = 1 m. γQH (where Q in m3/sec H in m and γ = 1000 kg/m3 75 From similarity laws for power

N (hp) =

H ∝

ηh

n2D2

or

Ν

D2 ∝

∝

H

ηh n 2

;

D5

∝

η n3D5 or D5 ∝

 H   2    ηh n  N ηn 3

5/ 2

∝

H 5/ 2 η5h/ 2 n5

185

MODEL ANALYSIS

Combining both by removing ‘D’ N ηn 3

∝

H 5/ 2 ηH 5 / 2 2 ∝ 5 / 2 5 or n ηh n η5h/ 2 n5 10,000

90% x=

10,

000 x=

1 ,0

00

1,000 85% x=

10 0

80% 100 x=

10

70% 10

0

100

200

300

400

Specific speed (n s)

Fig. 8.2. (a) Efficiency change for different sizes

x = 10

x = 100

x = 1,000

Fig. 8.2. (b) Size of the unit (x)

x = 10,000

186


n = nS if H = 1 m and N = 1 hp, Designating η as ηs (efficiency of the series) and ηh as ηhs. where suffix ‘s’ indicates one series of pumps having same value of ns.

 ηs .1 ηh5 / 2 N  ns . =  5/ 2  n  ηhs .1 ηH 

1/ 2

=

η s ηh 5 / 4 ηhs

N

ηH

...(8.9)

5/ 4

If efficiencies are same for all pumps of same series i.e., for one value of ns, n N

ηs = η and ηhs = ηh. So, ns = Substituting the value N =

...(8.10)

H 5/ 4

γQH 1000 = . Q.H 75 75

3.65n Q 1000 n Q . = 3 / 4 75 H 3/ 4 H Characteristics linear dimension in pump is the diameter D. So replacing l and D ns =

Struhaul’s No. Sh = Eular’s No. Eu = The flow rate

V V or n = S h .D nD p . Since p = γH or γ = ρg, ρV 2

Eu =

gH V2

or H =

V2 . Eu g

Q = AV ns =

where, K =

...(8.11)

3.65n Q

= 3.65

H 3/ 4

K A .V g3/ 4 V . 3/ 2 3/ 4 = . Sh Eu3 / 4 S h .D V ( Eu )

...(8.12)

A 3.65 A g 3 / 4 which is constant for one series of pump, since is constant for one series D D

n s = 40 ÷ 80

80 ÷ 150

150 ÷ 300

300 ÷ 600

400 ÷ 600

Fig. 8.3. Impeller shapes for different ns

600 ÷ 1200

D

D 6T

D

D D 6T

D6T

D2

D1

D0 D1

D1

D0

D2

D2

D2 D0

D 0 = D1

of pump. The specific speed, ns is a function of similarity of Struhaul’s and Eular’s numbers and hence similarity of Struhaul’s, Reynold’s and Froude numbers. Each value of ns designates one series of pump, which has its own operating region at which overall efficiency is maximum and hence the form, shape of pump of one series will be same for one ns value. But forms and shapes will be different for each series. Specific speed ns completely defines the characteristics of one series. (Fig. 8.3) shows impeller shape for each value of ns.

1200 ÷ 2000

187

MODEL ANALYSIS

The test results of a model pump i.e., a pump from one series having one value of ns, can be used for developing other pumps in the same series (same ns value), if it is brought out in a non-dimensional form. In pump industries, unit head (KH), unit discharge (KQ), and unit power (KN), are the nondimensional parameters used to study the pump characteristics of one series. Quantity of flow, Q = ηV .πDB.Cm For geometrically similar pump, B ∝ D and hence, DB ∝ D2. For kinematic similarity in pump, Cm ∝ u and u= πDn. (n-speed is rps). So, Q ∝ ηVπD2 Dn or Q ∝ nD3 KQ =

Q nD3

is constant for one series of pump and is called unit discharge.

Similarly total head, H =

Cu u .ηh. Since, Cu α u α nD, H α n2D2 g

KH = Power N =

KN =

H is constant for one series of pump and is called unit head. n D2 2

γQH γ ∝ nD3 n2 D2 α n3 D5 75 75

N

is constant for the series of pump and is called unit power. Test results of the n D5 model pump conducted at different speeds are reproduced in these three non-dimensional parameters namely σ, η, KH , KN = f (KQ). This is called universal characteristics of pump and remains same for one series of pump i.e., for pumps having same ns but with different n, Q and H. Substituting values KQ and KH in specific speed ns equation, Q = KQ n D3, H = KH n2 D2. 3

ns =

3.6560n KQ nD3 2

2 3/ 4

(K H n D )

=

219 KQ ( K H )3 / 4

...(8.13)

8.1.1 Real Fluid Flow Pattern in Pumps Real fluid flow pattern prevailing in pumps, at all regions of operations i.e., below optimum, optimum and above optimum regions, is complicated and is far different from the theoretical flow pattern. This necessitates proper streamlining of flow passages, and bringing the same while manufacturing. Also identical flow pattern should be maintained for pumps of same specific speed, but possessing different sizes. One of the effective methods adopted, in practice is MODEL ANALYSIS based on mechanical similarity of real fluid flow. Complete mechanical similarity cannot be achieved. For example, Reynold’s number (Re) of flow cannot be maintained same at identical points of operation for different pumps of same specific speed. This inturn changes the frictional coefficient (hf) and correspondingly the hydraulic losses and efficiency. This is overcome by operating the pumps in automodelling region, where frictional coefficient remains same for all Reynold’s numbers.

188


Also absolute value of surface roughness purely depends upon the manufacturing techniques ε adopted. This value will remain same independent of the pump size. So the relative roughness   will D be higher for small pumps and lower for larger pumps. As a result, the frictional coefficient ( f ) will be lower for larger pumps, and higher for smaller pumps even though the pumps operate at same Reynold’s number. Hydraulic losses will be higher and hydraulic efficiency will be lower in smaller pumps and vice versa in larger pumps. This scale effect is taken into account by using theoretical equations with practical experimental coefficients. Figs. 8.4, 8.5, 8.6 and 8.7 show the increase in efficiency of same pump when relative roughness is reduced. N H kW M 50

H N

40 30

2 1

20 η%

0 10 80

η

0

60 40 20 0

0

25

50

75

100

125

3

Q,m /hr

Fig. 8.4. Effect of surface roughness on pump performance (1) Original from foundry casting (2) After smoothening the flow passage

Head (H) m and efficiency % 12 18

60

40

6m

20

0

3

6

9

12

Discharge (Q) LPS

Fig. 8.5. Effect of improving surface finish of the impeller shrouds on pump performance – – – Machined and polished —— Rough surface 0.5 mm grain size

189

MODEL ANALYSIS

Pump type 2 × 1½ SB 33

N (hp) input H (m) total head % efficiency

Effect due to roughness change in impeller surface

1 5 10

2 10 20

3 15 30

4 20 40

5 25 50

6 30 60

Due to reduction in grain size of moulding sand by 50%

Discharge lps

Fig. 8.6. Effect of roughness on performance – – – – Reduced roughness ——— Original

190


Pump type 2½ × 2 SB 26

hp input 20 m total head % efficiency

Effect to reduction in relative roughness in impeller passage

50 30 10

1

2 5 20

3

4 10 40

5

6 15 60

70

by reducing the grain size of the moulding sand by 15%

Discharge lps

Fig. 8.7. Effect of roughness on performance – – – – Fire sand ——— Regular sand

191

MODEL ANALYSIS

Convergent flow takes place in turbines whereas divergent flow prevails in pumps. Laws applied to turbine cannot be applied to pumps. The relative values of volumetric and mechanical losses are more in pumps. As per model analysis, the total head of a pump increases with the square of the speed of pump, theoretically, but practically a little lower. This equation is defined, based on the assumptions that efficiency of model and prototype are same when operated at identical points. Actually, when speed increases, cavitation characteristics of pump reduces, which inturn reduces the efficiency to a certain extent. This is confirmed by many authors. So also viscosity of the pumping liquid influences on hydraulic efficiency.

8.2 SIMILARITY OF HYDRAULIC EFFICIENCY Head loss

Hence,

Since

V2 V2 ∆H ∝ = λ. 2g 2g ηh = 1 –

∆H V2 = 1 – λ. = 1 – const. λ Hm 2 gH m

V2 is same for model and for prototype. 2 gH m

Prof. Nikuradse |67| stated that under auto model region of operation, the frictional coefficient λ can be expressed as, λ =

1 2

...(8.14)

2

...(8.15)

R  1.74 + 2 log  ε 

and hydraulic efficiency (ηh) can be written as, ηh =

Const. R  1.74 + 2 log  ε

Since absolute value of surface roughness (ε) is constant as it depends upon the manufacturing process, whatever may be the pump size, a general form of hydraulic efficiency can be expressed as, ηh =

A ( B + log . D)2

...(8.16)

Prof. A. A. Lomakin has suggested that A = 0.42 and B = – 0.172 ηh =

0.42 (log D1nom − 0.172) 2

...(8.17)

192


where, D1 nom is the reference inlet diameter calculated as per the equation (8.15) and is expressed in mm based on the equation (8.17). Hydraulic efficiency for prototype from model efficiency can be written as, ηhp

 log D1nom.m − 0.172   = 1– (1– ηhm)    log D1nomp − 0.172 

2

...(8.18)

In Fig. 8.8 the curve is drawn as per equation (8.17) for the diameter 350 mm. Practically, the drop in efficiency is found more due to non auto model effect. Graph B, ∆ηh = ηhp − ηhm is calculated as per equation (8.18) for the same diameter 350 mm. This graph gives an idea of change in hydraulic efficiency between prototype and model. This value can be taken for actual design. ∆ηh% 4,5 4,0 3,5

ηh% 90

A

80 70 60

B

3,0 2,5 2,0 1,5 1,0 0,5

50 40 30 20 10 10

20

40 60

100

200

400

600 1000

2000

4000 6000 10000

Dmm

Fig. 8.8. Hydraulic efficiency of pumps (A) and increase in hydraulic efficiency between prototype and model (B)

8.3 SIMILARITY OF VOLUMETRIC EFFICIENCY Volumetric efficiency,

or,

ηV = 1 ηV

Qa Qa + ∆Q

= 1+

∆Q Qa

where, Qa is the actual quantity of flow and ∆Q is the leakage flow through clearance, which can be expressed as ∆Q = φAL

2 g ∆H L

where, φ = the flow coefficient, A—area of the clearance and ∆HL is the pressure drop across the clearance. Flow coefficient φ depends upon the linear dimensions of the clearance and frictional coefficient. Assuming the frictional coefficient is constant for prototype and for model and since absolute values of clearance dimension are same for model and prototype. φ p = φm Suffix ‘p’ refers prototype and m refers model.

193

MODEL ANALYSIS

For geometrical similarity, linear dimensions of model and prototype are proportional i.e., Ap = K2 Am, where K is constant of proportionality for linear dimension. The pressure drop ∆HL is proportional to the total head of the pump i.e., ∆HL ∝ H ∝ n2 D2. So,

∆H Lp

Hp

2 ∆H Lm = H m = K

∆Q p ∆Qm

=

 np     nm 

2

φp . Ap 2 g ∆H Lp φm . Am 2 g ∆H L m

2  np  n 2 p  3 K =   = K  nm  n  m np . ∆Qm K3 nm ∆Q p = 1+ =1+ np Qp K3 Qm nm

K2

Therefore,

1 ηVP

= 1+

1 ∆Qm = nVm Qm

or

...(8.19)

ηVP = ηVm

Thus, volumetric efficiency of model and prototype remain same, when clearance dimensions are same for model and for prototype and the flow through the clearance is fully turbulent. If the leakage clearances are different, these values are determined as per equation (8.19). Correspondingly, volumetric efficiency of prototype will slightly change from that of model. Since, clearance change will be negligibly small, it is usually neglected.

8.4 SIMILARITY OF MECHANICAL EFFICIENCY Mechanical efficiency, ηm is expressed as ηm =

γQth H m γQth H m + ∆N m

...(8.20)

where, ∆Nm is the total mechanical losses consisting of losses due to disc friction ‘∆Nd’, losses in stuffing box, ∆NS and losses in bearing, ∆NB. Losses in bearings are proportional to square of speed (n2) and losses in stuffing box is proportional to speed n (Equation 6.8). Losses due to disc friction occupies considerably a longer percentage of mechanical losses whereas losses in bearings and stuffing box are very small and hence it is neglected. Total mechanical losses are taken as disc friction losses only. Equation (8.20) can be written as 1 ∆N d 1 + ηm = γQth H m

∆Nd ∝ γn3 D5 = Cf γn3 D5

...(8.21)

194


where, Cf is the frictional coefficient. Assumining Cfp = Cfm

∆N dp ∆ndm

=

C fp γ p n3p D5p

=

C fm γ m n3m D5m

γ p Qthp H mp

Np

γp

N m = γ m Qthm H mm = γ m

Power ratio,

K5

K5

γ p n3p

...(8.22)

γ m nm3  np     nm 

3

...(8.23)

∆N dp 1 = 1 + ηmp γ p Qthp H mp K 5 γ p n3p

= 1+

γ m n3m 3

∆N dm

γ p  np  k   γ m Qthm H mm γ m  nm  5

=

1 ηmm

...(8.24)

Hence, ηmp= ηmm. i.e., Mechanical efficiency of model and prototype remains same. So also disc friction losses for model and for prototype has the same power ratio. Combining all the three efficiencies, overall efficiency η will be

η p = ηmp ηvp ηhp ηm = ηmm ηvm ηhm

=

ηvp .ηhp ηvm .ηhm

...(8.25)

ηvp and ηvm remain same for same clearance ratio for model and for prototype. If not the volumetric efficiency differs. Prof. A.A. Lomakin has recommended the following empirical law to determine the mechanical efficiency and volumetric efficiency in terms of the specific speed (ns ) of the pump. Volumetric efficiency,

1 = 1 + 0.68 ns–2/3 ην

Mechanical efficiency,

820 1 = 1+ 2 ηm ns

where, ns =

3.65 n Q H 3/ 4

.

...(8.26)

9 CAVITATION IN PUMPS

9.1 SUCTION LIFT AND NET POSITIVE SUCTION HEAD (NPSH) V

p0 Cs z s,0

hs hfs Z su

Pat

Zsu

Allowable suction lift (HS) is referred as the vertical height difference between pump axis and water level in suction sump. Referring to Fig. 9.1, suction lift hs = Z0s – Zsu. Reference line for the calculation of suction lift, to determine cavitation characteristics for different pump installations is given in Fig. 9.2. For horizontal pumps, pump axis is always taken as reference line. All measurement measured above pump axis are referred as ‘delivery’ and all measurements measured below pump axis are called ‘suction’. Energy in suction line i.e., from suction sump (su) to the impeller inlet edge of the pump (o) remains constant. So also the energy in delivery line is constant. Mechanical energy is converted into hydraulic energy in impeller and added to the available energy at impeller ‘blade’. Referring to Fig. 9.1, the energy equation between suction sump and the impeller inlet, before the inlet edge of the blade can be written as

sump [su]

Fig. 9.1. Suction lift determination

C 2su p0 psu C2 + Z0 + 0 + hf (s – 0) = + Zsu + 2g γ γ 2g p0 C02 C 2su p + = su + (Zsu – Z0 ) + – hf (s – 0) γ 2g 2g γ Since, (Zsu – Z0 ) = – hs, and Csu the velocity of fluid in suction sump, is zero

p0 C 20 γ + 2g

=

psu – ( hs + hf s) γ

...(9.1)

In order to have a perfect cavitation free operation, the suction pressure ( p0) must be greater than the vapour pressure (pvp) of the pumping fluid at the pumping temperature i.e., p0 ≥ pvp or 195

pvp p0 – ≥0 γ γ

196


i.e., must be always positive. Substituting this condition in equation (9.1). The Net Positive Suction

 p0 − pvp C 02  + head of the pump (NPSH)p = Hsv =  will be greater than zero or always positive. 2g  γ 

Pump centerline

Pump centerline and datum elevation

Datum elevation outer diameter of inlet edge (b) Single suction-vertical

Pump centerline

Datum elevation Center line of the outlet edge (c) Double suction vertical

Fig. 9.2. Reference level for suction head measurement

i.e.,

 p0 − pvp C 20 + Hsv =  γ 2g 

  ≥ 0 

Combining equation (9.1) and (9.2)

 psu − pvp   – ( hs + hfs) ≥ 0 Hsv =  γ  

or

...(9.2)

 psu − pvp    ≥ ( hs + hfs) γ    psu − pvp  Taking   as (NPSH)A i.e., net positive suction head available and (hs + hf s) (NPSH)R γ  

i.e., net positive suction head required, the condition required for cavitation free operation will be

197

CAVITATION IN PUMPS

 psu − pvp  (NPSH)A > (NPSH)R. Rearranging equation (9.2) and taking   = Hsu – Hvp suction lift (hs) γ   will be hs = Hsu – Hvp – Hsv – hfs ...(9.3) For safe operation of pump, i.e., for net (NPSH)p, a reserve in Hsv is added and is written with a coefficient φ . Normally φ = 1.15 to 1.4 and safe suction lift will be hs = Hsu – Hvp – φHsv – hfs

...(9.4)

If the sump is open to atmosphere Hsu = Hatm. Atmospheric pressure at any altitude ‘∆’ can be written as  ∆  Hatm. = Hatm0 –   where Hatm0 is the atmospheric pressure at sea level. Hatm0 = 10.336  900 

MWC = 760 mm of mercury column. If the pumping liquid is other than water patm0 = γw Hw = γl Hl or

Hw =

where Sl is the specific gravity =

γl H l = Sl Hl γw

γl . Suffix ‘w’ refers to water and ‘l’ refers to liquid. γw

hsw = H atm0 −

∴

Corresponding liquid column hsl =

∆ − H vp − φH sv − h fs 900

...(9.5)

hsw sl

TABLE 9.1: Atmospheric pressure at different altitudes ∆

0

500

1000

2000

Hatm

10.336

9.7

9.2

8.1

The value of Hvp depends upon the temperature of the pumping fluid. It increases when the temperature is increased. Fig. 2.9 gives the vapour pressure value at different temperatures for water. Table 9.1 gives the atmospheric pressure at different altitudes. Net positive suction head ((NPSH)p = Hsv) of a pump is defined as the total pressure at stagnation condition at inlet of the pump above the vapour pressure of the pumping fluid at the pumping temperature.

 p C2   p   =  st  , stagnation Referring to the inlet of the pump and since  +  γ 2g   γ      Condition,

p0 C 20 p + Z0 + = ost + Z0 γ 2g γ

198


Absolute flow is axisymmetric in suction pipe i.e., from the fluid level in suction chamber, (point ‘su’) to the impeller inlet edge (point 1), whereas relative flow is axisymmetric on the impeller blade surface, i.e., from inlet edge, (point 1) to outlet edge (point 2). At the impeller edge (point 1), both absolute and relative flows are axisymmetric. Writing down the Bernoulli’s equation between point ‘O’ and point 1, i.e., points immediately before the impeller inlet edge and on the inlet edge of the impeller blade, and since absolute flow is axisymmetric,

C02 C12 p0 p1 + Z0 + = + Z1 + + hf (0 – 1) 2g 2g γ γ

...(9.6)

Point of min. Pr. on blade inlet (suffix ‘x’)

Inlet edge of blade (suffix ‘I’)

Measuring point

Impeller eye (suffix ‘o’)

In the same manner, writing down the Bernoulli’s equation between point ‘1’ and point ‘x’ [Fig. 9.3 and Fig. 9.6 (b)] on the impeller blade and since relative flow is axisymmetric.

2

Suction tank pressure hsc (or)

hfs

C0 2g

2

C1 2g

Blade loading

hs

si d

e

2

C 2g

Su

p γ

h at

P r.

hvp

on c ti

sid

e

Cavitation

Fig. 9.3. NPSH determination and cavitation inception at inlet

p1 ( w12 − u12 ) px ( w2 − u x2) + Z1 + = + Zx x + hf (1 – x) γ 2g γ 2g

...(9.7)

Referring the inlet velocity triangle, w21 = C21 + u21 – 2u1 Cu1 (or)

C2 u C w12 − u 21 = 1 – 1 u1 2g g 2g

...(9.8)

Combining equations (9.7 and 9.8) and rearranging

p1 ( wx2 − u x2 ) C2 C u p + z1 + 1 – u1 1 = x + Zx + + hf (1 – x) γ 2g 2g g γ

...(9.9)

199

CAVITATION IN PUMPS

Combining equations (9.6 and 9.9)

C 20 p0 + Z0 + 2g γ

=

 w2 – u x2  u1Cu1 px + Zx +  x  + g + hf (0 – x) γ  2g 

...(9.10)

For cavitation free operation, minimum pressure px ≥ pvp the vapour pressure. At minimum pressure

 wx2 – u x2   wx2 – u x2  px = px (min), velocity   =   . Adding (– pvp) on both sides of equation (9.10) and  2g   2 g max rearranging.

( p0 − pvp ) γ

+

( px − pvp ) C02 wx2 − u x2 uC + − + + 1 u1 + h f (0− x ) Z Z ( ) = 0 x 2g γ 2g g

...(9.11)

( p0 − pvp )

C 02 uC But + = Hsv and (Zx– Z0) is taken as = 0, since it is very small, 1 u1 = 0 for 2 g γ g normal entry in pumps. px min= pvp . Under critical condition for cavitation free operation. Equation (9.11) will be  wx2 – u x2  + hf (0 – x) (Hsv)cr =    2 g max

...(9.12)

Since point (0) and point (x) are very near to each other at suction.

p0 C 20 + Z0 + γ 2g

=

p0 st p + Z 0 = xst + Z x γ γ

...(9.13)

Combining equations (9.10) and (9.13)

 w2 − u x2  pxst p uC + Zx = x + Zx +  x + 1 u1 + h f (0− x )  γ γ g  2g   pxst − px  ∆p0   = γ = ∆h0 = γ  

 wx2 – u x2  u1Cu1  2 g  + g + hf (0 – x)  

 w2 – u x2  (∆h0)max =  x + hf (0 – x)   2 g max

...(9.14)

...(9.15)

Comparing equations (9.12) and (9.15), it can be written as (Hsv)cr = (∆h0)max where, (∆h0) max is called as Maximum Dynamic Depression. It is evident that (Hsv)cr is a function of kinetic energy of the flow at suction. Hence, dynamic similarity law can be applied between model and prototype values of (Hsv). hf (0 – x) is neglected, since it is very small because of convergent flow pattern between points 0 and x.

200


Dynamic depression can also be expressed in some other form. All functions, as far as cavitation is concerned, take place at suction side and on the inlet edge of the blade (from point 0 to point ‘x’ on the blade). Referring inlet velocity triangle under normal entry condition Cu0= Cu1= 0, C0 = Cm0, C1 = C m1, u 20 + C2m0= w02 and u12 + C 2m1= w12. Due to vane thickness flow velocity increases Cm1 = K1Cm0 where, K1 is the vane thickness coefficient ( ∆ h0)max

w2x − u 21 w2x − w21 + C 12 wx2 − u x2 = = = 2g 2g 2g =

Cm21 w2x − w21 + 2g 2g

2   w  2  w21  Cm1  Cm2 0 x − + 1  =   w   C  2 g  2 g m0  1

 Cm1  Taking, m=    Cm 0  Substituting this value in equation (9.16) (∆h0)max

2

...(9.16)

2

  wx   and n =    – 1    w1  

C 2m 0 w21 =n +m 2g 2g

...(9.17)

...(9.18)

Experiments conducted on different pumps by different authors, indicate that m = 1.0 to 1.2 and n = 0.3 to 0.4. Since, m and n are velocity ratios; similarity laws can be applied. Values of m and n remain constant for pumps of same specific speed.

σ ) THOMA’S CONSTANT 9.2 CAVITATION COEFFICIENT (σ Prof. Thoma |97| has defined cavitation coefficient (σ) as σ=

H svcr = ( ∆h0 )max H

which is a non-dimensional number. Substituting H =

H sv H

...(9.19a)

Cu 2u2 g ,

 wx2 − u x2   2g  =  C u   u2 2   g 

indicates that σ represents velocity ratios, which is constant for model and prototype of same specific speed, i.e., σm= σp . However, this coefficient has certain drawbacks. For example, two pumps having identical inlet conditions but different outer diameters, Hsv will remain same but H will differ and hence the value σ

201

CAVITATION IN PUMPS

changes. This is overcome by defining another non-dimensional expression, called Cavitation Specific Speed (C). Moscow Power Institute | 58 | recommends a relation between σ and ns as ( ns ) 4 / 3 ...(9.19b) 4700 Based on intensive experimental investigation on cavitation on axial flow pumps, Leningrad Polytechnic Institute |105| recommends the following equation to determine σ :

σ=

2   4 δm   2 + + w (1 β) 1   π l ∞ –u       σ= 2 gH

...(9.19c)

where, β—curvature and δm—maximum thickness.

9.3 CAVITATION SPECIFIC SPEED (C) Professors Rudnoff | 104 |, Wislicenus | 133 |, Watson | 103 | and Karrassik | 54 | defined cavitation specific speed (C). 5.62n Q

C=

( H sv )3 / 4

n Q  , or Hsv = 10   C 

4/3

...(9.20)

This expression is similar to that of specific speed and hence called cavitation specific speed (C). Normally, pump speed is selected based on cavitation specific speed. Increase in speed for the given head and discharge of a pump, reduces the size of the pump. Due to reduction in area, the flow velocity increases, which inturn increase the main friction losses and increased secondary flow losses. The cavitational property reduces considerably. In order to improve cavitational property, flow passage especially suction side of the pump must be improved and well designed for better streamlined flow. This can be done only by proper construction and efficient manufacturing technology. Since improvement in manufacturing of pump has its own limitations, for example, surface roughness cannot be reduced below certain limit unless costlier manufacturing processes are adopted. That’s why cavitation specific speed (C) has a narrow range of operation unlike normal specific speed which ranges theoretically from ‘0’ to ∞, practically from 10 to 2500. Cavitation specific speed (C) ranges from 800 to 1100. To improve C above 1100, improved manufacturing and construction techniques must be adopted. Pump cost also considerably increases. For normal design ‘C’ can be taken as 900 to 1100 depending upon the manufacturing process available and speed is determined. For special pumps C is selected as C = 1200 to 1500.

9.4 CAVITATION DEVELOPMENT When pressure at the point ‘x’ (Figs. 9.3 and 9.10) on the leading side of the impeller blade of the pump, falls below vapour pressure of the liquid for the prevailing temperature of the pumping liquid, the pumping liquid becomes vapour, 1 cc of liquid in the form of water, when converted into vapour,

202


occupies approximately 1780 cc of water vapour. Since the space available in between impeller blades is very small, pressure instantaneously raises to a very high value. This pressure rise makes the vapour to condense to liquid. Now the pressure falls below the vapour pressure and the liquid changes into vapour. Likewise the pressure changes from high positive to high vacuum instantaneously, many times in a second. The pressure rise is approximately in the order of 100 to 300 atmospheres. This sudden high instantaneous fluctuating pressure rise gives a heavy hammer blow on impeller blades, like shock waves. When pressure exceeds elastic limit of the material of the blade, metal is gradually removed from the blade. This pressure fluctuation followed by metal erosion and subsequent corrosion is called cavitation. Due to cavitation, impeller blades, shrouds, especially at inlet leading edge as well as other parts of pump like suction side of casing get damaged. Flow does not follow streamlined or axisymmetric pattern. Hydraulic losses increase; hydraulic efficiency and overall efficiencies considerably decrease. Huge noise and heavy vibrations are produced. Life of the pump reduces. Under severe cavitation condition, pump fails to work. At high vacuum, oxygen present in the fluid is released from the liquid, gets reacted with the material of the impeller and other parts of the pump. The metal is converted into metal oxide. This metal oxide, in the form of powder being weak, is carried away by the flowing fluid. Thus, corrosion adds to the erosion in reducing the metal thickness increasing the roughness of the surface. No metal is resistant to cavitation. Low strength metals gets corroded at a faster rate, whereas high strength materials gets corroded at a slower rate. Phosphor bronze gun metal have more elastic and anti corrosive property but possess low strength and smooth surface. Cast iron, malleable iron possess high strength but gets corroded at a faster rate. Stainless steel SS304 and SS316 an anti corrosive and high strength material is also used for pumps having more cavitating characteristics. Carpenter, Alloy 20 Ni hard, Ni resist materials possess still higher strength and high anti corrosive quality. Initial stage of cavitation does influence on parameters of pumps namely head, discharge, power, efficiency and speed. When cavitation increases the rate of drooping down property of H-Q curve is noticed. Entire system becomes unstable when pump runs under severe cavitation. Pump cannot be run at this condition. Rate of flow, total head, power, efficiency and speed drops down suddenly and fluctuates. Fig. 9.4 (a) shows a typical performance characteristics of pump under normal and at cavitation operating conditions. (H-Q) and (η-Q) curves start droping down suddenly at certain flow rate when η hs = 0.5 m

H H, N, η

Cavitation

Normal

Head (m)

40 H

hs = 2 m h

30

s

=

7

m

η

m

1

Fig. 9.4. (a) Pump performance under normal and cavitating condition

Critical NPSH

10

50

.5

s

10

Q

60

=5

h

70

m

hs

20

N

=3 .5

80

20

2

4

30 40 50 Flow Q [L/S]

NPSH

8

4 2

60

Critical NPSH (m) Efficiency %

50

70

Fig. 9.4. (b) The effect of cavitation on a centrifugal pump performance (effect of suction lift hs and NPSH)

CAVITATION IN PUMPS

203

cavitation occurs. No further increase in flow is possible. When suction lift (hs) increases or NPSH decreases (H-Q) and (η-Q) curves drop down more and more at a lower flow rate (Q) than the previous value Fig. 9.4 (b). So also power discharge (N-Q) curve also drops down. The point, where it starts droping down suddenly, indicates the inception of cavitation.

9.5 CAVITATION TEST ON PUMPS Cavitation test is the process of the determination of the point of osciliation in Q, H, N, η, n, when suction lift (hs) or NPSH (Hsv) or dynamic depression (or anti cavitating reserve) ∆h is changed from maximum to minimum, when pump is running at one point of H-Q curve [Figs. 9.5(a) and 9.5(b)]. For every operating point of the pump, there is one value of Hsv below which cavitation starts. Cavitation test ends, the moment (Hsvcr), the critical value of Hsv or ∆h or hs is determined for all selected point of operation. A curve joining all Hsv or hs values, obtained for different operating points gives the complete characteristics (Hsv) = f (Q) or C or σ = f (KQ) (Fig. 9.11). In closed test rigs, cavitation test is conducted by reducing the pressure in the space above water level in the closed reservoir with the help of a vacuum pump. Fig. 9.6 illustrates a schematic sketch of a cavitation test set up and Figs. 9.7 and 9.8 show the actual cavitation test rigs for centrifugal pumps and axial flow pumps. Essentially a cavitation test rig consists of a closed tank to which suction and delivery pipe lines are connected. The delivery pipe has a venturimeter to measure the flow through the pump, a gate valve to control the flow rate and a tapping point to measure the delivery head of the pump. The suction pipe has a tapping point to measure the suction head of the pump and another tapping point to measure the temperature of water. A mercury manometer is connected to the delivery and suction head measuring points to measure the total head of the pump. Another mercury manometer is connected to the venturimeter to measure the flow rate of the pump. All the measuring points are located with sufficient upstream and downstream straight pipes (3D to 6D where D is the diameter of the pipe) before and after all flow obstructions. A vacuum pump is connected to the closed tank to change the vacuum in the tank. A mercury manometer is connected to the suction tapping point to measure the vacuum at the inlet of the pump. Pump, to be tested is kept at the adjustable test bed. A variable speed DC dynamometer is connected to the pump through a flexible coupling. Speed is measured by a tachometer. Torque output from the DC dynamometer to the pump is measured by a dial indictator. Proper cooling arrangement is provided at the stuffing box to avoid air entry into the pump through stuffing box and at the same time keep the stuffing box at low temperature. Additional supply of water to the tank and removal of water from the tank are carried out by separate gate valves. This arrangement is essential to keep the water temperature constant as water gets heated due to constant circulation. The temperature of water is measured by a thermometer fitted at the suction pipe. Cavitation test is conducted as per the method suggested here. From the load test performance graph, (i.e., H-Q and η-Q graphs) a few operating points are selected very near to maximum efficiency point for (NPSH)p determination value (points 1, 2, 3, 4, 5 in Fig. 11.3). Pump is started with gate valve in closed condition for radial flow centrifugal pumps, whereas gate valve in opened condition for mixed and axial flow pumps. Speed is adjusted to run always at constant speed. The gate valve is adjusted so that the pump runs at point 1. After attaining steady flow condition, suction head i.e., vacuum before the impeller, total head, quantity power, speed are taken and efficiency is calculated. All the readings are entered in a tabular form (Table 9.2).

204


TABLE 9.2: NPSH (Hsv) determination for H = ...., Q = ...., N = ...., η = ..... S.No.

Suction head hs

Total head H

Flow rate Q

Power N

Effy η

Speed n

(NPSH)p Hsvp

Net positive suction head of the pump, Hsvp is calculated by the formula Hsvp =

psu − pvp γ

– (Hs + Hfs)

 psu  p − hs − h fs  – vp =  γ  γ  = Vacuum manometer reading – Vapour pressure Now, the suction head ‘hs’ is increased by operating the vacuum pump and the vacuum in kept constant at one level. All values, mentioned above, are measured and entered in Table 9.2. In the same manner, suction head ‘hs’ is increased further in step by step and experiment is conducted until unsteady condition is attained. Discharge, total head, power and efficiency remains same, up to critical value of (Hsvp). At Hsvp critical, all readings suddenly drop down and fluctuates. Pump runs with noise and vibration. This indicates that the pump is running under severe cavitation. No further increase in vacuum is possible and no further test on pump could be conducted. The vacuum is reduced and the pump is brought to the normal operating condition. Now, by adjusting the flow control valve, point ‘2’ is set in the test. The experiment is repeated as mentioned earlier until (Hsvp) critical point is reached. Likewise the experiment is repeated for points 3, 4, 5, i.e., for all selected points. A graph H, Q, n, η = f(Hsv) (or) hs (or) ∆h is drawn taking values from the conducted test results for all points from Table 9.2. One such graph is given for one operating point in Fig. 9.5 (a). Since exact point of the beginning of severe cavitation could not be determined, 1 to 2% drop in the values of normal flow rate, total head, power and efficiency i.e., 98 to 99% of normal flow rate, total head, power and efficiency is taken as (Hsvp) critical and this is the value of (NPSH)p of the pump at operating point 1 in load test curve of the pump. In the same manner, from the tests conducted (NPSH)p at other selected points (points 2, 3, 4, 5) are determined. All (NPSH)p values are now plotted on the load test graph to get Hsv = f (Q) curve. The minimum most point in this curve is the best point of operation of the pump for cavitation free operation, which corresponds to maximum value of ‘hs’(Fig. 9.5). Best cavitation free operating point need not be the best efficiency point of operation. For long life of the pump, it is always better to run the pump at best cavitation free operating point, than at best efficiency point. In open test rigs, the cavitation test is conducted by closing the gate valve at suction line, keeping the delivery gate valve at one position constant throughout the test. The experiment is repeated for different positions of delivery gate valve. Critical values of Hsvcr or (∆hcr or hsmax) depends upon the type of impeller i.e., specific speed (ns) of the pumps. For low specific speed pumps ns < 100, H, Q, η, N curves remain constant with the decrease of Hsv (or increase hs) until critical point is reached. At critical point i.e., when cavitation starts,

205

CAVITATION IN PUMPS

all these values suddenly drop, i.e., horizontal lines change to vertical lines in the graph. When ns is increased, i.e., ns = 100 to 350, these values H, Q, η, N gradually reduce until critical point is reached and then suddenly drops. In axial flow pumps ns > 450, there will not be a sudden drop after critical point instead it will be gradual. Correct critical point, infact, cannot be determined. 55

H

50 N, kW 40 45 35 30 40 25 20 35

Q

Q H η

H

N

Q

lit/sec 30

η

25

15

80 75

10

70

20

1 to 2% ∆h (or) h s (or) H sv

H svmin

Q

η%

η

H s(cr)

65 1

2

(a) Schematic diagram Hm

80

50 40

Q. lit/sec 190 180 170 η,% 80 70 60 H,m 1,2 1 1,1 1,0

η

30 20 N,kW 15 10 60

N

10

0

5

6

7

Hs m

H

70

5

4

(b) Centrifugal pump

60 η%

3

Q

L/S 15

H sv(min)

Q

10 5

1 2

4

6

8

10

12

7 3

2

4

5

6

9

2

3

4

5

6

H sv

(c) Centrifugal pump

(d) Axial flow pump

Fig. 9.5. Actual cavitation characteristics of pump 2

1

8

p1

∆h

pa ps 1

Fig. 9.6. Schematic diagram of cavitation test rig 1. Control valve, 2. Flow measurement

7

10 11 8 H s,m

206

Control valve Collecting tank

Orifice meter Delivery

Manometer flow measurement h Thermometer

Manometer vacuum

Speed measurement

Manometer total head

D.C. Dynamometer

Vacuum pump

Pump for test

Fig. 9.7. Cavitation test rig for centrifugal pump

Power measurement


Suction

1

2

3

5

4

14

13

12

8

B

15

16 17

B

18

CAVITATION IN PUMPS

B 3

19 A

7

21

C

9

6

10

B

11 B

A

B

B

B

B

B

20

C

23 A

B-Reference level for all measurement

24

22 Size 15 m × 6 m

Fig. 9.8. Cavitation and load test rig for axial flow pump Power measurement Speed measurement Torque arm DC Dynamometer Pump under test (Axial flow) Delivery pipe Manometer for flow measurement Inlet cone (suction)

9. 10. 11. 12. 13. 14. 15. 16.

Manometer—total head measurement Vacuum manometer Base level-manometer Vacuum box Suction pipe Suction chamber Control value Vacuum chamber

17. 18. 19. 20. 21. 22. 23.

Vacuum chamber level indicator Vacuum tank Air removal pump Circulating pump Water supply pump Outlet tank Venturimeter

24.

Inlet tank

207

1. 2. 3. 4. 5. 6. 7. 8.

208


The gradual drop in pumps of higher specific speed is due to the decrease in efficiency at a faster rate than in low specific speed pumps even before reaching the critical point of Hsv. That’s why critical values of Hsv in these pumps are always determined from the efficiency graph η = f (Hsv), instead of from the graph H, Q, N, = f (Hsv). Referring to the consolidated graphs (η-Q, H-Q) of different pumps of different specific speed (Fig. 11.4) that for low specific speed pumps the η-Q and H-Q curves are more inclined towards horizontal lines i.e.,more flat, whereas for high specific speed pumps the (H-Q) and (η-Q) curves are more inclined towards Y-axis, which indicates that the percentage drop of efficiency in high specific speed pumps are more than the drop in efficiency for low specific speed pumps. For the same range of (Qmin to Qmax) operating region, efficiency variation in high specific speed pumps are more than that in low specific speed pumps. This effect changes, the H, Q, η, N = f (Hsv) graph. Even under cavitation in such high specific speed pumps, there may not be noise heard or vibration presence felt or even cavitation erosion seen. That’s why, these pumps are not economical when operated under cavitation. The change in the appearance of cavitation and subsequent erosion in pumps depends upon the impeller construction. In pumps with smaller specific speed, flow passages are radial. The length of the flow passage depends upon the blade angle β, number of blades (Z) and the diameter ratio (D2/D1). At the time of cavitation, the pressure at the inlet edge at the suction side of the blade, will be equal to vapour pressure (hvp). For any further reduction in total head or increase in flow rate, this low pressure (vapour pressure) area spreads over the entire area across the channel. No further reduction in pressure is possible. Flow cannot be increased any further even the vacuum is increased further since the pressure side and suction side pressure and hence the difference in pressure remains same, which is equal to the difference between inlet pressure and vapour pressure, which exists across the complete flow passage width between blades at inlet. In impellers of high specific speed the passage area between two successive blades are wider but with shorter lengths of blades. The vapour pressure will not cover completely the entire area at suction side instead only partly, as a result of which further drop in pressure is necessary so that vapour pressure can cover the whole passage area at inlet which results in for a higher flow rate. Normally in axial flow pumps, two successive blades do not overlap. Hence, the drooping tendency exists for more area before cavitation starts. Even at the time of cavitation, there exists a flow passage between two vapour pressure regions, the area being approximately equal to the area prevailing between blades at full closed condition of blades. In this passage the pressure is larger than vapour pressure and there exist a flow even after cavitation, which results in the gradual drop instead of sudden drop at critical point.

 C02  In pumps of smaller specific speed the term   is more predominant than  2g   w21 does not have any importance. In pumps of higher specific speed, the term   2g  C 20 than   2g

 w21   2g

 w12  . In fact 2g 

  is more predominant 

 w2 , since 1 depends upon the pump head (H) [and hence the speed (n)] and number of 2g 

209

CAVITATION IN PUMPS

blades (Z). The relative velocity w1 at inlet reduces when speed (n) is reduced or the total head (H) is reduced or when number of blades are increased. In pumps of medium specific speed, maximum flow rate for the given suction head at inlet can be increased by increasing the blade length i.e., extending the blade into the impeller eye area at inlet and rounding off the inlet edge (Fig. 9.10). This increases the inlet area and reduces the inlet velocity (C1). The blade, instead being purely radial at inlet becomes a double curvature type, due to change in the inlet diameter D1 from hub to periphery. At outlet, however, the blade is radial. 3 2

b2

D2

DS = D0

b1

D0 = Ds

Fig. 9.9 Impeller with cylindrical vanes (pure radial)

2

D D1

D1 DH

D2

x

o

DH

Fig. 9.10 Impeller with vanes extended into impeller eye at inlet (Double curvature at inlet)

In multistage pumps such as feed water pumps and in condensate pumps, the cavitation effect is taken care of only for the Ist stage. The reduction in H, Q, Ν, η = f (Hsv or ∆h) curve is at a lower rate than in single stage pumps. The reduction in these curves is due to the presence of vapour pressure at inlet due to release of air and vapour, at low boiling point. The deciding factor for cavitation inception is not the absolute value of unit hydraulic energy but the value above the vapour pressure at inlet for the pumping liquid conditions. The value of unit hydraulic energy above vapour pressure is called Dyamic Depression or anti cavitating reserve of suction (∆h) for the pumping liquid at pumping temperature. When pumping liquids of high temperature such as boiling water by feed water pump or by condensate pump, this anti cavitating reserve is attained by providing higher suction pressure or higher suction head. Sometimes in the graphs H, Q, N, η = f (Hsvp) efficiency curve alone slightly raises and then drops down Fig. 9.5 (c), under critical cavitation, while all other curves drops down from normal values [Figs. 9.5 (b) and 9.5 (c)]. Under critical cavitation condition, maximum relative velocity at the inlet edge of the impeller blade occurs at point x (Fig. 9.6). Flow separation also takes place, and the losses increase. As a result, efficiency drops down. Sometimes, flow separation and vortex formation does not take place at the point of maximum relative velocity even under critical cavitation condition. This result is slight increase in efficiency before sudden drop of efficiency. Similar to equal efficiency ‘O’ curves equal ‘C’cavitation specific speed ‘O’curves are also drawn on universal characteristics. Fig. 9.11 gives one such curve. Figs. 9.12 and Fig. 9.13 give the normal places in impellers of different pumps, where cavitation usually occurs and places of cavitation erosion, that usually occurs in axial flow pumps.

210


0

KH 0,10

C=

η

% = 60

n

s=

80

900

1200

4 50 0 0 0

70 75

0,08

1000

30 0

80

0,06

60 0

1500

7 80 0 0 0 83 900 85 0 8 8 = 95 0 87 86

1700 2000 2500

85

0,04

83 80

ϕ = + 20°

75 70 60

0,02 η = 50%

0,3

0,4

5000

ϕ = +10°

4000 ϕ = 0°

ϕ = – 7,5°

0,2

ns = 3000

3500

ϕ = – 5°

0,5

0,6

0,7

0,8

0,9

1,0

1,1

KQ

η and C’ O-curves Fig. 9.11. Universal characteristics of axial flow pump in KH-KQ co-ordinates with ‘η Centrifugal pump Axial flow

Suction side inlet Axial flow

Outlet

Inlet Centrifugal pump

Peripheral radial clearance

Double suction

Single suction Mixed flow

Centrifugal pump

Suction tongue

Volute tongue

Impeller and diffuser inlet

Fig. 9.12. Places affected by cavitation in different pumps

Volute

211

CAVITATION IN PUMPS

1½ – 2 <1 2–3 6–8 Inlet

4 Outlet

75°

2

up to 4

mm

Erosion depth 5 mm 3 mm 90° up to 4

Fig. 9.13. Erosion in axial flow pumps due to cavitation

9.6 METHODS ADOPTED TO REDUCE CAVITATION (a) Increasing Suction Pipe Size Referring to equation (Eqn. 9.2) Hsvp, the net positive suction head of pump can be positive if the frictional losses hfs is reduced. Increasing the suction pipe diameter (ds) reduces the suction pipe velocity

Cs2 at suction, thereby Hsvp (Cs). This considerably reduces the frictional looses (hfs) and velocity head 2g can be made positive. This is done in all pumps mostly in agricultural, chemical process pumps, etc. The suction pipe size is always greater than the delivery pipe diameter. (b) Reducing the Suction Lift Referring to equation (9.2), Hsvp can be positive if suction lift (hs) is reduced. This is done by lowering down the pump with respect to the fluid level in suction sump. If suction lift reduction is insufficient the pump level can be brought down below the fluid level in suction tank. Suction lift will be negative (– hs). This is called suction head or positive suction. This is followed in industries, such as furnance, oil pumping, boiler feed pumps, chemical process pumps etc. (c) Increasing the Suction Tank Pressure In case of pumping high temperature gaseous fluids at ordinary pressures and temperatures, vapour pressure of the fluids will be very high. The term (p su – p vp) becomes negative. In order to overcome this condition, suction tank is closed and the tank pressure is increased above the vapour pressure of the fluid at the pumping temperature. This procedure is adopted in chemical industries, especially where high temperature fluid pumping is carried out. (d) Increasing the Width at Inlet of the Impeller Similar to that of increasing the suction pipe size, the impeller inlet width is considerably increased in axial direction. This method has been proved experimentally a good viable method. The minimum

212


pressure at point X on the inlet edge of the blade is found to be higher than the vapour pressure which avoids cavitation (Fig. 9.14). Cavitation specific speed ‘C ’ considerably increases. The area at inlet

 π  A0 (= πD0 b1) ≈ 2.5 to 3.0 A1  = ( D02 bh2  . By  4  increasing the breadth, b1 the percentage of quantity of flow (Q) passing through every point of inlet edge, with respect to the total quantity (Qt) entering the Q inlet v =   is less. Also the blade velocity u0  Qt  decreases. The inlet edge is extended into the Fig. 9.14. Increased inlet width for nonimpeller eye area. Inlet edge of the blade is inclined cavitating operation instead of horizontal. The blade velocity u0 from hub to periphery increases as a result of which, the blade becomes a two dimensional and twisted, instead of purely cylindrical at inlet. Due to reduced blade velocity at periphery, maximum relative velocity wmax also reduces by which the cavitational characteristics of the pump ‘C’ increases. At the periphery of the inlet edge, a reverse flow prevails and axial vortex exists before the inlet edge, which also improves the cavitation property. The inlet edge is made sharp, because of this, flow separation at inlet edge exists. Axisymmetric potential flow no longer prevails. Due to this effect, minimum pressure is higher than vapour pressure. Unsteady vortex reversed flow at inlet mixes with the main flow and goes back to the blade inlet. There is a constant and continuous exchange of energy in the liquid. The cavitation specific speed ‘C ’ increases even up to 2200 to 2500, from normal value of 800 to 900. However, hydraulic losses increase the flow velocity Cm does not change uniformly from inlet to outlet edge. (e) Provision of Inlet Guide Blades Maximum dynamic depression ∆hmax is determined by the maximum relative velocity prevailing at the entrance edge (Eqn. 9.18). Even by providing an angle of attack at inlet, a reduction of local maximum relative velocity could not be achieved. However, it can be shown that by providing a prewhirl before the inlet edge i.e., provision of Γ1 a circulation at inlet, a reduction in w1max is obtained. This is achieved by providing inlet guide blades, (Fig. 9.15) before impeller eye at approach channel. Prewhirl increases ∆hmax by

Fig. 9.15. Fitting with inlet guide blades

u1Cu1 as per the total head equation. Total head, however, should not be changed due to g this prewhirl addition. Such Prewhirl is obtained by adding a circulatory flow to the main axial flow. This circulating motion is achieved by the impeller rotation at a velocity u0 . Moreover, when inlet width is higher a reverse flow also occurs at the periphery of blade inlet. Liquid coming out of the blade

an amount

213

CAVITATION IN PUMPS

H,N η, %

η

Axial clearance = 0.1D 0

80 3 70 60 2

H

50

1 2

40 30 1

20 1 10

2 100

200

Q, L/S

Fig. 9.16. Influence of inlet guide blade on performance (1) without inlet blade (2) with inlet guide blade

at periphery receives considerable circulatory motion and mixes with the main flow entering the impeller. Fluid is rotated and thus a prewhirl is developed. Experiments show that C increases up to 1200 to 1500 from normal value of 900–1000. Hydraulic efficiency is achieved by bringing back to normal velocity at impeller blade outlet. If by the prevailing manufacturing process, surface finish of the impeller could not be improved to get a good cavitational property, Prewhirl can be adopted by fitting inlet guide blades. Cavitation specific speed ‘C ’ is improved to a considerable value in cylindrical or radial blades. ( f ) Provision of Axial Prewhirl Impeller (Inducers) Provision of axial Prewhirl impeller, before the main impeller increases the pressure and also gives a tangential component of velocity i.e., a circulatory motion. The reverse flow at the periphery of the main impeller blade is shifted to the axial Prewhirl impeller (Fig. 9.18).

Inducer

Fig. 9.17 Pump fitted with inducer

Fig. 9.18. Inducers

214


Cavitation characteristics is well improved from C = 900–1000 to 1200–1500 and in special cases C is increased up to 2500 to 3000. In Figs. 9.19, 9.20 and 9.21, a graph σ = f (ns) as recommended by Prof A.J. Stepanoff | 112 | is given, which can be used for design.

Specific speed, double suction 500

1000

2000

4000

6000 10,000 20,000

Hydraulic institute values of σ Single suction Double suction

35

71

500

1000

142

285

428

713

1427 Metric

2000 4000 6000 10,000 20,000 British

Specific speed, single suction

Fig. 9.19. Cavitation constant σ versus specific speed for b.e.p. as recommended by A.J. Stepanoff

215

CAVITATION IN PUMPS

0.4 0.3

0.

0.

0.2 0.15

σ

0.8 80 5 0. 90

70

ηλ

0.10 0.09 0.08 0.07 0.06 0.05 0.04

η λ, % 70 75 80 85 88 90 92

0.03 0.02 0.015

25 30

40 50 607080 100

150 200 300

Fig. 9.20. Cavitation constant σ versus specific speed (metric units) for different efficiencies. To convert ns to English units multiply by 14.15; ηλ is hydraulic efficiency (Rutschi) 0.40

9¼′′ impeller

0.35 6′′ 3. stage 8½ ′′

0.25

0.20

3′′ 8. stage 2430 r.p.m.

2920 3540

0.15

b.e.p.

Cavitation constant, σ

0.30

b.e.p.

0.10

3550 r.p.m. 9½′′ 3300 r.p.m. 9½′′ 2880 r.p.m. 9½′′

0.05 106

0 0

500 35

1000 71

1500

142

2000 142 Specific speed

2500 178

3000 214

3500 British 250 Metric

Fig. 9.21. Cavitation constant σ versus specific speed for a constant speed head capacity curve

10 AXIAL FLOW PUMP 10.1 OPERATING PRINCIPLES AND CONSTRUCTION Basically, axial flow pump consists of an impeller with impeller blades rotating inside a concentric cylindrical annular housing. The impeller is followed by a diffuser with blades. A bell mouth shaped suction inlet precedes the impeller. In some of the pumps, suction blades are provided with or without suction hub. Suction hub has a sleeve bearing to support the pump shaft. The diffuser is followed by a straight pipe and a bend or only bend depending upon the site conditions. Pump shaft is supported by a main ball or roller bearing along with thrust bearing kept at the top of the delivery bend as well as by (bush) sleeve bearing supports kept at the diffuser and at the bend before the stuffing box. The stuffing box is located at the delivery bend in between the sleeve bearing and main bearing. In some of the pumps, the pump shaft is also supported by a sleeve bearing at the suction hub, if the impeller weight is high and it cannot be overhung as cantilever support. Stuffing box

Delivery casing Diffuser

Impeller

Suction casing

Fig. 10.1. Axial flow pump 216

217

AXIAL FLOW PUMP

Seal

Sleeve bearing

Outlet bend

Diffuser blades Sleeve bearing Impeller housing

Variable pitch impeller

Approach pipe

Fig. 10.2. Axial flow pump (vertical)

Impeller blades of axial flow pumps have double curvature form at inlet and at outlet due to the change in diameter from hub to periphery. Absolute flow before and after the impeller and relative flow along the impeller passage are axisymmetric and potential. There is no radial mixing. Under this condition, each streamline is parallel to the axis of the pump. Fluid passes parallel to the pump axis i.e., along the streamline. At any streamline u1 = u2 and Cm1= Cm2. Flow lines are in the form of a concentric tube or circular cylinder. By spreading out the cylindrical tube, a plane surface is obtained, wherein blades are located at equal distant apart starting from infinity and ending at infinity. The distance between two successive blades is called pitch (t) and is equal to t = 2πr/Z, where ‘r’ is the radius of the streamline of the cylinder and Z is the number of blades in the cylinder stream tube surface. Basically, the principle of operation of the pump is the force of interaction and energy transfer from the impeller blades to the fluid. There is no centrifugal force in these pumps. Energy transfer takes place purely from kinetic energy to pressure energy i.e., diffuser effect. But diffuser pattern of flow has its own limitations. Angle of divergence (x) should not exceed 8° to 10° and a strictly smooth streamlined

218


flow passage must exist, otherwise, flow separation at the boundary layer and corresponding flow mixing between streamlines take place. Axisymmetric and potential flow no longer prevails. Thats why, the design of axial flow pump is more complicated than the design of centrifugal pump. For the given flow rate (Q), axial flow pumps possess, lesser dimensions and size, than all other pumps. These pumps are adopted for low head and high discharge conditions. The surface of the blades and flow passages are manufactured with high smoothness i.e., with very low surface roughness. Even at high speed, these pumps give a very high efficiency due to less friction and less area of contact of fluid with pump parts. However, at partial flow conditions, these pumps give a lower value of efficiency than centrifugal pumps due to high secondary flow prevailing at all elements of the pump.

10.2 FLOW CHARACTERISTICS OF AXIAL FLOW PUMP Basic requirement of hydraulic design of axial flow pump is to determine the dimensions and shape of all flow passages of pump to get high efficiency at all regions of operation for the given value of total head (H), flow rate (Q) and cavitation characteristics (s). Theoretically, maximum efficiency is obtained when the flow is axi-symmetric and potential (irrotational and vortex free) Under potiential flow condition, circulation (Gb) along the contour of blades of axial flow pump is directly proportional to flow rate (Q) and the angular velocity of the fluid (ωf), not only at optimum regions, but also at non-optimum regions of operation. Due to this, circulation and hence, total head remain constant at all streamlines from hub to peripery.

10.3 KUTTA-JOWKOVSKI THEOREM Forces acting on the blades of axial flow pump can be determined from the fluid flow over a plane cascade system, obtained by spreading over the cylindrical section of flow passage of axial flow pump (Fig. 10.3). β

l

u

t=

2 πr z

α

δm

α

t

Fig. 10.3. Profile in a cascade system

Flow in the radial direction i.e., perpendicular to the cylindrical section does not exist i.e., Cr the radial velocity is equal to zero. Flow is purely axial at all cylindrical sections, i.e., at all streamlines flow pattern form is a plane flow or a two dimensional flow. The blades may be moving with a circumferencial velocity ‘u’ perpendicular to the axis, which is equal to the peripheral velocity of the impeller blades or stationary as in diffuser blades. Accordingly, relative velocity ‘w’ should be considered for a moving blade system, whereas absolute velocity ‘C’ must be considered for a stationary blade system. Assuming flow is incompressible, i.e., density ‘ρ’ is constant, the circulation ‘Γb’ around one

219

AXIAL FLOW PUMP

w 2m = wm

p 2 β2 b

w 2u

C

t w 1u

t

w 2u A

Pu

D

B

β∞

w1

P

a Pm = Pz

w 1u β1

d

w∞

w 1m = w m

w1

F w2

wm

β∞ β 2 β1

E

p1

C

Fig. 10.4. Pressure and velocity acting on a cascade system

blade of the plane cascade system can be determined by the closed contour abcd enclosing the blade. Referring to Fig. 10.4 lines ab and cd are same lines located symmetrically with respect to the blade, at a distance ‘t’, called pitch. Linear integral along these two flow lines are equal, but opposite in direction, hence gets cancelled. Lines ‘bc’ and ‘da’ are parallel to the direction ‘u’ flow lines are opposite in direction. Circulation Gb around the blade is Γb =

∫

wds cos( w, s )

abcd

=

∫ wds cos(w, s) + ∫ wds cos (w, s) + ∫ wds cos(w, s) + ∫ wds cos(w, s)

ab

=

bc

cd

da

∫ wds cos(w, s) + ∫ wds cos(w, s)

bc

da

Γb = (w1u– w2u) t, taking unit width perpendicular to the paper and anticlock-wise direction as positive.

...(10.1)

Flow through the blade ∆ Q = htw1m– htw2m where ‘h’ is the height of the blade from hub to D − dh , where D is the outer diameter, dh is the hub diameter and wm is the axial flow 2 velocity. Net force ‘P’ acting on the blade system can be resolved into two components, tangential (Pu) and axial (Pz). Applying momentum,

periphery =

Pu = ρ∆Q ( w1u– w2u) = ρht w1m (w1u – w2u) = ρhwmΓb Pz = ρ∆Q (w2m– wm) + ht (p2 – p1) = ht (p2 – p1)

 ...(10.2)  

Since w1m = w2m i.e., axial flow velocity is constant throughout the impeller blade, and applying Bernoulli’s equation between points ‘1’ and ‘2’ and since Z2 = Z1, where Z is the level from a reference point, the pressure

220


p2 – p1 =

ρ 1 ρ (w21 − w22 ) = ( w12u + w1m2 – w22u – w2 m2 ) 2 2

=

ρ ( w12u − w22u ) 2

=

ρ ( w + w2u ) ( w1u − w2u ) 2 1u

ρht ( w + w ) ( w − w ) 1u 2u 1u 2u ...(10.3) 2 since the blade velocities are same at inlet and at outlet (uo= u1= u2= u3), due to axial flow. In order to

Therefore,

Pz =

 w1u + w2u obtain the value  2 

  , inlet and outlet velocity triangles are combined into one.  C u2 – C u1 2 A

C u2 – C u1 2 D

B w∞

C∞

C1

∆β

w2

w1

C2

C m= C z

w u∞ = w u2 + w u1 2

α1 α2

α∞

β1 β∞

β2

C

w u1

C u1 C u2

w u2 u1 = u2

Fig. 10.5. Combined velocity triangle

Referring to Fig. 10.5 the vectors AC = w1 and CB = w2. Vector CD = w∞ is the geometrical average of vectors AC and CB . Point D is the middle point on the line CD. Blade angles are β1 at inlet, β2 at outlet and correspondingly β∞ for vector velocity w∞, i.e., for the line CD . From the velocity triangle. (Fig. 10.5) wm ( w1u + w2u ) = FD = wu∞ = tan β∞ 2

Hence,

PZ = ρhwu∞ Γ b Pu = ρhwm∞ Γ b

and Since,

w1m = w2m= wm= wm∞

...(10.4)

221

AXIAL FLOW PUMP

Total force on one blade, P = Pu + Pz = ρhw∞ Γ b

...(10.5)

wu∞ + wm∞ = w∞

Since,

For unit height, (h =1) Pu = ρhwm∞ Γ b Pm = ρhwm∞ Γ b and P = ρw∞ Γ b

...(10.6)

w∞

Z

Pu π 2

u

β∞ β∞

P

Pz

Fig. 10.6. Forces acting on a blade in a cascade system under ideal fluid flow condition

Force P in equation (10.6) is independent of the pitch ‘t’ of the blade. It remains same even when pitch increases to infinity (∞) i.e., when cascade blade system changes to isolated blade. Also, circulation ‘Γb’ remains constant inspite of increase in flow due to increase of pitch ‘t’. The magnitude and direction of the flow w∞ and β∞ remain same, at all points, i.e., before the blade from infinity, on the blade and after the blade up to infinity. Equation (10.6) is called Kutta-Jowkovski equation. This equation also indicates that the lift force P is perpendicular to the flow direction. However, exact location of the force P on the blade can be determined only by experimental investigation. The equation (10.6) can also be applied for conditions where the flow velocity Cm is not constant. Lift force (Yp), the force perpendicular to the flow direction of w∞, P = ρw∞ Γb. Drag force, Xp parallel to the flow direction of w∞ is zero. Total head,

Hm =

(Cu 2u2 − Cu1u1 ) u (Cu 2 − Cu1 ) = g g

Since, u 1 = u2 = u From velocity triangle, Cu = (u – wu) (Cu 2 – Cu1 ) = u2 − wu 2 − u1 + wu1 = ( wu1 − wu 2 ) =

Hm =

u Γb g t

∴ Circulation on one blade, Γ b =

Total circulation on all blades,

Γb t

2πgH m gH m t gH m 2πr . = = Z Zω u u 2πrgH m 2πgH m = Γ = ZΓ b = u ω

...(10.7)

222


10.4 REAL FLUID FLOW OVER A BLADE While determining the force Pz in equation (10.5), Bernoulli’s equation (10.3) was written for an ideal fluid flow. Equation for a real fluid flow can be written as

p1 p2 w22 − u22 w2 – u12 + Z1 + 1 = + Z2 + + h f (1− 2) (= h fp ) γ γ 2g 2g where, hfp = hf (1–2) is the profile loss in impeller. Since, u1 = u2, Z1= Z2 in axial flow pumps and γ = ρg p2 – p1 =

1 ρ (w12 – w22) – γhfp 2

ρth fp

(w12 – w22) – ρthfp = PZ – γthfp 2 = ρhwu∞ Γb – γthfp

and actual force,

Rz =

for unit height, (h = 1)

Rz = ρwu∞ Γb – γthfp

...(10.8)

i.e., the real axial force RZ is reduced and is less than the ideal axial force Pz by frictional losses γthfp. The tangential force Ru = Pu= Γwz∞ Γb remains same, since the head developed remains unaltered. Z

X u

Rz

Ru = Pu β ∞+λ

γthfp

R

λ

β∞

Y Pz

P

Fig. 10.7. Hydrodynamic forces acting on a blade of cascade system due to real fluid flow condition

The total force, R = ( Ru + R z ) turns towards the flow direction of ‘w ∞’ by an angle ‘λ’ (Fig. 10.7), where λ is the angle of incidence. Resolving the total force R into two forces, one parallel and another perpendicular to the flow direction of ‘w∞’ Lift force, Yp = R cos λ and Drag force, Xp = R sin λ and tan λ =

Xp Yp

.

Loss of head (h fi = h fp ) , the profile loss in impeller is the work done by drag force Xp per unit weight of the fluid (γtwz∞) along the direction of the velocity w∞ i.e., hfi= hfp =

X p w∞ γtwZ ∞

=

R sin λ w∞ R sin λ = γt sin β∞ γtw∞ sin β∞

...(10.9)

223

AXIAL FLOW PUMP

Head developed, Hm is the work done by the force Ru per unit weight of the fluid Hm =

Impeller efficiency,

ηi =

uRu uR sin (β∞ + λ) = γtw∞ sin β∞ γtwz∞ H m − h fp Hm

= 1−

= 1–

...(10.10)

h fp Hm

w∞ sin λ ⋅ u sin (β∞ + λ )

...(10.11)

When angle of incidence λ increases, the drag force Xp increases. Correspondingly losses increase and efficiency reduces. From aerodynamics of airfoils the lift (Yp) and drag (Xp) of an airfoil are given by Yp = Cy. ρ

w∞2 .l 2

...(10.12)

w∞2 .l 2 where ‘Cy’and ‘Cx’ are the coefficients of lift and drag respectively and ‘l’ is the chord length of profile.

X p = Cx ρ

Manometric head,

Hm =

uR sin (β∞ + λ) uY sin (β∞ + λ ) = Cm γt cos λ γtw∞ sin β∞

...(10.13)

Combining equation (10.12) and (10.13) and rearranging Cy

l 2 gH m Cm cos λ ⋅ ⋅ = 2 t u sin (β∞ + λ ) w∞

...(10.14)

10.5 INTERACTION BETWEEN PROFILES IN A CASCADE SYSTEM Thin Straight Plates in Cascade System Interaction between straight, thin plates in a cascade system was studied by many authors. Most important is the work done by Prof Shlihandl | 67 |. Circulation Γ, given by plate cascade, is determined by the formula. Γpl.ca = Γpl . Lp = Lp π l w∞ sin i where, Lp is the coefficient accounting for the deviation in flow of thin plate in a cascade system with respect to the flow of thin isolated plate. In Fig. 10.6, the values of Lp determined by Shlihandl |67| as a function of angle of attack (αp) and relative pitch t/l are given. The coefficient Lp depends only upon the geometric parameters of the cascade system. This graph can be used to determine the camber line of the real profile as first approximation. While determining the value of the coefficient Lp, the following condition must be observed. 1 t  β t lplate = 2lprof . (or)   plate =   prof and α plate= α + . l 2   2 l

224


Lp 2,4 2,2 2,0 10

°

10 15

15

α

=

1,8 1,6

0

20

20

25

1,4

30

1,2

35 1,0

40 50 60 90

0,8 25 30 35 40

0,6 0,4 0,2 0

0,2

0,4

0,6

0,8

1,0

1,2

( , tl ) α

Fig. 10.8. Coefficient L = f

1,4

1,6

1,8

2,0

t/l

for thin straight plate

10.6 CURVED PLATES IN A CASCADE SYSTEM Flow over a curved thin plate cascade system adopted for axial flow machines as given by Prof. E.N. Voznisenski | 130 | and further developed by V.F. Pekin | 84 |, A.F Lisohin | 65 | L.A. Semanoff | 66 | and N.A. Kolokolsoff | 84 | is given below. While deriving the cascade theory, it is assumed that (1) Profile thickness is neglected i.e.,the profile is a thin plate and (2) the curvature of the plate is an arc of a circle. (Fig. 10.9) C u2

y

w2

x

ds

w

w

y

l

C2 u

T

y0

s

α

r t

T w1 u

β

x0 T

C 1 = C m1 C u1 = 0

Fig. 10.9. Thin curved plate in the form of an arc of a circle in a cascade system (mean of a thick profile)

225

AXIAL FLOW PUMP

Two methods are adopted for the cascade design namely, (i) Flow parameters are determined for the given boundary conditions (called direct method) and (ii) Geometrical characteristics of the developed cascade are determined as per the given flow parameters under infinite conditions (called indirect method). The integration method, given by Prof. Voznisenski, | 130 | determines the flow conditions of the cascade system over a thin curved plate in the form of an arc of a circle i.e., direct method. Prof. Pekin and Prof. Kolokolsoff developed systematic procedure to design the pumps by Indirect method. Under integration method, the thin profile is considered as a vortex passage consisting of two flows : (1) Plane flow over a cascade with equal velocity everywhere and (2) a vortex flow due to the presence of circulation distributed over the entire length of the plate. Net flow function ψ(t) at a point (s) situated at a distance ‘t’ from the starting point is the sum of both vertices ψ (t) = ψ0(t) + ψ1(t) where, ψ0 (t) is the plane undisturbed flow and ψ1(t) flow due to circulation and is given as t

ψ1(t) =

dΓ

∫ 2π

ln r(s, t)

0

d Γ= d Γ (s) = v(s) ds is the circulation dΓ v(s) = —vortex intensity at the point s on the plate, r—distance from the considered point on ds the profile to the adjacent point with an elementary circulation d Γ(s) t

So,

ψ (t) = ψ 0 (t ) +

1 v (s) ln r(s, t) ds = constant 2π ∫0

...(10.15)

However, the absence of infinite velocity at the outer end of the profile i.e., postulate Chapligin indicates ν(l) = 0 The flow equation for the cascade system, from an isolated curved thin blade is obtained by modifying the integral equation (10.15) available for isolated profiles.The undisturbed plain uniform steady flow function ψ0 is determined, from the geometrical average velocity w∞, instead of undisturbed velocity from infinity before the blade inlet edge C∞ and simple function ‘ln r’ in the integral equation vds ln r , a more 2π complicated flow function developed by the vortex flow i.e., circulation located on the elementary distance ds of all profiles in cascade system is used i.e.,

(10.15) for the vortex flow located on isolated blade, which is determined by d ψ1 =

lim νds dψ1 = n → ∞ 2π

k = +a

∑

k =− a

ln ri

...(10.16)

where, ri is the distance between flow point Z, where the flow function ψ is determined and the points ‘s’ on each blade of the cascade system. Infinitesimal summation of logarithm leads to infinitesimal transformation under logarithm, which can be expressed as trigonometric function.

226


The final form of flow function as given by Prof. Pekin in his paper | 84 | when l =1 i.e., relative t pitch, T0 = is l 1 l v( s) ln K ds ψ (t) = ψ 0 (t ) + ...(10.17) 2π ∫ 0 where

K= l

and

∫ 0 v(s)ds

=

sin 2

π π ( xt − xs ) + sh 2 ( yt − ys ) T0 T0

Γ1 l0

...(10.18)

β , the length of the curved plate with unit chord length and curvature β. The general sin β0 form of the integral equation (10.17) is a function of various parameters given by v(s) = f (T0, s, α, β, w∞x, w∞y, C) where, α is the angle between the cascade axis (direction of blade velocity u) and the direction of velocity w∞ i.e., α1. Taking into account equation (10.18), the above function can be rewritten as

where l0 =

Γ1 = f1(T0, α, β, w∞, C) l0

where C is the constant of integration and can be obtained from the condition v(l) = 0 i.e., by applying Postulate Chapligin. Γ1 = f2(T0, α, β, w∞) ...(10.19) So, l0 The disturbed flow due to the introduction of circulation on all the blades of the cascade system, changes the flow direction by an angle ∆ α = α – α1. Angle ∆ α takes into account the influence of flow on the profile by other profiles in the cascade system. The final form of the equation (10.19) is written as Γ = f3 (T0, α, β, ∆α) ...(10.20) W∞ l0 ∆ α is the angle between the direction of velocity w∞ and the chord of the profile. Equation (10.20) is the final form of the integral equation (10.17) and (10.18) by which the flow over a cascade system is determined by direct method. Prof. V.F Pekin and Prof. N.A. Kolokolsoff gave a systematic calculation for a cascade flow under an additional condition of shockless entry (i.e., δ = 0) as given by Prof. I.N. Voznicenski, which provides a simplified approach for the design as well as to develop pumps with better load and cavitational characteristics. Shockless entry i.e., δ = 0 indicates no circulation at inlet edge, which is written as v (0) = 0. The condition v(l) = v (0) = 0 leads one and the same design procedure for pump and turbine. The equation (10.20) can be written as L=

Γ1 = f1 (T0,α, β) w∞ l0

and, ∆α = f2 (T0, α, β) ...(10.21) Equation (10.21) is solved by step by step integration of equations (10.17) and (10.18) by providing (n +2) linear equations with (n +2) unknown values and with (n +1) points on the blade curve, determined by geometric parameters of the cascade.

AXIAL FLOW PUMP

15° 10°

Γ1 w ∞ l0 β

20°

2.0 25°

30° 1.5 35° 50° 55° 60°

1.0 40°

65° 70°

45°

80° 0.5

T0 = t l

0 0.5

1.0

1.5

2.0

3.0

227

Γ1 Fig. 10.10. w l β = f (T0, α ) ∞ 0

2.5

228


Calculations were done for n = 4, 6, 8. It is found that, for a wide range of curvature β, the value

Γ1 depends upon only T0 and α. A graph has been drawn χ = f (T0, α) (Fig 10.10). Also, it is w∞ l0β found that ∆α is +ve for all operating ranges of cascade parameters. For small ranges of α and β under χ=

T >1, which corresponds to peripheral section of the blade of axial flow pumps of l higher specific speed such as ship propellers, ∆ α change is very small, not exceeding 1°. ∆ α increases when curvature β increases and when α and T0 decreases (i.e., where l/t increases). When α < 34 to 40°, ∆ α mostly depends upon β and T0 (Fig. 10.11). When α > 45°, which is mostly for diffusers and hub sections of impeller blades, ∆ α value increases up to 15° and depends not only upon α and T0, but also β. Following figures (Figs. 10.12 to Fig. 10.22) illustrate these variations for β changing from 20 to 40°, at the interval of 2°.

the condition T0 =

∆α = f(T0 , β)β

5° t = 0,75

∆α

t = 0,8 4° t = 0,85 t = 0,9 t = 0,95 3°

t = 1,0 t = 1,05 t = 1,1 t = 1,15 t = 1,2

2°

1°

7° 8°

10°

12°

14°

16°

18°

20°

22°

Fig. 10.11. ∆α = f (T0, β )

24°

26°

t = 1,25 t = 1,3 t = 1,35 t = 1,4 t = 1,45 t = 1,5 t = 1,6 t = 1,7 t = 1,8 t = 1,9 t = 2,0 t = 2,2 t = 2,5 t = 3,0 28° β

229

AXIAL FLOW PUMP ∆α ° = f(α , T0, β )

T0= 0,5 4 3

0,6 0,7

2

0,9 1,0 1,25 1,5 2,0 3,0

1 0

45

50

60

70

α°

Fig. 10.12. ∆α for β = 20° ∆α ° = f(α , T0, β )

5

T0= 0,5

4 3 2 1 0

0,6 0,7 0,8 0,9 1,0 1,25 1,5 2,0 3,0 50

60

70

α°

Fig. 10.13. ∆α for β = 22° ∆α ° = f(α , T0, β )

6

T0= 0,5

5 0,6 4 0,7 3 2 1 0

0,8 0,9 1,0 1,25 1,5 2,0 3,0 50

60

Fig. 10.14. ∆α for β = 24°

70

α°

230


∆α° = f(α, T0, β)

7

T0 = 0,5

6 0,6 5 0,7 4

0,8 0,9

3

1,0

2

1,25

1

1,5 2,0 3,0

0 40

50

60

α°

70

Fig. 10.15. ∆α for β = 26°

9

T 0= 0,5

8 7 6 5

0,6

0,7

4

0,8 0,9

3

1,0

2

1,25 1,5

1

2,0 3,0

0 40

45

50

60

Fig. 10.16. ∆α for β = 28°

70

α°

231

AXIAL FLOW PUMP ∆α ° = f(α ,T0, β )

T 0 = 0,5

10 9 8

0,6

7 6

0,7

5

0,8

4

0,9 1,0

3 2 1 0 40

1,25 1,5 2,0 3,0 45

50

60

70

α°

Fig. 10.17. ∆α for β = 30° ∆α ° = ( α,T 0, β) 12 T = 0,5 0

11 10 9 0,6 8 7 0,7 6 0,8 5 0,9 4 3

1,0 1,25

2

1,5

1

2,0 3,0

0 40

50

60

Fig. 10.18. ∆α for β = 32°

70

α°

232

ROTODYNAMIC PUMPS (CENTRIFUGAL AND AXIAL) ∆α ° = f(α ,T0, β ) T0 = 0,5 13

12 11 10 0,6 9 8 0,7

7

0,8

6

0,9

5

1,0

4

1,25

3

1,5

2

2,0 1 3,0 0 40

50

60

70

Fig. 10.19. ∆α for β = 34°

α°

∆α ° = f(α ,T 0,β )

15 T 0=0,5

14 13 12 11

0,6

10 9

0,7

8 7

0,8

6

0,9

5

1,0

4 1,25 3 1,5

2

2,0

1 0

3,0 40

50

60

70

Fig. 10.20. ∆α for β = 36°

α°

233

AXIAL FLOW PUMP ∆α° = f(α, T0, β)

T0 = 0,5

18 17 16 15

∆α° = ( α, T 0, β)

16 T0= 0,5

14

15

13

14

12

0,6

13

11

12

0,7

0,6

10

11 9

10 0,7

9

8 7

8 0,8 7

6

0,9 6

1,0

1,25 4

1,25

3

1,5

2

2,0

3 2

1 0

0,9

5

1,0 5 4

0,8

1

3,0 40

50

60

70

α°

0

Fig. 10.21. ∆α for β = 38°

1,5

2,0 2,5 3,0 50

60

70

α°

Fig. 10.22. ∆α for β = 40°

10.7 EFFECT OF BLADE THICKNESS ON FLOW OVER A CASCADE SYSTEM The profile, determined as per the method given by Prof. Voznisenski, is dressed over the camberline or the middle line with finite vane thickness (δ). However, vane thickness reduces the flow area of the passage and correspondingly increases the flow velocity. The coefficient of area reduction can be expressed as (Fig. 10.23). wm A X = C = A − ∆A m

234


t

l

A

∆A

δm α

χ) Fig. 10.23. For the determination of flow area reduction coefficient (χ due to dressing of thick profile on thin curved plate

where, A = t. l sin α and ∆A =

l

∫ 0 δ. dl

and δ is the blade thickness measured normal to the chord

length l. The integral

l

∫ 0 δ. dl can be approximately determined by applying Simpson’s rule at three points,

under the condition that (δm) the maximum thickness is at the middle. Since, the blade thickness area (∆A) is very small when compared to flow passage area (A), ∆A ≈

So,

χ=

2 l 4δ m = δm . l 3 b

t . l sin α 1 = 2 2 δm tl . sin α − δ m l 1− 3 3 t sin α

...(10.22)

10.8 METHOD OF CALCULATION OF PROFILE WITH THICKNESS IN A CASCADE SYSTEM Most of the design procedures for pumps are carried out by direct method. The camber line is determined and then dressed by a known profile. Thickness effect is corrected by repeating the design until the velocity wm remains constant. The profiles are selected from the airfoil test results from Gottingen profiles data or from NASA profiles data or from profile data of USSR research institute. Most convenient design is obtained by conformal transformation of the profile from the given parameters of the pump, by which theoretical blade profile with thickness is obtained. In most cases, conformal transformation is done from the cylinder, either isolated or cascade system. Hodograph principle is also used while adopting conformal transformation. This process simplifies the design procedure for flow over a profile, where in the velocity of flow remains constant. Application of Hodograph method by which, theoretical, flow velocity is also known which enables to find the flow of real fluid i.e., flow with aerodynamic wake after the blade. Most accurate method of getting the blade profile with thickness is the addition of vortex or circulation, sources and sinks in the plane uniform steady flow over the blade system. Earlier vortex or circulation (Γ) is added to the plain flow, to get thin profile of the cascade system. Prof. A.F. Lisohin | 65 | and Prof. L.A. Simonoff | 66 | gave a systematic method to get profiles with thickness i.e., by

235

AXIAL FLOW PUMP

providing source (+q) and sink (–q) along with circulation ( Γ ), distributed over the profile. The skeleton line or otherwise called as camberline or middle line, becomes a streamline inside the profile, where there is no cross flow perpendicular to this camberline. Addition of a source very near to the inlet edge and a sink very near to the outlet edge of the skeleton profile, enables to get the flow with profile thickness. The rounded inlet edge and outlet edge with proper thickness are obtained by proper selection and distribution of source (+q) and sink (–q). Outer edge contour of the profile which encloses the camberline is also a streamline. Profile shape is obtained by the group of streamlines starting from source kept at inlet and ending at sink kept at the outlet. The sum of intensity of source and sink is equal to zero. Mathematically expressing these as +

Qin + Qout +

l 2

∫ q(s)ds

=0

...(10.23)

l − 2

Circulation, source and sink are selected in such a way that the combination of these, with the plane flow gives rise to the profile of required specification to meet the head and discharge of the axial flow pump. In order to get this, two conditions are observed. 1. The magnitude and direction of the infinite velocity w∞ or C∞ before and after the profile remain same. 2. Closed contour encircling the skeleton of the profile is the profile as per the required parameters such as magnitude and location of maximum profile thickness, radius of the roundness of inlet and outlet edges of the profile etc. First condition is fulfilled by proper distribution of circulation of vortex on the skeleton of the profile as per the integration law +

Γ1 =

1 2

∫ γ ( s)ds

...(10.24)

1 − 2

where, Γ1 is the circulation around one profile. Second condition is attained by proper distribution of source and sink q (s) on the profile skeleton to fulfil the condition as per the equation (10.23) Absence of cross flow across the skeleton yields to an expression ...(10.25) w∞ + ν* = 0 where, w∞ =

 w + wu 2  wm2 +  u1   2

2

is the geometrical average of relative velocity of the plane flow and ν* the induced velocity at the point considered on the skeleton of one profile, due to the disturbance created by adjacent profiles in the cascade system. Summation of these induced velocities, located on other profiles kept at a distance of ‘t’ from the profile and the integration of these along the skeleton ‘s’ is +

νu* =

1 2t

l 2

∫ −

l 2

γ ( s ) sh Z 1+ q ( s )sin u1 ds chZ1 − cos u1

236


ν*m =

1 2t

+l / 2

∫

–l / 2

– γ ( s ) sin u1 + q (s ) shZ1 ds chZ1 – cos u1

...(10.26)

2π ( Z 0 − Z ) and u1 = 2π (u0 − u ) t t Due to complexity of the above equation, the integral is carried out by step by step method.

where ,

Z1 =

Selection of γ (s) and q (s) Distribution of circulation γ (s), source and sink q (s) along the profile is carried out to get uniform and even distribution of pressure and velocity over the profile from inlet edge to the outlet edge. Prof. Ginsberg |37| has suggested, the following, for the distribution of circulation and source and sinks. For γ (s) he suggested γ (s) = A0 γ0 (s) + A1γ1 (s) + A–1 γ–1 (s) + A2 γ2 (s) + A–2 γ–2 (s) + A* γ* (s) _ where, γ0 (s) = 1 − s 6 _  0, when − 1 ≤ s ≤ 0  γ1 (s) =    s (1 − s ), when 0 ≤ s ≤ 1  − s ( s + 1), when − 1 ≤ s ≤ 0  γ–1 (s) =   0, when 0 ≤ s ≤ 0  

...(10.27)

(2 s − 1)(1 − s ), when 0.5 ≤ s ≤ 1  γ2 (s) =   0, when − 1 ≤ s ≤ 0.5   −(1 + 2s )(1 + s ), when − 1 ≤ s ≤ −0.5 γ–2 (s) =   0, when − 0.5 ≤ s ≤ 1   1− s 1+ s

γ* (s) =

y

+3 + 5 X 2 +2 +1 x 0 –

5 2

X

–1 –2

–3

Fig. 10.24. Profile coordinates

237

AXIAL FLOW PUMP

y s

n 0

) q (s

γ (s

x

l

)

α

t

t

Fig. 10.25. For profile calculation—Distribution of circulation γ (s), source and sink q (s)

z

ζ1 Inlet edge ζ2

n Outlet edge

ρ2

ρ**

h2

δm

h1 ρ*

+

δm 2 C S

– l =s 2

Point sink

Point source

0

s – l 2

ρ*

– δm

B

2

ρ *) and Fig. 10.26. For profile calculation of inlet edge radius (ρ ρ **) and profile thickness (δ δ m) outlet radius (ρ y + l 2

l 2

n

αK δ+m 2

δm

– l 2

S

–

δm 2

Fig. 10.27. For profile calculation

ρ **

+

l s=+ 2

x

l 2

238


s (s is the point selected on the profile (–3, –2.5, –2, –1, 0, +1, +2, 2.5, +3) and l is the l length of the profile. Centre of the skeleton line (mean line) is the origin. Profile length l is given as l  l  − to +  (Fig. 10.27). 2  2 –

where, s =

z

dV′

dV

γ ′n

dv′ qn

dVγ′

dVq′

+ l 2

dV

γ ′s

K(x K, zK) nk

N(x, y)

dVb

M(SK)

θ

αK

K

dV′

SK

x

N

o S

l – 2

N(s)

(a)

(b) w ∞s + V s′′

z n z

+V

w

l/2

M

x

o

w∞z+V′′

– l 2

z

dVcr

′′ x

x ∞

w ∞x+V ′′x

w ∞n + V′′n

α

x

(c)

Fig. 10.28. Determination of induced velocity on the skeleton of the profile

239

AXIAL FLOW PUMP

The value, A* γ* (s) corresponds to the flow with angle of attack over the cascade A* = 0 for flow without angle of attack. γ(s) 1.0

1– s 6

γ0

0.9 0.8 0.7 0.6 0.5 0.4

– s(1 + s)

s (1 – s)

0.3 0.2 0.1

– 1.0

γ–2

γ–1

γ1

γ2

1.0

Fig. 10.29. Theoretical distribution of circulation λ (s)

Distribution of sources and sinks q (s) is given by Prof. Ginsburg | 37 | as j =k

q (s) =

∑

j =0

BJ s j

Selection of number of points ‘j’ depends upon the selection of profile shape at the outlet edge, whether rounded edge or sharp edge is required. For rounded edged profile

q ( s ) = B0 + B1 s + B2 s 2 + B * + B **

...(10.28a)

where, B* is the intensity of the point source selected at the inlet edge of the profile (s = – l/2) and B** the intensity of the point sink selected at the outlet edge of the profile (s = – l/2) in order to get rounded edges, ρ* the radius at inlet edge and ρ** the radius at the outlet edge. Coefficients A0, A1, ..., B0, B1, B2, B*, B** are selected to get the designed profile.

sm l δm vane solidity, . Maximum thickness ratio and = sm location of maximum t l/2 (l / 2) thickness on the mean line. The following equations are applied: Taking

(i) 2 B0 +

2 B2 + B * + B ** = 0 3

(ii) B0 (1 − sm ) +

1 1 δ B1 (1 − sm2 ) + B2 (1 − sm3 ) + B ** = 2 χ m w∞ 2 3 l

(iii) B0 + B1sm + B2 sm2 = 0 (iv) B*=

8π ρ * . w∞ cos β 3 l

(v) B**=

8π ρ ** . w∞ cos β 3 l

...(10.28b)

240


Equation (10.28a & b) is given based on the condition that the sum of all the sources and sinks located on the mean line is equal to zero i.e., +l/2

i.e.,

l l q ( s )ds + B* + B** = 0 2 2 –l/2

∫

...(10.28c)

On integration of the above equation (10.28c), Ist equation is obtained. Second equation is obtained based on the following condition: +l/2

∫

q ( s )ds = χδmw∞

–l/2

where, χ is the coefficient of area reduction due to profile thickness and is determined approximately as χ=

1

...(10.28d) 2 δm 3 t sin α The third equation is obtained based on the condition q (sm) = 0 i.e., change from source to sink takes place at the point where maximum thickness is located (sm). Fourth and fifth equations approximately determines the value ρ∗ and ρ∗∗ from the condition that sum of all velocities from the source at inlet edge (ρ∗) and sink at the outlet edge (ρ∗∗) and the main flow w1 at inlet and w2 at outlet is zero. The velocity due to circulation, other sources and sinks are neglected,

1−

l * B 2 w1 = at inlet 2π r1 l ** B 2 for outlet. w2 = 2πr2 Flow due to B* across the line h, the distance between the profile thickness at inlet and the mean B* i.e., line will be 4 From Fig. 10.26, we can write w1 h 1 =

ζ1 4 2 ≈ = r1 2π 3

from which Taking, ∴

l B . 2 4

ρ* ≈ h1

ζ1 2 = 3 ρ* B* = 2 .

8π ρ * 8π ρ * 2πζ1 w1 = w1 = . .w∞ cos β∞ 3 l 3 l l

241

AXIAL FLOW PUMP

In the similar way,

B** =

8π ρ ** . .w∞ cos β∞ 3 l

Solving the equations, sm sm (2 . sm ) s (2 + sm ) K3 K2 – B0 = 2 (1 − s 2 ) 2 K1 – m 2 m (1 – sm ) 2 (1 + sm )

B1 = B2 =

– (1 – 3 sm2 ) 2 2

(1 – sm ) − 3 sm 2 (1 −

sm2 ) 2

2(1 + 2 sm )

K1 −

(1 + sm )

2 2

K2 +

2 (1 − 2 sm ) (1 − sm ) 2

K3

3 3 K3 K2 + 2 (1 – sm ) 2 (1 + sm )

K1 −

B* = 2K2 and B** = –2K3. K1 = 8x

where,

K2 =

δm w∞ ρ

4π ρ * w∞ cos β∞ 3 l

4π ρ** w∞ cos β∞ 3 l For sharp edged outlet edge of the profile q(s) is written as

K3 =

q ( s ) = B0 + B1s– + B2s– 2 + B3s– 3

....(10.28e)

point source B* is applied at inlet edge of the profile and instead of radius ρ** at the outlet edge, the included angle of the outlet edge is determined. The coefficients Bj are determined as (i) 2B0 +

2 B + B* = 0. 3 2

(ii) B0(1 + sm ) +

1 1 1 δm B1 ( 1 − sm2 ) + B2 (1 – sm3 ) + B3( 1 − sm4 ) = 2x w . 2 3 4 l ∞

(iii) B0 + B1 sm + B2 sm2 + B3 sm 3 = 0 8π ρ * w 3 l ∞ (v) B0 + B1 + B2 + B3 = – w∞ tan θ

(iv) B* =

and

B0 = B1 = B2 =

sm (1 + sm ) (1 − sm )

3

K1 –

sm sm K2 – K3 (1 + sm ) (1 − sm ) 2

1 + 2 sm

1 + 4 sm + sm2 (1 + sm ) K + K2+ K3 1 2 (1 − sm ) 2 (1 − sm ) (1 + sm ) − 3 sm 2 (1 + sm ) (1 − sm )

3

K1 –

3 sm 3 K3 K2 + (1 − sm )2 (1 + sm )

242

+y –a –a – x ––ab ––ab + x –b –b –y

0.0 46

0.0465

0.0 485

4 .0

9

4. 0

0. 4 2 0 .4 0 .3 4 0. 3 5 0. 3 4 0.32

t = 396 Γy vx = Γa = t 2π r 2

Γ vy = Γb = t 2π r 2

Fig. 10.30. Nomogram for the determination of coefficients ‘a’ and ‘b’

0. 5 5 0.7

0. 5 1 0 .5 2 0. 5 0. 5 3 4 0. 5 6 0.5 8 0. 6 0.6 6


0.0 48 0.0 5

8

4.05

0.0 40 4. 0 4 .0 6 7

4.0 2 4 .03 4 .0 4

0.0 469 4.01

4.0

0.0467

243

AXIAL FLOW PUMP

B3 =

1 + 3 sm (1 + sm ) (1 − sm ) 2

3

K1 +

2 2 K2 – K 2 (1 − sm )2 3 (1 + sm)

B* = 2K2 and

K1 = 8x

δm w∞ l

4π ρ * w 3 l ∞ K3 = w∞ tan θ. In order to select the coefficients A0, A 1.... the values of γ (s) is substituted in equation

K2 =

Γ1= ∫ γ ( s )ds for one profile

  Γ1 A1 + A−1 A2 + A−2 π + + + A *  = 0. A0 + 1.098  − 12 48 2  l  Major circulation is from the value A0 γ0(s). All other values only change the intensity of circulation distribution at individual positions of the mean line, mostly near the outlet in order to get uniform, smooth velocity distribution. Fig (10.29) illustrates the distribution of γ0, γ1.... Values A0, A1... must be properly selected in order to avoid uneven changes in profile shape, unevenness in pressure and velocity distribution from inlet to outlet of the profile, since these coefficients do influence much on the profile configuration. Usually, these values are suitably altered after obtaining the results from Ist approximation. One such pressure distribution is shown in Fig. 10.57. The main advantage of this process is that, profile shape can be suitably modified to get better cavitational characteristics. Experimental verification shows that very good results are obtained in turbines and pumps. However, this process is used only, when a good cavitational property is required, since this method is a tedious and lengthy process.

10.9 (A) PUMP DESIGN BY DIRECT METHOD (JOWKOVSKI’S METHOD, LIFT METHOD) X . sin (β∞ + λ ) = sin β∞ cos λ + cos β∞ . sin λ. For normal Y entry Cu= 0, so, ∆Cu= Cu2 – Cu1= Cu2. In general, it can be written as Cu.

Referring to the Fig. (10.7) tan λ =

Equation (10.14) can be written as CyR .

2 (∆Cu ) sin β∞ gH m . 2sin β ∞ l = = u.Cm (1 + cot β ∞ .tan λ) Cm (1 + cot β∞ tan λ ) t

...(10.29)

l is the vane solidity, T0 = t is the relative pitch t l Equations (10.9), (10.11), (10.14), (10.29) are the basic equations for axial flow pump design by method of Lift and Drag. The same procedure is followed for axial flow diffuser design also. Prof. N.E. Jowkovski developed this method for ship propellers at Moscow University, USSR. Later, his students developed the practical design procedure for the design of axial flow pump. In this design, the profiles

where, t =

244


for each radius of impeller blade or diffuser blade are selected from the known test results of many profiles, tested in wind tunnels. i.e., Test results of different profiles for which the Lift and Drag coefficients (Cy and Cx) as a function of angle attack ‘δ’ are known under infinite velocity condition before and after, the profile and for isolated profile. For pump calculations Cyc of casade system = 0.8 to 0.85 Cymax of the isolated profile. Cymax is taken from wind tunnel test results of isolated profile Cy,Cx = f (δ) (Figs. 10.31 and 10.32) corresponding angle of attack δ is used for the selected radius of the blade. From the combined velocity triangle and the value of Cyc, using equation (10.29), vane solidity l/t is determined for the selected radius. The value of C X is determined by the expression tan λ = x . Normally, λ will be selected as 3° to 5°. While determining Cy the angle of incidences (λ), it is assumed that Cyc for cascade and Cy for isolated profile will have the same influence on performance of pump impeller. l

Angle of attack ‘δ’ and vane solidity t are the deciding parameters for the selected radius of the impeller blade. Chord length is selected based on the profile strength and constructional possibility. From the known values of l and (l/t), pitch ‘t’ can be determined. Profile form or profile shape for all sections remains unchanged. Thus, the geometric parameters of the profile are obtained, based on the lift of the known profile that’s why this method is called lift method of design of axial flow machine. Drawback of this method is that the assumption is made that Cyc = Cyi which is not correct. This method proves to be successful for low head pumps with smaller angle of blade rotation (5° to 8°), where l/t << 1 i.e., vanes are spaced at a larger distance between them i.e., t is large, number of vanes Z is less (≈3). But for higher head range, Cyc < Cyi, correction factors must be applied. Fluid flow in pumps, designed by lift method with correct value of angle of attack (δ) and l/t, is found to be a flow without separation. However, correct value of (Cyc /Cyi) could not determined. Also drag coefficient Cxc ≠ Cxi . It is not possible to predict exact value of hydraulic efficiency (eqn. 10.11). In order to overcome this drawback, test results of compressor (diffuser type) cascades, tested in wind tunnels are taken for design of pumps. Instead of finding Cyc Cxc λ, from the test results, the magnitude and direction of velocity before and after the cascade are simultaneously changed during the test. Pressure and velocity on the profile are measured. Cyc and Cxc are calculated by using the equation for which, values w1, w2, β1, β2, are already known and by constructing the combined velocity triangle the values ∆wu (=∆Cu), Cu, β∞ are also known. Cyc is determined from equation (10.14). But λ depends on Cy and Cx which makes difficult to determine correct value of λ. That’s why, Cyc is determined under the condition (λ = 0). Since for smaller values of λ the difference between Cyc and Cyi is negligibly small. Equation (10.29) when λ = 0 becomes ∆Cu l C yc = 2 sin β ...(10.30) t Cz From the known l, t, β∞ Cm and ∆Cu, CyR is calculated. In the similar manner Cx is determined, λ is determined from Cyc and Cxc. This value is substituted again in equation (10.14). By repeating this process correct values of Cyc, Cxc could be determined. Sometimes wind tunnel test results of profiles are also given in the graphical form Cyc and Cxc= f (i), where ‘i’ is the angle between the tangent to the camberline at entrance and the direction of w1 (Fig. 10.31). Results are also given in the form of the flow deviation between inlet and outlet of the profile ∆β = β2 – β1, as a function of ‘i’ i.e., ∆β , Cxc = f (i) (Fig. 10.32).

245

AXIAL FLOW PUMP

C yc

Cx

1,0

0,25

∆β°

0,20

40

0,8 C yc 0,6

0,15

C xc

30

0,4

0,10

20

0,2

0,05

10

– 10 – 5 0 5 (C y, C x) cascade = f(i)

10

i°

Cx ∆βmax ∆β

∆β

0,075 C xp 0,050 0,025

– 20 – 10

Fig. 10.31. Cyc and Cxc for profile in cascade system as a function of angle of incidence (i)

0,100

x

i*0

+10

i°

Fig. 10.32. ∆β ∆β, Cx = f (i) Allowable flow deviation and drag coefficient as a function of (i)

For calculations ∆β = 0.8 ∆β(max) is taken, in order to have a flow without separation. Tests conducted on different cascade systems in wind tunnels show that the velocity ∆β (determined as 0.8 ∆βmax) and corresponding values of Cyc and Cxc mainly depend upon l/t, β∞, whereas the influence of blade curvature, blade thickness (maximum) are less. Angle (i), changes 5% depending upon the curvature of the profile. Some other test results are also given as ∆β and Cyc as a function of β∞or β2 and l/t i.e., ∆β (β2 – β1) = f (β∞ or β2) for different valves of (l/t) (Figs. 10.33 and 10.34) i.e., Cyc Cxc = f (i) ∆β, Cxc = f (i). Graphs given in Figs. 10.31, 10.32, 10.33 and 10.34 are experimental results on profile, when working in a cacade system. Drag coefficient, Cxc and the value λ are higher than the values of isolated profile (both for impellers and diffusers). C yc

l = 0,5 t

1,2 1,1

1,0

1,0

1,5

0,9 2,0 0,8

2,5

0,7 0,6 0,5 0,4 20

25

30

35

40

45

50

55

l  Fig. 10.33. (Cyc) = f  , β ∞  t 

60

65

70

β°∞

246

ROTODYNAMIC PUMPS (CENTRIFUGAL AND AXIAL) ∆β*

i = ±5°

2,0

5

36 33

l = 2,5 t

∆β* = 0,8∆βmax

1,

42 39

1, 0

30 27

0,5

66

24 0,5

21 18 15 12 9 6 3 20

30

40

50

60

70

80

90

β°2

l  Fig. 10.34. ∆β ∆β* = f  , β 2  t  ctg λ l = 2,0 t l = 1,0 t

0,060 0,055

l = 0,666 t

0,050 0,045 0,040 0,5

1,0

1,5

2,0

2,5

Fig. 10.35. tan λ = f (cot β ∞ and

3,0

ctg β∞

l ) t

In order to determine the correct value of hydraulic loss, exact value of Cxc must be determined. Cxc again depends upon the angle α. Since Cxc > Cxi angle ∠ic >∠ii. Hydraulic losses consist of (1) Profile losses, which purely depends upon boundary layer on the profile surface and the wake formation after the profile (2) End losses which depends on boundary layer on walls, which encloses the cascade system (at periphery and at hub) and due to the clearance between casing and impeller and (3) Secondary losses, arising due to cross flow existing at the channel passage due to pressure difference prevailing between leading and trailing edges of the blades both in axial and in radial directions. Due to the presence of casing, the flow is brought to rest at the casing surface. Centrifugal force is developed and boundary layer is increased which complicates the flow further. Boundary layer at the hub is increased. In variable pitch axial flow units, the radial clearance is increased due to blade rotation, which increases the end losses or annular losses. It is essential to bring the diffuser inlet edge very close to the impeller outlet

247

AXIAL FLOW PUMP

edge, which will reduce the profile losses due to aerodynamic wake at the outlet of the impeller, as well as shock losses at the entry to the diffuser.

10.9 (B) DESIGN OF AXIAL FLOW PUMP AS PER JOWKOVSKI’S LIFT METHOD—ANOTHER METHOD (i) For the given value of Q, H and hs the speed ‘n’ is determined as

 H sv  nQ =    10 

3/ 4

...(10.31)

 C2  Hsv = (Hat – Hvp) –  hs + h fs + s  2g   The suction head hs and frictional loss are equal to zero (i.e., hs = 0, hfs = 0), since axial flow pumps work under submerged condition. Hat = barometric pressure 10.336 MWC Hvp vapour pressure for water at 15°C = 0.336 MWC and Cs is the eye velocity = C1, the absolute velocity of the liquid at entry. Under normal entry condition C1 = Cm1 = Cm, where Cm1 is the meridional or flow velocity at entry. (ii) Suction specific speed C is selected as C = 800 to 1200 for preliminary calculation. Correct value is obtained after the design. The speed (n) is calculated as per equation (10.31), which gives a relation between (Hsv) and specific speed (ns). Prof. Suhanoff | 108 | recommends that, for cavitations free operation, the Dynamic Depression (∆h) can be expressed as  C12   w21  + n ∆h = m    2g   2g   

where,

where

C1 — average absolute velocity at inlet w1 — average relative velocity at inlet m — the experimential coefficient, which is defined as the ratio of actual velocity (C1) to average velocity (C1av ) at inlet =1.02. n — the experimental coefficient, which is defined as the ratio of actual velocity (w1) to average of relative velocity (w1av) at inlet = 0.2. The value of ‘n’ should not be less than 0.2 and it depends upon the specific speed (ns). Dynamic Depression (∆h) depends upon the impeller inlet diameter, the velocity on the blade to inlet, and suction conditions. This equation is applicable only when flow is a nonseparated flow or near to that. Under separated flow condition the coefficient ‘m’ and ‘n’ depend upon the angle of attack. In axial flow pumps, flow separation on the blade at inlet is due to pressure drop below vapour pressure. Writting down the equation between point (1) and (X) at inlet.

p1 px w 2 − u x2 w2 − u12 + Zx + x + Z1 + 1 = + 1hf(1 – x) γ γ 2g 2g since hf (1–x) = 0, Zx = Zl and u1 = ux for axial flow pumps w2 – wx2 p1 px + 1 = γ 2g γ

248


Cavitation starts when wx = wmax i.e., px = pmin.

p1 w2 – w2max pmin + 1 = γ 2g γ

Hence the above equation can be return as

The coefficient ‘∆h’ is a characteristic coefficient of dynamic depression on the profile, which depends upon the flow conditions, the form of profile and its geometrical parameters  δm f m t   l , l , l , which depend upon the location of the profile in the blade system. That’s why   cavitational characteristics depend upon the pump constructon to a considerable extent especially for axial flow pumps of high specific speed.

Thoma’s cavitation coefficient ‘σ’ is determined from the equation as σ =

ns4 / 3 and Hsv = σH 4700

from which the speed ‘n’ can be calculated. 15°

αsle

14° 13° 0,8 12° 11° 1,0 10° 9° 8°

t

1,5 0,5

7°

0,6

6°

0,7

5°

0,8 0,9

4° 3° 2°

1,0 1,2 1,4 1,6 1,8 2,0

1° 0 0,01

0,02

0,03

0,04

0,05

0,06

0,07

0,08

 fm t  α sle + 1°) Fig. 10.36. α sle = f  ,  for the profile of a cascade system—for shockless entry (α l l

249

AXIAL FLOW PUMP

Cavitation calculation also depends upon the relation between the average force to the maximum force on the impeller. i.e.

K =

pav pvmax

pv max = p1 – pmin, the maximum pressure depression on the profile, when compared to the pressure on the profile. 2

 w1  C  C  p = = 0 – hs –  1  = Hsv –  1  and pav = γ Cy  . γ γ γ  2g   2g   2g  Knowing the lift force of the blade, the area of the blade for cavitation free operation can be calculated as

pv max

p1 – pvp

Z1 =

2

2

A ∆r pav

K is a function of profile form. As per Suhanoff K = 0.65, as per Rudinoff K = 0.55. K ≈ 0.9 for low specific speed pumps and K = 1.67 for profiles developed by Moscow Power Institute, Russia. The value of K >1 indicates that the load on the pressure side (concave side) of the profile is more. (iii) Impeller diameter D = K 3 Q n , where K = 4.5 to 5.4 and sometimes up to 6. For axial flow pumps, higher value is selected. Value 6 is selected under special circumstances. If the value K is small, cavitation effect will be earlier due to smaller eye diameter which leads to higher flow velocity at inlet. Outer diameter is always selected for economical flow velocity i.e., as minimum flow velocity as possible to reduce the profile losses and cavitation. At the same time, higher value of outer diameter increases the overall size of the pump. (iv) Hub diameter dh is taken as dh = 0.35 to 0.6 Di for ns = 1100 to 800. However, hub diameter should be selected to accommodate the impeller blade turning mechanism. Although cylindrical hub is normally used for pumps of higher specific speed, sometimes concial hub is adopted to get a better control on total head. Mostly the area ratio (A2/A1) = 0.85 to 0.9. (v) For better efficiency, flow velocity Cm is selected as Cm = 0.74

2gH or Cm = (0.25 ± 0.05)

ω Ri . The value 0.25 is for periphery and Ri is the selected radius. Hydraulic efficiency, η h = 0.86 to 0.89 and Impeller efficiency, ηi = 0.92 to 0.94. Hi Head developed by the impeller is calculated as Hm = and Hi = ηi Hm. Therefore, ηh H = Hm hh = Hi

ηh . η1

(vi) Calculations are carried out as per Euler’s and Jowkovski’s formula. A relation between hydraulic efficiency ηh and impeller effiency ηi is given by | 131 |

250


ηh =

1 1 + ξd

 A  C2 + ξ0  3  m 2 2g  A2  2 g

Cu22

ηi

...(10.32a)

Where ξd is the coefficient of friction for the diffuser and ξ0 is the coefficient of friction forthe outlet channel. A3 is the area of the diffuser inlet and A2 is the area of the impeller outlet. ξd = 0.36 and ξ0 = 0.17 are the experimental coefficients. Another expression is

1

ηh =

–4 3

1 + 2240 ηd ns

2 3

4 3

4 3

ηi

...(10.32b)

K a + ξ0 × 00014ns K a

where Ka is the coefficient, Ka = 0.25 to 0.37; 0.25 is for ηh = 0.94 and 0.37 for ηh = 0.91. (vii) Based on the experimental results, the angle subtended by the vane in plan should be approximately 85° for peripheral profile and 115° for hub profile. (viii) Number of blades Z = 3 to 6 for ns = 1500 to 450. (ix) If the blade curvature is too much, which normally occurs at hub sections, a flow separation occurs especially for a diffuser passage at an early stage. At the same time the blade should have a minimum curvature and should not be a straight blade. Minimum curvature occurs at peripheral section of the blade. Blade curvature must always be selected, so that correct value of Cy is attained without any flow separation. Based on the experimental results the empirical value of the relative maximum blade curvature recommended is

fm = 7% for hub and 2% for periphery. Relative maximum l

δm will be 10% at hub and 3% at periphery. The change of maximum blade l curvature and maximum blade thickness for other sections is selected such that smaller variation at the top half of the blade and larger variation at the lower half of the blade is

thickness

δm at hub is selected based on the strength requirements and at l periphery as low value as possible to avoid undue vibration as well as facility to cast in foundry. However, the danger of cavitation is more at the periphery, especially very near to axial clearance between impeller and casing. If the blade thickness is reduced too low, the force on the profile and cavitation increases steeply to a maximum and steeply decreases on both sides of the blade. If the blade thickness is increased, the suction effect (hs) reduces for a narrow range of angle of attack. If it is decreased suction effect reduces for a wide range of angle of attack. That’s why, the blade thickness must be properly selected. (x) All profiles of different sections are linked in such a way that their centre of gravity is in a radial line and passes through the axis of the rod connecting the blade and turning mechanism. This point will be mostly the centre point of maximum thickness and is usually at 0.4 to 0.5l depending upon the profile.

attained. Blade thickness

251

AXIAL FLOW PUMP

Force calculations, determination of Cy and selection of profile are carried out by the following method : Axial velocity, Cm = 0.74 2gH or Cm = (0.25 ± 0.05) ω Ri, 0.25 is for periphery and Ri is the selected radius. Tangential velocity, Cu2 =

H gH m , Hm = η and ηh = 0.85 to 0.87. u h

Pressure difference between the blade inlet and outlet

C u2 2 p2 – p1 = Hi – ∆h = 2g γ

and Hi = ηi Hm.

Axial component of the total hydro dynamic force will be

∆Rz = (p2 – p1) (2πRi), where Ri ∆r

is the radius of the streamline selected. Tangential force acting on the blade is ∆Ru ∆r

= γ (2πRi)

Cu 2 Cm g

Resultant force, ∆R = ∆r

2

 ∆Rz   ∆Ru   r  + r   ∆   ∆ 

2

 ∆Ru  Cm .  . Angle β∞ = tan–1  Angle, δ = tan–1  Cu 2  ∆ R   z   u– 2 The geometrical average relative velocity, w∞ is

w∞ =

Cm sin β∞

The deviation angle, λ = (δ – β∞). Normal force acting on the blade is dY =

cos λ dRz . cos δ

Number of impeller blades are selected Z = 3 to 6. Allowable maximum depression on the profile ∆pvmax γ

= Hsv –

C12 . 2g

Average depression will be ∆pav = K∆pv max.

...(10.33)

t = 1,6 l

t = 1,4 l

Γc 2,8

t = 1,0 l

t = 1,2 l

Γ1 2,6

252

3,0

t = 0,8 l

α = 12° α = 12°

2,4

α = 12° α = 12°

2,2 2,0 1,8

16°

16°

16°

16°

20°

20°

20°

24° 28° 32°

24° 28° 32° 36°

1,6

1,0

24°

0,8

28°

32°

20°

0,6 0,4 0

0,01

0,02

0,03

0

0,01

t = 1,6 l

2° αoc – α01

0



 l



16° 20° 24° 28° 32°

0,04

0,01

0

0,05

0,03

0,02

0,04

0,05

0,01

0

32

°

0,02

36° 0,03

0,04

0,05

fm l

 fm t  , , α  for cascade  l l 

, , α  and m2 = α 0c − α 01 = ψ 2  l

t = 1,4 l

α = 12° 16° ° 0 2 ° 24 8 ° 2 ° 32

1°

 fm t

= ψ1 

0,03

0,02

24° 28° 32° 36°

t = 1,2 l

α = 12°

16° 20° 24° 28°

t = 1,0 l

α = 12°

– 1°

α = 12°

16° 20° 24° 28°

32°

t = 0,8 l α = 12° 24° 28°

32°

32°

36° 36° 36°

– 2° 0

0,01

0,02

0,03

0

0,01

0

0,03 0,04

0,02

Γc Γ1

0,01

 fm t



 l



=f

0,02

0,03 0,04

0

0,05

0,02

0,03 0,04

 fm t  , , α  for cascade  l l 

, , α  and m2 = (α 0c − α 01 ) = f  l

0,01

Fig. 10.37

0,05

0

0,01

0,02

0,03 0,04

0,05

fm l


Γ1

0 0,01

0,04

0,03

0,02

Γc

α = 12°

28 2 4 ° °

1,4 1,2

253

AXIAL FLOW PUMP

The value of K is suitably selected from 0.55 to 1.7 depending upon the profile selected. Prof. Erimena | 31 | proved that, the pressure at the concave surface of the blade decides the load on the profile, under normal working condition. Length of the profile l is determined as l =

dY dr

l 1 and can be calculated. ∆pav t Z γ

f  Relative maximum curvature  m  is determined from experimental results approximately  l   δm  7% at hub and 2% at periphery. Relative maximum thickness   is determined as 10%  l   δm  f  at hub and 3% at periphery. The change of  m  and   between hub and periphery  l   l  are carried out as smaller at higher half of the blade and larger at the lower half of the blade. The coefficient of lift (Cy) is determined as

Cy =

dY dr

1 w∞2 2g

For the selected

Zly

fm  f t l and , the value of αsle is determined from the graph αsle = f  ,  t l  l l

(Fig. 10.36) for a shockless flow. The angle of chord with respect to the axis (u-direction) θ = β1 + α sle. The value α = θ – β∞. For the obtained value of α,

fm t , , the values of m1 l l

and m2 are obtained. The value of Cy is calculated as fm   + α + m2  Cy = 0.096 m1 100 l   I and α is calculated as

αI =

Cy 0.096

–

100 f m l

...(10.33a)

...(10.33b)

While doing so initially the value of α is taken from the calculation to find m1 and m2. Then the determined αI is used to find new values of m1 and m2. The calculations are repeated three to four times until Cy obtained from the graph is equal to previous Cy value. For the given Cy of the cascade pmin and the correct value of hs is calculated as pmin

C y  w∞  =   1.6  w1 

2

254


 n Q   and C = 3/ 4  H a – hs    10    (xi) The mean line of the profile is an arc of a circle. The radius of this arc h1s

and

 n  = 10 –    90 

R =

2

 Cm21  2g 

 w12 – pmin   2g

f l2 + m 8 fm 2

...(10.34)

(xii) The Radial clearance is 0.001 Di (should not exceed 0.25 mm). (xiii) Distance between blade outlet and diffuser blade inlet is 0.15 Di.

Diffuser Calculations (i) Absolute angle α2, tangential component of absolute velocity Cu2 and meridional velocity Cm2 at the outlet of the impeller are known, from which the inlet conditions of the diffurser can be determined. Taking Cm3 = 1.05 to 1.07Cm2, in order to account for profile thickness Cm 3 . of diffuser blade, and since, Cu3 = Cu2, tan α3 = Cu 3 l (ii) In order to get complete conversion of Cu2 into pressure, the value   is always selected t l as > 1.5. t

(iii) From the experimental analysis | 131 | it has been established that an additional angle (∆) must be added over and above 90° for the diffuser blade angle at oulet in order to make the flow tangential to the mean line and the flow can be purely axial at the outlet of the diffuser. The following table (10.1) gives the value of (∆) for the selected l/t value. TABLE 10.1 l t

0.7

1.0

1.5

2.0

2.5

3.0

∆°

20.5

19.5

16.9

11.5

12.5

10.5

(iv) The mean line of the profile is an arc of a circle. fm l (v) The blade curvature and vane solidity are selected to get proper angle of divergence t l 2ε for the flow passage between two diffuser blades and also to get constant axial velocity at all sections between inlet and outlet as well as desired velocity distribution of Cm3 along the radius before and after the diffuser. About 2ε = 6° at periphery and 8° at hub is recommended which provides constant height (H) along the meridional plane. Further, the fm curvature should be selected so as to get sufficient value of Cy under non-separated l flow condition.

255

AXIAL FLOW PUMP

(vi) Selection of number of diffuser blades is normally (Zi + 1), where Zi is the number of impeller blades. However, number of diffuser blades should be selected such that the inlet flow passage is a square. The

l l 1 – sin α3 for diffuser is determined as = . Length of the t 2 tan ε t

blade l =

∆A ∆rZC y

w12 2g

...(10.35)

From the known coefficient of lift (Cy), the profile and its characteristics can be obtained. From the profile characteristics, the pressure pmin can be found out. An approximate value of Cy = 1.65 pmin. The distance between impeller outlet edge and diffuser inlet edge is recommended as 0.15Di, where Di is the impeller diameter. Angle subtended by the diffuser blade in plan is found to increase 1.6 times at periphery and 2 times at hub than that of impeller blade for ns = 450 to 750.

10.10 FLOW WITH ANGLE OF ATTACK Indirect method suggested by Prof. Lisohen | 65 | , inspite of complicated and tedious process, gives a very good agreement between theory and experiment. Hence, this method is used only when there is an absolute necessity to improve cavitational characteristics of pumps for which entire process has to be repeated again with corrections applied to the velocity distribution and the shape of the profile already available from I set of calculation. The direct method suggested as per Lift method as well as by Prof N. E. Voznisenski and Prof. Pekin gives a flow on thin profile for shockless entry without any angle of attack [v(0) = 0]. For a flow with angle of attack these processes do not give good results especially for cavitational characteristics. For axial flow pumps, ux = u1, Zx = Z1. hf(1 – x) = 0. ∴

Then,

px w 2 − wx2 p1 + 1 = γ γ 2g 2 2 pmin w1 − wmax p1 + = γ γ 2g

Circulation Γ for a flow over a cylinder can be written as Γ = 4πa.V∞ sin α, where a is the radius of cylinder, α is the angle of attack i.e., angle between the direction of the velocity vector V∞ and the horizontal line passing through the centre of the cylinder which is the profile or cascade axis (= direction of blade velocity ‘u’), V∞ is the infinite velocity or undisturbed velocity before and after the blade. The above equation can be written for a curved plate as Γ∗ = πlw∞ sin α, where l = 4a, the chord length of the profile, w′∞ is the new infinite velocity of flowing fluid before and after the blade with an angle of attack α. Normally α is very small (< ±5°), so sin α ≈ α and hence, Γ∗= πlw∞′α. Taking L as the ratio of circulation of profile in cascade to isolated profile ...(10.36) Γc∗ = L.π lw′∞α

256


Value of ‘L’ can be found from the graph (Fig 10.8). Referring to the combined velocity triangle, (Fig. 10.5), β∞ is the angle of flow over profile for no circulation. With ‘α’ as the angle of attack, the flow will be at an angle β′∞ = (β∞ + α). The new w′∞ and w′u∞ are reduced since blade velocity u and meridional velocity Cm remain same. The procedure for the calculation of flow over a thin profile with angle of attack is as follows: Circulation Γb for a flow with angle of attack ‘α’ can be written as Γb =

2πgH m ηh zω

...(10.37a)

l

α

w ′∞

t

w∞

α β′∞

w′∞′

β∞

Fig. 10.38. Velocity triangle for a flow with angle of attack

For the selected value of angle of attack α ≤ ±5°, circulation Γ* with angle of attack is Γ* = Lπlw′∞α ...(10.37b) Values β′∞ and w′∞ are determined from the combined velocity triangle. Value L is determined from the graph (Fig. 10.8) for the given value of α. Hence, the circulation Γ without any angle of attack is ...(10.38) Γ = Γb – Γ* Geometrical average velocity from the velocity triangle (Fig.10.38) w′∞ = w∞cos α. Geometrical average angle β∞ for a flow without angle of attack is β∞ = β′∞ – α ...(10.39) From the known values of w∞, Γ and β∞, lift method or Prof. Voznisenski’s method can be applied for design of axial flow pump.

10.11 CORRECTION IN PROFILE CURVATURE DUE TO THE CHANGE FROM THIN TO THICK PROFILE Methods suggested in lift method and by Prof. Voznisenski for the design of axial flow pumps, give a thin profile in the form of an arc of circle. In real practice, blade system has thick profiles with definite thickness instead of thin profile which is called camberline in profiles. Due to this additional thickness, flow area in between two blades, in the cascade system reduces, which results in change in relative velocity from inlet edge to the outlet edge of the profile. Flow velocity and the quantity of flow also change. Correction factors are applied on blade curvature of the thin profile, designed by lift method or Prof. Voznisenski’s method, in order to overcome this drawback, and the performance of pump remains unaltered.

257

AXIAL FLOW PUMP

Blade thickness is always selected based on the strength and durability of hub section of impeller blade, where the thickness is higher and based on technology in manufacture for the peripheral section of impeller as well as for the diffusers, where the thickness is small. Prof. S. M. Beelosirkovski, Prof. A. C. Genevski, Prof. Polovski, Prof. E. L. Bloch | 9, 105 | developed method to overcome the drawback of change in performance due to the dressing of thin camber line with thick profiles. This work was reworked by Prof. A.N. Papir | 85, 86, 87, 105 | by the following procedure: Profiles in cascade system consist of: (i) diverging passage type used in mixed and axial flow pumps, where the relative and meridional flow velocities reduce from inlet to outlet and another, (ii) converging passage type used in mixed and axial flow turbines, where the relative and meridional flow velocities increase from inlet to outlet. Apart from that, hydrodynamic machines are classified as: (i) machines with high aspect ratio (l/t > 1.2 to 1.4) and (ii) machines with low aspect ratio (l/t < 0.5 to 0.7). In high aspect ratio machines, fluid velocity on the blade is practically independent of the changes in fluid velocity before the blade system. The direction of fluid velocity is practically same as the blade angle at outlet, whatever may be the circulation. In low aspect ratio machines, the fluid velocity on the blade depends upon the fluid velocity before the blade system i.e., depend upon the circulation around the profile or the load on the blade | 105 |. This means that in high aspect ratio profile system, the fluid velocity direction at outlet is independent of change in angle of attack and lift of the profile, where in low aspect ratio units, it mostly depends upon the angle of attack and lift. Based on the above factors the influence of profile dressing on a thin camber line, on pump performance is found to be a function of two factors: (1) The change in the interactive force of thick profile, when compared to that of thin profile, under ideal fluid flow conditions and (2) Effect of viscosity on velocity distribution along the profile. Prof. A. N. Papir has developed an expression l  ∆f = f  , β2  , which is given in a graphical form (Fig. 10.39). A short description is given below. t  c Outlet blade angle of thin profile under real fluid flow condition is given as cot β2 = A cot β1 + B ...(10.40)

where, β1 and β2 are the inlet and outlet flow angle measured with respect to the blade velocity ‘u’ and A and B are constants and are a function of geometrical parameters and lift in a cascade system.

1l C cos β0 4 t yi A= 1l 1+ C cos β0 4 t yi

...(10.41)

1l C yi sin β0 2t B= 1l 1+ C yi cos β0 4t

...(10.42)

1−

and

259

AXIAL FLOW PUMP

β0 is the flow angle under zero circulation β0 =

π β −α+ 2 2

C yi is the coefficient of lift under angle of attack i = 0  dcy  Cyi =   di  i =0 The coefficient A and B for a thick profile will be different and can be obtained by an approximate formula

and

l d At = A + a .   m t l

...(10.43)

l  d Bt = B + b  , α, β m t  l

...(10.44)

l δ l  Functions a   and b  , α,β  , account for the thickness of profile m = δ m, relative thickness t l t  ratio which determines approximately the change in circulation in thick profile with respect to the circulation in thin profile.

The result of Prof. Papir’s analysis is given in graphical form (Fig. 10.39), with θ2 = the outlet blade angle) where θ2 is designated as indirect blade angle in x-axis and

π − β2 ( β2 is 2

∆f in y-axis where c

fm d relative blade curvature of the camberline (thin profile) and c = m relative maximum l l ∆f 1 β thickness. The function c = tan . In the graph +ve direction is for pumps and –ve direction is for 2 2 fm =

∆f , is c independent of angle of attack i.e., not depending upon the angle of direction ∆β but depends upon only

turbines. From the graph it is evident that for high aspect ratio l/t > 1.2 to 1.4 correction factor,

∆f l the flow direction at outlet, whereas for low aspect ratio  < 0.5 to 0.7  the correction factor is c t  practically independent of outlet flow direction but mostly depends upon the angle of deviation ∆β i.e., depends upon lift force and angle of attack. For turbine cascade system the dependence with ∆β starts earlier than for pump cascade system i.e., already when l/t = 1.

10.12 EFFECT OF VISCOSITY The result of viscous effect on flow is the development of boundary layer at the surface. Under non-separated flow condition the real fluid flow is on the surface of the thick profile instead of on the

260


surface of thin profile. Normally, at inlet, the profile thickness is always more at the convex surface of the profile than at the concave side. As a result, the deviation in flow direction of viscous fluid when compared to the flow direction of ideal fluid, will be always with lesser angle of curvature i.e., ∆β is less in the cascade. Circulation in real fluid will be less than that in ideal fluid. This deviation will be larger in pump, (i.e., divergent flow) than that in turbines (i.e., convergent flow). As a result, boundary layer thickness at the convex surface will be higher in pumps than in turbines. Effect of viscosity and subsequent reduction in hydraulic efficiency can be accurately calculated from the boundary layer thickness in profile | 32, 64, 78 |. With sufficient accuracy, the effect of viscosity on circulation can be taken as | 8, 16 |. KΓ =

Γ = 0.86 to 0.93 Γth

KΓ increases when l/t increases. For pumps KΓ ≈ 0.9 and for turbines KΓ ≈ 0.95. It is essential to take Γth = 1.1 Γ, where Γ is the circulation actual calculated as per the equation (10.7).

10.13 SELECTION OF IMPELLER DIAMETER AND SPEED Flow velocity at suction eye under optimum condition is given as C0 = Cm0 = (0.06 to 0.08) as For axial flow pump, where, d =

Cm =

3

Qn 2

...(4.24)

4Q _ πD (1 − d 2)

...(10.45)

2

dh , d is the hub diameter. Combining the above two equations and rearranging D h

Q = (0.06 to 0.08)

π (1 − d 2 )3 / 2 .n.D3 4

where, n is speed in rpm. Using non-dimensional parameter KQ =

Q nD3

...(10.46) (where n is in rps) in the above

equation Q=

1 K Q nD 3 60

(

)

π  2  KQ = 60 (0.06 to 0.08) 1 − d  4  

3/ 2

...(10.47)

Taking an average of 0.066 for the coefficient, which is practically used for all pumps, _ 0.7 (1 − d 2 ) 3 / 2 KQ = For axial flow pumps hub ratio d = 0.4 to 0.6 . KQ is 0.32 to 0.54. Under maximum efficiency condition, KQ = 0.4 to 0.5.

261

AXIAL FLOW PUMP

Mostly speed is determined for better cavitational characteristics for which cavitational specific speed (C) is used C=

n Q  H sv     10 

...(10.48)

3/ 4

Expressing C in terms of K Q

60 KQ n3 D3 C=

2/3 

or

1/ 2

H  C  sv   10  nD 2/3 60 KQ1/ 3

3/ 4

 H sv   10   

1/ 2

H  C 2 / 3  sv   10  = 15.3 KQ1/ 3

≈ constants

Since, C, KQ , H sv are very nearly constant for pumps. Taking C = 1000,

...(10.49) H sv = 1 K Q = 0.45 for 10

most of the pumps and K Q = 0.6 for very high specific speed pumps, nD ≈ 8.4 for normal units, nD ≈ 7.3 for very high specific speed units (n is in rev/sec). Correspondingly uperi (= πDn) = 26 to 27 m/sec and ≈ 33 if C is higher and, uperi = 23 m/sec for very high specific speed units.

10.14 SELECTION OF HUB RATIO Free vortex design is adopted while designing axial flow pumps. Circulation and total head developed at all radii is constant, i.e., Cu r = constant. For potential flow and for normal entry Cu1 = 0. Blade curvature (β) and geometric average blade angle (β∞) increase from periphery to hub. Blade becomes a twisted blade with more twist at hub angle β∞ at hub and less at periphery. Karl Pfliderer | 97| has suggested that outlet should not exceed 90°. Based on this he gave an expression 3

 1.09ηh   _1   1 =  −     d   d  max  1+ p  max

3/ 2

1  n 2 S tgβ 0a  365 

...(10.50)

where, p is the head correction coefficient due to finite number of blades in impeller and β0a is the inlet blade angle at hub section. However, based on the experimental results on a number of axial flow pumps, Prof. Papir | 84 | has developed an expression for hub ratio selection, which is given below Specific speed,

ns =

3.65n(rpm) Q 3/ 4

=

219n(rps) Q

H H 3/ 4 π – Flow rate through impeller, Q = D 2(1 – d 2) C m. From velocity triangle, flow velocity 4 Cu 2   πd h n πd h n – = . D . = πDn.d (n in rps). Cm=  u −  tgβ∞. Blade velocity at hub section, uh= 2 60 D 60  

262


Tangential velocity Cu2 at hub, Cu2h =

Q=

gH m _ . Combining all the above equations, flow rate ηh πDn.d

_ π 3 D n (1 − d 2 ) 4

Substituting the value of unit head, KH =

 gH m  d − 2ηh π 2 n 2 D 2 d 

  tgβ ∞ 

Hm Q in the above 2 2 and unit discharge, KQ = n D nD 3

equation  gK H  π2 1 – d 2 d − tgβ∞ 4 2ηh π 2 d   Substituting this in specific speed equation

KQ =

ns =

(

219π 2

)

 gK H  (1 − d 2 )  d − 2ηh π 2 d   KH

3/ 4

...(10.51)

tgβ∞

...(10.52)

A graph KH= f (ns) (Fig. 10.40) is drawn based on the test results of different specific speeds (ns= 450 to 1600) having ηmax ≥ 85% taken from universal characteristics. The values KQ ranges from 0.4 to 0.6 in these pumps. However, KQ is taken as constant and = 0.5 for all pumps and β∞ for hub is taken as 38° although it ranges from 35° to 40°. These values are substituted in equation (10.51) and a graph dh = f (ns) is drawn (Fig. 10.41). Experimental results are also indicated in this graph. Dotted line indicates the recommendation given by K. Pfliderer | 97 |. Fig. 10.42 gives the combination of above two graphs (Figs. 10.40 and 10.41). It gives a relation dh = f (KH)opt (KH)opt 0,2

0,1

0 500

1000

Fig. 10.40. (KH)opt = f (ns)op

1500 (n s) opt

263

AXIAL FLOW PUMP

dh D

0.8 0.7 0.6 0.5 0.4 0.3

d D min as per eqn. 10.45

0.2 0.1 200 400

600

800

1000

1200 1400

1600 1800 2000 2200

2400

2600 2800

3000 (n s) 0

 dh  Fig. 10.41. d =   D  = f (ns )opt dh D 0,6 0,5 0,4 0,3 0,05

0,10

0,15

0,20

(KH)opt

as per equation as per Pfliderer

Fig. 10.42. d =

dh D

= f (K H )opt

l

10.15 SELECTION OF  t  — ASPECT RATIO AT PERIPHERY peri A major part of losses occur in impeller due to high velocity of flow and the divergent passage. Profile loss is the sum of frictional losses and losses in divergent passage. Aspect ratio plays a very important role. Frictional losses increase when (l/t) ratio increases but loss due divergence decrease. It is

264


necessary to find optimum value of (l/t) for which the sum of these two losses are minimum. Based on the equation (10.1), we can write for a divergent passage as wu2 – wu1 =

Γ ZΓ = t 2π r

...(10.53)

Bernoulli’s equation for real fluid flow through impeller passage will be ρ 2 – w2 ) = γh (wu1 f u2 2 where, hf is the combined losses in impeller passage. For constant head at all radii in impeller passage, under optimum condition

∆p = (p1 – p2 ) +

2πgH gt ( H + h f ) = ηh ωZ u

Γ=

...(10.54)

...(10.55)

Since, H = Hm. ηh = (H + hf) ηh From the cascade analysis, the force due to losses i.e., drag force ‘X’ will be X = t sin β∞ and ∆p = γt hf sin β∞ and lift will be

Y=γ

w∞ (H + hf) t + γthf cos β∞ u

w∞2 w∞2 Using coefficients X = Cx ρ l, Y = Cy ρ 2 2

l = ρ w∞ Γ

and

3 w∞2 l l w∞ = Cx wz = w∞ sin β∞, hf = Cx t 2 g sin β ∞ t 2 gwz

and

Cy = ∴ For normal entry,

...(10.56)

2 gH 2Γ 2(Cu 2 − Cu1 ) 2Cu = = ηu lw∞ w∞ (l / t ) w∞ (l / t ) h

...(10.57)

Cu1 = 0 ∆Cu = Cu2 – Cu1 = Cu2 = Cu hf =

hf H

3

= Cx

l w∞ t 2 gHwz

...(10.58)

Equation (10.58) shows that losses are the function of aspect ratio l/t and the relative velocity w3∞ . The geometrical average relative velocity w∞ will be maximum at periphery. Hence, major percentage of losses in impeller of axial flow pump occurs at periphery of the impeller passage. Losses in impeller consist of profile losses arising due to friction in impeller passages and subsequent wake formation at the outlet of the impeller cascade system and non profile or secondary losses arising out of secondary flow in impeller passage due to pressure difference between leading side and trailing side of blades as well as due to clearance between casing and impeller blades. Since, flow in impeller passage is under fully developed turbulent region, where ‘f ’ is independent of Reynold’s number, the losses depend upon Cy and Cx, a relation between Cx and Cy can be written as Cy = a C 2x + bCx + C ...(10.59)

265

AXIAL FLOW PUMP

where a, b, c are constants, depend upon the geometry of blade system. Substituting the values of Cy and Cx from equation (10.12) and rearranging hf =

1 wm

 2 gHw∞ t Cu w∞2 w∞3 l  + + b C . .   a 2 l gH 2 gH t   ηh u

Using non-dimensional coefficient, KQ =

Q nD

3

1 Cu   w∞ =  u −  cosβ = nD 2   ∞

and KH =

...(10.60)

H 2

n D2

 1 gK H  π −  2ηh π  cosβ∞ 

...(10.61)

Substituting these values in equation (10.59) hf =

 π − gK H  l 2 gK π  1 a 2 2 H +b 4 KQ  ηh π cos β∞  2π ηh  t 2ηh π cos 2 β ∞   1 gK H  l  − π +c  3 2 gK H cos β ∞  2π ηh  t 

 gK H   π − 2η π  h

2

3

...(10.62)

l  In axial flow pumps, KQ is mostly constant for all ns values. Taking, KQ = constant h f = f  , K H  t 

Differentiating the equation (10.62) up to first approximation with respect KH and equating it to zero,

_ dh f dK H

=

− Taking,

 gK H  bg − π−   2 2 π η π cos β∞  ηh π cos2 β∞ h  2ag

η2h

2

2 gK H2

L = π–

gK H 2πηh

M = π+

gK H πηh

N = π–

gK H πηh

S =

2

 gK H   gK H  π− π+    3 2π ηh   2π ηh  cos β∞ 

C

g 2 cos 2 β ∞ K H2 η2h

2

π LM

T=

⋅

 gK H  l  π − 2πη  t h 2

l   = 0 t

...(10.63)

b C

4 g 2 K H2 cos3 β∞ N a ⋅ C ηh2 π2 L2 M

...(10.64)

266


Equation (10.63) can be written as

l = −S +   t opt I

S2 + T

...(10.65)

Differentiating the equation (10.63) up to second approximation and equating it to zero

l 2 cosβ ∞ =   t opt II  π 2 ηh  1  gK  – 2  H

a c

...(10.66)

Equations (10.65 and 10.66) are more or less found to be same. From the test results of pumps having η > 85% it is found a b = 8.15 and = – 15 c c From fundamental equations u = π Dn, where n is the speed in rps

gH gnD Cu = = K D n 2π η 2 π ηh H 2 h CZ =

tan β∞ =

4nDKQ 4Q _ = 2 πD (1 − d h ) π (1 − d 2h ) 2

4 KQ CZ = C gK H  2  u− u π 1 − dh  π − 2 2π ηh  

(

)

...(10.67a)

The value KQ for axial flow pumps ranges from 0.4 to 0.6. Hydraulic efficiency, (ηh) is more or less constant for all pumps (≈ 90%). Substituting these values in equation (10.67), β∞ changes from 14° to 18°. At higher values 24° to

l 34°. Taking β∞ is constant for all ranges of KQ and KH equation (10.66) leads to  t  is directly  opt proportional to (KH)opt i.e., a straight line variation. Practically, for each value of KH , there exists a

l  range of β∞, but this variation is negligible. A relation   = f (KH)opt is drawn in Fig. 10.43. Possible  t opt variations in angle β∞ is also indicated with dotted line in this figure. It is seen that this graph coincides

l with the values of l/t of tested pumps.   can be selected from this graph to get better cavitational  t opt characteristics.

267

AXIAL FLOW PUMP

t

peri

β∞

1,4 1,2 1,0 0,8 0,6 0,4 0,2

0.05

0.10

0.15

0.20

(K H)opt

  = f (KH )opt , β ∞ at η Fig. 10.43.  t  max condition peri

l Professor Wislicenus | 133 | has recommended the selection of vane, solidity   as t r2 l Z = log ...(10.67b) t r1 2sin β∞

Prof. A.J. Stepanoff |112| has given a chart for the selection of hub ratio, aspect ratio and number of blades as a function of specific speed (ns). (Fig. 10.44). Hub ratio 1,1

30 0. 35 0.

0. 40

45

50

0.

0.

55

65

60

0.

0.

0.

1,0

2v

–

0.

4v

3v

3 –

5

2v

0.

0,7

–

0.

3v

33 –

0.

2v

0 .4

3v

3v

–

–

7 57

5 63

4v

0,6

0.

Vane spacing//tt

–

5

–

0,8

29

5

42

40

0.

0.

70

0.

0. 0,9

25

0,5

–3 v

5

0,4

4 424

495

3

565 636 706 778 848 919 990 Specific speed

vanes 2 1272 1201 1060 1131

Fig. 10.44. Hub ratio number of vanes, and l/t ratio for axial flow pumps (as per Stepanoff)

268


10.16 CALCULATION OF HYDRAULIC LOSSES AND HYDRAULIC EFFICIENCY Total hydraulic losses, (hf) is the sum of hydraulic losses in approach channel (hfa), in impeller (hfi), in diffuser (hfd) and in discharge channel (hf 0) hf = hfa + hfi + hfd + +hf 0 Relative values are hf =

hf H

=

h fa + h f i + h fd + h f 0 H

= h fa + h fi + h fd + h f 0

Losses in approach and discharge pipes are calculated as

C02 hfa = Ka 2g

Cd2 and hf(0) = K0 2g

where, Ka and K0 are the coefficients at inlet and at outlet respectively and Co and Cd are the velocities in approach and delivery pipes respectively. hfi + hfd are the losses in impeller and diffuser, respectively which are classified as losses in blade system and also called as internal hydraulic losses. Prof. Staritzky | 105, 121, 122, 123, 124 | has shown in his paper that for the given value of head coefficient (KH or ψ), the relative head loss (hf) in pump depends upon the flow coefficient (KQ or φ) and has a minimum condition. Losses in impeller are of two types: (1) Profile losses arising due to friction on the blade and wake formation at the outlet of the blade system and (2) Non-profile or secondary losses arising due to the circulatory flow in blade passage, and cross flow through clearance between casing and impeller developed due to pressure difference between convex and concave side of the blade. Non-profile losses are negligible (≈ 5% of total impeller losses) when compared to profile losses, relative head loss in cylindrical section for any radius of the impeller blade will be, h fi

C2 (1 + tan 2 β ∞i ) w∞ sin λi = = u tan β∞i (tan β∞i . cot λi + 1) u sin (β∞i + λ i )

...(10.68)

where λi average incidence angle for the blade as a whole and the blade velocity ‘u’ corresponds to the peripheral section. Similarly, relative head loss in diffuser can be written as h fd

(

)

C2 1 + tan 2 α∞d w∞ sin λd = = u sin (α∞i + λ d ) u tan α∞d tan α∞d . cot λ d + 1

(

From the velocity triangles, (Fig. 10.45) tan β∞i =

Cm 2 2Cm and tan α∞d = Cu 2 u u− 2

From fundamental equation, Cu2 =

gH m and Cma= Cmi = Cm2, Cu1d = Cu2i ηh u

)

...(10.69)

269

AXIAL FLOW PUMP

wu2

Cu2

u

w1

C4

Cm4

C2 C m2

w2

C1

Cm3

Cm1

u

Cu3

(a) Impeller

(b) Diffuser

Fig. 10.45. Flow through impeller and diffuser of axial flow pump

Using non-dimensional coefficients, KQ =

Q nD

3

and KH =

H 2

n D2

(n is in rps), the above equation

can be modified as tan β∞i = tan α∞d = Taking the value ρ =

gK H 1 − 2π 2 ηh

hfi =

8ηh KQ

...(10.70)

2π ηh − gK H 2

8ηh KQ

...(10.71)

gK H , the above equation can be written as 1  4KQ  ρ+  2  ρ π 

2

...(10.72)

1 4 K Q ctgλi 1+ ⋅ ρ π2

1  4 KQ  1 − ρ  π 2  = 1 4 KQ 1+ . ctgλd (1 − ρ) π 2

2

(1 − ρ) 2 +

h fd

Losses in the approach and discharge channel can be expressed as h f ( ap ) =

h f ( ap ) H

= ζap

Cm2( ap ) 2 gH

...(10.73)

270


hf 0 =

and,

where,

Cm(a) =

and,

Cme =

h f (e) H

= ζ0

Cm2 (0)

...(10.74)

2 gH

Cmi Ai = xi Cm(i) Aap Cmi Ai = xeCmi Ae

...(10.75)

Ai Ai = xa , = x0 , Ai , Aa , A0 are the area of flow of impeller passage, approach channel A0 Aa immediately before the impeller inlet and the discharge channel immediately after impeller exit. Cmi is the flow velocity in impeller. These two expressions can be expressed in non-dimensional form as

where,

h f ( a + 0) = (ζa χ2a + ζ0 χ20 )

8ηh KQ2

...(10.76)

g π2 K H

h f = h fi + h fd + h fa + hf 0 ; ηh = 1 – h f

It can be seen that ηh =f (KQ, KH, cot λi , cot λ d χa, χ0, ζa, ζ0 ηh ) Based on the experimental results on pumps of η > 85% the values are taken as χa= χ0= 1, ζ0 + ζa = 0.124, ηhΙ = 0.86, cot λi = 25.6, cot λ d = 46.16. Since, ηh appears in the equation on both sides, the equation is solved by trial and error method until two successive values of ηh are equal. Prof. Staritzky graphically gave this equation in coordinates KH – KQ, since all performance graph for pumps are always represented in these coordinates only (Figs. 10.46, 10.47 and 10.48). h, η 0,9 0,8 0,7

η

0,6 0,5 0,4 0,3 0,2 0,1 0

hK h0 10h a 0,5

1,0

1,5

2,0 K Q

l — ≈1 Fig. 10.46. Hydraulic loss and efficiency as a function of (KQ) h , η = f (KQ) for    t  peri

271

AXIAL FLOW PUMP

85

80

%

0,22

0

n

s

=

60

0

50

0

KH 0,24

η

H

=

0,20 0,18

100 ηr

on

m

0,16

0

0,14

120

0,12 0,10

0

0 140 0 0 16 0 200

80%

0,08 0,06 75%

0,04

70%

0,02 0

0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0 1,1 1,2 1,3 1,4 K Q

  ≈1 Fig. 10.47. Universal graph KH = f (KQ , η , ns) for   t peri

s=

00

80 10

0,10

%

0

70

50 0 n 60

85 η s=

0

0

KH 0,14 0,12

2 n s= 1

00

15

20 00

%

0,08

80

0,06

70%

2600

60%

0,04 0,02 0

00 1700

3000

3500 4000 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0 1,1 1,2 1,3 1,4 1,5 1,6 1,7 1,8 1,9 KQ

  ≈ 0.3 when KH = 1 Fig. 10.48. Universal characteristics KQ = f (KH , η , ns) for   t peri

10.17 CALCULATION OF PROFILE LOSSES USING BOUNDARY LAYER THICKNESS δ **| 67, 105, 106| 10.17.1 Notations and Abbreviations S2

B

—

∫ w( s )

ds value of integration

S1

C Cm D

— — — —

Constant of integration (with suffix) Suction specific speed (without suffix) Axial (meridional) velocity Outer diameter

272


d d G, G1 g H He hs hf

— — — — — — — —

KH

— Unit head =

KQ

— Unit discharge =

1 n p p1

— — — —

pmin

— Minimum pressure (non—dimensional)

Q q Re r s — s t u w

— — — — — — — — —

w

—

—

Z α — U β Γ γ δ∗ δ∗∗

— — — — — — — —

δ∗∗ η

Hub diameter Hub ratio A parameter depends on Re** Acceleration due to gravity (= 9.81 m/sec2) Total head of the pump Ratio of two boundary layer thicknesses Suction head Loss of head H (n — in rev/sec) n D2 2

Q (n — in rev/sec) n D3

Chord length Speed (rpm) Pressure Pressure loss to friction Discharge, flow rate Source and sink Reynold’s number Radius Distance of any point on the profile from inlet edge (s/l), Non-dimensional distance on the profile Pitch Vane velocity Relative velocity of fluid (velocity of fluid on the vane) w Non-dimensional relative velocity = w1∞ Number of vanes Absolute angle Non-dimensional velocity ratio. Vane angle Circulation Specific weight of the liquid Displacement thickness Momentum thickness

δ ** — Non-dimensional momentum boundary layer thickness l — Efficiency

—

273

AXIAL FLOW PUMP

χ ρ σ v τ

— Ratio of transition end point to laminar point on the profile — Density — Cavitation constant (Thoma’s constant) — Kinematic viscosity — Shear stress

SUFFIXES 1 2 u f lam, 1 tr, t tu, e P I, III, V i m, min

— — — — — — — — — — —

Inlet conditions to the vane Outlet conditions to the vane Tangential component theoretical (average) values Frictional losses End of laminar region, Laminar region End of transition region, Transition region End of turbulent region, Turbulent region Profile losses Streamlines from hub to periphery Any streamline Minimum conditions

Hydraulic losses in axial flow pumps are due to: (1) friction over the blades (viscous flow), and the aerodynamic wake after the blades and (2) vortices in other places such as: (a) leakage losses through the radial clearances, (b) losses in flow passages between vanes due to the pressure differences between concave and convex surface of vanes and (c) losses in the hub of the impeller due to increased boundary layer thickness (Fig. 10.49). Secondary flow through clearance

Periphery

Secondary flow through hub

Hub

Fig. 10.49. Type of secondary flow in cascade of axial flow pump

Losses due to friction on blades and due to wake formation after the blades are grouped as profile losses, whereas losses due to other effects are classified as non-profile losses or secondary losses, because of the finite length of blades.

274


Since profile losses form major percentage of the hydraulic losses (about 96%), non-profile losses are usually neglected and hence hydraulic losses are assumed to be equal to profile losses. Experimental investigations also confirm that the error in such assumption is very negligible ≈ 1% | 13, 105 |. Several methods for calculating profile losses in a cascade system of fluid machinery are available; some taking into account the profile losses in the cascade system only, | 74, 75, 114 |, and some other taking into account the profile losses in the cascade and in the wake, | 67, 92,115 |. A comparison of the above method shows that the method suggested by Prof. Loisanski gives more accurate results than other methods | 14,106,142 |.

10.17.2 Determination of Profile Losses and Hydraulic Efficiency At the outlet end of the profile, the two boundary layers, coming out from concave and convex surfaces are separated from one another up to a certain distance (Fig. 10.50) from 2 to ∞. At point ∞ these two layers join together. Neglecting the non-uniformity of the velocity distribution in this region (∞ – ∞), it can be written as δ2* =1 p2 = p2∞; δ*2 = δ2**, H2 = ** δ2 a re rm re a w o f ni su U res p

F

ti ra pa a e s e w ar lo

2∞ (K )

w

t

2

w

2∞

y

2

on

Flow at the outlet edge

1

p1∞

w

1∞

w

e

e Blade exit edge

Fig. 10.50. Flow at the outlet from the profile in a cascade system (boundary layer development at wake) p – 0,4

1

– 0,2 0 0,5

l 1,0

0,2 0,4

2

0,6 0,8 (1) Pressure side (2) Suction side Average value Instantaneous value

Fig. 10.51. Pressure distribution along the profile in a cascade system

275

AXIAL FLOW PUMP

Considering the flow of fluid (Fig. 10.7), the component of force PZ due viscosity is reduced to RZ by an amount γthf . TABLE 10.2: Values of α and

 w 2e   w  2∞

α 1 – α – He 2

α

Hcr = 1.3 0.9 0.92 0.94 0.96 0.98 1.0 1.1

0.190 0.154 0.116 0.078 0.040 0 – 0.21

δ**2 δ** 2e

1–α α 1+ 2

δ**2 δ2e

2.8

0.687 0.747 0.808 0.871 0.935 1.0 1.347

0.544 0.631 0.721 0.812 0.905 1.0 1.504 2

0.74 0.79 0.84 0.89 0.942 1.0 1.352

 we  δ**2  w  α = 1–   ** = 1 –  e   w2∞  δ 2 e  w2∞ 

Hcr = 1.3

2.8

0.718 0.769 0.823 0.879 0.938 1.0 1.35

0.663 0.722 0.786 0.853 0.924 1.0 1.45

H e +5 2

Von Karman’s momentum equation, applied to the boundary layer, can be written as w2

τ0 dw d δ** + (2δ**+ δ* ) w = ρ dS dS

...(10.77)

The expression for the complete pressure loss can be written as p′ = γhf = ρw22∞

δ2** t sin β2∞

...(10.78)

The value of all quantities in section 2–2 (Fig. 10.50) of the equation (10.78) must be expressed through the values of the outlet end of the profile. From impulse momentum equation a relation between δ**e and δ**2 can be obtained as | 143, 10 | 1–α–

 w  where, α = 1 –  e   w2∞  and, we = wcritical.

α2 δ2** 1− α He < ** < 2 α δe 1+ 2

2

and He =

δe* δe**

...(10.79)

...(10.80)

276 ROTODYNAMIC PUMPS (CENTRIFUGAL AND AXIAL)

 δ**  2 Fig. 10.52. He curve = α = f  **  for He = 2.8 to 1.3  δ2e 

277

AXIAL FLOW PUMP

From the analysis of different authors, | 14, 92 |, it is found He changes considerably. For nonseparated flow with high Reynold’s numbers and for convergent flow passages He = 1.4 to 1.3. For conditions very near to flow separation, He = 2.0 to 2.8. Square and Young | 140 | obtained, from the experiments conducted on isolated profiles with smaller relative thickness and with smaller lift coefficient, a relation between α and expressed as δ2** δ** e

 w  =  e   w2∞ 

δ2** δ** e

which can be

He + 5 2

...(10.81)

Equations 10.79, 10.80 and 10.81 were analysed for the values 0.9 <

we < 1.1 and 1.3 < He< 2.8. w2∞

(Table 10.2) From the analysis it was found that the line He = 1.3 and line He = 2.8 lie in between the lines as per equation 10.79 when α is small i.e., (we /w2∞) → 1, all lines are very close together and hence, He can be assumed as 1.4 for practical purposes. This was also confirmed by Zicin and Mologen | 143 |. Table 10.2 gives the values as per equations 10.79, 10.80 and 10.81 (Fig. 10.52 gives graphical representation of the above equations). Combining equations 10.78 and 10.81 and taking He=1.4, the pressure loss can be expressed as 3.2

p′ = γhf = ρ (w2∞

 we  δe**  .  w2∞  t sin β2∞

)2. 

p′  w  = (w2∞)2 .  e  γ  w2∞  Hydraulic efficiency of the pump (ηh) will be

3.2

hf =

ηh = ηprofile =

H H + hf

or

δe** gt sin β2∞ H − hf H

...(10.82)

...(10.83)

...(10.84)

It is evident from the equation 10.88, that the values of δe∗∗, the momentum boundary layer thickness at the outlet end of the profile, should be determined in order to calculate the loss of head (hf) and subsequently the hydraulic efficiency (ηh). A method to determine the momentum boundary thickness at the end of the profile (δ∗∗ e ) for different conditions of flow, is given below.

δ **) 10.17.3 Determination of Momentum Boundary Layer Thickness (δ (a) Basic assumptions made for the changeover from one region to another The nature of flow of fluid over the vane changes from laminar at inlet edge of the profile to transition and then to the turbulent at the end of the profile. The momentum thickness at the end of the profile (δe** ) should also be calculated according to the prevailing condition. If the flow starts from laminar at inlet stagnation point and ends with turbulent region, it is necessary to find the momentum **) and thickness at the end of laminar region (δ** l ) then momentum thickness end of transition region (δtr

278


** finally the momentum thickness at the end of the vane (δtu ) . During this process, it is assumed that at the end of

at the beginning of

region

region

δl ** (laminar) = δ tr** (transition) at the end of

at the beginning of

region

region

δtr** (transition) = δtu**

...(10.85)

(turbulant)

(b) Determination of End Points (i) End point of laminar region: The method suggested by Prof. E.M. Minski | 79 | has been – taken for the determination of the end point of laminar region (s l ), which gives the relation between this point, Reynold’s number based on inlet conditions (Rel) and the point of minimum pressure on the profile ( Sm ) , which is taken from the known non-dimensional pressure distribution curve of the profile. It can be written as — Sl

= Sm + S

...(10.86)

The graph (Fig. 10.53) S = f (Rel) is drawn from various experimental data collected by him. The Reynold’s number is calculated by the equation w .l Rel = 1∞ ...(10.87) ν and the non-dimensional pressure is calculated as 0.30

0.25

S

0.20 0.15 0.10

0.05 12 0 2 × 10

6

4 × 10

6

—

6 × 10

6

8 × 10

6

10 × 10

6

14 R** e =

16

w. v

Fig. 10.53. S = f (Re**) determination of end of laminar region (Minski)

18

20 × 10

6

279

AXIAL FLOW PUMP

p =

u 2 − w2 2 gH

...(10.88)

(ii) End point of transition region: Zicin and Mologin | 143 | suggested a relation between the end point of transition region (Str) and that of laminar region (Si) as a function of pressure ratio, ( p ). It is written as  Str  χ=   =f(p)  Sl  p p = m = p0

pmin + 1+

( p1 / γ ) w21∞ / 2 g ) ( p1 / γ )

...(10.89)

( w12∞ / 2 g )

p1 C2 = 10 – hS – m , pmin = 1– w2max γ 2g From the experiments of Zicin and Mologin at different conditions, a relation betweem pm and χ is given (Fig. 10.54), from which χ can be determined and hence Str . 3 x

4

2.5 3

2

1.5

1 2

1.0 0.4

0.45

0.5

0.55

0.6

0.65

0.7

pm 0.75

0.8

0.85

0.9

0.95

1.0

Fig. 10.54. χ = f (pm) determination of end of transition region –

(1) Concave surface—diverging passage (3) Convex surface—converging passage

(2) Convex surface—diverging passage (4) Concave surface—converging passage

(c) Determination of momentum thickness for different conditions of flow ( δ** ) Substitutitng these parameters into the impulse momentum equation (10.77) Taking, and

f= φ=

w′δ** G w τ0 ρw 2

G

...(10.90a) ...(10.90b)

280


wδ** . ν Assuming that ‘f ’ is a function of velocity distribution on the profile, and the pressure gradient, influences

where, f is the form factor, φ is the frictional coefficient, G is an expression depending on Re** =

‘f ’ only through G which depends only on Re** . The expression for impulse momentum equation can be rewritten in the following form | 14, 143, 67, 99, 135, 142 |. F ( f ) = a – bf where, a and b are constants depending upon the type of flow. For laminar flow, a = 0.44; b = 5.8; For transition flow, a = 0.9; b = 6.5; For turbulent flow, a = 1.17; b = 4.8 if Re > 5 × 106 a = 1.25; b = 4.8 if Re < 5 × 106 Combining 10.90 and 10.91, we get Re** Taking,

a wδ** = = νG w(b − 2) ν

S

(b −1)

∫w

...(10.91)

. dS

G1 = G . Re** G1 =

a ν w(b −2)

...(10.92)

0

...(10.93) S

∫w

(b −1)

. dS

...(10.94)

0

The expression for δ** can be obtained depending upon G1 for each region of flow. (i) Laminar region: For laminar flow as per Prof. Loisanski | 67 | G = R** e ν G1 ...(10.95) w The above equation has been rewritten substituting its values to suit the present work | 105, 106 |

and hence,

δ** =

 0.44 δ** =  5.8 wl Rel 

Sl

∫w

4.8

. dS

...(10.96)

0

Rel** = wl δl ** Rel

...(10.97)

where, Rel** is the Reynold’s number based on δ** l at the end of laminar region. (ii) Transition region: Zincin and Mologen | 142 | has determined a relation for G as G = 1259 ( Re**) – (1/10) Correspondingly

δ** =

ν  G1  w 1259 

...(10.98)

(10 / 9)

...(10.99)

281

AXIAL FLOW PUMP

The above expression is transformed into a convenient form for the present work as | 105, 106 |  Rel =  1259 wtr 4.5 

** Retr

(10 / 9)

 St   5.5 0.9 ∫ w . d S + Ctr     Sl  

...(10.100)

where, Ctr, the constant of integration, is determined from the parameters at the end of laminar region as per the condition 10.79 Ctr = wl 5.5 . δl ** . Gtr (

...(10.101) )

( ) −1/10

Gtr = 1259 Rel**

and

...(10.102)

The value of δtr** will be

δtr** =

Retr** Rel wtr

...(10.103)

If the flow starts from transition region directly Ctr = 0. (iii) Turbulent region: For turbulent region, as per Prof. Loisanski | 67 | suggested that For Re< 5 × 106

( )1/ 4

G = 79.5 Re** δ**

and

ν  G1  = w  79.5 

...(10.104)

(4 / 5)

...(10.105)

If, Re > 5 × 106 1/6 G = 153.2 (R**) e

and

δ**

...(10.106)

ν  G1  (6 / 7) = w 153.2 

...(10.107)

Above expressions are rewritten to suit the present work | 105, 106 | For Re > 5 × 106 Stu =1

4/5  w3.8 . d S + Ctu  Re** = ...(10.108) 2.8  1.25 ∫  79.5 wtu  Str  where, Ctu is the constant of integration, and is determined from the condition (equation 10.85) and calculated from the conditions at the end of transition region as

Re1

and

Ctu = w3.8 . δtr** . Gtr

...(10.109)

** 1/ 4 Gtu = 79.5 ( Retr )

...(10.110)

** Retu δtu** = R w el tu

...(10.111)

282


=

2°

m+1

n

m

=

4 m

=

5 0.9

0.9 =

0.8

0.8

0.7

0.7

0.6

0.6

log G 1

n

3

0.5

0.4

0.4

m

=

4 m

0.3

=

5 m

=

6 0.3

–

m 0.2

0.2

n

=

1 n

=

2 n

=

3 0.1

0.1

m

n

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

n+1

log (R** e )

( )10

** Fig. 10.55. log G1 (Re**) = f (log Re**) for Gtr = 1259 Re

9

If, Re > 5 × 106   Stu =1  Rel   3.8 Re** =  1.17 ∫ w . dS + Ctu   153.2 w2.8   Str  

Ce = wtr 3.8 . δ** . Gtu tr where, and

Gtu = 153.2 ( Retr** ) 1/ 4

δtu**

** Retu = Rel wtu

6/7

...(10.112)

...(10.113) ...(10.114) ...(10.115)

283

AXIAL FLOW PUMP

3.0 4 2.9

2.8

log G 1

3 2.7

2.6

2

2.5 1 2.4

2.3

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

log R** e

(

Fig. 10.56. log G (Re**) = f (log (Re**)) for Gtu = 153.2 R** etu

)6

1

Entire procedure for the calculation of profile losses has been tabulated and given in chapter 13. This procedure has been experimentally verified with an actual pump test. Hydraulic efficiency actual is 1% less than the hydraulic efficiency calculated as per the procedure given above.

10.17.4 Computer Programme A computer programme in C++ has been developed to determine the profile losses using above mentioned equations and is given in chapter design with an example.

10.18 CAVITATION IN AXIAL FLOW PUMPS In Fig. 10.57, theoretical pressure distribution over a peripheral section profile of an impeller blade is given for one value of Q, H, n, hs (or Hsv), which corresponds to one point on the graph Q, n, H, η, N = f (hs or Hsv) (Fig. 11.3), under non-cavitational condition. Under constant H, Q, n, N, η, any change in hs changes the overall pressure distribution from inlet to outlet of the pump. Since, total head H remains constant, change in hs, increases the suction head and reduces the delivery head. A part of delivery head is transferred into suction head. From the figure, it is seen that a sudden reduction in pressure prevails at the suction side of the profile very near to the blade inlet, more or less at one point.

section III P

r = 168 mm

0,9

P 0,7

0,8

0,7

0,5

0,6

0,6

0,4

0,5

1,1307

0,4

2,1160

1,5850

0,5

0,2 0,2

+5

–2 5 2

–1

0

0,4

–5 2

0,4 –3

0,8

0,6

0,7

+1

2 +2 +3

–3

–2

–1

0

section V P 0,9

0,8

0,9

+1

+ 25 +2 +3

0,4

0,2 –

–0,1 –3

5 2 –2 –1

0

–0,2

–0,1

–0,2

–0,3

–0,2

–0,5 –0,6

–0,4 –0,5 –0,6 –0,7 –0,8

–0,7 –0,8

–0,3 –0,4 –0,5 –0,6 –0,7 –0,8

as per original selection of γ(s) after redistribution of γ(s)

Fig. 10.57. Pressure distributions along the profiles at sections I, III, V

+1

+2 +3


–0,4

+ 52

0,4

–0,1

–0,3

2,1160

1,5850

1,1307

section I

284

r = 122.8 mm

r = 145.4 mm

285

AXIAL FLOW PUMP

Since this sudden decline in pressure is acting only on a very small area of the blade, total head or lift on the profile is not affected. When hs is further increased, the low pressure acting area at the suction side of the blade gradually increases but it does not affect lift on the profile.The total head remains constant. Further increase in suction head reduces the delivery pressure below the required pressure to provide the required total head, with the result, the total head reduces. Reduction in pressure at suction side increases the relative velocity. Since outlet velocity remains same, the diffuser effect of the passage increases, which in turn increases frictional losses, due to the high velocity and eddy losses due to increased diffuser effect. Efficiency also reduces. In centrifugal pumps, the rate of drop in pump parameters is almost sudden. When specific speed of pump increases the rate of drop is less. In axial flow pump, this drop will be gradual (Fig. 10.58) i.e., complete flow separation takes place in low specific speed pumps, whereas partial separation only takes place, at higher specific speed pumps. This is due to the fact that even under cavitation in high specific speed pumps, lift force exists and hence partially head is developed. Pumps working under such condition are called super cavitating pumps. 190 180 170

Q

η,% 80 70 60

H,M 1,2 1,1 1 1,0

7 3

2

4

6

5

8

9 10

1

2

3

4

5

6

7

8

11 H s(m)

Fig. 10.58. Cavitation test results on an axial flow pump

Always pumps design is carried out for a non-separated flow condition, since flow passage is a divergent passage. Experiment shows that separation of flow exists in pumps. Increase in the length of profile although reduces the diffuser effect but increases friction. So an optimum profile length for minimum loss must be established. A systematic analysis conducted by Prof. Howell, | 42, 43 |on diffuser l type cascade system enables to select optimum value of   as a function of outlet blade angle ‘β2’. In t axial flow pump relative velocity ‘w’ is higher at periphery. Impeller friction losses are more, cavitation effect is more predominant at periphery. l  l   is selected as per Howell’s graph (Fig.10.34) and   t peri is selected from A.N. Papir’s t hub l graph. (Fig. 10.43)   for intermediate radii are selected by interpolation in order to get a smooth t change over from hub to periphery. Circulation Γ and the total head are proportional to the angle of deviation of flow in impeller blade ∆β = β2 – β1 Γ = (Cu2 – Cu1) t = t Cm (cot β2 – cot β1)

286


More the value ∆β, higher the value of total head. But increase in total head, increases the angle of divergence of flow passage. In order to provide a flow without separation, divergence should be limited

l must be selected based on to less than 10°. Correspondingly, ∆β must be reduced. That’s why,    t  hub ∆β. Experimental investigation shows that a perfect non-separated flow can be achieved when ∆βnom= 0.8 ∆βmax, where, ∆βmax is the value given by Prof. Howell, ∆βnom can be calculated from the geometric parameters of the pump. Howell’s graph can be used under fully turbulent condition of flow where frictional coefficient ‘f ’ is independent of Reynold’s number Re i.e., w1l ≥ (2 to 2.5), 105 ν Thoma’s constant σ is determined as

Re =

σ=

Critical value,

σcr =

( wx2 − ux2 )

...(10.116)

2g

( wx2 − u x2 )max 2 gH

=

2 wmax − u2 2 gH

...(10.117)

‘u’ is constant in axial flow pump for the given streamline from inlet to outlet. Normally for axial flow pump   is selected <1.0 to 1.1, taking into consideration, profile curvature and maximum  t  peri

l

thickness under cavitation free operation. Maximum relative velocity on the profile under zero angle of attack is given by Prof. A.A. Lomakin | 70 | as 4 dm  wmax = (1 + β)  1 + π l 

  w∞ 

...(10.118)

where, dm is the maximum profile thickness and β is the angle between chord and the tangent to the curvature at the outlet. Experimental results confirm the validity of this equation even up to 1.5 to 2° of angle of attack. For higher angle of attack, σcr increases proportionately. Cavitational characteristics is always determined by (l/t), value at periphery. Since, maximum thickness (dm) the angle of deviation ∆β and (l/t) at peripheral section are smaller than these values at other sections, the profile can be assumed as equivalent to a thin straight plate. The coefficient of lift for thin plate can be written as β  Cy = 2 π sin  + α  ≈ πβ 2 

Total head,

H=

uC y lw∞2 sin (β∞ + λ ) 2 gtCZ cos λ

...(10.119)

...(10.120)

287

AXIAL FLOW PUMP

4nDKQ 4Q = π(1 − d 2 ) πD 2 (1 − d 2 )

For velocity,

Cm =

Relative velocity,

1  gK H  w∞ = nD  π −  2πηh  cos β∞ 

...(10.121)

...(10.122)

From the above equation, it is seen that σcr = f (D, d , β∞, λ, —

l , n, KH, KQ ) t

—

For axial flow pumps, KQ= 0.4 to 0.6, d = 0.4 to 0.6, KH = 0.04 to 0.22 Angle β∞ is a function of KQ for constant values of KH . Substituting the values of KQ and KH in the equation (10.61), (β∞)av for peripheral section is found to be (β∞)av ≈ 20° for KH = 0.22 to 0.15 (β∞)av ≈ 23° for KH = 0.09 to 0.08 (β∞)av ≈ 26° for KH = 0.055 to 0.045 Substituting equations (10.113), (10.114), (10.115) and (10.116) in equation (10.110) and the l  value β∞ from the above determined value, a relation between σcr= f  , K H  is established taking an t  average value of KH = 0.09 to 0.08 for all the three values of β∞. 2  l 0.555  l 0.7351 K 0.2419 σcr = +   + H    t t KH

...(10.123)

Constants can be determined if necessary for each value of β ∞ . Taking into consideration

l already established relation   = f (KH) (Fig. 10.43), A graph σcr= f (KH) is drawn (Fig. 10.59).  t  peri σ 4,0

3,5 3,0 2,5 2,0 1,5 1,0 0 0,05

0,10

0,15

0,20

Fig. 10.59. σ = f (KH) for axial flow pump

0,25 (KH)opt

288


Actual test results on different axial flow pumps are also marked in this graph, which shows that

l

should be theoretical equation coincides with actual results. However, it is also found that    t  peri

l

selected a little more than the value obtained from the graph   = f (KH) for better value of σcr. But  t  peri maximum efficiency will shift to higher side of head (lower side of quantity) from the optimum value.

l

= f (KH). The amount of shift in efficiency can be found from the graph    t  peri While designing the impeller blade, apart from peripheral section, other sections must also be verified with equations (10.113), (10.114), (10.115) and (10.116) (Fig. 10.59) and corrected if necessary so that cavitation characteristics of the pump at all sections are within the allowable range. A.J. Stepanoff | 112 | has recommended the selection of l/t, d and number of impeller blades for axial flow pump based on specific speed. This recommendation can also be used for axial flow pumps Fig. 10.34 (b) l/t, zi, d = f (ns) for axial flow pumps as per AJ. Stepanoff. Moscow Power Institute, Russia has recommended a relation between (σ) and ns as σ =

( ns ) 4 / 3 4700

based on the anticavitating property at inlet.

10.19 RADIAL CLEARANCE BETWEEN IMPELLER AND IMPELLER CASING dm

Mutual interaction between two successive impeller blades, when they rotate inside a housing, having a clearance between Radial Blade impeller and impeller housing is of clearance + importance due to the presence of cavitation in clearance. Pressure difference between δ convex and concave surface of the blade casing induces a flow through the radial clearance. The disturbance created by one blade may Fig. 10.60. Erosion and cavitation at radial clearance induce further disturbance in the following blade, since all the blades are moving at constant angular velocity. Due to high velocity of flow of fluid through the clearance, metal erosion and clearance cavitation occurs. Increase in radial clearance increases cavitation and also increases end losses i.e., hydraulic losses. Experiments conducted by different authors |84, 88| show that this radial clearance (Fig. 10.60) recommended as δ = (0.10 to 0.15)l, where l is the chord length of the profile. Some authors recommend 0.001D, where D is the impeller diameter. In multistage units, however, this radial clearance should accommodate technological considerations in manufacture and assembly of pumps. Minimum requirement for technological consideration will be δ the radial clearance should be 0.15 to 0.2 mm independent of pump size. In multistage units, where diffuser of the previous stage comes before the second stage impeller. The clearance is doubled i.e., δ = (0.2 to 0.25) l or 0.002D.

289

AXIAL FLOW PUMP

However, it should be remembered that provision of guide vanes at approach pipe i.e., before the first stage impeller reduces the performance both in energy and cavitation. 0.7

maximum efficiency, n

0.6

1

0.5

2 s

0.4 0.3

dh D 0.2

0.1 0

0.002 0.004 0.006 0.008 0.010 0.012 relative gap width s/d 2

Fig. 10.61. Recommended radial clearance in axial flow pumps

10.20 CALCULATION FOR AXIAL FLOW DIFFUSERS In axial flow pumps, the inlet conditions of diffuser blade depends upon the outlet conditions of flow from impeller blade of the pump. At the outlet of the diffuser, the flow is always axial. The kinetic energy of the fluid coming out of the impeller is converted into pressure energy and the spiral motion of fluid at the outlet of the impeller is brought to pure axial motion. Cm6 = C6 and Cu6 = 0 The outlet axial velocity (Cm3) from the impeller will be equal to the axial velocity at the diffuser inlet (Cm4) before the inlet edge. Cm3 = Cm4 On the blade at inlet of the diffuser the axial velocity Cm5 will be more than Cm4 due to vane thickness. Taking thickness factor ‘Kd’ for the diffuser inlet Cm5 = Kd Cm4 The tangential component of absolute velocity of fluid at the outlet of the impeller (Cu3) is equal to the tangential component of absolute velocity at the inlet of the diffuser (Cu4) before and on the diffuser blade inlet. Cu3 = Cu4 = Cu5 Due to the absence of the blade velocity u4= u5 = u6, combined velocity triangle for a diffuser will be as follows (Fig. 10.62).

290


C1′

r

C 2 = Cz

C ′∞

r

∆β′

β′2

β′∞ β′1

C ′u1

Fig. 10.62. Combined velocity triangle for diffuser

From the velocity triangle (Fig. 10.62), C∞5 =

C  Cm25 +  u 5   2 

2

2Cm5 tg α∞ = C u5

and

Applying the basic theory of cascade system to the diffuser blades, the parameters and coefficient of lift for hub and periphery sections are determined. Applying Jowkovski’s theorem for diffuser profile (Fig. 10.63) Rud = ρCm∞ Γ1d = ρCm∞d tCu5 ...(10.124) Referring Fig. 10.63, yd Rd = cos λ and Rud = Rd sin (α∞d + λd) d

R ′v

x′

β′∞

λ′ β′∞

R ′Z

R′ y′

Fig. 10.63. Forces acting on axial flow diffuser

291

AXIAL FLOW PUMP

Rud = yd

sin (α∞d + λ d ) cos λ d

= ρcycd C2∞d ld

∴

Cycd

....(10.125)

sin (λ ∞d + λ d ) = ρCmd∞ td Cu5 cos λ d

2 sin 2 α ∞d cos λ d Cm 5 ZCu 5 cos λ d ld = C = sin (α + λ ) td ∞d d ∞d sin (α ∞d + λ d )

...(10.126)

From velocity triangle (Fig. 10.62) for hub section, Cm 5 tan α5 = C and α6 = 90° u5

∴ ∆ α = α6 – α5 = 90° – α5 From the values ∆α and α6 from Howell’s diagram (Fig. 10.34) the l/t value is determined. Taking λd = 0 as first approximation Cyd is determined from graph (Fig. 10.34). Taking (Zd) number of diffuser vanes, the length of profile (ld) is established. Number of diffuser vanes Zd is normally selected as (Zimp+1), and λd ≤ 0.2D, similar to that equations (10.9) and (10.11) for impeller blades, Hydraulic loss for a diffuser will be determined as

tan λ d C ycd C 2∞d . ld xd C∞d ρC xcd . C∞2d ld hfd = γt C = = 2γtd sin α ∞d 2 g sin α∞d td d md

...(10.127)

Since α6 = 0, the diffuser blade length will be longer than impeller blades and also due to more number of blades, hydraulic loss in diffuser will be higher than that in impeller. Based on experimental results, circulation Γd in diffuser will be normally (0.75 to 0.8) times of impeller circulation. This in turn reduces hydraulic loss and compensates for the increase in hydraulic loss due to increased length. A better diffuser design will be when (α∞d) between hub and periphery is 5° to 6°. (l/t) for diffuser under normal value of α∞d can be reduced by increasing the pitch of the blade about 25%. This can be achieved by reducing number of diffuser blades.

10.21 AXIAL THRUST Axial component of the hydrodynamic force acting on the blade (PZB) can be determined from Jowkovski’s theorem (Equation 10.5). Elementary axial force (∆PZB) acting on elementary strip of ‘dr’ at a radius ‘r’ can be written as ∆PZB = Z ρΓ1 w∞u dr R

Total force,

PZB = Ζ Γ1ρ

∫ w∞u dr

rh

292


r Mu Pu

dr

Pz

P

r Pz R z

l

b

α

Mu

Mz

c

a y

u

Pu

Mz

Fig. 10.64. Axial thrust on axial flow pump

From velocity triangle, w∞u = u – Cu2 =

Cu 2 2

gH m uηh

Z Γ1ω 2πg

and Hm =

Combining all the equations and integrating for the entire blade PZB

= ∫ ∆Pz =

=

πR2 γ

= πR2 γ

2πγH m ηh

R



∫  r − η

rh 

gH m  2   dr h 2ω r 

2 H m   rh  gH R 1 −   − 2 2m − ln  ηh   R  r  ω R ηh

gK 1 Hm  − 2 − H2 − ln  1 − d dh  ηh  ηh π

...(10.128)

Axial force between hub diameter and shaft diameter PZh = π (rh2 – rs2 ) γ

Hm ηh

Since, rS << rh it can be neglected. PZh = πr2h + γ Total axial force,

Hm ηh

PZ = PZB +PZh = π R2 γ

...(10.129) Hm ηh

 gK H 1 − ln  1 − 2 dh   ηh π

...(10.130)

11 TESTING, PERFORMANCE EVALUATION AND REGULATION OF PUMPS 11.1 INTRODUCTION A graphical representation of the relation between basic parameters of pump is called characteristics of pump. Fluid flow through pump passages at non-optimum region is complicated. All available analytical methods, to determine the correct pattern of flow passages do not give exact solutions. Only by the analysis of experimental results, flow pattern can be determined. Inspite of this, the exact relation between the parameters of pump is not available. Most useful method of understanding the interaction between pump parameters and further analysis of the flow pattern is by a graphical representation of the relation between the parameters. Pump characteristics are represented by total head (H), power absorbed N, suction condition HSV (σ (or) ∆h (or) C) and efficiency (η) as a function of flow rate (Q) at rated speed (n). η, H, N, HsvC (or σ) = f(Q, n) These graphs are also presented in non-dimensional parameters such as unit head (KH) and unit power (KN) as a function of unit discharge (KQ). More detailed analysis is also carried out by graphical representation of non-dimensional parameters viz. suction characteristics (σ) and efficiency (η) as a function of (KQ) i.e., KH , KN , η, σ = f (KQ, n).

11.2 PUMP PERFORMANCE—RELATION BETWEEN TOTAL HEAD AND QUANTITY OF FLOW Total head

Hth = (1 + p)Hm = (1+ p)

Ha

hh

=

(1 + p ) (Cu 2u2 − Cu1u1 ) . ηh g

For normal entry Cu1 = 0 and from velocity triangles. (Fig. 4.1) Cu2 = u2 – Cm2cot β2 = u2 – H th = Hm = 1+ p

Q cot β 2 πDB

 u22 Qu2 cot β2   g − πD B .g    2 2 293

...(11.1)

294


Diameter (D2) and breadth (B2) are constants for the pump to be tested. Equation (11.1) represents a drooping down straight line characteristics with Hm in y-axis and Q in x-axis. If the pump is tested at constant speed ‘n’, u2, the blade velocity is also constant, since D2 is constant. H–Q graph is a straight

H line inclined at an angle of β2 with respect to y-axis, since cot β2 = (Fig 11.1). Maximum head is Q 2 u2 , when Q = 0. Hmax = g Th eo (H

β2

–Q )c urv

n

e

C

B Cm α Q

Cu α H

H th

D

E

1. (H–Q) Theoretical infinite blades 2. (H–Q ) Theoretical-finite blades 3. (H–Q ) actual 2 h 11—frictional losses α Q h 12—secondary losses

β2

u2 g

Hm H

H th

2

1 u 0.586 g

2

Ha h f2 Q

3

hf1 3′

Fig. 11.1. Theoretical (H–Q) curve developed from outlet velocity triangle

If blade angle β2 is changed, keeping the speed ‘n’ constant, the slope of the H–Q line also changes (Fig. 11.2). Drooping down H–Q characteristic prevails so long β2 < 90° since cot β2 is positive. When β 2 = 90°, H–Q curve becomes horizontal and parallel to x-axis, since cot β 2 = 0. When β2 > 90° the H–Q curve slope is towards upward direction since cot β2 is negative. β2 < 90 is used for pumps and blowers. β2 = 90° is used for air compressors.

295

TESTING, PERFORMANCE EVALUATION AND REGULATION OF PUMPS

H

H

β2 = 90°

β2 < 90°

A D

D

H th = f(Q)

F

B H th = f(Q)

A F

H a = f(Q)

η.max

H0

N

E

B

η = f (Q)

H a = f(Q) QF

E

QN

C Q(w 2)

Q(w 2)

(a)

(b) H

B

β2 > 90°

H T = f(Q)

A F D

E H a = f(Q)

Q(w 2)

(c) Fig. 11.2. Theoretical and actual (H–Q) curve for different β 2 values at one speed

Although hydraulic efficiency (ηh) in equation 11.1 is assumed to be constant, practically it is changing. ηh is maximum only at the point where Hydraulic losses are minimum. This point is called maximum efficiency point. The head and discharge of pump for which it is designed will be at this maximum efficiency point. At this point, profile losses and secondary losses are minimum. At all other point of operation, secondary losses and shock losses increase which increases total hydraulic losses and hence reduces hydraulic efficiency. At low flow rates, recirculation losses also prevail and it increases when the flow rate ‘Q’ is further reduced, as a result, the theoretical H–Q curve which is a straight line changes to a drooping down curve (Fig. 11.2). Hydraulic losses are in general proportional to the square of flow rate. That’s why, actual H–Q curve is an approximate parabola. Actual pump characteristics consisting of total head (H), input power (N) and efficiency (η) as a function of flow rate (Q) for one speed (n), and for one value of β2 < 90° is represented in a graphical form (Fig. 11.3).

296


n = 2900 rpm D 2 = 236 mm

HM N hp 18 17 16 15 η% 60 50

HT

14

N

13 12

40

11

30

10

20

9

10

8

η

H vac

H vac

8 7

Reserve

7

6

H vac

6 1 0

2 5

3 10

4

5 15

6 20

5 lit/sec 7 8 9 10 3 Qm /hr 25 30 35

Fig. 11.3. (a) Actual pump performance H, η , N, Hsv = f (Q) for one value of outlet blade angle β 2 and one speed ‘n’ H, m 80 1

60

2

η 3

4

40 H

20 0

N hs m 6 4

hs

5

η, % 90 80 70 60 50 40 30 20 10 0 400 200 0

2 0 100

200

300

400

500

600

700

800 Q, lit/sec

Fig. 11.3. (b) Performance of centrifugal pump D = 700 mm, n = 960 rpm

In Fig. 11.4, the performance variation of H–Q, η–Q and N–Q for pumps of different specific speeds is given.

297


100 1

max

75

%η/η

1

50

4 6

5

7

7

6

25 0 250 7 200

6

%N/N

BEP

150 1 100 50

5 4 3 1

6 7

2

0

7

180

6

160

5

BCP

140

%H/H

5

4 3

120

2 100 1 80 60 0

25

50 75 % Q/Q BEP

100 125

150

Fig. 11.4. Performance of pumps at different specific speeds (ns)

(1) – ns = 65, (4) – ns = 210, (7) – ns = 650.

(2) – ns = 105, (5) – ns = 280,

(3) – ns = 155 (6) – ns = 400,

Referring to Fig. 11.4 it is seen that efficiency, η = f (Q) is not constant. It increases from zero flow rate to a certain value of Q and then decreases for further increase in flow rate. Head and quantity at maximum efficiency point is the best operating point for the pump. However, for safe operation, pump is operated at 90 to 95% of the maximum efficiency point.

298


Since the blade velocity u α Dn, for the given blade angle ‘β2’ of the pump, and if the diameter of the impeller ‘D2’ is increased, keeping the speed ‘n’ constant, H–Q curve shifts parallel upwards with respect the original (H–Q) curve and vice versa. The same pattern exists, if speed is increased keeping diameter constant. [Fig. 11.6 (a) and (b)]. Based on the model analysis and the loss analysis, an expression between total head (H), flow rate (Q), and speed (n) can be written as H = An2 + BNQ + CQ2

...(11.2)

This graph H = f (Q, n) will be a hyperbolic paraboloid with main axis coinciding with H0-axis and the peak at the origin. The symmetrical plane passing through H0-axis makes an angle φ with the plane (Q, H) (Fig. 11.5). B A+C

(H ,

n)

tan 2φ =

M

Hc

H c = f(Q)n = c H0

H b = f(Q) n=b

Hb

Ha B (Q

, n)

Qc

n=

c

n=

a

Qa φ

Qb

=a n=

f(Q ) n

b

Ha =

0 Q0

Fig. 11.5. (H–Q) curve for different speeds (3D diagram)

In Fig 11.5, a number of H–Q graphs for different speeds n = a, b, c are drawn. Each H–Q graph is a parabola for one value of speed ‘n’. Entire space diagram lies in the first quarter of H–Q graph. Line OM is a parabola and is obtained from the hyperbolic paraboloid of the symmetrical plane formed by the plane (Q–H) with an angle φ. Resultant section of the hyperbolic paraboloid consists of n = constant i.e., n = a, n = b, n = c and corresponding H–Q curves Ha = f (Qa) Hb= f (Qb) and Hc= f (Qc) etc. The plane H = constant is a hyperbola. When H = 0, Q = Qmax. The plane changes instead of hyperbola, into a straight line coinciding with asymtode and passing through origin (line OB in Fig 11.5). In Fig. 11.5 the 3D space diagram is

299


shown representing different (H–Q) curves for different speeds ‘n’. The same pattern of change prevails when diameter (D2) is changed at constant speed n [Fig. 11.6 (b)]. A set of congruent curved parabolic lines, (H–Q) curves, each for one speed n = a, b, c, are drawn in Fig 11.5. The line OM joining all peak points of (H–Q) curves in (Fig. 11.5) is same as OM in Fig 11.6. Hence, H–Q curve for any speed n = n can be drawn from the known H–Q curve for one speed from model analysis. Congruent property of the lines is maintained under non-separated flow and non-cavitating conditions.

20

0,10 0,20 0,30 0,4 0 0,5 0 0 ,6 0, 6 0 5 0,7 0 0 ,7 0, 5 0, 78 0,880 2 0,84

H,m 30

15

n 1 = 750

25

η=

0 n 2 = 600

10

n 3 = 500

5 0 0

n0 = 9

20

40

60

80

60 M r pm

5 0 . 8 84 0, 2 0 ,8 0 0 ,8,78 0 ,7 5 0 0 ,7 0 65 0, 6 0 0,

100 120

140 160 180

200

0,60 Q,lit/sec

(a) Centrifugal pump

Fig. 11.6. (a) Performance (H–Q) change due to speed change ‘n’ for the same diameter D2. Universal (H–Q) graph with equal efficiency ‘O’ curves Hm

η = 76 77 7 8 89

50

80 81

81 8

07 9

78

40 30

76

%

∆h

av 2

20

∆h

av 3

10 0

5

10 15 20 25 30

Q L/S

Fig. 11.6. (b) D1 > D2 > D3 (H–Q) performance due to diameter change for the same speed n.

300


=

6

00

65

00

C=

70

7 00

75

0,14

η

=5

s

0

η

=6 ns

0%

60

KH 0,18

0,12

800

ϕ=+12,5°

80

85 90

84

0

83

%

86 η=

87

85

0,10

82 84

82

81

1000

80

0,08 1100

C 15

=1

00

0,04

16

ns=5000 0,3

ϕ=0

ϕ=–5°

0,4

70 η=

00

0,5

0 0 0 1 90 0 0 20

0,02

ϕ=8.75°

0

18

65 60 55 50

40

%

1200 1300

0,06

0,2

75

0,6

ϕ =+5°

0,7

0,8 K Q

Axial flow pump

Fig. 11.6. (c) Axial flow pump (with ‘C’ curves)

In practice, pumps are operated for a range of total head and flow conditions. In order to achieve this, pump is run at different speeds ranging from minimum to maximum allowable speeds. Test results are plotted in one graph called universal graph. Universal characteristics of a pump consists of a series of (H–Q) curves for different speeds over which, η = constant, 0-curves as well as Hsv or hs or ∆h or σ = constant 0-curves are plotted. This graph [Fig. 11.6 (a), (b), (c)] provides a complete range of operation of pump to meet the head and flow rate of the site conditions, at any one of the speeds selected under high efficiency conditions. The method of drawing universal characteristics is given in Fig. 11.7. 1. H–Q curves of pump test conducted at different speeds are plotted in one graph. 2. Another set of graphs η = f (Q) for different speeds are drawn in one graph, selecting the scale of x-axis i.e., Q-axis same as that selected in H–Q curve. (Fig. 11.7)


301

0 04 = 1 60 n6 = 9 n 5 880 = n 4 00 =8 n 3 720 = n 2 = 640 n1

3. A horizontal line parallel to x-axis i.e., Q-axis is Hm drawn. This line meets all (η–Q) curves at two 50 60 70 C3 rpm points (one point before maximum efficiency n6 = 1040 C2 75 point, and another point after the maximum 40 78 n5 = 960 C1 efficiency point). This horizontal line represents 75 M3 n4 = 880 one value of efficiency at different speeds. 79 30 M1 70 4. All the meeting points (in η–Q curve) are now M2 R n3 = 800 20 n2 = 720 transferred to meet corresponding (H–Q) curve. G 5. All the points, so obtained in H–Q curve are joined 60 E B1 L together. This curve will be either in full 0-shape n1 = 640 10 or part of 0-shape. This O-curve is the equal A1 D efficiency curve, the value of which is obtained 0 100 from the corresponding horizontal line drawn in 100 300 η ,% η–Q graph. 80 6. In the similar manner, a number of equal efficiency N T A 70 curves can be drawn on (H–Q) curves. B 60 7. A line is drawn passing through the turning points of all 0 curves (O–C2 Fig. 11.7). This line is the 40 K best operating line for the pump. The centre point F S p of all 0-curves through which the best operating 20 line passes is the best point of operation (Point M2). At this point, the efficiency is highest. The 0 0 head, discharge, speed, and efficiency values can 100 200 300 Q, lit/sec be read from this universal graph. Corresponding power observed can be obtained from the N–Q Fig. 11.7. Construction of universal graph. Power can also be calculated from the characteristics of pump (H–Q) curve at universal graph taking η, H, Q, at maximum point different speeds and equal efficiency ‘O’ η –Q) curves curves obtained term (η M2 (OC3). 8. Boundary conditions of operating range are also drawn in some of the universal graphs, which limits the operating range of the pump lines OC and OC3. The boundary line at low flow rate (OC3) is determined based on efficiency considerations, i.e., based on economical power consideration, whereas the boundary conditions at higher flow rate is determined based on cavitation considerations, efficiency and percentage of overload in power allowance (line OC1). 9. A smooth line drawn joining all maximum efficiency points of (η–Q) graph, also gives the range of operating regions. This line is same as the line drawn in universal graph. However, optimum efficiency conditions need not be optimum cavitation conditions. Cavitation condition is more important than efficiency condition for safe, continuous operation of the pump.

11.3 PUMP TESTING Pump testing is carried out in special test stand, mostly at research institutes, manufacturing industries dealing with pumps etc. The main aim of pump testing is to obtain, complete characteristics of pump for commercial exploitation. Mostly the test is conducted to establish working characteristics i.e., to find the characteristics

302


H, N, η = f (Q) at rated speed. Universal characteristics is drawn by conducting tests at different speeds for further analysis and at the same time to confirm the correctness of the theoretical design with practical results. Tests are also conducted to find the pressure and velocity distribution before, after and on the elements of the pump to find the behaviour of the pump at various operating regions. This will enable to find hydraulic losses at each element, to find the stability of pump at specified regime, to find the influence of each element on other elements of pump. Tests are classified into development test and production test. Production tests are conducted on 1 in 10 pumps manufactured at 5 to 6 predetermined operating point such as 2 points before b.e.p, one or two points, at, or, very near to b.e.p and 2 points after b.e.p [Fig. 11.3 (b)]. These tests are conducted to maintain the quality of pumps. Development tests are intensive tests conducted on new designs and also at regions where results are not available for existing pumps, in order to find the bevaviour of the pump completely at all conditions of operation and also to verify and correct the theory to adopt efficient design procedure. Intensive tests are carried out on model pumps in order to develop efficient prototype units of higher capacity, which cannot be tested. (example, circulating pumps, condensate pumps etc.) Tests are also conducted with other fluid, instead of the original liquid. High capacity units are tested in air instead of water. Pumps used for pumping liquids other than water are also tested in water. The results obtained are corrected for original fluid pumping (example, oil pump, pulp, hot-water pumping, milk, acid, alkaline, distilled water pumping etc). Mainly two types of tests are conducted on pumps : 1. Test on open test rig to determine the load tests and 2. Tests on closed test rigs to determine load and cavitation tests, but mostly cavitation tests.

(a) Load Tests Conducted on Open Test Rig In Figs. 11.8 and 11.9 an open test rig is presented. pd = pat Delivery tank

hd

Delivery pipe

G

HT

H st = hs + hd X = XG + XV

XG

Pump

ps = pat Suction tank

V Suction pipe

Z1 hs

Xv

Foot valve and stainer

Fig. 11.8. Schematic diagram of an open test rig

303


7

H S

Scale

h2

3

h3

H

4

5

6 T

P

2

l

1

Fig. 11.9. Open test rig for load test on pumps

The pump to be tested (1) takes water from one end of the open type water tank (2) through suction pipe (5) and delivers through delivery pipe (3) to the other end of the water tank. The two ends of the tank is separated by a plate, which is fitted with a V-notch (4). Wave suppressors and flow straighteners are provided before V-notch, in order to ensure proper flow before reaching notch. Flow regulation valves are fitted, one at suction (6) and another at delivery pipe (7). One differential U-tube manometer (H) is connected to the pressure tappings at the suction and at delivery of the pump to measure the total head of the pump. Another U-tube manometer (S) is connected, one end to suction tapping and another open to atmosphere, to measure the vacuum at the inlet end of the pump. Mercury is used as manometric liquid. A water gauge is used at the tank before V-notch to measure the head over V-notch in order to calculate the flow rate of the pump. Another water gauge is used to find the water level at the other end of the tank, where suction pipe is attached. Sufficient height of water above suction entry is maintained to avoid air entry. The speed is maintained constant throughout the test. A tachometer is attached to the prime mover shaft to measure the speed. Input power to the pump is measured by the swinging field AC/DC dynamometer. Load test stand for axial flow pump testing is same as that of centrifugal pump test stand, except that suction and delivery pipe diameters and the collection tank size will be larger to accommodate high flow rate. For total head measurement, some other manometric liquid of slightly higher density than water other than mercury is used as manometric fluid or inclined tube manometers with slighlty higher density than water as manometric fluid are used. Before starting the pump for test, a thorough check is made on pump, coupling, prime mover, lubricating oil in pump and prime mover bearings, cooling of mechanical seal and stuffing box, sufficient level of manometric fluid in manometers to measure complete range of total head and suction head, suction regulating valve fully opened and sufficient water in water tank to keep suction entry point bell mouth under fully submerged condition as well as sufficient water height to avoid air entry into suction pipe. The delivery regulating valve will be in fully closed condition in case of centrifugal pump testing whereas in fully opened condition for axial flow and mixed flow pump testing. Manometer tappings are in closed condition to avoid sudden peak pressure reaching the manometer at the time of start. Sill level in V-notch is entered from the water gauge in tank.

304


The pump is started and brought to the speed, at which the pump is to be tested. The speed is maintained constant throughout the test by adjusting the speed regulator of the prime mover. The swinging field dynamometer is adjusted at this speed such that the torque arm is in horizontal position with weighs on the weighing pan. Pressure tappings for manometer are opened. The pump is run for sometime so that all readings i.e., tachometer reading, manometer readings, water gauge readings are at steady level. Under the steady level condition, head over V-notch for flow measurement, total head and suction head manometer readings, weight in the dynamometer pan and speed are noted and entered in a tabular form. Since delivery regulating valve is in closed condition, flow rate is zero and water level in tank is at the sill level. After entering all the readings in Table 11.1 flow rate, total head, input power, efficiency, unit discharge, unit head, and unit power are calculated using relevant equations. The results should be immediately entered, in a graph is drawn i.e., graphs H, Ni, η = f (Q) are plotted (Fig. 11.3) on the same graph sheet. TABLE 11.1: Test on centrifugal/mixed/axial flow pump Type of Test : Development/Production Test Pump No. Testing Stand No. I Impeller Diameter, D = mm IV II V-Notch Constant: K V III Speed (n) = rpm VI VII VIII (C3) H = (Sl – 1)

Inclined manometer limb angle θ° = ... R° Specific gravity of manometric liquid Sl = ... Arm length in Dynamometer (L) = ... m Temperature of liquid = ... C° Sill level reading = ... m

(C .2) h 1( mm)   =  (V) – 1) m of H2O  1000 1000  

(C .4) h2( mm )   =  (V) – 1) m of H2O 1000 1000   5/2 5/2 3 (C7) Q = K (h3) = (II) (C.6) m /sec

(C5) hs = (Sl – 1)

γQH 9.81 × 1000 × (C .7)(C 4) = kW const 1000 2πn × 9.81× W × L 2 × 3.14 × (III) (C.9)(VI) (C10) Ni = = kW 60 ×1000 60 × 1000

(C8) N0 =

(C12) KQ = (C13) KH = (C14)KN =

60 (C.7) 60Q 3 = III.(I)3 = nD (60) 2 H n2 D 2

(60)3 Ni n3 D 5

(60) 2 (C .3)

=

(III)2 (I) 2 (60)3 (C.10)

=

(III)3 (I)5

=

C1

Total head mano Rdg h1

Total head H

Vac. Suction mano head hs Rdg h2

mm

m of

mm of

m of

of Hg

H2O

Hg

H2O

C2

C3

C4

C5

Water level h3

Flow rate Q= k h3

Output power N0

Weight in Pan W

Input power Ni

mm

m3/sec

kW

Kg

kW

C6

C7

C8

C9

C10

Efficiency η

—

C11 =

C8 C10

Non-dimensional units

KQ

KH

KN

—

—

—

C12 C13

C14

Remarks

Valve condition

1

Full close

10

Full open


S.No.

305

306


Delivery regulating valve is slightly opened to regulate the flow rate to the next position. All readings are entered in the table 11.1 and all the values are calculated. In the same manner experiment is repeated from full closed position to full open position of the delivery regulating valve. It is necessary to take more number of readings near optimum efficiency point. In case of centrifugal pump, test is conducted from full close position to full open position, whereas in case of mixed and axial flow pumps, test is conducted from full open to partly closed condition until a break or surge in (H–Q) curve appears. Further test on mixed and axial flow pumps can be conducted only if the power capacity of the prime mover permits and pump runs under steady flow conditions without any oscillations in readings. Normally, test on centrifugal pumps are conducted at different speeds to get the universal graph. In axial flow pumps, tests are conducted for different impeller blade pitch (φ) at constant speed. In some of the test stands a thermometer is attached to measure the temperature of water before entry into the pump. It is essential to keep the temperature of water used for pump to remain same during the entire test period. For this, a separate valves are fitted to the water tank to allow additional water from other source and to drain water from the tank if the level of water in the tank is in excess.

(b) Pump Test in Closed Test Rig (Figs. 9.7 and 9.8) Cavitation test on pumps can be conducted only in closed test stands. Details of such tests and the graphical reference of the test results (performance of pump) are given in chapter 9. Load tests can also be conducted in closed test rigs by keeping the pressure in the tank equal to atmospheric pressure. This is done either by opening the tank to atmosphere or keeping the pressure in the space in the closed tank at atmospheric pressure.

11.4 SYSTEMS AND ARRANGEMENTS A system, in a pumping plant, consists of suction and delivery pipe lines along with all fittings such as sudden or gradual reducer or expander, all types of valves, such as gate valve, non-return valve, butterfly valve, etc. Tee or cross joints etc. A system is an already available pipe layout in the field to suit the actual conditions; for example, city water supply, multi storied buildings, layout in chemical, fertilizer and in power station etc. It consists of different lengths and diameters of pipes, fitted with various pipe fittings. The level between inlet and outlet end of the system may be equal or different. Total resistance offered by a system is the sum of major and minor losses in the system along with the level difference between inlet and outlet end of pipe Hsy = hs + hfs + hd + hfd+ Σhfmi for all fittings attached to this pipe line. Suffix s is for suction and d is for delivery. Major friction losses in pipe are expressed as hf =

flv 2 v2 and minor losses are expressed as hfmi = K , where K is the constant for that particular fittings 2 gd 2g attached to the main pipe. ∴

vs2 vd2 fld vd2 fls vs2 + + ΣKs + ΣKd Hsy = (hs + hd )+ 2g 2g 2 gd d 2 gd s

...(11.3)

307


Since,

4Q Q = A πd 2

v=

Hsy = (hs + hd ) + + Σ(Ks)

16 fls Q 2 2 gπ 2 d 5s

16Q 2

+

16 f d Q 2 2 gπ 2 d d5

+ Σ(Kd)

2 gπ 2 d 4s Since, flow rate Q is constant through both pipes

16Q 2

...(11.4)

2 gπd 4d

2

 dd  Q = asvs= ad vd , i.e., d2s vs = d d2 vd or vs =   . vd  ds  In equation (11.4) the values ls, ld , ds, dd, are constants for the already laid piping system, g, π, f are constant. Hence, it can be expressed as ...(11.5) Hsy = Hst +(Ksm + Kdm + Kmis + ΣKmid)Q2 = Hst + KQ2 where, K = Ksm + Kdm + ΣKmis + Σ Kmid 16 fls

Ksm =

2 gπ 2 d s5

, Kdm =

16 fld 2 gπ 2 d 5d

, Kmis =

K s 16 2 gπ 2 d s4

, Kmid =

Hp A–Operating point

2

A

2 C 2 – C1 = const. Q 2g

and static head (level difference between inlet and outlet points in a pipe line Hst = hs + hd

p v X1 X2

z1

p 1 hfs

X2 – X1 = X h f = hfs + hfd

Hst =

H st

z2

h fd

H p = H sy

p2

p2 – p1 + Z 2– Z 1 γ

H dy = h f +

2

H sy

Fig. 11.10. (Hp) pump and Hsv = Hst + Hdy and location of operating point (A)

K d 16 2 gπ 2 d 4d

308


+H Q–H

B

D

F hfd

HT = H

C Hd

H st

H

A h fs

0 Hs

P

Q E

–H

Fig. 11.11. System head for positive suction head

Equation (11.5) represents a parabola (Fig. 11.10) in Hst–Q coordinates. If Hst = 0, the parabola starts from origin. Parabola will be more curved upwards if length of pipeline is increased, pipe diameter is reduced, pipe fitting are increased, or pipe fittings are changed, gate valve partly opened, change of gate valve, with non-return valve or globe valve etc. which totally increase the value ‘K ’ for minor losses. If the system arrangement is reversed. The parabolic curve comes down. This parabolic curve for any system is an already available graph since all values are already available from site conditions, and constant K from reference manuals.

(a) Pump System Curve Construction In Fig. 11.11 (a), a pumping system working between suction tank A and the delivery tank B is shown. Tank B is at a higher level than Tank A. Both the tanks are above pump level i.e., the system in suction side is with positive suction (suction head). OE is the system curve for the suction pipe and is drawn in the reverse direction, below the water level in the suction tank (A). CD is the system curve for the delivery pipe and is drawn above the water level in the delivery tank B. The vertical distance between CD and OE is the total system head for the given flow rate point Q. Curve CF is the total system curve (i.e., OE + CD). The meeting point of this curve with the (H–Q) curve of the pump at F is the operating point. The vertical distance Hs gives the manometer reading in suction pipe and the vertical distance Hd is the manometer reading in the delivery pipe.

(b) System in Series If two systems A and B are kept in series, then total resistance of the systems is the algebraic sum of individual resistances of systems A and B. Since same quantity of flow passes through both pipes QT = QA = QB and HSYT = HSYA + HSYB In Figs. (11.12 and 11.13) equation (11.6) is illustrated in a graphical form.

...(11.6)

309


H

Hp

8 A

Sy

s te

m

he

ad

(H s

)

y

7=6+3

p 2 – p1 γ H st

p 2 – p1 γ

3. Total static head (H st ) = (2 + 1) 4. Delivery and suction velocity head difference 5. Delivery and suction pipe friction losses (h fd + h fs) 6. Total dynamic head (H dy ) 7. Total system head H sy = H st + H dy i.e. (H sy – Q) 8 . Pump head (H p) i.e. (H p – Q)

6=4+5

3=2+1

H st +

1. Static height between delivery and suction 2. Delivery and suction tanks pressure difference

5

2 1

4

Fig. 11.12. (Hsy) system head construction

Hp – Q H

Q = constant lines

A number of Q lines at frequent intervals, from origion to Qmax parallel to Y-axis i.e., Q = constant lines are drawn in the graph having HSY(A) – Q and HSY(B) – Q graphs (Fig. 11.13). The two ordinate lengths are added to get the point of HSYT = HSYA + HSYB for one value of Q. Similarly, other HSYT points for other Q are determined. A smooth curve joining these HSYT points give HSYT = f (Q) curve for combined series operation of two systems. The same procedure is followed for many systems connected in series based on the equation (11.6). If the H–Q curve of the pump is overlapped on the combined system graph, point A is the operating point, when both systems work. Several systems operating in series is equal to one system with higher system head.

HSYT — Q (including static heights)

R2 R1 + R2

R1 Hs2 Hs1

Hs1 + Hs2

R1 R2

R1 + R2

Hs1 + Hs2 + R1 + R2

A — Operating point

Q

HSYT = HSYA + HSYB, HSYA = HS1 + R1, HSYB = HS2 + R2 Fig. 11.13. Two systems in series

(c) Systems in Parallel If two systems 1 and 2 are operated in parallel (Figs. 11.14 and 11.16), the total quantity (QT) is the sum of individual quantities passing through each system and the total resistance is equal to individual

310


resistance of each system i.e., ...(11.7) Q T = Q1 + Q2 and HSYT = HSY1 = HSY2 Many horizontal lines, i.e., H = constant lines, are drawn at frequent intervals from origin to shut off head, in the graph where HSY1 – Q and HSY2 – Q curves are drawn. Individual horizontal distances between y-axis and HSY1 and HSY2 are added together to get QT = Q1 + Q2. The same procedure is followed for many systems in parallel operation i.e., QT = Q1 + Q2 + .... + Qn and HSYT = HSY1 = HSY2 = HSY3 = . . . . = HSYn In Fig. 11.14. systems operation is parallel is illustrated.

Q = Q 1+ Q 2 Q2 Q1

Q1

y2

A H sy = H sy1 + H sy2

q2

2

1 H sy2

H sy1 1 q1 q2

H sy2 2

q1

Hs

y1

H sy1 3

H sy = H sy1 = H sy2 Hs

Q = Q1 + Q2 Q2

H

2 Q

Q

q= q 1+q 2 (a) without static head

(b) with different static head

Fig. 11.14. Pump with two systems and operating points

If H–Q curves of the pump is inserted on this combined graph. Operating points where both systems are in operation is point A and operating points when one any system is in operation is indicated by points 2 and 3.

(d) Determination of Operating Point H–Q curve of the pump to be connected to the system (between suction and delivery lines, i.e., point ‘s’ and ‘d ’) is overlapped on the available HSY – Q curve, the meeting point ‘A’ is the point of operation of the pump. Point A changes for different H–Q curve of the pump, for example, H–Q curve for different speeds, with the same Hsy–Q curve or with the same pump but with different (Hsy–Q) curve (i.e., system curve with different fittings). Head and discharge at point A will be the operating parameters. This parameter need not be the optimum parameter of the pump. Power consumption and efficiency can be obtained from Ni–Q and η–Q curves for the pump at point ‘A’.

11.5 COMBINED OPERATION OF PUMPS AND SYSTEMS (a) Pumps in Series [Fig. 11.15 (a) and Fig 11.15 (b)] If two pumps are operated in series, then the total quantity is equal to the individual quantity of each pump. The same quantity passes through all pumps. Total head is the sum of heads developed by individual pumps i.e.,

311


Q T = Q1 = Q2 and HT = H1 + H2 ...(11.8) Same procedure is followed for many pumps operated in series i.e., Q T = Q1 = Q2 = ........ = Qn and HT = H1 + H2 + ........ + Hn In order to get the combined HT = f (Q) for combined operation in series, a number of Q = constant lines are drawn in the graph where individual H1 – Q and H2 – Q curves are already drawn. Total head of all pumps running together HT is obtained by adding individual total heads as H1 and H2 for the same Q. A number of points so obtained is joined together by smooth line, which gives HT – Q, for two pumps in series. If system line (Hsy – Q) is overlapped on this graph, point A′′ is the operating point when both pumps are operated in series. Point A is the operating point when individually any one pump is operated. Point ‘A’ will be the operating point, when only one system (instead of both) is operated with one pump. H Q – H (1 + 2) A′′

C1

R 1+2

d Q – H (1,2)

A′

A 2

2H 0

1

R 1,2

H0 P2 Q T = Q 1= Q 2

P1 HT = H1 + H2

H (1+2)

D

C

P2

2Hst H

H 1,2

s1,2

a 0

Fig. 11.15. (a) Pumps in series

D1

P1 Q

Q 1, 2

Q

(1 + 2)

Fig. 11.15. (b) Pumps in series (H–Q) curve

(b) Pumps in Parallel (Fig. 11.16) If two pumps are operated in parallel, then both pumps work under the same head. Total quantity (QT) is the sum of the quantities of the individual pumps i.e., HT = HA = HB and QT = QA + QB A number of H = constant lines are drawn in the graph where individual (H–Q) curves of pump C and D are available. For one value of H = constant line, QA and QB lengths are added together to get QT. In the similar manner a number of H = constant points are selected and QT is determined. A smooth line joining all these QT points give the combined graph (HT – Q). If a system curve (Hsy – Q) is overlapped on the combined (HT – Q) of the pump and individual (HA – Q) and (HB – Q) graphs of the pumps, the operating point 3 is when both pumps operate and points 1 and 2 are when any one pump is operated. Same procedure is followed for many pumps when operated in parallel. HT = HA = HB = ........ = Hn and

QT = QA + QB + ....... + Qn

312


px

H

for construction Q T = Q 1+ Q 2

p 1+ p 2 series

Q2 Q1 Hp 3 H sy Q1 Q2

2

1

p1 || p2 parallel

p2

p1

p1

Q sy < Q 1 + Q 2

operation

Q

Fig. 11.16. Two different pumps operated in parallel Q1 + Q2 = QT, H1 = H2 = HT

p2

0

Q

Fig. 11.17. Comparison of series and parallel operation

Regulation of pumps in series or parallel is also achieved by sudden switching off of one or two pumps. This method saves the power requirement to the pump.

B

(a) in series

A

(b) in parallel

Fig. 11.18. Schematic diagram for two pumps to operate (a) in series and (b) in parallel

11.6 STABLE AND UNSTABLE OPERATION IN A SYSTEM (a) Stable Operation In Fig. 11.19. (H–Q) curve of pump meets (Hsy–Q) curve of a pumping system at two operating points A and B.

313


C

A

QA

(H

A0 (H – Q) p p2 z2 Qsy

A

O,C

–

Q)

QB

p

∆H = H sy – H p

Drooping down characteristics

QA Q A0

Ho

∆H = H sy – H p

Fig. 11.19. Stable and unstable operation in pumping system

z2 – z1

O QB

K

B

P2 – P1 + z2 – z 1 g

A

stable ∆ H > 0 Q) y (H – s

H pc =

∆Q A

K

Q ) sy (H –

HC

C

∆H< 0 unstable H QK

stable

∆H > 0

∆QB

∆H < 0

unstable

H

Q

z 1, p 1 Q


H

H 0 = H max

H–

Q

(H – Q) p Rasing characteristics

Q

Fig. 11.21. Condition for stable operation H0 = Hmax

At both point A and B, condition Hsy =Hp is prevailing. But, actually point A is the stable operating point and point B is the unstable operating point. Let the quantity at point A is shifted to the higher side of the quantity by a smaller amount (∆QA). At the new point Hsy > Hp. ∆H = Hsy– Hp > 0. Pump supplies quantity at a lower pressure (Hp), whereas the system is at a higher pressure (Hsy). Pump cannot supply flow against high pressure of the system. The flow gradually reduces until pump pressure (Hp) is equal to system pressure (Hsy). Same situation occurs when flow is reduced by a small amount i.e., (– ∆QA). Now, Hp > Hsy , ∆H = Hsy– Hp < 0. Pump supplies water at a higher pressure than the system pressure. This difference in pressure causes an increase in quantity. This increase in quantity reduces the pump pressure (Hp) and increase the system pressure (Hsy), until both the pressures are equal i.e., until point A is attained. Entire process is automatically carried out. Pump can operate only at point A where the condition is Hp = Hsy. Hence, point A is a stable operating point.

(b) Unstable Operation Normal head-discharge curve for a centrifugal pump will have a shape of gradually rising characteristics, when quantity of flow is reduced from maximum to zero flow condition (Fig. 11.20). In some of the pumps due to heavy secondary losses at very flow rates, head-discharge curve of the pump, droops down instead of raising after attaining maximum head at certain quantity (point K) in Fig. 11.20.

314


Always shut off head (Q = 0) will be the highest head in (Hp– Q) curve for all pumps of good design. But due to high secondary flow at low flow rate, (Hp– Q) curve droops down and the shut off head will be lower than the normal value. In Fig. 11.20, point B is the meeting point of the (Hsy– Q) and (Hp– Q) curves. A small increase in quantity (+ ∆ QB) moves the operating point to the right hand side of the graph (+ Q direction). At this stage, system pressure Hsy is lower than the pump pressure (Hp), ∆H = Hsy– Hp < 0 (negative value) as a result pump supplies more quantity to the system, which makes the point to move further to the right hand side (+ Q direction). Point B cannot be reached at all. Pump supplies more and more quantity to the systems until point A in attained. If the quantity is slightly reduced (–∆QB), it can be seen that pump pressure (Hp) is lower than system pressure (Hsy). ∆H = Hsy – Hp > 0. Unstable

Stable

∆H = H sy – H p< 0 ∆H = H sy – H p > 0

H + ∆Q C

+ ∆Q A ρ

K C

ε

E M

B 1 HD

QA

HB

QC

H ′N

Hp Hc

O N

2

A D

Q


D A

B

∆Q

O

Ho

C H syo < H o

However, pumps can be operated at stable condition even when drooping down characteristics prevails at low flow rates. In Fig. 11.19, the shut off head of pump (at Q = 0 ) is lower than the shut off head of the system. In Fig. 11.23 pump shut off head is at a higher level than system shut off head. System curve and pump curve meet at point B, which is located at

H ∆H

The quantity of flow is reduced further until pump reaches the point (Q = 0). At this stage, pump runs with regulating valve in opened condition, but there will not be any flow in the system. Here again point B cannot be reached. Point B is the unstable point. This region is called unstable region curve OBK in Fig. 11.20. This unstable effect will exist in all pumps if (HP–Q) curve droops down at lower flow rates. The pump should not be operated in this region.

Fig. 11.23. Condition for stable operation at unstable region Ho > Hsyo

315


H

R

K

2′

5

O

K C C B

n 1 H1

Q ′ 2 Q A Q2 Q 5 Q1

Hsy0

H0

H syo = H po

∆H > 0

unstable region of pump characteristics. The condition ) sy H ∆Q B –Q of point B in Fig. 11.24 is same as the condition of (H point A in Fig. 11.20. Any small increase in + ∆QB (Curve OB) increases Hsy and decreases Hp i.e., ∆H = Hsy – Hp > 0 H po > H syo B (H (+ve). Due to higher resistance in system, quantity –Q O )p reduces. This reduction continues until point B is reached. Any small decrease in flow rate (–∆QB ) ∆ H = Hsy – H p C makes Hp > Hsy or ∆H = Hsy – Hp < 0. Higher pressure in pump increase the flow to the system until point B is reached. Therefore, point B is a stable point although it lies on the drooping down side of pump Q characteristics. Mathematically, stable condition will dH sy dH p Fig. 11.24. Condition for stable operation in > exist if dQ dQ unstable region An example of this condition is illustrated in Fig. 11.25. A E

H2 H 5 H max

Q (a)

F

(b) Q

Q

Fig. 11.25. Change of unstable operation into stable operation due to (a) static head change (b) due to system resistance change

Referring to Fig. 11.25, point ‘1’ is the stable operating point of the system. When delivery tank water level raises, the operating point, due to static head change to point ‘2’ and ‘2 ′ ’. Point ‘2’ is the stable operating point, whereas point ‘ 2′ ’ is the unstable operating point. This unstable region starts from point ‘5’ and above. Maximum height of water level in the tank can be up to point ‘K’. Further increase in water level leads to reverse flow (Refer section 11.7).

11.7 REVERSE FLOW IN PUMP H P R

K A N

O

Accumulator Tank p 2

C S

–Q

P

Qp

Q

Fig. 11.26. Reverse flow in pump

Q2 Tank

Q1

To system

316


In Fig. 11.26 a pumping system is illustrated where in the system i.e., suction and delivery pipe, remain same. A closed delivery tank (Condenser, Accumulator) is attached to the system. Quantity Qp enters the tank from pump whereas a quantity Q1 leaves the tank to another system. Quantities Qp and Q1 are independent of each other. If Qp > Q1, water level in the tank gradually raises. Correspondingly, pressure p2 also increases. Since system remains same, the pattern of dynamic part of system curve remains same, whatever may be the quantity Q2, increase in pressure p2 and raise in level Z2 in the tank. Change is only in the static head i.e., Hsy =

p2 − p1 C 2 − C12 + ( Z 2 − Z1 ) + 2 = H st + H dy γ 2g

C 2 − C12 p2 − p1 + ( Z 2 − Z1 ) and H dy = 2 2g γ Due to this, (Hsy–Q) curve raises parallel to the previous (H sy–Q)curve (Figs. 11.21 and Fig. 11.26). Point A moves towards left and point B towards right along (Hp–Q) curve. Still QA > QB. However, at one condition, the increase in static head makes (Hsy–Q) curve will be tangent to (Hp–Q) curve at point K (Fig. 11.26). Any further increase in pressure p2 or level Z2 raises (Hsy–Q) curve above (Hp–Q) curve. Operating point K moves to point R, where only further increase in H sy is possible. i.e., to the area of reversible flow. Flow moves from the tank to the pipe. This induces a reversible water wave. Pressure p2 reduces. Operating point R moves to point N. Because flow reduction is possible only along the curve R.N, any further reduction in pressure moves the operating point from N to A and then gradually to S, because only at point A, further reduction in pressure is possible. Thus, a cycle is completed. This cycle repeats so long as there is no change in the system. Pressure fluctuates at a faster rate. If the flow flucuation is large, the unstable condition starts even earlier when operating point is at A. Even a small disturbance in pressure can induce unstable condition, if operating point is very near to point K. Reverse flow prevails until Hst = H0 (at Q = 0) i.e., shut off head. When Hst < H0 pump starts pumping with operating point at A or below point A under stable condition. When QA < QK stable operation of the pump is not possible. where, Hst =

dH sy dQ

<

dH p dQ

from Q = 0 to Q = QK . Hence unsteady flow prevails between these two points.

Quantity ‘Q’ passing through the system will be either Q = 0 or Q = QK. At this region reverse flow takes place. Drooping down characteristics at low flow rate is the unstable region so long Hst > H0. It becomes stable when Hst < H0 and stable operations starts from the condition Hst < H0 to the condition Hst = H0. The same thing will happen even pumps are running in parallel or in series. The only difference is that combined characteristics must be studied. However, a raising characteristics of H–Q curve from Qmax to Q0 i.e., up to shut off head is the best H–Q curve for stable operation. Such pressure fluctuations quite frequently occurs in boiler feed water storage drums, condensers, accumulators and in pipes with elastic properties. This effect is more predominate in pumping compressible fluids such as gas, air especially pumping to storage cylinders. A series of sudden and high intensity fluctuations can create oscillations in the system. Constant flucuations of low intensity quite frequently appears in the delivery line. Disturbances created by


317

aerodynamic wake after the impeller blades, flow over volute tongue, flow over diffuser blades, uneven angular velocity of the rotor (in case of gas and air pumping) are a few instances of creation of disturbances. If the frequency of the disturbance does not coincide with the frequency of the system, the amplitude of such oscillations give a very low effect on performance even when the pump works near maximum head point (K) and pump work under stable condition. If the disturbance frequency is very nearly or exactly equal to system frequency, resonance effect is created and the pump will work in the unstable operation even when pump is working far away (point A) from maximum head point K. The frequency of operation of the system depends on dimensions of the delivery pipe and does not depend upon the speed of rotation. A reduction in accumulator energy increases the frequency of oscillation which inturn increases the amplitude of pressure fluctuations. The energy waves developed due to pressure fluctuations, either direct and indirect, combined together, which can create high intensity shock waves. However, increase in frequency of the amplitude of fluctuations increases a self breaking effect. A change in quantity of flow is created by the change in circulation, provided sufficient time is available. When total, combined self oscillating frequency of the system is reduced or in other words, when oscillating frequency increases, circulation time almost reaches the condition for one cycle of oscillation. This almost stops further increase in the amplitude of the oscillation, with the result, energy is dissipated under low accumulator energy. Circulation is inversely proportional to speed or tangential velocity of the blade. As a result, any pump working in the unstable region, when speed is reduced up to certain limit will work in stable condition at all flow rates. The limited speed will be higher where system conditions are low. Boiler feed pumps, compressors working under high speed has a very little time to adjust, will undergo unstable operation even when accumulator energy is small. Stable operation in boiler feed pumps can be obtained by having continuous raising characteristics from full open to full close i.e., without any drooping down characteristics at partial flow. Experimental investigation shows that reduction in number of impeller blades and low outlet blade angle β2 will provide a stable raising pump (H–Q) curve. Better results are also found when the impeller blades are extended into impeller eye at suction i.e., increasing length of the blade at suction side. (Fig. 9.10) High specific speed pumps have raising characteristics. Diverting part of fluid from delivery line to suction line is also one of the reasons for unstable operation for pump.

11.8 EFFECT OF VISCOSITY ON PERFORMANCE The performance of water pumps, when used for viscous liquid pumping, changes considerably from water performance. Many authors already worked on this area. Prof. A. A Burdhakoff, Prof. R.E. Sheshenko, Prof. D.A. Suhanoff, Prof. B.D. Baklanoff, Prof. M.D. Aizentein, Prof. A. T. Ippan | 4 |, Prof. A.J. Stepanpoff | 112 | are a few authors to be mentioned. In chemical industries, centrifugal pumps, of ns = 80 to 135, are used for pumping viscous liquids. Spiral casings are used, instead of diffusers, in most of the pumps, because, when pumping viscous liquids, the flow velocity in pump parts is less than that for water pumping. Coefficient of reaction ρ = 0.7 and number of blades Z = 5 to 6, depending upon the viscosity of the pumping liquid. Fig 11.27 gives the test results of a pump of ns = 85, for different viscous liquid pumping.

318


Hm 50 47 44

ν= 0 0 09 2 cm /se c w a te r 0 ,1 3 8

41 38 35 32

0 ,5 9 5 H = f(Q )

1 ,5 0

82

29

=

26

s

3 ,6 9

n

23

8 ,5 5

20

1 2,2 8

N kW

1 8,8 0

1 2,2 8 8 ,5 5

18

1 8,8 0 3 ,6 9

16 14

η =f (Q )

12

1 ,5 0 0 ,5 9 5 ν = 0,00 9 W a ter

10 8

η%

6

80

4

72 64

ν = 0 ,00 93 W ate r 0 ,1 3 8

η =f (Q )

0 ,5 9 5 56 48

1 ,5 0

40 32

ns

=

82

3 ,6 9 8 ,5 5 1 2,2 8

24 16

1 8,8 0

8 10

20

30

40

50

60

70

80

Fig.11.27. Actual performance variation due to change in viscosity of liquids test results conducted on the same pump ns = 85 n = 2875 rpm

Prof. D. A. Suhanoff mentioned that the major losses involved in viscous liquid pumping are the hydraulic losses due to friction and disc friction losses. Head loss in pumps is due to high friction, while power loss is due to high disc friction. So, the important factor to decide the effect of viscosity on pump

ωr 2 , since the theoretical head, hydraulic losses depend upon ν only Reynold’s number. However, shock losses do not depend on Reyonld’s number. Shut off head (Ho) for the viscous liquid pumping and for water pumping remain same i.e., Hov= Ho, when Q = 0. Due to complicated flow pattern prevailing in flow passages, analytical methods aided with experimental correction coefficients are used to convert water parameters into viscous parameters for pumps of ns = 50 to 130. The results are given in Figs. 11.28 and 11.29. performance is Reynold’s number Re =


2,5 m′

2,0 1,6 Example

1,2

0,6 n′

0,5 0,4

50 43 35 31 27 23 19 15

0,3

r2w

0,15 0,125

k ′ = 0,93 m ′ = 1,28

k′

0,8

13 12 11 10 9 8 7 6 5,5 5 4,5 4 3,5 3 2,5 2 1,5 1,2 1,0 0,8 0,7 0,6 0,5 0,4 0,3 0,25 0,2

n = 1450 rpm ν = 1.5 cm 2/sec

Re = 2,5.10 n ′ = 0,73

47 39 33 29 25 21 17

4

r = 16 cm

1,0

25 20 16 12 10 5 70 0,

1

50

c

30

2

/s e

20 05

cm

n = 960 rpm

16

=

0,

n = 1450 rpm

12

ν

n = 2900 rpm

4

10 8 6 5 4 3 2,5

5 6

8

10

20

Impeller radius 'r' in cm

30 40 50 1

2 10

3

5

1 10

2 4

5

1 10

2

5

5

2 2

Re =

r ω ν

10

6

319

Fig. 11.28. Nomogram k′′ , m′′ , η′ = f (Re) to convert water parameters to equivalent liquid parameters

2

1

320


Experimental analysis indicates, that, for pumps of low ns and impellers with both shrouds, the performance of viscous liquid pumping and that for water pumping remain same at maximum efficiency point. When viscosity increases, total head reduces. It is found, sometimes that, the total head, for viscous liquid pumping is slightly higher than that for water pumping at smaller viscosity ranges. This is due to the absence of axial vortex at slightly increased viscosity. Cu2 reduces at a little lower rate when compared to Cu2∞, which not only compensates for the increase in hydraulic friction losses and also increase in total head. However, this compensation cannot be made, when viscosity increases further. The similarity laws cannot be applied directly, due to insufficient data for viscous liquid pumping, since a change in speed, changes Reynold’s number. However, for low viscous liquids i.e., for liquids with high Reynold’s number having ν = 5000 stokes, the performance of viscous liquid pumping coincides with that of water pumping, even when speed changes in a wide range. Model analysis must be considered including Reynold’s number also. In order to compare the characteristics of different viscous pumping liquid, Prof. D.A. Suhanoff gave a relation between head coefficient k (= KH) =

 N  H , power coefficient m = (= KN) =  3 5  , 2  γn D  n D 2

and efficiency (η) as a function of Reynold’s number (Re). The result obtained by him are given in a nomogram k′, m′, η′ = f (Re) (Fig. 11.28). His results perfectly agree with the experimental results of different viscous fluid pumping for the same pump and same viscosity for different pumps. The variation in k, m and η = f (Re) is attributed to the increased friction losses and disc friction losses especially when Re < 7*103. At low Reynold’s number η reduces to a greater extent than ‘k’ and ‘m’. When Re > 3*105, calculation by dimensional analysis gave more accurate results. The error will be more, especially, when the speed is changed. Comparing the pump characteristics for different Reynold’s numbers (Re), when unit quantity KQ = Q/nD3 is constant, Prof. Suhanoff, suggested that relative efficiency ratio (η′), relative power ratio (m′), and relative total head ratio (k′) can be used for all practical purposes as a function of Reynold’s number (Re) i.e., η′, m′ and k′ = f (Re). These values are given as η′ =

ηv η

, m′ =

kv mv , k′ = k m

...(1)

The values with suffix (v) attribute to viscous liquid and without suffix for water. Fig. 11.28 is a consolidated nomogram, giving all the above mentioned details for easy calculation. This nomogram has been developed based on the experimental results from many authors. The procedure is as follows : (1) A few points, very near to the maximum efficiency point are selected on water performance graph (Fig. 11.3) of the pump and η, m and k are calculated. ωr 2 . ν (3) From Fig.11.28, the relative coefficients η′, m′ and k′ are found out for the obtained Reynold’s number. (4) Using equation (1) the values kv, mv, ηv are determined. (5) The head and power of the viscous liquid can be calculated from the formula Hv = kv n2D2 and Nv = mv γ n3 D5 and ηv = ηη′.

(2) Reynold’s number is calculated using the formula Re =

Qliquid H , K′ = liquid = K′Q 2/3 Qwater H Hwater KQ′ =

0,5

KQ

0,4 1,0 0,9 1,0

0,8 1

0,7 KC 0,6

0,8

3

Kη

0,5

0,9

2

0,7 4

K′η =

ηliquid ηwater

0,4

0,6

5

0,3

0,5

0,2

0,4

0,1 2 10

2

3

4

5 6 7 8 9 10 3

2

3

4 5 6 7 8 9 10 4 Re =

2

3

4 5 6 7 8 9 10 5

2

3

6 4 5 6 7 8 910

3 Q lit/sec. 10 D cm.νt cm2/sec.

321

′ , K′H′ , K′η′ , and K′C′ = f (Re) Fig. 11.29. Coefficients K′Q 1 — as per Ajenshtein (nsi = 51, 60 to 70); 2 — as per Suhanoff (nsi = 82 to 130); 3 — as per Ippen (nsi = 90 to 115); 4— as per Stepanoff (nsi = 82); 5 — as per Ajenshtein for cavitation (nsi = 60 to 100)

Cliquid Cwater

0,7

K′C =

KH

0,8

(6) Shut off head Ho, when Q = 0, is obtained from the pump performance graph for water. Hov = Ho. Another method of calculation is :

0,9


0,6

1,0

322


(1) Under constant speed, and the change in pumping liquid viscosity, the (H–Q) for water pumping changes, in such a way that the specific speed for both liquid pumping remains same (Fig. 11.29). ns =

3.65 n Q

Q  H =  Qv  H v 

= nsv =

3.65 n Q v

...(11.9) ( H v )3 / 4 H The (H–Q) curve droops down more when viscosity increases (11.27). Suffix (v) is for viscous liquid and without suffix is for water. From the equation (11.9), we get 3/ 4

3/ 2

...(11.10)

From the equation (11.10), it is seen that the coefficients, kH, kN, kη and kC (Fig. 11.29) obtained at maximum efficiency point, remain same for all other selected points of water pump performance. So all other values can be obtained from the above correction coefficient. (2) Shut off head (Ho) for water pumping is taken from the water pump performance graph. This is same for viscous fluid pumping also, since shut off head is independent of viscosity of pumping liquid. (3) From Fig. 11.27, it is seen that under viscous liquid pumping, the power required increases more or less by the same value for a wide range of flow rate. Power (N) is given by

γQH For water pumping N = constant . η For viscous liquid pumping, Nv =

γ v Qv H v constant ηv

Combining these two, we get N < Nν or Q H  γ  QH <  v ×  v v η  γ  ηv 

 gv   Qv H v  hv γv  g  ×  QH  > h = η′ or kQ kH γ > η′ Qv Hv and kH = Q H Combining equations (11.10) and (11.13), we get

Where

kQ =

...(11.11)

...(11.12) ...(11.13)

gv 5/2 ...(11.14) g k H > kη Equation (11.10) and (11.14) give a relation between the correction coefficient for head, flow rate and efficiency for viscous liquid pumping. Absolute values of these coefficients must, however, be determined experimentally. Fig 11.29 gives the values of these coefficients as a function of Reynolds number (Re).


Re =

323

Qnor , where Dequ = 4 D2 b2 k , where k is the correction coeficient (≈0.9) and also equal ( Dequ . ν )

to k = 1/K2, where K2 is the area restriction coefficient at outlet. In Fig. 11.29, another graph is also given containing the variation of the cavitation specific speed (C). The correction coefficient kc = Cν /C = f (Re). From the graph it is seen that when Re > 7.103 the coefficient kQ, kH are nearly equal to one (i.e. kQ = kH = 1), which indicates that, the increase in hydraulic losses between water pumping and viscous pumping is same i.e., the effect of viscosity is negligible. The drop in overall efficiency kη indicates that disc friction losses are increasing. At Re < 7.103, the hydraulic friction losses considerably increase, which is responsible for reduction in efficiency. The disc friction loss will be low. In practice, it is found that, pump performance comparison for viscous liquid pumping with that of water with respect to Reynolds’s number, give an error of ±5% for ns = 85 to 130.

11.9 PUMP REGULATION The process of changing the characteristics of system and that of pump to meet the output demand is called regulation of pumps. By this, the operating point of (Hsy–Q) and (Hp–Q) curves will meet the required output head and discharge. Regulation is done by several methods namely (1) Flow regulation by valve control system; (2) by transfer line control system; (3) by speed control of the pump; (4) by diameter control of the impeller of the pump; (5) by impeller and diffuser blade adjustments and (6) by changing the static height in suction and delivery tank.

(a) Regulation by Delivery Valve Control This method is simple, but with heavy hydraulic losses in the control valve. Regulation is done by a regulating valve fitted at the delivery pipe. Fig. 10.30 shows the performance by delivery valve regulation process. By adjusting the valve, either closing or opening, the (Hsy–Q) curve changes (curves oAx1, oAx2, oAx3). Each curve is for one position of gate valve opening. H

∆H c

η wx Ax2 Ax1

Ax3

A C

H sy Hp

H sy

Hp

η

ηHy

Q

Qx QA

Fig. 11.30. Regulation by delivery control valve

324


20

Load test on centrifugal pump — with delivery valve control – – with suction valve control ×

70 60

50 10 40 30 20

Power (N) kW. 5

10

(η) Efficiency %

Total Head (H) m

15

80

90

100 %

Due to the adjustment in valve, the system resistance is changed by which, the operating point moves either upwards or downwards along the (Hp–Q) curve. Hydraulic losses are higher due to obstruction created by the valve and pump efficiency reduces considerably. However, due to simplicity, this regulation is widely used. Flow regulation by suction valve, although possible, is not carried out for

0

5

10

15

20

Q. Flow rate lit/sec Fig. 11.31. Performance of pump (a) operation of delivery valve with full opening of suction valve (b) by operation of suction valve for one position of delivery valve opening

325


incompressible fluids like water, oil etc. due cavitation problem, since regulation of suction valve increase total suction head. However, for compressible fluid, suction valve control is carried out since density increases due to fall in pressure, which reduces the flow quantity. Moreover, unstable condition area moves toward left hand side of H–Q curve, more for air and less for gas. Fig. 11.31 illustrates the graph for valve regulation by suction control valve and by delivery valve control. In Fig. (11.30), A is the operating point when the delivery control valve is at full open condition. Points Ax1, Ax2, Ax3 are the operating points when control valve is gradually closed. Dotted lines corresponds to the (Hsy–Q) curves for different opening position of the suction valve.

(b) Regulation by Transfer Line Method In low specific speed pumps, the power of the pump reduces when control valve is gradually closed. Maximum power occurs when control valve is at full opened condition. But in axial flow pumps, power of the pump increases when flow rate is reduced or the control valve is gradually closed. In such installations a transfer line is always provided. The flow from the transfer line is sent back to the suction sump. When the control valve in the main line is gradually closed, part of the flow from the main line is transferred to transfer line. But pump supplies at constant flow rate, thereby power is kept constant and below overloading capacity. This method is adopted in axial flow pumps in general for all pumps, where N–Q curve has a raising characteristics, when flow (Q) reduces. In order to have minimum power at the time of starting, centrifugal pumps are always started with delivery gate valve in fully closed position while axial flow pumps are started with delivery gate valve in fully opened position. So a transfer line i.e., a parallel pipe is connected to the main supply line. This transfer line discharges the pumping fluid back to the suction tank. While starting, the transfer line valve is kept open and main line valve is fully or party closed. After starting, the main line valve is gradually opened and transfer line valve is gradually closed until required quantity in the main line is attained. Fig. 11.32 illustrates the regulation by transfer line. N H L

r D

H = f(Q)

N = f (Q) B R

F

K

M R+r

P

Nm

NK

C

A

R

H st

E

R r

r

A 0

Qc Q p QC

Qm Q

Qp

R – Main line r – Transfer line

QK QM = Qc + Qp

Fig. 11.32. Regulation by transfer line (r) OEPD-Transfer line, ACKB-main line(R), OEMF-combined line (R + r)

326


In order to maintain a steady flow, especially at low flow rate, recirculation method of regulation is often followed instead of regulating by gate valve. A part of the main flow from the delivery line is brought back to suction reservoir in order to maintain constant level. The suction reservoir is also constructed with greater depth. Referring to Fig. 11.33 the flow will be safe so long as the level in the reservoir is above the line ‘a–a’ i.e., above the entrance of suction pipe. For the safe operation of pump without cavitation, the water level in the suction reservoir should be maintained at ‘h2’. If the liquid level falls below the value ‘h1’, the recirculation line is automatically activated and the operating curve shifts from curve R to R + r. Out of total quantity ‘QH’ a quantity QC passes through the recirculation line and the balance through the main line (line r). The recirculation quantity maintains constant level of water in the sump and pump also operates safely. Recirculation line operates until the liquid level in the reservoir raises to ‘h2’. At this level recirculation line is closed. Between the point h1 and h2 pump operation is safe. In Fig. 11.32, parabola AB is the main line characteristics and parabola OD the characteristics of transfer line. OEF is the combined system characteristics (AB + OD) with transfer line opened. Parabola, LKM is the characteristics of pump. Under transfer line (OD) in closed condition, the operating point is K. QK is the quantity through the main line AB. When transfer line is opened, operating point is M. QC is the quantity passing through the main line and QP is the quantity passing through transfer line. Total quantity QM = QC + QP. The power NM consumed is less than power NK. H

R

R+r

QH

H

Q onm

Q

h1

h2

QC

r

a a

Fig. 11.33. Pump characteristics during constant recirculation through transfer line

In Fig. 12.38 most commonly adopted circulation pump in ships is illustrated. For light weight provision, the outlet bend is provided with flow ribs and the inlet is provided with a main circular cylinder suction entry alongwith a side pipe entry. Due to the flow from the side entrance, unequal pressure and velocity distribution prevails at the inlet, which induces cavitation, noise, vibration. In order to improve the performance, inlet vanes are provided. Referring to Fig. 11.34 curve R is the H–Q curve of the axial flow pump, when flow is through the main entrance, this curve is less steeper. H–Qa curve is the performance of the same pump, when flow is through the side entrance. This curve R1 and R2 are more steeper due to heavy secondary losses at inlet. Experimental investigation | 4 | shows that when the speed is reduced from n = 100% to n = 60%

327


severe cavitation prevails, when flow takes place through side channel. The flow Qa at n = 60% is found to be ≈15 to 20% of normal flow QH. Steady flow prevails up to the suction reservoir height up to ‘hs1’ where the inflow to the reservoir is equal to outflow from the pump. Steady flow exists during the height hs1 to hs2. The flow becomes unsteady if the height in the suction reservoir falls below hs1. Q – Ha

R2 R1

Q–H

n = 100%

n = 60%

R hs2

hs1 Qa

QH

Fig. 11.34. Axial flow pump performance when side channel emergency line is in operation under speed change

(c) Regulation by Speed Control Already known that the change in speed of the pump, moves the (Hp–Q) curve of the pump parallely up or down depending upon the increase or decrease in speed. Since the system is not altered, The (Hsy–Q) curve remains same. As per model analysis, total head increases as square of the speed and quantity increases in proportion to speed.

H

H H1

n3

A xn

η

B2 B3

Ay

A1 B1

ny

Ayn

Ax

n2

H2 H3

n1

A2

n

A3 nx

R2 η

H st

Hy

ηx = ηxn

C R1

Q3

Q2

Q1

Q

η

Q

Qx QA Qy

Fig. 11.35. Regulation by changing speed

Fig. 11.36. Regulation by changing speed

328

8 IHPN 20 Hm 80 % η


— O.D 225 φ O.D 222 φ . O.D 219 φ × O.D 215 φ ×× O.D 211 φ

222

4 × 3 5 ms6

225

225

6 15 60

219 215 211

225 222 219

225

4 10 40

222 219 215

2 5 20

211

0

8

16 Q LPS

24

Fig. 11.37. (a) Effect of changing diameter on performance

32

329


I II III IV V

I

with with with with with

OD OD OD OD OD

of 250ϕ of 247 ϕ of 244 ϕ of 241 ϕ of 238 ϕ

η

III

N(hp) 20 80 %

H(m)

II

Tested Tested Tested Tested Tested

IV V I

6 15 60

II

III

III

4 10 40

V

I

2 5 20

II III IV V

0

5

10

15

Q LPS

Fig. 11.37. (b) Effect of changing diameter on performance

330


50 % 55 % 60 %

22

65

(H p–Q)

18

60%

%

20

% 70 % 75

16

55

%

14

50

–Q ) (H s y

φ 220 φ 210 φ 200 φ 190 φ 180

8

0

4 240 144

φ 250

φ 240 φ 230

12 10

%

8 480 28.8

12 720 43.2

16 960 57.6

32 LPS 20 24 28 1200 1440 1680 1920 LPM 72 86.4 100.8 115.2 m 3/hr

Quantity Q

Head (m) Efficiency %

5.2 4.5 18 3.8 15

50

3.1 12

60

40

2.3 9

Quantity (Q) H s

60

68

Fig. 11.38. (H–Q) curve for different outer diameter D2 at one speed ‘n’

30

G∝D

Qu

an

tit

3

c yA

Theory

tua

l

Efficiency %

H∝D

2

Head

200

220 Diameter (D 2)

240mm

Fig. 11.39. Effect of impeller outer diameters (D2) reduction on pump performance

331


β′ 2

H

C ′ u2 u2

D 2m

D 1m

d1

A Hd E Hc

B

D 2i

D 2c

b2

O

α′2

C′2 u2

D C α′2

F Q

O

C m2 u2

B

C2

Q c Q d,

∆D 2

Fig. 11.40. For the calculation of probable diameter reduction

K

C ′2 C2

D – D ′2 ∆D =k 2 2 2

1,2 0,8 0,4

20

40

60

80

100

120

140 n s

Fig. 11.41. Experimental coefficient k for turning impeller diameter with respect to η s as per “RUTSCHI”||

n1ηv1 H n31 ηm 2 n 21 ηh1 Qn1 N n1 n1 = , = 2 and = 3 n2 ηv 2 H n2 ηmi Qn 2 n 2 ηh 2 N n2 n2

In Fig. 11.35 (Hp–Q) curves for three different speeds are drawn, over which the (Hsy–Q) curve C B3 B2 B1 is overlapped. The operating points are B3 B2 and B1 on each speed curve. It can be seen that although by model analysis points are obtained, efficiencies are not equal. At point ‘B1’ efficiency is maximum. But efficiencies at other two point B2 and B3 are less than maximum efficiency. The (Hsy–Q) curve must be altered in order make the pump to work at maximum efficiency at all speeds. However, the loss in efficiency due to speed change is negligible when compared to other methods i.e., regulation by delivery valve control method, as well as changing the impeller diameter keeping speed constant. This method of regulation is economical. There is no limit, provided prime mover capacity is sufficient to run the pump at such speed. Danger of cavitation is avoided completely. This method is adopted only for a single stage units. For pumps in series or parallel, valve control method is always carried out keeping pump speed constant for all pumps.

(d) Regulation by diameter control (Outer diameter trimming to meet the desired head and discharge) Instead of change in speed, which requires a prime mover with speed changing facilities to a greater extent, which involves huge cost, a change in diameter of impeller at constant speed is carried out in chemical industries. As per model analysis,

332


D13 ηv1 H1 D12 ηh1 D51 ηm 2 Q1 N1 , = 3⋅ = 2⋅ and = 5 ⋅ D2 ηv 2 H 2 D2 ηh 2 D2 ηm1 Q2 N2

Here also the (Hp–Q) curve for other diameters, changes parallely with respect to original (Hp–Q) curves. Loss of efficiency in changing the diameter is very high when compared to the loss of efficiency is changing the speed. Maximum efficiency also reduces for other diameters. This is due to the fact that flow at outlet is not streamlined flow when outer diameter is trimmed. There will be fully developed, separated flow at outlet edge of the impeller followed by an aerodynamic wake. Secondary flow increases considerably. Because of this, flow in diffuser and in outlet element is also a fully developed separated flow. As a result, hydraulic efficiency reduces to a greater extent, volumetric efficiency also reduces to a smaller extent. Mechanical efficiency very slightly decreases due to reduced disc friction loss. Overall efficiency reduces to a greater extent. However, in some impellers when impeller diameter is reduced very slightly, the efficiency improves slightly due to improved, better, flow pattern at the outlet of impeller at the passage between impeller, and diffuser, at the diffuser and at outlet elements. Figs. (11.37, 11.38) show the performance of pump with different diameters but with the same speed. The system curve is also shown in the same graph. Comparing the curves Figs. 11.35, 11.36, 11.37 and 11.38, it is evident that efficiency drop in changing diameter is high when compared to the efficiency drop in changing speed. In Fig. 11.41, given the experimental results giving the limit up to which the diameter of impeller can be safely reduced to get better performance i.e., without sacrifice in efficiency. If the same impeller with different trimmed outer diameters is tested in the same spiral casing or diffuser, it is found, than ηmax, for each trimmed diameter lies on the corresponding point, such that 3

D  D  Q1 H ª  1  and 1 ª  1  Q2 H2  D2   D2 

2

The change of Q in proportion to D3 is due to the change of Cm2 and the flow area ‘A2’ perpendicular to Cm2 in direct proportion to the impeller diameter. Referring to Fig. 11.40, the H–Q curve for the original diameter is curve AB. The required values of H and Q are given by point ‘C’. A line is drawn from the origion ‘O’ to pass through ‘C’ and then to meet the original H–Q curve AB at point ‘D’. This indicates that 2

 D2c   Qc   H c    ª  ª   D2d   Qd   H d  This method has been suggested by Berjourn of France |176| and later accepted by other countries. Although theoretically H ∝ D2 and Q ∝ D3 practically, the outer diameter ‘D2’ must be trimmed to a value slightly higher than the value calculated as per the law H ∝ D2, in order to compensate for the quantity which follows Q ∝ D3. However, it is found that, this correction is not necessary if the law suggested by Berjourn is applied.

(e) Regulation by Static Head Change (1) Static Head Fluctuation in Delivery Reservoir In Figs. 11.42 and 11.43, pump (H–Q) curve with a drooping down characteristics regulation by static head change is shown. Water level in the suction tank is taken as (origin, O–O, i.e., x–axis–

333


S H st1

H st

H st2

B1

B2

Q-axis) reference line. Pump 1 supplies water through the delivery pipe 2 to the reservoir 3. Water level in the delivery reservoir corresponds to the static height H, H1, H2 and H3 which is obtained by the meeting point of system curves AE, A1E2, A2E2 and A3E3 , with pump (H–Q) curve. Points B, B1, B2 and C are the corresponding operating points. Pump operation will be stable between heads H and H2 i.e., between points B and B2. For further increase in height H2 to H3 by reducing the flow rate QB2 to QB3, the system curve A1E1 raises parallel to x-axis and will meet the pump (H–Q) curve at two point. (see Fig. 11.43). This corresponding to the level in the reservoirs a2b2 and above. The flow rate is 0 < Q < QB2 the pump operation becomes unstable in this region. When the H H1 pump head is raised further, the system curve A2, E2 O goes up and will touch the H–Q curve at one point a2 H2 C. The moment pump operating point reaches point O C, the operating point momentarily shifts to point A2 and there will not be any flow. Since the system a1 head is higher, the operating point further shift towards left side (Fig.11.26) and will be operated at C point A (Fig. 11.26), where the flow will be reversed. This sudden change creates huge noise vibration and water hammer which depends upon the flow rate and C length of the pipe. Flow in positive direction will Q start only when the water level goes below a1, b1 Fig. 11.42. Regulation by static head change which corresponds to the head H1. Pump operation will be always unstable between H2 and H3 and above. H A3

E3 E2

C

3 E1

B2

A2

B1

E

A1

a3 a2

B3 B2

a1

B1

B Tank

Q –H

a

2

H

H1

H2

H3

A

To the system 1

0 Q lim

QB π

Q

Fig. 11.43. Static head fluctuation in delivery reserve

334


In order to avoid such unstable operation, pump (H–Q) curve must be a gradual raising characteristics up to Q = 0, i.e., without any drooping characteristics at low flow rate or other arise the curve A2–C will be a raising curve as a continuous curve of C B2–B1 . This fluctuation of water level in delivery tank normally exists in systems, where condensers and accumulators are fitted in the delivery line.

(2) Regulation with two different (H–Q) Curves of the Pump

H

In Fig. 11.44, a system operation with two different Q–H (H–Q) curves, one with less drooping characteristics H stmax A max B indicated by (H–Q)B. Continuous line and another with H stmin B min higher drooping characteristics (H–Q)A indicated by dotted A line are shown. It is evident that the higher drooping QBmin (H–Q)A curve (dotted line) is recommended and accepted QAmin for automatic regulated operation by a pressure control Q amax relay, if the fluctuations are large and quite frequent, Q bmx because of smaller difference between Qa max and Qa min for the given Hmax to Hmin. These values are also very close Q to the regular flow rate. But the range between Qb max and Qb min is large for the same head valuation Hmax to Hmin for the curve of B (continuous line curve (H–Q)B. It is necessary Fig. 11.44. Regulation by two different to bring down the value (Hmax – Hmin) in order to get safe (H–Q) curves of the pump automatic regulation. If the operating point (Hmin) is at the right side of maximum efficiency point, the pump may be overloaded due to power increase or pump may be unsteady due to cavitation. Hence, necessary care should be taken in fixing the normal regular operating point (Hmin) especially when automatic relay is fitted. Pump automatically starts at (Hmin) and automatically stops at (Hmax). Normally this pressure difference will be 2 to 3 kg/cm2.

(3) Static Head Change in Suction Reservoir In Fig. 11.45, two different (H–Q) curves of pump are given (H–Q)a and (H–Q)b. Curve CE is the system curve. Static height Hs2 corresponds to low water level in the suction reservoir, and Hs1 corresponds to the high water level in the suction reservoir. Taking the low water level in the suction tank as the normal operating condition, the operating point is B with parameters HB, QB and NB. When water level reaches the higher level, the head will be HA and the parameters are QA, HA, NA, corresponding to point A. The power increase ∆N = NA – NB. Prime mover must be capable of taking this excessive power, if not, regulation is carried out by operating the control valve until QA reduces to Q′B and NB′ . The difference in head HA–HB = h3 is the loss created by the control valve. The control valve system, although simple but not efficient. The efficiency loss will be more. That’s why when any system operates under larger and frequent suction or delivery reservoir level variations, system will be operated with another H–Q curve with larger drooping characteristics by this the change in Q, N, will be considerably small than in the lesser drooping characteristic of (H–Q). However, best and efficiency method is always changing the speed of the pump. The same system is adopted when delivery tank fluctuations take place.

335


(Q – H)a (Q – H)b

h3

E

A B

C HA H S1 HB

H S2

0

Q

QB

01

QA Q–N ∆N

Q–η

NA NB 0

Q

Fig. 11.45. Static head change is suction reservoir

(4) Regulation by Impeller Blades and Inlet Guide Blade Rotation

W 1aC

C 2a = C 1a

2a

W

a

W∞

1x

C 2x

W

W

C 2x = C 1x

∞x

2∞

W

A small rotation of inlet guide blades before impeller blades, to some extent, changes the input energy to the impeller, which in turn changes the total head of the pump for the same quantity of flow. Efficiency almost remains same in this process. However, regulation can be done only to a very small range. Efficiency and head drop very much for further changes. Hence this method is not adopted in pumps. But the blade rotation is widely used with variable pitch adjustable impeller blade axial flow pumps. It is not possible in fixed impeller blade propeller pumps.

2a

u α1a α1x

Fig. 11.46. Regulation by impeller blade pitch control

336


When impeller blade is rotated with respect to its own axis, the angle of attack is changed. If angle of attack is increased, area between two blades, axial velocity, tangential velocity, total head, circulation, lift, quantity of flow increase. The geometrical average velocity w∞ absolute and blade angles also change. In Fig. 11.46 continuous line is for the original value and dotted line is for increased angle of attack. Reducing angle of attack by rotating the impeller blade in opposite direction reduces all values mentioned above including efficiency. At maximum efficiency, total head increases slightly because total head depends upon the curvature of the profile. As already explained in chapter 10, it was shown that hydraulic efficiency ηh is equal to ηh = 1 −

w∞ sin λ u sin (β∞ + λ)

Relative velocity w∞ reduces only a little for a larger value of reduction in angle of attack. Value sin (β∞ + λ) reduces to a greater extent, with the result, hydraulic efficiency drops when angle of attack is reduced or when impeller blade is rotated in the opposite direction (–∆β∞ ). Up to a reduction of β∞ = 23° to 25°, efficiency drop is only very little. Futher reduction in angle of attack reduce efficiency significantly.

11.10 EFFECT OF THE PUMP PERFORMANCE WHEN SMALL CHANGES ARE MADE IN PUMP PARTS 1. Filing the trailing side of the outlet edge of the impeller blade raises (H–Q) curve (Figs.11.47, 11.48, 11.49 and 11.50) to a certain extent. Efficiency may remain same or may increase. But normally decreases. 2. Filing the leading side of the outlet edge of the impeller blade drops (H–Q) curve. Efficiency also drops (Fig. 11.51 and 11.52). 3. Instability in (H–Q) curve at low Q can be removed shifting the inlet edge of the impeller blade towards eye. 4. Thinning the inlet edge by filing both sides of the blade especially more on the leading side, less on the trailing side and rounding off the inlet edge reduces drooping nature at unstable region. 5. Increasing the eye diameter D0 near inlet edge of the impeller blade and sometimes shifting the inlet edge inside improves NPSH. 6. Increasing the blade length at hub and decreasing the blade length at periphery of the inlet edge, in other words, increasing the inclination of inlet edge with respect to outer shroud improves NPSH. 7. Filing the volute tongue removes the unstable nature at low Q, but Q reduces at higher flow rate. 8. Increase in volute area may increase the overall efficiency but optimum efficiency shifts to higher Q. Reduction in volute area will reduce overall efficiency and the optimum η shifts to lower Q.

337


Original vane thickness

Vane thickness after ‘underfiling’ A B

Metal removed

w2 w 2F

C2 C 2F C u2

β2 β 2F

C m2F C m2

C u2F

u2 (a) QF = QHF > H

β2 β 2F

6 12 18 24m Total head (H) and efficiency %

Fig. 11.47. Underfiling of outlet tip of pump blade (Filing the training ride of outlet of impeller blades)

60

40

20 60

40 C m2

C m2F

C u2 = C u2F u2 (b) HF = H QF > Q

Fig. 11.48. Discharge velocity triangles for underfilled vanes

Blunt exit tips Tapered exit tips 20 0

3

6 Discharge (Q)

9

12 L/s

Fig. 11.49. Effect of tapering-off the trailing side of the impeller exit tips

338


120 100 80

Efficiency %

Head – Per cent of normal

Head – Cap

60

Eff % 0

10

40 20 0

0

20

40

60

80

100

120

140

Capacity, Per cent of normal Q1 = Q

Q = Capacity normal Q 1 = Capacity after ‘underfiling’

A B

Head – capacity normal Head – capacity after ‘underfiling’

B = Vane spacing normal A = Vane spacing after ‘underfiling’

Fig. 11.50. Changing pump performance by impeller underfiling C A

d dF

d = dF Removed by overfiling

Overfiled

Removed by underfiling

E Underfiled B

D

Fig. 11.51. Underfiling and overfiling vane tips w1 c m1

c m1F c 1

w 1F c1F

β1

β1F

c u1 c1

w0 δ

δF

Fig. 11.52. Underfiling and overfiling vanes

12 PUMP CONSTRUCTION AND APPLICATION

12.1 CLASSIFICATION Pumps are classified into following groups : : Centrifugal, Mixed, Axial (1) Pump types (2) No. of stages : Single, Two, Multistage (3) Type of connection : Series of parallel (4) Type of construction : Horizontal or Vertical (5) Type of casing : Volute or diffuser type (6) Type of pumping liquid : Water, Milk, Acid, etc. (7) According to usage : Agricultural, Domestic, Chemical, Boiler feed pumps, Circulating pumps, Condensate pumps, Borewell pumps, Cryogenic pumps etc. Some of the most commonly used pumps are discussed here.

12.2 PUMPS FOR CLEAR COLD WATER AND FOR NON CORROSIVE LIQUIDS Pump used for pumping non-corrosive oil or clear water such as domestic, irrigation, industrial, drainage etc., are of either vertical or horizontal construction. The main shaft mounted with impeller is supported by one or two ball or roller and/or thrust bearings as well as one bush bearing support depending upon the type of usage. Pumps for agriculture use are mostly with one bearing with bush bearing support. Impeller is mounted on the shaft as cantilever. Heavy duty pumps and high speed pumps have two bearing supports along with one or two bush bearing supports. The ball or roller bearings are lubricated by oil or special grease, while bush bearings are lubricated by pumping fluid itself. Stuffing box provided will have labyrinth packing of carbon powder impregnated to asbestos packing to avoid air entry into the pump through stuffing box. Pumping liquid from the delivery is circulated through lantern ring provided at the stuffing box. Axial thrust is taken care of by ball or angular contact or roller bearings. In case of high axial thrust, a thrust bearing along with ball bearing is provided. In some of the pumps balancing holes or rear vanes are provided in order to take care of axial thrust. In multistage pumps balancing discs are provided or opposed impeller construction is used to balance the axial thrust. Figs. 12.1, 12.2 and 12.3 illustrate different types of pumps used for pumping clear cold water or for non-corrosive solutions. In this book only a few pumps of special applications and their constructional features are discussed. 339

340


38

Fig. 12.1. (a) End suction single stage pump with single bearing support and with closed impeller. Casing support—Back pullout type

Fig. 12.2. Single stage centrifugal pump (splitcasing, bracket supported)

341

PUMP CONSTRUCTION AND APPLICATION

(a)

(b)

Fig. 12.3. Single stage double ball bearing high speed pump

Fig. 12.4. (a) End suction single stage pump with semi open type impeller and with double bearing support

Fig. 12.4. (b)

342


Fig. 12.5. Pump with mechanical seal and with cooling arrangement

Fig. 12.6. Vertical single stage radial type centrifugal pump ns = 170

343


5 4

1

6 3

2

Fig. 12.7. In line pump 1. Rear bracket, 2. Casing, 3. Wearing ring, 4. Stuffing Box, 5. Bearing housing, 6. Impeller

Fig.12.8. Two stage domestic pump with centrifugal type impeller

344


13 4

6

5

11

10

12

1

14 15

2

16

7 8 9

Fig. 12.9. Two stage domestic pump with centipetal type impellers

Fig. 12.10. Horizontal double suction pump for high head ns = 250

345


Fig. 12.11. Horizontal opposed impeller 2-stage pump ns = 180

4

1

2

3

5

6

7

8

Fig. 12.12. Multistage pump with double entry with opposed impellers

9

346


Fig. 12.13. Multistage opposed impeller pump with intermediate channel connecting two numbers of two stage pump

12.3 OTHER PUMPS (a) Sump Pump Radial type centrifugal pumps can operate theoretically up to 10.00 m suction but practically up to 7 m suction when pumping water is at normal temparature and atmospheric pressure at suction sump. If the sump water level is more than 10 m depth, centrifugal pumps cannot take water from suction sump. The pump has to be lowered down in such a way that suction conditions are well within the limit for safe cavitation free operation. Hence, in open wells, pump and motor are lowered down for safe operation, mostly in agricultural, domestic areas as well as in some industries. Sump pump is one type of pump, wherein the pump is lowered down such that pump is immersed in water. The delivery pipe from volute of the pumps delivers water at floor level or to a delivery tank. The prime mover is an electric motor directly connected to pumps through a long shaft. This is enclosed by a concentric tube. The shaft is supported by bush bearing support very near to the impeller and another bush bearing supports near pump coupling at the top. If the shaft is too long, intermediate supports are provided. Axial thrust bearing is provided in between pump coupling and top bush bearing support. The bearing is lubricated by a separate lubricating grease or lubricating oil. A stuffing box is situated immediately above the lower bush bearing. Labyrinth packings are provided according to the type of liquid pumped such as acids, alkaline, neutral, distilled water, or ordinary water. Lower bush bearing is lubricated by pumping liquid, if it is water or by water if pumping liquid is other than water through a lantern ring. Top bush bearing and intermediate bush bearings are lubricated by lubricating oil. Sump pump is mostly a volute type radial centrifugal pump. Figs.12.14, 12,15, 12.16 and 12.17 illustrate different types construction of sump pumps. Sump pumps are used for a delivery height of 10 to 15 m while suction head will be (0.2 to 0.5 m) of positive suction.

347


20 3 8 = — HOLES 4 267 320

M

57 292

35

E

A B

30°

G D

34 5

280 & 380 MANHOLE

0 60

0 30

40 12

A

DISCHARGE

M

‘J’ NO. OF HOLES ‘K’ SIZE OF HOLES

0 MAX

0 MAX D 20

C 8

20

C 12

P P

Fig. 12.14. Sump pump wet pit type

Fig. 12.15. Sump pump dry pit type

348

Fig. 12.16. Sump pump vertical (channel pump)


Fig. 12.17. Sump pump vertical (channel pump)

(b) Deepwell Turbine If the volute type centrifugal pump is replaced by a diffuser pump, the pump is called a deepwell turbine pump. When the depth of suction well is more than 15 m, the cost of sump pump is increased considerably, due to increased length of shaft, enclosing the tube and delivery tube. Pump cost is reduced by modifying volute into a diffuser by which overall size is reduced. Pumping water is sent through the shaft enclosing pipe, by which separate delivery pipe is avoided. A separate delivery bend at the ground level is fitted to the shaft enclosing pipe, in order to divert water [Fig 12.18 (c)]. Top bush bearing, the stuffing box and thrust bearing are now located at the bend to make the shaft pass through box and straight to the prime mover, whereas water is diverted from vertical direction to horizontal direction. Top bush bearings and intermediate shaft supports are lubricated by pumping liquid itself. The bush bearing at the bottom is located at diffuser. However, in some of the pumps assembly, all bush

349


bearings are lubricated by a separate lubricating oil. But, lubricating oil mixes with the pumping liquid. Hence, this system is provided where lubricating oil mixing with pumping water does not affect the quality of water at end use. The depth of pump can be increased by adding more number of stages. Pumps can operate at a greater depth. Present pumps are operated for a depth of 150 to 200 m for clear cold water pumping. Although pump efficiency is very high about 82 to 88%. Overall efficiency of these pumps will be very low about 30 to 35% due to power loss in shaft, connecting prime mover at the ground level and pumps at the bottom, immeresed inside the sump. These pumps are used not only for open wells but also for bore wells.

145

δ

3 φ 250

180

185

580 290

φ 340

2600

1

φ 340

147

2

129

φ 340

(a)

(b) Fig. 12.18. Deepwell turbine pump

350


1

2

3

4 5

6 7

8 9 10 11 12 13 14 16 15

17

(c) Fig. 12.18. Deepwell turbine pump

351


η,% 80

H ,M 30 H

26

70

22

60 η

18

50

14 5

Fig. 12.19. Submersible pump

7

9

11

13

40 1 5 Q ,lit/se c

Fig. 12.20. Performance of the submersible pump

Fig. 12.21. Impeller, diffuser and return guide vanes

The pump can be radial, mixed or axial flow type. Fig. 12.18 illustrates deepwell centrifugal, and mixed flow pumps. The prime mover can be electric motor with direct or indirect drive or I.C. engine with belt drive

352


(c) Submersible Pumps In order to avoid the heavy transmission loss in deep well pumps, the prime mover is also located along with the pump at the bottom as a mono block construction. The transmission shaft and the stuffing box are eliminated. The electric motor is located at the bottom below the pump, in the bore well. Both pump and motor are submerged in water. Water passes over the electric motor and then to the pump suction. The electric motor is completely sealed and is cooled by the pumping liquid. The motor is completely sealed and filled with lubricating oil for lubrication of bearings. Water cannot enter the motor. Bush bearing supports are provided at the diffuser of each stage and are lubricated by pumping liquid itself. Figs. 12.19, 12.20 and 12.22 illustrate the submersible pump. Axial thrust is taken care of by an axial thrust bearing kept at the bottom of the motor. In some motors, axial thrust pad is used which acts as axial thrust balancing device. A long delivery pipe connected to the pump delivery takes water from pump to the delivery point. Since pumps are connected to electric motors only, speed of the pumps will be synchronous speed 1500, 3000 rpm. These pumps cannot be operated at different speeds as in the case of deepwell pumps. Both turbine pumps and submersible pumps are manufactured to suit different borewell sizes. Due to higher value in length-diameter ratio of the prime mover when compared to ordinary motors, the efficiency of Fig. 12.22. Submersible pump view prime mover is less when compared to ordinary motors, since the diameter of the prime mover is controlled by the borewell size. Depending upon the power required to run the pump, the length of the motor is altered. Due to non-provision of correct length diameter ratio for optimum efficiency, these motors always have lower efficiency.

(d) Oil Filled Motor Pumps In case of open well, however, the electric motors can be made for optimum length diameter ratio so that motor can run at optimum efficiency. The pump is mounted at the bottom of the electric motor. Both the pump and motor are suspended by chain and submerged in water at the suction sump. The pumps is a radial or mixed flow pump. The delivery from the pump passes over the motor, which acts as a cooling liquid for motor. Motor is completely sealed and filled with oil for bearing lubrication. Two ball bearings, one at the top of motor, another at the bottom of the motor are provided. Pump impeller is mounted on the extension of the motor shaft. Instead of stuffing box, oil seals or mechanical seals are used at the shaft to get a leak proof joint between impeller and motor. Delivery pipe for these pumps may be rigid metallic or plastic pipes or a flexible plastic pipe. This arrangement makes these pumps to be situated at any point in open well. These pumps are mostly single stage or two stage diffuser pumps. Water from diffuser passes over the motor at the outer periphery of the motor and then converges into the delivery pipe after the motor. Normal operating height of these pumps will be 10 m to 15 m. (Fig. 12.23).

353


2

5

4

3

6

7

1

8

9

Fig. 12.23. Oil filled motor pump 1. Impeller, 2. Delivery channel, 3. Casing, 4. Seal, 5. Compression spring, 6. Stationary ring, 7. Bronze ring, 8. Rotating spring, 9. Suction stainer

Fig. 12.24. Mixed flow pump (horizontal)

Fig. 12.25. Moulding

Submersible pumps and deepwell turbine pumps are used for irrigation, industrial, circulating, mining in open well type as well as for borewells. These pumps occupy major percentage in application than any other high capacity pumps. Two types are used : (1) Transmission type and (2) Borewell type. Transmission type consists of three parts: (1) pump is located at the bottom, (2) driving part and supporting part located at the ground level and (3) the intermediate supports located at frequent intervals of the delivery pipe. Because of its unique application, the pumps and supporting parts have a special construction. These pumps must have a limited outer diameter and must be in cylindrical form at outside to suit the borewell. Due to this restriction, these pumps have a number of stages connected in series, depending upon the pumping head. Normal operating head varies from 25 to 400 m. Some of the special designs have head up to 600 m. Pump itself has three sections: (1) the entry to the 1st stage through suction net fitted to the 1st stage, (2) impeller radial or mixed and (3) the diffuser with return guide blades for smooth entry to the 2nd stage suction or in general to the next stage suction. Every stage is symmetrical to each other. Each stage assembly consists of vaneless suction entry, impeller, with both shrouds closed type for radial type and semi open mixed type with only rear shroud for mixed flow type. Axial flow type units are also used but mixed flow type units occupy a major percentage of deepwell or submissible pumps.

354


Fig. 12.26. 30° coxe angle diffuser with impeller for vortex design (short curve construction for plane vane developments)

12.4 AXIAL FLOW PUMPS Axial flow pumps are used as circulating pumps for stationary conditions and for irrigation systems. Quantity of the pump ranges from 0.2 to 18 m3/sec, total head from 1 m to 22 m. Pumps work always under positive suction head. The impeller will be always under drowned condition. Mostly pumps are operated in vertical condition. Horizontal pumps are rarely used. In both systems, pump is immersed in water or pumping liquid as such total head is very low, pumps cannot operate with suction. Propeller pumps have fixed impeller blade system. Variable pitch impeller blade pump is combination of 5 to 7 propeller pumps where in impeller vanes can be adjusted to different blade positions. Figs. 12.27, 12.28 and12.36 illustrate vertical variable pitch axial flow pump. Construction of all axial flow pumps will be same, but with modifications to suit the site conditions after pump at the delivery line. The rotating mechanism for impeller blades is located inside the impeller hub. A long rod passes through the shaft and extends a little beyond pump coupling. By rotating this rod, impeller blades are rotated. Pump has to be stopped for blade adjustment. Now-a-days, pump impeller blades can be adjusted and regulated by a separate regulating mechanism by which pump need not be stopped for impeller adjustment. Suction end of the pumps forms the shape ‘bell mouth’ in order to provide smooth entry without flow separation at inlet. Impeller is mounted on the shaft as a cantilever element. Impeller is located inside a cylindrical housing. The contour in axial direction will be curved to suit the impeller blade rotation and at the same time minimum clearance is maintained at all positions of impeller blade. The impeller is followed by an axial diffuser which has a bush bearing to support the shaft which passes through the diffuser. Diffuser is followed by a bend or followed by a straight pipe and then bend to suit the site conditions. The shaft supporting the impeller at one end as cantilever is supported by two bush bearing supports and one axial thrust bearing. One bush bearing is located at diffuser, which is lubricated by pumping water itself while another bush bearing is kept at the bend. In case of intermediate pipe between bend and diffuser, intermediate bush bearing supports are provided in between bend and straight pipe. These bearings are also lubricated by pumping water itself. A stuffing box with labyrinth packing is provided after the bush bearing at the bend to avoid leakage of water from bend to atmosphere. Axial thrust bearing is provided at the top, above stuffing box and below pump coupling to take care of axial thrust. Bearing is lubricated by a separate lubricating oil or grease. The body of the pump is supported by external supports in such way that pump is immersed inside the suction sump. Axial pump construction

355


is very simple, easy to maneuver, occupy smaller floor space. Efficiency ranges from 85 to 92%, Fig. 11.6 (c) illustrates universal characteristics of an axial flow pump. Although propeller pumps are operated at optimum efficiency condition without any deviation, due to sharp fall in efficiency at other regions, variable pitch pumps are used for a wide range of operation without any appreciable loss in efficiency. The only drawback in these pumps is that pumps must be started only at full open condition for minimum power consumption at the time of starting the pump. Hence, in some of the installations, a transfer line connecting the pump delivery and suction sump is provided along with main line. At the time of start, main line will be closed and transfer line will be opened. Gradually transfer line will be closed while main line will be opened, until required flow in main line is achieved. This type of arrangement keeps the pump operated at high efficiency. In order to reduce the height of the pump in transport type units, the outlet bend is made as 90° short bend with a number of ribs to guide water. (Figs. 12.27, 12.28, 12.29 and 12.30). 9

10

8

7

6 5

4 3 2

1

Fig. 12.27. Vertical axial flow low depth-circulating pump 1. Suction mouth, 2. Impeller housing, 3. Impller blade, 4. Diffuser blade, 5. Shaft bush at diffuser, 6. Delivery bend, 7. Shaft, 8. Bearing, 9. Coupling, 10. Delivery flange

356


Fig. 12.28. Vertical axial flow high depth circulating pump SECTIONAL VIEW OF AP-5

570 PCD φ

1620

1415

Fig. 12.29. Horizontal axial flow propeller pump

250

750 PCD

700φ

470φ

750

500


357

Introduction of such short bend at the outlet of the pump creates uneven pressure and velocity distribution as well as increased hydraulic losses, which reduces the total head of the pump. Provision of radial ribs at the bend reduces the effect of such drawbacks. However, experimental investigation on such bends with ribs reduces efficiency by 7 to 9%. In some of the pumps, a cylindrical enclosure to ensure a smooth flow at bend is provided around the shaft and the shaft is not in contact with the pumping liquid. This also reduces overall efficiency by 10 to 12%. In transport type axial flow pumps, a side entrance apart from main entrance at suction is provided. Suction will not be a bell mouth shaped, instead a straight cylindrical piece with additional side Fig. 12.30. Condensate pump (double opening. This suction element will have a number of suction) axial ribs and impeller nose. (Fig.12.38). Flow from the main entrance is slightly deviated by the side entry if liquid which is streamlined by axial suction ribs, kept radially connecting the impeller nose and suction casing. When flow is only from the side opening, pump works only at 15 to 20% of normal working. This system is adopted in ships. Performance of pumps is found to be better if the ribs at the 90° bend placed in a streamlined position in the form of a profile. It is found that the performance of the pump remains same as that of ordinary construction i.e., with long low head bend, but the total height of the pump is considerably reduced. The hydraulic losses remain same even at 1.1 times normal flow rate.

12.5 CONDENSATE PUMPS Condensate pumps are either single stage or two stage radial type centrifugal pump. Two stage units are common. Selection of number of stages depend upon the parameter of the pumps and its characteristics. Based on the methods adopted for pump regulation, construction of pump is selected. Condensate pumps can be either vertical (used in ships) or horizontal (for land use). Vertical pumps are selected based upon the suction head available. Pumps of turbodynamic type are of horizontal type because required section head can be easily obtained only in such horizontal type. The size of the pump is also considerably smaller. Pumps are always erected very near to condenser in such a way that pump is always filled with water. Flow velocity in suction will be 0.5 to 1.0 m/sec and in delivery 2 to 3 m/sec. Top most point of suction flange of the pump is always kept in line with the highest water level in the condenser. This line is the level where boiled water and steam mixture exists at the boiling temperature for the pressure prevailing at the pump suction. Also liquid and vapour are at separated condition. In the absence of water, steam mixing line or insufficiency with respect to the impeller size the liquid entering the pump will be a liquid vapour mixture which will produce unstable, with noise and non-uniform working of impeller especially at limiting condition of operation.

358


4 2 3

2

1

2

Fig. 12.31. Single stage condensate pump 1. Impeller, 2. Stuffing box bush, 3. Lantern ring for seal, 4. Lantern ring for air entry pretention

359


3

2

4

1

Fig. 12.32. Two stage condensate pump 1. Impeller, 2. Intermediate bush bearing support, 3. Upper bearing support, 4. Transfer pipe

Fig. 12.33. Multistage horizontal opposed impeller with external crossover pipe connection condensate pump

360


Fig. 12.34. Two stage horizontal opposed impeller singe entry and double exit condensate pump

A single stage condensate pump is given in Fig.12.31. Impeller 1 is located at the bottom and the suction flange is at the top. Shaft is supported by three bush bearings 2 lubricated by water taken from delivery line through a filter in order to avoid construction materials, sand etc. to get into the units bearing clearances. A ball bearing support is provided at the top of the pump. Axial thrust is balanced by vent holes provided at the impeller back shroud. Stuffing box is provided above the bush bearing and condensate water is circulated through lantern rings provided near bush bearing. Another lantern ring at the middle of the packing material is provided through which air is sent to get perfect sealing. A two stage condensate pump is shown in Fig.12.32. First stage impeller is provided at the bottom of the pump. The suction flange is located at a higher level in order to keep impeller always immersed in water and also air entry through the stuffing box is stopped. Second stage impeller is located above the first stage impeller and in the opposed position to balance axial thrust. First stage impeller develops1/3 of total head and second stage impeller develops 2/3 of total head. Pumps are also designed to develop equal head by each impeller. Shaft is supported by two bearings: (1) Lower bearing is a bush bearing (2) Located inside casing between two stages and lubricated by hot water. Top bearing (3) is either bush bearing or angular contact ball or roller bearing to take care of the excess unbalanced force. If an axial thrust bearing is used at the bottom, the top bearing will be bush bearing. Fluid coupling can be used in case ball bearing is used at the top. If not flexible coupling is used. Wearing ring at impeller is provided in case the pump rings near are at cavitating zone. The clearance will be more than the normal. Mostly clearance will be filled up with a mixture of vapour and liquid. Stuffing box is provided at the second stage impeller. If the pump runs under low head or near cavitation region, lantern ring if provided. If pump operates at higher head, lantern ring need not be provided. To avoid air entry through stuffing box, condensate water is circulated. In some design, water from external source is circulated in order to


361

circulate water even when pump is not running. Packing materials must be checked periodically and changed since packing materials become hard at a faster rate especially at high temperatures. Casings are made from bronze, Impellers are made from monel metal (1/3 Nickel and 2/3 Copper), and shaft is from stainless steel material. Condensate pumps operate over a wide range of operating area. Radial and axial thrusts must be balanced in order to avoid fatigue stress on the shaft. Owing to the position, second stage impeller, located near delivery point will be running at positive pressure, whereas first stage impeller is mostly at the suction pressure. Axial and radial thrusts of second stage is higher than that at first stage especially when operated at low discharge condition. To avoid high value of axial and radial thrust, first stage impeller will be a double suction impeller and second stage will be two identical impellers located opposite to each other by which axial thrust is perfectly balanced and pumps can work at a wide range of operation. To avoid radial thrust, volute casing of first stage will be a concentric circular passage in some of the condensate pumps. (Figs. 11.33 and 11.34). Condensate pump operates under 94 to 97% vacuum at a temperature of 25 to 30°C. Although temperature is low, yet liquid will be very near to vapour state due to high vacuum, due to low vapour pressure at low temperature. Thats why, condensate pumps are located below the condensate in order to ensure only liquid entry at the suction of pump. Pumps will be always filled with water (No priming is necessary) and will not have any vapour mixing. Positive pressure will prevail at impeller inlet, around 0.5 to1.0 m head. Due to high vacuum at inlet, cavitation becomes most important aspect in condensate pumps, than overall efficiency. Cavitational coefficient C ranges from 1500 to 2500 in condensate pumps. Stuffing boxes are properly cooled from external cold water supply. Two labyrinth rings are usually provided, one at the middle and another very near to the impeller to avoid air entry into the condenser. Condensate itself is circulated through labyrinth packings. In some designs condensate supply to the labyrinth near impeller and air to the labyrinth at the middle are supplied at low pressure, since stuffing box always works at low pressure (1.0 to 1.5 ata). Condensate pumps are of horizontal or vertical type. Figs.12.30, 12.33 and 12.34 show a horizontal type condenser used for land operations such as boiler plants for power stations, industries etc., condensate is taken from the delivery line and passed through stuffing box labyrinth to avoid air entry into pump not only at working condition but also at idle condition thereby correct vacuum is always maintained. Where pump is working at very near to cavitation condition, or at cavitation condition air bubble, released from the liquid is collected at the top chamber of the section chamber and taken out by a vacuum pump. Ball or roller bearings are lubricated by consistent lubricant.

12.6 FEED WATER PUMPS Feed water pumps, used in power stations to supply almost boiled water to boilers are, low discharge and very high head pumps running at very high speeds. Feed water enters the pump from deaerator and is delivered into boiler at nearly boiling temperature. In order to avoid cavitation at the inlet of the first stage, impeller inlet is designed with large area. Positive suction pressure is always maintained at inlet of the first stage impeller. Impellers of all other stages are same type, since all other stages work under above atmospheric pressures. Feed water pumps must run continuously in parallel with other similar pumps at all conditions and under stable conditions. Pump characteristics (H–Q) must be a drooping down characteristics from

362


Q = 0 to Q = Qmax. Due to high temperature, of operation, all pumps parts must be manufactured with high accuracy. Proper clearances must be ensured so that pump can run smoothly without undue vibration even at high temperature of pumping liquid passing through the parts. Power capacity of pump will be very high in the order of 12 MW to 35 MW. Delivery pressures are very high such as 200 to 300 ata. Speed ranges from 3000 rpm to 25000 rpm. Total head ranges from 400 to 650 m. Feed water pumps must possess : (1) high quality manufacture and stable, continuous operation, (2) high efficiency, (3) high quality, quick and efficient regulation, (4) less weight for the given specification and (5) must be brought to operating condition i.e., to meet the demand in a very short time (15–20 sec.). Under very high speed of operation, in order to avoid cavitation, booster pumps are used before feed water pumps. High speed steam turbines or electric motors are used as prime movers. Feed water pumps are mostly multistage stage type pumps. Diffuser blades, return guide vanes are made in one disc. Impellers and diffuser and return guide blade discs are mounted one after another on the shaft, inside a drum. Concentricity is maintained between impellers and diffusers, when assembly is carried out inside the drum. Axial thrust is balanced by the balancing disc, which is a separate assembly made after assembly of all impellers, diffusers as well as inlet and outlet flow passages. Stuffing boxes are cooled by external supply of water for which, special annular ring is provided. Bearings are lubricated by consistent lubricants. Since electric motors operate at a relatively low speed, more number of stages are used. High speed turbine run pumps have less number of stages. Booster pumps are used to have a sufficient suction head at the first stage impeller inlet of the feed water pump. Mostly these pumps are double suction type pumps. Since distilled water is at almost boiling temperature, oxygen released from water chemically react with the material of the pump and erodes the material gradually. In Figs. 12.35 and 12.36 feed water pumps are illustrated.

Fig. 12.35. Multistage pump with side suction for high head pump

363


Fig. 12.36. High temparature horizontal type feed water pump

12.7 CIRCULATING PUMPS Circulating pumps are used for cooling main liquids. In power plant, circulating pump is used to cool the condensate. Pumps draw water from a pond, sent to the condensate, where it receives heat from the condensate under indirect method. Hot water from the condensate is brought to the pond and sprayed in cooling towers. Water is cooled by the atmospheric air and reaches the pond back. Circulating pumps are normally fitted near the pond which is located far away from power station. These pumps work under a low head involving main friction in long delivery pipe and resistance offered by the condense but handles large quantity of water. These pumps must work consistently in parallel with other similar pumps. High capacity double suction pumps or high capacity axial flow pumps are normally used as circulating pump. Centrifugal Pumps are used more than axial flow pumps due to high power consumption at part load by axial flow pumps. Pumps work under positive suction head. Flow velocity in pipe will be approximately 2.5 to 3.5 m/sec. Variable speed electric motors are often used for flow regulation. Two pumps in parallel are always used as circulatory pumps. Second pump will be operated only when

Fig. 12.37. Vertical double suction circulating pump and ns = 270

364


the load is increased. Flow from the circulating pump to condenser is determined from the quantity of steam to be condensed into liquid at the boiling temperature i.e., removal of latent heat, under normal load. A reserve of 10 to 15% extra will be added. If the circulating pump supplies water to other cooling systems such as oil cooling etc., proportionate increase in discharge must be taken into account. Total head of the circulating pump will be the resistance offered by condenser tubes and connecting pipes between pump and condenser. Total static head will be very small in the order of 0.5 to 1 m. Total (dynamic and static) head will be around 7 to 9 MWC. The speed of rotation will be always higher around 3000 rpm. Quick and fast regulation is essential for circulating pumps since quantity must be sufficiently enough only to remove latent heat of the condensate. The quantity of the condensate depends upon the output. Hence, as and when output load changes, quantity of circulating pump must be changed accordingly without delay. Change in flow is achieved by increasing the speed of the pump, since in this method efficiency of the pump is higher and maintained more or less same during other speeds. Flow control by regulating valve adjustment is not recommended since it involves heavy loss in efficiency and corresponding increase in power of the prime mover. Vertical double suction pumps, which posses higher suction characteristics and also minimum axial thrust. Moreover, vertical pumps occupy smaller area and provides a smooth lined inlet passage. (Fig. 12.37).

5

4 9 8 7 3

6

1 2

Fig. 12.38. Vertical axial flow circulating pump

Stuffing box is provided at the top of the shaft and in contact with suction volute. In order to avoid air entry into the pump, water is taken from the delivery side of the pump and is passed through lantern ring kept at the middle of the labyrinth packing in the stuffing box. Bush bearings at both ends of the shaft give proper alignment and lubricated by lubricating oil.


365

Fig. 12.38 illustrates a vertical axial flow pump. As against normal pumps, the pump shaft is enclosed by a circular cylinder, to avoid, the liquid to come in contact with the shaft. Bush bearings at the end of the bend and at the diffuser (9) keeps the shaft aligned. Inlet guide blades (1) with the impeller nose (2) provides proper flow direction at inlet. Impeller (6) is followed by a diffuser (8) and a bend with guiding ribs to guide the circulating water as well as to distribute the load above the bend to the bottom through these ribs.

12.8 BOOSTER PUMPS Booster pumps are used in a system where feed water passes through deaerator and the deaerator could not be located at correct height due to the conditions in site in order to provide sufficient positive suction at inlet to condensate pump or feed water pump, so that these pumps can work without any cavitation. Because of booster pumps, dearerators could be located at a great height (about 10 to12 m). Because of positive suction for cavitation free operation, feed water pumps and condensate pumps can work at high speed without any cavitation effect. Operation of the system with deaerators at a height of 3 to 4 m is more rational than using a low speed feed water pumps. The cost of economiser will be low, when a booster is used, because, low pressure economiser need be used before the feed water pump. Moreover, the feed water pump can be located very near to the boiler which reduces the length of high pressure delivery pipe line erected between pump and boiler. Since high speed feed water pumps can be used, deaerators can be kept at a greater height, low cost low pressure economiser can be used and high pressure feed water delivery pipe length can be considerably reduced, by using a booster pump. All modern power stations use booster pumps before feed water pump as well as before condensate pumps. A greater advantage is that a very high speed turbine run (speed ranging from 10000 to 25000 rpm) single stage feed water pump, instead of high speed (3000 rpm) electric motor run multistage centrifugal feed water pump can be used. Always booster pump capacity will be at least 50% more than maximum flow rate of feed water pump, since at full load operation ≈ 30 to 35% of condensate drawn from deaerator is recirculated in deaerator through recirculating line. Such recirculation of condensate through deaerator improve the quality of feed water. This process improves the function of high pressure boilers. The positive suction head at suction of the pump, taking water at boiling temperature from deaerator is determined from the static height of the deaerator above pump and the losses in the suction pipe of the pump. Providing enough positive suction at inlet of the pump is very important especially when the liquid is at a high temperature, (102 to 150°C), because even 1°C change in temperature changes suction head by 0.6 to 0.7 MWC. It is essential to test the model of the suction pipe connecting deaerator and booster pump, for minimum loss condition and reproduce the same to the natural prototype unit. In order to get positive suction at inlet of the pump for the boiling liquid pumping, instead of increasing the absolute pressure the boiling pressure is decreased by reducing the temperatures of the liquid at entry of the pump. In order to achieve, a cooling unit is provided between deaerator and pump. Condensate from the condenser passes through the cooling tubes to deaerator. Feed water from deaerator to pump passes through the space between tubes of the cooler and then to the pump. Introduction of cooler decreases the effectiveness of the heat cycle of the plant. Less the effectiveness achieved, if

366


temperature reduction is larger. Although use of cooler is not economical, it becomes essential for conditions when there is no possibility of using a booster pump or it is essential to get more positive suction if after using booster pump. The quantity for a booster pump is determined as the sum of quantity passing through the feed water pump, recirculating quantity in deaerator and the excess reserve quantity estimated (10 to 40% of Q). Deaerator quantity should never be less than the feed water pump capacity. The head for the booster pump is the sum of all losses from booster pump to feed water pump, static height difference between these two pumps. Pressure needed at the suction end of feed water pump is determined taking into account the partial evaporation at inlet of the feed water pump, for example, if the temperature of liquid is 107°C, but the temperature at the inlet of feed water is 115°C due to preheat the increase in head will be 0.8 kgf/cm2. This pressure increase should also be accounted for booster pump head calculation. Normally booster pumps operate at 20 m to 60 MWC depending upon the construction, specific speed of feed water pump, relative height (static) difference between booster pump and feed water pump, pressure at by pass line if any. Operating speed is determined by the suction head available. Booster pumps and condensate pumps are similar in construction. Both pumps draw water from closed suction tank under pressure. Booster pumps are single or two stage pumps. First stage impeller will be a double suction type depending upon the flow rate required. In most cases, booster pump with deaerator is taken as one unit for pumping hot water. A recirculation line from the delivery of the pump to the deaerator is also provided if necessary. Booster pumps are always connected and run along with either condensate pumps or feed water pumps as one unit in order to reduce the length of pipeline between booster pump and other pumps, which in turn reduces the friction loss thereby increase the total head. Also total space occupied is less. However, initial cost will be more. Condensate pumps are low power pumps. For higher economy and increased effectiveness in operation, booster pump and condensate pumps are connected as single unit. Steam turbine is used as prime mover. Both these pumps are assembled on the same shaft since total head and quantity pump for these two pumps are nearly same. Speed of these pumps are selected for anti cavitation characteristics in both these pumps. Connecting pipeline is a complicated inside construction. This model is tested for losses. Improved model only is used in prototype site construction. Fig. 12.39 shows the condensate and booster pump together. Booster pump is located above condensate pump and it is a two stage unit. By providing opposed impeller construction, axial force is reduced almost to zero. Ist stage impellers have wide inlet end to take care of the positive suction available. Head developed by each stage impeller is equal to 50% of total head. Booster pump bearings are lubricated by the condensate at 25°C whereas pumping temparature of fluid in booster pump will be ≈ 105°C and condensate is deaerated to remove oxygen. High speed single stage condensate booster pumps are also used. But oxygen removed deaerated water at high temparature is used. Axial thrust in such cases is done by balancing holes and balancing disc. Top bearing is a ball and roller bearing arrangement (3) lubricated by lubricating oil supplied by the screw type oil hub which supplies oil from the casing. Oil is cooled by an air cooler (5).

367


3 4 5

2 1

Fig. 12.39. Condensate booster pump

Feed Water Booster Pumps Booster pumps are considered as first stage pump of feed water pump. These two pumps are combined together and driven by turbine. The suction pipe between booster pump and feed water pump is considerably reduced. Due to vast difference in speed, feed water pump is directly connected to the prime mover and booster pump through a gear drive in order to run at a reduced speed. A one way nonreturn valve is fitted in the pipeline connecting feed water suction and booster pump delivery.

368


1

5

Outlet

4

Feed water entry 2 3

Fig. 12.40. Feedwater booster pump

11

9

7

2

4

6 5

3

14 1

18

16 15

17

20 19

21

26

24

22 23

25

30

28 27

29

32 31

34 33

36 35

38 37

40 39

12

13


8

10

41 42

43 44 45

46

Fig. 12.41. Non-dog pump with S-type impeller (impeller is semi-open type)

369

370


In Fig. 12.40 a combined feed water booster pump is given. Booster pump runs at a lower speed through a gear reducer box. (1) Both pumps are centrifugal, vertical type and spiral casing type impeller. (2) It is a double suction impeller bush bearing. (3) Ball bearings (4) and roller bearings (5) are used to support the shaft. Axial thrust is almost zero due to the adoptation of double suction impeller. Air entry through stuffing box is avoided by the lantern ring provided at the middle of labyrinth packing. Lantern rings get water supply from the delivery pipe of pump; through connecting tube.

12.9 PUMP FOR VISCOUS AND ABRASIVE LIQUIDS Pumps are used not only for pumping water, oil, acids, alkaline, neutral, high temparature liquids but also for pumping liquids with solids, paper pulp, molasses and such similar chemical solutions, and even pumping abrasive liquids. These pumps are used to transport such liquids from one place to another. Pump performance normally do not change so long as the solids in suspension does not exceed 7% of the total quantity of pumping liquid performance of such pumps remain same as that of water. However, when abrasiveness of the liquid is very high, consistency of the pumping liquid exceeds 7% or the space between rotary and stationary members must be larger. The wearing outer ring clearance must be flushed with water for trouble free operation of the pump. The pump must be made in a special design. These pumps are called ‘channel pumps’ since flow passages are in the from channel. Fig. 12.42 gives one such pump used for slurries. Typical construction of these pump is that the flow passages are wider, volute casing is made in the form of a concentric circle instead of spiral shape, without any change in area of cross-section, wearing ring and stuffing box are cleaned by water supplied from external source to avoid solid in suspension entering the clearance, thereby cooling the internal due to rubbing.

Fig.12.42. Slurry pump for low concentration

371


(a)

(c)

(b) Fig. 12.43. Slurry pump for high concentration (closed impeller)

Fig. 12.44. Pump for low concentration paper pulp (semi open type impeller)

Fig. 12.45. Pump for high concentration paper pulp

372


In Fig.12.43 more free spaces between impeller and casing is provided. The impeller blades are made in S-form with only two blades. Wearing ring clearances near stuffing box is guarded by the self pumping action of the blade, which is done by the holes made at the rear shroud of the impeller. Fig. 12.44 is the pump for pumping paper pulp. Impeller is a semi open type with 2 to 3 blades only. To avoid the abrasion of paper pulp with space between water and stationary parts, separate covers are attached so that the main casing will not be affected. The stuffing box is provided with lantern ring, through which clear water is circulated. Fig. 12.45 is given pump assembly of thick pulp which is used in textile mills, paper pulp and other industries where thick consistent fluids are to be pumped. Impellers are made in the form of a screw and semi open type. The inlet edge is made a little sharp for smooths entry of the fluid. Mostly anti corrosive steel is used such as stainless steel SS316 or liquid contact area. Fig. 12.46 is a pump used for abrasive solution mixer with water pumping in places, such as power stations, mines etc. Here also a special lining is provided around the casing

Fig. 12.46. Pump for abrasive liquids

Fig. 12.47. Single stage vertical high temperature oil pump (for petroleum liquid)

373


especially at the space between rotating and stationary area of pump to avoid erosion of main casing. Mostly materials used are anti abrasive steel such as maluable iron or steel mixed with manganese. Fig.12.47 pump is a multistage pump used in petroleum industries.These are very high pressure pump work up to 1000 m and the liquid temperature is around 400°C. Provision for thermal expansion is provided at all spaces between rotary stationary elements. The pump is always started when the pump elements temperature is equal to pumping fluid temperature.

Fig. 12.48. Multistage pump for high temperature oil pumping

4567321 Flow Diagram

Fig. 12.49. Multistage drum (Barrel) type water flow pump with opposed impellers and with double casing and double volute

In order to achieve, pump is preheated by filling the pump with the pumping fluid initially at the running temperature. After attaining normal conditions, pump will be started. These pumps should not be started without preheating. These pump are also used for pumping chemically reactive solutions, that could easily catch fire. Pump materials are selected to have anti corrosive property. Stuffing boxes are properly cooled and sealed by circulating clean water through lantern rings or by suitable liquids which

374


will not affect the quality of the pumping liquid when it mixes with pumping fluid through leakage in stuffing box. In some of them, mechanical seals are used instead of stuffing box arrangements. These seals are cooled by external source. In some of the constructions, double suction pumps are used instead of diffuser type multistage pumps. To avoid cavitation, the first stage impeller is specially designed. An axial pre whirl pump is used before the first stage impeller. Bearings are cooled by external oil supply, which acts as a lubricant as well as coolant for bearing. Axial thrust is balanced by balancing disc. In some pumps (Fig.12.47) opposed impellers are used to equalise the axial thrust. Entire assembly is carried out inside an outer casing. Hence, these pumps have two enclosures, one to accommodate impeller, diffuser, return guide passage of each stage. The entire stage assembly is accommodated inside second enclosure which contains suction and delivery months. Different types of pumps used for pumping clear cold water or non-corrosive solutions.

Fig. 12.50. Light weight high speed engine driven monoblock pump

In Fig. 12.50 a light weight (casing 1½ kg and impeller 1/2 kg) portable pump is made up from aluminium alloy for anti-corrosive properties. The pump runs at 6000 rpm. The impeller is especially designed for cavitation free operation at inlet. The vanes at inlet are of double curvature design. The pump is coupled to a petrol engine as a monoblock. This pump is used for agricultural and for industrial application.

13 DESIGN OF PUMP COMPONENTS Design No. D1-A DESIGN OF A SINGLE STAGE CENTRIFUGAL PUMP 1. Specification Total Head (H) = 13.8 m (42 feet) Quantity (Q) = 16.0 lit/sec (210 gpm), Speed n = 1800 rpm, Size 80 mm × 65 mm (3′′ × 2½′′)

2. General dimensions Specific Speed (nS) =

3.65n Q (H )

3/ 4

=

3.65 × 1800 × 0.016 = 115.2 (13.8)3 / 4

It is a radial type centrifugal pump. Norminal diameter, D1nom = 4.5 × 103

Hydraulic efficiency,

ηh = 1 –

= 1–

Volumetric efficiency, ηV =

3

Q = 4.5 × 103 × n

3

0.016 = 93.1 mm. 1800

0.42 (log D1 nom − 0.172)2 0.42 = 0.87 or 87%. (log 93.1 − 0.172) 2 1

1 + 0.68 ns

−2 / 3

=

1 1 + 0.68 (115)−2 / 3

= 0.968 = 96.8%.

Assuming mechanical efficiency, ηm = 0.96. Overall efficiency, η = ηh .ηV .ηm = 0.87 × 0.968 × 0.96 = 0.808 or 81%. Output power, No =

γQH 9.81 × 1000 × 0.016 × 13.8 = = 2.17 kW (2.91 hp). const. 1000 375

376


Input power,

Ni =

No 2.17 = = 2.68 kW (3.59 hp) η 0.81

Assuming an overload of 15%, Input power Ni = 1.15 × 2.68 = 3.1 kW = 4.12 hp. Torque,

T=

Ni 3.1 × 60 = = 0.0165 kN.m. ω 2 × π × 1800

Taking the shaft material as ‘En8’ (Mild Steel), Ultimate Stress (fm) = 35 N/mm2 and taking factor of safety (FS) as 2 for uniform speed of rotation. Working Stress, (f S ) =

Shaft diameter,

(dS) =

fm 35 17.5 × 1000 × 1000 = = 17.5 N/mm2 = = 1.75 × 104 kN/m2 2 1000 FS 3

16T = πf S

3

16 × 0.0165 π × 1.75 × 104

= 0.01687 ≈ 17 mm.

Taking fatigue stress (bending and shear) into account, minimum shaft diameter dS is taken as 25 mm, dS = 25 mm. Hub diameter, dh = 1.25 dS = 1.20 × 25 = 30 mm.

3. Inlet dimensions Theoretical discharge,

(Qth) =

0.016 Q = = 0.0165 m3/sec 0.968 ην

For the suction pipe diameter, DS = 76.4 mm, eye diameter of the impeller is taken as 76.4 mm (3′′), the axial velocity at impeller eye (C0 ) is C0 =

4 × 0.0165 4Qth = = 3.6 m/sec. 2 π × (0.0764) 2 πDS

The diameter of the inlet edge of the impeller blade is taken as 90 mm. The flow velocity before the inlet edge of impeller blade (Cm0) is Cm0 = 0.06 3 Qth n 2 = 0.063 0.0165 × (1800)2 = 2.26 m/sec. Inlet breadth,

B1 =

0.0165 × 1000 Qth = = 25.8 ≈ 26 mm. πD1Cm1 π × 0.09 × 2.26

Taking, K1= 1.4, Cm1 = K1 Cm0= 1.4 × 2.26 = 3.164 mps u1 =

πD1n π × 0.09 × 1800 = = 8.49 m/s ≈ 8.5 m/s. 60 60

Assuming normal entry, (Cu1 = 0), Inlet blade angle ‘β1’ will be C  3.164 = 0.3722  = 20.42° β10 = Arc tan  m1 = 8.5  u1 

Allowing an angle of attack δ ≈ 4.5°, β1 = β10 + δ = 20.42 + 4.5 ≈ 25°

377

DESIGN OF PUMP COMPONENTS

4. Outlet dimensions Manometric head,

Taking,

H m=

Cu 2 =

13.8 H = = 17.23 m. 0.87 ηh

Cu 2 = 0.5 u2

− Cu 2 u2 Cu 2 u 2 H Hm = = = g g ηh First Approximation u2 =

D2 =

Outer diameter, Taking,

9.81 × 17.23 = 18.45 m/s. 0.5

gH m = Cu 2

60u2 60 × 18.45 = = 196 mm. π × 1800 πn

Cm3 = 0.8 Cm0 = 0.8 × 2.26 = 1.81 m/sec.

Taking,

K2 = 1.2 and sin β2 = sin β1 .

Outlet blade angle, Outlet flow velocity, No. of blades,

w1 = 1.18. w2 1.2 K 2 w1 Cm3 . . = Sin 25 × × 1.18 × 0.8 = 0.3419. 1.4 K1 w2 Cm 0

β2 = 19.99° ≈ 20°

Cm 2 K 2 Cm3 1.2 = = × 0.8 = 0.687 Cm1 K1Cm 0 1.4

Cm2 = 0.687 Cm1 = 0.687 × 3.164 = 2.18 m/sec. Z = 6.5.

1 + = 6.5

D2 + D1  β + β2  . sin  1  D2 − D1  2   25 + 20  196 + 90 sin   = 6.71, Z is taken as 7. 196 − 90  2 

ψ = 0.6 (1 + sin β2 ) = 0.6 × 1.3420 = 0.8052. p=

2ψ ⋅ Z

1 r  1−  1   r2 

H∞ = (1+p) Hm = (1.2915) 17.23 = 22.26 m.

2

=

2 × 0.8052 × 7

1  90  1−    196 

2

= 0.2915

378


Second Approximation

Outlet blade velocity,

Cm 2 u2 = + 2 tan β2 2.18 = = 2 tan 20

Outlet diameter,

D2 =

2

 Cm 2    + gH ∞  2 tan π2  2

 2.18    + 9.81 × 22.26 = 18.07 m/sec  2 tan 20 

60u2 60 × 18.07 = = 0.191 m. or 192 mm. πn π × 1800

D2 Ist approximation (D2I = 196 mm) and D2 IInd approximation (D2II = 191 mm). Closely agrees. Final value of outer diameter D2 is taken as D2 = 200 mm. Cm3 =

Outlet breadth,

B2 =

Cm 2 2.18 = = 1.82 mps. 1.2 K2 Qth 0.0165 = = 0.0144 mm. ≈ 15 mm. πD2Cm3 π × 0.2 × 1.82

(d) Verification for flow coefficients: K1 =

K2 =

1 1 = = 1.414. Zδ1 7 × 0.0005   1− 1−  πD1 sin β1  π × 0.09 × sin 25  1 1 = = 1.195 Zδ 2  7 × 0.0005  1−  1− πD2 sin β2  π × 0.2 × sin 20 

w1 =

3.164 Cm1 = = 7.49 m/sec. sin β1 sin 25

w2 =

2.18 Cm2 = = 6.37 m/sec. sin β2 sin 20

S. No.

r

Cm

b

w

mm

m/sec

mm

m/sec

Cm w

mm

2πr Z mm

δ

δ t

t=

1

45

2.26

25.8

7.49

0.3017

5

40.3919

0.1238

2

50

2.22

23.65

7.39

0.3004

5

44.8798

0.1114

3

60

2.14

20.45

7.19

0.2976

5

53.8558

0.0928

4

70

2.06

18.2

6.98

0.2951

5

62.8319

0.07958

5

80

1.98

16.5

6.78

0.2920

5

71.8078

0.0696

6

90

1.89

15.4

6.58

0.2872

5

80.7838

0.0619

7

100

1.81

14.5

6.37

0.2841

5

89.7598

0.0557

 Cm δ  +   w t = sin β 0.4255 0.4118 0.3904 0.3747 0.3616 0.3491

tan β

1 B = r tan β

∆r =

Bi + Bi+ 1 =x 2

(ri+1 – ri) 25.2 24.3 22.98 22.0 21.2 20.43 19.87

0.4702 0.4519 0.4241 0.4041 0.3879 0.3725 0.3613

Bi + Bi+ 1 × ∆r 2 = ∆θ

47.26 0.005

45.759

0.228

0.01

41.7745

0.4177

0.01

37.3248

0.3733

0.01

33.7885

0.3379

0.01

31.0262

0.3103

0.01

28.7518

0.2875

44.25 39.2989 35.3506 32.2263 29.8260 27.6776

θ = ∑∆θ rad

deg

0

0

0.228

13°

0.6457

37°

1.0190

58.4°

1.3569

77.8°

1.6672

95.5°

1.9547

112°

379

0.3398

β


TABLE DIA-1: Vane development for radial type centrifugal pump (a) Ist method Z=7

380

TABLE D1A-2: Another method S.No.

1

r

Cm

b

w

sinβ =

β

mm

m/sec

mm

m/sec

KCm ω

dia.

45

3.164

25.8

7.49

0.4224

25°

tan β

0.4660

B=

x=

1 r tan β

Bi +1 + Bi 2

50

3.075

23.65

7.39

0.4161

24.6°

0.4575

60

2.896

20.45

7.19

0.4028

23.8°

0.4400

2.717

18.2

6.98

0.3893

22.9°

0.4226

80

2.538

16.5

6.78

0.3743

22°

0.4037

90

2.359

15.4

6.58

0.3585

21°

0.3840

100

2.18

14.5

6.37

0.3423

20.0

0.3643

0.01

27.45

 Cmb − Cm  δ (mm) =   t(mm). w  

Cmb1 = K1 Cm1, Cmb2 = K2 Cm2

0.01

deg

mm

0

0

4.8

0.228

13.1°

5.2

0.636

36.5°

5.7

0.995

57°

6.01

1.319

75.6°

5.91

1.619

92.8°

5.76

1.961

112.4°

5.22

0.3584

0.3238

0.30

28.9352 28.1926

7

0.01

rad

0.408

30.9636 29.9494

6

0.01

δ

0.228

33.8043 32.3839

5

0.01

θ = ∑∆θ

0.282


70

0.005

37.8788 35.8415

4

= ∆θ

43.7159 40.7973

3

x × ∆r

47.6872 45.7015

2

∆r


381

Design No. D1-A1 Computer Programming in C++ FOR RADIAL TYPE CENTRIFUGAL PUMP IMPELLER AND VOLUTE / * as per 19.11.98 * / # include < studio. h> # include < conio. h> # include include < process.h> float Q, H, n ; float K1,K2 float eff, eff_h, eff_ vol, eff_ m; float n_s Dn, n_i, N_i max. d_s, d_h, Q1_1, H_m ; double CO, B1,B2,B3 ; double temp, t1,Z1 ; double U1,U2 ; float Z,t,fs, Cu2_dash ; int c, count = 1, nn ; float delta, d] t, g = 9.8 1 ; float si, p, H_inf ; float beta 1, beta2 ; float D1, D21, J ; float Cm1, Cm11, Cm2, Cm21 ; float A,B, k1, D22, K21, tp ; float Doll, Do12, Co11, co 12, Km1, Km2 ; float X, R, Y, U21, U22, beta, Cm, T, W1, W2 ; float rI [10] ; float Fs, Ys, W, G [10], C, D, E, F ; float AVgG, drI, dO SdO, SdrI, Gby 2pi ; float RR [19], nb [19] ; float BB [19], AVgB, dR ; float dQ, Qi ; float R2, R3, Row, K ; int theta [ ] = {0,45,90,135,180, 225, 270, 315, 360} ; Void print_ heading ( ) ; Void head_impeller () ; Void head_imp 1 ( ) ; Void imp 11er () ; Void head_volute () ; Void volute () ; Void head_volcir () ; Void volute_ cir () ; Void main ()

382


{ clrscr () ; printf (“CENTRIFUGAL PUMP DESIGN - RADIAL TYPE | N”) ; / * INTIAL SPECIFICATION * / Q= H= n= printf (“/n/n INITIAL SPECIFICATIONS / N”) ; printf (“Q = %f m3/sec/n”, Q); printf (“H = %f m/n”, H) ; printf (“n = %f rpm/n” n,) ; / * BASIC PARAMETERS * / N_S = CEIL (3.65* n* sqrt (double (Q) )) / pow (double (H), 0.75)) ; printf (“Enter the value of a (4.0 to 4.5) : ” ) ; scanf (“%f ”, &A) ; Dn = ceil (A * 1000* pow (double) Q/n, 0.3333)) ; temp = log 10 (Dn) ; eff_h = 1– (0.42/ pow ((temp – 0.172), 2)) ; eff_vol = 1.0/ (1+0.68* pow (n_s, –2.0/3.0)) ; printf (“Enter the value of Mechanical Efficiency (0.92 to 0.98) : ” ) ; scanf (“%f ” & eff_m) ; eff = eff_ h*eff – vol* eff – m; printf (“/n/n BASIC PARAMETER /n”) ; printf (“n_s = %g/n”, n_s) ; printf (“Dn = %f mm/ n”, Dn) ; printf (“eff_h = %f/n” eff_h) ; printf (“eff_vol = %f\n”, eff _ vol); printf (“overall efficiency = %f/n”, eff) ; n_i = (9.81 * Q * H) / (eff) ; N_imax = 1.1* N_i ; printf (“power input Ni = %f KW/n” N_i) ; printf (“Ni maximum = %f KW/n”, N_imax) ; /* SHAFT& HUB DIAMETERS * / T = (N_imax * 60 * 1000) / (2*3.14*n) ; printf (“Enter the value of Ys in Kgf/sq.mm : ”) ; scanf (“%f ”, & Ys); printf (“/n Enter the value of Fs :”) ; scanf (“f ” & Fs) ; fs = (YS * 9.81*1000000) / FS ; tp = (16*T) / (3.14*fs) ; d_s = pow (double) tp, 0.333 ) ; d_h = 1.25* d_s ;


printf (“/n SHAFT AND HUB DIAMETER /n”) ; printf (“T = %f Nm / n”, T) ; printf (“Shaft diameter d_s = %f m /n”d_s) ; printf (“Hub diameter d_h = %f m/n”, d_h) ; fflush (stdin) ; getch () ; clrscs () ; / * INLET DIMENSIONS * / Q1_1 = Q/ eff_vo1 ; /* first approximation * / printf (“Enter the value of B (0.06 to 0.08) : ” ) ; scanf (“%f ”, &B) ; CO = B * pow (Q1_1*n*n), 0.333) ; temp = (4*Q1_1) / (3.14*CO) ; DO11 = sqrt (temp) ; DO12 = sqart (tempt +(d_h*d_h)) ; printf (“/n/ tDo11 = %f m”, DO11) ; printf (“/n Enter new DO11 ; ”) ; scanf (“%f ” & DO11) ; printf (“/n/tDO12 = %f m” Do12) ; printf (“/n Enter new DO12 : ”) ; scanf (“%f ”, & DO12) ; CO11 = (4* Q1_1) / (3.14 * (DO11* DO11)) ; CO12 = (4* Q1_1) / (3.14* (DO12*DO12)) ; printf (“/n Final value of DO12 is : %f m”, DO12) ; printf (“/n Enter the value of J (0.8 to 1.03) : ” ) ; scanf (“%f ”, & J) ; D1 = J * DO12 ; printf (“/n Enter the value of Km1 ; ” ) ; scanf (“%f ”, &Km1) ; Cm1 = Km1 * sqrt (2*g*H) ; B1 = Q1_1/(3.14*D1*Cm1) ; U1 = (3.14* D1* n) / 60 ; temp = Cm1/U1 ; betal = atan (temp) * (180/3.14) ; printf (“betal = %f degrees”, betal) ; printf (“/n Enter the value of dit in m: ”) ; scanf (“%f ”, &d1t) ; printf (“/n Enter the value of Z : ” ) ; scanf (“%f ”, & Z) ; betal = betal*(3.14/180); K1 = 1 / (1-(Z*d1t) / (3.14* D1*sin(betal))) ;

383

384


Cm11 = K1*Cm1 ; getch () ; clrscr () ; printf (“/n/n INLET DIAMENSIONS /n”) ; printf (“Q 1_1 = %f m3 /sec / n”, Q 1_1) ; printf (“DO11 = %f m/n”, DO11) ; printf (“DO12 = %f m/n”, DO12) ; printf (“CO = %f m sec/n”, CO) ; printf (“CO11 = %f m/sec/n”, CO11) ; printf (“CO12 = %f m/sec/ n”, Cm1) ; printf (“Cm1 = %f m/sec/n”, Cm1) ; printf (“B1 = %f m/n”, B1) ; printf (“U1 = %f m/sec/n”, U1) ; printf (“D1 = %f m sec/n” D1) ; printf (“betal = %f degrees /n”, betal* 180/3.14) ; printf (“K1 = %f /n”, K1) ; getch () ; clrscr () ; print_heading () ; printf (“%d %s.5f %8.5f %8.5f %8.5f %8.5f/n”, count, Cm1, tan (betal), betal* (180/3.14), sin (betal), K1) ; while (( fabs(Cm-Cm11)> 0.001) && (count<10)) { count++ ; Cm = Cm11 ; temp = atan (temp) ; / * in radians * / t1 = 3.14*D1*sin (betal) ; K1 = 1/ (1-((Z*d1t) /t1)) ; printf (“%d %8.5f %8.5f %8.5f %8.5f/n” count, Cm11, tan (betan), btetal* (180/3.14), sin (betal), K1) ; Cm11 = K1* Cm1 ; {; printf (“%d %8.5f %8.5f %8.5 %8.5f %8.5f/n” count, Cm11, tan (betal), betal* (180/3.14)), sin (betal), K1) ; printf (“-------------------------------------------------------- /n”) ; /* OUTLET DIAMENSIONS * / H_m = H/eff_h ; printf (“/n Enter the value of Cu2_dash (0.5 to 0.8) : ”) ; scanf (“%f ” & Cu2_dash ) ; U21 = sqrt (g*H_m) / Cu2_dash) ; D 21 = (60* U21) / (3.4*n) ; printf (“/n Enter the value of × (0.8 to 0.95) : ” ) ;


scanf (“%f ” & X) ; Cm21 = × * Cm 11 ; printf (“/n Enter the value of K2 (1.1 to 1.25) : ”) ; scant ( “%f, & K2) ; printf (“/n Enter the value of R = W1 /W2 (1.1 to 1.15) : ”) ; scanf ( “f ” & R) ; printf (“/n Final vlue of betal from table : % f degrees / n”, betal* 180/ 3.14) ; temp = R * × * S in (betal) ; beta 2 = as in (temp) ; Z1 = 6.5 * (D2+D1) / (D21-D1) * sin (betal+beta 2) / 2.0) ; getch () ; clrscr () ; printf (“/n/n OUTLET DIMENSIONS : /n” ) ; printf (“Hm = %6.3 f m/n” H_m) ; printf (“U21 = % 8.4 f m/sec/n”, U21) ; printf (“D21 = %f m/n”, D21) ; printf (“Cm21 = %f m/sec/n”, Cm21) ; printf (“beta 2 = %f degrees/ n”, beta 2* 180/3.14) ; printf (“Z1 = %f m/n”, Z1) ; printf (“Enter the value of Y (0.6 to 0.65) : ”) ; scanf ( “%f ”, &Y) ; Si = Y +0.6* sin (beta 2) ; temp = pow ( (D1/D21), 2.0) ; p = (2*Si/Z) * (1/ (1-temp) ; H_inf = (1+p) * H_m ; printf (“Si = %f / n”, Si) ; printf (“p = %f / n”, p) ; printf (“H_inf = %f m/n”, H_inf) ; /* SECOND APROXIMATION * / temp = Cm21 / (2* tan (beta 2) ; U22 = temp + sqrt (temp* temp + (g*H_inf) ) ; D22 = (60*U22) / (3.14*n) ; Cm2 = Cm21/K2’ B2 = Q1_1/(3.14* D21* Cm2) ; K21 = 1/ (1_(Z*dlt) / (3.14* d21* sin (beta2))) ; printf (“/n/n SECOND APPROXIMATION : /n”) ; printf (“U22 = %f m/sec/n”, U22) ; printf (“D22 = %f m/n”, D22) ; printf (“B2 = %f m/n”, B2) ; printf (“K21 = %f /n”, K21) ; getch () ; clrscr ;

385

386


printf (“/n/st/t/ FINAL VALUES /n/n”) ; printf (“/n/stZ = %f /n”, Z) ; printf (“/n/tB1 = %f m/n”, B1) ; printf (“/tB2 = % f m/n”, B2) printf (“/n/tCm1 = %f m/sec/n”, Cm1) ; printf (“t Cm2 = %f m/sec/ n”, Cm2) ; printf (“t DO11 = %f m /n”, DO11) ; printf (“t DO12 = %f m /n”, DO12) ; printf (“t D1 = %f m /n”, D1) ; printf (“t D21 = %f m/n”, D21) ; printf (“t D22 = %f m/n”, D22) ; printf (“n/t betal = %f degrees /n”, betal* (180/3.14) ; printf (“/tbeta 2 = %f degrees /n”, beta 2* (180/3.14) ; printf (“/tK1 = %f /n”, K1) ; printf (“/tK21 = %f /n”, K21) ; impeller () ; volute_cir () ; volute () ; } /*End of main prgm*/ /* VANE DEVELOPMENT (SINGLE CURVAURE) * / void impller () { int i ; count = 1 ; sdr I = drI = 0, nn = 7 ; printf (“/n Enter the no of values (7 to 10) : ”) ; scanf (“%d”, % nn) ; printf (“/n Enter the value of rI in m :” ) ; for (i = 0 ; i

387

W = W1 + (2* (W1–W2) / (D1– D21) * SdrI) ; t = (2*3.14* rI [i]) /Z ; D = d1 t/t ; temp = cm/w + D ; beta = as in (temp) ; F = tan (double) beta) ; G [i] = 1 (rI [i]* F) ; B = Q1_1/ (2*3.14* rI [i]* Cm) ; printf (“/n%1d %6.4f %7.5f %7.4f %7.5% %6.3f %7.5f %7.5%”, count, rI [i]), SdrI, Cm, W,t, (beta*180/3.14), F,B) ; count++ ; } printf (“/n------------------------------------------------------------------- /n”) ; getch () ; clrscr () ; head_imp1 () ; SdO = dO = 0 ; for (i = 0 ; < nn ; i ++) { AVgG = (G [i] + G [i+1]) /2 ; drI = rI [i+1]-rI [i] ; dO = AVgG * drI ; printf (“/n %6.2f %5.2f ”, G [i], SdO*180/3.14) ; if (i < nn –1) printf (“/n/t/t/t %6.2f %5.3f %5.3%f ”, AVgG, drI, dO) ; SdO = SdO + dO ; } printf (“/n----------------------------------------------/n”) ; return ; } void volute () { int i ; W = (2*3.14*n) /60 ; R2 = D21/2 ; Gby 2pi = (g*H_m) W ; printf (“/n/mn Enter the value of R3 (1.03 to 1.05) : ”) ; scanf (“%f ”, & R3) ; B3 = B2 + 0.05* D21 ; R3 = R3 * R2 ; printf (“/n Enter the value for R : ”) ; for (i = 0; i< 10 ; i ++) scan f (“% f ”, & nb [i]) ; print f (“/nEnter the values for B : ”) ; for (i = 0 ; i < 10 ; i ++) scanf (“%f ”, &nb [i] ; printf (“/n/n R3 = %f ”, R3) ;

388


printf (“/nB3 = %f ”, B3) ; getch () ; clrscr () ; Qi = count = 0 ; head_volute () ; for (i =0 ; <10 ; i++) { count++ ; BB [i] = nb [i] /RR [i] ; BB [i+1] = nb [i+1]/ RR [i+1] ; dR = RR [i+1]-RR [i] ; printf (“/n %2d % 6.4f % 6.4f %6.4f/t/t %7.5f ”, count, RR [i], nb [i], BB [i], Qi) ; if (i = = 9) ; { exit (1) ; } AVgG = (BB [i] + BB [i+1] /2 ; dQ = Gby 2pi * aVgB * dR ; Q i = Qi + dQ ; printf (“/n/t/t/ %6.4f %6.4f %7.5f ”, AvgB, dR, dQ) ; if (i = = 9) { getch () ; clrscr () ; head_volute () ; } } getch () ; clrscr () ; return ; } void volute_ cir () { int i ; count = 0 ; K = (720 *9.81*H_m*60)/(2*n*Q) ; printf (“/nk value is : %f ”, K) ; getch () ; clrscr () ; head_volcir () ; for (i= 0 ; < 9 ; ++0 { count ++ ; Qi = (theta [i] * Q )/360 ; Row (theta [i] /K + sqrt (2 * theta [i] * 0.135)/K) ; printf (“/n %1d %3d %8.6f %6.4f ”, count, theta [i], Qi, Row) ; }


getch () ; return ; } void pirt_heading () printf (“/n/n”) ; printf (“--------------------------------------------------------/n) ; printf (“Sr. No. Cm1 tan (betal) betal sin (betal) Ki /n”) ; printf (“--------------------------------------------------------/n ”) ; return ; } void head_impeller () printf (“n/n/”) ; printf (“-----------------------------------------------------------------------------------\n”); printf (“SNo rI SdrI Cm w t beta tan (beta) B\n”); printf (“------------------------------------------------------------------------------------\ n”); } void head _ imp1 () printf (“\n\n”) printf (“-------------------------------------------------------------------------------------/n”) ; printf (“ G Avg (G) drI do sdo /n”) ; printf (“------------------------------------------------------------------------------------ /n”) ; return ; } void head_volute () printf (“/n Volute Trapezoidal - free vortex /n”) ; printf (“/n/n ) ; printf (“-----------------------------------------------------------/n”) ; printf (“S.no r b B avgB dr dq gi/ n”) ; printf (“ ------------------------------------------------------- /n”) ; return ; } void head_ volcir () { printf (“/n volute circular - free vortex /n”) ; printf (“/n/n”) ; printf (“---------------------------------------------------------/n”) ; printf (“S.no theta qi row /n”) ; printf (“------------------------------------------------------- /n”) ; return ; }

389

390


Fig. D1-A1. Pump assembly

391

(H) m

Specification : H = 13.8 m (42 ft) Q = 16 lit/sec (210 gpm) η = 1800 rpm N not to exceed 3.73 kW (5 hp) British Size : 75 mm × 64 mm (3 ′′ × 2½ ′′)

H-Q 1800 rpm

75 80 3.0 3.2 15 16

85 3.4 17

90 % Efficiency (η) % Input (N) kW 18 19 20 Total Head


m 00 16 Q η-

18

00

50 2.0 10

55 2.2 11

60 2.4 12

H-Q 1440

ηQ

65 2.6 13

ηQ

14

rp

40

70 2.8 14

H-Q 1600

18

40 1.6 8

45 1.8 9

Q N-

1 60

30 1.2 6

35 1.4 7

N -Q

00

0

0 rp

m

0 0 0

5 0.2 1

10 0.4 2

15 0.6 3

20 0.8 4

25 1.0 5

N-

44 Q1

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

Flow Rate (Q) lit/sec

Fig. D1-A2. Performance curves

18

19

20

392


90

φ

0.00 28.3° 0.10 8P9 1 3 °1 8 ′

3 7° 4 8′ 59 °4 8′

79

°5

4′

98°

18 ′

Fig. D1-A3. (b) Impeller plan

393

8


7

61.6

6

53.9

5

46.2

3

4

38.5 15.4 7.7

26

Detail on volute sactions 148

30.8 23.1

1

2

Fig. D1-A4. Volute casing

394


Fig. D1-A5. Pump photo

395


Design No. D1-B DESIGN OF A MULTISTAGE CENTRIFUGAL PUMP 1. Specification Flow rate Q = 100 m3/hour, Delivery pressure p2 = 75 ata, Suction pressure ps = 4 ata, Pumping liquid : water at 100°C, Specific weight of water ‘γ’ at 100°C = 958 kg/m3. Pump is run by a turbine at a speed n between 7300–7800 rpm.

2. Selection of Pump Type 1. Total head of the pump, Hth =

p2 − p1 75 − 4 = × 104 = 741 m. 958 γ

2. Selecting the speed ‘n’ as 7500 rpm.  100  Head per stage is : H/Stage = 0.0165n4/3 Q2/3 = 0.0165 . 75004/3 ×    3600 

3. Number stages, Zst =

= 220 m.

741 = 3.21. 220

Taking number of stage, Zst = 3, H/Stage =

4. Specific speed, ns =

2/3

3.65n Q H 3/ 4

=

741 = 247 m. 3

3.65 × 7500 0.0278 = 74 (247)3 / 4

It is a radial type centrifugal pump. 5. Taking cavitational specific speed, C = 800 5.62n Q

C=

(HSV)3/4 =

( H SV )

3/ 4

=

5.62 × 7500 × 0.0278 = 800 ( H SV )3 / 4

5.62 × 7500 × 0.0278 = 8.78 m. 800

HSV = (8.78)4/3 = 18.126 m. Vapour pressure at 100°C for water

pvp γ

= 1.03 ata.Taking hfs= 4 m,

and hs= 0 m Flooded Suction and Flow veocity at inlet C = 8 m/s (HSV) nett =

p1 − pvp γ

 C12  h + h +   fs – s 2 g  

 4 − 1.03 82  × 104 − 0 − 4 – =   = 23.175 m. 2 × 9.81   958

396


(HSV) nett > HSV. So, the pump will be safe against cavitation at this speed and all elements of all 3 stages can be designed identically.

3. Basic Parameters 3

6. Nominal diameter at inlet, Dinom = 4.25 × 103

3

0.0278 = 98 mm. 7500

0.42 0.42 = 1 – = 0.854. (log 98 − 0.172) 2 (lgDinom − 0.172) 2

7. Hydraulic efficiency, ηh = 1 – 8. Volumetric efficiency, ηv =

Q = 4.25 × 103 × n

1 1 + 0.68ns

1 −2 / 3

=

1 + 0.68 (74) – 2 / 3

= 0.96.

However, ηv = 0.90 due to high temperature pumping and 3 stages of pumping. 9. Overall efficiency η from the figure for nS = 74 and D0 = 98 mm, η = 0.78. 10. Power, Ni =

γQH 9.81 × 958 × 0.0278 × 741 = = 248.2 kW 1000 × 0.78 η

Taking 15% extra to allow for overloading, Input power Ni = 1.15 × 248.2 = 285.5 kW Final value of Ni = 290 kW. 11. Shaft Diameter (dS) Taking material for shaft as SS 316, ultimate strength = 40.4 N/mm2 Due to high temperature of operation, factor of safety is taken as 4. Working stress is = Torque T =

dS =

40.4 = 10.1 N/mm2 4 60 N 60 × 290 × 1000 = = 369.2395 N.m. 2πn 2 × π × 7500 3

16T = πf S

3

16 × 369.2395 π × 10.1 × 106

= 0.057 m = 57 mm, taken as 60 mm

12. Impeller hub diameter at inlet (dh) dh = 1.3 dS = 1.3 × 60 = 78 mm can be taken as 80 mm dh = 80 mm. 13. Theoretical quantity, Qth =

0.0278 Q = = 0.03062 m3/s 0.90 ηv

14. Velocity of water at impeller eye, C0 = 0.06 15. Impeller eye diameter, D0 = as 105 mm.

4Qth + d h2 = πC0

3

2 3 Qth n 2 = 0.07 0.03062(7500) = 8.39 m/sec.

4 × 0.03062 + (0.08)2 = 105.1 mm can be taken π × 8.39

397


4. Inlet Dimensions 16. Inlet diameter D1 is taken as D1 = 110 mm. u1 =

πD1n π × 0.11 × 7500 = = 43 m/sec. 60 60

17. Taking flow velocity before the inlet edge of the blade C0m is equal to flow velocity at impeller eye C0 = 8.39 m/s and coefficient of contraction at inlet due to blade thickness K1 = 1.5. Flow velocity at inlet Cm1 = K1 × C0m = 1.5 × 8.39 = 12.585 m/sec. 18. Taking normal entry at inlet. Blade angle at inlet β1  Cm1 12.585  = = 0.2927  = 16.3°. β1 = Arc tan  43  u1 

Taking angle of attack at inlet ∆β1 = 8.7° β1 = 16.3 + 8.7 = 25°. 19. Check up for K1 K1 =

Relative velocity at inlet w1 =

1 1 = = 1.489 Zδ 8 × 0.06   1− 1−  πD1 sin β1  π × 0.11 × sin 25°  Cm1 12.585 = = 29.78 m/s. sin β1 sin 25

Qth 0.3062 20. Inlet breadth, b1 = πD C = = 0.01018 m or 10.18 mm; but taken as 11 mm. π × 0.11 × 8.39 1 0m

Final values : Cm0 = 8.39 m/s, K1 = 1.5, Cm1 = 12.585 m/s, w1 = 29.78 m/s, β1 = 25°, D1 = 110 mm b1 = 11 mm.

5. Outlet Dimensions H 247 21. Hm = η = = 289.2 ≈ 290 mm. 0.854 h  Cu 2   = 0.5 coefficient of reaction ρ = 1– 1 Cu 2 = 1 – 1 × 0.5 = 0.75 22. Taking Cu 2 =  u  2  2 2

Taking

0.5u 2 0.5 × u 2 .290 = .u= g 9.81

Hm =

Cu 2 u 22 g

D2 =

60u2 60 × 75.43 = = 0.1921≈ 192 mm πn π × 7500

=

9.81 . 290 = 75.43 mps. 0.5

w1 = 1.15 K2 = 1.27 Cm3 = 0.9 Cm0 = 0.9 × 8.39 = 7.551 m/s, w2

Cm2 = K2 Cm3 = 1.27 × 7.551 = 9.5898 m /sec.

398


23. sin β2 = sin β1

w1 1.27 K Cm3 = 2 = sin 25° × 1.15 × × 0.9 = 0.3703 1.1 w2 K1 Cm 0

β2 = 21.74° ≈ 22°. 24. Z = 6.5

D2 + D1 (β 2 +β1 ) 0.192 + 0.110° (25 + 22) sin = 6.5 × × sin = 9.46 D2 − D1 2 0.192 − 0.110° 2

However, number of impeller blades Z1 is taken as 8. 25. ψ = 0.65 + 0.6 sin β2 = 0.65 + 0.6 sin 22° = 0.8748. 26. p =

2ψ ⋅ Z

1 D  1−  1   D2 

2

=

2 × 0.8748 ⋅ 8

1  110  1−   192 

2

= 0.3256

27. Hth = (1 + p) Hm = (1.3256) 290 = 384.4 m.

Cm 2 28. u2 = ± 2 tan β2  9.5898  =   +  2 tan 22 

29. D2 =

2

 Cm 2    + gH th .  2 tan π2  2

 9.58987    + 9.81 × 384.4 = 74.41 mps .  2 tan 22 

60 × 74.41 60u2 = π× = 0.190 m, but final diameter D2 is taken as 192 mm. 7500 πn

Q 0.03062 = = 6.723 ×10–3 m = 7 mm. 30. Breadth, b2 = πD C π × 0.192 × 7.551 2 m3

Check up: w2 =

w1 29.78 = = 25.8957 m/s 1.15 1.15

Cm3 = 0.9 Cm0 = 0.9 × 8.39 = 7.551 Cm2 = K2 × Cm3 = 1.27 × 7.551 = 9.5898 m/s sin β2 =

Cm 2 9.5898 = = 0.3703 25.8957 w2

β2 = 21.74° ≈ 22° K2 =

1 1 = = 1.27. 8 × 0.006 Zδ 1− 1− π × 0.192 × sin 22 πD2 sin β2

399


6. Leakage Through the Suction Side Wearing Ring 31. ρ =

Hp

H m or Hp = ρ Hm = 0.75 × 290 = 217.5 m.

This pressure prevails at the outlet of the first stage impeller (point J in Fig. D1B-1). Taking wearing ring diameter D1 = 120 mm (i.e., thickness is 7.5 mm), pressure head at the entrance of wearing ring at suction side when liquid passes through the passage between the front shroud of the impeller and suction side casing. 32.

Hi = Hp –

ω2 2 (r2 – ri2 ) 8g

2 2 (2π × 7500)2   0.192  −  0.120       = 217.5 – 602 × 8 × 9.81   2   2  

= 217.5 – 44.14 = 173.36 m. Taking simple straight type wearing ring where in the length of the wearing ring is li = 40 mm and clearance is bi = 0.2 mm and frictional coefficient for the flow through the clearance λs= 0.065, the coefficient of discharge. 33. µS =

1 λ1

li + 1.5 2bi

1 = 0.3536. 0.065 × 0.04 + 1.5 2 × 0.0002

34. Leakage flow through the wearing ring clearance at suction end (HS = 4 ata = 4 × 10.366 = 41.344 m). qS = µS . 2 π ri bi 2 g ( H i − H s ) = 0.3536 × 2 × π × 0.06 × 0.0002 × = 1.3569 ×

10–3

m3/s

2 × 9.81 × (173.36 − 41.344)

= 1.3569 lit/s = 4.89 m3/hr.

7. Leakage Through the Shaft Bush Clearance between First Stage and Second Stage Impellers 35. Total head at point G, Total head at point F,

HG = H1 = 247 m. HF = Hp –

ω2 2 2 (r 2 – r h) 8g 2

1  2π × 7500  2 2 = 217.5 –   8 × 9.81 (0.097 – 0.04 ) 60  

= 217.5 – 59.86 = 157.64 m.

400


Pressure drop, ∆H = HG – HF = 247 – 157.64 = 89.3615 m Taking clearance between shaft and bush bearing bis = 0.2 mm and length of bush bearing lis = 40 mm and λ2 = 0.065.

1 = l2 λ2 + 1.5 2b2

1

36.

µ2=

37.

q2 = µ2 2π r1S b2 2g ∆H 2

40 0.065 + 1.5 2 × 0.2

= 0.3535

= 0.3535 × 2 × π × 0.04 × 0.0002 × = 0.7441 ×

10–3

m3/s

2 × 9.81 × 89.3615 or 0.74.41 l/s = 2.68 m3/h.

8. Determination Balancing Disc Dimensions 38. Total Axial force by 3 Impellers p1   2 r12 + rh2  ω2   F 1 = γ π (ri2 – rh2 )  H p –  r2 – 2  2 g    2   1 0.062 + 0.042   2 × π × 7500   2 = 985 × π (0.062 – 0.042) (217.5 × 3) −  0.097 −    8 × 9.81  2 60    

= 3716.45 kg = 36458.35 N. 39. Taking, allowable pressure in the disc, pd = 20 kg/ cm2 π (R2d – rd2 ) = Taking,

F1 3716.45 = = 185.82 cm2 20 pd

rd = rh +10 = 40 + 10 = 50 mm. π (Rd2 – 52 ) = 185.82 Rd =

185.82 + 52 = 9.17 cm. ≈ 90 mm π

Ld = 90 – 50 = 40 mm. Taking radial clearances between shaft hub and casing after the last stage of the impeller S1 = 0.25 mm, length of the clearance L1 = 100 mm and the axial clearance between stationary and rotating balancing disc S2 = 0.2 mm and taking λ3 = λ4 = 0.065. 40. Coefficient of flow µ3 for the bush bearing =

1 = l1 λ3 + 1.5 2 S1

1 100 0.065 + 1.5 2 × 0.25

= 0.26

401


J

E

A

C L2

rBm

δ

l G

F

B

X S1

rg

ri

r2

bi

S2

Rg

li

rBm L1

Fig. D1B-1. Balancing disc

41. Coefficient of flow, µ4 =

1 = ld λ4 + 1.5 2S2

1 40 + 1.5 0.065 2 × 0.2

= 0.3535

42. Total head at the delivery end of last stage of impeller (point A)

pS γ

HA = (H1 + H2 + Hp ) –

= 247 + 247 + 217.5 –

4 = 707 m. 0.958

43. Total head (at point B) before the shaft bush HB = HA –

ω2 2 2 (r – r ) 8g 2 h

2

1  2 × π×7500  (0.0972 – 0.042) = 647.14 m. 44. HB = 707 –   60   8 × 9.81

Flow through the clearance in balancing disc S2 and the flow through the clearance in the bush S1 are equal i.e., 45. q3 = µ3 2 πrh S1 2 gH B – H X = µ4 .2π rd S2 2 g ( H X − H E ) ∴

HB − H X H X − HE

=

µ4 . 2πrd S2 0.3535 × 0.05 × 0.0002 µ3 2πrh S1 = 0.26 × 0.04 × 0.00025 = 1.3596

(HB – HX ) = 1.8486 (HX – HE) HB + HE = 2.8486HX . Pressure at the balancing chamber C is equal to the sum of the suction pressure and frictional losses in the connecting pipe between chamber C and impeller eye. Normally pipe size will be 25 mm to 40 mm to reduce the frictional losses. Since, the flow through the pipe is very small about 1 to 1.5 lit/sec and the pipe size is very large, flow velocity is connecting pipe will be

402


negligibly small. Taking frictional losses to be 1 to 2 m, which can be verified afterwards, and taking hf = 2 m, Total head at the balancing chamber HE =

4 × 104 + 2 = 43.75 m. 958

Substituting this value in the above equation HX = ∴

647.14 + 43.75 = 242.5406 m. 2.8486

q3 = µ3 × 2 π rh × S1 2 g ( H B − H X ) = 0.26 × 2 × π × 0.04 × 0.00025 ×

2 × 9.81(647.14 − 242.54)

= 1.4555 × 10–3 m3/sec = 1.4555 l/s = 5.24 m3/hr. 46. Taking length of the connecting pipe lp = 3 m, diameter dp = 25 mm and f = 0.015. hf =

=

fLQ 2 .16 2 gπ 2 d 5 0.015 × 1.46 × 1.46 × 3 1000 × 1000 × π2 × (0.025)5 × 2 × 9.81

= 0.51 m.

which indicates that the assumption is safe. 47. Volumetric efficiency =

100 = 90.8% which is also safe. 100 + (1.3569 + 1.4555) × 3600 1000

9. Axial Thrust due to Centrifugal Force created due to the Turning of Liquid at Impeller Inlet 49. Reaction due to the turning force developed due to centrifugal force in one impeller F21 =

958 × 0.03062 × 8.39 γQth .C0 = = 25 kg. 9.81 g

For 3 impeller F2 = 25 × 3 = 75 kg. 50 Additional axial force due to wear in the wearing ring at suction side F3 = π

(r22

–

r12

u22 )γ 8g

 r 22  r 22 r 22 + r 21 ln + − 2  2 2 2 2  r1 2r 1  r2 − r1 

= π (0.0962 – 0.062)

958 × 74.892 8 × 9.81

2   0.0962 0.0962 + 0.062  0.096  ln + − 2   = 1598 kg.   2 2   2 × 0.062  0.06   0.096 − 0.06 

Total axial force = 3716.45 + 1598 – 75 = 5239 kg.

r

b

S.No.

m

m

1

0.055

0.011

2

0.06

3

C′m

w

t

δ

sin β

mps

Cm w

m

m

δ t

8.39

29.78

0.2817

0.0432

0.006

0.1389

0.4206

0.010

8.290

29.32

0.2827

0.0471

0.006

0.1274

0.4101

0.07

0.009

8.0903

28.396

0.2849

0.055

0.006

0.1091

0.3940

4

0.08

0.0077

7.8906

27.47

0.2872

0.0628

0.006

0.0955

0.3828

5

0.09

0.0074

7.6907

26.548

0.2897

0.0707

0.006

0.0849

0.3746

6

0.096

0.006

7.551

25.896

0.2916

0.0762

0.006

0.0787

0.3702

β°

25°

tan β

0.4636

=

Bi =

Q 2πrb

1 r tan β2

Bi +Bi + 1 2

24.2°

0.4497

37.06

23.2°

0.4287

33.32

35.19

22°

0.4143 0.4040

0.01

0

0

0.1907

10.93°

0.5426

31.09°

0.8601

49.28°

1.1485

65.8°

1.3086

74.98°

0.3519 0.3175

28.84

0.01

0.2884

30.17 27.5 25.86

deg

0.1907

0.01

0.006

rad

0.16

403

0.3986

0.005

Cm δ + w t

θ

∆θ c = x × ∆s

31.75

26.68 21.74°

∆S

39.22 38.14

22.5°

=x

=


TABLE D1B-1: Vane development

404


10. Design of Diffuser 51. Flow rate ‘Q’ through the diffuser Q = Q + q2 + q3 = 100 + 2.68 + 5.24 = 107.92 m3/hr. 52. Radius of the inlet edge of the diffuser r4 = r2 + 3 mm = 96 + 3 = 99 mm. 53. Flow velocity at the inlet of the diffuser is taken as equal to the flow Cm2 velocity at the impeller exit after the blade. C m3 Cm4 = Cm3 = 7.3376 m/s. 54. Tangential component, 7.4 rα ==7.4 αr22

Cm 2 Cu4 = Cu3 = Cu2 = u2 – tan β 2 = tan α3 =

33

C u4 = C u3 = C u2

π × 0.192 × 7500 0.9 × 9.226 – = 75.3 – 18.3874 = 57.0108 m/sec 60 tan 24.31° Cm 4 7.3376 = = 0.1287 Cu 4 57.0108

α3 = 7.4°

55. Taking unevenness in the velocity distribution at the inlet of the diffuser µ = 1. Diffuser blade thickness δ4 = 6 mm, No of diffuser blades zd = 5. Blade angle at the inlet of the diffuser α4 will be tan α4 = µ tan α3 +

δ4 t4

= 1 × 0.1287 +

1 + µ 2 tan 2 α3 6×5 2 π × 198 1 + 1 × 0.1287

= 0.1773 α4 = 10.05 ≈ 11° 56. Since initial portion of the diffuser must be a logarithmic spiral log ri =

tan α 4 0.1773 θ° + log r4 = ri + log 99 = 0.00134θ° + 1.995 132 132

TABLE D1B-2: Development of diffuser vane θ i°

0

15

30

45

60

75

0.00134θ°i

0

0.0201

0.0402

0.0603

0.0804

0.1005

log ri

1.995

2.0151

2.0352

2.0553

2.0754

2.0955

r1 mm

99

103.6

108.5

113.6

119

126

405


57. Width of the diffuser passage at inlet a4 a4 =

=

r4 cos α 4

 π sin 2α4   e zd − 1 – S4  

π sin 22  99   e 5 –1 – 6 = 20.76 ≈ 21 mm. cos11    

58. Width of the diffuser passage in axial direction b4 b4 = 1.75 × b2 = 1.75 × 7 = 12.25 ≈ 13 mm For best efficiency condition a4 = b4 . In this design, this is not possible since in order to achieve, number of diffuser blades must be increased, which will increase hydraulic loss. 59. Taking outlet velocity at diffuser C5 = 12.5 m/s and b4 = b5 = 13 mm, width of the canal at diffuser outlet a4 a4 =

Qth 107.92 × 1000 = = 36.89 ≈ 39 mm Z1c5b5 3600 × 5 × 12.5 × 0.013

11. Return Passage 60. Return passage is constructed in such a way that the velocity C5 gradually reduces to C0, when the liquid reaches the impeller eye area of the next stage. Taking the axial width b in the return passage remains constant throughout i.e., b5 = br = 13 mm and taking a = 45 mm at radius R = 136 mm. Velocity C will be C=

0.03 = 10.26 m/sec. 5 × 0.013 × 0.045

61. External radius of the diffuser r5 = 1.62 r4 = 1.62 × 99 ≈ 160 mm. The diffuser and return passage is in the Fig. D1B-6.

12. Mechanical Loss Taking Rd = r2, Loss of power due to friction in impeller and balancing disc. 62. Nr = 4 × 0.97 × 10–3 × ν0.2 ρ r24.6 ω2.8 × 0.746 = 4 × 0.97 × 10–3 (0.27 × 10–6)0.2 = 97.5 × 0.0954.6 × 7852.8 × 0.746 =34.7 kW. Taking pressure before the stuffing box pSB = 0.6 kgf/cm2, length lSB = 9 cm, coefficient of friction f = 0.01, loss in stuffing box. 63. NSB =

2 × π × 0.01 × 32 × 9 × 785 × 6 × 0.746 2πfr 2SB lSB ωpSB × 0.746 = 2.4 kW. 7500 7500

64. Loss of friction due to shaft rotation in water is taken as 2.4 kW. Total mechanical losses = 34.7 + 2.4 + 2.4 = 39.5 ≈ 40 kW.

406


Fig. D1B-2. Multistage pump for hot water pumping

6

140 φ 130 φ 120 φ

180 φ

160 φ

192 φ

γ

φ 80

φ 105

φ110

11

Fig. D1B-3. Impeller meridional section

407


w 30 ϕ° 100

w

Cm

25 10 9

80 20

8 Cm

7 60 15

6 5 4

5

s = 5 mm

3 20

s 2 = 2 mm

40 10

2 1 ri = 55 61

68

75

82

r 89 r2 = 95

Fig. D1B-4. Impeller parameters distribution along the blade 2

t2 = 7 4 ,5

R6

5

=

43

,2

φ1 10 φ1 2 0 φ1 30 φ φ1 1 40 60

R5 R4

X X X

X

°

X

80 φ1 1 9 2 φ

7 4 .9 7

31.0 9° 49 .28° 65 .80 °

R3 R2 R1

10 .93 °

t1

X

Fig. D1B-5. Vane development in plan

408


R1 = 18φ

17.5 φ 17 φ

8.5 φ 9φ 9.5 10 φ 10.5 11 φ 11.5 12 φ

16.5 φ 15φ 14φ 13φ

r

12φ R 0 42.5 45 47.5 50 52.5 55 57.5

R1 12 13 14 15 16.5 17 17.5 18

R2 12 11.5 11 10.5 10 9.5 9 8.5

11.5°

Fig. D1B-6. Impeller passage check up method

34 a 4=

r = 12 8

45

R= 13 6 φ 32 0

24

,3

r =1,5 1 8 12 11 = 15 5 r= R =1 11 r = R 109 r= ° 75

8

° 45° 30° 60 15°

φ 19

r=1

04

r = 99

Fig. D1B-7. Vaned diffuser and return guide vane passages

409


Design No. D2 SPIRAL CASING DESIGN Available Data for the Impeller b1 = 18.5 mm β1 = 37° b2 = 8.5 mm β2 = 24°

D1 = 80 mm D2= 238 mm

Pipe size 76 mm (3′′) × 65 mm ( 2½′′ ) Hm =

nS =

H = 13.73 m n = 1430 rpm

ηh = 0.85 Q = 0.01403 m3/sec

2 × π × 1430 2πn = = 149.75 rad/sec 60 60

ω=

H 13.73 = = 16.16 m ηh 0.85 3.65n Q (H )

=

3/ 4

3.65 × 1430 0.01403 = 87 (13.73)3 / 4

D3 = 1.04 D2 = 1.04 × 238 = 248 mm b3 = b2 + 0.05 D2 = 85 + 0.05 × 238 = 8.5 + 11.9 = 20.4 ≈ 21 mm TABLE D2-1 : Free vortex Cur = Constant circular cross-section K=

720πg H m 720 × π × 9.81 × 16.16 . = = 170674.34 ω 149.75 × 0.01403 Q

θ (m) K

θ ° (deg)

2r3

θ K

ρ mm

a3 = ρ + r3 mm

0

0

0

0

124

45°

2.637 × 10–4

8.086 × 10–3

8.35 ≈ 8.5

132.35

90°

5.273 × 10–4

0.01144

11.96 ≈12

135.96

135°

7.910 ×10–4

0.01401

14.79 ≈ 15

138.79

180°

10.546 × 10–4

0.01617

17.23 ≈ 17.5

141.23

225°

13.183 × 10–4

0.01810

19.4 ≈ 19.5

143.4

270°

15.82 × 10-4

0.01981

21.4 ≈ 21.5

145.4

315°

18.456 ×10–4

0.02140

23.25 ≈ 23.5

147.25

360°

21.093 × 10–4

0.02285

24.96 ≈ 25

148.96

410


2

3

4

5

6

φ 50 dia

47 φ

43 φ

35 φ

30 φ

24 φ

17 φ D 3 = φ 248 dia

1

39 φ

C ur = constant (free vortex) circular cross-section

7

8

b 3 = 21 Detail of volute sections 1 to 8

Fig. D2-1. Volute cross-sections

TABLE D2-2 : Constant velocity CV = constant and circular cross-section From the Fig. (5.5) velocity coefficient φV for volute (constant throughout) for nS = 87 φV = 0.42. Flow velocity in volute cross-sections, CV = φV Qθ m3/sec

Area A m2

2gH = 0.42

2 × 9.81 × 13.73 = 6.9 mps.

radius ρ mm

a3 = ρ + r3 mm

0

0

0

124

1.754×10–3

2.55 × 10–4

9.0

133

3.508× 10–3

5.1 ×10–4

12.75 ≈ 13

136.75

5.262 × 10–3

7.65 × 10–4

15.6 ≈ 16

139.6

7.015 × 10–3

10.2 × 10–4

18.0

142

8.769 × 10–3

12.75 × 10–4

20.15 ≈ 20.5

144.15

10.523 × 10-3

15.3 × 10–4

22.05 ≈ 22.5

146.05

12.277 × 10–3

17.85 × 10–4

23.85 ≈ 24

147.85

14.03 × 10–3

20.4 ×10–4

25.5

149.5

411


D 3 = 248 mm φ

51φ

48 φ

45 φ

41 φ

36 φ

32 φ

26 φ

18 φ

Constant velocity C v = constant circular cross-section

b 3 = 21mm Detail of volute sections 1 to 8

Fig. D2-2. Volute cross-sections

D2-A. Spiral Casing Design under Cur = Constant and Trapezoidal CrossSection Data : Q = 150 m3/hr = 0.0417 m3/sec, H = 18 m, ηh = 0.91 n = 1450 rpm , D2 = 260 mm, b2 = 25 mm ω=

Hm =

2πn 2 × π × 1450 = = 152 rad/sec. 60 60 gH m 18 9.81 × 19.78 H Γ = = 19.78 m, = = = 1.27 m2/sec. ω 152 ηh 0.91 2π

r3 = (1.03 to 1.05) r2 = 1.04 × 130 = 135.2 taken as 135 mm b3 = b3 + 0.05 D2 = 25 + 0.05 × 260 = 38 mm. ∆Q =

ΓC Bi + B1 + 1 × ∆ r m2/sec. 2 2π

412


TABLE D2-3

Point No:

r mm

b mm

1.

135

38

2. 3.

140 145

38 42

Bi =

b r

0.307

5.

155

50

0.323

7. 8. 9. 10.

165 170 175 180

58 62 66 70

190

78

0.411

200

86

0.430

15.

205

90

0.439

17. 18. 19.

215 220 225

98 100 100

0.005

0.00189

0.00353 0.00542 0.315

0.005

0.00200

0.330

0.005

0.00210

0.345

0.005

0.00219

0.358

0.005

0.00227

0.371

0.005

0.00236

0.383

0.005

0.00243

0.395

0.005

0.00251

0.00742 0.00952 0.01171 0.01398 0.01634 0.01871 0.02128 0.405

0.005

0.00257

0.416

0.005

0.00264

0.425

0.005

0.00270

0.02385

0.421

14.

94

0.298

0.00175

0.389

12.

210

0.00178

0.377

0.400

16.

0.005

0.365

74

82

0.280

0.02649 0.02919 0.435

0.005

0.00276

0.444

0.005

0.00282

0.452

0.005

0.00287

0.455

0.005

0.00289

0.450

0.005

0.00286

0.03195

0.448

0.03477

0.456

0.03764

0.455 0.444

Q= Σ∆Q 0

0.352

185

195

0.00175

0.338

11.

13.

0.005

0.290

46

54

0.276

x × ∆r

0.271

150

160

∆r

0.281

4.

6.

∆Q =

Bi + Bi +1 =x 2

0.04053 0.04339

413

38 mm

Fig. D2-3. Volute design Cur = constant trapezoidal cross-section

Q360° = 0.0417 m 3/sec Q 360°

0.02 0.01 130

140

150

160

170

Q90° = 0.001043m /sec

3

=

volute tongue

r3 = 135 mm

++

=

–

r90° = 172 mm

180 r180° = 192 mm

190

Q 180° = 0.03128

Q270° = 0.03128 200

210

++

r270° = 212 mm

Q360° = 0.047 220

225 mm

r360° = 228 mm

∆r = 5 mm

r

0.03

0.04

Q 0.05


Diffuser part of volute: Outer radius at r90° = 173 mm. For easy assembly of pipe with the delivery flange, the height of delivery flange must be well above the height of casing at r90° (see Fig. D24) i.e., length of the diffuser past of spiral casing (L) should be well above the volute L > r for Q90 + metal thickness (δ).

414


Delivery flange thickness = 10 mm r90 = 172 mm δ = 8 mm Dd 10 150 mm δ = 8 mm L

r90

r3

Q 360

r4

r4

X

r360

Fig. D2-4

Taking the distance between bottom side of the delivery flange and the outer side of the casing as 150 mm. Diffuser length, L = 172 + 8 + 150 + 10 = 340 mm. The distance X = (r360° – r4) = 228 – 140 = 88 mm. Assuming delivery velocity, Cd = 4 mps. (4 to 5 mps). Ad = pipe dia dd=

4 Ad = π

Q 0.0417 = = 0.01043 m2 Cd 4.0

4 × 0.01043 = 115 mm. π

But nearest standard pipe size available is 127 mm and Cd =

4Q 4 × 0.0417 = 3.2 m/sec and 2 = πd d π × 0.127 2

for pipe size of dd = 102 mm Cd = 5.10 m/sec. Angle of divergence for Ist pipe tan ε° =

(d d − X ) 127 − 88 = = 0.0574 ε° = 3.3°; for the second 2 × 340 2L

102 − 88 = 0.0206 and ε° = 1.2°. Since, ε° value is less than allowable value (4° to 5°), 2 × 340 anyone of the pipes can be used.

pipe is tan ε° =

D2-B. Spiral Casing Design with CV = Constant and Trapezoidal Cross-section Given, Q = 150 m3/sec = 0.0417 m3/sec, H = 18 m, ηh = 0.91 n= 1450 rpm, D2 = 260 mm, b2 = 25 mm. Hm –

H 18 = = 19.78 m ηh 0.91

415


nS =

3.65 × 1450 0.0417 (18)3 / 4

≈ 124

From the graph (Figs. 5.7 and 5.8) for nS = 124 the velocity coefficient. φ V = 0.32 to 0.34,

CV = 0.32 to 0.34 2 × 9.81 × 18 = 6.01 to 6.39 m/sec. CV is taken as 6.2 mps. The angle of divergence for volute α = 45° selected (range is 35° to 45°) Q360 = Q = 0.0417 m3/sec. r3 = (1.03 to 1.05), r2 is taken as 1.04 × 130 = 135.2 ≈ 135 mm. b3 = b2 + (0.05 D2) = 25 + 0.05 × 260 = 38 mm. Width at the tongue of the volute is b4 = b3 = 38 mm. TABLE D2-4: Determination Qθ, the flow at different angles θ, Qθ = S. No.

Angle θ

1 2 3 4 5 6 7 8 9

0 45 90 135 180 225 270 315 360

Qθ =

Q360 × θ0 360

Q360 360

0 0.00521 0.01043 0.01564 0.02085 0.02606 0.03127 0.03648 0.0417

TABLE D2-5: Construction of volute area under CV = constant CV = φ v 2 gHV = 6.2 mps, b3 = 38 mm = b4 , R4 = R3 + 5 = 140 mm S. No.

h mm

1 2 3 4 5 6 7 8

10 20 30 40 50 60 70 80

9

90

α   Aθ = hθ  hθ tan + b3  2  

Qθ = Aθ × CV m3/sec.

46.28 54.57 62.85 71.14 79.42 87.70 95.99 104.27

421.42 925.68 1512.78 2182.72 2935.50 3771.12 4689.58 5690.80

2.6128 × 10–3 5.7392 × 10–3 9.3792 × 10–3 0.01353 0.01820 0.02338 0.02908 0.03528

112.56

6775.02

0.04201

a0 = 2hθ tan

α + b3 2

Diffuser part of volute casing will be same as that of previously designed for Cu r constant design.

416

Sketch of the spiral casing design C v = constant Trapezoidal cross-section

a

h, a 120 mm

h

b

110

α

100 a θ(

+

for

ck c he

only up a

+

)

h

= +

– –

Tongue

1 Q 8 Q 45

1 Q 4 Q 90 Q 360

3 Q 8 Q 135

1 Q 2 Q 180

5 Q 8 Q 225

3 Q 4 Q 270

7 Q 8 Q 315

Q 360

Fig. D2-5. Volute design—constant velocity and trapezoidal cross-section

Q Q 360


+

417


D2-C. Design of Suction Volute π π (D02 – D2h ) = (0.1582 – 0.072) = 0.0158 m2 4 4

Area at section,

9 – A9 =

Area at section,

8 – A8 = 0.59 A9 = 0.0079 m2

Area at section,

6 – A6 =

3 A = 0.0059 m2 8 9

Area at section, 4 – A4 = 0.25 A9 = 0.00395 m2 Area at section, 2 – A2 = 0.125 A9 = 0.00198 m2

a

b

2

4

0

6

a

b

8

9

R8

R9

9

10

A9 = 0.0158m2 R9 = 0.32 m, A8 = 0.0079m2 R8 = 0.2133m A6 = 0.0059m2 A4 = 0.00395 m2 R4 = 0.16m A2 = 0.00198 m2 Tongue R0 = 0.11 to 0.13m Fig. D2-6. Suction volute

From Figs. 5.7 and 5.8, kVS = 3.7, mSV = 0.16 R8 = (kVS – 1)

D0 2

= (3.7 – 1)

0.158 = 0.2133 m 2

R9 = 1.5 R8 = 1.5 × 0.2133 = 0.32 m R4 = 0.75 R8 = 0.75 × 0.2133 = 0.16 m Tongue,

R0 = (0.5 to 0.6) R8 = (0.5 to 0.6) 0.2133 = 0.11 to 0.13 m

Velocity of liquid in suction volute, CVS = mSV 2 gH = 0.16 2 × 9.81 × 8 = 3 m/sec.

418


Design No. D3 DESIGN OF AXIAL FLOW PUMP As per Prof. Voznisenki’s method, Distribution of circulation Γ over the camber line (or mean line) of the profile, which is in the form of an arc of circle.

Pump Specification 1. Total head, (H) = 3.66 m (12 ft), Flow rate, (Q) : 136.8 lit/sec (1800 gpm) speed 1440 rpm. Recommended power 7.5 kW (10 hp British). Pipe size 250 mm × 250 mm (10′′ × 10′′)

3.65n Q

2. Specific Speed, nS =

H

3/ 4

3. Impeller eye velocity, C0= 0.07

3.65 × 1440 × 0.1368

= 3

(3.66)3 / 4 3

Qn 2 = 0.07

≈ 735 i.e., Axial flow pump.

0.1368 × (1440)2 = 4.6 m/sec.

_ d 4. From figure 10.41, for nS = 735 hub ratio d = h = 0.5 D 5. Outer diameter of impeller, D0 D0 =

(

4Q

π 1− d

2

)C

4 × 0.1368

=

π(1 − 0.52 ) 4.6

0

= 0.226 m or 226 mm.

_ 6. Hub diameter, dh = d × D0 = 0.5 × 0.226 = 0.113 m or 113 mm. 7. Unit head, KH =

H 2

n D

2

=

3.66 2

 1440  2   × (0.226)  60 

= 0.125

From figure 10.40, for ns = 735 (KH )opt = 0.125 8. Unit quantity, KQ =

Q nD 3

=

0.1368 = 0.493.  1440  3   (0.226)  60 

Referring to Figs. 10.47 and 10.48, for KH = 0.125 and for KQ = 0.493. ηh = 0.83 9. Selection of streamlines rI =

dn 113 + 0.02 D0 = + 0.02 × 226 = 61 mm 2 2

rII =

rI + rIII 85 + 61 = = 73 mm 2 2

rIII =

109 + 61 rI + rV = = 85 mm 2 2

419


rIV =

109 + 85 rIII + rV = = 97 mm 2 2

rV =

D0 226 – 0.02 D0 = + 0.02 × 220 = 109 mm. 2 2

 l TABLE D3-1: Selection of   for impeller  t

S. No.

Details/Section

I

V

1

radius mm

61

109

2*

C0 (m/sec)

4.6

4.6

3

u=

9.2

16.42

4

 C0  β1 = Arc tan   deg  u 

26.6°

15.7°

5

 gH  Cu2 =   m/sec  ηh u 

4.7

2.64

6

 C0   β2 = Arc tan   u − Cu 2 

45.7°

18.5°

7

∆β = (β2 – β1 )

19.1°

2.8°

8***

l    for 0.8 ∆β*** = from Howell t  (Fig. 10.34)

2.0

9**

l from Papir (Fig. 10.43)   t peri

πDn (m/sec) 60

deg

0.76

*Co is assumed as constant for all 5 sections. **

l t

for other sections are selected to have uniform variation between section I and section V  l  l  l   = 1.43,   = 1.11,   = 0.89. t II t III t IV

*** ∆β is given as ∆α in graph.

420


TABLE D3-2: Impeller design No. of impeller blades Zi = 4, η v = 0.95, K = 1.056 S. No.

Details/Section 2π r = (mm) Z = 4 Z

I

V

95.8

171

192

130

0.451

0.451

10

t=

11

l  l   × t mm t 

12*

Γ1 =

13*

Cm = 1.055 C0 m/sec

4.85

4.85

14

 Γ1     4πri 

2.355

1.318

15

 Γ w∞u = u –  1  4πri

6.84

15.102

16

 Cm   β∞ = Arc tan   wu∞ 

35.35°

17.82°

17

w∞ =

9.39

15.85

18

(Γ1/W∞ l)

0.284

0.219

19

(Γ1/W∞ lβ) from Fig. 10.10

1.27

1.98

20

Curvature β

0.223 13.33°

0.111 6.34°

21

∆β (∆α in graph) Fig. 10.11

1.67°

0.05°

22

β∞ + ∆β

37.01°

17.87°

23

(Γ1/W∞ lβ) from Fig 10.10

24

Curvature β

Γ 60 gH = η nZ m2/sec Z h

  m/sec 

Cm sin β ∞

in radians in degrees


* Γ1 is constant and is same for all sections.

1.24

1.98

0.229 13.17°

0.111 6.34°

421


25

l cosβ∞

153.2

123.7

26

l sinβ∞

115.2

40.0

27

 Cm   α2 = Arc tan   Cu 2 

44.3°

71.53°

28****

δm = (Blade thickness) mm

6.0

4.0

29

_  δm  δm =    l 

0.03133

0.0308

30

_ β 1 tan f = 2 2

0.0579

0.0277

0.28

0.13

0.00877

0.00400

0.06667

0.0317

2 Arc tan (2 f1 )

15.4°

7.22°

35

sin β1

0.2628

0.1283

36

 l   mm R=   2sin β1 

364

506

37

θ

30.5°

14.4°

38

4   m = (1 + β)  1 + δm  π  

1.3194

1.170

39

u2

84.64

269.62

40

(m w2∞ –u2)

8.2

24.35

41

σ=

0.114

0.339

33

∆f from Fig. 10.39 δm __ ∆f _ __ f1 = f + ∆f

34

β1 = (Curvature) impeller (deg)

31 32

( mw

2 ∞−

u2

)

2 gH

****Profile thickness changes uniformly from hub to periphery. δmII = 5.5 mm, δmIII = 5.0 mm, δmIV = 4.5 mm.

422


Fig. D3-1. Impeller and blade

Fig. D3-2. Templates for impeller and diffuser

80 η% 0.4 Kh

AXIAL FLOW PUMP TEST REPORT

KQ – η

Design speed : 144 rpm Size 10′′ × 10′′ – Design point Note : Q 3 KQ = ηD H KH = 2 2 nD KN = 3H 5 nD 3 Q – Quantity m /sec n – Speed RPS D – Diameter : 0.226 m H – Head m N – Power Metric hp

2 40 0.2

3KN 60 0.3

Pipe size (10′′ × 10′′) 250 × 250 mm Design data Test result H 3.66 m 3.65 m Q 136.8 lps 146.14 lps N 10 hp (British) 8.3 hp (British) 6.19 kW η 7.40 kW 88% (Maximum) KH 0.124 0.124 at optimum η KQ 0.494 0.528 at optimum η hs 735 760 Note : Test results are given for maximum η point

KH – KQ

1 20 0.1

KN – KQ

0 0 0

0.1

0.2

0.3

0.4

0.5

0.6

Fig. D3-3. Performance graph

0.7

KQ

423


TABLE D3-3: Diffuser design Γ diff = Γ imp = 1.804,

Zd  l ≥2   t Zi

0.162 K Q2 + 0.246 ηh2 K H2 ≥ 2, Zd is taken as 4

S. No.

Details/Section

I

V

1.

radius ri mm

61

109

2.

β3 deg

44.3°

60.2°

3.

β4 deg

90°

90°

4.

∆β = β3 – β4 deg

45.6°

29.8°

5.*

l from Howell for 0.8 ∆β* t

2.3

6.

l from Papir t

7.

t=

8.

2πr , Z = 4 mm Z

0.75

95.8

171

l l =   × t mm t

221

128

9.**

Cm m/sec

4.85

4.85

10.**

Γ = m2/sec

1.803

1.803

11.

Γ Cu 2 = 4πr 2 i

2.355

1.318

12.

 C  α∞ = Arc tan  m   Cu 2 / 2 

76.4°

82.3°

13.

sin α∞

0.972

0.989

14.

 Cm   C∞ =   sin α ∞ 

4.99

4.91

15.

(Γ1 /(C∞ l)

0.164

0.287

16.

(Γ1 /(C∞lβ) from Fig. 10.10

0.77

1.35

17.

β Curvature

0.2125 12.17°

0.2125 12.17°

in radians in degress

* ∆β is given as ∆α in Figure. ** Cm and Γ are same for all other sections.

424


18.∗∗

δm (Blade thickness) mm

6

6

19.

 δm  δ =  l 

0.0272

0.0468

20.

f =

0.0535

0.0535

21.

∆f / δm from Fig. 10.39

2.0

0.2

22.

∆f

0.0544

0.0024

23.

fd = f +∆ f

0.1079

0.0558

24.4°

12.8°

268

289

β 1 tan   2 2

24.

βd = 2 Arc tan ( fd ) deg

25.

R=

l mm 2sin β d

 l * For other Sections   is selected to get Uniform Variation from hub to periphery. t l l  l   = 1.631,   = 1.2,  t  = 0.933 t II t III IV

** Blade thickness is same for other sections also.

l , δm for section II, III, IV are determined to get uniform change from hub to t Γ1 l  periphery. The values = f  To ,  and the values ∆β (∆α in graph Fig. 10.19) from the graph  w∞ lβ t ∆α = f (To , β). The correction process is repeated until two successive values of (β) and ∆β (∆α) are same.

Values of

The camberline of the profile is drawn in the form of an arc of a circle with radius (R). This thin profile is dressed by a thick profile Table D3-4 for which the profile coordinates are known. The characteristics curves Cy , Cx , = f (λ) are determined from actual tests conducted in wind tunnels i.e., for flow conditions under non-separated and separated flow conditions. One such profile recommended by Moscow Power Institute, Moscow, for hydraulic machines |104| is taken for all section of impeller and diffuser blade. The non-dimensional coordinates of the profile is given in Table D3-5.

425


TABLE D3-4: Coordinates of Impeller Profile Section I

Section V

—

x

—

y

—

x

—

0

0

0

0

0.48

0.441

0.325

0.296

0.96

0.588

0.650

0.3915

1.44

0.687

0.975

0.458

1.92

0.795

1.3

0.53

2.4

0.882

1.625

0.588

4.8

1.216

3.25

0.8095

9.6

1.549

6.5

1.031

14.4

1.8

9.75

1.20

19.2

1.987

13.00

1.322

28.8

2.29

19.5

1.525

38.4

2.52

26.0

1.679

48.0

2.712

32.5

1.806

57.6

2.850

39

1.897

67.2

2.945

45.5

1.96

76.8

2.995

52.0

1.997

86.4

3.00

58.5

2.00

96.0

2.945

65

1.96

115.2

2.70

78

1.8

134.4

2.256

91

1.502

153.6

1.6280

104

1.084

163.2

1.275

110.50

0.850

172.8

0.8890

117.00

0.592

182.4

0.4880

123.5

0.325

186.2

0.3180

126.10

0.2115

190.0

0.1468

128.7

0.0977

192

0.0720

130.0

0.048

y

426


TABLE D3-5: Non-dimensional coordinates for airfoil (Moscow Power Institute, USSR) S.No.

X=

X l

Y=

Y = 0.45 Ymax

1

0

0

2

0,0025

0,1468

3

0,005

0,1958

4

0,0075

0,2290

5

0,01

0,2650

6

0,0125

0,2938

7

0,025

0,4050

8

0,05

0,5160

9

0,075

0,6000

10

0,10

0,6620

11

0,15

0,7630

12

0,20

0,8400

13

0,25

0,9040

14

0,30

0,9490

15

0,35

0,9810

16

0,40

0,9980

17

0,45

1,0000

For round edge profiles*

18

0,50

0,9820

0.9820

19

0,60

0,8950

0.8998

20

0,70

0,7420

0.7516

21

0,80

0,5280

0.5424

22

0,85

0,4080

0.4248

23

0,90

0,2770

0.2962

24

0,95

0,1410

0.1626

25

0,97

0,0832

0.1058

26

0.99

0,0254

0.0489

27

1,00

0

0.024

* Extra thickness is added on the convex side of the profile.

427


Design No. D4 CORRECTION FOR PROFILE THICKNESS BY INCREASING β) BLADE CURVATURE (β Correction for profile thickness can be carried out either by thickness correction factor ( χ ) or by the correction of curvature (β). In this design, correction by curvature addition (β) is carried out. Table D4-1 gives the design calculation for hub section of a diffuser. Same procedure is followed for other sections. TABLE D4-1 S. No.

Detail/Section

I

1.**

radius (r) mm

140

2. +

l t

1.10

3.

t l

0.91

4.

t=

5.

l mm

6.*

Γ=

2πgH m 2/sec KΓ ω

3.9077

7.*

Γ1 =

Γ 2 m /s Z

0.6513

8.*

Cm =

9.

Cu 3 2

10.

α∞ = tan–1

11.

C∞ = Cm/sin α∞

6.6023

12.

(Γ1/C∞l)

0.7

13.

(Γ1/C∞lβ) From Fig. 10.10

1.22

2πr .Z = 6 mm Z

125.8 138.4

(

4Q

π D 2 – d h2

)

m/sec

6.2385

2.17

Cm Cu 2 2

70.8°

428


radians degrees

0.574 32.85

14.

β

15.

∆α from Fig. 10.39

2.35

16.

αΙ = α∞ + ∆α

73.15°

17.

Γ1/C∞ lβ from Fig. 10.10

1.22

18.

δm

6.0

19.

δm = (δm/l)

0.043

20.

fm =

β 1 tan 2 2

0.1473

21.

α4

90

22.

θ

0

23.

(∆f / δm ) From Fig. 10.39

– 0.002

24.

(∆f m ) I

– 0.0001

25.

(fm)I = f m + ∆f m

0.1472

26.

β0 deg

32.8°

27.

sin β

0.5417

28.

R= l/2 sin β mm

127.8

* Value is same for all other sections. ** Radius for other sections are determined in the same manner as done for impeller at design D3. + Vane solidity (l/t) is determined is the same manner as done for impeller in design D3. The same profile given in Table D3-4 can be used for diffuser also.

429


Design No. D5 CALCULATION OF CORRECTION FOR BLADE THICKNESS USING χ) THICKNESS COEFFICIENT (χ TABLE D5-1 Data available Z = 7, Cm = 4.669 m/sec., Γ 1 = 0.56 S. No.

Detail/Section

I

1.

radius r1 mm

122.8

2.

Thickness δm(mm)

15.0

3.

Pitch t =

4.

β°∞

5.

K=

2 3

6.

χ=

1 (Fig. 10.11) 1− K

7.

CmI = χ Cm m/sec

8.

β∞Ι = tan

9.

w∞ = CIm /sin β∞ m/sec

9.6128

10.

l mm

214

11.

(Γ1 / ω∞ l)

0.2723

12.

(Γ1 /ω∞ l β ) From Fig. 10.8

1.28

13.

βΙ

14.

∆β (∆α from Fig. 10.9)

2πr mm Z

110.3 31.1833

δm t sin β∞

–1

CmI C u − u2 2


0.1751

1.2124 5.6860

36.2667

0.2127 12.2177

1.45

430


15.

β∞1 = (β∞ + ∆β )

37.7167

16.

w∞Ι = CmI /sin β∞Ι

9.2945

17.

(Γ1/w∞1 l)

0.2816

18.

(Γ1 / ω∞Ι l β ) From Fig. 10.8

1.22

19.

in radians βΙΙ in degrees

0.2232 12.8°

All necessary data are taken from impeller design D6. Calulations for other section are done in the same manner. Comparing the corrections made (1) by thickness correction (x) and by (2) by curvature correction, it is evident that final curvature (β) done by method (2) will be always more than that obtained by method (1).

431


Design No. D6 DESIGN OF AXIAL FLOW PUMP A : Design of camber line by Prof. Voznisenski and Prof. Pekin method for the impeller and diffuser.

1. General Specific Speed (ns) for axial flow pumps ranges from 500 and above. For high head and low discharge axial flow pumps, where ns < 500, (1) Hydraulic loss is higher due to high head operation and (2) Cavitational characteristics must be considered for a cavitation free operation. In order to achieve higher cavitational characteristics for low ns pumps, the impeller hub should be in conical form, which provides larger area at inlet and low flow velocity. Since, the Average inlet flow velocity (Cm1) reduces, the overall diffuser effect considerably reduces. Moreover, control of total head will be higher in conical hubs than in cylindrical hubs of impeller. But flow becomes three dimensional. Three dimensional design methods for axial flow pumps are not yet available. Available two dimensional design method with correction factors, experimentally determined, are adopted. In this design, a two dimensional approach is applied. Prototype pump parameters are : Total Head (H) = 22 m, flow rate (Q) = 100 m3/sec. Allowable suction lift is = –2 m. pumping liquid is water at 10°C. Since, cavitational characteristics must be higher the cavitational specific speed (Ccr) is taken as Ccr = 1100. The cavitational coefficient (φ) is taken as φ = 1.15 (Refer equation 9.4) Allowable HSV =

( H at − HVP ) − ( hS + h fs ) φ (for water at 10° C) ≈ 0.336 and hfs = 0 for submerged

Taking Hat= 10.336 MWC and HVP condition of the pump, 10.336 − 0.366 + 2 HSV = = 10.43 m 1.15 H SV 10.43 = = 0.475 σ= H 22 Allowable critical speed (ncr)

H  Ccr  SV   10  (ncr) = Q Specific speed

3.65n Q

(ns) =

H

3/ 4

3/ 4

=

 22  1100    10  = 100

3/ 4

≈ 125 rpm.

3.65 × 125 × 100 ≈ 450 (22)3 / 4

2. Determination of General Parameter Impeller eye velocity, C0 = 0.07

3

Qn 2 = 0.07

3

100 × 1252 = 8.08 m/sec.

_ _ d  From Fig. (10.41 ) d = f (ns), hub diameter d =  h  = 0.66 for ns = 450  Di 

432


Impeller diameter,

Di =

Unit Head,

(KH) =

KQ =

Unit Discharge, From Fig. (10.40) (KH)opt

4Q π (1 − d ) × C0 H 2

n D

4 × 100

=

2

π (1 − 0.66 2 ) × 8.08

22 2

Q 3

2

 125  2   .(5.31)  60 

=

= 5.31 m.

= 0.184

100 = 0.328 125 × (5.31)3

nD = f (ns) , (KH)opt = 0.22

Accordingly the (KQ) value will be KQ =

ns ( K H )3 / 4 = 219

450 × (0.22)3 / 4 = 0.44. 219

Hence, final values of the model pump will be (taken from the above prototype) KH = 0.22, _ KQ = 0.44, C = 1100, ns = 450 and σ = 0.475, d = 0.66. Model pump to be tested in the laboratory must be similar in all respect to the prototype pump. In the laboratory, model pump diameter (Dim) should be Dim = 0.35 m and the speed of the model pump (nm) must be nm = 800 rpm to suit the laboratory test rig. Accordingly model pump will have the following specifications: 2

H = Hm =

KH n2m

D3

m

 800  = 0.22 ×   × (0.35)2 = 4.78 m.  60 

 800  Q = Qm = KQ nm D3m = 0.44 ×   × (0.35)3 = 0.251 m3/sec.  60 

3. Calculation of Impeller Dimensions _ Hub diameter, dh = d × D = 0.66 × 0.35 = 0.231 m. Calculations are carried out for 5 Sections. The radius of each section is given as rI =

dh 0.231 + 0.02 Di = + 0.02 × 0.35 = 0.12.28 m. 2 2

rII =

rI + rIII 0.1228 + 0.1454 = = 0.1341 m. 2 2

rIII =

0.1228 + 0.168 rI + rV = = 0.1454 m. 2 2

rIV =

rIII + rV 0.1454 + 0.168 = = 0.1567 m. 2 2

rV =

Di 0.350 – 0.02 Di = – 0.02 × 0.35 = 0.168 m. 2 2

433


The Meridional velocity (Cm) is assumed to be constant at all radii as well as from inlet to outlet of the impeller Cm =

4Q _ 4 × 0.251 = = 4.669 m/sec. πDi2 (1 − d 2 ) π × 0.35 (1 − 0.662 )

IV III II

rii = 134.1 m

ri = 122.8 mm

riii = 145.4 mm

riv = 156.7 mm

rv = 168 mm

dh = 231 mm

D = 350 mm

I

Fig. D-6.1. Selection of radius of different design sections

Fig. D3-1 illustrates different stream line sections selected for the impeller design of the model pump. Model calculations for hub section is given below in Table D-3.1 calculations for other sections are to be carried out in the similar manner. Blade velocity, uI =

πDI n π × 0.1228 × 800 = = 10.2918 m/sec. 60 60

Γ act Assuming axial entry at inlet, Cu1 = 0 and coefficient of circulation KΓ = Γ = 0.9 th

Tangential velocity at outlet,Cu2 =

gH 9.81 × 4.78 = = 5.08 m/sec. uKΓ 10.2918 × 0.9

Blade angle at inlet,

4.669   Cm = β1 = tan–1   = 24.5°.  u 10.2918 

Blade angle at outlet,

 Cm  4.669 = β2 = tan–1   = 42°  u − Cu 2 10.2918 − 5.08 

Flow deviation,

∆βmax = β2 – β1 = 42 – 24.5 = 17.5°

l is selected from the graph (Fig. 10.34) as per Prof. A.R. Howell |43| and Vane Solidity    t  hub

l is selected from the graph (Fig. 10.43) as per Prof. A.N. Papir |84|, for which Nominal value of   t peri (∆β) for this design is selected as (∆β)nom = 0.9 (∆β)max. Since the graph given (Fig. 10.34) by Prof. Howell is for (∆β)nom = 0.8 (∆β)max, (∆β)nom for this design is (∆β)nom =

17.5 × 0.8 = 15.57°. From 0.9

434


l (Fig. 10.34)   = 2.0 as per Prof. Howell, but taken as 1.94 in order to get smooth variation from  tI l l hub to periphery and   = 1.36 as per Prof. A.N. Papir for KH = 0.22.   for other section (II, III,  tV t

IV) are obtained, in order to have a smooth variation of mean line curvature (β) from hub to periphery. Blade thickness (δm) is selected, based on the strength. δmI = 15 mm and δmV = 6 mm Thickness for other sections are determined to get smooth variation from hub to periphery. Maximum thickness (δm)max is located at 0.45 times the profile length (l). δm 15 = 0.1 × l 214 = 0.0015 m or 1.5 mm. Similarly the outlet edge rounding off radius for the profile, ρ** = (0.00065 to 0.001) D = 0.0009 × 0.35 = 0.00032 m = 0.32 mm. Number of blades (Zi) for the impeller is taken as Zi = 7. −

The inlet edge rounding off radius for the profile, ρ* = 0.15, δ m = (0.1),

 l TABLE D-6.1: Selection of vane solidity    t S. No.

Details/Section

I

1.

radius r mm

2.

u=

3.

Cu2 =

gH m/s uKΓ

5.08

4.

Cm =

4Q m/s π( D − d h2 )

4.669 for all sections

5*

β°2 tan–1

6*

 Cm  β°2 tan–1   u − Cu 2 

42°

7.

 l   Howell t I

2.0

8.

 l   Papir t

9.

 l   Selected t

2πrn m/s 60

2

Cm u

* in figure 10.11, β1 and β2 are given as α1 and α2.

122.8 10.2918

24.5°

V

1.36

1.94

1.36

435


TABLE D-6.2: Impeller dimensions S. No.

Details/Section

I

2πr mm Z

1.

t=

2.

 l l =   × t mm t

110.225 213.98 (214)

 Cm  β°∞ = tan–1  Cu 2   u −  2  β∞ = α∞ in figure

3*

4.

Curvature (β)

5.

dm =

32.72°

11.67°

dm l S Sm = m (l / 2)

6.

0.07 – 0.1 for all sections

7.

ρ* mm

1.5

8.

ρ** mm

0.32

* β∞ is given as α∞ in figure 10.11.

2.0 1.8 β

1.6

10°

1.4

9°

1.2

15

8°

1.0

14

7° — t

δm

l/t

β

l

13

δm

6°

12

0.08

5°

11

0.07

10

0.06

9

0.05

8

0.04

7

0.03

6

0.02

δm δm =

100

δm

2

150

Fig. D-6.2. Pump parameters

175

436


TABLE D-6.3: Design of impeller cascade system of axial flow pump (AFP 450) As per method suggested by Prof. Voznisenski and Prof. Pekin Design of thin profile (mean line for the thick profile) in the form of an Arc of circle. S. No.

Details/Section

I

1.

radius ri mm

122.8

2.

 l Vane Solidity    t

1.94

3.

 t Relative Pitch    l

0.5155

4.

Pitch t =

5.

l length l =   × t mm t

6*

Circulation Γ =

7.

Γ1 =

Γ 2 m /sec Z

0.56

8.

ui =

πDi n m/sec 60

10.2918

9.

Cm =

10.

Γ1 Cu 2 = m/sec 2πr1 2

11.

w∞ u = u –

Γ1 m/sec 4πri

7.7523

12.

β°∞ = tan–1

Cm (deg) w∞u

31.1833

13.

w∞ =

2πr1 (mm) Zi =7 Z

2πgH m2/sec K Γω

4Q m/sec πD 2i (1 − d 2 )

Cm m/sec sin β ∞

110.30

213.98 (214)

3.92

4.6690

2.5415

9.0591

437


14.

15**

 Γ1   w l 

0.2889

 Γ1   w lβ  from Fig. 10.8

1.45

∞

∞

16.

β


0.1992 11.117

17.

∆β° (∆α)

1.5333°

18.

βΙ = (β∞ + ∆β )

32.7166°

Γ1 ( w∞lβ)

1.42

19∗∗


0.2035 11.67°

20.

β

21.

 4 δm  m = ( 1 + β) 1 +  π l 

1.311

22.

(m.w∞ )2

140

23.

u2

106

24.

σ=

m.w∞2 − u 2 2 gH

Note: * Value is same for all sections. ** Symbol (∆β) is given as (∆α) in the graph 10.11.

0.375

438

Comp-1/Newage/Engineering/Pump-13w.pm6.5—2.11.07

Section I : Type AFP 450 TABLE D-6.4 α = 32° 43′′ ; β = 11° 40′′ = K

–3

SK

–1

1.

αK + α + βSK

2.

1 l 700 = 0.20365; R = = 4.91050; = 1,94; C = 196.75 × β t 3437.35

–5/2

–2 –0.866025

21° 03′

21° 27′

22° 37′

sin αK

0.35918

0.36569

0.38456

3.

cos αK

0.93327

0.93074

4.

20 – 2K

0.18131

5.

3K – 30

6.

–0.5

0

1 0.5

2 0.866025

5/2 0.96593

1

38° 33′

40° 49′

0.45218

0.54049

0.62320

0.67965

0.69445

0.69946

0.92310

0.89193

0.84135

0.78206

0.73353

0.71954

0.71468

0.17480

0.15593

0.08831

0

–0.08271

–0.13916

–0.15396

–0.15897

0.09192

0.08939

0.08175

0.05058

0

–0.05929

–0.10782

–0.12179

–0.12667

u = R (4)

0.89032

0.85836

0.76569

0.43365

0

–0.40615

–0.68335

–0.75602

–0.78062

7.

Z = R (5)

0.45137

0.43895

0.40143

0.24837

0

–0.29114

–0.52945

–0.59805

–0.62201

8.

(6)2 + (7)2

0.99641

0.7474

0.2497

0.2497

0.7473

0.9963

9.

check up (8 = 9)

1–

0.2498

0.7481

0.9965

2 3 3β – 4 64

0.9965

0.7481

43° 59′

3

32° 43′

β2 12

26° 53′

0

44° 23′

1 β2 – 4 192 0.2498

10.

u = C. (6) mm

339.8

327.6

292.3

165.5

0

–155.0

–260.8

–288.6

–298.0

11.

Z = C. (7) mm

172.3

167.6

153.2

94.8

0

–111.4

–202.1

–228.3

–237.4

12.

u2 + Z2

1452×102

1354×102

364.3×102

1089×102

1354×102

1452×102

1089×102

363.8×102


8.12.07

–0.96593

–1

 l   = 381.70. t

TABLE D6-5 Values taken from Nomogram K

i

–3

–5/2

–2

–1

0

1

2

5/2

3

Σ

aki –3

0

0.06

0.025

0.149

0.470

0.446

0.452

0.454

0.455

2.457

0

0.094

0.405

0.449

0.446

0.449

0.450

2.253

0

0.172

0.425

0.440

0.438

0.438

1.536

0

0.186

0.360

0.394

0.400

–0.164

0

0.135

0.180

0.195

–1.442

0

0.034

0.046

–2.203

0

–2.451

–2

–0.025

–0.015

–1

–0.149

–0.134

–0.094

0

–0.470

–0.457

–0.405

–0.172

1

–0.446

–0.446

–0.449

–0.425

–0.186

2

–0.452

–0.450

–0.446

–0.440

–0.360

–0.135

3

–0.455

–0.454

–0.450

–0.438

–0.400

–0.195

–0.046

–0.013


Section I : Type AFP 450

bki –3

0

0.016

0.241

0.200

0.089

0.076

0.073

0.072

0.831

0

0.172

0.251

0.092

0.079

0.075

0.075

0.633

0

0.212

0.172

0.095

0.087

0.085

0.016

0

0.186

0.182

0.159

0.153

–0.198

0

0.132

0.156

0.161

–0.179

0

0.036

0.048

–0.556

0

–0.677

–2

–0.064

–0.047

–1

–0.241

–0.222

–0.172

0

–0.200

–0.215

–0.251

–0.212

1

–0.089

–0.089

–0.092

–0.172

–0.186

2

–0.076

–0.076

–0.079

–0.095

–0.182

–0.132

3

–0.072

–0.072

–0.075

–0.085

–0.153

–0.161

–0.048

–0.011

439

0.064

440

Section I : Type AFP 450 TABLE D-6.6 Values ∆ aki and ∆ bki C1 = K

–3

–5/2

−βt β = –0.03343; C2 = – . C1 = 0.00114 πl 6

–2

–1

0

1

2

+5/2

3

Σ

1.

C1 cos αK

–0.03120

–0.03112

–0.03086

–0.02982

–0.02813

–0.02614

–0.02452

–0.02405

–0.02389

–

2.

C1 sin αK

–0.01201

–0.01223

–0.01286

–0.01512

–0.01807

–0.02083

–0.02272

–0.02322

–0.02338

–

3.

C2 cos αK

0.00106

0.00106

0.00105

0.00102

0.00096

0.00089

0.00084

0.00082

0.00082

–

4.

C2 sin αK

0.00041

0.00042

0.00044

0.00052

0.00062

0.00071

0.00078

0.00079

0.00080

–

∆ aKi = (1) –[(4) ( sk − si ) ] –0.031

–0.031

–0.031

–0.030

–0.029

–0.027

–0.026

–0.026

–0.025

–0.256

–2

–0.031

–0.031

–0.031

–0.030

–0.029

–0.027

–0.026

–0.026

–0.025

–0.256

–1

–0.031

–0.031

–0.031

–0.030

–0.028

–0.027

–0.025

–0.025

–0.025

–0.253

0

–0.031

–0.031

–0.030

–0.030

–0.028

–0.027

–0.025

–0.025

–0.024

–0.251

1

–0.031

–0.031

–0.030

–0.029

–0.028

–0.026

–0.025

–0.025

–0.024

–0.249

2

–0.030

–0.030

–0.030

–0.029

–0.028

–0.026

–0.025

–0.025

–0.024

–0.247

3

–0.030

–0.030

–0.030

–0.029

–0.028

–0.026

–0.024

–0.024

–0.024

–0.245

∆ bKi = (2) + [(3) ( sk − si ) ] –3

–0.012

–0.013

–0.013

–0.015

–0.017

–0.21

–0.021

–0.022

–0.022

–0.156

–2

–0.013

–0.013

–0.013

–0.015

–0.017

–0.21

–0.021

–0.022

–0.022

–0.156

–1

–0.013

–0.013

–0.013

–0.015

–0.017

–0.21

–0.022

–0.022

–0.022

–0.158

0

–0.013

–0.014

–0.014

–0.016

–0.018

–0.021

–0.022

–0.023

–0.023

–0.164

1

–0.014

–0.014

–0.014

–0.016

–0.019

–0.021

–0.023

–0.023

–0.024

–0.167

2

–0.014

–0.015

–0.015

–0.017

–0.019

–0.021

–0.023

–0.023

–0.023

–0.170

3

–0.014

–0.015

–0.015

–0.017

–0.019

–0.021

–0.023

–0.023

–0.023

–0.170


–3

TABLE D-6.7 Calculation of the values ϕ Ki and ψ Ki K i

–3

−

5 2

–2

–1

0

1

2

+

5 2

3

Σ

ϕ Ki = (2aKi + ∆ aKi ) –3

–0.031

–0.019

+ 0.019

0.268

0.911

0.865

0.878

0.882

0.885

4.658

–2

–0.081

–0.061

–0.031

0.158

0.781

0.871

0.866

0.872

0.875

4.250

–1

0.329

–0.299

–0.219

–0.030

0.316

0.823

0.855

0.851

0.851

2.819

0

–0.971

–0.945

–0.840

–0.374

–0.028

0.345

0.695

0.763

0.776

–0.579

1

–0.923

–0.923

–0.928

–0.879

–0.400

–0.026

0.245

0.335

0.366

–3.133

2

–0.934

–0.930

–0.922

–0.909

–0.748

–0.296

–0.025

0.043

0.068

–4.653

3

–0.940

–0.938

–0.930

–0.905

–0.828

–0.416

–0.116

–0.050

–0.024

–5.147



ψ Ki = –(2bKi + ∆ bKi ) 0.012

–0.019

–0.115

–0.467

–0.383

–0.157

–0.131

–0.124

–0.122

–1.506

–2

0.140

0.107

0.013

–0.329

–0.485

–0.163

–0.137

–0.128

–0.128

–1.110

–1

0.495

0.457

0.357

0.015

–0.407

–0.323

–0.168

–0.152

–0.148

0.126

0

0.413

0.444

0.516

0.440

0.018

–0.351

–0.342

–0.295

–0.283

0.560

1

0.192

0.192

0.198

0.360

0.391

0.021

–0.241

–0.289

–0.299

0.525

2

0.166

0.167

0.173

0.207

0.383

0.285

0.023

–0.049

–0.073

1.282

3

0.158

0.159

0.165

0.187

0.325

0.343

0.119

0.045

0.023

1.524

441

–3

442

Section I : Type AFP 450 TABLE D-6.8 Calculation of the values Aj ; Bj ; and other connected values  Γ1 A1 + A−1  A0 = –1.098  – + = 2.8736 ; 12   l

Γ1 = 2.6171 ; l

χ=

S m = –0, 1 ;

K2 =

B0 =

B2 =

K1 = 8 χ δ m w∞ = 6.1588 ;

4π ρ* 4π ρ** w∞ cos β = 0.2659 ; K3 = w∞ cos β = 0.0567. 3 l 3 l Sm 2(1 − S m2

)

2

− (1 − 3S m2 ) 2(1 − S m ) 2

– 3S m 2 (1 – Sm2 )

K1 −

S m (2 + S m ) S (2 − S m ) K2 − m K3 = – 0.66829 + 0.14957 + 0.01733 = – 0.50139 2(1 + S m ) 2 2(1 − S m ) 2

K1 +

2(1 + 2 S m ) 2(1 − 2S m ) K2 + K 3 = – 2.94040 + 0.49856 + 0.11025 = –2.33159 2 (1 + Sm ) (1 − Sm ) 2

K1 +

3 3 K2 + K 3 = 2.00481 – 1.24641 + 0.11813 = 0.87653 (1 – S m ) 2 (1 – S m ) 2

B * = 2K2 = 0.53180 = 0.53180 B ** = – 2K3= – 0.1134 = – 0.11340 Σ (check up) K A0 = 2.8736

– 1.60388 – 0.06648 + 0.1323 = – 1.53805

–3

– 5/2

–2

–1

0

1

2

5/2

3

A0 Vγ0

–1.9312

–1.7900

–1.4300

–0.5908

0

0.5908

1.4300

1.7900

1.9312

v′ = Σ

–1.9312

–1.7900

–1.4300

–0.5908

0

0.5908

1.4300

1.7900

1.9312


B1 =

1 = 1.2124 ; 2 δm 1− 3 t sin α

TABLE D-6.9 K

–3

–2

–1

0

1

2

3

Σ

l A = 5.57478 t 0

P10 (l/t) A0

0.00097

0.27524

0.62986

0.72645

0.62986

0.27524

0.00097

2.53859

Pi = ΣP10 (l/t) A0 + ......

Pi = Σ

0.00097

0.27524

0.62986

0.72645

0.62986

0.27524

0.00097

2.53859

qi = Σq10 (l/t) B0 + .....

qi = Σ

0.29633

0.26614

0.19769

–0.12841

–0.31925

–0.23130

–0.07921

+0.00199

QK = ΣQk0 B0 + ......

K

–3

–5/2

QK

0.13300

0.31012

–2 0.42858

–1

0

0.70506

1

0.74383

2

0.49110

5/2

0.18458

0.08965

2

+ 5/2

Σ

3 0.02825

3.11417



TABLE D-6.10 ∞ + v′′ ′′ Calculation of velocity components w∞ i

K

–3

–5/2

–2

–1

1

0.00097 0.27524 0.62986 0.72645 0.62986 0.27524 0.00097 Σ1

Σ

3

Pi ϕ Ki

Pi –3 –2 –1 0 1 2 3

0

0.0045 1.1698 1.7756 –0.4206 –1.9734 –1.2807 –0.0050 – 1.7743

– 1.7299

–1,5959

–1.0516

–0.0641

–0.9113

1.4299

1.5540

1.5907

Pi ψ Ki

Pi

–0.7298

Σ –0.0015 –0.3055 0.0794 0.4068 0.3307 0.3529 0.0015

Σ2

0.8171

0.8069

0.7757

0.5220

–0.0251

–0.4114

–0.5375

–0.5409

–0.5426

0.8648

443

–3 –2 –1 0 1 2 3

444

Section I : Type AFP 450 TABLE D-6.11 i

K

–3

–5/2

–2

–1

0

1

2

5/2

3

Σ

q i ϕ Ki

qi –3

0.29633

1.3803

–2

0.26614

1.1311

–1

0.19769

0.5573

0

–0.12841

0.0744

1

–0.31925

1.0002

2

–0.23130

1.0762

3

–0.07921

0.4077

Σ3

0.6141

0.6245

0.6451

0.7261

0.9102

0.7163

0.5072

0.4508

0.4330

5.6272

–3

–0.4463

–2

–0.2954

–1

0.0249

0

–0.0719

1

–0.1676

2

–0.2965

3

–0.1207 Σ4

–0.0266

–0.0563

–0.1426

–0.4571

–0.5645

W∞ u = 7.7503

–0.2085

–0.0024

+ 0.0371

+ 0.0474

CZ = 4.669

v′′x = Σ4 –Σ1

1.7477

1.6736

1.4533

0.5945

0.5004

–1.1198

–1.4323

–1.5169

–1.5433

v′′y = Σ2 + Σ3

1.4312

1.4314

1.4208

1.2481

0.8851

0.3049

–0.0303

–0.0901

–0.1096


qi ψ Ki

qi

TABLE D-6.12: Determination of angle α K K

–3

–5/2

–2

–1

0

1

2

5/2

3

1.

w ∞u + vx′′

9.4980

9.4239

9.2036

8.3448

7.2429

6.6305

6.3180

6.2334

6.2070

2.

Cm + vy′′

6.1211

6.1213

6.1107

5.9380

5.5750

4.9948

4.6596

4.5998

4.5803

3.

v′y

–1.9312

–1.7900

–1.4300

–0.5908

0

0.5908

1.4300

1.7900

1.9312

4.

(1)2 + (2)2

127.6798

126.2802

122.0469

104.8955

83.6417

68.9116

61.6290

60.0134

59.506

5.

(4) – (3)2

123.9503

123.0761

120.0020

104.5464

83.6417

68.5625

59.5841

56.8093

55.7765

6.

(5)

11.1333

11.0940

10.9546

10.2248

9.1456

8.2803

7.7191

7.5372

7.4684

7.

(1) × (3)

–18.3425

–16.8688

–13.1612

–4.9301

0

3.9173

9.0347

11.1576

11.9870

8.

(2) × (6)

68.1480

67.9097

66.9403

60.7149

50.9867

41.3584

35.9679

34.6696

34.2075

9.

(7) + (8)

49.8055

51.0409

53.7791

55.7848

50.9867

45.2757

45.0026

45.8274

46.1945

10.

(9) ÷(4) = sin αK

0.39008

0.40419

0.44064

0.53181

0.60958

0.65701

0.73022

0.76362

0.77630

11.

cos αK

0.92078

0.91468

0.89769

0.84686

0.79273

0.75388

0.68320

0.64569

0.63038

12.

(1) × cos αK

8.7456

8.6199

8.2620

7.0669

5.7472

4.9986

4.3165

4.0248

3.9128

13.

(2) × sin αK

2.3877

2.4742

2.6926

3.1579

3.3984

3.2816

3.4025

3.5125

3.5557

14.

Wx + v′′x = (12) + (13)

11.1333

11.0941

10.9546

10.2248

9.1456

8.2802

7.7190

7.5373

7.4685

15.

(10) × (14)

4.3429

4.4841

4.8270

5.4377

5.5750

5.4402

5.6366

5.7556

5.7978

16.

(3) × (11)

–1.7782

–1.6373

–1.2837

–0.5003

0

0.4454

0.9770

1.1558

1.2174

17.

(15) –(16)

6.1211

6.1214

6.1107

5.9380

5.575

4.9948

4.6596

4.5998

4.5804



445

446

Section I : Type AFP 450/2 TABLE D-6.13 Calculation of velocity (v′x′ ) K

–3

nK = nK /(l/2) j

0.01

0.02

–5/2

–2

0

0

1 0

Aj 0

2.8736

0 0

1 0

2 0

2.5 0

3 0.01

0.02

Aj uγ j –0.1459

– 0.1956

–0.6227

–1.0925

–1.4255

–1.4368

–1.4255

–1.0925

–0.6227

–0.1459

–0.1956

–0.0877

–0.02102

–0.3236

–0.4473

–0.5026

0.5383

–0.1043

–0.7114

–1.5980

–1.3436

–0.1012

0.0340

0.2583

0.5072

0.6018

1 –1 Σ Aa Bj

0

–0.50139

0.5026

0.4473

0.3236

0.2102

0.0877

0

1

–2.33159

–1.5980

–1.3436

–0.7114

–0.1043

0.5383

0.7422

2

0.87653

–0.6018

–0.5072

–0.2583

–0.0340

0.1012

0

*

0.53180

**

–0.11340

ΣBb  ∑ Bb + ∑ Aa vx′  ∑ B b − ∑ Aa

Bj uqj

–

–

2.4840

0.6318

0.1693

0.0846

0.0564

0.0454

0.0431

0.0423

0.0423

0.0090

0.0090

0.0092

0.0097

0.0120

0.0181

0.0361

0.1347

0.5297

–

–

–1.6882

–0.3945

1.8471

0.7134

0.9085

0.8449

0.4419

–0.1004

–0.2039

–1.4958

–1.2021

–1.8341

–1.5901

1.2244

–0.3791

–0.5170

–0.5919

–0.9836

–1.1929

–0.8266

–1.6417

–1.3977

–1.5423

–1.1989

2.4698

1.8059

2.3340

2.2817

1.8674

0.9921

0.4188

–1.3499

–1.0065


j

TABLE D-6.14 Calculation of velocities and thicknesses K

nk = nk /(l/2)

–3 0.01

wx + v′′x

11.1333

v′x

0.02

–5/2

–2

0

0

–1 0

0

1

2

2.5

0

0

0

0

3 0.01

0.02

11.0941

10.9546

10.2248

9.1456

8.2802

7.7190

7.5373

7.4685

–1.8341 –1.5901

1.2244

–0.3791

–0.5170

–0.5919

–0.9836

–1.1929

–0.8266

–1.6417

–1.3977

–1.5423 –1.1989

2.4698

1.8059

2.3340

2.2817

1.8674

0.9921

0.4188

–1.3499

–1.0065

w = wx

9.2992

9.5432

12.3182

10.5755

9.7078

8.5537

7.2966

6.5261

6.7107

5.8268

6.0708

+ v′′x + v′x

9.5910

9.9344

13.5639

12.7605

12.5588

11.4273

10.1476

8.7111

7.9561

6.1186

6.4620

0.13300

0.31012

0.42858

0.70506

0.74383

0.49110

0.18458

0.08965

0.0143

0.0139

0.0252

0.0409

0.0726

0.0870

0.0673

0.0283

0.0134

0.0049

0.0047

0.0139

0.0134

0.0229

0.0336

0.0561

0.0651

0.0484

0.0212

0.0113

0.0046

0.0044

QK d (y) =


Section I : Type AFP 450/2

0.02825

QK 1 2  w (0 ) + w ( y )

447

448

Section I : Type AFP 450/2 TABLE D-6.15 Calculation of velocity v′x′ (up to 4 digits) K nK = nK/ l/2

–5/2

–2

–1

0

0.02

0.03

0.03

0.04

0.05

0.05

0.06

0.07

0.08

0.06

0.07

0.08

0.09

–0.5590

–0.5517

–1.0048

–0.9799

–0.9571

–1.3494

–1.3346

–1.3201

–1.3057

0.3118

0.3007

0.2082

0.2068

0.2050

0.0873

0.0872

0.0870

0.0868

0

0

0

0 0.6432

Aj 2.8736

–1.3754

–1.3653

–1.3551

–1.3449

ΣA a Bj –0.50139 –2.33159

–0.6776

–0.6353

–0.1288

–0.1326

–0.1346

0.4833

0.4731

0.4632

0.4535

0.6749

0.6642

0.6536

0.87653

–0.2529

–0.2400

–0.0522

–0.0563

–0.0596

0.0307

0.0700

0.0734

0.0699

0

0

0

0

0.53180

1.8475

1.3993

0.6016

0.5801

0.5545

0.1676

0.1669

0.1660

0.1651

0.0843

0.0842

0.0841

0.0840

0.0092

0.0092

0.0097

0.0097

0.0097

0.0120

0.0120

0.0120

0.0120

0.0180

0.0180

0.0179

0.0179

ΣΒb

1.2380

0.8339

0.6385

0.6077

0.5750

0.8303

0.8162

0.8016

0.7873

0.7772

0.7664

0.7556

0.7551

ΣΒb + ΣAa

0.6790

0.2822

–

–0.3722

–0.3821

–0.5185

–0.6184

–

–

0.5995

–0.5898

ΣΒb – Σba

1.7970

1.3856

1.6433

2.1526

2.1317

1.5876

–

– –2.1797

2.1508

1

– 2

–

+5/2

0.04

0.05

0.06

0.07

0.02

0.03

0.01

0.02

–1.3643

–1.3494

–1.3346

–1.3201

–1.0318

–1.0048

–0.5767

–0.5590

–0.0874

–0.0873

–0.0872

–0.0870

–0.2093

–0.2082

–0.3203

–0.3118

0.4938

0.4833

0.4731

0.4632

–0.1229

–0.1288

–0.7072

–0.6776

–0.0846

–0.0807

–0.0770

–0.0734

0.0471

0.0522

0.2602

0.2529

0.0564

0.0564

0.0563

0.0563

0.0454

0.0454

0.0431

0.0431

0.0359

0.0357

0.0356

0.0354

0.1318

0.1283

0.4877

0.3940

0.4141

0.4074

0.4008

0.3945

–0.1079

–0.1111

–0.2365

–0.2999

–0.9338

–0.9256

–1.1397

–1.1159

–0.8132

–0.8584

0.9239

0.8937

0.3402

0.2596

1.7784

1.7568


–0.11340

TABLE D-6.16 Calculation of velocities and thickness −5 2

K

nK

= nKl/2

wx + ν′′x

–2

0

+1

+

+2

5 2

0.02

0.03

0.04

0.05

0.07

0.08

0.08

0.09

0.06

0.07

0.02

0.03

0.01

0.02

0.02

0.03

0.03

0.04

0.05

0.06

0.06

0.07

0.040

0.050

0.02

0.03

0.01

0.02

11.0941

ν′x

–1

10.9546

10.2248

9.1456

8.2802

7.7190

7.5373

0.6790

0.2622

–0.3722

–0.3821

–0.5185

–0.6184

–0.5995

–0.5898

–0.9238

–0.9256

–1.1397

–1.1159

–0.8132

–0.8584

1.7970

1.3856

1.6433

1.5876

2.1797

2.1508

2.1526

2.1317

1.7784

1.7568

+0.9239

0.8937

0.3402

0.2596

11.7731

11.3763

10.5824

10.5725

9.7063

9.6064

8.5461

8.5558

7.3464

7.3546

6.5793

6.6031

6.7241

6.6789

12.8911

12.4797

12.5979

12.5422

12.4045

12.3748

11.2982

11.2773

10.0586

10.0370

8.6429

8.6127

7.8775

7.7969


Section I : Type AFP 450/2

w = wx + ν′′x + ν′x

0.31012

QK d (y) =

0.42858

0.70506

0.74383

0.49110

0.18458

0.08965

0.0258

0.0262

0.0405

0.0405

0.0726

0.0730

0.0870

0.0870

0.0672

0.0666

0.0282

0.0281

0.0134

0.0134

0.0235

0.0238

0.0338

0.0339

0.0565

0.0566

0.0655

0.0655

0.0486

0.0487

0.0213

0.0213

0.0113

0.0114

QK 1

w 2

(0) + w ( y )

TABLE D-6.17 Values of A1 and A–1 Section Coefficients

I

III

V

9.50

6.00

6.00

A–1

–8.00

–8.80

–6.00 449

A1

450

Section I : Vari. A—AFP 450/2 TABLE D-6.18 Calculation of coefficients Aj, Bj and other related values (up to 4 digits)  Γ1 A1 + A−1   l + 12  = 2.7368;  

Γ1 = 2.6171; l

A0 = –1.098

A1 = 9.5;

A–1 = – 8.00; K1 = 5.600;

B0 =

B1 =

K3 = 0.035

Sm S (2 + S m ) S (2 − S m ) K1 − m K2 − m K 3 = –0.60766 + 0.14962 + 0.01069 = –0.4474 2 2 2 2(1 − S m ) (1 + S m ) (1 − Sm ) 2

− (1 − 3Sm2 ) 2(1 −

Sm2 ) 2

− 3S m 2(1 − S

2

m)

2

2(1 − 2Sm )

K1 +

2 (1 + 2Sm )

K1 +

3 3 K2 + K 3 = 1.82291 –1.24688 + 0.07292 = 0.6490 2 (1 + Sm ) (1 − Sm ) 2

(1 + Sm )

2

K2 +

(1 + Sm ) 2

K3 = –2.67361 + 0.49875 + 0.06806 = –2.1068

B* = 2K2 = 0.53200 = 0.5320 B** = – 2K3 = –0.07000 = 0.0700 Σ (check up) = –1.45835 – 0.06650 + 0.08167 = –1.4432

All values up to 5 digits K

–3

–5/2

–2

–1

A0 = 2.7368

A0 Vγ0

–1.8393

–1.7056

–1.3616

–0.5627

A1 = 9.5

A1 Vγ1

–0.1719

–0.1874

–0.1897

–0.2662

A–1 = –8.00

A–1 Vγ –1

0.6366

0.6992

0.7417

–1.3746

–1.1938

–0.8096

νy = Σ

0 –0.8289

0 0 –0.7560

1

2

+5/2

3

Σ

0.5627

1.3616

1.7056

1.8393

10.9384

0

0.8808

0.8303

0.7560

0.8959 0.7544

–0.6366

–0.2241

–0.1598

–0.1578

–0.1448

–1.8926

0.3386

2.0826

2.3781

2.4505


B2 =

K2 = 0.266;

TABLE D-6.19 K

–3

–2

–1

0

+1

+2

Σ

+3

l A = 5.3094 t 0

Pi0

l A t 0

0.0009

0.2621

0.5999

0.6920

0.5999

0.2621

0.0009

2.4178

l A = 18.4300 t 1

Pi1

l A t 1

–0.0085

0.0196

–0.0317

0.1378

0.5095

0.1482

–0.0049

0.7680

l A = –15.5200 t –1

Pi –1

0.0041

–0.1231

–0.4291

–0.1160

0.0267

–0.0165

0.0072

–0.6467

Pi

Σ

–0.0035

0.1586

0.1391

0.7138

1.1361

0.3918

0.0032

2.5391

0.22825

0.17166

–0.11431

–0.29545

–0.22122

–0.05968

l A t –1


Section I : Vari. A—AFP 450/2

l B = –0.8680 t 0 l B = –4.0872 t 1 l B = 1.2591 t 2 l * B = 1.0321 t l ** B = –0.1358 t qi

ΣQK0 B0 +...

0.29068

Σ

–3

–5/2

–2

–1

0

1

2

5/2

3

Σ

0.1330

0.3044

0.4057

0.6439

0.6772

0.4472

0.1587

0.0672

0.0175

2.8548

451

QK

Σ

452

Section I : Vari. A—AFP 450/2 TABLE D-6.20 Calculation of velocity components (wx + vx′′ ′′) K i

–3

–5/2

–2

–1

0

1

2

5/2

3

Σ

qi ϕ Ki

qi

–3

–0.0035

–0.016

–2

0.1586

0.674

–1

0.1391

0.392

0

0.7138

–0.413

1

1.1361

–3.559

2

0.3918

–1.823

3

0.0032 –2.1692

–2.1417

–2.0536

–1.6047

–0.6055

0.3490

1.0175

1.1955

1.2503

–4.762

P1ψ K –3

–0.0035

0.0053

–2

0.1586

–0.1762

–1

0.1391

0.0175

0

0.7138

0.3997

1

1.1361

0.5965

2

0.3918

0.5023

3

0.0032

0.0049 1.3500

Σ2

0.6695

0.6816

0.7137

0.7563

0.4760

–0.1842

–0.5532

–0.5990

–0.6107

1.3500


Σ1

TABLE D-6.21 Calculation of values of ϕ Ki and ψ Ki i

K

–3

–5/2

–2

–1

0

1

2

5/2

3

Σ

qi ϕ Ki

qi –3 –2 –1



0 1 2 3 Σ3

0.5624

0.5717

0.5905

0.6664

0.8336

0.6501

0.4603

0.4088

0.3926

5.1364

–0.1878

–0.0001

–0.0359

–0.0454

–1.2631

qiψ Ki –3

0.29068

–2

0.22825

–1

0.17166

0

–0.11431

1

–0.29545

2

–0.22122

3

–0.05968 Σ4

–0.0297

–0.0566

–0.1348

–0.4218

–0.5136

W∞u = 7.7520 ;

Cz = 4.6690

2.1395

2.0851

1.9188

1.1829

0.0919

–0.5368

–1.0176

–1.1596

–1.2049

3.4993

v′′y = Σ2 + Σ3

1.2319

1.2533

1.3042

1.4227

1.3096

0.4659

–0.0929

–0.1902

–0.2181

6.4864

453

v′′x = Σ4 –Σ1

454

Section I : Vari. A—AFP 450/2 TABLE D-6.22 Values within the bracket indicates when α = 45° α k) Determination of angle (α – 5/2

–2

–1

0

1

2

5/2

3

No. of digits

1. W∞u + v′′x

9.8915

9.8371

9.6708

8.9349

7.8439

7.2152

6.7344

6.5924

6.5471

3

2. Cz + v′′y

5.9009

5.9223

5.9732

6.0917

5.9786

5.1349

4.5761

4.4788

4.4509

3

–1.3745

–1.1938

–0.8096

–0.8289

–1.3926

0.3386

2.0826

2.3781

2.4505

4

4. (1)2 + (2)2

132.6624

131.8422

129.2035

116.9413

97.2704

78.4263

66.2928

63.5194

62.6750

3

5. (4) – (3)2

130.7729

130.4171

128.5480

116.2542

95.3311

78.3116

61.9556

57.8640

56.6700

3

5

11.4356

11.4201

11.3379

10.7821

9.7638

8.8494

7.8712

7.6069

7.5280

4

7. (1) × (3)

–13.5969

–11.7435

–7.8295

–7.4061

–10.9234

2.4431

14.0251

15.6774

16.0437

3

8. (2) × (6)

67.4803

67.6333

67.7235

65.6813

58.3739

45.4408

36.0194

34.0698

33.5064

3

9. (7) + (8)

53.8834

55.8898

59.8940

58.2752

47.4505

47.8831

50.0445

49.7472

49.5501

3

3. ν′y

6.

10. (9) : (4) = sin αK

0.406169

0.423914

0.463563

0.498328

0.487820

0.610549

0.754900

0.783180

0.790587

5

11. cos αK

0.913802

0.905702

0.886064

0.866988

0.872946

0.792009

0.655839

0.621793

0.612363

5

12. (1) cos αK

9.0389

8.9005

8.5689

7.7475

6.8473

5.7143

4.4167

4.0991

4.0092

4

13. (2) sin αK

2.3967

2.5106

3.7690

3.0356

2.9165

3.1351

3.4545

3.5077

3.5188

4

11.4356

11.4201

11.3379

10.7821

9.7638

8.8494

7.8712

7.6068

7.5280

4

15. (10) × (14)

4.6448

4.8411

5.2558

5.3730

4.7630

5.4030

5.9420

5.9575

5.9515

4

16. (3) × (11)

–1.2561

–1.0812

–0.7174

–0.7187

–1.2156

0.2682

1.3659

1.4787

1.5006

4

5.9009

5.9223

5.9732

6.0917

5.9786

5.1348

4.5761

4.4788

4.4509

4

14. wx+ v′′x = (12)+(13)

17. (15) – (16) = (2)


–3

K

nK = nK /(l/2) j 0 1 –1

v′x

0.01

0.02

–5/2

–2

–1

0

1

2

5/2

3

0

0

0

0

0

0

0

0.01

0.02

–1.3684 0 0 –1.3684

–1.3577 –1.1875 0 –2.5452

–1.0405 –0.5511 0 –1.5916

–0.5930 –0.1563 0 – 0.7493

–0.1390 –0.0310 0.0004 – 0.1696

–0.1863 –0.0489 0.0010 – 0.2342

0 0.6706 0 0.0847 0.0111 0.766 –0.6024 2.1344

–0.0782 0.4864 –0.0749 0.0564 0.0223 0.412 –2.1332 2.9572

–0.1876 –0.0942 0.0251 0.0454 0.0832 0.128 –1.7196 1.4636

– 0.2888 –0.6428 0.1913 0.0431 0.3270 –0.370 –1.1193 0.3793

– 0.4485 –1.4439 0.4456 0.0423 0 –1.404 –1.5736 –1.2344

– 0.3991 –1.2141 0.3756 0.0423 0 –1.195 –1.4292 – 0.9608

Aj 2.7368 9.5000 –8.0000 ΣAa

j 0 1 2 * **

–3

TABLE D-6.23 Calculation of velocity (v′′x).(up to 4 digits)

Ajuγ j –0.1390 –0.0006 0.0261 –0.1135

–0.1863 –0.0012 0.0412 –0.1463

–0.5930 0 0.1316 –0.4614

–1.0405 0 0.4641 –0.5764

Bj – 0.4474 –2.1068 0.6490 0.5320 –0.0700 ΣBb ΣBb+ ΣAa ΣBb– ΣAa

–1.3577 0 1.0000 –0.3577



Bj uqj 0.4485 –1.4439 –0.4456 0 0.0056 –1.435 –1.5485 –1.3211

0.3991 –1.2114 –0.3756 0 0.0056 –1.185 –1.3313 –1.0387

0.2888 –0.6428 –0.1913 2.4849 0.0057 1.945 1.4836 2.4064

0.1876 –0.0942 –0.0251 0.6320 0.0060 0.7063 0.1299 1.2827


0.0782 0.4864 0.0749 0.1693 0.0074 0.816 0.4583 1.1737

TABLE D-6.24 Calculation of velocities and thicknesses

K

nK = nK /(l/2) wx + vx′′ w = wx + vx′′ + v x′ QK

0.01

0.02

11.4356

–5/2

–2

–1

0

1

2

5/2

0

0

0

0

0

0

0

3 0.01

11.4201

11.3379

10.7821

9.7638

8.8494

7.8712

7.6069

9.8871

10.1043

12.9037

11.4678

11.2404

9.1614

6.7162

6.1516

6.4876

5.9544

10.1145

10.3969

13.8265

12.6206

11.9558

11.8982

11.8066

9.3348

7.9862

6.2936

0.3044

0.4057

0.6439

0.6772

0.4472

0.1587

0.0672

0.1330

0.02 7.5280 6.0988 6.5672 0.0175

0.0135

0.0132

0.0242

0.0354

0.0577

0.0740

0.0652

0.0256

0.0104

0.0029

0.0029

0.0131

0.0128

0.0227

0.0323

0.0539

0.0571

0.0383

0.0171

0.0084

0.0027

0.0027

455

QK d(y) = 1 [w(0) + w( y ) ] 2

–3

K

nK j 0 1 –1

j 0 1 2 * **

0.02

–2 0.03

–1

0.03

0.04

0.05

0 0.06

Aj

0.05

0

+1

+1

0.06

0.07

0.08

0.03

0.04

0.06

0.07

–1.3099 –0.1656 0.1456 –1.3299

–1.3003 –0.1791 0.1574 –1.3220

–1.2906 –0.1909 0.1678 –1.3137

–1.3137 –1.0546 0.0075 –2.3608

–1.2993 –1.0287 0.0100 –2.3180

–1.2711 –0.9793 0.0150 –2.2354

–1.2572 –0.9557 0.0174 –2.1955

0 0.6002 0 0.0843 0.0111 0.6956 –0.6264

0 0.5906 0 0.0841 0.0111 0.6858 –0.6279

0.0781 0.4559 –0.0656 0.0564 0.0222 0.3908 –

–0.0780 0.4462 –0.0626 0.0564 0.0221 0.3841 –

–0.0778 0.4275 –0.0570 0.0564 0.0220 0.3711 –1.8643

–0.0776 0.4185 –0.0543 0.0563 0.0219 0.3648 –1.8307

A j uγ j

2.7368 9.5000 –8.0000 ΣAa

–0.5324 –0.0025 0.1602 –0.3747

–0.5254 –0.0036 0.1735 –0.3555

–0.9569 –0.0043 0.4430 –0.5182

–0.9333 –0.0056 + 0.4372 –0.5017

–1.2851 –0.0142 0.8822 –0.4171

–1.2711 –0.0170 0.8609 –0.4272

–1.3197 0.1502 0.1320 –1.3379

Bj

Bj u q j

–0.4474 –2.1068 0.6490 0.5320 –0.0700 ΣBb ΣBb + ΣAa ΣBb –ΣAa

0.2782 –0.6123 –0.1872 1.8482 0.0057 1.3326 0.9579 1.7073

0.2683 –0.5741 0.1777 1.3998 0.0057 0.9220 0.5670 1.2770

0.1858 –0.1164 –0.0386 0.6018 0.0060 0.6386 0.1204 1.1568

0.1845 –0.1200 –0.0417 0.5803 0.0060 0.6091 0.1074

1.1108

0.0779 0.4367 0.0598 0.1677 0.0074 0.7495 0.3324

0.0778 0.4275 0.0570 0.1669 0.0074 0.7366 0.3094

0 0.6106 0 0.0845 0.0111 0.7152 –

0 0.6098 0 0.0844 0.0111 0.7053 –

1.1666

1.1638

2.0531

2.0352

–

–

+ 2

2.7516

1.7021

–

+ 5/2

0.01

0.02

0.03

0.01

0.02

–1.0105 –0.5192 0.0013 –1.5284 –0.1874 –0.1037 0.0304 0.0454 0.0827 –0.1326 – +1.3958

–0.9827 –0.5112 0.0025 –1.4914 –0.1868 –0.1111 0.0349 0.0454 0.0813 –0.1363 –1.6277 1.3551

–0.9569 –0.5040 0.0038 –1.4571 –0.1858 –0.1164 0.0386 0.0454 0.0792 –0.1390 –1.5961 –

–0.5492 –0.1662 0.0013 –0.7141 –0.2858 –0.6390 0.1929 0.0431 0.3010 –0.3878 –1.1019 0.3263

–0.5324 –0.1823 0.0025 –0.7122 –0.2782 –0.6122 0.1872 0.0431 0.2432 –0.4169 –1.1291 0.2953


v′x =

TABLE D-6.25 Calculation of velocities (v′x′ )

–5/2

= nK/(l/2)

456


TABLE D-6.26 Calculation velocities and thickness K

nK

–3

= nK/(l/2)

0.02

wx + v′′x

–5/2 0.03

0.03

11.4201

v′x

w = wx + v′′x + v′x

QK QK 1 2

W ( 0) + W ( y) 

0.04

–1 0.05

11.3379

0

0.06

1

2

2,5

0.07

0.08

0.06

0.07

0.02

0.03

0.01

0.02

0.05

0.06

0.03

0.04

0.01

0.02

0.00

0.01

10.7821

9.7638

8.8494

7.8712

0.5670

0.1204

0.1074

0.3324

0.3094 –0.6294 –0.6279 –1.8643 –1.8307 –1.6277 –1.5961 –1.1019 –1.1291

1.7073

1.2770

1.1568

1.1108

1.1666

1.1638

2.0531

2.0532

2.7516

1.7021 +1.3958

1.3551

0.3263 0.2953

12.372

11.9871 11.4583 11.4453 11.1145 11.0915

9.1370

9.1359

6.9851

7.0187

6.2435

6.2751

6.5050 6.4780

13.127

12.6970 12.4947 12.4487 11.9487 11.9459 11.8169 11.7990 11.6010 11.5515

9.2670

9.2263

7.9860 7.9330

0.4057

0.6439

0.6772

0.4472

0.1587

+3

7.6069

0.9579

0.3044

d(y) =

–2



0.0672

0.0239

0.0245

0.0354

0.0354

0.0576

0.0577

0.0740

0.0740

0.0653

0.0651

0.0256

0.0255

0.0103 0.0102

0.0225

0.0231

0.0323

0.0324

0.0539

0.0539

0.0571

0.0572

0.0382

0.0383

0.0171

0.0172

0.0084 0.0084

w

9.99

12.22

11.45

11.10

9.14

7.00

6.26

6.50

6.03

(Final Value)

10.26

13.01

12.48

11.95

11.81

11.56

9.24

7.94

6.43

457

458

Section I : Vari. A—AFP 450/2 TABLE D-6.27 Calculation of coordinates of skeleton (Relative values) K

–3

–5/2

–2

–1

cos α i

0

∑

+1

+2

+ 5/2

+3

0.42209

0.68791

0.75166

0.77269

–0.27643

0.38565

MKi cos α i

0.88606 0.86699 0.87295 0.79201 0.65584 0.61236 –0.87540

x′′K

–0.84439

–0.75500

–0.43596

y′′k

MKi sin α i –0.48238

–0.46823

–0.42374

–0.24384

0

0.26795

0.51840

0.59533

0.62216

xK′ = x3′′ − xK′′

1.64809

1.61708

1.52769

1.20865

0.77269

0.35060

0.08478

0.02103

0

yK′ = y3′′ − yK′′

1.10454

1.09039

1.04590

0.86500

0.62216

0.35421

0.10376

0.02683

0

0.40617 0.46356 0.49833 0.48782 0.61055 0.75490 0.79059


sin α i

0

∆S * =

K

TABLE D-6.28 Calculation of relative coordinates of profile

2δ = 0.0087; 3 K −3

∆S **

2δ = 00018; 3 K +3

∆S * cos α–3 = 0.0080 ;

∆S ** cos α+3 = 0.0011;

∆S * sin α–3 = 0.0035 ;

∆S ** sin α+3 = 0.0014 ;

*

–3

–5/2

–2

–1

0

1

2

5/2

3

1.

xK′

1.6481

1.6171

1.5277

1.2087

0.7727

0.3506

0.0848

0.0210

0

2.

yK′

1.1045

1.0904

1.0459

0.8660

0.6222

0.3542

0.1038

0.0268

0

0.40617 0.91380 0.0134 0.0130 0.0054 0.0122 0.0053 0.0119

0.42391 0.90570 0.0242 0.0223 0.0103 0.0219 0.0097 0.0206

0.46356 0.88606 0.0354 0.0324 0.0164 0.0314 0.0150 0.0287

0.49833 0.86699 0.0577 0.0539 0.0288 0.0500 0.0269 0.0467

0.48782 0.87295 0.0740 0.0572 0.0361 0.0646 0.0279 0.0499

0.61055 0.79201 0.0652 0.0383 0.0398 0.0516 0.0234 0.0303

0.75490 0.65584 0.0256 0.0172 0.0193 0.0168 0.0130 0.0113

0.78318 0.62179 0.0103 0.0084 0.0081 0.0064 0.0066 0.0052

0.79059 0.61236 0.0029 0.0027 0.0023 0.0018 0.0021 0.0016

3. 4. 5. 6. 7. 8. 9. 10.

sin αK cos αk d1 d2 (3) × (5) (4) × (5) (3) × (6) (4) × (6)

**

11.

x1

= (1) + (7) + 0.0011

1.6572

1.6546

1.6285

1.5452

1.2386

0.8099

0.3915

0.1052

0.0302

0.0034

12.

y1

= (2) –(8) + 0.0014

1.1094

1.0937

1.0699

1.0159

0.8174

0.5590

0.3040

0.0884

0.0218

–0.0004

0

13.

x2

= (1) –(9) + 0.0011

1.6439

1.6085

1.5138

1.1829

0.7459

0.3283

0.0729

0.0155

–0.0010

0

y2

= (2) + (10) +0.0014

1.1178

1.1124

1.0760

0.9141

0.6735

0.3859

0.1165

0.0334

0.0030

0.0107 0.0241 0.000695 0.000697

0.0200 0.0425 0.002206 0.002209

0.0314 0.0601 0.00460 0.00460

0.0557 0.0967 0.01245 0.01245

0.0640 0.1145 0.01721 0.01721

0.0632 0.0819 0.01070 0.01071

0.0323 0.0281 0.00183 0.00183

0.0147 0.0116 0.000351 0.000350

0.0044 0.0034 0.0000309 0.0000313

14. 15. 16. 17. 18.

(11) –(13) (14) –(12) (15)2 +(16)2 [(5) + (6)]2 = (17)

=

′ x–3

+

∆S

y*

=

y−′ 3

+

∆S * sin α–3

*

cos α–3 + +

∆S **

cos α+3 = 1,6575

∆S **

sin α+3 = 1,1091

459

x*



460

Section I : Vari. A—AFP 450 TABLE D-6.29 Coordinates of the profile Section rI =122.8 mm

*

–3

1

2

5/2

3

**

9.35

9.63

12.42

21.33

54.13

100.00

144.76

175.40

183.42

186.29

186.65

157.20

154.66

148.88

127.64

100.00

72.71

49.65

42.52

40.15

40.19

5.03

7.25

11.94

14.04

11.07

4.58

2.00

X1

7.87

8.17

11.05

20.11

53.40

100.00

145.93

177.91

186.35

189.34

189.71

Y1

152.72

151.25

148.93

143.76

124.56

100.00

76.21

55.86

49.61

47.57

47.64

4.49

6.39

10.41

12.21

9.30

3.85

1.80

6.48

6.78

9.75

18.97

52.75

100.00

146.88

179.88

188.63

191.72

192.07

Y1

147.10

145.87

143.78

139.17

121.85

100.00

79.07

60.91

55.35

53.57

53.67

3.86

5.43

8.78

10.24

7.58

3.17

1.59

X1

5.19

5.46

8.52

17.90

52.18

100.00

147.61

181.30

190.27

193.42

193.74

Y1

142.04

141.06

139.21

135.11

119.50

100.00

81.30

64.79

59.74

58.15

58.26

3.15

4.38

7.05

8.12

5.92

2.55

1.38

X1

4.00

4.20

7.35

16.90

51.69

100.00

148.11

182.18

191.27

194.44

194.72

Y1

137.52

136.83

135.23

131.59

117.52

100.00

82.90

67.51

62.79

61.31

61.43

2.36

3.24

5.21

5.86

4.31

1.98

1.18

R Rounding

Section

I

II

III

IV

V

off Radius

leading

1.41

1.24

1.05

0.83

0.59

trailing

0.31

0.30

0.28

0.26

0.25


X1

R rV =168.0 mm

0

158.88

R rIV =156.7 mm

–1

YI

R rIII = 145.4 mm

–2

X1 R

rII = 134.1 mm

–5/2

TABLE D-6.30 Theoretical pressure and velocity distribution on the profile

r = 122.8 mm;

w w2

p=

u −w 2 gH 2

2

u2 = 1.1309 2 gH

u2 = 105.9512;

u = 10.2918 m/sec; –3

–5/2

–2

–1

0

+1

+2

9.99

12.22

11.45

11.10

10.26

13.01

12.48

11.95

99.8840

149.3284

131.1025

105.1794

169.2601

0.0648 0.0082

9.14

7.00

6.26

6.50

6.03

11.81

11.56

9.24

7.94

6.43

123.2100

83.5396

49.0000

89.1876

42.2500

36.3199

155.7504

142.8085

139.4761

133.6336

85.3776

63.0436

40.9651

–0.4630

–0.2685

–0.1842

0.2499

0.6079

0.7126

0.6799

0.7432

–0.6757

–0.5315

–0.3933

–0.3578

–0.2955

0.2196

0.4580

0.6936

+5/2

+3

w2max − u 2 σ= 2 gH

= 0.5315

+5/2

+3


Section I : Pump—AFP 450

2gH = 93.6880

SECTION III r = 145.4 mm;

K w w2

u2 = 1.5850 2 gH

u2 = 148.4937;

u = 12.1858 m/sec;

–3

–5/2

–2

–1

0

+1

+2

11.75

13.77

12.84

12.42

10.35

8.85

8.50

8.93

8.48

12.05

14.73

14.14

13.50

13.37

13.27

11.52

10.46

8.92

136.1002

189.6129

142.4729

216.9729

164.8656

154.2564

107.1225

78.3225

72.2500

79.7449

72.0139

199.9396

182.2500

178.7569

176.0929

132.7104

109.4116

79.5180 461

u 2 − w2 2 gH

0.1323

–0.4289

–0.1747

–0.0615

0.4416

0.7490

0.8138

0.7338

0.8163

0.0643

–0.7309

–0.5491

–0.3603

–0.3230

–0.2946

0.1685

0.4172

0.7342

w2max − u 2 σ= 2 gH

= 0.5491

+5/2

+3

462

p=

SECTION V r = 168.00 mm;

w

w2

u −w 2 gH

2

u = 14.0799 m/sec;

–3

–5/2

–2

13.77

15.16

14.07

u2 = 2.1160 2 gH

= 198.2436;

–1

0

14.20

13.62

12.05

10.73

10.76

11.46

11.14

16.16

15.70

15.12

14.99

15.05

13.67

12.91

11.55

189.6129

229.8256

201.6400

185.5044

145.2025

115.1329

115.7776

131.3316

124.0996

197.9649

261.1456

246.4900

228.6144

224.7001

226.5025

186.8689

166.6681

133.402

0.0921

–0.3371

–0.0363

0.1360

0.5661

0.8871

0.8802

0.7142

0.7914

0.0030

–0.6714

–0.5150

–0.3242

–0.2824

–0.3016

0.1214

0.3370

0.6921

σ=

w2max − u 2 = 0.5150 2 gH

+1

+2


p=

2

u2

463


Section I

Section III

Section IV

r (s)

r (s)

r (s)

5

5

5

4

4

4

3

3

2

2

2

1

1

1

3

–2

–1

0

1

2

3 –3–2,5–2

–1

0

1

2 2,5 3 –3–2,5–2

–1

Fig. D-6.3. γ (s) distribution W 16

16 V

15 14

III

13 12

I

11 10

14 13 V

1.0

2.0

3.0

4.0

5.0 6.0 7.0 10 III 9 8

8.0

9.0

10

I

7 6

–3

–2 1 –2.5 –2 –1.0

–1 –0.5

0

+1

+2

+0.5

Fig. D-6.4. Velocity distribution on the profiles

+3 +2.5 + 1 2 +1.0

0

1

2 2,5 3

464

–1.0 –3

–2.5

–2

2

u –w 2gH

2

+1.0


+2.5

p=

1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2

+0.5

+2 +3

0.1 –0.5 +1 –0.1 –0.2

–1

–0.3 –0.4 –0.5 –0.6 –0.7 –0.8 –0.9

Fig. D-6.5. Pressure distribution along the profiles at I, III, V section

465


1

2

4

3

6

φ470

φ350

5

1. Conical outlet at diffuser 2. Diffuser hub 3. Diffuser blades. 4. Impeller hub. 5. Impeller blades 6. Impeller inlet cone Fig. D-6.6. Axial flow pump cross-section assembly (Cylindrical)

Fig. D-6.7. Impeller (cylindrical)

466


Fig. D-6.8. Diffuser

Fig. D-6.9. Pump assembly (Conical impeller)

467


(a)

(b)

Fig. D-6.10. Impeller (conical hub) (a) front view (b) side view kH 0.30

0.25 % 60 00 =8 0 C 90 68% 1000 % 65 00 % 11 69 % 70

η = 55%

% 71

1200 η = 50%

0.20

72%

C = 1300 η = 72%

1300

71% 0 120

0.15

70%

69% 68%

1100 65% = ns

1000

0 40

900 800 700

0 50

0.10

° ϕ = +15

60% 55%

50%

600 500

° ϕ = +10

0 60 700

η = 45%

800 900 1000 1100 1200 0 0 3 1 0 ° 0 14 ϕ = –10 1500

0.05

0

0.05

0.10

0.15

0.20

0.25

ϕ = +5°

ϕ = –5°

0.30

0.35

ϕ = –0°

0.40

0.45

0.50

0.55

0.60

0.65 K Q

Fig. D-6.11. Universal characteristics of axial flow machine as pump (ns = 450) (cylinder hub)

468


0.30

kH

0.25

0 40

0 50

10 00

00 14

1200

70

00 13

0.20

110 0

65

72

0.15

0 40

75 700

74 72

70

00 11 00 10

0 50

600

80 0

= ns

00 0 13 120

75

90 0

74

00 ηs = 8

65

900

700

600

0.10

1000

60

C= 0 90

0 80

700

1100 1200

800

55

70 0

900

1300 1400 1500

1000

0.05

1100

1300 1500

HSY – Q

1

1200 1400

5° 5°

2

φ°

0°

10°

. H SY – Q

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Fig. D-6.12. Universal characteristics of axial flow machines as pump (ns = 550) – conical hub

0.55

0.6

KQ

469

1.1


ϕ = –10°

1.0

ϕ = 0°

ϕ = –5°

ns

=2

00

ϕ = +5°

0.9

ϕ = +10°

n s = 25

0

55

η=

50

0.8

ϕ = +15°

0

0.6

η

=

60

0.7

n s = 30

45

η=

65

350 76

75

73 7 η4 =

2 71 = 7 η

400

0.4

n

s

=

20

65

0

0.5

70

η=

0 25

65

0.3

500

60

0.2

0 30 0 35 400

55 600

0.1

500

600

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Fig. D-6.13. Universal characteristics of axial flow machine as turbine (ns = 450) (cylindrical hub)

1.1

1.2

1.3

470


ϕ = –10°

ns

=2

ϕ = –5°

00

ϕ = 0°

ϕ = 5° ϕ = +10°

48

ns = 250

0.9 η = 52

0.8 56

0.7

n s = 30

0

20

0

η = 60

64

n

s

=

0.6

350 0.5

68 400 0 25

0.4

η = 72

76

0 30

500

0.3 0 35 0 4 0

0.2

η = 80

600 700

500

800 600 700

800

0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

kQ

Fig. D-6.14. Universal characteristics of axial flow machine as turbine ns = 550 (conical hub)

471


ϕ=

0 –1

ϕ=

1.0

° –5

°

ϕ = 0°

ns

=

° +5 ϕ=

0 20

ϕ=

+1

0°

n s = 25

0

60

ϕ = +15° 0.9 45

65

=

55

60

50

0.8

η

=

η

65

300

0.7 70 72

0.6 74 75 76

350 0.5

00 =2 ns

74

72

5 70 6

400

75

0.4 50 =2 ns

ns = 500 60

0.3 00 =3 ns

0.2 = ns 00 =4 n s 500 = ns 00 = 6 00 n s = 7 800 ns = ns

0

0.1

0.2

700

η = 72

64

70

0.3

ϕ = –5° 0.4

800

68

ϕ = +15°

64 60 56 52 48

ϕ = –10°

600

72

60

0 35

0.1

55

50 70

58

ϕ = 0° 0.5

ϕ = +10° ϕ = +5°

0.6

0.7

0.8

0.9

1.0

1.1

Fig. D-6.15. Universal characteristics of axial flow pump-turbine unit ns = 450 (cylindrical hub)

1.2

1.3

472


Q = 0°; K Q = 0.316; K H = 0.170

Q, lit/sec 180

1% Q

170 160 H, M

4 3 2 1 M, kr 11 10 9 8

Hs(max)

7 6

0

1

2

3

5

4

6

7

8

9

Fig. D-6.16. Actual cavitation test results of axial flow pump

Design 1200 1100

+

+

+

1000

Experiment

C

115.5

122.8

168

145.4

Radius

175

Fig. D-6.17. Comparison of design and experimental values of C

TABLE D-7.1 Profile losses calculation C 2∞ C 2Z = = 1,1111 2g 2g

v = 1.07 10–6 m2/sec; I S. No.

Section

1.

CZ m/sec

2.

Wu2∞m/sec

3.

tg β2∞ =

4.

sin β2∞

5.

Convex

III Concave

Convex

V Concave

Convex

Concave

4.6690 5.2118

7.8758

10.3599

0.695851

0.592828

0.450680

0.667255

0.509959

0.410896

l chord length (mm)

213.98

207.65

205.22

6.

t pitch (mm)

110.3

130.6

150.9

7.

w1∞ =

11.3010

13.0499

14.8328

8.

w2∞ m/sec

6.9973

9.1556

11.3630

9.

w 22∞

48.9622

83.8250

129.1178

0.619175

0.701583

0.766072

2.2600 × 106

2.5325 × 166

2.8448 × 106

0.210

0.202

0.189

0.0238

0.0220

0.0199

CZ m Wu 2∞

C 2z + u 2 m/sec

m/sec

10.

w2∞ : w1∞

11.

Re =

12. 13.

—

w1∞l ν

S (Re) from graph 10.53

— Smin


Design No. D7

473

wmax w1∞

14.

U max =

15.

pmin =1 – U 2 max = 1 – (14)2

16.

SH = S + Smin = (12) +(13)

17.

1.1512

1.0813

1.1380

1.0662

1.0881

1.0214

–0.3253

–0.1692

– 0.2950

– 0.1368

– 0.1840

–0.0433

0.2338

0.2240

p1 C12∞ = 10 – SH – γ 2g —

0.2089

10.8889

18.

Pm

0.5042

0.5626

0.4256

0.4958

0.3993

0.4707

19.

rx from graph 10.54

1.295

1.4000

1.3300

1.4200

1.3500

1.4300

20.

S K = rx × S H

0.3028

0.3273

0.2979

0.3181

0.2820

0.2987

21.

B=

0.3433

0.1278

0.2978

0.1947

0.2592

0.1350

1.0580

0.9870

1.0400

0.9570

1.0250

0.8800

1.2390

0.9517

1.1608

0.8463

1.0983

0.6160

2.3911 × 106

2.2306 × 106

2.6338 × 106

2.4236 × 106

2.915 × 106

2.5034 × 106

0.1219

0.0591

0.1129

0.1012

0.1038

0.09710

0.27549 × 106

0.13357 × 106

0.28592 × 106

0.25629 × 106

0.29529 × 106

0.27623 ×106

SH

∫U

4.8

dS

0

—

22.

UH

23.

UH

24.

— UH Re = (11). (22)

25.

θ =

26.

GeH = Reθ = (11). (25)

27.

ReH** =

G1H = (26)

5.2487 × 102

3.6549 × 102

5.3471 × 102

5.0625 × 102

5.4341 × 102

5.2558 ×102

28.

δH** =

ReH** (27) = (24) ReU H

2.1951 × 10–4

1.6385 × 10–4

2.0302 × 10–4

2.0888 × 10–4

1.8636 × 10–4

2.0995 ×10–4

29.

δ**H = δ**H. l = (28). (5)

0.04697 × 10–3

0.03506 × 10–3

0.04216 × 10–3

0.04337 × 10–3

0.03822 × 10–3 0.04309 ×10–3

3 ,8

0.44 B 0.44.(21) = 3.8 (23) U H

30.

U H5.5

31. 32.

1.3630

0.9305

1.2405

0.7860

1.1456

0.4960

GH=1259 ( Re** )1/10

673

698

672

675

671

673

CH = (28) (30) (31)

0.2014

0.1064

0.1692

0.1108

0.1433

0.0701

33.

— UK

1.0530

0.945

1.0300

0.9170

1.0170

0.9000

34.

U K4.5

1.2620

0.7743

1.1420

0.6775

1.0790

0.6225

35.

B=

dS

0.0939

0.0836

0.0964

0.0710

0.0782

0.0561

36.

0.9 B + CH = 0.9 (35) + (32)

0.2859

0.1816

0.2560

0.1747

0.2137

0.1206

37.

G1K =

0.5120 ×106

0.5301 ×106

0.5677 ×106

0.6530 ×106

0.5634 ×106

0.5511 ×106

38.

G1K : 1259

0.4067 ×103

0.4211 ×103

0.4509 ×103

0.5187 ×103

0.4475 ×103

0.4377 ×103

39.

792.9

824.1

889.2

1040

881.6

860.4

40.

Rek**= (38)10/9 — Re UK = (11). (33)

2.3798 ×106

2.1357 ×106

2.6085 ×106

2.3223 ×106

2.8932 ×106

2.5603 ×106

41.

δ **K =

0.3332 ×10–3

0.3859 ×10–3

0.3409 ×10–3

0.4478 ×10–3

0.3047 ×10–3

0.3361 ×10–3

42.

δ**K= δ**K . l = (41). (5)

0.071298×10–3

0.082575×10–3

0.07079×10–3

0.09299×10–3

43.

U 3.8 K

1.2170

0.8065

1.1178

0.7192

1.0661

0.6700

44.

**1/ 6 GK = 153.2 ReK

466.03

469.10

475.23

487.64

474.31

472.47

45.

CK = (43) (41) (44)

0.1890

0.1460

0.1811

0.1571

0.1541

0.1064

46.

U KP

0.5509

0.6503

0.7490

47.

2.8 U KP

0.1885

0.3000

0.4460

SK

∫U

5.5

SH

(11)(36) (34)

(39) δ **K = (40) ReU K

0.062531×10–3 0.068974×10–3

l

∫U

3.8

dS

0.7313

0.2383

0.7023

0.2476

0.7399

0.3048

49.

1.17 B +CK =1.17 (48) +(45)

1.0446

0.4763

1.0028

0.4468

1.0198

0.4630

50.

G1KP =

(49)(11) (47)

12.5241 ×106

5.6865 ×106

8.4658 ×106

3.7718 ×106

6.5048 ×106

2.9532 ×106

51

G1KP : 153.2

0.8175 ×105

0.3712 ×105

0.5526 ×105

0.2462 ×105

0.4246 ×105

0.1928 ×105

0.1625 ×105

0.08256 ×105

0.1161 ×105

0.05808 ×105

0.09262 ×105

0.04709 ×105

48.

B=

SK

52.

** ReKP

= (51)6/7

1.245 ×106

1.647 × 106

2.13 × 106

53.

Re U KP = (11). (46)

54.

δKP** =

55.

δ**KP = δKP** l = (54) (5)

56.

wKP : w2∞ = (46) : (10)

0.8897

0.9269

0.9777

57.

(56)3/2

0.6880

0.7842

0.9305

58.

(9) (57) [(55)Convex+ (55)Concave]

0.1424

0.1439

0.1626

59.

gt sin β2∞

0.7220

0.6534

0.6083

60.

hf = (58) : (59)

0.197

0.222

0.266

61.

η p=

96.3

95.6

94.7

** ReKP (52) = (53) U KP Re

HT − h f HT

%

0.01306

0.00663

0.00704

0.003522

0.00435

0.00221

0.002793

0.001426

0.001469

0.000732

0.000893

0.000453

(average) (ηhp) = 95.6%


477

TABLE D-7.2: Computer programme in C++ for profile loss calculation for impeller # include < stdio. h> # include < math. h> # include < Conio. h> # define Z 7.0 # define pi 22/7 char hip ; void main () } int i ; float xy [10], xy 2 [10], hf1 [10], B1_inf [10]; float eff [10], eff [10], effy = 0 effy = 1 = 0, efft, efft 1; float CxCtu [10], CvCtu [10], CxRetust 2 [10], CvRetust 2 [10], CvBtu [10], CxUtu [10], CvUtu [10]; float Cxde1 tatust_ tr [10], Cvde1 tatust_2 [10], Cvde1 tatist 2 [10], Cxde1 tatust2 [10] ; float CvCtr [10], CxCtr [10], CvRetrst 2 [10], CxRetrst 2 [10] ; float CxB_1 [10], CxU_1 [10], CxPU_1 [10], CxUR_1 [10], Cxthetal [10], CxRe1 [10], Cxde1tast 2_1 [10], Cxde1 tast 21 [10] ; float CvB_1 [10], CvU_1 [10], CvUR_1 [10], CvX1 [10], CxRe1 st 2 [10], Cvde1 tast2_1 [10], Cvde1 tast 21 [10]; float Nu = 1 1.07 e-9 = 9.81 ; float Cz, HS, P1_r, H,W1_inf [10], t [10] ; float U [10], r [10], B2_inf [10] ; float Wt ; float W2_inf [10], del_t_xx, hw ; float Wu2_ inf [10], Ret_xx, U_t, Re1 [10], U_t1,B, Ce ; float W1,max U1_max, P1_max ; float P1 P1m, P1_min ; float b, Bb, U_1, d_s, U1-t ; float U 1_b, Reb_xx, del_b_xx ; float U11, Cb, Ree_ xx, del_e, del_e ; float x, n de1_xx, del1_e_xx ; float s [10], s_min [10], s1 [10] ; float 1b, ub B_b, Q Xb, ReH_xx ; float de11_bxx, de11_e_xx ;

478


float Ct, k, Gt, T ; float St1, Ut1, B_t, sb1 ; float X_t, Ge, de11_t xx, Xe ; float de1_e_xx, de11_cv ; float de1_ t_xx_cv ; float Ret_xx_cv, U1_max_cv, P1_max_cv ; float W1_max_cv, U1_max_cv.P1_max_cv ; float P1_cv, P1m_cv, P1_t_cv : float B_b_cv, Reb_xx_cv, del_b_xx_cv ; float U1_b_cv, Reb_xx_cv, de1_b_xx_ cv ; float U11_cv, Cb_cv, Ree_xx_cv, Be_cv, U1_e_cv, del_e_cv ; float de1_xx_cv, U_cv, 1b1,ib1 ; float Q_cv, Xb_cv, Reh,xx_cv ; float de11_b_xx_cv, de11_e_xx_cv ; float Ct_cv, k_cv, Gt_cv ; float Sr1_cv, Ut1_cv, B_t_cv ; float X_t_cv, Ge_cv, de11_t_xx_c, Xe_cv ; int ch ; fflush (stdin) ; clrscr () ; / * data available * / printf (” /n this program calculates the profile losses in axial flow /pumps /n/n”) ; printf (” this is impleted in c language /t/t/n/n”) ; printf (” /n give input parameters /n”) ; printf (” /n ******** general detailss ******** ”) ; printf (” /n give the values for Cz, Hs, n, P1 r, H/n”) ; scanf (“%f %f %f %f f ”, & Cz, & Hs, & n, & P1_ r, & H) ; for (i = 1 ; i < = 3 ; i ++ ) { printf (” enter the value of r %d”, i) ; scanf (“%f ” & r [i]) ; U [i] = (( 2.0* pi * r [i]* n) /60.0) ; W1_inf [i] = sqrt (( Cz* Cz) +(U [i] * U [i] )) ; B1_ inf [i] = atan (Cz/ U [i] ) ; printf (” B1_inf [%d] = %f /n ” , i, B1_ inf [i]) ; printf (” / nU [%d] = %f /n W1_inf [%d] = %f /n “, i U [i], i, w1_ inf [i] ; printf (” / ngive the value of wu2_inf [i] :”, i) ; scanf (“%f ”, & Wu2_inf [i]) ; B2_inf [i] = atan (Cz/Wu2_inf [i]) ;


479

printf (” /nB2_inf (%d] = %f ”, i, (B2_inf [i] )) ; printf (” /n sin (B2_ inf [%d] = %f ”, i, sin (B2_inf [i] * (180/pi ))) ; printf (” / ngive the value for 1%d”, i) ; scanf (“%f ”, & 1 [i] ) ; t [i] = (2* pi *r [i] ) /Z ; printf (” /n value of 1%d/ T%d : %f/ n”, i, i, I1 [i] / t [i] )) ; printf (” /n value of W1_inf [%d] ** 2/2g is : %f ”, i, (W1_inf [i] * W1_inf [i] / (2*g ))) ; W2_inf [i] = Cz/ sin (B2_inf [i] * (180 /pi)) ; printf (” /n the value of W2_ inf [%d] is : %f ”, i, W2_inf [i] ; printf (” /n the value of W2_ inf [%d] square is : %f ”, i, W2_ inf [i] * W2_ inf [i] ; printf (” /n sthe value of W2_ inf {%d] / W1_ inf [%d] is : % f ”, i,i, W2_ inf [i] / W1_inf [i] ; Re1 [i] = (( W1_ inf [i] * 1 [i] )) / Nu ; printf (” /n The value of Re1 %d = = %e “, Re1 [i] ; getch () ; clrscr () ; } printf (” /n ******** laminar region ********* ”) ; printf (” /n Give the values or s1, s2, s3 : /n”) ; scanf (“%f %f %f ”, & s [1], & s_min [2], & s_ min [3] ) ; for (i = 1 ; i < = 3 ; i ++ ) } s1 [i] = s [i] + s_ min [i] ; printf (” /n The value of s1 [%d] = %f ”, i, s1 [i] ) ; } for (i = 1 ; i < = 3 ; i ++) { { printf (” / n CONVEX / N”) ; printf (” / n Enter the values for CxB_1 and CxU_1/n) ; scanf (“%f f ”, & CxB_ [i], & CxU_1 [i] ; CxPU _1 [i] = pow (CxU_1 [i], 3.8) ; CxUR_1 [i] = CxU_1 [i] * Re1 [i] ; Cxthetal [i] = 0.44 * CxB_ 1 [i] ; Cxx1 [i] = Re 1 [i] * Cx thta [i] ; CxRe1 st2 [i] = sqrt 9Cxx1 [i] ; Cxdel tast 2_1 [i] = CxRe1 st 2 [i] / CxUR_1 [i] ; Cxdel tast 21 [i] = Cxde 1 tast 2_1 [i] * 1 [i] ; Cxtr [i] = 1259 * pow (Cxre1 st 2 [i], –0.2) * Cxde tast2_1 [i] * pow (CxU-1 [i], 5.5) ;

480


printf (” Enter the value for CxBtr and CxUtr/n”) ; scanf (“%f %f ”, & Cx Btr [i], & CxUtr [i] ; CxRetrst 2 [i] = pow (((( 0.9 * CxBtr [i] + CxCtr [i]) * Re1 [i]) / (1259 * pow (CxUtr [i], 4.5 ))), 1.1111) ; Cxde1 tast2_tr [i] = CxRetrst2 [i] / Re1 [i] * CxUtr [i] ; Cxde1 tast 2tr [i] = Cxde1 tast 2_tr [i] * 1 [i] ; CxCtu [i] = 153.2 * pow (CxRetrst 2 [i], 0.167) * Cxde1 tast2_tr [i] * pow (CxUtr [i] = 3.8) ; printf (” Enter the value for CxUtku and CxBtu /n”) ; scanf (“% f% f ”, & CxUtu [i], & CxBtu [i]) ; CxRetust2 [i] = pow ((((1.17 * CxBtu [i] * CxUtu [i] ) ; Cxdel tatust2 [i] = Cxde tatust _2 [i] * 1 [i] ; } printf (” CONCAVE/n”,) ; scanf (“%f %f ”, & CvB_1 [i], & CvU_1 [i] ; CvPU_1 [i] = pow (CvU_1 [i], 3.8) ; CvUR_1[i] = CvU_1 [i] * Re1 [i] ; Cvtheta [i] = 0.44 * CvB_1 [i] /CvUR_1 [i] ; CvRe1 st2 [i] = CvRe1 st 2 [i] / CvUR_1 [i] ; Cvde1 tast2_1 [i] = Cvdel1 tast 2_1 [i] * 1 [i] ; CvCtr [i] = 1259 * pow (CvRelst2 [i], –0.1 * Cvde1 tast 2_1 [i] * ow (CVU_1 [i], 5.5) ; printf (” Enter the values for CvBtr and CvUtr /n”) ; scanf (“%f %f ”, & CvBtr [i], & CvUtr [i]) ; CvRetrst 2 [i] = pow (((( 0.9 * CvBtr [i] + CvCtr [i] * Re1 [i] / (1259 * pow (CvUtr [i], 4.5)), 1.1111) ; Cvde1 tast2_tr [i] = CvRetrst2 [i] * CvUtr [i] ; Cvde1 tast 2tr [i] = Cvde1 tast 2_tr [i] * 1 [i] ; CvCtu [i] = 153.2 * pow (CvRetrst 2 [i], 0.167 * Cvdel tast 2_tr [i] * pow (CvUtr [i], 3.8) ; printf (” Enter the value for CvUtu and CvBtu/ n”) ; scanf (“%f %f ”, & CvUtu [i], & CvBtu [i]) ; CvRetust 2 [i] = pow (((( 1.17 * CvBtu [i] + CvCtu [i] * Re1 [i] ) / (153.2 * pow (CvUtu [i], 2.8)), 0.8571) ; Cvde 1 tatust_2 [i] = Cv Retust 2 [i] / ( CvUtu [i] * Re1 [i] ) ; Cvde1 tatust 2 [i] = Cvde1 tatust_2 [i] * 1 [i] ; } } for (i = 1 ; i < 3 ; 1 ++) { printf (” CONVEX/n”) ; printf (” CxRe1st2 [%d] = %f /n”, “, i, CxRe1st2 [i] :


481

printf (” Cxdel tast 2_1 [%d] = %f /n”; i, Cxde1 tast 2_1 [i] ; printf (” Cxde1 tast 21 [%d] = %f /n”, i, Cxde1 tast 21 [i] ; printf (” CxCtr [%d] = %f /n”, i, CxCtr [i] ; printf (” CxRetrst 2 [%d] = %f /n”, i, CxRetrst2 [i]) ; printf (”Cxde1 tast2_tr [%d] = %f /n”, i, Cxde1tast 2_tr [i] ; printf (” Cxdel tast2 tr [%d] = %f /n”, i Cxde1tast 2tr [i] ; printf (” CxCtu [%d] = % f/n”, i, CxCtu [i] ; printf (” CxRetust 2 [%d] = %f /n”, i, CxRetust 2 [i] ; printf (” Cxde1tatust_2 [%d] = %f /n”, i, Cxde1 tatust_2 [i] ; printf (” Cxde1 tatust 2 [%d] = %f /n”, i, Cxde1tatust2 [i] ; printf (” /n CONCAVE /n”) ; = %f /n”, i, CvRe1st2 [i] ; printf (”CvRe1st 2 [%d] = %f/n”, i, Cvde1 taust2_1 [i] ; printf (” Cvde1tast 21 [id] = %f /n”, i, Cvde1tast 21 [i] ; printf (” CvCtr [%d] = %f /n”, i, CvCtr [i] ; printf (”CvRetrst 2 [%d] = %f /n”, i, CvRetrst2 [i] ; printf (” Cvde1tast2_tr [%d) = %f /n”, i, Cvde1tast2_tr [i] ; printf (” Cvde1tast 2tr [%d] = %f /n”, i, Cvde1tast2tr [i] ; printf (” CvCtu [%d] = %f /n”, i, CvCtu [i] ; printf (” CvRetust 2 [%d] = %f / n”, i, CvRetust 2 [i] ; printf (” Cvde1tatust_2 [%d] = %f /n”, i, Cvde1tatust_2 [i] ; printf (” Cvde1 tatust2 [%d] = %f /n”, i, Cvde1tatust 2 [i] ; } Clrscr () ; gpt i = 1 ; i < = 3 ; i ++ ) { xy1 [i] = pow (CxUtu [i] * W1__inf [i] /W2 inf [i], 3.2) * Cxde1 tatust 2 [i] ; xy2 [i] = pow (CvUtu [i] * W1__inf [i] /W2 inf [i], 3.2) * Cvde1 tatust 2 [i] ; hf1 [i] = (pow (W2_inf [i], 2) * (xy1 [i] + xy2 [i] )) / (g* t [i] * sin (B2_inf [i] * (180 /pi))) ; printf (“hf1 [%d] = %f /n”, i, hf1 [i] ; eff [i] = (H-hf1 [i]) / H ; eff [i] = H / (H+hf1 [i] ; printf (“eff [%d] = %f /t/teff1 [%d] = %f /n”, i, eff [i], i eff1 [i] ; effy = effy + eff [i] ; effy1 = effy1 + eff1 [i] ; } efft = effy / 3. efft1 = effy 1/3 ; printf (“efficiency = % f /t/t, Efficiency 1 = %f ”, eff, efft1) ; printf (“/n/n/n PROGRAM END /n”) ;

482


Design No. D8 DESIGN OF AXIAL FLOW PUMP—AS PER METHOD SUGGESTED BY PROF. N.E. JOWKOVSKI Given, H = 3 m; Q = 0.27 m3/sec; n = 900 rpm. Based on the experimental results already available, Hydraulic efficiency, ηh is assumed = 0.87. Impeller efficiency ηi = 0.94, K = (5.03 to 5.25) selected 5.24 afterwards. No. of Impeller blades Zi = 4. Suction specific speed, C = 1150.

(1)

Impeller Design Hm =

H 3 = = 3.4483 ηh 0.87

Hi = ηi . Hm = 0.94 × 3.4483 = 3.2414 m. ω= 3

Impeller outer diameter, Di = K

3

Q = 5.24 n

dh d = D = 0.5; 1

Hub diameter,

(2)

2πn 2 × π × 900 = = 94.2478 rad/sec. 60 60 0.27 = 0.350 m. 900

dh = 0.5Di = 0.175 m.

Suction Conditions Cm = C0 =

C=

4Q π( D12

n Q  H SV   10   

3/ 4

− d h2

)

=

= 1150 =

4 × 0.27 = 4.167 m/sec. π (0.352 − 0.1752 ) 90° 0.27  H sv   10   

3/ 4

HSV = 3.0128 m.

HSV = Hat – HVP – hS or hS = Hat – HVP – HSV = 10.336 – 0.336 – 3.0128 = 6.9872 m.

C2 ∆PV max 4.167 = HSV – m = 3.0128 – = 2.1278 m. 2g 2 × 9.81 γ Selecting the anticavitating profile developed by Moscow Power Institute for hydraulic Machines, Moscow K = 1.6 taken from its characteristics

Pav ∆P = K . V max = 1.6 × 2.1278 = 3.4045 m. γ γ

483


 f  Change of the relative curvature   in the radial direction is made such that, the load on the l  outer half of the blade (middle to periphery) is low and the load is high between hub and middle. The Impeller design and diffuser design are given in the following tabular form for hub suction only. The procedure for other sections will be same. However, the details for all other section are given whereever necessary.

TABLE D-8.1: Impeller Design S. No.

Details/Section

1.

radius rI = 94 mm, rII = 118, rIII = 143, rIV = 168

2.

u=

3*.

Cm1 = Cm2 = Cm = C0 m/s =

πDn m/s n = 900 rpm. 60

πD12

4Q (1 − d 2 )

I 94 8.8592 4.167

constant at all sections

4.

C  β1 = tan–1  m  (deg)  u 

25.1°

5.

w1 = (Cm /sin β1 ) m/sec

9.8232

6.

(w21 /2g) m

4.9182

7**.

 gH m  m/sec Cu2 =   u 

3.8183

8.

 Cm 2  α2 = tan–1  C  deg  u2 

47.5°

9.

β2 = tan–1

10.

w2 =

11.

(w22/2g) m

12.

 Cm  β∞ = tan–1  u − Cu 2  deg   2 

13.

w∞ = (Cm / sin β∞ ) m/s

8.1024

14.

(w2∞/2g) m

3.346

Cm 2 deg (u2 − Cu 2 )

Cm 2 m/sec. sin β 2

39.6°

6.5373 2.1782

31°

484


15.

 C 2u 2  (p2 – p1) = γ  H i − kg/m2 2 g  

2498.3

16.

dRz = (p2 – p1) 2 π ri kg dr

1475.56

17.

dRu γ = dr g Cu2 Cm 2 π ri

957.8

18.

 dRu  δ = tan–1   dRz 

33°

19.

λ = δ – β∞ deg

2.05

20.

lift force , Y =

21.

length, l =   /  Z  dr  

22.

t=

23.

 t Relative pitch, T =   l

24.

 fm    × 100 l

dy cos λ dRz = dr cosδ dr

1758.27

 dy   dpav  γ  mm.

129.28 (130).

2πr Z

147.66

1.1421

7.0

Curvature selected for other Sections II = 6.4, III = 5.5, IV = 3.0 25.

 δm   × 100 l 

Relative Profile Thickness δ m =  

10.0

For other sections II—7.8, III—5.5, IV—3.0 26.

 dy  CY =    dr

27.

α (δ in Fig.) from Fig. 10.36 angle of attack deg

10.3°

28.

βeI =

41.25

29.

 Γ ca  m1 =  from Fig. 10.37  Γ1 

30.

m2 = (α0Ca – α01 ) from Fig. 10.37

γZl

w∞2   2g 

β∞ + α.

1.01

0.7675 – 0.6

485


31. 32.

θ = (β1 + α) α1 = (θ – β∞)

35.4 4.4°

33.

 l2 f +  mm. R=  f 8 2  m 

237

34.

pmin =

CY  w∞2  1.6  w12 

0.429

35.

w12 Cm2 – pmin HS (check) = Hat – HVP – 2g 2g

36.

(check) C =

n ν  H SV    10 

3/ 4

=

900 × 0.27 (0.3)3/ 4

7

1153

TABLE D-8.2: Design of diffuser z=7 Details/Section

S. No.

1. 2. 3.

radius ri mm. same as for Impeller Cm3= 1.065 Cm2 (constant for all other sections) Γα = Cu2 r m/s (constant for all sections) pav 1 ⋅ γ t sinβ 2

4.

K1 = 1 +

5.

α3 = tan –1 C

Cm 3

(deg)

94 4.4379 4.4379

1.036

49.3°

u2

6.

angle of divergence (2ε)°

7.

l  1 − sin α3  =   2tan ε  t

8.

t=

9.

l =   × t mm t

2πr mm Z = 7 Z  l

I

8° 1.73 84.37 146

486




α3   2

10.

Blade Height, H = lcos  45° −

11.

∆ from Table deg

12.

β = 45 –

13.

Checkup ε = tan

14.

θ = (α3 + β)

15.

R=

16.

 δm   h     in figure and Text  l l

0.055

17.

δ (h) mm.

12.5

15.7°

1 (α – ∆) 2 3 –1

136.9

1 − sin α3    l  2  t  

l 2sin β

28.2

4° 77.5° 238.2

The mean line (or) the camber line is an arc of a circle of Radius ‘R’. The mean line is dressed with the thick profile for which the entire characteristics are known (from the wind Tunnel Test).

APPENDIX I

EQUATIONS RELATING Cy,

y max , δ ° FOR DIFFERENT PROFILES l y max + 0.092 δ° l

1. For Profiles 428, 682, 364, 480 CY = 4.8

... (1)

2. For Profiles 408, 490, 436, 387 CY = 4.4

ymax + 0.092 δ° l

... (2)

3. For Profiles 622, 623, 624, 384 CY = 4.0

ymax + 0.092 δ° l

... (3)

4. For Profiles (Camber line is arc of a circle) 608, 609, 610 CY = 5.0 5. For Profile munk – 6 (260) CY = 1.3

d max + 0.106 δ° l

6. For NACA Profile 23012 CY = 1.08 1.3

... (5)

d max + 0.106 δ° l

... (6)

where, dmax = Maximum Thickness 7. For Symmetrical Profile No. 443 CY = 0.095 δ°

16–0.09 16–0.06

487

ymax + 0.092 δ° ... (4) l

... (7) t

X

488


Polar Curves

Aerofoil Sections

Details of Aerofoil Sections Fig. AP-1.1

x

0

1.25

2.5

5.0

7.5

10

15

20

30

40

50

60

70

80

90

95

100

 y0 364   yu

0,85

4,05

5,45

7,30

8,60

9,65

11,00

11,85

12,50

12,10

11,10

9,50

7,55

5,35

2,90

1,55

0,10

0.85

0,00

0,05

0,35

0,55

0,65

1,05

1,30

1,70

1,85

1,80

1,55

1,25

0,90

0,45

0,20

0,10

 y0 384   yu

4,15

7,25

8,95

11,45

13,40

14,95

17,15

18,55

19,17

19,15

17,55

14,95

11,80

8,05

4,15

2,15

0,00

4,15

2,25

1,55

1,10

0,80

0,55

0,30

0,15

0,00

0,00

0,00

0,00

0,00

0,00

0,00

0,00

0,00

 y0 387   yu

3,20

6,25

7,65

9,40

10,85

11,95

13,40

14,40

15,05

14,60

13,35

11,35

8,90

6,15

3,25

1,75

0,15

3,20

1,50

1,05

0,55

0,25

0,10

0,00

0,00

0,20

0,40

0,45

0,50

0.45

0.30

0.15

0.05

0,15

 y0 408   yu

1,15

2,95

3,80

5,00

6,00

6,70

7,70

8,40

9,05

8,95

8,40

7,45

6,25

4,95

3,45

2,50

0,75

1,15

0,25

0,00

0.20

0,40

0.65

1,00

1,20

1,30

1,30

1,20

1.05

0,85

0,60

0.30

0.10

0,75

 y0 417   yu

0,65

2,50

3,75

5,05

6,25

7,05

8,15

8,85

9,30

9,15

8,55

7,55

6,25

4,50

2,40

1,20

0,00

0,65

0,05

0,25

0,70

1,10

1,50

2,20

2,55

3,65

3,90

3,65

3,20

2,50

1,70

0,80

0,40

0,00

 y0 428   yu

1,25

2,75

3,50

4,80

6,05

6,50

7,55

8,20

8,55

8,35

7,80

6,80

5,50

4,20

2,15

1,20

0,00

1,25

0,30

0,20

0,10

0,00

0,00

0,05

0,15

0,30

0,40

0,40

0,35

0,25

0,15

0,05

0,00

0,00

 y0 436   yu

2,50

4,70

5,70

7,00

8,10

8,90

10,05

10,25

11,00

10,45

9,55

8,20

6,60

4,60

2,45

1,25

0,00

2,50

1,00

0,20

0,10

0,05

0,00

0,00

0,00

0,00

0,00

0,00

0,00

0,00

0,00

0,00

0,00

0,00

0,00

0,60

0.85

1,15

1,45

1,60

1,90

2,15

2,50

2,50

2,35

2,05

1,60

1,15

0.65

0,30

0,00

2,55

5,10

6,15

7,65

8,85

9,80

11,25

12,10

12,85

12,60

11,60

10,00

7,85

5,45

2,85

1,45

0,00

2,55

0,80

0,30

0,05

0,00

0,10

0,45

0,70

1,10

1,45

1,55

1,50

1,25

0,85

0,40

0,20

0,00

 y0 443   yu  y0 480   yu

APPENDIX

TABLE AP-I.1: Profile coordinates

489

3,60

4,60

5,95

7,00

7,70

8,65

9,20

9,60

9,05

8,55

7,45

6,05

4,40

2,50

1,45

0,15

2,00

0,85

0,50

0,15

0,00

0,00

0,20

0,40

0, 95

0,80

0,80

0,60

0,40

0,15

0,00

0,05

0,15

 y0 587   yu

0.60

1,65

2,10

2,90

3,60

4,15

5,15

5,85

6,55

6,60

6,10

5,40

4,50

3,45

2,35

1,80

1,05

0,60

0,10

0,00

0,05

0,15

0,30

0,60

0,70

0,85

0,80

0,45

0,20

0,00

0,05

0,55

0,85

1,05

 y0 593   yu

3,00

5,50

6,50

7,85

8,90

9,75

10,95

11,50

12,00

11,70

10,85

9,45

7,65

5,50

3,00

1,65

0,00

3,00

1,80

1,35

0,85

0,55

0,40

0,25

0,15

0,10

0,00

0,00

0,00

0,00

0,00

0,00

0,00

0,00

 y0 622   yu

2,40

3,25

4,50

5,45

6,15

6,60

7,30

7,70

8,00

7,80

7,10

6,15

5,00

3,55

1,95

1,15

0,20

2,40

1,45

1,05

0,60

0,35

0,25

0,15

0,05

0,00

0,00

0,00

0,00

0,00

0,00

0,00

0,00

0,00

 y0 623   yu

3,25

5,45

6,45

7,90

9,05

9,90

10,95

11,55

12,00

11,70

10,65

9,15

7,35

5,15

2,80

1,60

0,30

3,25

1,95

1,50

0,90

0,35

0,20

0,10

0,05

0,00

0,00

0,00

0,00

0,00

0,00

0,00

0,00

0,00

 y0 623   yu

4,00

7,15

8,50

10,40

11,75

12,85

14,35

15,30

16,00

15,40

14,05

12,00

9,50

6,60

3,55

2,00

0,50

4,00

2,25

1,65

0,95

0,60

0,40

0,15

0,05

0,00

0,00

0,00

0,00

0,00

0,00

0,00

0,00

0,00

 y0 682   yu

2,50

4,55

5,55

8,00

8,05

8,90

10,00

10,65

11,20

10,90

10,05

8,65

6,90

4,85

2,55

1,35

0,00

2,50

1,05

0.60

0.25

0,10

0,00

0,05

0,20

0,55

0,75

0,80

0,85

0,75

0,6

0,35

0,15

0,00

NACA  y0 0,00  23012  yu 0,00

2,67

4,61

4,91

5,80

6,43

7,19

7,50

7,55

7,14

6,41

5,47

4,36

3,08

1,68

0,92

0,00

–123

–1,71

–2,26

–2,61

–2,92

–3,50

–3,97

–4,46

–4,48

–4,17

–3,67

–3,00

–2,16

–1,23

–0,70

0,00

0,00

1.98

2,81

4,03

4,94

5,71

6,82

7,55

8,22

8,05

7,26

6,03

4,58

3,06

1,55

0,88

0.00

0,00

–1,76

–2,20

–2,73

–3,03

–3,24

–3,47

–3,62

–3,70

–3,90

–3,94

–3,82

–3,48

–2,83

–1,77

–1,08

0,00

MUNK 6  y0  MUNK  yu


2,00

490

 y0 490   yu

491

APPENDIX

TABLE AP-I.2: Profile Co-ordinates (NACA) Profile No.

16-006

16-009

65-009

65-010

66-006

0

0

0

0

0

0

1,25 2,5

0,646 0,903

0,969 1,354

0,717 0,956

1,124 1,571

0,693 0,918

5,0

1,255

1,822

1,310

2,222

1,257

7,5

1,516

2,274

1,589

2,709

1,524

10,0 15 20 30

1,729 2,067 2,332 2,709

2,593 3,101 3,498 4,063

1,824 2,197 2,482 2,852

3,111 3,746 4,218 4,824

1,752 2,119 2,401 2,782

40 50 60 70

2,927 3,000 2,997 2,635

4,319 4,500 4,376 3,952

2,998 2,900 2,518 1,935

5,057 4,870 4,151 3,038

2,971 2,985 2,815 2,316

80 90

2,099 1,259

3,149 1,888

1,233 0,510

1,847 0,749

1,543 0,665

95

0,707

1,061

0,195

0,354

0,262

100

0,060

0,090

0,000

0,150

0,000

Leading edge Radius

0.176

0.396

0.240

0.666

0.223

L

y D

θ

X

y 0

25

50

75

100

Fig. AP-I.2

TABLE AP-I.3 x

2,5

5,5

10

15

20

25

40

45

40

45

50

55

y

0,29

0,48

0,86

1,26

1,68

2,10

2,53

2,95

3,35

3,74

4,09

4,41

x

60

65

70

75

80

85

90

95

97,5

98,5

99,0

99,5

y

4,68

4,85

4,97

4,99

4,87

4,59

4,06

3,08

2,26

1,78

1,47

1,05

492


Y 5 0 –5 0

10

20

30

40

50

60

70

80

90

100 x

Fig. AP-I.3

TABLE AP-I.4 x

0

0,50

0,75

1,25

2,50

5,00

7,50

10

15

y

0

0,752

0,890

1,124

1,571

2,222

2,709

3,111

3,746

x

20

25

30

35

40

45

50

55

60

y

4,218

4,570

4,824

4,982

5,057

5,029

4,870

4,570

4,151

x

65

70

75

80

85

90

95

100

y

3,627

3,038

2,451

1,847

1,251

0,749

0,354

0,150

Blade design

w1

Nomenclature β1

u

∆β( θ) = β2 – β1 θ

β1

i

xf

l

βe = β∞ + δ β2 β2

Point of maximum camber

t w2 γ

Cascade symbols

β1 = blade inlet angle β2= blade outlet angle θ = blade camber angle = β2 – β1 βe = setting or stagger angle = β∞ + δ t = pitch (or space) w 1 = Relative velocity of water of inlet w 2 = Relative velocity of water at outlet i = incidence angle = α1 – αi δ = angle of attack = βe – β∞ l = chord r = deviation angle

Fig. AP-I.4. Blade nomenclature

0.1 0 –0.1

d/2

L.E

d max

T.E.

xd Chord (L) or length of the camber line

0

0.5

1.0

Fig. AP-1.5. Thickness distribution of an aerofoil (base profile)

493

APPENDIX

A Impeller

βi

0 2.08 3.00 3.58 4.01 4.55

D R

θ = β2 – β1 β2

4.90

β1 + β2

βe

0 2.63 3.12 3.66 4.06 4.88 4.89 5.02 4.79

4.98 4.76 4.30

B Centre of camber arc

diffuser

4.31

3.70

3.72

2.91

3.00

2.02

2.15 1.20 0.68 0 Lower surface y %l

1.05 0.60 0 Upper surface x %l

o

Camber-line length, l

o

1.25 2.5 5.0 7.5 10 15 20 25 30 40 50 60 70 5.40 80 90 95 100

1.98 2.80 4.09 5.05 5.86 7.08 7.88 8.34 8.50 8.29 7.65 6.71 3.95 2.16 1.16 (.09)

L‘W’r 0 –.94 –1.21 –1.37 –1.36 –1.26 –1.01 –.76 –.60 –.50 –.43 –.29 –.15 – 05 –.00 –.02 –.05 (–.09)

L.E. Rad : 0.89 slope of radius through end of chord : 4/15

0 20 40 60 80 100 percent of chord

.12 .11

48 44

.10

40

.09

36 cD0

.08

32

.07

28

.06

24

.05

20

.04

16

.03

12 d0

.02

8

.01

4

0

–0

–.1

c∞0/4

–4

–.2

–8

–.3

–12

–.4 –.4.2

AIR Foil: N.A.C.A.4309. RN:3080 00 Date 4-13-31 Test :V.D.T.563 –16 corrected to infinite aspect ratio 0.2

.4

.6 .8

Fig. AP-I.7

1.0 1.2 1.4 1.61 Lift Coefficient .Cy

1.8

Angle of attack for infinite aspect ratio

θ

percent of chord

Up’r

Drag Coefficient Do. profile-drag coefficient CDo

Sta

20 10 0 –10

λ (degrees)

Fig. AP-I.6. Blade nomenclature and coordinates of a profile

494

.10

0

20 40 60 80 100 percent of chard 40

.09 .08 .07 .06 .05

36 32 28 24

c∞

20

.04

16

.03

12

.02

8 4

.01 a0

0

0 –.1

–4

c ∞0/4 –.2 –.3 –.4

λ (degrees)

– 0 0 1.25 3.45 –1.53 2.5 4.67 –2.13 5.0 6.44 –2.75 7.7 7.70 –3.00 10 8.88 –3.11 15 10.58 –2.93 20 11.81 –2.67 25 12.64 –2.29 30 13.15 –1.91 40 13.25 –1.25 50 12.46 –.76 60 11.10 –.34 70 9.16 –.04 80 6.70 .09 90 3.72 .04 95 2.01 –.05 100 (.16) (–.16) 100 0 – L.E. Rad : 248 slope of radius through end of chord : 6/20

20 10 0 –10


Up’r L‘W’r

Drag Coefficient D o. Profile-drag coefficient C Do

Sta

per cent of chord


–8 –12

AIR Foil:N.A.C.A.6415. RN:30600 α Date 9.3.31. Test :V.D.T. 661 –16 corrected to infinite aspect ratio

–.4.2 0.2 .4 .6 .8 1.0 1.2 1.4 1.61 1.8 Lift Coefficient .Cy

0

20 40 60 80 100 percent of chard

.12 .11 .10 .09 .08

48 44 40 36 32

c∞

28

.07 .06

24

C∞

.05

20 16

.04

12

.03 .02 .01

0 –.1 –.2 –.3

8

a0

4 c ∞0/4 C m0/4

–0 –4 –8 –12

AIR Foil:N.A.C.A.2306. RN:30800 ∞ Date 9-24-31 Test :V.D.T. 680 –16 corrected to infinite aspect ratio

–.4 –.4.2 0.2 .4 .6 .8 1.0 1.2 1.4 1.61 1.8 Lift Coefficient .Cy

Fig. AP-I.9

λ (degrees)

20 10 0 –10


Up’r L‘W’r

– 0 0 1.25 1.10 –.73 2.5 1.70 –.95 5.0 2.43 –1.15 7.7 3.01 –1.22 10 3.48 –1.22 15 4.18 –1.18 20 4.65 –1.09 25 4.91 –1.04 30 5.00 –1.00 40 4.88 –.94 50 4.49 –.81 60 3.92 –.65 70 3.19 –.48 80 2.30 –.33 90 1.26 –.19 95 .68 –.13 100 (.16) (–.16) 100 0 – L.E. Rad : 0.40 slope of radius through end of chord : 2/15

Drag Coefficient D o. Profile-drag coefficient C Do

Sta

per cent of chord

Fig. AP-I.8

APPENDIX II

ISI STANDARDS Mathematical signs and symbols ...... These are mainly taken from international standard (S.I. units). Sign of Symbol

∆

Meaning

Remarks Using E = mc2

Corresponds to

1g — ∆ = 9 × 1020 erg

≈

approximately equal to

j

asymtotically equal to

~ is also used

~

proportional to

∝ is also used

a

mean value of α

n   p

binomial coefficient

π

product

∆x

delta x = finite increment of x

δx

delta x = variation of x

dn f

n ( n − 1) ( n − p + 1) 1 × 2 × 3 × ... × p

f (n()n()x )

differential coefficient of order n of f (x)

δf ( x, y,.......)  δf    δx  δx  y

partial differential coefficient of f (x, y...)

fx(x, y...) and fx′

with respect to x when y,... are held constant

(x, y, ...) are also used

ln x, loge x

Natural log of x

lg x, log x, log10 x

Common log of x

lbx, log2 x

Binary logarithm of x

dx n

495

496


Mechanics Quantity

Symbol

Mass

m

Kilogram

kg

Gram

g

Tonne

t

Remarks

Metric carat

1 metric carat = 200 mg

Metric Technical unit of mass

Ref : 1 metric technical unit of mass is the mass that acquires an acctn., of 1 m/s2 under the influence of a force equal to 1 kg force = 9.80666 kg (exactly).

Density

ρ, e

Relative density

d

Ratio of the density of asubstance to the density of a reference under conditions that should be specified for both substances. When the reference substance is water, the name specific gravity is often used in English.

Specific volume

v

volume / mass

Momentum

p

mass × velocity

Moment of momentum, angular momentum Moment of inertia Kilogram per cubic metre Gram per milli litre

b, P0 Pa I, J kg/m3 g/ml

Kilogram metre per second

kg. m/s

Kilogram metre squared

kg. m2

Weight

Vector product of radius and the momentum of a particle

G (P, W)

Specific weight

γ

Moment of force

M

1 g/ml = 999.972 kg/m3

It is noteworthy that the weight arises not only from the resultant of the gravitational forces existing at the place where the body is, but also from the local centrifugal forces. The effect of atmospheric buoyancy is excluded and consequently the weight defined is the weight in vacuo.

497

APPENDIX

Bending moment

M

Torque, Moment of couple

T

Pressure

p

Normal stress

σ

Shear stress

τ

Force

F

Newton

N

1 N is a force which, when applied to a mass of 1 kg produces 1 m/s2 acceleration.

dyne

dyn

1 dyn = 10–5 N (exactly)

Sthene

Sn

1 sn =103 N (exactly) = 1 kN 1 sn is a force to a mass of 1t, acceleration, 1 m/s2

Kilogram force

kgf

Newton per cubic metre

N/m3

Newton metre

N.m

bar

bar

1 kgf = 9.80665 N (exactly)

1 bar = 105 N/m2 = 106 dyne/cm2= 1 hp Z (exactly) barye

∆l

linear strain

εe

ε= l 0

Shear strain (shear angle)

γ

Volume strain (bulk strain)

θϑ

Poisson’s ratio, Poisson’s number

µ, ν

Young’s modulus

E

E=

σ ε

Shear modulus (modulus of rigidity)

G

G=

τ γ

Bulk modulus (modulus of compression)

K

K=

−p θ

498

Normal atmosphere


1 atm = 101325 N/m2

atm

1 torr = 1/ 760 atm = 133.322 N /m2

torr technical atmosphere

1 at =1 kgf /cm2 = 98066.5 N/m2 = 0.967841 atm

at

compressibility

χ,K

Second moment of area (axial moment)

I, Ia

Second polar moment of area

Ip, J

Section modulus

Z, W,

χ=–

1 dV V dp

I ϑ

Coeff. of friction

µ , (f)

Dynamic viscosity

η, µ

Kinematic visosity

ν

ν=

Poise

P

1 P = 0 1 N. S/m2 = 1 dyn.s/cm2 = 1 g/cm.s

Stokes

St

1 St = 1 cm2/sec

Work

A, W

Energy

E, W

Potential energy

Ek, K, T

Surface tension

σ , (γ)

Erg

dVx dZ

µ ρ

E0, U, V, φ

Kinetic energy

Joule

τxz = η

J erg

Kilowatthour

kWh

electron Volt

ev

15 degree C caloric

Cal15

1 cal, is the heat required to warm 1 g of air free water 1 cal = 4.1885 J from 14.5 degree C to 5.5° C at 1 atm

I.T. Caloric

calIT

1 calIT = 4.1868 J

499

APPENDIX

thermo chemical caloric cal (thermochem)

1 calIT (thermochem) = 4.1840 J

Power

P

Thermic

th

1 th is the heat required to warm through 1 degree a body having a mass of 1 t and a specific heat equal to that of water at 15 degree C and at a pressure of 1 normal atmosphere 1 th = 4.1855 M.J

litre atmosphere

l atm

1 l atm is the work done on a piston by a fluid under a pressure of 1 atm when the volume swept by the piston is 1 l 1 l atm =101.328 J

Watt erg per second

W erg/s

metric horse power cheval vapour litre

1 W = 1 J/S 1 erg/s = 10–7 W (exactly) 1 metric horse power = 75 kgf. m/s = 735.499 W

l

Heat thermodynamic, temperature, absolute temperature

T, θ

customary temperature

t, θ, ϑ

linear expansion coefficient

α, λ

α=

1 dl . l dT

cubic expansion coefficient

α, β, γ

γ=

1 dV V dT

pressure coefficient

β

β=

1 dp . p dT

heat, quantity of heat

Q

heat flow rate

φ, (q)

density of heat flow rate

q (φ)

thermal conductivity

λ, (K)

coeff. of heat transfer

h, K, u, α

Heat flow rate divided by area

500

heat capacity thermal diffusivity

ratio of the specific heat capacities


c a, (α, x, k) γ, κ, k

entropy

S

specific entropy

s

internal energy

U, (E)

enthalpy

IH

free energy

F

Gibbs function

G

specific internal energy

u, (e)

specific enthalpy

h, (i)

specific free energy

f

specific Gibbs function

g

latent heat

L

specific latent heat

l

minute

min

day

d

year

a

radians per second

rad/s

kilometre per hour

km/h

length

l

breadth

b

height

h

thickness

d, δ

radius

r

diameter

d

length of path

s

a=

λ ρCp

501

APPENDIX

angle (plane angle)

α, β, γ, φ, θ etc.

solid angle

Ωω

area

A (S)

volume

V (v)

time

t

angular veliocity

ω

angular acceleration

α

velocity

u, v, w, C

acceleration

a

frequency

f, γ

rotational frequency

ν

angular frequency

ω

wavelength

λ

wave number

C speed of propagation of light in vacuum

σ, (γ)

circular wave number

k

damping coefficient

δ

logarithmic decrement

Λ

attenuation coefficient

α

phase coefficient

β

propagation coefficient

γ

velocity of sound

C

k = 2πσ

Turbines and Pumps specific speed

nS

absolute velocity at inlet

C1

C2 at outlet

vane velocity at inlet

u1

u2 at outlet

relative velocity at inlet

w1

w2 at outlet

whirl component at inlet

Cu1

Cu2 at outlet

flow component at inlet

Cm1

Cm2 at outlet

vane angle at inlet

β1

β2 at outlet

absolute angle at entry

α1

α2 at outlet

APPENDIX III System (SI) : Basic Units Quantity

Unit

Symbol

length (L) mass (M) time (t) amount of substance temperature (T) electric current luminous intensity plane angle

metre kilogram second mole kelvin ampere candela radian

m kg s mo1 k A cd rad

solid angle

steradian

sr

System (SI) : Derived Units Quantity

Unit

Symbol

Alternative Unit

In Basic Unit

force (F)

newton

N

kg m/s2

energy (E)

joule

J

Nm

kg m2 /s2

power

watt

W

J/s

kg m2 /s3

N/m2

kg/ms 2

pressure

pascal

Pa

frequency

hertz

Hz

s –1

electric charge

coulomb

C

As

electrical potential

volt

V

W/A–J/C

kg (m2/s3A)

capacitance

farad

F

C/V

s4 A2/(kg m2)

electrical resistance

ohm

V/A

kg m2/(s3A2)

magnetic flux

weber

Wb

Vs

kg m2/(s2A) kg/(s2A) kg m2/(s2A2)

density

testla

T

Wb/m 2

Inductance

Henry

H

Wb/A

Multiples and Submultiples of Units Multiplying Factor 1 000 000 000 000 1 000 000 000 1 000 000 1 000 100 1 0

= = = = = =

1012 109 106 103 102 101 502

Prefix

Symbol

tera giga mega kilo hecto deka

T G M k h d

503

APPENDIX

0.1 0.01 0.001 0.000 001 0.000 000 001

= = = = =

10–1 10–2 10–3 10–6 10–9

0.000 000 000 001 = 10–12

deci centi milli micro nano

d c m µ n

pico

P

Conversions of Length, Area, Volume and Angular Measurement To Convert length length length length area area area area volume volume volume volume volume volume volume plane angle plane angle

From

To

inch foot metre metre sq. inch sq. foot sq. metre sq. metre cubic inch cubic foot gallon cubic metre cubic metre cubic metre cubic metre degrees radian

metre metre inch foot sq.metre sq. metre sq. inch sq. foot cubic metre cubic metre cubic metre litre cubic inch cubic foot gallon radian degrees

Multiply By 0.0254 0.3048 39.3701 3.28084 0.00064516 0.0929030 1550.00 10.7639 0.000016387 1 0.028317 0.0037854 1000. 61023.74 35.315 264.173 0.0174533 57.2958

Angular Measure

2π Radians are equivalent to 360 degrees. One radian is equivalent to 57.3 degrees. Conversions of Mass, Density and Specific Volume To Convert

mass mass density density specific volume specific volume

From

1b kg 1b/ft3 kg/m3 ft3/1b m3/kg

To

kg 1b kg/m3 1b/ft3 m3/kg ft3/1b

Multiply By

0.453592 2.20462 16.0185 0.0624280 0.0624280 16.0185

504


Density of Water and Air Density of water = 1000 kg/m3 Density of air = 1.2 kg/m3 Conversions of Velocity, Acceleration and Force To Convert velocity velocity velocity velocity velocity velocity acceleration acceleration force force

From

To

fps fpm mph m/s m/s m/s ft/s 2 m/s 2 1b Newton

m/s m/s m/s fps fpm mph m/s 2 ft/s 2 Newton, 1b

Multiply By 0.304800 0.00508000 0.447040 3.28084 196.850 2.23694 0.304800 3.28084 4.44822 0.224809

Unit of Force, the Newton 1 N = 1 kg m/s2

Mass and Weight Standard Acceleration due to Gravity is 9.806 65 m/s2. Conversions of Pressure, Flow Rate and Angular Velocity To Convert pressure pressure pressure pressure pressure pressure presseure pressure pressure pressure volume flow volume flow volume flow volume flow

rate rate rate rate

From psi psf ft. of water (4 C) in. of water (4 C) in. of Hg (15.6 C) Pa (N/m2) Pa (N/m2) Pa (N/m2) Pa (N/m2) Pa (N/m2) cfm cfs gpm gpm

To Pa (N/m2) Pa (N/m2) Pa (N/m2) Pa (N/m2) Pa (N/m2) psi psf ft. of water (4 C) in. of water (4 C) in. of Hg (15.6 C) m 3/s m 3/s m3/s L/s

Multiply By 6894.76 47.8803 2988.98 249.082 3376.85 0.000145038 0.0208854 0.000334562 0.00401474 0.00029613 0.000471947 0.02831685 0.0000630902 0.0630902

505

APPENDIX

volume flow rate volume flow rate volume flow rate volume flow rate mass flow rate mass flow rate mass flow rate mass flow rate mass flow rate mass flow rate rotative speed

m3/s m3/s m3/s L/s 1b/s 1b/min 1/b/hr kg/s kg/s kg/s rpm

cfm cfs gpm gpm kg/s kg/s kg/s lb/s 1b/min 1b/hr rad/s

rotative speed

rad/s

rpm

2188.88 31.3147 15850.3 15.8503 0.453592 0.00755987 0.000125998 2.20462 132.277 7936. 64 0.104720 9.54929

Conversions of Temperature From

To

F R C K C

R F K C F

add 459.67 subtract 459.67 add 273.15 subtract 273.15 multiply by 9/5 and add 32

F

C

subtract 32 and multiply by 5/9

Conversions of Energy and Power To Convert

From

To

energy energy energy energy power power power power power power power

ft–1b Btu. (Int. Stream Table) J J Btu/hr hp (550 ft /1b/s) tons of refrig. boiler hp W W W

J J ft–1b Btu W W W W Btu/hr hp (550 ft –1b/s) tons of refrig.

1.355818 1055.06 0.73756212 0.0009478133 0.2930667 745.6999 3516.8 9809.5 3.41219 0.001341022 0.0002843494

power

W

boiler hp

0.0001019

1J=1Nm 1 W = 1 J/s

Multiply By

506


Conversions of Enthalpy, Specific Heat and Entropy To Convert

From

To

enthaply

Btu/1b

J/kg

enthalpy

J/kg

Btu/1b

specific heat

Btu/ (1b F)

J/(kg K)

specific heat

J/(kg K)

Btu (1b F)

entropy

Btu/ (1b. R)

J (kg K)

entropy

J/ (kg K)

Btu/(1b R)

Multiply By 2326.009 0.00042990095 4186.816 0.000238845 4186.816 0.000238845

specific heat of liquid (water) = 4190 J / (kg K) specific heat of air (constant pressure) Cp = 1 000 J / (kg K) specific heat of air (constant pressure) Cv = 712 J/ (kg K) Rate of heat transfer for air : kW=1.2 (volume flow ratio, m3/s) (temperature change, C) Rate of heat transfer for water : kW = 4 190 (volume flow rate, m3/s) (temperature change, C) Conversion of Viscosities To Convert

From

To

viscosity

1b/ (ft s)

pa s

Multiply By 1.48816

viscosity

1b/(ft hr)

pa s

0.000413378

viscosity

centipoise

pa s

0.00100

viscosity

pa s

1b / (ft s)

viscosity

pa s

1b / (ft hr)

2419.09

0.671971

viscosity

pa s

centipoise

1000.0

ft 2 /s

m2 /s

m2 /s

ft2/ s

kinematic viscosity

0.092903

kinematic viscosity

10.7639

507

APPENDIX

Viscosities and Densities of Dry Air and Water Temp.

Air (standard atmospheric pressure)

Water

C

Viscosity Pa s

Density kg/m3

Viscosity Pa s

Density kg/m3

10

10.768 × 10–6

1.3414

–

–

0

17.238 × 10–6

1.2922

1.789 9 × 10–3

999.84

10

16.708 × 10–6

1.2467

1.310 4 × 10–3

999.70

20

18.178 × 10–6

1.2041

1.007 8 × 10–3

998.21

30

18.648 × 10–6

1.1644

0.802 8 × 10–3

995.64

40

19.118 × 10–6

1.1272

0.655 6 × 10–3

992.22

50

19.588 × 10–6

1.0924

0.552 3 × 10–3

988.04

Absolute Roughness of Some Materials Materials

Roughness,

m

Riverted steel

0.0009 – 0.009

Concrete

0.0003 – 0.003

Cast iron

0.00026

Galvanized iron

0.00015

Asphalted cast iron

0.00012

Commercial steel or wrought iron

0.000046

Drawn tubing

0.0000015

Conversions of Heat Transfer Units To Convert

From

To

Conductivity Conductivity Conductivity Conductivity Convection coefficient and U-value heat flux heat flux

Btu/(hr ft F) Btu in/(hr ft F) W/(m K) W/(m K) Btu/(hr ft2 F) W/ (m2/K) Btu/(hr ft2) W/m2

W/(m k) W/(m K) Btu in (hr ft F) Btu in/(hr ft2 F) W/(m2 K) Btu/(hr ft2 F) w/m2 Btu/(hr ft2)

Multiply By 1.730742 0.1442285 0.577787 6.93344242 5.678286 0.176109481 3.154603 0.316997

508


Thermal Conductivities of Air and Water at Standard Atmospheric Pressure Temperature

Conductivity of Air

Conductivity of water

C

W/(m K)

W/(m K)

0

0.024 4

0.549

10

0.025 1

0.573

20

0.026 0

0.595

30

0.026 7

0.614

40

0.027 5

0.632

50

0.028 2

0.644

Properties of Common Liquids at Atmospheric Pressure and at 20 C Liquid

Specific Gravity (s)

Bulk modulus of Elasticity

Vapour pressure (p) KPa

Surface Tension (N/M) in contact with air

Ethy1Alcohol

0.97

1.21

5.86

0.0226

Benzene

0.88

1.03

10.00

0.0289

1.59

1.10

13.10

0.0267

Carbon Tetrachloride Kerosene

0.81

–

–

0.023 to 0.032

Mercury

13.57

26.20

0.00017

0.51

Crude oil

0.85 to 0.93

–

–

0.023 to 0.038

Lubricating oil

0.85 to 0.88

–

–

0.023 to 0.038

1.00

2.2

2.45

0.074

Water


S. No.

Author Name

Books

1.

Anderson, H.H.

Centrifugal Pumps III Edition. Trade and Technical Press Ltd., Morden Surrey SM45EN. England, 1980.

2.

Anderson, H.H.

Prediction of Head, Quantity and Efficiency in pumps ASME 22nd Fluid Conference, New Orleans 1980.

3.

Addison, H.

Centrifugal and other Rotodynamic pumps. Chapman and Hall Ltd., London.

4.

Artsikoff, A.P. and Vornoff, V.N.

Supplementary machines for ships, ship construction 1963. (Russian)

5.

API Standards, 610

Centrifugal pump for general Refinery Service, 8th Edition American Petroleum Institute, Washington D.C. 1995.

6.

Abromavich, S.F.

Application of N.E Joukovski’s method and Research on flow over cascade of profiles with finite thickness. Engineering collection VIII 1950 (Russian)

7.

Betz, A.

Introduction to the theory of flow machines, Pergamon Press, Oxford and London 1966.

8.

Belik, L.

Secondary flows in blade cascades of Axial Turbo Machines and the possibility of reducing its unfavourable effects. Inst, of JSME symposium of Fluid Mechanics and Fluidics. Tokyo Sept. 1972.

9.

Beelasirkovski, S.M. Polonski, A.C and Ginievski, A.E.

Force and Moment of Aerodynamic Characteristics of cascade of thin profiles. Industrial Aerodynamics. Vol. 22, 1962 (Russian)

10.

Blox, E.A. and Ginievski, A.E.

Vortex free flow of cascade in the form of circle and its usage for the calculation of Hydrodynamic Calculations. Industrial Aerodynamics Vol. 20 1961 (Russian)

11.

Bogdanovski, V.E.

Influence of radial Clearance in Impeller on the working of axial impellers. Journal of Institute of Hydraulic Machines Vol. 22 Moscow 1958 (Russian)

12.

The characteristics of 78 related Airfoil Sections from Tests in the variable density wind Tunnels Report No. 460 NACA. 509

510


13.

Csanady, G.T.

Theory of Turbo Machines, McGraw Hill 1964.

14.

Chung, P.K.

Separation of Flow 1970.

15.

Cohen H. Rogers G.F.C. and Saravanamuthoo, HII

Gas Turbine Theory Vth edition, Longmen group 1988

16.

Conference on Caviation Edinburgh 1974. I Mech. E.

17.

Conference on Pumps and Compressors for off-shore oil and gas, Aberden 1976 I Mech. E.

18.

Conference on Pumps or Nuclear plants Bath 1974, I Mech. E.

19.

Cornish, R.J.

Investigation of Leakage through fine Annular Clearances with inner and outer Boundary Water Stationary and Rotating – Manchester University 1924.

20.

Church, A.H.

Centrifugal Pumps and Blowers – John Wiley and Sons Inc. 1962.

21.

Dixon, S.L.

Fluid Mechanics Thermodynamics of Turbo machinery Pergamon Press 1975.

22.

Dunkan, J.A.

Review of Cascade Data on Secondary Losses in Turbines – J. Mech E. Sci. 12, 1970.

23.

Denney, D.F.

Vortex Formula in pump sump.

24.

Deiz, M.E and Samailovich, G.S.

Fundamental aerodynamics of Axial Turbo Machines Moscow Publishing House 1959 (Russian)

25.

Doozik, S.A.

Profiling of Subsonic Axial flow compressor blades with Mach No. < 1 (Russian)

26.

Dorfmann, L.A.

Indirect method for cascade system. Applied Mathematics and Mechanics, Vol. 5 1954 (Russian)

27.

Doomoff, V.E.

Calculation of Centrifugal pump stages with axial inducers before 1st stage impeller for high anticavitating property – Thermal Energy Vol. 6 1959 (Russian)

28.

Eckart, B.

Axial Compressors and Radial Compressors. Springer Verlag. Berlin 1957 (German)

29.

Dr. Ing Bruno Eck

Fans 1st English Edition – Pergamon Press, New York 1975.

30.

Ethinberg, E.A.

Investigation of Hydraulic losses in variable pitch propeller turbines. Energy Machine Construction Vol. 4, 1961 (Russian)

511


31.

Erimena, A.S.

Investigation of Profile Cavitation in Axial flow pump model Institute of Hydraulic Machines, Moscow Report No. HC490, 1954 (Russian)

32.

Foyers

Pumped Storage plants. Conference scaling for performance prediction in rotodynamic machines stirling 1977, I Mech. E.

33.

Fluid Machinery : Failures, Prediction and Prevention 1980, I Mech. E.

34.

Florjancic, D.

Net positive suction heads for feed pumps, Sulzer report 1984.

35.

Florjancic, D.

Influence of gas and Air admission, the behaviour of single and multistage pumps Sulzer Research No. 1970.

36.

Fluid Dynamics Techniques in the analysis and design of turbo machinery – ASME Journal of Fluid Engineering Sept. pp. 315-352.

37.

Ginievski, A.C.

Influence of Viscosity of fluid on the circulation around profiles of a hydrodynamic cascade system Vol. 9 Moscow Publishing House, Moscow 1957, (Russian)

38.

Ginzburg, B.L and Starilstav, G.

Application of A.F. Lisohin’s Method for the calculation of high specific speed axial flow pumps. Report of Leningrad Polytechnic, Energy Machine Construction No. 2, 1953 (Russian)

39.

Goflin, A.P.

Aerodynamic calculation of flow passages of Axial compressors for land use. Report of Central Steam Turbine Institute Book 34. Moscow Publishing 1959. (Russian)

40.

Guthovski, E.V.

Investigation of Hydrodynamic forces acting on Axial Turbines – Report of Leningrad Polytechnic, Energy Machine construction No. 193, Moscow Publishing 1958. (Russian)

41.

Garner, H.O.

The development of Turbulent Boundary layer ARC RM 2133-1944.

42.

Howell, A.R.

Proceedings of the I Mech E London Vol. 153, 1945 WEI No.12.

43.

Howell, A.R.

Proceedings of I Mech E No. 2 1945 and Vol. 163 (WEI 60) 1950 pp 235-248.

44.

Hydraulic Institute Standards for Centrifugal, Rotary and Reciprocating pumps Parsippany N.J. 1994.

45.

Heala C. Cameron

Hydraulic Data 18th edition. Ingersol dresser pumps liberty corner N.J. 1996.

512


46.

IS 5120 – 1977

Technical Requirements for Rotodynamic special purpose pumps.

47.

John, Tuzson

Centrifugal Pump Design. John Wiley and Sons Inc., New York 2000.

48.

Joukovski, N.E.

Calculation of flow over cascades of Turbomachine, Moscow Publishing House, 1960. (Russian)

49.

Joukovski, N.E.

Theory of ships, on flowing water and the reaction by flowing water – collection of papers Vol. IV ONTI 1937. (Russian)

50.

Joukovski, N.E.

The Reactive forces of flowing liquid over fixed blade and moving blade. Collection of papers Vol. IV ONTI 1937 (I & II) (Russian)

51.

Kovats, A. and Desmur, G.

Centrifugal and Axial Flow Pumps and compressors pergamon press Oxford 1964.

52.

Krasnoff, E.

Aerodynamic Translation for NASA 1976 TT 74-52006 NASA TTF 765 Moscow (Russian)

53.

Krause, E.

Application of Numerical Techniques in Fluid Mechanics. The Aeronautical Journal Research Association S. London.

54.

Karassik, I.J. and others

Pumps Hand Book – McGraw Hill International Edition, New York 1986.

55.

Karelin, V.A.

Cavitation inception in Centrifugal and Axial flow pumps – Moscow Publishing Institute, 1963 (Russian)

56.

Kovalev, N.N.

Hydraulic Turbines, Moscow Publishing Institute, 1961 (Russian)

57.

Kolton, A.U. and Ethinberg, E.A.

Fundamental Theory and Hydrodynamic Calculation of Hydraulic Turbines. Report of Leningrad Polytechnic, Energy Machine Construction No. 10 1966 (Russian)

58.

Kochin, N.I.

Hydrodynamic Theory of Cascade System. Hydraulic Energy Institute, Moscow 1949 (Russian)

59.

Kale, R. and Sreedhar, B.

A theoretical relationship between NPSH and Erosion rate for a centrifugal pump Vol. 190, ASME Feb. 1994, p. 243.

60.

Val, S. Labanoff and Robert R. Ross

Centrifugal Pumps (Design and Application) 2nd edition Jaico Publishing House, Mumbai-2003.

61.

Lev, Nelic

Centrifugal and Rotary Pumps Fundamental with applicant CRC Press, New York, 1999.

513


62.

Lakshmiarayana, B.

An Assessment of Computational Fluid Dynamics Technique in the analysis and design of Turbo Machinery ASME Journal of Fluid Engg. Sept. 1991, pp. 315-352.

63.

Lexicon

Centrifugal pumps. Pumps and Valves KSB III edition Frankenthal, Sept. 1997.

64.

Levish, C.P.

Aerodynamics of Centrifugal compressors. Moscow Publishing House, Moscow 1966 (Russian)

65.

Lisohin, A.F.

Calculation of Impeller Blades of Axial Flow Turbines (Cascade System for Profiles with finite thickness) Report of Leningrad Polytechnic, Energy Machine Construction. Technical Hydromechanics No. 5, Moscow Publishing House, 1953.

66.

Lisohin, A.F. and Simonoff, L.A.

Calculation of Kaplan type Turbine runner for the selected vortex distribution. Moscow Pub. House, 1931.

67.

Loitsanski, L.G.

Mechanics of liquids and gases – Government Technical Publishing House, 1959.

68.

Lomakin, A.A.

Methods of Determining highly economical Axial flow pumps, Turbines and Ventilators. (Russian)

69.

Lomakin, A.A.

Centrifugal and Axial flow pumps, Moscow Publishing House, 1966. (Russian)

70.

Lomakin, A.A.

Conditions of dimensional analysis for the determination of cavitation in models of Hydraulic machines. Report of Leningrad Polytechnic Hydraulic Machine construction No. 215 Moscow Publishing House, 1961. (Russian)

71.

Minski, E.M.

Approximate Method of calculating the point of change over from Laminar to Turbulent No. 71940.

72.

Myles, D.J.

Analysis of Impeller and Volute losses in Centrifugal Pumps Proc. I Mech. E Vol. 184, 1969-70.

73.

Moody, L.F.

Propeller Type Turbines Tran. ASCE Vol. 89, 1976.

74.

Mac-Gregor, G.A.

Two Dimensional losses in Turbine blades. Journal of Aero Science Vol. 19 No. 16, 1952.

75.

Markoff, N.M.

Calculation of Profile losses in compressor cascades under non separated flow of gas, Journal of energy, 1948.

76.

Mihaeloff, A.K. and Malishenoff, V.V.

Construction and Calculation of High Pressure Centrifugal Pumps. Moscow Publishing House, 1971 (Russian)

514


77.

MC. Nally W.D & Sokol, P.M.

Review Computational Methods for Internal flows with emphasis on Turbo Machines. ASME Journal of fluid Engg. Vol. 107, March pp. 6-22.

78.

Muggi, F.A., Eisele, K. Casay M.V. Gulich, J. & Schachensimann, A.

Flow Analysis in Diffusers. ASME Journal of Fluid Engineering Dec. 1997, pp. 978-985.

79.

Muggi, F.A and Schachenmann, A.

Comparison of three Navier Stokes codes with LDA measurements on an Industrial Radial Pump Impeller Pumping Machinery Symposium ASME Feb. Vol.154, 1993, pp. 247-252.

80.

Neumann, B.

The interaction between geometry and performance of a centrifugal pump – Mechanical Engineering Publication Ltd., London, 1991.

81.

Nixon, R.A. and Spencer, E.A.

Model Testing of High Head Pumps I Mech E Symposium on Model testing on Hydraulic Machinery, Cranfield, April, 1968.

82.

Nelik, L.

How much NPSH is enough. Pumps and Systems, March 1955.

83.

Papir, A.N.

Axial flow Blades for ships (Fundamental theory and calculation – Ship Building Publication, Leningrad 1965) (Russian)

84.

Papir, A.N.

Calculation of variable pitch propellers for ships, Sci & Tech Pub. Leningrad Polytechnic. Report of Energy Machine Construction No. 10, 1958, (Russian)

85.

Papir, A.N.

Conditions for Maximum efficiency for ships propulsion and pump selection. Sci & Tech. Inform. Bulletin Energy Machine construction report No. 6, 1959 (Russian)

86.

Papir, A.N.

Influence of Vane Solidity of cascade of Impeller blades of Axial flow pumps on load and cavitation effects – Energy Bulletin No. 11, 1961 (Russian)

87.

Papir, A.N. and Isaeff, U.M.

Flow in radial clearance of Hydraulic Machine and its effect on the working of impellers (Russian)

88.

Papir, A.N. and Gryanko, L.P.

Vane Pumps (Rotodynamic) Machine construction, Leningrad 1975 (Russian)

89.

Pevzner, B.M.

Centrifugal and Axial flow pumps for ships – Publication of Ship Construction 1964 (Russian)

90.

Pernik, A.D.

Problems in Cavitation, Ship Construction 1963 (Russian)

515


91.

Povx, L.

Methods of calculation of losses in cascades with real fluids. Leningrad Polytechnic Report, Technical Hydro Machines No. 5, Moscow Publishing House, 1953 (Russian)

92.

Proscura, G.F.

Hydrodynamics of Turbo Machines, Moscow Publishing House, 1954 (Russian)

93.

Pump Users Handbook

Gulf Publishing Company, Houston Texas, 1980.

94.

Pumping Viscous liquids without Damage using flexible Impeller – World Pumps No. 461 Feb. 2005.

95.

Pumps—Principles and Practice

Jaico Publishing House, Mumbai-23, 2nd Impression 2004.

96.

Pfliderer Carl

Diekreisel pumpen fur Flussigkeiten and gase Springer. Verlag Berlin 1955.

97.

Prandtl, L.

Stromungslehre 2, Aufl S. 323. Brownschweig.

98.

Pohlhausen

Zur naherungg Weisen Integration der differential gleiahungder Lmineren Reibungs Schicht Zamm. 1.252-268, 1921.

99.

Prandtl, L. and Tietjens, O.U.

Applied Hydro and Aerodynamics. Dover publication Inc. New York, 1934.

100.

Pearsall, I.S.

Pumps for Low suction pressures. Symposium on pumping problems university college Sivansea. I Mech E, I Chem E March, 1970.

101.

Pullak, F.

Pump Users’ Handbook.

102.

Ross, R.R.

Theoretical Predictions of NPSHR for cavitation free operation of centrifugal pumps.

103.

Rudinoff, C.C.

Design of propeller pump as per the theory of N.E Joukovski and experimental verification – Report of Moscow Institute of Hydraulic Machines, No. 18, 1938 (Russian)

104.

Sahu, G.K.

Pumps, New Age International (P) Ltd., Publishers, 2000.

105.

Srinivasan, K.M.

Comparative Analysis of Design of Axial Flow Pumps. Ph.D. Thesis (1966) – Deptt. of Hydraulic Machines, Leningrad Polytechnic, Leningrad USSR (Russian)

106.

Srinivasan, K.M.

Losses in Axial Flow Pumps, Journal of PSG College of Technology, Coimbatore, India, 1968.

107.

Srinivasan, K.M. and RKKR Govindarajan

Vortices on mixed flow pump performance. International Conference on Fluid Machines, PSG College of Technology, Coimbatore, India. November 1961.

516


108.

Suhanoff, D.A.

Propeller type pumps Report No. 1 Hydro Energy Publication Moscow 1934. (Russian)

109.

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Steam and Gas Turbines, McGraw Hill, 1927.

110.

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111.

Spannake

Kreiselraderals pumpen and Turbinen Bd1, Berlin, Springer, 1931.

112.

Stepanoff, A.J.

Centrifugal and Axial Flow Pumps, John Wiley and Sons Inc., London 1993.

113.

Stepanoff, A.J. and Stahl, H.A.

Dissimilarily laws in centrifugal pumps and blowers ASME paper 60, WA, 145.

114.

Stepanoff, G.U.

Hydro dynamics of Turbo machinery cascades (Russian) Go.Izd. Phy. Math. Lit. Moscow 1962.

115.

Schlichting, H.

Boundry Layer theory 7th edition “McGraw Hill” New York 1979.

116.

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117.

Specification for Horizontal suction centrifugal pumps for chemical process. ANSI/ASME B 73. IM. 1991. Standard ASME, New York 1991.

118.

Smailovich, G.S.

Calculation of Hydro Dynamic cascades, Applied Mathematics and Mechanics, Vol. 2, 1950 (Russian)

119.

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Methods of Dimensional Similarity in Technical Areas Govt. Pub. 1957 (Russian)

120.

Smirnoff, I.N.

Hydraulic Turbines. Govt. Pub. 1956 (Russian)

121.

Staritski, V.G.

Calculation of Interaction between Cascades of Impeller blades and diffuser blades in Axial flow Machines (Russian)

122.

Staritski, V.G.

Investigation of unequalness of flow parameters very near to diffusers of Axial flow pumps No. 193. Moscow Pub. House, 1958 (Russian).

123.

Staritski, V.G.

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124.

Stefanovski, V.A.

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517


125.

Shalnev, K.K.

Clearance Cavitation in Axial flow pumps, Collection of Academic Report, Vol. 141, 1953. (Russian)

126.

Traupel, N.

Die Berechrung Der Potentialstromung durch schanfelgitter. Techn Runaschan Sulzer. IV No. 1, 1945 and No. 2, 1948. (German)

127.

Thwaites, B.

Incompressible Aerodynamics. Oxford University Press, London, 1960.

128.

Troskoalanski, A.T and Lozarkaiawich, S.

Impeller Pumps, Pergamon Press 3rd edition (Wydawnictwa Naukowo – Techniczne Warszawa, 1973). (Polish)

129.

Truscott, G.F.

A Literature Survey on Abrasive wear in Hydraulic Machinery British Hydromechanic Research Association, London, UK. TN 1079, 1970.

130.

Voznisenski, I.N.

Life, action and selection of the work in the area of Hydraulic Machine development and automatic regulation. Moscow Pub. House, 1952. (Russian)

131.

Vaithasherski, D.A.

Hydraulic losses in cascades of impellers of Propeller Turbines. Report of Moscow Institute of Hydro Machines Vol. 24, 1959 (Russian)

132.

Von Doenoff, A.E. and Tetervin, N.

Determination of general relation for the behaviour of Turbulent Bundary layer NAA Ref. No.777, 1943.

133.

Wislicenus, G.F.

Fluid Mechanics of Turbo Machinery Mc-Graw Hill, New York, 1947.

134.

Worster, R.C.

Flow in Volute of centrifugal pumps and Radial form of Impeller BHRA RR543, 1956.

135.

H. Walz. A

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138.

Warring, R.H.

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INDEX Abrasive liquids, 372 Absolute flow, 38 Airfoil, 43 Angle of attack, 255 Asbestos packing, 149 Aspect ratio, 263 Atmospheric pressure, 197 Axial flow diffusers, 289 Axial flow low depth-circulating pump, 355 Axial flow pump, 216 Axial force, 164, 165 Axial prewhirl, 213 Axial thrust, 165, 167, 168, 170, 291 Axisymmetric flow, 48, 115

Back pullout type, 340 Balance the radial thrust, 180 Balancing axial thrust, 169 Balancing disc, 169 Balancing drum, 170, 172 Balancing hole, 169, 171 Bearing, 170 Bernoulli’s equation, 35, 36 Blade inlet edge, 69 Blade thickness, 233

Coriolis force, 36 Correction in profile curvature, 256 Curved plates, 224

Deepwell turbine, 349 Design, 134 Determination, 177 Different forms of energy, 1 Different types of stuffing box arrangements, 150 Different types of wearing rings, 156 Diffuser section of volute, 139 Diffuser, vaneless passage, return guide vanes, 143 Direct method, 243 Disc friction, 147 Diverging cone part of the diffuser, 142 Double curvature, 113 Double entry with opposed impellers, 345 Double exit, 360 Double suction, 169, 344 Dynamically, 182 Effect of blade thickness and normal entry, 72 Effect of finite number of blades, 74 End suction single stage, 340 Energy equation, 34 Erosion due to cavitation, 211 Error triangles, 126 Eular’s equation of motion, 36 Eular’s number, 182

Cascade system, 52 Cavitation coefficient, 200 Cavitation inception, 198 Centrifugal force, 36 Centrifugal type impeller, 343 Centripetal type, 344 Change in bend at inlet, 168 Channel, 348 Chapligin’s profile of finite thickness, 59 Circular volute, 137, 138 Circulating pump, 363, 364 Circulatory motion, 44 Comparison of hydraulic efficiency, 162 Condensate nooster pump, 367 Condensate, 357 Connection construction, 339 Constant velocity, 135, 138

Feed water pump, 363 Feedwater booster, 368 Flow area reduction, 234 Fluid energy, 1 Forms of casings, 130 Froude number, 182 Geometrically, 182 High/low concentration paper pulp, 371 High depth circulating pump, 356 High head pump, 362 High temperature, 373 Horizontal, 356 519

520


Hub ratio, 261 Hydraulic efficiency, 106, 191, 274 Hydraulic losses, 161 Hydraulic, 147

Radial vanes, 170 Rear shroud, 170 Reducing the suction lift, 211 Reference level for suction head measurement, 196 Relative flow, 40 Relative rotary motion, 44 Reynold’s number, 182 Rotodynamic pumps, 2

Impeller eye diameter, 89 Impeller shroud, 61 Impeller, diffuser and return guide vanes, 351 In line, 343 Increasing the suction tank pressure, 211 Increasing the width at inlet of the impeller, 211 Inducers, 213 Inlet guide blades, 212 Intermediate channel, 346 Isolated profile, 53

Jet pump hydraulic ram, 2 Jowkovski’s method, 243

Kinematically, 182 L eakage flow between two stage of a multistage pump, 159 Leakage flow through wearing rings, 154 Lift method, 243

Maximum dynamic depression, 199 Mechanical efficiency, 106, 147, 193 Mechanical seal, 342 Momentum boundary layer thickness, 277 Multistage drum (Barrel type), 373 Multistage horizontal opposed impeller, 359 Net positive suction head, 195 Non-dog, 369 Opposed impeller, 345 Performance of different specific speeds, 297 Petroleum liquid, 372 Places affected by cavitation, 210 Plates in cascade system, 223 Point by point method, 112 Positive displacement pumps, 2 Profile losses, 271 Profile, 55 Pump performance, 293 Pumps—Non-corrosive liquids, 339

Radial blades, 169 Radial clearance, 288 Radial force, 164, 177

Scale effect, 188 Scheme for spiral casing, 133 Secondary flow, 51 Semi-open impellers, 167 Shapes of volute cross-sections, 131 Similarity, 182, 191, 192, 193 Single entry, 360 Single stage condensate pump, 358 Single stage double ball bearing, 341 Slurry pump, 370, 371 Spiral casing, 130 Spiral part of diffuser passage, 141 Struhaul’s number, 182 Submersible, 351 Suction casing at inlet, 144 Suction lift, 195 Sump pump dry/wet pit, 347 Super cavitating pumps, 285

Thoma’s constant, 200 Trapezoidal cross-section, 134 Trapezoidal, 135 Two half volutes, 181 Two spiral passage, 181 Two stage condensate pump, 359 Two stage domestic pump, 343 Two stage horizontal opposed impeller, 360 Two volutes, 181

Vane under equal velocity construction, 120 Vector diagram for Coriolis component, 36 Velocity at the impeller eye, 69 Velocity distribution, 61 Velocity triangle, 67, 220 Vertical single stage, 342 Vertical, 364 Volumetric, 147 Volute casing, 130, 177 Vorticity, 49

Rotodynamic Pumps

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