Screw Compressor Basics

Screw Compressor Basics

Screw Compressor Modelling, Design and Use

Professor N. Stosic Chair in Positive Displacement Compressor Technology Centre for Positive Displacement Compressor Technology City University London, U.K.

INTRODUCTION Screw Compressor Today Highly competitive market, specially in air compression and refrigeration Continuous improvement: more compact, efficient and cost effective compressors New rotor generation, rotors optimized for certain compressor duty, specialized design Scope for innovation, improvement and development

INTRODUCTION

Basics

View from Front and Top

Top and front: Admission Bottom: Change of volume Bottom and Rear: Discharge

View from Bottom and Rear

INTRODUCTION

Basics

INTRODUCTION

Swedish company SRM, pioneer and leader Screw compressor profiles, Symmetric, Asymmetric, ‘D’ and ‘G’ Screw compressor design Screw compressor technology Licence system left many screw compressor manufacturers at margins of research and development

INTRODUCTION

Many companies started their own development Gardner Denver, Atlas Copco, Compair, Kaeser, GHH Trane, Ingersol-Rand, Hitachi, Fu Sheng, Hanbel, Refcomp Holroyd Many more or less successful rotor profiles Many more or less successful screw compressor designs Need exists to concentrate efforts in R&D

INTRODUCTION

Centre for Positive Displacement Compressor Technology

Improved methods of analysis Experimental validation Design of critical components Complete machine design Product development Training in machine design

INTRODUCTION

Methods Applied Advanced Computerized design tools Machine process modelling 2-D and 3-D Computational Fluid Dynamics Modern experimental technique Computerized data acquisition

Users Renown and new companies, large and small manufacturers in the U.K and abroad

SCREW COMPRRESSOR GEOMETRY Before modelling the physical process, the rotor lobe profiles must be defined together with the remaining parameters with which the rotor and housing geometry can be fully specified.

Rotor profile: x and y coordinates, pressure angle Helix/lead angle, rotor length Interlobe, end and radial clearance Suction/discharge ports

SCREW COMPRESSOR GEOMETRY

General case: non-parallel and non-intersecting

Given profile r1 = r1 (t ,θ ) = [ x1 , y1 , z1 ] = [ x01 cosθ − y01 sin θ , x01 sin θ + y01 cos θ , p1θ ]

∂y ∂x ∂y ∂r1  ∂x1 ∂y1   ∂x01  = , ,0 =  cosθ − 01 sin θ , 01 sin θ + 01 cosθ ,0 ∂t ∂t ∂t ∂t  ∂t ∂t   ∂t  ∂r1  ∂x1 ∂y1  = , ,0 = [− y01 , x01 ,0] ∂θ  ∂θ ∂θ 


General case: non-parallel and non-intersecting Meshed profile r2 = r2 (t ,θ , τ ) = [ x2 , y2 , z2 ] = [ x1 − C , y1 cos Σ − z1 sin Σ, y1 sin Σ + z1 cos Σ ] = [ x02 cos τ − y02 sin τ , x02 sin τ + y02 cos τ , p2τ ] ∂r2 = [− y2 , x2 , p2 ] = [ x02 sin τ + y02 cos τ , x02 cos τ − y02 sin τ , p2 ] = ∂τ  p1θ sin Σ − y1 cos Σ, p2 sin Σ + ( x1 − C ) cos Σ, p2 cos Σ − ( x1 − C ) sin Σ 

Meshing condition  ∂r1 ∂r1  ∂r1  ∂r1 ∂r1  ∂r2  ∂ × ∂θ  ⋅ ∂τ = −  ∂ × ∂θ  ⋅ ∂τ = 0  t   t 

∂y  ∂y ∂x   ∂x  C − x1 + ( p1 − p2 ) cot Σ   x1 1 + y1 1  + p1  p1θ 1 + ( p2 − C cot Σ ) 1  = 0 ∂t  ∂t ∂t   ∂t 


General case: non-parallel and non-intersecting corresponds to the rotor – hobbing tool relation Special cases: p2=0, rotor - plate milling tool, grinding tool relation Σ=0, screw compressor rotors


Rotor profile, Σ=0, i=p2/p1, k=1-1/i Meshing condition dy01  C C   ky01 − sin θ  + kx01 + cosθ = 0 dx01  i i 

Meshed profile

θ i θ y02 = x01 sin kθ + y01 cos kθ + C sin i

x02 = x01 cos kθ − y01 sin kθ − C cos

Rack profile x0 r = x01 cos θ − y01 sin θ y0 r = x01 sin θ + y01 cos θ − r1θ

Numerical solution of the meshing condition

Task: to find θ for a zero function Simple iteration method: Fast and reliable, but valid only for certain function Additional complication In certain areas two or more θ are the zero function Only one is valid, additional values found by half interval method


Demonstrator profile

‘N’ ROTOR PROFILE


- Rack generation procedure - Straight line on the rack - involute rotor contact - Small torque transmitted - Large displacement - Short sealing line - Strong gate rotor,

SCREW COMPRRESSOR THERMODYNAMICS Differential approach: Set of differential equations solved simultaneously Equations of continuity, momentum and energy Preintegrated equations inadequate and inaccurate if high leakage rate and heat transfer is involved

SCREW COMPRESSOR THERMODYNAMICS

Internal Energy dU dV ω = m! in hin − m! out hout + Q − ω p dθ dθ m! in hin = m! suc hsuc + m! l , g hl , g + m! oil hoil m! out hout = m! dis hdis + m! l ,l hl ,l

Continuity ω

dm = m! in − m! out dθ

m! in = m! suc + m! l , g + m! oil

m! out = m! dis + m! l ,l

Leakage Flow Momentum wl2 dx dp + f =0 wl dwl + ρ 2 Dg m! l = ρl wl Ag = Ag

p22 − p12  p2  a  ζ + 2 ln  p1   2

m! = ρ wA


Oil injection

dToil ho Ao (T − Toil ) = ω moil coil dθ

k=

ω moil coil ω d S coil = 6ho ∆θ ho Ao ∆θ

Toil =

T − kToil , p 1+ k

Numerical solution, Runge-Kutta IV order solver U − ( mcT )oil V U (θ ), m(θ ), V (θ ), v = , U = ( mu ) + ( mu )oil , u = m m Ideal Gas T = (γ − 1)

u R

p=

RT v

Real gas p=f1(T,v) u=f2(T,v)

Wet vapour

u = (1 − x ) u f + xug

v = (1 − x ) v f + xvg


Compressor integral parameters m = min − mout

Wind =

m! = mz1n / 60 V! = 60m / ρ 0 F1n + F2 n ) Lnz1 ρ ( m! t = 60 m! ηv = m! t

Psin d

P = V!

Wt ηt = Wind Wt = RT1 ln

∫

Vdp

cycle

Wind z1 n Pind = 60 V Wsind = ∫ dp m cycle

Wa ηa = Wind p2 p1

Wa =

γ R (T2 − T1 ) γ −1


Calculation of pressure loads On compressor rotors Radial forces B

Rx = − p ∫ dy = − p ( yB − y A ) A

B

Ry = − p ∫ dx = − p ( xB − x A ) A

Rotor torque B

B

(

T = p ∫ xdx + p ∫ ydy − 0.5 p xB − x A + yB − y A A

A

2

2

2

2

)


Bearing reactions and rotor deflections d 2δ M = 2 EI dz

SCREW COMPRESSOR OPTIMIZATION

Optimization variables and target function Single stage: Rotor variables: r0 Female rotor addendum r1 Male rotor lobe radius r2 Male rotor tip radius r3 Female rotor tip radius Compressor variables: Built-in volume ratio Operation variables: Shaft speed Oil flow Injection position Oil temperature 9 Variables

Multistage: 9 Variables x Number of stages + Interstage pressures Target function: Specific power combined with compressor price F=w1L+w2C

Box constrained simplex method for efficient and reliable multivariable optimization f ( x1 , x2 ,..., xn ) F = w1 L + w2C gi ≤ xi ≤ hi , i = 1, n gi ≤ yi ≤ hi , i = n + 1, m

yn+1,…,ym

f ( x h ) = max f ( x1 ), f ( x 2 ),..., f ( x k ) f ( x g ) = min f ( x1 ), f ( x 2 ),..., f ( x k ) 1 k i r l i l = + α − ( ) x x x x ≠ , x= x x x ∑ j k − 1 i =1 x r ( new ) = 0.5  x r ( old ) + cx + (1 − c) x h  + ( x − x h )(1 − c)(2 R − 1)

  nr c=  1 + − n k  r r 

nr + kr −1 nr

r0 [mm] r1 [mm] r2 [mm] r3 [mm] Built-in volume ratio Rotor speed [rpm] Oil flow [lit/min] Injection position [o] Oil temperature [o]

Oil Free Oil Flooded Refrigeration

Dry Oil-Flooded Refrig. 2.62 0.74 0.83 19.9 17.8 19.3 6.9 5.3 4.5 11.2 5.5 5.2 1.83 4.1 3.7 7560 3690 3570 12 8 65 61 33 32

EXAMPLE OF CALCULATION 5-6-128 mm Oil-Flooded Air Compressor

7 m3/min, max 10 m3/min at 8 bar (abs) 5-14 bar (abs), max 15 bar (max)

EXAMPLES OF CALCULATION

Rotors optimized for oil flooded operated air compression

5/6-128 mm, L/D 1.65 Displacement 1.56 l/rev Interlobe sealing line 0.13 m Blow-hole area 1.85 mm2


CAD Interface: Compressor ports


Experimental verification of the model


Compressor in the test bed


Comparison of the calculated and test results Flow-Power


EXAMPLES OF 3-D CFD CALCULATION Majority of design problems can be solved by the one-dimensional approach, some of them require the two dimensional calculation, however, there are situations where 3-D CFD must be applied Such are Oil flow distribution, Fluid-Solid Interaction

Grid generation - 1 • Grid topology - polyhedral - O - grid - H - grid - C - grid

• Cell shape

Grid generation - 2 • Grid topology strongly affects accuracy, efficiency and ease of calculation • Full structured block generated hexahedral 3D-O mesh • Screw compressor sub-domains: - Male rotor - Female rotor - End clearances Rotor connections, clearances, leakage paths - Suction port - Discharge port - Suction and discharge receivers Automatic discretization process: - The rack generating procedure - Rack - a rotor with an infinite radius - Divides working domain in two parts male and female rotor,

Comet

Screw Compressor FSI calculations

Mathematical model for screw compressor is based on conservation laws of continuity, momentum, energy, concentration and space: d ρ dV + ∫ ρ ( v − v s ) ⋅ ds = 0 dt V∫ S

ρ = ρ ( p, T ),

! − 2 µ div vI − pI T = 2µ D 3 T = 2η D + λ div wI −

d ρ vdV + ∫ ρ v( v − v s ) ⋅ ds = ∫ T ⋅ ds + ∫ fb dV dt V∫ S S V

(3λ + 2η )α (T − Tr ) I

d ρ hdV + ∫ ρ h( v − v s ) ⋅ ds = ∫ q h ⋅ ds + dt V∫ S S

∫ sh dV + ∫ ( v ⋅ grad p + S : grad v) dV − ∫ pvs ⋅ ds +

V

V

S

d ρ co dV + ∫ ρ co ( v − v s ) ⋅ ds = ∫ q co ⋅ ds + ∫ Sco dV dt V∫ S S V d dV − ∫ v s ⋅ ds = 0 dt V∫ S

e = e( p , T )

d pdV dt V∫

q h = κ grad T q co = ρ Do grad co

Closed by constitutive relations and equation of state and accompanied by turbulence model.

d ρ kdV + ∫ ρ k ( v − v s ) ⋅ ds = ∫ q k ⋅ ds + ∫ ( P − ρε )dV , S S V dt ∫V ε ε2 d ρε dV + ∫ ρε ( v − v s ) ⋅ ds = ∫ qε ⋅ ds + ∫ (C1 P − C2 ρ + C3 ρε ∇ ⋅ v)dV ∫ V S S V dt k k

Thermodynamic properties of real fluids - p-v-T equation compressibility factor z - z is assumed to change linearly with pressure err<2% - Antoine equation for saturation temperature

p = z ⋅ RT = z ( p) ⋅ RT ρ

z = p ⋅ B1 + B2 Tsat =

A2 A1 − log p

- Clapeyron equation for latent heat

 1 1  dPsat hL = T ⋅  −  ⋅  ρ v ρl  dTsat

- Specific heat for constant pressure

c pv = C0 + C1 ⋅ T + C2 ⋅ T 2 + C3 ⋅ T 3 ρ=

- Density of mixture - Coefficient in the pressure correction equation

1 1 − co2 co2 − ρv ρl

 dρ   1 ρ v ⋅ b1  ρ v − Cρ =   = ⋅ dp zRT z  ρ  T 

Multiphase flow

Euler-Lagrange approach

d (mo ho ) dh dmo ! = mo o + hol = Qcon + Q! mass dt dt dt

- Oil is assumed to be a passive ‘species’ - Mass calculated from the concentration Oil drag force influence concentration

mo = m ⋅ Co 1 f drag = − ρ Ao Cdrag v o − v ( v o − v) 2

- Energy source due to heat transfer between working fluid and oil is:

dT T k − T k −1 ! ≈ mo C po Qcon = mo C po δt dt

- Liquid phase is assumed to be an active ‘species’ m ⋅ C pm ⋅ (T − Ts )

- Mass source evaporated/condensed mass

m! L =

- Energy source energy of evaporation/condensation

m − mol dmL Q! mass = hL ≈ hL ol = hL m! L δt dt

hL s

Boundary conditions - Wall boundaries with wall functions are introduced on the housing and rotors. - Compressor positioned between suction and discharge receivers of small volume - Inlet & outlet receivers and oil port are treated as boundary domains:

- Mass equation corrected by mass source to maintain constant pressure

− p V ⋅ρ p  dm  ≈ const ⋅ m! add =   δt pconst  dt  p =const

- Energy equation corrected by energy source to update energy balance

 dm!  = m! add ⋅ hadd Q! add = hadd  add  dt   p = const

Screw Compressor performance - Volume flow (inlet and outlet)

tend

∑

V! = 60 ⋅

t =tstart

- Mass flow (inlet, outlet and oil)

m! =

tend

∑ V!

t = t start

- Boundary forces

(t ) f

⋅ ρ (t )

[kg

i =1

sec ]

Fx = pb * Axb ; Fy = pb * Ayb ; Fz = pb * Azb I

- Restraint Forces and Torque

I

(t ) V!f = ∑ v fi S fi

(t ) V!f  m3 min  ,

FrS = ∑ FrS (i ),[ N ]; i =1

I

Fa = ∑ Fa (i ),[ N ]; i =1

I

FrD = ∑ FrD (i ), [ N ] i =1

I

T = ∑ T (i ), [ Nm] i =1

- Compressor shaft power

P = 2 ⋅ π ⋅ n ⋅ (TM + TF ) [W ]

- Specific power

Pspec = P ! V ⋅1000

- Efficiency Volumetric and adiabatic

! ηv = V

Vd

;

 kW  3  m min 

ηi = Pad

P

Oil injected - Pressure in axial section

Oil injected - Pressure and velocity

Oil injected - Pressure 3D view

Real fluid - Ammonia – pressure

Experimental verification – P-α diagram

Oil injected – Deformation Pressure1

Pinl=1 b Pout=7 b n=5000 rpm tinl=20 oC tout=40 oC

Oil injected – Deformation Pressure-1

Pinl=1 b Pout=7 b n=5000 rpm tinl=20 oC tout=40 oC

Oil injected – Deformation Pressure-2

Pinl=1 b Pout=7 b n=5000 rpm tinl=20 oC tout=40 oC mag=20,000x

Oil injected – Deformation Temperature

Pinl=1 b Pout=3 b n=5000 rpm tinl=20 oC tout=150oC mag=1,000x

Oil injected – Deformation Pressure+Temperature

Pinl=30 b Pout=90 b tinl=0 oC tout=40 oC

n=5000 rpm mag=2,000x

DESIGN EXAMPLES

Oil-free air compressor Oil-flooded air compressor Retrofit rotors of an air compressor Refrigeration compressor

DESIGN EXAMPLES

Compressors for Oil-Free Air Delivery Design aims: Delivery: 350-700 and 700-1000 m3/h Working pressure: 1-2.5 (2.7) bar Volumetric efficiency 90 % + Low specific power Simple, reliable and compact machine

DESIGN EXAMPLES

3/5 lobe rotors Large displacement Convenient gear ratio 5/3=1.67

DESIGN EXAMPLES

Features of 3/5 ‘N’ Rotors - Highest possible displacement - Higher delivery - Better volumetric efficiency - Better adiabatic efficiency - Stronger gate rotor - Long durability - High reliability - Easy manufacturing - Easy compressor assembly - Reasonable noise

DESIGN EXAMPLES

XK18 Screw Compressor General Arrangement

DESIGN EXAMPLES

New design, fully customized by the manufacturer

New rotors, New, improved compressor New concept, better screw compressor compared with competition

DESIGN EXAMPLES

Comparison of Test Results R1- GHH C80 R2- Drum D9000 R3- Mouvex Typhoon R4- GHH CS-1000

DESIGN EXAMPLES

Screw Compressor Family for Oil-Flooded Operation Design aims: Delivery: 0.6-60 m3/min Working pressure: 5-13 bar Volumetric efficiency 90 % + Low specific power Simple, reliable and compact machine

DESIGN EXAMPLES

4/5 Rotors

Reasonable efficiency for moderate pressure ratios, at least the same as for 4/6 rotors Improved capacity and efficiency Lower power consumption Reduced manufacturing cost

Five compressors, rotor diameters: 74, 102, 159, 225 and 285 mm, L/D 1.55

DESIGN EXAMPLES

Performance of the Compressor Family

DESIGN EXAMPLES

Proven design, fully customized to the the manufacturer’s needs

New rotors, New, improved compressor

DESIGN EXAMPLES

Test Results, Compressors 73 and 159 mm

DESIGN EXAMPLES

Retrofit ‘N’ Rotors

5.4 % more displacement 6.5 % higher delivery 4 % better volumetric efficiency 2.5 % better adiabatic efficiency 75 % less torque on the gate rotor

DESIGN EXAMPLES

Asymmetric Rotors, the most common screw compressor rotors

‘N’ Rotor retrofit for more efficient screw compressors

DESIGN EXAMPLES

Design of a semihermetic compressor for air-conditioning and refrigeration based on ‘N’ rotors • • • • •

Semihermetic, convertible to open All existing refrigerants Modern, better than competition 5/6-102 mm L/D=1.55 Efficient rotors for air condition and refrigeration

DESIGN EXAMPLES

5/6-102 mm L/D=1.55 compressor

• • • • • • • •

Theoretical displacement 0.72 l/rev Gear-box 1500-10000 rpm Delivery 0.920-5.80 m3/min R-22 Air condition 5/40 oC 60-380 kW COP 4.20 at 3000 rpm Refrigeration -5/30 oC 33-210 kW COP 3.95 at 3000 rpm

DESIGN EXAMPLES

Refrigeration Application

Application of Design Optimization: A Family of Two-Stage Compressors

Range: 22-250 kW 8-16 bar (abs)

2nd

Two-stages: 19 Variables Target Function: Minimum specific power And only specific power 1st

Number of frames VFD vs gearbox Stage size Stage speed Stage rotor profile

CONCLUSIONS The screw compressor is a mature product at the millenium meeting point. Orchestrated efforts of a large number of companies driven by market forces resulted in the compact and efficient compressor machine. Every detail counts today. A small difference will give a small, but distinctive improvement which may be used as an individual advantage. Optimization opens hidden spots still left for better compressors. They result in stronger but lighter rotors with higher displacement, more compact and more efficient compressor machines at all areas of their application.

EPILOGUE The Centre for Positive Displacement Compressor Technology carries out research and provide a service to manufacturing companies in all aspects of compressor design and development. Through the everyday activity the Centre is gaining experience in the compressor research, development and design. Some of the recent experience is given in this presentation.

Screw Compressor Basics

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