Static and Dynamic Analysis of a Piping System By
Victor Robles Nieves A Thesis Submitted in Partial Fulfillment of the Requirements for the Degree of:
Master of Science In Mechanical Engineering University of Puerto Rico Mayagüez Campus December 2004
________________________________ Basir Shafiq, Ph.D. Member, Graduate Committee
_________________ Date
_________________________________ Frederick Just, Ph.D. Member, Graduate Committee
_________________ Date
_________________________________ Oswald Uwakweh , Ph. D. Representative of Graduate Studies
_________________ Date
_________________________________ Jia Yi, Ph.D. Chairman, Graduate Committee
_________________ Date
_________________________________ Prof. Paul Sundaram, Ph.D. Chairperson of the Department
_________________ Date
ABSTRACT An Investigation of flow-induced vibration is presented in this thesis. Three finite elements models for the pipe system were developed: a structural finite element analysis model with multi-support system for frequency analysis, a fluidstructure interaction (FSI) finite element model and a transient flow model for waterhammer induced vibration analysis in a fluid filled pipe. The natural frequencies, static, dynamic and thermal stresses, and the limitation of the pipeline system were investigated. The investigation demonstrates that a gap in a support at the segment k has a negative effect on the entire piping system. It was determinated that the first natural frequency of the whole system occurs at 2.07 Hz, and the second at a frequency of 5.65 Hz. Resonance vibration for the first mode shape was found at a flow rate of 40 lbm/s, and resonance vibration for the second mode shape occurs at a flow rate of 275lbm/s. In the warterhammer analysis, the limit maximum flow rates were determinated based on the rate of a rapid closure of the isolation valve. A study of the fluid transient in a simple pipeline was performed. Results obtained from FE model for fluid-structure interaction was compared with a model without considering fluid-structure interaction effects. The results show notable differences in the velocities profile and deformation due to the fluid-structure interaction effects.
ii
RESUMEN Una investigación de vibración inducida por fluido es presentada en esta tesis. Tres modelos de elementos finitos para las tuberías fueron desarrollados: un modelo estructural de elementos finitos con múltiples soportes para un análisis de frecuencias, un modelo de elementos finitos de fluido estructura y un modelo de fluido transiente para análisis de golpe de ariete en una tubería llena de fluido. Las frecuencias naturales, estreses dinámicos, estáticos y termales, y las limitaciones de la tubería fueron investigados. Al inspeccionar la tubería, se encontró un espacio entre el segmento K y su soporte. Los resultados indicaron que el espacio encontrado en este segmento tiene un efecto negativo en toda la tubería. Se determino que la primera y la segunda frecuencia natural del sistema completo ocurre a 2.07 Hz y a 5.65 Hz respectivamente. Para la primera forma de vibración fue encontrada resonancia a una razón de flujo de 40 lbm/s, y para la segunda forma de vibración a una razón de 275 lbm/s. Para el análisis de golpe de ariete, el límite máximo de flujo fue determinado basado en la razón de cerrado de la válvula de aislamiento. Se completó un estudio de flujo transiente para una tubería simple. Los resultados obtenidos del modelo de elementos finitos para el caso de interacción fluido-estructura fueron comparados con el modelo sin el efecto de la interacción. Se identificaron diferencias significativas entre los modelos.
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ACKNOWLEDGMENTS The author
wish to thank the Mechanical Engineering Department of the
University of Puerto Rico and NASA for their financial support; especially Dr. Travis for the opportunity to be a part of a summer intern at NASA facilities, Dr. Castillo and Dr. Just for their supports and helps. Special thanks to my advisor of the thesis, Dr. Yi Jia, who has treated me with honesty and provided wise advises for the completion of the work during all my master studies. The graduate students for their friendship and Virmarie Zengotita, who has been with me since the beginning of my graduate studies providing support and encourage. Finally my mother, even when fiscally can’t be with me, her lessons and values are always present. .
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TABLE OF CONTENTS LIST OF FIGURES ............................................................................................. viii LIST OF TABLES...................................................................................................x NOMENCLATURE .............................................................................................. xi CHAPTER 1 INTRODUCTION .............................................................................1 1.1 Introduction....................................................................................................1 1.1.1 Flow Induce Vibration ............................................................................2 1.1.2 The Analysis ...........................................................................................5 1.2 Literature Reviews .........................................................................................6 1.3 Objective ........................................................................................................9 CHAPTER 2 STATIC ANALYSIS.......................................................................11 2.1 Finite Element Model ..................................................................................11 2.1.1 Assumptions..........................................................................................14 2.1.2 Stress Calculation based on ASME B31.1............................................15 2.2 Static Analysis .............................................................................................16 2.2.1 Thermal Deformation...........................................................................17 2.3 Results..........................................................................................................19 2.3.1 Static Stress Analysis Results ...............................................................19 2.3.2 Results of Thermal Stress Analysis ......................................................21 2.4 Discussion ....................................................................................................24 2.5 Chapter Conclusions ....................................................................................25 CHAPTER 3 WATERHAMMER INDUCED TRANSIENT FLOW ANALYSIS...........................................................................................................27 3.1 Transient Flow Analysis ..............................................................................27
v
3.1.1 Governing Equation ..............................................................................29 3.1.2 Boundary conditions .............................................................................30 3.1.3 Numeric Discretization .........................................................................33 3.1.4 Transient Investigation Results and Discussion ...................................34 3.1.5 Valve Programming of Close-Open......................................................36 3.2 Specific Applications ...................................................................................40 3.2.1 Results and Discussion for Waterhammer Pressure Analysis ..............43 3.3 Conclusions..................................................................................................47 CHAPTER 4 RESONANT FREQUENCY ANALYSIS ......................................49 4.1 Resonant Analysis........................................................................................49 4.1.1 Governing Equations and boundary conditions....................................50 4.2 Results and Discussions...............................................................................51 4.3 Conclusions..................................................................................................56 CHAPTER 5 TURBULENCE INDUCED VIBRATION.....................................58 5.1 Turbulence induce vibration ........................................................................58 5.2 Results and Discussions...............................................................................64 CHAPTER 6 FE Model of Fluid-Structure Interaction .........................................66 6.1 Fluid Structure Interaction ...........................................................................66 6.2 FEM Analysis ..............................................................................................67 6.3 Finite Element Models.................................................................................67 6.3.1 Material Properties................................................................................70 6.3.2 Element Types ......................................................................................70 6.3.3 Mesh......................................................................................................71 6.3.4 Boundary Conditions ............................................................................71 6.4 Results..........................................................................................................71 vi
6.5 Conclusions..................................................................................................76 CHAPTER 7 SUMMARY AND CONCLUSIONS ..............................................78 7.1 Summary ......................................................................................................78 7.2 Conclusions..................................................................................................79 7.3 Future works ................................................................................................82 REFERENCES ......................................................................................................83 APENDIX ..............................................................................................................88
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LIST OF FIGURES Figure 1.1: Acoustic wave in pipes..........................................................................4 Figure 1.2: Diagram of the structural study .............................................................6 Figure 2.1: Elastic straight pipe elements ..............................................................12 Figure 2.2: 3D Structural model geometry ............................................................13 Figure 2.3: Different piping supports.....................................................................14 Figure 2.4: Pipe with gap .......................................................................................17 Figure 2.5a: Stress vs. length for gap space case...................................................19 Figure 2.5b: Stress vs. length for gap correction case ...........................................19 Figure 2.6: Gap locations in segment K.................................................................20 Figure 2.7: Stresses distribution along the piping system......................................21 Figure 3.1: Transient flow model...........................................................................31 Figure 3.2: Typically close-open curve [55]..........................................................32 Figure 3.3: Wave pressure for different dt.............................................................35 Figure 3.4: Detail of numerical noise effect ..........................................................35 Figure 3.5: Effect of friction loss...........................................................................36 Figure 3.6: Effect of time of close .........................................................................37 Figure 3.7: Effect of bulk Elasticity Modulus .......................................................38 Figure 3.8: Effect of time of close in the maximum pressure................................39 Figure 3.9: Effect of initial velocity in the wave pressure.....................................40 Figure 3.10: Wave in close duct ............................................................................41 Figure 3.11: Isolation valve and tank location.......................................................43 Figure 3.12: Length vs. stress/allowance no failure is predicted at this flow........45 Figure 3.13: Possible failure is presented at segment F and E...............................45 viii
Figure 3.14: Possible failures for segments A, B, C, D, and F ..............................46 Figure 3.15: Failures for almost all segments........................................................47 Figure 4.1: NASA diagram configuration from previous investigation[1]............50 Figure 4.2: Discretization of the system ................................................................52 Figure 4.3: Resonances per segment at different flow rates for the mode shape 1 ...........................................................................................................53 Figure 4.4: Possible resonances per segment at different flow rates for mode shape 2 ...........................................................................................................54 Figure 4.5: Possible resonances per segment at different flow rates for mode shape 3 ...........................................................................................................54 Figure 4.6: Natural frequency mode shape 1 of a complete systems.....................55 Figure 4.7: Vibration modes shape 2 for the complete system..............................55 Figure 5.1 Comparison of convective velocity predicted by Chen and Wambsganss and Bull [5] ..............................................................................61 Figure 5.2: Boundary layer type of turbulence power spectral density [5] ...........63 Figure 5.3: Longitudinal joint acceptances [5] ......................................................63 Figure 6.1 Fluid structure interaction loop flow chart ...........................................68 Figure 6.2: Geometry of free flowing channel.......................................................68 Figure 6.3: Geometry of channel with obstruction ................................................69 Figure 6.4: Average percent difference at different flows.....................................73 Figure 6.5: Velocities profile at the first iteration..................................................73 Figure 6.6: Velocities profile at the second iteration.............................................74 Figure 6.7: Velocities profile at the third iteration ................................................74 Figure 6.8: Velocities profile at the fourth iteration ..............................................75 Figure 6.9: Velocities profile at the fifth iteration ................................................75 ix
Figure 6.10: Velocities profile at the sixth iteration ..............................................76
LIST OF TABLES Table 2.1: Pipe Specifications................................................................................13 Table 2.2: Thermal Maximum Displacement for 0 Gap........................................22 Table 2.3: Thermal Maximum Displacement for 0.25 Gap...................................22 Table 2.4: Maximum Axial Rotation Due to Temperature Changes for 0 Gap.....23 Table 2.5: Maximum Axial Rotation Due to Temperature Changes for 0.25 Gap.................................................................................................................23 Table 2.6: Stress Due to Temperature Changes for 0 Gap ....................................24 Table 2.7: Stress Due to Temperature Changes for Gap Case...............................24 Table 2.8: System Maximum.................................................................................25 Table 3.1: E2 Facilities Technical Data of Pipe and Content ................................38 Table 3.2: Transient Pressures ...............................................................................44 Table 4.1: Fluid Excitation Frequencies by Others [1]..........................................53 Table 5.1: Uc, Frequency Parameters and Joint Acceptances ...............................64 Table 5.2: PSD and RMS Responses.....................................................................65 Table 6.1: Dimensions Free Flowing Channel ......................................................69 Table 6.2: Dimensions Channel with Obstruction.................................................69 Table 6.3: Material Properties................................................................................70
x
NOMENCLATURE ∇
= Divergent
ac
= Cross-sectional Area
Ac
= Corroded Cross-sectional Area
C
= Speed of Sound
Ca
= Corrosion Allowance
Cs
= Damping of Structure
Cv
= Damping due to Water
Di
= Inner Diameter
Do
= Outer Diameter
E
= Modulus of Elasticity
ΣF
= Total Force
Fa
= Axial Force
fn
= Natural Frequency
fs
= Vortex Shedding Frequency
g
= Gravity Force or Gravitational Force
I
= Moment of inertia of pipe cross section
i
= Intensification Factor
ii
= In-plane Stress Intensification Factor
io
= Out-of-plane Stress Intensification Factor
K
= Pipe Stiffness
L
= Length
M
= Structure Mass
m
= Mass intensity xi
ma
= Mass Added due water
Ma
= Torsion Moment
Mi
= In Plane moment
Mo
= Out of plane moment
mt
= Total Mass
n
= mode shape number
Pd
= Design Pressure
P
= Pressure
Pa
= Axial force from internal pressure
Po
= Applied load
r
= ratio of circular frequency
Re
= Reynolds Number
Sa
= Axial Stress
SB
= Bending Stress
Se
= Expansion Stress
Sh
= Strouhall Number
SH
= Hoop Stress
SL
= Longitudinal Stress
Ss
= Sustained Stress
So
= Sustained plus Occasional Stress
Ssm
= Maximum Shear Stress
Ssh
= Secondary Shear Stress
ST
= Torsion Stress
t
= Time
thk
= Pipe Wall Thickness xii
T1
= Low Temperature
T2
= High Temperature
V
= Flow velocity
Z
= Section Modulus of Effective Section Modulus
α
= Coefficient of Linear Expansion
γ
= 2α
β
= Coefficient of Volumetric Expansion
wf
= Natural frequency
w
= Applied frequency
∆
= deflection
εth
= Thermal Strain
υ
= Poisson Ratio
ρs
= Structure Density
ρw
= Water Density
σth
= Thermal Stress
W
= Strain Energy density function
C10 = Mooney-Rivlin constant C01 = Mooney-Rivlin constant ∆1
=The
principal stretch ratio in the unaxial direction
xiii
CHAPTER 1 INTRODUCTION 1.1 Introduction One of the major problems during the rocket-engine test at NASA is the vibration experienced from the exhaust plume on its components. Flow induced vibration occurs when the natural frequency, fn of the line transporting the propellant and fluid flow are the same or near 1.4. This matching of the two frequencies produces a condition known as resonance, this behavior in many cases yields to failure of components or collapse of an entire system. Another problem of this piping system is the operation of valve. The effect of suddenly stopping or accelerating a fluid by closing and opening a valve may induce a waterhammer overpressure. Is this overpressure is enough the pipeline may fail or deform. The temperature operational condition of this piping system is also of concern. This piping system operates at extremely low temperature. If the temperature of an object is changed in the structure, the object will experience length or area deformation thus volume changes. The magnitude of this change will depend on the coefficient of linear expansion. This drastically temperature changes create additional stress in the piping system. Two know investigation has been previously done in this facilities, Castillo [1] created a model to study the acoustic induce vibration, he obtain results of noise and frequency. Also shed vortices solutions, by calculating the vortex-shedding frequency, which is characterized by the Strouhal number. He also obtain critical velocities that may cause buckling of the pipelines. His models were based on a 1D mass spring model. It was performed to study the natural frequencies and
1
critical flows velocities at resonance, it main focus was on the fluid flow. He discretized the pipe system in straight segments, the problem of his model is that not considers the boundary conditions and support configurations. Also, it did not provide stresses and strain results in all axis. The other know work was performed by Indine, inc, they created a fluid dynamic model using EASY5 software to simulated the transient pressure and flow state at each point in the feedline. Furthermore a detailed time simulations of valve motions was presented. The modeling methodology discretized the feedline into a series of capacitance and flow nodes. These models allowed assessment of waterhammer pressure oscillations associated with valve opening and closing operations as well as pressure oscillation forces on propellant line. A problem of their model is that the pressure response effect was not applied to the piping system. This new investigation is focus on the structure, it consider the effect created by the support as well as other boundary conditions. In the investigation the vibration effect caused by vortex shedding and turbulence flow were consider as well the water hammer effect on the structure and the thermal stress. For this purpose a finite elements model was created. Furthermore, a fluid-structure interaction (FSI) finite element general model and transient timer response general model were develop.
1.1.1 Flow Induce Vibration Transporting liquids through piping systems is a common practice. The term piping system is not new; practically every person has used one. For the general public there is very little understanding of the phenomenon behind the use of piping systems. In some applications, like power plants, the failure of piping 2
systems can cause severe economic losses and in worst cases the loss of human lives. Some of the design or operation factors that may cause failures in piping systems are: incorrect support, transient pressure changes, flow induced vibration and thermal stresses. Several standard codes have been developed to regulate the design and fabrications of piping systems. There are various type of phenomena that may induce vibration on components; vortex shedding, turbulence, water hammer, acoustic among others. Vortex shedding occurs when the flow past an obstacle such as cylinder, sphere or any other disturbing object; resulting in vortices behind the cylinder. These vortices move downstream of the pipeline at a frequency, fs, if the conditions are appropriate these excitation frequencies may induce vibration. When the fluid velocity exceeds any but the smallest values characteristic of “seepage” flows, eddies will form even if the surface of the flow channel is perfectly smooth. The flow is said to be turbulent after it has achieve a specific Reynolds number. Turbulence flow in most application is desired; a typical application is to increase the efficiency of a heat exchanger. The force generated by the turbulence flow has the characteristic of being random. With the appropriate conditions this force will induce pipe vibration, this type of vibration is call turbulence induce vibration. Water hammer normally occurs during the opening or closing of valves, and it generates an acoustic wave that propagates upstream and downstream of the system. Figure 1.1 shows a diagram illustrating this phenomenon. Notice that this acoustic wave may indeed contribute to changes in the thermodynamic properties of the tank (i.e. thermodynamic equilibrium). This transient phenomenon manifest as a big noise coming out of the pipe. This is what is heard sometimes when the 3
water faucet is suddenly open or close. Fluid flow through valves, bends and orifices generates turbulence as the flow passes through the obstacle. This in turns radiates acoustic waves (of velocity Ua and pressure Pa) upstream and downstream of the valve. Thus, as the area of the valves and flow meters changes subsequently the acoustic waves. This is because the waves have an acoustic pressure that acts against the surface of the pipe. Consequently, the fluid flow and the solid surface are coupled through the forces exerted on the wall by the fluid flow. The fluid forces cause the structure to deform, and as the structure deforms it then produces changes in the flow. As a result, feedback between the structure and flow occurs: action-reaction. This phenomena is what is call fluid structure interaction. Because of the interaction between the fluid flow and the solid surface the equations of motions describing the dynamics are coupled. This makes the problem more challenging, and even worse when the flow is turbulent. In addition, this means that the Navier-Stokes equation and the structure equation for the solid surface must be solved simultaneously with their corresponding boundary conditions.
V a lve
A co u stic W av e
Flo w T an k
. Figure 1.1: Acoustic wave in pipes 4
1.1.2 The Analysis Steady flow and waterhammer analyses could provide information on the liquid behavior under operational conditions. Static pipe stress and structural dynamics analyses give insight to the corresponding behavior of the piping system; whereas the fluid analysis yields stream pressures; the structural dynamic analysis provide dynamics stress, reaction forces and resonance frequencies. Figure 1.2 shows the structural analysis element with its corresponding analyses. In the static analysis maximum stresses and displacements were found for the complete system, the weakest elements of the piping system with different support configurations were identified. For the dynamic analysis waterhammer pressure waves were applied to the system as internal pressure loads, as a result the maximum flow rate that the system can resist before failure was identify within its corresponding stress. For the thermal analysis stress caused by a temperature change were studied. As outcome thermal stresses and displacements were obtained. For the resonance vibration analysis the first and second natural frequencies of the piping system were identified using a finite element program and compared with the applied frequency from the fluid resulting from vortex shedding to identify possible resonance at different flow rates.
5
3D Solid Model
Static
Dynamic
Vibration
Maximum Stress
Maximum Displacement
Pressure History
Natural frequencies
Complete System
Complete System
Valve Closing Time
Complete System
Various Flows
Segments
Segments
Excitations Frequencies
Figure 1.2: Diagram of the structural study
1.2 Literature Reviews This section services as a literature review about previous works done by other researchers, which has been used as reference sources, support and background for this research. Many papers and books have been consulted, but most of them are briefly mentioned and some of them are discussed along the thesis. The papers with more significant contribution to the field are discussed here. Investigation of the flow induced vibration at the NASA Facility has been conducted by InDyne, Inc. They. Created a fluid dynamic model using EASY5 software to have simulated the transient pressure and flow state at each point in the feedline. Furthermore a detailed time simulations of valve motions was presented. The modeling methodology discretized the feedline into a series of capacitance and flow nodes. These models allowed assessment of waterhammer pressure oscillations associated with valve opening and closing operations as well as pressure oscillation forces on propellant line. Castillo [1] created a model to
6
study the acoustic induce vibration, he obtain results of noise and frequency. Castillo [1] obtained the shed vortices, by calculating the vortex-shedding frequency, which is characterized by the Strouhal number. He also obtain critical velocities that may cause buckling of the pipelines. Chiba [33];[34];[35];[36] extensively studied piping response using multiple support system generally under the action of seismic conditions for both linear and non linear behaviors under the action behaviors. Vayda [37], presented his research on the dynamic behavior of piping systems under the influence of support to pipe gap with the seismic conditions and the nonlinearity of the system Lockau,Haas and Steinweder [38] presented their work on piping and support design due to high frequency excitation as the criterion. Morgan [30] studied the propagation of axis-metric waves through fluid filled cylindrical elastic shells. The dependence of phase velocity on various physical parameters of the system was analyzed. However their results were restricted to real wave numbers and to circumferential modes of zero order. Thomson [29] introduced the effects of Poisson’s ratio and included flexural and axial wave motion and evaluated the phase velocities of the first three axisymmetric “fluid” waves. Blevins, [5] in his book “Flow-Induced Vibration”, presents an equation to estimate the values for the frequency of the vortex shedding, fs =
SV di
(1.1)
where S is the Strouhal number, V is the flow velocity and di the inner pipe diameter. He
proposed
that
for
the
high
Reynolds
number
ranges, 7
5.43 X 10 5 ≤ Re D < 6.06 X 10 6 , a Strouhal number of about 0.41 is appropriate. This is the range of Reynolds number used in our case. Blevins [26] gives a brief discussion of the application of dimensional analysis to flow-induced vibration. A.S. Tijsselin [9],[10] has done extensive literature reviews about Fluid Structure interaction problems with cavitation. He presents one dimensional basic equations by integration of general three-dimensional equations for fluid dynamics and structural linear elasticity. He solved by the method of the characteristics the governing equation, formulated as a hyperbolic set of fourteen first order partial differential equation. He simulated vaporous cavitation numerically. Taylor [41] offers an alternative way to measure the damping ratio by measuring the power supplied to maintain a steady-state, resonant vibration of the structure. J.M. Cuschieri [31] investigated the transmission of vibrational power from the piping system to the supporting structure using power flow and structural mobility methods. This approach can be applied to isolate straight pipe sections as well as a number of subsections joined together by components that can be represented by structural mobility terms. Kumar [27] derived the frequency equation for vibrations of a fluid-filled cylindrical shell using the exact three-dimensional equations of linear elasticity. These equations were analyzed quantitatively to study the flexural vibrations (n=1) of empty and fluid-filled shells of different thickness. The effect of fluid was negligible for vibrations of thick shells. As the thickness of the shell decrease, the presence of fluid gave extra modes of vibrations. T. Repp [13] Presents a simulation that shows an overall good agreement for the average pressure amplitude of a straight pipe in comparison to the analytical results obtained with the extended Joukowksy equation. He found that In the case of the bended pipe the pressure amplitude of the extended 8
Joukowsky equation seems to be too conservative. Samsury [28] discussed the phenomenon of liquid-structure coupling in fluid-filled pipes, which results in plane axial waves in the fluid getting converted to flexural beam vibrations of the pipe. A mathematical analysis of liquid-structure coupling in a liquid-filled elbow is presented. Morgan [32] studied the propagation of axis-metric waves through fluid filled cylindrical elastic shells. The dependence of phase velocity on various physical parameters of the system was analyzed. However their results were restricted to real wave numbers and to circumferential modes of zero order. M. K. Au-Yang [16],[20],[21],[23],[24] Reviewed and put onto a firm mathematical basis of the theoretical development of the acceptance integral method to estimate the random vibration of structures subject to turbulent flow. He derived closedform solutions for the joint acceptances for spring-supported and simply supported beams. K.T. TRUONG [22] in his paper evaluated dynamic stresses of a Pipe Line, presented a fast and reliable way to evaluate the harmonic dynamic stresses of a simply supported pipeline from the data collected on the field. He also offers a basic understanding to solve quickly vibration problem when and where the computer software is not accessible. Paidoussis, M.P.; Au-Yang, M.K. and Chen, S. S., [38] in 1988, studied leakage flow induced vibration. He collected technical papers, most of them dealing with numerical analysis or testing of specific components.
1.3 Objective The objective of this thesis is to conduct an investigation of flow induce vibration, the research will be extended to an specific propellant pipeline at NASA facilities. As outcomes, maximum flow rate that may cause resonance and 9
vibration amplitudes, based on transient flow analysis, will be identified. The scope extended to the fluid structure interaction phenomena, general application programs will be created In order to achieve these major goals. The following are specific objectives: 1. Create a finite element model for static structural analysis of the specific application at the NASA facilities. 2. Obtain maximum flow rate and maximum pressure solutions to prevent pipeline failure during operation. 3. Investigate the transient waterhammer phenomena. 4. Develop a general application subroutine that enables the study of fluid structure interaction.
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CHAPTER 2 STATIC ANALYSIS 2.1 Finite Element Model The model is created based on the actual pipe configuration; it is a 3dimensional model, which has the capability of simulating different boundary conditions for given problem. This model was created in the commercial software PipePack, which is a part of Algor® software. The structural analysis performed by this software is in compliance with various industrial standards piping codes. In our case the code that was applied is the ASME B31.1 power piping code. This model is intended to only simulate static fluid flow, and for a structural analysis. Simulating the effects of fluid flow will be presented in separate analysis in the following chapters. Finite element analysis is an advance method that divides the structure in small elements and applied it corresponding boundary conditions to solve a complex problem [6]. The type of element used can be described as following: an uniaxial element with tension-compression, torsion, and bending capabilities. The element has six degrees of freedom at two nodes: translations in the nodal x, y, and z directions and rotations about the nodal x, y, and z axes. Figure 2.1 shows the characteristic of the element used.
11
Figure 2.1: Elastic straight pipe elements
The entire pipeline has 14 straight segments, 13 elbows, 2 valves and a reducer as shown in Figure 2.2. The segments are named with letter that goes from A to N. The model has the capability of return values every 4 inches. For sections A to the beginning of section I the pipelines have an external diameter of 6.625 inches and an internal diameter of 4.209 inches, this leads to a thickness of 1.208 inches. From sections I to n the outside diameter is 4.5 inches with an inside diameter of 2.86 inches and a thickness of .820 inches. The material of the pipeline is Austenitic stainless grade (301-309) with a density of 0.2899 lb/cu in. The fluid inside of the piping is liquid oxygen with a density of .0411 lb/cu in. Table 2.1 summaries the properties of the pipelines.
12
A I
2
2 H F
E
B 2
3
J
3
G
1
D
C
K L
N 3
M
Figure 2.2: 3D Structural model geometry
Table 2.1: Pipe Specifications 6 in section
4 in section
Material
Austenitic 304 stainless steel
Austenitic 304 stainless steel
Outer Diameter (in)
6.625
4.50
Inner Diameter (in)
4.209
3.68
Thickness (in)
1.208
0.82
Inside Fluid
Liquid Oxygen
Liquid Oxygen
13
The pipeline has three types of supports as shown in Figure 2.3; the first one is a one-way support, constraining the movement in the negative Y axis. The second one is a 4 way constrain support, it has a .25 inches of gap for the x and for the positive Y axis, for the negative Y axis the displacement is constrained. The last type of supports constrains the movement in the negative Y direction.
Type 1
Type 2
Type 3
Figure 2.3: Different piping supports
2.1.1 Assumptions The weight of the tank which is at section A was not included in the analysis because all its weight is sustained by its own separate supports. This part of the piping was considered rigid and modeled with an anchor. At section 3, a T connection was considered welded under ANSI B16.9. The section connected in the T has two valves and was not considerate after the valves refer to Figure 2.2. For the location of the T, there is a flow meter that its weight was also not considerate at segment F because the additional weight of this segment compared to the piping is minimal. The supports were treated as rigid elements. The anchors
14
at sections F and N were modeled as rigid in all directions. The weight of the valve at section I is considered to be held by its own supports, thus is not include in the 3D model. All the analyses were modeled at an ambient temperature of 85°F witch is typical for the geographic location.
2.1.2 Stress Calculation based on ASME B31.1 Cylindrical pressure vessel and pipes carrying fluids at high pressure develop stresses with values that are dependent upon the radius of the element under consideration. The pressure inside of the cylinder acts on the wall of the same, as a result a stress acting uniformly over the area is created. This stress is the hoop stress and is calculated with the following equation. ⎧⎪⎡ ⎫ ⎤ Do − 0 .4 ⎬ SH = P ⎨⎢ ⎥ ⎪⎩⎣ 2(thk − ca) ⎦ ⎭
(2.1)
Were P is the internal pressure exerted by the fluid, Do is the exterior diameter, thk is the thickness and ca is the corrode cross-sectional area, in our case no corrosion is expected for the stainless steel, thus the ca value becomes 0. Another stress created due to pressure is the longitudinal stress, this stress is created along the pipe, and it will depend on the geometry of the pipe as well on support or any stress intensity factor in the pipe. The stress is calculated with the following equation.
[(i Mi) SL = i
2
+ (io Mo) 2 Z
]
1
2
+
Fa + Pa Ac
(2.2)
15
where Mi is the in-plane moment, Mo is the out of plane moment Z is the section Modulus of Effective Section Modulus, Fa is the axial force, Ac is the cross-sectional area and Pa is the axial force from internal pressure and is calculated with the following equation. Pa =
π 4
{P[D0 − 2(−ca)]2 }
(2.3)
In this equation the pipe is assumed without corrosion thus ca becomes 0.
2.2 Static Analysis The static analysis serves as a starting point, where the weakest elements are going to be identified and the cause of it. As well, the support with the reaction that creates to the piping system will be analyzed. It’s not expected any failure or critical stress at this point. After an inspection of the pipeline it was found a space or gap between certain supports and the pipeline as shown in Figure 2.4. The gap means that the pipelines in certain areas do not touch the support, therefore their weight load is supported by segments of others areas. The reason for the space is unknown but it might design to accommodate thermal expansions or a construction error. The analyses were made with the gap and without the gap to see if the gap makes any significant difference in the system stresses.
16
Figure 2.4: Pipe with gap
2.2.1 Thermal Deformation One of our goals in this investigation is to study the effect of the temperatures changes. If the temperature of an object is changed in the structure, the object will experience length or area thus volume changes. The magnitude of this change will depend on the coefficient of linear expansion, α, which is widely tabulated for solids. The coefficient of volumetric expansion, β, which is used extensively with liquids and gasses. Changes in temperature affect all dimensions in the same way. In this case, thermal strain is handled as strain due to an applied load. For example, if a bar is heated but is constrained the stress can be calculated from the thermal strain and Hooke’s law.
σ th = Eε th
(2.4)
where E is the modulus of elasticity and ε th is the thermal strain, the length L, area A, and volume V, strain are calculated with the following equations. 17
∆L = αL0 (T2 − T1 )
(2.5)
∆A = γA0 (T2 − T1 )
(2.6)
γ ≈ 2α
(2.7)
∆V = βV0 (T2 − T1 )
(2.8)
β ≈ 3α
(2.9)
Algor calculate the thermal stress using the restrained and unrestrained conditions, PipePlus determines the restrained or unrestrained status according to the Y coordinate for each segment of pipe. A positive Y coordinate value represents an aboveground (unrestrained) segment. A negative Y coordinate value represents a buried (restrained) segment. For the restrained segments the sustained stress is calculated with the following equation [14]: S L = Eα (T 2 − T 1)
(2.10)
For the unrestrained segment the expansion stress is calculated with the following equations [14]
[
S E = Sb2 + 4 St2
Where;
]
1
[(i M ) + (i M ) ] = 2
Sb
(2.11)
2
i
i
Z Mt St = 2Z
2
0
0
1
2
(2.12)
(2.13)
18
2.3 Results 2.3.1 Static Stress Analysis Results Maximum Stress/Allowance ratio with gap 0.12 Segment A Segment B
0.1
Segment C Stress/Allowance
Segment D 0.08
Segment E Segment F
0.06
Segment G Segment H Segment I
0.04
Segment J Segment K
0.02
Segment L Segment M
0
Segment N Segments
Figure 2.5a: Stress vs. length for gap space case
Maximum Stress/Allowance ratio gap correction 0.035 Segment A Segment B
0.03
Segment C Stress/Allowance
0.025
Segment D Segment E
0.02
Segment F Segment G
0.015
Segment H Segment I
0.01
Segment J Segment K
0.005
Segment L Segment M
0
Segment N Segments
Figure 2.5b: Stress vs. length for gap correction case
19
Figures (a) and (b) show the peak static stress/allowance ratio value for segments from A to M for both cases, with the gap and without the gap in section k, refers to Figure 2.6 for location of this segment. The stress to allowance ratio is the division of the maximum allowance stress per ASME code B31.1 and the actual maximum actual stress per segment. These values were obtained using the Algor finite element program.
I
J
Gap
K
L
Figure 2.6: Gap locations in segment K
20
545 psi
.25in gap
0 in gap
70 1789 PSI 1789 psi
220
Figure 2.7: Stresses distribution along the piping system
Figure 2.7 shows the stresses distribution along the piping system. It also shows the peak stress for the case with gap and without gap, and the location of the same. The left side of the Figure is the case without the gap and the right one is the case with the gap.
2.3.2 Results of Thermal Stress Analysis The following Tables 2.2 to 2.7 summarize some of the results. Although the system experiences some displacement due to temperature change, the maximum effect can be seen in the stresses, particularly for those where the gap of .25 inches is present
21
Table 2.2: Thermal Maximum Displacement for 0 Gap Maximum Displacement for 0 GAP
Temperature °F
X (inches)
Segment
Y (inches)
Segment
Z (inches)
Segment
-100
0.276
D
0.158
J
0.340
K
-200
-0.406
B
0.233
J
-0.490
I
-300
-0.531
B
0.330
L
-0.703
I
-400
-0.562
B
0.378
L
-0.748
I
Table 2.3: Thermal Maximum Displacement for 0.25 Gap Maximum Displacement .25 GAP
Temperature °F
X (inches)
Segment
Y (inches)
Segment
Z (inches)
Segment
-100
0.276
J
0.159
K
0.332
K
-200
-0.406
M
0.234
K
-0.490
K
-300
-0.531
M
0.330
K
-0.703
K
-400
-0.562
M
0.378
K
-0.748
K
The results from Tables 2.2 and 2.3 reveal that the structure experiences some displacement due to temperature changes, but the changes in all directions are almost identical. In Tables 2.4 and 2.5 present rotation experience by the zero and with the .25 inches gap cases. However, after -300 F degrees the two cases are identical.
22
Table 2.4: Maximum Axial Rotation Due to Temperature Changes for 0 Gap Maximum Rotational 0 GAP
Temperature °F
X (inches)
Segment
Y (inches)
Segment
Z (inches)
Segment
-100
-0.145
J
-0.233
K
-0.16
K
-200
0.249
M
-0.369
K
-0.22
K
-300
0.398
M
K
-0.245
K
-400
0.444
K
-0.226
K
M
-0.486 -0.487
Table 2.5: Maximum Axial Rotation Due to Temperature Changes for 0.25 Gap Maximum Rotational .25 GAP
Temperature °F
X (inches)
Segment
Y (inches)
Segment
Z (inches)
Segment
-100
-0.119
J
-0.252
K
-0.124
H
-200
0.243
M
-0.383
K
-0.197
K
-300
0.398
M
-0.486
K
-0.245
K
-400
0.444
M
-0.487
K
-0.226
K
The stresses due to temperature changes are shown in Tables 2.6 and 2.7 for the gap and elimination of the gap cases, respectively. Unlike the previous tables of displacement and rotation the results for the stresses are different for the zero gaps. Only at a temperature of -300 °F both stresses are the same, but for the other temperature cases the difference is evident. 23
Table 2.6: Stress Due to Temperature Changes for 0 Gap Stress 0 GAP Temperature °F
Maximum (psi)
Stress/Allowance
Segment
-100
7137
0.16
F
-200
10710
0.25
F
-300
12326
0.28
C
-400
12640
0.62
C
Table 2.7: Stress Due to Temperature Changes for Gap Case Stress .25 GAP Temperature °F
Maximum (psi)
Stress/Allowance
Segment
-100
7602
0.17
F
0.26
F
-200
11188
-300
12332
0.28
C
-400
13641
0.62
C
2.4 Discussion In Figure 2.5, it can be seen that a stress peak point at segment L. This peak value means that there are some factors increasing the stress in this location. Some factors that may contribute to the increment of the stress are supports, tees or anchors. In this particular case it was found that a .25 inches gap in two supports located at section K creates an increase of stress. The reason is that the entire 24
segment does not touch the support. With the gap correction the stress reduces, it is true not only in section L but also in almost all other segments, for comparison cases, Figure 2.7 can be referenced. With the .25 in gap the maximum static stress is 1789 psi that is a ratio of stress/allowance of .10. With the gap correction the maximum stress is only 545 psi, which is in segment I with a Stress/Allowance of .03. This number looks insignificant at this moment later when the fluid pressure is taken into account this increase in stress becomes more significant. Again the intention at this moment is to identify which are the weakest elements and its cause. Table 2.8 summaries the findings.
Table 2.8: System Maximum With Gap
With No Gap
Segment L
Segment I
Maximum Stress
1789 psi
545 psi
Stress/Allowance
0.10
0.03
2.5 Chapter Conclusions For the static stress case the gap on the supports at segment k has a negative effect in the piping, increasing the stress and displacement in almost all the segments. The more vulnerable segments of the pipe system are F, G, H and I. Correction of the gap definitely reduce the stress in almost half for the static case. None of the segments are close to the stress/allowance ratio of 1.0, and is expected that with the addition of the fluid pressures loads to the system, the stress will be affected by this gap also the natural frequencies and the pressure history. At this 25
phase of the investigation the more vulnerable segments were found. For the thermal case the stress and displacement with the gap of .25 in and zero gaps were analyzed in a range from -100 to -400 °F degrees, (the temperature were chosen to be in concordance of liquid oxygen properties). The maximum stress and displacement were found. However; the more important values correspond to the -200°F which are the operational values for liquid oxygen. For the .25 in the maximum stress and displacement was found 11,188 psi with a corresponding stress/allowance ratio of .26. For the zero gap 10,710 psi with a stress/allowance of .25 were obtained. Both maximum stresses were found at segment F. However, these values are in the acceptable range and no action is required at this phase of the investigation.
26
CHAPTER 3 WATERHAMMER INDUCED TRANSIENT FLOW ANALYSIS 3.1 Transient Flow Analysis In this chapter an investigation of transient flow due to waterhammer was performed. An extension to the specific case study was achieved. The main focus is to analyze the effect of suddenly stopping or accelerating a fluid by closing and opening a valves. The investigation was concentrated in the behavior of a control valve as a potential source for excessive pressure and the possible violent pipe vibration. Simulation of different opening and closing times of a simple valve is also treated. Therefore, the model used for the study of the valve behavior was a simple pipeline that connects two reservoirs. For application purpose the maximum waterhammer pressure was analysis for the NASA piping system and applied to the piping model discussed in previous chapters. The classical formulation of water hammer problem was applied and a numerical code has been developed. Then, the effect of closing the valve at different times was analyzed as a special parameter to control the maximum pressure. Contemplation of pressure attenuation is also performed by applying a model for frictional losses. The specific case of the NASA facilities is discussed in section 3.6. Classical equations that describe this problem can be found in the literature [42]. Develop of these equations and simplifications can be found in Chaundhry and Etal works. In this investigation the formulation includes the nonlinear terms. Contemplation of friction losses are estimated by applying a coefficient of
27
pressure drops in this investigation. In many cases this coefficient is calculated experimentally or the use of empirical correlations. Attributions of frictional losses are found in the valve and pipe, as consequence of fluid contractions and shear stress. Commonly in the case of a valve, a discharge a coefficient of pressure drop is applied to represent frictional losses. A simple valve can be idealized as a flow orifice; therefore modeling the same can be represented with a flow orifice study. Thus, the coefficient of discharge for the valve case is assumed the same as in the orifice. For the case of flow through an orifice Sisavath and etal [43] develop different models, the application of this model can be extended to pressure drop in a valve. For the case of friction losses due to shear stress the most useful model is the frictional loss, which depends on the velocity (Darcy-Weisbach formula). These models were compared with others models and the unsteady friction were classified by Bergant and etal [44] [45]. Also, they investigated the Brunone [46] models in detail and compare with results of laboratory measurements for water hammer cases with laminar and low Reynolds number in turbulent flows. Assumptions of pure liquid all the time without presence of air or bubbles is made, therefore the cavitation effect is not considered in this model. Under this assumption, the study of air valve [50] and entrapped bubbles [51] are not considered. Another assumption is that the pipe has thick wall. The method chosen to solve the system of equation is the MacCormak technique. This numerical technique discretized the resulting partial differential equation in the space and time domain [48]. For transient part the method of characteristic is applied, this is the preferred method to solve the time integration 28
[42] [49]. Limitation of this technique is when nonlinear terms are included, therefore the MacCormack predictor corrector method is used to solve for the nonlinear terms.
3.1.1 Governing Equation Under typical pipeline operating conditions of the fluid accelerates and holds suddenly, it is necessary to analyze the transient condition by solving the NavierStoke equation and the momentum equation. In order to derive the mathematical model for this problem some simplifications will be taking into account: axisymetric flow, no sterling flow and 1D model. Under these assumptions it is possible to neglect the viscous term in the momentum equation, but the friction losses is contemplated using unsteady model. Another consideration is that the fluid is essentially compressible and the pipe is considered flexible. The derivations of these equations are straightforward and can be found in standard references [42] [52].
∂p ∂V + ρa 2 =0 ∂t ∂x
(3.1)
∂V 1 dp fV V + + =0 ∂t ρ dx 2D
(3.2)
Where f is the friction factor. Bergant and etal [46], based on experimental results recommend the original Brunoe model as an effective model. Brunoe [47] model for the friction factor is:
29
f = fq +
kD ⎛ ∂V ∂V ⎞ − ⎜ ⎟ V V ⎝ ∂t ∂x ⎠
(3.3)
Where f q is the quasi-steady friction factor, which is plotted for different Reynolds number ( Re = VD ν ) and relative roughness ( e = k s D ) in the Moody Diagram [i]. Also the relative roughness can be found tabulated for different pipe diameters and materials. The coefficient k is called the Brunoe’s friction coefficient. It can be predicted analytically using Vardy’s [53] shear decay coefficient C * :
k=
C* 2
(3.4)
The Vardy’s [54] shear decay coefficient C * is given by:
0.00476
7.41 Re
log(14.3 / Re 0.05 )
Laminar Flow
(3.5a)
Turbulent flow
(3.5b)
3.1.2 Boundary conditions As mention earlier the objective of the investigation is to analyze the effect of opening and closing a valve located in a pipeline, therefore the following boundary condition assumption is made; Independent of flow oscillation is 30
assumed that the reservoir of the pipe line will maintain constant level, thus maintaining constant inlet hydraulic pressure Pi The boundary condition is expressed mathematically as:
pn
inlet
= ρgH = Pi
(3.6)
Where the subscript n indicate time at the instant n . Therefore, the boundary condition for the velocity at the valve, under steady state conditions, is known and also the volumetric flow rate. Using the discharge coefficient, the volumetric flow rate is:
Q0 Where the subscript
0
valve
= C d A0 2 p 0 / ρ
(3.7)
refers to steady state conditions, C d is the discharge
coefficient and A0 is the area of open valve. An schematic of the model with it corresponding boundary conditions is shown in Figure 3.1.
Figure 3.1: Transient flow model The volumetric flow as a function of valve steam depends on the type of valve and is specified by the manufacturer. Commonly the volumetric flow plotted as a percent of the maximum volumetric flow at the acting pressure of the system [54] [55] Figure 5.2 shows a typically close-open curve [56] for a valve.
31
Figure 3.2: Typically close-open curve [55]
Assuming that last relationship is valid for transient conditions, the velocity at the valve at time n is:
Vn
valve
= C d F (l )
AT A0
2
pn
ρ
n
= Vv
(3.8)
Where, F (l ) is the percent of caudal obtained from Figure 3.2 and l the stem position. AT the area of valve totally open and A0 the area of the valve partially open, according with the stem position l . As an initial condition a constant velocity profile and pressure in the pipe is used:
V 0 = V initial
p0 = Pinitial
(3.9)
32
3.1.3 Numeric Discretization The system of equations to solve is:
∂p ∂V + ρa 2 =0 ∂t ∂x
(3.10)
∂V 1 dp fV V + + =0 ∂t ρ dx 2D
(3.11)
V ( x,0) = Vinitial
(3.12)
p ( x,0) = Pinitial
(3.13)
p(inlet , t ) = P
(3.14)
V (valve, t ) = V
(3.15)
Then, for the spatial and temporal discretization the MacCormak is used. The MacCormak method is a two step predictor corrector finite different. The MacCormak method can solve linear partial differential equations (PDE), nonlinear PDE and system of PDE [48]. In the MacCormak method, the predicted provisional
values
are
obtained
using
first
order
forward
difference
approximations:
pin +1 = pin − ρa 2
Vi n +1 = Vi n −
∆t n (Vi +1 − Vi n ) ∆x
∆t ∆t n ( Vi p in+1 − p in ) − Vi n f ρ∆x 2D
(3.16)
(3.17)
In the second final step, a first order backward difference approximations based on the provisional values is used.
33
p in +1 =
Vi n +1 =
)
(
1⎡ n ∆t n +1 ⎤ p i + p in +1 − ρa 2 Vi − Vi n−1+1 ⎥ ⎢ 2⎣ ∆x ⎦
(
(3.18)
)
⎤ ∆t 1⎡ n ∆t n Vi pin +1 − pin−+11 ⎥ − Vi n f Vi + Vi n +1 − ⎢ 2⎣ ρ∆x 2D ⎦ (3.19)
The MacCormak method is conditionally stable and convergent. The stationary condition given by the Courant number less than one. For this system of equation it is necessary to satisfy the courant conditions in the two equations.
Cn1 = ρa 2
∆t ∆t ≤ 1 and Cn1 = ≤1 ρ∆x ∆x
(3.20)
Here, when ∆x is imposed, the ∆t can be found from the last equation.
3.1.4 Transient Investigation Results and Discussion To integrate the equations a Fortran code has been developed. As a test case, the instantaneously totally close behavior for the valve was performed to validate convergence. The nodal point selected for the spatial discretizacion was chosen as 500 for all the cases. The ∆t was chosen as: 0.03, 0.04 and 0.05 second. The frictional term for the test case was neglected. The pressure distributions for different times in the adjacent point to the valve are shown in Figure 3.3 as a result. Pressure fluctuation in each step corner of Figure 3.3 are not smooth, this is attributed to the noise effect. Figure 3.4 shows the details of the numerical noise effect of Figure 3.3 after 80 seconds.
34
Figure 3.3: Wave pressure for different dt
Figure 3.4: Detail of numerical noise effect 35
Based on previous result, ∆t and ∆x was selected. After the selection of step and time, a simulation considering the friction factor is performed. Figure 3.5 shows the results of the simulation for the pressure distribution adjacent to the point of the valve. The simulation is performed for both with and without friction case. Friction effect can be appreciated as a decrement of pressure along time. The effect of pressure losses can be seen when comparing with previous case.
Figure 3.5: Effect of friction loss
3.1.5 Valve Programming of Close-Open A study of time of closing was performed with the same parameter that in the test case. The valve studied is of a linear type. The behavior of the pressure for different time of closing is shown in Figure 5.6. 36
Figure 3.6: Effect of time of close Figure 3.6 shows that the maximum pressure as a function of valve closing time, the faster the valve is close the higher and the abrupt the change on pressure is. The fluid bulk modulus of elasticity is the other parameter that may contribute to the maximum pressure. For different Bulk modulus of elasticity the maximum pressure as a function of valve closing time is plotted in Figure 3.7.
37
Figure 3.7: Effect of bulk Elasticity Modulus
3.1.6 Case study To be in concordance with the NASA facilities, the material of the pipeline chosen was austenitic stainless steal and the fluid content is liquid oxygen. The applicable properties are shown in Table 3.1. To have liquid oxygen at ambient temperature a pressure of 2.5 GPa was assumed.
Table 3.1: E2 Facilities Technical Data of Pipe and Content Pipeline Properties
For Oxygen
Density
1137.64 kg/m3
1000 kg/m3
Outer Diameter (m)
0.1682
N/A
Inner Diameter (m)
0.1069
0.1069
Thickness (m)
.0306
N/A
Bulk Modulus of elasticity
1.93E15
1000
Length (m)
1
1
Operating temperature
30° C
30° C
38
Figure 3.8: Effect of time of close in the maximum pressure For this specific problem the spatial grid was made using ∆x = 0.0002 and the time grid with a ∆t = 0.0000002. This grid was chosen to minimize numerical fluctuation according with the previous analysis. Figure 3.8 show the maximum pressure as a function of initial velocity for different closing time. For all the cases it can be seen a linear behavior between maximum pressure and the initial velocity. In this Figure is evident that the faster the close time higher is the pressure. Also, the difference between closing the valve at .05 and .1 second is minimal thus, .01 second may be taken as the critical value. As expected the fluid will tend to increase it pressure at higher velocities.
39
Figure 3.9: Effect of initial velocity in the wave pressure Using a valve closing time of .01 seconds for different initial velocity, the pressure behavior is calculated in time, as shown in Figure 3.9. It can be seen a higher pressure in the first millisecond, the same is attenuated as time pass due to pipe friction.
3.2 Specific Applications As shown in Figure 3.9 transients flows has a peak maximum pressure value, thus if failure due to an overpressure could occur it will happen at this value. For the specific application our interest is to determinate flow limit due to a waterhammer maximum pressure. Therefore, a different approach will be made for this section. Analyzing the energy conservation for the case rapid valve closure or open. The diminution of Kinetic energy will transform in a compression work for the
40
fluid that will cause the fluid to full fill the pipe. As a result an over pressure is created. If the valve is rapidly open a depression or a negative transient pressure is obtained.
c
Flow Velocity Pressure
Moving wave front V V +dV P
P+d P
Figure 3.10: Wave in close duct If the flow velocity at the downstream end is changed from V to V + dV, thereby changing the pressure from P to P + dP. This change in pressure will produce a pressure wave that will propagate in the upstream direction. The pressure on the upstream side of this wave is p, whereas the pressure on the downstream side of this wave is p + dp. It is possible to transform the unsteady-flow situation to a steady-flow situation by letting the velocity reference system move with the pressure wave. Then creating a control volume at the interrupted area the momentum equation is solved as following: x2
d ∑ F = dt ∫ ρ wVAc dx + ( ρ w AcV 2 ) out − ( ρ w AcV 2 ) in x1
(3.21)
First because the flow is steady, the first term on the right-hand side of the momentum equation is zero. Referring to equation 3.21, and introducing the force and velocity into equation:
pAc − ( p + dp) Ac = (V + c + dV )( ρ w + dρ w )(V + c + dV ) Ac − (V + c) ρ w (V + c) Ac
(3.22) 41
By simplifying and discarding terms of higher order, this equation becomes
− dp = 2 ρ wV dV + 2 ρ w dVc + dρ w (V 2 + 2Vc + c 2 )
(3.23)
The general form of the equation for conservation of mass for one-dimension flows may be written as x2
0=
d ρ w Ac dx + ( ρ wVAc ) out − ( ρ wVAc ) in dt x∫1
(3.24)
Having steady flow the first term on the right hand side of equation 3.24 is zero and introducing the velocities the equation becomes 0 = ( ρ w + dρ w )(V + c + dV ) Ac − ρ w (V + c) Ac
(3.25)
Simplifying this equation, dρ w =
− ρ w dV V +c
(3.26)
− ρ w dV c
(3.27)
Because the fluid velocity v<< c dρ w =
Now, by substituting equation 3.27 into equation 3.23, discarding terms of higher order, and simplifying dP = − ρ w * dV * c
(3.28)
This equation is commonly named Joukowosky equation it predicts a pressure due to suddenly change in flow of a fluid. For the sound velocity c if the conduits of the walls are assumed to be slightly deformable instead of rigid, then the speed of sound would take the following form.
42
K
ρ
C= 1+
(3.29)
KD eE
3.2.1 Results and Discussion for Waterhammer Pressure Analysis The proposed facility maximum flow is 275 lbm/sec, because the structural analysis intends to study the limitation of the piping system this flow will be considered as the maximum theoretical flow. The inner pipe diameter recalling from previous chapter is 4.209 in. The total line length from the tank bottom to the isolation valve is 1131.7 in. Valve location is shown in Figure 3.11.
Tank Location
Isolation valve
Figure 3.11: Isolation valve and tank location
The flow velocity at a mass flow rate of 275 lbm/sec is V =
275 275 in = = 456 ρ w Ac .0433 × 13.914 sec
(3.30)
43
The theoretical maximum pressure surge for this flow velocity using a specific speed of sound C of 39,015 ft/s is:
Pmax =
ρw × c ×V g
=
.0433 × 39015 × 456 = 1994 psi 386
(3.31)
The pressures plotted in Table 3.2 are the pressures at the run valve, which is at steady state flow. At steady state the local pressure is the tank pressure minus flow friction losses. At a valve totally closure, flow is stopped therefore friction losses becomes zero. Since the friction losses are zero it can be assumed that the local pressure will be the peak surge pressure at the valve plus the tank pressure. The tank pressure is 8000 psia.
Table 3.2: Transient Pressures Velocity ft/s
Pressure (mpa)
Pressure (psia)
Pressure + Tank Pressure (psia)
16.40
5.937
861
8861
32.80
11.870
1722
9722
38.00
14.250
2067
10067
49.21
17.810
2583
10583
65.61
23.750
3445
11445
The pressure plotted in Table 3.2 are applied to the structural model created in Algor, this pressure do not consider as fluctuating over time, therefore considered as a constant pressure simulating failure by peak transient pressure. Algor has the capability of simulate loadings to the structure and study the effect along structural elements; it doesn’t have the capability of create a pressure wave running in the fluid.
44
Flow 16.4 ft/s 1
0.9
0.8
Stress/allowance
0.7 Segment Segment Segment Segment Segment Segment Segment Segment Segment
0.6
0.5
0.4
0.3
0.2
0.1
0 Segments
Figure 3.12: Length vs. stress/allowance no failure is predicted at this flow
With a 16.4 ft/s and a sudenly close no failure is predicted as shown in Figure 3.12, but it clearly can be seen that the stress per segment is close to the limit therefore, this can be considered a caution situation. Flow 32.8 ft/s 1.2
1 Segment A Stress/Allowance
Segment B 0.8
Segment C Segment D Segment E
0.6
Segment F Segment G
0.4
Segment H Segment I 0.2
0 Segments
Figure 3.13: Possible failure is presented at segment F and E.
45
A B C D E F G H I
With a flow of 32.4 ft/s and instantaneous closure of the isolation valve, possible failure for segments A, B, D, and F is predicted. As shown in Figure 3.13 most of the segments also are close to their limits. This may be considered as the maximum allowed flow in the case of an instantaneous closure of the isolation valve.
Flow 38 ft/s 1.2
1 Segment A stress/allowance
Segment B 0.8
Segment C Segment D
0.6
Segment E Segment F Segment G
0.4
Segment H Segment I 0.2
0 Segment
Figure 3.14: Possible failures for segments A, B, C, D, and F
With a flow rate of 38 ft/s which is the maximum flow rate proposed by the facility it is clear as shown in Figure 3.14, that with a suddenly close of the test valve almost all segments are on they limit and most of them are over their limits. No flow over this value is recommended based on an emergency situation, some cushion devices should be added for prevention.
46
Flow 49.21 ft/s 1.2
1 Segment A Sterss/Allowance
Segment B 0.8
Segment C Segment D
0.6
Segment E Segment F Segment G
0.4
Segment H Segment I 0.2
0 Segment
Figure 3.15: Failures for almost all segments
For this case failures of almost all segments is evident as shown in Figure 3.15, this flow velocity should be avoided and there is no reason for study higher flow values.
3.3 Conclusions A study of the fluid transient in a simple pipeline is done. For that reason, a Fortran code is developed to integrate the governing partial differential equation using MacCormak method. The behavior analyzed for different test cases the incremental time is performed based on these results. After that, using manufacturer information, the time of close and open is also analyzed as a parameter to control the crest of wave pressure. The result obtained indicates that with adequately time of operation of the wave crest no reach the pressure of failure. Maximum pressures caused by rapid closure of isolation valve are obtained for various flow using standard book equations. The transient flow pressure wave was 47
applied to the piping system before the isolation valve. Failure prediction is obtained for several segments. It is found that for the case of a valve rapid closure, possible failure will occur at a flow rate of 38 ft/s which is the maximum flow the facility is planning to run. Some pressures reducer is recommended before the test valve in order to reduce the impact of the traveling wave.
48
CHAPTER 4 RESONANT FREQUENCY ANALYSIS 4.1 Resonant Analysis Flow induced vibration is to a large extend, an operational problem that has on worst cases direct impact on public safety. Vibration in piping systems consists of the transfer of momentum and forces between piping and the contained liquid during flow. Excitation mechanisms may arise by rapid changes in flow and pressure or may be initiated by mechanical action of the piping. The resulting loads impart on the piping are transferred to the support mechanisms such as hangers, thrust blocks, etc. Special attention has to be taken when this phenomena is present. Free vibration occurs when a system is displaced from its static position and left free to oscillate. Under free vibration the system oscillates at its natural frequencies. The natural frequencies are dynamic characteristics of the system specified by its stiffness and inertia properties. Natural frequencies are calculated with modal analysis. Forced vibrations are classified into periodic and nonperiodic. In a periodic vibration, the response repeats itself at a regular time interval, called period T. Harmonic excitation is a sub-class of periodic vibration and is referred hereafter as an analytical approach for the present investigation. The resonance effect can be described as a non stable vibration. Resonance will take effect when the exiting frequency is near 1.4 the natural frequency [15][16]. Figure 4.1 shows the segments that the excitation frequencies need to be calculated
49
Figure 4.1: NASA diagram configuration from previous investigation[1]
4.1.1 Governing Equations and boundary conditions With the objective to derive the mathematical model for this problem some simplifications was taking into account: the piping system is idealized as a group of 1D beams and no damping is considered. Under these assumptions it is possible to consider the system as an undamped single degree of freedom (SDOF) system that is subjected to a harmonic force P (t) with amplitude Po and circular frequency, then the equation of motion is given by [14] ..
M Y + ky = P0 sin ωt
(4.1)
As one of boundary conditions, the beams is fixed at both ends. Solving for the displacement response, maximum displacement and natural frequency is 50
straight forward and the development of these set of equations can be found in text books [5] [14] The displacement response of the system is given by [5]: y (t ) = A cos ωt + B sin ωt +
P0 1 sin ωt k 1− r2
wf
r=
(4.2)
(4.3)
w
Then the solution for maximum displacement for an un-phased harmonic analysis is
δ dyn =
Po 1 k 1− r2
(4.4)
And the natural frequencies are: wf =
n 2π 2 L2
EI m
(4.5)
4.2 Results and Discussions The natural frequencies for the first 3 mode shapes were calculated per segment and compared with the excitation frequency from the fluid. Also the first 2 mode shapes were determined with the consideration of the pipeline as a whole. The natural frequencies and their corresponding mode shape were determinate using ALGOR and analytical equations. The excitation frequencies from the fluid obtained from previous research [1] did not cover all the segments. In order to determine resonance, therefore, the natural frequencies obtained in this work were fluid excitation frequencies for the entire segments. Recalling from previous discussion if the excitation frequencies are equal or between 1.4 the natural frequencies, resonance will occur. After calculating the 51
natural frequencies per segment, the natural frequencies were compared with the excitation frequencies from the fluid, Table 4.1 illustrates the excitation frequencies from the fluid obtained in [1]. The shaded cells represent possible flows rate that may cause resonance for the complete system. To determinate if resonance is present for individual segments, the natural frequencies were calculated for the first three mode shapes per segment, Figure 4.2 illustrates the beam discretization approach. Figures 4.3 to 4.5 show the relationship of fluid excitation frequency and natural frequencies for each segment. Note that any segment at the frequency ratio between the value of 1.4 or .5 is considered in resonance, two lines are included in each graphics representing the upper and lower limit. The fluid excitation frequencies were calculated at flow rate of 40, 113 and 275 lbm/sec .
11
9
7
1 2
6
10
12
8
3
5 13
4
14 15
17 16
Figure 4.2: Discretization of the system
52
Table 4.1: Fluid Excitation Frequencies by Others [1] VORTEX SHEDDING CALCULATIONS (Hz) Segment
fs_(113lbm/s)
Fs_(275lbm/s)
Fs_(40(lbm/s)
1
597.379005
1453.70771
2.11401778
2
74.6723757
181.713464
1.05700889
3
22.1251483
53.8410262
0.70467259
4
9.33404696
22.7141829
0.52850445
5
4.77903204
11.6296617
0.42280356
6
2.76564354
6.73012828
0.3523363
7
1.74162975
4.23821489
0.30200254
8
1.16675587
2.83927287
0.26425222
9
3.48451039
8.47947379
0.99881594
Resonanse for Mode Shape 1 2
1.8
1.6
1.4
Ratios
1.2
1
fs 113 fs 275 fs 40
0.8
0.6
0.4
0.2
0 0
1
2
3
4
5
6
7
8
9
10
Segments
Figure 4.3: Resonances per segment at different flow rates for the mode shape 1
53
Resonanse for Mode Shape 2 2
1.8
1.6
1.4
Ratios
1.2
1
fs 113 fs 275 fs 40
0.8
0.6
0.4
0.2
0 0
1
2
3
4
5
6
7
8
9
10
Segments
Figure 4.4: Possible resonances per segment at different flow rates for mode shape 2
Resonanse for Mode Shape 3 2
1.8
1.6
1.4
Ratios
1.2 fs 113 fs 275 fs 40
1
0.8
0.6
0.4
0.2
0 0
1
2
3
4
5
6
7
8
9
10
Segments
Figure 4.5: Possible resonances per segment at different flow rates for mode shape 3
54
Figure 4.6: Natural frequency mode shape 1 of a complete systems
Figure 4.7: Vibration modes shape 2 for the complete system
55
4.3 Conclusions The natural frequencies considering the system as a whole were obtained using ALGOR and compared with the excitation frequency obtained from the fluid at different flow rates. The results yields that the first natural frequency for the whole system will occurs at 2.07 Hz, and the second at a frequency of 5.65 Hz with its corresponding mode shapes. It is very clear in Table 4.1 that almost the entire feed line might suffer from the resonance effect as calculated by finite element analysis. The first mode is in resonance at a flow rate of 40 lbm/s, whereas, resonance for the second mode may occur at a flow rate of 275 lbm/s. Figures 4.6 and 4.7 show the displacement results for the first and second mode shapes. Notices that for the first mode shape the greatest displacement and stresses is near the end of the piping system whereas, for the second mode shape the segments near the center might be in resonance. It is expected that for the third mode the elements near the tank could have the greatest displacement, of course it may take place at higher frequencies. With the analytical method approach the pipeline was discretized in straight segments between supports as shown in Figure 4.1, the segments were idealized as simple supported beams which are more appropriate for piping [16,18]. In addition to explore more in deep the possibility of resonance, the natural frequencies were calculated for the first three mode shapes. Examining possible resonance was studied for the first mode shape in segments three, four and six. For the second mode shape resonance was studied only in segment number two and for the third mode shape resonance was predicted for segment one and two as shown in Figures 4.3, 4.4 and 4.5.
56
The results in this investigation reveal higher natural frequencies for all segments than the anticipated in previous research [1]. The discretization made in previous investigation was along straight pipe, which making the segments longer and more susceptible to vibration than analyzing between supports. Supports will tend to increase the stiffness of the segment. Taking into account only the straight segments will underestimate the additional stiffness that comes from these segments.
57
CHAPTER 5 TURBULENCE INDUCED VIBRATION 5.1 Turbulence induce vibration When the fluid velocity exceeds any but the smallest values characteristic of “seepage” flows, eddies will form even if the surface of the flow channel is perfectly smooth. The flow is said to be turbulent after it has achieve a specific Reynolds number. Turbulence flow in most application is desired; a typical application is to increase the efficiency of a heat exchanger. The force generated by the turbulence flow has the characteristic of being random. To study this type of phenomena probabilistic method has to be applied. This will eliminate any attempt for a detailed time history response. The approach to solve this problem is by calculating the root mean square values of the responses. With this calculation is possible to predict potential damage to the piping system. At this days is still not feasible to determinate the turbulent forcing function by numerical techniques. To study this phenomena is required a combination of experimental data an analytical techniques. The experimental data are used to determinate fluid parameters and analytical approach for the solid behavior. In simple word the experimental data obtained from the fluid is applied to the structure to predict it behavior under know conditions. The most widely used method to solve this type of problem is the acceptance integral method first formulated by Powell [17]. Chen and Wanbsganss [18] followed this method to estimate the parallel flow induce vibration of nuclear fuel roads and Chyu and Au-Yang [19] applied this method to estimate the response of panels exited by boundary layer turbulence. Au-Yang [20] applied this method to 58
estimate the response of reactor internal component excited by the coolant flow and again to cross-flow-induced vibration of a multiple span tube [21]. As previously mention the ultimate goal is to determinate the vibration root mean square amplitudes. To determinate the root-mean square (rms) response the following equation formulated by Powell [17] is often used. → →
AG p ( fα )ψ α ( x ) J αα ( fα ) ⎛→⎞ y ⎜ x⎟ =∑ 2 3 64π 3mα fα ζ α ⎝ ⎠ α 2
2
(5.1)
→
where J αα is the joint acceptance. The joint acceptance is a measurement of the matching in space between the forcing function and the structural mode shape. The same is tabulated in flow induce vibration text books [16] and is included in this work for reference, see Figure 5.3, The term G p ( fα ) is the structural
fluctuating power spectral density (PSD) due to boundary layer type of turbulence. Equation 5.1 is general and applicable to one dimensional as well as two dimensional structures in either; parallel flow or cross-flow. This equation is derived under many simplifying assumptions, of which the most import ants are that the cross modal contribution to the response is negligible, and the turbulence is homogeneous, isotropic and stationary. To characterize the turbulent forcing function three parameters are required: The convective velocity Uc, which determines the phase relationship of the forcing function at two different points on the surface of the structure; the correlation length λ , which determines the degree of coherence of the forcing function at two different points on the surface of the structure; and finally the power spectral density function, Gp, which determines the energy distribution as a function of the frequency of the forcing function. These three fluid parameters are
59
obtained by model testing and scaling. In this thesis existing data from the literature will be applied to turbulence induced vibration estimates. Based on data obtained from turbulent flows, Chen and Wambsganss [18] derived the following empirical equation for the convective velocity as a function of frequency: − 2. 2 ( Uc = 0.6 + 0.4e V
ωδ * V
)
(5.2)
Bull [22] suggested a slightly different equation: −.89ωδ Uc = .59 + .3e V V
Where
δ
*
*
(5.3)
is the displacement boundary layer thickness for boundary layer flow
or in our case the “hydraulic radius” in confined internal flow. Both equations show that except at very low frequencies, the convective velocity is fairly independent of the frequency, being equal to approximate 0.6 times the free stream velocity . In confined flow channels in which very high turbulence is generated or by flow in 90 degree channels, Au-Yang and Jordan [23], Au-Yang [24] found, in two separate experiments, that the convective velocity is about the same as the mean free stream velocity. Uc ≈V
60
Figure 5.1 Comparison of convective velocity predicted by Chen and Wambsganss and Bull [5]
For this investigation the flow is internal in a pipe, therefore the boundary layer can’t grow indefinitely. In small pipes and narrow flow channels, the boundary layer will fill up the entire cross section of the flow channel. In that case the displacement boundary layer thickness which is a fluid mechanical parameter is the hydraulic radius of the flow channel.
δ* =
DH = RH 2
(5.4)
The most important fluid mechanic parameter that characterizes the turbulence forcing function is the power spectral density (PSD). And can be obtained with the following empirical equation, which was derived based on data from a scale model test Au-Yang and Jordan [23]. ⎡Φ ( w) ⎤ G p ( f ) = 2πρ 2V 3δ * ⎢ 2PP 3 * ⎥ ⎣ρ V δ ⎦
(5.5)
61
In this equation the displacement boundary layer thickness δ * is the hydraulic radius. The quantity in [] is plot in the ordinate of Figure 5.2, the data of this Figure is unreliable in the low-frequency region, market “effective range.” For low frequency, turbulent flow without cavitations the fallowing equation applies [24] Gp ( f ) = .155e − 3 F , 0 < F < 1 2 3 ρ V RH
(5.6)
=.027e-1.26F, 1 ≤ F ≤ 5 where F = fRH/V
(5.7)
For turbulent flow with light cavitations
− x −4 GP ( f ) ) ,1.0} = min{20 F − 2 ( 2 3 ρ V RH RH
(5.8)
where x is the absolute value of the distance from the cavitation source such as an elbow or a valve.
62
Figure 5.2: Boundary layer type of turbulence power spectral density [5]
Figure 5.3: Longitudinal joint acceptances [5]
63
5.2 Results and Discussions For the turbulence induce vibration, the analysis was performed at the maximum flow rate proposed by NASA. The root-mean square (rms) response was obtained at this flow. The highest flow was chosen due to the reason that has the higher energy and representing the worst case. The results were limited to the segments which were fluid information was available. For simplicity the analysis was assumed without cavitation although cavitation may be experienced due to the nature of the system. It was found that segments 2, 3 and 4 experience the most significant vibration. The higher displacement may be attributed at the lower natural frequencies that characterize these segments. The specific segments have the lower frequencies because they are the longest comparing with the others. The stiffness of these segments can be increased if additional supports are added thus increasing the natural frequencies. Table 5.1: Uc, Frequency Parameters and Joint Acceptances
1
Natural Frequency (Hz) 61.181
Uc (in/s) Convective Velocity 417.130
2
19.921
3
4fL1 /Uc
Jmm
J´nm
ωδ*/V
59.255
0.010
1.000
0.193
493.204
28.597
0.010
1.000
0.063
46.784
432.455
49.980
0.010
1.000
0.147
4
14.565
513.828
23.471
0.010
1.000
0.046
5
717.158
393.950
214.811
0.001
1.000
2.260
6
8.058
544.733
16.467
0.010
1.000
0.025
7
78.79
406.410 1
69.018
0.010
1.000
0.248
8
195.507
394.154
112.100
0.001
1.000
0.616
9a
5660.833
393.950
603.516
0.001
1.000
17.842
9b
90.951
421.639
58.155
0.010
1.000
0.177
Segment
64
Table 5.2: PSD and RMS Responses
1
Natural Frequency (Hz) 61.181
2
19.921
4.000E-05
5.635E-04
0.211
3
46.784
4.000E-05
5.635E-04
0.211
4
14.565
4.000E-05
5.635E-04
0.211
5
717.158
3.500E-06
4.931E-05
0.020
6
8.058
4.000E-05
5.635E-04
0.211
7
78.791
2.000E-05
2.818E-04
0.149
8
195.507
2.000E-05
2.818E-04
0.047
9a
5660.833
2.000E-08
2.818E-07
0.001
9b
90.951
1.500E-05
1.304E-04
0.121
Segment
Normalized PSD
Gp (psi^2 / Hz)
Yrms (in)
2.000E-05
2.818E-04
0.149
65
CHAPTER 6 FE Model of Fluid-Structure Interaction 6.1 Fluid Structure Interaction Normally when it is desired to obtain the fluid velocity in a pipe, equations are applied with the assumption of no wall deformation. If the walls deform, the deformation will affect fluid thus creating a fluid structure interaction. This chapter concentrates on applying iterative method to develop a fluid structure interaction model. The solutions presented takes into account the interaction of the solid. Several plots presented were compared in the percent difference if the interaction between the solid and fluid is not taking into account. Considering the behavior of the fluid structure interaction, the fluid will applied a pressure to the pipe walls, and if the pressure is strong enough to cause pipe deformation, this deformation will decrease the velocity along the pipe. The pressure drop caused by a velocity decrease will change the pipe diameter again. This phenomenon is what we are calling fluid structure interaction. To solve fluid structure interaction problem, a subroutine was created using ANSYS. To achieve a solution, an iterative subroutine was created. This subroutine combines the solution of the fluid and applied the results to the structure until a criterion of convergence is accomplished. Two models were created one a 2D channel we no obstruction that will serve as the base model. The other will be the same channel with an obstruction; this obstruction can easily by a valve half open. The obstruction will locally increase the pressure, creating a large deformation compare to the base model. Different velocities were applied to compare both cases. 66
6.2 FEM Analysis The procedure to solve the couple problem is essentially obtaining the solution first form the dynamic fluid analysis, except that this solution are going to be saved in what is call a physic environment. Then the program calls the solid physic environment which at this point is only a solid with boundary conditions. Then the pressure solution obtained from the fluid physic is applied to the internal walls of the solid pipe. Finally a static analysis for the solid is performed, obtaining deformations and stresses results.
6.3 Finite Element Models Two models were created; a free flowing channel and a channel with 50% obstruction in its cross-sectional area shown in Figures 6.2 and 6.3. In order to compare the results, both geometries created were essentially the same for the analyses. Soft materials were chosen, thus, it is expected that the pipe will deform significantly enough to affect the fluid velocities. Tables 6.1 and 6.2 summarizes the dimensions used for the straight pipe.
67
Figure 6.1 Fluid structure interaction loop flow chart
Structure
Fluid
Structure
Figure 6.2: Geometry of free flowing channel
68
Table 6.1: Dimensions Free Flowing Channel Dimensions
Thickness
0.003175 m
Length
.5 m
Fluid cross section
.05 m Structure
Fluid
Obstruction
Structure
Figure 6.3: Geometry of channel with obstruction
Table 6.2: Dimensions Channel with Obstruction Dimensions
Thickness (m)
0.003175
Length (m)
.5 m
Fluid cross section (m)
.05 m
Obstruction Length
Half of the fluid cross section
Obstruction location
.25 m
69
6.3.1 Material Properties The structural analyses require the definition of Young’s modulus of elasticity and Poisson’s ratio. The modal analysis in addition to the previous properties also required to define the density.
For the Computational Fluid Dynamic CFD
analysis it is necessary to input density and viscosity of the working fluid. Material properties used in both analyses are resumed in Table 6.3.
Table 6.3: Material Properties Properties
Fluid density (kg/m3)
1000
Fluid Viscosity (kg-s/m)
.00046
Young's modulus for rubber (Pa)
2.82E+009
Poisson ratio rubber
0.49967
Mooney-Rivlin Hyperelastic constant 1
2.93E+005
Mooney-Rivlin Hyperelastic constant 2
1.77E+005
6.3.2 Element Types The sequential coupled field analysis requires a combination of solid and fluid elements. For the structural analysis the element chosen was HYPER 74. This element has the ability to accommodate nonlinear behavior being ideal to obtain stresses and pressures results. This element is also compatible with some fluid elements. For the fluid environment the element chosen was FLUID 141. This element is ideal for pressure and velocity solutions.
70
6.3.3 Mesh The procedure to mesh the areas in both types of analyses was practically the same and it yielded very similar meshes. It was not desired to free mesh the created volumes causing a mesh that would degenerate the geometry or that would be inconsistent, uneven or inconsistent. To prevent this from happening the mesh was done in a constant area basis except in the case with the obstruction were the mesh was finer near this area. This resulted in a regular mesh that was even and very similar between the two analyses.
6.3.4 Boundary Conditions Since both models involve a modal analysis, the displacement boundary conditions on the two were the same. This is a very important requirement since the boundary conditions affect greatly the results and without this similarity a comparison between them would not be effective. The displacement boundary conditions were placed at both ends of the channel in order to simulate supports acting on the outside of the channel. The channel was considered fixed at both ends. The other boundary condition applied was related to the fluid and the same were applied to the channel internal surface area. The velocities of the fluid near the internal walls of the channel were set to 0 and the pressure at the end of the channel was set to 0. The pressure will ensure flow in the desired direction.
6.4 Results The results presented are focus in to demonstrate that for a fluid flowing in a highly deformable environment the dynamic deformation of the pipe will have an
71
effect in the behavior of the fluid. Therefore, the result presented in this section is a comparison between a non fluid structure interaction and a fluid structure iteration solution. To present this, a graphic was created showing average percent difference of the velocity results obtained for each node of the channel. The procedure to calculate the percent difference was the fallowing. First a nodal solution was run to solve for the velocity this solution was saved and the same was used as the non fluid structure interaction solution. Then a subroutine was created using ANSYS. In the subroutine an iterative method was created were the pressure solution from the fluid was applied to the walls of the channel and the deformation created from the pressure was used to solve the fluid until velocity values converge. Then each velocity solution per node was compared with the non fluid structure interaction and for each of them the percent difference was calculated. Finally an average of the all the percent difference was calculated. This procedure was applied for both cases the channel with the obstruction and the free flowing one for flows from .01 m/s to .1 m/s. Figure 6.4 resumes the results. To physically see how the results change per iteration a sequence of pictures examples for the specific case of fluid flowing at .04 mps are presented, see Figures 6.5 to 6.10. Note how the maximum velocities change from the first iteration and the second one. This is expected due to the higher deformation will occur in this iteration.
72
25%
20%
P e rce n t E rro r
15%
10%
Obstruction No Obstruction 5%
0% 0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
Velocity M/S
Figure 6.4: Average percent difference at different flows
Figure 6.5: Velocities profile at the first iteration
73
Figure 6.6: Velocities profile at the second iteration
Figure 6.7: Velocities profile at the third iteration
74
Figure 6.8: Velocities profile at the fourth iteration
Figure 6.9: Velocities profile at the fifth iteration
75
Figure 6.10: Velocities profile at the sixth iteration
6.5 Conclusions Two models were created; a free flowing channel and a channel with 50% obstruction in its cross-sectional area. For both cases, a non-fluid structure interaction solution was compared against a fluid-structure interaction solution. Various flow cases were study, after comparing the non interaction with the interaction solution, it was notice a percent difference up to 25% in the fluid velocity. This may be attributed to the fact that when the fluid applies pressure to the rubber channel the rubber channel deforms, this deformation decrease the velocity at which the fluid is traveling. Because the axial velocity has decrease now the pressure applied by the fluid also decrease, at this point the rubber channel tries to gets is steady state form. Now the cross-sectional area has decrease again and by consequence the fluid increase again its velocity and the 76
pressure applied to the wall of the channel also increase. This phenomenon continues until a convergence is achieved. For the fluid structure interaction investigation, two models were created; a straight pipe and a straight pipe with 50% obstruction in its cross-sectional area. For both cases, a non-fluid structure interaction solution was compared against a fluid-structure interaction solution. Various flow cases were studied, after comparing the non interaction with the interaction solutions; it was noticed that a percent difference up to 25% in the fluid velocity. This is attributed to the fact that when the fluid applies pressure to the pipe, it deforms. This deformation decrease the velocity at which the fluid is traveling. As the axial velocity decreased, the pressure applied by the fluid also decrease. Since the cross-sectional area decreases again, by consequence the fluid increase its velocity again and the pressure applied to the wall of the pipe also increase. This phenomenon continues until a convergence is achieved. Comparing both models, the fluid structure interaction in pipe with 50% obstruction is more significant than the pipe with no obstruction. The main reason is that the obstruction creates a local increase of pressure leading to a deformation in this specific area.
77
CHAPTER 7 SUMMARY AND CONCLUSIONS 7.1 Summary The investigations of static and dynamic analysis of a piping system at NASA are presented in this thesis. For general purpose application; transient and fluidstructure interaction research were performed. The research conducted for the NASA facilities were; structural, thermal, water hammer, resonance and turbulence induced vibration analysis. Three finite elements models for the pipe system and segments at NASA facilities were developed: a structural finite element analysis model with multi-support system for frequency analysis, fluid-structure interaction (FSI) finite element model and transient flow model for waterhammer induced vibration analysis in a fluid filled pipe. The natural frequencies, static stress and the limitations of the pipeline system were determined. A simple chart characterizing the relation between stress and location along the length of the pipeline was developed for all segments. In the warterhammer case, the limit maximum flow rates were determinate based on the rate of a rapid closure of the isolation valve. A study of the fluid transient in a simple pipeline was performed. The behaviors of different test cases analyzed were completed based on these results. Subsequently, the time of valve close and open was analyzed as a parameter to control the crest of wave pressure. A fluid-structure interaction FE model was developed and compared with a model without considering fluid-structure interaction effects. The results show notable differences in the velocities profile and deformation. For comparison purpose, the percent difference of velocities and deformation were illustrated. 78
7.2 Conclusions In the structural analysis the gap on the supports at segment k has a negative effect on the piping system, the gap increases the stress and displacement in almost all the segments. The more vulnerable segments of the pipe system are found to be segments F, G, H and I. As results of the structure analysis, correction of the gap could reduce fifty percent of the maximum stress in the pipeline system. However, none of the segments are close to the critical stress and allowance ratio of 1 although it is expected that the dynamic analysis could be affected by this gap. The thermal stress analyses were conducted for the gap and non-gap support at certain segments. The analyses were performed in a range from -100 to –400 °F degrees as system content and 85°F as surrounding temperature. The results yield values of stresses and displacement. However, the -200°F is consistent to the facilities operational conditions. For the support with gap, the maximum stress was found as 11,188 psi with a stress/allowance ratio of .26, and for the support without gap, the maximum stress 10,710 psi with a stress/allowance ratio of .25. Both maximum stresses were found at segment F. However, these values are in the acceptable range and no action is required. Maximum waterhammer pressures caused by rapid closure of isolation valve were studied for various flows. The peak values of transient flow pressure were generated in the piping system before the isolation valve. Possible failures were predicted for several segments. It is clear that for the case of a valve rapid closure, possible failure might occur at a flow rate of 38 ft/s, which is the maximum flow
79
at which the facilities operate. Some pressures reducers placed before the test valves are recommended in order to reduce the impact of the traveling waves. A study of the fluid transient in a simple pipeline was completed. The governing partial differential equations were integrated using the MacCormak method, and the behavior of the system was analyzed for different test cases. The incremental time was performed based on these results. Then, using material and fluid properties, the time of valve close and open was analyzed as a parameter to control the crest of wave pressure. The results obtained indicate that with adequate time of valve operation, the wave crest will not reach to the pressure of failure. For the vibration resonance analysis, the natural frequencies of the system as a whole were obtained and compared with the excitation frequency of fluid at different flow rates. The results conclude that the first natural frequency of the whole system occurs at 2.07 Hz, and the second at a frequency of 5.65 Hz with its corresponding mode shapes. Resonance for the first mode shape was found at a flow rate of 40 lbm/s, whereas, resonance for the second mode occurs at a flow rate of 275 lbm/s. For the first mode shape the greatest displacement and stresses are near the end of the piping system, and for the second mode shape the segments near the center could be in resonance. It is expected that for the third mode the elements near the tank might have the greatest displacement, of course it takes place at higher frequencies. With the analytical method approach the pipeline was discretized in straight segments between supports. In addition to explore the possibility of resonance, the natural frequencies were calculated for the first three mode shapes. Examining possible resonances were predicted for the first mode shape in segments three, four and six. For the second mode shape resonance was predicted only in segment 80
number two and for the third mode shape resonance was predicted for segment one and two. The results in this investigation reveal higher natural frequencies for all segments than the anticipated in previous research [1]. The reason can be found in the idealization of the system. The discretization made in previous investigation was along straight pipe making the segments longer and more susceptible to vibration than analyzing between supports. Supports will tend to increase the stiffness of the segment. Taking into account only the straight segment will under estimate the additional stiffness that comes from them. For the fluid structure interaction investigation, two models were created; a straight pipe and a straight pipe with 50% obstruction in its cross-sectional area. For both cases, a non-fluid structure interaction solution was compared against a fluid-structure interaction solution. Various flow cases were studied, after comparing the non interaction with the interaction solutions; it was noticed that a percent difference up to 25% in the fluid velocity. This is attributed to the fact that when the fluid applies pressure to the pipe, it deforms. This deformation decrease the velocity at which the fluid is traveling. As the axial velocity decreased, the pressure applied by the fluid also decrease. Since the cross-sectional area decreases again, by consequence the fluid increase its velocity again and the pressure applied to the wall of the pipe also increase. This phenomenon continues until a convergence is achieved. Comparing both models, the fluid structure interaction in pipe with 50% obstruction is more significant than the pipe with no obstruction . The main reason is that the obstruction creates a local increase of pressure leading to a deformation in this specific area. 81
7.3 Future works The turbulence induces vibration solution, comprise analytical techniques and experimental data. Due to no experimental data was available for the facilities, data available in textbook was use to estimate the vibration response. For future works is recommended to measure the turbulent forcing function and apply the method of Chapter 5 In the transient investigation presented in Chapter 3, pressure analysis in a time response fashion was performed without considering the fluid-structure interaction effect. However, the analysis was made using the material properties of stainless steel. As a future work, the fluid-structure interaction effect should be investigated to determine the dynamic pressure response.
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with Cavitations, Ph. D. Thesis, Delft University of Technology, Faculty of Civil Engineering, [10] A.S. Tijsseling(1993) Fluid Structure Interaction In Case of Water hammer with Cavitations, Ph. D. Thesis, Delft University of Technology, Faculty of Civil Engineering, Communications on Hydraulic and Geotechnical Engineer, Report No. 93-6, Delft, The Netherlands, 1993, ISSN 0169-6548 [11] YI Jia, Frederick Just-Agosto and Luciano Castillo (2001) Investigation of the Flow induced Vibration in the E2 Test Facility proposal. [12] G.W. Housner 1952 “Bending vibration of a Pipe Line Containing Flowing Fluid” Journal of applied mechanics. [13] Thomas Repp (1998) Fluid dynamics Waterhammer Simulations With Consideration of Fluid Structure Interaction [14] Biggs,J.M,”Introduction to Structural Dynamics”, McGraw Hill Publisheng 83
Company [15] Leonard Meirovitch, Fundamental of Vibrations Mc Graw Hill higher education (2001) [16] Au-Yang,M.K, “Flow-Induced Vibration of Power and Process Plant Components”, ASME Press, Professional Engineering Publishing, NY (2001). [17] Powell, A 1958, “ On the Fatigue Failure of Structure Due to Vibration Exited by Random Pressure Fields, Jurnal of Acoustical Society of America, Vol 30 No 12, pp 130-1135 [18] Chen, S.S. and Wambsganss, M. W., 1970 “Response of a Flexible Rod to Near Field Flow Noice,” in Proceeding of Conference on Flow Induced Vibration in Reactor Components, Argonne National Laboratory Report ANL-7685, pp 5-31. [19] Chyu, W.J. and Au-Yang M.K., 1972 Random Response of Rectangular Panels to the Pressure field Beneath a Turbulent Boundary Layer in Subsonic Flow, NASA TN D-6970 [20] Au-Yang, M. K., 1975 “Response of Reactor Internals to Fluctuating Pressure Forces,” Journal Nuclear Engineering as Design, Vol. 35 pp 361375. [21] Au-Yang, M.K.,2000, “The Joint and Cross Acceptance in Cross-Flow Induced Vibration, Part I Theory and Part II – Charts and Applications,” ASME Transaction, Journal of Pressure Vessel Technology, Vol 122, pp 349361. [22] Bull, M.K., 1967, “Wall-Pressure Associate with Subsonic Turbulent Boundary Layer Flow,” Journal of Fluid Mechanic, Vol.28, part 4, pp. 719754. [23] Au-Yang, M.K. and Jordan, K.B., 1980, “Dynamic Pressure Inside a PWRA Study Based on Laboratory and Field Test Data, “Journal Nuclear Engineering as Design, Vol 58, pp 113-125 [24] Au-Yang M.K., Brenneman, B. and Raj, D. 1995 “Flow induce Vibration Test of an Advance Water Reactor Model, Part I Turbulence Induced Vibration Test,” Journal Nuclear Engineering and Design,Vol 157, pp 93-109 84
[25] K.T. TRUONG, Evaluating dynamic Stresses of a Pipe Line, This paper is provided to Piping design.com by Ultragen [26] Paidoussis, M.P.; Au-Yang, M.K. and Chen, S. S., 1988, editors, Proceeding: International Symposium on Flow-Induced Vibration and Noise, Vol. 4, Flow-Induced Vibration due to Internal and Annular Flow, ASME Press, New York. [27] R. Kumar 1971 Acustica 24, pp.137-146. Flexible vibrations of fluid-filled circular cylindrical shells [28] D.R. Samsury 1974 Research Report-Naval Ship Research and Development Center, Bethesda, Md. 20034. Liquid-structure coupling in pipes [29] W.T. Thomson 1953 Proceedings of First U.S. National Congress on Applied Mechanics, pp. 922-933. Transmission of pressure waves in fluid filled pipes. [30] T.C. Lin and G.W. Morgan 1956 The Journal of the Acoustical Society of America 28(6), pp.1165-1176. Wave propagation through fluid contained in a cylindrical, elastic shell. [31] J.M. Cuschieri 1988 Journal of the Acoustical Society of America 83(2), pp.641-646. Excitation and response of piping systems. [32] A.L. Lewis and D.R. Roll 1997 Tappi Journal, pp.76-82. Pulsing and vibration caused by control valve induced acoustic resonances. [33] Chiba,T., Koyanagi,R., Ogawa, N. and Minowa, C., (1989), A Test and Analysis of the Multiple Support Piping System, Journal of Pressure Vessel Technology, Vol. 111, pp 291p299 [34] Chiba,T., Koyanagi,R., Ogawa, N. and Minowa, C., (1990), Dynamic Response Studies of Piping Support System, Journal of Pressure Vessel Technology, Vol 112, pp 39-45 [35] Chiba,T., Koyanagi,R., Ogawa, N. and Minowa, C., (1990) Response Characteristic of Piping System Supported by Visco Elastic and ElastoPlastic Damper, Journal of Pressure Vessel Technology, Vol. 112, pp 34-38 [36] Chiba, T and Koyanagi, R., (1988), An Experimental Study of the Response of Multiple Support Piping System, Res Mechanica, Vol.25, pp. 145 – 157
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[37] Vayda, J.P., (1981), influence of Gap Size on the Dynamic Behavior of Piping System, Journal of Nuclear Engineering and Design, Vol. 67, pp 145164 [38] Lockau, J., Haas, E. and Steinweder, F., (1984), The influence of HighFrequency Excitation on Piping and Support Design, Journal of Pressure Vessel Technology, Vol. 106, pp. 175 -187 [39] Paidoussis, M.P.; Au-Yang, M.K. and Chen, S. S., 1988, editors, Proceeding: International Symposium on Flow-Induced Vibration and Noise, Vol. 4, Flow-Induced Vibration due to Internal and Annular Flow, ASME Press, New York [40] Taylor, C. E.; Pettigrew, M. J., Dickinson, T. J. and Currie, I. G., Vidalou, P, 1997 “Vibration Damping in Multispan Heat Exchanger Tubes,”
4th International Symposium on Fluid Structure Interactions, Vol II, edited by M.P. Paidoussis. ASME Special Publication AD-Vol. 53-2, pp. 201-208 [41] ASME, 1979, Steam Tables, Fourth Edition, ASME Press, New York [42] M. C. haudhry, Hydraulic Engineering, Jhon Wile & Sons Inc., New York, 1998 [43] S. Sisavath, X. Jing, C. C. Pain and R. W. Zimmerman, “Creeping Flow Through an Axisymmetric Sudden Contraction or Expansion”, Transactions of the ASME, 124, pp. 273-278. [44] A. Bergant and A. Tijsseling, “Parameters Affecting Water Hammer Wave Attenuation, Shape and Timing” [45] A. Bergant, A. R. Simpson J. Vitkovsky, “Developments in unsteady pipe flow friction modeling”. J. of Hydraulic Research, Vol. 39, NO. 3, pp. 249257, 2001. [46] B. Brunone, U.M. Golia, , and M. Greco, “Some remarks on the momentum equation for fast transients”. Int. Meeting on Hydraulic Transients with Column Separation, 9th Round Table, IAHR, Valencia, Spain, pp. 140-148,1991. [47] J. D. Hoffman, Numerical Methods for Engineers and Scientists, McGrawHill Inc. Singapore, 1993. 86
[48] H. M. V. Samani and A. Khayatzadeh, “Transient flow in pipe networks” J. of Hydraulic Research, Vol. 40, NO. 5, pp. 637-644, 2002. [49] D. Stephenson, “Effect of Air Valve and Pipework on Water Hammer Pressures”, J. of Transportation Engineering, pp. 101-106, March 1997. [50] M. A. Chaiko and K. W. Brickman, “Models for Analysis of Water Hammer in Piping with Entrapped Air”, Transactions of ASME, Vol. 124, pp. 194-204, 2002. [51] R. S. Gupta, Hydrology & Hydraulic Systems, Waveland Press Inc., Illinois, 1989. [52] M. Moran and H. Shapiro, Fundamentals of Engineering Thermodynamics, JHON WILE & SONS, INC., New York, 2000. [53] A. E. Vardy, and J. M. B. Brown, “On turbulent, unsteady, smooth-pipe flow”. Proc., Int. Conf. on Pressure Surges and Fluid Transients, BHR Group, Harrogate, England, pp. 289-311, 1996. [54] B. Fitzgerald and C. Linden, “The Control Valve’s Hidden Impact on the bottom Line (Part 1) Hidden Impact”, Valve Manufacturers Association, 2003. [55] C.S. Beard, Control Valves, Intrument Publishing company, Pittsburgh, 1957. [56] D. R. Yang, “Actuator And Control Valve Selection”, Lecture III, Dept. of Chemical and Biological Engineering Korea University, 2001.
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APENDIX *******************************program**************************** 1388,1389,1384,1385,1386,1387,1375,1383,1382,1381,1380,1379,1378,238,1377,1376 668,667,666,665,664,663,662,661,660,659,657,658,656,655,669 /BATCH /COM,ANSYS RELEASE 5.7.1 UP20010418 12:33:43 05/23/2002 /input,menust,tmp ,'',,,,,,,,,,,,,,,,1 !User interface on /GRA,POWER /GST,ON /PLO,INFO,3 /COL,PBAK,ON,1,BLUE /VIEW, 1 ,1,1,1 !view /ANG, 1 /REP,FAST /PREP7 /prep7 shpp,on !element shape checking !!Assign element ET,1,142 ET,2,58
! 3D fluid element static mesh ! 3D solid element
CYL4,0,0,.0534543, ,.0841375
!!Geometry !Hollow Cylinder, inner diameter outer !Fluid Area
FLST,2,4,3 !specify data required for a picking operation NFIELD, NARG, TYPE, Otype, LENG FITEM,2,7 !key points in active coordinate system FITEM,2,8 !key points in active coordinate system FITEM,2,6 !key points in active coordinate system FITEM,2,5 !key points in active coordinate system A,P51X !Area arbitrary through key points K, ,0,0,0, !key point in point 0,0,0 K, ,0,0,-1.524, !key point K, ,0,.6,-1.524, !key point LSTR, 9, 10 !straight line through key point LSTR, 10, 11 !straight line through key point !* LFILLT,10,9,.18, , !line fillet, line ,line ,radius FLST,2,2,5,ORDE,2 FITEM,2,1 FITEM,2,-2 FLST,8,3,4 FITEM,8,9 FITEM,8,11 FITEM,8,10 VDRAG,P51X, , , , , ,P51X FLST,2,2,6,ORDE,2 FITEM,2,3 FITEM,2,5 VADD,P51X FLST,2,2,6,ORDE,2 FITEM,2,4 FITEM,2,6 VADD,P51X
88
SAVE
! Assign Properties to each volume FLST,5,2,6,ORDE,2 !To select volumes FITEM,5,1 FITEM,5,7 VSEL,S, , ,P51X !volume selected vatt,2,2,2 !assign properties to p51x,elemento,material,real constant FLST,5,2,6,ORDE,2 FITEM,5,2 FITEM,5,3 VSEL,S, , ,P51X vatt,1,1,1 ALLSEL,ALL ALLSEL,ALL FLST,5,4,6,ORDE,4 FITEM,5,1 FITEM,5,2 FITEM,5,3 FITEM,5,7 CM,_Y,VOLU VSEL, , , ,P51X CM,_Y1,VOLU CHKMSH,'VOLU' CMSEL,S,_Y !* VSWEEP,_Y1 !* CMDELE,_Y CMDELE,_Y1 CMDELE,_Y2 !* SAVE
!select Volume P51x !assign properties to p51x,elemento,material,real constant
!Mesh Creation !Select all !Select all
! CM,Cname, Entity - Groups geometry items into a component Y=Volume
!select Volume P51x !Groups geometry items into a component Y1 = volume !Check Mesh named volu ! Selects a subset of components and assemblies !create mesh for Y1 volume !Deletes a component or assembly definition. !Deletes a component or assembly definition. !Deletes a component or assembly definition.
!!Fluid Environment et,1,142 et,2,0 !!CFD Conditions and Type of fluid flda,solu,flow,1 !Flotran executions option for flow 1 flda,solu,turb,1 ! Flotran executions option for flow 1 turbulent flda,iter,exec,100 !Flotran Iterations flda,outp,sumf,10 !Output Summary frequencies !! CFD Property Information flda,prot,dens,constant flda,prot,visc,constant flda,nomi,dens,998 flda,nomi,visc,8.94e-7 flda,conv,pres,1.E-8 FLST,5,1,5,ORDE,1 FITEM,5,2 CM,_Y,AREA ASEL,R, , ,P51X
! lbf/in^3 (density - water) ! Lb/in*s (viscosity of water) ! Tighten pressure equation convergence !Y = Area ! Selects a subset of areas Select R from P51x
89
CM,_Y1,AREA !Y1 = Area CMSEL,S,_Y ! Selects a subset of components and assemblies CMDELE,_Y !Deletes a component or assembly definition. !* !* !* DA,_Y1,VX,0,1 !Specify fluid velocity in volumes X Axis inlet DA,_Y1,VY,0,1 ! Specify fluid velocity in volumes y Axis inlet DA,_Y1,VZ,-5,1 !Specify fluid velocity in volumes z Axis inlet !* CMDELE,_Y1 !* FLST,5,12,5,ORDE,6 FITEM,5,7 FITEM,5,-10 FITEM,5,17 FITEM,5,-20 FITEM,5,27 FITEM,5,-30 CM,_Y,AREA ASEL,R, , ,P51X CM,_Y1,AREA CMSEL,S,_Y CMDELE,_Y !* !* !* DA,_Y1,VX,0,1 !Specify fluid velocity in volumes X Axis DA,_Y1,VY,0,1 !Specify fluid velocity in volumes Y Axis DA,_Y1,VZ,0,1 !Specify fluid velocity in volumes Z axis !* CMDELE,_Y1 !* FLST,5,1,5,ORDE,1 FITEM,5,32 CM,_Y,AREA ASEL,R, , ,P51X CM,_Y1,AREA CMSEL,S,_Y CMDELE,_Y !* /GO ! Reactivates suppressed printout. !* DA,_Y1,PRES,0,1 !Specify Pressure at outlet of piping !* CMDELE,_Y1 !* SAVE alls /title,Fluid Analysis physics,write,fluid,fluid physics,clear !Creating the Solid Enviroment et,1,0 ! The Null element for the fluid region et,2,58 ! assign element 45 to material 2 mp,ex,2,2.82e9 !psi Young's modulus mp,nuxy,2,0.49967 ! Poisson's ratio tb,mooney,2
90
tbdata,1,0.293E+6 !Mooney-Rivlin Constants tbdata,2,0.177E+6 FINISH /SOLU !Enters the solution processor. FINISH /PREP7 !Define the options for the structure analysis. Loads>>analysis options NLGEOM,0 !Large deformation off NROPT,AUTO, , !Newton raphson method (program chosen) LUMPM,0 !Use lumped mass approximation turned off EQSLV, , ,0, !Equation solver type PRECISION,0 !Single precision off MSAVE,0 !Memory safe off PIVCHECK,1 !pivot check on PSTRES,ON !prestress on TOFFST,0, !Temperature difference 0 ! Boundary conditions for the solid /REPLOT FLST,5,1,5,ORDE,1 FITEM,5,1 CM,_Y,AREA ASEL,R, , ,P51X CM,_Y1,AREA CMSEL,S,_Y CMDELE,_Y /GO DA,_Y1,ALL,0 CMDELE, Y1 FLST,5,1,5,ORDE,1 FITEM,5,31 CM,_Y,AREA ASEL,R, , ,P51X CM,_Y1,AREA CMSEL,S,_Y CMDELE,_Y /GO DA,_Y1,all,0 CMDELE, Y1 /title,structural analysis finish /solu antype,static nlgeom,on cnvtol,f,,,,-1 physics,write,struc,struc physics,clear save /REPLOT SAVE SAVE ! Couple fluid Solution *SET,loop,5 ! Maximum allowed number of loops *do,i,1,loop ! Execute fluid -> structure solutions /solu physics,read,fluid ! Read in fluid environment
91
*if,i,ne,1,then !!,IF,VAL1,Oper,VAL2, If I not equal to 1 then flda,iter,exec,100 ! Execute 100 global iterations for *endif ! each new geometry solve ! FLOTRAN solution fini ! end of fluid portion physics,read,struc ! Read in structures environment /assign,esav,struc,esav ! Files for restarting nonlinear structure elementsave !!Assign,Ident,Fname,Ext,Dir /assign,emat,struc,emat *if,i,gt,1,then ! Structural restart loop If i is grater than a parsave,all ! Save parameters for convergence check resume ! Resume DB - to return original node positions parresume ! Resume parameters needed for convergence check /prep7 antype,stat,rest !Analysis type Static, restart fini *endif /solu /Enter Solution preprocesor solc,off asel,s,,,7,10 ! Select proper areas to apply fluid pressures asel,a,,,17,20 asel,a,,,27,30 nsla,,1 !nsla,Type,NKEY - Selects those nodes associated with the selected areas esel,s,type,,2 !,ESEL Type,Item,Comp,VMIN,VMAX,VINC,KABS - Selects a subset of elements. ldread,pres,last,,,,,rfl ! Apply pressure surface load from Flotran alls rescontrol,,none ! Do not use multiframe restart for nonlinear solve *if,i,eq,1,then save ! save original node locations at the first run *endif fini /prep7 *SET,mkey,2 ! Select level of mesh morphing for fluid dvmorph,2, ,mkey dvmorph,3, ,mkey ! Perform morphing of volume 2 alls fini /assign,esav /assign,emat *enddo save
! Nodal coordinates of deformed geometry are saved
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