Static dynamic-analysis-of-piping-system

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. Date
Member, Graduate Committee
_________________________________ _________________
Frederick Just, Ph.D. Date
Member, Graduate Committee
_________________________________ _________________
Oswald Uwakweh , Ph. D. Date
Representative of Graduate Studies
_________________________________ _________________
Jia Yi, Ph.D. Date
Chairman, Graduate Committee
_________________________________ _________________
Prof. Paul Sundaram, Ph.D. Date
Chairperson of the Department

ii
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 fluid-
structure 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.

iii
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.

iv
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.
.

v
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

vi
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

vii
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

viii
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

ix
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
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

x
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

xi
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

xii
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

xiii
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
fw = 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

1
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

2
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

3
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

4
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.
.
Figure 1.1: Acoustic wave in pipes
Flow
Valve
Tank
Acoustic W ave

5
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.

6
Segments
Dynamic
3D Solid Model
Maximum
Stress
Complete
System
Segments
Maximum
Displacement
Complete
System
Various Flows
Pressure
History
Natural
frequencies
Excitations
Frequencies
Valve Closing
Time
Complete
System
Static Vibration
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

7
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,
di
SV
fs = (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,

8
65
1006.6Re1043.5 XDX <≤ , 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

9
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 closed-
form 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

10
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.

11
CHAPTER 2 STATIC ANALYSIS
2.1 Finite Element Model
The model is created based on the actual pipe configuration; it is a 3-
dimensional 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.

12
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.

13
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
1
3
2
2
3
2
3
A
B
CD
E
F
G
I H
J
K
M
N
L

14
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.
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
Type 1 Type 2 Type 3

15
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.
⎪⎩
⎪
⎨
⎧
⎭
⎬
⎫
−⎥
⎦
⎤
⎢
⎣
⎡
−
= 4.0
)(2 cathk
Do
PSH (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.
[ ]
Ac
PaFa
Z
MoiMii
SL oi +
+
+
=
2
1
22
)()(
(2.2)

16
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.
[ ]{ }2
0 )(2
4
caDPPa −−=
π
(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.

17
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.
thth Eεσ = (2.4)
where E is the modulus of elasticity and thε is the thermal strain, the length L,
area A, and volume V, strain are calculated with the following equations.

18
)( 120 TTLL −=∆ α (2.5)
)( 120 TTAA −=∆ γ (2.6)
αγ 2≈ (2.7)
)( 120 TTVV −=∆ β (2.8)
αβ 3≈ (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]:
)12( TTESL −= α (2.10)
For the unrestrained segment the expansion stress is calculated with the
following equations [14]
[ ] 2
1
22
4 tbE SSS += (2.11)
Where;
( ) ( )[ ]
Z
MiMi
S ii
b
2
1
2
00
2
+
= (2.12)
Z
M
S t
t
2
= (2.13)

19
2.3 Results
2.3.1 Static Stress Analysis Results
Maximum Stress/Allowance ratio with gap
0
0.02
0.04
0.06
0.08
0.1
0.12
Segments
Stress/Allowance
Segment A
Segment B
Segment C
Segment D
Segment E
Segment F
Segment G
Segment H
Segment I
Segment J
Segment K
Segment L
Segment M
Segment N
Figure 2.5a: Stress vs. length for gap space case
Figure 2.5b: Stress vs. length for gap correction case
Maximum Stress/Allowance ratio gap correction
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
Segments
Stress/Allowance
Segment A
Segment B
Segment C
Segment D
Segment E
Segment F
Segment G
Segment H
Segment I
Segment J
Segment K
Segment L
Segment M
Segment N

20
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.
Figure 2.6: Gap locations in segment K
Gap
J
K
I
L

21
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
70
.25in gap0 in gap
1789 PSI
1789 psi
220
545 psi

22
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.

23
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
-0.486
K -0.245 K
-400 0.444
M
-0.487 K -0.226 K
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.

24
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
-200
11188
0.26 F
-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

25
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

26
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.

27
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

28
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

29
[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 Navier-
Stoke 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].
02
=
∂
∂
+
∂
∂
x
V
a
t
p
ρ
(3.1)
0
2
1
=++
∂
∂
D
VfV
dx
dp
t
V
ρ (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:

30
⎟
⎠
⎞
⎜
⎝
⎛
∂
∂
−
∂
∂
+=
x
V
t
V
VV
kD
ff q
(3.3)
Where qf is the quasi-steady friction factor, which is plotted for different
Reynolds number ( νVD=Re ) and relative roughness ( Dke s= ) 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 :
2
*
C
k =
(3.4)
The Vardy’s [54] shear decay coefficient *
C is given by:
00476.0 Laminar Flow (3.5a)
)Re/3.14log( 05.0
Re
41.7
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

31
assumed that the reservoir of the pipe line will maintain constant level, thus
maintaining constant inlet hydraulic pressure iP The boundary condition is
expressed mathematically as:
iinletn PgHp == ρ
(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:
ρ/2 000 pACQ dvalve
=
(3.7)
Where the subscript 0 refers to steady state conditions, dC is the discharge
coefficient and 0A 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.

32
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:
v
nnT
dvalven V
p
A
A
lFCV ==
ρ
2)(
0 (3.8)
Where, )(lF is the percent of caudal obtained from Figure 3.2 and l the stem
position. TA the area of valve totally open and 0A 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:
initialVV =0 initialPp =0 (3.9)

33
3.1.3 Numeric Discretization
The system of equations to solve is:
02
=
∂
∂
+
∂
∂
x
V
a
t
p
ρ (3.10)
0
2
1
=++
∂
∂
D
VfV
dx
dp
t
V
ρ
(3.11)
initialVxV =)0,( (3.12)
initialPxp =)0,( (3.13)
Ptinletp =),( (3.14)
VtvalveV =),( (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:
( )n
i
n
i
n
i
n
i VV
x
t
app −
∆
∆
−= +
+
1
21
ρ
(3.16)
( ) n
i
n
i
n
i
n
i
n
i
n
i V
D
t
fVpp
x
t
VV
2
1
1 ∆
−−
∆
∆
−= +
+
ρ (3.17)
In the second final step, a first order backward difference approximations
based on the provisional values is used.

34
( )⎥⎦
⎤
⎢⎣
⎡
−
∆
∆
−+= +
−
+++ 1
1
1211
2
1 n
i
n
i
n
i
n
i
n
i VV
x
t
appp ρ
(3.18)
( ) n
i
n
i
n
i
n
i
n
i
n
i
n
i V
D
t
fVpp
x
t
VVV
22
1 1
1
111 ∆
−⎥
⎦
⎤
⎢
⎣
⎡
−
∆
∆
−+= +
−
+++
ρ
(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.
12
1 ≤
∆
∆
=
x
t
aCn ρ and 11 ≤
∆
∆
=
x
t
Cn
ρ
(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.

35
Figure 3.3: Wave pressure for different dt
Figure 3.4: Detail of numerical noise effect

36
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.

37
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.

38
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

39
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.

40
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

41
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.
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:
∫∑ −+=
2
1
22
)()(
x
x
incwoutcwcw VAVAdxVA
dt
d
F ρρρ (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:
cwcwwcc AcVcVAdVcVddVcVAdpppA )()())()(()( ++−+++++=+− ρρρ
(3.22)
Flow Moving wave front
Velocity
Pressure
V V +dV
P P+d P
c

42
By simplifying and discarding terms of higher order, this equation becomes
)2(22 22
cVcVddVcdVVdp www ++++=− ρρρ (3.23)
The general form of the equation for conservation of mass for one-dimension
flows may be written as
∫ −+=
2
1
)()(0
x
x
incwoutcwcw VAVAdxA
dt
d
ρρρ (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
cwcww AcVAdVcVd )())((0 +−+++= ρρρ (3.25)
Simplifying this equation,
cV
dV
d w
w
+
−
=
ρ
ρ (3.26)
Because the fluid velocity v<< c
c
dV
d w
w
ρ
ρ
−
= (3.27)
Now, by substituting equation 3.27 into equation 3.23, discarding terms of
higher order, and simplifying
cdVdP w **ρ−= (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.

43
eE
KD
K
C
+
=
1
ρ
(3.29)
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.
Figure 3.11: Isolation valve and tank location
The flow velocity at a mass flow rate of 275 lbm/sec is
sec
456
914.130433.
275275 in
A
V
cw
=
×
==
ρ
(3.30)
Isolation
valve
Tank
Location

44
The theoretical maximum pressure surge for this flow velocity using a specific
speed of sound C of 39,015 ft/s is:
psi
g
Vc
P w
1994
386
456390150433.
max =
××
=
××
=
ρ
(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.

45
Flow 16.4 ft/s
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Segments
Stress/allowance
Segment A
Segment B
Segment C
Segment D
Segment E
Segment F
Segment G
Segment H
Segment I
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
0
0.2
0.4
0.6
0.8
1
1.2
Segments
Stress/Allowance
Segment A
Segment B
Segment C
Segment D
Segment E
Segment F
Segment G
Segment H
Segment I
Figure 3.13: Possible failure is presented at segment F and E.

46
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
0
0.2
0.4
0.6
0.8
1
1.2
Segment
stress/allowance
Segment A
Segment B
Segment C
Segment D
Segment E
Segment F
Segment G
Segment H
Segment I
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.

47
Flow 49.21 ft/s
0
0.2
0.4
0.6
0.8
1
1.2
Segment
Sterss/Allowance
Segment A
Segment B
Segment C
Segment D
Segment E
Segment F
Segment G
Segment H
Segment I
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

48
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.

49
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 non-
periodic. 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

50
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]
tPkyYM ωsin0
..
=+ (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

51
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]:
t
rk
P
tBtAty ωωω sin
1
1
sincos)( 2
0
−
++= (4.2)
w
w
r
f
= (4.3)
Then the solution for maximum displacement for an un-phased harmonic
analysis is
2
1
1
rk
Po
dyn
−
=δ (4.4)
And the natural frequencies are:
m
EI
L
n
w f 2
2
2
π
= (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

52
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 .
Figure 4.2: Discretization of the system
1
2
3
4
5
7
12
6
9
8
11
10
13
14
1517
16

53
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
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
0 1 2 3 4 5 6 7 8 9 10
Segments
Ratios
fs 113
fs 275
fs 40
Figure 4.3: Resonances per segment at different flow rates for the mode
shape 1

54
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
0 1 2 3 4 5 6 7 8 9 10
Segments
Ratios
fs 113
fs 275
fs 40
shape 2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
0 1 2 3 4 5 6 7 8 9 10
Segments
Ratios
fs 113
fs 275
fs 40
shape 3

55
Figure 4.6: Natural frequency mode shape 1 of a complete systems
Figure 4.7: Vibration modes shape 2 for the complete system

56
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.

57
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.

58
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

59
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.
( )
∑
→→
→
=⎟
⎠
⎞
⎜
⎝
⎛
α ααα
ααααα
ζπ
ψ
323
2
2
64
)()(
fm
fJxfAG
xy p
(5.1)
where αα
→
J 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 )( αfGp 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

60
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
*
4.06.0 Vc
e
V
U
ωδ
−
+= (5.2)
Bull [22] suggested a slightly different equation:
V
e
V
Uc
*
89.
3.59.
ωδ−
+= (5.3)
Where *
δ 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≈

61
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.
H
H
R
D
==
2
*
δ (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
= *32
32 )(
*2)(
δρ
δπρ
V
w
VfG PP
p (5.5)

62
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]
10,155.
)( 3
32
<<= −
Fe
RV
fG F
H
p
ρ
(5.6)
=.027e-1.26F,
1 5≤≤ F
where
F = fRH/V (5.7)
For turbulent flow with light cavitations
}0.1,)(20min{
)( 42
32
−− −
=
HH
P
R
x
F
RV
fG
ρ
(5.8)
where x is the absolute value of the distance from the cavitation source such
as an elbow or a valve.

63
Figure 5.2: Boundary layer type of turbulence power spectral density [5]
Figure 5.3: Longitudinal joint acceptances [5]

64
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
Segment
Natural
Frequency
(Hz)
Uc (in/s)
Convective
Velocity
4fL1 /Uc Jmm J´nm ωδ*/V
1 61.181 417.130 59.255 0.010 1.000 0.193
2 19.921 493.204 28.597 0.010 1.000 0.063
3 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 1406.410 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

65
Table 5.2: PSD and RMS Responses
Segment
Natural
Frequency
(Hz)
Normalized
PSD
Gp
(psi^2 / Hz)
Yrms (in)
1 61.181 2.000E-05 2.818E-04 0.149
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

66
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.

67
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.

68
Figure 6.1 Fluid structure interaction loop flow chart
Figure 6.2: Geometry of free flowing channel
Fluid
Structure
Structure

69
Table 6.1: Dimensions Free Flowing Channel
Dimensions
Thickness 0.003175 m
Length .5 m
Fluid cross section .05 m
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
Fluid
Structure
Obstruction
Structure

70
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.

71
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

72
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.

73
0%
5%
10%
15%
20%
25%
0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
Velocity M/S
PercentError
Obstruction
No Obstruction
Figure 6.4: Average percent difference at different flows
Figure 6.5: Velocities profile at the first iteration

74
Figure 6.6: Velocities profile at the second iteration
Figure 6.7: Velocities profile at the third iteration

75
Figure 6.8: Velocities profile at the fourth iteration
Figure 6.9: Velocities profile at the fifth iteration

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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

77
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.

78
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 fluid-
structure 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.

79
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

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