Steam Bubble Collapse, Water Hammer and
Piping Network Response
Volume II. Piping Network Response to
Steam Generated Water Hammer
by
R. Gruel, W. Hurwitz, P. Huber and P. Griffith
Energy Laboratory Report No. MIT-EL 80-018
June 1980
IIMII IllIgiI lM
H1ih
N11 -
l,
I
Steam Bubble Collapse, Water Hammer and
Piping Network Response
Volume II. Piping Network Response to
Steam Generated Water Hammer
by
R. Gruel, W. Hurwitz, P. Huber and P. Griffith
Department of Mechanical Engineering
and
Energy Laboratory
Massachusetts Institute of Technology
Cambridge, Massachusetts 02139
sponsored by
Boston Edison Company
Consumers Power Company
Northeast Utilities Service Company
MIT'Energy Laboratory Report No. MIT-EL 80-018
June 1980
-2-
Foreword
Work on steam bubble collapse, water hammer and piping network response
was carried out in two closely related but distinct sections.
Volume I of
this report details the experiments and analyses carried out in conjunction
with the steam bubble collapse and water hammer project. Volume II details
the work which was performed in the analysis of piping network response to
steam generated water hammer.
Table of Contents
Volume II
Piping Network Response to Steam Generated Water Hamer
Page
Foreword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . .
I.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . .
II. Experimental Apparatus . . . . . . . . . . . . . . . . . . . .
2
3
5
III.
Experimental Procedure . . . . . . . . . . . . . . . . . . . .
12
IV.
Theoretical Analysis
. . .....................
..
Wave Propogation Model ...
...............
.
Governing Equations . . . ................
.
.
Formulation in Finite Difference Form . . . . ...
13
14
14
15
Representation of Piping Network . . . . . . . . .
Cavitation Model . . . . . ................ . .
Boundary Conditions . . . . . ..............
. ..
Deflection Model . . . . . . . . . . . . . . . . . . .
16
19
23
24
Bubble Collapse Model . .......
35
V.
Results
. . . . . . .
.
..........................
.
38
Wave Propagation Experimental Resuls ....
.......
Wave Propogation Theoretical Resu s . . . . . . . .
Displacement Experimental Results . . . . . . . . ....
..
38
..
44
Displacement Theoretical Results
VI.
.............
Conclusions
49
. . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . .
. . . .
.
49
58
111
I N
.,
4,11111
-3I Introduction
Conventional and nuclear power steam systems require the transport
of high pressure, high temperature steam and water through complex piping
networks.
During transient phases of operation, steam and subcooled
water can be present simultaneously in a variety of piping and reservoir
configurations.
Under certain conditions, instabilities in the two
phase flows can give rise to water hammer events.
Most notable are
the water hammers experienced in the feed pipes to steam generator
spargers in pressurized water nuclear reactors.
Little is currently
known about the mechanisms involved in the evolution of water hammer.
Creare (1977) offers a possible description of how instabilities in the
feedwater pipe of a PWR steam generator can lead to the formation of an
isolated steam bubble between two columns of subcooled water.
Under such
conditions, the steam inside the bubble rapidly condenses, causing bubble
collapse and impact of the two water columns.
This leads to the formation
of high amplitude pressure waves which propagate through the piping
network, causing deflections and possible damage to the pipes and their
supports.
This study combined with the work done by Gruel (1980) attempts to
analyze the events which occur after the formiation Of an isolated steam
vapor bubble.
Experimental and theoretical models are developed to
investigate the condensation process leading to steam bubble collapse,
the mechanisms involved in the propagation of pressure waves through
pipes, and the resultant fluid-structure interactions.
The focus of this study is on experiments run with a small scale
water hammer generator (Gruel 1980) which can be used to form an isolated
steam bubble between two columns of water under stable conditions.
By
-4opening a valve, the steam is brought into contact with cold water.
steam bubble collapses and a water hammer transient results.
The
The sub-
sequent wave propagation and structural interactions can be measured
in a length of pipe connected to the water hammer generator.
Three theoretical models are developed to help understand three
separately defined processes involved in the experiments.
The first
is a model of the condensation and initial collapse of the steam
bubble.
Gruel (1980) uses this model in the analysis of data taken
for this phase of the study.
The second model uses the method of
characteristics (Wylie & Streeter 1978) to describe the propagation
of pressure waves through the experimental piping network.
Finally, a
simple deflection model is used to help understand the fluid-structural
interactions associated with the water-hammer transient.
-5II Experimental Apparatus
The experimental apparatus consisted of two components.
was a water hammer generator (Gruel 1980) (fig. 1).
The first
The second component
consisted of a freely supported U-shaped section of pipe (the test
section), connected horizontally to the base of the water hammer
generator (fig. 2).
The overall network represented a system in which
water hammer transients could be generated by the water hammer generator
and subsequent system responses could be measured in the test section.
The properties for all of the materials used in the network are listed
in Table 1.
The details of the water hammer generator (fig 1.) are fully
described by Gruel (1980).
The main features include the cold water
reservoir, the steam ports and the quick action pneumatic valve.
Immediately prior to each test, steam was circulated through the steam
ports in the section of pipe below the cold water reservoir.
was isolated by closing the input and output valves.
The steam
Activating the
pneumatic valve initiated the water hammer transients by bringing the
cold water in the reservoir into contact with the hot steam.
The steam
bubble rapidly condensed resulting in a depressurization of the vapor
cavity.
A column of water from the reservoir accelerated into the
network and generated high pressure transients on impact with the
stationary steam-water interface.
These pressure transients propagated
through the test section where the network responses were measured.
Three geometrically identical test sections, (fig. 2) built from
three different materials (see Table 1): 3/4" carbon steel pipe, 3/4"
copper tubing and 3/4" CPVC pipe, were used in the experiments.
Pipe
unions were incorporated for the easy instalation of the test sections
-6-
Pressure Ports for
Operating Piston
Reservoir Pressure
Port
Water Inlet
*
Cold Water
Reservoir
4-Pneumatic
Valve
Water Level
0
Piston
.23m
Steam Pressure
' Gage
T2 Thermocouple
Port
.65m
Steam Inlet
Valves
TI Thermocouple
Port
Steam Outlet
xan Section
.27m
.20m
Steam -Water
Interface
_I .02m
Pressure Transducer
P2
-Outlet to
Test Section
Pressure Transducer
P
Fig. 1 Water Hammer Generator
Fig. 2
Piping Network Test Section
Table 1:
Properties of Materials Used in Piping Network
Test Sections
Water Hammer Generator
6" steel
Reservoir
1 1/2" Steel
Pipe
1 1/2" Lexan
Pipe
3/4" Steel
Pipe
3/4" Copper
Tubing
3/4" CPVC
Pipe
.1778
.0483
.0444
.0267
.0222
.0267
Inner Diameter (ID)
(m)
.1524
.0407
.0381
.0208
.0189
.0188
Wall Thickness (e)
(m)
.0127
.0038
.0032
.0029
.0017
.0039
Elasticity Modulus (E)
(kPa)
207 x 106
207 x 106
2.76 x 106
207.0 x 106
119.0 x 106
2.9 x 106
Material Density (p)
(Per Unit Length kg/m)
51.4
4.033
0.490
1.683
0.954
0.490
(7.806 x 103 )
(7.806 x 103)
(1.200 x 103)
(7.806 x 103)
(8.908 x 103)
(1.55 x 103
1360.5
1368.4
456.6
1390.9
1318.0
"682.9
Outer Diameter (OD)
(m)
(kg/m 3 )
*Wave Speed (c)
(filled with water)
0:
oo
(m/s)
Pwater
-1/2
1000 kg/m
Cwater = vTT
+
=
1440 m/s
C. water
Pwater
Ee
ID)
M
I
1III
il
II
I
III l1 II
into the network.
inmEE~ hIYInIYInmIIIYY
m
__
-9The copper and CP.VC test sections were supported
vertically at the end of the "U's" to prevent them from sagging under
their own weight.
To minimize the effect on horizontal displacements,
the support was provided by attaching a long thin wire from the ceiling
to the "U".
Pressure waves propagating through the test section created
unbalanced horiztonal forces when they passed by the two elbows of the
"U" (see deflection model).
Resultant displacements of the pipe were
measured at the two locations indicated in fig. 2, one at the end of
the "U" (x = L) and the other at the halfway point (X = L/2). In addition
the pressures were measured at two of the three pressure ports provided
in each of the test sections (fig. 2).
Instrumentation
The instrumentation used in the experiments. consisted of devices
for measuring the temperatures, pressures and displacements of the
piping network.
Two types of measurements were made; first the steady
state initial conditions and then the transient responses due to water
hammer.
The initial steam and cold water temperatures and pressures were
recorded immediately before each test.
Two iron-constantan thermocouples
were used for the temperature measurements (fig. 1).
Initial pressures
were measured with two bourbon pressure gages, one located in the
steam line (fig. 1) and the other (not shown in the figure) behind the
reservoir.
The transient pressure and displacement histories of the test sections
were recorded in each test, using three piezoelectric pressure transducers
and one displacement transducer.
Two of the pressure transducers were
-10-
used for high pressure measurements and were located at positions P
1
and P3 (fig. 2).
The third, a low pressure transducer was used to
record the steam bubble depressurization history (P2 % fig. 2).
The
displacement transducer was located at either of the two points (D1 or
D2 ) shown in fig. 2. The pressure and displacement signals were led
through various signal conditioners to a storage oscilloscope.
All
oscilloscope traces were recorded using an oscilloscope camera.
Table 2
gives a complete list of instrumentation used in the experiments.
A total of 5 different transients (PlA'
P2' P3 , Dl and D2 ) were
recorded for each set of experimental conditions.
However, the oscillo-
scope was only capable of recording two traces at a time.
This required
the repetition of tests under identical conditions to compile one
complete set of data.
For each run, the pressure P1A was recorded along
with one other variable.
This provided an unambiguous "time scale"
reference as well as an assesment of the run to run repeatability of the
pressure transients (which was, overall, very good.
See results).
nl
ilmmm
Moll
",Ill
l. . . .
Iii
-11
Table 2:
-
Instrumentation
Pressures
2 Kistler Type 6606A5000 Piezokompac
Pressure Transducers: scale factor = -1mv/psi,
Range = 0-5000 psi Response time = 3 -is
1 Sundstrand Model 206 Piezotron Pressure
Transducer: scale factor = 96mv/psi, Range = 0-80 psi
Response time = 3 ps
Temperatures
2 Omego Type J 3/16" Iron-Constantan Thermocouples
1 Omega Type MCJ-J Electronic Ice Point
Displacements
1 HR-DC 500 Shaevitz Displacement Transducer
1 R-C filter for filtering high frequency noise
1 PSM120 DC Power Supply (For transducer)
Recording
Hewlett Packard 3440A Digital Voltmeter W/Amplifier
Tektronix 434 Dual Beam Storage Oscilloscope
-12III
Experimental Procedure
The method outlined in Gruel (1980) for operating the water hammer
generator was followed.
Before each experimental run, the reservoir and
steam initial conditions were carefully monitored.
All of the experimental
tests in this study were run with the steam pressure (Ps) set at 170 kPa
absolute (10 psig).
In each case, saturation conditions were met,
corresponding to a steam temperature (T1 ) of 115
0
C (239 IF).
The water
temperature of the reservoir (T2 ) was maintained by circulating water
through the piping network between runs.
Due to the day to day changes
in the water supply temperature, the reservoir temperature varied
between 20 - 300 C. Three different reservoir pressures (back pressure
P ); 170 kPa, 240 kPa and 310 kPa, were used in the experiments.
The
back pressure was regulated by pressurizing the reservoir with nitrogen.
Together with the 3 different test sections, a total of nine experimental
conditions were run.
each of the cases.
the oscillograms.
Appendix A contains some sample oscillograms for
Appendix B contains the numerical data obtained from
II
*mmIImIII.YmIi nIa
-I
I1111
.1,1
I,1111
111
,
11
14
111
1111
-13-13-
IV Theoretical Analysis
The analysis of water hammer in the piping network has been divided
into three parts.
The first involves an evaluation of the processes
involved in the initial depressurization and collapse of the steam
bubble.
At the end of this section, a numerical model is proposed
that can be used to determine the time dependencies of the governing
parameters.
Gruel (1980) uses this model in an evaluation of his
experimental data collected on steam bubble collapse.
The second and
third parts of the water hammer analysis involve a study of the pressure
wave propagation and structural responses that occur after the initial
bubble collapse and water column impact.
These are considered separately
below. The wave propagation model assumes that the structural response
excited by the pressure transients does not significantly affect the
wave propagation mechanisms.
The deflection model uses the output from
the wave propagation model to predict the deflections of the test
section.
This proceudre should be accurate when the pressure transients
occur over a period that is short compared with-the vibration period of
the excited structure.
This was the case for the conditions of the
experiments, which can be verified by the oscillograms.
IIIIIYY
-14Wave Propagation model
The method for analyzing the propagation of pressure waves in
liquid filled pipes are well documented (e.g. Wylie & Streeter 1978).
To model wave propagation in the experimental network, a finite
difference form of the method of characteristics was used.
Pressure
and fluid ve'ocity histories were calculated at various locations in
the network, for times after the initial steam bubble collapse.
The
following sections will describe the pertinent details of the model
including the governing equations, the nodal representation of the piping
network, the model for cavitation and the treatment of the boundary
and initial conditions.
Governing Equations
The governing equations for compressible flow through constant
area ducts can be written as follows (Wylie & Streeter 1978):
;
+ V. x +
Dt
continuity:
momentum:
9V + V 3V _
3t
ax
aP
eq. of state:
where
p
V=
0
x
aP
p x
(1)
fVV(2)
2D
=
(3)
P = fluid pressure
V = fluid velocity
p = fluid density
f = Darcey-Weisbach friction factor
D = pipe diameter
S= bulk modulus of elasticity
With the proper stress-strain relationships for a thin walled elastic
11.1
~
1011111111
JI
julYY
I__
~
__
liffi
-15pipe, the continuity equation can be rewritten as:
1 p + V P + C2 V = 0
ax
p ax
p Bt
1
with
2
C2
where
1
1
/p
Ee/pD
(4)
(5)
C = wave speed of fluid in pipe
E = pipe modulus of Elasticity
e = pipe wall thickness
(A linearized analysis of eqs. 1-3 demonstrates that C is the fluid
wave speed corrected for the elasticity of the duct (Wylie & Streeter
1978).
Formulation in Finite Difference Form
The momentum and continuity equations (Eqs. 2, 4) represent two
quasi-linear hyperbolic equations in two unknowns.
For liquid flows,
one can assume that V << C, and can therefore neglect the convective
acceleration terms in equations 2 & 4. By choosing the proper
trajectories (or characteristics) in space and time, the characteristic
equations are formed as follows:
1 dP
pC dt
for
d
+
- 0
dx = C
(6b)
dt
1 dP
pC
d+d dV + f v V = 0
and
dt
pC dt
for
dx
-
(6a)
20
dt
20
C
(7a)
(7b)
dt
The values of +C (wave speed) represent the slope of the characteristics
in the x-t plane on which the corresponding compatibility equations are
vaild.
With the addition of the boundary and initial conditions,
-16-
equations 6 & 7 can be solved
numerically in finite difference form.
Representation of Pipe Network
Fig. 3 and Table 3 outline the procedure for developing a nodal
representation of the experimental network, with the steel test section,
usedin the finite difference calculations.
Each element, located
between two nodes (fig. 3c) representsa constant area segment of the
piping network and has a constant wave speed.
The wave speed and
area, however, are allowed to change from element to element.
There-
fore, a series of identical elements represents one uniform section
of pipe in a network.
The representation of a network composed of
many sections of pipe can be simplified if sections of pipe with similar properties are represented by "equivalent" sections (fig. 3b).
The wave transit times and the overall fluid momentum are properly
accounted for by defining "equivalent" wave speeds and pipe diameters:
L.
n, L
C_ = E
3
i
L.
L
2
3
where
L
(8)
1
nj
nJ LLi
i D.2
(9)
1
= length of jth equivalent section representing n
actual sections.
The values for the "equivalent" wave speeds and diameters can be
readily calculated if the actual lengths of the sections are maintained
(see Table 3).
The number of computational elements in the piping network is
determined from the "equivalent" representation.
Continuity in the
calculations is maintained by choosing the elements such that the
-17H
I
W II
SZ.02
,.
x
-=
L
13 6
(a) ACTUAL
L
=.104
|.0
PIPING NETWORK
L
i®
1
I
I
®
I
I x
r =.136
I
(b) EQUIVALENT NETWORK
-C-=.
L "IO4
Reservoir
End
~ ~
12 3
2
Pressure Transducer
P (x/L=.172)
IA
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i
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10101
(
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1
15 i16 17' 18 19 201
10~I 12. 13 13l4,
7~II
3 5 ,
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g"
g00ei0g
l
IIl
I
-1= 1.
L
.I,
I
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1*
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0*0*0*01
I
i
I
I
I
I
.104 IP,
x = IA
I
I
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t
IIi
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Closed
End
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Section
I
' 31
I
I
NODAL REPRESENTATION
Fig. 3 Development of Nodal Representation
(steel test section)
I
I
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I
iI
0101016*I
I
II
I
L1.0
Elbow
(A= 646
L
I
3233134 35
3
Pressure Traonsducers
and P2
(x/L=.160)
(c)
I
2 12/ 2312425126!27l28 29,30
I I
I
Section # I Section #2
x
I,
Elbow
=.558)
L
Pressure Tronsducer
P
-18Table 3 Development of Nodal Representation (steel test section)
Actual Piping Network
(Overall Length = 8.43m)
Pipe
Section
Length (Li)
(m)
1
2
6" Reservoir
1 1/2" Steel
3
4
1 1/2" Lexon
1 1/2" Steel
5
3/4" Steel
.23
Wave Speed (ci)
(m/s)
1361
.65
diameter (d i )
(m)
.1524
.0407
1368
457
.27
.51
6.77
1368
.0381
.0407
1391
.0208
Equivalent Piping Network
Section
Pipe
1
Reservoir &
1 1/2"Steel
2
Lexan
3
1 1/2"Steel
& 3/4" Steel
Length (L.) Wave Speed (c.)
(m/j)
3
8) I
.88
1366
.27
7.28
diameter (dj)
.0467
457
.0381
1400
.0214
Nodal Representation
Section
Length (L.)
Pipe
Adjusted
Diameter (d.) wavespeed
(m)
Steel
Lexon
Steel
.88
.27
7.28
L.
J = 0002 s
At N.
C.
33J
At=
(m)
(m/s)
.0467
.0381
.0214
1460
445
1400
# Divisions
(Ni)
-19wave transit time across each is a constant:
L.
At =
= constant
---
(10)
N.J C.
J
N = number of elements in "equivalent" section J.
where
Since the N 's must be integers, it
is unlikely that suitable values
can be found to satisfy eq. 10 exactly for all sections, given that
the L.'s and C 's are fixed.
This problem is avoided by modifying the
wave speeds by small amounts (say less than 10%) to assure that
equation (10) is satisfied.
The constant value for At is chosen by
taking into account both physical and computational factors.
The final representation of the piping network has nodes located
between each of the representative elements and at the system boundaries.
This results in a total of N.
3 + 1 nodes for each "equivalent" section
In addition, each junction between sections is represented
(fig. 3c).
by two nodes.
As the calculations proceed, the pressures and velocities
are determined at each node by the appropriate characteristics from the
bordering elements.
At the system boundaries, a boundary condition is
substituted for one of the characteristic equations.
Cavitation Model
In its pure form, the method of characteristics predicts negative
as well as positive absolute pressures in the piping network.
Realistically,
cavitation nust occur when the absolute pressure in the fluid drops
below the vapor pressure.
This state is identified and accounted for
in the wave propagation model by using a form of the continuity
equation.
In the calculations, when cavitation is indicated at a node
(P < Pvapor ), the pressure is artificially set equal to the vapor pressure,
-20which then serves as an internal "boundary condition" for the rest of
the system.
Growth and collapse of the cavity is calculated using
the following equation:
= E
(Vu - V) A At
(11)
t
where
-V
= cavity size
Vu = upstream velocity at the cavitating node
V = downstream velocity at the cavitating node
A = area of pipe
The upstream and downstream velocities are calculated from their
respective characteristics and the summation is taken over all time
for which the cavity exists.
can be calculated in two ways.
The time at which cavitation vanishes
Observations of the experimental
water hammers suggest that two types of cavitation must be considered.
The first type of cavitation involves the separation of the water
column and the formation of a vapor void over the entire cross section
of the pipe.
For this case, the disappearance of the cavity is cal-
culated to occur at the time when the cavity size, V, vanishes.
At
that instant, an accoustic overpressure is produced of amplitude:
AP = pc (Vu - V)
(12)
A second type of cavitation consists of a distributed void, without
localized separation.
The void volume is again calculated using
eq. 10 and in addition, the characteristics solution for the nodal
pressure is continued.
The cavity is taken to vanish when either the
computed cavity size, f, goes to zero, or when the pressure at the
cavitating node is calculated to be greater than the vapor pressure,
AMIII
III Al1
ll llm
lIMINIM M
Ill,
11Al
-
.
whichever occurs first.
-21This second type of cavitation models the
formation of a void consisting of very small vapor bubbles distributed
uniformly throughout the "cavity".
In this case the characteristics
solution is still valid, although the calculations will be somewhat in
error because they fail to account for the change in wave speed caused
by the presence of the vapor bubbles.
This error is equivalent to an
unaccounted for increase in the length of the piping network.
If the
cavitation region is small in length, compared to the overall length
of the system, the error will be small.
For better accuracy, Wylie &
Streeter (1978) outline a procedure which accounts for both variable
wave speeds and column separation at all computational modes in the
network.
The cavitation model is completed by specifying where to allow for
column separation and where to assume "bubbly" cavitation.
Fig. 4
illustrates some of the numerical results for the model calculations
corresponding to the steel test section.
If column separation is
allowed to occur everywhere in the network (Nseparation = 33), the
solution in fig. 4a results.
The pressure spikes occuring after the
second depressurization represent the random collapse of the cavities
formed at nodes throughout the piping network.
By allowing column
separation only at a few nodes in the region of the initial column
impact, the solutions in figs. 4b, c & d result (Ns = 8, 4, 1 respectively).
High speed films taken of several water hammer events (Gruel 1980) show
that column separation does in fact occur in this region.
Comparison
of the experimental results with fig. 4 suggests that allowing column
separation elsewhere in the system model does not accurately represent
the actual events.
4000
iI
I
,
lI
.
i
,
g
3500
3000
AC
a- 250
11
WJ2000L)
i-i,
Lu 1500
1000
C1
a1000
500
"
0
0.00
0
500002
0 04
0.0G
0.09
0.10
TIME
0.12
0.1e
0.20
~~ALA&
0.00 0.02
0.04 0.06
0.06 010
(S)
(c) NS =
4000
0.14 0.16
TIME
(b)
33
0.2 044
0.6
048
AB
.
0.20
CS)
N =8
r
3500
-
3000
a 2500
W 2000
(i
1500
0.00
0.02
0.04
0.06
0.09
0.10
TIME
0 12
0.14
01G
0.18 0.20
0.0
0.02
0.04 0.06
08
CS)
(c) Ns= 4
00
TIME
(d)
Fig. 4 Numerical Results with Column
Separation at Ns Nodes.
(steel Wee!tes,
test secti.on;
P
= 310 kPa)
s%-.tio;
P0 =30ka
2
CS)
Ns= I
04
0.2
-MI
iE1
INWil
H
I
-23Boundary and Initial Conditions
At the ends of the piping network, boundary conditions are substituted for one of the characteristic equations.
For the closed end,
the boundary condition is zero fluid velocity.
At the reservoir end,
kinetic energy losses are taken into account.
For flow between the
piping network and the reservoir, the effects of convective acceleration
and friction on pressure are:
po
Pe
Pe
where
=
o
(
+ K)
pV2
(13)
(V > 0)
(V < 0)
(14)
Pe = Pressure at entrance to piping network
P = Reservoir pressure
V = fluid velocity out of reservoir
K = entrance loss coefficient.
The transient calculations begin at the instant of the initial
column impact (t = 0), following steam bubble collapse.
The initial
conditions for the model correspond to the system pressures and velocities
at that time.
The pressures are set equal to the experimental initial
conditions for the reservoir (P ) and steam (Ps).
The fluid velocities
are everywhere zero, except for the velocity of the water column at
impact.
This velocity is computed from the experimental results,
using Eq. 6 in the following form:
V21
impact
pcT
(15)
I = impulse of initial pressure spike
T = time duration of initial pressure spike
Appendix B contains the necessary data compiled from the experimental
runs.
-24-
Deflection Model
Many structural codes have been developed to determine piping
network responses to specified forcing functions (see, for example NUPIPE
and PIPERUP).
The model developed here is not intended to duplicate or improve
the work done in these codes, but instead is to provide the basis for a
simple analysis of pipe deflections in the experimental network due to
water hammer tests.
In the experimental network, the unrestrained "U" shaped pipe (fig. 2)
is modeled as a pair of cantilevers joined at the free ends.
The end
shear created by the cross piece is neglected (an assumption justified
by the small lateral displacements) as is the added mass of the cross
piece (which is small compared to the total mass of the test section).
The forces producing lateral displacements of the test section result
from unequal pressure forces acting at the two elbows (fig. 2). The
magnitude of the forces are equal to the pressures inside the pipe at
the elbows, less the atmospheric pressure outside, multiplied by the
pipe cross sectional area.
Since the pressure forces at the elbows act
in opposite directions, no unbalanced forces, result when the elbow pressures are equal.
For a given time, the total force on the test section
is written as:
F(t) = [(PA(t) - PB(t)] A
where PA and PB refer to the pressuresat elbow A and elbow B respectively
(see fig. 2).
By convention, forces and displacements in the direction
of elbow A have been defined as positive.
The forces acting at elbow A
are then always positive and the forces on elbow B are always negative.
-25Due to the presence of the cross piece, F(t) is evenly divided between
the two cantilevers.
It is assumed that the two cantilevers vibrate in
phase, therefore the displacements can be determined by looking at a
single cantilever excited by a force of F(t)/2 concentrated at the end.
The governing equation and boundary conditions for a cantilever
beam are:
y
+
Wx1 2 at2
a
ax
with
and
f(xt)
(17)
El
a2 = EI
pA
(18)
Y(x=O) = 0
--Y (x=L) = 0
ax
xx=o)
ax
where
0
-Y
ax
(x=L) = 0
(19)
E = modulus of elasticity
I = cross section moment of inertia
p = material density
A = cross section area
f(x,t) = forcing function in x and t
Since the pipe was filled with water, the product pA was taken to be equal
to the sum of the masses of the pipe and the water, per unit length.
Damping effects are neglected, a simplification which appears to be
consistent with observations made during the time periods of interest
(the first few cycles of vibration).
The solution to eq. 17, for forced vibrations, can be expressed as
a series expansion of the normal modes of Vibration (Graff 1975):
-26Vn(x)
2
Y(x,t) =
n)
'n
pLn=l
where
iI
Yn(u)du
n0f
j
f(u,T) sin cn (t-T) dT (20)
Y (x) = normal modes of vibration
n = natural frequencies of vibration
Fig. 5 illustrates the first four modes of vibration for a cantilever
beam.
The corresponding frequencies of vibration are given by:
n
where the
=
aB n2
(21)
n's are solutions to the frequency equation:
cos
nL cosh BnL = -l
(22)
The first few roots of eq. 21 are:
6 L = 1.875
a3L = 7.855
(23)
2L = 4.695
84L = 10.996
Given that the forces on the test section are concentrated at the ends
of the cantilevers, they can be represented, consistant with eq. 16, as
f(x,t) = F(t) 6(x-L)
where
(24)
6(x-L) = 1 for x = L
= 0 for x
L
Substituting eq. 24 into eq. 20, the solution becomes:
Y(x,t) = 2
pAL nl
Y (x) Y (L) ft
n
n(L)
F(T) sin w (t-T) dT
Wn
n
(25)
By specifying F(T), eq. 25 can be solved to obtain the transient pipe
deflections at any position x.
The forces on the test section, given by eq. 16, can be determined
2.0
1.5 1.0 0.5 0.0
-0.5 Z
-1.0
-
-1.5
-
-2.0
0.0
0.1
0.2
0.9
0.4
0.5
0.6
0.7
0.9
X/L
Fig. 5 Normal Modes of Vibration for a Cantilever Beam
0.9
t0
-28from the pressures calculated from the wave propagation model.
For the
nodal representation of the steel test section, elbow A is represented
by node #22 and elbow B is represented by node #25 (see fig. 2 and
fig. 3).
Fig. 6a shows some sample pressure histories computed at these
nodes from the wave propagation model.
The very short duration of the
pressure spikes (compared with the natural period of vibration of the
test section: Eq. 21) suggests that the pressures can be accurately
represented by a series of impulses.
The trapezoidal rule was used to
integrate the pressure histories, in time, to obtain the impulse
representations.
Fig. 6b illustrates the impulse representations for
the corresponding pressure histories in fig. 6a.
The magnitudes of the
impulses are equal to the areas of the pressure spikes they represent
and the application times correspond to the midpoints of the integrated
time intervals.
The reference pressure for the calculations was chosen
so that only the high pressure spikes were included in the integrations.
The large magnitude pressure spikes contribute to most of the force
applied to the pipe structure.
Numerically, the reference pressure is
immaterial since the force on the test section is proportional to the
difference in the elbow pressures and not their magnitudes.
The magnitudes
of the initial pressure impulses are actually little changed (-10%) for
reference pressures chosen between 100 kPa (atmospheric pressure) and
500 kPa.
When higher reference pressures are used, the times of
application of the impulses shift closer to the time of maximum pressure
for the pressure spikes.
It is apparent that the impulse representation
becomes more inaccurate for times after the initial pressure spikes when
the pressure fluctuations are of the same order of magnitude as the
system
natural frequencies.
However, only the first few pressure spikes
Elbow (8)
Elbow (A)
r%
EL2500
a- 2500
(a)
W 2000
W-2000
n
(I)
u 1500
a.
Lu 1500
Er
Er
a-
0.00
0.02
0.04
o0.06
010 0.12
0.0.
TIME
0.14 0.16
CS)
0.18
0.00
0.20
(0)
0.02
0.04 0.06
0.0
0.10
0.12 0.14
Elbow PressureS
16
0.19 0.20
CS)
TIME
7
(b)
0.00 0.02
0.04 0.06 0.09
0A0 0.12
0.14 0.16
0.19G 0.20
0.00
0.02 0.04
0.06
0.09 0.0
0.2 0.14
0.6
0.19
0.20
TIME CS)
TIME (S)
(b) inteqgrted Elbow Pressures
Fig. 6 Reduction of Pressures to Impulses (Steel Test Section; Po
=
310 kPa) Preference = 500 kPa.
-30-
have a significant effect on the subsequent test section deflections.
For a single impulse of the form
F(t) = F6 (t-T)
(26)
applied to the end of a contilever, eq. 25 becomes:
Yn(L)
-oY n(x)
An=l
n
F
Y(xt) =
(27)
sin wn (t-r)
(A factor of 2 has been removed from the numerator to account for the
fact that the forces are evenly divided between the two cantilevers.)
Eq. 26 can now be summed over all of the impulses occuring at the two
elbows to obtain the test section deflection history.
show
Figs. 7 and 8
the deflection results for the steel test section.
These solutions
are based on the impulses occuring within the first 50ms after the initial
water column impact (t=0).
The reference pressure (PR) used to integrate
the pressure histories was 100 kPa for fig. 7 and 500 kPa for fig. 8.
Table 4 lists the impulse data.
In the figures, the solutions correspond
to the superposition of up to the firstfour modes of vibration.
The
maximum amplitudes for PR = 500 kPa (fig. 8) are almost 50% less than
those for PR = 100 kPa (fig. 7).
Table 4 shows that the difference in
the impulse magnitudes cannot account for the change in displacements.
However, there is a large change in the application times for some of
the impulses.
Since the pressure forces at the elbows appose one another:
F
Y ~pAWn
For small
T1
(m
(sin
n
(t-T1 ) - sin (wn(t-T 2 ))
and T2
~ F
max
F
F T TA)
pAL (B -- T
where F is an impulse.
(28)
TB - TA is roughly equal to the wave transit time
2.5E-3
-
2.0 1L5 -
0 1.0
1.5
-
S0.5
-
z
I-
S-0.5
-1.0
H
o -1.5
-2. 0
-2.5
-3.OE-3
-0.;2
0.0
0.2
0.4
0.6
0.8
i.0
-0.2
0.0
0.2
0.4
TIME CS)
06
0.9
.0
06
0=3
1.0
TIME CS)
(a) I M de
2 Modes
(b)
8E-3
-0.2
0.0
0.2
0.
0.6
0.9
1.
-0.2
0.0
TIME (S)
(c)
Fig. 7
3 Modes
Superposition of Natural Modes of Vibration (Steel
Preference = 100 kPa.
0.4
0.2
TIME (S)
(d)
4 M*oes
Test Section, P = 310 kPa)
2.0E-3
1 5E-2 r
1.5
0.5
z
U'
-0.5 Li
0.0
-1.5
-
0.2
-1
00
0.2
0.4
0
.
LO
ILf
-1.5E-3
-02
0.0
0.2
0.4
0&
0.
1.0
-1.0 -2.OE-3
-0.2
(b) TIME
2
0.0
0.4
0.2
-02
0.0
0.2
Mode
0.4
(c)
06
0.9
LO0
0.6
0.9
1.0
TIME CS)
TIME CS)
(o) I
CS)
(b)
0.6
0.9
1.0
-0.2
0.0
02
2 Nkxles
0.4
TIME
CS)
TIME (S)
3
Modes
(d) 4 Modes
Fig. 8 Superpositi on of Natural Modes of Vibration (Steel Test Section; Po = 310 kPa)
PReference = 500 kPa.
Table 4
Impulse Data for Steel Test Section
(P0 = 310 kPa)
Elk~
reference
= 100 kPa
Preference
= 500 kPa
El b w R
A
SA
Impulse
Time
(ms)
Magnitude
(kPa.s)
Duration
(ms)
3.2
7.2
6.2
3.6
7.24
6.8
10.8
7.4
5.2
9.2
6.5
3.4
-1.6
-
-
-
12.6
0.9
2.8
-
16.8
20.6
43.2
0.7
0.1
0.3
3.4
0.6
0.4
16.6
20.0
43.8
0.7
0.2
0.3
-0.2
-0.6
0.6
45.2
3.0
3.0
45.8
3.0
2.8
0.6
0.4
3.0
48.4
0.2
0.4
47.8
0.2
0.4
-0.6
4.4
6.9
3.0
5.0
7.0
3.0
0.6
9.6
6.6
3.0
9.0
6.4
3.0
-0.6
12.6
0.5
0.8
13.2
0.5
0.8
0.6
15.6
0.2
0.2
16.0
0.4
0.6
0.4
20.8
0.1
0.2
20.2
0.1
0.2
-0.6
43.2
44.8
48.4
0.3
2.9
0.2
0.4
2.0
0.4
43.8
45.4
47.8
0.3
2.87
0.2
0.4
2.0
0.4
0.6
0.6
-0.6
B
Time
(ms)
Impulse
Magnitude
(kPa-s)
________________________________
Duration
(ms)
TB - TA
(ms)
0.4
0.6
________________________________,__,
________________________________
_______________________________
between the two elbows.
-34Eq. 28 shows that the maximum deflection is very
sensitive to the application time assigned to each impulse.
This can
account for the differences seen in figs. 7 and 8 (see Table 4).
to the experiments, fig. 8 shows much better agreement.
section includes further discussion of this observation.
Compared
The results
-35Bubble Collapse Model
The preceding analysis was concerned with the system responses
occuring after the initial steam bubble collapse.
In this section, a
bubble collapse model is introduced that can be used to describe the
events leading to water column impact.
Fig. 9 illustrates a representation of the water hammer generator
at a time shortly after the opening of the reservoir valve.
The initial
length of the vapor cavity is k and the position of the moving water
column is represented by x. The basic conservation equations can be
applied to the control volume drawn in fig. 9. The continuity and
momentum equations are:
pdV +
continuity: -
pVdA = 0
(27)
c.v.
C.V.
momentum:
EF = -
v.
(pV)
dV +
p (V*dA) V
(28)
In terms of the quantities in fig. 9, eqs. 27 and 28 reduce to the
following forms:
V
dx
(29)
dt
P=
pt
(Vx)
- 2
(30)
dt
where
AP = P - Ps
and
P + pgh - (l1+K)
ps = steam density
p
= water density
AP = overall pressure drop
pV2
Ps
(31)
-36-
P0
h_..!.:.....:: ... ..... , .i ! i:. '".. :.
T1
I
Reservoir
..
Moving
Water Colu m n
X=O
I
SI.:. :;..."
I..
.".
I." .
Control Volume
Cavity
I
I
'TP
IIII~SIq
I
I
-...
Stationary
Steam - Water
Interface
Statio'
Water C lumn
.. .
Fig. 9:
•... . *
Control Volume for Steam Bubble Collapse Analysis.
-37In eq. 31, ps and AP are functions of time.
For the very fast
transients involved, conduction heat transfer from the steam bubble
is negligible.
The enthalpy flux out of the steam bubble is almost
entirely due to condensation.
The energy equation for the steam
bubble can be written as follows:
a
p edY
8t ; s s
=
-
dV
lch + Ps dts
c
(32)
e = specific energy of steam
where
h = specific enthalpy of steam
Vs = volume of the steam bubble
and
mc = mass condensation rate of the steam.
Assuming an ideal gas and a uniform temperature profile in the
steam bubble, eq. 32 becomes:
(33)
dX
d
S(c T1 sA (Y -x) + mcCpTl " Ps A
= 0
(33)
where c and cV are the constant specific heats.
The condensation rate, mc, can be defined as:
m=
c
(34)
d (P A( x:)
dt
s
o-x:))
The equation of state is:
Ps = psRT 1
(35)
where R is the universal gas constant.
Eqs. 29, 30, 31, 33, 34 and 35 represent 6 equations in seven
unknowns (x, V, AP, Ps' Ps, T1 and
c ) all of which are functions of
time.
A seventh equation is obtained by specifying the condensation
rate
c. Appendix C presents the equations in dimensionless form and
lists the appropriate variables.
The equations are solved numerically
-38using a Runge-Kutta integration technique.
Gruel (1980) uses this
method in an analysis of his experimental data.
Creare (1977) presents
solutions to a somewhat simpler set of equations, assuming an isothermal
bubble collapse and various constant condensation rates.
-38V Results
Appendix A contains sample oscillograms taken for each set of
conditions tested in the experiments.
The tables in appendix B contain
the numerical data complied from all of the recorded test runs.
ponding theoretical calculations are included for some cases.
CorresExperimen-
tal reproducibility is discussed at the end of this section and in
appendix D.
Wave Propagation Experimental Results
Fig. 10 compares the pressure histories (PA' t > 0) for the three
test sections, with the back pressure (P ) equal to 170 kPa.
compares the corresponding traces for P = 310 kPa.
Fig. 11
The most apparent
differences between corresponding oscillograms in figs. 10 and 11 are
the magnitudes of the initial pressure spikes, which range from 3000 kPa
for Po = 170 kPa to 5000 kPa for Po = 310 kPa.
For a given back pressure,
the early features of the pressure histories are remarkably similar for
the different test sections.
In the expanded time scales of figs. 12 and
13, it is clear that the initial pressure spike amplitudes and structures
are nearly identical.
The initial pressure spikes attain a maximum
average peak pressure about which is. superposed other oscillating
pressure variations.
The average peak pressures range from 2200 kPa
for PO = 170 kPa to 3000 kPa for P0 = 310 kPa, indpendent of the test
section.
The tables in appendix B contain the impulse magnitudes and
time durations for several initial pressure spikes from which the
average peak pressures can be determined.
The similarity of the initial
pressure spikes is expected since they are determined solely by the
-39-
5000
Steel Pipe
4000
3000
PIA
(kPo)
(a)
2000
I000
170I
....
0
' r n O
7"qT
1
40
80
120
160
200
160
200
t(ms)
5000 Copper Tubing
4000 -
3000-
PIA
(k Pa)
(b)
20001000
170
0
40
80
120
t (ms)
5000 CPVC
Pipe
4000 -
3000
PIA
(c)
(kPo)
2000 I000 170
0
Fig. 10
F
'I'
40
80
!
120
t (ms)
I'T
160
20 0
Experimental Pressure Histories for 3 Test Sections
(P = 310 kPa).
-40-
5000 Steel Pipe
4000
PIA
(a)
(kPo)
3000 2000 1000 170
0
40
80
120
t(ms)
0
40
80
160
200
160
200
160
200
200
5000Copper Tubing
4000-
(b)
PIA
(k PO)
30002000 ILJUU -
170
120
t(ms)
5000 CPVC Pipe
4000PIA
(c)
(kPo)
300020001000
J,
170
0
Fig.
40
80
120
t (ms)
Experimental Pressure Histories for 3 Test
(P = 310 kPa).
Sections
-41-
5000 Steel Pipe
4000 3000 -
PIA
(a)
(k Po)
2000 I000 170
0
10
20
30
t(ms)
40
50
10
20
30
t (ms)
40
50
5000 -1
Copper Tubing
4000-
(b)
PIA
(kPo)
30002000I000170-
CPVC
Pipe
5000
4000
3000(c)
PIA
(kPo)
2000 1000
170
0
I
10
'
I
20
'
I '
30
I
40
'
I
50
t (ms)
Fig. 12
Experimental Pressure Histories for 3 Test Sections
(expanded time scale, P0 = 170 kPa).
-42-
5000 Steel Pipe
40003000-
PlA
(a)
(kPa)
20001000170
-T
0
I0
20
30
t(ms)
40
50
10
20
30
t(ms)
40
50
1O
20
30
40
50
.,Mm..
5000 Copper Tubing
4000-
(b)
PIA
(k Pa)
300020001000170-
CPVC Pipe
500040003000-
(c)
PIA
(kPo)
2000I000170
-
0
Fig. 13
Experimental Pressure HistotreSor 3 Test Sections
(expanded time scale, P = 310 kPa).
-43characteristics of the water hammer generator and are not affected by
the wave propagation dynamics through the test section.
The reflected spikes from the closed end of the test section are
recorded shortly after the initial pressure spikes (t = 10 ms for the
steel and copper test sections, t : 17 ms for the CPVC test section).
The time intervals between the initial impact spikes and the reflected
spikes are determined by the acoustic wave propagation times through
twice the length of the test section (At = 5 ms for steel and copper,
At = 10 ms for CPVC).
Immediately after the reflected pressure spikes,
the oscillograms show periods of very low pressure that correspond to
water column spearation in the region of the initial water column impact.
For the higher back pressure cases (P = 310 kPa) shown in figs. 11 and
13, the periods of cavitation and column spearation are shorter than
those for the lower back pressure cases (P = 170 kPa) shown in figs. 10
and 12 (Atcavitation
cavitation
= 310 kPa, Atcavitation
15 ms for P0oaitto
40 ms for
As expected, these cavitation periods appear to be
Po = 170 kPa).
unaffected by the test section wave propagation dynamics.
tion is terminated by a recollapse
the cavitation.
Column separa-
of the vapor bubble generated by
A second water column impact occurs, analogous to the
one following the original steam bubble collapse.
For example, the
second impact time is at t = 60 ms for the steel test section (fig. 10a)
and at t - 35 ms for the CPVC test section ,(fig. 13c).
As before, the
impact pressure amplitudes increase with increasing back pressures.
The
pressure history after thesecond water column impact consists of cyclic
cavitations and bubble collapses qualitatively similar to the initial
transients.
The transients finally die out and the pressures in the
test section end at the reservoir back pressure.
-44The experimental reproducibility of the pressure transients was
generally excellent.
Appendix D includes some typical oscillograms
recorded for successive runs under identical conditions.
The initial
impact peak pressures generally varied less than 10% for a given set
of experimental conditions.
same repeatability.
The average peak pressures had about the
The time periods between initial and reflected
spikes were also consistent within about 10%.
The time-durations of
the cavitations following the reflected pressure spikes varied less than
20%.
Wave Propagation Theoretical Results
Theoretical results were obtained for the steel and CPVC test
sections.
The necessary inputs to the computer model are described in
the section on theoretical analysis.
A sample of the input is shown in
table 5 for the CPVC test section (P0 = 310 kPa).
The first two sections
of the CPVC nodal representation are identical to those for the steel
test section (see fig. 3).
cases.
The initial conditions ara similar for both
The only empirical inputs are the final water column impact
velocity and the specification of the regions in which column separation
is allowed to occur.
Eq. 15, with the experimental data in appendix B,
is used to calculate the impact velocity.
This equation appears to lead
to a good agreement between the calculated peak pressures and the
experimental average peak pressure for the initial impact pressure spikes.
Figs. 14 and 15 illustrate the numerical results obtained for the steel
and CPVC test sections (Po = 310 kPa) with the corresponding experimental
tests.
Gruel (1980) outlines a calculation of the impact velocity using
the bubble collapse model summarized in appendix C.
There is still,
--
~~IIIYIYYIII
r ,
~IIIII~
' ~YYIIYII
0I
il ll I
-45TABLE 5:
Sample Input for Wave Propgation Model
(CPVC Test Section; P = 3i 10 kPa; Ps (initial) = 170 kPa)
0N
Nodal Representation
Section
5
Material
Steel
Lexan
Steel
CPVC
CPVC
Length (L)
(m)
Adjusted Wave #Division (Ni)
Diameter (D.)
speeds(CJ) (m/s)
(m)
.88
.27
.0467
.0381
1366
457
.69
.0309
3.30
.0188
1374
689
3.30
.0188
689
Total # odes = 62 (two nodes per junction)
Initial Condi ions - (Refer to fig.
Pressures
Po
=
310 kPa (Reservoir Pressure)
0
Pl =P2 = P3 = P4 = P5 = P6 = P7 = 310 kPa (neglecting kinetic
Energy Losses)
All other Pn = 170 kPa (steam pressure)
Velocities (from eq. 15, appendix B)
I = 20.3 kPa - s
Initial Impulse:
Time interval:
Vimpact
=
T
=
.005 s
2 (20.3 x 103)
(I x 103)(1391)(.005)
= 5.84 m/s
V1 =V 2 = V3 = V4 = V5 =V 6 = V7 =5.84 m/s
Vu
= Vu2 = Vu3 = Vu4 =Vu 5 = Vu6 = Vu7 = 5.84 m/s
All other Vn and Vun = 0
Cavitation Data
1) Initially there is no cavitation anywhere in the network.
2) Column separation is only allowed in nodes 5, 6, 7 and 8.
(b) Theory
(a) Experiment
5000
5000
PIA
(kPa)
Pressure at entrance
to test section
4000
4000
3000
)-
3000
2000
2000
100C
1000
)-a
17C
40
0
80
10
160
204
0
t(ms)
(kPo)
Pressure at entrance
to test section
(expanded time scale)
120
160
200
40
50
500
400
4000
3000
3000
2000-
2000
W
1000
170-
80
1 (mS)
5000
PIA
40
TT
0
I0
Fig. 14:
20
30
t (ms)
1000
40
0
10
20
30
t (ms)
Theoretical Pressure Results Compared to Experiment for the
Steel Test Section (P = 310 kPa).
(0)
(b)
Experiment
Theory
50004000PIA
(kPa)
3000
Pressure at entrance
to test section
2000
1000
1700
0
I'so
80 '1 120
t (ms)
40
200
t (ms)
5000-
5000
4000-
4000
IA
(k Pa)
Pressure at entrance
to test section
(expanded time scale)
1
3000-
3000
2000
2000
F'
1O00-
I000
170
1
0
C --S i 10
1
1 I 1
20
t(ms)
Fig. 15:
30
I
1
40
ioo
1A
S1 i
1
10
20
IL1
4*Ai.
AI
30 40
t (ms)
I
I
50
Theoretical Pressure Results Compared to Experiment for CPVC
Text Section (P = 310 kPa).
-48however an empirical input to that model which involves the estimation
of the heat transfer rate from the steam bubble to the water column.
The numerical results in figs. 14 and 15 show good agreement with
experiment during the early phases of the transients.
The agreement is
better for the steel test section, perhaps because the nodal representation used for the steel test section more closely approximated the
actual system.
After the first two cavitations, the agreement between
theory and experiment is poor. This decay phase of the pressure
transients is of less interest and has little effect on the induced
test section deflections.
-49Displacement Experimental Results
Figs. 16-19 show experimental results for the displacement histories
of the 3 test sections.
These traces include the responses at locations
D1 (figs. 16, 17) and D2 (figs. 18, 19) (see fig. 2) for back pressures
of P = 170 kPa and 310 kPa.
The displacement magnitudes for the
respective test sections increased with increasing back pressure.
The
ringout frequencies were fairly constant although the observed amplitudes
of higher mode vibrations varied at different points (see for example
D2 (x = L/2) for P0 = 310 kPa).
For a given back pressure, the displace-
ment frequencies and amplitudes varied considerably with the different
test section materials.
Figs. 20 and 21 plot the first mode displacement
frequencies and amplitudes against theoretically motivated parameters
characterizing the test section properties.
For all of the experimental
runs, the initial displacements were in the negative direction (towards
elbow B, fig. 2).
Free constant amplitude vibrations corresponding to
the ring out of the test section were observed for times greater than
about 20 ms.
This supports the assumption that the important part of
the pressure transients occur within a time period that is much smaller
than that for the test section vibrations.
Displacement Theoretical Results
Numerically calculated displacement histories have been derived using
the wave propagation calculations and the deflection model outlined in the
previous section.
The wave propagation model provides the impulse data,
similar to that shown in table 4, for the two elbows.
pressure of 500 kPa was used in the integrations.
A reference
The test section
parameters defined for eq. 17 are listed in table 6. Fig. 22 shows the
-50theoretical results for the steel and CPVC test sections using a single
mode analysis at the position x = L (D1 ).
The agreement with experiment
is shown in figs. 20 and 21, where the calculated maximum amplitudes
and first mode frequencies are plotted with the experimental results.
Other calculations are included in appendix B. The predicted frequencies
appear to agree very well with experiment.
The computed maximum dis-
placements show good agreement for the steel test section but overpredict
the CPVC test section displacements by over 100%.
-51-
2.0Steel Pipe
1.0DI
(a)
0-
(mm)
-1 .0
-2.0
0.4
0.2
7.5
1.0
0.8
0.6
t(s)
Copper Tubing
5.0 2.50I
(b)
0-
(rMM)
-2.5
-5.0
-
-7.5
0
I
0.2
'
I
0.4
15.0CPVC
'
I
0.6
t (s)
'
I
0.8
'
I
1.0
Pipe
10.0
5.0
DI
(rMM)
(c)
0
-5.0
-I0.01
-15.0
Fig. 16
0-
0.4
0.8
1.2
t(s)
1.6
2.0
Experimental Deflection Histories for 3 Test Sections
(x = L, P0 = 170 kPa).
-52-
2.0Steel Pipe
(a)
1.0-
DI
(rMM)
-1.0
-2.0
0
0.4
0.2
0.6
1.0
0.8
t(s)
7.5Copper Tubing
5.0
2.5-
D0
(b)
(rMM)
0-
-5.0
-7.5
I
0
I
0.4
0.2
15.0CPVC
Pipe
0.6
t (s)
0.8
1.0
10.0 5.O-
(c)
0
DI
(mm)
-5.0 -10.0
-15.0
T
I
0.4
0
Fig. 17
Experimental
(x = L, Po
=
-
I
0.8
I
,
1.2
t (s)
1.6
1
1
2.0
Deflection Histories for 3 Test Sections
310 kPa).
-53.0 -
Steel Pipe
0.5D2
(a)
O-
(mm)
-0.5
I I
0.2
I I I I I 1 1
1.0
0.8
0.6
0.4
t (s)
I
0.2
I
3.0Copper Tubing
2.0I.O D2
(b):
(m m)
-1.0
-20.
-3.00
0
0.4
7.5
CPVC
I
0.6
'
0.8
1.0
t (s)
Pipe
5.0)
2.5(c)
D2
(mm)
0-
-2.5
-5.0
-7.5
I
0
Fig. 18
0.4
I
I
I 1
0.8
1.2
t (s)
1
1.6
2.0
Experimental Deflection Histories for 3 Test Sections
(x = L/2 Po = 170 kPa).
-541.0-
Steel Pipe
0.502
(a)
0-
(mm)
-0.
-1.01
I
0.2
0
I
II
I
I
0.4
0.6
0.8
1.0
0.8
1.0
'1
t(s)
3.0
Copper Tubing
2.0
.0 (b)
D2
(mm)
0-
-I.0 -2.0-3.0-
0.2
0.4
0.4
0.8
0.2
1
0.4
7.5
CPVC
0.6
0.6
t(s)
0.8
1.0
Pipe
5.0
2.5
02
(c)
0
,.
'
(mm)
-2.5
-5.0
-7.5
0
Fig. 19
1.2
t (s)
1.6
2.0
Experi mental Deflection Histories for 3 Test Sections
(x = L /2 P0 = 310 kPa).
pA
W VS.
20EeorSteel
Experiment
20
I
Wl
W
(S1)
'5
Copper
15
,
CPVC
50 --
S
,
Fig. 20:
3.73
I
(71
I
1,
1
5
10
,
I5
I5
A (S)
First Mode Vibration Frequencies for Test Sections.
16.99
I
I
20
25
20
Ymox
(MX1-3
(mxO )
15
I0
5
.04
.02
Fig. 21:
.06
.08
.10
.12
.14
pALW,
Test Sections for Similar Experimental Conditions
of
Maximum Displacements
(P
= 310 kPa, x = L).
-56Table 6
2)
(N/m
(m4 )
Test Section Parameters for Deflection Model.
Steel Test Section
207.0 x 10
CPVC Test Section
2.9 x 109
1.576 x 10-8
-8
1.881 x 10
(kg/m)
2.058
.7152
(m2/s)
39.81
8.733
(m)
2.87
2.87
(m- 1)
.6533
.6533
1 .636
1 .636
2.737
2.737
3.831
3.831
16.991
3.728
106.5
23.36
298.2
65.42
M4)
(m-
)
(mn1)
-l )
(m(m- l
)
(s-1
(s - )
(s-
)
(s-1)
584.4
128.2
(a)
(b)
Experimen
Theory
-A
-
1.01
Steel Pipe
DI
(rMM)
1.0 -O
0-
-I.0-
I.O -
-2.0 -T
0
0.2
0.4
I
15.0
0.6
t(s)
I
1
0.8
~-- J
-0.2
-0.2
L-
I
I
1.0
|
I
0.2 0.4 0.6 0.8
t (s)
1.0
,
CPVC Pipe
10.0 -
20.0
5.0DI
(MMr)
1i0.0
O-
0
-5.0 -
-10.0
-10.0-15.0-
-20.0
I
0.4
I
0.8
I
1.2
t (s)
Fig. 22:
1.6
'
I
2.0
-0.5
0
0.5
1.0
1.5
2.0
t (s)
Theoretical Displacement Results Compared to Experiment (Po = 310 kPa,
o
x = L).
-58-
VI Conclusions
The purpose of this study was to investigate various features
of water hammer transients generated by steam bubble collapse.
The
experimental work provided a body of data on water hammer pressure
transients and piping system responses under carefully controlled
repeatable conditions.
The simple experimental system was designed
to test all of the phenomena believed to be important in real piping
networks:
the collapse of an isolated steam bubble in subcooled
water, the propagation and reflection of acoustic pressure waves,
the cyclic cavitation and reimpact of the water column, and the
development of unbalanced loads inside the pipes leading to pipe
deflections.
The theoretical analyses illustrate the types of
computations necessary to explain the phenomena observed at the
different stages of the experimental transients.
Anticipating and predicting the extent of water hammer damage
to piping networks is the ultimate goal of this type of research.
A better understanding of the initiating mechanisms that lead to
steam bubble entrapment and the condensation processes that govern
steam bubble collapse is still required.
Creare (1977) provides
some limited discussion of a possible initiating mechanism.
Gruel
(1980) has investigated the dynamics of steam bubble collapse in
the water hammer generator. More work is required, however, before
these studies can be confidently applied to full scale systems.
The wave propagation model developed in part IV has shown that
-59pressure wave propagation and cavitation can be predicted quite
adequately, although accurate modeling of the cavitation dynamics
presents some difficulties.
The manner in which the pressure
transients induce unbalanced forces is qualitatively well understood.
The simple deflection analysis presented here produced
adequate and generally conservative agreement with experiment.
More sophisticated codes exist, however (for example NUPIPE and
PIPERUP) that could be used in this stage of the analysis.
In this study, the uncoupling of the wave propagation analysis
from the piping network structural response calculations was an
acceptable approximation of the experimental system.
This is an
important simplification since it allows the use of well established
acoustic wave propagation models to determine the pressure transient
"forcing functions."
A similar approach for full scale conditions
is probably possible, but requires a more formal justification
based on conditions corresponding to the full scale wave propagation
speeds and piping network natural frequencies.
-60References
A-Moneim, M.T. and Chang, Y.W. 1979 "Comparison of ICEPEL
Predictions with Single-Elbow Flexible Piping System Experiment"
Transactions of the ASME,Vol. 101 (May).
Creare, Inc. 1977 "An Evaulation of PWR Steam Generator Water
Hammer" U.S. Nuclear Regulatory Commission.
Cybernet Services, " PRTHRUST/PIPERUP Pipe Force and Whip Analysis"
and "NUPIPE II/THREAT Piping Analysis Programs" Control Data.
Graff, Karl F. 1975 "Wave Motion in Elastic Solids".
University Press.
Ohio State
Gruel 1980 Master's Thesis, Massachusetts Institute of Technology,
"Steam Bubble Collapse and Water Hammer in Piping Systems:
Experiment and Analysis"
Wylie, E. Benjamin and Streeter, Victor L. 1978 Fluid Transients.
McGraw-Hill International.
Timoshenko, S., Young, D.H., Weaver, Jr.W.,1974 Vibration Problems
in Engineering. John Wiley and Sons.
-61Appendix A:
Sample Oscillograms for each Set of Experimental
Conditions.
The following figures include some sample oscillograms for
each of the experimentally run test conditions.
Throughout the
text of the report, other oscillograms are illustrated to present
a larger cross section of the test data.
-625000
4000 -
PIA
(kPa)
3000 -
Pressure at entrance 2000to test section
(expanded time scale)
1000
170-:
0
I
I I
30
20
10
I
t
50
40
t (ms)
5000- 1
4000-
PIA
(kPa)
3000-
Pressure at entrance
to test sect ion
2000
0001
-120
(kPa)
Depressurization
-80
-40
U
170
"-
I
T
I
Fig.
Al:
I170
0
40
150
80
1
t (ms)
100
/
0
160
200
Steel Test Sec tion
).
(P0 = 170 kPa)
50
50u
12000P3
(kPa)
Pressure at closed
end
8000-4000170 !1
0
40
80
120
t (ms)
160
600
800
DI
(mm)
Pipe Displacement
at X= L
(compressed time scale)
200400
t (ms)
200
-635000 4000PIA
3000(k Pa)
Pressure at entrance 2000
to test section
(expanded time scale)
1000
170
b
t'ws
3!0
4b
5000"
4000
PIA
(k Po)
3000
Pressure at entrance
to test section
2000
I000
-80
-40C
I)IM0 0
170
P2
150
(kPa)
Depressurization 100
0 A 80
l0
1160
200
t (ms)
Fig. A2: Steel Test Section
(P0 = 240 kPa).
50
0 -1
12000
P3
(kPa)
Pressure at closed
end
8000
4000
I170
0
Dj
40
80
160
120
t (ms)
200
1.0-
(mm)
Pipe Displacement
of X=L
(compressed time scale)
-1.0I
0
1 I
200
I
I
I
400
600
t (ms)
f
I
800
I
-6450004000PIA
3000(kPa)
Pressure at entrance 2000to test section
(expanded time scale)
1000
170- r-
r
0
I
10
I
I
I
20
30
-- F--T---T
40
50
t (ms)
5000
4000
PIA
(k Pa)
Pressure at entrance
to test section
3000
2000
1000
)-T
-80
P2
-4C
170
10
40
0
170150
160
2
t (ms)
Fig. A3:
(kPa)
Depressurization t00
Steel Test Section
(P0 = 310 kPa).
500-
12000
P3
(kPa)
Pressure at closed
end
80001
4000-
1700
40
80
120
t (ms)
6IO
200
660
400
t (ms)
800
DI
(mm)
0-0
Pipe Displacement
at X= L
(compressed time scale)
200
-655000
40004
PIA
(kPa)
Pressure at entrance
to test section
(expanded time scale)
300020001000
1700
20
10
t (m.s)
Ms
30
40
50
t
50004
000
PIA
(k Pa)
Pressure at entrance
to test section
0002000
I000
-120
-'-i
-80
-40
170 4'
0
" 40
80
170
P2
(kPa)
S150
Fig. A4:
-100
Depressurization
10
t (ms)
160
200
Copper Test Section
(Po = 170 kPa).
12000
P3
(kPa)
Pressure at closed
end
8000
4000170-'
0
I-
40
I
I I
80
i
120
t (ms)
160
400
660
t (ms)
800
200
DI
(mm)
Pipe Displacement
at X L
(compressed time scale)
-1.0-
200
1000
-565000 4000IA
(k Pa)
Pressure ofatentrance
to test section
(expanded time scale)
30002000 -
1000170-
II
0
I
I
10
I
50
40
30
20
t (ms)
5000-4000PIA
(kPa)
(kPO)
Pressure at entrance
to test section
300020001000-
-80
P2
-40
170 0
17 0 L
40
1501
120
80
160
200
t (ms)
0,
Fig. A5:
(kPa)
100
Depressurization 00
Copper Test Section
(P = 240 kPa).
0
12000
P3
(kPo)
8000
Pressure at closed
end
4000-
170J
0
I
40
' I
80
I
t
120
t (ms)
'1 21
' 0
160
200
DI
(mm)
Pipe Displacement
at X L
(compressed time scale)
0
200
600
400
t (ms)
800 1000
-6750004000PIA
3000
(kPa)
Pressure at entrance
2000
to test section
(expanded time scole)
1000
170
0
10
20
.t
30
(ms)
40
50
50004000
PIA
(kPa)
Pressure at entrance
to test section
3000
2000
1000
-80
-40
170
S40
10I
t (ms)
8
170-
P2
150-
Fig. A6:
(kPa)
Depressurization 100
160
200
Copper Test Section
=
S(P 310 kPa).
12000 P3
(kPo)
Pressure at closed
end
80004000170- +
0
I
40
I
I
80
I
I
120
t (ms)
I
160
i
200
DI
(mm)
Pipe Displacement
atof
X= L
(compressed time scale)
0-
-I0
200
S
400
600
t(ms)
800
1000
-6850004000PIA
(k Pa)
3000-
Pressure at entrance 2000to test section
(expanded time scale)
1000170
i
Il0
lR
20
"
30
t (ms)
40
5000
4000
PIA
(k Pa)
Pressure at entrance
to test section
3000
2000
1000
-120
Lq
-80
-40 170o
40
80
-170
-150
P2
(kPa)
Depressurization
l0
t (ms)
Fig. A7:
160
200
CPVC Test Section
(P 0 = 170 kPa).
-100
12000 -
P3
(kPa)
8000-
Pressure at closed
end
4000170-F
0
40
80
120
t (ms)
160
200
10.0
DI
(mm)
Pipe Displacement
at X L
(compressed time scale)
5.0
0
- 5.01
10.r1
0
11
4
1
1I
.8
t
I
1.2
(s)
1.6
2.0
-6950004000PIA
3000
(k Pa)
Pressure at entrance
2000
to test section
(expanded time scale)
100030 40
50
t (ms)
5000
4000
Pt A
(kPa)
3000
Pressure at entrance
to test section
2000
1000
-80
-4 0
--
I 7A
Icy
jt
170-
pu
60
150
410
'4'o
18 10
'ab'
Fig. A8:
(kPa)
Depressurization I00-
-- -1
----
't60'
O
'1 i.o'
(ms)
200
CPVC Test Section
= 240 kPa).
(P
12000
P3
(kPoa)
Pressure at closed
end
8000
4000
l70-
0
40
80
120
t (ms)
160
200
10.0-t
DI
(mm)
Pipe Displacement
at XL
(compressed time scale)
-5.C
-
lo.oF
0
1
4
1I
6
.8
1
1
1.2
(S)
1
1
1.6
-
2.0
-7050004000PIA
(k Pa)
Pressure at entrance
to test section
(expanded time scale)
20001000-
60
17
io'30
40
50
t (ms)
5000
4000-
PIA
(kPa)
3000-
Pressure at entrance
to test section
20001000-
-80
I
I
-40
I
170-P
0
I
40
1 0
80
160
200
t ( ms)
170
P2
150
(kPa)
Depressurization 100
CPVC Test Section
(P = 310 kPa) .
Fig. A9:
50
0
12000P3
(kPa)
Pressure at closed
end
80001
4000
10
DI
(mm)
Pipe Displacement
at X=L
(compressed time scale)
0
40
80
4
.8
m120
t (ms)
160
200
1.6
2.0
5.0
0
-5.00
t
1.2
(5)
-71Appendix B Experimental Test Data
Table Bl :
Table B2:
Table B3:
-72Table BI:
P = 170 kPa
0
x=L
Initial
Depress.
Material (kpa-s)(s
Initial
Impulse
(kpa-s)(s)
Steel 1) 7.51 .126
2)
12.5
11.6
.005
Copperl)8.16 .112 10.7
2) 7.70 .110 4.8
.005
Theory
3)
4)
.994 18.81
16.99
.327 18.62
16.99
3.48
1.945 12.32 1.636 13.23
.004 1.964 12.69 1.964 13.37
2.055 12.57 1.945 13.51
2.409 12.82
3.227 12.82
2.727 12.82
5)
6)
7.93 .111
10.7
.005
Theory
CPVC 1) 7.83 .113 12.5
2) 7.31 .116 12.2
3)
Average
Theory
.336 18.48
.318 18.76
.836 19.04
7.51 .126 12.1
Average
Impact Velocity
Y
max
max m
(m/s)
(ms)
(mm) (s- 1) (mm) (s-1)
.005 1.109 18.21
.005 1.036 19.19
3)
Average
x=L/
2.388 12.67 1.848 13.37
10.00
10.00
.006 4.727
5.67 2.727
5.67
.006 7.000
3.455
5.45 3.773
5.39
5.64
5.061
5.50 3.250
3.73
5.65
3.73
.006
7.57 .115
12.4
7.70 .116
11.94 .005
3.08
3.56
Overall
Average
-
-
-
-
3.43
-73Table B2:
= 240 kPa
i
Material
Initial
Depress.
(kpa-s)(s)
Steel 1) 5.81
.076
2)
Initial
Impulse
(kpa-s)(s)
y
max
(mm)
(s-1)
x=L/2I
Y
Impact
max
Velocity
(mm) (s-'I
(m/s)
15.6
.006
1.164
18.08
.473
18.35
15.7
.005
1.000
18.76
.427
18.48
.450
18.42
x=L
3)
Average
Theory
18.62
5.81
.076
15.6
.006
1.082
16.99
Copperl) 6.99
.006
1.818
12.32
1.745
12.44
2) 6.14
.084
14.1
.008
2.600
12.57
1.636
12.57
3) 6.40
4)
.083 15.7
.007
2.691
2.255
13.09
12.57
1.527
2.364
13.09
13.51
5)
3.091
12.32
6)
3.045
12.69
7)
3.270
12.44
2.681
12.57
1.818
12.90
6.51
.084
15.1
.007
Theory
10.00
.084 15.7
.007
7.727
5.35
4.454
5.61
2) 6.53
3)
.084
17.2
.006
8.364
6.818
5.35
5.41
4.682
5.32
.084
16.5
.007
7.636
5.37
4.568
5.47
6.37
Theory
Overall
Average
3.73
6.35
.083 15.6
.006
4.34
10.00
1) 6.20
Average
4.49
16.99
.085 15.4
Average
CPVC
18.49
4.73
3.73
4.48
-74Table B3:
= 310 kPa
x=L
Material
Initial
Depress.
(kpa-s)(s)
Initial
Impulse
(kpa-s)(s)
x=L/2
y
max
(mm)
(s-1)
max
(mm) (s-1
1.127
18.76
.591
l Impact
elocity
)
(m/s)
Steel 1)
4.57 .064 16.0
2)
3)
18.8
.008 1.182
18.76
18.21
.509 18.21
4.57 .064 17.4
.008 1.155
1.1
18.58
16.99
.550 18.42
.65
16.99
.007
2.945
13.09
1.818
11.86
2) 6.40 .084 15.7
.009 2.182
12.32
2.309
12.08
3) 5.88 .072 18.8
.007 1.582
12.96
2.436
12.20
16.6
.008 1.164
12.69
2.200
11.86
5)
6)
3.227
12.96
2.400
13.44
3.818
12.57
7)
4.364
12.44
3.307
12.72
2.233
12.29
Average
Theory
Copper 1) 5.22 .079 19.0
4)
Average
5.83 .078 17.52
.008
.008
10.00
Theory
CPVC 1)
2)
.007 10.182
.009 8.545
5.27
5.27
10.090
5.24
5.52 .080 20.3
.008 9.606
25.0
5.26
3.73
5.52 .077
.008
4.90 .080 19.6
6.14 .080 20.9
3)
Average
Theory
Overall
Average
18.18
18.62
5.01
5.04
10.00
4.682
4.363
5.417
4.523
8.5
5.499
3.73
5.580
5.84
5.23
-75Appendix C Development of Bubble Collapse Model
The governing equations for steam bubble collapse, developed
in Section IV, can be summarized as follows (see fig. 9)
p2
AP
dx2
P=
P x dt 2
Momentum:
d__x
cdt
v
-x)) + m c T = PA dx
s dt
c p 1
S
Condensation rate:
AP
Pressure Drop:
(C4)
RT
1
S
d
dt
= -
c
=
(C3)
o
s
P =p
Eq. of state:
(C2)
T p A(
(c
d
Energy:
(Cl)
V
dx
dt
Continuity:
(C5)
P=+P gh - (1+K)- p V2 _P(C6)
P
Pe
(pA
5 ( o-x))
These equations can be nondimensionalized by defining the following
dimensionless variables (Creare 1977):
x
x*
-
=
t*
o
P*
Vat
Ps
T*=
AP
AP
AP*
P0S
P0
P
psoRT
o
1
p
o
(C7)
C* = Vg
Va
V* = Va
V
P
s
so
12
where Va = ( )
Pk
m
and Vg
=
Ap
(C8)
Ap s
Here C* is a dimensionless condensation rate which is determined
by the variables defined in eq. C8.
pso is the initial density
-76of the steam in the steam bubble.
Substitution into the governing
equations yields:
dx*
Continuity:
=
V*
V
(9
(C9)
dt
AP* = x* d2 x*2
dt*
Momentum:
Energy'.
d
(CI0)
(p* (1-x*) T*) + yC* T* = (y-1)
b
Enrg.dt*
P* dx*
(C)
dt*
c
where y = p
C
v
(C12)
Eq. of State: P* = p* T*
Condensation rate:
Pressure Drop:
C*
(C13)
d (p* (l-x*))
(l+K) V*2 - p*
AP* = 1 -
(C14)
2
Equations C9 - C14 represent 6 equations in 7 unknowns.
They
can be solved numerically be specifying C* as a function of time.
The Runge-Kutta numerical integration method can be applied if the
derivitives of each of the unknowns are specified.
The following
equations represent the necessary relationships:
dx*= V*
Position:
Posiion-dt*
Velocity:
Delcity:
Density:
= AP
x*
Vdt*
- C*
do* = 0*V*l-x*
dt*
(Cl5)
(C16)
(C17)
-77Temperature:
dT*
dt*
p* T* V* - T* (l-x*) dP*
dt*
-y C* T* + (y-l) P* V*
(c18).
p* ( -x*)
Pressure:
Pressure Drop:
dP*
dAP*
dt*
Condensation rate:
d
(p* T*)
-
V* dV*
dt*
(C19)
dP*
dt*
(C20)
C* to be specified
Some numerical solutions to these equations can be found in
Gruel (1980).
-78Appendix D Experimental Repeatability.
Generally, the run to run experimental repeatability was about
the same for the different experimental conditions,
Typical results
are presented here for the copper test section.
Figs. Dl and D2 show several pressure traces for the copper
test section recorded on two different time scales.
The traces
shown in (a) and (b) of the two figures demonstrate the typical
run to run repeatablility.
The "largest" run to run discrepencies,
for nominally identical conditions, are shown in traces (c) of
figs. Dl and D2.
Several factors may have played a role in explaining the
occasional run discrepancies evidenced by the differences between
traces (a) and (c) of figs. Dl and D2.
Residual air bubbles in the
test section could have significantly reduced the acoustic wave
speeds.
Air in the steam bubble could have had a similar effect
on the impact pressure magnitudes.
The boundary condition at the
downstream end of the test section may have occasionally deviated
from an ideal "closed end."
Any leakage or compliance in the
downstream valve (fig. 2) could have reduced the magnitude of
the reflected pressure spikes (at t = 10 to 15 ms).
The greater
time differences between the initial and reflected spikes in
figs. Dl and D2 (c), as compared to the corresponding intervals
in figs. Dl, D2 (a) and (b), could be explained by slower wave
speeds caused by air in the system.
I
-79The displacement data exhibited about the same level of
repeatability as the corresponding pressure histories. Fig. 03
show a representative set of traces for a fixed set of experimental
conditions.
Traces (a) and (b) illustrate the typical run to run
repeatability for the displacements at the end of the test section
(x = L).
In most cases, primarily the first modes of vibration
were excited.
Occasionally, the second mode of vibration was
superposed on the
03 (c).
1 st
mode excitations, as illustrated in fig.
Fig. 03 (d) shows a case where the displacements were
dominated by only the second mode.
The trace in fig. D3 (d)
generally occured when the pressure histories were of the type
shown in figs. Dl (c) and D2 (c).
The amount of second mode
ring out in the test section could have been affected by the
degree of damping arising from pipe connections or from friction
in the displacement transducer.
The overall displacement amplitudes
appear to have been dependent on the magnitude of the pressure
spikes.
-80-
5000
4000
()
pA
(kPo)
3000
2000
1000
170 --
120
t(ms)
160
200
120
t(ms)
160
200
120
160
200
40
80
0
40
80
0
40
80
0
5000
4000
(b)
PIA
3000
(kPO)
2000
1000
I000-I
170 --
5000'
4000
(c)
PIA
3000
(kPo)
2000
I000
170
t (ms)
Fig. DI:
Reproducibility of Pressure Data (Copper
Test Section, PO- = 310 kPa).
-815000 4000 (l)
PIA
(k Po)
300020001000170-
i0
'
I
20
5000-
PiA
(kPo)
I
30
t(ms)
40
40
50
50
1 7
4000-
(b)
'
30002000-
.A
1000170
30
5000.
4000-
3000
(c)
PIA
(kPo)
2000
1000
170
Fig. D2:
t (ms)
Reproducibility of Pressure
Data (expanded time scale;
Copper Test Section, P = 310
0
(0)
0!
(mm)
(b)
7.5
7.5-
5.0
5.0 -
2.5
2.5-
0
0-2.5
-2.5
-5.0
-5.0w
-7.5
-7.5
0
0.4
0.2
0.6
t(s)
0.8
0
1.0
0.2
0.4
0.6
0.8
1.0
0.8
1.0
t(s)
3.0-
D,
(rMM)
5.0-
2.0
2.5-
I1.0
0
0
-
-2.5 -
-1.0
-5.0-
-2.0
-7.5-
-3.0
I
0.2
Fig. D3:
'
I
0.4
I
0.6
t(s)
0.8
1.0
0
0.2
0.4
0.6
t (s)
(d)
(c)
Reproduci bility of Displacement Data (Copper Test Section, Po
x = L).
=
310 kP