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Analytical prediction on the pump-induced pulsating pressure in a reactor coolant pipe
Int. J. Pres. Ves. & Piping 52 (1992) 417-425
Analytical Prediction on the Pump-induced Pulsating Pressure in a Reactor Coolant Pipe K y e B o c k L e e , In Y o u n g I m & S a n g K e u n L e e Thermal Hydraulic Design Department, Korea Atomic Energy Research Institute, PO Box 7, Daeduk-Danji, Daejeon 305-606, Korea (Received 3 December 1991; accepted 22 December 1991)
ABSTRACT An analytical method is presented for predicting the amplitudes of pump-induced fluctuating pressures in a reactor coolant pipe using a linear transformation technique which reduces a homogeneous differential equation with non-homogeneous boundary conditions into a nonhomogeneous differential equation with homogeneous boundary conditions. A t the end of the pipe, three types of boundary conditions are considered--open, closed and piston-spring supported. Numerical examples are given for a typical reactor. Comparisons of measured pressure amplitudes in the pipe with model prediction are shown to be in good agreement for the forcing frequencies.
1 INTRODUCTION Flow-induced vibrations have led to many problems in reactor operation. These types of vibrations may be caused by vortex shedding, turbulence flow and pump-induced pressure fluctuation. Because of the general complexity of this problem, we restrict our analysis to the pressure fluctuations of the reactor coolant caused by p u m p pulsation. The pressure fluctuations generated at the p u m p discharge can interact with the reactor internals and produce damaging vibrations. The boundary condition at the p u m p end consists of a time d e p e n d e n t harmonic function describing the p u m p discharge pressures at different 417 Int. J. Pres. Ves. & Piping 0308-0161/92/$05.00 (~) 1992 Elsevier Science Publishers Ltd, England. Printed in Northern Ireland
418
Kye Bock Lee, In Young lm, Sang Keun Lee
pump forcing frequencies. The resolting boundary value problem consists of a homogeneous differential equation with nonhomogeneous time dependent boundary conditions. Pressure pulsations in a PWR were first studied by Penzes, ~ and later by Bowers and Horvay 2 and Cepkauskas. 3 They treated the problem of pulsations by introducing a body force concept, but they neglected to recognize the restrictions on the body force necessary to make the boundary conditions homogeneous. Fisher et al. 4 demonstrated an improved technique in solving one-dimensional wave propagation problems with time dependent boundary conditions. This technique introduces a transformation equation in terms of a new variable and auxiliary functions. The mathematical description is recast in the form of a nonhomogeneous differential equation. The boundary conditions become homogeneous by restricting the auxiliary functions at the boundaries. Due to the homogeneity of the resulting boundary conditions, the transformed system of equations is amenable to solution by separation of variables. Lee and Chandra 5 analyzed the pump-induced pulsating pressure in a reactor coolant pipe using this technique, but they missed the constraints on the auxiliary functions to make the boundary conditions homogeneous. Thus their solution cannot satisfy the boundary condition at the pipe end for the piston-spring supported end case. By considering the missing constraints on the auxiliary functions in the previous study we have developed an improved analytical model which satisfies all boundary conditions.
2 ANALYTICAL MODEL
2.1 Governing equation In the analytical model considered compressible and inviscid. Since the smaller than the static pressure, the used to reduce the Navier-Stokes equation: v2
here, the fluid is assumed to be pressure pulsations considered are small perturbation method can be equations to the following wave
C~o
where P = pressure pulsation, and Co = sonic velocity.
Pump-induced pulsating pressure in a reactor coolantpipe
419
If the variation of the pressure pulsations in the radial and circumferential direction is ignored, eqn (1) reduces to d2o~JP=0
(2)
The excitation for the system occurs at the pump discharge (x = 0). The boundary condition is P Ix=0= PD COS toot
(3)
where PD is the pressure wave at the pump discharge, and top is the associated pump frequency. 2.2 Boundary conditions The following three types of boundary conditions are considered at
x=t: (a) open end
PIx=L=0
(4a)
OPox x=L = 0
(4b)
(b) closed end
(c) piston-spring supported end
lOP -~x
Ap821)] K Ot2 Jx=a = 0
(4c)
Here, A is the pipe cross-sectional area, p is the fluid density, and K is the empirically determined spring constant. The geometry considered in this study is shown in Fig. 1. 2.3 Linear transformation Equations (2), (3) and (4) describe the boundary value problem with the time dependent non-homogeneous boundary condition. To render the boundary condition homogeneous the following equation is used:
P(x, t) = Q(x, t) + gl(x)fl(t) + g2(x)f2(t)
(5)
where gdx), g2(x) are auxiliary functions to be defined later, and Q is the transform function. Substitution of eqn (5) into eqns (3) and (4c)
Kye Bock Lee, In Young Ira, Sang Keun Lee
420
t~ i
L
TM
,,d v]
Pump x
I (a)
(b)
(c) Fig. 1.
The geometry of each boundary condition. (a) open end; (b) closed end; (c) piston-spring supported end.
yields the following homogeneous boundary conditions for Q(x, t):
3Q Ox
O(0, t) = 0
(6)
A p 92Q x=t. K at 2 =0
(7)
If we set gl(O) = 1,
g,(L) = 0
and f,(t) = PD COS (opt,
f2(/) = ff COS (opt
(8)
then g2(x) and f2(t) should satisfy gdO) = O,
gz(L) = 1
and g;(L)f2(t) +
g2(L)fz(t) + g;(L)f,(t) = 0
(9)
These are the only constraints placed upon the auxiliary functions gl(X), g2(x) and they are arbitrary within 0 < x < L. Equation (2) is
transformed to a nonhomogeneous differential equation: Ox 2
C~ Ot2
;=~
g'{f"-Co g']~i
(10)
P u m p - i n d u c e d pulsating pressure in a reactor coolant p i p e
421
Natural frequencies of the water can be obtained from the homogeneous expression of eqn (10), using the separation of variables: O~X.
Ox2 X.Ix=o = O,
~o~
+ ~ x o =o
(11)
Co
-OX. -+ Ox
ApwZ. X.Jx=L = 0 K
(12)
Solving eqn (11) with the boundary conditions of eqn (12), we obtain the frequency equation and normal modes: tan
L -t A Copw. - 0 (-On
X. = sin ~oox
(14)
Because of the boundary condition (eqn (12)), the following orthogonality relations are obtained. For n 4=m:
1£ L C-~o
~X n X m d x --
Xn(L)Xm(L
) : 0
foLax. aXmax 0 dx
dZX. ~oL ax2 "xmd~
dX.( L )
dx
Xm ( L ) =
(15)
O.
Solution of the nonhomogeneous eqn (10) is sought in the form:
Q(x, t)= ~ X.(x)T.(t)
(16)
n=l
Substituting eqn (16) into eqns (7) and (10), multiplying both sides by Xm(X), integrating from 0 to L and utilizing the orthogonality relations of the eigenfunctions yields: 1 gi'fi )} Xn dx 7". + ~oZT. = 12' £/" {i~l _ ( g~'fi-- C--~o
(17)
where
Neglecting the natural vibration modes by assuming the existence of damping, the solution of eqn (17) from zero initial conditions is
422
Kye Bock Lee, In Young Ira, Sang Keun Lee
obtained as follows:
2
T.(t) = (Lto.)-' E
f,
t t L f,(r) sin to.(t - r)} dr g,X. - g~X.]o
i=1
to. L dx> Co2 f~(tlfo giX.
(18)
Using eqns (5), (14), (16) and (18), the following expression for P(x, t) is obtained as follows: o¢
P(x, t) = ~ Pr)/3. + od3. n=a
Iln
. to. Co
. COS (-Opt. S l n - - x
+ gl(x)fl(t) + gz(x)f2(t)
(19)
L C o sin 2to.L I,. = 2- + 4to. Co
(20)
where
3n --
C2{
L2to.2+C~ 2.164 to.
(~nn)(1 - c o s (t'CoOnt~ (J)nZ~ / - L s i n Co J
~ ~ C2
sin Co J
[ L2(.On [
/3.=Lm.+C02L Co ( 1 2 2
Co(1
to.
(O)p)2 /
to°c
(-d)nh
oAto~L)] ~ //
to.L
( t o ~ _ to~)
,to., J 12n COS
~=pD
(21)
\O')n/ .} c o s -Co
-c°s--~o)(2"841 +2"164
Coto n COS
+
(to~_to2)
(22)
to.L Co
Y" *l.coto. (to~"-04)
n=l
Z n=l
w.L I3n C O S Co
(23)
I,°Coto° (to~- to"~)
g,(x) = 1 - -
X
L
(24)
423
Pump-induced pulsating pressure in a reactor coolant pipe TABLE 1
Solutions for Open End and Closed End Boundary Conditions Boundary condition
Open
Mode shape
Natural frequency
Auxiliary function
(x.)
(to.)
(g(x))
ton sin--x Co
n:t --Co
x 1---
L
Forced response P(x, t)
® 2
2.,_1= • ~ ~
L
t
P~ { , ]2 1-t°)P/t \tOn! J
nJ~x
• cos toot. sin - - ~
(n=1,2,3 ......
Closed
ton sin--XCo
n:r 2L--C°
~4
:rx cos--L
)
P~
-- • n=, n:t { 1 - (~-~) } t o P2 n:rx . cos topt. sin 2--ff (n=1,3,5 ......
)
TABLE 2
Calculated Pressure vs. PVMP Pressure Plant condition (Maine Yankee)
Frequency ( Hz )
20
40
100
200
PVMP pressure (psia(kPa))
0-072 (0.496)
0.074 (0.510)
0.257 (1.772)
0.126 (0-869)
Calculated pressure (psia(kPa))
0.113 (0-778)
0.192 (1.323)
0-121 (0.835)
0.197 (1.357)
PVMP pressure (psia(kPa))
0-057 (0-393)
0.022 (0.152)
0-131 (0.903)
0.199 (1.372)
Calculated pressure (psia(kPa))
0.170 (1-171)
0.080 (0.551)
0-225 (1-757)
0.223 (1-537)
Cold (T = 121 °C)
Hot (T = 278 °C)
Kye Bock Lee, In Young lm, Sang Keun Lee
424
g2(x) --
/3(e - 1) - 1
e x/L -
(e - 1) 2
/3(1
-
+
e 2) (e
+
/3(e -- e 2)
(e
-
-
4-
e2
1) 2
e-x/L
(e a + 1)
(25)
1) 2
-
pAw~L
/3 -
PD
K
3 NUMERICAL
+
0{
(261
RESULTS
Analytical values of reactor vessel inlet nozzle pressures at different forcing frequencies are calculated for an empirically d e t e r m i n e d nondimensional spring constant Ko = 40.0 (Ko = K L / A C o p ) . Solutions for the closed end and open end boundary conditions are given in Table 1. In Table 2 the calculated pressure pulsation and their corresponding PVMP (precritical vibration monitoring program) 6 data are tabulated. As can be seen from the table, the a g r e e m e n t b e t w e e n calculated and PVMP data is good. Axial variations of the pressure field for different b o u n d a r y conditions are shown in Fig. 2. The pressure values are normalized to a pump discharge pressure. Figure 3 represents the axial pressure distribution for different p u m p forcing frequencies. Since Ko depends upon the acoustic characteristics; it is d e t e r m i n e d using the available test data. To study the sensitivity of the nondimensional spring constant Ko, m o r e data should be accumulated.
Forcing Frequency:
200 Hz
2tJ3 LI
1~
0N
<
-1p
z
~
ISTON - S P R I N G
-2-
5
CLOSED i
i
DISTANCE FROM PUMP DISCHARGE (X/'L)
Fig. 2. Pump-induced pressure distribution (~ov = 200 Hz).
Pump-induced pulsating pressure in a reactor coolant pipe
425
3-
2O0
1-
c-', Ld
O-
IJJ t:d 12_
2 0 Hz
N
<
0 z
-1-2-3 DISTANCE FROM PUMP
DISCHARGE ( X / L )
Fig. 3. Pressure distribution for different pump forcing frequencies in the case of piston-spring supported end boundary condition.
4 CONCLUSIONS A n analytical model is presented to solve the problem of propagation of pump-induced pressure pulsations through the inlet pipe of a P W R . Pressures are calculated at the reactor vessel inlet nozzle and c o m p a r e d with the P V M P data. Comparisons are shown to be in good a g r e e m e n t for forcing frequencies. REFERENCES 1. Penzes, L. E., Theory of pump-induced pulsating coolant pressure in pressurized water reactors, Nucl. Eng. & Design, 27 (1974). 76-88. 2. Bowers, G. & Horvay, G., Forced vibrations of a shell inside a narrow water annulus. Nucl. Eng. & Design, 34 (1975) 221-31. 3. Cepkauskas, M. M., Acoustic pressure pulsations in pressurized water reactors. Trans. of the 5th Int. Conf. on S M I R T , (Structural Mechanics in Reactor Technology), vol F, Paper No F4/2, Berlin, 1979, pp. 1-9. 4. Fisher, H. D., Cepkauskas, M. M. & Chandra, S., Solution of time dependent boundary value problems by the boundary operator method. Int. J. Solids and Structures, 15 (1979) 607-14. 5. Lee, L. & Chandra, S., Pump-induced fluctuating pressure in a reactor coolant pipe. Int. J. Press. Ves. & Piping, 8 (1980) 407-17. 6. Anon., Maine Yankee PVMP, Combustion-Engineering Inc., CENPD-93.
BIBLIOGRAPHY 1. Au-Yang, M. K., Response of reactor internals to fluctuating pressure forces. Nucl. Eng. & Des., 35 (1975) 361-75. 2. Bohm, G. J., Analytical problems associated with core support structure of PWR, Nucl. Eng. & Des., 18 (1972) 305-21.
Analytical Prediction on the Pump-induced Pulsating Pressure in a Reactor Coolant Pipe K y e B o c k L e e , In Y o u n g I m & S a n g K e u n L e e Thermal Hydraulic Design Department, Korea Atomic Energy Research Institute, PO Box 7, Daeduk-Danji, Daejeon 305-606, Korea (Received 3 December 1991; accepted 22 December 1991)
ABSTRACT An analytical method is presented for predicting the amplitudes of pump-induced fluctuating pressures in a reactor coolant pipe using a linear transformation technique which reduces a homogeneous differential equation with non-homogeneous boundary conditions into a nonhomogeneous differential equation with homogeneous boundary conditions. A t the end of the pipe, three types of boundary conditions are considered--open, closed and piston-spring supported. Numerical examples are given for a typical reactor. Comparisons of measured pressure amplitudes in the pipe with model prediction are shown to be in good agreement for the forcing frequencies.
1 INTRODUCTION Flow-induced vibrations have led to many problems in reactor operation. These types of vibrations may be caused by vortex shedding, turbulence flow and pump-induced pressure fluctuation. Because of the general complexity of this problem, we restrict our analysis to the pressure fluctuations of the reactor coolant caused by p u m p pulsation. The pressure fluctuations generated at the p u m p discharge can interact with the reactor internals and produce damaging vibrations. The boundary condition at the p u m p end consists of a time d e p e n d e n t harmonic function describing the p u m p discharge pressures at different 417 Int. J. Pres. Ves. & Piping 0308-0161/92/$05.00 (~) 1992 Elsevier Science Publishers Ltd, England. Printed in Northern Ireland
418
Kye Bock Lee, In Young lm, Sang Keun Lee
pump forcing frequencies. The resolting boundary value problem consists of a homogeneous differential equation with nonhomogeneous time dependent boundary conditions. Pressure pulsations in a PWR were first studied by Penzes, ~ and later by Bowers and Horvay 2 and Cepkauskas. 3 They treated the problem of pulsations by introducing a body force concept, but they neglected to recognize the restrictions on the body force necessary to make the boundary conditions homogeneous. Fisher et al. 4 demonstrated an improved technique in solving one-dimensional wave propagation problems with time dependent boundary conditions. This technique introduces a transformation equation in terms of a new variable and auxiliary functions. The mathematical description is recast in the form of a nonhomogeneous differential equation. The boundary conditions become homogeneous by restricting the auxiliary functions at the boundaries. Due to the homogeneity of the resulting boundary conditions, the transformed system of equations is amenable to solution by separation of variables. Lee and Chandra 5 analyzed the pump-induced pulsating pressure in a reactor coolant pipe using this technique, but they missed the constraints on the auxiliary functions to make the boundary conditions homogeneous. Thus their solution cannot satisfy the boundary condition at the pipe end for the piston-spring supported end case. By considering the missing constraints on the auxiliary functions in the previous study we have developed an improved analytical model which satisfies all boundary conditions.
2 ANALYTICAL MODEL
2.1 Governing equation In the analytical model considered compressible and inviscid. Since the smaller than the static pressure, the used to reduce the Navier-Stokes equation: v2
here, the fluid is assumed to be pressure pulsations considered are small perturbation method can be equations to the following wave
C~o
where P = pressure pulsation, and Co = sonic velocity.
Pump-induced pulsating pressure in a reactor coolantpipe
419
If the variation of the pressure pulsations in the radial and circumferential direction is ignored, eqn (1) reduces to d2o~JP=0
(2)
The excitation for the system occurs at the pump discharge (x = 0). The boundary condition is P Ix=0= PD COS toot
(3)
where PD is the pressure wave at the pump discharge, and top is the associated pump frequency. 2.2 Boundary conditions The following three types of boundary conditions are considered at
x=t: (a) open end
PIx=L=0
(4a)
OPox x=L = 0
(4b)
(b) closed end
(c) piston-spring supported end
lOP -~x
Ap821)] K Ot2 Jx=a = 0
(4c)
Here, A is the pipe cross-sectional area, p is the fluid density, and K is the empirically determined spring constant. The geometry considered in this study is shown in Fig. 1. 2.3 Linear transformation Equations (2), (3) and (4) describe the boundary value problem with the time dependent non-homogeneous boundary condition. To render the boundary condition homogeneous the following equation is used:
P(x, t) = Q(x, t) + gl(x)fl(t) + g2(x)f2(t)
(5)
where gdx), g2(x) are auxiliary functions to be defined later, and Q is the transform function. Substitution of eqn (5) into eqns (3) and (4c)
Kye Bock Lee, In Young Ira, Sang Keun Lee
420
t~ i
L
TM
,,d v]
Pump x
I (a)
(b)
(c) Fig. 1.
The geometry of each boundary condition. (a) open end; (b) closed end; (c) piston-spring supported end.
yields the following homogeneous boundary conditions for Q(x, t):
3Q Ox
O(0, t) = 0
(6)
A p 92Q x=t. K at 2 =0
(7)
If we set gl(O) = 1,
g,(L) = 0
and f,(t) = PD COS (opt,
f2(/) = ff COS (opt
(8)
then g2(x) and f2(t) should satisfy gdO) = O,
gz(L) = 1
and g;(L)f2(t) +
g2(L)fz(t) + g;(L)f,(t) = 0
(9)
These are the only constraints placed upon the auxiliary functions gl(X), g2(x) and they are arbitrary within 0 < x < L. Equation (2) is
transformed to a nonhomogeneous differential equation: Ox 2
C~ Ot2
;=~
g'{f"-Co g']~i
(10)
P u m p - i n d u c e d pulsating pressure in a reactor coolant p i p e
421
Natural frequencies of the water can be obtained from the homogeneous expression of eqn (10), using the separation of variables: O~X.
Ox2 X.Ix=o = O,
~o~
+ ~ x o =o
(11)
Co
-OX. -+ Ox
ApwZ. X.Jx=L = 0 K
(12)
Solving eqn (11) with the boundary conditions of eqn (12), we obtain the frequency equation and normal modes: tan
L -t A Copw. - 0 (-On
X. = sin ~oox
(14)
Because of the boundary condition (eqn (12)), the following orthogonality relations are obtained. For n 4=m:
1£ L C-~o
~X n X m d x --
Xn(L)Xm(L
) : 0
foLax. aXmax 0 dx
dZX. ~oL ax2 "xmd~
dX.( L )
dx
Xm ( L ) =
(15)
O.
Solution of the nonhomogeneous eqn (10) is sought in the form:
Q(x, t)= ~ X.(x)T.(t)
(16)
n=l
Substituting eqn (16) into eqns (7) and (10), multiplying both sides by Xm(X), integrating from 0 to L and utilizing the orthogonality relations of the eigenfunctions yields: 1 gi'fi )} Xn dx 7". + ~oZT. = 12' £/" {i~l _ ( g~'fi-- C--~o
(17)
where
Neglecting the natural vibration modes by assuming the existence of damping, the solution of eqn (17) from zero initial conditions is
422
Kye Bock Lee, In Young Ira, Sang Keun Lee
obtained as follows:
2
T.(t) = (Lto.)-' E
f,
t t L f,(r) sin to.(t - r)} dr g,X. - g~X.]o
i=1
to. L dx> Co2 f~(tlfo giX.
(18)
Using eqns (5), (14), (16) and (18), the following expression for P(x, t) is obtained as follows: o¢
P(x, t) = ~ Pr)/3. + od3. n=a
Iln
. to. Co
. COS (-Opt. S l n - - x
+ gl(x)fl(t) + gz(x)f2(t)
(19)
L C o sin 2to.L I,. = 2- + 4to. Co
(20)
where
3n --
C2{
L2to.2+C~ 2.164 to.
(~nn)(1 - c o s (t'CoOnt~ (J)nZ~ / - L s i n Co J
~ ~ C2
sin Co J
[ L2(.On [
/3.=Lm.+C02L Co ( 1 2 2
Co(1
to.
(O)p)2 /
to°c
(-d)nh
oAto~L)] ~ //
to.L
( t o ~ _ to~)
,to., J 12n COS
~=pD
(21)
\O')n/ .} c o s -Co
-c°s--~o)(2"841 +2"164
Coto n COS
+
(to~_to2)
(22)
to.L Co
Y" *l.coto. (to~"-04)
n=l
Z n=l
w.L I3n C O S Co
(23)
I,°Coto° (to~- to"~)
g,(x) = 1 - -
X
L
(24)
423
Pump-induced pulsating pressure in a reactor coolant pipe TABLE 1
Solutions for Open End and Closed End Boundary Conditions Boundary condition
Open
Mode shape
Natural frequency
Auxiliary function
(x.)
(to.)
(g(x))
ton sin--x Co
n:t --Co
x 1---
L
Forced response P(x, t)
® 2
2.,_1= • ~ ~
L
t
P~ { , ]2 1-t°)P/t \tOn! J
nJ~x
• cos toot. sin - - ~
(n=1,2,3 ......
Closed
ton sin--XCo
n:r 2L--C°
~4
:rx cos--L
)
P~
-- • n=, n:t { 1 - (~-~) } t o P2 n:rx . cos topt. sin 2--ff (n=1,3,5 ......
)
TABLE 2
Calculated Pressure vs. PVMP Pressure Plant condition (Maine Yankee)
Frequency ( Hz )
20
40
100
200
PVMP pressure (psia(kPa))
0-072 (0.496)
0.074 (0.510)
0.257 (1.772)
0.126 (0-869)
Calculated pressure (psia(kPa))
0.113 (0-778)
0.192 (1.323)
0-121 (0.835)
0.197 (1.357)
PVMP pressure (psia(kPa))
0-057 (0-393)
0.022 (0.152)
0-131 (0.903)
0.199 (1.372)
Calculated pressure (psia(kPa))
0.170 (1-171)
0.080 (0.551)
0-225 (1-757)
0.223 (1-537)
Cold (T = 121 °C)
Hot (T = 278 °C)
Kye Bock Lee, In Young lm, Sang Keun Lee
424
g2(x) --
/3(e - 1) - 1
e x/L -
(e - 1) 2
/3(1
-
+
e 2) (e
+
/3(e -- e 2)
(e
-
-
4-
e2
1) 2
e-x/L
(e a + 1)
(25)
1) 2
-
pAw~L
/3 -
PD
K
3 NUMERICAL
+
0{
(261
RESULTS
Analytical values of reactor vessel inlet nozzle pressures at different forcing frequencies are calculated for an empirically d e t e r m i n e d nondimensional spring constant Ko = 40.0 (Ko = K L / A C o p ) . Solutions for the closed end and open end boundary conditions are given in Table 1. In Table 2 the calculated pressure pulsation and their corresponding PVMP (precritical vibration monitoring program) 6 data are tabulated. As can be seen from the table, the a g r e e m e n t b e t w e e n calculated and PVMP data is good. Axial variations of the pressure field for different b o u n d a r y conditions are shown in Fig. 2. The pressure values are normalized to a pump discharge pressure. Figure 3 represents the axial pressure distribution for different p u m p forcing frequencies. Since Ko depends upon the acoustic characteristics; it is d e t e r m i n e d using the available test data. To study the sensitivity of the nondimensional spring constant Ko, m o r e data should be accumulated.
Forcing Frequency:
200 Hz
2tJ3 LI
1~
0N
<
-1p
z
~
ISTON - S P R I N G
-2-
5
CLOSED i
i
DISTANCE FROM PUMP DISCHARGE (X/'L)
Fig. 2. Pump-induced pressure distribution (~ov = 200 Hz).
Pump-induced pulsating pressure in a reactor coolant pipe
425
3-
2O0
1-
c-', Ld
O-
IJJ t:d 12_
2 0 Hz
N
<
0 z
-1-2-3 DISTANCE FROM PUMP
DISCHARGE ( X / L )
Fig. 3. Pressure distribution for different pump forcing frequencies in the case of piston-spring supported end boundary condition.
4 CONCLUSIONS A n analytical model is presented to solve the problem of propagation of pump-induced pressure pulsations through the inlet pipe of a P W R . Pressures are calculated at the reactor vessel inlet nozzle and c o m p a r e d with the P V M P data. Comparisons are shown to be in good a g r e e m e n t for forcing frequencies. REFERENCES 1. Penzes, L. E., Theory of pump-induced pulsating coolant pressure in pressurized water reactors, Nucl. Eng. & Design, 27 (1974). 76-88. 2. Bowers, G. & Horvay, G., Forced vibrations of a shell inside a narrow water annulus. Nucl. Eng. & Design, 34 (1975) 221-31. 3. Cepkauskas, M. M., Acoustic pressure pulsations in pressurized water reactors. Trans. of the 5th Int. Conf. on S M I R T , (Structural Mechanics in Reactor Technology), vol F, Paper No F4/2, Berlin, 1979, pp. 1-9. 4. Fisher, H. D., Cepkauskas, M. M. & Chandra, S., Solution of time dependent boundary value problems by the boundary operator method. Int. J. Solids and Structures, 15 (1979) 607-14. 5. Lee, L. & Chandra, S., Pump-induced fluctuating pressure in a reactor coolant pipe. Int. J. Press. Ves. & Piping, 8 (1980) 407-17. 6. Anon., Maine Yankee PVMP, Combustion-Engineering Inc., CENPD-93.
BIBLIOGRAPHY 1. Au-Yang, M. K., Response of reactor internals to fluctuating pressure forces. Nucl. Eng. & Des., 35 (1975) 361-75. 2. Bohm, G. J., Analytical problems associated with core support structure of PWR, Nucl. Eng. & Des., 18 (1972) 305-21.