Analysis of waterhammer induced by steam-water counterflow in a long horizontal pipe

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INT. COMM. HEAT MASS TRANSFER Vol. 19, pp. 507-518, 1992 Printed in the United States

Copyright© 1992 Pergamon Press Ltd.

ANALYSIS OF WATERHAMMER INDUCED BY STEAM-WATER COUNTERFLOW IN A LONG HORIZONTAL PIPE Moon-Hyun Chum and Ho-Yum Nam

Korea Advanced Institute of Science and Technology Department of Nuclear Engineering Taejon, Korea

(Communicated by J.P. Hartnett and WJ. Minkowycz) ABSTRACT Improved analytical models have been proposed that can predict the lower and upper limits of the waterhammer region for given flow conditions by incorporation of recent advances made in the understanding of phenomena associated with the condensationinduced waterhammer into existing methods. Present models are applicable for steamwater counterflow in a long horizontal pipe geometry. Both lower and upper bounds of the waterhammer region are expressed in terms of the 'critical inlet water flow rate' as a function of axial position. Waterhammer region boundaries predicted by present and typical existing models are compared for particular flow conditions of the waterhammer event occurred at San Onofre Unit 1 to assess the applicability of the models examined. The result shows that present models for lower and upper bounds of the waterhammer region compare favorably with the best performing existing models. Introduction Condensation-induced waterhammer (CIWH) is the most damaging form of waterharnmer and its diagnosis is extremely difficult because of the complex nature of the underlying phenomena that occur at the steam-water interface. A review of the literature revealed that a number of comprehensive studies on the phenomena associated with waterhammer events were performed : The earlier studies [1,2,3] identified basic mechanisms for waterhammer initiation. Descriptions about how these mechanisms occurred in nuclear power plants, and what could be done to avoid these initiating events were also made. However, there exist only a few quantitative analysis on CIWH, in particular. The main purpose of the present work is to develop improved models for lower and upper bounds where waterhammer is predicted to occur in a long horizontal pipe by incorporating the current advances made in the understanding of interfadal transport phenomena into existing analytical methods proposed by earlier workers [1,4,5,6]. Specific modifications and improvements made over the existing models will be subsequently delineated. 507

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M.-H. Chun and H.-Y. Nam

Vol. 19, No. 4

Analytical Models for Lower and Upper Bounds The physical phenomena of waterhammer induced by steam-water counterllow in a long horizontal pipe may be described by reference to the flow geometry shown in Fig.1 : The counterflow of steam interfering with inlet water flow can form a water slug and thereby isolate a steam bubble up-stream of the water slug. Rapid condensation of the isolated steam bubble by continuous inflow of subco~ed water causes acceleration of the water slug, giving rise to a waterbammer in the pipe [1]. All of the basic waterhammer event classes involve the five sequence of events or stages : (1) slag formation, (2) isolated void formation, (3) slag acceleration, (4) void collapse, and (5) impact. Inferring from these basic physical phenomena of waterhammer and observations made by previous workers [1,3,4,5,6] two analytical models to estimate the critical inlet water flow rates, one for the lower limit and the other for the upper limit of the waterhammer region are derived here. Lower Bound The annlytical method to derive the lower boundary where condcrlsation induced water hammer is predicted to occur in a long horizontal pipe is similar to that of Bjorge and Griffith [4] except the following : (1) In the expression of the steam condensation rate, the effect of heat trAn~er from the steam to the pipe wall is included, whereas this effect was not considered in the work of others

[4].

(2) In an effort to incorporate current advances made in the undeman~n__g of interf~al transport phenomena, the latest available correlations of interfacial friction factor and interracial heat trander coefficient are used. The flow geometry for analysis is shown in Fig.1 : Consider a countercurrent stratified steam-water flow in an inclined pipe. To simplify the analysis, the governing equations, which describe the interaction between the steam and liquid phases, are based on the following assumptions : (1) The flow is steady, incompressible, turbulent, and one-dimensional, (2) The mass transfer is due to condensation, that is, the flow is nonentraining. (3) The actual time varying film thickness is replaced with a temporal mean film thickness. (4) The steam is saturated and the saturation temperature remains constant along the pipe. (5) The pressure variation along the flow direction is negligible. The applicable equations of continuity, momentum, and energy balance for both phases are

•

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WATERHAMMER

I N D U C E D BY S T E A M - W A T E R C O U N T E R F L O W

Continuity : t~ 1 = PlUlAl t~ =

Momentum

(1)

PsUs(A-Al) =

p,usA s

(2)

:

--~-~(i~llUl) - ui(~-~-~) =

" A V 'dPi ~--

- plgAid

.

Energy

+

d81 dx

(3)

,riSi - ,ri~1

dP s

d~n.

dx (m,u,)

ptgAt sinO

"

~(--~)

=

- A, ~

- pa, A, She

+

,,S, +

"i~

(4)

:

d(t~llil)

dfi~

where

d~ iS - -d~

i ' d_x_ +

-

dx

(5)

diI = CI~dT P

Define dimensionless variables as follows :

~I" ~"

~1 --

D

;

X

X*

=

As/(132/4)

--

(6b)

•

~* = ~ o

; ~l" =

TI*

; TS e

--

p,gA~

(6a)

;

D

(60

~r~o +is~

.~.

; Ti

pc,A,

- -

ctPlgAI

(6d)

%" ~- (T,-%o) / (T,-T~

(6e)

q" =

(60

[hiSi(Ts-Tl)+hwASs(Ts-TwA)lAl / (mlSiiS)

4 , = ( i-,~) a

p,%= • ,I, Plttl 2

is-i ~ ~o -

(I-,,)

'

a

-

iS

gA l

gpl2Al 3

(6g)

U1

is+is-i L ; ~

iS

Fr 2 -

u, iri~ ; ~2 -

gAl3

(6h)

iS (6i)

509

510

M.-H. Chun and H.-Y. Nam

Vol. 19, No. 4

Governing Equations : Substituting Eqs.(1) and (2) into Eqs.(3),(4), and (5) and using dimensionless variables defined above, following dimensionless governing equations can be obtained [4] :

d S l ' = 2F2q'(0-1) - xi" - r 1" - xs" - 0 11-Fr2(1+ d~)] dx---"

(7)

dx"

(8)

-

q" (

ms" -

)(

fiho

+

)

132

l~llo

(9)

[~1

Boundary Conditions : The applicable boundary conditions are : T 1" = 0, rns" = 0

at x* = 0

1-F~(I+d~) = 0

at x* -

L D

(10) (11)

In the above dimemionless governing equations (7),(8), and (9) there are three dependent variables 81", TI" , and ~ ' .

Before these equations can be solved simultaneously to obtain

the three unknowns, five quantities such as "q, -ri, xs, hi and hwA must be specified using appropriate constitutive relations. Constitutive Relation : Constitutive relations to obtain TI, %, Ti, hi, and hWA are as follows : The liquid wail shear stress (Xl) and steam wall shear stress (To are obtained from the basic relation between frictional stress (T) and friction factor (f) :

fpu 2 T -

(12)

8

where the friction factor for turbulent flow could be approximated by f = 0.316 Re ° ' ~

(13)

In Eq.(13) Reynolds numbers for liquid and steam are defined as Re I -

P # l I ~ , Re s -

pstql)~

(14 a,b)

where I)111 and D ~ are given by

r~-

4A 1

s~

,r~-

4A s

s,+s~

(15 a,b)

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W A T E R H A M M E R I N D U C E D BY S T E A M - W A T E R C O U N T E R F L O W 511

The interfacial shear stress 'ri in a condensing flow is calculated as a linear superposition of the interface shear stress with no condensation and the suction parameter 24 us

= y-0,u,

st

dml

(16)

dx

For the nearly horizontal (0=4 °) countereurrent steam-saturated water flow in rectangular channels (aspect ratio=5), Kim et al. [7] give the following correlation for the adiabatic interfacial friction factor : fla = 0.560 x 10-5 Ret+0.084

(17)

The local interfaciai condensation heat transfer coefficient h i is obtained from the correlation of Bankoff et. al. [8] for countercurrent stratified flow of steam and cold water at atmospheric pressure in a flat plate geometry at an inclination of 4° from the horizontal Nui = 0.63 x 10-2 Res°'9 Rel°'Ts Pr °'al

(18)

The local film condensation heat transfer coefficient on the wall hwA can be obtained from the modified Nusselt correlation hWA = [ pl(Pl-ps)gkl3 if$' ]0.25 4x p,I(Ts-TwA) where ifg' = ifs+3/8 Cpl (Ts-TwA)

(19)

(20)

In the above equations, TwA is approximated by (1",+T])/2. Criterion for the Condensation-Induced Waterhammer Initiation : In the present work, a localized water slug formation is assumed to lead to a 'steam bubble collapse induced water hammer' in a long horizontal pipe, in particular. To predict the lower boundary for waterhammer initiation, the following criterion of stratified-slug flow boundary for a horizontal or nearly horizontal circular pipe given by Taitel and Dukler [9] is used : 1-(~ NTD = (

a

p, )(

Pl

)(Si__ g

)

u2 • 2 > 1.0 AI(1-BI ) - -

(21)

Upper Bound Bjorge and Griffith [4] noted that the region where waterhammer is predicted to occur is bounded by the stratified flow instability limit of Eq.(21) and the criterion for the minimum water flow rate necessary to run the pipe full expressed by Block et. al. [4] : • 2 16 mlo

2p?gD

-

0.25

(22)

512

M.-H. Chun and H.-Y. Nam

Vol. 19, No. 4

For the case examined in the present work (see Fig.2), however, it is found that Eq. (22) grossly overpredicts the upper limit of the waterhammer region and this equation is not useful for practical applications. Therefore, an approximate method is used here to derive an alternative upper bound based on the following observations of earlier investigators [3,6] : When the injected water flow rate is high enough, the reduction rate in the steam volume due to refilling is approximately equal to that resulting from condensation on the steam-water interface and the pipe wall. At this point, the net steam flow into the pipe is approximately zero and is not large enough to generate a water slug• In other words, if the inlet water flow rate exceeds the condensation rate (i.e., mlo > fiat), then no net steam will flow into the pipe, and waterhammer will not occur. Thus, an alternative upper bound may be obtained from the following relations : The mass of steam is balanced by the condensation mass rate. d(PsVs) dt

= - r~c

(23)

Then, an approximate expression for an upper bound criterion to avoid waterhammer (24) where =

hiSiL(Ts-Tl)+ hwASsL(Ts-TwA) •

ifg

(25)

Equation (25) assumes that the steam is condensed on both the water surface and the pipe wall. The approximate average heat transIer coefficients hi and hwA are obtained from the result of Brucker and Sparrow [10] and the modified Nussclt correlation, respectively. The approximate value of the average steam-water interface condensation coefficient is hi -~ 104 W/m2-oC (1800 Btu/hr-ft2-*F)

(26)

whereas that for the pipe wall is hwA = 0.943 [ gPl(Pt'Ps)kl3 lfg ]o.25

(27)

L gi(T,-TWA) where ifg' = ifg+0.68 C#(Ts-TwA )

(28)

By simple geometrical considerations, one can relate A ! and Si to the water depth 81 and the pipe diameter D. The water depth 81, on the other hand, can be obtained from the solution of the following equations : Fr2gAi3 2 ~ - - " D (81"-81"2)0'5 = 0

(29)

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W A T E R H A M M E R INDUCED BY STEAM-WATER C O U N T E R F L O W 513

where Ql can be determined by rhl _

ifs+ c~(Ts'T~

• lo

(30)

%+ c~(T.-T~)

Equation (29) is derived from the Froude number defined by Eq.(6i), whereas Eq.(30) is obtained from the mass and energy balances for the control volume of both phases. To find the liquid depth and the critical inlet water flow rate (ink,) that satisfy the upper bound criterion of Eq.(24) and Eq.(29) can be solved by the Newton-Raphson method. Numerical Results and Discussion Solutions for Lower and Upper Bounds The lower bound for waterhammer initiation in a long horizontal pipe is the stratifiedslug flow boundary of Eq.(21), whereas the upper bound is given by Eq.(24). The region where waterhammer is predicted to occur is thus bounded by these two criteria. To obtain the lower bound, a solution for the flow field (i.e., 81", Tl*, and ~ * ) has been first obtained by solving the system of Eqs.(7), (8), and (9) using Eqs.(10) and (11) as boundary conditions. Then Eq.(21) is used to calculate the stability parameter NTD for each node. This procedure is repeated by varying the inlet water flow rate (mk~) until the calculated maximum Taitel-Dukler parameter (NTD) is just greater than 1.0 at any node. That is, localized water slug formation, which is assumed to be leading to a steam bubble collapse-induced waterhammer, is predicted when NTD >

1.0 at any node.

An approximate upper bound for the 'critical inlet water flow rate' that satisfy Eq.(24), on the other hand, has been obtained from the solution of Eqs.(29) and (30) by the NewtonRaphson method. Results and Discussion Present and typical existing numerical models for lower and upper bounds have been applied to the waterharnmer event of San Onofre Unit 1 [6]. The main reason for chosing this event is to assess the applicability of the present and existing models by testing whether each model can explain the main question : Why a severe waterhammer occurred only in the main feedwater line B even though all three feedwater lines (A,B, and C) of the San Onofre Unit 1 were partially or completely voided and then refilled during the event ? Waterhammer region boundaries predicted by various models for San Onofre event conditions are presented in Fig.2. This figure shows a range of 'critical inlet water flow rates' as a function of feedwater pipe length. As noted by previous investigators [3,6], the A and C loops (each about 38.10 m) are about half the length of line B (-60.96 m).

514

M.-H. Chun and H.-Y. Nam

Vol. 19, No. 4

Therefore, the A and C loops were completely filled at a high auxiliary feedwater flow rate ( - 9 . 4 5 kg/s) while the B loop was filling at a low auxiliary feedwater flow rate (1.58 kg/s) at the time of the waterhammer. These data points are indicated in Fig.2 and this figure shows that the A and C loops were operated well outside the region bounded by lower and upper critical inlet water flow rates. The B loop, on the other hand, falls within the lower and upper bounds where waterhammer may occur. Figure 2 also shows that the lower bound for critical inlet water flow rates obtained from the present model is slightly lower than that from Bjorge's 'absolute stability limit' [4] whereas the upper bound for critical inlet water flowrate predicted by present method is slightly higher than that by Chiu et al. [6]. Profiles of dimensionless variables as a function of dimensionless axial position x*, on the other hand, are given in Fig.3. These profiles are obtained for San Onofre waterhammer event from the present lower bound model. The curve for NTD versus x* in Fig.3, in particular, shows that the Taitel and Dukler stability parameter (NTD) goes through a maximum value at about x* = 40. The location of this maximum N3.D value (about x~10.13 m from the inlet) is the point where a waterhammer is predicted to occur. The effects of major system parameters (i.e., pipe diameter D, inlet water temperature Tlo, and system pressure p) on the 'critical inlet water flow rate' for waterhammer initiation, on the other hand, are shown in Fig.4. From this figure, increasing the system pressure, decreasing the inlet water temperature, and decreasing the pipe diameter are each seen to reduce the 'critical inlet water flow rate' for waterhammer initiation. Conclusions Improved analytical models have been derived here that can predict the initiation of waterhammer induced by steam-water counterflow in a long horizontal pipe. That is, two analytical models to estimate the 'critical inlet water flow rate' for given flow conditions, one for the lower limit and the other for the upper limit of the waterhammer region are developed. Waterhammer region boundaries obtained by the present and typical existing models are compared for particular flow conditions of the waterhammer event occurred at San Onofre Unit 1 to assess the applicability of the present and existing models. The result of this assessment indicates that present models for lower and upper bounds of the waterhammer region compare favorably with the best performing existing models. It may be concluded, therefore, that present models can be used with more confidence to find the critical conditions for waterhammer initiation (or to find the region where waterhammer can be induced so that the waterhammer can be avoided by proper variation of input parameters) in a long horizontal pipe where a countercurrent flow of steam and water exists.

Vol. 19, No. 4

WATERHAMMER INDUCED BY STEAM-WATER COUNTERFLOW 515

Acknowledgments Authors gratefully acknowledge

the support

of the Korea Science & Engineering

Foundation through the Center for Advanced Reactor Research at the Dept. of Nuclear Engineering, KAIST. Nomenclature A

flow

area,

m2

liquid specific heat, J/kg-K D

diameter, m

f

friction factor

Fr

Froude number, Eq.(6i)

g

gravitational acceleration, 9.80665 m/s2

h

heat transfer coefficient, W/m2-K

lfg

enthalpy of vaporization, J/kg as that defined in Eq.(20)

i kl

enthalpy, J/kg

L

length of pipe, m

thermal conductivity of liquid, W/m-K mass condensation rate of steam, kg/s mass flow rate, kg/s

N~

Taitel-Dukler stability parameter, Eq.(21)

Nui

Nusselt number for local interracial condensation, Eq.(18)

P P~

pressure, Pa

QI

volumetric flow rate of liquid phase, m3/s

q"

dimensionless condensation rate, Eq.(6f)

Re

Reynolds number, Eq.(14a)

S

perimeter, m

liquid Prandtl number

T

temperature, K

U

velocity, m/s

Vs

volume of steam, m 3

X

axial position, m void fraction, Eq.(6b) dimensionless temperature difference, Eq.(6h)

132 ~o

dimensionless temperature difference, Eq.(6h) dimensionless temperature difference, Eq.(6h) depth, m

516

M.-H. Chun and H.-Y. Nam

Vol. 19, No. 4

pipe inclination, radians viscosity, kg/s-m density, kg/m3 shear stress, N/m2 dimensionless group, Eq.(6g) dimensionless group, Eq.(6g) Subscripts 1

liquid

s

steam

h

hydraulic

ia

adiabatic interracial

i

interracial

1o

liquid at inlet

so

steam at outlet

WA

pipe wall

Superscripts overbar represents average values *

dimensionless quantities References

1. ~ o u , Y., and Cniffith, P., "Avoiding Steam-Bubble-CoUapse-Induced Water Hammer in Piping Systems," EPRI Report EPRI NP-6447, 1989. 2. U.S. Nuclear Regulatory Commission, "Evaluation of Waterhammer Occurrence in Nuclear Power Plants," NRC Report NUREG-0927, Revision 1, 1984. 3. Izenson, M.G., Rothe, P.H., and Wailis, G.B., '~iagnosis of Condensation - Induced Waterhammer," NRC Report NUREG/CR-5220, Vol.1 and 2, 1988. 4. Bjorge, R.W., and Griffith, G., "Initiation of Waterhammer in Horizontal and Nearly Horizontal Pipes Containing Steam and Subcooled Water," ASME Journal of Heat Transfer,

Vol.106, 1984, pp.835-840. 5. Jackobek, A.B., and Griffith, P., 'Investigation of Cold Leg Water Hammer in a PWR Due to the Admission of Emergency Core Cooling (ECC) During a Small Break LOCA," NRC Report NUREG/CR-3895, 1984. 6. Cb.iu, C., Tuttle, D., and Serkiz, A.W., "Water Hammer in a PWR Horizontal Feedwater Line," ANS Transactions, Vol.52, 1986, pp.589-590.

Vol. 19, No. 4

WATERHAMMER INDUCED BY STEAM-WATER COUNTERFLOW 517

7. Kim, H.J., Lee, S.C., and Bankoff, S.G., '1-Ieat Transfer and Interracial Drag in Countercurrent Steam-Water Stratified Flow," Int. J. Multiphase Flow, Vol.ll, No.5, 1985, pp.593-606. 8. Bankoff, S.G., and Kim, H.J., "Direct-Contact Condensation of Steam on cold Water in Stratified Countercurrent Flow," NRC Report NUREG/CR-4415, 1985. 9. Taitel, Y., and Dukler, A.E., "A Model for Predicting Row Regime Transitions in Horizontal and Near Horizontal Gas-Liquid How," AIChE Journal, Vol.22, No.l, 1976, pp.47-55. 10. Brucker, G.G., and Sparrow, E.M., "Direct Contact Condensation of Steam Bubbles in Water at High Pressure," Int. J. Heat Mass Transfer, Vot.20, 1977, 1313.371-381.

,,

~,,," ,,

~, ~,,

" / _ J '" • .~

~j~"

\

~qZ,'////-~

/

FIG. 1

Model Used for Analysis of Steam-Water Counterflow Induced Waterhammer

518

M.-H. Chun and H.-Y. Nam

n

Vol. 19, No. 4

~

~ Loop h2kC( L=3& Ira. Ih/o=gAsiq~) I0

~5.N~

~ -3~.530~,/~)

T, • 5 3 9 . 8 K

Tto =299.8K 8

Symbols~ T~is w~k

B)~'getRd.'~l ~

adlag~okR lc ~

f.5I

2

0 0

21)

60

40

80

Feedwa~a" Pipe L,aagth(m)

FIG. 2 Comparison of Waterhammer Region Boundaries Predicted by Various Models for San Onofre Unit 1 Event 1,2

3.0

....

, ....

, ....

, ....

, ....

, ....

, ....

,__.___/NTD _= 1.0

L-| 5.2 - 76.2m I ~ I~ -O.~3m, I~ ~1.305m,I~ nI).381m 1.0

PI ~ 4 . 8 ~ 1 ~ P2~x621~11

2.5

(Tt,)U.28g.Tg.O't,)2=299.SK

GroupA

c~e I IN, ('~)11 t'm$ylnbol~ * ¢aS~2 I ~ Ct~,)tJ

--Cao~C(I2~ •

Z

,._" 0.8

, * cme4IP2,(T~)21 cale 5 [P*,(T~)d

2.0

GroupB

ca~ 6 I1~(g,)ll cme 7 lP" fr~)2] ca~ g [el,frto)2]

i 0.6

GtoepC • t~sel0[P2'(T/°)l[

~---G~I3)II iI III

0.4

t

1).2

0.0

:",,. o5

511

Illl

15(1

2(10

250

D~a~mCr.ss Axud Poatma(x*)

FIG. 3 Profiles of Dimensionless Variables Predicted by Present Lower Bound Model for San Onofre Waterhammer Event

• c'asel2[P2'(T/°'21

O.(I

~Grou~Att~

,

* ,

t

50

I

I

.............................

II!1)

151)

I

2~1

I

2.51)

I

I

?~)

350

L/D

FIG. 4 Effects of Pipe Diameter, Inlet Water Temperature, and System Pressure on the Critical inlet Water Flow Rate

Received May 22, 1992