Derivation of mechanistic critical heat flux model for water based on flow instabilities

Int. Comm. Heat Mass Transfer, Vol. 23, No, 8, pp. 1109-1119, 1996 Copyright © 1996 Elsevier Science Ltd Printed in the USA. All rights reserved 0735-1933/96 $12.00 + .00

Pergamon

PII S0735-1933(96)00092-9

DERIVATION

OF MECHANISTIC

CRITICAL HEAT FLUX MODEL

FOR WATER BASED ON FLOW INSTABILITIES Soon Heung Chang, Yun Ii Kim and Won-Pil Baek Department of Nuclear Engineering Korea Advanced Institute of Science and Technology 373-1, Kusong-dong, Yusong-gu, Taejon, 305-701, Korea

(Communicated by J.P. Hartnett and W.J. Minkowycz) ABSTRACT The critical heat flux (CHF) is a major parameter which determines the cooling performance and therefore the prediction of CHF with accuracy is of importance for the design and safety analysis of nuclear power plant (NPP). In this study, the mechanistic CHF model and correlation for water are derived based on flow excursion (or Ledinegg instability) criterion and the simplified two-phase homogeneous model. The relationship between CHF for the water and the principal parameters such as mass flux, heat of vaporization, heated length-to-diameter ratio, vapor-liquid density ratio and inlet subcooling is derived on the developed correlation. The developed CHF correlation predicts very well at the applicable ranges, 1 < P < 40 bar, 1,300 < G 27,00 kg/m2s and inlet quality is less than -0.1. The overall mean ratio of predicted to experimental CHF value is 0.988 with standard deviation of 0.046. Copyright © 1996 Elsevier Science Lid

Introduction

The occurrence of CHF limits the high cooling capability and can ultimately leads to the burnout of a heated surface and its destruction. Therefore, the accurate prediction of the CHF is of importance for design and safety analysis of NPP. Until now, most of the existing CHF correlations have been developed under stable flow conditions [1, 2]. However, the flow trend of NPP can be changed from the stable state to the unstable state in transient and accident conditions, such as a loss-of-coolantaccident (LOCA). It is generally known that boiling channels are subject to various flow instabilities such as flow excursion, pressure drop oscillations and density-wave oscillations [3]. In particular, the temporary flow reduction due to flow excursion may be the cause of premature burnout as a consequence. Under certain conditions, the CHF mechanism is influenced by the coupling and the 1109

1110

S.H. Chang, Y.I. Kim and W.-P. Baek

Vol. 23, No. 8

existence of hydrodynamic instabilities such as excursive flow instability (Ledinegg instability) and oscillatory instabilities [4]. Ishii et al. [5] investigated the flow excursion CHF condition of liquid metals under low flow from the relevant experimental data. Chang et al. [6], especially, studied on the CHF for liquid metal in consideration of flow excursion under low heat flux-low flow conditions. But, the study and investigation on the accurately predictable correlation and the mechanistic understanding of the CHF due to flow instability is still not sufficient in condition of water. This study aims to derive the predictable correlation and to increase the understanding of the CHF due to flow excursion. To derive the flow excursion CHF correlation, a homogeneous two-phase frictional pressure drop correlation is used. By using the homogeneous two-phase frictional pressure drop correlation and the Ledinegg instability criterion, the relationship betwee,, CHF for water and the parameters has been derived.

Flow Excursion

General Considerations This is the most interesting form of static instability which is one where a small change in one of the independent variables leads to a large change in one of the dependent variables. The flow excursion was first analyzed by Ledinegg as one of the flow instabilities which are occurred in two-phase flow systems. The criterion for the instability is as follows <

c3APext

where ~xPe~I : the variations of pressure drop induced by the external circuit when inlet flow rate is varied. ~xPint : the variations of pressure drop that would be induced by two-phase flow (with constant power and Pout)

This means that the flow excursion occurs when the slope of the internal pressure-drop vs flowrate curve becomes smaller (more negative) than the external pressure-drop vs flow-rate curve (or pump characteristic pressure head). This is represented in the shape of"S" curve as shown in Fig. 1. If the flow is changed from the onset of boiling point B to the high quality point C because of the large liquid to vapor density ratio, a two-phase system may encounter with the CHF due to flow excursion. They often cause a sudden reduction or/and oscillation of flow rate with a small increase in the heat

Vol. 23, No. 8

MECHANISTIC CRITICAL HEAT FLUX MODEL

1111

flux. The temporary reduction of the flow rate due to instability causes total liquid starvation which may lead to a premature burnout as a consequence.

Internal

AP

Chatact~/

Pressure Head

A 4 4

Heated section: I, ;

q,,

F w

/

Nigh

Low

Quality Boiling I Quality

All Liquid

Qo

Q

Noooil length: ~. ~

q 4 - t

"__

FIG. 2

FIG. 1 Flow Excursion and AP Characteristic Curve

Schematic of Test Section Geometry

CHF due to Flow Excursion The flow excursion CHF for water has been studied by Mishima et al. [7, 8] by analyzing their experimental CHF data for round tube. But, the study is limited on the their experimental loop. The objective of this study is to obtain a general correlation for CHF due to flow excursion of water. So, the basic geometry under consideration is given as Fig. 2 to simplify the derivation of the CHF correlation. The flow excursion stability criterion is as follows 6~APe.w < C~int

According to the Ledinegg criterion, the above criterion have to be satisfied to prevent the premature burnout at the two-phase systems.

Development and Assessment of Correlation

Derivation of the Model Based on homogeneous frictional pressure drop models the internal pressure drop can be expressed approximately as A~int where

=

PfiU2fi~ D

[2

+ OL (lh -

A)] + g p ~ l h - g A p f (l h - 2 / 2) - gAp-~(l h - 2)

(1)

1112

S.H. Chang, Y.I. Kim and W.-P. Baek

Vol. 23, No. 8

2 = AhipslujD 4q

(2)

p~u fi D

0.316

and Rei - - - " I

(3)

~ o : Homogeneous two-phase multiplier

(4)

-a--

(5)

no = ~

1

1

1+ 1 - x Pg

l+l-xe/2Pg xe/2 Pl

x Pl Xe =( 4qlh

t.pliu~D

~ 3 1- -

(6)

hsg

The CHF ( q~nF ) can be derived by solving for the flow excursion condition

~ A~int = 0 and

du~

seeking the solution for the equation. The partial derivative of internal pressure drop can be expressed as follows;

f fo [~, +~2o(l h _ 2 ) ] + ,ofiu2fi 2-'-D 1 eu---'7 g3~ [3" + ~2°(lh - "g)] duoa - p~(2u~ )2-D't

~APin t

+::6

+

lax

(7)

:.--7, a-d

+ gAPs 7 d u-"'-~-t gAp ~

<~x

(lh - 2 ) + gAp'a : u:

where, ,.-

duff d -a dUff

4q (

u•' duff

ufi \---~

/

,, 4).25

)

-

1 - 0.5Xe Pg jog .~ -2. OXe C3Xe - - t Z X e )'2-- , = 0.5x e 71-2 PS crUfi Oufi

= t 1+

s 4ufi 4ql h 1 p~hygD u2fi

Therefore the above equation reduces to 8APin t

ffo [A"

7

du~ - i p ~ u ~ - ~

+~(TAPi A

1

fro

0~ [ +

+¢2°(lh-A)l+p~u~' 2 D 0 u y

. C~2o (l--¢2)+~u~(lh

_.,~.}]

(8)

+Apc~)_gAp_~fi(lh_).) N

In Eq. (8), the 3rd and 4th term (-103) are negligible in comparison to the 1st and 2nd term (-105). Therefore Eq. (8) is rearranged as

Vol. 23, No. 8

e At]int

MECHANISTIC CRITICAL HEAT FLUX MODEL

f So [7 2

2

( l h ~b~° L - 3. )+ t;t~° ,~u~ ua(lh "' - 3.) ~u~ = p~ufi2"-D/4

({fit2°_If)A4

]

J

1113

(9)

To satisfy the condition of ~ Z ~ i n t = 0 to Eq. (9), it can be rewritten as

Ou:

2

2

11

7¢L(z h _3.)+ 4~L ,,,: (th - 3.)-(~L,, - T ) 3 . =o 4

(lO)

,~u:

The two-phase multiplier is the function of pressure and exit quality, so the partial derivative of

Dua

Dx~

(11)

Du:

OXeufi(lh_3.)_(¢2o l l 3. :x, Ou: --~-) =o

762(/h_A)+~2o

a)

Two-phase multiplier

( 4% ) for the

(12)

linear equation of x e

eJL = "40 + AlXe

(13)

DC~2L°= A,

(14)

By substituting Eq. (13) and (14) into Eq. (12), the equation can be rewritten as,

7.,.+

[.,.

7 (AlXe + AoXlh GDAh 4q i ) _ .41 ~ ( t h

GDAh, + Ao ) 1 1 ] GDAh/ 4q )--[ (Alxe 4 J 4q =0

Since, __~ce ~: u~ =

: o (16)

4/hq

GDhlg

-47(AlXe + Ao)(Xe • -GDhfg. --T~) ,4q Since, (1h -

+

) = x,

-

.

4lhq ~xe " "GDhfg)-V(AlXe---7----1 ~lJnfg 4q L

Z~l ~

ll'l GDhfg + Ao) - -~--J(xt •-~-~q ) = 0

(17)

aDh:~ 4q

4lhq 7(4x~,+~x,)-4x, cZ:,h: The exit quality is expressed as

1 1] Ah~ 0 - [(4x, + 4~)- T/-~-=

(18)

1114

Xe

S.H. Chang, Y.I. Kim and W.-P. Baek

Vol. 23, No. 8

41hq Ah~ 4 GDhfg hfg=-C q-t

~

m

41hq GDhfg

4

4

~q=(

Oh:~

Since, C = ~

lh / D'

(19)

q-t)+t a~

(20)

t = --

hyg

[A,(4-t)2 + AO(q-t)]-EAl(4-t)][(4-t)-tl-EA,(4-t)+(A L

o -1-~)]t = 0

(21)

q

-~Al(--~q-t) +-~ Ao(--~q

(22)

3 A,(cq-t)2+(7 Ao-2A,)(cq-t)+(Ao-l~)t=O

(23)

Finally, the above equation can be rearranged as follows

(4q-t)2+

q-t)--~l(4Ao-ll)t=O

(7Ao - 8 A l ) (

(24)

The solution of the Eq. (24) may be given in the following form, 4

--~q- t =

-a_+ a U ~ - 4 b

(25)

2

where, 1 6/=--¸

3Aj

(7Ao - S A l ) , b = -

q=~-

t÷

1

3A1

(4Ao - 11)t

(26)

(27)

2

b) Two-phase multiplier ( ~ o ) for homogeneous model

(28)

L

°:gJL

&J

The two-phase multiplier is simplified as follows (29)

L

% J

Vol. 23, No. 8

MECI[~NISTIC CRITICAL HEAT FLUX MODEL

TABLE 2 Parameter Ranges o f CHF Experimental Data for Water

TABLE 1 Comparison o f the Two-Phase Frictional Multiplier for the Homogeneous Model Steam-Water System Pressure, bar (psia)

Steam

quality

1.01 (14.7)

20.59 (100)

1115

Parameters 34.48

(500)

Range

Pressure (MPa)

0.1 - 4.0

Mass flux (kg/m2s)

1,300 - 27,000

Yoby wt.

Eq. (28)

Eq. (29)

Eq. (28)

Eq. (29)

Eq. (28)

Eq. (29)

0.0

1.0

1.0

1.0

1.0

1.0

1.0

Inlet quality (-)

< -0.1

0.5

9.0

8.8

1.4

1.4

1.2

1.2

Heated length (m)

0.025 - 0.86

I.O

17.0

16.2

1.8

1.8

1.7

1.4

Tube diameter (m)

0.001 - 0.0108

1.5

25.0

23.3

1.6

33.0

30:2

2.2 2.5

1.9

2.0

2.2 2.6

3.4

1.8

Number of data

21

L 2o #j by Eq. (30) , 4 0 = l , Al =

og

(31)

2ojs

By substituting Eq. (3 I) into Eq. (26), Eq. (27) can be rewritten as

q=C{2t-a- a2~-4b}

(32)

Finally, from the two-phase multiplier for the homogeneous model and Ledinegg instability criterion the relationship between CHF and the principal parameters is derived and can be expressed as q = 0.125

Gh~ 12t-a-a2~-4b} lh/O{

(33)

where ~ t _-AJ~ a = 14 v/

h~

8 b=14 oy t

3t,jg 3'

3ojg

Assessment of Correlation The accuracy of the present model is evaluated by comparing model predictions with the experimental data reported in the literature. Also to assess the reasonability o f the present model approach, the other CHF correlations for water are used here for reference purposes. The overall mean accuracy ratio of the present model is calculated as following;

1116

S.H. Chang, Y.I. Kim and W.-P. Baek

Mean = qp,~a

qm~as

Vol. 23, No. 8

and the error in applying the developed correlation to each experimental data point

was found as defined by

e

-qpr~d qmeas

qmeas

where, qmeas is t h e critical heat flux obtained from the experimental (or measured) data and qpred is t h e critical heat flux calculated from the developed correlation For groups o f data, the mean error, RMS error and standard deviation o f the error from the mean

were also calculated as; i=N

i=N

i=1

/ i=1

~1/2

To assess the developed CHF correlation based on flow excursion mechanism, a series of data for water were selected from KAIST CHF data base [9]. The parameter ranges of the selected data are 1 < P < 40 bar, 1,300 < G < 27,000 kg/m2s and xi(~ h ~ ) < -0.1 as shown in Table 2. TABLE 3 Experimental Conditions for Selected Data P

lh

D

G

Ah~

x~

xc

(kP~

(m)

(m)

(kg/m2s)

(lO/kg)

(-)

(-)

0.0650 0.0650 0.1220 0.0780 0.0250 0.0250 0.0250 0.3440 0.3440 0.3440 0.3440 0.4000 0.1140 0.4000 0.4000 0.4000 0.4000 0.7620 0.4320 0.4320 0.8600

0.0013 0.0013 0.0024 0.0031 O.OOlO O.OOlO 0.0010 0.0060 0.0060 0.0060 0,0060 0.0100 O.O011 O.OlO0 O.OlO0 0.0100 0.0100 0.0103 0.0108 0.0108 0.0108

101.0 101.0 lOi.O 101,0 103,0 103.0 103.0 101.0 101.0 101.0 lOl.O 2059.0 2137.0 '3040.0 3040.0 3040.0 3040.0 3448.0 3861.0 ,. 3861.0 3896.0

16546.0 26311.0 13494.5 4109.4 2793.0 3729.0 5749.0 1768.0 1722.0 1748.0 1226.0 1292.0 6780.0 1191.0 1196.0 1348.0 1300.0 5953.0 4068.0 4068.0 4068.0

312.0 312.0 317.0 321.0 319.0 319.0 319.0 293.0 252.0 168.0 293.0 801.0 380.0 890,0 890,0 890,0 890.0 193.0 192.0 192.0 369.0

-0.138 -0.138 -0.140 -0.142 -0.141 -O.141 -0.141 -0,129 -O.lll -0,074 -0,129 -0.425 -0.202 -0.497 -0.497 -0.497 -0,497 -0.109 -Odll -O.lll -0.214

0.007 -0.020 0.020 -0.020 0,012 0.008 0.007 -0.010 -0.004 O.Oll 0.016 0.018 -0.019 0.018 0.017 -0.013 -0.005 0.020 0,012 0.012 0.018

q~ (kW/m 2) 27129.0 35015.0 24669.0 11514.0 9685.0 12587.0 19275.0 2082.0 1823.0 1476.0 1757.0 6723.0 5836.0 6842.0 6863.0 7287.0 7143.0 4574.0 5408.0 5380.0 5110.0

q~d (kW/m 2) 25840.0 41090.1 21412.2 13205.6 9097.8 12146.7 18726.6 2261.3 1894.3 1281.9 1568.1 6598.1 6589.3 6839.0 6867.7 7740.5 7464.9 4040.4 5103.3 5103.3 4929.5

Vol. 23, No. 8

MECHANISTIC CRITICAL HEAT FLUX MODEL

1117

Results and Discussion

To compare the prediction accuracy of the present model with the other CHF correlations, Bowring [2] and Katto [1] correlations of water are selected. In Table 4 these correlations are presented and the prediction accuracy is compared. The overall mean accuracy ratio of Bowring and Katto's correlations are 1.10 and 1.614 with the standard deviation of 0.314 and 0.935, respectively. Also, the various comparisons of each prediction on mass flux, pressure, inlet subcooling and exit quality are presented in Fig. 3 through Fig. 7. From the results, the present mechanistic CHF correlation for water shows that the predictions agree well with the CHF data of water.

TABLE 4 Comparison of the Accuracy for Water CHF Correlations Correlations

,//

I [ • Pr~"iZ o

Standard

Mean

z,

deviation

.it

Present :

A

2

Gh

• A

A

0.046

0.988

/m

Bowring : 1.110

q= ,4'+0250G~

0.314

i ~ ' 0

o

C'+lh

°

0

I0000 15t~0 ~

Kauo :

o

o

o

25~0 ~

36~0 ~

45~tI

M N ~ l r ' e d CHF,

1.614

0.935

q = qco ( 1 + K --~--)

4,0

FIG. 3 Comparison of Predicted and Measured CHF Values

.~.

4.0

r-1

3.s ~, 3.0

: 8cwt~ A

A

Z5

iiloo.o_ t.5

0

A

~

•

..

l

¢

o

A --

% o 0

5000

10000

2-5

A

,,ti

o

~.

~=-

1

3,° A

15000

o 20000

25000

M a s s Flt~<, kg/rn2s

FIG. 4 Prediction Accuracy on Mass Flux

t

o

Q0 0 30000

o

r-

~

dm xho pressure, kPa

"

~oo

FIG. 5 Prediction Accuracy on Pressure

1118

S•H. Chang, Y.I. Kim and W.-P. Baek

4.0

4.0

'I

3.5

AA

A

,

3.0

Vol. 23, No. 8

•

Pr~

o A

Bowrtng i K.t~aO

Present Bo~ng A

3.0 o

2.5 & o

~

o& o oA

1.5

o

z5 2.0-

OA

1.5-

A

o

tO

0

0 o

A

OQ

10 ! o

0.5

g

0.5-

0.0

0.0

1000

0

~,012

Inlet subooding, k.l~,g

o o o

~

0.@0

0.bl

o~

Exit quality

FIG. 6 Prediction Accuracy on Inlet Subcooling

FIG. 7 Prediction Accuracy on Exit Quality

Conclusions

(1) From the simplified two-phase homogeneous frictional pressure drop and the Ledinegg instability(or Flow Excursion) criterion, the relationship between CHF for the water and the principal parameters such as mass flux, heat of vaporization, heated length-to-diameter ratio, vapor-liquid density ratio and inlet subcooling is derived. (2) The present mechanistic model to predict the CHF of water agree well with the extensive KAIST CHF data. The overall mean ratio of predicted to experimental CHF value is 0.988 with standard deviation of 0.046• (3) Finally, the CHF correlation of water based on Ledinegg instability has been derived as follows; q = 0.125 Ghyg

12t- a - a2~-~-4b} lh/Dt

where _ _

t= ~

hs~

, a=

u/ 3o/~

14

8

3

,

b

of 3o/~

14 =--

t

The applicable range of the present CHF correlation is roughly recommended as : 0.1 < P <40 bar, 1,300 < G 27,00 kg/mZs and inlet quality is less than -0.1.

From the above study several further studies are recommended as follows: (1) Consideration of the another two-phase frictional multiplier to develop CHF correlation with an high accuracy.

Vol. 23, No. 8

(2)

MECI-UkNISTIC CRITICAL HEAT FLUX MODEL

1119

Construction of the CHF data occurred by flow excursion to validate the present correlation. Acknowled~,ment

The authors wish to express great appreciation to the Center for Advanced Reactor Research (CARR) at the KAIST and the Korea Science and Engineering Foundation(KOSEF) for their financial support. Nomenelatu re

A D

f

g G

/h P q O Z

Ahi Ap A0f

cross-sectional area (m 2) hydraulic diameter (m) friction factor of liquid gravitational acceleration (m/s :) mass flux (kg/m2s) heat of vaporization (J/kg) heated length (m) pressure (Pa) heat flux (W/m2) inlet liquid velocity (m/s) specific volume (m~/kg) axial coordinate inlet subcooling (J/kg) density difference(py-p~), (kg/m3) density change(13pfiAT), (1/*C)

~o la xi xe

two-phase multiplier viscosity (N-s/m 2) inlet quality exit quality

Subscripts ext f fi g int meas pred

external fluid inlet fluid vapor internal measured predicted

References

I.

Y. Katto and H. Ohno, Int. J. Heat Mass Transfer ~

2.

R.W. Bowring, A simple but accurate round tube, uniform heat flux, dryout correlation over the

1641, (1984).

pressure range 0.7 - 17 MN/m 2 (100-2500 psia), AEEW-R 789, (1972). 3.

J.A. Boure, A.E. Bergles and L.S. Tong, Nucl. Eng. & Design., 25, 165, (1973).

4.

M. Ledinegg, Die Warme, 61, 8, (1938).

5.

M. Ishii and H.K. Fauske, Nucl. Sci. Eng., ~

6.

S.H. Chang and Y.B. Lee, Nucl. Eng. & Design., 148, 487, (1994).

7.

K. Mishima, H. Nishihara and I. Michiyoshi, Int. J. Heat Mass Transfer 28[6], 1115, (1985).

8.

K. Mishima, Boiling burnout at low flow rate and low pressure conditions, Ph.D. Thesis, Kyoto

131, (1983).

Univ., Japan, (1984). 9.

S.H. Chang et al., The KAIST CHF Data Bank, Interim Report, KAIST-NUSCOL-9401, (1994). Received June 7, 1996