A critical heat flux approach for square rod bundles using the 1995 Groeneveld CHF table and bundle...

Nuclear Engineering and Design 197 (2000) 357–374

A critical heat flux approach for square rod bundles usingthe 1995 Groeneveld CHF table and bundle data of heat

transfer research facility

Min Lee *Department of Engineering and System Science, National Tsing Hua Uni6ersity, 101, Section 2, Kuang Fu Road, Hsinchu, Taiwan,

ROC

Received 12 January 1998; received in revised form 3 September 1999; accepted 27 September 1999

Abstract

The critical heat flux (CHF) approach using CHF look-up tables has become a widely accepted CHF predictiontechnique. In these approaches, the CHF tables are developed based mostly on the data bank for flow in circulartubes. A set of correction factors was proposed by Groeneveld et al. [Groeneveld, D.C., Cheng, S.C., Doan, T., 1986.1986 AECL-UO Critical Heat Flux lookup table. Heat Transf. Eng. 7(1–2), 46] to extend the application of the CHFtable to other flow situations including flow in rod bundles. The proposed correction factors are based on a limitedamount of data not specified in the original paper. The CHF approach of Groeneveld and co-workers is extensivelyused in the thermal hydraulic analysis of nuclear reactors. In 1996, Groeneveld et al. proposed a new CHF table topredict CHF in circular tubes [Groeneveld, D.C., et al., 1996. The 1995 look-up table for Critical Heat Flux. Nucl.Eng. Des. 163(1), 23]. In the present study, a set of correction factors is developed to extend the applicability of thenew CHF table to flow in rod bundles of square array. The correction factors are developed by minimizing thestatistical parameters of the ratio of the measured and predicted bundle CHF data from the Heat Transfer ResearchFacility. The proposed correction factors include: the hydraulic diameter factor (Khy), the bundle factor (Kbf), theheated length factor (Khl), the grid spacer factor (Ksp), the axial flux distribution factors (Knu), the cold wall factor(Kcw) and the radial power distribution factor (Krp). The value of constants in these correction factors is differentwhen the heat balance method (HBM) and direct substitution method (DSM) are adopted to predict the experimentalresults of HTRF. With the 1995 Groeneveld CHF Table and the proposed correction factors, the average relativeerror is 0.1 and 0.0% for HBM and DSM, respectively, and the root mean square (RMS) error is 31.7% in DSM and17.7% in HBM for 9852 square array data points of HTRF. © 2000 Elsevier Science S.A. All rights reserved.

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1. Introduction

The critical heat flux (CHF) condition is char-acterized by a sharp reduction in the local heattransfer coefficient (HTC), which results from the

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M. Lee / Nuclear Engineering and Design 197 (2000) 357–374358

replacement of liquid by vapor adjacent to theheat transfer surface. The deterioration of theheat transfer mechanism could lead to a suddentemperature increase in the heat transfer surface.The critical heat flux forms a very importantlimitation in the safe operation of nuclear powerreactors. The technical importance of the CHFcondition has led to the development of a varietyof empirical correlations. These empirical CHFcorrelations are valid over relatively narrowranges and cannot be extrapolated to conditionsfar beyond the database. To overcome this prob-lem, look-up table methods (Doroshchuk et al.,1975; Groeneveld et al., 1986, 1996; Bestion,1990) are proposed to predict CHF in circulartubes. Critical heat flux lookup tables are derivedby statistically averaging experimentally observedCHF values within each pressure, mass flux, andquality interval, and can be used to predict CHFin circular tubes over a wide range of conditions.The CHF tables are developed based mostly onthe data bank for flow in circular tubes. A varietyof correction factors have been proposed byGroeneveld et al. (1986, 1992) and Bobkov et al.(1995, 1997) to extend the application of CHFtables to other situations including flow in rodbundles. Critical heat flux look-up table methodshave become widely accepted CHF predictiontechniques. The CHF approach of Groeneveld etal. (1986) is extensively used in the thermal hy-draulic analysis of nuclear reactors.

Groeneveld et al. (1996) point out that CHFlook-up table methods have the following advan-tages over correlations or semi-empirical CHFmodels:1. accurate prediction;2. the widest ranges of applications;3. ease of use (no fluid properties are needed);4. ease of updating;5. correct parametric and asymptotic trends.

In 1996, Groeneveld et al. proposed a newCHF table (the 1995 Groeneveld CHF Table).The table has been developed based on 22 946data from circular tubes. The new table predictsthe data with an average error of 0.69% and aroot mean square (RMS) error of 7.82% whenheat balance method (HBM) is adopted. In thepresent study, correction factors are developed to

extend the application of the new CHF table toflow in rod bundles of square array. The correc-tion factors are developed by minimizing thestatistical parameters of the ratio of the measuredand predicted bundle CHF data (Fighetti andReddy, 1983a) from the Heat Transfer ResearchFacility (HTRF).

2. Heat balance method and direct substitutionmethod in predicting CHF

There are two types of CHF approach (Heizlarand Todreas, 1996a). In the first type (Type I),CHF is an explicit function of local quality. In thesecond type (Type II), the local quality is elimi-nated by incorporating the heat balance along thechannel up to the point of interest. When variousCHF approaches are compared against one an-other and test data, or are used to predict CHFmargins, two methods (Heizlar and Todreas,1996a) can be adopted in the CHF calculations.They are the direct substitution method (DSM)and the heat balance method (HBM). Groeneveldet al. (1986, 1996) refer to these two methods asthe ‘constant dryout quality approach’ and the‘constant inlet subcooling approach’, respectively.In DSM, the local equilibrium quality (or cross-sectional average thermodynamic quality) at eachaxial position of interest is calculated using theheat balance equation. This local quality is thendirectly substituted into the CHF approach toobtain CHF. In HBM, CHF at axial position ofinterest is calculated by varying the magnitude ofthe heat flux input into the channel, with the fluxshape preserved, until the critical heat flux givenby the CHF approach at this given location isreached. In other words, one is looking for theintersection of the heat balance curve and theCHF curve.

CHFs predicted by Type II CHF approachusing DSM and HBM are identical. However,when DSM and HBM are used to compare theType I CHF approach with test data, HBM givesbetter results than DSM (Inasaka and Nariai,1996). Celata (1996) argues that HBM is the onlycorrect method that can be used to evaluate theexperimental data. He points out the inconsis-

M. Lee / Nuclear Engineering and Design 197 (2000) 357–374 359

tency between the CHF and the exit quality whenusing DSM because one relies on the CHF whichis ‘different from that necessary in the heat bal-ance to give the assumed quality’. Groeneveld(1996) also favors the use of HBM in assessing theerrors of Type I CHF approach. However, Siman-Tov (1996) takes the opposite position. Accordingto Siman-Tov’s view, only DSM is correct forcomparison of correlation with experimental databecause it involves the comparison of the DSMcorrelation predicted CHF to experimental heatflux, under the same thermal hydraulic conditions,while HBM compares heat fluxes at different ther-mal hydraulic conditions.

From the application point of view, when theprediction of CHF is required under a givengeometrical configuration, inlet thermal hydraulicconditions and system pressure, the quality atCHF is not known and the only possible way toapply the Type I CHF approach is HBM. How-ever, when CHF approaches are implemented intothe system thermal hydraulic analysis code; suchas RELAP/Mod3 (Carlson et al., 1990), DSM isused to compare the local heat flux with a pre-dicted CHF based on a code calculated quality.

As suggested by Heizlar and Todreas (1996b) andWeisman (1996), it is important to obtain bothDSM and HBM statistics of Type I CHF ap-proach to allow the application of predictive pro-cedure in nuclear reactor accident analysis and toallow realistic comparisons.

The CHF table delineates an explicit depen-dency of CHF on local quality and, therefore, theCHF look-up table method is classified as a TypeI approach. Which method, DSM or HBM, ismore justified when the Type I CHF approach isused in the prediction of CHF is still a verycontroversial issue. However, given that the ap-proach might be used in either method, bothHBM and DSM are given equal consideration inthe present study. When the 1995 GroeneveldCHF table is used with the HBM set of correctionfactors, the approach is classified as a Type IIapproach. When the 1995 Groeneveld CHF tableis used with the DSM set of correction factors, theapproach is classified as a Type I approach.

3. HTRF bundle critical heat flux data bank

Researchers at the HTRF at Columbia Univer-sity have carried out CHF tests (Fighetti andReddy, 1983a,b) and collected more than 11 000data points from 235 separate test sections. Thetest sections are designed to cover a wide range ofgeometry with triangular-pitch bundles for heavywater reactors and with square-pitch rod arraysfor both pressurized water reactors (PWRs) andboiling water reactors (BWRs). The bundle sizecovered the ranges of 3×3 to 6×6 square pitchand 19-, 28-, and 37-rod triangular pitch. The testsections can be classified into 16 different bundlearrangements. The parameter ranges and geomet-ric data from the HTRF CHF data bank aresummarized in Table 1.

There are 11 077 data in the HTRF data bank.However, only 10 918 data points have sufficientinformation for the assessment of CHF ap-proaches. Among these data, there are 420 datafrom Test Sections 151, 167, 168, 169 and 170which include the effect of rod bowing. The rodbowing could change the local thermal-hydrauliccondition and have a significant impact on CHF

Table 1Parameter ranges and geometry data of HTRF critical heatflux data bank

Range or specification

Parameter1.276–17.0Pressure (MPa)54.2–6050Mass flux

(kg/mg2·s)Inlet enthalpy 6.98–1140

(kJ/kg)3.6–3940Average heat flux

(kW/m2)

GeometrySquare array 3×3–6×6Triangular array 19, 27, 37 rods

0.762–4.267Heated length(m)

Diameter of 9.5–19.8heated rod(mm)

Axial flux distri- Uniform and non-uniform (17 differentbution distribution)


(Hill et al., 1975; Markowski et al., 1977; Macduffand Fighetti, 1983). These 420 data points withrod bowing are not included in the presentassessment.

According to the mechanism of CHF, it shouldoccur downstream of the location of onset ofnucleate boiling (ONB) and before the locationwhere the water is completely evaporated. Theadequacy of the data points in the HTRF bundleCHF data bank was checked using energy balanceat the locations of ONB and CHF. The energybalance equation is:

X(zCHF)=Xin+N · p · D

& zCHF

0

q¦(z) dz

GAfhfg

(1)

where X is average quality; Xin, inlet quality; N,number of heated rods; D, diameter of heatedrod; q¦(z), heat flux; G, mass flux; Af, flow cross-section of test section; hfg, evaporation heat ofwater; zCHF, the location of CHF observed in theexperiment.

The calculated X(zCHF) for all the data pointsshould be less than 1.0.

For boiling to occur, the wall temperature, Tw,has to be greater than the saturation temperature,Tsat, of fluid. The wall temperature can be calcu-lated as:

Tw(z)=Tin+N · p · D ·

& z

0

q¦(z) dz

GAf · Cpf

+q¦(z)

hfo

(2)

and

hfo=0.023�G · Dhy

mf

n0.8�Cpf · mf

Kf

n1/3 Kf

Dhy

(3)

where hf0 is the single phase heat transfer coeffi-cient; Tw(z), wall temperature; Tin, water inlettemperature; Cpf, specific heat of water; Dhy, hy-draulic diameter; Kf, thermal conductivity of wa-ter; mf, viscosity of water.

At the location of ONB, the calculated walltemperature of Eq. (3) should be equal to the walltemperature calculated based on the incipientboiling or fully developed subcooled boiling. Thewall temperature can be calculated with the equa-tion proposed by Bergles and Rohsenow (1963):

Tw(zONB)=Tsat+0.056� q¦(zONB)

1082 · P1.156

n0.463 P0.0234

(4)

where Tw is the wall temperature (K); q¦, heat flux(W/m2); P, pressure (bar), 1BPB138 bar; Tsat,saturation temperature of water (K); zONB, loca-tion of ONB.

If the calculated ONB is located downstream ofthe observed CHF location, the data point isconsidered to be unreasonable.

Through the above calculations, it is found that14 data points of Test Section 35 are unreason-able. It is determined that all the 31 data points insection 35 are discarded. During the assessment, itis found that the data points number 60 through85 of Test Section 30 have unusually high ratiobetween the predicted and the measured CHF. Itis decided that these 26 consecutive points are notconsidered in the assessment. In summary, 10 411data points were considered in the present assess-ment. Among these data points, there are 9582square array data points and 559 triangular arraydata points.

4. Correction factors of using CHF look-up tablesin prediction CHF of rod bundles

The AECL-UO CHF table (Groeneveld et al.,1986) and the 1995 Groeneveld CHF table(Groeneveld et al., 1996) are derived for upwardflow in a uniformly heated 8-mm tube. When thetable is used to predict CHF in other geometryand flow conditions, correction factors are re-quired. First of all, a correction factor is requiredto account for the difference between the CHFvalues of an 8-mm tube and a tube with D-mmdiameter. After a extensive literature survey,Groeneveld et al. (1986) suggest that the correc-tion factor for tube diameter has the followingform:

q¦CHF(D ; P, G, x)q¦CHF−Table(8 mm; P, G, x)

=�D

8�n

(5)

where D is tube diameter in mm. In the derivationof the AECL-UO CHF table (Groeneveld et al.,1986), the exponent n is set equal to −1/3 for2BDB16 mm. When the tube diameter is


greater than 16 mm, the CHF ratio is assumed tobe 0.79. In the derivation of the 1995 GroeneveldCHF table (Groeneveld et al., 1996), the exponentn is set equal to −1/2 for 3BDB25 mm.

Bobkov et al. (1995) also use Eq. (5) to accountthe effect of tube diameter. Their analysis indicatethat the influence of other parameters (such aspressure, mass flux, and quality) on the tubediameter effect is small and can be ignored. Theyhave used both −1/2 and −1/3 for the exponentn in their analyses (Bobkov et al., 1995, 1997).

When the CHF table is used to predict CHF ofa non-circular geometry (such as concentric andeccentric annular channel, and subchannel in rodbundles), the conventional hydraulic diameter canbe used in place of tube diameter. Bobkov et al.(1995) suggest that the difference in CHF causedby the non-uniformity of the geometry shouldalso be taken into consideration. The proposedcorrection factor is a complicated function ofpressure, mass flux, steam quality, and geometrycharacteristics.

Bobkov et al. (1997) compare the 1995Groeneveld CHF table with the FEI bundle dataof hexagonal rod bundles with a triangular rodarrangement. They find that, after taking the ef-fect of tube diameter, geometry non-uniformity,and spacers of rod bundles into consideration,there is an additional and pronounced dependenceof the relative value of CHF on mass flux, steamquality, and pressure. Therefore, a new substan-tial correction function should be introduced. Itsvalue ranges from 0.2 to 1.3. These dependences isnot observed for flow in an isolated channel.Bobkov et al. call the phenomenon the bundleeffect.

In view of this, Bobkov et al. propose that theCHF in sparsely spaced (with pitch-to-diameterratio greater than 1.16) hexagonal rod bundlescan be described as:

q¦CHF(Dhy; P, G, X)q¦CHF−Table(8 mm; P, G, X0)

=�Dhy

8�n

Kbf(P, G, X)

(6)

where Kbf is the correction factor for bundleeffect, and

X0=X+0.05(C0+1)2(1−0.8P/Pcr)

C0=1.047(s/D)−1

where D is the diameter of the heated rod and s isthe rod pitch.

In Eq. (6), Dhy, the hydraulic diameter of thetriangular array is defined as:

Dhy=1.103D [(s/D)2−1]

For the selected data array of sparsely spaced rodbundles, Kbf can be calculated as:

Kbf(P, G, X)

=�

a+bP

Pcr

+c� P

Pcr

�2

+d� P

Pcr

�3n× [0.90+0.5(X−0.4)]

The coefficients of the polynomial are as follows:

a=0.361+0.138g−0.010g2

b= −1.590−0.584g+0.269g2

c=4.8473+5.004g−1.367g2

d= −2.46−5.32g+1.24g2

where g=G/1000, with G in kg m−2 s−1, and Pcr

is the critical pressure. Using the above relation-ship, Bobkov et al. (1997) succeed in describingthe entire selected data array with a maximalmean deviation of 5% and a maximal root-mean-square data spread of 24%. The test conditionsfor these data are: pressure from 1 to 20 MPa,steam quality from 0.4 to 0.8, mass flux from 150to 5000 kg m−2 s−1.

In the Groeneveld et al. (1986) CHF approach,the bundle CHF is calculated as:

q¦CHF=q¦CHF−Table · CHFmul (7)

where q¦CHF−Table is determined through a CHFtable and CHFmul is the product of seven modifi-cation factors (K1 through K7) which are functionsof variables related to the heat surface geometryand operating conditions. The modification fac-tors use in the Groeneveld et al. CHF approach(1986) are listed in Table 2. K1 in Table 2 is thecorrection for hydraulic diameter and K2 is thecorrection factor for the bundle effect.Groeneveld et al. also include the effect of gridspacer (K3), heated length (K4), and non-uniform


Table 2Correction factors of AECL-UO CHF table

FormFactor

K1, subchannel or tube cross-section factorFor 0.002BDhyB0.016 m: K1=

�0.008

Dhe

�1/3

For Dhy\0.016 m: K1=0.79

K2, bundle factor K2=min[0.8, 0.8 exp(−0.5·X1/3)]

K3, grid spacer factorK3=1+1.5K0.5 ·

� G

1000

�0.2

· exp�−0.1

Lsp

Dhy

�K4, heated length factor

L/Dhy]5: K4=exp�Dhy

Le2a

�a evaluated from homogeneous model

a=X/[X+rg/rl(1−X)]

XB0: K5=1.0K5, axial flux distribution factor

X\0: K5=� 1

(z−zX=0)·& z

zX=0

q¦(z %) dz %n,

q¦(z)

Stratified flow (GBG1): K6=0.0K6, horizontal flow factor

Nonstratified flow (G\G2): K6=0.0

Intermediate (G1BGBG2): K6=G−G1

G2−G1

To find G1 and G2 use Taitel and Dukler’s flow regime map

K7, vertical flow factor GB−400 kg m−2 s−1 or X=0.0: K6=1.0

−50BGB10 kg m−2 s−1: CHF=CHFG=0,X=0 · (1−a)C1

aB0.8: C1=1.0

a\0.8: C1=0.8+0.2rl/rg

a+(1−a)rl/rg

10BGB100 kg m−2 s−1

−400BGB−50 kg m−2 s−1

"linear interpolation

heat flux distribution (K5) in their corrections.The proposed correction factors are based ona limited amount of data not specified in theoriginal paper. A systematic literature searchof the effect of flow geometry on CHF hasbeen carried out by Groeneveld et al. (1992). Ithas been identified in their work that CHF isaffected by heated equivalent diameter, hy-draulic equivalent diameter, inter-element gap,element/pressure gap, curvature and subchannelshape.

5. CHF prediction of bundle experiment usingCHF look-up table

The CHF tables delineate an explicit depen-dency of CHF on local quality and, therefore, theCHF look-up table method is classified as a TypeI CHF correlation. In the present assessment,both DSM and HBM are adopted to predict theHTRF bundle data. The following iteration stepsare taken to predict CHF in rod bundles whenHBM is adopted in the assessment.


Critical heat flux always occurs at the exit ofthe heated section when heat flux is uniformlydistributed axially. The average exit quality, Xexit,of a heated section with an average heat flux ofq¦ave (average of radial power distribution) can beobtained from an energy balance calculation (Eq.(1), with z set equal to the heated length). CHFfor a given q¦ave can be determined through tablelookup and calculation of the correction factors.That is:

q¦CHF=q¦CHF−Table(P, G, Xexit)KCorrection-factors

(8)

It is assumed that the condition of CHF isreached when the maximum heat flux of the rodbundles, q¦max, is equal to q¦CHF where q¦max isdefined as

q¦max=q¦ave·Pradial, max (9)

Pradial, max is the maximum radial peaking factorof the heated rods. In the calculation, the CHFfor a given geometry and operation condition isdetermined iteratively by varying q¦ave.

For data points with non-uniform axial heatflux distribution, the actual axial location of CHFis unknown. Conceptually, q¦CHF(z) as a functionof heated length can be calculated and comparedwith the axial heat flux profile. The CHF occursat the location where the q¦CHF(z) profile firsttouches the actual heat flux profile, q¦(z). In thepresent analysis, the steps outlined in the previousparagraph determine the values of q¦CHF at theselected location zi. The calculated q¦CHF(zi) isrelated to the bundle average heat flux q¦ave(zi) by

q¦ave(zi)=q¦CHF(zi)/Paxial(zi) Pradial, max (10)

where Pradial(zi) is the axial peaking factor atlocation zi. This procedure is repeated for allselected locations. It is assumed that CHF occursat the locations where q¦ave(zi) has the smallestvalue among all the locations calculated.

When the DSM is used to predict CHF ofHTRF test sections with non-uniform heat fluxdistribution, q¦ave(zi) is determined by a directtable look-up based on the cross-sectional averagethermodynamic quality obtained from energy bal-ance. It is also assumed that the CHF occurs at alocation where q¦ave(zi) is the smallest among allthe locations calculated.

6. Correction factors for bundle CHF data ofHTRF

In the present study, correction factors are in-cluded to account for the effect of hydraulic di-ameter, heated length, bundle effect, cold wall,grid spacer, radial power distribution, non-uni-formly axial heat flux distribution. These correc-tion factors are incorporated sequentially into theCHF calculations based on the characteristics ofthe test section of HTRF. Even thoughGroeneveld et al. (1996) do not suggest the use ofthe correction factors of the AECL-UO CHFtable in conjunction with the 1995 CHF table,these factors are chosen as the starting point ofthe derivation. The appropriate forms of correc-tion factors suggested in the literature are usedwhenever possible. The form of the correctionfactors is kept as simple as possible. The form ofthe correction is chosen based on the results ofHBM calculation. The constant in the correctionfactors is then adjusted separately for HBM andDSM.

The correction factors are developed by mini-mizing the statistical parameters of the ratio ofthe predicted and measure bundle CHF data ofthe HTRF in different pressure, mass flux, andsteam quality ranges. The definitions of thesesstatistical parameters are:

Ratio of predicted and measured CHF:

RCHF=q¦CHF, predicted

q¦CHF, measured

(11)

Average of RCHF,I:

AVGCHF=1N

SN

i=1RCHF, i (12)

where N is the number of data points. Root MeanSquare of RCHF,I:

RMSCHF='1

NSN

i=1(RCHF, i−1)2 (13)

Standard Deviation of RCHF,I:


STDCHF=' 1

N−1SN

I=1(RCHF, i−AVGCHF)2

(14)

AVGCHF and RMSCHF represent the accuracyand the uncertainty, respectively, of the approach.AVGCHF minus one is the relative error of theapproach. STDCHF represents the spread of ratioof the predicted and measured CHF around theaverage ratio.

6.1. Hydraulic diameter factor

In the derivation of the 1995 CHF table,Groeneveld et al. (1996) use Eq. (5) with n= −1/2 to modify the tabulated value for a tube ofdiameter between 3 and 25 mm. It is not men-tioned what factor is used to correct the data fora tube diameter less than 3 mm or greater than 25mm. In the bundle CHF data bank of the HTRF,the hydraulic diameter is within the range of 6–16mm Eq. (6) with n= −1/2 is used to correct forthe hydraulic diameter effect. That is:

Khy=� Dhy

0.008�−1/2

(15)

where Dhy is the hydraulic diameter in m.

6.2. Heated length factor

A preliminary calculation of the 3108 datapoints of square array with uniform axial heatflux distribution shows that the heated lengthfactor (K4 in Table 2) has some marginal impacton the average ratio of the predicted and mea-sured CHF. The incorporation of heated lengthfactor has a tendency to increase the value of thepredicted CHF slightly. However, the impact ofthe heated length factor on the root mean squareof the ratio of the predicted and measured CHFof the HTRF bundle data is insignificant. Theheated length factor suggested by Groeneveld etal. (1986) will be used in the following calculation.

Khl=exp�Dhy

Le2a

�; L/Dhy]5 (16)

while a is the void fraction and is evaluated usinga homogenous model.

6.3. Bundle factor

A detailed analysis of the 3108 data points ofsquare array with uniform axial heat flux distribu-tion and without cold wall effect shows that thebundle factors in Table 2 have a tendency tounder-predict the bundle CHF at low quality(when quality\0.0) and to over-predict at highquality. It also has a tendency to under-predictthe bundle CHF at high mass flux and to over-predict at low mass flux. It also over-predicts thebundle CHF data when pressure is less than 3MPa.

Based on the trends observed, a new empiricalbundle factor is proposed:

Kbf=A×B

A=min[c1, c1 exp(−1.12X)]

B=exp[−0.073+0.035(G/1000)] (17)

where X is the average thermodynamic quality,and G is mass flux (kg m−2 s−1). Based on thestatistics of the results using HBM and DSM, thevalue of C1 is set equal to 0.98 and 1.02, respec-tively. The improvement of the approach in theprediction of bundle CHF data using Eq. (17) asthe bundle factor can be seen form the com-parison of the results of Case A and Case B inTable 3.

6.4. Correction factor for cold wall effect

There are cold walls (unheated), e.g. guide tube,thimble tube, water rod, channel box etc., in thefuel bundles of a reactor. A thin layer of liquidfilm forms on the cold wall, which affects the voiddistribution within the fuel bundle. As shown inCase C of Table 3, without a correction for thecold wall effect, the 1995 Groeneveld CHF tablehas a tendency to over-predict the 4544 datapoints of square array with cold wall by 4.2% inDSM and by 0.7% in HBM.

In W-3 (Tong, 1967) correlation, the cold walleffect correction factor (Tong, 1972) is a functionof the ratio of hydraulic diameter and equivalentheated diameter, pressure, mass flux and quality.In the CENPP (1977) correlation, the correctionfactor for the cold wall has the form of Dhe/Dm,

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Table 3Statistics of the ratio of the predicted and measured HTRF bundle CHF using DSM and HBM

Direct substitution method Heat balance methodCase Modification factor Data type No. of points

STD AVG RMS STDRMSAVG

0.860 0.277 0.239 0.948 0.137 0.126A Khy, Khl, and K2 in Table 2 Square, uniform, no cold wall 31080.248 1.001 0.121 0.1210.248Khy, Khl, and KbdB 1.0003108Square, uniform, no cold wall

0.324Khy, Khl, and Kbd 0.321 1.007 0.107 0.107Square, uniform, cold wall 4544 1.042CD 0.307Khy, Khl, Kbd, and Kcw 0.999 0.107 0.107Square, uniform, cold wall 4544 0.994 0.307

0.285 1.000 0.113 0.1130.285E 1.0007652Square, uniformKhy, Khl, Kbd, and Kcw

0.281 1.000 0.111 0.111F Khy, Khl, Kbd, Kcw, and Ksp Square, uniform 7652 1.000 0.2810.276 1.001 0.105 0.1050.276G 1.0007652Square, uniformKhy, Khl, Kbd, Kcw, Ksp, and Krp

H 0.459Khy, Khl, Kbd, Kcw, Ksp, and Krp 1.191 0.386 0.336Square, non-uniform 2200 1.199 0.5000.430 1.000 0.319 0.3190.4302200I Khy, Khl, Kbd, Kcw, Ksp, Krp, and Knu Square, non-uniform 1.000

J 0.317Khy, Khl, Kbd, Kcw, Ksp, Krp, and Knu 0.999 0.177 0.177Square, uniform and non-uniform 9852 1.000 0.317


where Dm is the equivalent heated diameter of thesame subchannel without cold wall. When there isno cold wall, Dhe/Dm=1.0. In the EPRI-1(Fighetti and Reddy, 1983b) correlation, the coldwall correction factor is a function of mass flux.The following correction factor is proposed toconsider the cold wall effect:

Kcw=q¦CHF, CW

q¦CHF

=�Dhy

Dhe

�0.115

(18)

where Dhy is the hydraulic diameter, Dhe is theequivalent heated diameter, and q¦CHF, CW is theCHF when the cold wall effect is considered.Through trial and error, it is determined that C2 is1.0 in DSM and is 1.03 in HBM. When the coldwall effect factor is incorporated (Case D in Table3), the RMS is 30.7% in DSM and is 10.7% inHBM for data points of square array with coldwall (4544 data points). As shown in Case E ofTable 3, the RMS is 11.3% in HBM and is 28.5%in DSM for 7652 data points of square array withuniform axial heat flux distribution after the in-corporation of correction factors Khy, Kbf, andKcw.

6.5. Grid spacer factor

The presence of a grid spacer perturbs thevapor blanket formed during the process of de-parture from nucleate boiling and increases theCHF. However, the grid spacer disrupts the liquidfilm and decreases the CHF when the quality ishigh. The grid spacer correction factor proposedby Tong (1969) is a function of thermal diffusioncoefficient, pressure, quality, and distance be-tween the grid spacer. Based on the HTRF bundledata, the grid spacer correction factor in EPRI-1(Fighetti and Reddy, 1983b) correlation is a linearfunction of loss form coefficient of the gridspacer.

Theoretically, it can be expected that the impactof the grid spacer on the CHF decreases as thelocation of CHF moves away from the gridspacer. An assessment with 7652 HTRF datapoints of square array and uniform axial heat fluxdistribution shows that the grid spacer correctionfactor proposed (K3 in Table 2) by Groeneveld etal. (1986) has a tendency to over-predict under

high Lsp/Dhy and under-predict under low Lsp/Dhy.Statistically, the bundle CHF data in the HTRFdata bank do not show any dependency of spacereffect on Lsp/Dhy. In the present study, a new gridspacer correction factor is proposed.

Ksp=0.86(1+0.155K1.5) (19)

The difference between the results of Case Eand Case F in Table 3 shows the impact of thegrid spacer on the predictability of bundle CHFdata of square array with uniform axial heat fluxdistribution. The RMS is 28.1% in DSM and is11.1% in HBM for 7652 data points of squarearray with uniform axial heat flux distribution.

6.6. Radial power distribution factor

The radial power distribution can affect thevoid distribution within the bundle and, therefore,it also changes the value of CHF. In WSC-2(Bowring, 1979) correlation, a factor is includedto consider the effect of radial power distributionon the CHF.

An assessment using 7652 HTRF data points ofsquare array and uniform heat flux distributionshowed that the 1995 Groeneveld CHF table hasa tendency to over-predict the CHF whenPradial,max is large. Therefore, the following correc-tion factor is proposed to consider the impact ofradial power distribution on CHF:

Krp=2.51 exp(−0.82Pradial,max) (20)

A comparison of the results of Case F and CaseG in Table 3 shows the impact of the radialpeaking correction factor on the predictability ofbundle CHF data of square array with uniformaxial heat flux distribution. The RMS is 27.6% inDSM and is 10.5% in HBM for 7652 data pointsof square array with uniform heat fluxdistribution.

6.7. Non-uniform heat flux distribution correctionfactor

The impact of non-uniform axial heat flux dis-tribution on the prediction of CHF has been avery important issue in the research of CHF. TheF factor was proposed by Tong (1969) to considerthe impact of axial flux distribution on CHF.


F=q¦CHF, uniform

q¦CHF,non-uniform

=C

q¦(zCHF)[1−exp(−ClCHF)]

×& ZCHF

Z ONB

q¦(z) exp[−C(zCHF−z)] dz (21)

and

C=0.44[1−X(zCHF)]7.9

(G/106)1.72

where G is mass flux (lb h−1·ft−2); lCHF=zCHF−zONB (in−1); q¦CHF, uniform, CHF with uni-form axial heat flux distribution; q¦CHF, non-uniform,CHF with non-uniform axial heat flux distribu-tion; zCHF, axial location of CHF; zONB, axiallocation of onset of nucleate boiling.

An alternative F factor has been proposed byCollier and Thom (1995)

F %=C

q¦(zCHF)[1−exp(−ClCHF)]& ZCHF

0

q¦(z) exp

[−C(zCHF−z)] dz (22)

The difference between Eq. (21) and Eq. (22) isthe lower limit of integration and the variable inthe function of heat flux distribution.

The correction factor Cnu for non-uniform heatflux proposed by Bowring (1979) has the follow-ing form

Cnu=1+ (Y−1)/(1+G) (23)

Y=� 1

zCHF

& ZCHF

0

q¦(z) dzn,

q¦(zCHF) (24)

In the CHF approach proposed by Groeneveldet al. (1986), the correction factor K5 had thefollowing form:

K5=q¦CHF,non-uniform

q¦CHF,uniform

=� 1

zCHF−zX=0

& ZCHF

ZX=0

q¦(z) dzn,

q¦(zCHF)

(25)

zX=0 was used as the lower limit of integrationto simplify the calculation. The correction factorused in RELAP5/MOD3 (Carlson et al., 1990) isthe reciprocal of Eq. (25).

Eqs. (21)–(25) are used to predict the 2200HTRF data points with non-uniform heat fluxdistribution. From these calculations, it has beenconcluded that none of the proposed correctionfactors can effectively improve the predictabilityof the 1995 Groeneveld CHF table for the HTRFsquare array bundle CHF data with non-uniformaxial heat flux. A new form of correction factor isproposed to consider the effect of non-uniformaxial heat flux distribution on CHF.

Knu=q¦CHF, non-uniform

q¦CHF, uniform

=C3[1+ (Y %−1)e2.66X]

(26)

where

Y %=1

zCHF−zX=0

& ZCHF

ZX=0

q¦(z) dz/q¦(zCHF)

Through trial and error, it is determined thatC3 is 0.728 in DSM and is 0.592 in HBM. Theresults of Case H and Case I in Table 3 show theimpact of factor Knu on the predictability ofGroeneveld et al. CHF approach for the squarearray data points with non-uniform axial heat fluxdistribution.

Case J in Table 3 summarises the statistics of9852 square array HTRF data points when thecorrection factors Khy, Khl, Kbf, Kcw, Ksp, and Knu

are incorporated to correct the CHF value ob-tained from 1995 Groeneveld CHF table.

6.8. Bundle factor for triangular array

The above derivations are based on the data ofsquare array in HTRF data bank. It is found thatthe CHFs are over-predicted when these correc-tion factors are applied to 559 data points withtriangular array. To improve the performance ofthe 1995 Groeneveld CHF table in predicting theCHF of triangular array, the constant C1 in factorA of Eq. (16) needs to change to 0.486 in DSMand to 0.419 in HBM. It is important to point outhere that there are only 559 triangular array datapoints in the HTRF data bank. The number isnot statistically large enough to give reliableresults.


Table 4Correction factors of the 1995 Groeneveld CHF table

FormFactor

Khy, subchannel or tube For 0.002BDhyB0.025 m:cross-section factor

Khy=� Dhy

0.008

�−1/2

Khl, heated length factor L/Dhy]5: Khl

=exp�Dhy

Le2a

�a evaluated from homogeneous model

a=X/[X+rg/rl(1−X)]

Kbf

, bundle factor Kbf=A · B

A=min[c1, c1 exp(−1.12X)]

B=exp�−0.073

+0.035� G

1000

�nfor square array: for triangular array:

c1=1.02(DSM) c1

=0.486(DSM)

c1=0.98(HBM) c1

=0.419(HBM)

Kcw

, cold wall factorKcw=C2

�Dhy

Dhe

�0.115

C2=1.00(DSM)

C2=1.03(HBM)

Krp=2.51 exp(−0.82Pradial,Krp

, radial flux distributionfactor max)

Ksp

, grid spacer factor K3=0.86(1+0.155K1.5)XB0: Knu=1.0Knu, axial flux distribution

factorX\0: Knu

=C3 [1+(Y %−1) e2.66X]

Y %=� 1

(z−zX=0)·& z

zX=0

q¦(z %) dz %n,

q¦(z)

C3=0.728(DSM)

C3=0.592(HBM)

6.9. Summary

Based on the statistics of the ratio of the pre-dicted and the measured HTRF bundle CHF datausing the 1995 Groeneveld CHF table, a set ofcorrection factors is proposed to extend the appli-cation of the CHF table in the prediction of CHFin rod bundles. The proposed correction factorsare given in Table 4. In summary, the correctionfactors included are: the hydraulic diameter factor(Khy), the heated length factor (Khl), the bundlefactor (Kbf), the grid spacer factor (Ksp), the coldwall effect factor (Kcw), the radial heat flux distri-bution factor (Krp), and the axial heat flux distri-bution factor (Knu). The constant of thecorrection factors might be different when theCHF table is used in DSM and HBM. In order toobtain better results, the constant in bundle factor(Kbf) is different for square and triangular arrays.However, the number of triangular array data inthe HTRF CHF data bank is not large enough togive a statistically meaningful value for the con-stant in the triangular bundle factor.

The statistics for the prediction of the HTRFbundle CHF data using the 1995 GroeneveldCHF table and the proposed correction factorsare summarized in Tables 5–8 for HBM andDSM. In these tables, the detailed numerical com-parisons for test sections of different characteris-tics are given. The results using the 1986 CHFapproach of Groeneveld et al. (AECL-UO CHFtable) are also summarized in these tables forcomparison.

As summarized in Tables 5 and 6, the averageerrors in the prediction of the HTRF square arraybundle data using 1995 Groeneveld CHF tableare 0.1 and 0.0% for HBM and DSM, respec-tively, when the corresponding set of correctionfactors is applied. Compared with the results us-ing the CHF approach of Groeneveld et al.(1986), the RMS error in the prediction of theHTRF bundle data using the proposed approachdecreases significantly. For square array, theRMS error changes from 22.9 to 17.7% in HBM,and changes from 41.3 to 31.7% in DSM. Theimprovement in the prediction of triangular arraybundle data using the new approach is also verysignificant. However, there are only 559 triangular


Table 5Statistics using HBM to predict HTRF bundle CHF data

No. of data points AECL-UO CHF tableArray type 1995 Groeneveld CHFAxial heat-flux distributiontable

AVG RMS STD AVG RMS STD

Square Uniform 7652 0.994 0.132 0.132 0.999 0.105 0.1052200 1.224 0.419 0.354Non-uniform 1.000 0.319 0.3199852 1.045 0.229 0.225Total 0.999 0.177 0.177

UniformTriangular 490 1.345 0.561 0.443 1.000 0.347 0.34869 1.721Non-uniform 0.923 0.581 1.007 0.303 0.305

559 1.391 0.617 0.478 1.001 0.342Total 0.342

Table 6Statistics using DSM to predict HTRF bundle CHF data

Array type Axial heat-flux distribution No. of data points AECL-UO CHF table 1995 Groeneveld CHFtable

AVG RMS STD AVG RMS STD

7652 0.970 0.386 0.385 1.000Square 0.276Uniform 0.2762200 1.197 0.496 0.456Non-uniform 1.000 0.430 0.430

Total 9852 1.021 0.413 0.413 1.000 0.317 0.317Triangular Uniform 490 1.745 1.212 0.957 1.000 0.560 0.560

69 2.036 1.287 0.769Non-uniform 0.968 0.303 0.304559 1.781 1.222 0.940 0.996 0.535 0.535Total

Fig. 1. Histogram of relative error of Groeneveld et al. CHFapproach in predicting HTRF bundle data using the heatbalance method.

Fig. 2. Histogram of relative error of Groeneveld et al. CHFapproach in predicting HTRF bundle data using the directsubstitute method.


Tab

le7

Pre

dict

ion

accu

racy

usin

gH

BM

topr

edic

tH

TR

Fbu

ndle

data

1995

Gro

enev

eld

CH

Fta

ble

Arr

ayty

peN

o.of

data

poin

tsA

xial

heat

-flux

dist

ribu

tion

AE

CL

-UO

CH

Fta

ble

�Err

�B0.

5�E

rr�B

0.75

�Err

�B1.

0�E

rr�B

0.1

�Err

�B0.

2�E

rr�B

0.5

�Err

�B0.

75�E

rr�B

1.0

�Err

�B0.

1�E

rr�B

0.2

0.99

91.

000.

713

0.95

00.

999

0.89

30.

999

0.99

80.

999

0.58

576

52U

nifo

rmSq

uare

0.91

30.

959

0.29

80.

546

0.92

10.

971

Non

-uni

form

0.98

622

000.

298

0.53

00.

811

0.98

00.

991

0.62

10.

859

0.98

20.

993

0.95

60.

996

0.52

10.

812

Tot

al98

520.

645

Uni

form

0.76

70.

900

0.10

20.

300

0.88

00.

986

1.00

490

0.23

90.

447

Tri

angu

lar

0.29

0N

on-u

nifo

rm0.

580

0.78

30.

348

0.65

20.

971

0.97

10.

971

690.

000

0.10

10.

744

0.88

60.

132

0.34

30.

891

0.98

40.

601

0.99

60.

404

0.20

955

9T

otal

Tab

le8

Pre

dict

ion

accu

racy

usin

gD

SMto

pred

ict

HT

RF

bund

leda

ta

Axi

alhe

at-fl

uxdi

stri

buti

onN

o.of

data

poin

tsA

rray

type

AE

CL

-UO

CH

Fta

ble

1995

Gro

enev

eld

CH

Fta

ble

�Err

�B0.

1�E

rr�B

0.2

�Err

�B0.

5�E

rr�B

0.75

�Err

�B1.

0�E

rr�B

0.1

�Err

�B0.

2�E

rr�B

0.5

�Err

�B0.

75�E

rr�B

1.0

Uni

form

7652

0.28

00.

537

0.94

0Sq

uare

0.98

50.

993

0.39

10.

696

0.97

60.

994

0.99

8N

on-u

nifo

rm22

000.

243

0.47

00.

798

0.89

00.

934

0.22

70.

447

0.86

80.

937

0.96

6T

otal

9852

0.27

20.

522

0.90

90.

964

0.98

00.

354

0.64

10.

952

0.98

20.

991

Tri

angu

lar

Uni

form

490

0.12

40.

247

0.48

20.

584

0.65

50.

073

0.22

00.

631

0.88

00.

933

Non

-uni

form

690.

058

0.07

20.

130

0.33

30.

580

0.36

20.

623

0.92

80.

971

0.98

6T

otal

559

0.11

60.

225

0.43

80.

553

0.64

60.

129

0.27

00.

667

0.89

10.

939


Fig. 3. (a) Performance of the Groeneveld et al. CHF approach in different mass flux intervals using the direct substitute method.(b) Performance of the Groeneveld et al. CHF approach in different pressure intervals using the direct substitute method. (c)Performance of the Groeneveld et al. CHF approach in different quality intervals using the direct substitute method.


Fig. 4. (a) Performance of the Groeneveld et al. CHF approach in different mass flux intervals using the heat balance method. (b)Performance of the Groeneveld et al. CHF approach in different pressure intervals using the heat balance method. (c) Performanceof the Groeneveld et al. CHF approach in different quality intervals using the heat balance method.

array data points, and the values of constants inthe correction factors need further justification.

As summarized in Tables 7 and 8, for 9852square array bundle data points, there are 62.1%of data points having an error less than 10% and0.4% of data points having an error greater than100% when the proposed approach is used in

HBM. There are 35.4% of data points havingerror less than 10% and 0.9% of data pointshaving an error greater than 100% when the pro-posed approach is used in DSM. Figs. 1 and 2display the histogram of the relative error of thepredicted and the measured bundle CHF ofsquare array using HBM and DSM, respectively.


The results using the CHF approach ofGroeneveld et al. (1986) are also displayed forcomparison.

Figs. 3 and 4 display the spread of the relativeerror of the predicted and measured HTRFsquare array CHF using the CHF approach ofGroeneveld et al. (1986) and the proposed ap-proach in different mass flux, pressure and qualityintervals. In these figures, the standard deviationis shown as a line extending above and below theaverage to show the spreading of the relativeerror. As shown in Fig. 3a and Fig. 4a, theperformance of the proposed CHF approach isreasonable good in different mass flux intervals.As shown in Fig. 3b and Fig. 4b, the performanceof the proposed CHF approach is also reasonablegood in different pressure intervals. However, theproposed approach has a tendency to over-predictthe data points when pressure is less than 3 MPa.As shown in Fig. 3c, the proposed approach has atendency to over-predict the data when the qual-ity is less than 0.0 and when the quality is greaterthan 0.8 in HBM. From Fig. 4c, it can be seenthat the proposed approach has a tendency toover-predict the data when quality is less than 0.0and under-predict the data when quality is greaterthan 0.8.

7. Conclusions

The bundle HTRF CHF data have been usedto develop a set of correction factors to extend theapplication of the 1995 Groeneveld CHF table toflow in square rod bundles. There are 11 077 datapoints in the HTRF CHF data bank. Due tovarious reasons discussed in Section 3, only 10 411data points are included in the present study.Among these data points, there are 9852 squarearray data points and 559 triangular array datapoints. The correction factors developed in thepresent study are: the hydraulic diameter factor(Khy), the heated length factor (Khl), the bundlefactor (Kbf), the grid spacer factor (Ksp), the coldwall effect factor (Kcw), the radial heat flux distri-bution factor (Krp), and the axial heat flux distri-bution factor (Knu). These factors are incor-porated sequentially into the CHF calculations

using the 1995 Groeneveld CHF table to minimisethe relative and RMS error of the predicted andthe measured CHF at different pressures, massflux and quality intervals. Two separate sets ofcorrection factors are developed for HBM andDSM. The derivations of the correction factorsare based on the data of square array in HTRFdata bank. It is found that the CHFs are over-predicted when these correction factors are ap-plied to 559 data points with triangular array. Theaverage error in the prediction of the HTRFsquare array bundle data using the proposed ap-proach is 0.1 and 0.0% for HBM and DSM,respectively, when the corresponding set of cor-rection factors is applied. Compared with theresults using the CHF approach of Groeneveld etal. (1986), the RMS error in the prediction of theHTRF bundle data using the proposed approachdecreases significantly. For square array, theRMS error changes from 22.9 to 17.7% in HBMand from 41.3 to 31.7% in DSM.

In the proposed approach, the cross-sectionalaverage thermodynamic quality is used to deter-mine the CHF value through table look-up.Therefore, the proposed approach cannot be usedto calculate CHFs in the subchannel analysis ofnuclear reactors.

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