Enthalpy transfer between PWR fuel assemblies in analysis by the lumped subchannel model

Nuclear Engineering and Design 53 (1979) 165-186 © North-Holland Publishing Company

ENTHALPY TRANSFER BETWEEN PWR FUEL ASSEMBLIES IN ANALYSIS BY THE LUMPED SUBCHANNEL MODEL

Chong CHIU *, Pablo MORENO, Neil TODREAS and Robert BOWRING Department of Nuclear Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA

Received 12 December 1978

Assessment is made of the error in enthalpy predictions in PWR cores resulting from various assumptions in the size of homogenized calculation regions. A coefficient is developed from the differential conservation equations and numerically determined to minimize these errors. It was found that these errors are only of significance for regions comprising three to seven subchannels, region sizes which may be employed in coupled neutronic-thermal hydraulic calculations. A correlation for coefficient values for a range of PWR conditions is presented.

1. Introduction

A PWR core might contain up to tens of thousands of fuel rods with even more coolant subchannels between them. Thermal hydraulic analysis of the core considering each subchannel individually would clearly be impracticable because of the excessive computer running time and array sizes involved. Instead, the "lumped" parameter approach is adopted in which subchannels are considered in groups.

The core is divided into a number of regions which for convenience might coincide with one or more fuel assemblies. The thermal-hydraulic conditions within a region are considered to be uniform radially. They are computed taking into account the mass, energy and momentum transfer across the region boundaries using a subchannel code such as THINC [1], COBRA [2], or HAMBO [3], regarding each region as if it were a subchannel. Differing strategies for def'ming suitable regions for a core analysis exist. Alternative approaches are (a) the two-pass chain method [4] and (b) the single-pass concentric nodes method [5]. In the first, a preliminary whole-core analysis gives the boundary conditions for the region of interest which is analyzed in detail superimposing those boundary conditions. In the second, the boundary conditions for the region of

* Present address: Combustion Engineering, Inc., 1000 Pros- pect Hill Road, Windsor, CT 06095, USA.

interest are self-generated by considering the rest of the core as a number of concentric lumped regions.

A main consideration in both lumped parameter approaches is to represent in a realistic way the mass, energy and momentum transferred across a region boundary. For example, in fig. 1 regions 1 and 2 are separated by a row of rods through which energy is interchanged. The energy gained by or lost from a region is driven by the enthalpy difference immedia- tely across the dividing row of rods. For radially uniform conditions within a region, the difference would be (Hi - / / 2 ) where H 1 and//2 are the average enthalpies of the region. On the other hand, one could argue that a linear enthalpy gradient through the clusters might be more realistic, in which case the enthalpy difference would be (Hi - H2)/N where N = number of rows of rods in each region. Thus, one may define an enthalpy transport coefficient NH as the effective number of rods through which the enthalpy gradient occurs such that the driving enthalpy difference is (H1 - H2)/N~t; in the first case, NH = 1 and in the second NH = N. If the two regions were divided into the constituent subchannels, the value Of NH could be calculated more precisely from the gradient through the regions; in general, the "true" value would be between 1 and N.

Since the interassembly diversion cross-flows in a PWR are normally fairly small, errors in the calculation of pressure drop and radial variation in flow

165

166 C. Chiu et al. / Enthalpy transfer between PWR fuel assemblies

arising from the use of incorrect transport coefficients are unlikely to be large. Errors in the radial variation in enthalpy could however be significant, therefore it is this aspect of the problem which is considered here. In this initial formulation, all calculations were done without subcooled boiling. It is first shown that the other transport coefficients are insignificant (i.e. they may be set to unity) compared to the enthalpy coefficient. The values of NH were computed for a number of cases from which an empirical expression for NH was derived. It is shown that under normal PWR oper- ating conditions and considering the limits of fully homogenized bundles or single subchannels, corrections from application of the NH concept are not significant. However, for homogenized regions of three

to seven subchannels, which may be necessary to consider in coupled neutronic-thermal/hydraulic analyses, corrections in region enthalpy of 10-15% result from application of NH factors.

2. Formulation of the equations for transport coefficients

The introduction suggested the form of the enthalpy transport coefficient. In this section we examine the forms of all the coefficients with respect to the conservation equations used in lumped parameter analysis methods. We desire to confirm the form of

O0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 O0 0 0 O0 0 0 O0 ,~, 0 0 ~ O0 0 0 O0 0 0 O0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Linear Radial Model (N H = N) " ' ~ "True" Divided Subchannel Model (I < NH< N)

H I ~ ~--~I

"''~. ~ ]~_Uniform Radial Model (NH = i)

Radial Dis ta--nce ~.

Fig. 1. Radial enthalpy distribution through two "lumped" regions according to linearly varying, subchannel distribution and uniform models.

C Chiu et al./Enthalpy transfer between PWR fuel assemblies 167

the enthalpy transport coefficient presented in the introduction and to determine what additional coefficient formulations would be required for a perfect representation.

The development is two dimensional (axial and one lateral direction). The general methodology for the analogous 3D problem was also developed but because of its complexity, it has been applied only to a specific case [6]. Section 6 indicates the applicability of the 2D results developed here to the general 3D problem.

The two-dimensional solution is developed with reference to the geometry of fig. 2 which shows an illustrative N X N rod assembly (N odd) surrounded by neighboring assemblies. Distinction should be made between rows of rods comprising the assembly, N, and rows of subchannels from the assembly center to the boundary, N', since the lumped parameter methods relate subchannel interactions. The two-dimensional solution is developed by selecting adjacent rows of N' subchannels from the assembly center to the boundary where for N odd, N' = ½N + ½. Note that the subchannel which includes the edge region is half the size of the interior subchannels. For N even, N ' = ~N where

now the subchannel which includes the edge region is equal to the interior subchannel. The selected adjacent rows shown in fig. 3(a) are assumed to be bounded by adiabatic, impervious boundaries. The homogenized representation of fig. 3(a) is shown in fig. 3(b) which has flow area, wetted perimeter, heat flux and mass flow equivalent to that of the row o f N ' subchannels.

We require that at every axial position of the homogenized region, the total flow, the energy content and the pressure drop from inlet should be the same as those obtainable by averaging the values of the individ- ual subchannels. These conditions are met if the mass, heat and momentum transfer across the boundary between the adjacent strips at every axial position are the same in both cases. In calculating the boundary transport, we use subchannel values of the pressure and enthalpy differences in the multisubchannel case but the difference in the averaged values in the homogenized representation.

For the homogenized representation calculation we require a method of calculating the lateral transport in terms of what would have been the local parameter difference while knowing only the average differences.

ASSEMBLY 3

0 0 0 0 0 0 0 0 0

Assembly 1

O 0 O 0 O 0

0 0 0 0 0 0

ASS~4BLY 4

0 0 0 0 0 0 0 0 0

Assembly 2

O 0 O 0 O 0

Fig. 2. N X N rod assemblies in square array, N = 5.


Channel Subscripts Let£ers - Adiabatic Boundary ' l l l l l ; l l l l

a

,/ .,IA, H,,,,//,b,,////,,%t//tJJ,/,v,'tJ'''~"'!'L 1

I--N' Ch .... is --1~ N' Ch .... is

b

b" C.A~BL L LI------ CHA~L R "t

Fig. 3. Subchannel and homogenized region representation of rows of N' channels.

2.1. Problem solution by analysis of the differential form of the conservation equations

The form of the coefficients which transform average (homogenized representation) parameter differences to local (multichannel representation) differences are derived in the Appendix from the differential expression of the conservation equations. These forms are as follows.

Coefficient for enthalpy in the energy equation:

HL - HR. N . = he _ h ~ ' (1)

coefficient for axial velocity in the axial momentum equation:

U L - UR. (2) Nu= R c _ U D '

coefficient for transverse momentum convection in

the transverse momentum equation:

NTU = a(UWL'R)/ax', (3) E

i~=A (O~ Wi, i+ 1 )

coefficient for pressure in the transverse momentum equation:

= PL - PR. (4) NTp PA - PF '

coefficient for friction and form losses in the transverse momentum equation:

C (I4/L,R/P[,R) WL, R N T F = =

E E ~ C i ~ Wi, i+lWi, i+l i=A i=A P~,i+l

(5)

where p* is the density of fluid at the donor channel for the diversion crossflow. (Refer to fig. 3 for subscripts A, B, C, D, E and F; L and R.)

2.2. Problem solution by analysis of the difference (i.e. COBRA IllC) form of the conservation equations

In ref. [6] it was demonstrated that the coefficients of eqs. ( 1 ) - (5 ) derived from the differential equations are not equally reducible from the difference equations forms always employed in code computations (i.e. COBRA IIIC). However, it was demonstrated that the simple, albeit approximate, forms of coupling coefficients summarized in the previous section, will yield satisfactory results for most practical reactor conditions. The most limiting assumptions in the difference equations concern the diversion cross-flow terms. Therefore, results utilizing these approximate coefficients for analysis of conditions of severe flow and/or power upset conditions can be in significant error.

3. Cases investigated

(a) The prime case of interest to PWRs is that of adjacent heat bundles with different linear power ratings and different inlet mass flow rates. Within each bundle the linear power and mass flow rate are taken as constant. We identify this case as the power/

C. Chiu et al. / Enthalpy transfer between PWR fuel assemblies 169

flow upset conditions. This case has been studied for the condition leading to the maximum cross flow which occurs when the higher power and lower inlet flow exist together in the same bundle. Other variables in this case are the power and flow ratios between assemblies. These ratios were both taken as 1.2, the maximum anticipated in practice.

(b) A degenerate version of the previous case is that of linear power upset but the same inlet mass flow to each assembly. This case was studied since cross-flow is reduced versus the previous case and is only due to radial enthalpy gradients between assemblies. This case is called the power upset condition.

In addition, the following two cases were examined to develop a fuller understanding of the form of the transport coefficients. They are of interest in themselves since experiments based on these cases may be performed as they are considerably less complex and costly than heated, multipin assembly tests. These cases are as follows.

(c) Unheated rods with all subchannels in each of the two adjacent assemblies at a uniform inlet enthalpy. Each assembly, however, has a different inlet en- enthalpy. Both assemblies have the same inlet mass flux to each subchannel. This case is called the enthalpy upset condition.

(d) Same as case (c) except the upset condition is in inlet mass flux plus the inlet enthalpy. This case is called the enthalpy/fiow upset condition.

For each case of interest, a range of bundle sizes, N' =f(N) and subchannel mixing rates, fl, were investigated spanning the range of interest to PWR application. The input parameters K and S/L for the cross- flow resistance and the control volume of the cross- flow are kept constant as 0.5.

4. Method of analysis and results

4.1. Method of analysis

The transport coefficients were derived by analysis of the subchannel representation of the regions shown in fig. 3(a), using a version of the code COBRA IIIC [2], further developed at MIT [7]. The calculation of the enthalpy transport coefficient is described below; the method of deriving the other transport coefficients was similar.

The enthalpy transport coefficient is defined as

HL(Z)--HR(Z) NH - hc(z )_ hD(Z ) . (1)

The values of hc(z ) and hD(z) are calculated in the standard version of the code and printed. Extra computations are required to obtain HL(z) and HR(z) from the radial-average of (m" h) from the subchannels making up the left- and right-hand strips. Thus Nt-t may be computed at each axial position and printed.

The effectiveness of the use of the transport coefficients was then checked by running the same case, but using the homogenized representation [fig. 3(b)] with the values of NH(z) derived from the subchannel run.

The predicted values of Nt~ are reported in sections 4.2-4.4. The first study was undertaken on a typical power upset case to determine the need to utilize all five coefficients. It was concluded (see section 4.2) that utilization of only the enthalpy coefficient, Nn(z), was sufficient; therefore the subsequent work was focused on unheated and heated bundles and assessed the effectiveness of the enthalpy coefficient applied alone.

4.2. Effectiveness of transport coefficients OVH, NU, NTp, NTF and NTU) in the power upset case

The total enthalpy transport across the boundary between homogenized regions can be broken into three components. These components listed below are consistent with the COBRA representation of the overall enthalpy transport process as due to turbulent interchange and diversion cross-flow; the heat content of the latter depends upon both its enthalpy and mass flow:

(a) turbulent interchange, (b) enthalpy of the diversion cross-flow, and (c) mass flow rate of the diversion cross flow. In this section the three components of the homo-

genized representations are examined and compared with those of the multisubchannel representation in order to evaluate the effectiveness of using several coupling coefficients simultaneously.

The first component, the enthalpy change in the hot zone due to turbulent interchange only, gives us an ideal of the effectiveness of each transport coefficient for the homogenized representation on the energy transport due to turbulent mixing. The second

170 C Chiu et aL / Enthalpy transfer between PWR fuel assemblies

component is the enthalpy increment in the hot zone due to the diversion cross-flow transport. This will illustrate the effectiveness of each transport coefficient on the energy transport by the diversion cross- flow. The third component is the total diversion cross- flow across the boundary in the homogenized representation. In the case we examine here, the diversion cross-flows at the boundary always move toward the cold zone for any channel axial position, and therefore it is considered convenient to utilize the total diversion cross-flow through the boundary as a parameter to illustrate the effectiveness of the coupling coefficients on the transverse momentum transport.

The results are shown in table 1. Part (a) tabulates the components which are the enthalpy change with respect to the inlet enthalpy of the hot zone due to the turbulent interchange only. As can be observed in this table, Ntt(z) in the homogenized computation is the most effective of the transport coefficients in matching the homogenized results to that of the

multisubchannel computation (-3.012 on line 4 of table 1 compared to -3 .096 on line 1). All the other transport coefficients show negligible effects on this parameter (unchanged at -3.012 on lines 2 and 3). Part (b) tabulates the second component for the multi- subcharmel representation and homogenized representation, i.e. AhD, c and z2kHD,c, respectively. It is worth noting that the results of the homogenized representation using all the transport coefficients have the best agreement with that of the multisubchannel representation (0.251 on line 2 compared to 0.244 on line 1). However, utilization Of NH(Z) yields tolerable accuracy and particularly significant improvement compared to taking N n = 1.0. Part (c) tabulates the integral value of diversion cross-flow across the central boundary for both the multisubchannel representation and the homogenized representation. The conclusion we can draw from part (c) is similar to that from part (b). From these results we conclude that utilization of the enthalpy coefficient, Ntt, only should be an efficient, effec-

Table 1 Comparison of the (A) hot zone enthalpy increments due to diversion cross-flow only, (B) hot zone enthalpy increments due to turbulent interchange only, (C) total diversion cross-flow across the boundary, respectively, between the multisubchannel representation and the homogenized representation with different combination of coupling coefficients for power upset case

Line Multisubchannel representation (A) (B) (C) A,~ T.I. A,~D. C . fWc,DdZ (Btu/lbm) (Btu/lbm) (Ibm/h)

i -3 .096 0.244 0.1758

Line Homogenized representation (A) (B) (C)

A/'/T.I. Error ~d-/D.C. Error fWL,RdZ Error (Btu/lbm) (%) (Btu/lbm) (%) (Ibm/h) (%)

2 NH, N U, NTU, NTF, NTp -3 .012 -2 .7 0.251 3.0 0.1776 2.2 3 HH(Z),Nu(z) -3 .012 -2 .7 0.253 3.8 0.1798 2.3 4 NH(Z) -3 .012 -2 .7 0.261 7.1 0.1825 3.8 5 NH = 1.0 -7 .07 128.4 0.001 -99.6 0.1120 -36.3

'SJ/D.C.- ~ D . C . ~ r / T . I . - Z:U~T.I. fWL,R dz - fWC,D dz Error -= or or

AhD.c. AhT.I. fWc,Dd2

Bundle geometry N = 11 Energy conditions / t in -- 600 Btu/lbm L = 144 in. q " = 0.2 MBtu/h ft 2

H R = 1.0 Flow conditions G = 2.66 Mibm/h ft 2 PR = 1.5

F R = 1.0 Input coefficients t3 = 0.02

C. Chiu et al. / Enthalpy transfer between PWR fuel assemblies 171

tive approach. In the next section this approach is pur- sued and the resultant accuracy determined for first unheated, then heated bundles.

4.3. Calculation of coupling coefficients for unheated bundles

4.3.1. Enthalpy upset case The numerical values of Nit are evaluated from the

results of multichannel computation utilizing a step- shaped inlet enthalpy upset. It is suggested that two half-sized subchannels C and D always be utilized to obtain the subchannel parameters h c and h D. The reason is that the required difference, h c - h D, in the definition of Nit is generally poorly approximated by the enthalpies of the subchannel B and E [refer to fig. 3(a) for subscripts B, C, D and E]. For instance, if we compute NH by the following relation:

HL - HR (6) NH(Z)- 1,,

~ n B - h E)

the error of the enthalpy rise involved in the homogenized region case will be 45% higher than that using Nit evaluated by the parameters of the half-sized subchannels C and D.

It should be noted that the numerical solution of COBRA llIC imposes a limitation on/3 due to enthalpy fluctuation when the energy transport is assumed to occur only by the turbulent interchange, which is more restrictive in the half-size channel computation than in the full-sized computation. The maximum allowable value for this case is 0.048 [6]. Therefore /3 = 0.04 is picked as the upper bound of/3 in our approach.

Calculated values of NH(z) versus axial elevation for a typical case is shown in fig. 4. The dips and humps of NH at low elevation positions for/3 = 0.04 are due to enthalpy fluctuation by turbulent interchange, as mentioned previously. Note that the fluctuation amplitude o fN n will not be damped out if values of/3 larger than 0.048 are used. The validity of these Nit values was confirmed by comparing the hot zone enthalpy of the homogenized case with that of the multisubchannel case at each axial node.

Comparison between various forms o f N H. The results of hot zone enthalpy in the homogenized case using the following approximations for Nit are next com-

pared:

NH(z),

Nit, where average is taken over channel length L,

Nit =1.0, allz,

Nit = N, all z,

where

AHT. i. (7) Nt-I = ~ AHi, T.I,/NHi

i

i = subscript of elevation node and AHT.L = enthalpy rise of the "homogenized representation" due to turbulent interchange.

The results are illustrated in fig. 5. The enthalpy changes of the homogenized case by using Nn(z) and N . coincide with that of the multisubchannel case. On the other hand, NH = 1 and NH = N have eerors of enthalpy rise equal to -203 and 57.4%, respectively. compared with the results of the multisubchannel representation. This example demonstrates the need for utilizing a value of Nit other than 1 for application in enthalpy and enthalpy and flow upset cases.

Two methods are investigated to illustrate the behavior of N . versus/3, N and the axial position of the subchannel. One is curve-fitting of our predicted Nit(z) results, the other is the averaging of tile Nit(z) over the axial enthalpy increment of the hot zone to give an N . . Each provides a convenient way to incorporate the NH concept into design practice.

Curve fits. The Nn (z) predictions can be represented using a continuous function f(z) versus channel position z:

az f ( z ) = t . 0 + - - . ( 8 )

b+z

The values of a and b can be evaluated by fitting two values of Nit with smallest deviation from the true value at energy elevation node. The values of a and b for the nine cases we have analyzed are tabulated in table 2.

Averaged value of Nit(z), i.e. Art4. From the definition of Nit [refer to eq. (7)], is it expected that the enthalpy of the hot zone at any axial positionz in the homogenized representation predicted using Nt-t will

1 7 2 C Chiu et al. / Enthalpy transfer between PWR fuel assemblies

5.C

4.

Z

3.0

1.0~ 0

..... i ' ' ! I I I I I ! I I

Bundle Geometry N = ii

L = 144" Flow Conditions G = 2.66 Mlbm/hr-ft 2

FR= 1.0

Energy ConditiDnsH -- H. = 600 Btu/ibm In

-q"= 0 HR= 1.22

! ;

PR = 1.0

Input Coeff~nts ~ = Variable

Y / ~-/ / _~f

I . t 1 |

50 i00 150 Axial Channel Location, Inches

Fig. 4. Nil(Z) v e r s u s c h a n n e l l e n g t h f o r e n t h a l p y u p s e t c a s e , N = 1 1 .

m

m

m

m

m

m

u

m

m

m

m

coincide with that in the multisubchannel case. The -Nt/for/~ = 0, 0.005, 0.02 and 0 .04 ;N = 2, 5, 11 and 23, and L = 144 in. are tabulated in table 3. For/3 = 0.02, NH for a range of channel lengths N = 2, 5, 11, 17 and 23 are plotted in fig. 6. It is interesting to no- tice that 2Vhr increases with channel length due to the

development of the enthalpy prof'de along the channel.

Furthermore, the Nt/curve becomes asymptotic as N increases for certain fl and N. The phenomenon is due to the average parameters h L and ~/R as well as HL and H a which become less dependent on the sub-

C Chiu et al. / Enthalpy transfer between PWR fuel assemblies 173

656

65C

. ~ , ~ ~ ~ t ~ , N ~

\ oo. -~oc~ ~ 1 z,A

,-o

640 O

e~ r~

Q)

o N

o

6 3 0

57.4%

-203.9%

. Bundle Geometry N = ii

\ L = 144"

Flow Conditions G = 2.55 Mlbm/hr-ft 2

F = 1.0 - R

Energy Conditions H = Hin=600 Btu/ibm

q" =0

H R = I. 22 620 I PR = 1.0

Input Coefficients 8 = 0.02 K=0.5 S / L = 0 . 5

I I

, , , , I , , , . , I 0 5O

Axial Channel Location, Inches

I I I

i00 150

Fig. 5. Comparison between results of multisubehannel representations and homogenized representations for enthalpy upset case.

channel parameters of the center half-sized subchannels, i.e. h c and h D when N is large.

is convenient to define the ratio between these cases in terms of a multiplier, RH, where

4.3.2. Enthalpy and f low upset case

For the step inlet flow upset with the higher flow rate in the hot zone, together with the inlet enthalpy upset, N H will increase as the flow ratio increases. It

NH (enthalpy and fl0 w upset) R/¢ = NH (¢nthalpy upset)

(9)

Fig. 7 demonstrates that this multiplication factor for R H = 1.22 is roughly directly proportional to FR,

174 C Chiu et al. / Enthalpy transfer between PWR fuel assemblies

Table 2 Coefficients for NH(Z) = 1.0 + az/(b + z) (z in in.) under enthalpy upset condition

;3 N

2 5 11 23

0.005 a 0 3.66 6.11 6.75 b 183 275 269

0.02 a 0 2.04 4.36 5.96 b 9.22 34.0 52.4

0.04 a 0 1.93 4.79 8.09 b 0.31 22.7 54.8

Bundle geometry N = variable L = 144 in.

Flow conditions G = 2.66 Mlbm/h ft 2

F R = 1.0

Energy conditions H = / t i n = 600 Btu/lbm q = 0 H R = 1.22 PR = 1.0

Input coefficients # = variable K = 0.5 s/t. = 0.5

Table 3

/VH for different # and different N over z = 0 - 1 4 4 in. under enthalpy upset condition

# N

2 5 II 23

0 1.00 ~ 1.00 ~ 1.00 ~ 1.00

0.005 1.00 1.77 1.93 2.03

0.02 1.00 2.40 3.03 3.32

0.04 1.00 2.57 3.65 4.23


Flow conditions G = 2.66 Mlbm/h ft 2

~.~R = 1:0 Energy coefficients H = Hin = 600 Btu/Ibm

~" --0 H R = 1.22 PR = 1.0

Input coefficients # = variable K = 0.5

s/z. = o . 5

4.0

3.0

2.0

1.0

Bundle GeOmetry

Flow Conditions

Energy Conditions

Input Coefficients

N = Variable

L = Variable

= 2.66 x 106 ibm/hr

F R = 1.0

= Hin =600Btu/Ibm

q" = 0

H R = 1.22

PR = 1.0

3 = 0 . 0 2

K = 0.5

S/L = O.5

I I I I I

- - j

a I i ! I

6 i0 14 18 22

NLu~ber of Channels, N

Fig. 6. N H versus N for enthalpy upset case.

26

Bundle Geometry N = Ii

L = 144"


FR= V a r i a b l e

Energy Conditions H---H. =600 Btu/lbm in

q"= 0

H = 1.22 R P = 1.0

R Input Coefficients ~ = 0.02

K = 0.5

S/L = 0.5

1.2 o

" a ~ e

Z

O i.i 1.2

F R

Fig. 7. Multiplication factor versus F R for enthalpy and flow upset case.

C Chiu et al. /Enthalpy transfer between PWR fuel assemblies 175

where FR is defined as

F R = Fhot/Fcold, ( 1 1)

Fho t = inlet flow of the hot zone in the homogenized representation, and

Fco~d = inlet flow of the cold zone in the homogenized representation.

The comparison between the results of the homogenized cases using NH(Z) and NH = 1 and the results of the multisubchannel case is illustrated in fig. 8. It

660

Bundle Geometry

Flow Conditions

Energy Conditions

Input Coefficients

N = ii

L = Variable

= 2.66 Mlbm/hr-ft 2

F = 1.22 R

~ H. =600 Btu/Ibm in

q"= 0

H = 1.22 R

PR = 1.0

8 = 0.02

K=0.5

s/L = o.5

,-,I

4a Izl A

lq 0

,-I

m

e, 0

6 5 0

640

N H = NH(Z )

+13%

NR 1.0 N -190%

630

620 0 50 iO0 150


F~. 8. Comparison of multisubchannel repre~ntation and homogen~od repre~ntations with enth~py and flow up~t.

176 C Chiu et al. / Enthalpy transfer between PWR fuel assemblies

should be noticed that the result of the homogenized representation, using only one coupling coefficient N H, is not as good as that in the enthalpy upset case, fig. 4.

The 13% error for the enthalpy rise of the homogenized region case can be explained as follows.

(1) Assumption (A.1.12a) in the Appendix, made to derive the N~t in terms o f N H and known parameters, becomes invalid when flow upset goes up, i.e. diversion cross-flow becomes significant.

(2) Under the large diversion cross-flow condition in this case the neglect o f N U, NTF, NTp and NTU adversely affects the results.

(3) Error exists due to the finite difference approximations made in COBRA IIIC computations.

4.4. Calculation o/coupling coefficients for heated bundles

4.4.1. Power upset case The numerical values of Nn are evaluated for three

values of Nand three values of/3, i.e. N = 5, 11 and 23; /3 = 0.005, 0.02 and 0.04. The maximum in this case is limited to 0.048 when half-sized subchannels are used to obtain the subchannel parameters required in the evaluation of Nn. Because the exit enthalpy difference between the homogenized representation and the multisubchannel representation in this case is less sensitive to NH than that for enthalpy upset case, N n can be evaluated by eq. (6) without the half-sized subchannels in the center region and yields good results of the homogenized representation.

Values Of NH(Z) versus axial elevation for a typical case, N = 11, are shown in fig. 9. The dips and humps o f N H at low elevation positions which were present in the analogous plots of the enthalpy upset case do not appear in these figures. The reason is that the heat added from the rods overcomes the small axial enthalpy fluctuation due to the turbulent interchange. How- ever, the computational restriction for the enthalpy fluctuation due to the turbulent interchange still imposes an upper limit on 13, i.e. 0.048, in the homogenized calculation with half-sized subchannels in the center region.

The validity of the recommended NH values was also confirmed by comparing the hot zone enthalpy of the homogenized representation with that of the multisubchannel representation at each axial elevation.

Bumdle Geometry N = 12

L = Variable


F R = 1 . O

Energy Conditions Hin-- - 600 Btu/Ibm

q" = 0.04 MBtu/hr-ft 2

H R = 1.0

PR=I.5

Input Coefficients ~ = Variable

K = 0.5

S/L = O. 5

4.0

3.0

v 2.0

Z =

1 . 0

' ' ' ' I ' ' w , I ' ' ' '

50 i00 150

Axial Channel Location, inches

Fig. 9. Nil(z) versus channel length for power upset case, N = ]I.

Comparison between various forms o f Nn. The results of the hot zone enthalpy in the homogenized representation using the following approximations for NH are compared with use o f N v [3]:

NH(z)

NH =1.0, all z

NH = N, all z

where/VH is defined by eq. (7). The results are illustrated in fig. 10 for a typical N,

/3 combination. The predicted hot zone enthalpies for the homogenized representation using NH(z) and NH coincide with that of the multisubchannel representation. However, results for Nn = 1 and NH -- N have errors of enthalpy rise equal to -3 .2 and 2.1%, respectively. Note that for the power upset case the error in


625 I I I I I I l l

i 2.1%

< ~620 m

O

>, ~615

c -

O 610

605

-3.2%

Multi-subchannel Representation and Homogenized Represen-

tation for NH(Z) a n d N H

Bundle G~met~ N = II L = Vari~le


F R = I . 0

Energy Conditions H. = 600 Btu/ibm In

q" = 0.04 Btu/hr-ft 2

H R = 1 . 0 P =1.5

Input Coefficients 8 R= 0,02

K=0.5

S~ = 0.5

600

0 20 40 60 80 I00 120 140 160

Axial Channel Location, inches

Fig. 10. Comparison between results of homogenized representations and multisubchannel representation for power upset case,

hot side enthalpy for the homogenized representation is not as sensitive to the form of NH as in the unheated bundle cases.

The same methods are employed here to illustrate the behavior of Nn versus/3, N and the axial position of the subchannel as those discussed for unheated bundles.

Curve~ts. The NH(z) can be presented by using a continuous function f(z) versus channel position z for /3= 0.005, 0.02 and 0 .04;N = 5, 11,23:

CZ - - ( 1 1 ) f ( z )=a +b + z"

The values of a, b and c can be evaluated by fitting three values of Nn with the smallest error deviating from the true value at every elevation node. The values of a, b and c for the nine case (three/3 and three N) are tabulated in table 4.

Average value of NH. The ~/H for/3 = 0, 0.005, 0.02 and 0.04;N=_2, 5, 11 and 23 are tabulated in table 5. For/3 = 0.02, N H for a range of channel lengths and N = 2, 5, 11 and 23 are plotted in fig. 11. As we can observe, NH saturates faster when N increases than it does in the unheated bundle case. This is because the transverse power profile is the dominant contributor to the transverse enthalpy profile which determines the value of NH-

178 C Chiu et al. / E n t h a l p y transfer between PWR fue l assemblies

Table 4 Coefficients for NH(Z) = a + cz/(b + z) under power upset condition, where z is the channel location in inches

N

2 5 11 23

0.005

0.02

0.04

a i .0 0.96 0.95 0.94 b 1.0 314 383 384 c 0.0 2.60 3.6 3.9 a 1.0 0.82 0.75 0.73 b 1.0 48.5 70.6 81.1 c 0.0 1.94 3.4 4.0 ~.a a 1.0 0.096 0.47 1.2 b 1.O 10.8 38.8 124 c 0.0 2.60 4.00 5.00


tqow condit ions G = 2.66 Mlbm/h ft 2

F R =1.0 Energy condit ions //in = 600 Btu/lbm

" = 0.04 MBtu/h ft 2

H R = 1.00 PR = 1.5

Input coefficients ~ = variable K =0 .5 s/t~ : 0 . 5

2.(

1.0

Bundle ~ometry N = Variable

L = Variable

Flow Conditions G = 2.66 Mlbm/ft2-hr

FR = 1.0

Energy Conditions --Hin-- 600 Btu/ibm

2 i 3" = 0.04 MBtu/ft -hr

HR=I.O

PR =1.5

Input Coefficients 8 = 0.02

K =0.5

S/5= 0.5

v . I I

L = 144" ~ e

L = 86.4" ~ .

L = 46.1"

L = 17.3"

~ e o , _ e

I I , I I I 6 i0 14 18 22

N

Fig. 11. N H versus N and L for power upset case.

Table 5

N H for different N and # under power upset condit ion z = 144 in.

N

2 5 11 23

0 1.00 ~ 1,00 ~ 1.00 ~ 1.00 0.005 1.00 1.51 1.61 1.66 0.02 1.00 2.0 2,49 2.67 0.04 1.00 2.33 3.05 3.20


F low condit ions G = 2,66 Mlbm/h ft 2 F R : 1.o

Energy condit ions / / in = 600 Btu/lbm q" = 0.04 MBtu/h ft 2 H R = 1.0 PR = 1.5

Input coefficients ~ = variable K = 0.5 S/L : 0.5

1.4

1.2 1

1.0

Bundle Geometry

Flow Conditions

Energy Conditions

Input Coefficients

' ' I

N = iI

L = 144"

= 2.66 Mlbm/hr-ft 2

F = Variable R

Hin=600 Btu/Ibm

q" = 0.04 MBtu/hr-ft 2

H R = 1.0

PR = 1.5

= 0.02 K = 0.5

S/L = O.5

I ' | ' 1 ' I '~ I

~ p

°'~ , J I i , i , I 0.7 1.0 1,5

Fig. 12. Rp versus F R for power and flow upset case.


4.4.2. Power and flow upset case For the step inlet flow upset, together with the

step linear heat generation rate, NH for the homogenized representation will increase as the flow ratio increases. This phenomenon can be expected from the results of the enthalpy and flow upset case (section 4.3.2). The multiplication factor for this case is

defined as

/VH (power and flow upset) Rp - /V/_/ (power upset) (12)

and is plotted versus F R in fig_l 2. The validity of NH(z) and NH was confirmed by

comparing the multisubchannel results with those of

Bundle Geometry

Flow Conditions

Energy Conditions

Input Coefficients

N = ii L = 144"

= 2.66 Mlbm/hr-ft 2

F R = 1.22

Hin = 600 Btu/ibm

3"= 0.04 MBtu/hr-ft 2

H R = 1.0

PR=I.5

8 = 0.02 K=0.5

s/L = 0.5

622

621

=~"16 2 0

0

+E

619

c

c O 618

617

616

615

l I

Homogenized Representation

N H = N

\

' +1.4% I

t 1 Homogenized Representation

N H = NH.(Z) and NH as w e l l - as the

Mul ti-subchannel Representation

/~ Multi-subchannel Representation

• Homogenized

Representation

Homogenized Representation N H = 1.0

I I '1 I 0 50 I00 150


Fig. | 3. Va|idity of NH(Z) for power and flow upset case in the subchanne] exit region.

180 C Chiu et el. / Enthalpy transfer between PIeR fuel assemblies

the homogenized representation, as shown in fig. 13. However, results for NH = 1 and N H = N have errors of enthalpy rise equal to -2.7 and 1.4%, respectively.

5. Correlations for NH(7,) and/VH The numerical values o f N H for the two-dimen-

sional subchannel layout under different operational conditions considered in this study, are summarized in table 6.

For subchannels with axial length less than 12 ft, NH(z) gives detailed values at each axial position up to the exit, whereas NH has been calculated for only certain specific axial locations(figs. 6 and 11). How- ever, it is easy to incorporate NH into computer codes. Two empirical correlations for/VH for the cases discussed in this paper are represented as follows by eqs. (13) and (14) in which L is in inches. The equations can be used directly without resorting to the tables and figures listed above.

{ I1 / N - 2 " + " ° ° ~ /V/-t = 1 + In + 4 2 0 0 [ - - - ~ )

^I 43[L(in') ~l'sq~ × IJ" [ ' - i ~ ] ] IRH" (13)

Eq. (13) determines the value of/V n to -+5% for the

Table 6 NH(Z)

Enthalpy Power upset upset

No flow upset Fig. 4 (N= 11) Fig. 9 (N= 11) Table 2 (best Table 4 (best

fit curves) fit curves)

NH

Enthalpy Power upset upset

No flow upset Fig. 6 (8 = 0.02) Table 3 (z = 144 in.)

Flow upset Fig. 7 (multi-

plication factor)

Fig. 11 (3 = 0.02) Table 5 (z = 144 in.)

Fig. 12 (multiplication factor)

power upset case (i.e. RH = 1) and to -+15% for the power and flow upset case.

- . { I / N - 2 , 3 . S ~ l ( ° . ° ' s + ~ ) NH = l + l n 1+353~: - - -~)

X 31.1/L(in.)~l'S]/R ~ 4 - - / ? _]j p. (14)

Eq. (14) determines the values o f N n to +5% for the enthalpy upset case (i.e. Rp = 1) and to +5% for the enthalpy and flow upset case.

These correlations are valid for the conditions tested with N < 23, 3 < 0.04, L < 144, in. PR < 1.5 and F a < 1.2. The factors RH and Rp are multiplication factors for enthalpy upset and power upset, respectively, which can be obtained from figs. 7 and 12, respectively. Under no flow upset condition, RH = 1.0 and Rp = 1.0.

6. Discussion

For PWR application, eq. (13) which determines /VH for the power and flow upset case is applicable. It should be recalled that the limits on the power upset (maintained axially) and flow upset (imposed at inlet) conditions between adjacent assemblies are 1.5 and 1.2 respectively. Conditions outside this range would require further investigation. The representation in eq. (13) of the enthalpy coefficient as an axially averaged quantity is a choice made to balance accuracy and computational convenience. It should be recognized that if a value of.~ H is used as a constant for a given channel length, the enthalpies at axial positions below this length will be subject to error, which can be esti- mated from the results presented.

The error inherent in analysis of PWR enthalpy conditions when transport coefficients are not employed (i.e. NH = 1.0) was computed as described below using conservative simplifying estimates; the results are presented in table 7.

Conditions of no flow upset were assumed. The error was defined as the difference between the hot zone enthalpy rises for the homogenized and multisubchannel representations divided by the hot zone multisubchannel enthalpy rise. This definition follows directly from the fact that the homogenized representation is the approximation to the true multisubchan-

C Chiu et al. /Enthalpy transfer between PlttR fuel assemblies 181

Table 7 Expected errors in 2D homogenized region enthalpy for power upset case

N

2 3 4 5 7 9 11 15 23

0.0 0 0 0 0 0 0 0 0 0

0.005 0 -0.66 -0.61 -0.53 -0.40 -0.33 -0.27 -0.206 -0.135

0.02 0 -3.4 -3.6 -3.2 -2.3 -2.0 -1.72 -1.3 -0.86

0.04 0 -7.83 -8.5 -7.56 -5.31 -4.63 -3.9 -2.83 -1.83

0.06 0 -13.3 -14.4 -12.3 -9.34 -7.4 -6.12 -4.53 -2.36

1 -1/Nn Note: This table is built by using error% =

PRNAx/(PR- 1)LaS- 1/N n' where L = 12 ft, A s = 0.00519 ft 2 PR = 1.5, S = 0.22" in.

{ [" [N- 2\3"SBl(O.OlS+~ -l } ,n l+ L353 T) t.,j ±5% [eq-(14)].

nel representation. The error was derived [6] as

error in Ah = (1 - N ~ 1 PRNAS/(PR - 1)L/3S- ~ 1 . (15)

For a given geometry eq. (15) indicates that this error is a function of the number of subchanneis homogenized, No and the parameter/3, which characterizes the turbulent mixing between the subchanneis. Table 7 illustrates that for heated bundles a maximum error of about 14.4% occurs at large/3 and intermediate N, i.e.

= 0.06 and N = 4. For the chain and single pass pro- cedures of analysis, the error is minimal because the nodal layouts are such that N ' values are either very high or equal to 1. However, increasing interest [8] is developing on methods for coupled neutronic-thermal hydraulic analysis. Here more accurate prediction of channel enthalpy is desired for intermediate sized, ho- mogenize d regions, although coupled calculations have not been performed to date to assess either the degree of accuracy required or whether it can be achieved solely by proper sizing of the homogenized regions. Consequently, derivation and prediction of transport coefficients is of potential application in the future development of these coupled analysis methods.

The behavior of Nn(z) and the error in hot zone enthalpy rise per eq. (15) have a characteristic depen-

dence on/3, the turbulent mixing intensity, N, the number of subehannels homogenized, and z, the axial length. In the absence of diversion cross-flow the thermal interaction between two homogenized regions studied here is analagous to the transient con. duction behavior of two finite slabs of equal conductivity suddenly brought into perfect contact. The power upset case represents slabs with the same initial temperature but different energy generation rates. The enthalpy upset case represents slabs with different initial temperatures but zero energy generation rates.

Based on this analogy Nu(z) tends toward an equilibrium value with increasing z faster for the power upset case and faster for smaller N for either case. For an N, an increased value of/3 promotes a more rapid exchange of enthalpy across the boundary which decreases the local enthalpy gradient h c - h D while affecting less the region averaged enthalpy difference HL - HR. Therefore at any z, under otherwise identi- cal conditions, the magnitude of NH is directly proportional to/3. A complete development of the conduc. tion analogues to these cases in both the 2D and 3D formulations is presented in ref. [6].

The characteristics of the traLa_sport coefficient Nn, together with the d._ef'mition of Nn, suggest some rea- sonable values for N H under exceptional conditions [6]. They are as follows:


(i) Since IV H approaches an equilibrium value as N increases (figs. 6 and 11) NH for N > 23 has the same value as for N = 23.

(ii) For the homogenized representation with two homogenized rows of subchannels with uneven subchannel numbers, NH can be obtained by the following formula:

under the criteria

t t

NL - NR <0 . 3 , rnin(N[.

where

NH L =/VH value for N = 2N L ,

/VH R ~/VH value for N = 2NR

and N~, N~ = the number of subchannels comprising the left and right side homogenized regions, respectively.

(iii) Since )VH is not a function of power level and PR (HR also) as confirmed analytically, NH can be taken as a constant value through the period of a power transient (power excursion of shutdown transient) when NH is used in the transient computation.

With regard to the overall utility of these transport coefficients to compensate for homogenizing effects, it should be recognized that the most difficult source of error to deal with comes from the diversion cross- flow effect. This arises from the inability to formulate the transport coefficients to compensate for these nonlinear effects. The resulting errors are somewhat higher than they might be with a more complicated procedure using all five coefficients due to the deci- sion to utilize a single transport coefficient, N H, and introduce factors for the flow upset, R H and Rp. This choice was made to provide a simpler design procedure. Care should therefore be taken in applying these results outside the range of conditions investigated. The flow upset cases introduce cross-flows and hence are subject to the most error. Subcooled boiling conditions would also introduce some cross-flow but the magnitudes are not typically large_The enthalpy/flow upset case prediction using NH = NH had significant error (13%- fig. 8) while the power/flow upset had

negligible error for the upset ranges examined. This difference results from the fact that for the unheated bundles (enthalpy/flow upset) the energy transferred by the cross-flow was significant compared to the total enthalpy change whereas for the heated bundles (power/flow upset) this is not the case.

Finally, the additional effect of exchange processes between the subject region and homogenized regions in the second lateral direction on these results should be addressed. This was referred to as the 3D problem in section 2. As mentioned earlier, we have developed the general methodology and applied it to a specific case [6] of N-- 3, a condition where errors in the 2D case are themselves significant. The result indicated that

(i) enthalpy errors in the 3D case without application of transport coefficients were greater than for an analogous 2D problem;

(ii) N H values from the 2D problem exhibited the same trends but different numerical values from those from the 3D problem; and

(iii) the NH values from 3D problems are sensitive to power levels in regions adjacent to the boundary of interest.

From these results we concluded that practical 3D recommendations for N H would require development of a means to consolidate the large number of boundaries having unique neighboring region power profiles that would be encountered in analysis of any one core. However, our results on the two-dimensional problem gave valuable information regarding nominal behavior of the NH parameter and insights useful for future development of 3D coefficients should they be needed for coupled neutronic-thermal hydraulic analyses.

Appendix: derivation of the coupling coefficients

In this appendix the coupling coefficient, NH, is derived in eqs. (A.1.1)-(A.1.12). The multisubchannel layout and homogenized representation layout can be seen in figs. 2 and 3, respectively. For N odd, subchannels C and D can be regarded as half channels. For N even, subchannels C and D can be regarded as full channels. The derivations presented here for these coupling coefficients are valid for N either even or odd.

C Chiu et aL / Enthalpy transfer between PWR fuel assemblies 183

A.I. Derivation o f N H

The steady state energy equation for adjacent subchannels i and ] following COBRA IIIC (ref. [2, eq. (A.6)]) can be written as:

N N

a x - q t - .= ( t i - t / ) ei, / - .= ( h i - h / ) w i , /

N

- ~ wi,/h ° , ]=!

where

wi,/h* = wi,]hi, if wi, i > O,

wi,/h ° = wi,/hi, if wi, i < 0.

(A.I.1)

(A.1.2a)

(A.1.2b)

Note that wi,/> 0 means the direction of diversion cross-flow is from channel i to channel/, and wi, j < 0 means the direction of diversion crossflow is from channel ] to channel i.

We consider a multisubchannel layout shown in fig. 2 and write the energy equation for each subchannel (see table A. 1). Also, relationships exist for each pair of channels between the energy carried by diversion flows, conductivity factors ci, i and energy carried by turbulent interchanges. For example, for the channel pair A, B, the following relationships hold:

CA,B = CB,A, (A. 1.4a)

' - ' ( A . 1 . 4 b ) WA, B - WB, A,

WA, s = -wa , A, (A.1.4c)

WA,Bh* =WA,BhA I if WA,B > 0 (i.e. WB,A < 0) ' WB,Ah* WB,AhA ) '

(g.1.4d)

WA'nh* =WA'nhat ifwA, B < 0 (i.e. WB, A >0 ) . WB,Ah* WB,AhB ~ '

(A.1.4e)

Therefore, a simple relationship between WA,Bh* and wB,Ah* can be derived from eqs. (A.1.4c), (A.1.4d) and (A.1.4e):

WA,Bh* =--WB,Ah* , for WA, n <> 0. (A.1.5)

Define the left-hand region composed of subchannels A, B, and C as region L, and similarly, D, E, and F as

region R. Add the energy equations for region L and region R, whence utilizing the relations of eqs. (A.1.4a), (A.1.4b) and (A.1.5) we obtain the following two energy equations:

C C ahirni- ~ q~- (t c - tD)CC, D

i=A ax i=A

-- (h C - hD)W~, D - h*WC,D, (A.1.6a)

F E ahimi = ~ q ; _ (t D - tC)CD, c

i= D aX i=D

- ( h D - h c ) w ~ , c - h ' w v , c . (A.1.6b)

On the other hand, we can express the energy equation for the homogenized regions of fig. A.3 in (vol. II) [6] directly as

~ n LM L , ( TL - T R ax - QL - ~ NH / CL'R

(A.1 .Ta)

~HRMR , [TR-- TL~ ~ - - QR - ~ ~ 1 CR,L

H R - H L , H * N H WR'L -- ~ H WR'L" (A.1.7b)

Since

C ahimi - ~HLML (A.1.8a)

i=A ~X ~X '

F ahimi = ~HRMR (A.1.8b)

i = D ~X a x '

C

QL - i~= Aq; (A.1.8c)

F

Qrt = i_~D.= q~, (A.1.Sd)

[4/~,R _ s - WC,D, (A.1.8e)

WL, R = WC,D, (A.1 .St')

184

Table A. 1 Equations (A.1.3a) to (A.1.3f)

C. Chiu et al. / Enthalpy transfer between PWR fuel assemblies

amAh A

ax

a m # a

ax

amch C

ax

amDhD

ax itmEh E

ax

amFhF

ax

_ _ = qk - (t A - tB)CA, B - (h A - hB)w'A, B - WA, Bh* ,

=q~ - (t B - tc )eKA - (t B - tA)CB, A - (h B - h c ) w k c - (h B - hA)WkA -- WB, ch* - WB,Ah* ,

=q~ - (t c - ttgCc, o - (t c - ta)Cc, a - (h c - hD) w~, D - (h c - h a) W~,B - w c , o h * - WC,Bh* ,

- qD - ( tD- tE)CD, E - ( tD- tc)Co,c - (hD - hE)W'D,E - (hD - hC)WD,C - WD, Eh* - wD,C h*,

= qE - (tE - tF)CE, F - (rE - tD)CE, D - (hE - hF)W~F - (hE - hD) WE, D - WF~Fh* - WE, Dh*,

= q~ - (t F - tF) CF, E - (h F - hE) WF, E - WF, Eh*.

(A.1.3a)

(A.1.3b)

(A.1.3c)

(A.l .3d)

(A.1.3e)

(A.1.3f)

then the two sets of equations are equivalent if:

N~/--- TL - TR (A. 1.9a) t C -- t D '

NH = HL - Ha (A. 1.9b) hc hD '

N ~ - h ~ , (A. 1.9c)

where

H* _ H L if WL, R > 0,

h* h C'

H* _MR if WR,L < 0.

h* h D '

If we assume that the specific heat at each elevation is constant , then

N ~ = A m . (A. I .10)

Also the subchannel enthalpy h* can be expressed as a function o f N n in the following manner by virtue of the definitions o f HL, HR and NH [eq. (A. 1.9b)] and the assumption o f a symmetric enthalpy profile with respect to the central boundary:

½(h C + h D ) = ½ ( H L +MR) , ( A . l . l l a )

h* = ½(H L + H R ) + [/4 L -- ~(H L +HR)]/Nlt ,

if We, D > 0, (A.1.1 l b )

and

h" = ½ (HL + HR) - [HR 1 - ~(H L + H R)]INH,

i f w c , D < 0. (A.1.1 l c )

Therefore, we can obtain N ~ in terms Of NH and other known quantities:

H * _ H L

N ~ =--~- i (/./L + H R ) + [ H L _ ½(HL +HR)] /NH,

if Wc, D < 0. (A.1.1 2b)

From the above derivation we have defined the coupling coefficients required in the energy equations (A.1.7a) and (A.1.7b). By virtue o f eqs. (A. I .10) and (A. 1.12a and b) these coefficients are all expressable in terms of the single coefficient NH.

A.2. Transport coefficients N u , NTF, NTI, and NTU

Analogous methods can be applied to the axial and transverse momentum equations (ref. [2, eqs. (A.11) and (A. 17)]) to derive the corresponding transport coefficients. They are not given here because the other coefficients may be neglected (i.e. set to uni ty) compared to NH. However, they may be found in ref. [6].

C Chiu et al./Enthalpy transfer between PWR fuel assemblies 185

N o m e n c l a t u r e

Ai As Ak

ci,/

CL,R

Ci

C

Ax Ah

AH

F =

=

g =

G i =

d --

h i =

h* =

~k =

Bin ~"

H R =

g

L = /71 i =

=

N = NH, Nu, NTp, NTF, JVTu =

N ~ _-

= cross-section area for subchannel i = cross-section area for any subchannel = cross-section area for homogenized

subchannel k Pi = thermal condition coefficient for sub- Pk

channels i and ] PR = thermal conduction coefficient for q[

homogenized subchannels L and R = cross-flow friction force for subchan-

nel i Qk = cross-flow friction force for homoge-

nized subchannels RH = axial elevation increment = axial change of radially averaged Rp

enthalpy in the multisubchannel representation S

= axial change of radially averaged enthalpy in the homogenized representation Uk axial friction force per unit length ratio of axial flow in adjacent assemblies average flow rate wi, / gravitational constant mass flux of channel i WE,R averaged mass flux enthalpy for subchannel i w'i, / effective enthalpy carried by diversion cross-flow I4/[, R radially averaged multisubchannel enthalpy p* average homogenized enthalpy of strips R and L

inlet enthalpy Subscripts homogenized enthalpy for region k i ratio of inlet enthalpies of adjacent k assemblies cross-flow resistance coefficient channel length Variables flow rate for subchannel i J flow rate for homogenized region k total number of rods

total coefficients averaged transport coefficient total number of subchannels equal to

~N for N even 1 1 ~N + ~ for N odd

= total number of subchannels comprising the left and right side homogenized regions, respectively.

= pressure for subchannel i = pressure for homogenized region k = ratio of powers in adjacent assemblies = heat addition per unit length, for sub-

channel i = averaged heat addition per unit length = heat addition per unit length for ho-

mogenized region k = ratio of NH in the enthalpy and flow

upset case to the enthalpy upset case = ratio of NH in the power and flow

upset case to the power upset case = rod spacing = effective averaged velocity for adja-

cent channels = effective momentum velocity for

homogenized region k = effective averaged velocity for homo-

genized region L and R = diversion cross-flow between adjacent

subchannels = diversion cross-flow between homoge-

nized region L and R = turbulent interchange between adja-

cent subchannels i and ] = turbulent interchange between adja-

cent homogenized region L and R = density carried by the diversion cross-

flow = turbulent mixing parameter

subchannel identification number homogenized region identification numbers (L and R)

axial elevation node along the subchannels.

R eferen ces

[1] P.T. Chu, H. Chelemer and L.E. Hochreiter, THINC-|V, An Improved Program for Thermal-Hydraulic Analysis of


Rod Bundle Cores, WCAP-7956 (1973). [21 D.S. Rowe, COBRA IIIC: A Digital Computer Program

for Steady State and Transient Thermal-Hydraulic Anal- ysis of Rod Bundle Nuclear Fuel Elements, BNWL-1695 (1973).

[3] R.W. Bowring, HAMBO; A Computer Program for the Subchannel Analysis of the Hydraulic and Burnout Char- acteristics of Rod Clusters. Part I, General Description, AEEW-R524 (1967).

[4] J. Weisman and R. Bowring, Nucl. Sci. Eng. 57 (1975) 255.

[5] P. Moreno, J. Liu, E. Khan and N. Todreas, Nuct. Eng. Des., to be published.

[6] P. Moreno, C. Chiu, R. Bowring, E. Khan, J. Liu and N. Todreas, Methods for Steady State Thermal/Hydraulic Analysis of Pressurized Water Reactor Cores, MIT-EL- 76-006, Original March (1977), Revision 1, July (1977).

[7] R. Bowring and P. Moreno, COBRA IIIC/MIT Com- puter Code Manual, to be issued as an EPRI Report.

[8] R. Bowring, J. Stewart, Robert Shober and Randal Sims, MEKIN: MIT-EPRI Nuclear Reactor Core Kinetics Code (1975).

Enthalpy transfer between PWR fuel assemblies in analysis by the lumped subchannel model

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