Ann. nucl. Energy, Vol. 10, No. 8, pp. 395~403, 1983 0306 4549/83 $3.00+0.00 Printed in Great Britain. All rights reserved Copyright :t;) 1983 Pergamon Press Ltd.

POWER-REACTOR PERFORMANCE EVALUATION USING NODAL/MODAL ANALYSIS

H. W. GRAVES JR

7723 Curtis Street, Chevy Chase, MD 20815, U.S.A.

(Received 10 January 1983)

Abstract- -An analytical procedure for calculating the 3-D power distribution in LWRs based on two-group coarse-mesh nodal analysis is described. The use of source-sink analysis of the fast-neutron source distribution yields a very etticient eigenvalue equation. Neutron coupling between nodes is evaluated using a modal analysis of the neutron-source shape within each node based on the multiplication and transport properties of the source node and its immediate neighbors. Both the core and fuel-assembly geometry and the neutronic characteristics of LWRs are particularly well suited to this type of analysis.

INTRODUCTION

Operations analysis and incore fuel management of LWRs involve many decisions that can be made more efficiently and accurately if they are supported by the results of a reliable power-distribution simulator. In operations analysis, such a simulator can predict cycle lifetime, the effects of control-rod movement on power distribution, and the interaction between reactor operating variables. It is of value in fuel management in selecting the number and enrichment of fuel assemblies to be loaded at the beginning of a reactor cycle, and the arrangement of these assemblies in the core along with the partially-spent fuel assemblies that are retained.

Although a power-distribution simulator need not encompass the spatial detail required in fuel-assembly design calculations, it should account for the effects of all variables at the disposal of the reactor operator, be sufficiently accurate to differentiate between alternative cources of action, be relatively easy to use once it has been set up for a particular system and make efficient use of both computer-t ime and computer-storage requirements.

Considerable progress has been made in the last ten years in the development of efficient 3-D power- distribution calculation techniques (Askew, 1980). Procedures for such calculations tend to fall into two general categories: calculations using a Green's function approach that treat the reactor as an array of neutron sources and sinks (Askew and Mott , 1980; Delp et al., 1964) and calculations based on the evaluation of partial neutron currents and region- averaged fluxes in a coarse-mesh array (Finneman et al., 1977 ; Shober et al., 1977). The intent of this article is

395

to describe a procedure that uses source-sink nodal analysis in which the source distribution in each node is treated as a superposition of simple modal source distributions for which inter-nodal coupling can be more easily evaluated.

DERIVATION OF THE TWO-GROUP NODAL EQUATIONS

In nodal analysis, the reactor is subdivided into an array of nodal volumes, usually equal volumes. Selection of assembly-wise node spacing is ad- vantageous since it yields good physical correspon- dence between the analytical model and the operating reactor. Each volume is represented by a set of space- and energy-averaged neutronic parameters that describe its neutron multiplication and transport properties (Kollas and Henry, 1976). There are a number of advantages in using a nodal approach based on the evaluation of neutron sources rather than neutron fluxes and currents. In a thermal reactor, all fission neutrons are born in the fast-neutron energy group, so that evaluation of the reactor eigenvalue, ~, and the fast-neutron source distribution can be performed with a relatively simple matrix equation.

Si = ~ Aiflj, (1) j - t

where

Si = fission-neutron source at node i ; Aq = fission neutrons produced at node i due to a

unit fission-neutron source at nodej .

The summation in equation (1) includes the source

396 H . W . GRAVES JR

node and all other surrounding nodes that make a significant contribution to the fission source at node i. In one-group nodal analysis, the evaluation of the matrix elements of the eigenvalue equation is quite straightforward. In this case,

A i j = W~ik~oi, (2)

where

VCji = probability that a neutron produced at nodej will be absorbed at node i (neutron coupling coefficient);

k® = neutron multiplication factor at node i.

For two neutron energy groups, evaluation of the matrix elements is considerably more complex since it must account for neutron transport in both energy groups, fission in both energy groups and neutron coupling between the groups (neutron thermalization). A derivation of the eigenvalue equation for the fast- neutron source distribution for two neutron energy groups has been developed previously (Graves 1973, 1979). The matrix elements in the two group equation are expressed as follows"

Ai j =. kiF Wj iF + k~T W~ip Wj iF q_ kiT Wjlp w~jF

+k~ ~ w~,p~W~, (3) k = l

where

k F = fast-neutron multiplication factor; p = thermalization probability;

k T = thermal-neutron multiplication factor; F _ _ W~i-coupl ing coefficient for fast-neutron

transport; T__ Wji - coupling coefficient for thermal-neutron

transport;

It is worthwhile to briefly summarize the physical significance of each of the terms in equation (3). The first term represents neutrons produced by fast fission at the reference node due to neutrons born at both the reference node and the n nearby nodes from which fast- neutron transport is significant. The second term accounts for thermal fission neutrons from the same neutron-source terms. In this term the thermalization probability (pi) and thermal non-leakage probability (W T) are required. The third and fourth terms represent fast-neutron production by thermal fission caused by thermal neutron in-leakage to the reference node. In the third term, neutrons are thermalized in the same nodal volume in which they were produced, while the fourth term represents neutrons that are born at one node, thermalized at a nearby node and leak to the reference node as thermal neutrons.

It is important to note that the summation limits for the various source terms may have different values. In LWRs, there is a significant difference in the mean-free- path for absorption and removal between fast and thermal neutrons. The value for fast neutrons is between 6 and 8 cm. This is about one-third the transverse dimension of a fuel assembly. The value for thermal neutrons is usually 1.5-2.5 cm, which is about one-tenth of the transverse fuel-assembly dimension, so that fast-neutron transport is considerably more important in a reactor-simulation model than thermal- neutron transport. Because of this difference, one- group and modified one-group coarse-mesh models based on fast-neutron transport properties have met with reasonable success in simulating neutronic behavior in LWRs. The primary disadvantage of these models is their inability to simulate a nodal volume that is a net source for fast neutrons and a net sink for thermal neutrons, a condition that occurs frequently in LWRs.

Since fuel-assembly dimensions are large relative to mean neutron path lengths, neutron coupling may be restricted to a relatively small number of nodes. Figure 1 is a sketch ofa 2-D array of fuel assemblies. Of the 24 assemblies surrounding the neutron-source assembly (S), four are referred to as facing assemblies (F), four as diagonal assemblies (D) and the remaining 16 as peripheral assemblies (P). Table 1 shows the value of coupling coefficients to surrounding nodes in such an array for a uniform source in a uniform medium based on a fine-mesh diffusion-theory calculation. The second column gives typical values for fast-neutron transport, while the third column presents typical values for thermal neutrons. For fast neutrons, for instance, all but 3~o of the neutrons are absorbed in the source- assembly node (S), the four facing assemblies (F) or the four diagonal assemblies (D). Although the coupling coefficient values are a function of both neutron-source shape and neutron-absorption properties, the infor-

P

D F D

F S ~ F

D F D

Fig. 1.5 x 5Fuel-assembly array.

Calculation of 3-D power distribution in LWRs 397

Table 1. Neutron coupling in a uniform medium

Dimensionless thickness (d/L)

Fast Thermal neutrons neutrons

3.0 8.0

Source absorption 0.48 0.75 Absorption in facing nodes 0.38 0.23 Absorption in diagonal nodes 0.11 0.02 Absorption in peripheral nodes 0.03

mation in Table 1 provides a reasonable basis for evaluating summation limits in the eigenvalue equation (equation (1)). If we confine fast-neutron coupling in 2-D to the eight surrounding nodes and force a neutron balance by transferring peripheral absorptions to facing and diagonal nodes, the error in coupling to these nodes will be comparable to uncertainties in diffusion parameters, so that this would appear to be a reasonable course of action. Using the same argument, confinement of thermal-neutron coupling to four facing nodes would also produce relatively small errors. The neutron balance is maintained within each energy group if the value for self-coupling, W,, satisfies the following relationship :

W . = 1-- ~ W O. (4) j=l

EVALUATION OF COUPLING COEFFICIENTS

Although the large dimensions of an LWR fuel assembly in terms of neutron mean-free-paths is advantageous in limiting the number of nodes for which couplingis significant, it also requires that the model for evaluating coupling coefficients takes account of the spatial distribution of the neutron source within each nodal volume. A technique for evaluating source- shape-dependent coupling coefficients has been developed that requires relatively small computer running time and also has a reasonably sound basis in reactor physics principles. This technique involves the representation of the space-dependent source in the nodal volume by a superposition, or sum,of a number of simple analytical functions. This technique is best illustrated by considering the evaluation of coupling coefficients in a 1-D problem. The analytical evaluation of coupling coefficients for this case is straightforward. For a uniform source region of half-thickness 'a' embedded in an infinite medium, the diffusion-theory value for the 1-D coupling coefficient between region i and each adjacent semi-infinite region j, w o, is:

Li [ ZiLi ] - i wij = 2aa coth (a/L 3 + ~ . j , (5)

where

E = macroscopic absorption (plus removal) cross section for neutrons in energy group (cm- 1);

L = neutron diffusion length for neutrons in energy group (cm).

Based on the arguments presented in the previous section, we will confine neutron transport to adjacent assemblies of equal thickness. In this case, a more symmetric relationship is :

~a Li ~ ! E)~'iLiLj ~] 1 Wij = [coth (a~Li)+ coth (a/Lj) . (6)

This expression is not only consistent with the assumed zone of influence, but also has the summetry required to satisfy the time-honored reciprocity relationship. In these equations, the subscripts i andj refer to the source and sink nodes, respectively. Although these equations apply to a uniform source, similar analytical re- lationships can be derived for sinusoidal, hyperbolic and exponential source shapes.

In the diffusion-theory solution of a two-group, multiregion, 1-D problem, the neutron flux in a uniform region of thickness 2a that serves as a net source of neutrons can be written as follows :

~b(x) = C1 e-X/L+c2 sin B(x+h)+C 3 e -(2a-x)/L.

(7)

If the region is a net sink for neutrons, the above expression becomes :

dp(x) = C1 e-:'/L +C2 sinh B(x +h)+C 3 e ~2,-x)/L

(8)

In these equations, the origin is at the left region boundary, L is the diffusion length in the region, B is the material buckling, and the coefficients C, and h are derived from matching flux and current boundary conditions at the left and right region boundaries. These relationships form the basis for selecting the analytical form of the functions used as components for the space-dependent neutron source in each energy group in each nodal volume.

We may think of the terms in equations (7) and (8) as modal components of the flux or source shape. The exponential terms are spatial transient modes. The magnitude of their coefficients is primarily a function of the mismatch in neutron-absorption and diffusion properties between the regions on either side of the boundary. The central term will be referred to as the fundamental mode since it depends only upon the multiplication and transport properties within the nodal volume.

The generic characteristics of LWRs permits

398 H.W. GRAVES JR

considerable simplification in the above relationships. Since the only source for thermal neutrons is thermalization from the fast-neutron energy group, the thermal-neutron source has the same shape as the fast flux. In an LWR, although the fast absorption and diffusion properties of the fuel assembly are influenced to some extent by fissile and control content, they are much more dependent upon the volume fractions of water, structural material and fertile material. Since these volume fractions do not vary significantly between fuel assemblies of the same design, the transient terms in equation (7) are quite small and may be neglected in the thermal-source calculation. The fast-neutron source, which is derived from both fast and thermal fission, will have both a fundamental mode term and two transient mode terms. The magnitude of the transient-term coefficients will depend primarily on the discontinuity in the thermal absorption cross section at each boundary. Figure 2 illustrates a composite source shape in 1-D. Note that the transient components can be positive or negative. The latter condition occurs when the thermal absorption cross section is greater on the external side of the region boundary, and is illustrated in Fig. 2. IfSa, Sb and Sc are the volume-integrated source values for the funda- mental and the two transient modes, respectively, and a coupling coefficient is evaluated for each modal component, superposition can be used to derive a coupling coefficient for the complex source shape, yielding the following :

w~ = (Sawai~+SbWbi~+ScwcO/(S,+Sb+Sc) (9)

A solution of the 1-D problem will then depend only on developing a reasonably accurate procedure for evaluating the coefficients (or their ratios) in equations (7) and (8).

The coefficients C2 and h in equations (7) and (8) are

S(x)

Transient Modes

F°°d¢"2?2'°' 1 Fig. 2. Source nodes in 1-D.

primarily dependent upon the nuclear properties of the source node and its immediate neighbors, so that evaluation of neutron coupling parameters is a local, rather than a global, problem.

COUPLING COEFFICIENTS IN 2- AND 3-D

In view of the fact that most thermal-neutron transport is to facing assemblies due to the short thermal diffusion length relative to fuel-assembly dimensions, it seems reasonable to use a 1-D approximation for thermal-neutron coupling. This is clearly no t -adequa te for fast-neutron coupling, however. The simplest way of extending the 1-D procedure is to assume separability of the source components ; i.e. in a 2-D problem :

S(x, y) = Sl(x)S2(y). (lo)

In a uniform medium for which 1-D coefficients, w~i, have been evaluated, the 2-D coefficients, W0, would then be :

Wij = wij(1-2w~j) for facing nodes (11)

and

W~j = w~ for diagonal nodes. (12)

The efficacy of separability has been investigated for a uniform source in a uniform medium (Graves, I973), and a linear correction for equations (11) and (12) has been found to yield reasonable agreement with a 2-D calculation over the dimensional range of interest. This 'modified separability' model has the following form :

Wii = wi j (1 -Clwi j ) forfacingnodes (13)

W~j -- C2w~ for diagonal nodes. (14)

This separability approximation will yield reasonably good results under conditions where the fundamental mode shape is nearly uniform, when it is the dominant source component and when the transport and absorption properties of the surrounding nodal volumes are close to those of the source node. It will not be as accurate if the fundamental mode shape has a large variation across the nodal volume and when the nearby node properties are significantly different. The greatest loss in accuracy will therefore take place in peripheral assemblies bounded on one or two sides by the core baffle and reflector. Although equations (13) and (14) are not as accurate for transient-mode source components, the magnitude of these source com- ponents relative to the fundamental component makes this error less significant than that associated with peripheral nodes, since the transient component at each

Calculation of 3-D power distribution in LWRs 399

face is usually less than 2 3% of the total source component.

It should be noted that the specific model used to evaluate neutron coupling coefficients is not critical to the simulation concept described in this paper. The salient feature of the proposed application of source- sink nodal analyses to LWRs is the evaluation of coupling coefficients on a local basis using non-iterative techniques, and the use of these coefficients in the iterative solution of a coarse-node matrix equation to solve for the global power sharing between nodes. If this procedure is compared with fine-mesh calculations, a relatively complex global problem, that requires a costly iterative solution procedure, has now been transformed into a very simple iterative global problem and a series of non-iterative local problems of intermediate complexity (evaluation of coupling between nodes). Compared with two-group, coarse- mesh procedures based on the evaluation of surface currents and average fluxes, the need for matrix inversion has been eliminated and the number of equations in the iteration matrix is reduced by at least a factor of two for the same number of fuel nodes since the number of rows in the matrix equation is confined to the number of fuel-bearing (fast-neutron source) assemblies.

APPLICATION TO AN ENGINEERING SIMULATOR

Although the two-group equations and the general technique for evaluating inter-nodal coupling de- scribed in the previous sections provide a sound basis for developing a practical reactivity and power- distribution simulator, several problems must be resolved before the procedure becomes useful for engineering analysis. First, a quantitative technique for evaluating the fundamental and transient source term coefficients in 2- and 3-D must be established. In addition, the inaccuracies associated with the separability correction at the core-reflector interface must be accounted for. Finally, the effects of non- uniformities introduced by control material, structural inhomogeneities and non-uniform depletion within individual fuel assemblies should be evaluated. A 3-D reactor-simulation program based on 'nodal/modal ' analysis called P O D N O D has been developed and benchmarked to a variety of applications, two of which will be described later in this section. This program utilizes the following engineering approximations to resolve the above problems :

(a) The buckling dependence of the fundamental source mode is accounted for by making a linear

(b)

(c)

(d)

correction to the 1-D uniform source coupling coefficient ; i.e.

wij = ~ij[1 + K 1 B ~ + K 2 ( B 2 - B ~ ) ] , (15)

where O~j is the 1-D coefficient for a uniform source (equation (6)) and values o fK 1 and K z are derived from generic analytic solutions to l-D, two-group, three-region diffusion-theory problems.

The magnitude of the integrated transient-mode fast-neutron source, ST, relative to the integrated fundamental-mode source, S F, was also derived from two-group, three-region diffusion-theory problems and is of the form :

. / / ) 2 a 2 k T p ~ l/wij(IF, a2i--Ya2j)~ ), (16)

where K 3 is a constant derived from the 1-D generic study.

A study of fast-neutron source shapes in peripheral assemblies has resulted in isolation of four different shape classifications for fast-neutron coupling to facing nodes, and four shape classifications for coupling to diagonal nodes. The geometries associated with each of these eight classifications are shown in Fig. 3. For each of these geometries, a coupling correction to the modified separability relationship presented in equations (13) and (14) can be developed. Since the fundamental source shape in peripheral assemblies is most strongly influenced by geometry and baffle and reflector nuclear properties, it is hypothesized that these corrections are relatively insensitive to fuel properties. If this is the case, then once an eight- element correction vector is established for a core with a given set of reactor-vessel internals, this vector is valid for a wide range of fuel arrangements. The individual corrections can be obtained by normalizing to either more rigorous calculations or to experimental data.

As depletion proceeds, the global reactor power shape will change. Power-shape changes will also take place within individual assemblies. Two physical situations where these changes can noticeably affect the neutron coupling coefficients are in peripheral assemblies and in assemblies containing fixed burnable poison. The funda- mental source shape in peripheral assemblies has the form (cf. equation (7)) :

S(x) = K sin B(x + h). (17)

In this equation, the coefficient 'h' corresponds to a

400 H.W. GRAVES JR

BR4

~ R ~ BR8 Fig. 3. Fuel-assembly geometries associated with fast-neutron reflector coupling vector.

reflector savings. As depletion proceeds, the burnup imbalance created by non-uniform power generation between the outside and inside edges of peripheral assemblies creates a shift in power to the outside edge. The effect of this shift on core- reflector fast-neutron coupling can be simulated by using a fuel-burnup-dependent reflector savings correction. This effect is illustrated in Fig. 4. A small linear burnup-dependent correction to the reflector

~ CORE REGION

Uniform Fuel Source Shape

Depleted Fuel Source Shape

x

REFLECTOR REGION

Coolant Flow Baffle

Fig. 4. Non-uniform depletion effects in peripheral assemblies.

savings has been found to be quite adequate to account for this physical phenomenon.

Fixed burnable poisons also distort the fast- neutron source shape within assemblies. Since the bulk of the burnable poison absorber is consumed during the reactor operating cycle, this distortion is burnup dependent. In operating cycles that contain burnable poison, this effect has been successfully simulated using a correction to the coupling coefficient that is linearly dependent upon the effective control cross section ; i.e.

W/j o ¢ = Wq [1 +K(Y~,2/Y:a2), ] (18)

where W/° is the poison-free coupling coefficient, and the linear coefficient, K, is based on normalization to more rigorous calculations or experimental data.

Although the above modifications to the coupling- coefficient model may be disparaged by the purist as empirical 'tuning', such tuning or normalization is an integral part of engineering simulation. If the normalization coefficients have a well-defined physical significance and are subject to straightforward evaluation, their existence in a simulation model provides more flexibility for potential users to adapt the model to physical situations that may not have been considered in its initial development.

Calculation of 3-D power distribution in LWRs 401

RE GINNA OPERATING DATA

A 3-D version of PODNOD equipped with moderator density, fuel doppler and equilibrium xenon-poisoning feedback has been applied to two operating cycles of the RE Ginna PWR operated by the Rochester Gas & Electric Co. Calculated values of assembly average power have been compared with measured results derived from the plant incore instrumentation system (Mecredy, 1978). The Ginna reactor is a 121-assembly reactor that uses chemical shim to compensate for reactivity loss due to depletion. The core and reflector regions are separated by a 3 cm stainless-steel coolant-flow baffle that is explicitly considered in the PODNOD calculation.

The power-distribution measuring system for the Ginna reactor consists of a set of movable fission detectors that traverse the full length of up to 36 fuel assemblies. The detectors are located in guide tubes positioned near the center of each fuel assembly. During a traverse of the fuel assembly, readings are taken at 55 positions along the active fuel length : an axial spacing of approx. 7 cm. The system has the capability ofdriving each movable detector into a common thimble to provide inter-normalization between the detectors.

The average power generation in each assembly at each axial position is obtained using calculated normalization factors based on fine-mesh 2-D diffusion-theory calculations. These factors apply to the assembly containing the detector thimble, as well as the assemblies in a region several rows around the measurement location. The detector thimbles are distributed in the core in an arrangement such that no core position is more than one diagonal assembly pitch from a measurement location. Comparison of integrated assembly power values from symmetric positions and for repeated measurements in similar systems would indicate that the measurement accuracy (la) is in the range of 1.0-1.5%. Three full core maps from cycles 8 and 9 of Ginna were used for comparison. Figures 5-7 compare the measured and calculated values of integrated assembly power for the 19 assemblies in one octant of the core at the beginning and end of cycle 8 and at the beginning of cycle 9. The agreement between calculation and measurement is of the same order as the uncertainty in the measured results. These results tend to confirm the hypothesis that the core-reflector coupling vector is relatively insensitive to changes in fuel-multiplication distri- bution. Figure 8 compares the core-averaged power generation at each axial measuring position from the end-of-cycle-8 measurement with the corresponding 12-node axial distribution from the nodal calculation. The sharp dips in the experimental trace identify the

1.024 1.166 1.078 0.997 1.037 1.170 1.088 0.993 0.013 0,003 0.009 - 0 . 0 0 4

0.993 1.058 1.053 1.000 1.072 1.055 0,007 0.01 3 0.002

1.002 1.198 1.014 1.201 0.011 0.003

Calc. va lue~ 1.028 Meas. value--* 1.042 Error = ( M - C ) / M ~ 0.013

r.m.s, error = 0.011 Max. error = 0.020

1.203 1.187

- 0 . 0 1 4

1.139 1.127 0.011

1.018 1.006

- 0 . 0 1 2

1.061 1.072 0.010

Fig. &Cycle 8 -BOCintegrated assembly

1.072 0.816 1058 0.822

- 0 . 0 1 4 0.007

1.148 0.667 1.1 45 ! 0.667 0.003 0.001

0.956 0.944

- 0 . 0 1 4

0.663 0.656

- 0 . 0 1 0

power values.

1.049 1.175 1.102 1.016 1.052 1.068 1.092 1.003 0.003 - 0 . 0 0 6 - 0 . 0 0 9 - 0 . 0 1 3

1.026 1.086 1.066 1.016 1.078 1.055

- 0 . 0 1 0 - 0 . 0 0 7 - 0 . 0 1 0

1.026 1.178 1.019 1.178

- 0.007 0.000

Calc. va lue~ 1.031 Meas. va lue~ 1.037 Error = ( M - C ) / M ~ 0.006

r.m.s, error = 0.013 Max. error = 0.036

1.1 62 1.1 58

- 0,003

1.118 1.109

- 0.008

1.021 1.001

- 0 . 0 2 0

1 047 1.067 0.019

1 0 5 0 0.822 1.035 0.832

- 0 . 0 1 5 0.013

1.100 0.680 1.109 0.692 0.008 0.017

0,941 0.950 0.010

0.684 0.693 0.014

Fig. 6. Cycle 8 EOC integrated assembly power values.

0.915 1.086 0.877 0.928 0.903 1.050 0.867 0.929

- 0 . 0 1 3 - 0 . 0 3 4 -0 .011 0.001

0.943 0.998 1.087 0.927 0.991 1.101

- 0 . 0 1 7 - 0 , 0 0 7 0.013

0.969 1.240 0.988 1.235 0.020 - 0.004

Calc. va lue~ 1.049 Meas. va lue~ 1.058 Error = ( M - C ) / M ~ 0.009

r.m.s, error = 0.01 3 Max. error = 0.034

1.209 1.225 0.013

1.216 1.224 0.006

1.1 O0 1.103 0.003

1.225 1.203

- 0 , 0 1 8

0.976 0.740i 0.986 0 . 7 4 3 0.011 0.004

1.068 0.611 1.081 0.617 0.012 0.009

0.970 0,974 0.004

0.707 0.696

- 0 . 0 1 6

Fig. 7. Cycle 9 BOC intezrated assembly oower values.

402 H . W . GRAVES JR

1.2

1.0

no 0.8

5 × <

.~ 0.6

+f 0

I

- ."" - " ""I ,.'"I """"t."l o e - o • • • • ° • •

J- ~ Measured Points /

= = . = = _ ] C a l c u l a t i o n (12" overages) /

• g

I I I I I I 0 20 30 40 50 60

R e l a t i v e A x i a l P o s i t i o n

Fig. 8. Cycle 8 EOC average axial power distribution.

location of fuel-rod spacer grids. These grid regions are not separately identified in the calculation, but are volume-averaged over the fuel-core height. Figure 9 compares calculated values of critical boron concen- tration at full-power equilibrium xenon operating conditions with plant measurements. Both reactivity and power-distribution agreement are excellent.

I A E A BENCHMARK PROBLEM

Although the Ginna operating data tend to validate most aspects of the nodal/modal model, the use of

soluble-poison control precluded evaluation of the model under conditions in which a large discontinuity in thermal absorption exists between adjacent assemblies. The 2-D IAEA benchmark problem (Weiss, 1977) provides an extreme test of the effects of control insertion on power distribution. It mocks up a 177- assembly reactor with two fuel zones of different enrichment, nine inserted control rods and a light- water reflector. Although the neutronic properties of the core are somewhat unrealistic in that there is no fast fission and no strongly-absorbing baffle between the core and reflector assemblies, the variation of ko~ between controlled and uncontrolled assemblies is probably greater than would exist in a realistic system. The results of this application yield an eigenvalue difference between P O D N O D and the IAEA reference calculation of 0.002, a mean relative power difference of 0.7%, and a maximum power difference of 1.5%. Figure 10 shows the fuel- and control-assembly arrangement as well as a comparison of PODNOD and reference relative power values. In this case, the reference calculated results were used to normalize the core reflector coupling correction vector, so that the agreement at the core-reflector boundary only serves to indicate that the nodal/modal model is capable of accurate normalization. This vector has little effect on the interior distribution, however, and the close agreement in the vicinity of control assemblies tends to validate the inter-assembly buckling correction (equation (15)) derived from 1-D diffusion-theory calculations. Since the control poison is uniformly distributed within the assembly, no control correction (equation (18)) was made to coupling coefficients in controlled assemblies.

(000

800

c 600 8 g

g ~, 40o m

-6

200 [_)

-•% Measured Data

\_ -

® °° o o ° o

_ " ~ . o e " x ~ cycle 9 - cyc le 8 ~ o ~ . ~

- - e Oo

- I I ° ° o nl 2.0 4 0 6 0 8 0

C y c l e Burnup (MWd kg - I )

0 . 7 4 5 1.315 1 . 4 6 1 1.217 0.609 0.934 0 . 7 4 6 1.308 1.454 1.210 0.610 0.935 0.001 0.005 0.005 - 0.006 0.002 0.001

1.445 1.473 1.320 1.072 1.043 1.435 1.479 1.314 1.070 1,036

-0.007 0.004 -0.005 0.002 -0.007

1.447 1.334 1.181 1.071 1.469 1.348 1.179 1.071 0.015 0.008 --0.001 0.000

1.192 0.966 0.901 1,192 0.966 0.906 0.000 0.000 0.006

P O D N O D value ~ 0 .471 0.692 Reference value ~ 0 .471 0.685 Error ~ 0.000 -0.010

r.m.s, error = 0.007 Max. error = 0.015

0.942 0.934

-0.008

0.951 0.951 0.000

0.963 0.976 0.013

0.851 0.847

- 0.004

0.601 0.598 0.004

0.751 0.755 0.006

0.743 0.736 0.009

0.683 0.692 0.012 i

Fig. 9. Critical boron letdown curve. Fig. 10. IAEA benchmark problem.

Calculation of 3-D power distribution in LWRs 403

SUMMARY AND CONCLUSIONS

A reactor-neutronics simulation model based on the source-sink nodal calculation of the fast-neutron source distribution can yield quite accurate values for power distribution and reactivity parameters in an LWR with very small central memory requirement and very little processing time. The modal analysis described is based on a modified separability assumption to expand 1-D coupling coefficients to 2- and 3-D. Although it is necessary to modify the core-to- reflector fast-neutron coupling coefficients by normal- ization to more detailed calculations or experiment, the normalization factors appear to depend primarily on reflector (and core baffle) properties and reactor internals geometry since the same normalization vector applies to successive operating cycles with different fuel properties. A 3-D evaluation of a large PWR (193 assemblies) in eighth-core symmetry with an axial node spacing of 30 cm requires approx. 60K words of central memory. A sequence of 3-D calculations with full spatial feedback for such a reactor requires about 60 ms of Univac 1180 CPU time for each fully-converged time

step. The physical characteristics of LWRs are particularly well suited to this type of analysis from the standpoint of both fuel-assembly geometry and reactor neutronic characteristics. The suitability of this simulation procedure to LWRs will make it a prime candidate for applications with mini- and microcom- puters with a limited core and peripheral storage capability.

REFERENCES

Askew J. R. (1980) Proc. Specialists' Meet. on Calculation qf 3- Dimensional Rating Distributions in Operating Reactors, OCDE, 1979.

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