Ann. nucl. Energy, Vol. 12, No. 7, pp. 327-338, 1985 0306-4549/85 $3.00 +0.00 Printed in Great Britain. All rights reserved Copyright © 1985 Pergamon Press Ltd

N E U T R O N I C S A N D T H E R M A L H Y D R A U L I C S M O D E L L I N G OF THE HARWELL MATERIALS TESTING

REACTORS D I D O A N D P L U T O - - I

N E U T R O N I C S ANALYSIS

T. D. BEYNON and M. JAVADI

Department of Physics, University of Birmingham, Birmingham B 15 2TT, England

(Received 20 December 1984)

Abstraet--A detailed 2-D cylindrical diffusion theory neutronics model is presented for the Harwell reactors DIDO and PLUTO, based on the WlMS-E program. The model for these highly asymmetric reactors allows for the presence of the various control systems, experimental rigs and fuel burnup. Comparisons made with measurements of burnup and of radial and axial flux distributions validate the approach.

1. INTRODUCTION

Since the startup of the two Harwell Materials Testing Reactors (HMTRs), DIDO and PLUTO, there has been a growing demand for theoretical modelling principally to calculate neutron fluxes, and neutron- induced reaction rates in the reactors' experimental facilities. Additionally, there is a requirement to determine the limits for the safe operation of these reactors and to gain a better understanding of the physics of the systems. A reactor model generally has to be designed for a particular purpose and system for which it is to be used. The main object of the present work has been to model the temporal behaviour of the reactors operating up to full power and to develop the mathematics and associated computer programs to obtain maximum generality combined with economic computer usage, acceptable numerical accuracy and realism when compared with measurements. For these and other reasons, which will be discussed later, a 2-D diffusion model for neutronics and a lumped- parameter mathematical model for point-reactor kinetics for dynamics modelling has been developed.

Early neutronics models for the HMTRs were based on few-group 1-D and 2-D diffusion theory (Wade, 1966) and were successful in predicting integral quantities such as temperature coefficients and neutron generation times. 2-D multiregion calculations by Needham (1969, unpublished), based on the MuF'r- SOFOCATE code (Leslie, 1963 : Amster and Suarez, 1957), used one thermal group and control absorbers ; rigs and burnup were ignored. The model gave reasonable radial flux distributions but was unable to reproduce

12:7-.4.

the fast flux distributions. Some 3-D analysis of DIDO (Hopper, 1979), again ignoring absorbers and experimental rigs, has been performed to examine the feasibility of 3-D modelling for these reactors.

In the present paper, the results of a detailed neutronics study are presented for fuel-element modelling, and for a detailed 2-D cylindrical (r,z) whole-reactor model which allows for experimental rig configuration, control assemblies and fuel burnup. Predictions of radial and axial flux distributions, both thermal and fast, are compared with experiment.

Part II of the study is described in a subsequent paper which extends the modelling to include reactivity coefficient calculations for the fuel, coolant and moderator. Together with heat-transfer modelling, a complete thermal hydraulics model is developed and compared with experimental measurements.

2. THE HARWELL MTR~ A BRIEF

DESCRIPTION

The two HMTRs, DIDO and PLUTO, were started up in 1956 and 1957 respectively and, like many other materials testing reactors, are used extensively for irradiation experiments and neutron beam studies. DIDO and PLUTO are virtually identical reactors, fuelled by highly enriched U, cooled and moderated by D20 and currently operated at 25.5 MW(th).

The core of the DIDO reactor is a heavy-water/U lattice of 60.95 cm height and 85.98 cm equivalent cylindrical diameter, containing 25 fuel elements in rows of 4, 6, 5, 6 and 4 elements on a 15.24 cm lattice

327

328 T.D. BEVNON and M. JAVADI

pitch. The fuel elements are supported by the reactor A1 tank, 2 m in diameter, which is filled with heavy water which acts as coolant, moderator and reflector. To decrease the neutron leakage from the reactor, the space between the AI tank and reactor steel tank is filled with graphite 61 cm thick.

The reactor is mainly controlled by six Cd/Eu plates clad in stainless steel which swing in the gaps between rows of fuel elements: these plates are called coarse control arms (CCAs). A fine control responding system is also provided for the reactor by one fine control rod and three safety rods which operate vertically outside the core. These rods consist essentially of Cd absorber. There are also five retractable Co rigs which can be raised during the second half of the cycle to compensate for burnup of 235U.

In order to make maximum possible use of the

reactor for irradiation experiments, there are 28 vertical and 20 horizontal holes in the reactor. In addition to these, there is a central hole inside each fuel element in which an experiment may be loaded.

Figure ! is a cut-away drawing of the horizontal plane of the reactor, showing its main component parts.

DIDO operates using the MK5/7 fuel element which is an improved version of earlier designs, with a higher heat-transfer surface. There are generally two types of these elements, the unpoisoned fuel element, normally referred to as MK5/7A, and the poisoned fuel element, normally referred to as MK5/7B, MK5/7C and MK5/7D depending on the type of poisons used in these fuel elements. The poisons are used to compensate for both short-term reactivity changes due to Xe and Sm and long-term reactivity changes due to fuel burnup as the reactor cycle progresses.

AIITANK

5 V G R - 2

~ G C 2 Q 10 VGR-2 6~

A-I SR-1 2V6

2VL "6--7 2Dv; GRAPHITE 2V8 SR-3

2V2 )-1 D-E 2V9 ~'~ FCR

2Vl ~R2 E-1 E-4 ~ ~ /

O1 O

Fig. 1. Sketch of the horizontal section of the DIDO reactor at the centreline of the core, with fuel rods, control rods and experimental holes indicated. Regions A-E are fuel elements, O-labelled regions are D20 outlets, SR denotes safety rods, FCR is a fine control rod, and the remaining regions are experimental holes and neutron

beam tubes.

Neutronics analysis of DIDO and PLUTO 329

The MK5/7 element consists of four concentric fuelled tubes. Each fuel tube consists of 20/80 by weight U/AI alloy clad in AI and is of axial height 66.04 cm. The fuel tubes are tapered over the last 5.7 cm at each end in order to smooth the axial power profile. In order to maintain correct coolant flow over the fuel tubes, a rig or thimble is always fitted into the centre of the fuel elements. When there is no irradiation experiment a flux-scanning rig is fitted into the centre of the fuel element.

The dimensions and material contents of the fuel element are given in Table 1.

It is evident that the H M T R s are markedly spatially asymmetric and are consequently obvious candidates for a full (3-D) diffusion theory neutronics analysis. However, these reactors have a large number of experimental facilities in the core and in the heavy- water reflector, both of which are of equal interest and importance in a neutronics analysis. In practice, a 3-D diffusion calculation offers no significant advantages over carefully modelled 2-D calculations since the rigs and experimental holes cannot be adequately modelled by diffusion theory due to its inherent limitations. Moreover, the methods of geometrical and materials homogenizat ion required for diffusion theory appli- cations are essentially the same for both 2-D and 3-D calculations. Of course with a 3-D calculation it is easier to apply appropriate boundary conditions around rigs to model them in diffusion theory but since typical rigs are not black absorbers these boundary conditions are

Table 1. The geometrical and U specifications of the MK 5/7 fuel element

Regions and U content Radial dimensions (cm)

Inner Outer

Flux scan rig 2.540 2.699 23sU : 43.8 g IC 3.009 3.055 234U :0"52 g FS 3.055 3.126 236U:2.12g 23sU: ll.90g OC 3.126 3.177 a3sU:49.3 g IC 3.487 3.533 234U :0.58 g FS 3.533 3.604 236U:2.38g 23sU : 13.40 g OC 3.604 3.654 235U:55.8g IC 3.964 4.010 2~4U:0.66 g FS 4.010 4.081 2 3 6 U ; 2.70 g OC 4.081 4.132 2asU: 15.16 g 235U : 56.1 g IC 4.442 4.488 234U : 0.67 g FS 4,488 4.559 23sU:2.71 g 23Su: 15,24 g OC 4,559 4.609 Outer tube 4.850 4.992 D20 as a moderator

in the unit cell 4.992 8.598

IC = inner cladding; FS = fuel section; OC = outer cladding.

not accurately known. Finally, economic incentives strongly favour the use of 2-D modelling when one is primarily interested in predicting reactor performance parameters over a wide range of operating conditions and rig configurations.

Consequently, the whole-reactor model described in this paper is based on 2-D diffusion theory modelling in a cylindrical (r, z) geometry using the WlUS-E program (Askew et al., 1982). Space prevents a full discussion of the methodology developed and the interested reader should refer to the detailed report by Javadi (1983).

3. FUEL-ELEMENT MODELLING

To implement a diffusion theory model, a computat ional model of the MK5/7 type fuel element must be developed for a cell calculation which is the most widely used method for obtaining effective cross- sections in homogenized regions for global neutron- diffusion calculations. In this method we consider each fuel element as located in an infinite lattice of identical cells. The local flux distribution within a cell is then calculated using a collision probability method and used to obtain spatially weighted homogenized cross- sections.

The resulting theoretical model can also be used for the following calculations :

(i) new developments in the performance of the fuel elements ;

(ii) reaction rate and y-ray heating calculations in the rigs ;

(iii) simulation of burnup calculations for use in global critical-reactor modelling.

The fuel elements in HMTRs, like many other reactors, are arranged in a periodic manner, so that at least the central part of the core can be regarded as being made up of a number of identical unit cells. To simplify the flux calculations in a unit cell, it is first assumed that the cell is infinitely long. For a unit cell with a cylindrical geometry the actual square pitch boundary can be replaced by a cylindrical boundary so that the flux in the unit cell becomes a function of one space co-ordinate (i.e. radial r), and a 1-D geometry model of the unit cell can be used.

In order to choose the opt imum number of regions in the unit cell, a series of flux calculations have been made using an exact collision probability method ( W F L U R I G ) and an approximate collision probability method (WPERS), both available in WlMS-E. The flux calculations, obtained using the full 69 energy groups in WIMS-E, have been used to compute the infinite medium multiplication constant (ko~) for the cell. The object of this exercise is to determine an opt imum spatial zone

330 T. D. BEYNON and M. JAVADI

Table 2. Region optimization of the unit cell

No. of No. of Method of energy spatial No. of CPU time

solution groups groups meshes (min) k® (k +~- k~)/k +~ x 100

WFLURIG 69 25 40 7.42 1.8701 -- (=k~)

WPERS 69 25 40 1.56 1.8727 0.14 WPERS 69 21 40 1.50 1.8726" 0.13" WPERS 69 13 16 0.4 1.8707" 0.03 * WPERS 69 13 15 0.38 1.8706" 0.027* WPERS 69 13 13 0.33 1.8704 0.016"

*Indicates fuel/cladding homogenization.

configuration for subsequent use with WPERS. The results are listed in Table 2 from which the following two important points emerge.

(a) Treating each of the four fuel tubes as a homogeneous mixture of U and AI will give better results for WPERS than treating the cladding as a separate region.

(b) Despite the fact that it is generally true that by increasing the number of meshes one obtains better results, this is not the case for the WPERS module. Here the best result has been obtained by choosing quite coarse meshes in each region.

The reason for these two anomalies lies in the kind of approximation used in the WPERS collision proba- bility method to accelerate the calculation (Javadi, 1983). Table 3 summarizes the geometrical dimensions and material properties of the final theoretical model of the unit cell used in all our subsequent reactor modelling.

4. ENERGY G R O U P OPTIMIZATION

Performing whole-reactor calculations with the 69 neutron energy groups available in WlMS-E can prove expensive in computer resources and a condensation to a few-group set is desirable. The arbitrariness of this procedure can be reduced by observing the following constraints, relevant to this particular reactor study, to produce an optimum number of energy groups as a subset of the WIMS-E library, using the same neutron energy boundaries.

Firstly, the level of Pu concentrations in both DIDO and PLUTO is sufficiently low that the groups in WlMS-E which are chosen to ensure adequate defin- ition of neutron capture in the thermal resonance in 2 3 9 p u (0 .29 eV) and 24"°Pu (1 .0 eV) can be combined. Secondly, as a result of the neutron absorption in many of the rigs and control absorbers, together with the high leakage rates associated with the experimental beam

holes, large thermal flux gradients occur in many parts of the core and reflector. This, together with the requirement for an adequate treatment of the D up- scattering, demands that a reasonable number of thermal groups be used. The WIMS-E boundary at 0.625 is included, below which Cd absorbers are assumed black.

Finally, the importance of fast neutrons for the irradiation of certain rigs, and the importance of distinguishing the prompt- and delayed-neutron fission spectra for calculating an effective delayed- neutron fraction for kinetics modelling, demands a reasonable fast-group representation. The WIMS-E boundary at 821 keV is particularly convenient since it coincides with earlier work (Beynon et al., 1981) on defining effective fast-neutron diffusion coeffÉcients in D20. Moreover, the low content of 23au in the highly enriched fuel element of the HMTRs allows all the groups in the 238U-resonance region (9.12 keV-4.0 eV) to be combined into one group for global reactor calculations.

Having chosen group boundaries at 821 keV, 9.12 keV, 4.00 eV and 0.625 eV on the basis of the above general points, the choice of the number of groups and the other group boundaries is still somewhat arbitrary, especially in the thermal region. Because the number of groups and group structure required depends on the adequacy of the spectrum calculations that are used to condense the cross-sections, a I-D ( r ,~) reactor model whose spectrum was used to prepare the few group cross-sections--has been used to select a compatible few-group structure from the 69 energy groups. The final group optimization has been based on a number of key parameters, namely keff, 238U(n,y)/235U(n,f) reaction-rate ratios and the core/D20 reflector ratios for 59Co(n,7)6°Co and 58Ni(n,p)58Co. These parameters are first evaluated using the full 69-group WIMS-E set with the (r, oo) model for the reactor and the unit-cell model for the fuel element. Using these for a comparison basis, a 7-group

Neut ron ics analysis of D I D O and P L U T O

Table 3. The geometry and material specifications of the theoretical unit cell with the MK5/7 fuel element

331

Outer Average radius No. of temperature Number densities

Region (cm) meshes (°C) ( x 10- ~4)

I 2.54 2 60 H: 1.5742 x 10 -4 D: 6.5432 × 10 -2 O : 3.2795 × 10- 2

2 2.699 1 61 AI : 6.0238 x 10- z 3 3.009 1 61 H: 1.5733 x 10 -4

D : 6.5396 x 10- 2 O:3.2777 x 10 -2

4 3.177 1 100 AI: 5.9298 x 10 -2 234U : 7.4915 × 10 -6 235U:6.2929x 10 4 238U: 1.6883 x 10 -4 2 3 6 U : 3.0296 x l0 5

5 3.487 1 65 H : 1.5700 x 10 -4 D:6.5258x 10 2 O:3.2707x 10 2

6 3.654 1 100 AI : 5.9322 × 10- 2 234U : 7.3045 × 10 -6 235U:6.1358 x 10 4 238U : 1.6462 × 10 -4 236U : 2.9540 x 10- 5

7 3.964 1 65 The same as region 5 8 4.132 1 100 AI: 5.9323 x 10 2

234U : 7.2924 × 10 -~ ~35U:6.1256 x 10 -4 238U: 1.6434 × 10 -4 23~U:2.9490x l0 5

9 4.442 1 65 The same as region 5 l0 4.609 1 100 A1:5.9415 x 10 -2

234U :6.5580 X 10 -6 2 3 5 U : 5.5087 x 10 4 23su : 1.4779 × lO 4 236U:2.6521 x 10 -5

11 4.850 1 62 H : 1.5724 x 10 -4 D: 6.5460 x 10 -2 O:3.2759 x 10 -2

12 4.992 1 70 A1:6.0137 x 10 -2 13 8.598 2 70 H: 1.5656x 10 4

D : 6.5078 x 10- 2 O:3.2617 x 10 -2

Table 4. Upper energy boundaries of the optimized 15- group set

Group No. Upper energy boundary (eV)

1 10 TM

2 8.21 x 105. 3 6.734 x 10 a 4 9.118 x 103* 5 367.26 6 9.877 7 4.0* 8 1.30 9 0.625*

10 0.320 11 0.250* 12 0.180 13 0.050* 14 0.030 15 0.005

*upper values for the seven-group set.

s t r u c t u r e fo r (r, z) w h o l e - r e a c t o r m o d e l l i n g a n d a 1 5-

g r o u p s t r u c t u r e fo r fuel cell c a l c u l a t i o n s w a s o b t a i n e d ,

w i t h t h e e n e r g y b o u n d a r i e s s h o w n in T a b l e 4.

5. T H E T H E O R E T I C A L (r-z) M A P

O F T H E D I D O R E A C T O R

T h e ( r , z ) t h e o r e t i c a l m o d e l o f t h e D I D O r e a c t o r

c o n s i s t i n g o f 11 r e g i o n s is d e p i c t e d in F ig . 2. T h e r e g i o n s

h a v e b e e n c h o s e n a s a r e s u l t o f a s t u d y o f t h e

e n g i n e e r i n g d r a w i n g s fo r D I D O a n d its r i g l o a d i n g s . A

r e g i o n is d e f i n e d as a h o m o g e n i z e d m i x t u r e o f m a t e r i a l s

in t h a t v o l u m e o f t h e r e a c t o r . T h e h o m o g e n i z a t i o n o f

m a t e r i a l s in t h e c o r e ( r e g i o n 1 in F ig . 2) is b a s e d o n a

f l u x - v o l u m e a v e r a g i n g ( u s i n g t h e c o l l i s i o n p r o b a b i l i t y

m e t h o d ) , a n d in t h e r e f l e c t o r s is b a s e d o n p u r e v o l u m e

332 T . D . BEYNON and M. JAVADI

-_ q _ - = r z . , . ~ - t = ~ r 3

2O

19

5 18

17 __ -- --

16

15

14

1

14

15

21

2 2

2 3

I I I I I I I I I I I I I I I I 1 I I I I I I I I I I I I I I 12 13 14

28

27

26 I

29

I I

t 1 I I I 16

10

i i I I I I 1 I I I I I

I I 2

I I I I I I I I I I I I 17 I e

I I J I

I I I

I

I

I I I

I 19

30 1 (3

24

25

31

11

3 4

I I I I I I I I I I i 9 1 I I

I I I I

I I

I 132 133 t I I I I I I

I I I I I I 141 I I I I I I I I l ie 113 I I I I I I I I I I I I I I I

I I I 1 1 1 I I

I I I I

I I I I I I I I I I I I I I I t I I I 35 136

Fig. 2. The (r, z) model of the DIDO reactor, consisting of 11 main regions ( ) and a number of subregions (- --). (The material and geometrical specifications are given in Table 5.)

Neutronics analysis of D I D O and P L U T O

Table 5. The material and geometrical specifications of the (r, z) reactor model

Region 2 Region 3 Region 4 T: 343 K T : 343 K T : 480 K AI: 8.0982 x 10 -4 AI : 3.1917 x 10-2 C : 7.3000 x 10-2 D: 6.2210 x 10 -2 H : 1.4908 x 10 -4 O:3.1180 x 10 -2

Region 5 Region 6 Region 7 T:343 K T:343 K T:343 K A1 : 3.3326 × 1 0 - 3 A1:6.9110 × 1 0 - 3 AI : 1.7691 x 10 -a D:6.1650 x 10 -2 D:5.7011 x I0 -2 D:4.7693 × 10 -3 H : 1.4832 x 10 -4 H : 1.3716 x 10 -4 H : 1.1474 x 10- 5 O : 3.0899 x 10- 2 O : 2.8574 x 10- 2 O : 2.3904 x 10- 3

C : 5.9359 × 10- 2

Region 8 Region 9 Region I0 T:343 K T:480 K T:343 K AI : 7.8536 x 10- 5 C : 7.5340 × 10- 2 AI : 1.3754 x 10- 3 D: 6.4780 x 10-2 D :6.2521 x 10-2 H : 1.5585 x 10 -4 H : 1.5041 × 10 -4 0:3.2468 × 10 -2 O:3.1336 x 10 -2

Region 11 Dimensions (cm) T = 4 8 0 K hi = 80.19 , h2 = 5.72 C:7.5340 x 10 -2 h 3 = 49.53, h 4 = 35.00

h 5 = 8.06, r, = 42.99 r 2 = 57.34, r 3 = 60.33

t = 2.54, h 6 = 62.86

All number densities are in units of atoms, barn t cm- t.

333

ave rag ing . T h e ma te r i a l s , n u m b e r dens i t i es a n d

d i m e n s i o n s o f t he ref lector r eg i ons a re g iven in T a b l e 5.

H a v i n g ca l cu l a t ed t he s p e c t r u m in each r eg ion one

can i m p r o v e t he m o d e l l i n g by d i v i d i ng each r eg ion in to

a few s u b r e g i o n s a c c o r d i n g to h o w the s p e c t r u m is

c h a n g i n g in t h a t reg ion . A s u b r e g i o n is def ined as a pa r t

o f a r eg ion in w h i c h the g r o u p - a v e r a g e d c o n s t a n t s (i.e.

c ro s s - s ec t i ons a n d d i f fus ion coefficients) r equ i r ed to

so lve the d i f fus ion e q u a t i o n are the s ame . I t s h o u l d be

n o t e d t h a t by th i s de f in i t ion a r eg ion w h i c h cons i s t s o f a

h o m o g e n i z e d m i x t u r e o f m a t e r i a l s m a y still be d iv ided

in to different r eg ions ( sub reg ions ) d u e to dif ferences in

s p e c t r a u sed to co l l apse the l ib ra ry c ros s - sec t i ons to

f e w - g r o u p c ross - sec t ions . T h e n u m b e r o f s u b r e g i o n s

a n d thei r b o u n d a r y l ines h a v e been d e t e r m i n e d by

d i rec t o b s e r v a t i o n w h i c h def ines s u b r e g i o n s w h e r e the

s p e c t r u m is c h a n g i n g rapid ly , a n d by i t e ra t ive

t echn iques . T h e b o u n d a r i e s o f s u c h s u b r e g i o n s a re

s h o w n by d a s h e d l ines in Fig. 2. H e n c e f o r t h all s u b r e g i o n s will be referred to as reg ions .

6. GENERATION OF GROUP-AVERAGED

CONSTANTS

T o g e n e r a t e the g r o u p - a v e r a g e d c o n s t a n t s for the

s even e n e r g y g r o u p s de f ined in Sec t ion 4, t h ree different

m e t h o d s h a v e been inves t iga ted . T h e s e a re as fol lows.

Method I

T h e spa t i a l ly s m e a r e d s p e c t r u m ca l cu l a t ed in 69

ene rgy g r o u p s u s i n g t he col l i s ion p r o b a b i l i t y m e t h o d in

a n ' a v e r a g e fuel e l e m e n t ' c a n be u sed to g e n e r a t e the

g r o u p - a v e r a g e d c o n s t a n t s for all t he r eg ions in the

mode l l ing . B e c a u s e t he s p e c t r u m in the core is very

different f r o m the ref lectors th is m e t h o d is n o t expec ted

to give g o o d resu l t s if u sed to g e n e r a t e g r o u p - a v e r a g e d

c o n s t a n t s for the ref lectors .

Method 2

(i) T h e s p e c t r a c a l cu l a t ed in 69 e n e r g y g r o u p s in a

1 - D cyl indr ica l (r, oo) w h o l e - r e a c t o r m o d e l u s i n g

the col l i s ion p robab i l i t y m e t h o d (i.e. W P E R S )

c a n be u sed to g e n e r a t e t he g r o u p - a v e r a g e d

c o n s t a n t s for the core, t he rad ia l h e a v y - w a t e r

a n d g r a p h i t e ref lectors a n d t he r eac to r A1 t a n k

( reg ions 1-15 in Fig. 2).

(ii) I n a s imi la r f a s h i o n to C a s e (i), two 1-D axial (z, oo) w h o l e - r e a c t o r m o d e l s c a n be u s e d to

g e n e r a t e t he g r o u p - a v e r a g e d c o n s t a n t s for t o p

a n d b o t t o m ref lec tors ( reg ions 16-20, a n d 21 -25

in Fig. 2)

G e n e r a t i o n of the g r o u p - a v e r a g e d c o n s t a n t s for

r eg ions 2 6 - 3 6 is pa r t i cu l a r ly difficult, b e c a u s e no 1-D

w h o l e - r e a c t o r m o d e l c an be set up for t hese regions .

F o r t u n a t e l y , it h a s been f o u n d t ha t the use o f different

334 T.D. BEYNON and M. JAVADI

spectra to generate the group-averaged constants for these regions has a negligible effect on the overall reactor modelling and therefore the spectrum from Case (i) can be used.

Method 3

Due to the inherent limitations and approximations of the (r, z) reactor model, especially in modelling the CCAs, the experimental rigs and the 6H and 10H experimental holes, it may be acceptable to use more approximate methods to generate the group-averaged constants than Method 2. The idea in this method is to use the spectra calculated in Case (i) of Method 2 to generate the group constants for all the regions.

The difference between the reactivity of the system calculated by this method and Method 2 is about 0.07%, which is not significant for the HMTRs which have about 24% excess reactivity. However, Method 2 has been adopted for the subsequent whole-reactor modelling.

It is important to correct for the effects of anisotropic scattering in D when calculating the diffusion coefficients for the regions containing heavy water. The method and data described by Beynon et al. (1981) have been used for this purpose.

The overall schematic flow diagram for the global reactor modelling is shown in Fig. 3.

7. CRITICAL-REACTOR MODELLING

The final reactor model has been designed to include the reactivity effects associated with the coarse control arms, experimental rigs and fuel and fission-product burnup.

7.1. Experimental rig modelling

An accurate representation of the various rigs in the HMTRs would probably require a Monte Carlo whole- reactor model. A simpler method has been used, depending on whether the particular rig is in the core, or in the heavy-water reflector.

This method simply uses a 1°B concentration based on the calculated worth of the B burnable poison in MK5/7B cores (using the (r, z) model) to give values for the B concentration to be added to a 2.54 cm radius central facility for the required reactivity absorbed by the rigs. This method was also used to calculate the 1 o B concentration as a burnable poison for end-of-cycle (EOC) and beginning-of-cycle (BOC) modelling to be added to the outer tube of a MK5/7 fuel element.

In the heavy-water reflector the rigs were homogenized by simply volume averaging to absorb about the correct fraction of the total reactivity, typically 2~o.

7.2. CCA representation

The CCAs in normal operation absorb less than 2% at BOC, and less than 0.5% at EOC, of the 24% excess reactivity.

The major inadequacy of the (r, z) model is that the semaphore arm geometry of the CCAs cannot be represented explicitly, and other approximate methods should be adopted. The methods tried were using annuli of black absorbers, a homogenized mixture of absorbers in the top reflector and a homogeneous mixture of absorbers in annular form in the top reflector and in the core. Of these methods the third proved to be more accurate in representing the CCAs than the other two. The inner and outer radius and the height of the annulus in the third method were determined from the position of the CCAs with respect to the core centre (Javadi, 1983).

7.3. Burnup modelling of an average fuel element

The most accurate way of dealing with the burnup effect is to use a whole-reactor calculation in which the irradiation history of each fuel element is followed interactively with all other fuel elements within the reactor. This is obviously very expensive in computer time and storage and consequently this problem has been overcome using an approximate synthesis approach. In this method each lattice cell arrangement is considered separately but with some attempt to take into account the effects of the environment by applying leakage terms either in the form of input bucklings or by using a multicell option. In this latter case a small number of different fuel cells are linked together in a simulation of the reactor or a part of the reactor. The first option has been used in our burnup modelling.

The HMTRs operate on a 28-day cycle consisting of 24 days at full power (25.5 MW), followed by 4 days' shutdown. On average each fuel element spends three cycles in the reactor, those near the core centre averaging less and those near the edge more than three cycles. Examination of the reactor operating records reveals that over many cycles the mean burnup at EOC of fuel at all positions across the core (estimated from the coolant temperature differences) is constant for the PLUTO reactor at 31.5% and has the values 35.6, 33.7, 30.2 and 30.9% for four regions in the core of the DIDO reactor. This implies that there is no need to model fuel elements individually and the 'average fuel element' representation will be an excellent burnup model for the 'average PLUTO reactor' and an acceptable model for the 'average DIDO reactor'.

Having set up an average fuel element with zero burnup, the WIMS-E Module W-BRNUP (Gubbins and Sherwin, 1973) was used to simulate the fuel and fission-

Neutronics analysis of DIDO and PLUTO 335

WHEAD ]

WPRES and WRES I

Input material and geometrical data for the unit cell

Do shielded resonance calculations

SET-UP REFLECTOR MATERIAL AND GEOMETRY FOR 1-D MODEL OF THE REACTOR

Use spectrum I from 1-D model I of the reactor I to generate I few-group cross- sections for CCAs

D[ WPERS

1 WSMEAR

t WMERGE

l WMERGE

Calculate fluxes in di f ferent unit cells in the core

Perform a spatial con- densation of cross- section data and f lux distr ibution

Merge the core smeared data together to form one set of data ( i .e. one interface)

Merge the core and the ref lector data together to form one data set

WPERS

WCOND

CORRECT THE DIFFUSION COEFFICIENTS FOR THE EFFECT OF ANISOTROPIC

SCATTERING OF NEUTRONS IN D20

b[ WSNAP

CALCULATE RADIAL AND AXIAL BUCKLINGS

Calculate fluxes in 1-D model of the reactor, and i f i t is necessary make the spectrum cr i t ica l using buckling search option

Use spectrum from I-D model of the reactor to generate few-group cross- sections, to be used in r-z diffusion calculations

Calculate normal and adjoint fluxes

Iterate the calculations as many times as necessary unti l there is insigni- f icant change in the reactor parameters ( i .e.

kerr)'

Fig. 3. Computational flow diagram for the (r, z) whole-reactor modelling.

product burnup. The burnup calculations were carried out for three cycles each consisting of 24 days at full power, followed by 4 days shutdown (4 W) for cycles 1 and 2. The calculated mean burnup at EOC for the DIDO reactor was 31.8~o compared with the experimental value of 32.6~o, the mean of the four values quoted above. The burnup data at the BOC and EOC for each cycle was averaged and used to set up average fuel elements for a BOC and EOC whole-reactor

model. These averaged fuel elements were then used to start the critical whole-reactor model.

8. CALCULATIONS AND MEASUREMENTS OF FLUXES

In order to assess the performance of the (r,z) neutron-diffusion models developed, calculations of radial and axial scalar flux distributions have been

ANE 12:7-B

336 T .D. BEYNON and M. JAVAD!

,~ ~CORE =i= D2 0 -.--i- GRAPHITE -! I 0

Group Energy ~ ronge

~ ~ 1 9.118 keV < E -.< 10 MeV o \3 2 1.15 eV< E-. < 9.118 keY c 3 O~ < E~ < 1.15 eV

5 2 u')

(M

J

Z 0 c r

h l Z 0 50 100 150

DISTANCE FROM CORE CENTRE ( cm)

Fig. 4. Radial scalar fluxes in three energy groups, evaluated at the level of the core centreline.

made and compared with measurements. Typical radial scalar fluxes in the three indicated energy groups are shown in Fig. 4 plotted at the level of the core centreline. However, these flux distributions cannot be compared directly with available measurements since in the HMTRs the 'Westcott flux' (measured using 59Co(n, ?)6°Co detectors) and the 'effective fission flux' (measured using 5SNi(n,p)SSCo detectors) are used to ~-- describe' thermal ' and 'fast' fluxes, respectively. For this '~ reason the 'edit' option of the W1MS-E code is used to 'E express the scalar neutron fluxes calculated by the (r, z) =0 diffusion model in terms of 'Westcott flux' and 'effective fission flux'. All results have been normalized to 25.5 z~

t ~ M W reactor power.

txl The following important points should be taken into

account in making a correct interpretation of the × measured fluxes. D d

i,

1. In the HMTRs there are generally large variations of neutron flux distributions due to changes of fuel, experiments and CCA positions. Therefore, any

Z measurement of neutron flux can be compared

J with calculation only if the modelling has been < done for the particular reactor loading at that time. a:

w Although this modelling can in principle be x achieved, the (r,z) model which has been developed corresponds to an average reactor loading. For this reason it is preferable to use the measured neutron flux values which have been obtained from the analysis o/ 'many measurements. Fig. This is in fact a common practice at Harwell to

estimate the flux distributions in the reactors for design work, rig loadings etc.

2. Since the (r,z) diffusion model provides fluxes which are radially smeared and because the reactors are very asymmetric, all the measure- ments that are at the same radial distance from the core centreline have been averaged to make them suitable for comparison with the calculations.

3. Corrections should be made to the calculated fluxes representing measurements made in vertical experimental facilities in the graphite reflectors to account for the fact that, since they were measured at the centre of a voided thimble, a considerable amount of streaming will result in a reduced thermal flux.

These are typical examples of the care required in making a comparison between the calculations and experimental measurements. With full allowance for the points mentioned above, the comparison of theoretical and measured radial and axial flux distributions are given in Figs 5-8. All the experimental values have been normalized to the theoretical centre flux in each case.

As can be seen from Fig. 5, experiment and theory agree well for radially distributed thermal fluxes. The agreement between the theory and experiment for the

6 5 - - CORE- -~ - D20 4 I

3

2 X ----... X X-~x "--x. x _x / ~ < ~

1014 X

=!= GRAPHITE -i I i

x Experimentol - - Theoreticol

X I 0 ~3

1012 I I 0 50 100 150

DISTANCE FROM CORE CENTRE (cm)

5. Measured and calculated thermal flux radial distributions at the level of the core centreline.

Neutronics analysis of DIDO and PLUTO 337

-,,--- CORE ~ ! ",, D20 - !..,,-- GRAF' HI T E - i 1014 =-~-~ x ..~.~.~ I I [

X~,,

X Exper iment

I I013 - - Theoret ical (with ~n modi f ied d i f fus ion

(~E ~ coef f ic ients) o - - - Theoreticel (with conventionol

di f fusion coefficients)

x 1012 t I

X :)j 1011 b- \\ X

Z 0 ~

W ~'Zl 1010 \ \ \ \ \ \ \ \

if) h

10 9 I ~ I 50 100 150

D I S T A N C E F R O M C O R E C E N T R E ( cm)

Fig. 6. Measured and calculated fast flux radial distributions at the level of the core centreline.

fast fluxes derived using conventional diffusion coefficients is poor in the D 2 0 and graphite reflector, but good in the core (cf. Fig. 6). With diffusion

-coefficients modified for anisotropic scattering in D 2 0 (Beynon et al., 1981) the agreement between theory and

- - Theoreticol (h = 61 cm, 0=22 *) X 1.3 X Experiment ( 8 = 22% error + 2%)

( h is the core height) _J h %1 ~- × ~

I - X ~ x 0.7 w ~x

W 0.5

,~ 0.3 J W r r 0.1 I I I I I I I I

- 4 0 - 3 0 - 2 0 10 0 10 20 30 40 ,50

AXIAL DISTANCE FROM CORE CENTRE (cm)

Fig. 7. Axial distribution of the measured and calculated thermal (Westcott) fluxes. Each set has been normalized to unity at the core centre and corresponds to a positioning of the

CCAs at an angle 0 = 22 ° to the horizontal.

1.2

X 1.0 / LL 0.8 }--

~ 0.6 b_

W 0.4 >

<[ 0.2 ..-J W rY I I i

- 4 0 - 3 0 - 2 0 -10

- - Theoretical ( 8 = 2 2 * ) x Experiment (8=22* ,e r ro r_+30%)

x f X ~

I I I I I 0 10 20 30 40

DISTANCE FROM CORE CENTRE (cm)

Fig. 8. Axial distribution of the measured and calculated fast fluxes. Each set has been normalized to unity at the core centre and corresponds to a positioning of the CCAs at an angle

0 = 22 ° to the horizontal.

experiment for the fast fluxes in the D20 is expected to improve, as is illustrated in Fig. 6.

The axial thermal and fast flux distributions are given in Figs 7 and 8, respectively. The axial thermal fluxes (and to some extent the fast fluxes) are affected by the pos i t ion ing .of the CCAs. These results demonstrate that the general agreement between the theory and experiment is good.

9. DISCUSSION AND CONCLUSIONS

The analysis described in this paper has aimed at producing a neutronics model of the H M T R s , D I D O and P L U T O . The study has been part of a larger program to produce a whole-reactor thermal hydraulics model of both the H M T R s which has been supported by a parallel experimental program. An additional aim has been to provide a reactor model which can be used with an acceptable degree of confidence to analyse and predict properties of various experiments and rig irradiations in these reactors.

The complexi ty of the H M T R s has dictated a pol icy of compromise, between computing costs on the one hand and the demands of detailed physics model l ing on the other. The comparisons made with available measurements of fuel burnup and flux distributions have certainly justified this approach. At the same time, the model l ing has indicated that perhaps future flux measurements in the H M T R s should have better energy resolution than has been the practice, in order to test the present model l ing more stringently.

Acknowled(tements M. Javadi is indebted to the UKAEA for financial support over the period of this study. Both authors acknowledge their debt to various staff at AERE, Harwell, particularly to V. S. Crocker for his encouragement, and to K. P. Nicholson, G. Constantine and N. P. Taylor for many useful discussions.

338 T .D. BEYNON and M. JAVAD1

REFERENCES

Amster H. and Suarez R. (1957) Westinghouse Report No. WAPD-TM-39.

Askew J. R. et al. (1982) UKAEA Report No. AEEW-R1315 and references therein.

Beynon T. D., Hopper E. D. A. and Azhar A. (1981) A T K E - K 37, 276.

Gubbins M. E. and Sherwin J. (1973) UKAEA Report No. AEEW-R902.

Hopper E. D. A. (1979) J. Br. nucl. Energy Soc. 19, 215. Javadi M. (1983) Ph.D. Thesis, Univ. of Birmingham, U.K. Leslie D. C. (1963) J. nucl. Energy 17, 293. Wade B. O. (1966) Proc. IAEA Study Gp on Research Reactor

Experimental Techniques, p. 31. Editura Acaderniei Republicii Socialiste Romania, Bucharest.