0 UNITED KINGDOM A T O M l C ENERGY A U T H O R I T Y .

Enqul r l es about copy r l ght and reproduct Ion should be addressed ta the W ~ n f r l t h Secretar iat , i t o m ~ c Energy Establ lsnment. Wlnfrith

OORCHESTER. Oorset. England. -

UNCLASSIFIED

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MONTE-CARLO DEPLETION CALCULATIONS IN A SLAB

A F COURSE M J HALSALL J L HUTTON

This report describes a study in which a Monte-Carlo (MC) technique has been applied to the depletion of a!l-D slab, and the results compared with those from the equivalent deterministic calculation. Estimates of the uncertainties on the MC results

- have been obtained.

The calculations included cases where the distribution of k-infinityin the slab was made to vary linearly and quadratically with irradiation. One calculation simulated the withdrawal of a control rod which was partially inserted initially. As a rather extreme example of rod movement a calculation was completed in which the rod was partially inserted or withdrawn at alternate depletion steps for two thirds of the time and then withdrawn until the end of life.

Some smoothing of source distributions has been achieved by the use of orthogonal polynomials, although this is intended as no more than an indication of the recibction in uncertainty which may be possible.

Plots of source distributions at several times during the depletion have been produced. Further plots illustrate the magni- tude ofthe terms of the orthogonal polynomial and so suggest how many are required for an adequate fit.

actor Phy AEE Winf rith

May 1983

W10678

.sics Division

'CONTENTS

SUMMARY

INTRODUCTION

BASIC MODEL PROBLEM AND DETERMINISTIC SOLUTION

2.1 The Model Problem 2.2 The Deterministic Solution

MONTE CARLO MODEL

3.1 Tracking 3.2 Depletion

CALCULATIONS

RESULTS

5.1 Computing Times 5.2 Reactivities 5.3 Irradiation 5.4 Source Distributions

SMOOTHING

EXTRAPOLATION TO A MORE REALISTIC DEPLETION MC

CONCLUSIONS~

REFERENCES

TABLE 1 - CPU Times TABLE 2 - Preliminarv Monte-Carlo Burn-UD Calculations *

a ~eactivities

TABLE 3 - End of Life Reactivity Differences TABLE 4 - Preliminary Monte-Carlo Burn-up Calculations

Irradiations

TABLE 5 - Percentage Mean Difference Between MC Irradiation 10 Estimates and Exact Values

- TABLE 6 - Effects of Smoothing on Source Uncertainty 13

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(ii)

APPENDIX 1

Source Distributions

APPENDIX 2

Orthogonal Polynomials

APPENDIX 3

Smoothed Monte-Carlo and Exact Source Plots

Page

16

33

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UNCLASSIFIED

1. INTRQDUCTION

An important aspect of the analysts of a reactor core is the determination of its end-of-life 5rradiation or cycle length. The usual method of estimatimg this 2s Based on determintstic depletion calculations for the dffferent assembly typeswithin the reactor. It is well known that in Yonte Carlo calculations of a single integral quantity, for example k-effective, tlie cost of the calculation for a given accuracy of result is not strongly dependent on the geometrical complexity of the problem or on the number of eneray groups of the nuclear data. It is therefore at least plausible to suppose that the addition of time as an extra dimension would not carry too heavy a cost penalty, and that it may therefore be feasible to use a Monte Carlo method to do whole core depletion calculations in order to determine cycle length with considerable accuracy. This memorandum describes the use of a Monte Carlo (MC) method to simulate the deple- tion of a simple fuel slab and so test the hypothesis. For such a model it is possible to compare the results of a MC calculation with those from an exact solution, and thus assess the accuracy of its estimates of end-of-life reactivity and irradiation.

Four different model problems have been simulated by determin- istic and MC calculations for the one dimensional slab. All of these are intended to demonstrate the ability of a MC method to cope with changes in material composition as a result of the burn-up of fuel and absorber or because of rod movement.

In the first two cases there is no rod movement, the only changes being the result of fuel or absorber depletion. One of these simply models the burn-up of the fuel by making the local multiplication decrease linearly with irradiation. In this case it is anticipated that local statistical errors in power distribution will tend to

- 'burn out'. In other words, a local high flux at one time step will give too high an irradiation at that position and this will lead to a lower value of local multiplication and a compensation in the source strength for the next time step.

A more complicated case simulates the effect on local multipli- cation of the simultaneous depletion of fuel and a heavy absorber such as gadolinium. Initially the presence of the absorber holds down the multiplication. As the absorber burns out the multiplication rises to a maximum until the continuing depletion of the fuel reduces it again. This has been modelled by making local multiplication a quadratic function of irradiation which reaches its maximum value about a third of the way through the calculation. The interest in this model is to see whether the increasing value of multiplication has a significant destabilising influence on the MC calculation. A . local statistical error in power distribution is likely to be magnified during that part of the depletion when multiplication is increasing with irradiation.

The third case is a rather exteme model of large and rapid rod oscillations and is therefore in one sense a severe test of the method although it might be anticipated that the integral effects of the motion will be well predicted. The fourth case contains both the quadratic dependence of local multiplication on irradiation and a steady rod withdrawal as burn-up process, which is intended to demon- strate a more realistic reactor behaviour.

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These four cases are used to test the ability of the PIC method to produce solutions that are in acceptable agreement with an 'exact' solution and to demonstrate that the cost of obtaining such solutions is comparable with the cost of obtaining a single start-of-life eigenvalue.

Plots of the source distribution at several depletion steps are presented for both the exact and MC calculations, in order to illustrate how well the MC simulation has responded to changes in local multiplication and rod movement of the model p:roblems.

2. B A S I C MODEL PROBLEM AND D E T E R W I N I S T I C S D L U T I O N

2.1 The Model Problem

The problem studied consists of a uniform fuel 'paste, inTinite in the y and z direction, finite and unreflected in the x direction, and divided into 20 equal meshes. The neutrons are all assumed to have the same velocity, the scattering cross section is zero, the absorption cross section Za and diffusion coefficient D are uniformly

-~

constant and the fission yield cross section vZ+ varies in a speci- - fied manner. A particular assumption made for ease of computation is that all source neutrons start from the centre of the mesh in which they are produced. This assumption is made consistently in both the deterministic and the MC calculations.

The scalar flux @(x) therefode satisfies the equation:

where on the RHS of this equation the source term in mesh i is strictly

and hence

2 where K = Za/D, and we assume a value of K = 10/E where I! is the height of the slab,

k(x) = vZf(x)/Za the local multiplication and depends on x through the specified variations in vXf

and X is the eipenvalue.

This can be solved as an integral equation:

A(x) = X J G(x',x) A(x') dx' where A(x) = Xa4(x)

K -K/x-x' / and G(x',x) = k(x') 7 e

from the standard expression for the diffusion kernel from an infinite plane source

2.2 The Deterministic Solution

The deterministic solution is obtained by evaluating Gk',x) bearing in mind the assumption that neutrons start from the centre of the mesh in which they are born. This removes any spatial vari- ation of A(x') from the integral so we need only find

for appropriate ranges of x.

For mesh point i the probability that a source neutron will be absorbed in the same mesh point is

and the probability of absorption in mesh point j is

H Where x = (j-i-%lE i

An initial flat guess of the absorption distribution Ai followed

by a calculations of tke subsequent distribution A using these j

values of Pii constitutes the first iteration. Repeated iterations - without acce~eration converge rapidly and a convergence criterion of 10-6 was adopted for the 'exact' deterministic solution.

The simulation of moving control rods and depletion is achieved simply by varying vEf(x) in accordance with a simple recipe based on

time and number of absorptions in each mesh (see section 4 ) . The above deterministic calculation was repeated at each time step.

a 3. MONTE CARLO MODEL

3.1 Tracking

Referring to the integral form of the diffusion equation in Section 2, the Green function G(x1,x) can be represented by a Monte Carlo process (i.e. by discrete transitions from x! to x.). Then the integral equation becomes I' 3

A(x.) = A E 3

Cx! ,x.) A(x;) tracks i Gmc 1 3

where Gmc (x;,x.) = k(x!) P(X;,X.) 1 1 1 .

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P is the probability of a particle starting at x! ending at x and 1 j

is randomly selected from the probability distribution.

The track length is given by

1 x . . = jx.-x! 1 = -- log jzI K Z a random number in the range O<Z<1 11 3 1

For simplicity we assume discretization to N mesh ce:ntre points x! 1'

that is, we assume that all neutrons start at the ce:ntre of the mesh in which they are born. Then the mesh j of the next absorption is given by

i is the mesh of the absorption in the last generation.

m is the sign of Z i.e. m = +1 Z>O m = -1 Z<O

and d is the highest integer less than

H is the height of the slab

For the present problem

The solution of the diffusion euuation outlined above resembles the solution of the transport equation used in MONIZ (1). The solution is obtained by iteration. That is a guess of the function A(x) is input. This will give the source strength in each mesh and conse- quently a further estimate of A(x) if the neutrons are tracked to the next collision. h is then estimated as the ratio of the source strengths.

Then we require

where M is some integer which is equal to the number of neutrons tracked per batch.

The number of neutrons set off from each mesh should be propor- tional to ~(x,). However, as the values of ~ ( x , ) are not integers,

L I

a zandom selection procedure is applied to the fractional part of S(x.) to select integers ISi as close as possible in value to s(xi)

1 -

for the number of neutrons started in mesh i, with the added condition that

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Hence this procedure will give both an estimate of 'A and the eiqen- function &(xi) .

Because of the similarity to the MONK procedure this model should be capable of illuminating some of the aspects of using Monte Carlo on depletion or other reactor physics calculations. Using this method has the great advantage of speed and hence cost. The primary reasons for the increase in speed are:

(1) Simple geometry - Homogeneous in 1-D (2) Simple tracking - There is only one collision between

successive source selections as opposed to a50 in a realistic NONK calculation. No scattering process has to be modelled.

( 3 ) Only one group - Only a very simple selection of type of next event is used in this calculation.

3.2 Depletion

As the fuel is irradiated it depletes and hence k(x) will change. In realistic problems spatial transport does not vary much with burnup and in this simple model, K, the inverse diffusion length is constant.

Defining irradiation as I(x)

Then it is assumed that k(x,t) wili be a function of both I and rod position R(t) .

Thus the source S and consequently the absorption rate will also be a function of space and time.

In outline, the solution of the depletion equationsthen becomes a 2-Dimensional problem in space-time. The first approximation to an iteration procedure (which is adopted in this study) is to treat the two dimensions separately. This will be analogous to the conventional deterministic calculation. Thus the calculation proceeds as follows:

(1) For a given time or irradiation calculate k (x,t) and hence A(x,t). This later step may take a number of iterationsofthe Monte Carlo solution procedure.

(2) With that value of A(x,t) evaluate I (x,t) at the next time step for a given At and then repeat (1).

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In the calculations presented in this study 30 time steps were used. To estimate the initial yalue of A(x,o) 10 stages of 1000 neutronswereused and between each time step either one step of 1000 neutrons or 10 stages of 100 neutrons were used. In cases where the ~- variation between tkme steps was small e . small rod movenents) no settling stages between estimates of A were required. But with rapid or large rod movements this was no longer adequate and 5 settling stages were used. Hence in these cases only the last 5 stages of the 10 stages between each time step were used for the estimate which consequently had a larger standard error.

It must be emphasised that this procedure is only the first attempt at a Xonte Carlo depletion. Other aspects of the problem that still have to be investigated are:

(1) Need for a two pass system as above

(2) Convergence tests following large changes in data or geometry.

(3) Optimum method of estimating source variation with time.

However as this note is primarily intended to denons-trate the flex- ibility of MC Burnup, these points can for now be regarded as refinements of the method.

4. CALCULATIONS

Four depletion calculations have been completed.

No Rods

k(x) varies linearly with irradiation i.e.

k(x,t) = k(x,o) - 0.01 I(t) k(x,o) = 1.1 At = 1.0

NO Rods

k(x) varies quadratically with I. This was intended as a simulation of burn-out of a poison cell.

k(x,t) = k(x,o) - 0.001 (I(t) - 10) 2

k(x,o) = 1.2 At = 1.0

Oscillating Rods

k(x) varies as above but in the bottom half of the core an extra absorber is inserted at alternate time steps for the first 20 time steps and then withdrawn.

2 k(x,t) = k(x,o) - 0.001 (I(t) - x > H/2 k(x,t) = k(x,o) - 0.001 (I(t) - 10) - X6(t) x < ~ / 2

k(x,o) = 1.2 B(t) = 1 fort during even

= o otherwise At = 1.0 X = .10 (rod strength)

steps from 2 to 20

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( i v ) Withdrawing Rods

k ( x ) v a r i e s with. i r r a d i a t i o n as i;n C i A l , bu t t h e rod i n i t i a l l y i n s e r t e d i n t h e bottom h a l f of t h e c o r e i s then s t e a d i l y withdrawn u n t i l f u l l y o u t a t t h e end of 20 s t e p s .

t m = i n t e g r a l p a r t t < 20 A t

= l o t z 20 A t

I n cases (i) and (ii) 1 0 0 0 neu t rons were t racked i n a s i n g l e s t a g e a t each d e p l e t i o n s t e p whereas i n (iii) and ( i v ) 1 0 s t a g e s of 100 neu t rons eachwere t r acked and only t h e f i r s t 5 s t a g e s used t o e s t i m a t e A. I n a l l c a se s t h e f u e l l i f e was d iv ided i n t o 30 d e p l e t i o n s t e p s . To g e t t h e quoted u n c e r t a i n t i e s on t h e Xonte Car lo r e s u l t s i n each of t h e c a s e s , 5 independent dep le t ion c a l c u l a t i o n s

a were c a r r i e d o u t , each s t a r t i n g wi th a d i f f e r e n t random number.

5. RESULTS

5.1 Computing T i m e s

The IBM 3081 cpu times f o r t h e s e c a s e s are given i n Table 1.

TABLE 1

CPU Time ( s e c s )

iii 4 . 4 i v 4 . 3

The smal l i nc rease i n c o s t f o r ca ses (iii) and ( i v ) i s due t o t h e need t o c a r r y o u t 10 s t a g e s a t each burn-up s t e p when rod movements a r e p r e s e n t . Although no more neutrons a r e t racked a s a r e s u l t , t h e source d i s t r i b u t i o n has t o be re-normalised f o r t h e nex t s t a g e and t h e r e i s evidence t h a t t h i s i s t h e cause of t h e e x t r a c o s t of t h e s e runs .

Each run c o n s i s t s of t r a c k i n g 200,000 neutrons . Hence t h e c o s t i s -2 .0 s e c s pe r 1 0 0 , 0 0 0 neu t rons . I n a t y p i c a l MONK ca l cu l - a t i o n t h e t ime f o r t r a c k i n g 100,000 neutrons would be about 5000 seconds.

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This factor of 2500 would be made up of the fo>lowing:-

There are -50 collisions/absorption in MONK as opposed to 1 collision/track in this calculation. ':his gives a factor of -50 in cost.

MONK uses weighted tracklng so that the mean weight of a neutron in a track is ~0.5. This in turn means that there are %2 tracks/absorption, i.e. a fac-tor 2 in cost.

Using Woodcock tracking in zones containing heavy absorbers leads to many pseudo collisions in MONK This increases the cost by a factor -5.

MONK calculations are 69 group transport theory calcul- ations with complicated geometry which add:^ a factor of -2.5 to the cost per collision.

Finallythere is a factor -2 for the relation between the processing time for a simple code used in this exercise and a general, complicated code such as MONK.

5.2 Reactivities

See Table 2 on Page 9.

Table 2 shows MC reactivity estimates throu~hout life and their exact deterministic equivalents. At start of life MC reactivity estimates differ from the exact values by 0.19% which is within one standard error in all cases. At the end of life the differences averaged ever the last 3 time steps are as follows.

TABLE 3

Case

From the results in Table 3 the difference between MC estimates and the exact value are again of the order of 0.1% with an error of ~0.4%. The difference of case (ii) is larger than one standard devi- ation but is not significant given the small sample of results.

i

i i

iii

iv

Me an

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Percentage Difference Between MC and Exact Values

- 0.1 0.6

- 0.1 0.0

+ 0.1 + .3

Standard Errors on

MC Estimate %

TIME

0 1 2 3 4 5 6 7 8 9

10 1 1 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

EXACT

1 .O29 1.018 1.007 0.996 0.986 0.975 0.964 0.954 0.943 0.933 0.922 0.912 0.901 0.891 0.880 0.870 0.860 0.. 850 0.839 0.829 0.819 0.809 0.799 0.789 0.779 0.769 0.759 0.749 0.739 0.729 0.719

' 'TABLE 2 . . . . . . . . . . . . . . . . ..

PRELIMTNARY XONTE-CARLL'BURN-UP'CAI;CULATIONS

EXACT

1.029 I .049 1.067 1.082 1 .O9S 1.105 1.112 1.117 1.119 1.118 1.115 1.109 1.101 1.091 1.078 1 .O63 1.047 1 .O28 I .008 0.987 0.964 0.940 0.915 0.889 0.862 0.834 0.805 0.776 0.743 0.714 0.681

EXACT

1.029 1.010 1.067 1.044 1.095 1.066 1.112 1.078 1.118 1.078 1.114 1 .O68 1 . I00 1.049 1.077 1.020 1.047 0.984 1.010 0.942 0.968 0.945 0.919 0.893 0.865 0.836 0.807 0.776 0.745 0.714 0.681

EXACT

0.990 1.018 1.037 1.059 1.071 1.087 I .093 1.102 1.103 1.105 1.101 1.098 1.089 1.081 1 .O68 1.056 1.039 1.023 1.004 0.983 0.963 0.939 0.915 0.889 0.862 0.834 0.805 0.776 0.745 0.714 0.682

(i) k-infinity is a linear function of irradiation.

(ii) k-infinity is a quadratic function of irradiation.

(iii) The same as (ii) but with periodic rod movement until time = 20. Rod out thereafter.

(iv) The same as (ii) but an initially inserted rod is gradually withdrawn until fully out at time = 20. Rod out thereafter.

Uncertainties on Monte-Carlo (MC) results are percentages.

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The errors quoted for these values are similar to the errors for tracking %60,000 neutrons in XONK but in this case are based on a sample of %15,000 neutrons. This indicates that a significantly higher variance per neutron will result from scaling these results to WONK. Thus it is ex~ected that for a depletion calculation using the same number of neutrons and based on the MONK code, the errors in final reactivity will be of the order of '0.6% if no smoothing of the reactivity curve is carried out.

5.3 Irradiation

See Table 4 on Page 11.

Table 4 shows the XC irradiation estimates at end of life and the eq-uivalent exact values. These values are summarised in Table 5.

TABLE 5

PERCENTAGE MEAN DIFFERENCE BETFTEEN NC IR3ADIATION ESTIMATES AND EXACT VALUES

iii 2.1

iv 1.4

Me an 1.7

Mean Standard Error on MC Estimates

Again the differences are all of the order of the standard error indicating an agreement between the YC and exact calculation to about +1.5%.

5.4 Source Distributions

Estimates of the source distribution from both the exact and the MC solution of the depletion problem are available. These results will give more detail of how well the MC calculations are coping with the time dependent variation in data due to depletion and more particularly rods. In depletion calculations, in the main, the mechanisms lead to negative feed back and hence stable solutions. Hence it is expected that at an interval after a rod ,movement the MC calculations will tend to line up with the exact solution. This may not be the case shortly after the rod movement. This behaviour will be evident from an analysis of the time dependence of the source distribution.

MESH

1 2 3 4 5 6 7 8 9

10 I I 12 13 14 15 16 17 18 19 20

EXACT

15.59 21 .oo 25.59 29.36 32.37 34.70 36.44 37.67 38.45 38.83 38.83 38.45 37.67 36.44 34.70 32.37 29.36 25.59 21 .oo 15.59

Mean S.E.

TABLE 4

PRELMINARY NONTE-CARLO BURN-UP CALCULATIONS

EXACT

17.53 23.29 27.73 30.90 33.04 34.41 35.25 35.74 35.99 36.11 36.11 35.99 35.74 35.25 34.41 33.04 30.90 27.73 23.29 17.53

TRRADIATIONS

(ii) (iii) (iv) MC+S .E. - EXACT

Cases (i), (ii), (iii) and (iv) are as defined in Table 2.

Uncertainties on Monte-Carlo (MC) results are percentages.

.a

EXACT

17.40 23.16 27.62 30.83 33.01 34.41 35.29 35.83 36.14 36.31 36.32 36.19 35.88 35.33 34.42 32.98 30.78 27.56 23.12 17.39

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Graphical representat2,ons of the source distributions are given in Appendix 1. These plots show the lnrtial source distribution and its value at several later stages of depletion for the 4 cases described in Section 4. The plots also show the uncertainty assoc- iated with each point on the source distribution. Tn all cases the uncertainty associated with the inital source distribution is smaller than for later stages. This is because the lnitial distribution is sampled from either 1,000 or, in the rod cases, 500 neutrons.

Examination of the results indicates that with the procedure chosen the MC source distribution does follow the time variation of the exact solution reasonably accurately, given the uncertainties on the point values. There is some indication from earlier runs that the MC source lags behind after a rod movement especially in the first few stages. The ~rocedure adopted to throw away the first 5 stages after a rod movement has largely removed this lag. However this is clearly a point for further investigation.

On the whole the results of these calculations are encouraging, indicating that a MC depletion code is feasible.

6. SNOOTHTNG

Given the nature of the MC depletion calculation it would be attractive if the estimates of source and eigenvalue at different points in time could be smoothed out to reduce noise. It is assumed that the variation of any time dependent estimator is 'smooth' in time. That is the higher order derivatives with respect to time are relatively small. To investigate this aspect of the depletion calculations a preliminary look at aspects of the smoothing problem has been carried out.

In this study only the time dependence is considered when smoothing. Eventually the aim will be to construct a. time-space smoothing algorithm but that is outside the scope of this initial look at the problem.

To smooth out the estimates of source and eigenvalue the approach adopted is based on orthogonal polynomials. This does not preclude other ways of attackingthe problem but is probably the most instruc- tive way of studying the smoothing effects. a

Consider a MC estimate of source which is a function of time and space, S(t,x). Then we can write:

points

TO smooth out S(t,x) we restrict the expansion to N terms.

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Then

3 0 and Rn(x) = C S(t,x)

time steps

Similarly

K 1 Pn(t) = time steps

k (t)

the number of time steps then this If N is small compared to I smoothinq procedure should have a marked effect on the noise assoc-

iated with each estimate.

To obtain N the expansion for the exact solution was studied. It was found that for terms higher than 5 the value of Rn and Kn was very small. Two smoothing procedures were considered. They were for N=5 and N=9. It was expected that the N=5 case should be adequate but the N=9 case was run to test this assumption. The results of these smoothing calculations are presented graphically in Appendix 2 and 3. In Appendix 2 the values of Rn(x) are plotted for both the MC and the exact fitting and the two shown to be in good agreement. In Appendix 3 results for the smoothed estimates of both source at a point time and point power as a function of time are presented and shown to be in reasonable agreement with the exact solution for both the N=5 and the N=9 calculations.

The application of smoothing does indeed reduce the uncertainties associated with the source distribution as shown in Table 6.

TABLE 6

EFFECTS OF SMOOTHING ON SOURCE UNCERTAINTY

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Final Uncertainty %

5.5

4.2

Smoothing Degree (N)

9

5

Reduction Factor on Uncertainty

1.6

2.1

This reduction factor for the N=5 case amounts to a factor 4.4 on variance and hence cost.

Similar reductions were also obtained for the smoothing out of the time dependence of reactivity and hence fuel lifetime. That is, the uncertainties quoted in Table 1 are reduced by '1.2 on s~.oothina.

7. EXTRAPOLATION TO A NORE REALISTTC DEPLETION MC

Because of the simplicity of the model used in this study any extrapolation to a complex reactor situation should be treated with caution. However, from Section 5.2 it was evident that the uncert- ainty in a non-depletion MONK calculation per neutron tracked is similar to the values quoted for Snap shot calculations for this model. In fact the uncertainty in k-eff per neutron tracked increased by approximately a factor 2 in MONK relative to this simple model.

Hence, as a first approximation, it i.s reasonable to expect that the uncertainties found in this study will be increased by %2 on going to a realistic reactor situation. This would equate to the following if '1.200,000 neutrons were tracked:- a -

'(a) Variation of k-eff with time to %0.4%.

(b) Irradiation distribution (20 mesh points) at end of life to '1.3%.

(c) Axial source distribution (20 mesh points) to '1.8% at any time.

This latter figure could presumably be reduced by smoothing in space by a further factor of four but this has yet to be demonstrated.

8. CONCLUSIONS

The study reported in this note was intended as a preliminary look at the problem of Monte Carlo depletion. From the results of the study the following conclusions can be drawn:

The calculations on a simple 1-D slab indicate that, for a cost comparable with that of a start of life calculation, it is possible to carry out a Monte- Carlo depletion calculation to a reasonable accuracy.

Both reactivities and irradiation at different depletions have low uncertainties and are in good agreement with an exact solution of the problem. This is true for several combinations of dependence in k-infinity on irradiation and rod movements.

A preliminary attempt at reducing the noise assoc- iated with the Monte Carlo estimates has been made' using a smoothing procedure based on orthogonal polynomials. The results have shown that significant reductions in uncertainty at little extra cost are possible. This indicates that a further investigation of this aspect should be profitable.

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( 4 ) The results are on the whole encouraging, indicating that developing a depletion Monte Carlo code is a reasonable objective with presently available computing power.

1. J L HUTTON 'Modifications to the Monte Carlo Neutronics Code '?l0UIONK1 AEEW - P, 1 2 4 9 ( 1 9 7 9 )

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SOURCE DISTRIBUTIONS

These plots illustrate source distributions in the slab at start of life (time = 0) and after several depletion steps (tine = 10, 20 and 30) for each of the 4 cases described in Section 4. The piecewise continuous line represents the exact distribution given by the collision-probability method. The mesh numbers mark the mean value of the source in that mesh obtained from the Monte Carlo calculation and the vertical bars represent the estimated uncert- ainty on the mean.

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SOURCE DISTRIBUTION AT TIME = 0.0

I I I I I I I I

MESH

CASE i

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CASE i

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SOURCE DISTRIBUTION AT TIME = 20.0

I I I I I I I I

0.00 1 I I I I I I I I 1

0 2 4 6 8 ' 10 12 14 16 18 20

MESH

CASE i

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MESH

CASE i

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SOURCE DISTRiBUTION AT T I M E = 0.0

I I I I I I I I I 1

MESH

CASE ii

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CASE ii

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SOURCE D I S T R I B U T I O N A T T I M E = 20.0

I I I I I I I I I

MESH

CASE ii

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CASE ii

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SOURCE DISTRIBUTION AT T IME = 0.0

I I I I I I I I

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MESH

CASE iii

NESH

CASE iii

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SOURCE DISTRIBUTION AT TIME = 20.0

MESH

CASE iii

AEEW - M 2028

MESH

CASE iii

AEEW - M 2 0 2 8

SOURCE DISTRIBUTION AT TIME = 0.0

I I I I I I I I I 1

MESH

CASE i v

AEEW - M 2028

CASE iv

AEEW - M 2 0 2 8

SOURCE D I S T R I B U T I O N AT T I M E = 20.0

I I I 1 I I I I I 1

MESH

AEEW - M 2 0 2 8

. ~ . .

CASE iv

CASE iv

AEEW - M 2 0 2 8

", ' . "I)

. . i. j'

APPENDIX 2

ORTHOGONAL POLYNOMIALS

These plots display the first 6 terms of a 9-degree polynomial fitted to the time distribution of the sources in each mesh resulting from both the exact and PIC calculations. The terms of the poly- nomial were evaluated at end of life and on the plots are labelled as modes whose sum gives the source. Letters D and M on the plots refer to deterministic and YC calculations respectively. In some meshes the exact and MC modes are so close that the two letters are indistingiushable.

AEEW - M 2028

MODE 1

MESH

AEEW - M 2028

MODE 2

I I I I

AEEW - M 2 0 2 8

MODE 3

I I

MESH

AEEW - M 2028

~. : ~. . .

MODE 4

I I I

AEEW - M 2 0 2 8

NODE 5

I I I I

MESH

AEEW - M 2 0 2 8

AEEW - M 2 0 2 8

APPENDIX 3

SMOOTHED MONTE-'CARLO 'AND EXACT SOURCE PLOTS

Both the smoothed Monte-Carlo and exact source distributions in these plots are from Case iv results. The plots are in two sets of nine plots each. The first nine show Monte-Carlo results which have been smoothed in time by means of 5th degree orthogonal ploynomials, and the second nine by means of 9th degree polynomials. In each set of nine plots the first four display the results as source distributions in the 1-D slab at a few depletion times, and the remainder the same results as time distributions in 5 meshes of the slab.

AEEW - M 2028

SOURCE DISTRIBUTION AT TIME = 0.0

I I I I I I I I I

MESH

AEEW - M 2028

SOLIRCE D I STH 1 BUT i O N A T TI ME L-I 1 0.0

I 1 I I I I I I 1

MESH

AEEW - M 2 0 2 8

SOURCE DISTRIBUTION AT TIME = 20.0

I I I I I I I I I 1

MESH

AEEW - M 2028

SOURCE D I S T R I B U T I O N A T T I M E = 30.0

MESH

AEEW - M 2028

SOURCE DISTRIBUTION IN MESH = 1

TIME

AEEW - M 2028

AEEW - M 2028

SOURCE n I ' 3 T R I BUT1 ON IN MESH = 1 0

I I I I I 1

TIME

AEEW - M 2028

AEEW - M 2028

SOURCE DISTRIBUTION

1 I I

IN MESH = 20

I

AEEW - M 2 0 2 8 .

0.10

0 . 0 8

0.06

U_i 0

2 r ~ O . 0 4 m 0 LLI N H

A < ro. 02 LY 0 Z

SOURCE D I S T R I B U l i C I N A T T I M E = 0 . 0

I I I I I I I I 1

0.00 I 0 2 4 6 8 10 '1 2 14 ' 6 18 20

MESH

- AEEW - M 2 0 2 8

SOURCE DISTRIBUTION AT TIME = 10.0

0 2 4 6 8 10 12 14 16 18 20

MESH

AEEW - M 2028

i < t o . 02 a 0 Z

0.00

SOURCE DISTH!BUI IOIY A T T I M E = 20.0

I I I I I I I

MES t i

AEEW - M 2 0 2 8

SOURCE DISTRIBUTION A T T I M E = 30.0

I I I I 1

MESH

AEEW - M 2028

SCJRCE DISTRIBUTION IN MESH = 1

I I I

T I M E

AEEW - M 2 0 2 8

SOURCE DISTRIBUTION IN MESH = 5

I I I I

T I M E

AEEW - M 2028

T I M E

AEEW - M 2028

SOURCE DISTRIBUTION IN MESH = 15

I I I I I

AEEW - M 2028

SOURCE nl5TRIBiJTIoN I N MEYH = 20

TIME

AEEW - M 2028.