AE-279 UDC 539.125.523.348 621.039.51.12

O) IN CM LU < Calculations of Neutron Flux Distributions by Means of Integral Transport Methods

I. Carlvik

AKTIEBOLAGET ATOMENERGI

STOCKHOLM, SWEDEN 1967

AE -279

CALCULATIONS OF NEUTRON FLUX DISTRIBUTIONS BY MEANS OF INTEGRAL TRANSPORT METHODS

I Carlvik

SUMMARY

Flux distributions have been calculated, mainly in one energy group, for a number of systems representing geometries interesting for reactor calculations. Integral transport methods of two kinds were utilised, collision probabilities (CP) and the discrete method (DIT). The geometries considered comprise the three one-dimensional geo ­ metries, plane, spherical, and annular, and further a square cell with a circular fuel rod and a rod cluster cell with a circular outer boundary. For the annular cells both methods (CP and DIT) were used and the re­ sults were compared. The purpose of the work is twofold, firstly to demonstrate the versatility and efficacy of integral transport methods and secondly to sefrve as a guide for anybody who wants to use the methods.

Printed and distributed in May 1967 LIST OF CONTENTS

Page

1. Introduction 3

2. On the relation between CP and DIT 6

3. Methods for calculating collision probabilities and 10 transport kernels

4. Plane geometry 13

5. Spherical geometry 17

6. Annular geometry 20

7. Boundary conditions in annular geometry 23

8. Anisotropic scattering in annular geometry ' 26

9. Cluster geometry 27

10. Rectangular geometry 29

11. Conclusions 31

12. References 32

Figures

Tables - 3 -

1. INTRODUCTION

During the last five or ten years there has been an increasing interest in integral transport methods, that is methods for calculating neutron flux distributions that are based on the integral form of the transport equation. These methods are suitable for systems, where the characteristic dimension is less than say, ten mean free paths. In such small systems, e.g. reactor lattice cells, they are often superior in accuracy and speed of computation to other transport methods such as the spherical harmonics method and the Carlson method, which are based on the integro-differential equation which represents the usual form of the Boltzmann equation. Considerable effort has been devoted to the development of in­ tegral transport methods at AB Atomenergi. In fact methods of this kind have been incorporated in all current cell codes produced at the establishment. Even the standard cell burnup code, REBUS, which utilises the Westcott formalism, determines the intracell thermal flux distribution by means of a complete integral transport calculation. Two different formulations have been studied. The first one is the most common form of integral transport methods, the one that uses flat source collision probabilities. It will be called CP for short. The second method uses a point-wise representation of space, and it de­ scribes the neutron transport by means of transport kernels. This form together with the use of Gaussian integration over the space co ­ ordinate was first suggested by Kobayashi and Nishihara [ l] . We have suggested that the method be called DIT for Discrete Integral Transport. The next two paragraphs contain some general remarks on in­ tegral transport methods, and give a short survey of methods for cal ­ culating collision probabilities and transport kernels. However, the main content of the paper is a collection of sample calculations using either CP or DIT. The purpose is first of all to demonstrate the ver ­ satility and efficacy of the methods, but it is also hoped that the col ­ lection can serve as a guide for users of the codes, and therefore such things as the effect of varying the distribution of points and regions are investigated. The relationship between the accuracy achieved and the computer time used is studied in some detail, since it is this which - 4 -

determines the usefulness of a method. All computing times given refer to calculations on an IBM 7044 computer. In a complete cell programme using integral transport methods there are three main parts which determine the speed of a calculation, 1) the evaluation of the transport matrices, one for each energy, 2) the solution of the system of linear equations, and 3) the administration of the programme and particularly the handling of a large amount of cross section data. Only the first part will be considered here. The main in­ terest is directed to the method used for evaluating transport matrices. It is then superfluous to keep the energy dependence, and consequently most of the calculations refer to one-group problems with an external source. The major part of the calculations also refer to one-dimensional systems, plane, annular, and spherical. In all three geometries examples of DIT calculations are shown. The annular cases considered have also been calculated by means of CP, and the results are compared. The important parameter is the number of space points or regions neces­ sary to reach a certain accuracy. This is of particular importance in a multi-group code, both for the speed of part 2) above and for the necessary capacity of the fast memory. It is shown that in general CP is superior if the flux variations in the system are small and vice versa. Integral transport methods are most advantageous when it is pos ­ sible to assume isotropic scattering. If anisotropic scattering must be considered, it is necessary to carry one more flux moment in the calcu ­ lation for each added term in the expansion of the scattering cross section. There are also other difficulties, for example in annular geometry the azimuth angle varies along a neutron path, and if CP is used, this is somewhat in contradiction to the flat source assumption. It is, however, possible to use suitable averages over segments of neutron paths [ 2] . In the DIT method this difficulty does not exist", although there are others related to the problem of finding suitable diagonal elements. The DIT method has been applied to linear anisotropic scattering in annular geo ­ metry, and sample calculations are shown. In plane geometry there are no particular difficulties with anisotropic scattering in the DIT method. - 5 “

It may be fruitful to attack two- or three-dimensional problems by means of a generalised DIT-method, but so far the applications have been to one-dimensional geometries with the exception that problems including so called pole-fluxes in annular geometry could be classified as two-dimensional. Another ’’restricted two-dimensional ” system, which we hope to be able to work on in the future is the annular system with an axial buckling, that is with an axial cosine variation of the flux. A main dif­ ficulty in this problem is to develop a fast computer routine for the generalised Bickley function

e~x cosh u ------cos (y sinh u) du J cosh u o

In more complicated two-dimensional geometries CP seems to be the only alternative to Monte Carlo methods. Fuel elements for heavy water reactors usually consist of rod bundles. This cluster geometry is very difficult to handle by means of conventional methods, and most cell programmes consider only a homogeneous mixture of fuel, cladding, and coolant. However, using collision probabilities it is by no means impossible to manage calcu ­ lations in cluster geometry. It has even been possible to make multi­ group calculations and obtain good agreement with measured detailed activation distributions inside the rod cluster [ 3] . One-group calcu ­ lations on a cluster are shown here;,, and the accuracy that can be achieved is demonstrated. Another two-dimensional case, which we have considered, is that of a one-group flux distribution in a square lattice cell with a single fuel rod. The resulting distribution is compared with the one obtained in a calculation for the corresponding circular Wigner-Seitz cell. The results give a contribution to the discussion of suitable boundary conditions in the Wigner-Seitz cell. Beyond the particular geometrical configurations investigated in this report there are others that could be well worth studying. For example,when considering annular cells with several tubes and voided - 6 -

regions surrounding the fuel one could suspect that the approximations to the Bickley functions used here would not be sufficient. Another case is an annular system with azimuthal flux variation, for which an important question is what boundary conditions should be used under various conditions. It would, however, take too much time and money to cover all possible types of system. The ultimate test of a computational method is a comparison with experiments. There are no such comparisons in this paper. This is because only the transport in space is considered. A comparison with experiments demands a full physical model, and what should be checked in that context is the physical model, not the numerical accu ­ racy of the calculations inside the model. Therefore, the various parts of the calculations should be checked in accuracy, one at a time, by comparisons with more exact calculations as is done in this paper. In this manner one makes sure that an agreement with measured results has something to say about the physical model, rather than indicating that errors in the model happen to be compensated by numerical errors in the calculations. The author tried to make the material as complete as possible, and it could not be avoided that the report contains a multitude of nu­ merical results. It may be appropriate to apologise for the great number of tables in the report.

2. ON THE RELATION BETWEEN CP AND DIT

The two formulations are naturally closely related; they differ only in_that they employ different schemes for the integration over space. Since CP is unwieldy in treating anisotropic scattering, all neutron sources, both external sources, scattering sources, and fission sources, are assumed to be isotropic in this paragraph to make the discussion more perspicuous. If $ (jr) is the flux density and ^ (r) the total source density (in­ cluding scattering and fission) both for a certain energy, the spatial part of the transport equation is [ 4 J - 7 -

(r) = ^ dr* 4* (r• ) >- P(l,r' ) 2.1 4ir I r - r1 all space where

I r - r» I

P (£>£') = J S (r' + s,) ds 2.2

is the optical distance between jr and jri. £ (r) is the macroscopic total cross section at _r. The system under consideration is assumed to consist of a number of homogeneous regions. The DIT method uses an integration formula of Gaussian type for the integral in equation 2.1. If i and j are used for numbering the points of the point set, the equation can be written

4>. = S 4>. V. T. . 2.3 1 j J J

Vj is the product of the volume of one of the homogeneous regions and a suitable weight, and it can be formally interpreted as a subregion of the corresponding homogeneous region. In a general three-dimensional system one has

2.4 2 Ti,j 4ir |r. - r. -i -j

In a two-dimensional system T^ ^ is calculated as an average over two curves and in a one-dimensional system as an average over two surfaces. The approximation scheme is a little different in the CP method. The homogeneous regions are divided into subregions, which are chosen in such a way that the source density ip varies little inside a subregion. The size of a subregion is denoted by W, and k and t - 8 -

are used for numbering the subregions. Then the approximation to equation 2.1 is

e'P

Equation 2. 5 is integrated over subregion k to give the average flux in W,

J dr (r) = J dr J dr. P 2.6

The collision probability ^ from l to k can be obtained from

equation 2. 6 as ^ for ip^ = in subregion l and ip = 0 in all

other subregions. The result is the following well-known expression

2. 7 l ~ W. *k

Combining equations 2.6 and 2.7 one obtains the usual spatial equation in the CP formalism.

2.8

An alternative form is obtained if equation 2.8 is divided by skwk

kj l *k = S vp, W, 2.9 2kWk

The last equation is completely equivalent to equation 2.3 of the DIT method. A computer routine devised for solving group fluxes and - 9 -

possibly also an eigenvalue for one of the methods can be used for the other as well. In practical problems it often happens that certain regions, e.g. gas channels, can be considered as voided. This does not cause any difficulty in the DIT method, since T. . can always be calculated. If 1» J CP in the form of equation 2. 8 is used, the result of the calculation does not give any flux density in the voided regions, because the pri­ marily calculated quantity, the collision density S^^^is zero in the void region. If, however, the equations for the group fluxes are written in the form of equation 2. 9, the calculation gives the flux den­ sities also in the empty regions. It must be observed that if £, = 0, Pk l also P. . = 0, and the matrix element v *.y — must be calculated as a k, c 3bi_W,_ k k limit. It should be born in mind, when a multi-group code is written, p that both matrices, T. .of equation 2. 3 and ^ of equation 2. 9 are symmetric. A few words can be said about what the approximations mean in the explicit physical model. The assumption in the CP method is the flat source approximation. Consider neutrons that have collided in a certain subregion. Some of them are scattered and start again, but before they are started, they are spread out uniformly over the sub­ region. This means a neutron transport from points with higher flux to points with lower flux, and consequently the CP method will tend to give too small flux variations in a system. The collision probabilities utilised in the well-known THERMOS code [ 5] are not the usual flat source collision probabilities, but in­ stead they refer to neutrons that start from a central point of a sub­ region. In the physical picture neutrons that have collided inside a certain region are concentrated to the central point before they start again. This should make the coupling between the subregions too small, and it has also been observed that THERMOS tends to give too high flux variations. There is another disadvantage with this type of collision pro ­ bability. In a system where one knows that the flux density is almost constant in a large zone it is possible to use very large subregions if - 10 -

the flat source collision probabilities are used. However, in the THERMOS scheme the subregions must still be so small that the escape probability from the central point is representative for the escape pro ­ bability from the subregion. It is not so easy to see how the approximation in the DIT method works. However, at least if the number of space points is small, it seems that the coupling between various parts of the system should be too small in the calculation, so that the flux variation over the system should be too large. This has also been found in the calculations in an ­ nular geometry, where CP and DIT are compared. The moderator to fuel flux ratios converge from above in the DIT method and from below in the CP method.

3. METHODS FOR CALCULATING COLLISION PROBABILITIES AND TRANSPORT KERNELS

A key problem in the application of integral transport methods is the calculation of the collision probabilities or the transport kernels. Various methods have been proposed. They will not be discussed in de­ tail here, but a short survey will be given. A majority of the literature on the subject deals with the calcu ­ lation of collision probabilities in annular geometry. Formulae for multi-region collision probabilities (in some papers only two or three regions) have ibfeen given by Takahashi [ 6], Millier [ 7] , Kiesewetter [ 8] , Di Pasqbdmtonio [ 9] , Pennington [ 10] , and Carlvik [ 11 ] among others. Application of the correct formulae must necessarily use up some computer time and memory space. Considerable efforts have been devoted to the derivation of approximate collision probabilities that are easy to calculate but still accurate enough to allow the calcu ­ lation of flux densities of sufficient accuracy. A frequently used approximation is the assumption that neutrons that have crossed a boundary between two adjacent annuli have a cosine distribution. With this assumption it is possible to uncouple any an ­ nulus from the others except the two adjacent ones, and only the self - 11 -

collision probability and certain transmission probabilities have to be calculated for each annulus. This method has been employed by Mtiller and Linnartz [ 12] , and they have derived polynomial approximations that give the probabilities conveniently. The same assumption has been used by Kalnaes et al. [ 13] . With this assumption - that neutrons have a cosine distribution after crossing a region boundary - it seems probable, although it has not been proved, that if the number of regions is increased, the solu ­ tion should converge towards the DPq solution, that is the solution of the double spherical harmonics method of order zero. The assumptions regarding the angular distribution are less severe in the approximation scheme of Bonalumi [ 14]. Bonalumi's method has been developed further by Jons son [ 15], and it has also been generalised to linearly anisotropic sources by Hyslop [ 16]. A drawback with this kind of approximation is that one cannot be certain that the solution converges towards the correct solution, when the number of regions increases. If accurate results are desired, it is necessary to use the exact formulae. This was done by Honeck [ 5] in the THERMOS code for the particular type of collision probabilities discussed in paragraph 2. The same method for calculating collision probabilities was employed by Stamm'ler [ 17]. The probability that neutrons starting at a certain point collide in the different annuli is calculated by means of a Gaussian integration over the azimuth angle. The integration scheme developed by Wexler for the MINOTAUR code [l8] calculates the ordinary volume-to-volume collision proba ­ bilities. It is more elaborate, and it takes into account the discon ­ tinuities of the derivative of the integrand. Kobayashi and Nishihara [ 1 ] employ the straight forward in­ tegration over the azimuth angle for the calculation of transport kernels in annular geometry. The method we have used for the calculations in the present paper is also based on the exact formulae, but the scheme for per­ forming the integration over the azimuth angle is different from the one used by other authors and seems to be more efficient. In fact we have found that the gain in computing time when using an approximate - 12 -

method is not very large. Our scheme has been described in reference [4] for collision probabilities and in reference [ 11 ] for transport kernels. The former reference regarding collision probabilities is a little brief, but the latter one is more detailed, and the modification of the scheme when going from transport kernels to collision probabilities is obvious. . There has been less interest in collision probabilities in spheri­ cal geometry, because this geometry is not so important from a practi ­ cal point of view. In plane geometry collision probabilities and transport kernels can be expressed directly in exponential integrals, and no par ­ ticular schemes are needed to calculate them. The two-dimensional geometry of a square or hexagonal cell with a cylindrical fuel rod is more complicated, and calculations will neces­ sarily take longer than for one-dimensional geometries. Honeck [ 19] has considered this geometry. He calculates collision probabilities for neutrons starting at a representative point in each region by Gaussian integration over the azimuth angle as in the annular case. The division of the cell into subregions is quite general. Two-region collision probabilities for the same geometry have been calculated, both from exact expressions by Fukai [ 20] and by means of approximative assumptions by Sauer [ 21 ]. Sauer extended his method by including a third region, the cladding of the rod. Fukai generalised his method to a multi-region cell consisting of concentric annuli plus one more region made up of the remaining corners of the cell [22]. Willis has used a Monte Carlo technique to calculate collision probabilities in the three-region cell, fuel, cladding, moderator [23]. The general method developed independently by Beardwood, Clayton and Pull [24] and by Carlvik [ 11 ], which was originally intended for cluster geometry, can be used for square or hexagonal cells as well. For the calculation of collision probabilities sets of parallel, equidistant lines are drawn over the system in a number of directions, intersections with region boundaries are determined, and collision probabilities are obtained as combinations of Bickley functions of optical distances between intersections. - 13 -

Even for cluster geometry, simpler methods have been pro ­ posed. Leslie and Jons son [ 25] developed a scheme, which relies on the use of approximate analytic expressions for the collision proba ­ bilities in the cluster. The Bickley functions are vital for integral transport theory in general cylindrical geometry, one-dimensional or two-dimensional. The original paper by Bickley and Nayler [ 26 ] gives the most im­ portant formulae and a short table. Larger tables for the closely re­ lated functions Kj have been given by Muller [ 27] . However, what is needed in computer codes is numerical approximations that can be used for quick subroutines. Such approximations can be found in papers by Fukai [ 20] , Danielsen et al. [ 28] and Clayton [ 29] . The approximations we have used are due to Tellander [ 30] .

4. PLANE GEOMETRY

The calculations in plane geometry were made by means of the DIT code PLUS LA, which calculates the flux distribution in one group for an external source. The programme allows anisotropic scattering, but the calculations reported here assume isotropic scattering. The basic functions for the calculation of transport kernels in plane geometry are the exponential integrals. With isotropic scattering C00 -xu only the first order function E^ (x) = ^ —-— du is needed. Approxi- 1' mations for E, were taken from the Handbook of Mathematical Functions 1 [ 31 ] , formulae 5. 1.53 and 5. 1. 56. The maximum absolute error of the - 7 - 8 approximation is 2x10 in the interval 0 < x < 1 and 2x10 for 1 < x < oo. In plane geometry, in contrast to annular or spherical geometry, the transport kernel is expressed directly in known functions (the expo ­ nential integral) and no integrations are needed. On the other hand in a lattice one must perform a summation over all cells in the lattice. The FLUSLA code terminates the summation, when the optical distance between the field point and the source point is larger than a prescribed number, OPMAX. The values 6, 8, and 10 have been used in the cal ­ culations. - 14 -

The effect of the uncertainty in the exponential integral can be in­ vestigated in the following manner. Consider a case, where the source density ip at any point is equal to the local macroscopic cross section. The flux density at a point x is then equal to

+oo y +oo * (x) = -| J Ei { • J £ (t) dt I j- £ (y) dy = -| J E4 (| u | ) du = E2 (0) = 1

-oo X -eo (4.1)

The maximum absolute error caused by the uncertainty in E^ and by the fact that the integration is terminated at OPMAX is

1 OPMAX , oo Mmax( x)= ( 2 x 10'7 dT + J 2 x 10"8 dT + j (t) d T m X o 1 OPMAX

= 2 x 10"7 + (OPMAX - 1) x 2 x 10'8 + E^ (OPMAX) (4. 2)

Thus in this case the relative error in the flux density is less than 3.2x10 ^ if OPMAX=6, and less than 4.2x10 8 if OPMAX=10. In practical cases the source density varies, but this calculation shows that the error caused by the inaccuracy of E^ can in general be neglected. Two series of lattices were studied. The first one, typical for light water reactors, has been considered by Theys [ 32 j and the second one, typical for fast reactor lattices of the Z PR-III type, by Meneghetti [ 33 ]. Geometrical data and macroscopic cross sections of the lattices are given in figure 4.1. Neither the paper by Theys nor a later paper by Ferziger and Robinson [ 34], in which the same lattices were considered, quote an absorption cross section for the light water moderator, so we thought that the absorption cross section had been neglected. Later we found [35] that the calculations of these authors had used the value =0.0195. The corresponding change in the flux ratio is quite small, 0.00002 in the thinnest cell and 0. 0093 in the thickest one, and it does not in any way influence the conclusions of the general investigations of accuracy and computing times. However, for the comparison with the - 15 -

result of other authors we repeated a series of calculations, so that the data for the results of table 4.4 and figure 4.4 are the same as those used by the other authors. In the calculations on the four lattices of Theys the distribution of the space points and the value of OPMAX were varied in order to study the effect on accuracy and computing time. First the four lattices were calculated with 12 space points in the half-cell and with the space points divided between the two regions in various ways. For the thickest lattice a similar series was done with 16 space points. The resulting flux ratios, that is the ratio average flux in moderator to average flux in fuel, are given in table 4.1 and shown graphically in figure 4. 2. The flux ratio converges towards the correct value from above. Thus the most suitable distribution of the 12 points is 3 in the fuel and 9 in the moderator for the first three lattices, and 2 and 10 respectively for the thickest lattice. In the case of 16 points in the thickest lattice the best division is 3 points in the fuel and 13 in the moderator. How ­ ever, the division is not critical.for the result. A simple way to distribute points would be to assign to each region a number of points proportional to the optical thickness of the region. The ratio between the optical thicknesses of the two regions is a little over 11, the same for all four lattices. However, in the cases investigated the ratio.between the number of points in the optimum distributions is 3 to 5, low in thin cells and high in thick cells. A qualitative conclusion is that the distribution of points between regions should be more even than proportional to the optical thicknesses, the more even the thinner the cells. Then the influence of OPMAX was investigated in calculations on the thickest and the thinnest cell with 8, 12, 15, and 20 space points. Three values for OPMAX were used, 6, 8, and 10. The cal ­ culated flux ratios are given in table 4. 2. The difference in the flux ratios for OPMAX=6 and OPMAX=10 is about 0.00012 for the thin cell and 0. 00006 for the thick cell. The difference for OPMAX=8 and OPMAX=10 is about 0.000004 for the thin cell, but for the thick cell it happens that the same number of cells is considered for both values -16-

of OPMAX, so the calculations are identical. These results indicate that OPMAX=10 can be used with confidence, and also that OPMAX=6 is sufficient in most practical calculations. Finally the total number of space points was varied. All four Theys lattices were calculated with 8, 12, 15, and 20 points and with OPMAX=10. The results are given in table 4.3 and the convergence of the flux ratio with the number of points is also shown in figure 4. 3. The calculation with 20 points gives something like four correct decimals for the thin cell (optical thickness 1.7744) and three correct decimals for the thick cell (optical thickness 7. 0976). However, the computing time is considerably longer for the thin cell. The lattices of Theys have also been calculated by Ferziger and Robinson [ 34] by means of the eigenfunction expansion method. A com ­ parison of the results is shown in table 4.4 and in figure 4.4, where also results by P^ calculations and by the method of Theys (similar to the Amoyal-Benoist method for annular geometry) are shown. The DIT results agree with the results from the eigenfunction expansion method, and an Sg -calculation by K. Lathrop (not shown in the figure) is very close. As could be expected the flux ratios of the diffusion theory are much too low, and use of the method of Theys means a considerable im­ provement over diffusion theory. The Meneghetti lattices were calculated with 5, 10, and 20 space points and with OPMAX=6, 8, and 10. The calculated flux ratios are given in table 4. 5 and shown graphically in figure 4. 5. The conclusions that can be drawn are similar to those from the calculations on the Theys lattices. Finally a few words should be said about computing times. The number of -functions that has to be calculated for a case is proportional to the squarfe of the number of points, and also about proportional to the ratio OPMAX/(optical thickness of the cell). In a case with many points and a thin cell the computing time is about proportional to this number, since the calculation of E^-functions takes the major part of the time. In the opposite case, few points and a thick cell, other parts of the calcu ­ lation dominate the time. The computing times in the tables show this general behaviour. However, one can also see that the computing time - 17 -

for a certain accuracy does not vary much from cell to cell, A thick cell needs more points than a thin one, but this is compensated by the fact that fewer cells have to be taken into account. An accuracy of 0.1 % can be obtained in about 1 second for all cells. If an accuracy of this order is considered as sufficient, it is possible to calculate the transport matrices for e. g. a 30-group cal ­ culation in less than 30 seconds. If desired this time could be reduced considerably if a simpler E ^ - routine were used. The one used here contains a rational expression with altogether 6 or 10 terms in nu­ merator and denominator and further either a log function or an expo ­ nential function. A maximum absolute error of something like 5x10 ^ in the E^-function would be more compatible with a relative error of 0.1 % in the flux, and this could certainly be achieved by a faster routine.

5. SPHERICAL GEOMETRY

No spherical one-group DIT code exists, and the calculations in this paragraph were made by means of the multi-group DIT code FLUB AG. The routine for evaluating the transport kernels was de­ veloped at AB Atomenergi, but the solution of the fluxes is achieved by means of a slightly changed version of the British PIP routine [ 36]. FLUB AG can accomodate 16 energy groups and 40 space points, and it is possible to make either an eigenvalue calculation or a fixed source calculation. Calculations in spherical geometry are of interest for fast reactors rather than for reactor lattice cells. However, integral transport methods are not suitable for large systems, so the FLUB AG code is of interest mainly for small fast reactors. In fact the code was composed mainly because only some small changes were needed to convert the corresponding annular multi-group code to a spherical one. The basic function needed for the calculation of transport ker­ nels in spherical geometry is the ordinary exponential function, so no special functions had to be built in. - 18 -

It was not possible to analyse one-group problems in detail re­ garding computing times with FLUBAG as was done in plane geometry and also in annular geometry. When a simple one-group problem is calculated by means of a multi-group code, the real computing time is short as compared to the time spent on administration and link changes. ' Two types of calculations in spherical geometry will be ac ­ counted for. The first is the classical test problem for a spherical transport code, the determination of the critical parameter c 2g + y (c =---- s------) for a bare homogeneous critical sphere in one energy ^tot group. Three spheres were calculated with the radii 1, 2, and 4 mean free paths. For each radius calculations were performed with 4, 6, 8, 10, 12, 16, and 20 space points. The results are given in table 5.1 and shown graphically in figure 5.1. The values labelled "exact" in the table were calculated by means of a variational method [ 37] . The latter is equivalent to the usual variational method using powers of the radius as test functions* although in this case Legendre polynomials were used instead. In this way it was easy to use a high order approximation. Figure 5.1 shows the c-values from the FLUBAG calculations, and also the c - value s from the variational method as functions of the order of the polynomial used. For this particular problem the varia ­ tional method shows a much better convergence, but also the DIT method gives c with an accuracy better than 0.1 % with less than 10 points. Of course, the DIT method shows its strength in more com ­ plicated systems, containing regions with different compositions. I may be of interest to compare the flux distributions obtained. This is done for the sphere of radius 2 m. f. p. in figure 5.2. The full line gives the distribution obtained in a variational calculation with a polynomial of the order 10. Flux values are also given for FLUBAG calculations with 8, 12, and 16 points. The points agree quite well with the curve, although the 8- point values are a little high near the center of the sphere. In spherical geometry, as well as in annular, the transport matrix is evaluated by means of integrations. The accuracy is deter- - 19 -

mined by two parameters IA and IM, which are' defined in reference [4] The number of Gauss points in the integrations varies between IA and IM according to the size of the interval. Standard values in FLUB AG are IA=2, IM=5. The effect of the parameters was investigated in a series of calculations for all three spheres. Each sphere was calculated with 10 and 20 Gauss points, and the result for IA=2, IM=5 was compared with the result for IA=4, IM=10. This comparison is shown in table 5. 2. The largest change in c is 4x10 so the values IA=2, IM=5, which were used in the calculations of table 5.1 should be sufficient for the 5 decimals given there. In another series of calculations the effective multiplication factor of the fast zero-power reactor FR-0 was calculated. FR-0 is similar to the ZPR-III reactor or the British Vera reactor. In this particular loading the core of radius 17 cm contains 20 % enriched uranium and some structural material, and the reflector of radius 47 cm consists of copper. The calculation was performed in three energy groups. The group data, table 5.3, were provided by Haggblom [38]. In the first series of calculations the space-points, 20 in all, were divided in different ways between core and reflector. The results are given in table 5.4 and figure 5. 3. The best division is 7 points in the core and 13 in the reflector. Then a series of calculations was performed with different total numbers of points, table 5.5 and figure 5.4. Some of the values were also calculated with higher accuracy in the transport kernels, IA=4, IM=10, instead of IA=2, IM=5. Apparently also for this larger system IA=2, IA=5 is sufficient. The use of the more accurate integration gives a change in of the order of 10 The accuracy in k^^ that can be obtained with the number of points available in the code, 40, is about 10 \

Values for k^^ obtained by means of two other methods are given in table 5.5. Also the computing times of the FLUB AG calcu ­ lations are included. The code HATTEN was developed by Haggblom [ 38] . It uti­ lises an integral transform of the P^-equations. - 20 -

The radius of the system considered here is 22 mean free paths with group 3 cross sections and 9.4 mean free paths with group 1 cross sections. The calculations show that the DIT method can indeed be used for problems of this size, but it probably represents the limit, and for larger systems one should use differential methods, S or P or a method such as HATTEN.

6. ANNULAR GEOMETRY

The application of DIT to annular geometry has been studied in more detail than the application to the other two one-dimensional geo ­ metries. The calculations were performed by means of the one-group code FLURIG-4. It is rather general, and it allows the study of the following things, although only one at a time: a) linear anisotropic scat ­ tering, b) azimuthal flux variation, c) the effect of a perfectly reflecting cell boundary (the standard boundary condition being a white boundary). With isotropic scattering and a white boundary the albedo of the boundary may have any value, even negative. As in the case of spherical geometry the transport kernel is evaluated by means of integrations, and the accuracy in the integrations is determined by the parameters IA and IM. The basic functions in the annular case are the Bickley functions [ 26 ] . In our calculations we have used approximations to the Bickley functions Ki^, Ki^» and Ki^, that have been derived by Toll and er [ 30 ] . They are very fast but have a limited accuracy, the maximum absolute -5 error being 3x10 . The effect of the inaccuracy of the routines for the Bickley functions can be estimated in the same manner as was used for the in­ accuracy of the routine for the exponential integrals in the plane case. Using the kernel for a line source instead of the kernel for a plane source, equation 4.1 for the plane case is now replaced by - 21 -

2ir oo r oo 4> (°) = ^ da ^ 2“ Ki4 J £ (ri) dr« S(r) rdr = J Ki1 (p) d p =

= Ki^ (o) = 1 (6.1)

The approximations for the Bickley functions assume that Ki (x) = 0 for x > 10. Thus one obtains for the maximum absolute error i n the flux

10 00 (o) = j 3x10 ^dp + j Ki^ (r) dp = 3 x 10 ^ + Ki^ (10) = a4> 10 -4 = 3. 2 x 10 (6. 2)

The true error is on the average considerably smaller since - 5 3x10 is the maximum absolute error of the Bickley approximations, and the error varies between the two extreme values as the argument varies. If one tries to converge the solution of a calculation by in­ creasing the number of points, one could expect that the result would start oscillating, when the error caused by other sources comes down -4 below say 10 . However, some of our results seem to have converged to an accuracy of about 10 ^ and no oscillations have occurred, so the errors have probably been well smoothed out in the integrations. The accuracy of the Bickley approximations is certainly sufficient for most practical applications. Some calculations of disadvantage factors in annular geometry have been reported earlier [ 39]. For the present more detailed in­ vestigation we have chosen two particular cells. The first, which is a typical heavy water cell is the same as the heavy water cell considered in the previous paper. The second one is the third of the six Thie cells [40], also considered in the previous paper. Data for both cells are given in table 6. 1 First both cells were calculated using 10 and 20 radial points in all, but with the points divided in different ways between regions. These calculations gave the disadvantage factors of table 6. 2 and table 6.3. The results are also shown in figure 6.1 to 6.3. It was - 22 -

found for the heavy water cell that the most accurate flux ratios were obtained with equal number of points in the fuel region and in the moderator. As pointed out in paragraph 2, the DIT method gives an overall flux variation in the cell that is too large, and consequently the moderator to fuel flux ratio, which represents the overall flux varia ­ tion, has a minimum for a certain distribution of the points in figure 6.1. On the other hand the shroud to fuel flux ratio has a monotonic variation, figure 6.2. This can be explained as follows. The thin shroud region will be coupled most closely to that of the two main regions with the most points. As a consequence the flux in the shroud will be too close to the fuel flux, that is too low, if most of the points are in the fuel region, and vice versa. The minima for the flux ratio in the light water cell are shallow, figure 6. 3, and also for this cell we used in the rest of the calculations the same number of points in the fuel and in the moderator, although the minimum does not come exactly at that point. Next the effect of varying the accuracy parameters IA and IM was investigated, and the results are shown in tables 6.4 and 6.5. The flux ratios from calculations using IA=0, IM=5, which are the values that were used in most annular calculations, do not differ by more than 0. 03 % from more accurate values. Finally calculations were performed with an increasing num­ ber of radial points, tables 6.6 and 6.7. The convergence of the flux ratios is shown graphically in figures 6.4 and 6.5. Similar calculations were also performed by means of a collision probability code FLURIG-2, and results are given in tables 6. 8 and 6. 9, and they are drawn in the same diagrams as the DIT results for comparison, figures 6.4 and 6.5. The abscissa is the number of points for the DIT calculation and the number of regions for the CP calculation. The CP code FLURIG-2 does not print out the computing times as the DIT code FLURIG-4 does, but equal numbers of points and regions give about equal computing times. Looking now at figures 6.4 and 6.5, one can see that both methods converge well. For equal numbers of points or regions the - 23 -

DIT calculations are more accurate in the case of the heavy water cell. For the light water cell the CP results are more accurate, except if one goes to a very large number of points or regions. The explanation of this is the flat source approximation of the CP method. The approxi ­ mation is good, if the source is nearly constant, and this is what happens in the light water cell, where the disadvantage factor is only 1.14. The total variation of the flux over the cell, /<& . , which max' min can be found from the calculations, is about 1.35. For this cell even a two region CP calculation gives the flux ratio with an accuracy of 0. 2 %. In the heavy water cell the total flux variation is considerably larger, about 2.5, and the flat source approximation is worse. One can see from tables 6. 6 and 6. 7 that an accuracy of 0.1 % in the flux ratios can be obtained by means of the DIT method in about 1.5 seconds for the heavy water cell, and in less than 0.5 seconds for the light water cell. This is the same order of magnitude as the 1.0 second found in the plane case. One would think that the plane cal ­ culation would be much faster, because there are no integrations in the calculation of the transport kernel. One fact that compensates partly for this difference is the summation over cells that is needed in the plane case but not in the annular case, if a white boundary is used. Another point is the fact that the approximations we have used for the basic functions, exponential integrals and Bickley functions respective ­ ly, are much more accurate and consequently slower in the plane case than in the annular case.

7. BOUNDARY CONDITIONS IN ANNULAR GEOMETRY

The annular Wigner-Seitz cell is an approximation to a square, triangular or hexagonal lattice cell. The correct boundary condition of the true cell in an infinite lattice is perfect reflection of neutrons that reach the boundary. It was earlier customary to retain the pro ­ perty of perfect reflection, when the true boundary was replaced by the circular boundary in the process of forming the Wigner-Seitz cell. Newmarch showed [ 41 ] that this boundary condition gives serious errors in the flux ratio for small cells, and that it even gives wrong — 24 —

flux ratio in the limit, when the size of the cell approaches zero. The effect has been discussed in relation to specific cells by Thie [40] . Several other authors have considered the problem of the boundary con ­ dition in Wigner-Seitz cells [ 20, 42-45] . It is now generally agreed that it is better to use a ’'white" bounda ­ ry in the Wigner-Seitz cell, that is neutrons returning through the bounda ­ ry have a cos 6 distribution, with 9 equal to the angle between the neutron direction and the inward normal of the boundary surface [46, 47, 48] . We will not discuss this point in detail here, but it will be illus­ trated by some FLURIG-4 calculations and by two-dimensional collision probability calculations in a square cell. The former calculations treat the two annular cells considered in the preceding paragraph, but the latter calculation refers only to the light water cell. The calculations on the two annular cells were performed by means of FLURIG-4 using the option of perfectly reflecting boundary, table 7.1 and 7. 2. Calculations with a white boundary were described in paragraph 6, and the two-dimensional calculations on the light water cell are described in paragraph 10. The flux ratios obtained in the calcu ­ lations are shown in table 7.3. The values in the table from ABH, and Sn calculations were given by Thie [40] . The results for the heavy water cell are very little dependent on the boundary conditions. This indicates that the boundary is not important in a cell with a radius of several mean free paths. In the light water cell on the other hand there is a large difference in flux ratio between the calculations for the two boundary conditions. Assuming that the two-dimensional calculation is correct one finds that the use of a white boundary makes the calculated flux ratio about 1 % too small, while the use of a perfectly reflecting boundary makes it about 11 % too large. Thus the assumption that the white boundary is better is confirmed for this cell. The flux distributions of the two cells with the two boundary con ­ ditions are shown in figures 7.1 and 7.2, and it can be seen there how the use of another boundary condition does not affect the distribution very much in the heavy water cell but causes a great change in the light water cell. 25 -

The figures also show clearly that neither boundary condition gives a zero flux gradient at the boundary. In both cells a white boundary gives a negative slope and a perfectly reflecting boundary gives a positive slope. Thus the supposition of Fukai [48]., that the white boundary in integral transport theory should give a zero flux gradient at the boundary, is not confirmed. If the computing times of tables 7.1 and 7.2 are compared with those of tables 6.6 and 6. 7, it is found that the calculations using a white boundary are considerably faster. With perfect reflection it is necessary to follow the neutron path through a number of reflections, and many more Bickley functions have to be calculated. Kobayashi and Nishihara [ 1 ] summed the contributions of the consecutive re­ flections, but with that method it is no longer possible to use Bickley functions. One more numerical integration must be carried out, and there is no gain in speed. Thus not only does the white boundary give better flux ratios, but it also allows a faster calculation. Finally a few words will be said about our findings concerning the variation of the flux ratio in close packed lattices, when the den­ sity of the moderator is decreased. Clendenin [ 42 ] assumed that the flux ratio should decrease monotonically with the decreasing moderator density. Weiss and Stamm'ler [ 47] found in calculations with K7- TRANSPO that the flux ratio has a minimum and then in­ creases again for very low moderator density. They also gave a proof of the existence of this effect utilising approximate expressions for collision probabilities. We calculated all six lattices of Thie and added two more with lower moderator density [ 39] . The results of the K7-TRANSPO calculations and the FLURIG-4 calculations are shown in table 7.4. Although there is a slight difference in the values for the sixth lattice, our results confirm the conclusion of Weiss and Stamm'ler about the minimum in the flux ratio. The existence of a minimum is not in contradiction to the fact that the flux ratio converges towards unity when the dimensions of the cell decrease. In the calculations described above the fuel rod radius and the fuel density were constant. It is clear that even with no moderator present the flux ratio is larger than unity, if the neutron source is located in the empty space between the rods. — 26 -

8. ANISOTROPIC SCATTERING IN ANNULAR GEOMETRY

Neutron scattering in the laboratory system is in general ani ­ sotropic, but to take due account of more than the isotropic part of scattered neutrons in a calculation often implies complications. It is customary to use the so called transport correction, that is the scattering cross section is reduced by the factor (1-p), where |i is the average of the cosine of the scattering angle. This measure is unam- bigous in a one-group problem, but in multi-group problems it is not immediately clear how one single transport correction could be applied. We shall not, however, discuss this last problem, but instead we shall study how the transport correction works in one-group cell calculations. The annular code FLURIG-4 allows linear anisotropic scattering. This option was used for both the cells defined earlier. The one-group data refer to a thermal group, and in order to demonstrate the effect of anisotropic scattering {1 of the moderator (and of the coolant in the heavy water cell) was varied rather arbitrarily from 0 to i/3 in the heavy water cell and from 0 to 2/3 in the light water cell. The transport cross section was kept constant, that is Sgo was varied so that (l-|i)l!go was constant. In this way the calculations would give a constant flux ratio, independent of |i, if the transport correction gave exact results. The results of the calculations are given in tables 8.1 and 8.2. Regarding accuracy it is found that IA=0, IM=5 is sufficient for four correct decimals in the flux ratio. However, the convergence with the number of points is slower when |i is larger, so the flux ratios obtained are less accurate than before. Figure 8.1 shows the variation of the flux ratios with (i. The ratios are not constant, but the variation is less than 2 %. Anisotropic scattering has been taken into account in multi­ group calculations on light water cells by Takahashi (2). He used gene ­ ralised first-flight collision probabilities in the code FIRST II, which can be considered as a generalisation of the THERMOS code to cover linear anisotropic scattering. Takahashi calculated the disadvantage factor for dysprosium activation in a series of uranium-water lattices. In all cases results from the calculations with linear anisotropy and calculations using the so called Honeck's transport correction agree quite well, the difference being less than 1.5 % - 27 -

Thus - regarding the effect of linearly anisotropic scattering in the thermal region - both Takahashi's calculations and ours indicate that for practical reactor calculations the use of transport corrected cross sections in a method assuming isotropic scattering should be sufficient. As said before the inclusion of linear anisotropic scattering in integral transport theory necessitates the calculation of the neutron current. The current can of course be calculated also for isotropic scattering. The result of a calculation of the current in the standard light water cell is shown in figure 8. 2. The current has been multiplied by 2 rr r to give the total incurrent at a certain radius. ' Since there is no external neutron source in the fuel the total in­ current at any r should be equal to the absorption inside r. To check the accuracy we added a very thin region between the fuel and the moderator (thickness 0. 001 cm). The total incurrent at that point was 0.81244. The absorption in the fuel rod obtained from the flux values was 0.81230. The difference is only about 0.02 %. The calculation used five points in the fuel and five points in the moderator. Outside the rod the incurrent decreases again. The white boundary implies a zero current at the boundary, and as can be seen in the figure the current extrapolates well to the value zero at the boundary.

9. CLUSTER GEOMETRY

Two types of two-dimensional geometry have been dealt with, both by means of collision probabilities. The first one is the cluster geometry typical for heavy water lattices. Two codes have been written for this geometry, the one-group code CLUCOP, which calculates the flux distribution for a fixed source, and the multi-group code CLEF. The calculations described here were made by means of CLUCOP. The thermal flux distribution in a Marviken boiler cell was chosen as a sample case. The geometrical details of the cell are shown in figure 9.1. The division in flat flux subregions is typical for a cluster calculation, although the full generality of the CLUCOP code has not been required, radial and azimuthal subdivision of both the pins and the coolant are allowed. “ 28 -

Complete geometrical and cross section data of the cell are given in tables 9.1 and 9.2. The accuracy of the collision probabilities is determined by the number of linear (L) and angular (V) steps in the integrations that give the collision probabilities [ 11 ] (see also paragraph 3 of the present re­ port). In the sample case the CLUCOP type integration is performed up to radius 7.2 cm (inner radius of the shroud), so the size of the linear interval is 7. 2/L. Because of the symmetry of the cluster, the angular integration is performed over 36°, and the size of the angular interval is 36°/V. The time needed for the calculation of a collision probability matrix is roughly proportional to LxV, and to the square root of the number of regions. A series of calculations was made with different values of L and V. Table 9.3 shows what values were used and what the computing times were in each case. The mean fluxes of the regions are given in table 9. 4. Figure 9.2 shows the flux distribution in the cluster and part of the moderator along a radius that passes through one rod from each ring. The flux values were taken from the calculation with V=l6, L=32. The figure shows what detail in the flux picture can be achieved in a calculation of this kind, for exemple the flux depression in the fuel re­ lative to the canning and the coolant, and the flux gradient in the fuel rods. The most accurate calculation with V=l6, J-.= 32 was taken as "exact" and the relative "errors" of the mean fluxes from t]ie other calculations were determined. Table 9.5 shows the four largest "errors" for each calculation together with the number of the region in which it occurs. The maximum absolute "error" in the calculation of the next highest accuracy, V=8, L=24, is 3.7 %. In calculations of lower accuracy there are errors as large as 12 %. The large errors all occur in canning regions (regions number 2, 6, 7, 12, 13, 18, and 19 are canning), which are small and not so well covered in the in­ tegration. From this series of calculations one can draw the following con ­ clusions. A calculation wifh rather small L and V gives surprisingly - 29 -

good results, sufficient for many practical applications. When the accuracy of integration is improved, the computing time increases ra ­ pidly, and it is expensive to follow the convergence towards flux values of very high accuracy. Unfortunately,it is not practical to proceed to smaller sub­ regions in the cell to study if the division of figure 9.1 is adequate. It should be possible to decrease the size of the subregions in one part of the cell at the expense of lost detail in other parts, for example to in­ vestigate the azimuthal flux variation in the shroud. However, the comparison between multi-group calculations and measured thermal activations by Jons son and Pekarek [ 3 ] indicate that the division used is sufficient for practical purposes. Of course, this does not apply to flux distributions in the resonance region.

10. RECTANGULAR GEOMETRY

The second two-dimensional geometry is an x-y-geometry with a rectangular unit cell. However, the diagonal of the rectangle is as ­ sumed to be an axis of symmetry, so the elementary cell is in fact a triangle, which is either half of a square or one twelfth of a hexagon. A one-group code, BOCOP, calculates the flux distribution in a cell of this type. A rectangular mesh is used for dividing the cell into sub­ regions, and the prescribed source density is constant in each sub­ region. All calculations described in this paragraph were made by means of BOCOP. Collision probabilities between regions are calculated by means of the technique described by Carlvik [ 11 ] . The accuracy of the col ­ lision probabilities is determined by the interval between the lines, Ay, and the number of directions considered, Ncp. Since the boundaries are perfectly reflecting, it must be specified how many mean free paths the neutrons shall be followed, OPMAX, just as in the code FLUSLA for plane geometry. The flux ratio $ /$ was calculated for the third Thie cell, m r that is the light water cell of paragraph 6. The cross section data are given in table 6.1 The correct square cell boundary was used, but the - 30 -

circular boundary of the fuel rod had to be approximated by straight line segments. Three meshes were used as shown in figures 10.1 to 10.3. For each mesh the polygon representing the fuel rod was chosen so that the corners of the polygon were all at the same distance from the circle, and so that the area of the rod was conserved. The rest of the mesh was chosen to give small regions where the flux gradient was thought to be large, and large regions where it was thought to be small. The computing time for a case is almost exactly proportional to Ncp and to ^. It is further roughly proportional to the square root of the number of regions in the mesh, and it is dependent upon OPMAX in about the same way as is the computing time of FLUSLA. Computing times for two-dimensional calculations are neces­ sarily much longer than for one-dimensional geometries, so it is not the accuracy of the Bickley routines, but rather the economical factor, \ which sets a limit to the accuracy that can be obtained. It can be mentioned that 7. 3 minutes were used for the most accurate 31 region calculation of table 10.1 (Ay=0. 035, Ncp=l6). Calculated flux ratios with various numbers of regions, various Ncp and Ay are given in table 10.1. OPMAX=6 was used in the calcu ­ lations, and a few check calculations with OPMAX=9 showed that the value 6 was sufficient. Figures 10.4 and 10.5 show the convergence of the flux ratio in 6 and 15 region calculations respectively. Each curve in the diagrams corresponds to a constant value of Ncp/Ay, that is the points on a certain curve represent about equal computing times. The abscissa is a measure of how the time is used for the calculation of collision proba ­ bilities, to the right sparse lines in many directions, to the left dense lines in few directions. On can see from the diagrams that the optimum value of the Ncp parameter j^log 0.14 - log is somewhere around 0 or 1. This 4 & Ay ]* suggests that one could best see the convergence of the flux ratio by following the corresponding zig-zag-path in table 10.1, as is shown in figure 10. 6. We estimate that a reasonable extrapolation in this figure gives $ m/=1.150+0. 005, which is the value in table 7.3 of paragraph 7. - 31 -

The flux ratio is not very sensitive to the number of regions, 6, 15, or 31. This is because the overall variation of the flux density in the cell is small (compare the calculation with collision probabilities of paragraph 6, where only two regions gave the flux ratio with an accura ­ cy of 0. 2 %).

11. CONCLUSIONS

We hope that most of what can be learned from the present work has been pointed out in the preceeding paragraphs. Only a few summing inferences will be drawn here. It has been found that integral transport methods are very useful for calculating neutron flux distributions in several types of reactor lattice cells. A typical computing time for one-group flux ratios in a cell with a fixed source is one second on IMB 7044 for an accuracy of 0.1 %, if the calculation is performed in a one-dimensional geometry and with assumed isotropic scattering (with transport corrected cross sections). Integral transport methods are also applicable to systems with more complex geometry, although computing times will necessarily be longer. These methods are, in general, to be preferred to Monte Carlo methods, if the geometry is not too complicated. — 32 —

12. REFERENCES

1. KOBAYASHI, K and NISHIHARA, H The solution of the integral transport equations in cylindrical geometry using the Gaussian quadrature formula J. Nucl. Energy A/B JJ3 (1964) 513-522 ,

2. TAKAHASHI, H The generalized first-flight collision probability in the cylindri- calized lattice system Nucl. Sci. Eng. 24 (1966) 60-71

3. JONSSON, A and PEKAREK, H Multi-group collision probability theory in cluster geometry. Comparison with experiments Paper to ’’Reactor Physics in the Resonance and Thermal Regions ’’ ANS National Topical Meeting, San Diego 7-9.2.1965

4. CARLVIK, 1 Integral transport theory in one-dimensional geometries. 1966. (AE-227)

5. HONECK, H C The distribution of thermal neutrons in space and energy in reactor lattices. Part I: Theory Nucl. Sci. Eng. 8 (i960) 193-202

6. TAKAHASHI, H First flight collision probability in lattice systems Reactor Science (J. Nucl. Energy, A).dJL (I960) 1-15

7. MOTHER, A Stosswahrscheinlichkeiten in Zylindergeometrie Nukleonik^4 (1962) 53-56

8. KIESEWETTER, H Zur Berechnung der Stosswahr scheinlichkeiten in regularen Stabgittern Kernenergie j6 (1963) 106-113

9. Di PASQUANTONIO, F Calculation of first-collision probabilities for heterogeneous lattice cells composed of n media Reactor Sci. Tech. (J. Nucl. Energy A/b) 17 (1963) 67-81

10. PENNINGTON, E M Collision probabilities in cylindrical lattices Nucl. Sci. Eng. 19 (1964) 215-220

11. CARLVIK, I A method for calculating collision probabilities in general cylindrical geometry with applications to flux distributions and Dane off factors U.N. Int. Gonf. on the Peaceful Uses of Atomic Energy, Geneva, 3. 1964. Vol. 2. Geneva 1965, p. 225 - 33 -

12. MULLER, A and LINNARTZ, E Zur Berechnung des the r mi sc hen Nutzfaktors einer zylindrischen Zelle aus mehreren konzentrischen Zonen Nukleonik _5 (l 963) 23-27

13. KALNAES, O, NELTRUP, H and 0LGAARD, P L A recipe for heavy-water lattice calculations. 1964 (Ris/ Report No. 81)

14. BONALUMI, R Neutron first collision probabilities in reactor physics Energia Nucleare _8 (l961) 326-336

15. JONSSON, A One-group collision-probability calculations for annular systems by the method of Bonalumi Reactor Sci. Techn. (J. Nucl. Energy A/b) 17 (1963) 511-518

16. HYSLOP, J First-collision probabilities associated with linearly anisot ­ ropic flux distributions in cylindrical geometry Reactor Sci. Techn. (J. Nucl. Energy A/B) _17 (1963) 237-244

17. STAMM'LER, R J J K-7 THERMOS (Neutron thermalization in a heterogeneous cylindrical!/ symmetric reactor cell). 1963 (KR-47)

18. WEXLER, G Blackness and collision probabilities in complex fuel elements II. 1963 (TNPG/PD1. 571)

19. HONECK, H C The calculation of the thermal utilization and disadvantage factor in uranium water lattices Light Water Lattices, Vienna 1962, IAEA Technical Reports Ser. No. 12) 233-265

20. FUKAI, Y First-flight collision probability in moderator-cylindrical fuel ' system Reactor Sci. Techn. F7 (1963) 115-120

21. SAUER, A Thermal utilization in the square lattice cell J. Nucl. Energy A/B _18 (1964) 425-447 22. FUKAI, Y Validity of the cylindrical cell approximation in lattice cal ­ culations J. Nucl. Energy A/B 18 (1964) 241-259 - 34 -

23. WILLIS, I J A method for determining mean thermal flux ratios in a reactor lattice using first-flight collision probabilities J. Nucl. Energy A/B 1_9 (1965) 7-16

24. BEARD WOOD, J E, CLAYTON, A J and PULL, I C The solution of the transport equation by collision probability methods Proceedings of the Conference on the Application of Computing Methods to Reactor Problems, May 1965 (ANL-7050)

25. LESLIE, D C and JONSSON, A The calculation of collision probabilities in cluster-type fuel elements Nuclear Sci. Eng. _23 (1965) 272-290

26. BICKLEY, W G and NAYLER, J A short table of the functions Ki (x) from n=l to n=l6 Phil. Mag. Ser. 7, 20 (1935) 349-347

27. MULLER, G M , z«x Table of the function Kj (x)=— \ u nK (u) d u. 1954 (HW-30323) n xn Jo °

28. DANIELSEN, T M, HA VIE, T and STAMM'LER, R J J On the numerical calculation on Bickley functions. 1963 (KR-55)

29. CLAYTON, A J Some rational approximations for the KL and Ki, functions J. Nucl. Energy A/B, _18 (1964) 82-84

30. TOLLANDER, B Private communication

31. ABRAMOWITZ, M and STEGUN, I A (eds) Handbook of mathematical functions with formulas, graphs and mathematical tables. Wash. D. C. 1964 (National Bureau of Standards, Appl. Mathematics Ser. 55)

32. THEYS, M H Integral transport theory of thermal utilization factor in in­ finite slab geometry Nucl. Sci. Eng. J7 (i960) 58-63

33. MENEGHETTI, D Discrete ordinate quadratures for thin slab cells Nucl. Sci. Eng. 14 (1962) 295-303

34. FERZIGER, J H and ROBINSON, A H Transport calculations of the disadvantage factor Trans. Am. Nucl. Soc. _7 (1964) 12-13 — 35 —

35. ROBINSON, A Private communication (1966)

36. PULL, I C The solution of equations arising from the use of collision pro ­ babilities. 1963 (AEEW-M 355) CLAYTON, A J The programme PIP1 for the solution of the multigroup equations of the method of collision probabilities. 1964 (AEEW-R 326)

37. CARLVIK, I Monoenergetic critical parameters and decay constants for small spheres and thin slabs, 1967 (AE-273) ,

38. HAGGBLOM, H Private communication

39. CARLVIK, I On the use of integral transport theory for the calculation of the neutron flux distribution in a Wigner-Seitz cell Nukleonik 8 (1966) 226-227

40. THIE, J A Failure of neutron transport approximations in small cells in cylindrical geometry Nucl. Sci. Eng. 9 (1961) 286-287

41. NEWMARCH, D A Errors due to the cylindrical cell approximation in lattice calculations. I960 (AEEW-R 34)

42. C LEND ENIN, W W Effect of zero gradient boundary conditions on cell calculations in cylindrical geometry Nucl. Sci. Eng. 14 (1962) 103-104

43. POMRANING, G C The Wigner-Seitz cell: a discussion and a simple calculational method Nucl. Sci. Eng. 1J (1963) 311-314

44. SIMMS, R, KAPLAN, I, THOMPSON, T J and LANNING, D D The failure of the cell cylindricalization approximation in closely packed uranium/heavy-water lattices Trans. Am. Nucl. Soc. _7 (1964) 9 - 36 -

45* HONECK, H C The calculation of the thermal utilization and disadvantage factor in uranium/water lattices Nucl. Sci. Eng. _18 (1964) 49-68

46. HONECK, H C Some methods for improving the cylindrical reflecting boundary condition in cell calculations of the thermal neutron flux Trans. Am. Nucl. Soc. j> (1962) 350-351

47. WEISS, Z and STAMM'LER, R J J Calculation of disadvantage factors for small cells Nucl. Sci. Eng. 19 (1964) 374-377

48. FUKAI, Y Zur Randbedingung mit isotroper Reflexion bei der Berechnung des zylindrischen Zellenrandes Nukleonik J7 (1965) 144-152 I : Fuel Moderator j !No source Constant .source j 1 jXt = 0.717 It= 2.33 | ii o o

jX«=0320 r L a b ! r a = 0.1 b= 3.5 * a j i , 02 I 0.3 i j 0.4 1 Theys' lattice

[Constant ! 1 j source Non - source region ; j region i I, = 0.20 It= 0.07 1^0.12 Ia= 0.028 i L a . . b i o = 0.32 b= 7 x a 3.2 i Meneghetti ’s lattice

Figure 4.1 Data for slab lattices. Half-cell dimensions are shown. Fuel half 1.0982 thickness

0.1 cm 1.0980

1.2332

0.2 cm 1.2330

1.4126 /

1.412 2

1.411 8

1.6420

1.6410

1.6400

2 3 4 5 Fuel —1------1------1------1------10 9 8 7 Moderator

Figure 4. 2 Flux ratios in the lattices of Theys. Variation of the distribution of points. 1.099 Fuel half 1.098 thickness 0.1 cm 1.234 0.2cm 1.232

1.414

0.3cm 1.412

1.644

1.642 0.4 cm 1.640

1.638

Number of points

8 12 15 20 J I I I—

Figure 4. 3 Flux ratios in the lattices of Theys. Variation of the number of points. f

Ferziger- Robinson

Diffusion theory

. Fuel half thickness

Figure 4. 4

Flux ratios in the lattices of Theys by various methods. 1.123 \ Thin cell

1.121 e

2.815

2.810

2.805 *s $ns 2.800 Thick cell

2.795

2.790

Number of points 2.785 5 10 20 J______I------L_

Figure 4. 5 Flux ratios in the lattices of Meneghetti. 1.989_| \ Radius CRIPASS in m.fp.

1.988. FLUBAG 1.0

1.987. /

1.397, \ 1.396. \CRIPASS • — .o....— o •— 2.0 /■ FLUBAG 1.395.

1.394. /

1.140,

\ 1.139. X^RIPASS 4.0 1.138 . FLUBAG

1.137 . Number of points or order of polynomial

0 10 20

Figure 5.1 Critical parameter* of homogeneous sphere*. FLU BAG 8 2.0 . v points 12 v \ 16 \ CRIPASS

1.5

1.0 .

0.5 .

X r. 0.0 0.0 0.5 1.0

Figure 5. 2 Flux distribution in a critical homogeneous sphere of radius 2 mean free paths. 1.025 „

1.020 _

Number of 1.015 „ points core

14 13 12 11 10 9 8 reflector

Figure 5. 3 of FR-0 calculated by means of FLUBAG Variation of the distribution of points. 1.04 .

1.03 .

1.0 2 . FLUBAG

S4------HATTEN

Number of points in FLUBAG calculation

Figure 5. 4 °f FR-0 calculated by means of FLUBAG, S4, and HATTEN. 1.750 10 points

V 1.750

1.745 fuel region 3 4 5 6 H------1------1------1------6 5 4 3 moderator

1742 20 points

1740 fuel region 7 8 9 10 11 12 h------h 12 11 10 9 8 7 moderator

Figure 6. 1 Flux ratios 4* /9, in the heavy water cell, m i 1 Variation of the distribution of points. \

10 points

1.330 __ Number of points fuel region

moderator

1.338 ^ 20 points

fuel region

moderator

Figure 6. 2 Flux ratio 4>g /4^ in the heavy water cell. Variation of the distribution of points. 1.144 I 10 points 1.142 .

1.140 _ Number of points fuel region 3 4 5 6 7 H------1------1------1------1------7 6 5 4 3 moderator

1.142 _ 20 points

1.140 _

fuel region 7 8 9 10 11 12 13 H------1------1------1------1------h------h 13 12 11 10 9 8 7 moderator

Figure 6. 3 Flux ratio in the light water cell. Variation of the distribution of points. 1.80.

1.70

o

1.60. number of space points or regions

—I------r- 0 10 20 30

1.30. /• CP

number of space points or regions

Figure 6.4

Convergence of flux ratios in the heavy water cell. £m

1.150.

1.14 5.

number of space points or regions

1.13 5.. —i------r~ 0 10 20

Figure 6.5

Convergence of the flux ratio in the light water cell. Reflecting

Shroud White

Moderator Canning Coolant

Radius

Figure 7. 1 Flux distribution in the heavy water cell with white and perfectly reflecting boundary. Reflecting

White

Moderator

0.6 cm

Radius

Figure 7. 2 Flux distribution in the light water cell with white and perfectly reflecting boundary. 1.74 _

1.73 „

1.72 „

$ 5 f f

Heavy water cell

1.13 .

1------1------1------1------1------1------1— 0/9 1/9 2/9 3/9 4/9 5/9 6/9 Light water cell

Figure 8.1 Flux ratios in the heavy water and light water cells. Variation with while keeping the transport cross section constant. , <|>(r) 2TTrJ(r)

Moderator

0.9

0.81244 0.8

0.6 = 0.81230

0.4

0.2

0.0

Radius

Figure 8. 2 Flux and incurrent in the light water cell.

T3 3 O

CO

7 z cn c *c

4 -> -4 —' c V)

Figure 9. 2 Flux distribution in the inner part of the cluster cell 0.5715

0.2348

CO o in o CN CO

d d d 0.5715

Figure 10.1 6 region mesh of square cell. 0.5715

0.1894

CD on CD o o o O o cxi CXj

o o' o O d 0.5715

Figure 10. 2 15 region me eh of square cell. 0.5715

0.2522

0.1627

CXI o o cr> CO CD If) CO CO o o CM CM CO

o CD CD CD CD CD o 0.5715

Figure 10. 3 31 region mesh of square cell. 014 Ay

Figure 10. 4 Flux ratio in square cell calculated for 6 regions. Figure 10. 5 Flux ratio in square cell calculated for 15 regions. Figure 10. 6 Convergence of the flux ratio in the square cell. Table 4.1

Flux ratios ^/♦j. i* the lattices of Theys

Variation of the distribution of points.

5 Computing Half thickness 0PMAX Number of points ^moderator of fuel time Total Fuel Mode ­ *fuel (cm) rator (millimins)

0.1 10.0 12 2 10 1.098163 39 3 9 1.098082 40 4 8 1.098116 38 5 7 1.098198 40

0.2 12 2 10 1.233120 23 3 9 1.232969 25 4 8 1.233066 24 5 7 . 1.233278 23

0.3 12 2 10 1.411996 20 3 9 1.411839 19 4 8 1.412117 19 5 7 1.412644 19

0.4 12 2 10 1.639596 16 3 9 1.639620 17 4 8 1.640382 17 5 7 1.641704 16

0.4 16 2 14 1.638765 28 3 13 1.638343 27 4 12 1.638433 28 5 11 1.638669 28 Table 4.2

Flux ratios in the lattices of Theys

Variation of OFMAX

Half thickness OFMAX Number of points ^moderator Computing of fuel time Total Fuel Mode ­ ^fuel (cm) rator (millimins)

0.1 6.0 8 2 6 1.098542 14 8.0 1.098426 16 10.0 1.098423 18

6.0 12 3 9 1.098201 29 8.0 1.098086 36 10.0 1.098082 39

6.0 15 3 12 1.098155 43 8.0 1.098039 55 10.0 1.098036 6l

6.0 20 4 16 1.098122 77 8.0 1.098006 98 10.0 1.098002 106

0.4 6.0 8 2 6 1.644370 8 8.0 1.644311 8 10.0 1.644311 8

6.0 12 3 9 1.639676 15 8.0 1.639620 17 10.0 1.639620 17

6.0 15 3 12 1.638574 22 8.0 1.638516 25 10.0 1.638516 25

6.0 20 4 16 1.638006 38 8.0 1.637947 43 10.0 1.637947 42 Table 4.3

Flux ratios in the lattices of Theys

Variation of the number of points

Half thickness OFMAX Number of points ^moderator Computing of fuel time Total Fuel Mode ­ ^fuel (cm) rator (millimins)

0.1 10.0 8 2 6 1.098423 18

12 3 9 1.098082 39

15 3 12 1.098036 6l

20 4 16 1.098002 106

0.2 8 2 6 1.233815 12

12 3 9 1.232969 24

15 3 12 1.232832 37

20 4 16 1.232743 64

0.3 8 2 6 1.413813 9

12 3 9 1.411839 19

15 3 12 1.4ll44l 29

20 4 16 1.411217 50

0.4 8 2 6 1.644311 8

12 3 9 1.639620 17

15 3 12 1.638516 25

20 4 16 1.637947 42 Table 4.1+

Flux ratios $ m/?£. in the lattices of Theys

by various methods

Method Half thickness of fuel (cm) 0.1 0.2 0.3 0.4

Diffusion appr. 1.03 1.11 1.24 1.43

Theys (ABH) 1.08 1.20 1.36 1.58

Lathrop , Sg 1.090 1.231 1.410 1.632

Ferziger-Robinson 1.094 1.227 1.401 1.623 Eigen-function expansion

Carlvik , BIT 1.0979 1.2318 1.408 1.629 Table 4.5

Flux ratios 4>g/i^g in the lattices of Meneghetti

Half thickness OPMAX Number of points *S^NS Computing time Source Non-source S N-S Total region region region region (millimins)

0.32 2.24 10 5 3 2 1.122025 21 8 1.122059 19 6 1.122393 15

10 10 6 4 1.121083 87 8 1.121118 73 6 1.121452 57

10 20 12 8 1.121011 360 8 1.121046 299 6 1.121379 234

3.2 22.4 10 5 3 2 2.813663 5 8 2.813668 6 6 2.814217 5

10 10 6 4 2.790584 15 8 2.790590 15 • 6 2.79H32 13

10 20 12 8 2.788610 56 8 2.788615 52 6 2.789158 46

Table $.1

Es + vEf Critical parameter c = for homogeneous spheres tot

IA IM Number Radius in m.f.p. of points 1.0 2.0 4.0

2 5 4 1.98426 1.39107 1.13078

6 1.98705 1.39425 1.13584

8 I.9878 O 1.39515 1.13730

10 1.98808 1.39549 1.13781

12 1.98821 1.39564 1.13807

16 1.98831 1.39577 1.13829

20 1.98834 1.39582 1.13837

Extict (CRIPASs) 1.98839 1.39587 1.13847 Table 5.2 Zs + vl Critical parameter c = —z------for homogeneous spheres tot Variation of IA and IM

Radius IA IM Number of points =. in m.f.p vtotu£f

1.0 2 5 10 1.988078 4 10 1.988077

2 5 20 1.988348 4 10 1.988344

2.0 2 5 10 1.395485 4 10 1.395489

2 5 ~ 20 1.395820 4 10 1.395820

4.0 2 5 10 1.137806 4 10 1.137805

2 5 20 1.138368 4 10 1.138367 Table 5.3

Data for three-group calculation on FR-0

Group nr Ztot vZf Fission ______spectrum i j = 1 j = 2 j = 3

Core ‘ 1 0.19836 0.07791 0.08552 0.07767 0.00623 0.574 outer ra ­ 2 0.28614 0.02766 0.26091 0.00765 0.393 dius 17 cm 3 0.50872 0.03915 0.47814 0.033

Reflector 1 0.20095 0 0.13609 0.06045 0.00356 outer ra ­ 2 0 dius U7 cm 0.30525 0.29819 0.00559 3 0.45010 0 0.44449 Table 5.4

Three-group calculation of keff of FR-0

Variation of the distribution of points

IA IM Number of points keff Total Core Reflector

2 5 20 6 lit 1.01768

7 13 1.01693

8 12 I.OI698

9 11 1.01766

10 10 1.01897

11 9 1.02106

12 8 1.02424 Table 5.5

Three-group calculations of kef^ of FR-0

Variation of the number of points and of IA and IM

IA IM Number of points keff Computing time Total Core Re­ flector (millimins)

2 5 12 4 8 1.03515 516 4 10 1.03514 538

2 5 15 5 10 1.02469 581

2 5 20 7 13 1.01693 751 4 10 1.01692 8l4

2 5 2k 8 16 1.01436 928

2 5 30 10 20 1.01236 124 4 10 1.01236 139

2 5 36 12 2k 1.01139 162

S 1.01230 4 Hatten 1.0105

Table 6.1

Data for annular cells

Heavy water cell Light water cell Region nr 1 2 3 1 2

Material Nat. U + can + Shroud Moderator Enriched Moderator coolant (DgO) D2° uranium H2°

Outer radius 7.2 7.35 13.5631 0.381 0.644869 (cm)

Etot O.ltlSpUO 0.095709 0.401884 0.780 1.0618

Zs (cm-1) 0.366520 0.082500 0.401817 0.387 1.053

Source dens, 0.67695 0 1.0 0.0 1.0 (cm-3 gec"l) Table 6.2

Flux ratios in the heavy water cell • DIT-method

Variation of the distribution of points IA=0, IM=5

Number of points Computing Y4 shroud ^moderator time Total Fuel Shroud Mode ­ ^fuel *fuel region rator (millimins)

10 3 1 6 1.33950 1.75045 12 10 4 1 5 1.33604 1.74721 11 10 5 1 4 1.33306 1.74776 10 10 6 1 3 1.32770 1.75237 10

20 7 1 12 1.33804 1.74135 4l 20 8 1 11 1.33783 1.74112 39 20 9 1 10 1.33765 1.74103 38 20 10 1 9 1.33743 1.74105 37 20 11 1 8 1.33714 1.74120 37 20 12 ; 7 1.33669 1.74151 35

21 7 2 12 1.33801 1.74135 44 21 8 2 11 1.33778 1.74112 43 21 9 2 10 1.33758 1.74104 4i 21 10 2 9 1.33733 1.74106 4l 21 11 2 8 1.33701 1.74121 40 21 12 2 7 1.33652 1.74152 39 Table 6.3

Flux ratios in the light water cell DIT-method

Variation of the distribution of points IA=0, IM=5

Number of points 2 Computing ^moderator time Total Fuel Moderator ^fuel (millimins)

10 3 7 1.14240 11 4 6 1.14145 11 5 5 1.14110 10 6 4 1.14103 11 7 3 1.14124 10

20 7 13 1.14083 4i 8 12 1.14079 4o 9 11 1.14077 4o 10 10 1.14075 39 11 9 1.14075 38 12 8 1.14075 38 13 7 1.14075 37 Table 6.4

Flux ratios in the heavy water cell DIT-method

Variation of Ik and IM

Number of points IA IM 5 Computing ^shroud moderator Total Fuel Shroud Mode ­ time *fuel *fuel region rator (millimins)

0 5 11 5 1 5 1.33556 1.74544 13 2 5 1.33550 1.74535 15 0 8 1.33546 1.74544 13 2 8 1.33552 1.74535 17

0 5 22 10 2 10 1.33751 1.74091 45 2 5 1.33751 1.74090 53 0 8 1.33722 1.74092 54 2 8 1.33722 1.74091 63 Table 6.5

Flux ratios in the light water cell DIT-method

Variation of IA and IM

Number of points IA IM ^moderator Computing time Total Fuel Mode ­ ^fuel rator (millimins)

0 5 10 5 5 l.lUllO 10 2 5 l.l4o86 13 0 8 l.lUllO 11 2 8 1.14085 14

0 5 20 10 10 1.14075 39 2 5 1.14073 43 0 8 1.14075 47 2 8 1.14073 55 Table 6.6

Flux ratios in the heavy water cell DIT-method

Variation of the number of points IA«0, IM*5

Number of points A _ ♦ . . Computing Tshroud moderator Fuel Shroud Mode ­ time region rator ^fuel ^fuel (millimins)

3 1 1 1 1.960U5 3.10756 It ' 5 2 1 2 1.32626 1.80053 It 7 3 1 3 1.32936 1.75907 6 9 k 1 k 1.333^5 1.7k9k9 9 13 6 1 6 1.33656 1.7k 339 16 17 8 1 8 1.33735 1.7kl60 27 21 10 1 10 1.33758 1.7k09l kl 22 10 2 10 1.33751 1.7k09l k5 26 12 2 12 1.33763 1.7k059 6k 30 lk 2 lk 1.33768 1.7k0k2 87 3k 16 2 16 1.33769 1.7k032 113 35 16 3 16 1.3377% 1.7k032 122 Table 6.7

Flux ratios in the light water cell DIT-method

Variation of the number of points IA=0, IM=5

Number of points (4 Computing moderator Total Fuel Moderator time ^fuel (millimins)

2 1 1 1.19564 3 4 2 2 1.14625 3 6 3 3 1.14262 5 8 1* 4 1.14152 7 12 6 6 1.14093 15 16 8 8 l.l4o8o 24 20, 10 10 1.14075 39 2k 12 12 1.14074 57 28 14 Ik 1.14073 78 32 16 16 1.14072 105 Table 6.8

Flux ratios in the heavy water cell CP-method

Variation of the number of regions

Number of regions *shroud ^moderator Total Fuel Shroud Mode­ *fuel *fuel region rator

3 1 1 1 1.24131 1.47747 7 3 1 3 1.29596 1.64345 9 4 1 4 1.30986 1.67765 11 5 1 5 1.31796 1.69691 17 8 1 8 1.32861 1.72135 22 10 2 10 1.33149 1.72773 30 14 2 14 1.33419 1.73359 Table 6.9

Flux ratios in the light water cell CP-method

Variation of the number of regions

Number of regions ^moderator Total Fuel Moderator ^fuel

2 1 1 1.13778 It 2 2 1.13929 6 3 3 1.14000 8 4 4 1.14028 10 5 5 1.14038 14 7 7 1.14058 20 10 10 1.14062 Table 7.1

Flux ratios in the heavy water cell Perfectly reflecting boundary DIT-method

Number of points IA IM A A Computing Tshroud moderator Total Fuel Shroud Mode ­ time ^fuel region rator *fUel (millimins)

0 5 13 6 1 6 1.33698 1.74982 32 22 10 2 10 1.33793 1.74730 97 34 16 2 16 1.33811 1.74670 251

2 8 34 16 2 16 1.33781 1.74670 382 Table 7.2

Flux ratios in the light water cell Perfectly reflecting boundary

DIT-method

Number of points Computing IA IM ^moderator time Total Fuel Mode ­ ^fuel rator (millimins)

0 5 12 6 6 1.28456 88 20 10 10 1.28432 283 32 l6 16 1.28428 802

2 8 32 16 16 1.28427 1270 Table 7.3

Flux ratios in the two annular cells

Influence of boundary conditions

Type of boundary Heavy water cell Light water cell A $ *shroud ^moderator moderator ^fuel ^fuel ^fuel

Circular white 1.3373 1.7W3 1.1U07

Circular reflecting DIT 1.338 1.7^7 1.281*3 ABH 1.155 . P1 1.036 P3 1.207 DSN-Sq 1.291 SNG-Sq 1.276 SNG-S^ 1.276

Square reflecting 1.15 Table 7.4

Flux ratios in the lattices of Thie

Lattice number % void K7-TRANSP0 FLURIG-4

1 0.25 1.157 1.158

2 0.50 1.149 1.152

3 0.25 l.l4l i.l4i

4 0.50 1.138 1.139

5 0.75 1.136 1.137

6 0.95 1.149 1.139

7 97.5 1.139

8 99.5 l.l4i

Lattice number 3 is the standard light water lattice of this report (table 6.1). Numbers 3 to 8 are geometrically equal. Numbers 1 and 2 differ only by a larger cell radius, 0.859825 cm. Table 8.1

Flux ratios in the heavy water cell

Anisotropic scattering

Number of points I Computing a IA IM shroud ^moderator Total Fuel Shroud Mode ­ time region rator ^fuel ^fuel (millimins)

1/9 0 5 22 10 2 10 1.33556 1.73719 143 34 16 2 16 1.33536 1.73499 402

2/9 0 5 22 10 2 10 1.33357 1.73388 l4l 34 16 2 16 1.33289 1.72950 400

3/9 0 5 22 10 2 10 1.33164 1.73114 l4l 2 8 22 16 2 16 1.33129 1.73114 180

0 5 34 10 2 10 1.33031 1.72392 397 2 8 34 16 2 16 1.32993 1.72393 498 Table 8.2

Flux ratios in the light water cell

Anisotropic scattering

Number of points I Computing IA IM moderator V Total Fuel Mode ­ time rator ffuel (millimins!

1/9 0 5 20 10 10 1.13924 110 2/9 1.13742 109 3/9 1.13515 108 4/9 1.13225 109 5/9 1.12839 107 6/9 1.12296 107

6/9 2 8 20 10 10 1.12293 138 0 5 32 16 16 1.12288 329 2 8 32 16 16 1.12287 413 Table 9.1

Geometrical data for rod cluster cell

Fuel rod, 1.2 % enriched UO,

outer radius 0.63 cm

Canning inner radius 0.63 It outer radius 0.735 M

Radii of rod rings ring nr 1 0.00 II 2 2.16 Tl 3 4.31 IV 4 6.4l II

Shroud tube inner radius 7.20 IV outer radius 7.35 II

Subregions, radii in coolant moderator

1.25 cm 7.50 cm 2.16 " 8.01 II 3.30 " 8.669 II 4.31 " 9.329 II 5.40 " 9.988 II 6.4l 10.648 II 11.307 II 11.967 It 12.499 II 13.031 n 13.563 tr Table 9.2

Cross section data for rod cluster cell

Material %a Zs Source density

Fuel 0.2235 0.37264 0.0

Canning 0.012973 0.071491 0.0

Coolant , moderator 0.000067 0.40182 1.0

Shroud 0.013209 0.08250 0.0 Table 9.3

CLUCOP calculations

Computing times in minutes

L 12 16 24 32 V

2 0.69 0.84 1.16

4 l.l4 1.46 2.07

8 2.07 2.69 3.92

16 10.12 Table 9*4

Region fluxes in the rod cluster cell

Region Area L=12 16 24 12 16 24 12 16 24 32 number (cm2) V= 2 2 2 4 4 4 8 8 8 16

1 1.2469 44.36 44.87 43.40 43.88 44.05 43.49 44.19 43.29 43.72 43.59 2 0.4.503 42.70 43.34 45.96 45.98 47.48 46.82 46.12 47.72 46.32 46.08 3 3.2116 47.74 49.81 48.77 49.32 49.36 49.16 49.4i 49.01 49.20 48.87 4 3.1172 46.47 45.50 46.59 47.31 48.62 47.06 47.27 47.37 46.83 47.19 5 3.1172 46.83 43.36 45.56 45.06 44.51 45.53 44.70 44.60 45.17 45.50 6 1.1257 47.91 55.96 52.73 50.75 51.97 51.38 48.89 52.11 50.91 50.61 7 1.1257 42.42 51.99 47.90 48.64 46.35 49.67 49.24 48.00 49.92 48.13 8 5.8131 50.72 51.44 50.11 50.35 50.88 50.66 50.60 50.69 50.59 50.31 9 15.004 53.91 55.35 54.4i 54.14 54.94 54.50 54.39 54.43 54.47 54.25 10 6.2345 54.91 54.92 54.72 54.05 54.91 54.20 54.11 53.70 54.44 54.31 11 “ 6.2345 51.49 52.99 52.25 51.63 51.43 51.58 51.70 51.30 51.76 51.73 12 2.2513 59.67 60.ll 58.45 58.29 60.69 59.70 60.21 60.57 59.52 59.27 13 2.2513 54.54 54.45 54.08 54.59 56.52 55.35 55.68 56.06 55.43 55.03 l4 15.968 56.72 57.63 56.90 56.92 57-80 57.27 57.17 57.13 57.13 57.02 15 24.46 63.06 63.52 63.10 62.86 64.02 63.15 63.09 63.01 63.34 63.12 16 9.3517 67.36 67.05 67.56 67.50 67.33 68.50 68.60 67.31 67.51 67.71 17 9.3517 62.30 62.82 62.22 63.12 62.28 62.86 61.92 62.86 62.73 62.48 18 3.3770 75.15 76.64 74.19 72.94 75.77 74.80 73.44 72.59 74.73 73.93 19 3.3770 68.36 67.74 66.11 62.54 69.11 65.87 65.26 65.38 66.62 65.96 20 25.054 68.45 68.97 68.19 67.93 68.82 68.28 68.03 67.95 68.30 68.19 21 20.739 76.99 77.79 77.57 77.38 77.95 77.45 77.38 77.31 77.42 77.31 22 6.8565 80.75 81.40 81.00 80.90 81.49 81.02 80.86 80.91 81.10 80.98 23 6.9979 82.51 83.10 82.63 82.54 83.11 82.63 82.46 82.49 82.67 82.54 24 24.850 86.60 87.17 86.60 86.61 87.19 86.71 86.54 86.56 86.74 86.62 25 34.531 92.44 93.00 92.52 92.45 93.05 92.56 92.39 92.41 92.58 92.47 26 37.318 97.71 98.28 97-79 97.72 98.32 97.83 97-66 97.69 97.86 97.74 27 39.992 101.96 102.52 102.04 101.97 102.57 102.08 101.91 101.93 102.10 101.99 28 42.788 J-05.34 105.90 105.42 105.35 105.95 105.46 105.29 105.31 105.48 105.37 29 45.454 107.95 108.51 108.03 107.96 108.56 108.07 107.90 107.93 108.09 107.98 30 48.257 109.86 110.42 109.94 109.86 110.47 109.98 109.81 109.83 110.00 109.89 31 40.891 111.02 111.58 111.10 111.03 111.63 111.14 110.98 111.00 111.17 111.05 32 42.669 111.59 112.15 111.67 111.60 112.20 111.71 111.55 111.57 111.74 111.62 33 44.447 111.65 112.22 111.73 111.66 112.27 111.78 111.61 111.63 111.80 111.69 Table 9-5

Relative errors in CLUCOP calculation

X\ L 12 l6 24 V \

Region "Error" Region "Error" Region "Error" number % number % number %

2 7 -11.9 6 +10.6 6 + 4.2 2 - 7.3 7 + 8.0 13 - 1.7 6 - 5.3 2 - 5.9 12 - 1.4 19 + 3.7 5 - 4.7 4 - 1.3

4 19 - 5.2 19 + 4.8 7 + 3.2 12 - 1.7 7 - 3.7 4 - 2.7 18 - 1.3 2 + 3.0 2 + 1.6 7 + 1.1 4 + 3.0 6 + 1.5

8 6 - 3.4 2 + 3.6 7 + 3.7 7 + 2.3 6 + 3.0 18 + 1.1 5 - 1.8 12 + 2.2 19 + 1.0 12 + 1.6 5 - 2.0 4 - 0.8 Table 10.1

Flux ratios for the third Thie lattice calculated by means of B0C0P

The three values refer to calculations with 6 regions 15 n 31 ii

0 1 2

Ay 0.14 0.07 0.035

1.1043 1.1023 1.1069 0 4 1.1076 1.1085 1.1089 1.1080 - -

1.1087 1.1307 1.1320 1 8 1.1266 1.1310 1.1349 1.1307 1.1324 -

1.1244 1.1419 1.1423 2 16 1.1266 1.1422 1.1449 - 1.1396 1.1457

1.1284 1.1376 1.1444 3 32 1.1251 1.1366 1.1471 ** — —

LIST OF PUBLISHED AE-REPORTS 244. Physics experiments at the Agesta power station. By G. Apelqvist, P.-A. Bliselius, P. E. Blomberg, E. Jonsson and F. Akerhielm. 1966. 30 p. Sw. 1—200. (See the back cover earlier reports.) cr. 8:-. 245. Intercrystalline stress corrosion cracking of inconel 600 inspection tubes in 201. Heat transfer analogies. By A. Bhattacharyya. 1965. 55 p. Sw. cr. 8:—. the Agesta reactor. By B. Gronwall, L. Ljungberg, W. Htibner and W. 202. A study of the ”384 ” KeV complex gamma emission from plutonium-239. Stuart. 1966. 26 p. Sw. cr. 8:-. By R. S. Forsyth and N. Ronqvist. 1965. 14 p. Sw. cr. 8:—. 246. Operating experience at the Agesta nuclear power station. By S. Sand- 203. A scintillometer assembly for geological survey. By E. Dissing and O. strom. 1966. 113 p. Sw. cr. 8:—. Landstrom. 1965. 16 p. Sw. cr. 8:—. 247. Neutron-activation analysis of biological material with high radiation levels. 204. Neutron-activation analysis of natural water applied to hydrogeology. By By K. Samsahl. 1966. 15 p. Sw. cr. 8:-. O. Landstrom and C. G. Wenner. 1965. 28 p. Sw. cr. 8:—. 248. One-group perturbation theory applied to measurements with void. By R. 205. Systematics of absolute gamma ray transition probabilities in deformed Persson. 1966. 19 p. Sw. cr. 82-. oad-A nuclei. By S. G. Malmskog. 1965. 60 p. Sw. cr. 8:-. k 249. Optimal linear filters. 2. Pulse time measurements in the presence of 206. Radiation induced removal of stacking faults in quenched aluminium. By noise. By K. Nygaard. 1966. 9 p. Sw. cr. 8:-. U. Bergenlid. 1965. 11 p. Sw. cr. 8:—. 250. The interaction between control rods as estimated by second-order one- 207. Experimental studies on assemblies 1 and 2 of the fast reactor FRO. Part 2. group perturbation theory. By R. Persson. 1966. 42 p. Sw. cr. 82-. By E. Hellstrand, T. Andersson, B. Brunfelter, J. Kockum, S-O. London 251. Absolute transition probabilities from the 453.1 keV level in 183W. By S. G. and L. I. TirSn. 1965. 50 p. Sw. cr. 8:—. Malmskog. 1966. 12 p. Sw. cr. 8:-. 208. Measurement of the neutron slowing-down time distribution at 1.46 eV 252. Nomogram for determining shield thickness for point and line sources of 8 and its space dependence in water. By E. Moller. 1965. 29 p. Sw. cr. :—. gamma rays. By C. Jonemalm and K. Malen. 1966. 33 p. Sw. cr. 8:—. 209. Incompressible steady flow with tensor conductivity leaving a transverse 253. Report on the personnel dosimetry at AB Atomenergi during 1965. By K. A. 8 magnetic field. By E. A. Witalis. 1965. 17 p. Sw. cr. :-. Edwardsson. 1966. 13 p. Sw. cr. 8:-. 210. Methods for the determination of currents and fields In steady two­ 254 Buckling measurements up to 250°C on lattices of Agesta clusters and on dimensional MHD flow with tensor conductivity. By E. A. Witalis. 1965. DaO alone in the pressurized exponential assembly TZ. By R. Persson, 13 p. Sw. cr. 8:-. A. J. W. Andersson and C.-E. Wikdahl. 1966. 56 p. Sw. cr. 6:—. 211. Report on the personnel dosimetry at AB Atomenergi during 1964. By 255 Decontamination experiments on intact pig skin contaminated with beta- K. A. Edvardsson. 1966. 15 p. Sw. cr, 8:-. gamma-emitting nuclides. By K. A. Edwardsson, S. HagsgSrd and A. Swens- 212. Central reactivity measurements on assemblies 1 and 3 of the fast reactor son. 1966. 35 p. Sw. cr. 8:—. FRO. By S-O. Londen. 1966. 58 p. Sw. cr. 8:-. 256. Perturbation method of analysis applied to substitution measurements of 213. Low temperature irradiation applied to neutron activation analysis of buckling. By R. Persson. 1966. 57 p. Sw. cr. 8:—. mercury in human whole blood. By D. Brune. 1966. 7 p. Sw. cr. 8:—. 257. The Dancoff correction in square and hexagonal lattices. By I. Carlvik. 1966. 214. Characteristics of linear MHD generators with one or a few loads. By 35 p. Sw. cr. 8:-. E. A. Witalis. 1966. 16 p. Sw. cr. 8:-. 258. Hall effect influence on a highly conducting fluid. By E. A. Witalis, 1966. 215. An automated anion-exchange method for the selective sorption of five 13 p. Sw. cr. 8:-. groups of trace elements in neutron-irradiated biological material. By 259. Analysis of the quasi-elastic scattering of neutrons in hydrogenous liquids. K. Samsahl. 1966. 14 p. Sw. cr. 8:—. By S. N. Purohit. 1966. 26 p. Sw, cr. 8:—. 216. Measurement of the time dependence of neutron slowing-down and therma- 260. High temperature tensile properties of unirradiated and neutron irradiated lization in heavy water. By E. Moller. 1966. 34 p. Sw. cr. 8:—. 20Cr—35Ni austenitic steel By R B Roy and B Solly. 1966. 25 p. Sw. 217. Electrodeposition of actinide and lanthanide elements. By N-E. Barring. cr. 8:-. 1966. 21 p. Sw. cr. 8:-. 261. On the attenuation of neutrons and photons in a duct filled with a helical 218. Measurement of the electrical conductivity of He* plasma induced by plug. By E. Aalto and A. Krell, 1966. 24 p. Sw. cr. 82-. neutron irradiation. By J. Braun and K, Nygaard. 1966. 37 p. Sw. cr. 8:-. 262. Design and analysis of the power control system of the fast zero energy 219. Phytoplankton from Lake Magelungen, Central Sweden 1960—1963. By T. reactor FR-O. By N. J. H. Schuch. 1966. 70 p. Sw. cr. 8:—. Willen. 1966. 44 p. Sw. cr. 8:—. 263. Possible deformed states in 115 ln and ,17ln. By A. Backlin, B. Fogelberg and 220. Measured and predicted neutron flux distributions in a material surround­ S. G. Malmskog. 1967. 39 p. Sw. cr. 10:-. ing av cylindrical duct. By J. Nilsson and R. Sandlin. 1966. 37 p. Sw. 264. Decay of the 16.3 min. mTa isomer. By M. Hojeberg and S. G. Malmskog. cr. 8:—. 1967. 13 p. Sw. cr. 10:-. 221. Swedish work on brittle-fracture problems in nuclear reactor pressure 265. Decay properties of ,<7Nd. By A. Backlin and S. G. Malmskog. 1967. 15 p. vessels. By M. Grounes. 1966. 34 p. Sw. cr. 8:-. Sw. cr. 10: -. 222. Total cross-sections of U, UOa and ThOi for thermal and subthermal 266. The half life of the 53 keV level in 197Pt. By S. G. Malmskog. 1967. 10 p. neutrons. By S. F. Beshai. 1966. 14 p. Sw. cr. 8:—. Sw. cr. 10:-. 223. Neutron scattering in hydrogenous moderators, studied by the time de­ 267. Burn-up determination by hight resolution gamma spectrometry: Axial and pendent reaction rate method. By L. G. Larsson, E, Moller and S. N. diametral scanning experiments. By R. S. Forsyth, W. H. Blackadder and Purohit. 1966. 26 p. Sw. cr. 8:—. N. Ronqvist. 1967. 18 p. Sw. cr. 10:-. 224. Calcium and strontium in Swedish waters and fish, and accumulation of 268. On the properties of the s,y2 ------>- d3y2 transition in mAu. By A. Backlin strontium-90. By P-O. Agnedal. 1966. 34 p. Sw. cr. 8:-. 225. The radioactive waste management at Studsvik. By R. Hedlund and A. and S. G. Malmskog. 1967, 23 p. Sw. cr. 10:—. Lindskog. 1966. 14 p. Sw. cr. 8:—. 269. Experimental equipment for physics studies in the Agesta reactor. By G. 226. Theoretical time dependent thermal neutron spectra and reaction rates Bernander, P. E. Blomberg and P.-O. Dubois. 1967. 35 p. Sw. cr. 10:-. in H2O and DzO. S. N. Purohit. 1966. 62 p. Sw. cr. 8:-. 270. An optical model study of neutrons elastically scattered by iron, nickel, 227. Integral transport theory in one-dimensional geometries. By I. Carlvik. cobalt, copper, and indium in the energy region 1.5 to 7.0 MeV. By B. Holmqvist and T. Wiedling. 1967. 20 p. Sw. cr. 10:-. 1966. 65 p. Sw. cr. 8:—. 228. Integral parameters of the generalized frequency spectra of moderators. 271. Improvement of reactor fuel element heat transfer by surface roughness. By B. Kjellstrom and A. E. Larsson. 1967. 94 p. Sw. cr. 10:-. By S. N. Purohit. 1966. 27 p. Sw. cr. 8:—. 229. Reaction rate distributions and ratios in FRO assemblies 1, 2 and 3. By 272. Burn-up determination by high resolution gamma spectormetry Fission pro ­ duct migration studies. By R. S. Forsyth, W. H. Blackadder and N. Ron­ T. L. Andersson. 1966. 50 p. Sw. cr. 8:—. qvist. 1967. 19 p. Sw. cr. 10:-. 230. Different activation techniques for the study of epithermal spectra, app ­ lied to heavy water lattices of varying fuel-to-moderator ratio. By E. K. 273. Monoenergetic critical parameters and decay constants for small spheres and thin slabs. By I. Carlvik. 24 p. Sw. cr. 10:-. Sokolowski. 1966. 34 p. Sw. cr. 8:-. 231. Calibration of the failed-fuel-element detection systems In the Agesta 274. Scattering of neutrons by an enharmonic crystal. By T. Hogberg, L. Bohlin and I. Ebosjo. 1967. 38 p. Sw. cr. 10:—. reactor. By O. Strindehag. 1966. 52 p. Sw. cr. 8:—. 232. Progress report 1965. Nuclear chemistry. Ed. by G. Carleson. 1966. 26 p. 275. The IAKI=1, E1 transitions in odd-A isotopes of Tb and Eu. By S. G. Malm­ Sw. cr. 8:—. skog, A. Marelius and S. Wahlborn. 1967. 24 p. Sw. cr. 10:-. 233. A summary report on assembly 3 of FRO. By T. L. Andersson, B. Brun­ 276. A burnout correlation for flow of boiling water in vertical rod bundles. By felter, P. F. Cecchi, E. Hellstrand, J. Kockum, S-O. Londen and L. I. Kurt M. Becker. 1967. 102 p. Sw. cr. 10:-. Tiren. 1966. 34 p. Sw. cr. 8:—. 277. Epithermal and thermal spectrum indices in heavy water lattices. By E. K. 234. Recipient capacity of Tvaren, a Baltic Bay. By P.-O. Agnedal and S. O. W. Sokolowski and A. Jonsson. 1967. 44 p. Sw. cr. 10:-. Bergstrom. 1966. 21 p. Sw. cr. 8:-. 278. On the d^2<-"^97/2 transitions in odd mass Pm nuclei. By A. Backlin and 235. Optimal linear filters for pulse height measurements in the presence of S. G. Malmskog. 1967. 14 p. Sw. cr. 10:-. noise. By K. Nygaard. 1966. 16 p. Sw. cr. 8:-. 279. Calculations of neutron flux distributions by means of integral transport 236. DETEC, a subprogram for simulation of the fast-neutron detection pro ­ methods. By I. Carlvik. 1967. 94 p. Sw. cr. 10:-. cess in a hydro-carbonous plastic scintillator. By B. Gustafsson and O. Aspelund. 1966. 26 p. Sw. cr. 8:—. 237. Microanalys of fluorine contamination and its depth distribution in zircaloy Forteckning over publicerade AES-rapporter by the use of a charged particle nuclear reaction. By E. Moller and N. Starfelt. 1966. 15 p. Sw. cr. 8:—. 1. Analys medelst gamma-spektrometri. Av D. Brune. 1961. 10 s. Kr 6:-. 238. Void measurements in the regions of sub-cooled and low-quality boiling. 2. Bestr&lningsforandringar och neutronatmosfar i reaktortrycktankar — nfigra P. 1. By S. Z. Rouhani. 1966. 47 p. Sw. cr. 8:—. synpunkter. Av M. Grounes. 1962. 33 s. Kr 6:—. 239. Void measurements In the regions of sub-cooled and low-quality boiling. 3. Studium av strackgransen i mjukt st8I. Av G. Ostberg och R, Attermo. P. 2. By S. Z. Rouhani. 1966. 60 p. Sw. cr. 82—. 1963. 17 s. Kr 6:-. 240. Possible odd parity in U8Xe. By L. Broman and S. G. Malmskog. 1966. 4. Teknisk upphandling inom reaktoromrSdet. Av Erik Jonson. 1963. 64 s. 10 p. Sw. cr. 8:-. Kr 8:-. 241. Burn-up determination by high resolution gamma spectrometry: spectra 5. Agesta Kraftvarmeverk. Sammanstallning av tekniska data, beskrivningar from slightly-irradiated uranium and plutonium between 400-830 keV. By m. m. for reaktordelen. Av B. Lilliehook, 1964. 336 s. Kr 15: —. R. S. Forsyth and N. Ronqvist, 1966. 22 p. Sw. cr. 8:—. 6. Atomdagen 1965. Sammanstallning av foredrag och diskussioner. Av S. 242. Half life measurements in 1S5 Gd. By S. G. Malmskog. 1966. 10 p. Sw. Sandstrom. 1966. 321 s. Kr 15: —. cr. 8:—. Additional copies available at the library of AB Atomenergi, Studsvik, Ny- 243. On shear stress distributions for flow in smooth or partially rough annuli. koping, Sweden. Micronegatives of the reports are obtainable through Film- By B. Kjellstrom and S. Hedberg. 1966. 66 p. Sw. cr. 8:-. produkter, Gamla landsvagen 4, Ektorp, Sweden.

EOS-tryckerierna, Stockholm 1967