3. NATURE OF PHYSICAL PROBLEM SOLVED
One-dimensional neutron and gamma-ray shield penetration calculations. For neutrons the removal- diffusion method is used. For gamma-rays there is the option of using either kernel integration with build-up factors or diffusion theory with a first flight correction. Linked calculations are possible in which the gamma-ray sources are calculated from the neutron fluxes. Neutron reaction-rates and gamma-ray dose and heating rates may be printed out.
A detailed description of the different modules follows:
MINNIE (WRS system module No. 17133) Reads in material compositions in various forms and converts them to standard format.
NADCON (WRS system module No. 23016) Calculates multigroup data for use in shielding calculations employing the method of Adjusted Diffusion Coefficients (with first flight correction). NADCON prepares macroscopic data for materials of specified compositions by mixing microscopic data from a library supplied with the module.
BADCON (WRS system module No. 17066) Derives Adjusted Diffusion Coefficient Data without the first flight correction, from the multigroup data produced by NADCON.
REMOVAL (WRS system module No. 42869) Calculates the uncollided flux and first-collision source in a slab geometry system containing a source distributed through some or all of the slabs. The removal fluxes and the removal sources in the output groups (which may differ from the removal groups) are expressed in terms of Chebyshev series.
The module can make use of a library of removal cross sections on magnetic tape or disc containing micro- scopic cross sections for a selection of elements in a number of removal groups. For example, there is a standard tape containing neutron data for 32 elements in 18 removal groups. The module can use this to calculate macroscopic removal cross sections for each material in the problem, or alternatively it can read these cross sections from the inputdata.
SCRIM (WRS system module No. 22947) Calculates the uncollided flux and first collision sources in a slab shield due to neutrons from a small cylindrical reactor core. The shield may be either on top of or beside the cylindrical core.
The kernel is integrated numerically using the Gauss quadrature technique. The amount of computing time required is dependent on the quadrature order which also determines the accuracy of the calculation. Tests have been carried out to discover the most suitable choice of quadrature order for the type of problem for which SCRIM was originally designed, that is for a core similar to that of the LIDO reactor. The results of these tests are used by a routine that will automatically select the quadrature order.
Although the module will operate for field points inside the core, the method of integration is unsuitabl for accurate calculations at such points, even if high order quadrature is used. The calculation of removal fluxes within the core should therefore be avoided in general.
Removal fluxes and sources are calculated for each of a series of energy groups so that they can be fed into a multigroup diffusion calculation. Fluxes and sources in the removal group scheme must first be converted into the so-called output group scheme which is used in the diffusion calculation.
For each material used, group-averaged macroscopic cross sections are either input or calculated from a library of microscopic removal cross sections stored on magnetic tape or disc. The present tape contains data for 32 elements averaged over a standard 18 removal group scheme.
SREM (WRS system module No. 3348) Calculates the uncollided flux and first-collision source from a disc source in a slab geometry system, a line source at the centre of a cylindrical system or a point source at the centre of a spherical system. An energy group treatment is used. In each case the source particles are emitted isotropically. The source energy spectrum is defined group-wise, and a fission spectrum in 18 groups is built in for use if required.
For different applications of the module the cross sections used may be either total cross sections or some form of empirical removal cross sections. The module can make use of a library of removal cross sections on magnetic tape or disc containing microscopic cross sections for a selection of elements in a number of removal groups. For example, there is a standard tape containing neutron data for 32 elements in 18 removal groups.
The calculation is performed in two stages: the removal flux through the shield is calculated in a number of removal groups, and the removal flux and source are then obtained in an output group scheme which may be different. The removal source can be calculated by multiplying the removal flux in each group by the corresponding removal cross section and assigning each result to an output group. Alternatively, a full matrix of transfer cross sections may be used. The module produces a functional representation of the flux and source in each group and each region using Chebyshev series. This ensures that the execution time does not depend on the number of printout points requested, and that the module output is in a form from which values at any point may be calculated.
PARAMETERS (WRS system module No. 35836) Calculates multigroup neutron diffusion parameters for use in the age- diffusion method. Parameters are derived for materials of specified compositions from a library of microscopic data. This data is normally held on a magnetic tape and is written by an auxiliary program which processes files from the UKAEA Nuclear Data Library. For each material and each energy group the module calculates values of the diffusion coefficient D, the inverse slowing down length K and the absorption coefficient Alpha.
REGROUP (WRS system module No. 13774) Converts multigroup flux spectra from one energy group scheme to another. Several spectra can be processed during one execution of the module, and there is a facility for applying scaling factors to the converted spectra. There are two alternative modes of operation. In one mode a 1/E spectrum shape is assumed within each input group, while in the other a shape is deduced from the flux values in adjacent groups. In both cases the total flux is preserved in the conversion.
DIFFUSION (WRS module No. 7539) Solves a set of multigroup diffusion equations for a system represented in one-dimensional plane, cylindrical or spherical geometry.
The system is divided into regions each of which is assigned a material throughout which the diffusion length, the diffusion coefficient and the cross sections are uniform. The module may be supplied with group parameters appropriate to the continuous slowing down model namely the diffusion coefficient, the absorption parameter and the inverse slowing down length.
It then calculates the diffusion length and the cross section for transfer to the next lower group from simple relationships involving the above parameters.
There is provision to allow for buckling due to lateral leakage by supplying a buckling value for each group and region. Void regions may be included. The module may also be used to solve the adjoint problem.
In order to solve the differential equation for each group, a spatial mesh is superimposed on the system and the equation at each mesh point is expressed using finite difference approximations for the derivatives.
These, together with extra equations for the internal and external boundary conditions, form a set of linear equations which, being almost tri-diagonal, are easily solved by elimination and back substitution.
DOCTOR (WRS module No. 1827) Calculates any number of reaction rates given a set of multigroup fluxes at a number of spatial mesh points in one spatial dimension.
The flux values may be read from cards by the module, or may be in a data group which is already open when the module is called. For each energy group, values are supplied at a number of specified points in one spatial dimension, these points being divided into regions in which they are not necessarily spaced evenly. Average cross sections in each group can be supplied for each detector in the input data, or they may be in a data group which is already open. In the former case it is also possible to make use of a small library of cross sections which is incorporated in the module and which contains data for a number of neutron detectors in two standard group schemes. If these cross sections are used there is provision to adjust the values for the thermal group and the lowest epithermal group to allow for variations due to temperature. Results are printed out at the set of points at which the flux was given, and the contribution from each group may be included if desired.
GAMSOURCE (WRS module No. 38474) Calculates the source strength of gamma rays arising from neutron capture in a system represented by one-dimensional geometry.
The gamma ray sources are 'prompt' or 'delayed', 'prompt' including all gammas produced by neutron capture (but not from fission or inelastic scattering), 'delayed' comprising gammas released in the decay of irradiation products.
To calculate the prompt source, values of the average microscopic absorption cross section for each element present in a region are calculated for each neutron energy group from a cross section library obtained by processing cross sections from UKNDL. In this library the epithermal absorption cross section is represented by a 100 step histogram between 1 MeV and 0.07 eV.
Values for the thermal group are obtained by averaging over a Maxwellian spectrum.
The photons produced are assigned to prompt energy groups. Two standard prompt group schemes are available: one of 5, the other of 13 groups. For the 5 group scheme, the gamma spectra are those used by the COMPRASH code, while the 13 group spectra are those compiled by Side otham. The spectra may also be inputted if a different group scheme is used.
GAMSOURCE also allows independent gamma sources to be read in.
The method for delayed gamma sources is similar to that of the prompt gammas, but differs in that there are no standard group schemes. It is therefore necessary to supply values of the number of photons emitted in each group during the decay process. The module may generate a new mesh for the gamma sources.
GRAB (Gamma Ray Build-up) (WRS system module No. 60221) Calculates the gamma ray flux, dose or heating rate in a slab shield using the build-up factor solution.
Gamma ray sources of different energies may be distributed throughout the system. To summarize the method, an integration is performed over the source volume and source energy range of a kernel representing the contribution to the quantity of interest from uncollided photons, to which is applied a factor that takes into account scattered photons.
The module requires values of the absorption coefficient (i.e. total cross section) and the energy absorption coefficient in each energy group for each material. For five standard groups these may be calculated by the module from the material compositions using a stored set of values of the mass absorption coefficient and the mass energy absorption coefficient for a number of commonly used elements. The module uses the Taylor approximation to the build-up factor, and therefore requires values of the parameters which define the factor for each group. It should be noted that since only one set of values can be used, a choice of build-up factor must be made with regard to the quantity of primary interest and the materials present. For each region, results are produced at the points of a uniform mesh which may be specified by the user. A separate integration is performed for the flux at each output point, and the execution time for the module therefore depends on the total number of output points requested. Dose and heating rates are, however, calculated from the flux values, so that little extra time is required for these. The contributions from selected groups may be totalled according to specifications in the input data. For the first point of every region the module can also print out the contributions to the flux from the source in each region.
MIXGAM (WRS system module No. 1300) Generates macroscopic cross section sets and diffusion constants for specified material compositions for use in gamma ray diffusion calculations (incorporating a removal flight correction) from a library of microscopic cross sections. For the purpose of the first flight calculation the source is assumed to be composed of discrete energies; the diffusion calculation is performed in groups. The data supplied by the module for each material is as follows:
Total cross sections at the source energies.
Scattering cross sections from the source energies into each diffusion group.
Diffusion coefficient (D) and diffusion length (L) for each diffusion group.
Group to group transfer cross sections for the diffusion groups.
Energy deposition factors for each source energy and for each diffusion group.
The module extracts microscopic cross sections for the required elements from a library which contains data for a given set of source energies and a given diffusion group scheme. The standard library supplied with the module contains data for 33 elements. It employs 13 source energies and 32 diffusion groups. This data has been prepared using the GAMCON program which obtains cross sections from the data of Hubbel by reference to the subroutine HEITLER. The module can, however, handle any library of data prepared in the appropriate format.
The group diffusion length and consequent diffusion coefficient are obtained from standard formulae.
REMTWO (WRS system module No. 299) Calculates the uncollided flux due to a distributed source in a one-dimensional plane geometry system. A method is used in which the flux is approximated by a sum of diffusion equation solutions. From the calculated flux a first collision source may be obtained (externally) for use in the diffusion equations.
The REMTWO module may be used for a system which can be represented by a series of plane parallel sided slabs of infinite lateral extent, each composed of a uniform material. Some or all of these regions contain sources, which are of varying strength and have a range of energies. The source strength shows no lateral variation, and is therefore a function of a single spatial coordinate x. The source is defined as a function of x by values given at the points of a mesh which is uniform in each region, while a group treatment is used for the energy dependence. Two cases are distinguished by whether the source energy spectrum is the same throughout the system or varies with position.
In the latter case the spatial variation of source strength is defined separately for each group using a mesh which may be different for each group.
The cross section for each group is defined throughout the system by assigning a material to each region and a set of cross sections to each material. The uncollided flux in group g at position x is given by the integral over all volume of the source strength multiplied by exp(-t) (the number of mean free paths) divided by 4 Pi r**2.
When the integration over the lateral coordinates is performed in this expression, there remains an integral with respect to x in which the integrand involves the E1 function. By substituting an approximation of this function it is possible to express the flux throughout the system as the sum of three solutions of the diffusion equation. By this method, the module calculates flux values at the points of a mesh, and it can print out values at a specified spacing in each region. The mesh can be different from the source mesh and will be set automatically by the module if it is not supplied. The module does not solve the diffusion equations itself but sets the parameters which define them completely and then returns control to the calling program so that the solutions can be obtained externally.