Keno: A Monte Carlo Criticality Program

L. M. Petrie, K. B. Bekar, C. Celik, D. F. Hollenbach,1 C. M. Perfetti, S. Goluoglu,1 N. F. Landers,1 M. E. Dunn, B. T. Rearden

KENO is a three-dimensional (3D) Monte Carlo criticality transport program developed and maintained for use as part of the SCALE Code System. It can be used as part of a sequence or as a standalone program. There are two versions of the code currently supported in SCALE. KENO V.a is the older of the two. KENO-VI contains all current KENO V.a features plus a more flexible geometry package known as the SCALE Generalized Geometry Package. The geometry package in KENO-VI is capable of modeling any volume that can be constructed using quadratic equations. In addition, such features as geometry intersections, body rotations, hexagonal and dodecahedral arrays, and array boundaries have been included to make the code more flexible.

The simpler geometry features supported by KENO V.a allow for significantly shorter execution times than KENO-VI, while the additional geometry features supported in KENO-VI make the code appropriate for cases where geometry modeling is not possible with KENO V.a. In particular, KENO-VI allows intersections, body truncations with planes, and a much wider variety of geometrical bodies. KENO-VI also has the ability to rotate bodies so that volumes no longer must be positioned parallel to a major axis. Hexagonal arrays are available in KENO-VI and dohecahedral arrays enable the code to model pebble bed reactors and other systems composed of close packed spheres. The use of array boundaries makes it possible to fill a non-cuboidal volume with an array, specifying the boundary where a particle leaves and enters the array.

Except for geometry capabilities, the two versions of KENO share most of the computational capabilities and the input flexibility specific to most SCALE modules. They can both operate in multigroup or continuous energy mode, run as standalone codes, or integrated in computational sequences such as CSAS, TSUNAMI-3D, or TRITON. Both versions of the code are continually updated and are written in FORTRAN 90.

Computational capabilities shared by the two versions of KENO include the determination of k‑effective, neutron lifetime, generation time, energy-dependent leakages, energy- and region-dependent absorptions, fissions, the system mean-free-path, the region-dependent mean-free-path, average neutron energy, flux densities, fission densities, reaction rate tallies, mesh tallies, source convergence diagnostics, problem-dependent continuous energy temperature treatments, parallel calculations, restart capabilities, and many more.

1Formerly with Oak Ridge National Laboratory

ACKNOWLEDGMENTS

Many individuals have contributed significantly to the development of KENO. Special recognition is given to G. E. Whitesides, former Director of the Computing Applications Division, who was responsible for the concept and development of the original KENO code. He has also contributed significantly to some of the techniques used in both KENO versions. The late J. T. Thomas offered many ideas that have been implemented in the code. R. M. Westfall, retired from ORNL, provided early consultation, encouragement, and benchmarks for validating the code. The special abilities of J. R. Knight, retired from ORNL, contributed substantially to debugging early versions of the code. S. W. D. Hart was instrumental in implementing continuous energy temperature treatments. W. J. Marshall has provided substantial validation and quality assurance reviews. Appreciation is expressed to C. V. Parks and S. M. Bowman for their support of KENO and the KENO3D visualization tool. The late P. B. Fox provided many of the figures in this document. D. Ilas, B. J. Marshall, and D. E. Mueller consolidated the previous KENO V.a and KENO-VI manuals into this present form. The efforts of L. F. Norris (retired), W. C. Carter (retired), S. J. Poarch, D. J. Weaver (retired), S. Y. Walker and R. B. Raney in preparing this document are gratefully acknowledged.

The authors thank the U. S. Nuclear Regulatory Commission and the DOE Nuclear Criticality Safety Program for sponsorship of the continuous energy, source convergence diagnostics, and grid geometry features in the current version.

Introduction to KENO

KENO, a functional module in the SCALE system, is a Monte Carlo criticality program used to calculate \(k_{eff}\), fluxes, reaction rates, and other data for three-dimensional (3-D) systems. Special features include multigroup or continuous energy mode, simplified data input, the ability to specify origins for spherical and cylindrical geometry regions, a Pn scattering treatment, and restart capability.

The KENO data input features flexibility in the order of input. The only restrictions are that the sequence identifier, title, and cross section library must be entered first. A large portion of the data has been assigned default values found to be adequate for many problems. This feature enables the user to run a problem with a minimum of input data.

In addition to the features listed above, KENO-VI uses the SCALE Generalized Geometry Package (SGGP), which contains a much larger set of geometrical bodies, including cuboids, cylinders, spheres, cones, dodecahedrons, elliptical cylinders, ellipsoids, hoppers, parallelepipeds, planes, rhomboids, and wedges. The code’s flexibility is increased by allowing: intersecting geometry regions; hexagonal, dodecahedral, and cuboidal arrays; bodies and holes rotated to any angle and translated to any position; and a specified array boundary that contains only that portion of the array located inside the boundary. Users should be aware that the added geometry features in KENO‑VI can result in significantly longer run times than KENO V.a. A KENO-VI problem that can be modeled in KENO V.a will typically run about four times as long with KENO-VI as it does with KENO V.a. Therefore KENO-VI is not a replacement for KENO V.a, but rather an additional version for more complex geometries that could not be modeled previously.

Blocks of input data are entered in the form

READ XXXX input_data END XXXX,

where XXXX is the keyword for the type of data being entered. The types of data entered include parameters, geometry region data, array definition data, biasing or weighting data, albedo boundary conditions, starting distribution information, the cross section mixing table, extra one-dimensional (1-D) (reaction rate) cross section IDs for special applications, energy group boundaries for tallying in the continuous energy mode, a mesh grid for collecting flux moments, and printer plot information.

A block of data can be omitted unless it is needed or desired for the problem. Within the blocks of data, most of the input is activated by using keywords to override default values.

The treatment of the energy variable can be either multigroup or continuous. Changing the calculation mode from multigroup to continuous energy or vice versa is established by simply changing the cross section library used. All available calculated entities in the multigroup mode can also be calculated in the continuous energy mode. If the calculated entity is energy or group dependent, it is automatically tallied into the appropriate group structure in the continuous energy mode.

The KENO V.a geometry input consists of spheres, hemispheres, cylinders, hemicylinders, and cuboids. Although the origin of the cylinders, hemicylinders, spheres, and hemispheres is zero by default, they may be specified to any value that will allow the geometry to fit in the problem. This feature allows the use of nonconcentric cylindrical and spherical shapes and provides a great deal of freedom in positioning them. Another feature that expands the generality of the code is the ability to place the cut surface of the hemicylinders and hemispheres at any distance between the radius and the origin.

An additional convenience is the availability of an alternative method for specifying the array definition unit-location data. This method uses FIDO-like options for filling the array.

As mentioned above, KENO-VI uses the SGGP, which contains a much more flexible geometry package than the one in KENO V.a. In KENO-VI, geometry regions are constructed and processed as sets of quadratic equations. A set of geometric shapes (including all of those used in KENO V.a plus others) is available in KENO-VI, as well as the ability to build more complex geometric shapes using sets of quadratic equations. Unlike KENO V.a, KENO-VI allows intersections between geometry regions within a unit, and it provides the ability to specify an array boundary that intersects the array.

The most flexible KENO V.a geometry features are the “ARRAY-of-ARRAYs” and “HOLEs” capabilities. The ARRAY-of-ARRAYs option allows the construction of ARRAYs from other ARRAYs. The depth of nesting is limited only by computer space restrictions. This option greatly simplifies the setup for ARRAYs involving different UNITs at different spacings. The HOLE option allows a UNIT or an ARRAY to be placed at any desired location within a geometry region. The emplaced UNIT or ARRAY cannot intersect any geometry region and must be wholly contained within a region. As many HOLEs as will snugly fit without intersecting can be placed in a region. This option is especially useful for describing shipping casks and reflectors that have gaps or other geometrical features. Any number of HOLEs can be described in a problem, and HOLEs can be nested to any depth.

The primary difference between the KENO V.a and KENO-VI geometry input is the methodology used to represent the geometry/material regions in a unit. KENO-VI uses two geometry records (cards) to describe a region. The first record, called the GEOMETRY record, contains the geometry (shape) keyword, region boundary definitions, and any geometry modification data. Using geometry modification data, regions can be rotated and translated to any angle and position within a unit. The second record, the CONTENT record, contains the MEDIA keyword; the material, HOLE, or ARRAY ID number; the bias ID number; and the region definition vector. KENO-VI requires that a GLOBAL UNIT be specified in all problems, including single unit problems.

In addition to the cuboidal ARRAYs available in KENO V.a, hexagonal ARRAYs and dodecahedral ARRAYs can be directly constructed in KENO-VI. Also, the ability to specify an ARRAY boundary that intersects the ARRAY makes it possible to construct a lattice in a cylinder using one ARRAY in KENO-VI instead of multiple ARRAYs and HOLEs as would be required in KENO V.a.

Anisotropic scattering is treated by using discrete scattering angles. The angles and associated probabilities are generated in a manner that preserves the moments of the angular scattering distribution for the selected group-to-group transfer. These moments can be derived from the coefficients of a Pn Legendre polynomial expansion. All moments through the 2n − 1 moment are preserved for n discrete scattering angles. A one-to-one correspondence exists such that n Legendre coefficients yield n moments. The cases of zero and one scattering angle are treated in a special manner. Even when the user specifies multiple scattering angles, KENO can recognize that the distribution is isotropic, and therefore KENO selects from a continuous isotropic distribution. If the user specifies one scattering angle, the code selects the scattering angle from a linear function if it is positive between -1 and +1, and otherwise it performs semicontinuous scattering by picking scattering angle cosines uniformly over some range between –1 and +1. The probability is zero over the rest of the range.

The KENO restart option is easy to activate. Certain changes can be made when a problem is restarted, including using a different random sequence or turning off certain print options such as fluxes or the fissions and absorptions by region.

KENO can also compute angular fluxes and flux moments in multigroup calculations, which are required to compute scattering terms for generation of sensitivity coefficients with the SAMS module or the TSUNAMI-3D control module. Fluxes can also be accumulated in a Cartesian mesh that is superimposed over the user-defined geometry in an automated manner.

KENO can perform Monte Carlo transport calculations concurrently on a number of computational nodes. By introducing a simple master-slave approach via MPI, KENO runs different random walks concurrently on the replicated geometry within the same generation. Fission source and other tallied quantities are gathered at the end of each generation by the master process and are then processed either for final edits or subsequent generations. Code parallel performance is strongly dependent on the size of the problem simulated and the size of the tallied quantities.

KENO Data Guide

KENO may be run stand alone or as part of a SCALE criticality safety or sensitivity and uncertainty analysis sequence. If KENO is run stand alone in the multigroup mode, cross section data can be used from an AMPX [DG02] working format library or from a Monte Carlo format cross section library. If KENO uses an AMPX working format library, a mixing table data block must be entered. If a Monte Carlo format library is used, a mixing table data block is not entered, and the mixtures specified in the KENO geometry description must be consistent with the mixtures created on the Monte Carlo format library file.

If KENO is run stand alone in the continuous energy mode, a mixing table data block must be provided unless the restart option is used.

If KENO is run as part of a SCALE criticality safety or sensitivity and uncertainty analysis sequence, the mixtures are defined in the CSAS or TSUNAMI-3D input, and a mixing table data block cannot be entered in KENO. Furthermore, the mixture numbers used in the KENO geometry description must correspond to those defined in the composition data block of the CSAS or TSUNAMI-3D input. To use a cell-weighted mixture in KENO, the keyword “CELLMIX=,” followed by a unique mixture number, must be specified in the unit cell data of the CSAS or TSUNAMI‑3D sequence. Unit cell data are applicable only in the multigroup mode. The mixture number used in the KENO input is the unique mixture number immediately following the keyword “CELLMIX=.” A cell‑weighted mixture is available only in SCALE sequences that use XSDRN to perform a cell-weighting calculation using a multigroup cross section library. Table 136 through Table 149 summarize the KENO input data blocks. These input data blocks are discussed in detail in the following sections.

In order to run KENO parallel (standalone execution), the user must provide a name with the “%” prefix in the input file (=%kenovi). Control modules like CSAS, TRITON, and TSUNAMI-3D automatically initiate parallel KENO execution if the user provides the required arguments while running this code.

Table 136 Summary of parameter data.
_images/tab11.svg
Table 137 Summary of array data.
_images/tab21.svg
Table 138 Summary of biasing data.
_images/tab31.svg
Table 139 Summary of boundary condition data.
_images/tab41.svg
Table 140 Summary of boundary condition data specific to KENO-VI.
_images/tab51.svg
Table 141 Summary of geometry data in KENO V.a.
_images/tab6.svg
_images/tab6cont.svg
Table 142 Summary of geometry data in KENO-VI.
_images/tab7.svg
Table 143 Summary of mixing table data.
_images/tab8.svg
Table 144 Summary of plot data.
_images/tab9.svg
Table 145 Summary of starting data.
_images/tab10.svg
Table 146 Summary of volume data (KENO-VI).
_images/tab111.svg
Table 147 Summary of grid geometry data.
_images/tab12.svg
Table 148 Summary of energy group boundary data.

ENERGY

Format: READ ENERGY energy group boundaries END ENERGY

Enter upper energy boundary for each group in eV. The last entry is the lower energy boundary of the last group. For N groups, there are N+1 entries. Entries must be in descending order and in units of eV.

Table 149 Summary of reaction data.
_images/tab14.svg

Keno input outline

The data input for KENO is outlined below. Default data for KENO have been found to be adequate for many problems. These values should be carefully considered when entering data.

Blocks of input data are entered in the form:

READ XXXX input_data END XXXX

where XXXX is the keyword for the type of data being entered. The keywords that can be used are listed in Table 8.1.15. A minimum of four characters is required for a keyword, and some keyword names may be as long as twelve characters (READ PARAMETER, READ GEOMETRY, etc.). Keyword inputs are not case sensitive. Data input is activated by entering the words READ XXXX followed by one or more blanks. All input data pertinent to XXXX are then entered. Data for XXXX are terminated by entering END XXXX followed by two or more blanks. Note that multiple READ GRID blocks are used if multiple grid definitions are needed.

Table 150 Types of input data.

Type of data

First four characters

Parameters

PARA or PARM

Geometry

GEOM

Biasing

BIAS

Boundary conditions

BOUN or BNDS

Start

STAR or STRT

Energy

ENER

Array (unit orientation)

ARRA

Extra 1-D cross sections

X1DS

Cross section mixing tablea

MIXT or MIX

Plota

PLOT or PLT or PICT

Volumes

VOLU

Grid geometry

GRID

Reactions

REAC

a MIX and PLT must include a trailing blank, which is considered part of the keyword.

Three data records must be entered for every problem: first the SCALE sequence identifier, then the problem title, and then the END DATA to terminate the problem.

(1) KENO is typically run using one of the SCALE CSAS or TSUNAMI sequences, but it may also be run stand alone using KENO V.a or KENO-VI. The sequence identifier is specified using one line similar to:

=kenovi

This line may also include additional runtime directives that are described throughout the SCALE manual. For example:

=kenova parm=check

The following guidance generally assumes the user is running KENO stand alone. If KENO is to be run using of the other sequences (e.g., CSAS5), see the appropriate manual section for additional guidance.

  1. problem title

Enter a problem title (limit 80 characters, including blanks; extra characters will be discarded). A title must be entered. See Sect. 8.1.2.3.

  1. READ PARA parameter_data END PARA

Enter parameter input as needed to describe a problem. If parameter data are desired in standalone KENO calculations (i.e., non-CSAS), they must immediately follow the problem title. Default values are assigned to all parameters. A problem can be run without entering any parameter data if the default values are acceptable.

Parameter data must begin with the words READ PARA, READ PARM, or READ PARAMETER. Parameter data may be entered in any order. If a parameter is entered more than once, the last value is used. The words END PARA or END PARM, or END PARAMETER terminate the parameter data. See Title and parameter data.

(n1)…( n13) The following data may be entered in any order. Data not needed to describe the problem may be omitted.

(n1) READ GEOM all_geometry_region_data END GEOM

Geometry region data must be entered for every problem that is not a restart problem. Geometry data must begin with the words READ GEOM or READ GEOMETRY. The words END GEOM or END GEOMETRY terminate the geometry region data. See Geometry data.

(n2) READ ARRA array_definition_data END ARRA

Enter array definition data as needed to describe the problem. Array definition data define the array size and position units (defined in the geometry data) in a 3-D lattice that represents the physical problem being analyzed. Array data must begin with the words READ ARRA or READ ARRAY and must terminate with the words END ARRA or END ARRAY. See ARRAY Data.

(n4) READ BOUN albedo_boundary_conditions END BOUN

Enter albedo boundary conditions as needed to describe the problem. Albedo data must begin with the words READ BOUN, READ BNDS, READ BOUND, or READ BOUNDS, and it must terminate with the words END BOUN, ENDS BNDS, END BOUND, or END BOUNDS. See Albedo data.

(n3) READ BIAS biasing_information END BIAS

The biasing_information is used to define the weight given to a neutron surviving Russian roulette. Biasing data must begin with the words READ BIAS. The words END BIAS terminate the biasing data. See Biasing or weighting data.

(n5) READ STAR starting_distribution_information END STAR

Enter starting information data for starting the initial source neutrons only if a uniform starting distribution is undesirable. Start data must begin with the words READ STAR, READ STRT or READ START, and it must terminate with the words END STAR, END STRT or END START. See Start data.

(n6) READ ENER energy_group_boundaries END ENER

Enter upper energy boundaries for each neutron energy group to be used for tallying in the continuous energy mode. Energy bin data begin with the words READ ENER or READ ENERGY and terminate with the words END ENER or END ENERGY. The last entry is the lower energy boundary of the last group. The values must be in descending order. This block is only applicable to continuous energy KENO calculations. See Energy group boundary data.

(n7) READ MIXT cross_section_mixing_table END MIXT

Enter a mixing table to define all the mixtures to be used in the problem. The mixing table must begin with the words READ MIXT or READ MIX and must end with the words END MIXT or END MIX. Do not enter mixing table data if KENO is being executed as a part of a SCALE sequence. See Mixing table data.

(n8) READ X1DS extra_1-D_cross_section_IDs END X1DS

Enter the IDs of any extra 1-D cross sections to be used in the problem. These must be available on the mixture cross section library. Extra 1-D cross section data must begin with the words READ X1DS and terminate with the words END X1DS. See Extra 1-D XSECS IDs data.

(n9) READ PLOT plot_data END PLOT

Enter the data needed to provide a 2-D character or color plot of a slice through a specified portion of the 3-D geometrical representation of the problem. Plot data must begin with the words READ PLOT, READ PLT, or READ PICT and terminate with the words END PLOT, END PLT, or END PICT. See Plot data.

(n10) READ VOLU volume_data END VOLU

Enter the data needed to specify the volumes of the geometry data. Volume data must begin with the words READ VOLU or READ VOLUME and end with the words END VOLU or END VOLUME. See Sect.Volume data.

(n11) READ GRID mesh_grid_data END GRID

Enter the data needed to specify a simple Cartesian grid over either the entire problem or part of the problem geometry for tallying fluxes, moments, fission sources, etc. Grid data may be entered using the keywords READ GRID, READ GRIDGEOM, or READ GRIDGEOMETRY, and they are terminated with either END GRID, END GRIDGEOM, or END GRIDGEOMETRY. Multiple grids may be defined by repeating the READ GRID block several times, specifying a different mesh grid identification number for each so defined grid. See Sect. Grid geometry data for further information.

(n12) READ REAC reaction_data END REAC

Enter the data needed to specify filters for the reaction tally calculations. Reaction data must begin with the words READ REAC and terminate with END REAC. This block is only applicable to calculations in the continuous energy mode. See Reaction data.

(n13) END DATA must be entered

Terminate the data for the problem.

Procedure for data input

For a standalone KENO problem, the first data records must be the sequence identifier (e.g., =kenovi or =kenova) and the title. The next block of data must be the parameters if they are to be entered. A problem can be run without entering the parameters, which causes KENO to use default values for input parameters. The remaining blocks of data can be entered in any order.

BOLD TYPE specifies keywords. A keyword is used to identify the data that follow it. When a keyword is used, it must be entered exactly as shown in the data guide. All keywords except those ending with an equal sign must be followed by at least one blank.

small_italics correlate data with a program variable name. The actual values are entered in place of the program variable name and are terminated by a blank or a comma.

CAPITAL ITALICS identify general data items. General data items are general classes of data including

(1) geometry data such as UNIT INITIALIZATION and UNIT NUMBER DEFINITION, GEOMETRY REGION DESCRIPTION, GEOMETRY WORD, MIXTURE NUMBER, BIAS ID, and REGION DIMENSIONS,

  1. albedo data such as FACE CODES and ALBEDO NAMES,

  1. weighting data such as BIAS ID NUMBERS, etc.

Square brackets The square brackets, [ and ], are used to show that an entry is optional.

Broken line The broken line, |, is used as a logical “or” symbol to show that the entries to its left and right are alternatives that cannot be used simultaneously.

Title and parameter data

A title, a character string, must be entered at the top of the input file. The syntax is:

title a string of characters with a length of up to 80 characters, including blanks.

The PARAMETER block may contain parameter initializations for those parameters that need to be changed from their default value. The syntax for the PARAMETER block is:

READ PARA[METER] p1 … pN END PARA[METER]

or

READ PARM p1 … pN END PARM

p1 … pN are N (N greater than or equal to zero) keyworded parameters that together make up the PARAMETER DATA

The commonly changed parameters are TME, GEN, NSK, and NPG. Seldom changed parameters are NBK, NFB, XNB, XFB, WTH, WTL, TBA, BUG, TRK, and LNG.

The PARAMETER DATA, p1 … pN, consists of one or more of the parameters described below.

Floating point parameters

RND = rndnum input hexadecimal random number, a default value is provided.

TME = tmax execution time (in minutes) for the problem, default = 0.0 (no limit).

TBA = tbtch time allotted for each generation (in minutes), default = 10 minutes. If tbtch is exceeded in any generation, the problem is assumed to be looping. Execution is terminated, and final edits are performed. The problem can loop indefinitely on a computer if the system-dependent routine to interrupt the problem (PULL) is not functional. TBA= is also used to set the amount of time available for generating the initial starting points.

SIG = tsigma if entered and > 0.0, this is the standard deviation at which the problem will terminate, default = 0.0, which means do not check sigma.

WTA = dwtav the default average weight given a neutron that survives Russian roulette, dwtav default = 0.5.

WTH = wthigh the default value of wthigh is 3.0 and should be changed only if the user has a valid reason to do so. The weight at which splitting occurs is defined to be wthigh x wtavg, where wtavg is the weight given to a neutron that survives Russian roulette.

WTL = wtlow Russian roulette is played when the weight of a neutron is less than wtlow x wtavg. The wtlow default = 1.0/wthigh.

Note

The default values of wthigh and wtlow have been determined to minimize the deviation per unit running time for many problems.

MSH = mesh_size Length (cm) of one side of a cubic mesh for tallying fluxes. Default = 0.0. A positive non-zero value must be entered if MFX=YES and READ GRID input is not entered.

TTL = temperature_tolerance The continuous energy cross sections must be within the temperature_tolerance (in degrees Kelvin) of the requested temperature for the problem to run. A negative value specifies the use of the closest temperature to that requested. TTL is ignored when DBX is nonzero. The default = -1.0.

DBH = dbrc_high the energy cutoff (in eV) up to which the Doppler Broadening Rejection Correction (DBRC) method will be used on nuclides for which DBRC is enabled, and cross section libraries are available. DBH is only used in CE simulations. Default = 210.0 eV.

DBL = dbrc_low the energy cutoff (in eV) down to which DBRC will be used on nuclides for which DBRC is enabled and cross section libraries are available. Only used in CE simulations. Default = 0.4 eV.

Integer parameters

GEN = nba number of generations to be run, default = 203.

NPG = npb number of neutrons per generation, default = 1000.

NSK = nskip number of generations (1 through nskip) to be omitted when collecting results, default = 3.

RES = nrstrt number of generations between writing restart data, default = 0. If RES is zero, restart data are not written. When restarting a problem, RES is defaulted to the value that was used when the restart data block was written. Thus, it must be entered as zero to terminate writing restart data for a restarted problem.

NBK = nbank number of positions in the neutron bank, default = npb + 25.

XNB = nxnbk number of extra entries in the neutron bank, default = 0.

NFB = nfbnk number of positions in the fission bank, default = npb.

XFB = nxfbk number of extra entries in the fission bank, default = 0.

X1D = numx1d number of extra 1-D cross sections, default = 0.

BEG = nbas beginning generation number, default = 1. If BEG is greater than 1, restart data must be available. BEG must be 1 greater than the number of generations retrieved from the restart file.

NB8 = nb8 number of blocks allocated for the first direct-access unit, default = 1000.

NL8 = nl8 length of blocks allocated for the first direct-access unit, default = 512.

NQD = nquad quadrature order for angular flux tallies, default = 0, which means do not collect. Angular fluxes are typically only needed for TSUNAMI-3D calculations.

NGP = ngp number of neutron energy groups to be used for tallying in the continuous energy mode. If NGP corresponds to a standard SCALE group structure, then the SCALE group structure will be used. If it does not correspond to a standard structure, an equally spaced in lethargy group structure will be used. If nothing is specified for a continuous energy problem, the SCALE 238 group structure will be used.

PNM = isctr highest order of flux moment tallies, default = 0. Flux moments are typically only tallied for TSUNAMI-3D calculations.

CET = ce_tsunami_mode. mode for CE TSUNAMI (See TSUNAMI-3D manual).

0 = No sensitivity calculations

1 = CLUTCH sensitivity calculation

2 = IFP sensitivity calculation

4 = GEAR-MC calculation (with CLUTCH only)

5 = GEAR-MC calculation (with CLUTCH+IFP)

7 = Undersampling metric calculation

CFP = number_of_latent_generations

number of latent generations used for IFP sensitivity or \(F^{*}\left( r \right)\) calculations. Note:

  • If CET=1 and CFP= -1 then \(F^{*}\left( r \right)\) is assumed to equal one everywhere.

  • If CET=4 and CFP= -1 then \(F^{*}\left( r \right)\) is assumed to equal zero everywhere.

DBR = lusedbrc use Doppler Broadening Rejection Correction method. See Sect. 8.1.6.2.9 for more details. Only used in CE simulations. Default = 2.

0 = No DBRC

1 = DBRC for 238U only

2 = DBRC for all available nuclides (232Th, 234U, 235U, 236U, 238U, 237Np, 239Pu, 240Pu)

DBX = db_xs_mode

option for performing problem-dependent or on-the-fly Doppler Broadening. See Sect. 8.1.6.2.10 for more details. Default = 2.

0 = No problem-dependent or on-the-fly Doppler Broadening

1 = Perform problem-dependent Doppler Broadening for 1D cross sections only.

2 = Perform problem-dependent Doppler Broadening for both 1D and 2D (thermal scattering data) cross sections.

Alphanumeric parameter data

CEP = lcep key for choosing the calculation mode in stand alone KENO calculations. The parameter is set to the appropriate value by the calling sequence if not stand alone KENO. For stand alone KENO, enter NO for multigroup mode, or enter the continuous energy directory filename for the continuous energy mode. The directory file is the file containing pointers to files significant for the continuous energy run.

FNI = mode_in extra field in the input restart file name [restart_*mode_in*.keno_input] and [restart_*mode_in*.keno_calculated]. The default is an empty field.

FNO = mode_out extra field in the output restart filename [restart_*mode_out*.keno_input] and [restart_*mode_out*.keno_calculated]. The default is an empty field.

Logical parameter data … enter YES or NO

APP = lappend key for appending the restart data, default = NO.

FLX = nflx key for collecting and printing fluxes, default = NO.

FDN = nfden key for collecting and printing fission densities, default = YES.

ADJ = nadj key for running adjoint calculation, default = NO. Adjoint cross sections must be available to run an adjoint problem. If LIB= is specified, the cross sections will be adjointed by the code. If XSC= is specified, the cross sections must already be in adjoint order.

PTB = ptb key for using probability tables in the continuous energy mode, default = YES

PNU = lpromptnu key for using delayed or prompt ν in the continuous energy mode, default = NO – use total.

FRE = lfree_analytic key for using free analytic gas treatment, default = YES.

AMX = amx key for printing all mixture cross section data. This is the same as activating XAP, XS1, XS2, PKI, and P1D. If any of these are entered in addition to AMX, that portion of AMX will be overridden, default = NO.

XAP = prtap key for printing discrete scattering angles and probabilities for the mixture cross sections, default = NO.

XS1 = prtp0 key for printing mixture 1-D cross sections, default = NO.

XS2 = prt1 key for printing mixture 2-D cross sections, default = NO.

XSL = prtl key for printing mixture 2-D PL cross sections, default = NO. The Legendre expansion order L is automatically read from the cross section library.

PKI = prtchi print input fission spectrum, default = NO.

P1D = prtex print extra 1-D cross sections, default = NO.

FAR = lfa key for generating region-dependent fissions and absorptions for each energy group, default = NO.

GAS = lgas key for printing region-dependent fissions and absorptions by energy group, applicable only if FAR = YES. Default = FAR. GAS = YES prints region-dependent data by energy group. GAS = NO suppresses region-dependent data by energy group.

MKP = larpos calculate and print matrix k-effective by unit location, default = NO. Unit location may also be referred to as array position or position index.

CKP = lckp calculate and print cofactor k-effective by unit location, default = NO. Unit location may also be referred to as array position or position index.

FMP = pmapos print fission production matrix by array position, default = NO.

MKU = lunit calculate and print matrix k-effective by unit type, default = NO.

CKU = lcku calculate and print cofactor k-effective by unit type, default = NO.

FMU = pmunit print fission production matrix by unit type, default = NO.

MKH = lmhole calculate and print matrix k-effective by hole number, default = NO.

CKH = lckh calculate and print cofactor k-effective by hole number, default = NO.

FMH = pmhole print fission production matrix by hole number, default = NO.

HHL = lhhgh collect matrix information by hole number at the highest hole nesting level, default = NO.

MKA = lmarry calculate and print matrix k-effective by array number, default = NO.

CKA = lcka calculate and print cofactor k-effective by array number, default = NO.

FMA = pmarry print fission production matrix by array number, default = NO.

HAL = langh collect matrix information by array number at the highest array nesting level, default = NO.

BUG = ldbug print debug information, default = NO. Enter YES for code debug purposes only.

TRK = ltrk print tracking information, default = NO. Enter YES for code debug purposes only.

PWT = lpwt print weight average array, default = NO.

PGM = lgeom print unprocessed geometry as it is read, default = NO.

SMU = lmult calculate the average self-multiplication of a unit, default = NO.

NUB = nubar calculate the average number of neutrons per fission and the average energy group at which fission occurred, default = YES.

PAX = lcorsp print the arrays defining the correspondence between the cross section energy group structure and the albedo energy group structure, default = NO.

TFM = ltfm perform coordinate transform for flux moments and angular flux calculations, default = NO.

PMF = prtmore print angular fluxes or flux moments if calculated, default = NO.

CFX = nflx collect fluxes, default = NO.

UUM = lUnionizedMix use unionized mixture cross section, default=NO. Only used in CE simulations. See Sect. 8.1.6.2.3 for further details.

M2U = luseMap2Union store cross sections for each nuclide on a unionized energy grid, default=NO. Only used in CE simulations. See Sect. 8.1.6.2.3 for further details.

SCX = lxsecSave save CE cross sections to restart file, default=NO.

MFX = make_mesh_flux compute mesh fluxes on intervals defined by MSH above or by READ GRID data block, default = NO.

PMS = print_mesh_flux print mesh fluxes if computed, default = NO.

MFP = mean-free-path compute and print the mean-free-path of a neutron by region, default = NO.

HTM = html_output produce HTML formatted output for interactive browsing, sorting, and plotting of results, default = YES.

PMM = print_mesh_moments print the angular moments of the mesh flux, if computed, default = NO.

PMV = print_mesh_volumes print the volume of each mesh interval, if computed. Default = NO.

FST = lprint_FStar Create a .3dmap file that contains the F*(r) mesh used by a CE-TSUNAMI CLUTCH sensitivity calculation.

RUN = lrun key for determining if the problem is to be executed when data checking is complete, default = YES.

PLT = lplot key for drawing specified plots of the problem geometry, default = YES.

Note

The parameters RUN and PLOT can also be entered in the PLOT data. See Sect. 8.1.2.11. It is recommended that these parameters be entered only in the parameter data in order to ensure that the data printed in the Logical Parameters table is actually performed. If RUN and/or PLT are entered in both the parameter data and plot data, the results vary depending on whether the problem is run (1) stand alone, (2) as a restarted problem, (3) as CSAS with parm=check, or (4) as CSAS without parm=check. These conditions are detailed below.

KENO standalone and CSAS with PARM=CHECK

The values of RUN and/or PLT entered in KENO parameter data are printed in the Logical Parameters table of the problem output. However, values for RUN and/or PLT entered in the KENO plot data will override the values entered in the parameter data.

Restarted KENO

The values of RUN and/or PLT printed in the Logical Parameters table of the problem output are the final values from the parent problem unless those values are overridden by values entered in the KENO parameter data of the restarted problem. If the problem is restarted at generation 1, KENO plot data can be entered, and the values for RUN and/or PLT will override the values printed in the Logical Parameters table.

CSAS Without PARM=CHECK

The values of RUN and/or PLT entered in the KENO parameter data override values entered in the KENO plot data. The values printed in the Logical Parameters table control whether the problem is to be executed and whether a plot is performed.

Parameters that are either Integer or Logical

SCD= lScnvgDiag enable fission source convergence diagnostics (ScnvgDiag), default=YES. See Sect. 8.1.6.7for further details.

CDS = lcadis/lGridFissions accumulate neutron fissions to use as fission source in subsequent MAVRIC/Monaco shielding calculation or for visualization, default = NO

GFX = lGridFlux compute grid fluxes averaged over the volume of the mesh on intervals defined by a READ GRID data block, default = NO.

MFX = lMeshFlux compute mesh fluxes averaged over the volume of mixtures/materials in a mesh on intervals defined by MSH above or by READ GRID data block, default = NO.

CGD = lStarMesh grid ID for the F*(r) mesh for continuous energy CLUTCH sensitivity calculations. This mesh is defined in the READ GRID data block, default = NO.

Note

The KENO codes in previous SCALE versions allowed for only one mesh definition in the user input with either MSH parameter or READ GRID data block, and (2) calculation of a single mesh-based quantity, such as MFX (mesh fluxes) or CDS (fission source accumulation on a mesh), per KENO simulation.

The option to define multiple spatial meshes during a single simulation has been implemented in the KENO codes to add flexibility to mesh-based quantity calculations. The new implementation requires that each mesh definition in the READ GRID block should have a unique NUMBER (grid ID), which is used for mesh assignment. Users can assign any number of mesh-based quantities by setting the mesh parameters SCD, CDS, GFX, MFX and CGD to this grid NUMBER.

To support these former and new definition formats, the parameters SCD, CDS, GFX, MFX and CGD have been redesigned to allow either integer or logical entries. Integer entries are required if multiple mesh-based quantities are requested by the user. In this case, each integer entry must point to a grid ID specified in any READ GRID data block. See Sect. 8.1.2.14. for several examples for the use of these parameter definitions. These entries are detailed below.

SCD=yes Enable source convergence diagnostics using the fission source accumulation on the default mesh, which is 5 × 5 × 5 Cartesian mesh overlaying the whole problem geometry, generated automatically. See Sect. 8.1.6.7.

SCD=id Enable source convergence diagnostics using the fission source accumulation on the mesh defined with READ GRID data block with grid ID, id.

MFX=yes Compute mesh fluxes (fluxes averaged over each region volume in a voxel) on intervals defined by MSH above or by the first specified READ GRID data block.

MFX=id Compute mesh fluxes on intervals defined by the READ GRID data block with grid ID, id.

CDS=yes Accumulate fission sources on intervals defined by the first specified READ GRID data block.

CDS=id Accumulate fission source on intervals defined by the specified READ GRID data block with grid ID, id.

GFX=id Compute grid fluxes (fluxes averaged over a voxel volume) on intervals defined by the READ GRID data block with grid ID, id.

CGD=id Enable a mesh grid defined by the READ GRID data block with grid ID, id for CLUTCH \(F^{*}\left( r \right)\) calculations.

All of the above quantities may be requested in a single input using either the same or different grids. See Sect. 8.1.2.14 for further details.

I/O Unit Numbers

XSC = xsecs I/O unit number for a Monte Carlo format mixed cross section library. When LIB≠0, default = 14. To read a mixed cross section library from a Monte Carlo format library file or CSASI, XSC must be specified.

ALB = albdo I/O unit number for albedo data, default = 79.

WTS = wts I/O unit number for weights, default = 80.

LIB = lib I/O unit number for AMPX working format cross section library, default = 0.

SKT = skrt I/O unit number for scratch space, default = 16.

RST = rstrt I/O unit number for reading restart data, default = 0.

Enter a logical unit number to restart if BEG > 1.

WRS = wstrt I/O unit number for writing restart data, default = 0.

A non-zero value must be entered if RES > 0.

GRP = grpbs I/O unit number for an energy group boundary library, default = 77.

Example: READ PARAM NPG=203 FLX=YES END PARAM

Geometry data

The GEOMETRY_ DATA consists of a series of UNIT descriptions, one of which may be the GLOBAL UNIT. The UNIT is the basic geometry piece in KENO and often corresponds to a well-defined physical entity (e.g., a fuel pin). A UNIT, therefore, may consist of multiple material regions. Each UNIT has its own, local coordinate system. The UNITs are assembled to construct the problem’s global geometry for KENO. The GEOMETRY_ DATA must be entered unless the problem is being restarted. See Geometry for detailed examples.

UNITS

Geometric arrangements in KENO are achieved in a manner similar to using a child’s building blocks. Each building block is called a UNIT. An ARRAY or lattice is constructed by stacking these UNITs. Once an ARRAY or lattice has been constructed, it can be placed in a UNIT by using an ARRAY specification.

Each UNIT in an ARRAY or lattice has its own coordinate system. In KENO V.a, all coordinate systems in all UNITs must have the same orientation. This restriction is removed in KENO-VI. All geometry data used in a problem are correlated to the absolute coordinate system by specifying a GLOBAL UNIT. UNITs are constructed of combinations from several allowed shapes or geometric regions. These regions can be placed anywhere within a UNIT. In KENO V.a the regions are oriented along the coordinate system of the UNIT and do not intersect other regions. This means, for example, that a CYLINDER must have its axis parallel to one of the coordinate axes, while a rectangular parallelepiped must have its faces perpendicular to a coordinate axis. The most stringent KENO V.a geometry restriction is that none of the options allow geometry regions to intersect. In KENO V.a, each region in a unit must entirely contain each preceding region. The orientation, intersection, and containment restrictions are eliminated in KENO-VI. Fig. 51 shows some situations that are not allowed in KENO V.a, but are allowed in KENO-VI.

_images/fig15.png

Fig. 51 Examples of geometry allowed in KENO-VI but not allowed in KENO V.a.

For KENO V.a, unless special options are invoked, each geometric region in a UNIT must completely enclose each interior region. Regions may touch at points of tangency and may share faces. See Fig. 52 for examples of allowable situations.

_images/fig24.png

Fig. 52 Examples of correct KENO V.a units.

Special options are provided to circumvent the complete enclosure restriction in KENO V.a or to enhance the basic geometry package in KENO-VI. These options include ARRAY and HOLE descriptions. The HOLE option is the simplest of these and allows placing a UNIT anywhere within a region of another UNIT. In KENO V.a, HOLEs are not allowed to intersect the region into which they are placed; this restriction does not apply in KENO-VI (see Fig. 53). In both geometry packages, a HOLE cannot intersect the UNIT boundary. It is recommended that the outer boundary of a UNIT used as a HOLE should not be tangent to or share a boundary with another HOLE or a region of the UNIT containing the HOLE because the code may find that the regions are intersecting due to precision and round-off. Since a particle must check every region to determine its location within a UNIT, using HOLEs to contain complex sections of a problem may decrease the CPU time needed for the problem in KENO-VI. Inclusion of HOLEs increases run-time in KENO V.a, but in many cases cannot be avoided. An arbitrary number of HOLEs can be placed in a region in combination with a series of surrounding regions. The only restrictions on HOLEs are (1) when they are placed in a UNIT, they must be entirely contained within the UNIT, and (2) they cannot intersect other HOLEs or nested ARRAYs. HOLEs in KENO V.a cannot intersect an ARRAY; in KENO-VI, the HOLE cannot intersect the ARRAY boundary.

_images/fig34.png

Fig. 53 Example demonstrating HOLE capability in KENO.

Lattices or arrays are created by stacking UNITs. In KENO V.a, only rectangular parallelepipeds can be organized in an ARRAY. HEXPRISMs and DODECAHEDRONs are allowed in KENO-VI to construct triangular pitched or closed-packed dodecahedral ARRAYs, respectively. The adjacent faces of adjacent UNITs stacked in this manner must match exactly. See Arrays and holes for additional clarification and Fig. 54 and Fig. 55 for typical examples.

_images/fig43.png

Fig. 54 Example of triangular pitched ARRAY construction.

_images/fig52.png

Fig. 55 Example of ARRAY construction.

The ARRAY option is provided to allow for placing an ARRAY or lattice within a UNIT. In KENO-VI, an ARRAY is placed in a UNIT by inserting it directly into a geometry/material region as a content record. In KENO V.a, the ARRAY is placed directly in the unit like a CUBOID: it must be the first region in the UNIT, or the ARRAY elements must intersect with the smaller region. Subsequent regions in the UNIT containing the ARRAY must contain it entirely. In KENO-VI, the reverse is true: the region boundary containing the ARRAY must coincide with or be contained within the ARRAY boundary. Therefore, in KENO-VI the region boundary becomes the ARRAY boundary, with the problem ignoring any part of the ARRAY outside the boundary. A particle enters or leaves the ARRAY when the region boundary is crossed. In KENO V.a, only one ARRAY can be placed directly in a UNIT. However, multiple ARRAYs can be placed within a UNIT by using HOLEs. When an ARRAY is placed in a UNIT via a HOLE, the UNIT that contains the ARRAY (rather than the ARRAY itself) is placed in the UNIT. ARRAYs of dissimilar ARRAYs can be created by stacking UNITs that contain ARRAYs. In KENO-VI, it is possible to place multiple ARRAYs in a UNIT by placing them in separate regions. Also in KENO-VI, using HOLEs to insert ARRAYs allows the ARRAYs to be rotated when placed. See Fig. 56 for an example of an ARRAY composed of UNITs containing HOLEs and ARRAYs.

_images/fig61.png

Fig. 56 Example of an ARRAY composed of UNITs containing ARRAYs and HOLEs.

The method of entering GEOMETRY_DATA in the geometry data block follows:

READ GEOM GEOMETRY_ DATA END GEOM

UNIT initialization

The description of a UNIT starts out with the UNIT INITIALIZATION and is terminated by encountering another UNIT INITIALIZATION or END GEOM.

The UNIT INITIALIZATION has the following format:

[GLOBAL] UNIT u

u is the identification number (positive integer) assigned to the particular UNIT. It may be used later to reference a UNIT previously constructed that the user wishes to place in a HOLE, or it may be used in an ARRAY (see below for more details).

GLOBAL is an attribute that specifies that the respective UNIT is the most comprehensive UNIT in the KENO problem to be solved, the UNIT that includes all the other UNITs and defines the overall geometric boundaries of the problem. In general, a GLOBAL UNIT must be entered for each problem.

In KENO V.a, the GLOBAL specification is optional. If it is used, it can precede either a UNIT command or an ARRAY PLACEMENT_DESCRIPTION. If it is not entered and the problem does not contain ARRAY data, UNIT 1 is the default GLOBAL UNIT. If there is no GLOBAL UNIT specified and UNIT 1 is absent from the geometry description, an error message is printed. If the geometry description contains an ARRAY, KENO V.a defaults the global array to the array referenced by the last ARRAY PLACEMENT_DESCRIPTION that is not immediately preceded by a unit description. Otherwise, it is the largest array number specified in the array data (ARRAY Data).

Examples of initiating a UNIT:

  1. Initiate input data for UNIT No. 6.

UNIT 6

  1. Initiate input data for the GLOBAL UNIT which is UNIT No. 4.

GLOBAL UNIT 4

For each UNIT, the UNIT’s DESCRIPTION follows the UNIT’s INITIALIZATION. The DESCRIPTION is realized by combining the commands listed below. The basic principles for constructing a UNIT are different between KENO V.a and KENO-VI. A brief discussion of these principles, together with a few examples, is presented at the end of this section following the description of the basic input used to build the geometry of a UNIT. The keywords that may be used to define a UNIT in KENO are as follows:

shape

COM=

HOLE

ARRAY

REPLICATE (KENO V.a only)

REFLECTOR (KENO V.a only)

MEDIA (KENO-VI only)

BOUNDARY (KENO-VI only)

Shape

Shape is a generic keyword used to describe a basic geometric shape that may be used in building the geometry of a particular UNIT. The general format varies between KENO V.a and KENO-VI. In KENO V.a, the shape defines a region containing a material, so the user is required to provide both a material and a bias ID. In KENO-VI the shape is used strictly as a surface, which is later used to define the mono-material regions (using the MEDIA card). The user is therefore required to enter a label for this surface so that the shape can be referenced later.

KENO V.a:

shape m b d1dN [a1 …* [aM ]…]

KENO-VI:

shape l d1dN [a1 …* [aM ]…]

shape is a generic keyword that describes a basic predefined KENO shape (e.g., CUBOID, CYLINDER) that is used to build the geometry of the UNIT. The predefined shapes differ between KENO V.a and KENO-VI. See Appendix A for a description of the KENO V.a basic shapes and Appendix B for the KENO-VI shapes.

m is the mixture number of the material (positive integer) that fills the particular shape in KENO V.a UNIT description. A material number of zero indicates a void region (i.e., no material is present in the volume defined by the shape).

b is the bias identification number (bias ID, a positive integer) assigned to the particular region defined by the shape in the KENO V.a UNIT description.

l is the label (positive integer) assigned to the particular shape in the KENO-VI UNIT description. This label is used later to define a certain mono-material region within the UNIT.

d1dN represent the N dimensions (floating point numbers) that define the particular shape (e.g., radius of a sphere or cylinder). See Appendixes A and B for the particular value of N and how each shape is described.

a1aM are M optional ATTRIBUTES for the shape. The attributes provide additional flexibility in the shape description. The attributes that may be used with either KENO V.a or KENO-VI are described below (see shape ATTRIBUTES).

shape ATTRIBUTES

The ATTRIBUTES that can be used to enhance the shape description are CHORD, ORIG[IN], CENTER, and ROTATE (KENO-VI only).

The CHORD attribute

This attribute has different formats in KENO V.a and KENO-VI. The user will notice that it is more restrictive in KENO V.a. Only the HEMISPHERE and HEMICYLINDER shapes can be CHORDed in KENO V.a, but all 3-D shapes may be CHORDed in KENO-VI.

KENO V.a: CHORD ρ

KENO-VI: CHORD [+X=x+] [-X=x-] [+Y=y+] [-Y=y-] [+Z=z+] [-Z=z-]

\(p\)

is the distance ρ from the cut surface to the center of the sphere or the axis of a hemicylinder. See Fig. 57 and Fig. 58. Negative values of ρ indicate that less than half of the shape is retained, while positive values indicate that more than half of the shape will be retained.

+X=, -X=, +Y=, -Y=, +Z=, -Z=

are subordinate keywords that define the axis parallel to the chord. The “+” and “-” signs are used to define the side of the chord which is included in the volume. A “+” in the keyword indicates that the more positive side of the chord is included in the volume. A “-” in the keyword indicates that the more negative side of the chord is included in the volume.

x+, x-, y+, y-, z+, z-

are the coordinates of the plane perpendicular to the chord. For each chord added to a body, the keyword CHORD must be used, followed by one of the subordinate keywords and its dimension.

In KENO V.a, the CHORD attribute is applicable for only hemispherical and hemicylindrical shapes, not for SPHERE, XCYLINDER, YCYLINDER, CYLINDER, ZCYLINDER, CUBE, or CUBOID.

_images/fig71.png

Fig. 57 Partial hemisphere or hemicylinder; less than half exists (less than half is defined by ρ < 0).

_images/fig81.png

Fig. 58 Partial hemisphere of hemicylinder; more than half exists (more than half is defined by ρ > 0).

Fig. 59 provides two examples of the use of the CHORD option in KENO-VI.

_images/fig91.png

Fig. 59 Examples of the CHORD option in KENO-VI.

The ORIG[IN] attribute

The format is slightly different between KENO V.a and KENO-VI. Since the entries in KENO-VI are key worded, the user has more flexibility in choosing the order of these entries or in using default values. Only non-zero values must be entered in KENO-VI, but all applicable values, whether zero or non-zero, must be entered in KENO V.a.

KENO V.a: ORIG[IN] a b [c]

KENO-VI: ORIGIN [X=x0] [Y=y0] [Z=z0]

\(a\)

is the X coordinate of the origin of a sphere or hemisphere; the X coordinate of the centerline of a Z or Y cylinder or hemicylinder; the Y coordinate of the centerline of an X cylinder or hemicylinder.

\(b\)

is the Y coordinate of the origin of a sphere or hemisphere; the Y coordinate of the centerline of a Z cylinder or hemicylinder; the Z coordinate of the centerline of an X or Y cylinder or hemicylinder.

\(c\)

is the Z coordinate of the origin of a sphere or hemisphere; it must be omitted for all cylinders or hemicylinders.

X=, Y=, Z=

are the subordinate keywords used to define the new position of the origin of the shape. If the a subordinate keyword appears more than once after the ORIGIN keyword, the values are summed. If the new value is zero, the particular coordinate does not need to be specified.

x0, y0, z0

are the values for the new coordinates where the origin of the shape is to be translated.

The CENTER attribute

This attribute establishes the reference center for the flux moment calculations, which can be useful in TSUNAMI calculations. The syntax for this attribute is:

CENTER center_type [u] [x y z]

center_type

is the reference center value, as described in Table 151. The default value is global.

u

is the UNIT number to be used as a reference center for this region when the center_type is unit.

x, y, z

are the offset from the point specified by the center_type. The default is 0.0 for all three entries.

Table 151 Reference center values

center_type

Reference point

unit

Reference is defined as the origin of UNIT unit_number plus the offset defined by xy, and z.

global

Reference is defined as system origin—i.e., (0,0,0) point of the GLOBAL UNIT—plus the offset defined by x, y, and z.

local

Reference is defined as the origin of the current UNIT plus the offset defined by x, y, and z.

fuelcenter

Reference is defined as the center of all fissile material in the system plus the offset defined by x, y, and z.

wholeunit

When entered for the first region in a unit, the reference for all regions in the unit are defined as the origin of the current unit plus the offset defined by xy, and z.

The ROTATE attribute

This attribute can only be used in the KENO-VI input. It allows for the rotation of the shape or HOLE to which it is applied. If ORIGIN and ROTATE data follow the same shape or HOLE record, the shape is always rotated prior to translation, regardless of the order in which the data appear. Fig. 8.1.10 provides an example of the use of the ROTATE option. Its syntax is:

ROTATE [A1=a1] [A2=a2] [A3=a3]

A1=, A2=, A3=

are subordinate keywords to specify the angles of rotation of the particular shape with respect to the origin of the coordinate system. The Euler X-convention is used for rotation.

a1, a2, a3

are the values of the Euler rotation angles in degrees. The default is 0 degrees. If a subordinate keyword appears more than once following the ROTATE keyword, the values are summed.

_images/fig101.png

Fig. 60 Explanation of the ROTATE option.

Examples of shapes:

  1. Specify a hemisphere labeled 10, containing material 2 with a radius of 5.0 cm which contains only material where Z > 2.0 within the sphere centered at the origin, and its origin translated to X=1.0, Y=1.5, and Z=3.0. KENO V.a (no label, but material and bias ID are the first two numerical entries):

    HEMISPHERE 2 1 5.0 CHORD -2.0 ORIGIN 1.0 1.5 3.0

    or

    HEMISPHE+Z 2 1 5.0 CHORD -2.0 ORIGIN 1.0 1.5 3.0

    KENO-VI (no material; this is to be specified with MEDIA):

    SPHERE 10 5.0 CHORD +Z=2.0 ORIGIN X=1.0 Y=1.5 Z=3.0

  2. Specify a hemicylinder labeled 10, containing material 1, having a radius of 5.0 cm and a length extending from Z=2.0 cm to Z=7.0 cm. The hemicylinder has been truncated perpendicular to the X axis at X= −3 such that material 1 does not exist between X= −3 and X= −5. Position the origin of the truncated hemicylinder at X=10 cm and Y=15 cm with respect to the origin of the unit, and rotate it (in KENO-VI input) so it is in the YZ plane at X=10 and at a 45° angle with the Y plane.

    KENO V.a (no rotation possible, no label):

    ZHEMICYL+X 1 1 5.0 7.0 2.0 CHORD 3.0 ORIGIN 10.0 15.0

    KENO-VI (no material; this is to be specified with MEDIA card):

    CYLINDER 10 5.0 7.0 2.0 CHORD +X= −3.0 ORIGIN X=10.0 Y=15.0 ROTATE A2= −45

COM=

The keyword COM= signals that a comment is to be read. The optional comment can be placed anywhere within a unit definition. Its syntax is:

COM = delim comment delim

delim

is the delimiter, which may be any one of “ , ‘ , * , ^ , or !

comment

is the comment string, up to 132 characters long.

Example of comment within a UNIT:

COM=“This is a fuel pin”

HOLE

This entry is used to position a UNIT within a surrounding UNIT relative to the origin of the surrounding UNIT. HOLEs may share surfaces with but may not intersect other HOLEs, the BOUNDARY of the UNIT which contains the HOLE, or an ARRAY boundary. In KENO-VI, the BOUNDARY record of a UNIT placed in a HOLE may contain more than one geometry label, but all labels must be positive, indicating inside the respective geometry bodies. The syntax for HOLE is:

KENO V.a: HOLE u x y z

KENO-VI: HOLE u [a1 … [aM ]…]

u

is the unit previously defined that is to be placed within the HOLE.

x y z

is the position of the HOLE in the KENO V.a host UNIT.

a1 … aM

are optional KENO-VI ATTRIBUTES for the HOLE. The ATTRIBUTES can be ORIGIN or ROTATE and follow the same syntax previously defined for KENO-VI shape ATTRIBUTES. These ATTRIBUTES allow for the translation and/or rotation of the HOLE within the host region.

Examples of HOLE use:

Place UNIT 2 in the surrounding UNIT such that the ORIGIN of UNIT 2 is at X=3, Y=3.5, Z=4 relative to the origin of the surrounding UNIT.

KENO V.a: HOLE 2 3 3.5 4

KENO-VI: HOLE 2 ORIGIN X=3.0 Y=3.5 Z=4.0

ARRAY

When used within a UNIT description, this entry provides an ARRAY placement description. In KENO V.a, it always starts a new UNIT and generates a rectangular parallelepiped that fits the outer boundaries of the specified ARRAY. The specified ARRAY is positioned in the UNIT according to the most negative point in the ARRAY with respect to the coordinate system of the surrounding UNIT. Thus, the location of the minimum x, minimum y, and minimum z point in the array is specified in the coordinate system of the UNIT into which the ARRAY is being placed.

In KENO-VI, the ARRAY keyword is used to position an ARRAY within a region in a surrounding UNIT relative to the origin of the surrounding UNIT. When the subordinate keyword PLACE is entered, it is followed by six numbers that precisely locate the ARRAY within the surrounding UNIT as shown in the example below. The first three numbers consist of the element in the ARRAY of the UNIT selected to position the ARRAY. The next three numbers consist of the position of the origin of the selected UNIT in the surrounding UNIT. Higher level ARRAY boundaries may intersect lower level ARRAY boundaries as long as they do not intersect HOLEs in the UNITs contained in the ARRAY or in UNITs contained in lower level ARRAYs.

The syntax for the ARRAY card is as follows:

KENO V.a: ARRAY array_id x y z

KENO-VI: ARRAY array_id l1lN [PLACE Nx Ny Nz x y z]

array_id

is the label that identifies the array to be placed.

l1lN

is the REGION DEFINITION VECTOR. These are previously defined shape labels, and together they define the region in which the array array_id is to be placed. This is used only in KENO-VI.

Nx Ny Nz

are three integers that define the element in the ARRAY of the UNIT selected to position the ARRAY. This is used only in KENO-VI.

x y z

specify the position of the ARRAY in the UNIT.

In KENO V.a, the x, y, and z values are the point where the most negative x, y, and z point of the ARRAY is to be located in the UNIT’s coordinates.

In KENO-VI, the x, y, and z values are the point where the origin of the UNIT specified by Nx, Ny, and Nz is to be located in the shape specified by the REGION DEFINITION VECTOR.

Example of ARRAY use:

In KENO V.a, position the most negative point of ARRAY 6 at X = 2.0, Y = 3.0, Z = 4.0 relative to the origin of the containing UNIT.

ARRAY 6 2.0 3.0 4.0

In KENO-VI, position instead the origin of UNIT (1,2,3) of ARRAY 6 at X = 2.0, Y = 3.0, Z = 4.0 and specify the ARRAY boundary to be the region that is inside the geometry shapes labeled 10 and 20 and outside the geometry shape labeled 30 used to describe the surrounding UNIT.

ARRAY 6 10 20 −30 PLACE 1 2 3 2.0 3.0 4.0

REPLICATE and REFLECTOR

These keywords specific to KENO V.a are used to generate additional geometry regions having the shape of the previous region. The geometry keyword REFLECTOR is a synonym for REPLICATE. The desired weighting functions can be applied to those regions by specifying biasing data as described in Biasing or weighting data. The total thickness generated for each surface is the thickness per region for that surface times the number of regions to be generated, nreg.

The replicate specification is frequently used to generate weighting regions external to an ARRAY placement description. Thus an ARRAY placement description followed by a REPLICATE description would generate regions of a cuboidal shape. A cylindrical reflector could be generated by following the ARRAY placement description with a CYLINDER and then a REPLICATE. A HOLE cannot immediately follow a REPLICATE.

Extra regions using default weights can be generated by specifying the first importance region, imp, to be one that was not defined in the BIASING INFORMATION provided in a READ BIAS block. This capability can be used to generate extra regions for collecting information such as fluxes, leakage, etc.

Multiple replicate descriptions can be used in any problem. This capability can be used to model different reflector materials of different thicknesses on different faces.

The number of appropriate region dimensions needed for specifying REPLICATE is determined by the preceding region. For example, if the previous region were a SPHERE, one entry (i.e., t1) would be required. If the previous region were a CYLINDER, the first entry, t1, would be the thickness/region in the radial direction, the second entry, t2, would the thickness/region in the positive length direction, the third entry, t3, would be the thickness/region in the negative length direction, etc. The REPLICATE specification requirements for a CUBE are the same as for a CUBOID.

Syntax:

REPLICATE|REFLECTOR m b t1tN nreg

m

is the number of the material (non-negative integer) that fills the particular REPLICATE/REFLECTOR region in the UNIT description. A material of zero indicates a void region (i.e., no material is present in the volume defined by the shape).

b

is the bias identification number (positive integer) assigned to the particular region defined by the shape in the KENO V.a UNIT description. If the specified bias ID is defined in a READ BIAS block, the bias ID number will be incremented automatically, increasing one for each additional region up to nreg.

t1tN

represent the thickness (floating point number) per region for each of the N surfaces that define the particular shape. If the specified bias ID is one that is defined in the READ BIAS block, the region thicknesses should be consistent with the thicknesses used to generate the bias data being used. See 9-1-2-7.

nreg

is the number of regions (integer) to be generated.

Example:

Create five regions of material 4, each being 3 cm thick, outside a cuboid region (a cuboid has six dimensions). The inner-most of the five generated regions has a bias id of 2. The following four regions have bias id of 3, 4, 5, and 6.

MEDIA

This card is used in the KENO-VI input file to define the location of a mixture relative to the geometric shapes in the UNIT. Fig. 61 shows the input for a set of three intersecting SPHEREs in a CUBOID. The total volume data for a region in the problem may be entered as the last entry on the MEDIA card by using the keyword “VOL=” keyword immediately followed by the volume in cm3. The volume entered is the volume of the region in the unit multiplied by the number of times the unit occurs in the problem minus any volume excluded from the problem by ARRAY boundaries and HOLEs. The volumes for any or all regions may be entered. If the volume is entered here, this value will be used even if volumes are also entered as a file or calculated (See Volume data). Volumes not entered will be determined by the input specified in the VOLUME DATA block. If no volume is supplied, the KENO-VI default volume of -1 will be used. This only affects volume-averaged quantities, i.e., not keff.

Syntax:

MEDIA m b l1lN [VOL=v]

m

is the material (positive integer or zero for vacuum) that fills the region defined by MEDIA.

b

is the bias id for the material sector being defined.

l1lN

is the region definition vector (N integers). These are N previously defined shape labels that together define the material sector.

VOL=

is an optional sub-keyword used to input the material sector volume.

v

is the volume in cm3 of the material sector defined by the MEDIA card.

_images/fig111.png

Fig. 61 Example of the MEDIA record.

BOUNDARY

This card is used in KENO-VI to define the outer boundary of the UNIT. In KENO V.a, the outer boundary of the UNIT is implicitly defined by the last shape in the UNIT. Each UNIT must have one and only one BOUNDARY card.

Syntax:

BOUNDARY l1lN

l1lN

is the UNIT BOUNDARY DEFINITION VECTOR (N integers). These are N previously defined shape labels that together define the outer boundary of the UNIT. All entries must be positive for the UNIT being defined to be used subsequently as a HOLE.

ARRAY Data

The array definition data block is used to define the size of an ARRAY and to position UNITs (defined in the geometry data) in a 3-D lattice that represents the ARRAY being described. As many arrays as are necessary can be described in a problem, subject to computer storage limitations. In KENO V.a, only one ARRAY may be placed directly in a UNIT, but as many ARRAYs as are needed may be placed in the UNIT by using HOLEs. In KENO-VI, any number of arrays can be placed in any UNIT either directly or indirectly using HOLEs. There is no default global array. If a global array is desired it must be explicitly defined.

The ARRAY definition data is entered as:

READ ARRAY ARRAY DATA END ARRAY

The ARRAY_ DATA consists of ARRAY_PARAMETERS and UNIT_ORIENTATION_DESCRIPTION.

ARRAY Parameters

The ARRAY parameters that can be used in the definition of an ARRAY are:

ARA=

GBL=

NUX=, NUY=, NUZ=

PRT=

COM=

TYP= (KENO-VI only)

ARA=

The ARA= parameter defines a reference number for an ARRAY. It has no default in KENO-VI. In KENO V.a, if is missing, the default is 1.

Syntax:

ARA=a

a

is the reference number for the ARRAY. It has no default in KENO-VI. In KENO V.a, if is missing, the default is 1.

GBL=

This is used to input the number of the global array.

Syntax:

GBL=g

g

is the reference number for the global ARRAY. In KENO V.a it must not be entered more than once. The default is the largest value for a, the reference number for the ARRAY. In KENO-VI it is no default value and if entered more than once, the last value is used.

PRT=

This entry is used to enable printing the ARRAY of UNIT numbers.

Syntax:

PRT=print

print

is a logical constant which defaults to YES, indicating that the ARRAY of UNIT numbers is printed. If the value is NO, then a summary table is printed instead containing the number of times each unit is used in each array.

NUX=, NUY=, NUZ=

These entries are used to input the number of units in the X, Y, and Z directions, respectively.

Syntax:

NUX=nx NUY=ny NUZ=nz

nx ny nz

are the number of units in the X, Y, and Z directions, respectively. There is no default in KENO-VI. In KENO V.a, each of them defaults to 1.

TYPE=

This entry is used to specify the type of ARRAY and is specific to KENO-VI, where more than one type of arrays can be used. It cannot be used in KENO V.a.

Syntax:

TYP=atyp

atyp

type of array (cuboidal or square, hexagonal or triangular, rhexagonal, shexagonal, dodecahedral), default = cuboidal

COM=

This keyword is used to enter a comment.

Syntax:

COM=delim comment delim

delim

is a delimiter. Acceptable delimiters are “, ‘, * , ^ , or !.

comment

is the comment string. Maximum comment length is 132 characters.

ARRAY orientation data

There are two methods to enter the UNIT numbers constituting an ARRAY: LOOP and FILL.

LOOP Construct

The LOOP construct resembles a FORTRAN DO-loop construct. The arrangement of UNITs may be considered as consisting of a 3-D matrix of UNIT numbers, with the UNIT position increasing in the positive X, Y, and Z directions, respectively.

Syntax:

LOOP u ix1 ix2 incx iy1 iy2 incy iz1 iz2 incz END LOOP

u

is the UNIT identification number (a positive integer).

ix1

is the starting position in the X direction; ix1 must be at least 1 and less than or equal to nx of NUX=, NUY=, NUZ=.

ix2 is the ending position in the X direction; ix2 must be at least 1 and less than or equal to nx.

incx

is the number of UNITs by which increments are made in the positive X direction; incx must be greater than zero and less than or equal to nx.

iy1

is the starting position in the Y direction; iy1 must be at least 1 and less than or equal to ny.

iy2

is the ending position in the Y direction; iy2 must be at least 1 and less than or equal to ny.

incy

is the number of UNITs by which increments are made in the positive Y direction; incy must be greater than zero and less than or equal to ny.

iz1

is the starting position in the Z direction; iz1 must be at least 1 and less than or equal to nz.

iz2

is the ending position in the Z direction;, iz2 must be at least 1 and less than or equal to nz.

incz

is the number of UNITs by which increments are made in the positive Z direction; incz must be greater than zero and less than or equal to nz.

The syntax for ending the LOOP construct is:

END LOOP

The sequence u through incz is repeated until the entire ARRAY is described. If any portion of an ARRAY is defined in a conflicting manner, the last entry to define that portion will determine the ARRAY’s configuration. To use this feature, fill the entire ARRAY with the most relevant UNIT number and superimpose the other UNIT numbers in their proper places. An example showing the use of the LOOP option is given below. This 5 × 4 × 3 ARRAY of UNITs is a matrix of UNITs that has 5 UNITs stacked in the X direction, 4 UNITs in the Y direction, and 3 UNITs in the Z direction. X increases from left to right, and Y increases from bottom to top. Each Z layer is shown separately.

Given:

1 2 1 2 1         2 1 2 1 2       1 1 1 1 1
1 1 1 1 1         2 2 2 2 2       1 3 3 3 1
1 1 1 1 1         2 2 2 2 2       1 3 3 3 1
1 2 1 2 1         2 1 2 1 2       1 1 1 1 1
Z Layer 1         Z Layer 2       Z Layer 3

The data for this array could be entered using the following entries.

(1) 1 1 5 1 1 4 1 1 3 1 This fills the entire array with 1s.

(2) 2 2 5 2 1 4 3 1 1 1 This loads the four 2s in the first Z layer.

(3) 2 1 5 1 2 3 1 2 2 1 This loads the second and third rows of 2s in the second Z layer.

(4) 2 1 5 2 1 4 3 2 2 1 This loads the desired 2s in the first and fourth rows of the second Z layer.

(5) 3 2 4 1 2 3 1 3 3 1 This loads the 3s in the third Z layer and completes the array data input.

The second layer could have been defined by substituting the following data for entries (3) and (4):

(3) 2 1 5 1 1 4 1 2 2 1 This completely fills the second layer with 2s.

(4) 1 2 4 2 1 4 3 2 2 1 This loads the four 1s in the second layer.

When using the LOOP option, there is no single correct method of entering the data. If a UNIT is improperly positioned in the ARRAY or if some positions in the ARRAY are left undefined, it is often easier to add data to correctly define the ARRAY than to try to correct the existing data.

FILL Construct

The FILL construct enters data by stringing in UNIT numbers starting at X=1, Y=1, Z=1, and varying X, then Y, and then Z to fill the ARRAY. nx x ny x nz entries are required. FIDO-like input options specified in Table 152 are also available for filling the ARRAY.

Syntax:

FILL u1uN {END FILL}|T

u1uN are the N=nx x ny x nz UNIT numbers that make up the ARRAY

The syntax for ending the FILL construct is

END FILL

An alternative to end the UNIT data in FILL is by entering the letter T.

Table 152 FIDO-like input for mixed box orientation fill option.

Count

field

Option

field

Operand

field

Function

j

stores j at the current position in the array

i

R

j

stores j in the next i positions in the array

i

*

j

stores j in the next i positions in the array

i

$

j

stores j in the next i positions in the array

F

j

fills the remainder of the array with unit number j, starting with the current position in the array

A

j

sets the current position in the array to j

i

S

increments the current position in the array by i (This allows for skipping i positions; i may be positive or negative.)

i

Q

j

repeats the previous j entries i times (default value of i is 1)

i

N

j

repeats previous j entries i times, inverting the sequence each time. (default value of i is 1)

i

B

j

backs i entries. From that position, repeats the previous j entries in reverse order (default value of i is 1)

i

I

j k

provides the end points j and k, with i entries linearly interpolated between them (i.e., a total of i+2 points). At least one blank must separate j and k. When used for an integer array, the I option should only be used to generate integer steps—i.e., (k− j)/(i+1) should be a whole number

T

terminates the data reading for the array

Note

When entering data using the options in this table, the count field and option field must be adjacent with no imbedded blanks. The operand field may be separated from the option field by one or more blanks.

Example: Consider a 3 × 3 × 1 ARRAY filled with 8 UNIT 1s and a UNIT 2, as shown below.

1     1       1
1     2       1
1     1       1

The input data to describe this ARRAY could be entered as follows:

Option (1) 1 1 1 1 2 1 1 1 1 T

This fills the array one position at a time, starting at the lower left corner. The T terminates the data.

or

Option (2) F1 A5 2 END FILL

The F1 fills the entire array with 1s, the A5 locates the fifth position in the array, and the 2 loads a 2 in that position. The END FILL terminates the data.

Albedo data

Albedo boundary conditions are entered by assigning an albedo condition to each face of the outermost boundary. The default value for each face is vacuum. The default values are overridden only on faces for which other albedo names are specified. Albedo boundary conditions are applied only to the outermost region of a problem. In KENO V.a this geometry region must be a rectangular parallelepiped. The outer boundary can be any shape (or combination of shapes) in KENO-VI. Material-specific albedos (e.g., H2O and CONC) may not be used in continuous energy calculations.

KENO-VI users need to be aware that when a neutron reaches a surface with a vacuum albedo, that neutron is lost. If a model contains features that are reentrant, that is a neutron could exit the model and reenter the model on the other side of an unmodeled region, all neutrons passing through the problem boundary are lost when they reach the unmodeled region. Neutrons are not “transported” across unmodeled areas between reentrant surfaces. It is not possible to create a KENO V.a model with reentrant problem outer boundary surfaces.

The syntax for entering the albedo boundary conditions is:

READ BOUNDS fc1=a1 [fc2=a2… [fcN=aN ]…] END BOUNDS

fc1fcN are N face codes as defined in Table 153.

a1aN are the albedo types as defined in Table 154

Table 154 lists some material-specific albedo sets. Care must be exercised when using material-specific albedo types. These data sets were generated using a real problem, and they implicitly reflect the neutron energy spectrum, materials, and geometry from that model. Where neutron energy spectra, materials, and geometry vary from that model, the material-specific albedos may give significantly incorrect results. This may be checked by comparing results from a sample of calculations performed with both explicitly modeled reflectors and material-specific albedos. In general, use of material-specific albedos is not recommended.

Table 153 Face codes and surface numbers for entering boundary (albedo) conditions (continued in the next table).

Face code

Faces defined by face codes

+XB=

Positive X face

&XB=

Positive X face

−XB=

Negative X face

+YB=

Positive Y face

&YB=

Positive Y face

−YB=

Negative Y face

+ZB=

Positive Z face

&ZB=

Positive Z face

−ZB=

Negative Z face

ALL=

All six faces

XFC=

Both positive and negative X faces

YFC=

Both positive and negative Y faces

ZFC=

Both positive and negative Z faces

+FC=

Positive X, Y, and Z faces

&FC=

Positive X, Y, and Z faces

−FC=

Negative X, Y, and Z faces

XYF=

Positive and negative X and Y faces

XZF=

Positive and negative X and Z faces

YZF=

Positive and negative Y and Z faces

+XY=

Positive X and Y faces

+YX=

Positive X and Y faces

&XY=

Positive X and Y faces

&YZ=

Positive X and Y faces

+XZ=

Positive X and Z faces

+ZX=

Positive X and Z faces

&XZ=

Positive X and Z faces

&ZX=

Positive X and Z faces

+YZ=

Positive Y and Z faces

+ZY=

Positive Y and Z faces

&YZ=

Positive Y and Z faces

&ZY=

Positive Y and Z faces

−XY=

Negative X and Y faces

−XZ=

Negative X and Z faces

−YZ=

Negative Y and Z faces

YXF=

Positive and negative X and Y faces

ZXF=

Positive and negative X and Z faces

ZYF=

Positive and negative Y and Z faces

−YX=

Negative X and Y faces

−ZX=

Negative X and Z faces

−ZY=

Negative Y and Z faces

BODY= x

x is the body’s geometry label in the global unit (KENO-VI only)

SURFACE( ii )=

Boundary condition for surface number ii of body x (KERNO-VI) only

_images/tab18_Page_1.png
_images/tab18_Page_2.png
_images/tab18_Page_3.png
Table 154 Albedo names available on the KENO albedo library for use with the face codes *

DP0H2O

DPOH2O

DP0

DPO

12 in. (30.48 cm) double P0 water differential albedo with 4 incident angles

H2O

WATER

12 in. (30.48 cm) water differential albedo with 4 incident angles

PARAFFIN

PARA

WAX

12 in. (30.48 cm) paraffin differential albedo with 4 incident angles

CARBON

GRAPHITE

C

78.74 in. (200.00 cm) carbon differential albedo with 4 incident angles

ETHYLENE

POLY

CH2

12 in. (30.48 cm) polyethylene differential albedo with 4 incident angles

CONC-4

CON4

CONC4

4 in. (10.16 cm) concrete differential albedo with 4 incident angles

CONC-8

CON8

CONC8

8 in. (20.32 cm) concrete differential albedo with 4 incident angles

CONC-12

CON12

CONC12

12 in. (30.48 cm) concrete differential albedo with 4 incident angles

CONC-16

CON16

CONC16

16 in. (40.64 cm) concrete differential albedo with 4 incident angles

CONC-24

CON24

CONC24

24 in. (60.96 cm) concrete differential albedo with 4 incident angles

VACUUM

VOID

Vacuum condition

SPECULAR

MIRROR

REFLECT

Mirror image reflection

PERIODIC

Periodic boundary condition

WHITE

White boundary condition

* Material-specific albedos may not be used in continuous energymode

The BODY and SURFACE keywords are unique to KENO-VI. The face code BODY= refers to the body label in global unit input. For example, assume the GLOBAL UNIT boundary record in a KENO-VI input consisted of the following: BOUNDARY 10 −30 20. In this case BODY=10 would refer to the geometry record labeled 10, BODY=20 would refer to the geometry record labeled 20, and BODY=30 would refer to the geometry record labeled 30. All surface numbers following the BODY keyword apply to that body. The default value of BODY is the first geometry label listed in the GLOBAL UNIT boundary record.

All the face codes, listed in the first part of Table 153, except BODY= and SURFACE(ii)= were intended to apply only to cuboids (KENO V.a). However, when used with non-cuboidal surfaces (KENO-VI) they will fill in the first six surface positions of a body in the following order, +X, −X, +Y, −Y, +Z, −Z. The ALL face code will apply the listed boundary conditions to all surfaces of the body currently being considered.

Albedo boundary conditions may be entered on each GLOBAL UNIT boundary surface multiple times. The boundary condition that applies to the surface is the last one entered. If no boundary data are entered or if no albedo boundary condition is applied to a GLOBAL UNIT boundary surface, then the boundary surface is assumed to have a void or vacuum boundary condition. Any CHORDed surfaces that are GLOBAL UNIT boundaries will use the default (void) boundary condition, and it cannot be changed. This restriction may need to be considered when building the geometry of the GLOBAL UNIT.

Example:

Use a 24 in. concrete albedo boundary condition on the −Z face of a problem with a cuboidal boundary and use mirror image reflection on the +X and −X faces of the cuboid to represent an infinite linear array on a 2 ft. thick concrete pad.

READ BOUNDS −ZB=CON24 XFC=MIRROR END BOUNDS

Example:

Use a 24 in. concrete albedo boundary condition on the −Z face of a problem with a hexagonal boundary, and use mirror image reflection on all side faces of the hexprism to represent an infinite planar array on a 2 ft. thick concrete pad.

READ BOUNDS

SURFACE(1)=MIRROR SURFACE(2)=MIRROR SURFACE(3)=MIRROR

SURFACE(4)=MIRROR SURFACE(5)=MIRROR SURFACE(6)=MIRROR

SURFACE(7)=VACUUM SURFACE(8)=CONC24

END BOUNDS

Example:

The outer boundary of the global unit consists of a cuboid (body label 10) and a sphere (body label 20). The sphere is large enough to cut the corners of the cuboid leaving most of the cuboid intact. Use a 24 in. concrete albedo boundary condition on the −Z face of the cuboid to represent a 2 ft. thick concrete pad. Use the DP0H2O on the other surfaces to represent an infinite water reflector.

READ BOUNDS

BODY=10 ALL=DP0H2O −ZB=CON24

BODY=20 SURFACE(1)=DP0H2O

END BOUNDS

Warning

The user should thoroughly understand material-specific albedos (e.g., DP0H2O, CON24, etc.) before attempting to use these reflectors. Missapplication of these problem-specific albedo data can cause the code to produce incorrect results without obvious symptoms.**

Biasing or weighting data

The biasing data block is used (in only multigroup mode) to define the weight that is given to a neutron surviving Russian roulette. The average weight of a neutron that survives Russian roulette, wtavg, is defaulted to dwtav (WTA= in the parameter data [see Title and parameter data]) for all BIAS IDs and can be overridden by entering biasing information.

The biasing_information is used to relate a BIAS ID to the desired energy-dependent values of wtavg. This concept is similar to the way the MIXTURE ID, mat, is related to the macroscopic cross section data.

The weighting functions used in KENO  are energy-dependent values of wtavg that are applicable over a given thickness interval of a material. For example, the weighting function for water is composed of sets of energy-dependent values of wtavg for 11 intervals, each interval being 3 cm thick. The first set of wtavg’s is for the 0–3 cm interval of water, the second set of wtavg’s is for the 3–6 cm interval of water, etc. The eleventh set of wtavg’s is for the 30–33 cm interval of water.

To input biasing information, a BIAS ID must be assigned to correspond to a set of wtavg. Biasing data can specify a MATERIAL ID from the existing KENO V.a weighting library or from the AUXILIARY DATA input. The materials available from the KENO  weighting library are listed in Table 155.

The biasing_information is entered in one of the following two forms. The first set is said to input the CORRELATION DATA, while the second form is said to input the AUXILIARY DATA.

READ BIAS ID=m ib ie END BIAS

or

READ BIAS WT[S]=wttitl id s t1 i1 g1 w1,1w1,i1xg1ts is gs ws,1ws,isxgs END BIAS

ID= specifies that CORRELATION DATA will be entered next.

WT= or WTS= specifies that AUXILIARY DATA will be entered next.

m

is the identification (material ID) for the material whose weighting function is to be used. A material ID can be chosen from the existing KENO weighting library (Table 155) or from the auxiliary data input using the second form of the BIAS block as described later. If a material ID appears in both the KENO weighting library and the auxiliary data, the weights from the auxiliary data will be used.

ib

is the bias ID of the weighting function for the first interval of material m. The geometry record having the bias ID equal to ib will use the group-dependent weights from the first interval of material m.

ie

is the bias ID of the group-dependent weights from the (ieib + 1)th interval of material m.

wttitl

is an arbitrary title name (12 characters maximum), such as CONCRETE, WATER, SPECIALH2O, etc., to identify the material for which the user is entering data. Embedded blanks are not allowed.

id

is an identification number (material ID). The value is arbitrary. However, if the data are to be utilized in the problem, this ID must also be used at least once in the first form of the BIAS block.

s

is the number of sets of group structures for which weights will be read for this ID.

t1ts

are s thicknesses of each increment for which weights will be read for this ID.

i1is

are s numbers of increments for which weights will be read for this ID.

g1gs

are s numbers of energy groups for which weights will be read.

w1,i1xg1ws,isxgs

are s sets of weights, each set containing a number of weights equal to the product of number of increments times the number of groups for that set. The group index varies the fastest.

Table 155 IDs, group structure and incremental thickness for weighting data available on the KENO  weighting library.

Material

Material

ID

Group structure

for which weights

are available

Incrementa

thickness

Total

number of

increments available

Concrete

301

27

28

56

200

238

252

5

5

5

5

5

5

20

20

20

20

20

20

Paraffin

400

27

28

56

200

238

252

3

3

3

3

3

3

10

10

10

10

10

10

Water

500

27

28

56

200

238

252

3

3

3

3

3

3

10

10

10

10

10

10

Graphite

6100

27

28

56

200

238

252

20

20

20

20

20

20

10

10

10

10

10

10

aGroup-dependent weight averages are supplied for each increment of the specified incremental thickness (i.e., for any given material) the first ngp (number of energy groups) weights apply to the first increment of the thickness specified here, the next ngp weights apply to the next increment of that thickness, etc CAUTION–If bias IDs defined in the weighting information data are used in the geometry, the region thickness should be consistent with the incremental thickness of the weighting data in order to avoid overbiasing or underbiasing.

Warning

The user should thoroughly understand weighted tracking before attempting to generate and use auxiliary data for biasing. Incorrect weighting can cause the code to produce incorrect results without obvious symptoms.

Caution

1. Each set of AUXILIARY or CORRELATION data must be completely described in conjunction with its keyword. Complete sets of these data can be interspersed in an arbitrary order but data within each set must be entered in the specified order.

2. AUXILIARY DATA: If the same m is specified in more than one set of data, the last set having the group structure used in the problem is the set that will be utilized. When AUXILIARY DATA are entered, CORRELATION DATA must also be entered in order to use the AUXILIARY DATA.

3. CORRELATION DATA: If biasing data define the same bias ID (from the geometry data) more than once, the value that is entered last supersedes previous entries. Be well aware that multiple definitions for the same bias ID can cause erroneous answers due to overbiasing.

  1. Bias data may not be used in continuous energy mode.

Examples

  1. Use the first form of the BIAS block to utilize the water biasing factors in bias IDs 2 through 11. From Table 155, water has material ID m=500 and has bias parameters for 10 intervals that are each 3 cm thick.

READ BIAS ID=500 2 11 END BIAS

  1. Use the second form of the BIAS block to specify biasing factors for SPECIALWATER to be used in bias IDs 6 and 7. The SPECIALWATER biasing factors have a value of 0.69 for BIAS ID 6 and 0.86 for bias ID 7 in each energy group. Sixteen-group cross sections are being used. Each weighting region is 3.048 cm thick. The material ID is arbitrarily chosen to be 510. Note that the first form of the BIAS block must be entered to allow the second form of the BIAS block to be used for BIAS IDs 6 and 7.

READ BIAS WT=SPECIALWATER 510 1 3.048 2 16 16*0.69 16*0.86 ID=510 6 7 END BIAS

  1. An example of multiple definitions for the same bias ID follows:

READ BIAS ID=400 2 7 ID=500 5 7 END BIAS .

The data for paraffin (ID=400) will be used for bias IDs 2, 3, and 4, and the data for water (ID=500) will be used for bias IDs 5, 6, and 7. The paraffin data for bias IDs 5, 6, and 7 have been overwritten by water data.

Multiple definitions for the same bias ID are not necessarily incorrect, but the user should be cautious about using multiple definitions and should ensure that the desired biasing or weighting functions are used in the desired geometry regions.

  1. An example of how the bias ID relates to the energy-dependent values of weights is given below.

Assume that a paraffin reflector is to be used, and it is desirable to use the weighting function from the KENO weighting library to minimize the running time for the problem. Also assume that these weighting functions are to be used in the volumes defined in the geometry records having the bias ID (defined on a shape or MEDIA card for KENO V.a and KENO-VI, respectively) equal to 6, 7, 8, and 9. Correlation data are then entered and auxiliary data will not be entered.

The biasing data would be:

READ BIAS ID=400 6 9 END BIAS.

The results of these data are

(1) the group-dependent weights for the 0–3 cm interval of paraffin will be used in the volume defined by the geometry region having bias ID= 6.

(2) the group-dependent weights for the 3–6 cm interval of paraffin will be used in the volume defined by the geometry region having bias ID= 7.

(3) the group-dependent weights for the 6–9 cm interval of paraffin will be used in the volume defined by the geometry region having bias ID= 8.

(4) the group-dependent weights for the 9–12 cm interval of paraffin will be used in the volume defined by the geometry region having bias ID= 9.

Start data

Special start options are available for controlling the initial neutron distribution. The default starting distribution for an array is flat over the overall array dimensions, in fissile material only. The default starting distribution for a single unit is flat over the system, in fissile material only. See Table 156 for the starting distributions available in KENO. The syntax for the START block is:

READ START p1pN END START

p1pN are N initializations for the parameters listed below.

The starting information that can be entered is given below. Enter only the data necessary to describe the desired starting distribution.

NST = ntypst

start type, default = 0 Table 156 lists the available options under the heading, “Start type.”

TFX = tfx

the X coordinate of the point at which neutrons are to be started. Default = 0.0. Use for start types 3, 4, and 6.

TFY = tfy

the Y coordinate of the point at which neutrons are to be started. Default = 0.0. Use for start types 3, 4, and 6.

TFZ = tfz

the Z coordinate of the point at which neutrons are to be started. Default = 0.0. Use for start types 3, 4, and 6.

NXS = nbxs

the x index of the unit’s position in the global array. Default = 0. Use for start types 2, 3, and 6.

NYS = nbys

the y index of the unit’s position in the global array. Default = 0. Use for start types 2, 3, and 6.

NZS = nbzs

the z index of the unit’s position in the global array. Default = 0. Use for start types 2, 3, and 6.

KFS = kfis

the mixture whose fission spectrum is to be used for starting neutrons that are not in a fissionable medium. Defaulted to the fissionable mixture having the smallest mixture number. Available for start types 3, 4, and 6.

LNU = lfin

the final neutron to be started at a point. Default = 0. Each lfin should be greater than zero and less than or equal to NPG. Each successive lfin should be greater than the previous one. Use for start types 6 and 8.

NBX = nboxst

the unit in which neutrons will be started. Default = 0. Use for start types 4 and 5.

FCT = fract

the fraction of neutrons that will be started as a spike. Default = 0. Use for start type 2.

XSM = xsm

the −X dimension of the cuboid in which the neutron will be started. For an array problem, XSM is defaulted to the minimum X coordinate of the global array. If the reflector key RFL is YES, then and the outer reflector region is a cube or cuboid, XSM is defaulted to the minimum X coordinate of the outer reflector region. If RFL is YES and the outer region of the reflector is not a cube or cuboid, then XSM must be entered in the start data and must fit inside the outer reflector region. Available for start types 0, 1, 2, and 8.

XSP = xsp

the +X dimension of the cuboid in which the neutrons will be started. For an array problem, XSP is defaulted to the maximum X coordinate of the global array. If the reflector key RFL is YES, then and the outer reflector region is a cube or cuboid, XSP is defaulted to the maximum X coordinate of the outer reflector region. If RFL is YES and the outer region of the reflector is not a cube or cuboid, then XSP must be entered in the data and must fit inside the outer reflector region. Available for start types 0, 1, 2, and 8.

YSM = ysm

the −Y dimension of the cuboid in which the neutron will be started. For an array problem, YSM is defaulted to the minimum Y coordinate of the global array. If the reflector key RFL is YES, then YSM is defaulted to the minimum Y coordinate of the outer reflector region, provided that region is a cube or cuboid. If RFL is YES and the outer region of the reflector is not a cube or cuboid, then YSM must be entered in the start data and must fit inside the outer reflector region. Available for start types 0, 1, 2, and 8.

YSP = ysp

the +Y dimension of the cuboid in which the neutrons will be started. For an array problem, YSP is defaulted to the maximum Y coordinate of the global array. If the reflector key RFL is YES, then YSP is defaulted to the maximum Y coordinate of the outer reflector region, provided that region is a cube or cuboid. If RFL is YES and the outer region of the reflector is not a cube or cuboid, then YSP must be entered in the start data and must fit inside the outer reflector region. Available for start types 0, 1, 2, and 8.

ZSM = zsm

the −Z dimension of the cuboid in which the neutrons will be started. For an array problem, ZSM is defaulted to the minimum Z coordinate of the global array. If the reflector key RFL is YES, then ZSM is defaulted to the minimum Z coordinate of the outer reflector region, provided that region is a cube or cuboid. If RFL is YES and the outer region of the reflector is not a cube or cuboid, then ZSM must be entered in the start data and must fit inside the outer reflector region. Available for start types 0, 1, 2, and 8.

ZSP = zsp

the +Z dimension of the cuboid in which the neutrons will be started. For an array problem, ZSP is defaulted to the maximum Z coordinate of the global array. If the reflector key RFL is YES, then ZSP is defaulted to the maximum Z coordinate of the outer reflector region, provided that region is a cube or cuboid. If RFL is YES and the outer region of the reflector is not a cube or cuboid, then ZSP must be entered in the start data and must fit inside the outer reflector region. Available for start types 0, 1, 2, and 8.

RFL = rflkey

the reflector key. If the reflector key is YES, then neutrons can be started in the reflector. If it is NO, then all the neutrons will be started in the array. Enter YES or NO. Default = NO. Available for start types 0, 1, and 2.

PS6 = lprt6

the key for printing start type 6 input data. If the key is YES, then start type 6 data are printed. If it is NO, then start type 6 data are not printed. Enter YES or NO. Default = NO. Available for start type 6.

PSP = lpstp

the key for printing the neutron starting points using the tracking format. If the key is YES, then print the neutron starting points. If it is NO, then do not print the starting points. Enter YES or NO. Default = NO. Available for all start types.

RDU = rdu

the file from which ASCII start data are to be read for start type 6.

WS6 = ws6

the file to which ASCII start data are written.

MSS = filename.msl

the file from which ASCII start data are to be read. filename may include a valid pathname. Available for start type 9.

Table 156 Starting distributions available in KENO.

Start

type

Required

data

Optional

data

Starting distribution

0

None

NST

XSM

XSP

YSM

YSP

ZSM

ZSP

RFL

PSP

Uniform throughout fissile material within the volume defined by (1) the outer region of a single unit, (2) the outer region of a reflected array having the reflector key set true, (3) the boundary of the global array, or (4) a cuboid specified by XSM, XSP, YSM, YSP, ZSM, and ZSP.

1

NST

XSM

XSP

YSM

YSP

ZSM

ZSP

RFL

PSP

The starting points are chosen according to a cosine distribution throughout the volume of a cuboid defined by XSM, XSP, YSM, YSP, ZSM, and ZSP. Points that are not in fissile material are discarded.

2

NST

NXS

NYS

NZS

FCT

XSM

XSP

YSM

YSP

ZSM

ZSP

RFL

PSP

An arbitrary fraction (FCT) of neutrons are started uniformly in the unit located at position NXS, NYS, NZS in the global array. The remainder of the neutrons is started in fissile material, from points chosen from a cosine distribution throughout the volume of a cuboid defined by XSM, XSP, YSM, YSP, ZSM, ZSP.

3

NST

TFX

TFY

TFZ

NXS

NYS

NZS

KFS

PSP

All neutrons are started at position TFX, TFY, TFZ within the unit located at position NXS, NYS, NZS in the global array.

4

NST

TFX

TFY

TFZ

NBX

KFS

PSP

All neutrons are started at position TFX, TFY, TFZ within units NBX in the global array.

Starting distributions available in KENO (continued)

Start

type

Required

data

Optional

data

Starting distribution

5

NST

NBX

PSP

Neutrons are started uniformly in fissile material in units NBX in the global array.

6

NST

TFX

TFY

TFZ

LNUa

NXS

NYS

NZS

KFS

PS6

PSP

RDU

The starting distribution is arbitrarily input. LNU is the final neutron to be started at a point TFX, TFY, TFZ relative to the global coordinate system or at a point TFX, TFY, TFZ, relative to the unit located at the global array position NXS, NYS, NZS.

7

XSM

XSP

YSM

YSP

ZSM

ZSP

The starting points are chosen according to a flat distribution in the X- and Y-dimensions and a (1.0 − cos(z))2 distribution in the Z-dimension throughout the volume of a cuboid defined by XSM, XSP, YSM, YSP, ZSM, and ZSP. Points that are not in fissile material are discarded.

8

NST

ZSM

ZSP

FCT

XSM

XSP

YSM

YSP

Neutrons are started with flat distribution in X and Y, and a segmented distribution in Z, with the X-Y limits defined by XSM, XSP, YSM, YSP and the relative fraction in ZSP-ZSM defined by FCT. FCT must be the last thing entered for each segment.

9

NST

MSS

Mesh source from Sourcerer. The starting distribution is read from a previously created mesh source file declared with MSS=filename .msl, where filename may include a valid pathname. See Sourcerer section of SCALE manual for more details.

a When entering data for start 6, LNU must be the last entry for each set of data and the LNU in each successive set of data must be larger than the previous value of LNU. A set of data consists of required and optional data. The last LNU entered should be equal to the number per generation (parameter NPG= in the parameter input, Title and parameter data).

Extra 1-D XSECS IDs data

Extra 1-D cross section IDs are not required. They are allowed as input in order to simplify future modifications to calculate reaction rates, etc., as well as for compatibility with other SCALE codes. The syntax for the extra 1-D cross section data block is:

READ X1DS NEUTRON i1 …ix1d END X1DS

NEUTRON

is a keyword to indicate that the following ID identifies a neutron interaction.

i1ix1d

X1D 1-D identification numbers or keyword identifiers for the 1-D cross section to be used. These cross sections must be available on the mixture cross section library. X1D entries are expected to be read (see integer PARAMETER data).

Mixing table data

A cross section mixing table must be entered if KENO is being run stand alone and a Monte Carlo cross section format library is not being used in the multigroup mode, or KENO is being run stand alone in the continuous energy mode. If the parameter LIB= (Sect. 8.1.2.3) is entered, then mixing table data must be entered. A cross section mixing table is entered using the following syntax:

READ MIXT p1pN END MIXT

p1pN

are N parameters that might or might not be keyworded.

The possible parameters that can be used in a MIXT block are described below.

SCT = nsct

is used to input the number of scattering angles and only applies in multigroup mode. nsct is the number of discrete scattering angles, default = 1. The number of scattering angles specifies the number of discrete scattering angles to be used for the cross sections. If SCT is not set (i.e., SCT= −1), then the number of scattering angles is determined from the cross section library specified. The number of scattering angles defaults to (ncoef+1)/2, where ncoef is the largest Legendre polynomial order used in the problem. It needs to be entered only once for a problem. If more than one value is entered, the last one is used for the problem. For assistance in determining the number of discrete scattering angles for the cross sections, see Number of scattering angles.

EPS = pbxs

is used to enter the cross section message cutoff value, and it only applies in multigroup mode. pbxs is the value of the P0 cross section for each transfer, above which generated warning messages will be printed, default = 3 × 10−5. The primary purpose of entering this cutoff value is to suppress printing these messages when they are generated during cross section processing. For assistance in determining a value for EPS, see Cross section message cutoff.

MIX = mix

is used to input the identification number of the mixture being described. mix defines the mixture being described.

NCM = ncmx

is used to input the nuclide mixture IDs to be used for this mixture. ncmx defines the nuclide mixture ID. When MIX=mix is read, ncmx is defaulted to mix also. Then, as long as all the nuclides that need to be mixed into mix already have mix specified as their nuclide mixture (frequently the case when using SCALE), the user does not need to specify NCM. The most usual case where NCM must be specified is when the mixtures were specified as a different mixture number when they were created in SCALE as compared to the mixture number used for them in KENO. Cell homogenized mixtures also need NCM specified.

TMP|TEM = temperature

is used to input the desired temperature of the CE cross section data.

nucl

is the nuclide ID number from the AMPX working format cross section library.

XS=fname

is used to input the optional continuous energy cross section filename to override the default cross sections. fname is the name of the file.

dens

is the number density (atoms/b-cm) associated with nuclide ID number nucl.

The sequence “nucl NCM=ncmx [XS=fname] dens” may be repeated until the mixture defined by MIX=mix has been completely described.

The sequence “MIX = mix NCM = ncmx TMP|TEM = temperature nucl NCM=ncmx [XS=fname] dens” may be repeated until all the mixtures have been described.

Note

If a given nuclide ID is entered more than once in the same mixture, then the number densities for that nuclide are summed.

If a mixture number is used as a nuclide ID, then it is treated as a nuclide and the number density associated with it is used as a density modifier. (If the density is entered as 1, then the mixture is mixed in at full density. If it is entered as 0.5, the mixture is mixed in at one half of its full density.) A Monte Carlo formatted cross section library is generated on the unit defined by the parameter XSC=. If this data set is saved, subsequent cases can utilize these mixtures without remixing.

The entry XS=fname is optional. If a nuclide is entered more than once in a mixture and this entry is specified, then they must be the same (i.e., cannot use more than one continuous energy cross section sets for a nuclide in a given mixture). Different mixtures may have the same nuclide with different continuous energy cross section sets.

Plot data

Plots of slices specified through the geometry can be generated and displayed (1) as character plots using alphanumeric characters to represent mixture numbers, unit numbers or bias ID numbers or (2) as color plots which generate a PNG file using colors to represent mixture numbers, unit numbers or bias ID numbers. Color plots require an independent program to display the PNG file to a PC or workstation monitor or to convert the file to be displayed using a plotting device. The keyword SCR= is used to control the plot display method. SCR=YES, the default value, uses the color plot display method. SCR=NO uses the character plot display method. The value of SCR determines the plot display method for all the plots specified in a problem. If SCR= is entered more than once, the last entry determines the plot display method. In other words, all plots generated by a problem will be either character plots or color plots.

The plot data can include the data for any or all types of plots. A plot by mixture number is the default. The kind of plot is defined by the parameter PIC=. Character plots are printed after the volumes are printed and before the final preparations for tracking are completed. Plot data are not required for a problem, but theyb can be used to verify the problem description. The actual plotting of the picture can be suppressed by entering PLT= NO in the parameter data or plot data. This allows plot data to be kept in the problem input for reference purposes without actually plotting the picture(s). Entering a value for PLT in the plot data will override any value entered in the parameter data. However, if a problem is restarted, the value of PLT from the parameter data is used. The upper left and lower right coordinates of the plot must be specified relative to the origin of the problem. See Color plots for a discussion of plot origins and plot data.

Enter the plot data using the following syntax:

READ PLOT p1pN END PLOT

p1pN

are N parameters entered using keywords followed by the appropriate data. The plot title and the plot character string must be contained within delimiters. Enter as many picture parameters as necessary to describe the plot. Multiple sets of plot data can be entered. The parameter input for each plot is terminated by a labeled or unlabeled END. The labeled END cannot use the word PLOT as the first four characters of the label. For example, END PLT1 is a valid label, but END PLOT1 is not. If an unlabeled END is used, it cannot start in column 1.

The possible parameters that can be used in a PLOT block are described below.

TTL= delim ptitl delim

Enter a one-character delimiter delim to signal the beginning of the title (132 characters maximum). The title is terminated when delim is encountered the second time. Acceptable delimiters include “ , ‘ , * , ^ , or !. Default = title of the KENO case.

PIC= wrd The plot type, wrd, is followed by one or more blanks and must be one of the keywords listed below. The plot type is initialized to MAT; the default is the value from the previous plot.

MAT

MIX[T[URE]]

MEDI[A]

These keywords will cause the plot to represent the mixture numbers used in the specified geometry slice.

UNT

UNIT[TYPE]

These keywords will cause the plot to represent the units used in the specified geometry slice. In the legend of the color plot, the material number actually refers to the units.

IMP

BIAS[ID]

WTS

WEIG[HTS]

WGT[S]

These keywords will cause the plot to represent the bias ID numbers used in the specified geometry slice. In the legend of the color plot, the material number actually refers to the bias ID numbers.

TYP=

Enter the type desired. XY for an X-Y plot XZ for an X-Z plot YZ for a Y-Z plot Direction cosines do not need to be entered if TYP is entered.

Plot coordinates

Enter values for the upper left and lower right coordinates of the plot as described below. Data must be entered for all nonzero coordinates unless all six values from the previous plot are to be used.

Upper left coordinates

Enter the X, Y, and Z coordinates of the upper left-hand corner of the plot.

XUL=xul

is used to enter the X coordinate of the upper left-hand corner of the plot. Default = value from previous plot; initialized to zero if any other coordinates are entered.

YUL= yul

is used to enter the Y coordinate of the upper left-hand corner of the plot. Default = value from previous plot; initialized to zero if any other coordinates are entered.

ZUL= zul

is used to enter the Z coordinate of the upper left-hand corner of the plot. Default = value from previous plot; initialized to zero if any other coordinates are entered.

Lower right coordinates

Enter the X, Y, and Z coordinates of the lower right-hand corner of the plot.

XLR= xlr

is used to enter the X coordinate of the lower right-hand corner of the plot. Default = value from previous plot; initialized to zero if any other coordinates are entered.

YLR= ylr

is used to enter the Y coordinate of the lower right-hand corner of the plot. Default = value from previous plot; initialized to zero if any other coordinates are entered.

ZLR= zlr

is used to enter the Z coordinate of the lower right-hand corner of the plot. Default = value from previous plot; initialized to zero if any other coordinates are entered.

Direction cosines across the plot

Enter direction numbers proportional to the direction cosines for the AX axis of the plot. The AX axis is from left to right across the plot. If any one of the AX direction cosines is entered, the other two are set to zero. The direction cosines are normalized by the code.

UAX= uax is used to enter the X component of the direction cosines for the AX axis of the plot. Default = value from previous plot; initialized to zero if any other direction cosines are entered.

VAX= vax is used to enter the Y component of the direction cosines for the AX axis of the plot. Default = value from previous plot; initialized to zero if any other direction cosines are entered.

WAX= wax is used to enter the Z component of the direction cosines for the AX axis of the plot. Default = value from previous plot; initialized to zero if any other direction cosines are entered.

Direction cosines down the plot

Enter direction numbers proportional to the direction cosines for the DN axis of the plot. The DN axis is from top to bottom down the plot. If any one of the DN direction cosines is entered, the other two are set to zero. The direction cosines are normalized by the code.

UDN= udn is used to enter the X component of the direction cosines for the DN axis of the plot. Default = value from previous plot; initialized to zero if any other direction cosines are entered.

VDN= vdn is used to enter the Y component of the direction cosines for the DN axis of the plot. Default = value from previous plot; initialized to zero if any other direction cosines are entered.

WDN= wdn is used to enter the Z component of the direction cosines for the DN axis of the plot. Default = value from previous plot; initialized to zero if any other direction cosines are entered.

Scaling parameters

Enter one or more scaling parameters to define the size of the plot.

Note

If any of the scaling parameters are entered for a plot, the value of those that were not entered is recalculated. If none of the scaling parameters are specified for a plot, the values from the previous plot are used.

DLX= dlx

is used to input the horizontal spacing between points on the plot. Default = value from previous plot; initialized to zero if NAX or NDN is entered.

DLD= dld

is used to input the vertical spacing between points on the plot. Default = value from previous plot; initialized to zero if NAX or NDN is entered.

Note

If either DLX or DLD is entered, the code will calculate the value of the other. If both are entered, the plot may be distorted.

NAX= nax

is used to input the number of intervals to be printed across the plot. Default = value from previous plot; initialized to zero if DLX or DLD is entered.

NDN= ndn

is used to input the number of intervals to be printed down the plot. Default = value from previous plot; initialized to zero if DLX or DLD is entered.

Global scaling parameter

LPI= lpi

is used to input a scaling factor used to control the horizontal to vertical proportionality of a plot or plots. SCALE 4.3 and later versions allow lpi to be input as a floating point number. For an undistorted character plot, lpi should be specified as the number of characters down the page that occupy the same distance as ten characters across the page. For an undistorted color plot, lpi should be entered as ten times the ratio of the vertical pixel dimension to the horizontal pixel dimension. The default value of lpi is 8.0 for a character plot and 10.0 for a color plot. lpi=10 will usually display an undistorted color plot.

The value entered for lpi applies to all plot data following it until a new value of lpi is specified.

Note

Plot data must include the specification of the upper left corner of the plot and the direction cosines across and down the plot.

Additional data required to generate a plot are one of the following combinations:

1. the lower right corner of the plot, the global scaling parameter, LPI, and one of the scaling parameters (DLX, DLD, NAX, NDN).

2. the lower right corner of the plot, one of the scaling parameters related to the horizontal specifications of the plot (DLX or NAX), and one of the scaling parameters related to the vertical specification of the plot (DLD or NDN). LPI, even if specified will not be used.

3. NAX and NDN and any two of LPI, DLX, and DLD. If LPI, DLX, and DLD are all specified, LPI is not used.

The data required to generate a plot may be supplied from (1) defaulted values, (2) data from the previous plot, or (3) data that are specifically entered for the current plot.

Miscellaneous parameters

Enter miscellaneous parameters

RUN= run

is used to determine if the problem is executed or is terminated after data checking. A value of YES for run means the problem will be executed if all the data were acceptable. A value of NO specifies the problem will be terminated after data checking is completed. The default value of RUN is YES.

PLT= plt

is used to specify if a plot is to be made. A value of YES for plt specifies that a plot is to be made. If plot data are entered, PLT is defaulted to YES.

Note

The parameters RUN and PLT can also be entered in the PARAMETER data. See Title and parameter data. It is recommended that these parameters be entered only in the parameter data block in order to ensure that the data printed in the “Logical Parameters” table are what is actually performed.

SCR= src

This is used to determine the plot display method. The plot display method is specified by entering either YES or NO for src. The default value is YES. SCR=YES uses the color plot display method. SCR=NO uses the character plot display method. If SCR is entered more than once in a problem, the last value entered is the one that is used.

NCH= delim char delim

Enter only if plots are to be made utilizing the character plot display method (SCR=NO). Enter a delimiter (i.e., “ , ‘ , * , ^ , or !) to signal the beginning of character string char. The character string is terminated when the delim character is encountered the second time. Do not use the initial delimiter in the char string, as it will be read as terminating the string. char is a character string with each entry representing a plottable quantity (i.e., media {mixture} number, unit number, or bias ID). These are the characters that will be used in the plot. The first entry represents media, unit, or bias ID zero; the second entry represents the smallest media, unit, or bias ID used in the problem; the third entry represents the next larger media, unit, or bias ID used in the problem; etc. For example, assume PIC=MAT is specified, and 15 mixtures are defined in the mixing table, and the geometry data use only mixtures 3 and 7. By default, a blank will be printed for mixture zero, a 1 will be printed for mixture 3, and a 2 will be printed for mixture 7. If you wish to print a zero for a void (mixture 0), a 3 for mixture 3, and a 7 for mixture 7, enter NCH=‘037’.

The default values of CHAR are the following:

Quantity

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

SYMBOL

1

2

3

4

5

6

7

8

9

A

B

C

D

E

F

Quantity

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

SYMBOL

G

H

I

J

K

L

M

N

O

P

Q

R

S

T

U

V

Quantity

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

SYMBOL

W

X

Y

Z

‘#’

,

‘$’

‘−’

‘+’

‘)’

‘|’

‘&’

‘>’

‘:’

‘;’

Quantity

47

48

49

50

51

52

53

54

55

56

57

58

SYMBOL

‘⋅’

‘−’

“%”

“*”

“”“

“=”

“!”

“(“

“@”

“<“

“/”

0

CLR= n1 r*(*n1) g*(*n1) b*(*n1) … nN r*(*nN) g*(*nN) b*(*nN) END COLOR

this entry is used to define the colors to be used by the color plot. It may be entered only if plots are to be made utilizing the color plot display method (SCR=YES). After entering the keyword CLR=, 4 numbers are entered N times. The first number, n1, represents a media (mixture) number, unit number, or bias ID. The next three numbers, whose values can range from 0 through 255, define the red, green, and blue components of the color that will represent this n1 in the plot. The sequence of 4 numbers is repeated until the colors associated with all of the media (mixture) numbers, unit numbers, or bias IDs used in the problem have been defined. The smallest number that can be entered for ni is −1, representing undefined regions in the plot. An ni of 0 represents void regions; ni of 1 represents the smallest media, unit, or bias ID used in the problem; ni of 2 represents the next larger media, unit, or bias ID used in the problem, etc. The color plot definition data are terminated by entering the keywords END COLOR. A total of 256 default colors are provided in Table 157 Two of those colors represent undefined regions, ni=-1*, as black and void regions, and ni=0 as gray. The remaining 254 colors represent the default values for mixtures, bias IDs, or unit numbers used in the problem. If num is entered as −1, the next three numbers define the color that will be used to represent undefined regions of the plot. The default color for undefined regions is black, represented as 0 0 0. If ni is entered as 0, the next three numbers define the color that will represent void regions in the plot. The default color for void is gray, represented as 200 200 200. For example, assume a color plot is to be made for a problem that uses void regions and mixture numbers 1, 3, and 5. By default, the undefined regions (Index −1) will be black; void regions (Index 0) will be gray; the first mixture, mixture 1 (Index 1), will be medium blue; the next larger mixture, mixture 3 (Index 2), will be turquoise2; and the last mixture, mixture 5 (Index 3), will be green2. If these values are acceptable, data do not need to be entered for CLR=. If the user decides to define void to be white (255 255 255), mixture 1 to be red (255 0 0), mixture 3 to be bright blue (0 0 255), and mixture 5 to be green (0 255 0), then the following data could be entered:

CLR=0 255 255 255 1 255 0 0 2 0 0 255 3 0 255 0 END COLOR

In this example, the first number (0) defines the void, and the next three numbers are the red, green, and blue components that combine as the color white. The fifth number (1) represents the smallest mixture number (mixture 1), and the next three numbers are the red, green, and blue components of red. The ninth number (2) represents the next larger mixture number (mixture 3), and the next three numbers are the red, green, and blue components of bright blue. The thirteenth number (3) represents the next larger mixture number (mixture 5), and the next three numbers are the red, green, and blue components of green. The END COLOR terminates the color definition data. Because color data were not entered for ni of −1, undefined regions will be represented by the color black, the default specification from Table 157. The red, green, and blue components of some bright colors are listed below.

Display Color

red

green

blue

black

0

0

0

white

255

255

255

“default void gray”

200

200

200

red

255

0

0

green

0

255

0

brightest blue

0

0

255

yellow

255

255

0

brightest cyan

0

255

255

magenta

255

0

255

The 256 default colors are listed in Table 157.

Table 157 Default color specifications for the color plot display method
_images/tab22_Page_01.png
_images/tab22_Page_02.png
_images/tab22_Page_03.png
_images/tab22_Page_04.png
_images/tab22_Page_05.png
_images/tab22_Page_06.png
_images/tab22_Page_07.png
_images/tab22_Page_08.png
_images/tab22_Page_09.png
_images/tab22_Page_10.png
_images/tab22_Page_11.png

Energy group boundary data

Upper energy group boundary data in eV are entered to determine the groups into which the tallies will be collected in the continuous energy mode. For G groups G+1 entries are entered. The last entry is the lower energy boundary of the last group. The values must be in descending order. The parameter NGP is set equal to the number of entries−1. The syntax is:

READ ENERGY u1 …uG uG+1 END ENERGY

u1uG

are the upper energy limits of energy groups 1 … G, respectively.

uG+1

is the lower energy limit of energy group G.

Example:

READ ENERGY

2e7 1e5 1 1e−5

END ENERGY

Defines a 3-group structure with group 1 (2e+7 eV to 1e+5 eV), group 2 (1e+5 eV to 1 eV), and group 3 (1 eV to 1e−5 eV) and sets NGP=3.

Energy group boundary data are optional. Default values for the energy group boundaries in the calculations are determined as in the following order:

  1. Use energy group boundaries from READ ENERGY block if specified in the input. The number of entries in the READ ENERGY block is NGP+1.

  2. If only NGP is specified (in READ PARAMETER) in the input and NGP is equal to the number of energy groups in one of the SCALE neutron cross section libraries, the energy group structure from that library will be used.

  3. If only NGP is specified (in READ PARAMETER) in the input and NGP is not equal to the number of energy groups in one of the SCALE neutron cross section libraries, NGP equal lethargy bins will be used.

  4. Use SCALE 238 group structure as default, NGP=238.

Volume data

If volumes are needed (for calculating fission densities, fluxes, etc.), then the data necessary to determine them are entered. The syntax for this block is:

READ VOLUME p1pN END VOLUME

p1pN

are N parameters entered using keywords followed by the appropriate data.

The possible parameters that can be used in a VOLUME block are described below.

READVOL=vol

used to input the file name (up to 256 characters) of the file from which user-specified volumes are read. This is an optional parameter and only works for KENO-VI. The data are read in sections for each UNIT contained in the problem. First the keyword “UNIT” is read, followed by the UNIT number. For that UNIT the data for each region containing material in the order shown in the input is read as follows: the keyword “MEDIA” is read, followed by the mixture number, followed by the keyword “VOL=”, followed by the total volume for that region. Regions containing ARRAYs and HOLEs are skipped. An example of the data contained in a volume file is given later in this section.

TYPE=vcalc

used to determine the type of volume calculation. vcalc can have the values:

NONE: (only works in KENO-VI, where it is also the default) No volume calculation, volumes are set to –1.0 (only in KENO-VI).

TRACE: A trapezoidal integration will be performed (only in KENO‑VI).

RANDOM: A Monte Carlo integration will be performed.

NRAYS=ntotal

the number of intervals used in the trapezoidal integration (default 100,000). Used only with TYPE=TRACE (KENO-VI).

BATCHES=nloop

the number of batches to be used in the Monte Carlo integration (default 500). Used only with TYPE=RANDOM.

POINTS=nplp

the number of points per batch used in the Monte Carlo integration (default 1000). Used only with TYPE=RANDOM.

XP=xp

the plus X face of the encompassing cuboid.

XM=xm

the minus X face of the encompassing cuboid.

YP=yp

the plus Y face of the encompassing cuboid.

YM=ym

the minus Y face of the encompassing cuboid.

ZP=zp

the plus Z face of the encompassing cuboid.

ZM=zm

the minus Z face of the encompassing cuboid.

SAMPLE_DEN=sampleden

the density of sampling points per cm3 per batch. Used only with TYPE=RANDOM.

IFACE =fname

the face of the enclosing cuboid where the trapezoidal integration will be performed. Enter either XFACE, YFACE, or ZFACE. KENO-VI will integrate over the face with the smallest area by default. This allows specifying a different face. Used only for TYPE=TRACE (KENO-VI).

The volume parameters include specifying the type of calculation to determine the volumes and additional parameters needed for the selected type. In KENO-VI the default type is NONE (i.e., no volume calculation will be performed), and the volumes for regions not containing HOLEs or ARRAYs will be set to -1.0. No other data are needed for this type.

In KENO-VI the volume data may be entered for any or all regions within the geometry data by placing the keyword VOL= followed by the total volume of that region in the problem at the end of the MEDIA card. See Geometry data (Geometry Data) for more details.

For KENO-VI, in the same problem, volumes may be entered using a combination of three methods: (1) in the geometry data using VOL=, (2) read from a volume file, and (3) calculated. The calculated volumes (method 3) are obtained for both the regions, and the meshes are defined by a grid (such as in TSUNAMI runs). As for KENO V.a, the mesh volumes must always be calculated (i.e., there is no method to input the mesh volumes). If volumes are entered or calculated using more than one method, the following hierarchy is used to determine which volume is used for the regions.

  1. Volumes entered as part of a MEDIA card using VOL= are always used.

  2. Volumes read from the volume file are used if that volume for the region was not specified using VOL= following a MEDIA card.

  3. Calculated volumes are used if they are not specified using VOL= and if there is no volume file or data for that region on the volume file.

  4. Volumes that have not been set or calculated will be set to –1.0. This may result in negative fluxes and fission densities for these regions.

  5. Volumes are only calculated for regions containing material. Regions containing ARRAYs or HOLEs have no volume. Those volumes are associated with the UNIT contained in the ARRAY or HOLE.

In KENO V.a, the region volumes are always calculated by the code without the user’s intervention. This is possible because KENO V.a has no region intersections, so calculation of the volumes is always possible using analytical methods. The use of the (RANDOM) calculated volumes using the VOLUME block is then only justified when the user needs to calculate the volumes defined by a grid, such as for TSUNAMI calculations.

When volumes are calculated using either RANDOM or TRACE, then a file containing volumes and named _volxxxx (where xxxx is an 18-digit number with the leftmost unused digits padded with zeros) is created in the temporary directory. The program searches the temporary directory for a file name beginning with _vol. If it is not found, the volume file that is created is named _vol000000000000000000. If a file exists, then a new file will be created where the file number is the largest number associated with a previous volume file incremented by 1. The file is automatically copied to the user directory with the input file base name prepended to it, such as inputfile.volxxxx.volumes.

Below is an example of the VOLUME data block associated with a case in which volumes are being calculated using ray tracing. The number of rays used is set to one million, and if the outer unit volume is not a cuboid, then a cuboid will be placed around the global region prior to calculating volumes.

read volume
type=trace nrays=1000000
XP=10  XM=-15 YP=15  YM=-15  ZP=15  ZM=-15
end volume

Below is an example of the VOLUME data block associated with a case where volumes are being calculated using random sampling. The number of particles per batch is set to 100,000, and the number of batches used is set to 500. After being calculated, the volume data will be written to a file in the temporary directory as discussed above.

read volume
type=random points=100000 batches=500
XP=10  XM=-10  YP=15  YM=-15  ZP=25  ZM=15
end volume

Below is an example of the VOLUME data block associated with a case where volumes are both read in from the file VOLUME_DATA and calculated using random sampling. The number of particles per generation is set to 1,000,000, and the number of generations used is set to 500. The file VOLUME_DATA must be formatted as shown below. The calculated volume data are written in the temporary working directory to a file as discussed above. Calculating volume data for some volume regions and providing input volume data for others may be useful if only part of the volume data is known and the remaining data need to be calculated.

read volume
type=random points=1000000 batches=500
readvol=volume_DAta
end volume

Example volume file VOLUME_DATA:

UNIT 1
   MEDIA 1 VOL=110.0
   MEDIA 2 VOL=2435.8
   MEDIA 2 VOL=3242.9
UNIT 2
   MEDIA 2 VOL=342.8
   MEDIA 0 VOL=4235.0

Below is an example of a sample problem in which volumes are being calculated using random sampling. The number of particles per generation is set to 100,000, and the number of generations used is set to 500. After being calculated, the VOLUME data will be written as described above.

=CSAS6
SAMPLE PROBLEM WITH VOLUMES CALCULATED AND PRINTED TO FILE
v7.1-252n
READ COMP
URANIUM 1 DEN=18.76 1 293 92235 93.2 92238 5.6 92234 1.0 92236 0.2 END
END COMP
READ GEOMETRY
UNIT 3
COM='SINGLE UNIT CENTERED'
SPHERE 10 4.000
CUBOID 20 6P6.0
MEDIA 1 1 10
MEDIA 0 1 20 -10
BOUNDARY 20
UNIT 1
COM='SINGLE UNIT CENTERED'
SPHERE 10 5.000
CUBOID 20 6P6.0
MEDIA 1 1 10
MEDIA 0 1 20 -10
BOUNDARY 20
GLOBAL UNIT 2
COM='Global UNIT'
CUBOID 10 6P18.0
ARRAY 1 10 PLACE 2 2 2 3R0.0
BOUNDARY 10
END GEOMETRY
READ ARRAY ARA=1 NUX=3 NUY=3 NUZ=3 TYP=CUBOIDAL FILL
1 1 1 1 1 1 1 1 1 3 3 3 3 3 3 3 3 3 1 1 1 1 1 1 1 1 1 END ARRAY
READ VOLUME
TYPE=RANDOM POINTS=100000 BATCHES=500
END VOLUME
END DATA
END

Example volume file _volxxxx:

UNIT 1
   MEDIA 1 VOL=9423.45
   MEDIA 0 VOL=21678.6
UNIT 3
   MEDIA 1 VOL=2410.81
   MEDIA 0 VOL=13143.1

Below is an example of a problem with the volumes entered in the geometry data block using VOL= followed by the volume for all MEDIA type content records. Note that the keyword “VOL=” should never follow a HOLE or ARRAY content record.

=csas26
Sample problem with volumes input in geometry data
v7.1-252n
read comp
uranium  1 den=18.76 1 293 92235 93.2 92238 5.6 92234 1.0 92236 0.2 end
end comp
read geometry
unit 3
com='unit 3'
sphere   10  4.000
cuboid   20  6p6.0
media  1 1 10       vol=2412.7
media  0 1 20 -10   vol=13139.3
boundary 20
unit 1
com='UNIT 1'
sphere   10  5.000
cuboid   20  6p6.0
media  1 1 10       vol=9424.8
media  0 1 20 -10   vol=21679.2
boundary 20
global unit 2
com='GLOBAL Unit 2'
cuboid 10  6p18.0
array 1    10 place 2 2 2 3r0.0
boundary  10
end geometry
read array  ara=1 nux=3 nuy=3 nuz=3  typ=cuboidal fill
1 1 1 1 1 1 1 1 1 3 3 3 3 3 3 3 3 3 1 1 1 1 1 1 1 1 1  end array
end data
end

Grid geometry data

This data block is used to input the data needed to define a Cartesian grid for tallying purposes.

READ GRID N p1pL END GRID

N

mesh grid identifier, always entered.

p1pL

are L parameters chosen from the list below. The parameters are entered using keywords followed by the appropriate data, except for the grid identifier, which is always entered first as an integer.

N[UM]XCELLS=numx

number of cells in the x direction, default = 1.

N[UM]YCELLS=numy

number of cells in the y direction, default = 1.

N[UM]ZCELLS=numz

number of cells in the z direction, default = 1.

XMIN=xmin

minimum cell boundary in the x direction, default = 0.

XMAX=xmax

maximum cell boundary in the x direction, default = 1.

YMIN=ymin

minimum cell boundary in the y direction, default = 0.

YMAX=ymax

maximum cell boundary on the y direction, default = 1.

ZMIN=zmin

minimum cell boundary in the z direction, default = 0.

ZMAX=zmax

maximum cell boundary in the z direction, default = 1.

XPLANES=xplanes

the cell boundaries in the x direction followed by end, default = 0, 1 end.

YPLANES=yplanes

the cell boundaries in the y direction followed by end, default = 0, 1 end.

ZPLANES=zplanes

the cell boundaries in the z direction followed by end, default = 0, 1 end.

XLINEAR=numcellsx *xmin xmax

generate y-z planes from xmin to xmax creating numcellsx intervals.

YLINEAR=numcellsy *ymin ymax

generate x-z planes from ymin to ymax, creating numcellsy intervals.

ZLINEAR=numcellsz *zmin zmax

generate x-y planes from zmin to zmax, creating numcellsz intervals.

TITLE=title

optional title for this mesh grid. Only used in KENO if an error in the grid causes a debug print.

If numx, xmin, xmax are entered, then the code will calculate numx equally spaced cells in the x direction between xmin and xmax.

If xplanes is entered, then the code will count the number of unique xplanes, and order them from minimum to maximum, deleting any duplicates.

If the user inputs both sets of data, then the code will use the xplanes data.

If xplanes and xlinear are both entered, then the code will retain all unique planes from xplanes and all xlinear entries provided. The above also applies to Y and Z.

NOTE: The user MUST set the minimum and maximum values in each direction so that the actual geometry is totally covered by the mesh for mesh flux tally that is used in TSUNAMI sensitivity calculations.

KENO checks for and eliminates duplicate or nearly duplicate planes.

The user may specify multiple mesh grids; each must be defined in separate READ GRID blocks. In this case, each grid should have different N (grid ID number). See KENO Multiple Mesh and Mesh-based Quantity Specifications for details and samples.

Reaction data

The reaction data block is used to specify the type of tally (e.g., reaction rates, flux, and few group reaction cross sections) and the reaction/nuclide pairs in any mixture used in the problem for reaction tally calculations. This block is operational only with the continuous energy mode, and it provides the specifications for reaction rate, neutron flux, and reaction cross section tallies. See Reaction Rate and Few Group Micro Cross Section Calculations for more details. For multigroup KENO calculations, use KMART5 or KMART6, which are described in the KMART section of the SCALE manual.

A reaction data block consists of REACTION FILTERS, TALLY TYPE, ENERGY GROUP BOUNDARIES, and OUTPUT EDITS. These data types can be entered in any order. A combination of parameters for describing the REACTION FILTERS and TALLY TYPE must be entered for any reaction or cross section tally calculation. ENERGY GROUP BOUNDARIES and OUTPUT EDITS data are optional. Tally calculations can be performed for multiple reactions specified by the REACTION FILTERS. Only one energy grid, either specified with the data in ENERGY GROUP BOUNDARIES or from the READ ENERGY block or from the code defaults, is used for all reaction tally calculations. To provide data for the continuous energy depletion calculations, another energy grid can be specified and used for tallying only the mixture flux.

Enter REACTION DATA in the form:

READ REACTION REACTION FILTERS [TALLY TYPE] [ENERGY GROUP BOUNDARIES][OUTPUT EDITS] END REACTION

REACTION FILTERS

define a reaction map that is used in reaction tally calculations. The REACTION FILTERS must be entered in the following order; mixture data (MIX or MIXLIST) followed by nuclide data (NUC or NUCLIST) followed by reaction IDs (MT or MTLIST). Each filter is defined using a combination of the following keywords:

MIX=mixnum

Mixture number, no default value. Specified mixture number must exist in the mixing table and be used in the problem for a valid filter generation. A wildcard “*” can be used to define a filter applicable for all mixtures in the problem.

MIXLIST mixnum1 mixnum2mixnumNEND

A list of mixture numbers followed by end, no default values. Specified mixture numbers must exist in the mixing table and be used in the problem for a valid reaction tally calculation. Within each filter, use either MIX or MIXLIST, but not both.

NUC=nucid

Nuclide identifier, no default value. Specified nuclide must be a constituent of the mixtures used in this filter definition (specified with MIX or MIXLIST). A wildcard “*” can be used to define a filter applicable for all nuclides in each mixture in this filter definition. Nuclide identifiers are listed for all isotopes in the Standard Composition Library section of the SCALE manual (see Standard Composition Library).

NUCLIST nucid1 nucid2nucidNEND

A list of nuclide identifiers followed by end, no default values. Specified nuclides must be the constituents of the mixtures used in this filter definition (specified by MIX or MIXLIST). Within each filter, use either NUC or NUCLIST, but not both.

MT=mt

Reaction MT number, no default value. Specified reaction MT number should be available for the nuclides defined in this filter definition (specified by NUC or NUCLIST). Otherwise, the code skips the filter definition with this given reaction MT. A wildcard “*” can be used to define a filter with all reaction MTs. Valid SCALE library MT values are listed in the SCALE Cross Section Libraries section of the SCALE manual (see Appendix A of 11-1).

MTLIST mt1 mt2mtNEND

A list of reaction MT numbers followed by end, no default values. Specified MT numbers should be available for the nuclides defined in this filter definition (specified by MIX or MIXLIST). Otherwise, KENO skips that reaction specified in the filter for the reaction tally calculations. Within each filter, use either MT or MTLIST, but not both.

A reaction filter consists of either single or multiple mixture, nuclide and reaction definitions. A valid reaction filter starts with mixture specification, followed by nuclide specification, and ends with reaction specification. Mixture(s) must be specified with either MIX or MIXLIST keywords. Nuclide(s) in these mixtures must be entered with either NUC or NUCLIST, and reactions for each nuclide must be specified with either MT or MTLIST.

Mixture, nuclide, and reaction number are required for mixture average fluxes, even though the nuclide and reaction numbers are not used for the neutron flux tallies.

Multiple reaction filter definitions are allowed. KENO processes all the definitions and creates a reaction map based on them. The following examples demonstrate the reaction filter specifications for different problems. In these examples, reaction filters are specified based on the following composition data used in the problem:

compositions in the example problem

mixture

nuclides

10

92235, 92238, 8016

20

92238, 94239, 8016

30

92235, 92238, 8016

40

1001, 8016

100

1001, 8016, 5010, 5011

Example-1:

READ REACTION
         MIX=10 NUC=92235  MT=18
         
END REACTION

Defines a reaction filter used to tally only fission reaction (MT=18) of 235U in mixture 10.

Example-2:

READ REACTION
          MIX=10 NUC=92235  MT=*
          
END REACTION

Defines a reaction filter used to tally all available reactions of 235U in mixture 10.

Example-3:

READ REACTION
          MIX=10 NUC=92235  MTLIST 2 18 102 END
          
END REACTION

Defines a reaction filter used to tally the elastic scattering (mt=2), fission (mt=18), and capture (mt=102) reactions of 235U in mixture 10.

Example-4:

READ REACTION
          MIX=10 NUC=92235  MT=2
          MIX=10 NUC=92235  MT=18
          MIX=10 NUC=92235  MT=102
          
END REACTION

Defines a reaction filter used to tally the elastic scattering (MT=2), fission (MT=18), and capture (MT=102) reactions of 235U in mixture 10. Reaction filter definition in this example is identical to the filter definition given in Example-3.

Example-5:

READ REACTION
          MIX=10 NUC=*  MT=18
          
END REACTION

Defines a reaction filter used to tally the fission reaction (MT=18) of 235U and 238U in mixture 10. Code skips the reaction tally request for 16O since the requested reaction is not available for this nuclide in the data library.

Example-6:

READ REACTION
          MIX=10 NUC=92235  MT=18
          MIX=10 NUC=92238  MT=18
          
END REACTION

Defines a reaction filter used to tally the fission reaction (MT=18) of 235U and 238U in mixture 10. Reaction filter definition in this example is identical to the filter definition given in Example-5.

Example-7:

READ REACTION
          MIX=* NUC=8016  MT=102
          
END REACTION

Defines a reaction filter used to tally the capture reaction (MT=102) of 16O in all mixtures.

Example-8:

READ REACTION
      MIX=10  NUC=8016  MT=102
      MIX=20  NUC=8016  MT=102
      MIX=30  NUC=8016  MT=102
      MIX=40  NUC=8016  MT=102
      MIX=100 NUC=8016  MT=102
                      
END REACTION

Defines a reaction filter used to tally the capture reaction (MT=102) of 16O in mixtures 10, 20, 30, 40, and 100 respectively. Reaction filter definition in this example is identical to the filter definition given in Example-7.

Example-9:

READ REACTION
          MIXLIST 10 20 30 40 100 END NUC=8016  MT=102
          
END REACTION

Defines a reaction filter used to tally the capture reaction (MT=102) of 16O in mixtures 10, 20, 30, 40, and 100 respectively. Reaction filter definition in this example is identical to the filter definition given in Examples 7 and 8.

Example-10:

READ REACTION
      MIXLIST 10 20 30 END NUC=92238  MT=102
      MIX=20  NUC=94239 MT=18
      MIX=40  NUC=1001  MT=*
      MIX=*   NUC=8016  MT=2
      MIX=*   NUC=*     MT=27
      
END REACTION

Defines a complex reaction filter used to tally:

  1. Capture reaction (MT=102) of 238U in mixtures 10, 20 and 30 respectively,

  2. Fission reaction (MT=18) of 239Pu in mixture 20,

  3. All reactions of 1H in mixture 40,

  4. Elastic scattering reaction of 16O in all mixtures,

  5. Total absorption reaction of all nuclides in all mixtures.

Parameters of TALLY TYPE are logical parameters used to select quantities (reaction cross section, reaction rate, and mixture flux) that are tallied for the given problem. The user specifies any combination of these TALLY TYPEs once for all filters:

XSTALLY=lCEXSTally

Enter YES or NO. A value of YES specifies that reaction cross sections be tallied for the reactions listed in REACTION FILTERS. The default value of XSTALLY is NO. Computed reaction cross sections are saved in a file named BASENAME_keno_micro_xs.0 in RTNDIR, which is a SCALE environment variable for the directory from where the calculation was started. BASENAME is a SCALE environment variable that is the base name of the input file. (BASENAME is equal to “mytest” if the SCALE input name is “mytest.inp.”)

RRTALLY=lCERRTally

Enter YES or NO. A value of YES specifies that reaction rates be tallied for the reactions listed in REACTION FILTERS. The default value of RRTALLY is NO. Computed reaction rates are saved in a file named BASENAME_keno_micro_rr.0 in RTNDIR.

Note

KENO combines and saves reaction rate and reaction cross section tallies to the same file, named BASENAME_keno_micro_xs_rr.0 in RTNDIR, if both XSTALLY and RRTALLY are set to YES.

MIXFLX=lCEMixFlux

Enter YES or NO. A value of YES specifies that mixture fluxes are to be tallied for the mixtures listed in REACTION FILTERS. The default value of MIXFLX is NO. Computed mixture fluxes are saved in a file named BASENAME_keno_mixture_flux.0 in RTNDIR.

Mixture, nuclide, and reaction number are required for mixture average fluxes, even though the nuclide and reaction numbers are not used for the neutron flux tallies.

Example-11:

READ REACTION
          MIX=10 NUC=92235  MT=18
          XSTALLY=YES
          
END REACTION

Defines a reaction filter, which uses fission reaction (MT=18) of 235U in mixture 10, for tallying reaction cross sections.

Note

Computed data are saved in a file named BASENAME_keno_micro_xs.0

Example-12:

READ REACTION
          MIX=10 NUC=92235  MT=18
          XSTALLY=YES RRTALLY=YES
          
END REACTION

Defines a reaction filter, which uses fission reaction (MT=18) of 235U in mixture 10, for tallying reaction rates as well as the reaction cross sections.

Note

Computed data are saved in files named BASENAME_keno_micro_xs_rr.0

Example-13:

READ REACTION
        MIX=10 NUC=92235  MT=18
         XSTALLY=YES MIXFLX=YES
         
END REACTION

Defines a reaction filter, which uses fission reaction (MT=18) of 235U in mixture 10, for tallying reaction cross sections. In addition, mixture flux is tallied for mixture 10 given in this reaction filter.

Note

Computed data are saved in files named BASENAME_keno_micro_xs.0, and BASENAME_keno_mixture_flux.0, respectively.

ENERGY GROUP BOUNDARIES data define energy group structure other than the defaults for tallying both reaction cross sections/reaction rates and mixture fluxes.

ENER_XS e1 e2 e3 …END Upper energy boundary for each group. The last entry is the lower energy boundary of the last group. For N groups, there are N+1 entries. Entries must be in descending order. This may be specified once in the REACTION block and, if used, is applied to all cross section and reaction rate tallies.

ENER_FLX e1 e2 e3 …END Upper energy boundary for each group, default is NGP-group data. The last entry is the lower energy boundary of the last group. For N groups, there are N +1 entries. Entries must be in descending order. This may be specified once in the REACTION block and, if used, is applied to all mixture flux tallies.

Note

Default values for the energy group boundaries in reaction tally calculations are determined as in the order described in Energy group boundary data.

Example-14: (no READ ENERGY block, no NGP in READ PARAMETER block, no energy group

READ REACTION
        MIX=10 NUCLIST 92235 92238 END
         MTLIST 16 17 18 END
          MIXFLX=YES XSTALLY=YES
          
END REACTION

Default SCALE 238-group energy structure is used for tallying both mixture flux and reaction cross sections.

Example-15: (energy group bounds specified in READ ENERGY block)

READ ENERGY
         20.E6   0.6 1.E-4
END ENERGY
    
READ REACTION
         MIX=10 NUCLIST 92235 92238 END
         MTLIST 16 17 18 END
         MIXFLX=YES XSTALLY=YES
         
END REACTION

2-group energy structure given in READ ENERGY block is used for tallying both mixture flux and reaction cross sections.

Example-16: (NGP is set in READ PARAMETER block)

READ PARAMETER
               
               NGP=4
END PARAMETER
  
READ REACTION
         MIX=10 NUCLIST 92235 92238 END
         MTLIST 16 17 18 END
         MIXFLX=YES XSTALLY=YES
         
END REACTION

4-group energy group structure (4-equal lethargy bins) is used for tallying both mixture flux and reaction cross sections.

Example-17:

READ REACTION
      MIX=10 NUCLIST 92235 92238 END
      MTLIST 16 17 18 END
      MIXFLX=YES XSTALLY=YES
      ENER_XS 20.E6 1.E3 1.0 1.E-4 END
       
END REACTION

3-group energy group structure given in READ REACTION block is used for tallying reaction cross sections, and default SCALE 238 group energy structure is used for tallying mixture flux.

Example-18:

READ REACTION
      MIX=10 NUCLIST 92235 92238 END
      MTLIST 16 17 18 END
      MIXFLX=YES XSTALLY=YES
      ENER_FLX 20.E6 1.E3 1.0 1.E-4 END
      ENER_XS 20.E6 1.0 1.E-4 END
       
END REACTION

2-group energy group structure given in READ REACTION block is used for tallying reaction cross sections, and 3-group energy group structure given in READ REACTION block is used for tallying mixture flux.

Parameters of OUTPUT EDITS are logical parameters used to print reaction tallies and mixture fluxes in separate files. These parameters are optional parameters.

PRNTXS=lCEprintXS Enter YES or NO. A value of YES specifies that reaction cross sections tallies for each mixture be written in separate files in RTNDIR (BASENAME_keno_micro_xs_mix{mixnum}.0, mixnum is the mixture numbers specified in the reaction filters). The default value of PRNTXS is NO.

PRNTRR=lCEprintRR Enter YES or NO. A value of YES specifies that reaction rate tallies for each mixture be written in separate files in RTNDIR (BASENAME_keno_micro_rr_mix{mixnum}.0, mixnum is the mixture numbers specified in the reaction filters). The default value of PRNTRR is NO.

PRNTFLX=lCEprintMixFlux Enter YES or NO. A value of YES specifies that mixture flux tallies for each mixture be written in separate files in RTNDIR (BASENAME_keno_mixture_flux_mix{mixnum}.0, mixnum is the mixture numbers specified in the reaction filters). The default value of PRNTFLX is NO.

Notes for Keno Users

This section provides assorted tips designed to assist the KENO user with problem mockups. Some information concerning methods used by KENO is also included.

Data entry

The KENO data is entered in blocks that begin and end with keywords as described in Keno input outline. Only one set of parameter data can be entered for a problem. However, for other data blocks, it is possible to enter more than one block of the same kind of data. When this is done, only the last block of that kind of data is retained for use by the problem, except for the GRID block for which all blocks are retained.

Within data blocks, a number, x, can be repeated n times by specifying nRx, n*x, or n$x.

Numbers in engineering notation may be specified with or without an “E” between the base and the exponent. For example; 0.0011 may be specified as 1.1e-3 or as 1.1-3.

Multiple and scattered entries in the mixing table

In the following examples, assume 1001 is the nuclide ID for hydrogen, 8016 is the nuclide ID for oxygen, 92235 is the nuclide ID for 235U, and 92238 is the nuclide ID for 238U. If a given nuclide ID is used more than once in the same mixture, the result is the summing of all the number densities associated with that nuclide. For example:

MIX=1 92235 4.3e-2 92238 2.6e-3 1001 3.7e-2 92235 1.1e-3 8016 1.8e-2

would be the same as entering:

MIX=1 92235 4.41e-2 92238 2.6e-3 1001 3.7e-2 8016 1.8e-2

A belated entry for a mixture can be made as follows:

MIX=1 1001 6.6e-2 MIX=2 92235 4.3e-2 92238 2.6e-3 MIX=1 8016 3.3e-2

This is the same as entering:

MIX=1 1001 6.6e-2 8016 3.3e-2 MIX=2 92235 4.3e-2 92238 2.6e-3

Multiple entries in geometry data

Individual geometry regions cannot be replaced by adding an additional description. However, entire unit descriptions can be replaced by adding a new description having the same unit number. The last description entered for a unit is used in the calculation. For example, the following geometry descriptions are equivalent in KENO V.a and KENO-VI, respectively:

In KENO V.a:

READ GEOM  UNIT 1 SPHERE 1 1 5.0 CUBE 0 1 10.0 -10.0
UNIT 2 CYLINDER 1 1 2.0 5.0 -5.0  CUBE  0 1 10.0 -10.0
UNIT 1  CUBOID  1 1 1.0 -1.5 2.5 -2.0 5.0 -6.0  CUBE  0 1 10.0 -10.0
END GEOM

is the same as entering:

READ GEOM  UNIT 1  CUBOID 1 1 1.0 -1.5 2.5 -2.0 5.0 -6.0
CUBE  0 1 10.0 -10.0
UNIT 2  CYLINDER  1 1 2.0 5.0 -5.0  CUBE  0 1 10.0 -10.0 END GEOM

or

READ GEOM  UNIT 2 CYLINDER  1 1 2.0 5.0 -5.0  CUBE  0 1 10.0 -10.0
UNIT 1  CUBOID  1 1 1.0 -1.5 2.5 -2.0 5.0 -6.0  CUBE  0 1 10.0 -10.0
END GEOM

In KENO-VI:

READ GEOM
UNIT 1 SPHERE 10 5.0
CUBOID 20 10.0 -10.0 10.0 -10.0 10.0 -10.0
MEDIA 1 1 10
MEDIA 0 1 20 -10
BOUNDARY  20
UNIT 2
CYLINDER  10  2.0 5.0 -5.0
CUBOID  20  10.0 -10.0 10.0 -10.0 10.0 -10.0
MEDIA 1 1 10
MEDIA 0 1 20 -10
BOUNDARY  20
UNIT 1
CUBOID  10   1.0  -1.5  2.5  -2.0  5.0  -6.0
CUBOID  20  10.0 -10.0 10.0 -10.0 10.0 -10.0
MEDIA 1 1 10
MEDIA 0 1 -10 20
BOUNDARY  20
END GEOM

is the same as entering

READ GEOM
UNIT 1
CUBOID  10   1.0  -1.5  2.5  -2.0  5.0  -6.0
CUBOID  20  10.0 -10.0 10.0 -10.0 10.0 -10.0
MEDIA 1 1 10
BOUNDARY  20
MEDIA 0 1 -10  20
UNIT 2
CYLINDER  10   2.0   5.0 -5.0
CUBOID    20  10.0 -10.0 10.0 -10 10.0 -10.0
MEDIA 1 1 10
MEDIA 0 1 -10 20
BOUNDARY  20
END GEOM

or

READ GEOM
UNIT 2
CYLINDER  30  2.0  5.0 -5.0
CUBOID    40  6P10.0
MEDIA 1 1 30
MEDIA 0 1 -30  40
BOUNDARY  40
UNIT 1
CUBOID  20  1.0 -1.5 2.5 -2.0 5.0 -6.0
CUBOID  10  6P10.0
MEDIA 1 1 20
MEDIA 0 1 10 -20
BOUNDARY  10
END GEOM

The order of entry for UNIT descriptions is not important because the UNIT number is assigned as the value following the word UNIT. They do not need to be entered sequentially, and they do not need to be numbered sequentially. It is perfectly acceptable to enter UNITs 2, 3, and 5, omitting Units 1 and 4 as long as UNITs 1 and 4 are not referenced in the problem. It is also acceptable to scramble the order of entry as in entering UNITs 3, 2, and 5.

Default logical unit numbers for KENO

The logical unit numbers for data used by KENO are listed in Table 158.

Table 158 KENO logical unit numbers

Function

Parameter name

Unit number

Variable name

Problem input data (ASCII)

5

INPT

Problem input data (binary)

95

BIN

Program output (ASCII)

6

OUTPT

Albedo data

ALB=

79

ALBDO

Scratch unit

SKT=

16

SKRT

Read restart data

RST=

0a

RSTRT

34b

RSTRT

Write restart data

WRS=

0a

WSTRT

35c

WSTRT

Direct access storage for input data

8

DIRECT(1)

Direct access storage for supergrouped data

9

DIRECT(2)

Direct access storage for cross section mixing

10

DIRECT(3)

Mixed cross section data set

XSC=

14d

ICEXS

Group-dependent weights

WTS=

80

WTS

AMPX working format cross sections

LIB=

0a

AMPXS

Group boundary Library (KENO-VI)

GRP=

77

GRPBS

a Defaulted to zero.

b Defaulted to 34 if BEG= a number greater than 1 and RSTRT=0.

c Defaulted to 35 if RES= a number greater than zero and WSTRT=0.

d Defaulted to 0; if LIB= a number greater than zero, ICEXS is defaulted to 14.

Parameter input

When the parameter data block is entered for a problem, the same keyword may be entered several times. The last value that is entered is used in the problem. Data may be entered as follows:

READ PARAM  FLX=YES NPG=1000 TME=0.5 TME=1.0
NPG=50 TME=10.0 FLX=NO
NPG=500
END PARA

This will result in the problem having FLX=NO, TME=10.0, and NPG=500. It may be more convenient for the user to insert a new value than to change the existing data.

Certain parameter default values should not be overridden unless the user has a very good reason to do so. These parameters are as follows:

1. X1D= which defines the number of extra 1-D cross sections. The use of extra 1‑D cross sections—other than the use of the fission cross section for calculating the average number of neutrons per fission—requires programming changes to the code;

2. NFB= which defines the number of neutrons that can be entered in the fission bank (the fission bank is where the information related to a fission is stored);

3. XFB= which defines the number of extra positions in the fission bank;

4. NBK= which defines the number of neutrons that can be entered in the neutron bank (the neutron bank contains information about each history);

5. XNB= which defines the number of extra positions in the neutron bank;

6. WTH= which defines the factor that determines when splitting occurs;

7. WTA= which defines the default average weight given to a neutron that survives Russian roulette;

8. WTL= which defines the factor that determines when Russian roulette is played; and

9. LNG= which sets the maximum words of storage available to the program.

It is recommended that BUG=, the flag for printing debug information, never be set to YES. The user would have to look at the FORTRAN coding to determine what information is printed. BUG=YES prints massive amounts of sparsely labeled information. The user should only rarely consider using TRK=YES. This generates thousands of lines of well-labeled output that provides information about each history at key locations during the tracking procedure. All other parameters can be changed at will to provide features the user wishes to activate.

Cross sections

In multi-energy group mode, KENO always uses cross sections from a mixed cross section data file. The format of this file is the Monte Carlo processed cross section file. A mixed cross section file can be created by previous KENO run, or  by using an AMPX working format library and entering mixing table data in KENO.

Use a mixed cross section Monte Carlo format library

A mixed cross section Monte Carlo format library (premixed cross section data file) from a previous KENO case may be used. This file is specified using the parameter XSC=. If a mixing table data block is entered, the premixed cross section data file will be rewritten. Therefore, a mixing table should not be entered if a premixed cross section data file is used. The user should verify that the mixtures created by a previous KENO case are consistent with those used in the geometry data of the problem.

Use an AMPX working format library

When an AMPX working format library is used, it must be specified using the parameter LIB=, and mixing table data must always be entered. IDs used in the mixing table must match the IDs on the AMPX working format library. The user must provide a file with the correct cross sections with a name that matches the pattern ftNNf001, with NN being the number of the logical unit.

Number of scattering angles

The number of scattering angles is defaulted to 1 (defaulted to 2 when KENO-VI is run as part of the CSAS6 sequence). This stand alone default is not adequate for many applications. The user should specify the scattering angle to be consistent with the cross sections being used. The number of scattering angles is entered in the cross section mixing table by using the keyword SCT=. See Mixing table data.

The order of the last Legendre coefficient to be preserved in the scattering distribution is equal to (2 × SCT − 1). SCT=1 could be used with a P1 cross section set such as the 16-group Hansen-Roach cross section library, and SCT=2, for a P3 cross section set such as the SCALE 27-group cross section library. ENDF/B-V cross section libraries such as the 44-group or 238-group libraries contain many nuclides having P5 cross section sets. Isotropic scattering is achieved by entering SCT=0.

Cross section message cutoff

The cross section message cutoff value, pbxs, is defaulted to 3 × 10−5. Warning messages generated when errors are encountered in the PL expansion of the group-to-group transfers will be suppressed if the P0 cross section for that particular energy transfer is less than pbxs. The value of pbxs is specified in the cross section mixing table by using the keyword EPS=. See Mixing table data.

The default value of pbxs is sufficient to assure that warning messages will not be printed for most of the SCALE P1 and P3 cross section libraries. However, the v7.0-238n library may print a few errors if P5 cross sections are specified.

The error messages below were printed for a problem using the 238-group cross section library and pbxs = 3.0 × 10–5. If the default value of pbxs allows too many warning messages to be printed, a value can be determined which does not print the error messages from the printed messages by choosing a number larger than the P0 component on the first line, as shown below.

THE LEGENDRE EXPANSION OF THE CROSS SECTION (P0-PN) IS P0 P1 P2 … Pn

THE MOMENTS CORRESPONDING TO THIS DISTRIBUTION ARE M1 M2 … Mn

THE MOMENTS CORRESPONDING TO THE GENERATED DISTRIBUTION ARE M1 M2 … Mn

THE LEGENDRE EXPANSION CORRESPONDING TO THESE MOMENTS IS P0 P1 P2 … Pn

___ MOMENTS WERE ACCEPTED

For the following messages, EPS=6.9e−5 would cause all three messages to be suppressed. A value less than 5.615159e−5 and greater than 4.767635e−5 would suppress the second message, and a value less than 6.855362e−5 and greater than 5.615159e−5 would suppress the first two messages.

KMSG060       THE ANGULAR SCATTERING DISTRIBUTION FOR MIXTURE 2 HAS BAD MOMENTS FOR THE TRANSFER FROM GROUP 28 TO GROUP 72
                  1 MOMENTS WERE ACCEPTED
                THE LEGENDRE EXPANSION OF THE CROSS SECTION (P0-PN) IS
           5.615159E-05   1.155527E-06  -2.804013E-05  -1.732067E-06
THE MOMENTS CORRESPONDING TO THIS DISTRIBUTION ARE
           2.057870E-02   4.234578E-04   8.710817E-06
THE MOMENTS CORRESPONDING TO THE GENERATED DISTRIBUTION ARE
           2.057870E-02   4.235078E-04   8.710817E-06
THE LEGENDRE EXPANSION CORRESPONDING TO THESE MOMENTS IS
           5.615159E-05   1.155527E-06  -2.804011E-05  -1.732066E-06
                THE WEIGHTS/ANGLES FOR THIS DISTRIBUTION ARE
           9.999995E-01   5.268617E-07
           2.057881E-02  -1.973451E-01
THE MOMENTS CORRESPONDING TO THIS DISTRIBUTION ARE
           2.057870E-02   4.235078E-04   8.710817E-06


KMSG060       THE ANGULAR SCATTERING DISTRIBUTION FOR MIXTURE 2 HAS BAD MOMENTS FOR THE TRANSFER FROM GROUP 31 TO GROUP 75
                  1 MOMENTS WERE ACCEPTED
                THE LEGENDRE EXPANSION OF THE CROSS SECTION (P0-PN) IS
           4.767635E-05   7.834378E-07  -2.381887E-05  -1.174626E-06
                THE MOMENTS CORRESPONDING TO THIS DISTRIBUTION ARE
           1.643242E-02   2.700205E-04   4.451724E-06
                THE MOMENTS CORRESPONDING TO THE GENERATED DISTRIBUTION ARE
           1.643242E-02   2.700282E-04   4.437279E-06
                THE LEGENDRE EXPANSION CORRESPONDING TO THESE MOMENTS IS
           4.767635E-05   7.834378E-07  -2.381885E-05  -1.174627E-06
                THE WEIGHTS/ANGLES FOR THIS DISTRIBUTION ARE
           9.999858E-01   1.420136E-05
           1.643265E-02  -2.334324E-07
                THE MOMENTS CORRESPONDING TO THIS DISTRIBUTION ARE
           1.643242E-02   2.700282E-04   4.437279E-06


KMSG060       THE ANGULAR SCATTERING DISTRIBUTION FOR MIXTURE 2 HAS BAD MOMENTS FOR THE TRANSFER FROM GROUP
32 TO GROUP 74                                                                  (1)
                  1 MOMENTS WERE ACCEPTED                                     (2)
                THE LEGENDRE EXPANSION OF THE CROSS SECTION (P0-PN) IS            (3)
            6.855362E-05   1.341944E-06  -3.423741E-05  -2.011613E-06             (4)
                THE MOMENTS CORRESPONDING TO THIS DISTRIBUTION ARE                (5)
            1.957510E-02   3.831484E-04   7.601939E-06                          (6)
                THE MOMENTS CORRESPONDING TO THE GENERATED DISTRIBUTION ARE (7)
            1.957510E-02   3.832207E-04   7.502292E-06                          (8)
                THE LEGENDRE EXPANSION CORRESPONDING TO THESE MOMENTS IS      (9)
            6.855362E-05   1.341944E-06  -3.423740E-05  -2.011629E-06           (10)
                THE WEIGHTS/ANGLES FOR THIS DISTRIBUTION ARE                    (11)
            9.999056E-01   9.437981E-05                                       (12)
            1.957695E-02  -1.848551E-06                                         (13)
                THE MOMENTS CORRESPONDING TO THIS DISTRIBUTION ARE              (14)
                 1.957510E-02   3.832207E-04   7.502292E-06                     (15)

The user does not need to attempt to suppress all these messages. They are printed to inform the user of the fact that the moments of the angular distribution are not moments of a valid probability distribution. The original Pn coefficients and their moments are listed in lines 3–6 of the message. Lines 7–10 list the new corrected moments and their corresponding Pn coefficients.

The weights and angles printed in lines 11–13 were generated from the corrected moments. The last two lines of the message list the moments generated from those weights and angles. They should match line 8, which lists the moments corresponding to the generated distribution.

For most criticality problems, the first moment contributions are much more significant than the contributions of the higher order moments. Thus, the higher order moments may not affect the results significantly. The user may compare the original moments and corrected moments to make a judgment as to the significance of the change in the moments.

Mixing table

Mixtures can be used in defining other mixtures. When defining mixture numbers, care should be taken to avoid using a mixture number that is identical to a nuclide ID number if the mixture is to be used in defining another mixture. If a mixture number is defined more than once, it results in a summing effect.

The nuclide mixing loop is done before the mixture mixing loop, which performs mixing in the order of data entry. Thus, the order of mixing mixtures into other mixtures is important because a mixture must be defined before it can be used in another mixture. Some examples of correct and incorrect mixing are shown below, using 1001 as the nuclide ID for hydrogen, 8016 as the nuclide ID for oxygen, 92235 as the nuclide ID for 235U, and 92238 as the nuclide ID for 238U.

EXAMPLES OF CORRECT USAGE

1.  READ MIXT
      MIX=1 1001 6.6e-2  8016 3.3e-2
      MIX=2 1 0.5
    END MIXT

This results in mixture 1 being full-density water and mixture 2 being half-density water.

2.  READ MIXT
      MIX=1  2  0.5
      MIX=3  1  0.5
      MIX=2  1001  6.6e-2  8016  3.3e-2
    END MIXT

This results in mixture 1 being half-density water, mixture 2 being full-density water, and mixture 3 being quarter-density water. Because the nuclide mixing loop is done first, mixture 2 is created first and is available to create mixture 1, which is then available to create mixture 3.

3.  READ MIXT
      MIX=1   1001   6.6e-2   8016   3.3e-2
      MIX=2   92235   7.5e-4   92238   2.3e-2  8016   4.6e-2  1 .01
   END MIXT

This results in mixture 1 being full-density water and mixture 2 being uranium oxide containing 0.01 density water.

4.  READ MIXT
      MIX=1    1001  6.6e-2    8016  3.3e-2
      MIX=2  92235  4.4e-2  92238  2.6e-3
      MIX=1  1  0.5
    END MIXT

This results in mixture 1 being water at 1.5 density (1001 9.9e-2 and 8016 4.95e-2) and mixture 2 is highly enriched uranium metal.

EXAMPLES OF INCORRECT USAGE

1.        READ MIXT
      MIX=3  1  0.75
      MIX=1  2  0.5
      MIX=2  1001  6.6e-2  8016  3.3e-2
    END MIXT

Here the intent is for mixture 2 to be full-density water, mixture 1 to be half-density water, and mixture 3 to be 3/8 (0.75 × 0.5) density water. Instead, the result for mixture 3 is a void, mixture 1 is half-density water, and mixture 2 is full-density water. This is because the nuclide mixing loop is done first, thus defining mixture 2. The mixture mixing loop is done next. Mixture 3 is defined to be mixture 1 multiplied by 0.75, but since mixture 1 has not been defined yet, 0.75 of zero is zero. Mixture 1 is then defined to be mixture 2 multiplied by 0.5. If the definition of mixture 1 preceded the definition of mixture 3, as in item (2) under examples of correct usage, it would work correctly.

2.        READ MIXT
      MIX=1         1001   6.6e-2     8016     3.3e-2
      MIX=1001 92235   4.4e-2   92238   2.6e-3
      MIX=2   1001   0.5
    END MIXT

This results in mixture 1 being full-density water, mixture 1001 being uranium metal, and mixture 2 being hydrogen with a number density of 0.5 because 1001 is the nuclide ID number for hydrogen. When a mixture number is identical to a nuclide ID and is used in mixing, that number is assumed to be a nuclide ID rather than a mixture number. The intent was for mixture 1 to be full-density water, mixture 1001 to be uranium metal, and mixture 2 to be half-density uranium metal.

Geometry

In general, KENO geometry descriptions consist of (1) geometry data (Geometry data) defining the geometrical shapes present in the problem, and (2) array data (ARRAY Data) defining the placement of the units that were defined in the geometry data. The geometry data block is prefaced by READ GEOM, and the array data block is prefaced by READ ARRAY.

When a 3-D geometrical configuration is described as KENO geometry data, it may be necessary to describe portions of the configuration individually. These individual partial descriptions of the configuration are called UNITs. KENO geometry modeling is subject to the following restrictions:

  1. Units are composed of regions. These regions are created using

    geometric bodies and surfaces (shapes) that are previously defined.

    1. In KENO-VI the geometric bodies and surfaces may intersect. Regions are defined relative to the geometric bodies and surfaces in a UNIT. HOLEs provide a means of creating complex geometries in a UNIT and then inserting the UNIT into existing UNITs. For complex geometries the use of HOLEs may decrease the CPU time required for the problem.

    2. In KENO V.a, each geometry region in a UNIT must completely enclose all geometry regions which precede it. Boundaries of the surfaces of the regions may be shared or tangent, but they must not intersect. The use of HOLEs in KENO V.a provides an exception to this complete enclosure rule. The use of HOLEs in KENO V.a will increase the CPU time required for the problem.

  2. All geometrical surfaces must be describable in KENO V.a as spheres,

    hemispheres, cylinders, hemicylinders, cubes, cuboids, or as a set of quadratic equations in KENO-VI.

  3. When specifying an ARRAY, each UNIT used in a cuboidal

    ARRAY must have a CUBE or CUBOID as its outer region (this is the only option in KENO V.a); a hexprism as the outer boundary if it is a hexagonal, triangular, or shexagonal ARRAY; a rhexprism as the outer boundary if it is a rhexagonal ARRAY; and a dodecahedron as the outer boundary if it is a dodecahedral ARRAY. In addition, the outer boundary of a UNIT cannot be rotated or translated in KENO V.a.

  4. When several UNITs are used to describe an ARRAY, the

    adjacent faces of the UNITs in contact with each other must be the same size and shape.

  5. UNITs are placed directly into regions using HOLEs. As

    many HOLEs as will fit without intersecting other HOLEs, nested ARRAYs or HOLEs, or the UNIT BOUNDARY can be placed in a UNIT without intersecting each other. In KENO V.a, HOLEs cannot intersect any of the regions within the UNIT in which they are placed. HOLEs are described in more detail in Use of holes in the geometry, and nested HOLEs are described in Nesting holes.

  6. Multiple ARRAYs may be required to describe a complicated

    system. In KENO V.a, only one ARRAY may be placed directly into a UNIT. However, multiple ARRAYs may be placed in a UNIT by placing the ARRAYs in other UNITs and placing those UNITs in the original UNIT using HOLEs. Multiple ARRAYs may be placed directly into a UNIT in KENO-VI. These UNITs may then be used to create other ARRAYs, or they may be placed in other UNITs using HOLEs. UNITs placed in ARRAYs or HOLEs that are contained in other HOLEs or ARRAYs are said to be nested. The nesting level of a UNIT is the number of ARRAYs and HOLEs between the ARRAY or HOLE in the GLOBAL UNIT or ARRAY and the UNIT. Multiple ARRAYs are described in more detail in Multiple arrays.

The KENO V.a geometry package allows any applicable shape to be enclosed by any other applicable shape, subject only to the complete enclosure restriction. The KENO-VI geometry package allows any shape describable using quadratic equations to be enclosed or intersected by any other allowable shape. The implication of this type of description is that the entire volume between two adjacent geometrical surfaces contains only one mixture, HOLE, or ARRAY. A void is specified by a mixture ID of zero. If HOLEs are present in the volume between two surfaces, the volume of that region is reduced by the HOLE’s volume(s).

In KENO V.a geometry, if the problem requires several UNITs to describe its geometrical characteristics, each UNIT that is used in an ARRAY must have a rectangular parallelepiped as its outer surface. This restriction is relaxed in KENO-VI, where the outer surface may be a rectangular parallelepiped, a hexagonal prism, a 90º rotated hexagonal prism, or a dodecahedron, but all units placed in an array must use the same shape as their outer BOUNDARY. In order to describe the composite overall geometrical characteristics of the problem, these UNITs may be arranged in a rectangular ARRAY for KENO V.a geometry, or in either a rectangular, hexagonal, shexagaonal, rhexagonal, or dodecahedral ARRAY for KENO-VI geometry. This is done by specifying the number of units in the X, Y, and Z directions. If more than one UNIT is involved, data must be entered to define the number assigned to the ARRAY and the placement of the individual UNITs in the ARRAY. The ARRAY number, the number of UNITs in the X, Y, and Z directions, and the placement data are called array data (ARRAY Data).

In the KENO V.a geometry description, the surrounding regions of any shape may be placed around an ARRAY, and they may consist of any number of regions in any order subject to the complete enclosure restriction. ARRAYs are positioned relative to the UNIT ORIGIN by specifying the location of the most negative point in the array, i.e. the most negative X, most negative Y, and most negative Z corner of the ARRAY. In KENO-VI geometry, an ARRAY may be placed in a region of any shape if the region boundary is contained within the ARRAY or shares the ARRAY boundary and does not cut across nested ARRAYs or HOLEs. In this case, only the section of the ARRAY contained within the region is recognized by the problem. ARRAYs are positioned relative to the UNIT ORIGIN by placing the ORIGIN of a specified UNIT in the ARRAY at a specified location in the UNIT.

To create a geometry mockup from a physical configuration, the user should keep the restrictions mentioned earlier in mind. There may be several ways of correctly describing the same physical configuration. Careful analysis of the system can result in a simpler mockup and shorter computer running time. A mockup with fewer geometry regions may run faster than the same mockup with extraneous regions. The number of UNITs used can affect the running time, because a transformation of coordinates must be made every time a history moves from one UNIT into another. Thus, if the size of a UNIT is small relative to the mean free path, a larger percentage of time is spent processing the transformation of coordinates. Because all boundaries in a UNIT must be checked for crossings, it may be more efficient to break up complex UNITs into several smaller, simpler UNITs. The trade-off involves the time required to process more boundary crossings vs the time required to transform coordinate systems when UNIT boundaries are crossed.

Geometry dimensions: The use of FIDO syntax may help simplify the description of the geometry. For example, a 20 × 20 × 2.5 cm rectangular parallelepiped would have been described as CUBOID 1 1 10.0 −10.0 10.0 −10.0 1.25 −1.25 in KENO V.a and CUBOID 1 10.0 –10.0 10.0 –10.0 1.25 –1.25 in KENO-VI. By using the P option (see Table 8.1.17), the same rectangular parallelepiped could be described as CUBOID 1 1 4P10.0 2P1.25 in KENO V.a and or CUBOID 1 4P10.0 2P1.25 in KENO-VI, where the last 6 entries describe the geometry. The P option simply repeats the dimension following the P for the number of times stated before the P, and it reverses the sign every time. Therefore, 6P8.0 is equivalent to 8.0 −8.0 8.0 −8.0 8.0 −8.0.

Geometry comments: One comment can be entered for each UNIT in the geometry region data. Similarly, one comment can be entered for each ARRAY in the array definition data. A comment can be entered using the keyword COM=. This is followed by a comment with a maximum length of 132 characters. The comment must be preceded and terminated by a delimiter character. Acceptable delimiters include “ , ‘ , * , ^ , or !. One comment is allowed for each UNIT in the geometry region data. If multiple comments are entered for a UNIT, the last comment is used. The comment can be entered anywhere after the UNIT number where a keyword is expected (Geometry data). See the following example.

KENO V.a:

READ GEOM
UNIT 1
  COM=*SPHERICAL METAL UNIT*
  SPHERE 1 1 5.0
  CUBE 0 1 2P5.0
UNIT 2
  CYLINDER 1 1 5.0 2P5.0
  CUBE 0 1 2P5.0
  COM=!CYLINDRICAL METAL UNIT!
UNIT 3
  HEMISPHE+X 1 1 5.0
  COM='HEMISPHERICAL METAL UNIT'
  CUBE 0 1 2P5.0
UNIT 4
  COM=^ARRAY OF SPHERICAL UNITS^
  ARRAY 1 3*0.0
UNIT 5
  COM=“ARRAY OF CYLINDRICAL UNITS”
  ARRAY 2 3*0.0
UNIT 6
  COM='ARRAY OF HEMISPHERICAL UNITS'
  ARRAY 3 3*0.0
END GEOM

KENO-VI:

READ GEOM
UNIT 1
  COM=*SPHERICAL METAL UNIT*
  SPHERE 1  5.0
  CUBOID 2  6P5.0
  MEDIA 1 1 1
  MEDIA 0 1 -1 2
  BOUNDARY 2
UNIT 2
  CYLINDER 1 5.0 2P5.0
  CUBOID 2 6P5.0
  MEDIA 1 1 1
  COM=!CYLINDRICAL METAL UNIT!
  MEDIA 0 1 -1 2
  BOUNDARY 2
UNIT 3
  SPHERE 1  5.0 CHORD +X=0.0
  MEDIA 1 1 1
  COM='HEMISPHERICAL METAL UNIT'
  CUBOID 2  6P5.0
  MEDIA 0 1 -1 2
  BOUNDARY 2
UNIT 4
  COM='ARRAY OF SPHERICAL UNITS'
  CUBOID 1 6P15
  ARRAY 1 1 PLACE 2 2 2 3*0.0
  BOUNDARY 1
UNIT 5
  CUBOID 1 6P15.0
  COM='ARRAY OF CYLINDRICAL UNITS'
  ARRAY 2 1 PLACE 2 2 2 3*0.0
  BOUNDARY 1
UNIT 6
  COM='ARRAY OF HEMISPHERICAL UNITS'
  CUBOID 1 6P15.0
  ARRAY 3 1 PLACE 2 2 2 3*0.0
  BOUNDARY 1
GLOBAL UNIT 7
  COM='ARRAY OF ARRAYS'
  CUBOID 1 4P15.0 2P45.0
  ARRAY 4 1 PLACE 1 1 2 3*0.0
  BOUNDARY 1
END GEOM

One comment is allowed for each array in the array definition data. The rules governing these comments are the same as those listed above. However, the comment for an ARRAY must precede the UNIT arrangement description, and it can precede the ARRAY number (ARRAY Data). Examples of correct ARRAY comments are given below.

KENO V.a:

READ ARRAY
  COM='ARRAY OF SPHERICAL METAL UNITS'
  ARA=1 NUX=2 NUY=2 NUZ=2 FILL F1 END FILL
  ARA=2 COM='ARRAY OF CYLINDRICAL METAL UNITS'
  NUX=2 NUY=2 NUZ=2 FILL F2 END FILL
  ARA=3 NUX=2 NUY=2 NUZ=2
  COM='ARRAY OF HEMISPHERICAL METAL UNITS'
  FILL F3 END FILL
  ARA=4 COM='COMPOSITE ARRAY OF ARRAYS. Z=1 IS SPHERES, Z=2 IS CYLINDERS,   Z=3 IS HEMISPHERES'
  NUX=1 NUY=1 NUZ=3 FILL 4 5 6 END FILL
END ARRAY

KENO-VI:

READ ARRAY
  COM='ARRAY OF SPHERICAL METAL UNITS'
  ARA=1 NUX=3 NUY=3 NUZ=3 FILL F1 END FILL
  ARA=2 COM='ARRAY OF CYLINDRICAL METAL UNITS'
  NUX=3 NUY=3 NUZ=3 FILL F2 END FILL
  ARA=3 NUX=3 NUY=3 NUZ=3
  COM='ARRAY OF HEMISPHERICAL METAL UNITS'
  FILL F3 END FILL
  ARA=4 COM='COMPOSITE ARRAY OF ARRAYS. Z=1 IS SPHERES, Z=2 IS CYLINDERS, Z=3 IS  HEMISPHERES'
  NUX=1 NUY=1 NUZ=3 FILL 4 5 6 END FILL
END ARRAY

Some of the basics of KENO geometry are illustrated in the following examples:

EXAMPLE 1. Assume a stack of six cylindrical disks each measuring 5 cm in radius and 2 cm thick. The bottom disk is composed of material 1, and the next disk is composed of material 2, etc., alternating throughout the stack. A square plate of material 3 that is 20 cm on a side and 2.5 cm thick is centered on top of the stack. This configuration is shown in Fig. 62.

_images/fig121.png

Fig. 62 Stack of disks with a square top.

This problem can be described as a single UNIT problem, describing the cylindrical portion first. In this instance, the origin has been chosen at the center bottom of the bottom disk. The bottom disk is defined by the first cylinder description, and the next disk is defined by the difference between the first and second cylinder descriptions. Since both disks have a radius of 5.0 and a −Z length of 0.0, the first cylinder containing material 1 exists from Z = 0.0 to Z = 2.0, and the second cylinder containing material 2 exists from Z = 2.0 to Z = 4.0. When all the disks have been described, a void cuboid having the same X and Y dimensions as the square plate and the same Z dimensions as the stack of disks is defined. The square plate of material 3 is then defined on top of the stack. Omission of the first cuboid description would result in the stack of disks being encased in a solid cuboid of material 3 instead of having a flat plate on top of the stack. The geometry input is shown below.

Data description 1, Example 1.

KENO V.a:

READ GEOM
CYLINDER 1  1     5.0 2.0    0.0
CYLINDER 2  1     5.0 4.0    0.0
CYLINDER 1  1     5.0 6.0    0.0
CYLINDER 2  1     5.0 8.0    0.0
CYLINDER 1  1     5.0 10.0   0.0
CYLINDER 2  1     5.0 12.0   0.0
CUBOID 0 1 10.0 -10.0 10.0 -10.0 12.0 0.0
CUBOID 3 1 10.0 -10.0 10.0 -10.0 14.5 0.0

KENO-VI:

READ GEOM
GLOBAL UNIT 1
CYLINDER   1   5.0   2.0   0.0
CYLINDER   2   5.0   4.0   2.0
CYLINDER   3   5.0   6.0   4.0
CYLINDER   4   5.0   8.0   6.0
CYLINDER   5   5.0  10.0   8.0
CYLINDER   6   5.0  12.0  10.0
CUBOID     7  10.0 -10.0  10.0 -10.0  12.0  0.0
CUBOID     8  10.0 -10.0  10.0 -10.0  14.5  0.0
MEDIA 1 1 1
MEDIA 2 1 2
MEDIA 1 1 3
MEDIA 2 1 4
MEDIA 1 1 5
MEDIA 2 1 6
MEDIA 0 1 -1 -2 -3 -4 -5 -6 7
MEDIA 3 1 -7 8
BOUNDARY  8
END GEOM

An alternative description of the same example is given below. The origin has been chosen at the center of the disk of material 1 nearest the center of the stack. This disk of material 1 is defined by the first cylinder description, and the disks of material 2 on either side of it are defined by the second cylinder description. The top and bottom disks of material 1 are defined by the third cylinder, and the top disk of material 2 is defined by the last cylinder. The square plate is defined by the two cuboids. This description is more efficient than the previous one because there are fewer surfaces to check for crossings.

Data description 2, Example 1.

KENO V.a:

READ GEOM
CYLINDER 1 1  5.0   1.0 -1.0
CYLINDER 2 1  5.0   3.0 -3.0
CYLINDER 1 1  5.0   5.0 -5.0
CYLINDER 2 1  5.0   7.0 -5.0
CUBOID   0 1 10.0 -10.0 10.0 -10.0 7.0 -5.0
CUBOID   3 1 10.0 -10.0 10.0 -10.0 9.5 -5.0

KENO-VI:

READ GEOM
GLOBAL UNIT 1
CYLINDER 10  5.0   1.0 -1.0
CYLINDER 20  5.0   3.0 -3.0
CYLINDER 30  5.0   5.0 -5.0
CYLINDER 40  5.0   7.0 -5.0
CUBOID   50 10.0 -10.0 10.0 -10.0 7.0 -5.0
CUBOID   60 10.0 -10.0 10.0 -10.0 9.5 -5.0
MEDIA 1 1 10
MEDIA 2 1 -10 20
MEDIA 1 1 -20 30
MEDIA 2 1 -30 40
MEDIA 0 1 -40 50
MEDIA 3 1 -50 60
BOUNDARY  60
END GEOM

Example 1 can also be described as an ARRAY. Define three different UNIT types. UNIT 1 will define a disk of material 1, UNIT 2 will define a disk of material 2, and UNIT 3 will define the square plate of material 3. The origin of each UNIT is defined at the center bottom of the disk or plate being described. As mentioned earlier, only UNITs with a CUBE or CUBOID as their outer boundary can be placed in a cuboidal ARRAY. The geometry input for this arrangement is shown below.

Data description 3, Example 1.

KENO V.a:

READ GEOM
UNIT 1
CYLINDER  1 1  5.0   2.0  0.0
CUBOID    0 1 10.0 -10.0 10.0 -10.0 2.0 0.0
UNIT 2
CYLINDER  2 1  5.0   2.0  0.0
CUBOID    0 1 10.0 -10.0 10.0 -10.0 2.0 0.0
UNIT 3
CUBOID    3 1 10.0 -10.0 10.0 -10.0 2.5 0.0
END GEOM
READ ARRAY  NUX=1 NUY=1 NUZ=7 FILL 1 2 1 2 1 2 3 END ARRAY

Note

The ARRAY is assumed to be the GLOBAL ARRAY because only a single ARRAY is defined.

KENO-VI:

READ GEOM
UNIT 1
CYLINDER 1  5.0   2.0  0.0
CUBOID  2 10.0 -10.0 10.0 -10.0 2.0 0.0
MEDIA 1 1 1
MEDIA 0 1 -1 2
BOUNDARY 2
UNIT 2
CYLINDER  1  5.0   2.0  0.0
CUBOID    2 10.0 -10.0 10.0 -10.0 2.0 0.0
MEDIA 2 1 1
MEDIA 0 1 2 -1
BOUNDARY 2
UNIT 3
CUBOID    1 10.0 -10.0 10.0 -10.0 2.5 0.0
MEDIA 3 1 1
BOUNDARY 1
GLOBAL UNIT 4
CUBOID 1  10 -10 10 -10 14.5 0.0
ARRAY 1 1  PLACE 1 1 1 3*0.0
BOUNDARY 1
END GEOM
READ ARRAY ARA=1 NUX=1 NUY=1 NUZ=7 FILL 1 2 1 2 1 2 3 END ARRAY

If the user prefers for the origin of each unit to be at its center, the geometry region data can be entered as shown below. The array data would be identical to that of data description 3, Example 1.

Data description 4, Example 1.

KENO V.a:

READ GEOM
UNIT 1
CYLINDER  1 1  5.0   1.0 -1.0
CUBOID    0 1 10.0 -10.0 10.0 -10.0 1.0 -1.0
UNIT 2
CYLINDER  2 1  5.0   1.0 -1.0
CUBOID    0 1 10.0 -10.0 10.0 -10.0 1.0 -1.0
UNIT 3
CUBOID  3 1 10.0 -10.0 10.0 -10.0 1.25 -1.25
END GEOM

KENO-VI:

READ GEOM
UNIT 1
CYLINDER 1  5.0   1.0 -1.0
CUBOID  2   10.0 -10.0 10.0 -10.0 1.0 -1.0
MEDIA 1 1 1
MEDIA 0 1 -1 2
BOUNDARY  2
UNIT 2
CYLINDER 1  5.0   1.0 -1.0
CUBOID  2  10.0 -10.0 10.0 -10.0 1.0 -1.0
MEDIA 2 1 1
MEDIA 0 1 -1 2
BOUNDARY  2
UNIT 3
CUBOID 1 10.0 -10.0 10.0 -10.0 1.25 -1.25
MEDIA 3 1 1
BOUNDARY  1
GLOBAL UNIT 4
CUBOID 1 10 -10 10 -10 14.5 0.0
ARRAY 1 1 PLACE 1 1 1 0.0 0.0 1.0
BOUNDARY 1
END GEOM

Be aware that each UNIT in a geometry description can have its origin defined independent of the other UNITs. It would be correct to use UNITs 1 and 3 from data descriptions 3, and UNIT 2 from data description 4. The array data would remain the same as data description 3, Example 1. The user should define the origin of each unit to be as convenient as possible for the chosen description. Care should be taken when assigning coordinates to the UNIT used to PLACE the ARRAY in its surrounding region.

Another method of describing Example 1 as a bare array is to define UNIT 1 to be a disk of material 1 topped by a disk of material 2. The origin has been chosen at the center bottom of the disk of material 1. UNIT 2 is the square plate of material 3 with the origin at the center of the UNIT. The ARRAY consists of three UNIT 1s, topped by a UNIT 2, as shown below.

Data description 5, Example 1.

KENO V.a:

READ GEOM
UNIT 1
CYLINDER  1 1  5.0   2.0  0.0
CYLINDER  2 1  5.0   4.0  0.0
CUBOID    0 1 10.0 -10.0 10.0 -10.0 4.0 0.0
UNIT 2
CUBOID    3 1 10.0 -10.0 10.0 -10.0 1.25 -1.25
END GEOM
READ ARRAY  NUX=1 NUY=1 NUZ=4 FILL 3R1 2 END ARRAY

KENO-VI:

READ GEOM
UNIT 1
CYLINDER  1  5.0   2.0  0.0
CYLINDER  2  5.0   4.0  2.0
CUBOID  3   10.0 -10.0 10.0 -10.0 4.0 0.0
MEDIA 1 1 1
MEDIA 2 1 2
MEDIA 0 1 -1 -2 3
BOUNDARY  3
UNIT 2
CUBOID  1  10.0 -10.0 10.0 -10.0 1.25 -1.25
MEDIA 3 1 1
BOUNDARY  1
GLOBAL UNIT 3
CUBOID  1 10 -10 10 -10 14.5 0.0
ARRAY 1 1 PLACE 1 1 1 3*0.0
BOUNDARY  1
END GEOM
READ ARRAY  ARA=1 NUX=1 NUY=1 NUZ=4 FILL 3R1 2 END ARRAY

Example 1 can be described as a reflected ARRAY by treating the square plate as a reflector in the positive Z direction. One means of describing this situation is to define UNITs 1 and 2 as in data description 3, Example 1. The origin of the GLOBAL UNIT is defined to be at the center of the ARRAY. The corresponding input geometry is shown below.

Data description 6, Example 1.

KENO V.a:

READ GEOM
UNIT 1
CYLINDER  1 1   5.0   2.0  0.0
CUBOID    0 1  10.0  -10.0  10.0  -10.0  2.0  0.0
UNIT 2
CYLINDER  2 1   5.0   2.0  0.0
CUBOID      0 1  10.0  -10.0  10.0  -10.0  2.0  0.0
GLOBAL UNIT 3
ARRAY        1 -10.0  -10.0  -6.0
CUBOID     3 1  10.0  -10.0  10.0  -10.0  8.5  -6.0
END GEOM
READ ARRAY  NUX=1 NUY=1 NUZ=6 FILL 1 2 1 2 1 2 END ARRAY

KENO-VI:

READ GEOM
UNIT 1
CYLINDER  1  5.0   2.0  0.0
CUBOID  2   10.0 -10.0 10.0 -10.0 2.0  0.0
MEDIA 1 1 1
MEDIA 0 1 -1 2
BOUNDARY  2
UNIT 2
CYLINDER  1  5.0   2.0  0.0
CUBOID  2  10.0 -10.0 10.0 -10.0 2.0  0.0
MEDIA 2 1 1
MEDIA 0 1 -1 2
BOUNDARY  2
GLOBAL UNIT 3
CUBOID 1  10.0 -10.0 10.0 -10.0 6.0 -6.0
CUBOID 2  10.0 -10.0 10.0 -10.0 8.5 -6.0
ARRAY 1 1 PLACE 1 1 1 0.0 0.0 -6.0
MEDIA 3 1 -1 2
BOUNDARY  2
END GEOM
READ ARRAY ARA=1 NUX=1 NUY=1 NUZ=6 FILL 1 2 1 2 1 2 END ARRAY

The user could have chosen the origin of the ARRAY boundary to be at the center bottom of the ARRAY, in which case the geometry description for the GLOBAL UNIT would be:

KENO V.a:

ARRAY       1 -10.0  -10.0  0.0
CUBOID   3 1 10.0  -10.0  10.0  -10.0  14.5  0.0

or

ARRAY    1  -10.0  -10.0 0.0
REPLICATE  3  4*0.0  2.5  0 1

The reflector region at the top of the array can be added by using a CUBOID or by using a REPLICATE description in KENO V.a. Recall that there is no REPLICATE function in KENO-VI.

A simpler method of describing Example 1 as a reflected array is to define only one unit as in data description 5, Example 1. The square plate is treated as a reflector, as in data description 6, Example 1. The input for this arrangement is given below.

Data description 7, Example 1.

KENO V.a:

READ GEOM
UNIT 1
CYLINDER  1 1   5.0   2.0  0.0
CYLINDER  2 1   5.0   4.0  0.0
CUBOID      0 1  10.0  -10.0  10.0  -10.0  4.0  0.0
GLOBAL UNIT 2
ARRAY        1  -10.0  -10.0  0.0
CUBOID     3  1  10.0  -10.0  10.0  -10.0  14.5  0.0
END GEOM
READ ARRAY NUX=1 NUY=1 NUZ=3 END ARRAY

KENO-VI:

READ GEOM
UNIT 1
CYLINDER  1   5.0   2.0  0.0
CYLINDER  2   5.0   4.0  2.0
CUBOID   3   10.0 -10.0 10.0 -10.0 4.0 0.0
MEDIA 1 1 1
MEDIA 2 1 2
MEDIA 0 1 -1 -2 3
BOUNDARY  3
GLOBAL UNIT 2
CUBOID  1  10.0 -10.0 10.0 -10.0  12.0 0.0
CUBOID  2  10.0 -10.0 10.0 -10.0 14.5 0.0
ARRAY 1 1 PLACE 1 1 1 3*0.0
MEDIA 3 1 -1 2
BOUNDARY  2
END GEOM
READ ARRAY ARA=1 NUX=1 NUY=1 NUZ=3 FILL F1  END FILL  END ARRAY

EXAMPLE 2. Assume that the stack of six disks in Example 1 is placed at the center bottom of a cylindrical container composed of material 6 whose inside diameter is 16.0 cm. The bottom and sides of the container are 0.25 cm thick, the top is open, and the total height of the container is 18.25 cm. Assume the square plate of Example 1 is centered on top of the container.

The geometry input can be described utilizing most of the data description methods associated with Example 1. One method of describing Example 2 as a single UNIT is given below.

Data description 1, Example 2.

KENO V.a:

READ GEOM
UNIT 1
CYLINDER  1 1  5.0    1.0  -1.0
CYLINDER  2 1  5.0    3.0  -3.0
CYLINDER  1 1  5.0    5.0  -5.0
CYLINDER  2 1  5.0    7.0  -5.0
CYLINDER  0 1  8.0   13.0  -5.0
CYLINDER  6 1  8.25  13.0  -5.25
CUBOID    0 1 10.0  -10.0  10.0 -10.0 13.0 -5.25
CUBOID    3 1 10.0  -10.0  10.0 -10.0 15.5 -5.25
END GEOM

KENO-VI:

READ GEOM
GLOBAL UNIT 1
CYLINDER  1  5.0    1.0  -1.0
CYLINDER  2  5.0    3.0  -3.0
CYLINDER  3  5.0    5.0  -5.0
CYLINDER  4  5.0    7.0  -5.0
CYLINDER  5  8.0   13.0  -5.0
CYLINDER  6  8.25  13.0  -5.25
CUBOID    7  10.0 -10.0  10.0 -10.0 13.0 -5.25
CUBOID    8  10.0 -10.0  10.0 -10.0 15.5 -5.25
MEDIA 1 1 1
MEDIA 2 1 -1 2
MEDIA 1 1 -2 3
MEDIA 2 1 -3 4
MEDIA 0 1 -4 5
MEDIA 6 1 -5 6
MEDIA 0 1 -6 7
MEDIA 3 1 -7 8
BOUNDARY  8
END GEOM

In the above description, the origin is defined to be at the center of the disk of material 1 nearest the center of the stack. This disk is defined by the first cylinder description. The disks of material 2 above and below it are defined by the second cylinder description. The disks of material 1 above and below them are defined by the third cylinder description. The top disk of material 2 is defined by the fourth cylinder description. The void interior of the container is defined by the fifth cylinder description. The container is defined by the last cylinder description. The first cuboid description is used to define a void whose X and Y dimensions are the same as the square plate and whose Z dimensions are the same as the container. The last cuboid description defines the square plate. Omission of the first cuboid description would result in the container being encased in a solid cuboid of material 3. Thus, both cuboids are necessary to properly define the square plate.

Example 2 can be described as a reflected ARRAY. One of the descriptions uses only one UNIT and is similar to data description 7, Example 1. This description is shown below.

Data description 2, Example 2.

KENO V.a:

READ GEOM
UNIT 1
CYLINDER  1  1  5.0    2.0   0.0
CYLINDER  2  1  5.0    4.0   0.0
CUBOID    0 1  5.0   -5.0   5.0   -5.0  4.0  0.0
GLOBAL UNIT 2
ARRAY          1  -5.0   -5.0   0.0
CYLINDER  0  1  8.0   18.0   0.0
CYLINDER  6  1  8.25  18.0  -0.25
CUBOID      0  1  10.0  -10.0  10.0  -10.0  18.0  -0.25
CUBOID      3  1  10.0  -10.0  10.0  -10.0  20.5  -0.25
END GEOM
READ ARRAY  NUX=1 NUY=1 NUZ=3 END ARRAY

KENO-VI:

READ GEOM
UNIT 1
CYLINDER  1  5.0    2.0   0.0
CYLINDER  2  5.0    4.0   2.0
CUBOID    3  5.0   -5.0   5.0   -5.0  4.0  0.0
MEDIA 1 1 1
MEDIA 2 1 2
MEDIA 0 1 -1 -2 3
BOUNDARY  3
GLOBAL UNIT 2
CUBOID   1 5.0 -5.0 5.0 -5.0 12.0 0.0
ARRAY 1 1 PLACE 1 1 1 3*0.0
CYLINDER  2  8.0   18.0   0.0
MEDIA 0 1 -1 2
CYLINDER  3  8.25  18.0  -0.25
MEDIA 6 1 -2 3
CUBOID    4 10.0  -10.0  10.0  -10.0 20.5 18.0
CUBOID    5 10.0  -10.0  10.0  -10.0 20.5 -0.25
MEDIA 3 1 4
MEDIA 0 1 -3 -4 5
BOUNDARY  5
END GEOM
READ ARRAY ARA NUX=1 NUY=1 NUZ=3 FILL F1 END FILL END ARRAY

In the above data description, the first two cylinder descriptions define a disk of material 1 with a disk of material 2 directly on top of it. A tight fitting void cuboid is placed around them so they can be stacked three high to achieve the stack of disks shown in Example 1, Fig. 62. This array comprises the array portion of the geometry region description. The origin of the array boundary, a tight fitting cube or cuboid that encompasses the array, is defined by the ARRAY description. Everything after the ARRAY record is considered part of the reflector. The first cylinder after the ARRAY record defines the void interior of the cylindrical container. The next cylinder defines the walls of the container. The second-to-last cuboid defines a void volume outside the container from its bottom to its top and having the same X and Y dimensions as the square plate. The last cuboid defines the square plate of material 3 that is sitting on top of the container.

Another method to describe Example 2 is as an array composed of units that contain both the stack and container. This description requires a minimum of four units to describe the problem. This configuration is given below in data description 3, Example 2.

Data description 3, Example 2.

KENO V.a:

READ GEOM
UNIT 1
CYLINDER  6 1  8.25   0.25   0.0
CUBOID    0 1 10.0  -10.0   10.0 -10.0 0.25  0.0
UNIT 2
CYLINDER  1 1  5.0    2.0    0.0
CYLINDER  2 1  5.0    4.0    0.0
CYLINDER  0 1  8.0    4.0    0.0
CYLINDER  6 1  8.25   4.0    0.0
CUBOID    0 1 10.0  -10.0   10.0 -10.0 4.0   0.0
UNIT 3
CYLINDER  0 1  8.0    3.0   -3.0
CYLINDER  6 1  8.25   3.0   -3.0
CUBOID    0 1 10.0  -10.0   10.0 -10.0 3.0  -3.0
CUBOID    3 1 10.0  -10.0   10.0 -10.0 5.5  -3.0
END GEOM
READ ARRAY  NUX=1 NUY=1 NUZ=5 FILL 1 3R2 3 END ARRAY

KENO-VI:

READ GEOM
UNIT 1
CYLINDER  1  8.25   0.25   0.0
CUBOID  2   10.0  -10.0   10.0 -10.0 0.25  0.0
MEDIA 6 1 1
MEDIA 0 1 2 -1
BOUNDARY  2
UNIT 2
CYLINDER  10  5.0    2.0    0.0
CYLINDER  20  5.0    4.0    0.0
CYLINDER  30  8.0    4.0    0.0
CYLINDER  40  8.25   4.0    0.0
CUBOID  50   10.0  -10.0   10.0 -10.0 4.0   0.0
MEDIA 1 1 10
MEDIA 2 1 20 -10
MEDIA 0 1 30 -20
MEDIA 6 1 40 -30
MEDIA 0 1 50 -40
BOUNDARY  50
UNIT 3
CYLINDER  1  8.0    3.0   -3.0
CYLINDER  2  8.25   3.0   -3.0
CUBOID    3 10.0  -10.0   10.0 -10.0 5.5   3.0
CUBOID    4 10.0  -10.0   10.0 -10.0 5.5  -3.0
MEDIA 0 1 1
MEDIA 6 1 2 -1
MEDIA 3 1 3
MEDIA 0 1 4 -3 -2
BOUNDARY  4
GLOBAL UNIT 4
CUBOID  1  10.0 -10.0 10.0 -10.0 20.75 0.0
ARRAY 1 1 PLACE 1 1 1 3*0.0
BOUNDARY  1
END GEOM
READ ARRAY  NUX=1 NUY=1 NUZ=5 FILL 1 3R2 3 END ARRAY

In the above description, UNIT 1 is the bottom of the cylindrical container. The void CUBOID is only as tall as the bottom of the container, and its X and Y dimensions are the same as the square plate on top of the container. If all the UNITs in the ARRAY use these same dimensions in the X and Y directions, the requirement that adjacent faces of units in contact with each other must be the same size and shape is satisfied. This ARRAY is stacked in the Z direction, so all UNITs must have the same overall dimensions in the X direction and in the Y direction. UNIT 2 will be used in the ARRAY three times to create the stack of disks. It contains a disk of material 1 topped by a disk of material 2. The portion of the container that contains the disks and the cuboid that defines the outer boundaries of the unit are included in UNIT 2. UNIT 3 describes the empty top portion of the container and the square plate on top of it. The Z dimensions of UNIT 3 were determined by subtracting three times the total Z dimension of UNIT 2 from the inside height of the container [18.0 − (3 × 4.0) = 6.0]. This can also be determined from the overall height of the container by subtracting off the bottom thickness of the container and three times the height of UNIT 2 [18.25 − 0.25 − (3 × 4.0) = 6.0]. The origin of UNIT 3 is located at the center of this distance. For the KENO-VI input, a GLOBAL UNIT is also provided, UNIT 4, containing the ARRAY built with UNITs 1, 2, 3.

EXAMPLE 3. Refer to Example 1, Fig. 62, and imagine a HOLE 1.5 cm in diameter is drilled along the centerline of the stack through the disks and the square plate. In KENO V.a this HOLE would eliminate the possibility of describing the system as a single unit because the HOLE in the center of the alternating materials of the stack cannot be described in a manner that allows each successive geometry region to encompass the regions interior to it. Therefore, it must be described as an ARRAY. The square plate on the top of the disks is defined as a UNIT in the ARRAY. In the geometry description given below, the square plate is defined in UNIT 3. KENO-VI can easily describe this configuration as a single UNIT.

Data description 1, Example 3.

KENO V.a:

READ GEOM
UNIT 1
CYLINDER  0 1  0.75   2.0  0.0
CYLINDER  1 1  5.0    2.0  0.0
CUBOID    0 1 10.0  -10.0 10.0 -10.0 2.0 0.0
UNIT 2
CYLINDER  0 1  0.75   2.0  0.0
CYLINDER  2 1  5.0    2.0  0.0
CUBOID    0 1 10.0  -10.0 10.0 -10.0 2.0 0.0
UNIT 3
CYLINDER  0 1  0.75   2.5  0.0
CUBOID    3 1 10.0  -10.0 10.0 -10.0 2.5 0.0
END GEOM
READ ARRAY  NUX=1 NUY=1 NUZ=7 FILL 1 2 2Q2 3 END FILL END ARRAY

KENO-VI:

READ GEOM
GLOBAL UNIT 1
CYLINDER  1  0.75     7.0  -5.0
CYLINDER  2  5.0      1.0  -1.0
CYLINDER  3 5.0      3.0  -3.0
CYLINDER  4 5.0      5.0  -5.0
CYLINDER  5 5.0      7.0  -5.0
CYLINDER  6 8.0    13.0  -5.0
CYLINDER  7 8.25  13.0  -5.25
CUBOID  8  10.0  -10.0  10.0 -10.0 15.5  13.0
CUBOID   9 10.0  -10.0  10.0 -10.0 15.5 -5.25
MEDIA 0 1  1
MEDIA 1 1 -1  2
MEDIA 2 1 -1 -2  3
MEDIA 1 1 -1 -3  4
MEDIA 2 1 -1 -4  5
MEDIA 0 1 -5  6
MEDIA 6 1 -6  7
MEDIA 3 1  8
MEDIA 0 1 -7 -8  9
BOUNDARY   9
END GEOM

In data description 1, Example 3 above, KENO V.a input, UNIT 1 describes a disk of material 1 with a HOLE through its centerline. The first CYLINDER defines the HOLE, the second defines the rest of the disk, and the CUBOID defines the size of the UNIT to be consistent with the square plate so they can be stacked together in an ARRAY. UNIT 2 describes a disk of material 2 in similar fashion. UNIT 3 describes the square plate of material 3 with a HOLE through its center. The CYLINDER defines the HOLE and the CUBOID defines the square plate. These three UNITs are stacked in the Z direction to achieve the composite system. This is represented by FILL 1 2 2Q2 3. The 2Q2 repeats the two entries preceding the 2Q2 twice. Alternatively, this can be achieved by entering FILL 1 2 1 2 1 2 3 END FILL. The same ARRAY can also be achieved using the LOOP option. An example of the data for this option is:

LOOP 1 6R1 1 5 2 2 6R1 2 6 2 3 6R1 7 7 1 END LOOP.

UNIT 1 is placed at the X = 1, Y = 1, and Z = 1,3,5 positions of the ARRAY by entering 1 6R1 1 5 2. UNIT 2 is positioned at the X = 1, Y = 1 and Z = 2,4,6 positions in the ARRAY by entering 2 6R1 2 6 2. UNIT 3 is placed at the X = 1, Y = 1, Z = 7 position of the ARRAY by entering 3 6R1 7 7 1. See Sect. 8.1.2.5 for additional information regarding ARRAY specifications.

For the KENO-VI input, UNIT 1 contains the entire problem description. The first CYLINDER describes the 1.5 cm diameter hole through the stack. The next four CYLINDERs define the stack. The sixth and seventh CYLINDERs describe the void and container. The two CUBOIDs describe the top plate and surrounding global region of void. The MEDIA cards are used to place the materials in the appropriate regions.

EXAMPLE 4. Assume two large cylinders that are 2.5 cm in radius and 5 cm long are connected by a smaller cylinder that is 0.5 cm in radius and 10 cm long, as shown in Fig. 63. All three cylinders are composed of material 1. By starting the geometry description in the small cylinder, this system can be described as a single unit.

_images/fig131.png

Fig. 63 Two large cylinders joined axially by a small cylinder.

Data description 1, Example 4.

KENO-VI:

READ GEOM
GLOBAL UNIT 1
XCYLINDER  1  0.5  5.0  -5.0
XCYLINDER  2  2.5  5.0  -5.0
XCYLINDER  3  2.5 10.0 -10.0
MEDIA 1 1 1
MEDIA 0 1 2 -1
MEDIA 1 1 3 -2
BOUNDARY  3
END GEOM

KENO V.a:

READ GEOM
CYLINDER  1 1 0.5  5.0  -5.0
CYLINDER  0 1 2.5  5.0  -5.0
CYLINDER  1 1 2.5 10.0 -10.0
END GEOM

The origin is at the center of the small cylinder, which is described by the first cylinder description. The second cylinder description defines a void cylinder surrounding the small cylinder. Its radius is the same as the large cylinders, and its height (length) coincides with that of the small cylinder. The last cylinder description defines the large cylinders on either end of the small cylinder. In KENO V.a, because this problem does not specify otherwise, the length of the CYLINDERs is assumed to coincide with the Z axis. In KENO-VI, because the problem was created using XCYLINDERs, the long axes of the CYLINDERs coincide with the X axis.

EXAMPLE 5. Assume two large cylinders with a center-to-center spacing of 15 cm, each having a radius of 2.5 cm and length of 5 cm, are connected radially by a small cylinder having a radius of 1.5 cm, as shown in Fig. 64.

_images/fig141.png

Fig. 64 Two large cylinders radially connected by a small cylinder.

This system cannot be described rigorously in KENO V.a geometry because the intersection of the cylinders cannot be described. However, it can be approximated two ways, as shown in Fig. 65. The top approximation is described in data description 1, Example 5. The bottom approximation is described in data description 2, Example 5, and data description 3, Example 5. These may be poor approximations for criticality safety calculations.

_images/fig151.png

Fig. 65 KENO V.a approximations of cylindrical intersections.

Data description 1, Example 5.

KENO V.a:

READ GEOM
UNIT 1
CYLINDER  1 1 2.5 2.5 -2.5
CUBE      0 1 2.5 -2.5
UNIT 2
XCYLINDER 1 1 1.5 5.0 -5.0
CUBOID    0 1 5.0 -5.0 2.5 -2.5 2.5 -2.5
END GEOM
READ ARRAY NUX=3 NUY=1 NUZ=1 FILL 1 2 1 END ARRAY

UNIT 1 defines a large CYLINDER, and UNIT 2 describes the small CYLINDER. In both UNITs the origin is at the center of the CYLINDER. The large CYLINDERs have their centerlines along the Z axis and the small CYLINDER has its length along the X axis.

Data description 2, Example 5.

KENO V.a:

READ GEOMETRY
UNIT 1
CYLINDER 1 1 2.5 1.0 0.0
CUBOID 0 1 4P2.5 1.0 0.0
UNIT 2
ZHEMICYL-X 1 1 2.5 2P1.5 CHORD 2.0
CUBOID 0 1 2.0 3P-2.5 2P1.5
UNIT 3
ZHEMICYL+X 1 1 2.5 2P1.5 CHORD 2.0
CUBOID 0 1 2.5 -2.0 2P2.5 2P1.5
UNIT 4
XCYLINDER 1 1 1.5 2P5.5
CUBOID 0 1 2P5.5 2P2.5 2P1.5
UNIT 5
CUBOID 0 1 2P5.0 2P2.5 1.0 0.0
UNIT 6
ARRAY 1 3*0.0
UNIT 7
ARRAY 2 3*0.0
END GEOMETRY
READ ARRAY ARA=1 NUX=3 NUY=1 NUZ=1 FILL 1 5 1 END FILL
ARA=2 NUX=3 NUY=1 NUZ=1 FILL 2 4 3 END FILL
ARA=3 NUX=1 NUY=1 NUZ=3 FILL 6 7 6 END FILL
END ARRAY

The above geometry description uses ARRAYs of ARRAYs (see Multiple arrays) to describe the bottom approximation of Fig. 65. UNIT 1 defines a large CYLINDER that has a radius of 2.5 cm and a height of 10 cm inside a close-fitting CUBOID. This is used in both large CYLINDERs as the portion of the large CYLINDER that exists above and below the region where the small CYLINDER joins it. UNIT 5 is the spacing between the tops of the two large CYLINDERs and the spacing between the bottoms of the two large CYLINDERs. ARRAY 1 thus defines the bottom of the system: two short CYLINDERs (UNIT 1s) separated by 10 cm (UNIT 5 is the separation). UNIT 6 contains ARRAY 1.

UNIT 2 is the left hemicylinder that adjoins the horizontal CYLINDER, and UNIT 3 is the right hemicylinder that adjoins the horizontal CYLINDER. UNIT 4 defines the horizontal CYLINDER. ARRAY 2 contains UNITs 2, 4, and 3 from left to right. This defines the central portion of the system where the horizontal CYLINDER adjoins the two hemicylinders. These hemicylinders are larger than half CYLINDERs. UNIT 7 contains ARRAY 2. The entire system is achieved by stacking a UNIT 6 above and below the UNIT 7 as defined in ARRAY 3, the GLOBAL ARRAY.

Data description 3, Example 5.

KENO V.a:

READ GEOMETRY
UNIT 1
CYLINDER 1 1 2.5 1.0 0.0
UNIT 2
CYLINDER 1 1 2.5 1.0 0.0
CUBOID 0 1 17.5 -2.5 2P2.5 1.0 0.0
HOLE 1 15.0 0.0 0.0
UNIT 3
ZHEMICYL-X 1 1 2.5 2P1.5 CHORD 2.0
UNIT 4
ZHEMICYL+X 1 1 2.5 2P1.5 CHORD 2.0
UNIT 5
XCYLINDER  1 1 1.5 2P5.5
CUBOID 0 1  2P10.0 2P2.5 2P1.5
HOLE 3 -7.5 2*0.0
HOLE 4 7.5 2*0.0
END GEOMETRY
READ ARRAY
ARA=1 NUX=1 NUY=1 NUZ=3 FILL 2 5 2 END FILL
END ARRAY

The above geometry description uses HOLEs (see Use of holes in the geometry) to describe the bottom approximation of Fig. 65. UNIT 1 defines a large CYLINDER with a radius of 2.5 cm and a height of 1.0 cm. UNIT 2 defines the same CYLINDER within a CUBOID that extends from X = −2.5 to X = 17.5, from Y = −2.5 to Y = 2.5, and Z = 0.0 to Z = 1.0. The origin of the CYLINDER is at (0.0,0.0,0.0). Thus UNIT 2 describes the top and bottom of the CYLINDER on the left. UNIT 1 is positioned within this CUBOID as a HOLE with its origin at (15.0,0.0,0.0) to describe the top and bottom of the CYLINDER on the right. UNIT 3 is the left hemicylinder that adjoins the horizontal CYLINDER, and UNIT 4 is the right hemicylinder that adjoins the horizontal CYLINDER. UNIT 5 defines the horizontal CYLINDER with its origin at the center within a CUBOID that extends from X = −10.0 to X = +10.0, Y = −2.5 to Y = 2.5, and Z = −1.5 to Z = 1.5. UNIT 3 is positioned to the left of the horizontal CYLINDER, and UNIT 4 is positioned to the right of the horizontal CYLINDER by using HOLEs. The entire system is achieved by stacking a UNIT 2 above and below UNIT 5 as shown in the ARRAY data.

This same geometry description can be used with UNIT 2 redefined, having its origin defined so that it extends from X = −10 to X = 10, Y = −2.5 to Y = 2.5, and Z = 0.0 to Z = 1. In this instance, the geometry data would be identical except for UNIT 2. This alternative description of UNIT 2 is

KENO V.a:

UNIT 2
CYLINDER 1 1 2.5 1.0 0.0 ORIGIN -7.5 0.0
CUBOID 0 1 2P10.0 2P2.5 1.0 0.0
HOLE 1 7.5 0.0 0.0

This system can be easily described in KENO-VI geometry because intersections are allowed. The small CYLINDER is rotated in data description 1, Example 5.

Data description 1, Example 5.

KENO-VI:

READ GEOM
GLOBAL UNIT 1
CYLINDER   1  2.5  2.5 -2.5
CYLINDER   2  2.5  2.5 -2.5 ORIGIN Y=15.0
YCYLINDER  3  1.5 15.0  0.0
CUBOID     4  5.0 -5.0 17.5 -2.5 2.5 -2.5
MEDIA 1 1 1
MEDIA 1 1 2
MEDIA 1 1 3 -1 -2
MEDIA 0 1 4 -3 -2 -1
BOUNDARY  4
END GEOM

The first and second CYLINDERs define the two large CYLINDERs, and the third CYLINDER describes the small connecting CYLINDER. The two large CYLINDERs are oriented along the Z axis. The second large cylinder is translated so its origin is at position (0.0, 15.0, 0.0). The small CYLINDER is oriented along the Y axis. Region 1 consists of the material in the first large CYLINDER. Region 2 consists of the material in the second large CYLINDER. Region 3 consists of the material in the small CYLINDER but not in either of the large CYLINDERs. Region 4 is the BOUNDARY region.

EXAMPLE 6. Assume 2 small cylinders 1.0 cm in radius and 10 cm long are connected by a large cylinder 2.5 cm in radius and 5 cm long as shown in Fig. 66.

_images/fig16.png

Fig. 66 Two small cylinders joined axially by a large cylinder.

This problem is very similar to Example 4. It can be described as a single UNIT in KENO-VI, but not in KENO V.a where it must be described as an array. In KENO V.a, UNIT 1 defines the large cylinder, and UNIT 2 defines the small cylinder. The origin of each UNIT is at its center. The composite system consists of two UNIT 2s and one UNIT 1 as shown below. In KENO-VI, CYLINDER 1 defines the long thin cylinder, and CYLINDER 2 defines the short thick cylinder. The origin of each cylinder is at its center. In both KENO V.a and KENO-VI, the centerline of the cylinders lies along the Z axis.

Data description 1, Example 6.

KENO V.a:

READ GEOM
UNIT 1
CYLINDER  1 1 2.5 2.5 -2.5
CUBE      0 1 2.5 -2.5
UNIT 2
CYLINDER  1 1 1.0 5.0 -5.0
CUBOID    0 1 2.5 -2.5 2.5 -2.5 5.0 -5.0
END GEOM
READ ARRAY NUX=1 NUY=1 NUZ=3 FILL 2 1 2 END ARRAY

KENO-VI:

READ GEOM
GLOBAL UNIT 1
CYLINDER  1  1.0 12.5  -12.5
CYLINDER  2  2.5  2.5   -2.5
CUBOID    3  4P2.5 12.5 -12.5
MEDIA 1 1 1
MEDIA 1 1 2 -1
MEDIA 0 1 3 -2 -1
BOUNDARY  3
END GEOM

EXAMPLE 7. Assume an 11 × 5 × 3 square-pitched array of spheres of material 1, radius 3.75 cm, with a center-to-center spacing of 10 cm in the X, Y, and Z directions. The data for this system are given below.

Data description 1, Example 7.

KENO V.a:

READ GEOM
SPHERE 1 1 3.75
CUBE   0 1 5.0 -5.0
END GEOM
READ ARRAY NUX=11  NUY=5 NUZ=3 END ARRAY

KENO-VI:

READ GEOM
UNIT 1
SPHERE  1  3.75
CUBOID  2  6P5.0
MEDIA 1 1 1
MEDIA 0 1 2 -1
BOUNDARY  2
GLOBAL UNIT 2
CUBOID 10 55.0 -55.0 25.0 -25.0 15.0 -15.0
ARRAY 1 10 PLACE 6 3 2 3*0.0
BOUNDARY  10
END GEOM
READ ARRAY NUX=11  NUY=5 NUZ=3 FILL F1 END FILL END ARRAY

EXAMPLE 8. Assume an 11 × 5 × 3 rectangular-pitched array of spheres of material 1, whose radius is 3.75 cm and whose center-to-center spacing is 10 cm in the X direction, 15 cm in the Y direction, and 20 cm in the Z direction. The input for this geometry is given below.

Data description 1, Example 8.

KENO V.a:

READ GEOM
SPHERE  1 1 3.75
CUBOID  0 1 5.0  -5.0 7.5 -7.5 10.0 -10.0
END GEOM
READ ARRAY  NUX=11 NUY=5 NUZ=3 END ARRAY

KENO-VI:

READ GEOM
UNIT 1
SPHERE  1 3.75
CUBOID  2  5.0  -5.0 7.5 -7.5 10.0 -10.0
MEDIA 1 1 1
MEDIA 0 1 2 -1
BOUNDARY  2
GLOBAL UNIT 2
CUBOID  1  55.0 -55.0 37.5 -37.5 30.0 -30.0
ARRAY 1 1 PLACE 6 3 2 3*0.0
BOUNDARY  1
END GEOM
READ ARRAY  NUX=11 NUY=5 NUZ=3 FILL F1 END FILL END ARRAY

EXAMPLE 9. Assume an 11 × 5 × 3 square-pitched array of spheres of material 1 with a 3.75 cm radius and 10 cm center-to-center spacing in the X, Y, and Z directions. This array is reflected by 30 cm of material 2 (water) on all faces, and weighted tracking (biasing) is to be used in the water reflector. The array spacing defines the perpendicular distance from the outer layer of spheres to the reflector to be 5 cm in the X, Y, and Z directions. The geometry input for this system is given below.

Data description 1, Example 9.

KENO V.a:

READ GEOM
UNIT 1
SPHERE     1  1   3.75
CUBE         0  1   5.0   -5.0
GLOBAL UNIT 2
ARRAY               1  -55.0  -25.0  -15.0
REFLECTOR    2  2  6*3.0  10
END GEOM
READ ARRAY  NUX=11 NUY=5 NUZ=3 END ARRAY
READ BIAS ID=500 2 11 END BIAS

KENO-VI:

READ GEOM
UNIT 1
SPHERE  1  3.75
CUBOID  2 6P5.0
MEDIA 1 1 1
MEDIA 0 1 2 -1
BOUNDARY  2
GLOBAL UNIT 2
CUBOID  1  55.0  -55.0  25.0  -25.0  15.0  -15.0
CUBOID  2  58.0  -58.0  28.0  -28.0  18.0  -18.0
CUBOID  3  61.0  -61.0  31.0  -31.0  21.0  -21.0
CUBOID  4  64.0  -64.0  34.0  -34.0  24.0  -24.0
CUBOID  5  67.0  -67.0  37.0  -37.0  27.0  -27.0
CUBOID  6  70.0  -70.0  40.0  -40.0  30.0  -30.0
CUBOID  7  73.0  -73.0  43.0  -43.0  33.0  -33.0
CUBOID  8  76.0  -76.0  46.0  -46.0  36.0  -36.0
CUBOID  9  79.0  -79.0  49.0  -49.0  39.0  -39.0
CUBOID 10  82.0  -82.0  52.0  -52.0  42.0  -42.0
CUBOID 11  85.0  -85.0  55.0  -55.0  45.0  -45.0
ARRAY 1 1 PLACE 6 3 2 3*0.0
MEDIA 2 2 2 -1
MEDIA 2 3 3 -2
MEDIA 2 4 4 -3
MEDIA 2 5 5 -4
MEDIA 2 6 6 -5
MEDIA 2 7 7 -6
MEDIA 2 8 8 -7
MEDIA 2 9 9 -8
MEDIA 2 10 10 -9
MEDIA 2 11 11 -10
BOUNDARY  11
END GEOM
READ ARRAY  NUX=11 NUY=5 NUZ=3 FILL F1 END FILL  END ARRAY
READ BIAS ID=500 2 11 END BIAS

In the KENO V.a input, the ARRAY boundary defines the origin of the REFLECTOR to be at the center of the ARRAY. The 6*3.0 in the REFLECTOR description repeats the 3.0 six times. The REFLECTOR record is used to generate ten REFLECTOR regions, each of which is 3.0 cm thick, on all six faces of the ARRAY.

In the KENO-VI input, the basic UNIT used to construct the ARRAY is defined in UNIT 1. The ARRAY is positioned in UNIT 2 (the GLOBAL UNIT) using the ARRAY card and the PLACE option. The ARRAY is then surrounded by ten REFLECTOR regions, each 3.0 cm thick, on all sides.

The first bias ID is 2, so the last bias ID will be 11 if 10 regions are created. The biasing data block is necessary to apply the desired weighting or biasing function to the reflector. The biasing material ID is obtained from Table 155. IDs, group structure and incremental thickness for weighting data available on the KENO  weighting library. If the biasing data block is omitted from the problem description, the 10 reflector regions will not have a biasing function applied to them, and the default value of the average weight will be used. This may cause the problem to execute more slowly, and therefore require the use of more computer time.

EXAMPLE 10. Assume the reflector in Example 9 is present only on both X faces, both Y faces, and the negative Z face. The reflector is only 15.24 cm thick on these faces. The top of the array (positive Z face) is unreflected.

Data description 1, Example 10.

KENO V.a:

READ GEOM
UNIT 1
SPHERE    1  1   3.75
CUBE        0  1   5.0   -5.0
GLOBAL UNIT 2
ARRAY             1  -55.0   -25.0  -15.0
REFLECTOR  2  2  4*3.0    0.0   3.0  5
REFLECTOR  2 7  4*0.24   0.0  0.24  1
READ ARRAY NUX=11 NUY=5 NUZ=3 END ARRAY
READ BIAS ID=500 2 7 END BIAS

KENO-VI:

READ GEOM
UNIT 1
SPHERE  1 3.75
CUBOID  2 6P5.0
MEDIA 1 1 1
MEDIA 0 1 2 -1
BOUNDARY  2
GLOBAL UNIT 2
CUBOID  1  55.0  -55.0  25.0  -25.0  15.0  -15.0
CUBOID  2  58.0  -58.0  28.0  -28.0  15.0  -18.0
CUBOID  3  61.0  -61.0  31.0  -31.0  15.0  -21.0
CUBOID  4  64.0  -64.0  34.0  -34.0  15.0  -24.0
CUBOID  5  67.0  -67.0  37.0  -37.0  15.0  -27.0
CUBOID  6  70.24 -70.24 40.24 -40.24 15.0  -30.24
ARRAY 1 1 PLACE 6 3 2 3*0.0
MEDIA 2 2 2 -1
MEDIA 2 3 3 -2
MEDIA 2 4 4 -3
MEDIA 2 5 5 -4
MEDIA 2 6 6 -5
BOUNDARY  6
END GEOM
READ ARRAY  NUX=11 NUY=5 NUZ=3 FILL F1 END FILL  END ARRAY
READ BIAS ID=500 2 6 END BIAS

In the KENO V.a input, the first REFLECTOR description generates five regions around the ARRAY, each region being 3.0 cm thick in the +X, −X, +Y, −Y, and −Z directions, and of zero thickness in the +Z direction. This defines a total thickness of 15 cm of reflector material on the appropriate faces. The second REFLECTOR description generates the last 0.24 cm of material 2 on those faces. Thus, the total reflector thickness is 15.24 cm on each face of the array, except the top which has no reflector. Five reflector regions were generated by the first REFLECTOR description, and one was generated by the second REFLECTOR description; so, six biasing regions must be defined in the biasing data. Thus, the beginning bias ID is 2, and the ending bias ID is 7.

In the KENO-VI input, the first CUBOID in Unit 2 represents the boundary for the ARRAY. The next four CUBOIDs represent the first four regions around the ARRAY, each region being 3.0 cm thick in the +X, X, +Y, −Y, and −Z directions, and of zero thickness in the +Z direction. A total thickness of 12 cm of reflector material is on the appropriate faces. The last CUBOID represents the last 3.24 cm of material 2 on those faces. Thus, the total reflector thickness is 15.24 cm on each face of the array, except the top which has no reflector. The beginning bias ID is 2, and the ending bias ID is 6. The last region could either be larger or smaller than the recommended thickness to complete the reflector.

The biasing material ID and thickness per region are obtained from Table 155. The thickness per region should be very nearly the thickness per region from the table to avoid over biasing in the reflector. Partial increments at the outer region of a reflector are exempt from this recommendation. If a biasing function is not to be applied to a region generated by the REFLECTOR record, the thickness per region can be any desired thickness and the biasing data block is omitted.

EXAMPLE 11. Assume the array of Example 7 has the central unit of the array replaced by a cylinder of material 4, 5 cm in radius and 10 cm tall. Assume a 20 cm thick spherical reflector of material 3 (concrete) is positioned so its inner radius is 65 cm from the center of the array. The minimum inner radius of a spherical reflector for this array is 62.25 cm (\(\sqrt{55^{2} + 25^{2} + 15^{2} }\)). If the inner radius is smaller than this, the problem cannot be described using KENO V.a geometry.

Data description 1, Example 11.

KENO V.a:

READ GEOM
UNIT 1
SPHERE  1  1  3.75
CUBE  0  1  5.0  -5.0
UNIT 2
CYLINDER  4  1  5.0  5.0  -5.0
CUBE  0  1  5.0  -5.0
GLOBAL UNIT 2
ARRAY     1 -55.0  -25.0  -15.0
SPHERE  0  1  65.0
REPLICATE  3  2  5.0  4
END GEOM
READ ARRAY
   NUX=11 NUY=5 NUZ=3
   LOOP 1  1  11  1  1 5 1  1 3 1  2  6 6 1  3 3 1  2 2 1  END LOOP
END ARRAY
READ BIAS ID=301 2 5 END BIAS

KENO-VI:

READ GEOM
UNIT 1
SPHERE  1 3.75
CUBOID   2 6P5.0
MEDIA 1 1 1
MEDIA 0 1 2 -1
BOUNDARY  2
UNIT 2
CYLINDER 1 5.0 5.0 -5.0
CUBOID  2  6P5.0
MEDIA 4 1 1
MEDIA 0 1 2 -1
BOUNDARY  2
GLOBAL UNIT 3
CUBOID 1  55.0 -55.0 25.0 -25.0 15.0 -15.0
SPHERE 2 65.0
SPHERE 3 70.0
SPHERE 4 75.0
SPHERE 5 80.0
SPHERE 6 85.0
ARRAY 1 1 PLACE 6 3 2 3*0.0
MEDIA 0 1 2 -1
MEDIA 3 2 3 -2
MEDIA 3 3 4 -3
MEDIA 3 4 5 -4
MEDIA 3 5 6 -5
BOUNDARY  6
END GEOM
READ ARRAY
NUX=11 NUY=5 NUZ=3
LOOP 1  1  11  1  1 5 1  1 3 1 2  6 6 1  3 3 1  2 2 1 END LOOP
END ARRAY
READ BIAS ID=301 2 5 END BIAS

UNIT 1 describes the SPHERE and spacing used in the ARRAY. UNIT 2 defines the CYLINDER located at the center of the ARRAY. In KENO V.a, the ARRAY record defines the origin of the reflector to be at the center of the ARRAY, while in KENO-VI it defines the origin of the ARRAY to be at the center of the GLOBAL UNIT. The first SPHERE in the GLOBAL UNIT defines the inner radius of the reflector. The next four SPHERE and four MEDIA records of the KENO-VI input and the REPLICATE record of the KENO V.a input will generate four spherical regions of material 3, each 5.0 cm thick. The data for the BIAS block is generated in a similar manner to previous examples, except that concrete (ID=301) is used. The recommended reflector thickness is 5 cm; this thickness is incorporated explicitly in the KENO-VI model and with 4 repetitions of the 5 cm thick reflector via REPLICATE in the KENO V.a model. The first 10 entries following the word LOOP fills the 11 × 5 × 3 ARRAY with UNITs 1. The next 10 entries position UNIT 2 at the center of the ARRAY (X = 6, Y = 3, and Z = 2), replacing the UNIT 1 that had been placed there by the first 10 entries.

EXAMPLE 12. Assume a data profile such as fission densities is desired in a cylinder at 0.5 cm intervals in the radial direction and 1.5 cm intervals axially. The cylinder is composed of material 1 and has a radius of 5 cm and a height of 15 cm. The REPLICATE or REFLECTOR description can be used to generate these regions in KENO V.a. A biasing data block is not entered because default biasing is desired throughout the cylinder.

Data description 1, Example 12.

KENO V.a:

READ GEOM
CYLINDER  1 1 0.5 1.5 0
REFLECTOR 1 1 0.5 1.5 0 9
END GEOM

KENO-VI:

READ GEOM
GLOBAL UNIT 1
CYLINDER 1 0.5 1.5 0
CYLINDER 2 1.0 3.0 0
CYLINDER 3 1.5 4.5 0
CYLINDER 4 2.0 6.0 0
CYLINDER 5 2.5 7.5 0
CYLINDER 6 3.0 9.0 0
CYLINDER 7 3.5 10.5 0
CYLINDER 8 4.0 12.0 0
CYLINDER 9 4.5 13.5 0
CYLINDER 10 5.0 15.0 0
MEDIA 1 1 1
MEDIA 1 1 2 -1
MEDIA 1 1 3 -2
MEDIA 1 1 4 -3
MEDIA 1 1 5 -4
MEDIA 1 1 6 -5
MEDIA 1 1 7 -6
MEDIA 1 1 8 -7
MEDIA 1 1 9 -8
MEDIA 1 1 10 -9
BOUNDARY  10
END GEOM

EXAMPLE 13. (KENO-VI due to pipe junctions) Assume a cross composed of two Plexiglas cylinders (material 3) having an inner diameter of 13.335 cm and an outer diameter of 16.19 cm. The bottom and side legs of the cross are closed by a 3.17 cm thick piece of Plexiglas. From the center of the intersection, the bottom and side legs are 91.44 cm long and the top leg is 121.92 cm long. The cross is filled with a UO2F2 solution (material 1) to a height of 28.93 cm above the center of the cylinder intersection. The cross is then surrounded by a water reflector (material 2) that extends from the center of the intersection: 111.74 cm in the ±X directions, 20.64 cm in the ±Y directions, 29.03 cm in the +Z direction, and –118.428 cm in the –Z direction. A schematic of the assembly is shown in Fig. 67.

_images/fig17.png

Fig. 67 Plexiglas UO2F2-filled cross.

Data description of Example 13 (KENO-VI only).

READ GEOMETRY
GLOBAL UNIT 1
CYLINDER   10 13.335  28.93 -88.27
CYLINDER   20 13.335 121.92 -88.27
CYLINDER   30 16.19  121.92 -91.44
YCYLINDER  40 13.335  88.27 -88.27
YCYLINDER  50 16.19 91.44 -91.44
CUBOID     60 2P111.74 2P20.64  29.03 -118.428
CUBOID     70 2P111.74 2P20.64 121.92 -118.428
MEDIA 1 1 10
MEDIA 0 1 20 -10
MEDIA 3 1 30 -20 -40
MEDIA 1 1 40 -10
MEDIA 3 1 50 -40 -30
MEDIA 2 1 60 -30 -50
MEDIA 0 1 70 -30 -60
BOUNDARY  70
END GEOMETRY

EXAMPLE 14. (KENO-VI only because of rotation) Assume a Y-shaped aluminum cylinder (material 2) with a 13.95 cm inner radius and a 0.16 cm wall thickness is filled with a UO2F2 solution (material 1). From the center where the Y intersects the cylinder, the bottom leg is 76.7 cm long, the top leg is 135.4 cm long, and the Y leg is 126.04 cm long, canted at a 29.26-degree angle. The bottom of the bottom leg and the top of the Y leg are sealed with 1.3 cm caps. The Y cylinder is filled to a height of 52.8 cm above the center where the Y leg intersects the vertical cylinder. The cylinder is surrounded by a water reflector (material 3) that extends out 37.0 cm in the ±X direction, 100.0 and  −37.0 cm in the ±Y direction, and 135.4 and –99.6 in the ±Z direction. A schematic of the assembly is shown in Fig. 68.

_images/fig18.png

Fig. 68 Y-shaped UO2F2-filled aluminum cylinder.

Data description of Example 14 (KENO-VI only).

READ GEOMETRY
GLOBAL UNIT 1
COM='30 DEG Y CYLINDER'
CYLINDER 10 13.95 135.4 -75.4
CYLINDER 20 14.11 135.4 -76.7
CYLINDER 30 13.95 124.74  0.0  ROTATE A2=-29.26
CYLINDER 40 14.11 126.04  0.0 ROTATE A2=-29.26
CUBOID   50 2P37.0 100.0 -37.0 52.8 -75.4
CUBOID   60 2P37.0 100.0 -37.0 135.4 -99.6
MEDIA 1 1 10 50
MEDIA 2 1 20 -10 -30
MEDIA 1 1 30  50 -10
MEDIA 2 1 40 -30 -20
MEDIA 0 1 10 -50
MEDIA 0 1 30 -50 -10
MEDIA 3 1 60 -20 -40
BOUNDARY  60
END GEOMETRY

Use of holes in the geometry

Geometry tells how each KENO V.a geometry region in a UNIT must completely enclose all previously described regions in that UNIT and how KENO-VI geometry allows regions in a UNIT to intersect, thus eliminating the need for HOLEs. HOLEs can be used to circumvent the complete enclosure restriction in KENO V.a to some degree. In KENO-VI, they can be useful in simplifying the input of a problem and decreasing the total CPU time needed for a problem. A HOLE is a means of placing an entire UNIT within a geometry region. A separate HOLE description is required for every location in a geometry region where a UNIT is to be placed. The information contained in a HOLE description is (1) the keyword HOLE, (2) the UNIT number of the UNIT to be placed, and (3) any modification data needed to correctly position and rotate (in KENO-VI) the specified UNIT within the containing UNIT. In KENO V.a, a HOLE is placed inside the geometry region that precedes it. This excludes HOLEs … (i.e., if a CUBE geometry region is followed by four HOLE descriptions, all four HOLEs are located within the CUBE). In KENO V.a, HOLEs are subject to the restriction that they cannot intersect any other geometry region. HOLEs can be nested to any depth (see Nesting holes). It is not advisable to use HOLEs tangent to other HOLEs or geometry, because round-off error may cause them to overlap. It is not uncommon for a problem that runs on one type of computer to fail on another type using the same data. Therefore, it is recommended that tangency and boundaries shared with HOLEs be avoided. This may be accomplished by separating the otherwise collocated or tangent surfaces by a very small (i.e., 10-6 cm) distance.

In KENO V.a, tracking in regions that contain holes is less efficient than tracking in regions that do not contain holes. Therefore, holes should be used only when the system cannot be easily described by conventional methods. One example of the use of holes is shown in Fig. 69, representing nine close-packed rods in an annulus.

In KENO-VI, tracking in regions that contain HOLEs can be more efficient than tracking in regions that do not contain HOLEs because every region boundary in a UNIT must be checked for a crossing whenever a crossing is possible. Putting small but complex geometries in a hole will lessen the number of boundaries that need to be checked for possible crossings. However, the indiscriminate use of holes is not advised since the particle must change coordinate systems every time a hole is entered or exited. Therefore, holes should be used carefully and only when the system can be simplified significantly by their use.

EXAMPLE 15. One example of a unit that requires holes in KENO V.a is better described not using holes in KENO-VI as shown in Fig. 69, representing nine close-packed rods in an annulus. The large rods are 1.4 cm in radius and composed of mixture 3. The small rods are 0.6 cm in radius and composed of mixture 1. The inside radius of the annulus is 3.6 cm, and the outside radius is 3.8 cm. The annulus is made of mixture 2. The rods and annulus are both 30 cm long. The annulus is centered in a cuboid having an 8 cm2 cross section and a length of 32 cm. The black and gray areas in Fig. 69 are void.

_images/fig19.png

Fig. 69 Close-packed cylinders in an annulus.

Data description of Example 15.

KENO V.a:

READ GEOM
UNIT 1
CYLINDER 1 1 0.6 2P15.0
UNIT 2
CYLINDER 3 1 1.4 2P15.0
GLOBAL UNIT 3
CYLINDER 1 1 0.6 2P15.0
CYLINDER 0 1 3.6 2P15.0
HOLE 2 0.0 -2.0 0.0
HOLE 1 2.0 -2.0 0.0
HOLE 2 2.0 0.0 0.0
HOLE 1 2.0 2.0 0.0
HOLE 2 0.0 2.0 0.0
HOLE 1 -2.0 2.0 0.0
HOLE 2 -2.0 0.0 0.0
HOLE 1 -2.0 -2.0 0.0
CYLINDER  2 1 3.8 2P15.0
CUBOID    0 1 4P4.0 2P16.0
END GEOM

KENO-VI:

READ GEOM
GLOBAL UNIT 1
CYLINDER 1 0.6 2P15.0
CYLINDER 2 0.6 2P15.0 ORIGIN X=2.0  Y=-2.0
CYLINDER 3 0.6 2P15.0 ORIGIN X=2.0  Y=2.0
CYLINDER 4 0.6 2P15.0 ORIGIN X=-2.0 Y=2.0
CYLINDER 5 0.6 2P15.0 ORIGIN X=-2.0 Y=-2.0
CYLINDER 6 1.4 2P15.0 ORIGIN X=2.0
CYLINDER 7 1.4 2P15.0 ORIGIN Y=2.0
CYLINDER 8 1.4 2P15.0 ORIGIN X=-2.0
CYLINDER 9 1.4 2P15.0 ORIGIN Y=-2.0
CYLINDER 10 3.6 2P15.0
CYLINDER 11 3.8 2P15.0
CUBOID  12  4P4.0 2P16.0
MEDIA 1 1 1
MEDIA 1 1 2
MEDIA 1 1 3
MEDIA 1 1 4
MEDIA 1 1 5
MEDIA 3 1 6
MEDIA 3 1 7
MEDIA 3 1 8
MEDIA 3 1 9
MEDIA 0 1 10 -1 -2 -3 -4 -5 -6 -7 -8 -9
MEDIA 2 1 11 -10
MEDIA 0 1 12 -11
BOUNDARY  12
END GEOM

The first HOLE description in the KENO V.a input represents the bottom large rod. It takes UNIT 2 and places its ORIGIN at (0.0,−2.0,0.0) relative to the ORIGIN of UNIT 3. The second HOLE description represents the small rod to the right of the large rod just discussed. It places the origin of UNIT 1 at (2.0,−2.0,0.0) in UNIT 3. The third HOLE description represents the large rod to the right. It places the origin of UNIT 2 at (2.0,0.0,0.0) in UNIT 3. This procedure is repeated in a counterclockwise direction until all eight rods have been placed within the region that defines the inner surface of the annulus. The CYLINDER that defines the outer surface of the annulus is described after all the HOLEs for the previous region have been placed. Then the outer CUBOID is described. This example illustrates that a UNIT that is to be placed using a HOLE description need not have a CUBE or CUBOID as its last region. Note that including the central rod directly in UNIT 3 reduces the CPU time required for transport compared to the case of all 9 rods being inserted as HOLEs. It is also important that the 9 HOLEs are inserted after the void cylinder into which they are inserted. Entering the HOLEs in any other position in the input would generated region intersection errors. The order of the HOLE records in any given region is not important, as they can be interchanged with each other randomly. However, they must always appear immediately after the region in which they are placed.

The KENO-VI input does not need to use HOLEs. The first CYLINDER description in this case represents the middle small rod. The next four CYLINDER records describe the four remaining small rods surrounding the middle rod. The ORIGIN attribute is used to shift the origin of each CYLINDER to the appropriate location. The following four CYLINDER records represent the four large rods. Again, the ORIGIN attribute is used to shift the ORIGIN of each CYLINDER to the appropriate location. Only the nonzero dimensions need to be entered in the ORIGIN data. The tenth CYLINDER record is the void in the annulus that contains the rods. The last CYLINDER record defines the outer surface of the annulus. Finally, the CUBOID record describes the surrounding UNIT boundary.

In KENO-VI, holes may not extend across any array outer boundary, may not intersect with other holes, and may not cross the host UNIT outer boundary. Thus a hole may be placed so that it crosses several regions within an array. The hole description replaces the unit description within the hole domain. Since the holes are placed using the host UNIT coordinate system, the location of the hole record in the unit definition is not relevant.

An array of the arrangement shown in Fig. 69 can be easily described by altering the array description data. For example, a 5 × 3 × 2 array of these shapes with a center-to-center spacing of 8 cm in X and Y and 32 cm in Z can be achieved by using the following array data:

READ ARRAY  NUX=5 NUY=3 NUZ=2  FILL F3  END FILL  END ARRAY

or

READ ARRAY  NUX=5 NUY=3 NUZ=2  FILL 30*3  END FILL  END ARRAY

or

READ ARRAY  NUX=5 NUY=3 NUZ=2  LOOP  3 1 5 1 1 3 1 1 2 1  END LOOP END ARRAY

Nesting holes

This section illustrates how holes are nested. Holes can be nested to any level. Consider the configuration illustrated in Fig. 69 and replace the large rods with a complicated geometric arrangement. The resulting Fig. is shown in Fig. 70. Fig. 71 shows the complicated geometric arrangement that replaced the large rods of Fig. 69. Fig. 72 shows a component of the arrangement shown in Fig. 70.

_images/fig20.png

Fig. 70 Configuration using nested holes.

_images/fig211.png

Fig. 71 Complicated geometric arrangement by Unit 7.

_images/fig221.png

Fig. 72 Geometric component represented by Unit 4.

EXAMPLE 16. There is no predetermined preferred method to create a geometry mockup for a given physical system. The user should determine the most convenient order. To describe the configuration shown in Fig. 8.1.20 using nested HOLEs, it is likely most convenient to start the geometry mockup at the deepest nesting level, as shown in Fig. 8.1.22. The small CYLINDERs are composed of mixture 1, and they are each 0.1 cm in radius and 30 cm long. There are five small CYLINDERs used in Fig. 8.1.22. Their centers are located at (0,0,0) for the central one, at (0,−0.4,0) for the bottom one, at (0.4,0,0) for the right one, at (0,0.4,0) for the top one, and at (−0.4,0,0) for the left one. The rectangular parallelepipeds (CUBOIDs) are composed of mixture 2. Each one is 30 cm long and 0.1 cm by 0.2 cm in cross section. The large CYLINDER containing the configuration is composed of mixture 3, is 30 cm long and has a radius of 0.5 cm.

A possible geometry mockup for this system is described as follows in KENO V.a:

  1. define a small cylinder to be UNIT 1,

(2) define a small CUBOID with its length in the X direction to be UNIT 2,

(3) define a small CUBOID with its length in the Y direction to be UNIT 3,

(4) define UNIT 4 to be the large cylinder and place the CYLINDERs and CUBOIDs in it using HOLEs.

UNIT 1
CYLINDER 1 1 0.1 2P15.0
UNIT 2
CUBOID 2 1 2P0.1 2P0.05 2P15.0
UNIT 3
CUBOID 2 1 2P0.05 2P0.1 2P15.0
UNIT 4
CYLINDER  1 1 0.1 2P15.0
CYLINDER  3 1 0.5 2P15.0
HOLE 1 0.0 -0.4 0.0
HOLE 1 0.4 0.0 0.0
HOLE 1 0.0 0.4 0.0
HOLE 1 -0.4 0.0 0.0
HOLE 2 -0.2 0.0 0.0
HOLE 2 0.2 0.0 0.0
HOLE 3 0.0 -0.2 0.0
HOLE 3 0.0 0.2 0.0

The first cylinder description in UNIT 4 places the central rod,

the second cylinder description in UNIT 4 places the outer cylinder,

the first HOLE places the bottom CYLINDER,

the second HOLE places the CYLINDER at the right,

the third HOLE places the top CYLINDER,

the fourth HOLE places the CYLINDER at the left,

the fifth HOLE places the left CUBOID whose length is in X,

the sixth HOLE places the right CUBOID whose length is in X,

the seventh HOLE places the bottom CUBOID whose length is in Y, and

the eighth HOLE places the top CUBOID whose length is in Y.

A possible geometry mockup for this system is described as follows in KENO-VI:

  1. define UNIT 1 to contain the five small cylinders and four blocks,

  2. define UNIT 2 to contain the next two larger-sized cylinders and UNIT 1 as HOLEs, and

  3. define GLOBAL UNIT 3 to contain the large cylinders and UNIT 2 as HOLEs.

UNIT 1
CYLINDER  1 0.1 2P15.0
CYLINDER  2 0.1 2P15.0 ORIGIN Y=-0.4
CYLINDER  3 0.1 2P15.0 ORIGIN X=0.4
CYLINDER  4 0.1 2P15.0 ORIGIN Y=0.4
CYLINDER  5 0.1 2P15.0 ORIGIN X=-0.4
CUBOID    6 -0.3 -0.1  2P0.05 2P15.0
CUBOID    7 0.3 0.1  2P0.05 2P15.0
CUBOID    8 2P0.05 -0.3 -0.1  2P15.0
CUBOID    9 2P0.05 0.3 0.1  2P15.0
CYLINDER 10 0.5 2P15.0
MEDIA 1 1 1
MEDIA 1 1 2
MEDIA 1 1 3
MEDIA 1 1 4
MEDIA 1 1 5
MEDIA 2 1 6
MEDIA 2 1 7
MEDIA 2 1 8
MEDIA 2 1 9
MEDIA 3 1 10 -1 -2 -3 -4 -5 -6 -7 -8 -9
BOUNDARY  10

geometry record 1 places the central rod,

geometry record 2 places the bottom CYLINDER,

geometry record 3 places the CYLINDER at the right,

geometry record 4 places the top CYLINDER,

geometry record 5 places the CYLINDER at the left,

geometry record 6 places the left CUBOID whose length is in X,

geometry record 7 places the right CUBOID whose length is in X,

geometry record 8 places the bottom CUBOID whose length is in Y,

geometry record 9 places the top CUBOID whose length is in Y, and

geometry record 10 is the surrounding CYLINDER that defines the unit boundary.

In Fig. 71, the large plain cylinders are composed of mixture 1 and are 0.5 cm in radius and 30 cm long. The cylindrical component of UNIT 4 for KENO V.a or UNIT 1 for KENO-VI is the same size: an outer radius of 0.5 cm and a length of 30 cm. The small cylinders located in the interstices between the large cylinders are composed of mixture 2, are 0.2 cm in radius, and are 30 cm long. The annulus is composed of mixture 4, has a 1.3 cm inside radius and a 1.4 cm outer radius. The volume between the inner cylinders is void. The large cylinders each have a radius of 0.5 cm and are tangent. Therefore, their origins are offset from the origin of the UNIT by 0.707107. This is from X2 + Y2 = 1.0, where X and Y are equal.

For KENO V.a, define UNIT 5 to be the large plain cylinder, UNIT 6 to be the small cylinder, and UNIT 7 as the annulus that contains the cylinders. Its origin is at its center. The geometry mockup for this portion of the problem follows:

KENO V.a:

UNIT 5
CYLINDER  1 1 0.5 2P15.0
UNIT 6
CYLINDER  2 1 0.2 2P15.0
UNIT 7
CYLINDER  2 1 0.2 2P15.0
CYLINDER  0 1 1.3 2P15.0
HOLE      5 0.707107 0.0 0.0
HOLE      6 0.707107 0.707107 0.0
HOLE      4 0.0 0.707107 0.0
HOLE      6 -0.707107 0.707107 0.0
HOLE      5 -0.707107 0.0 0.0
HOLE      6 -0.707107 -0.707107 0.0
HOLE      4 0.0 -0.707107 0.0
HOLE      6 0.707107 -0.707107 0.0
CYLINDER  4 1 1.4 2P15.0

The first HOLE places the larger CYLINDER of mixture 1 at the right with its origin at (0.707107,0.0,0.0),

the second HOLE places the small CYLINDER in the upper right quadrant,

the third HOLE places the top CYLINDER that contains the geometric component defined in UNIT 4,

the fourth HOLE places the small CYLINDER in the upper left quadrant,

the fifth HOLE places the larger CYLINDER of mixture 1 at the left,

the sixth HOLE places the small CYLINDER in the lower lower left quadrant,

the seventh HOLE places the bottom CYLINDER that contains the geometric component defined in UNIT 4, and

the eighth HOLE places the small CYLINDER in the lower right quadrant.

The last CYLINDER defines the outer surface of the annulus.

For KENO-VI, UNIT 2 is the annulus that contains the cylinders.

KENO-VI:

UNIT 2
CYLINDER  1 0.2 2P15.0
CYLINDER  2 0.2 2P15.0 ORIGIN X=0.707107  Y=0.707107
CYLINDER  3 0.2 2P15.0 ORIGIN X=-0.707107 Y=0.707107
CYLINDER  4 0.2 2P15.0 ORIGIN X=-0.707107 Y=-0.707107
CYLINDER  5 0.2 2P15.0 ORIGIN X=0.707107  Y=-0.707107
CYLINDER  6 0.5 2P15.0 ORIGIN X=0.707107
CYLINDER  7 0.5 2P15.0 ORIGIN X=-0.707107
CYLINDER 10 1.3 2P15.0
CYLINDER 11 1.4 2P15.0
MEDIA 2 1 1
MEDIA 2 1 2
MEDIA 2 1 3
MEDIA 2 1 4
MEDIA 2 1 5
MEDIA 1 1 6
MEDIA 1 1 7
HOLE 1 ORIGIN Y=0.707107
HOLE 1 ORIGIN Y=-0.707107
MEDIA 0 1 10 -1 -2 -3 -4 -5 -6 -7
MEDIA 4 1 11 -10
BOUNDARY  11

In Unit 2 of the above KENO-VI geometry,

CYLINDER 1 places a small CYLINDER of mixture 2 at the origin,

CYLINDER 2 places the small CYLINDER of mixture 2 in the upper right quadrant,

CYLINDER 3 places the small CYLINDER of mixture 2 in the upper left quadrant,

CYLINDER 4 places the small CYLINDER of mixture 2 in the lower left quadrant,

CYLINDER 5 places the small CYLINDER of mixture 2 in the lower right quadrant,

CYLINDER 6 places the larger CYLINDER of mixture 1 at the right with its origin at (0.707107,0.0,0.0),

CYLINDER 7 places the larger CYLINDER of mixture 1 at the left with its origin at (0.0,0.707107.0.0),

CYLINDER 10 defines the inner surface of the annulus,

CYLINDER 11 defines the outer surface of the annulus and the UNIT boundary,

the first HOLE places the top CYLINDER that contains the geometric component defined in UNIT 1, and

the second HOLE places the bottom CYLINDER that contains the geometric component defined in UNIT 1.

To complete the geometry mockup, consider Fig. 70.

For KENO V.a geometry, define UNIT 8 to be a cylinder of mixture 2 having a radius of 0.6 cm and a length of 30 cm. Define UNIT 9 to be the central rod and the large annulus of 3.6 cm inner radius, 3.8 cm outer radius, and 30 cm length centered in a CUBOID having an 8 cm2 cross section and being 32 cm long.

KENO V.a:

UNIT 8
CYLINDER  2 1 0.6 2P15.0
UNIT 9
CYLINDER  2 1 0.6 2P15.0
CYLINDER  0 1 3.6 2P15
HOLE   7 2.0 0.0 0.0
HOLE   8 2*2.0 0.0
HOLE   7 0.0 2.0 0.0
HOLE   8 -2.0 2.0 0.0
HOLE   7 -2.0 2*0.0
HOLE   8 2*-2.0 0.0
HOLE   7 0.0 -2.0 0.0
HOLE   8 2P2.0 0.0
CYLINDER  4 1 3.8 2P15.0
CUBOID    0 1 4P4.0 2P16.0

In UNIT 9 of the KENO V.a description, the first CYLINDER defines the rod of mixture 2, centered in the annulus. The second CYLINDER defines the void volume between the central rod and the annulus.

The first HOLE places the composite annulus of UNIT 7 to the right of the central rod,

the second HOLE places a rod defined by UNIT 8 in the upper right quadrant of the annulus,

the third HOLE places the composite annulus of UNIT 7 above the central rod,

the fourth HOLE places a rod defined by UNIT 8 in the upper left quadrant of the annulus,

the fifth HOLE places the composite annulus of UNIT 7 to the left of the central rod,

the sixth HOLE places a rod defined by UNIT 8 in the lower left quadrant,

the seventh HOLE places the composite annulus of UNIT 7 below the central rod, and

the eighth HOLE places a rod defined by UNIT 8 in the lower right quadrant.

The last CYLINDER defines the outer surface of the annulus. The outer CUBOID is the last region.

For KENO-VI geometry, define UNIT 3 to be the central rod and four outer rods of 0.6 cm radius and 30.0 cm length, and the large annulus of 3.6 cm inner radius, 3.8 cm outer radius, and 30 cm length centered in a cuboid having an 8 cm2 cross section and a length of 32 cm.

KENO-VI:

GLOBAL UNIT 3
CYLINDER  1 0.6 2P15.0
CYLINDER  2 0.6 2P15.0 ORIGIN X=2.0 Y=2.0
CYLINDER  3 0.6 2P15.0 ORIGIN X=-2.0 Y=2.0
CYLINDER  4 0.6 2P15.0 ORIGIN X=-2.0 Y=-2.0
CYLINDER  5 0.6 2P15.0 ORIGIN X=2.0 Y=-2.0
CYLINDER 10 3.6 2P15.0
CYLINDER 11 3.8 2P15.0
CUBOID   12 4P4.0 2P16.0
MEDIA 2 1 1
MEDIA 2 1 2
MEDIA 2 1 3
MEDIA 2 1 4
MEDIA 2 1 5
HOLE  2 ORIGEN X=2
HOLE  2 ORIGEN Y=2
HOLE  2 ORIGEN X=-2
HOLE  2 ORIGEN Y=-2
MEDIA 0 1 10 -1 -2 -3 -4 -5
MEDIA 4 1 11 -10
MEDIA 0 1 12 -11
BOUNDARY  12

In UNIT 3 of the above KENO-VI description,

CYLINDER 1 defines the rod of mixture 2, centered in the annulus,

CYLINDER 2 places a rod of mixture 2 in the upper right quadrant of the annulus,

CYLINDER 3 places a rod of mixture 2 in the upper left quadrant of the annulus,

CYLINDER 4 places a rod of mixture 2 in the lower left quadrant,

CYLINDER 5 places a rod of mixture 2 in the lower right quadrant,

CYLINDER 10 defines the void volume between the central rod and the annulus,

CYLINDER 11 defines the outer surface of the annulus,

CUBOID 12 defines the unit boundary,

the first HOLE places UNIT 2 to the right of the central rod,

the second HOLE places UNIT  2 above the central rod,

the third HOLE places UNIT  2 to the left of the central rod, and

the fourth HOLE places UNIT  2 below the central rod.

This problem illustrates three levels of HOLE nesting. The total input data for the problem is given below. The geometry description accurately recreates the geometry arrangement of Fig. 70. The 2‑D color plot output is shown in Fig. 73.

_images/fig231.png

Fig. 73 Color plot of nested holes example problem.

KENO V.a:

NESTED HOLES SAMPLE

READ GEOM
UNIT 1
CYLINDER 1 1 0.1 2P15.0
UNIT 2
CUBOID 2 1 2P0.1 2P0.05 2P15.0
UNIT 3
CUBOID 2 1 2P0.05 2P0.1 2P15.0
UNIT 4
CYLINDER 1 1 0.1 2P15.0
CYLINDER 3 1 0.5 2P15.0
HOLE 1  0.0 -0.4 0.0
HOLE 1  0.4  0.0 0.0
HOLE 1  0.0  0.4 0.0
HOLE 1 -0.4  0.0 0.0
HOLE 2 -0.2  0.0 0.0
HOLE 2  0.2  0.0 0.0
HOLE 3  0.0 -0.2 0.0
HOLE 3  0.0  0.2 0.0
UNIT 5
CYLINDER 1 1 0.5 2P15.0
UNIT 6
CYLINDER 2 1 0.2 2P15.0
UNIT 7
CYLINDER 2 1 0.2 2P15.0
CYLINDER 0 1 1.3 2P15.0
HOLE 5 0.707107 2*0.0
HOLE 6 0.707107 0.707107 0.0
HOLE 4 0.0 0.707107 0.0
HOLE 6 -0.707107 0.707107 0.0
HOLE 5 -0.707107 0.0 0.0
HOLE 6 -0.707107 -0.707107 0.0
HOLE 4 0.0 -0.707107 0.0
HOLE 6 0.707107 -0.707107 0.0
CYLINDER 4 1 1.4 2P15.0
UNIT 8
CYLINDER 2 1 0.6 2P15.0
GLOBAL UNIT 9
CYLINDER 2 1 0.6 2P15.0
CYLINDER 0 1 3.6 2P15.0
HOLE 7 2.0 0.0 0.0
HOLE 8 2*2.0 0.0
HOLE 7 0.0 2.0 0.0
HOLE 8 -2.0 2.0 0.0
HOLE 7 -2.0 2*0.0
HOLE 8 2*-2.0 0.0
HOLE 7 0.0 -2.0 0.0
HOLE 8 2P2.0 0.0
CYLINDER 4 1 3.8 2P15.0
CUBOID 0 1 4P4.0 2P16.0
END GEOM
READ PLOT
  TTL='X-Y SLICE AT Z MIDPOINT. NESTED HOLES'
  XUL=-0.1  YUL=8.1 ZUL=16.0
  XLR=8.1 YLR=-0.1 ZLR=16
  UAX=1.0 VDN=-1.0  NAX=260 NCH=' *-.X' SCR=NO
END PLOT
END DATA
END

KENO-VI:

READ GEOM
UNIT 1
CYLINDER  1 0.1 2P15.0
CYLINDER  2 0.1 2P15.0 ORIGIN Y=-0.4
CYLINDER  3 0.1 2P15.0 ORIGIN X=0.4
CYLINDER  4 0.1 2P15.0 ORIGIN Y=0.4
CYLINDER  5 0.1 2P15.0 ORIGIN X=-0.4
CUBOID    6 -0.3 -0.1  2P0.05 2P15.0
CUBOID    7 0.3 0.1  2P0.05 2P15.0
CUBOID    8 2P0.05 -0.3 -0.1  2P15.0
CUBOID    9 2P0.05 0.3 0.1  2P15.0

CYLINDER 10 0.5 2P15.0
MEDIA 1 1 1 -6 -7 -8 -9
MEDIA 1 1 2 -8
MEDIA 1 1 3 -7
MEDIA 1 1 4 -9
MEDIA 1 1 5 -6
MEDIA 2 1 6 -1 -5
MEDIA 2 1 7 -1 -3
MEDIA 2 1 8 -1 -2
MEDIA 2 1 9 -1 -4
MEDIA 3 1 -1 -2 -3 -4 -5 -6 -7 -8 -9 10
BOUNDARY  10
UNIT 2
CYLINDER  1 0.2 2P15.0
CYLINDER  2 0.2 2P15.0 ORIGIN X=0.707107  Y=0.707107
CYLINDER  3 0.2 2P15.0 ORIGIN X=-0.707107 Y=0.707107
CYLINDER  4 0.2 2P15.0 ORIGIN X=-0.707107 Y=-0.707107
CYLINDER  5 0.2 2P15.0 ORIGIN X=0.707107  Y=-0.707107
CYLINDER  6 0.5 2P15.0 ORIGIN X=0.707107
CYLINDER  7 0.5 2P15.0 ORIGIN X=-0.707107
CYLINDER 10 1.3 2P15.0
CYLINDER 11 1.4 2P15.0
MEDIA 2 1 1
MEDIA 2 1 2
MEDIA 2 1 3
MEDIA 2 1 4
MEDIA 2 1 5
MEDIA 1 1 6
MEDIA 1 1 7
HOLE  1 ORIGIN Y=0.707107
HOLE  1 ORIGIN Y=-0.707107
MEDIA 0 1 10 -1 -2 -3 -4 -5 -6 -7
MEDIA 4 1 11 -10
BOUNDARY  11
GLOBAL UNIT 3
CYLINDER  1 0.6 2P15.0
CYLINDER  2 0.6 2P15.0 ORIGIN X=2.0  Y=2.0
CYLINDER  3 0.6 2P15.0 ORIGIN X=-2.0 Y=2.0
CYLINDER  4 0.6 2P15.0 ORIGIN X=-2.0 Y=-2.0
CYLINDER  5 0.6 2P15.0 ORIGIN X=2.0  Y=-2.0
CYLINDER 10 3.6 2P15.0
CYLINDER 11 3.8 2P15.0
CUBOID   12 4P4.0 2P16.0
MEDIA 2 1 1
MEDIA 2 1 2
MEDIA 2 1 3
MEDIA 2 1 4
MEDIA 2 1 5
HOLE 2 ORIGIN X=2
HOLE 2 ORIGIN Y=2
HOLE 2 ORIGIN X=-2
HOLE 2 ORIGIN Y=-2
MEDIA 0 1 10 -1 -2 -3 -4 -5
MEDIA 4 1 11 -10
MEDIA 0 1 12 -11
BOUNDARY  12
END GEOM
READ PLOT
  TTL='X-Y SLICE AT Z MIDPOINT. NESTED HOLES'
  XUL=-4.1  YUL=4.1 ZUL=0.0
  XLR=4.1 YLR=-4.1 ZLR=0
  UAX=1.0 VDN=-1.0  NAX=800
END PLOT
END DATA
END

Multiple arrays

EXAMPLE 17. Geometry demonstrates how UNITs are composed of geometry regions and how these UNITs can be stacked in an ARRAY. This same procedure can be extended to create multiple ARRAYs. Furthermore, ARRAYs can be used as building blocks within other ARRAYs.

Consider Sample Problem 19 from Appendix C. This problem is a critical experiment consisting of a composite array [Tho73][Tho64] of four highly enriched uranium metal cylinders and four cylindrical Plexiglas containers filled with uranyl nitrate solution. A photograph of the experiment is given in fig9-1c-3. The coordinate system is defined to be Z up the page, Y across the page, and X out of the page.

The Plexiglas containers have an inside radius of 9.525 cm and an outside radius of 10.16 cm. The inside height is 17.78 cm, and the outside height is 19.05 cm. Four of these containers are stacked with a center-to-center spacing of 21.75 cm in the Y direction and 20.48 cm in the Z direction (vertical). This arrangement of four Plexiglas containers can be described as follows: mixture 2 is the uranyl nitrate and mixture 3 is Plexiglas, so the Plexiglas container with its appropriate spacing CUBOID can be described as UNIT 1. This considers the ARRAY to be bare and suspended with no supports.

KENO V.a:

UNIT 1
CYLINDER  2 1 9.525  2P8.89
CYLINDER  3 1 10.16  2P9.525
CUBOID    0 1 4P10.875  2P10.24

KENO-VI:

UNIT 1
CYLINDER 1  9.525  2P8.89
CYLINDER 2  10.16  2P9.525
CUBOID  3   4P10.875  2P10.24
MEDIA 2 1 1
MEDIA 3 1 2 -1
MEDIA 0 1 3 -2
BOUNDARY  3

The ARRAY of four Plexiglas containers can be described as ARRAY 1 in the array data as follows:

ARA=1 NUX=1 NUY=2 NUZ=2 FILL F1 END FILL

The four metal cylinders, comprised of mixture 1, each have a radius of 5.748 cm and are 10.765 cm tall. They have a center-to-center spacing of 13.18 cm in the Y direction and 12.45 cm in the Z direction (vertical). Thus, one of the metal cylinders with its appropriate spacing CUBOID can be described as UNIT 2. This ARRAY is also considered to be bare and unsupported.

KENO V.a:

UNIT 2
CYLINDER 1 1  5.748 2P5.3825
CUBOID   0 1  4P6.59 2P6.225

KENO-VI:

UNIT 2
CYLINDER 1 5.748 2P5.3825
CUBOID   2   4P6.59 2P6.225
MEDIA 1 1 1
MEDIA 0 1 2 -1
BOUNDARY  2

The array of four metal cylinders can be described as ARRAY 2 in the array data.

ARA=2 NUX=1 NUY=2 NUZ=2 FILL F2 END FILL

Now two ARRAYs have been described. The overall dimensions of the ARRAY of Plexiglas containers are 21.75 cm in X, 43.5 cm in Y, and 40.96 cm in Z. The overall dimensions of the ARRAY of metal cylinders are 13.18 cm in X, 26.36 cm in Y, and 24.9 cm in Z.

In order to describe the composite ARRAY, these two ARRAYs must be positioned within UNITs and stacked together into one ARRAY. In order for them to be stacked into one ARRAY, the adjacent faces must match. This is accomplished by defining a UNIT 3 which contains ARRAY 1, the ARRAY of Plexiglas solution containers. The overall dimensions of this UNIT are 21.75 cm in X, 43.5 cm in Y, and 40.96 cm in Z. These dimensions are calculated by the code and need not be specified. UNIT 3 is defined as follows:

KENO V.a:

UNIT 3
ARRAY 1  3*0.0

KENO-VI:

UNIT 3
CUBOID  1 2P10.875 2P21.75 2P20.48
ARRAY 1 1 PLACE 1 1 1 0.0 -10.875 -10.24
BOUNDARY  1

The ARRAY of metal cylinders will be defined to be UNIT 4. However, this ARRAY is 17.14 cm smaller in the Y and 16.06 cm smaller in the Z dimensions than the ARRAY of Plexiglas UNITs. Therefore, a void region must be placed around the ARRAY in those directions so UNIT 4 and UNIT 3 will be the same size in Y and Z.

KENO V.a:

UNIT 4
ARRAY  2  3*0.0
REPLICATE  0 1 2*0.0  2*8.57  2*8.03  1

KENO-VI:

UNIT 4
CUBOID 1 2P6.59 2P13.18 2P12.45
CUBOID 2 2P6.59 2P21.75 2P20.48
ARRAY  2  1 PLACE 1 1 1 0.0 -6.59 -6.225
MEDIA 0 1 2 -1
BOUNDARY 2

Now that UNIT 3 and UNIT 4 have been defined, they must be placed in the global or universe ARRAY to define the physical arrangement of the eight pieces. This procedure is implemented via a GLOBAL ARRAY in KENO V.a, while KENO-VI uses a GLOBAL UNIT 3 as follows:

KENO V.a:

GBL=3  ARA=3  NUX=2  NUY=1  NUZ=1  FILL  4 3  END FILL

KENO-VI:

GLOBAL UNIT 5
CUBOID 1 34.93 0.0 43.5 0.0 40.96 0.0
ARRAY 3 1 PLACE 1 1 1 6.59 21.75 20.48
BOUNDARY  1

The description of ARRAY 3 in KENO-VI is identical to that shown for KENO V.a.

This completes the geometry description for the problem. The complete geometry input description for the problem is given below.

KENO V.a:

=KENOVA
SAMPLE PROBLEM 19 4 AQUEOUS 4 METAL ARRAY OF ARRAYS
READ PARAM  LIB=4  RUN=NO END PARAM
READ MIXT SCT=1
     MIX=1
  1092238     3.2275e-3
  1092235     4.4802e-2
     MIX=2
  20011023    5.81e-2
  2007014     1.9753e-3
  2008016     3.6927e-2
  20092235    9.8471e-4
  20092238    7.7697e-5
     MIX=3
  11006012    3.5552e-2
  11011023    5.6884e-2
  11008016    1.4221e-2
END MIXT
READ GEOM
UNIT 1
  CYLINDER  2 1 9.525 8.89 -8.89
  CYLINDER  3 1 10.16 2P9.525
  CUBOID  0 1 4P10.875 2P10.24
UNIT 2
  CYLINDER  1 1 5.748 2P5.3825
  CUBOID  0 1 4P6.59 2P6.225
UNIT 3
  ARRAY 1 3*0.0
UNIT 4
  ARRAY 2 3*0.0
  REPLICATE 0 1 2*0.0 2*8.57 2*8.03 1
END GEOM
READ ARRAY
  ARA=1 NUX=1 NUY=2 NUZ=2                FILL F1 END FILL
  ARA=2 NUX=1 NUY=2 NUZ=2                FILL F2 END FILL
  GBL=3 ARA=3 NUX=2 NUY=1 NUZ=1   FILL 4 3 END FILL
END ARRAY
  READ PLOT TTL='X-Y SLICE AT Z=10.24'
  XUL=-1.0    YUL=44.5 ZUL=10.24
  XLR=35.93 YLR=-1.0  ZLR=10.24
  UAX=1.0 VDN=-1.0 NAX=640 PIC=MIX END
  TTL='X-Z SLICE AT Y=10.875'
  XUL=-1.0 YUL=10.875 ZUL=41.96 XLR=35.93 YLR=10.875 ZLR=-1.0
  UAX=1.0 WDN=-1.0 PIC=MIX END  END PLOT
END DATA
END

KENO-VI:

=KENO-VI
SAMPLE PROBLEM 19 4 AQUEOUS 4 METAL ARRAY OF ARRAYS
READ PARAM
  LIB=4 FLX=YES FDN=YES NUB=YES SMU=YES MKP=YES MKU=YES FMP=YES FMU=YES
END PARAM
READ MIXT
  SCT=2
  MIX=1
      1092234 4.82717E-04
      1092235 4.47971E-02
      1092236 9.57233E-05
      1092238 2.65767E-03
  MIX=2
      2001001 5.77931E-02
      2007014 2.13092E-03
      2008016 3.74114E-02
      2092234 1.06784E-05
      2092235 9.84602E-04
      2092236 5.29386E-06
      2092238 6.19414E-05
  MIX=3
      11001001 5.67873E-02
      11006000 3.54921E-02
      11008016 1.41968E-02
END MIXT
READ GEOM
UNIT 1
  CYLINDER 1  9.525  2P8.89
  CYLINDER 2  10.16  2P9.525
  CUBOID   3  4P10.875  2P10.24
  MEDIA 2 1 1
  MEDIA 3 1 2 -1
  MEDIA 0 1 3 -2
  BOUNDARY  3
UNIT 2
  CYLINDER  1 5.748 2P5.3825
  CUBOID    2 4P6.59 2P6.225
  MEDIA 1 1 1
  MEDIA 0 1 2 -1
  BOUNDARY  2
UNIT 3
  CUBOID 1 2P10.875 2P21.75 2P20.48
  ARRAY 1 1 PLACE 1 1 1 0.0 -10.875 -10.24
  BOUNDARY  1
UNIT 4
  CUBOID 1 2P6.59 2P13.18 2P12.45
  CUBOID 2 2P6.59 2P21.75 2P20.48
  ARRAY  2  1 PLACE 1 1 1 0.0 -6.59 -6.225
  MEDIA 0 1 2 -1
  BOUNDARY
GLOBAL UNIT 5
  CUBOID 1 34.93 0.0 43.5 0.0 40.96 0.0
  ARRAY 3 1 PLACE 1 1 1 6.59 21.75 20.48
  BOUNDARY  1
END GEOM
READ ARRAY
  ARA=1 NUX=1 NUY=2 NUZ=2                 FILL F1 END FILL
  ARA=2 NUX=1 NUY=2 NUZ=2                 FILL F2 END FILL
  GBL=3 ARA=3 NUX=2 NUY=1 NUZ=1    FILL 4 3 END FILL
END ARRAY
READ PLOT TTL='X-Y SLICE AT Z=10.24'
  XUL=-1.0 YUL=44.5 ZUL=10.24
  XLR=35.93 YLR=-1.0 ZLR=10.24
  UAX=1.0 VDN=-1.0 NAX=130 NCH=' *.-' PIC=MIX END
  TTL='X-Z SLICE AT Y=10.875'
  XUL=-1.0 YUL=10.875 ZUL=41.96
  XLR=35.93 YLR=10.875 ZLR=-1.0
  UAX=1.0 WDN=-1.0 PIC=MIX END
END PLOT
END DATA
END

A plot of an X-Y slice taken through the bottom layer of the array is shown in Fig. 74. A plot of an X-Z slice taken through the +Y half of the array is shown in Fig. 75. These plots were used to verify the geometry mockup.

_images/fig241.png

Fig. 74 X-Y plot of mixed array.

_images/fig25.png

Fig. 75 X-Z plot of mixed array.

STORAGE ARRAY

EXAMPLE 18. Consider a storage array of highly enriched uranium buttons, each 1 in. tall and 4 in. in diameter. These buttons are stored on stainless steel shelves with a center-to-center spacing of 60.96 cm (2 ft) between them in the Y direction, and only one button on each shelf in the X direction. The shelves are 0.635 cm (1/4 in.) thick (Z dimension), 45.72 cm (18 in.) wide (X dimension), 609.6 cm (20 ft) long (Y dimension), and are 45.72 cm (18 in.) from the top of a shelf to the bottom of the shelf above it. Each rack of storage shelves is four shelves high, with the first shelf being 15.24 cm (6 in.) above the floor. The storage room is 586.56 cm (19.5 ft) in the X direction by 1293.44 cm (43 ft) in the Y direction with 365.76 cm (12 ft. ) ceilings in the Z direction. The walls, ceiling, and floor are composed of concrete, 30.48 cm (1 ft) thick. All the aisles between the storage racks are 91.44 cm (3 ft) wide. The racks are arranged with their length in the Y direction and an aisle between them. The arrays of racks are arranged with two in the Y direction and five in the X direction. Mixture 1 is the uranium metal, mixture 2 is the stainless steel, and mixture 3 is the concrete.

The metal button and its center-to-center spacing are described first. The void vertical spacing has arbitrarily been chosen to extend from the bottom of the button to the next shelf above the button. The shelf of stainless steel is described under the button.

KENO V.a:

UNIT 1
COM='METAL BUTTONS'
CYLINDER 1 1 5.08 2.54 0.0
CUBOID  0 1 2P22.86 2P30.48 45.72 0.0
CUBOID  2 1 2P22.86 2P30.48 45.72 -0.635

KENO-VI:

UNIT 1
CYLINDER 1 5.08  2.54  0.0
CUBOID   2   2P22.86   2P30.48  45.72   0.0
CUBOID   3   2P22.86   2P30.48  45.72  -0.635
MEDIA 1 1 1
MEDIA 0 1 2 -1
MEDIA 2 1 3 -2
BOUNDARY  3

ARRAY 1 creates an ARRAY of these buttons that fills one shelf. UNIT 2 then contains one of the shelves shown in Fig. 76.

_images/fig26.png

Fig. 76 Two racks of uranium buttons.

KENO V.a:

ARA=1 COM='SINGLE SHELF CONTAINING 10 METAL BUTTONS'
      NUX=1 NUY=10 NUZ=1 FILL F1 END FILL

UNIT 2
COM='SINGLE SHELF (1 X 10 X 1 ARRAY OF METAL BUTTONS ON A SHELF)'
ARRAY 1 3*0.0

KENO-VI:

ARA=1  NUX=1  NUY=10  NUZ=1  FILL  F1  END FILL

UNIT 2
CUBOID 1 45.72 0.0 609.60 0.0 46.355 0.0
ARRAY  1  1  PLACE  1  1  1  22.86  30.48  0.635
BOUNDARY  1

Note

The origin of UNIT 2 is on the bottom of the bottom shelf; it has been moved from the bottom of the button. The X and Y position of the origin is at the front, left-hand corner of the bottom of this lowest shelf.

Stack four UNIT 2s vertically to obtain one of the racks shown in Fig. 76. One rack is defined by array 2.

ARA=2 COM=’SINGLE RACK OF 4 SHELVES’

NUX=1 NUY=1 NUZ=4 FILL F2 END FILL

Generate a UNIT 3 that contains a rack of shelves and a UNIT 4 that is the aisle between the ends of the two racks in the Y direction.

KENO V.a:

UNIT 3
COM='SINGLE RACK (4 SHELVES TALL)'
ARRAY 2 3*0.0

UNIT 4
COM='CENTRAL AISLE UNIT SAME HEIGHT AS 4 SHELVES'
CUBOID 0 1 2P22.86 2P45.72 185.42 0.0

KENO-VI:

UNIT 3
CUBOID 1 45.72 0.0 609.60 0.0 185.42 0.0
ARRAY  2 1 PLACE 1 1 1 3*0.0
BOUNDARY  1
UNIT 4
CUBOID  1  2P22.86  2P45.72  185.42  0.0
MEDIA 0 1 1
BOUNDARY  1

Stack UNITs 3 and 4 together in the Y direction to create UNIT 5 which contains both racks in the Y direction and the aisle between them. This configuration is shown in Fig. 76.

KENO V.a:

ARA=3 COM='TWO RACKS END TO END WITH CENTRAL AISLE'
      NUX=1 NUY=3 NUZ=1 FILL 3 4 3 END FILL

UNIT 5
COM='SET OF TWO RACKS END TO END SEPARATED BY THE CENTRAL AISLE'
ARRAY 3 3*0.0

KENO-VI:

ARA=3  NUX=1  NUY=3  NUZ=1  FILL  3 4 3  END FILL

UNIT 5
CUBOID 1 45.72 0.0 1310.64 0.0 185.42 0.0
ARRAY 3 1 PLACE 1 1 1 3*0.0
BOUNDARY  1

Create a UNIT 6, which is an aisle 91.44 cm (3 ft) wide in the X direction and 1310.64 cm (43 ft) in the Y direction (full length of the room).

KENO V.a:

UNIT 6
COM='AISLE BETWEEN ADJACENT SETS OF TWO RACKS & CENTRAL AISLE (UNITS 5)'  CUBOID 0 1 91.44 0.0 1310.64 0.0 185.42 0.0

KENO-VI:

UNIT 6
CUBOID 1 91.44 0.0  1310.64 0.0  185.42 0.0
MEDIA 0 1 1
BOUNDARY  1

Stack UNITs 5 and 6 in the X direction to achieve the array of racks in the room. Then put the 6 in. spacing below the bottom of the racks, the spacing between the top of the top rack and the ceiling, and add the concrete floor, walls, and ceiling around the array. ARRAY 4 describes the array of racks in the room. ARRAY record (first CUBOID description in KENO-VI) encompasses this ARRAY, and the first REFLECTOR (second CUBOID in KENO-VI) descriptions are used to add the spacing between the top rack and the ceiling. The last two REFLECTOR (CUBOIDs 3 through 9 in KENO-VI) descriptions add the ceiling, walls, and floor in 5.0 cm increments to bias the concrete. A perspective of the room is shown in Fig. 77.

KENO V.a:

ARA=4 COM='ENTIRE STORAGE ARRAY'
      NUX=9 NUY=1 NUZ=1 FILL 5 6 3Q2 5 END FILL
GLOBAL
UNIT 7
COM='STORAGE ARRAY IN THE ROOM WITH WALLS, FLOOR AND CEILING'
ARRAY  4  3*0.0
REFLECTOR   0  1  4*0.0  165.1  15.24  1
REFLECTOR  3  2  6*5.0  6
REFLECTOR  3  8  6*0.48  1

KENO-VI:

GBL=4  ARA=4  NUX=9  NUY=1  NUZ=1  FILL 5 6  3Q2  5  END FILL

GLOBAL UNIT  7
CUBOID 1 594.36   0.0  1310.64   0.0  185.42   0.0
CUBOID 2 594.36   0.0  1310.64   0.0  350.52 -15.24
CUBOID 3 599.36  -5.00 1315.64  -5.00 355.52 -20.24
CUBOID 4 604.36 -10.00 1320.64 -10.00 360.52 -25.24
CUBOID 5 609.36 -15.00 1325.64 -15.00 365.52 -30.24
CUBOID 6 614.36 -20.00 1330.64 -20.00 370.52 -35.24
CUBOID 7 619.36 -25.00 1335.64 -25.00 375.52 -40.24
CUBOID 8 624.36 -30.00 1340.64 -30.00 380.52 -45.24
CUBOID 9 624.84 -30.48 1341.12 -30.48 381.00 -45.72
ARRAY  4 1 PLACE 1 1 1 3*0.0
MEDIA 0 1 2 -1
MEDIA 3 2 3 -2
MEDIA 3 3 4 -3
MEDIA 3 4 5 -4
MEDIA 3 5 6 -5
MEDIA 3 6 7 -6
MEDIA 3 7 8 -7
MEDIA 3 8 9 -8
BOUNDARY  9
_images/fig27.png

Fig. 77 Entire storage array in the room.

The complete input for this room is given below: The plots for this problem must be quite large in order to see all the detail because the array is sparse and the shelves are thin. Therefore, the plots for this system are not included as Fig.s. The user can generate the plots if it is desirable to see them. The nuclide IDs used in this problem are for the 16-group Hansen-Roach working format library, which is no longer distributed with SCALE.

KENO V.a:

=KENO5A
STORAGE ARRAY
READ PARAMETERS  FDN=YES LIB=41
END PARAMETERS
READ MIXT  SCT=1  MIX=1 92500 4.48006e-2  92800 2.6578e-3  92400 4.827e-4
92600 9.57e-5  MIX=2 200 1.0 MIX=3 301 1 END MIXT
READ GEOMETRY
UNIT 1
COM='METAL BUTTONS'
CYLINDER 1 1 5.08 2.54 0.0
CUBOID  0 1 2P22.86 2P30.48 45.72 0.0
CUBOID  2 1 2P22.86 2P30.48 45.72 -0.635
UNIT 2
COM='SINGLE SHELF (1 X 10 X 1 ARRAY OF METAL BUTTONS ON A SHELF)'
ARRAY 1 3*0.0
UNIT 3
COM='SINGLE RACK (4 SHELVES TALL)'
ARRAY 2 3*0.0
UNIT 4
COM='CENTRAL AISLE UNIT SAME HEIGHT AS 4 SHELVES'
CUBOID 0 1 2P22.86 2P45.72 185.42 0.0
UNIT 5
COM='SET OF TWO RACKS END TO END SEPARATED BY THE CENTRAL AISLE'
ARRAY 3 3*0.0
UNIT 6
COM='AISLE BETWEEN ADJACENT SETS OF TWO RACKS & CENTRAL AISLE (UNITS 5)'
CUBOID 0 1 91.44 0.0 1310.64 0.0 185.42 0.0
GLOBAL
UNIT 7
COM='STORAGE ARRAY IN THE ROOM WITH WALLS, FLOOR AND CEILING'
ARRAY 4 3*0.0
REFLECTOR 0 1 4*0.0 165.1 15.24 1
REFLECTOR 3 2 6*5.0    6
REFLECTOR 3 8 6*0.48  1
END GEOMETRY
READ ARRAY
ARA=1 COM='SINGLE SHELF CONTAINING 10 METAL BUTTONS'
      NUX=1 NUY=10 NUZ=1 FILL F1 END FILL
ARA=2 COM='SINGLE RACK OF 4 SHELVES'
      NUX=1 NUY=1 NUZ=4 FILL F2 END FILL
ARA=3 COM='TWO RACKS END TO END WITH CENTRAL AISLE'
      NUX=1 NUY=3 NUZ=1 FILL 3 4 3 END FILL
ARA=4 COM='ENTIRE STORAGE ARRAY'
      NUX=9 NUY=1 NUZ=1 FILL 5 6 3Q2 5 END FILL
END ARRAY
READ BIAS ID=301 2 8 END BIAS
READ START NST=5  NBX=5 END START
READ PLOT TTL='X-Z SLICE AT Y=30.48 WITH Z ACROSS AND X DOWN'
XUL=594.8 YUL=30.48 ZUL=-1.0 XLR=-0.5 YLR=30.48 ZLR=186.0
WAX=1.0 UDN=-1.0 NAX=640  END
TTL='Y-Z SLICE OF LEFT RACKS, X=22.86 WITH Z ACROSS AND Y DOWN'
XUL=22.86 YUL=1311.0 ZUL=-0.5 XLR=22.86 YLR=-3.0 ZLR=186.0
WAX=1.0 VDN=-1.0 NAX=640 END
TTL='X-Y SLICE OF ROOM THROUGH SHELF Z=0.3175 WITH X ACROSS AND Y DOWN'
XUL=-1.0 YUL=1312.0 ZUL=0.3175 XLR=596.0 YLR=-2.5 ZLR=0.3175
UAX=1.0 VDN=-1.0 NAX=320 END
END PLOT
END DATA
END

KENO-VI:

=KENOVI
STORAGE ARRAY
READ PARAMETERS  FDN=YES LIB=41  END PARAMETERS
READ MIXT  SCT=1
MIX=1 92500 4.48006e-2  92800 2.6578e-3  92400 4.827e-4  92600 9.57e-5
MIX=2 200 1.0 MIX=3 301 1
END MIXT
READ GEOMETRY
UNIT 1
CYLINDER 1  5.08  2.54  0.0
CUBOID   2   2P22.86   2P30.48  45.72  0.0
CUBOID   3   2P22.86   2P30.48  45.72  -.635
MEDIA 1 1 1
MEDIA 0 1 2 -1
MEDIA 2 1 3 -2
BOUNDARY  3
UNIT 2
CUBOID 1 45.72 0.0 609.60 0.0 46.355 0.0
ARRAY  1  1  PLACE  1  1  1  22.86  30.48  0.635
BOUNDARY  1
UNIT 3
CUBOID 1 45.72 0.0 609.60 0.0 185.42 0.0
ARRAY 2 1 PLACE 1 1 1 3*0.0
BOUNDARY  1
UNIT 4
CUBOID  1  2P22.86  2P45.72  185.42  0.0
MEDIA 0 1 1
BOUNDARY  1
UNIT 5
CUBOID 1 45.72 0.0 1310.64 0.0 185.42 0.0
ARRAY 3 1 PLACE 1 1 1 3*0.0
BOUNDARY  1
UNIT 6
CUBOID 1 91.44 0.0  1310.64 0.0  185.42 0.0
MEDIA 0 1 1
BOUNDARY  1
GLOBAL UNIT  7
CUBOID 1 594.36    0.0 1310.64    0.0 185.42    0.0
CUBOID 2 594.36    0.0 1310.64    0.0 350.52 -15.24
CUBOID 3 599.36  -5.00 1315.64  -5.00 355.52 -20.24
CUBOID 4 604.36 -10.00 1320.64 -10.00 360.52 -25.24
CUBOID 5 609.36 -15.00 1325.64 -15.00 365.52 -30.24
CUBOID 6 614.36 -20.00 1330.64 -20.00 370.52 -35.24
CUBOID 7 619.36 -25.00 1335.64 -25.00 375.52 -40.24
CUBOID 8 624.36 -30.00 1340.64 -30.00 380.52 -45.24
CUBOID 9 624.84 -30.48 1341.12 -30.48 381.00 -45.72
ARRAY  4 1 PLACE 1 1 1 3*0.0
MEDIA 0 1 2 -1
MEDIA 3 2 3 -2
MEDIA 3 3 4 -3
MEDIA 3 4 5 -4
MEDIA 3 5 6 -5
MEDIA 3 6 7 -6
MEDIA 3 7 8 -7
MEDIA 3 8 9 -8
BOUNDARY  9
END GEOMETRY
READ ARRAY
ARA=1 NUX=1 NUY=10 NUZ=1 FILL F1 END FILL
ARA=2 NUX=1 NUY=1 NUZ=4 FILL F2 END FILL
ARA=3 NUX=1 NUY=3 NUZ=1 FILL 3 4 3 END FILL
GBL=4 ARA=4 NUX=9 NUY=1 NUZ=1 FILL 5 6 3Q2 5 END FILL
END ARRAY
READ BIAS ID=301 2 8 END BIAS
READ START NST=5 NBX=5 END START
READ PLOT PLT=YES TTL='X-Z SLICE AT Y=30.48 WITH Z ACROSS AND X DOWN'
XUL=594.8 YUL=30.48 ZUL=-1.0 XLR=-0.5 YLR=30.48 ZLR=186.0
WAX=1.0 UDN=-1.0 NAX=640  END
TTL='Y-Z SLICE OF LEFT RACKS, X=22.86 WITH Z ACROSS AND Y DOWN'
XUL=22.86 YUL=1311.0 ZUL=-0.5 XLR=22.86 YLR=-3.0 ZLR=186.0
WAX=1.0 VDN=-1.0 NAX=640 END
TTL='X-Y SLICE OF ROOM THROUGH SHELF Z=0.3175 WITH X ACROSS AND Y DOWN'
XUL=-1.0 YUL=1312.0 ZUL=0.3175 XLR=596.0 YLR=-2.5 ZLR=0.3175
UAX=1.0 VDN=-1.0 NAX=320 END
END PLOT
END DATA
END

Arrays and holes

Use of holes in the geometry and Nesting holes describe the use of HOLEs, and Multiple arrays describes multiple ARRAYs and ARRAYs of ARRAYs. HOLEs can be used to place ARRAYs at locations in other UNITs. This section contains examples to illustrate the combined use of ARRAYs and HOLEs.

EXAMPLE 19. A SIMPLE CASK

This example consists of cylindrical mild steel container with an inside radius of 4.15 cm and a radial wall thickness of 0.45 cm. The thickness of the ends of the container is 1.27 cm, and the inside height is 10.1 cm. Highly enriched uranium rods 1 cm in diameter and 10 cm long are banded together into square bundles of four. These bundles are then positioned in the mild steel container as shown in Fig. 8.1.28. The rods sit on the floor of the container and have a 0.1 cm gap between their tops and the top of the container.

_images/fig28.png

Fig. 78 Uranium rods in a cylindrical container.

To generate the geometry description for this system, UNIT 1 is defined as one uranium rod and its associated square-pitch close-packed spacing region.

KENO V.a:

UNIT 1
CYLINDER  1 1  0.5  2P5.0
CUBOID    0 1  4P0.5  2P5.0

KENO-VI:

UNIT 1
CYLINDER 1 0.5  2P5.0
CUBOID   2   4P0.5  2P5.0
MEDIA 1 1 1
MEDIA 0 1 2 -1
BOUNDARY  2

From here, the geometry description diverges between KENO V.a and KENO-VI.

KENO V.a:

ARRAY 1 is defined to be the central square ARRAY consisting of four bundles of rods.

ARA=1 NUX=4 NUY=4 NUZ=1 FILL F1 END FILL

ARRAY 2 is defined to be a bundle of four rods.

ARA=2 NUX=2 NUY=2 NUZ=1 FILL F1 END FILL

ARRAY 2 is placed in UNIT 2. This defines the outer boundaries of an imaginary CUBOID that contains the ARRAY. It is convenient to have the origin of the ARRAY at its center, so the most negative point of the array will be (−1, −1,−5).

UNIT 2 ARRAY 2 -1.0 -1.0 -5.0

An ARRAY record is used to place ARRAY 1 in the GLOBAL UNIT. Then the cylindrical container is described around it and HOLEs are used to place the four outer bundles around the central ARRAY.

GLOBAL UNIT 3
ARRAY   1  -2.0  -2.0  -5.0
CYLINDER   0 1  4.15  5.1  -5.0
HOLE   2    0.0 -3.0  0.0
HOLE   2    3.0  0.0  0.0
HOLE   2    0.0  3.0  0.0
HOLE   2   -3.0  0.0  0.0
CYLINDER   2  1  4.6  6.37 -6.27

The first HOLE places the bottom bundle of four rods, the second HOLE places the bundle of four rods at the right, the third HOLE places the top bundle of rods and the fourth HOLE places the left bundle of rods.

KENO-VI:

Define UNIT 2 to be a void CUBOID with the same square pitch as the rod square pitch.

UNIT 2

CUBOID 1 4P0.5 2P5.0

MEDIA 0 1 1

BOUNDARY 1

ARRAY 1 is defined to be the central square 10 × 10 ARRAY consisting of 32 rods and 68 void positions that can be used to represent the array shown in Fig. 78.

ARA=1 NUX=10 NUY=10 NUZ=1 FILL 14*2 1 1 8*2 1 1 7*2 4*1 4*2 8*1 2 2 8*1 4*2 4*1 7*2 1 1 8*2 1 1 14*2 END FILL

ARRAY 1 is placed in UNIT 3. The first CYLINDER card defines the ARRAY BOUNDARY. Everything external to this boundary is not considered part of the problem. The positions in the ARRAY that do not contain rods are filled with cuboids consisting of void. The ARRAY boundary must either coincide with the outer boundary of the ARRAY or be contained within the ARRAY. An exterior void region is placed around the array boundary to coincide with the size of the interior radius of the container. The 10 × 10 ARRAY with the ARRAY boundary is shown in Fig. 79.

UNIT 3
CYLINDER 1  4.15 5.0 -5.0
CYLINDER 2 4.15 5.1 -5.0
ARRAY 1 1 PLACE 5 5 1   -0.5 -0.5 -0.0
MEDIA 0 1 2 -1
BOUNDARY  2
_images/fig29.png

Fig. 79 The 10 × 10 array of 32 uranium rods and 68 void cuboids with the array boundary.

The UNIT containing the ARRAY is now placed within the global unit using a HOLE content record. The location of the HOLE is determined using ORIGIN data to match the origin of the UNIT in the HOLE with an X, Y, Z position in the surrounding UNIT. In this problem, the origin of the UNIT must be at position (0,0,0). Since only nonzero data must be entered, ORIGIN data are not needed for this problem. The boundary region consists of the steel container.

GLOBAL UNIT 4
CYLINDER 2 4.6 6.37 -6.27
HOLE 3 ORIGIN X=0.0 Y=0.0 Z=0.0
MEDIA 2 1 2
BOUNDARY  2

The overall problem description is shown below. Two of the color plots used for verification of this mockup are shown in Fig. 80 and Fig. 81. The black outside border of the two color plots indicates volume outside the global unit. The plot can be extended just outside the global unit boundary to ensure that the entire problem is included in the plot. This results in a black area surrounding the actual problem.

KENO V.a:

=KENOVA
CASK ARRAY
READ PARAMETERS  FDN=YES LIB=41 GEN=10
END PARAMETERS
READ MIXT  SCT=1  MIX=1 92500 4.48006e-2  92800 2.6578e-3  92400 4.827e-4
92600 9.57e-5  MIX=2 100 1.0 END MIXT
READ GEOMETRY
UNIT 1
CYLINDER 1 1 0.5 2P5.0
CUBOID  0 1 4P0.5 2P5.0
UNIT 2
ARRAY 2 -1.0 -1.0 -5.0
GLOBAL UNIT 3
ARRAY  1 -2.0 -2.0 -5.0
CYLINDER 0 1 4.15 5.1 -5.0
HOLE 2 0.0 -3.0 0.0
HOLE 2 3.0 0.0 0.0
HOLE 2 0.0 3.0 0.0
HOLE 2 -3.0 0.0 0.0
CYLINDER 2 1 4.6 6.37 -6.27
END GEOM
READ ARRAY
ARA=1 NUX=4 NUY=4 NUZ=1 FILL F1 END FILL
ARA=2 NUX=2 NUY=2 NUZ=1 FILL F1 END FILL
END ARRAY
READ PLOT TTL='X-Z SLICE AT Y=0.25 WITH X ACROSS AND Z DOWN'
XUL=-5.0 YUL=0.25 ZUL=6.5 XLR=5.0 YLR=0.25 ZLR=-6.5
UAX=1.0 WDN=-1.0 NAX=640  END
TTL='X-Y SLICE AT Z=0.0 WITH X ACROSS AND Y DOWN'
XUL=-5.0 YUL=5.0 ZUL=0.0 XLR=5.0 YLR=-5.0 ZLR=0.0
UAX=1.0 VDN=-1.0 NAX=640 END
END PLOT
END DATA
END

KENO-VI:

KENO VI
CASK ARRAY
READ PARAMETERS  TME=1.0 FDN=YES LIB=41 GEN=10  END PARAMETERS
READ MIXT  SCT=1
MIX=1 92500 4.48006e-2  92800 2.6578e-3  92400 4.827e-4  92600 9.57e-5
MIX=2 100 1.0 END MIXT
READ GEOMETRY
UNIT 1
CYLINDER 1  0.5  2P5.0
CUBOID   2   4P0.5  2P5.0
MEDIA 1 1 1
MEDIA 0 1 2 -1
BOUNDARY  2
UNIT 2
CUBOID 1 4P0.5 2P5.0
MEDIA 0 1 1
BOUNDARY  1
UNIT 3
CYLINDER 1  4.15 5.0 -5.0
CYLINDER 2 4.15 5.1 -5.0
ARRAY 1 1 PLACE 5 5 1   -0.5 -0.5 -0.0
MEDIA 0 1 2 -1
BOUNDARY  2
GLOBAL UNIT 4
CYLINDER 2 4.6 6.37 -6.27
HOLE 3 ORIGIN X=0.0 Y=0.0 Z=0.0
MEDIA 2 1 2
BOUNDARY  2
END GEOM
READ ARRAY
ARA=1  NUX=10  NUY=10  NUZ=1  FILL  14*2 1 1 8*2 1 1 7*2 4*1 4*2 8*1 2 2 8*1 4*2 4*1 7*2 1 1
8*2 1 1 14*2  END FILL
END ARRAY
READ PLOT TTL='X-Z SLICE AT Y=0.25 WITH X ACROSS AND Z DOWN'
XUL=-5.0 YUL=0.25 ZUL=6.5 XLR=5.0 YLR=0.25 ZLR=-6.5
UAX=1.0 WDN=-1.0 NAX=640  END
TTL='X-Y SLICE AT Z=0.0 WITH X ACROSS AND Y DOWN'
XUL=-5.0 YUL=5.0 ZUL=0.0 XLR=5.0 YLR=-5.0 ZLR=0.0
UAX=1.0 VDN=-1.0 NAX=640 END
END PLOT
END DATA
END
=
_images/fig30.png

Fig. 80 X-Y slice of uranium rods in a cylindrical container.

_images/fig311.png

Fig. 81 X-Z slice of uranium rods in a cylindrical container.

EXAMPLE 20. A TYPICAL PRESSURIZED WATER REACTOR (PWR) SHIPPING CASK

A typical PWR shipping cask is illustrated in Fig. 82. The interior and exterior shell of the cask is carbon steel (mixture 7), and a depleted uranium gamma shield (mixture 6) is present in the annulus between the steel layers. The shipping cask contains seven PWR fuel assemblies. Each assembly is a 17 × 17 ARRAY of fuel rods with water holes. Each assembly is contained in a stainless steel (mixture 5) box. Each fuel rod is composed of 4% enriched UO2 (mixture 1) clad with Zircaloy (mixture 2). Rods of B4C clad (mixture 4) with stainless steel are positioned between the fuel assemblies. The entire cask is filled with water (mixture 3).

_images/fig321.png

Fig. 82 Typical PWR shipping cask.

To describe the geometry of the cask, some simple units are defined as shown in Fig. 83. UNIT 1 represents a fuel rod and its associated square pitch spacing region. UNIT 2 represents a water hole in a fuel assembly. UNITs 3, 4, and 6 represent the B4C rods with their various spacings, and UNIT 5 is a water hole that is used in association with some of the B4C rods.

_images/fig331.png

Fig. 83 Simple units.

KENO V.a:

UNIT 1
CYLINDER  1 1 .41148 365.76 0.0
CYLINDER  2 1 .48133 365.76 0.0
CUBOID    3 1 .63754 -.63754 .63754 -.63754 365.76 0.0

UNIT 2
CUBOID    3 1 .63754 -.63754 .63754 -.63754 365.76 0.0


UNIT 3
CYLINDER  4 1 .584 365.76 0.0
CYLINDER  5 1 .635 365.76 0.0
CUBOID    3 1 .9912 -.9912 2.2352 -1.27 365.76 0.0

UNIT 4
CYLINDER  4 1 .584 365.76 0.0
CYLINDER  5 1 .635 365.76 0.0
CUBOID    3 1 .9912 -.9912 1.2702 -2.235 365.76 0.0

UNIT 5
CUBOID    3 1 .9912 -.9912 1.7526 -1.7526 365.76 0.0

UNIT 6
CYLINDER  4 1 .584 365.76 0.0
CYLINDER  5 1 .635 365.76 0.0
CUBOID    3 1 1.1875215 -1.1875215 1.883706 -1.883706 365.76 0.0

KENO-VI:

UNIT 1
CYLINDER 1  .41148 365.76 0.0
CYLINDER 2  .48133 365.76 0.0
CUBOID   3  .63754 -.63754 .63754 -.63754 365.76 0.0
MEDIA 1 1 1
MEDIA 2 1 2 -1
MEDIA 3 1 3 -2
BOUNDARY  3

UNIT 2
CUBOID   1  .63754 -.63754 .63754 -.63754 365.76 0.0
MEDIA 3 1 1
BOUNDARY  1

UNIT 3
CYLINDER  1  .584 365.76 0.0
CYLINDER 2   .635 365.76 0.0
CUBOID    3   .9912 -.9912 2.2352 -1.27 365.76 0.0
MEDIA 4 1 1
MEDIA 5 1 2 -1
MEDIA 3 1 3 -2
BOUNDARY  3

UNIT 4
CYLINDER 1  .584 365.76 0.0
CYLINDER 2  .635 365.76 0.0
CUBOID   3   .9912 -.9912 1.2702 -2.235 365.76 0.0
MEDIA 4 1 1
MEDIA 5 1 2 -1
MEDIA 3 1 3 -2
BOUNDARY  3

UNIT 5
CUBOID 1   .9912 -.9912 1.7526 -1.7526 365.76 0.0
MEDIA 3 1 1
BOUNDARY  1

UNIT 6
CYLINDER 1  .584 365.76 0.0
CYLINDER 2  .635 365.76 0.0
CUBOID  3    1.1875215 -1.1875215 1.883706 -1.883706 365.76 0.0
MEDIA 4 1 1
MEDIA 5 1 2 -1
MEDIA 3 1 3 -2
BOUNDARY  3

UNITs 1 and 2 are stacked together into ARRAY 1 to form the ARRAY of fuel pins and water holes in a fuel assembly as shown in Fig. 84. This ARRAY is then encompassed with a layer of water and a layer of stainless steel to complete a fuel assembly in a storage cell (UNIT 7) as shown in Fig. 85.

_images/fig341.png

Fig. 84 Quarter section of fuel pin array.

_images/fig35.png

Fig. 85 Quarter section of fuel assembly.

KENO V.a:

ARA=1 NUX=17 NUY=17 NUZ=1 FILL
39R1 2 2Q3 8R1 2 9R1 2 22R1 2 4Q3 38R1 2 4Q3
Q51 22R1 2 Q10 Q9 2Q3 39R1                END FILL

UNIT 7
ARRAY 1 -10.83818 -10.83818 0.0
CUBOID    3 1 11.112495 -11.112495 11.112495 -11.112495 365.76 0.0
CUBOID    8 1 11.302238 -11.302238 11.302238 -11.302238 365.76 0.0

KENO-VI:

ARA=1 NUX=17 NUY=17 NUZ=1 FILL
39R1 2 2Q3 8R1 2 9R1 2 22R1 2 4Q3 38R1 2 4Q3
Q51 22R1 2 Q10 Q9 2Q3 39R1                END FILL
UNIT 7
CUBOID  1 10.83818 -10.83818 10.83818 -10.83818 365.76 0.0
CUBOID  2 11.112495 -11.112495 11.112495 -11.112495 365.76 0.0
CUBOID  3 11.302238 -11.302238 11.302238 -11.302238 365.76 0.0
ARRAY 1 1  PLACE 9 9 1  3*0.0
MEDIA 3 1 2 -1
MEDIA 5 1 3 -2
BOUNDARY  3

An array of UNIT 6s is created to represent the array of B4C rods that is positioned between the fuel assemblies. In KENO V.a, the array of B4C rods shown in Fig. 86 is contained in UNIT 8 for further use. KENO-VI geometry description does not need the placement of ARRAY 2 in a separate UNIT.

_images/fig36.png

Fig. 86 2 × 6 array of B4C rods.

KENO V.a:

ARA=2 NUX=2 NUY=6 NUZ=1    FILL F6        END FILL

UNIT 8
ARRAY 2 0 0 0

KENO-VI:

ARA=2 NUX=2 NUY=6 NUZ=1    FILL F6        END FILL

The next step is to create the central array of three fuel assemblies with B4C rods between them. In KENO V.a, this is done by stacking fuel assemblies in storage cells (UNIT 7) and B4C rod arrays (UNIT 8) into an array (ARRAY 3) and placing it in a UNIT (UNIT 9). In KENO-VI description, UNIT 7 (fuel assembly in storage cell) and the array of B4C rods (ARRAY 2) are directly placed into a UNIT (UNIT 8). The resultant geometry is shown in Fig. 87.

_images/fig37.png

Fig. 87 Central array.

KENO V.a:

ARA=3 NUX=5 NUY=1 NUZ=1    FILL 7 8 7 8 7 END FILL

UNIT 9    ARRAY 3 0 0 0

KENO-VI:

UNIT 8
CUBOID 4  -11.302238 -16.052324 11.302238 -11.302238 365.76 0.0
CUBOID 5    16.052324 11.302238 11.302236 -11.302236 365.76 0.0
CUBOID 6    38.052324 -38.052324 11.302236 -11.302236 365.76 0.0
HOLE 7
HOLE 7 ORIGIN X= -27.354562
HOLE 7 ORIGIN X= 27.354562
ARRAY 2 4 PLACE 1 1 1 -14.8648025  -9.418530 0.0
ARRAY 2 5 PLACE 1 1 1  12.4897595  -9.418530 0.0
MEDIA 3 1 6 -5 -4
BOUNDARY  6

UNITs 3, 4, and 5 are used to define the arrays of B4C rods that fit above and below the central array, as shown in Fig. 88.

_images/fig38.png

Fig. 88 Long B4C rod arrays.

ARA=4 NUX=39 NUY=1 NUZ=1   FILL 3 5 2Q2 3 4 2Q2 5 4 3 2Q2 5 3 4 2Q2 5 4 3 2Q2 5 2Q2 3
END FILL

ARA=5 NUX=39 NUY=1 NUZ=1   FILL 4 5 2Q2 4 3 2Q2 5 3 4 2Q2 5 4 3 2Q2 5 3 4 2Q2 5 2Q2 4
END FILL

In KENO V.a these ARRAYs are placed in UNITs (UNITs 10 and 11) for further use.

KENO V.a:

UNIT 10   ARRAY 4 0 0 0

UNIT 11   ARRAY 5 0 0 0

UNITs 9, 10, and 11 in the KENO V.a description, or UNIT 8 and ARRAYs 3 and 4 in the KENO-VI description, are stacked to form the central array with B4C rods as shown in Fig. 89.

_images/fig39.png

Fig. 89 Central array with long B4C arrays.

KENO V.a:

ARA=6 NUX=1 NUY=3 NUZ=1    FILL 11 9  10  END FILL

KENO-VI:

CUBOID 2  38.052324 -38.052324 14.807436  11.302236  365.76 0.0
CUBOID 3  38.052324 -38.052324 11.302236 -14.807436  365.76 0.0
HOLE 8
ARRAY 3 2 PLACE 20 1 1 0.0 13.054836  0.0
ARRAY 4 3 PLACE 20 1 1 0.0 13.054836  0.0

This completes the three central fuel assemblies and all the B4C rods associated with them. Next, UNITs 7 and 8 in KENO V.a geometry, UNIT 7 and ARRAY 2 in the KENO-VI geometry, are stacked together to form the array of two fuel assemblies separated by B4C rods as shown in Fig. 90. This is designated as ARRAY 7 and UNIT 12 in KENO V.a, and UNIT 9 in KENO-VI. The origin of UNIT 12 for KENO V.a is specified at the center of the array in the X and Y directions and the bottom of the fuel assemblies (Z=27.94 cm). The origin of UNIT 9 in the KENO-VI description is specified at the center of the B4C array in the X and Y directions and the bottom of the array in the Z direction.

_images/fig40.png

Fig. 90 Two fuel assemblies and B4C rods.

KENO V.a:

ARA=7 NUX=3 NUY=1 NUZ=1    FILL 7 8 7     END FILL

UNIT 12   ARRAY 7 -24.979519 -11.302238 27.94

KENO-VI:

UNIT 9
CUBOID 1 2.375043 -2.375043  11.302236 -11.302236 365.76 0.0
CUBOID 4  24.979519  -23.7919975 11.302238 -11.302238 365.76 0.0
ARRAY 2 1 PLACE 1 1 1 -1.1875215 -9.418530 0.0
HOLE 7 ORIGIN X=-13.67728
HOLE 7 ORIGIN X=13.67728
MEDIA 0 1 4 -1
BOUNDARY  4

In KENO V.a, UNIT 13 is simply a cylindrical lid that fits on top of the shipping cask. It is described relative to the origin of the shipping cask and is made of depleted uranium.

KENO V.a:

UNIT 13
CYLINDER  6 1 47.625 457.2 449.58

All necessary subassemblies that make up the shipping cask have been built. The shipping cask is completed by specifying the origin of the central section (ARRAY 6 in KENO V.a) (see Fig. 89) to be at the center of the array in X and Y and the bottom of the array in the Z direction. A cylinder of water defining the interior of the shipping cask is described around the array. In the KENO V.a geometry, a HOLE is used to place UNIT 12 (Fig. 90) below the array, and a second HOLE is used to place another UNIT 12 above the array. In the KENO-VI geometry, UNIT 9 is placed as a HOLE above and below the central array. Then a cylinder of steel is placed around the water, which is in turn encased by depleted uranium. The depleted uranium is then contained in the outer steel cylinder of the shipping cask. In KENO V.a description, a third HOLE is used to place the depleted uranium lid (UNIT 13) on the shipping cask. This completes the shipping cask description of Fig. 82

The geometry data for this shipping cask are shown below. The plot data have been included for verification of the geometry description. However, the plot generated by this data is quite large and is therefore not included in this document.

KENO V.a:

READ GEOM
UNIT 1
CYLINDER  1 1 .41148 365.76 0.0
CYLINDER  2 1 .48133 365.76 0.0
CUBOID    3 1 .63754 -.63754 .63754 -.63754 365.76 0.0
UNIT 2
CUBOID    3 1 .63754 -.63754 .63754 -.63754 365.76 0.0
UNIT 3
CYLINDER  4 1 .584 365.76 0.0
CYLINDER  5 1 .635 365.76 0.0
CUBOID    3 1 .9912 -.9912 2.2352 -1.27 365.76 0.0
UNIT 4
CYLINDER  4 1 .584 365.76 0.0
CYLINDER  5 1 .635 365.76 0.0
CUBOID    3 1 .9912 -.9912 1.2702 -2.235 365.76 0.0
UNIT 5
CUBOID    3 1 .9912 -.9912 1.7526 -1.7526 365.76 0.0
UNIT 6
CYLINDER  4 1 .584 365.76 0.0
CYLINDER  5 1 .635 365.76 0.0
CUBOID    3 1 1.1875215 -1.1875215 1.883706 -1.883706 365.76 0.0
UNIT 7    ARRAY 1 -10.83818 -10.83818 0.0
CUBOID    3 1 11.112495 -11.112495 11.112495 -11.112495 365.76 0.0
CUBOID    8 1 11.302238 -11.302238 11.302238 -11.302238 365.76 0.0
UNIT 8    ARRAY 2 0 0 0
UNIT 9    ARRAY 3 0 0 0
UNIT 10   ARRAY 4 0 0 0
UNIT 11   ARRAY 5 0 0 0
UNIT 12   ARRAY 7 -24.979519 -11.302238 27.94
UNIT 13
CYLINDER  6 1 47.625 457.2 449.58
ARRAY     6  -38.6568 -14.807438 27.94
CYLINDER  3 1 47.625 447.04 16.51
HOLE     12    0.0 -26.1097 0.0
HOLE     12    0.0  26.1097 0.0
CYLINDER  7 1 48.895 447.04 13.335
CYLINDER  6 1 59.06 447.04 3.81
CYLINDER  7 1 63.01 462.28 0.0
HOLE     13    0.0 0.0 0.0
END GEOM
READ ARRAY
ARA=1 NUX=17 NUY=17 NUZ=1 FILL
39R1 2 2Q3 8R1 2 9R1 2 22R1 2 4Q3 38R1 2 4Q3
Q51 22R1 2 Q10 Q9 2Q3 39R1                END FILL
ARA=2 NUX=2 NUY=6 NUZ=1    FILL F6        END FILL
ARA=3 NUX=5 NUY=1 NUZ=1    FILL 7 8 7 8 7 END FILL
ARA=4 NUX=39 NUY=1 NUZ=1   FILL 3 5 2Q2 3 4 2Q2 5 4 3 2Q2 5 3 4 2Q2
5 4 3 2Q2 5 2Q2 3                         END FILL
ARA=5 NUX=39 NUY=1 NUZ=1   FILL 4 5 2Q2 4 3 2Q2 5 3 4 2Q2 5 4 3 2Q2
5 3 4 2Q2 5 2Q2 4                         END FILL
ARA=6 NUX=1 NUY=3 NUZ=1    FILL 11 9  10  END FILL
ARA=7 NUX=3 NUY=1 NUZ=1    FILL 7 8 7     END FILL
END ARRAY
READ PLOT
TTL=? SHIPPING CASK IF-300 X-Y SLICE  ?
XUL=-63 YUL=63 ZUL=180 XLR=63 YLR=-63 ZLR=180
UAX=1 VDN=-1 NAX=350
PLT=NO
END PLOT

KENO-VI:

READ GEOM
UNIT 1
CYLINDER  1 .41148 365.76 0.0
CYLINDER  2 .48133 365.76 0.0
CUBOID  3    .63754 -.63754 .63754 -.63754 365.76 0.0
MEDIA 1 1 1
MEDIA 2 1 2 -1
MEDIA 3 1 3 -2
BOUNDARY  3
UNIT 2
CUBOID  1  .63754 -.63754 .63754 -.63754 365.76 0.0
MEDIA 3 1 1
BOUNDARY  1
UNIT 3
CYLINDER 1   .584 365.76 0.0
CYLINDER 2   .635 365.76 0.0
CUBOID   3    .9912 -.9912 2.2352 -1.27 365.76 0.0
MEDIA 4 1 1
MEDIA 5 1 2 -1
MEDIA 3 1 3 -2
BOUNDARY  3
UNIT 4
CYLINDER 1  .584 365.76 0.0
CYLINDER 2  .635 365.76 0.0
CUBOID  3    .9912 -.9912 1.2702 -2.235 365.76 0.0
MEDIA 4 1 1
MEDIA 5 1 2 -1
MEDIA 3 1 3 -2
BOUNDARY  3
UNIT 5
CUBOID 1   .9912 -.9912 1.7526 -1.7526 365.76 0.0
MEDIA 3 1 1
BOUNDARY  1
UNIT 6
CYLINDER 1  .584 365.76 0.0
CYLINDER 2  .635 365.76 0.0
CUBOID  3    1.1875215 -1.1875215 1.883706 -1.883706 365.76 0.0
MEDIA 4 1 1
MEDIA 5 1 2 -1
MEDIA 3 1 3 -2
BOUNDARY  3
UNIT 7
CUBOID 1  10.83818 -10.83818 10.83818 -10.83818 365.76 0.0
CUBOID 2  11.112495 -11.112495 11.112495 -11.112495 365.76 0.0
CUBOID 3  11.302238 -11.302238 11.302238 -11.302238 365.76 0.0
ARRAY 1 1  PLACE 9 9 1  3*0.0
MEDIA 3 1 2 -1
MEDIA 8 1 3 -2
BOUNDARY  3
UNIT 8
CUBOID 4  -11.302238 -16.052324 11.302238 -11.302238 365.76 0.0
CUBOID 5    16.052324 11.302238 11.302236 -11.302236 365.76 0.0
CUBOID 6    38.052324 -38.052324 11.302236 -11.302236 365.76 0.0
HOLE 7
HOLE 7 ORIGIN X= -27.354562
HOLE 7 ORIGIN X= 27.354562
ARRAY 2 4 PLACE 1 1 1 -14.8648025  -9.418530 0.0
ARRAY 2 5 PLACE 1 1 1  12.4897595  -9.418530 0.0
MEDIA 0 1 6 -5 -4
BOUNDARY  6
UNIT 10
CUBOID 1 1.1875215 -1.1875215  11.302236 -11.302236 365.76 0.0
CUBOID 4  23.7919975 -23.7919975 11.302238 -11.302238 365.76 0.0
ARRAY 2 1 PLACE 1 1 1 3*0.0
HOLE 7 ORIGIN X=-12.4897595
HOLE 7 ORIGIN X=12.4897595
MEDIA 0 1 4 -1
BOUNDARY  4
GLOBAL UNIT 11
CUBOID 2  38.052324 -38.052324 13.284638  11.302236  365.76 0.0
CUBOID 3  38.052324 -38.052324 11.302236 -13.284638 365.76 0.0
CYLINDER 6 47.625 419.10  -11.43
CYLINDER 7   48.895 419.10 -14.605
CYLINDER 8  59.06 419.10 -24.13
CYLINDER 9  47.625 429.26 421.64
CYLINDER 10  63.01 434.34 -27.94
HOLE 8
ARRAY 3 2 PLACE 20 1 1 0.0 12.293438  0.0
ARRAY 4 3 PLACE 20 1 1 0.0 12.293438  0.0
HOLE 9 ORIGIN Y=24.586876
HOLE 9 ORIGIN Y=-24.586876
MEDIA 3 1 6 -3 -2
MEDIA 7 1 7 -6 -3 -2
MEDIA 6 1 8 -7
MEDIA 7 1 9
MEDIA 6 1 10 -9 -8
BOUNDARY  10
END GEOM
READ ARRAY
ARA=1 NUX=17 NUY=17 NUZ=1 FILL
39R1 2 2Q3 8R1 2 9R1 2 22R1 2 4Q3 38R1 2 4Q3
Q51 22R1 2 Q10 Q9 2Q3 39R1                END FILL
ARA=2 NUX=2 NUY=6 NUZ=1    FILL F6        END FILL
ARA=3 NUX=5 NUY=1 NUZ=1    FILL 7 8 7 8 7 END FILL
ARA=4 NUX=39 NUY=1 NUZ=1   FILL 3 5 2Q2 3 4 2Q2 5 4 3 2Q2 5 3 4 2Q2
5 4 3 2Q2 5 2Q2 3                         END FILL
ARA=5 NUX=39 NUY=1 NUZ=1   FILL 4 5 2Q2 4 3 2Q2 5 3 4 2Q2 5 4 3 2Q2
5 3 4 2Q2 5 2Q2 4                         END FILL
ARA=6 NUX=1 NUY=3 NUZ=1    FILL 11 9  10  END FILL
ARA=7 NUX=3 NUY=1 NUZ=1    FILL 7 8 7     END FILL
END ARRAY
READ PLOT
TTL=? SHIPPING CASK IF-300 X-Y SLICE  ?
XUL=-63 YUL=63 ZUL=180 XLR=63 YLR=-63 ZLR=180
UAX=1 VDN=-1 NAX=350
PLT=NO
END PLOT

Triangular pitched arrays in KENO-VI

EXAMPLE 21.

Triangular pitched ARRAYs can be described in KENO-VI by defining the UNITs that make up the ARRAY as HEXPRISM and in the array data block setting TYP=TRIANGULAR, HEXAGONAL, SHEXAGONAL, or RHEXAGONAL. This includes close-packed triangular pitched arrays. Since the ARRAYs are constructed by stacking hexprisms, care must be taken to ensure that the ARRAY boundary is completely enclosed within the stacked UNIT. Below is an example of a triangular pitched ARRAY.

The first and second UNITs are the HEXPRISMs that make up the ARRAY. UNIT 1 is the fuel cell stacked in a triangular pitched or hexagonal lattice. UNIT 2 is a dummy UNIT used to fill in the ARRAY so the array boundary is contained within the stacked UNITs. Since the ARRAY is not moderated, UNIT 2 contains void. Fig. 91 shows an X-Y cross section schematic of UNITs 1 and 2.

_images/fig411.png

Fig. 91 Fuel cell and empty cell set up as hexprisms.

UNIT  1
COM='SINGLE CELL FUEL CAN IN HEXPRISM'
CYLINDER 10  10.16  18.288  0.0
CYLINDER 20  10.312 18.288 -0.152
HEXPRISM 30  10.503 18.288 -0.152
MEDIA  1  1  10
MEDIA  2  1  20  -10
MEDIA  0  1  30  -20
BOUNDARY  30
UNIT  2
COM='EMPTY CELL'
HEXPRISM  10  10.503  18.288  -0.152
MEDIA  0  1  10
BOUNDARY  10

UNIT 3 is the GLOBAL UNIT that contains the ARRAY and ARRAY BOUNDARY. The ARRAY is an unmoderated triangular pitched assembly of 7 cells. Unrotated triangular or hexagonal pitched arrays can be set up in two array configurations. The first configuration, selected using the TYP= followed by keyword HEXAGONAL or TRIANGULAR, stacks the UNITs so that the faces perpendicular to the X axis meet. Each consecutive row in the Y direction begins ½ of the face-to-face dimension farther over in the positive X direction than in the previous row. The second configuration, selected using the TYP= followed by keyword SHEXAGONAL, also stacks the UNITs so that the faces perpendicular to the X axis meet. However, for this type of ARRAY, the odd numbered rows in the Y direction (1, 3, 5, etc.) begin at the left edge of the ARRAY, and the even numbered rows in the Y direction (2, 4, 6, etc.) begin ½ of the face-to-face dimension to the right of the left edge of the ARRAY. Fig. 92 and Fig. 93 show X-Y cross section schematics of the assembly for the two different unrotated hexagonal ARRAY types.

GLOBAL UNIT 3
COM='7 CYLINDERS IN A CIRCLE WITH CYLINDRICAL BOUNDARY'
CYLINDER 10  32.000  18.288  -0.152
ARRAY  1  10 PLACE 3 3 1 3*0.0
BOUNDARY 10
READ ARRAY GBL=1 ARA=1 TYP=HEXAGONAL NUX=5 NUY=5 NUZ=1
FILL 7*2 2*1 2*2 3*1 2*2 2*1 7*2 END FILL  END ARRAY
_images/fig421.png

Fig. 92 Seven cylinders stacked in a HEXAGONAL array with cylindrical array boundary.

_images/fig431.png

Fig. 93 Seven cylinders stacked in a SHEXAGONAL array with a cylindrical array boundary.

The overall problem description is shown below. The cross section library would be generated in a separate CSAS-MG step. An X-Y cross section color plot used for verification of this mockup is shown in Fig. 94.

_images/fig44.png

Fig. 94 X-Y slice of 7 cylinders in a triangular pitch array.

Data description of Example 21.

=KENOVI
TRIANGULAR PITCHED ARRAY 7 PINS IN A CIRCLE
READ PARAMETERS LNG=20000 LIB=4 END PARAMETERS
READ MIXT SCT=2
MIX=1 NCM=8 92235 1.37751E-03  92238 9.92354E-05  8016 3.32049E-02
       9019 2.95349E-03  1001 6.05028E-02
MIX=2 NCM=14 13027 6.02374E-02
END MIXT
READ GEOMETRY
UNIT 1
COM='SINGLE CELL FUEL CAN IN HEXPRISM'
CYLINDER 10  10.16  18.288  0.0
CYLINDER 20  10.312 18.288 -0.152
HEXPRISM 30  10.503 18.288 -0.152
MEDIA  1 1 10
MEDIA  2 1 20 -10
MEDIA 0 1 30 -20
BOUNDARY  30
UNIT 2
COM='EMPTY CELL'
HEXPRISM 10  10.503 18.288 -0.152
MEDIA  0 1 10
BOUNDARY  10
GLOBAL UNIT 3
CYLINDER 10  32.000 18.288 -0.152
COM='7 CYLINDERS IN A CIRCLE WITH CYLINDRICAL BOUNDARY'
ARRAY  1 10 PLACE 3 3 1 3*0.0
BOUNDARY  10
END GEOMETRY
READ ARRAY  GBL=1 TYP=HEXAGONAL NUX=5 NUY=5 NUZ=1
FILL 7*2 2*1 2*2 3*1 2*2 2*1 7*2 END FILL  END ARRAY
READ PLOT
TTL='TRIANGULAR PITCHED ARRAY, 7 PINS IN A CIRCLE'
XUL=-33.0  YUL=33.0  ZUL=0.0
XLR=33.0  YLR=-33.0  ZLR=0.0
UAX=1  VDN=-1  NAX=640    END
END PLOT
END DATA
END

EXAMPLE 21a.

Another hexagonal ARRAY type involves stacking rotated hexprisms, which are hexprisms rotated 30°/ 90º so that the flat faces perpendicular to the Y axis now meet. Rotated hexagonal arrays are specified by setting TYP=RHEXAGONAL in the array data block. Because the ARRAYs are constructed by stacking hexprisms, care must be taken to ensure the array boundary is completely enclosed within the stacked UNIT. Below is an example of a rotated hexagonal pitched ARRAY.

The first and second UNITs are the rotated hexprisms that make up the ARRAY. They are specified using the geometry keyword RHEXPRISM. UNIT 1 is the fuel cell that is stacked in a rotated hexagonal lattice. UNIT 2 is a dummy UNIT used to fill in the ARRAY so that the ARRAY BOUNDARY is contained within the stacked UNITs. Since the ARRAY is not moderated, UNIT 2 contains void. Fig. 95 shows an X-Y cross section schematic of UNITs 1 and 2.

_images/fig45.png

Fig. 95 Fuel cell and empty cell set up as rotated hexprism (RHEXPRISM).

UNIT  1
COM='SINGLE CELL FUEL CAN IN HEXPRISM'
CYLINDER 10  10.16  18.288  0.0
CYLINDER 20  10.312 18.288 -0.152
RHEXPRISM 30  10.503 18.288 -0.152
MEDIA  1  1  10
MEDIA  2  1  20  -10
MEDIA  0  1  30  -20
BOUNDARY  30
UNIT  2
COM='EMPTY CELL'
RHEXPRISM  10  10.503  18.288  -0.152
MEDIA  0  1  10
BOUNDARY  10

UNIT 3 is the GLOBAL UNIT that contains the ARRAY and ARRAY BOUNDARY. The ARRAY is an unmoderated rotated hexagonal pitched assembly of 7 cells. The rotated hexagonal array type is specified in the array data block using TYP= with the keyword RHEXAGONAL. This ARRAY type stacks the UNITs so the faces perpendicular to the Y axis meet. In the odd numbered columns (i.e., 1, 3, 5, etc.), the UNITs are stacked so the columns begin at the lower edge of the array and in the even numbered columns (i.e., 2, 4, 6, etc.), the UNITs are stacked so the columns begin ½ of the face-to-face dimension above the lower edge of the ARRAY. Fig. 96 shows the X-Y cross section schematic of the assembly for the rotated hexagonal ARRAY type.

_images/fig46.png

Fig. 96 Seven cylinders stacked in a RHEXAGONAL array with a cylindrical array boundary.

GLOBAL UNIT 3
COM='7 CYLINDERS IN A CIRCLE WITH CYLINDRICAL BOUNDARY'
CYLINDER 10  32.000  18.288  -0.152
ARRAY  1  10 PLACE 3 3 1 3*0.0
BOUNDARY 10
READ ARRAY GBL=1 ARA=1 TYP=RHEXAGONAL NUX=5 NUY=5 NUZ=1
FILL 6*2 3*1 2*2 3*1 3*2 1*1 7*2 END FILL  END ARRAY

The overall problem description is shown below. The cross section library would be generated in a separate CSAS-MG step. An X-Y cross section color plot used for verification of this mockup is shown in Fig. 97.

_images/fig47.png

Fig. 97 X-Y slice of 7 cylinders in a rotated hexagonal pitched array.

Data description of Example 21a.

=KENOVI
TRIANGULAR PITCHED ARRAY 7 PINS IN A CIRCLE
READ PARAMETERS LNG=20000 LIB=4 END PARAMETERS
READ MIXT SCT=2
MIX=1 NCM=8 92235 1.37751E-03  92238 9.92354E-05  8016 3.32049E-02
       9019 2.95349E-03  1001 6.05028E-02
MIX=2 NCM=14 13027 6.02374E-02
END MIXT
READ GEOMETRY
UNIT 1
COM='SINGLE CELL FUEL CAN IN HEXPRISM'
CYLINDER 10  10.16  18.288  0.0
CYLINDER 20  10.312 18.288 -0.152
RHEXPRISM 30  10.503 18.288 -0.152
MEDIA  1 1 10
MEDIA  2 1 20 -10
MEDIA 0 1 30 -20
BOUNDARY  30
UNIT 2
COM='EMPTY CELL'
RHEXPRISM 10  10.503 18.288 -0.152
MEDIA  0 1 10
BOUNDARY  10
GLOBAL UNIT 3
CYLINDER 10  32.000 18.288 -0.152
COM='7 CYLINDERS IN A CIRCLE WITH CYLINDRICAL BOUNDARY'
ARRAY  1 10 PLACE 3 3 1 3*0.0
BOUNDARY  10
END GEOMETRY
READ ARRAY  GBL=1 TYP=RHEXAGONAL NUX=5 NUY=5 NUZ=1
FILL 6*2 3*1 2*2 3*1 3*2 1*1 7*2 END FILL  END ARRAY
READ PLOT
TTL='ROTATED HEXAGONAL ARRAY, 7 PINS IN A CIRCLE'
XUL=-33.0  YUL=33.0  ZUL=0.0
XLR=33.0  YLR=-33.0  ZLR=0.0
UAX=1  VDN==-1  NAX=640    END
END PLOT
END DATA
END

Triangular pitched Arrays in KENO V.a

Triangular pitched arrays can be described in KENO V.a geometry by properly defining the basic unit from which the array can be built. This includes close-packed triangular pitched arrays. Two geometry configurations are described below.

EXAMPLE 1. Bare Triangular pitched Array.

Fig. 98 illustrates a small close-packed triangular pitched ARRAY. Each rod in the ARRAY has a radius of 2.0 cm, and the pitch of the ARRAY is 4 cm. To create this ARRAY, describe five units as defined in Fig. 99.

Assume the rods described in the ARRAY are each 2.0 cm in radius and 100 cm tall. The rods are composed of mixture 1. The geometry descriptions for the first four UNITs are given below.

UNIT 1
ZHEMICYL-Y 1 1 2.0 50.0 -50.0

UNIT 2
ZHEMICYL+Y 1 1 2.0 50.0 -50.0

UNIT 3
ZHEMICYL-X 1 1 2.0 50.0 -50.0

UNIT 4
ZHEMICYL+X 1 1 2.0 50.0 -50.0
_images/fig48.png

Fig. 98 Bare triangular pitched ARRAY.

_images/fig49.png

Fig. 99 Units used to describe a bare triangular pitched ARRAY.

To describe UNIT 5, the origin of the UNIT is defined to be at its center. One of the hemicylinders is built into the box, and the other three are added as holes. In this example, the +X hemicylinder is built into the box, and the other hemicylinders are inserted as holes. (Because the +X hemicylinder is built into UNIT 5, UNIT 4 is not used in the problem.) The half dimension of the box in the X dimension is equal to the radius, 2.0 cm. The half dimension of the box in the Y direction is \(\frac{\sqrt{3}}{2}\) times the pitch (0.866025 * 4.0) or 3.46411 cm. UNIT 5 is described below.

UNIT 5
ZHEMICYL+X  1 1 2.0 50.0 -50.0 ORIGIN -2.0 0.0
CUBOID  0 1 2P2.0 2P3.46411 2P50.0
HOLE 1 0.0  3.46411   0.0
HOLE 2 0.0 -3.46411   0.0
HOLE 3 2.0            0.0

In the description of UNIT 5, the ZHEMICYL+X places the hemicylinder at the left of the box. HOLE 1 places the top hemicylinder, HOLE 2 places the bottom hemicylinder, and HOLE 3 places the hemicylinder at the right of the box.

Next, a UNIT 6 is defined that can be used to complete the lower rod of UNIT 5. A UNIT 7 is defined that can be used to complete the upper rod of UNIT 5. UNIT 8 is defined to complete the left rod of UNIT 5, and UNIT 9 is defined to complete the right rod of UNIT 5. UNIT 10 is defined to complete the corners of the overall ARRAY. The input data for these UNITs are given below and are illustrated in Fig. 100.

_images/fig50.png

Fig. 100 UNITs to complete the triangular pitched ARRAY.

UNIT 6
ZHEMICYL-Y       1 1 2.0 50.0 -50.0
CUBOID           0 1 2P2.0 0.0 -2.0 50.0 -50.0

UNIT 7
ZHEMICYL+Y   1 1 2.0 50.0 -50.0
CUBOID           0 1 2P2.0 2.0 0.0 2P50.0

UNIT 8
ZHEMICYL-X       1 1 2.0 50.0 -50.0
CUBOID           0 1 0.0 -2.0 2P3.46411 2P50.0

UNIT 9
ZHEMICYL+X   1 1 2.0 50.0 -50.0
CUBOID           0 1 2.0 0.0 2P3.46411 2P50.0

UNIT 10
CUBOID           0 1 2.0 0.0 2.0 0.0 2P50.0

Fig. 101 shows the arrangement of the UNITs to complete the ARRAY. The data to describe the ARRAY are shown below.

ARA=1 NUX=6 NUY=4 NUZ=1 FILL 10 4R6 10 8 4R5 9 1Q6 10 4R7 10 END FILL

The bottom row of the ARRAY is described by the data entries 10 4R6 10. The second row of the ARRAY is described by the data entries 8 4R5 9. The third row is filled by repeating the previous six entries (1Q6). It could also have been described by entering 8 4R5 9. The top row of the ARRAY is described by the data entries 10 4R7 10.

_images/fig511.png

Fig. 101 Completed ARRAY.

EXAMPLE 2a. Triangular Pitched ARRAY in a Cylinder

Fig. 102 illustrates a close-packed triangular pitched ARRAY in a cylinder. This array may be described by defining five basic units that are the same as those of Example 1 shown in Fig. 99.

_images/fig521.png

Fig. 102 Triangular pitched ARRAY within a cylinder.

UNIT 1
ZHEMICYL-Y 1 1 2.0 50.0 -50.0

UNIT 2
ZHEMICYL+Y 1 1 2.0 50.0 -50.0

UNIT 3
ZHEMICYL-X 1 1 2.0 50.0 -50.0

UNIT 4
ZHEMICYL+X 1 1 2.0 50.0 -50.0

To describe UNIT 5, the origin of the UNIT to be at its center is defined. One of the hemicylinders is built into the box, and the other three are added as HOLEs. In this example, the +X hemicylinder is built into the box, and the other hemicylinders are inserted as HOLEs. The half dimension of the box in the X dimension is equal to the radius, 2.0 cm. The half dimension of the box in the Y direction is \(\frac{\sqrt{3}}{2}\) times the pitch (0.866025 * 4.0) or 3.46411 cm. UNIT 5 is described below.

UNIT 5
ZHEMICYL+X 1 1 2.0 50.0 -50.0 ORIGIN -2.0 0.0
CUBOID     0 1 2P2.0 2P3.46411 2P50.0
HOLE 1  0.0  3.46411 0.0
HOLE 2  0.0 -3.46411 0.0
HOLE 3  2.0  0.0     0.0

In the description of UNIT 5, the ZHEMICYL+X places the hemicylinder at the left of the box. HOLE 1 places the top hemicylinder, HOLE 2 places the bottom hemicylinder, and HOLE 3 places the hemicylinder at the right of the box.

To describe the base ARRAY of the problem, UNITs 5 is stacked in a 4 × 2 × 1 array as shown in Fig. 103. The input data for the ARRAY are the following:

ARA=1 NUX=4 NUY=2 NUZ=1  FILL F5 END FILL
_images/fig53.png

Fig. 103 4 × 2 × 1 array to be placed within a cylinder.

Next, the ARRAY is placed within the cylinder. This is done by placing the ARRAY in a UNIT, defined here to be UNIT 6. The origin of the cylinder has been defined to be at the center of the ARRAY. The resulting geometry is shown in Fig. 104.

_images/fig54.png

Fig. 104 4 × 2 × 1 ARRAY within a cylinder.

UNIT 6
ARRAY 1 -8.0 -6.92822 -50.0
CYLINDER 0 1 12.4  2P50.0
CYLINDER 2 1 12.65 2P50.0

Next, the hemicylinders necessary to complete all of the half cylinders remaining in Fig. 104 are added. This is done by placing four UNITs 1 at the appropriate positions along the bottom of the ARRAY, four UNITs 2 at the top of the ARRAY, two UNITs 3 at the left of the ARRAY, and two UNITs 4 at the right of the ARRAY. The input data are shown below, and the resulting configuration is shown in Fig. 105. In UNIT 6, the first HOLE 1 places a UNIT 1 under the lower left UNIT of the ARRAY. The second HOLE 1 places a UNIT 1 under the next ARRAY UNIT to the right of the first one. This procedure is repeated for the next two lower ARRAY UNITs, thus completing the lower row of cylinders. Similarly, the first HOLE 2 places a UNIT 2 above the upper left UNIT of the ARRAY. The second HOLE 2 places a UNIT 1 to the right of the first one, etc., until the four cylinders at the top of the ARRAY are completed. The first HOLE 3 places a UNIT 3 at the lower left side of the ARRAY to complete that rod. The second HOLE 3 completes the rod above it. The first HOLE 4 completes the lower rod on the right side of the ARRAY. The second HOLE 4 completes the rod above it. The geometry data listed below result in the configuration shown in Fig. 105.

_images/fig55.png

Fig. 105 Partially completed triangular pitched ARRAY in a cylinder.

UNIT 6
ARRAY 1 -8.0 -6.92822 -50.0
CYLINDER 0 1 12.4  2P50.0
HOLE 1  -6.0 -6.92822 0.0
HOLE 1  -2.0 -6.92822 0.0
HOLE 1   2.0 -6.92822 0.0
HOLE 1   6.0 -6.92822 0.0
HOLE 2  -6.0  6.92822 0.0
HOLE 2  -2.0  6.92822 0.0
HOLE 2   2.0  6.92822 0.0
HOLE 2   6.0  6.92822 0.0
HOLE 3  -8.0 -3.46411 0.0
HOLE 3  -8.0  3.46411 0.0
HOLE 4   8.0 -3.46411 0.0
HOLE 4   8.0  3.46411 0.0
CYLINDER 2 1 12.65 2P50.0

To complete the desired configuration, a cylinder is defined, UNIT 7, and it is placed at the four appropriate positions as shown below. The first HOLE 7 places the cylinder of UNIT 7 at the left of the ARRAY, the second HOLE 7 places the cylinder at the top of the ARRAY, the third HOLE 7 places the cylinder at the right of the ARRAY, and the fourth HOLE 7 places the cylinder at the bottom of the ARRAY. The completed configuration is shown in Fig. 106. It is not necessary for UNIT 7 to precede UNIT 6. It is allowable to place UNIT 7 after UNIT 6 in the input data. Because the final configuration is defined in UNIT 6, it must be designated as the GLOBAL UNIT. The total geometry input for this example is listed below.

_images/fig56.png

Fig. 106 Completed triangular pitched array in a cylinder.

READ GEOM
UNIT 1
ZHEMICYL-Y 1 1 2.0 50.0 -50.0

UNIT 2
ZHEMICYL+Y 1 1 2.0 50.0 -50.0

UNIT 3
ZHEMICYL-X 1 1 2.0 50.0 -50.0

UNIT 4
ZHEMICYL+X 1 1 2.0 50.0 -50.0

UNIT 5
ZHEMICYL+X 1 1 2.0 50.0 -50.0 ORIGIN -2.0 0.0
CUBOID     0 1 2P2.0 2P3.46411 2P50.0
HOLE 1  0.0  3.46411 0.0
HOLE 2  0.0 -3.46411 0.0
HOLE 3  2.0  0.0     0.0

UNIT 7
CYLINDER 1 1 2.0 2P50.0

GLOBAL UNIT 6
ARRAY 1 -8.0 -6.92822 -50.0
CYLINDER 0 1  12.4  2P50.0
HOLE 1  -6.0  -6.92822 0.0
HOLE 1  -2.0  -6.92822 0.0
HOLE 1   2.0  -6.92822 0.0
HOLE 1   6.0  -6.92822 0.0
HOLE 2  -6.0   6.92822 0.0
HOLE 2  -2.0   6.92822 0.0
HOLE 2   2.0   6.92822 0.0
HOLE 2   6.0   6.92822 0.0
HOLE 3  -8.0  -3.46411 0.0
HOLE 3  -8.0   3.46411 0.0
HOLE 4   8.0  -3.46411 0.0
HOLE 4   8.0   3.46411 0.0
HOLE 7 -10.0   0.0     0.0
HOLE 7   0.0  10.39233 0.0
HOLE 7  10.0   0.0     0.0
HOLE 7   0.0 -10.39233 0.0
CYLINDER 2 1  12.65 2P50.0
END GEOM

READ ARRAY
ARA=1 NUX=4 NUY=2 NUZ=1  FILL F5 END FILL
END ARRAY

EXAMPLE 2b. Alternative Mockup of Triangular Pitched ARRAY in a Cylinder.

Consider the triangular pitched ARRAY shown in Fig. 102. Another method of describing this configuration is given below. Four basic UNITs are defined, as listed below. These are the same UNITs shown in Fig. 99.

UNIT 1
ZHEMICYL-Y 1 1 2.0 50.0 -50.0

UNIT 2
ZHEMICYL+Y 1 1 2.0 50.0 -50.0

UNIT 3
ZHEMICYL-X 1 1 2.0 50.0 -50.0

UNIT 4
ZHEMICYL+X 1 1 2.0 50.0 -50.0

UNIT 5 is the same as previously defined in Example 2a, and pictured in Fig. 99.

UNIT 5
ZHEMICYL+X 1 1 2.0 50.0 -50.0 ORIGIN -2.0 0.0
CUBOID     0 1 2P2.0 2P3.46411 2P50.0
HOLE 1  0.0  3.46411 0.0
HOLE 2  0.0 -3.46411 0.0
HOLE 3  2.0  0.0     0.0

To describe the basic array for the problem, UNITs 5 is stacked in a 4 × 2 × 1 as shown in Fig. 103. The input data for the ARRAY are the following:

ARA=1 NUX=4 NUY=2 NUZ=1  FILL F5 END FILL

Next, the ARRAY (ARRAY 1) is placed in UNIT 6, and UNITs 7 and 8 are defined to be placed to the left and right of it (see Fig. 107). UNIT 7 will complete the two rods at the left boundary of the ARRAY and will contain half of the far left rod in the completed configuration. In the description of UNIT 7, the ZHEMICYL+X is half of the far right rod and is located with its cut face at the left boundary of a box that is as tall as the entire ARRAY of Fig. 103. The first HOLE 3 in UNIT 7 completes the lower left rod of that ARRAY, and the second HOLE 3 completes the upper left rod. UNIT 8 is constructed in similar fashion to complete the two rods at the right of the ARRAY shown in Fig. 8.1.53. UNIT 8 is the mirror image of UNIT 7. UNITs 6, 7, and 8 are stacked in an ARRAY (ARRAY 2) to achieve the configuration shown in Fig. 107. The data to accomplish this are listed below.

_images/fig57.png

Fig. 107 Array to be placed in a cylinder.

UNIT 6
ARRAY 1 -8.0 -6.92822 -50.0

GLOBAL
UNIT 7
ZHEMICYL+X 1 1 2.0 50.0 -50.0
CUBOID     0 1 2.0 0.0 2P6.92822 2P50.0
HOLE 3 2.0 -3.46411 0.0
HOLE 3 2.0  3.46411 0.0

UNIT 8
ZHEMICYL-X 1 1 2.0 50.0 -50.0
CUBOID     0 1 0.0 -2.0 2P6.92822 2P50.0
HOLE 4 -2.0 -3.46411 0.0
HOLE 4 -2.0  3.46411 0.0

ARA=2 NUX=3 NUY=1 NUZ=1 FILL 7 6 8 END FILL

Next, ARRAY 2 is placed in the cylinder as shown in Fig. 108. The data are listed below.

_images/fig58.png

Fig. 108 Partially completed array in a cylinder.

UNIT 9
ARRAY 2 -10.0 -6.92822 -50.0
CYLINDER 0 1 12.4  2P50.0
CYLINDER 2 1 12.65 2P50.0

Now UNITs 10 and 11 are described and placed above and below the ARRAY. These UNITs are shown in Fig. 109. UNIT 10 is described to complete the two central rods at the top of the ARRAY of Fig. 108. Fig. 109 and Fig. 110 illustrate these UNITs. The ZHEMICYL-Y is placed at the top of the UNIT to describe half of the rod at the very top of the ARRAY. The first HOLE 2 completes the left center rod at the top of the ARRAY pictured in Fig. 108. The second HOLE 2 completes the right center rod at the top of the ARRAY. UNIT 11 is described in similar fashion. It is the mirror image of UNIT 10 and is placed below the ARRAY of Fig. 108. The resulting configuration is shown in Fig. 109.

_images/fig59.png

Fig. 109 Description of UNITs 10 and 11 for Example 2b.

UNIT 10
ZHEMICYL-Y 1 1 2.0 2P50.0
CUBOID  0 1 2P4.0 0.0 -3.46411 2P50.0
HOLE 2 -2.0 -3.46411 0.0 2P50.0
HOLE 2  2.0 -3.46411 0.0

UNIT 11
ZHEMICYL+Y 1 1 2.0 2P50.0
CUBOID  0 1 2P4.0 3.46411 0.0 2P50.0
HOLE 1 -2.0 3.46411 0.0
HOLE 1  2.0 3.46411 0.0

Now UNITs 10 and 11 are placed above and below the ARRAY of Fig. 108 to obtain the configuration shown in Fig. 110.

_images/fig60.png

Fig. 110 Partially completed triangular pitched ARRAY in a cylinder.

UNIT 9
ARRAY 2 -10.0 -6.92822 -50.0
CYLINDER 0 1 12.4  2P50.0
HOLE 10 0.0 10.39233 0.0
HOLE 11 0.0 -10.39233 0.0
CYLINDER 2 1 12.65 2P50.0

To complete the array, the remaining half cylinders must be entered as HOLEs, as shown below. The first HOLE 1 completes the half rod at the lower left of Fig. 110. The second HOLE 1 completes the half rod at the lower center, and the third HOLE 1 completes the half rod at the lower right of Fig. 110. Similarly, the first HOLE 2 completes the half rod at the upper left of Fig. 110. The second HOLE 2 completes the half rod at the upper center, and the third HOLE 2 completes the half rod at the upper right. HOLE 3 completes the half rod at the left of Fig. 110, and HOLE 4 completes the half rod at the right. The final geometry configuration is shown in Fig. 111. UNIT 9 must be specified as the GLOBAL UNIT because it defines the overall configuration.

_images/fig611.png

Fig. 111 Final configuration of triangular pitched ARRAY in a cylinder.

GLOBAL  UNIT 9
ARRAY 2 -10.0 -6.92822 -50.0
CYLINDER 0 1 12.4  2P50.0
HOLE 10  0.0 10.39233 0.0
HOLE 11  0.0 -10.39233 0.0
HOLE 1  -6.0  -6.92822 0.0
HOLE 1   0.0 -10.39233 0.0
HOLE 1   6.0  -6.92822 0.0
HOLE 2  -6.0   6.92822 0.0
HOLE 2   0.0  10.39233 0.0
HOLE 2   6.0   6.92822 0.0
HOLE 3 -10.0   0.0     0.0
HOLE 4  10.0   0.0     0.0
CYLINDER 2 1 12.65 2P50.0

The geometry data for Example 2b are given below.

READ GEOM

UNIT 1
ZHEMICYL-Y 1 1 2.0 50.0 -50.0

UNIT 2
ZHEMICYL+Y 1 1 2.0 50.0 -50.0

UNIT 3
ZHEMICYL-X 1 1 2.0 50.0 -50.0

UNIT 4
ZHEMICYL+X 1 1 2.0 50.0 -50.0

UNIT 5
ZHEMICYL+X 1 1 2.0 50.0 -50.0 ORIGIN -2.0 0.0
CUBOID     0 1 2P2.0 2P3.46411 2P50.0
HOLE 1  0.0  3.46411 0.0
HOLE 2  0.0 -3.46411 0.0
HOLE 3  2.0  0.0     0.0

UNIT 6
ARRAY 1 -8.0 -6.92822 -50.0

UNIT 7
ZHEMICYL+X 1 1 2.0 50.0 -50.0
CUBOID     0 1 2.0 0.0 2P6.92822 2P50.0
HOLE 3 2.0 -3.46411 0.0
HOLE 3 2.0  3.46411 0.0

UNIT 8
ZHEMICYL-X 1 1 2.0 50.0 -50.0
CUBOID     0 1 0.0 -2.0 2P6.92822 2P50.0
HOLE 3 0.0 -3.46411 0.0
HOLE 3 0.0  3.46411 0.0

UNIT 10
YZHEMICYL-Y 1 1 2.0 2P50.0
CUBOID  0 1 2P4.0 0.0 -3.46411 2P50.0
HOLE 2 -2.0 -3.46411 0.0
HOLE 2  2.0 -3.46411 0.0

UNIT 11
YZHEMICYL+Y 1 1 2.0 2P50.0
CUBOID  0 1 2P4.0 3.46411 0.0 2P50.0
HOLE 1 -2.0 3.46411 0.0
HOLE 1  2.0 3.46411 0.0

GLOBAL
UNIT 9
ARRAY 2 -10.0 -6.92822 -50.0
CYLINDER 0 1 12.4  2P50.0
HOLE 10  0.0  10.39233        0.0
HOLE 11  0.0  -10.39233       0.0
HOLE 1  -6.0    -6.92822      0.0
HOLE 1   0.0  -10.39233       0.0
HOLE 1   6.0    -6.92822      0.0
HOLE 2  -6.0     6.92822      0.0
HOLE 2   0.0   10.39233       0.0
HOLE 2   6.0     6.92822      0.0
HOLE 3 -10.0     0.0          0.0
HOLE 4  10.0     0.0          0.0
CYLINDER 2 1 12.65 2P50.0
END GEOM

READ ARRAY
ARA=1 NUX=4 NUY=2 NUZ=1  FILL F5 END FILL
ARA=2 NUX=3 NUY=1 NUZ=1 FILL 7 6 8 END FILL
END ARRAY

Dodecahedral pitched arrays

EXAMPLE 22.

Dodecahedral pitched ARRAYs can be described in KENO-VI by defining the UNITs that make up the ARRAY as dodecahedra and in the ARRAY data block setting TYP=DODECAHEDRAL. Since the ARRAYs are constructed by stacking dodecahedra, care must be taken to ensure the ARRAY boundary is completely enclosed within the stacked unit. Below is an example of a dodecahedral ARRAY that represents a close packed ARRAY of spheres.

The first and second UNITs are the dodecahedra that make up the ARRAY. UNIT 1 is the fuel sphere stacked in a dodecahedral lattice. UNIT 2 is a dummy UNIT used to fill in the ARRAY so the ARRAY boundary is contained within the stacked UNITs. Since the ARRAY is not moderated, UNIT 2 contains void. Fig. 112 shows an isometric, cross section view of UNITs 1 and 2.

_images/fig62.png

Fig. 112 Fuel cell and empty cell set up as dodecahedra.

UNIT  1
COM='SINGLE CELL FUEL CAN IN DODECAHDRON'
SPHERE       10   8.0
SPHERE       20   8.5
DODECAHEDRON 30  10.5
MEDIA  1  1  10
MEDIA  2  1  20  -10
MEDIA  0  1  30  -20
BOUNDARY     30
UNIT  2
COM='EMPTY CELL'
DODECAHEDRON 10  10.5
MEDIA  0  1  10
BOUNDARY     10

UNIT 3 is the GLOBAL UNIT that contains the ARRAY and ARRAY BOUNDARY. The ARRAY is an unmoderated triangular pitched assembly of 17 fuel spheres. Dodecahedral ARRAYs are specified by using TYP= followed by keyword dodecahedral in the array data block. The X and Y coordinates are stacked together as a square pitched ARRAY with the pitch equal to twice the dodecahedron radius. In the Z dimension, the odd Z planes (Z = 1, 3, 5, etc.) have the X and Y UNITs begin at the most negative edge of the ARRAY, while the even Z planes (Z = 2, 4, 6, etc.) have the X and Y UNITs begin one dodecahedron inscribed sphere radius in the positive direction from the most negative edge of the ARRAY. Also, the Z distance between the centers of the UNITs in successive Z planes is the square root of 2.0 times the dodecahedron radius (or the dodecahedron diameter divided by the square root of 2.0). Fig. 113 and Fig. 114 show X-Y cross sectional color plots of the assembly at an odd and even Z plane.

GLOBAL UNIT 3
COM='17 CLOSE PACKED FUEL SPHERES IN A CYLINDER'
CYLINDER  10  41.0  44.5   0.0
CYLINDER  20  42.0  44.5  -1.0
ARRAY  1  10 PLACE 3 3 1 3*0.0
MEDIA  3  20  -10
BOUNDARY 20

READ ARRAY GBL=1 ARA=1 TYP=DODECAHEDRAL NUX=5 NUY=5 NUZ=5
FILL 25*2
     6*2  2*1  3*2  2*1  12*2
     6*2  3*1  2*2  3*1  2*2  3*1  6*2
     6*2  2*1  3*2  2*1  12*2
     25*2 END FILL  END ARRAY
_images/fig63.png

Fig. 113 X-Y slice of dodecahedral array at even level Z = 2.

_images/fig64.png

Fig. 114 X-Y slice of dodecahedral array at odd level Z = 3.

The overall problem description is shown below. The cross section library would be generated in a separate CSAS-MG step.

Data description of Example 22.

=KENOVI
CLOSE PACKED DODECAHEDRAL ARRAY 17 FUEL SPHERES IN A CYLINDER
READ PARAMETERS LNG=20000 LIB=4 END PARAMETERS
READ MIXT SCT=2
MIX=1 NCM=8 92235 1.37751E-03  92238 9.92354E-05  8016 3.32049E-02
       9019 2.95349E-03  1001 6.05028E-02
MIX=2 NCM=14 13027 6.02374E-02
MIX=3 NCM=14 13027 6.02374E-02
END MIXT
READ GEOMETRY
UNIT  1
COM='SINGLE CELL FUEL CAN IN DODECAHDRON'
SPHERE       10   8.0
SPHERE       20   8.5
DODECAHEDRON 30  10.5
MEDIA  1  1  10
MEDIA  2  1  20  -10
MEDIA  0  1  30  -20
BOUNDARY     30
UNIT  2
COM='EMPTY CELL'
DODECAHEDRON 10  10.5
MEDIA  0  1  10
BOUNDARY     10
GLOBAL UNIT 3
COM='9 CLOSE PACKED FUEL SPHERES IN A CYLINDER'
CYLINDER  10  41.0  44.5   0.0
CYLINDER  20  42.0  44.5  -1.0
ARRAY  1  10 PLACE 3 3 1 3*0.0
MEDIA  3  1 20  -10
BOUNDARY 20
END GEOMETRY
READ ARRAY GBL=1 ARA=1 TYP=DODECAHEDRAL NUX=5 NUY=5 NUZ=5
FILL 25*2
     6*2  2*1  3*2  2*1  12*2
     6*2  3*1  2*2  3*1  2*2  3*1  6*2
     6*2  2*1  3*2  2*1  12*2
     25*2 END FILL  END ARRAY
READ PLOT
TTL='DODECAHEDRAL ARRAY, 4 SPHERES - Z LEVEL = 2'
XUL=-43.0  YUL=43.0  ZUL=14.85  XLR=43.0  YLR=-43.0  ZLR=14.85
UAX=1  VDN=-1  NAX=640    END  PLT0
TTL='DODECAHEDRAL ARRAY, 9 SPHERES - Z LEVEL = 3'
XUL=-43.0  YUL=43.0  ZUL=29.70  XLR=43.0  YLR=-43.0  ZLR=29.70
UAX=1  VDN=-1  NAX=640    END  PLT1
END PLOT
END DATA
END

Alternative sample problem mockups

The geometry data for KENO can often be described correctly in several ways. Some alternative geometry descriptions are given here for sample problems C.12 and C.13. (See Appendix C of the KENO manual.)

Sample Problem C.12, First Alternative

This mockup maintains the same overall unit dimensions that were used in sample problem C.12. In sample problem C.12, the origin of UNIT 1, the solution cylinder, is at the center of the unit; the origin of UNITs 2, 3, 4, and 5, the metal cylinders, are at the center of the cylinders. In this mockup, the unit numbers remain the same and the origin of each unit is at the center of the unit. In each unit the cylinder is offset by specifying the position of its centerline relative to the origin of the UNIT.

KENO V.a:

READ GEOM
UNIT 1
CYLINDER  2 1 9.525 8.89 -8.89
CYLINDER  3 1 10.16 9.525 -9.525
CUBOID    0 1 10.875 -10.875 10.875 -10.875 10.24 -10.24
UNIT 2
CYLINDER  1 1 5.748 9.3975 -1.3975 ORIG 4.285 4.285
CUBOID    0 1 10.875 -10.875 10.875 -10.875 10.24 -10.24
UNIT 3
CYLINDER  1 1 5.748 9.3975 -1.3675 ORIG 4.285 -4.285
CUBOID    0 1 10.875 -10.875 10.875 -10.875 10.24 -10.24
UNIT 4
CYLINDER  1 1 5.748 1.3675 -9.3975 ORIG 4.285 4.285
CUBOID    0 1 10.875 -10.875 10.875 -10.875 10.24 -10.24
UNIT 5
CYLINDER  1 1 5.748 1.3675 -9.3975 ORIG 4.285 -4.285
CUBOID    0 1 10.875 -10.875 10.875 -10.875 10.24 -10.24
END GEOM
READ ARRAY  NUX=2 NUY=2 NUZ=2 FILL 2 1 3 1 4 1 5 1 END ARRAY

KENO-VI:

READ GEOM
UNIT 1
CYLINDER 1  9.525 8.89 -8.89
CYLINDER