This section provides a resource to tools used by the criticality safety practitioner. Links to existing codes and programs are provided as well as reviews or discussions of the applicability or useability of those tools. In some cases, technical papers on criticality safety techniques are also provided here.

"How to Model Dry Powder Spills" is a technical paper that provides techniques for modeling dry powder or particulate spills. The paper focuses on the generation of models that conservatively bound conical spills. These models include the derivations of equations for stacked flat cylinders and hemispherical approaches to model spills using KENO-V.a as shown in the two examples.

Conical Spill Within Stacked Flat Cylinders

Conical Spill Within a Particular Hemisphere Modeling Approach

Along with the paper, the associated spreadsheet, included also, provides the equations in formulas, plots a comparison of the hemispherical modeling approaches, and provides the KENO-V.a input for the desired approach. The spreadsheet is broken into three different sheets, each dealing with different spill modeling capabilities.

The four different hemispherical modeling approaches are addressed by the first sheet of the spreadsheet and are as depicted in this figure.

Comparison of Different Hemispherical Modeling Approaches

Sheet 1 provides the equations for the hemispherical modeling approaches and compares them graphically and parametrically (comparing the volume and surface areas of each). The volume of the spill (in liters) and the angel of repose (in degrees) are entered in the top, left, yellow-colored cells of the spreadsheet. The hemisphere radius and the chord length as well as the resulting cone radius and height are displayed immediately to the right. A graph comparing the different approaches is shown below the data.

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The KENO-V.a input generated from this data is available below the comparison plot. The input can be directly cut and pasted into a KENO-V.a input deck (assuming that mixture 1 is fuel material and mixture 2 is water).

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Sheet 2 provides an input generator for a stacked cylinders model. The conical spill parameters (height and radius) are input to the top, left yellow-colored cells of the sheet along with the modeling parameters. (The conical spill height and radius can be calculated for a given volume and angle of repose using Sheet 1.) These modeling parameters use the individual stacked cylinder height (referred to as the "Increment"), which along with the conical height will determine the number of stacked cylinders required to model the spill, and the upper conical surface reflector thickness (referred to as "Ref Thickness"), which allows a water reflector to be modeled as part of the stacked cylinders. Four additional parameters are provided to set the starting unit number of the model, set the modeling approach to either use holes or an array, specify if the excess fuel at the outside corner of the cylinders should be considered as part of the reflector thickness (i.e., the excess fuel should be considered to bound any reflection effect since the excess fuel is located in the portion of the model where reflector should be, if a reflector thickness is used), and specify if the top-most reflector increments should be consolidated into the last fuel increment (i.e., decrease the number of unit to those required to model fuel). The sheet will generate the required input for the specified approach to the right (and below, if an array approach is used) of the specified data for up to 100 stacked cylinders.

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Sheet 3 provides a specialized hemisphere model for modeling square-pitched arrays of spills that have a limited surface area onto which to spill and, hence, the spills would co-mingle with neighboring spills flowing into each other producing a raised spill. This type of spill is approximated by a hemisphere modeled on top of a square cross-sectional area cuboid. The hemisphere radius is determined by the maximum cross-sectional area (entered as the "Maximum 1/2 Cuboid" in the sheet, which refers to half the maximum square side of the surface area that the individual square-pitched array unit can occupy). From that radius, the resulting hemipshere is formed using the specified angle of repose with the residual volume of the spill not contained in the hemipshere being modeled in the cuboid below. Similar to Sheet 1, the specialized model is compared to the normal conical spill and the normal "maintaining volume and angle of repose" hemisphere model. The input generated below the comparison plot, again, can be directly cut and pasted into a KENO-V.a input deck.

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In performing a techical review of calculations, the reviewer is sometimes required to verify container/system volumes (in different units from cc to liter, gal, and cu ft); isotopic, element, compound, and overal masses/weights (in grams or pounds); assuring that minimum volumes have been used, as well as assuring solution concentrations, H/X and C/X ratios, and amounts of moderator used within a model. This spreadsheet, originally generated for use with KENO-IV and the Hansen-Roach 16-group cross-section set^*, still provides some usefulness in evaluating today's KENO-V.a input deck models. Feel free to use the spreadsheet, modify it, or update it to current KENO-V.a and materials specifications.^** Unit dimensions and atom densities^*** and volume fractions of interest are entered on the right side of the spreadsheet. The resulting verified parameters are displayed in the middle or left side of the spreadsheet.

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^*Note the sigma-p and cross-section ID calculation set in the upper left corner of the spreadsheet.
^***If you do update it, please send us a copy so that we can update the version posted here.
^**Since atom densities are rarely entered directly in KENO-V.a input decks, they can be obtained from the mixture generation section of the KENO-V.a output.