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<li class="toctree-l1"><a class="reference internal" href="MAVRIC.html">MAVRIC: Monaco with Automated Variance Reduction using Importance Calculations</a></li>
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  <div class="section" id="csas6-control-module-for-enhanced-criticality-safety-analysis-with-keno-vi">
<span id="csas6"></span><h1>CSAS6:  Control Module for Enhanced Criticality Safety Analysis with KENO-VI<a class="headerlink" href="#csas6-control-module-for-enhanced-criticality-safety-analysis-with-keno-vi" title="Permalink to this headline"></a></h1>
<p><em>L. M. Petrie, K. B. Bekar, D. F. Hollenbach,</em><sup>1</sup> <em>S. Goluoglu</em><sup>1</sup></p>
<p>The <strong>C</strong>riticality <strong>S</strong>afety <strong>A</strong>nalysis <strong>S</strong>equence with
KENO-VI (CSAS6) provides reliable and efficient means of performing
<em>keff</em> calculations for systems that are routinely encountered in
engineering practice. In the multigroup calculation mode, CSAS6 uses
XSProc to process the cross sections for temperature corrections and
problem-dependent resonance self-shielding and calculates the <em>keff</em>
of three-dimensional (3-D) system models. If the continuous energy
calculation mode is selected no resonance processing is needed and the
continuous energy cross sections are used directly in KENO-VI, with
temperature corrections provided as the cross sections are loaded. The
geometric modeling capabilities available in KENO-VI coupled with the
automated cross-section processing within the control sequences allow
complex, 3-D systems to be easily analyzed.</p>
<p><sup>1</sup>Formerly with Oak Ridge National Laboratory</p>
<p>ACKNOWLEDGMENTS</p>
<p>The CSAS6 Criticality Safety Analysis Sequence is based on the CSAS
control module, and the KENO‑VI functional module, described in their
respective chapters. G. E. Whitesides is acknowledged for his
contributions through early versions of KENO. Appreciation is expressed
to C. V. Parks and S. M. Bowman for their guidance in developing CSAS6.</p>
<div class="section" id="introduction">
<h2>Introduction<a class="headerlink" href="#introduction" title="Permalink to this headline"></a></h2>
<p>Criticality Safety Analysis Sequence with KENO-VI (CSAS6) provides
reliable and efficient means of performing <em>keff</em> calculations for
systems that are routinely encountered in engineering practice,
especially in the calculation of <em>keff</em> of three-dimensional (3-D)
system models. CSAS6 implements XSProc to process material input and
provide a temperature and resonance-corrected cross-section library
based on the physical characteristics of the problem being analyzed. If
a continuous energy cross-section library is specified, no resonance
processing is needed and the continuous energy cross sections are used
directly in KENO-VI, with temperature corrections provided as the cross
sections are loaded.</p>
</div>
<div class="section" id="sequence-capabilities">
<h2>Sequence Capabilities<a class="headerlink" href="#sequence-capabilities" title="Permalink to this headline"></a></h2>
<p>CSAS6 is designed to prepare a resonance-corrected cross-section library
for subsequent use in KENO‑VI. In order to minimize human error, the
SCALE data handling is automated as much as possible. CSAS6 and many
other SCALE sequences apply a standardized procedure to provide
appropriate number densities and cross sections for the calculation.
XSProc is responsible for reading the standard composition data and
other engineering-type specifications, including volume fraction or
percent theoretical density, temperature, and isotopic distribution as
well as the unit cell data. XSProc then generates number densities and
related information, prepares geometry data for resonance self-shielding
and flux-weighting cell calculations, if needed, and (if needed)
provides problem-dependent multigroup cross-section processing. CSAS6
invokes a KENO-VI Data Processor to read and check the KENO-VI data.
When the data checking has been completed, the control sequence executes
XSProc to prepare a resonance-corrected microscopic cross-section
library in the AMPX working library format if a multigroup library has
been selected.</p>
<p>For each unit cell specified as being cell-weighted, XSProc performs the
necessary calculations and produces a cell-weighted microscopic
cross-section library. KENO-VI may be executed to calculate the <em>keff</em>
or neutron multiplication factor using the cross-section library that
was prepared by the control sequence.</p>
</div>
<div class="section" id="multigroup-csas6-limitations">
<h2>Multigroup CSAS6 limitations<a class="headerlink" href="#multigroup-csas6-limitations" title="Permalink to this headline"></a></h2>
<p>The CSAS6 control module was developed to use simple input data and
prepare problem-dependent cross sections for use in calculating the
effective neutron multiplication factor of a 3-D system using KENO-VI
and possibly XSDRNPM. An attempt was made to make the system as general
as possible within the constraints of the standardized methods chosen to
be used in SCALE. Standardized methods of data input were adopted to
allow easy data entry and for quality assurance purposes. Some of the
limitations of the CSAS6 sequence are a result of using preprocessed
multigroup cross sections. Inherent limitations in CSAS6 are as follows:</p>
<blockquote>
<div><p>1. Two-dimensional (2-D) effects such as fuel rods in assemblies
where some positions are filled with control rod guide tubes,
burnable poison rods and/or fuel rods of different enrichments. The
cross sections are processed as if the rods are in an infinite
lattice of rods. If the user inputs a Dancoff factor for the cell
(such as one computed by MCDancoff), XSProc can produce an infinite
lattice cell, which reproduces that Dancoff. This can mitigate some
two dimensional lattice effects.</p>
</div></blockquote>
<p>It is strongly recommended that the user perform CSAS6 calculations of
benchmark experiments similar to the problem of interest to demonstrate
the validity of the cross-section data and processing for that type of
problem.</p>
<div class="section" id="continuous-energy-csas6-limitations">
<h3>Continuous energy CSAS6 limitations<a class="headerlink" href="#continuous-energy-csas6-limitations" title="Permalink to this headline"></a></h3>
<p>When continuous energy KENO calculations are desired, none of the
resonance processing modules are applicable or needed. Moreover, the MG
limitations noted in the previous section are eliminated. The continuous
energy cross sections are directly used in KENO. An existing multigroup
input file can easily be converted to a continuous energy input file by
simply specifying the continuous energy library. In this case, all cell
data is ignored. However, the following limitations exist:</p>
<ol class="arabic simple">
<li><p>If CELLMIX is defined in the cell data, the problem will not run in
the continuous energy mode. CELLMIX implies new mixture cross
sections are generated using XSDRNPM-calculated cell fluxes and
therefore is not applicable in the continuous energy mode.</p></li>
<li><p>Only VACUUM, MIRROR, PERIODIC, and WHITE boundary conditions are
allowed. Other albedos, e.g., WATER, CARBON, POLY, etc. are for
multigroup only.</p></li>
<li><p>Problems with DOUBLEHET cell data are not allowed as they inherently
utilize CELLMIX feature.</p></li>
</ol>
</div>
</div>
<div class="section" id="input-data-guide">
<h2>Input Data Guide<a class="headerlink" href="#input-data-guide" title="Permalink to this headline"></a></h2>
<p>The input data for CSAS6 are composed of two broad categories of data.
The first is XSProc, including Standard Composition Specification Data
and Unit Cell Geometry Specification. This first category specifies the
cross-section library and defines the composition of each mixture and
optionally the unit cell geometry that may be used to process the
cross sections. The second category of data, the KENO-VI input data, is
used to specify the geometric and boundary conditions that represent the
physical 3-D configuration of the problem. Both data blocks are
necessary for CSAS6.</p>
<p>All data are entered in free form, allowing alphanumeric data,
floating-point data, and integer data to be entered in an unstructured
manner. Up to 252 columns of data entry per line are allowed. Data can
usually start or end in any column with a few exceptions. As an example,
the word END beginning in column 1 and followed by two blank spaces or a
new line will end the problem and any data following will be ignored.
Each data entry must be followed by one or more blanks to terminate the
data entry. For numeric data, either a comma or a blank can be used to
terminate each data entry. Integers may be entered for floating values.
For example, 10 will be interpreted as 10.0. Imbedded blanks are not
allowed within a data entry unless an E precedes a single blank as in an
unsigned exponent in a floating-point number. For example, 1.0E 4 would
be correctly interpreted as 1.0 × 10<sup>4</sup>.</p>
<p>The word “END” is a special data item. An “END” may have a name or label
associated with it. The name or label associated with an “END” is
separated from the “END” by a single blank and is a maximum of
12 characters long. <em>At least two blanks or a new line MUST follow every
labeled and unlabeled “END.” It is the user’s responsibility to ensure
compliance with this restriction. Failure to observe this restriction
can result in the use of incorrect or incomplete data without the
benefit of warning or error messages.</em></p>
<p>Multiple entries of the same data value can be achieved by specifying
the number of times the data value is to be entered, followed by either
R, *, or $, followed by the data value to be repeated. Imbedded blanks
are not allowed between the number of repeats and the repeat flag. For
example, 5R12, 5*12, 5$12, or 5R 12, etc., will enter five successive
12s in the input data. Multiple zeros can be specified as nZ where n is
the number of zeroes to be entered.</p>
<p>The purpose of this section is to define the input data in discrete
subsections relating to a particular type of data. Tables of the input
data are included in each subsection, and the entries are described in
more detail in the appropriate sections.</p>
<p>Resonance-corrected cross sections are generated using the appropriate
boundary conditions for the unit cell description (i.e., void for the
outer surface of a single unit, white for the outer surface of an
infinite array of cylinders, spheres, or planes). As many unit cells as
needed may be specified in a problem. A unit cell is cell‑weighted by
using the keyword CELLMIX= followed by a unique user specified mixture
number in the unit cell data.</p>
<p>To check the input data without actually processing the cross sections,
the words “PARM=CHECK” or “PARM=CHK” should be entered, as shown below.</p>
<blockquote>
<div><p>=CSAS6    PARM=CHK</p>
</div></blockquote>
<p>or</p>
<blockquote>
<div><p>#CSAS6    PARM=CHK</p>
</div></blockquote>
<p>This will cause the input data for CSAS6 to be checked and appropriate
error messages to be printed. If plots are specified in the data, they
will be printed. This feature allows the user to debug and verify the
input data while using a minimum amount of computer time.</p>
<div class="section" id="xsproc-data">
<h3>XSProc data<a class="headerlink" href="#xsproc-data" title="Permalink to this headline"></a></h3>
<p>The XSProc reads the standard composition specification data and the
unit cell geometry specifications. It then produces the mixing table and
unit cell information necessary for processing the cross sections if
needed. The XSProc section of this manual provides a detailed
description of the input data and processing options.</p>
</div>
<div class="section" id="keno-vi-data">
<h3>KENO-VI data<a class="headerlink" href="#keno-vi-data" title="Permalink to this headline"></a></h3>
<p><a class="reference internal" href="#tab2-2-1"><span class="std std-numref">Table 8</span></a> contains the outline for the KENO-VI input. The KENO-VI
input is divided into 13 data blocks. A brief outline of commonly used
data blocks is shown in <a class="reference internal" href="#tab2-2-1"><span class="std std-numref">Table 8</span></a>. Note that parameter data must
precede all other KENO data blocks. Information on all KENO-VI input is
provided in the KENO chapter of this document and will not be repeated
here.</p>
<span id="tab2-2-1"></span><table class="docutils align-default" id="id4">
<caption><span class="caption-number">Table 8 </span><span class="caption-text">Outline of KENO data</span><a class="headerlink" href="#id4" title="Permalink to this table"></a></caption>
<colgroup>
<col style="width: 25%" />
<col style="width: 25%" />
<col style="width: 25%" />
<col style="width: 25%" />
</colgroup>
<tbody>
<tr class="row-odd"><td><p><strong>Type of
data</strong></p></td>
<td><p><strong>Starting
flag</strong></p></td>
<td><p><strong>Comments</strong></p></td>
<td><p><strong>Termination
flag</strong></p></td>
</tr>
<tr class="row-even"><td><p>Parameters*</p></td>
<td><p>READ PARAMETER</p></td>
<td><p>Enter
desired
parameter
data</p></td>
<td><p>END PARAMETER</p></td>
</tr>
<tr class="row-odd"><td><p>Geometry</p></td>
<td><p>READ GEOMETRY</p></td>
<td><p>Enter
desired
geometry
data</p></td>
<td><p>END GEOMETRY</p></td>
</tr>
<tr class="row-even"><td><p>Array data</p></td>
<td><p>READ ARRAY</p></td>
<td><p>Enter
desired
array data</p></td>
<td><p>END ARRAY</p></td>
</tr>
<tr class="row-odd"><td><p>Boundary
conditions</p></td>
<td><p>READ BOUNDS</p></td>
<td><p>Enter
desired
boundary
conditions</p></td>
<td><p>END BOUNDS</p></td>
</tr>
<tr class="row-even"><td><p>Energy group
boundaries</p></td>
<td><p>READ ENERGY</p></td>
<td><p>Enter
desired
neutron
energy group
boundaries</p></td>
<td><p>END ENERGY</p></td>
</tr>
<tr class="row-odd"><td><p>Start data
or initial
source</p></td>
<td><p>READ START</p></td>
<td><p>Enter
desired
start data</p></td>
<td><p>END START</p></td>
</tr>
<tr class="row-even"><td><p>Plot data</p></td>
<td><p>READ PLOT</p></td>
<td><p>Enter
desired plot
data</p></td>
<td><p>END PLOT</p></td>
</tr>
<tr class="row-odd"><td><p>Grid
geometry
data</p></td>
<td><p>READ GRID</p></td>
<td><p>Enter
desired mesh
data</p></td>
<td><p>END GRID</p></td>
</tr>
<tr class="row-even"><td><p>Reaction</p></td>
<td><p>READ REACTION</p></td>
<td><p>Enter desire
reaction
tallies (CE
mode only)</p></td>
<td><p>END REACTION</p></td>
</tr>
<tr class="row-odd"><td><p>KENO-VI data
terminus</p></td>
<td><p>END DATA</p></td>
<td><div class="line-block">
<div class="line">Enter to
signal the
end of all</div>
<div class="line">KENO-VI
data</div>
</div>
</td>
<td></td>
</tr>
<tr class="row-even"><td><p>*Must precede
all other data
blocks in this
table.</p></td>
<td></td>
<td></td>
<td></td>
</tr>
</tbody>
</table>
</div>
</div>
<div class="section" id="sample-problems">
<h2>Sample Problems<a class="headerlink" href="#sample-problems" title="Permalink to this headline"></a></h2>
<p>This section contains sample problems to demonstrate some of the options
available in CSAS6. A brief problem description and the associated input
data for multigroup mode of calculation are included for each problem.
The same sample problems may be executed in the continuous energy mode
by changing the library name to an continuous-energy library. See
Appendix A (SECTIONREFERENCE) for additional examples.</p>
<div class="section" id="sample-problem-1-aluminum-30-degree-pipe-angle-intersection">
<h3>Sample Problem 1: Aluminum 30 Degree Pipe Angle Intersection<a class="headerlink" href="#sample-problem-1-aluminum-30-degree-pipe-angle-intersection" title="Permalink to this headline"></a></h3>
<p>The purpose of this problem is to calculate the k-effective of a system
composed of intersecting aluminum pipes, in the shape of a Y, filled
with a 5% enriched UO<sub>2</sub>F<sub>2</sub> solution. The
UO<sub>2</sub>F<sub>2</sub> solution at 299°K contains 907.0 gm/l of
uranium, no excess acid, and has a specific gravity of
2.0289 gm/cm<sup>3</sup>. The assembly is composed of a 212.1 cm long
vertical pipe and a second pipe that intersects the vertical pipe
76.7 cm from the outside bottom at an angle of 29.26 degrees with the
upper vertical pipe. Both pipes have 13.95 cm inner diameters and
14.11 cm outer diameters. The vertical pipe is open on the top and
1.3 cm thick on the bottom. The Y-leg pipe, in the YZ-plane, is
126.04 cm in length with the sealed end 0.64 cm thick. The assembly is
filled with solution to a height 129.5 cm above the outside bottom of
the vertical pipe. From the point where the pipes intersect the assembly
is surrounded by water 37.0 cm in the ±X directions, 100 cm in the
+Y direction, –37 cm in the –Y direction, to the top of the assembly in
the +Z direction, and –99.6 cm in the –Z direction.</p>
<div class="figure align-center" id="id5">
<span id="fig2-2-1"></span><a class="reference internal image-reference" href="_images/fig13.png"><img alt="_images/fig13.png" src="_images/fig13.png" style="width: 400px;" /></a>
<p class="caption"><span class="caption-number">Fig. 12 </span><span class="caption-text">Critical assembly of UO<sub>2</sub>F<sub>2</sub> solution in a 30°-Y aluminum pipe.</span><a class="headerlink" href="#id5" title="Permalink to this image"></a></p>
</div>
<div class="highlight-scale notranslate"><div class="highlight"><pre><span></span><span class="nf">=csas6</span>
<span class="err">sample</span> <span class="err">problem</span> <span class="m">1</span>  <span class="err">Y-</span><span class="m">30</span><span class="err">,</span> <span class="m">5</span><span class="err">%uo</span><span class="m">2</span><span class="err">f</span><span class="m">2</span><span class="err">,</span> <span class="m">907</span><span class="err">.</span><span class="m">0</span><span class="err">g/l,</span> <span class="m">128</span><span class="err">.</span><span class="m">2</span><span class="err">,</span> <span class="err">soln.</span> <span class="err">ht.</span>
<span class="err">v</span><span class="m">7</span><span class="err">-</span><span class="m">238</span>
<span class="n">read</span><span class="err"> </span><span class="n">comp</span>
  <span class="err">solution</span>
    <span class="err">mix=</span><span class="m">1</span>
    <span class="err">rho[uo</span><span class="m">2</span><span class="err">f</span><span class="m">2</span><span class="err">]=</span><span class="m">907</span><span class="err">.</span><span class="m">0</span> <span class="m">92235</span> <span class="m">5</span><span class="err">.</span><span class="m">0</span> <span class="m">92238</span> <span class="m">95</span><span class="err">.</span><span class="m">0</span>
    <span class="err">density=?</span>
    <span class="err">temperature=</span><span class="m">299</span><span class="err">.</span><span class="m">0</span>
  <span class="n">end</span><span class="err"> </span><span class="n">solution</span>
  <span class="err">al</span>        <span class="m">2</span> <span class="m">1</span><span class="err">.</span><span class="m">0</span> <span class="n">end</span>
  <span class="err">h</span><span class="m">2</span><span class="err">o</span>       <span class="m">3</span> <span class="m">1</span><span class="err">.</span><span class="m">0</span> <span class="n">end</span>
<span class="n">end</span><span class="err"> </span><span class="n">comp</span>
<span class="n">read</span><span class="err"> </span><span class="n">parameters</span>
  <span class="err">flx=yes</span> <span class="err">fdn=yes</span> <span class="err">far=yes</span> <span class="err">pgm=yes</span> <span class="err">plt=yes</span>
<span class="n">end</span><span class="err"> </span><span class="n">parameters</span>
<span class="n">read</span><span class="err"> </span><span class="n">start</span>
  <span class="err">nst=</span><span class="m">6</span> <span class="err">tfx=</span><span class="m">0</span><span class="err">.</span><span class="m">0</span> <span class="err">tfy=</span><span class="m">0</span><span class="err">.</span><span class="m">0</span> <span class="err">tfz=</span><span class="m">0</span><span class="err">.</span><span class="m">0</span> <span class="err">lnu=</span><span class="m">1000</span>
<span class="n">end</span><span class="err"> </span><span class="n">start</span>
<span class="n">read</span><span class="err"> </span><span class="n">geometry</span>
  <span class="err">global</span>
  <span class="err">unit</span> <span class="m">1</span>
    <span class="err">com=</span><span class="s">&#39;30 deg y&#39;</span>
    <span class="err">cylinder</span> <span class="m">10</span>  <span class="m">13</span><span class="err">.</span><span class="m">95</span>  <span class="m">135</span><span class="err">.</span><span class="m">4</span> <span class="err">-</span><span class="m">75</span><span class="err">.</span><span class="m">4</span>
    <span class="err">cylinder</span> <span class="m">20</span>  <span class="m">14</span><span class="err">.</span><span class="m">11</span>  <span class="m">135</span><span class="err">.</span><span class="m">4</span> <span class="err">-</span><span class="m">76</span><span class="err">.</span><span class="m">7</span>
    <span class="err">cylinder</span> <span class="m">30</span>  <span class="m">13</span><span class="err">.</span><span class="m">95</span>  <span class="m">125</span><span class="err">.</span><span class="m">4</span>   <span class="m">0</span><span class="err">.</span><span class="m">0</span>   <span class="err">rotate</span> <span class="err">a</span><span class="m">2</span><span class="err">=-</span><span class="m">29</span><span class="err">.</span><span class="m">26</span>
    <span class="err">cylinder</span> <span class="m">40</span>  <span class="m">14</span><span class="err">.</span><span class="m">11</span>  <span class="m">126</span><span class="err">.</span><span class="m">04</span>  <span class="m">0</span><span class="err">.</span><span class="m">0</span>   <span class="err">rotate</span> <span class="err">a</span><span class="m">2</span><span class="err">=-</span><span class="m">29</span><span class="err">.</span><span class="m">26</span>
    <span class="err">cuboid</span>   <span class="m">50</span>  <span class="m">2</span><span class="err">p</span><span class="m">37</span><span class="err">.</span><span class="m">0</span> <span class="m">100</span><span class="err">.</span> <span class="err">-</span><span class="m">37</span><span class="err">.</span><span class="m">0</span> <span class="m">52</span><span class="err">.</span><span class="m">8</span> <span class="err">-</span><span class="m">75</span><span class="err">.</span><span class="m">4</span>
    <span class="err">cuboid</span>   <span class="m">60</span>  <span class="m">2</span><span class="err">p</span><span class="m">37</span><span class="err">.</span><span class="m">0</span> <span class="m">100</span><span class="err">.</span> <span class="err">-</span><span class="m">37</span><span class="err">.</span><span class="m">0</span> <span class="m">135</span><span class="err">.</span><span class="m">4</span> <span class="err">-</span><span class="m">99</span><span class="err">.</span><span class="m">6</span>
    <span class="err">media</span>  <span class="m">1</span>  <span class="m">1</span> <span class="m">10</span>  <span class="m">50</span>
    <span class="err">media</span>  <span class="m">2</span>  <span class="m">1</span> <span class="m">20</span> <span class="err">-</span><span class="m">10</span> <span class="err">-</span><span class="m">30</span>
    <span class="err">media</span>  <span class="m">1</span>  <span class="m">1</span> <span class="m">30</span>  <span class="m">50</span> <span class="err">-</span><span class="m">10</span>
    <span class="err">media</span>  <span class="m">2</span>  <span class="m">1</span> <span class="m">40</span> <span class="err">-</span><span class="m">30</span> <span class="err">-</span><span class="m">20</span>
    <span class="err">media</span>  <span class="m">0</span>  <span class="m">1</span> <span class="m">10</span> <span class="err">-</span><span class="m">50</span>
    <span class="err">media</span>  <span class="m">0</span>  <span class="m">1</span> <span class="m">30</span> <span class="err">-</span><span class="m">50</span> <span class="err">-</span><span class="m">10</span>
    <span class="err">media</span>  <span class="m">3</span>  <span class="m">1</span> <span class="m">60</span> <span class="err">-</span><span class="m">20</span> <span class="err">-</span><span class="m">40</span> <span class="err">-</span><span class="m">10</span>
    <span class="err">boundary</span>  <span class="m">60</span>
<span class="n">end</span><span class="err"> </span><span class="n">geometry</span>
<span class="n">read</span><span class="err"> </span><span class="n">volume</span>
    <span class="err">type=random</span>  <span class="err">batches=</span><span class="m">1000</span>
<span class="n">end</span><span class="err"> </span><span class="n">volume</span>
<span class="n">read</span><span class="err"> </span><span class="n">plot</span>
  <span class="err">scr=yes</span>  <span class="err">lpi=</span><span class="m">10</span>
  <span class="err">ttl=</span><span class="s">&#39;y-z slice at x=0.0  through centerline of both pipes&#39;</span>
  <span class="err">xul=</span><span class="m">0</span><span class="err">.</span><span class="m">0</span>  <span class="err">yul=-</span><span class="m">39</span><span class="err">.</span><span class="m">0</span>  <span class="err">zul=</span><span class="m">137</span><span class="err">.</span><span class="m">0</span>
  <span class="err">xlr=</span><span class="m">0</span><span class="err">.</span><span class="m">0</span>  <span class="err">ylr=</span><span class="m">105</span><span class="err">.</span><span class="m">0</span>  <span class="err">zlr=-</span><span class="m">105</span><span class="err">.</span><span class="m">0</span>
  <span class="err">vax=</span><span class="m">1</span>    <span class="err">wdn=-</span><span class="m">1</span>
  <span class="err">nax=</span><span class="m">400</span>  <span class="n">end</span><span class="err"> </span><span class="n">plt</span><span class="err">0</span>
  <span class="err">ttl=</span><span class="s">&#39;x-y slice at z=26.0  slightly above point of separation&#39;</span>
  <span class="err">xul=-</span><span class="m">40</span><span class="err">.</span><span class="m">0</span> <span class="err">yul=</span><span class="m">105</span><span class="err">.</span><span class="m">0</span>  <span class="err">zul=</span><span class="m">26</span><span class="err">.</span><span class="m">0</span>
  <span class="err">xlr=+</span><span class="m">40</span><span class="err">.</span><span class="m">0</span> <span class="err">ylr=-</span><span class="m">40</span><span class="err">.</span><span class="m">0</span>  <span class="err">zlr=</span><span class="m">26</span><span class="err">.</span><span class="m">0</span>
  <span class="err">uax=</span><span class="m">1</span>     <span class="err">vdn=-</span><span class="m">1</span>
  <span class="err">nax=</span><span class="m">400</span> <span class="n">end</span><span class="err"> </span><span class="n">plt</span><span class="err">1</span>
  <span class="err">ttl=</span><span class="s">&#39;x-y slice at z=75.0  well above point of separation&#39;</span>
  <span class="err">xul=-</span><span class="m">40</span><span class="err">.</span><span class="m">0</span> <span class="err">yul=</span><span class="m">105</span><span class="err">.</span><span class="m">0</span>  <span class="err">zul=</span><span class="m">75</span><span class="err">.</span><span class="m">0</span>
  <span class="err">xlr=+</span><span class="m">40</span><span class="err">.</span><span class="m">0</span> <span class="err">ylr=-</span><span class="m">40</span><span class="err">.</span><span class="m">0</span>  <span class="err">zlr=</span><span class="m">75</span><span class="err">.</span><span class="m">0</span>
  <span class="err">uax=</span><span class="m">1</span>     <span class="err">vdn=-</span><span class="m">1</span>
  <span class="err">nax=</span><span class="m">400</span> <span class="n">end</span><span class="err"> </span><span class="n">plt</span><span class="err">2</span>
<span class="n">end</span><span class="err"> </span><span class="n">plot</span>
<span class="n">end</span><span class="err"> </span><span class="n">data</span>
<span class="n">end</span>
</pre></div>
</div>
</div>
<div class="section" id="sample-problem-2-plexiglas-cross">
<h3>Sample Problem 2: Plexiglas Cross<a class="headerlink" href="#sample-problem-2-plexiglas-cross" title="Permalink to this headline"></a></h3>
<p>The purpose of this problem is to calculate the k-effective of a system
composed of intersecting Plexiglas pipes, in the shape of a cross,
filled with a 5% enriched UO<sub>2</sub>F<sub>2</sub> solution. The room
temperature UO<sub>2</sub>F<sub>2</sub> solution contains 896.1 gm/l of
uranium, no excess acid, and has a specific gravity of
2.015 gm/cm<sup>3</sup>. The pipes have a 13.335 cm inner diameter and
16.19 cm outer diameter. The vertical pipe is open on the top and
3.17 cm thick on the bottom. The horizontal pipe ends are 3.17 thick.
The vertical pipe is 210.19 cm in length and filled with solution to a
height of 117.2 cm. The two horizontal legs, positioned in the XZ‑plane,
intersect the vertical pipe 91.44 cm from the outside bottom at an
89 degree angle with the upper section of the pipe. Each horizontal is
91.44 cm in length and filled with the above specified
UO<sub>2</sub>F<sub>2</sub> solution. A water reflector surrounding the
solution filled pipes extends out from the point where the pipes
intersect 111.76 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.</p>
<div class="figure align-center" id="id6">
<span id="fig2-2-2"></span><a class="reference internal image-reference" href="_images/fig22.png"><img alt="_images/fig22.png" src="_images/fig22.png" style="width: 400px;" /></a>
<p class="caption"><span class="caption-number">Fig. 13 </span><span class="caption-text">Critical assembly of UO<sub>2</sub>F<sub>2</sub> solution in a Plexiglas cross.</span><a class="headerlink" href="#id6" title="Permalink to this image"></a></p>
</div>
<div class="highlight-scale notranslate"><div class="highlight"><pre><span></span><span class="nf">=csas6</span>
<span class="err">sample</span> <span class="err">problem</span> <span class="m">2</span>  <span class="m">89</span><span class="err">-cross,</span> <span class="m">5</span><span class="err">%</span> <span class="err">uo</span><span class="m">2</span><span class="err">f</span><span class="m">2</span> <span class="err">soln,</span> <span class="err">plexiglass</span> <span class="err">pipes,</span> <span class="err">h</span><span class="m">2</span><span class="err">o</span> <span class="err">refl.</span>
<span class="err">v</span><span class="m">7</span><span class="err">-</span><span class="m">238</span>
<span class="n">read</span><span class="err"> </span><span class="n">comp</span>
  <span class="err">solution</span>
    <span class="err">mix=</span><span class="m">1</span>
    <span class="err">rho[uo</span><span class="m">2</span><span class="err">f</span><span class="m">2</span><span class="err">]=</span><span class="m">896</span><span class="err">.</span><span class="m">1</span> <span class="m">92235</span> <span class="m">5</span><span class="err">.</span><span class="m">0</span> <span class="m">92238</span> <span class="m">95</span><span class="err">.</span><span class="m">0</span>
    <span class="err">density=?</span>
    <span class="err">temperature=</span><span class="m">298</span><span class="err">.</span><span class="m">0</span>
  <span class="n">end</span><span class="err"> </span><span class="n">solution</span>
  <span class="err">plexiglass</span> <span class="m">3</span> <span class="m">1</span><span class="err">.</span><span class="m">0</span> <span class="n">end</span>
  <span class="err">h</span><span class="m">2</span><span class="err">o</span>        <span class="m">2</span> <span class="m">1</span><span class="err">.</span><span class="m">0</span> <span class="n">end</span>
<span class="n">end</span><span class="err"> </span><span class="n">comp</span>
<span class="n">read</span><span class="err"> </span><span class="n">param</span>
  <span class="err">plt=yes</span>
<span class="n">end</span><span class="err"> </span><span class="n">param</span>
<span class="n">read</span><span class="err"> </span><span class="n">geom</span>
  <span class="err">global</span> <span class="err">unit</span> <span class="m">1</span>
    <span class="err">cylinder</span>  <span class="m">10</span>  <span class="m">13</span><span class="err">.</span><span class="m">335</span> <span class="m">28</span><span class="err">.</span><span class="m">93</span> <span class="err">-</span><span class="m">88</span><span class="err">.</span><span class="m">27</span>
    <span class="err">cylinder</span>  <span class="m">20</span>  <span class="m">13</span><span class="err">.</span><span class="m">335</span> <span class="m">121</span><span class="err">.</span><span class="m">92</span> <span class="err">-</span><span class="m">88</span><span class="err">.</span><span class="m">27</span>
    <span class="err">cylinder</span>  <span class="m">30</span>  <span class="m">16</span><span class="err">.</span><span class="m">19</span> <span class="m">121</span><span class="err">.</span><span class="m">92</span> <span class="err">-</span><span class="m">91</span><span class="err">.</span><span class="m">44</span>
    <span class="err">cylinder</span>  <span class="m">40</span>  <span class="m">13</span><span class="err">.</span><span class="m">335</span> <span class="m">88</span><span class="err">.</span><span class="m">27</span> <span class="m">0</span><span class="err">.</span><span class="m">0</span> <span class="err">rotate</span> <span class="err">a</span><span class="m">1</span><span class="err">=</span><span class="m">90</span><span class="err">.</span> <span class="err">a</span><span class="m">2</span><span class="err">=</span><span class="m">89</span><span class="err">.</span>
    <span class="err">cylinder</span>  <span class="m">50</span>  <span class="m">16</span><span class="err">.</span><span class="m">19</span>  <span class="m">91</span><span class="err">.</span><span class="m">44</span> <span class="m">0</span><span class="err">.</span><span class="m">0</span> <span class="err">rotate</span> <span class="err">a</span><span class="m">1</span><span class="err">=</span><span class="m">90</span><span class="err">.</span> <span class="err">a</span><span class="m">2</span><span class="err">=</span><span class="m">89</span><span class="err">.</span>
    <span class="err">cylinder</span>  <span class="m">60</span>  <span class="m">13</span><span class="err">.</span><span class="m">335</span> <span class="m">88</span><span class="err">.</span><span class="m">27</span> <span class="m">0</span><span class="err">.</span><span class="m">0</span> <span class="err">rotate</span> <span class="err">a</span><span class="m">1</span><span class="err">=-</span><span class="m">90</span><span class="err">.</span> <span class="err">a</span><span class="m">2</span><span class="err">=</span><span class="m">89</span><span class="err">.</span>
    <span class="err">cylinder</span>  <span class="m">70</span>  <span class="m">16</span><span class="err">.</span><span class="m">19</span>  <span class="m">91</span><span class="err">.</span><span class="m">44</span> <span class="m">0</span><span class="err">.</span><span class="m">0</span> <span class="err">rotate</span> <span class="err">a</span><span class="m">1</span><span class="err">=-</span><span class="m">90</span><span class="err">.</span> <span class="err">a</span><span class="m">2</span><span class="err">=</span><span class="m">89</span><span class="err">.</span>
    <span class="err">cuboid</span>    <span class="m">80</span>  <span class="m">2</span><span class="err">p</span><span class="m">111</span><span class="err">.</span><span class="m">74</span> <span class="m">2</span><span class="err">p</span><span class="m">20</span><span class="err">.</span><span class="m">64</span> <span class="m">29</span><span class="err">.</span><span class="m">03</span> <span class="err">-</span><span class="m">118</span><span class="err">.</span><span class="m">428</span>
    <span class="err">cuboid</span>    <span class="m">90</span>  <span class="m">2</span><span class="err">p</span><span class="m">111</span><span class="err">.</span><span class="m">74</span> <span class="m">2</span><span class="err">p</span><span class="m">40</span><span class="err">.</span><span class="m">64</span> <span class="m">121</span><span class="err">.</span><span class="m">92</span> <span class="err">-</span><span class="m">118</span><span class="err">.</span><span class="m">428</span>
    <span class="err">media</span> <span class="m">1</span> <span class="m">1</span> <span class="m">10</span>
    <span class="err">media</span> <span class="m">0</span> <span class="m">1</span> <span class="m">20</span> <span class="err">-</span><span class="m">10</span>
    <span class="err">media</span> <span class="m">3</span> <span class="m">1</span> <span class="m">30</span> <span class="err">-</span><span class="m">10</span> <span class="err">-</span><span class="m">20</span> <span class="err">-</span><span class="m">50</span> <span class="err">-</span><span class="m">70</span>
    <span class="err">media</span> <span class="m">1</span> <span class="m">1</span> <span class="m">40</span> <span class="err">-</span><span class="m">10</span> <span class="err">-</span><span class="m">20</span>
    <span class="err">media</span> <span class="m">3</span> <span class="m">1</span> <span class="m">50</span> <span class="err">-</span><span class="m">40</span> <span class="err">-</span><span class="m">10</span> <span class="err">-</span><span class="m">20</span>
    <span class="err">media</span> <span class="m">1</span> <span class="m">1</span> <span class="m">60</span> <span class="err">-</span><span class="m">10</span> <span class="err">-</span><span class="m">20</span>
    <span class="err">media</span> <span class="m">3</span> <span class="m">1</span> <span class="m">70</span> <span class="err">-</span><span class="m">60</span> <span class="err">-</span><span class="m">10</span> <span class="err">-</span><span class="m">20</span> <span class="err">-</span><span class="m">50</span>
    <span class="err">media</span> <span class="m">2</span> <span class="m">1</span> <span class="m">80</span> <span class="err">-</span><span class="m">10</span> <span class="err">-</span><span class="m">20</span> <span class="err">-</span><span class="m">30</span> <span class="err">-</span><span class="m">40</span> <span class="err">-</span><span class="m">50</span> <span class="err">-</span><span class="m">60</span> <span class="err">-</span><span class="m">70</span>
    <span class="err">media</span> <span class="m">0</span> <span class="m">1</span> <span class="m">90</span> <span class="err">-</span><span class="m">20</span> <span class="err">-</span><span class="m">30</span> <span class="err">-</span><span class="m">80</span>
    <span class="err">boundary</span>  <span class="m">90</span>
<span class="n">end</span><span class="err"> </span><span class="n">geom</span>
<span class="n">read</span><span class="err"> </span><span class="n">volume</span>
   <span class="err">type=trace</span>
<span class="n">end</span><span class="err"> </span><span class="n">volume</span>
<span class="n">read</span><span class="err"> </span><span class="n">start</span>
  <span class="err">nst=</span><span class="m">6</span>  <span class="err">tfx=</span><span class="m">0</span><span class="err">.</span> <span class="err">tfy=</span><span class="m">0</span><span class="err">.</span> <span class="err">tfz=</span><span class="m">0</span><span class="err">.</span> <span class="err">lnu=</span><span class="m">1000</span>
<span class="n">end</span><span class="err"> </span><span class="n">start</span>
<span class="n">read</span><span class="err"> </span><span class="n">plot</span>
  <span class="err">scr=yes</span>  <span class="err">lpi=</span><span class="m">10</span>
  <span class="err">ttl=</span><span class="s">&#39; x-z slice at y=0.0 &#39;</span>
  <span class="err">xul=-</span><span class="m">113</span><span class="err">.</span> <span class="err">yul=</span><span class="m">0</span><span class="err">.</span> <span class="err">zul=</span>  <span class="m">48</span><span class="err">.</span>
  <span class="err">xlr=</span> <span class="m">113</span><span class="err">.</span> <span class="err">ylr=</span><span class="m">0</span><span class="err">.</span> <span class="err">zlr=-</span><span class="m">120</span><span class="err">.</span>
  <span class="err">uax=</span><span class="m">1</span><span class="err">.</span><span class="m">0</span>   <span class="err">wdn=-</span><span class="m">1</span><span class="err">.</span><span class="m">0</span>
  <span class="err">nax=</span><span class="m">400</span>  <span class="n">end</span><span class="err"> </span><span class="n">plt</span><span class="err">0</span>
  <span class="err">ttl=</span><span class="s">&#39; y-z slce at x=0.0 &#39;</span>
  <span class="err">xul=</span><span class="m">0</span><span class="err">.</span>    <span class="err">yul=-</span><span class="m">42</span><span class="err">.</span> <span class="err">zul=</span> <span class="m">122</span><span class="err">.</span>
  <span class="err">xlr=</span><span class="m">0</span><span class="err">.</span>    <span class="err">ylr=</span> <span class="m">42</span><span class="err">.</span> <span class="err">zlr=-</span><span class="m">120</span><span class="err">.</span>
  <span class="err">vax=</span><span class="m">1</span><span class="err">.</span><span class="m">0</span>   <span class="err">wdn=-</span><span class="m">1</span><span class="err">.</span><span class="m">0</span>
  <span class="err">nax=</span><span class="m">400</span>  <span class="n">end</span><span class="err"> </span><span class="n">plt</span><span class="err">1</span>
  <span class="err">ttl=</span><span class="s">&#39; x-y slice at z=0.0 &#39;</span>
  <span class="err">xul=-</span><span class="m">113</span><span class="err">.</span><span class="m">0</span> <span class="err">yul=</span> <span class="m">42</span><span class="err">.</span> <span class="err">zul=</span><span class="m">0</span><span class="err">.</span>
  <span class="err">xlr=</span> <span class="m">113</span><span class="err">.</span><span class="m">0</span> <span class="err">ylr=-</span><span class="m">42</span><span class="err">.</span> <span class="err">zlr=</span><span class="m">0</span><span class="err">.</span>
  <span class="err">uax=</span><span class="m">1</span><span class="err">.</span><span class="m">0</span>    <span class="err">vdn=-</span><span class="m">1</span><span class="err">.</span><span class="m">0</span>
  <span class="err">nax=</span><span class="m">400</span>  <span class="n">end</span><span class="err"> </span><span class="n">plt</span><span class="err">2</span>
<span class="n">end</span><span class="err"> </span><span class="n">plot</span>
<span class="n">end</span><span class="err"> </span><span class="n">data</span>
<span class="n">end</span>
</pre></div>
</div>
</div>
<div class="section" id="sample-problem-3-sphere">
<h3>Sample Problem 3: Sphere<a class="headerlink" href="#sample-problem-3-sphere" title="Permalink to this headline"></a></h3>
<p>This problem models an assembly consisting of a 93.2% enriched bare
uranium sphere, 8.741 cm in radius, having a density of
18.76 gm/cm<sup>3</sup>. Problem 3 models the assembly as a single bare
sphere. The second problem models the assembly as a hemisphere with
mirror reflection on the flat surface. The next three problems model the
sphere using chords. This set of four problems is designed to illustrate
the use of multiple chords in a problem.</p>
<div class="highlight-scale notranslate"><div class="highlight"><pre><span></span><span class="nf">=csas6</span>
<span class="err">sample</span> <span class="err">problem</span> <span class="m">3</span>  <span class="err">bare</span> <span class="m">93</span><span class="err">.</span><span class="m">2</span><span class="err">%</span> <span class="err">enriched</span> <span class="err">uranium</span> <span class="err">sphere</span>
<span class="err">v</span><span class="m">7</span><span class="err">-</span><span class="m">238</span>
<span class="n">read</span><span class="err"> </span><span class="n">comp</span>
  <span class="err">uranium</span>  <span class="m">1</span> <span class="err">den=</span><span class="m">18</span><span class="err">.</span><span class="m">76</span> <span class="m">1</span> <span class="m">293</span> <span class="m">92235</span> <span class="m">93</span><span class="err">.</span><span class="m">2</span> <span class="m">92238</span> <span class="m">5</span><span class="err">.</span><span class="m">6</span> <span class="m">92234</span> <span class="m">1</span><span class="err">.</span><span class="m">0</span> <span class="m">92236</span> <span class="m">0</span><span class="err">.</span><span class="m">2</span> <span class="n">end</span>
<span class="n">end</span><span class="err"> </span><span class="n">comp</span>
<span class="n">read</span><span class="err"> </span><span class="n">geometry</span>
  <span class="err">global</span> <span class="err">unit</span> <span class="m">1</span>
    <span class="err">sphere</span> <span class="m">10</span>  <span class="m">8</span><span class="err">.</span><span class="m">741</span>
    <span class="err">cuboid</span>   <span class="m">20</span>  <span class="m">6</span><span class="err">p</span><span class="m">8</span><span class="err">.</span><span class="m">741</span>
    <span class="err">media</span>  <span class="m">1</span> <span class="m">1</span> <span class="m">10</span>      <span class="err">vol=</span><span class="m">2797</span><span class="err">.</span><span class="m">5121</span>
    <span class="err">media</span>  <span class="m">0</span> <span class="m">1</span> <span class="m">20</span> <span class="err">-</span><span class="m">10</span>  <span class="err">vol=</span><span class="m">2545</span><span class="err">.</span><span class="m">3424</span>
    <span class="err">boundary</span> <span class="m">20</span>
<span class="n">end</span><span class="err"> </span><span class="n">geometry</span>
<span class="n">end</span><span class="err"> </span><span class="n">data</span>
<span class="n">end</span>
</pre></div>
</div>
</div>
<div class="section" id="sample-problem-4-sphere-models-using-chords-and-mirror-albedos">
<h3>Sample Problem 4: Sphere Models Using Chords and Mirror Albedos<a class="headerlink" href="#sample-problem-4-sphere-models-using-chords-and-mirror-albedos" title="Permalink to this headline"></a></h3>
<p>This problem models an assembly consisting of a 93.2% enriched bare
uranium sphere, 8.741 cm in radius, having a density of
18.76 gm/cm<sup>3</sup>. The problem models the assembly as a hemisphere
with mirror reflection on the flat surface.</p>
<div class="highlight-scale notranslate"><div class="highlight"><pre><span></span><span class="nf">=csas6</span>
<span class="err">sample</span> <span class="err">problem</span> <span class="m">4</span>  <span class="err">bare</span> <span class="m">93</span><span class="err">.</span><span class="m">2</span><span class="err">%</span> <span class="err">U</span> <span class="err">sphere,</span> <span class="err">hemisphere</span> <span class="err">w/</span> <span class="err">mirror</span> <span class="err">albedo</span>
<span class="err">v</span><span class="m">7</span><span class="err">-</span><span class="m">238</span>
<span class="n">read</span><span class="err"> </span><span class="n">comp</span>
  <span class="err">uranium</span>  <span class="m">1</span> <span class="err">den=</span><span class="m">18</span><span class="err">.</span><span class="m">76</span> <span class="m">1</span> <span class="m">293</span> <span class="m">92235</span> <span class="m">93</span><span class="err">.</span><span class="m">2</span> <span class="m">92238</span> <span class="m">5</span><span class="err">.</span><span class="m">6</span> <span class="m">92234</span> <span class="m">1</span><span class="err">.</span><span class="m">0</span> <span class="m">92236</span> <span class="m">0</span><span class="err">.</span><span class="m">2</span> <span class="n">end</span>
<span class="n">end</span><span class="err"> </span><span class="n">comp</span>
<span class="n">read</span><span class="err"> </span><span class="n">geometry</span>
  <span class="err">global</span> <span class="err">unit</span> <span class="m">1</span>
    <span class="err">sphere</span> <span class="m">10</span>  <span class="m">8</span><span class="err">.</span><span class="m">741</span>  <span class="err">chord</span> <span class="err">+x=</span><span class="m">0</span><span class="err">.</span><span class="m">0</span>
    <span class="err">cuboid</span>   <span class="m">20</span>  <span class="m">8</span><span class="err">.</span><span class="m">741</span> <span class="m">0</span><span class="err">.</span><span class="m">0</span>  <span class="m">8</span><span class="err">.</span><span class="m">741</span> <span class="err">-</span><span class="m">8</span><span class="err">.</span><span class="m">741</span> <span class="m">8</span><span class="err">.</span><span class="m">741</span> <span class="err">-</span><span class="m">8</span><span class="err">.</span><span class="m">741</span>
    <span class="err">media</span>  <span class="m">1</span> <span class="m">1</span> <span class="m">10</span>      <span class="err">vol=</span><span class="m">2797</span><span class="err">.</span><span class="m">5121</span>
    <span class="err">media</span>  <span class="m">0</span> <span class="m">1</span> <span class="m">20</span> <span class="err">-</span><span class="m">10</span>  <span class="err">vol=</span><span class="m">2545</span><span class="err">.</span><span class="m">3424</span>
    <span class="err">boundary</span> <span class="m">20</span>
<span class="n">end</span><span class="err"> </span><span class="n">geometry</span>
<span class="n">read</span><span class="err"> </span><span class="n">bounds</span>
  <span class="err">-xb=mirror</span>
<span class="n">end</span><span class="err"> </span><span class="n">bounds</span>
<span class="n">end</span><span class="err"> </span><span class="n">data</span>
<span class="n">end</span>
</pre></div>
</div>
</div>
<div class="section" id="sample-problem-5-sphere-models-using-chords-and-mirror-albedos">
<h3>Sample Problem 5: Sphere Models Using Chords and Mirror Albedos<a class="headerlink" href="#sample-problem-5-sphere-models-using-chords-and-mirror-albedos" title="Permalink to this headline"></a></h3>
<p>This problem models an assembly consisting of a 93.2% enriched bare
uranium sphere, 8.741 cm in radius, having a density of
18.76 gm/cm<sup>3</sup>. The problem models the assembly as a quarter
sphere with mirror reflection on the two flat surfaces.</p>
<div class="highlight-scale notranslate"><div class="highlight"><pre><span></span><span class="nf">=csas6</span>
<span class="err">sample</span> <span class="err">problem</span> <span class="m">5</span>  <span class="err">bare</span> <span class="m">93</span><span class="err">.</span><span class="m">2</span><span class="err">%</span> <span class="err">U</span> <span class="err">sphere,</span> <span class="err">quarter</span> <span class="err">sphere</span> <span class="err">w/</span> <span class="err">mirror</span> <span class="err">albedo</span>
<span class="err">v</span><span class="m">7</span><span class="err">-</span><span class="m">238</span>
<span class="n">read</span><span class="err"> </span><span class="n">comp</span>
  <span class="err">uranium</span>  <span class="m">1</span> <span class="err">den=</span><span class="m">18</span><span class="err">.</span><span class="m">76</span> <span class="m">1</span> <span class="m">293</span> <span class="m">92235</span> <span class="m">93</span><span class="err">.</span><span class="m">2</span> <span class="m">92238</span> <span class="m">5</span><span class="err">.</span><span class="m">6</span> <span class="m">92234</span> <span class="m">1</span><span class="err">.</span><span class="m">0</span> <span class="m">92236</span> <span class="m">0</span><span class="err">.</span><span class="m">2</span> <span class="n">end</span>
<span class="n">end</span><span class="err"> </span><span class="n">comp</span>
<span class="n">read</span><span class="err"> </span><span class="n">geometry</span>
  <span class="err">global</span> <span class="err">unit</span> <span class="m">1</span>
    <span class="err">sphere</span> <span class="m">10</span>  <span class="m">8</span><span class="err">.</span><span class="m">741</span>  <span class="err">chord</span> <span class="err">+x=</span><span class="m">0</span><span class="err">.</span><span class="m">0</span>  <span class="err">chord</span>  <span class="err">+y=</span><span class="m">0</span><span class="err">.</span><span class="m">0</span>
    <span class="err">cuboid</span>   <span class="m">20</span>  <span class="m">8</span><span class="err">.</span><span class="m">741</span> <span class="m">0</span><span class="err">.</span><span class="m">0</span> <span class="m">8</span><span class="err">.</span><span class="m">741</span> <span class="m">0</span><span class="err">.</span><span class="m">0</span> <span class="m">8</span><span class="err">.</span><span class="m">741</span> <span class="err">-</span><span class="m">8</span><span class="err">.</span><span class="m">741</span>
    <span class="err">media</span>  <span class="m">1</span> <span class="m">1</span> <span class="m">10</span>      <span class="err">vol=</span><span class="m">2797</span><span class="err">.</span><span class="m">5121</span>
    <span class="err">media</span>  <span class="m">0</span> <span class="m">1</span> <span class="m">20</span> <span class="err">-</span><span class="m">10</span>  <span class="err">vol=</span><span class="m">2545</span><span class="err">.</span><span class="m">3424</span>
    <span class="err">boundary</span> <span class="m">20</span>
<span class="n">end</span><span class="err"> </span><span class="n">geometry</span>
<span class="n">read</span><span class="err"> </span><span class="n">bounds</span>
  <span class="err">-xy=mirror</span>
<span class="n">end</span><span class="err"> </span><span class="n">bounds</span>
<span class="n">end</span><span class="err"> </span><span class="n">data</span>
<span class="n">end</span>
</pre></div>
</div>
</div>
<div class="section" id="sample-problem-6-sphere-models-using-chords-and-mirror-albedos-eighth-sphere">
<h3>Sample Problem 6: Sphere Models Using Chords and Mirror Albedos (Eighth Sphere)<a class="headerlink" href="#sample-problem-6-sphere-models-using-chords-and-mirror-albedos-eighth-sphere" title="Permalink to this headline"></a></h3>
<p>This problem models an assembly consisting of a 93.2% enriched bare
uranium sphere, 8.741 cm in radius, having a density of
18.76 gm/cm<sup>3</sup>. The problem models the assembly as an eighth
sphere with mirror reflection on the three flat surfaces.</p>
<div class="highlight-scale notranslate"><div class="highlight"><pre><span></span><span class="nf">=csas6</span>
<span class="err">sample</span> <span class="err">problem</span> <span class="m">6</span>  <span class="err">bare</span> <span class="m">93</span><span class="err">.</span><span class="m">2</span><span class="err">%</span> <span class="err">U</span> <span class="err">sphere,</span> <span class="err">eighth</span> <span class="err">sphere</span> <span class="err">w/</span> <span class="err">mirror</span> <span class="err">albedo</span>
<span class="err">v</span><span class="m">7</span><span class="err">-</span><span class="m">238</span>
<span class="n">read</span><span class="err"> </span><span class="n">comp</span>
  <span class="err">uranium</span>  <span class="m">1</span> <span class="err">den=</span><span class="m">18</span><span class="err">.</span><span class="m">76</span> <span class="m">1</span> <span class="m">293</span> <span class="m">92235</span> <span class="m">93</span><span class="err">.</span><span class="m">2</span> <span class="m">92238</span> <span class="m">5</span><span class="err">.</span><span class="m">6</span> <span class="m">92234</span> <span class="m">1</span><span class="err">.</span><span class="m">0</span> <span class="m">92236</span> <span class="m">0</span><span class="err">.</span><span class="m">2</span> <span class="n">end</span>
<span class="n">end</span><span class="err"> </span><span class="n">comp</span>
<span class="n">read</span><span class="err"> </span><span class="n">geometry</span>
  <span class="err">global</span> <span class="err">unit</span> <span class="m">1</span>
    <span class="err">sphere</span> <span class="m">10</span>  <span class="m">8</span><span class="err">.</span><span class="m">741</span>  <span class="err">chord</span> <span class="err">+x=</span><span class="m">0</span><span class="err">.</span><span class="m">0</span>  <span class="err">chord</span>  <span class="err">+y=</span><span class="m">0</span><span class="err">.</span><span class="m">0</span>  <span class="err">chord</span> <span class="err">+z=</span><span class="m">0</span><span class="err">.</span><span class="m">0</span>
    <span class="err">cuboid</span>   <span class="m">20</span>  <span class="m">8</span><span class="err">.</span><span class="m">741</span> <span class="m">0</span><span class="err">.</span><span class="m">0</span> <span class="m">8</span><span class="err">.</span><span class="m">741</span> <span class="m">0</span><span class="err">.</span><span class="m">0</span> <span class="m">8</span><span class="err">.</span><span class="m">741</span> <span class="m">0</span><span class="err">.</span><span class="m">0</span>
    <span class="err">media</span>  <span class="m">1</span> <span class="m">1</span> <span class="m">10</span>      <span class="err">vol=</span><span class="m">2797</span><span class="err">.</span><span class="m">5121</span>
    <span class="err">media</span>  <span class="m">0</span> <span class="m">1</span> <span class="m">20</span> <span class="err">-</span><span class="m">10</span>  <span class="err">vol=</span><span class="m">2545</span><span class="err">.</span><span class="m">3424</span>
    <span class="err">boundary</span> <span class="m">20</span>
<span class="n">end</span><span class="err"> </span><span class="n">geometry</span>
<span class="n">read</span><span class="err"> </span><span class="n">bounds</span>
  <span class="err">-fc=mirror</span>
<span class="n">end</span><span class="err"> </span><span class="n">bounds</span>
<span class="n">end</span><span class="err"> </span><span class="n">data</span>
<span class="n">end</span>
</pre></div>
</div>
</div>
<div class="section" id="sample-problem-7-grotesque-without-the-diaphragm">
<h3>Sample Problem 7: Grotesque without the Diaphragm<a class="headerlink" href="#sample-problem-7-grotesque-without-the-diaphragm" title="Permalink to this headline"></a></h3>
<p>The purpose of this problem is to calculate the <em>k</em><sub>eff</sub> of a system
composed of eight enriched uranium units placed on a diaphragm, with an
irregularly shaped centerpiece positioned in the center hole of the
diaphragm <a class="bibtex reference internal" href="#mihalczo-brief-1999" id="id1">[Mih99]</a> The assembly and centerpiece are shown in <a class="reference internal" href="#fig2-2-3"><span class="std std-numref">Fig. 14</span></a>,
which is Fig. 4 from Ref. 1. The eight units consist of an approximate
parallelepiped with an irregular top, a parallelepiped, and
six cylinders of various sizes. The centerpiece, which penetrates the
hole in the diaphragm, consists of a cylinder topped by a parallelepiped
topped by a hemisphere. The diaphragm is not modeled in this example.</p>
<div class="figure align-center" id="id7">
<span id="fig2-2-3"></span><a class="reference internal image-reference" href="_images/fig32.png"><img alt="_images/fig32.png" src="_images/fig32.png" style="width: 400px;" /></a>
<p class="caption"><span class="caption-number">Fig. 14 </span><span class="caption-text">Grotesque experimental setup.</span><a class="headerlink" href="#id7" title="Permalink to this image"></a></p>
</div>
<div class="highlight-scale notranslate"><div class="highlight"><pre><span></span><span class="nf">=csas6</span>
<span class="err">sample</span> <span class="err">problem</span> <span class="m">7</span>  <span class="err">keno-vi</span> <span class="err">grotesque</span> <span class="err">w/o</span> <span class="err">diaphragm,</span> <span class="err">ornl/csd/tm-</span><span class="m">220</span>
<span class="err">v</span><span class="m">7</span><span class="err">-</span><span class="m">238</span>
<span class="n">read</span><span class="err"> </span><span class="n">comp</span>
  <span class="err">uranium</span>   <span class="m">1</span> <span class="err">den=</span><span class="m">18</span><span class="err">.</span><span class="m">76</span> <span class="m">1</span> <span class="m">293</span> <span class="m">92235</span> <span class="m">93</span><span class="err">.</span><span class="m">2</span> <span class="m">92238</span> <span class="m">5</span><span class="err">.</span><span class="m">6</span> <span class="m">92234</span> <span class="m">1</span><span class="err">.</span><span class="m">0</span> <span class="m">92236</span> <span class="m">0</span><span class="err">.</span><span class="m">2</span> <span class="n">end</span>
  <span class="err">uranium</span>   <span class="m">2</span> <span class="err">den=</span><span class="m">18</span><span class="err">.</span><span class="m">76</span> <span class="m">1</span> <span class="m">293</span> <span class="m">92235</span> <span class="m">93</span><span class="err">.</span><span class="m">2</span> <span class="m">92238</span> <span class="m">5</span><span class="err">.</span><span class="m">6</span> <span class="m">92234</span> <span class="m">1</span><span class="err">.</span><span class="m">0</span> <span class="m">92236</span> <span class="m">0</span><span class="err">.</span><span class="m">2</span> <span class="n">end</span>
  <span class="err">uranium</span>   <span class="m">3</span> <span class="err">den=</span><span class="m">18</span><span class="err">.</span><span class="m">76</span> <span class="m">1</span> <span class="m">293</span> <span class="m">92235</span> <span class="m">93</span><span class="err">.</span><span class="m">2</span> <span class="m">92238</span> <span class="m">5</span><span class="err">.</span><span class="m">6</span> <span class="m">92234</span> <span class="m">1</span><span class="err">.</span><span class="m">0</span> <span class="m">92236</span> <span class="m">0</span><span class="err">.</span><span class="m">2</span> <span class="n">end</span>
  <span class="err">uranium</span>   <span class="m">4</span> <span class="err">den=</span><span class="m">18</span><span class="err">.</span><span class="m">76</span> <span class="m">1</span> <span class="m">293</span> <span class="m">92235</span> <span class="m">93</span><span class="err">.</span><span class="m">2</span> <span class="m">92238</span> <span class="m">5</span><span class="err">.</span><span class="m">6</span> <span class="m">92234</span> <span class="m">1</span><span class="err">.</span><span class="m">0</span> <span class="m">92236</span> <span class="m">0</span><span class="err">.</span><span class="m">2</span> <span class="n">end</span>
  <span class="err">uranium</span>   <span class="m">5</span> <span class="err">den=</span><span class="m">18</span><span class="err">.</span><span class="m">76</span> <span class="m">1</span> <span class="m">293</span> <span class="m">92235</span> <span class="m">93</span><span class="err">.</span><span class="m">2</span> <span class="m">92238</span> <span class="m">5</span><span class="err">.</span><span class="m">6</span> <span class="m">92234</span> <span class="m">1</span><span class="err">.</span><span class="m">0</span> <span class="m">92236</span> <span class="m">0</span><span class="err">.</span><span class="m">2</span> <span class="n">end</span>
  <span class="err">uranium</span>   <span class="m">6</span> <span class="err">den=</span><span class="m">18</span><span class="err">.</span><span class="m">76</span> <span class="m">1</span> <span class="m">293</span> <span class="m">92235</span> <span class="m">93</span><span class="err">.</span><span class="m">2</span> <span class="m">92238</span> <span class="m">5</span><span class="err">.</span><span class="m">6</span> <span class="m">92234</span> <span class="m">1</span><span class="err">.</span><span class="m">0</span> <span class="m">92236</span> <span class="m">0</span><span class="err">.</span><span class="m">2</span> <span class="n">end</span>
  <span class="err">uranium</span>   <span class="m">7</span> <span class="err">den=</span><span class="m">18</span><span class="err">.</span><span class="m">76</span> <span class="m">1</span> <span class="m">293</span> <span class="m">92235</span> <span class="m">93</span><span class="err">.</span><span class="m">2</span> <span class="m">92238</span> <span class="m">5</span><span class="err">.</span><span class="m">6</span> <span class="m">92234</span> <span class="m">1</span><span class="err">.</span><span class="m">0</span> <span class="m">92236</span> <span class="m">0</span><span class="err">.</span><span class="m">2</span> <span class="n">end</span>
  <span class="err">uranium</span>   <span class="m">8</span> <span class="err">den=</span><span class="m">18</span><span class="err">.</span><span class="m">76</span> <span class="m">1</span> <span class="m">293</span> <span class="m">92235</span> <span class="m">93</span><span class="err">.</span><span class="m">2</span> <span class="m">92238</span> <span class="m">5</span><span class="err">.</span><span class="m">6</span> <span class="m">92234</span> <span class="m">1</span><span class="err">.</span><span class="m">0</span> <span class="m">92236</span> <span class="m">0</span><span class="err">.</span><span class="m">2</span> <span class="n">end</span>
  <span class="err">uranium</span>   <span class="m">9</span> <span class="err">den=</span><span class="m">18</span><span class="err">.</span><span class="m">76</span> <span class="m">1</span> <span class="m">293</span> <span class="m">92235</span> <span class="m">93</span><span class="err">.</span><span class="m">2</span> <span class="m">92238</span> <span class="m">5</span><span class="err">.</span><span class="m">6</span> <span class="m">92234</span> <span class="m">1</span><span class="err">.</span><span class="m">0</span> <span class="m">92236</span> <span class="m">0</span><span class="err">.</span><span class="m">2</span> <span class="n">end</span>
  <span class="err">uranium</span>  <span class="m">10</span> <span class="err">den=</span><span class="m">18</span><span class="err">.</span><span class="m">76</span> <span class="m">1</span> <span class="m">293</span> <span class="m">92235</span> <span class="m">93</span><span class="err">.</span><span class="m">2</span> <span class="m">92238</span> <span class="m">5</span><span class="err">.</span><span class="m">6</span> <span class="m">92234</span> <span class="m">1</span><span class="err">.</span><span class="m">0</span> <span class="m">92236</span> <span class="m">0</span><span class="err">.</span><span class="m">2</span> <span class="n">end</span>
  <span class="err">uranium</span>  <span class="m">11</span> <span class="err">den=</span><span class="m">18</span><span class="err">.</span><span class="m">76</span> <span class="m">1</span> <span class="m">293</span> <span class="m">92235</span> <span class="m">93</span><span class="err">.</span><span class="m">2</span> <span class="m">92238</span> <span class="m">5</span><span class="err">.</span><span class="m">6</span> <span class="m">92234</span> <span class="m">1</span><span class="err">.</span><span class="m">0</span> <span class="m">92236</span> <span class="m">0</span><span class="err">.</span><span class="m">2</span> <span class="n">end</span>
  <span class="err">uranium</span>  <span class="m">12</span> <span class="err">den=</span><span class="m">18</span><span class="err">.</span><span class="m">76</span> <span class="m">1</span> <span class="m">293</span> <span class="m">92235</span> <span class="m">93</span><span class="err">.</span><span class="m">2</span> <span class="m">92238</span> <span class="m">5</span><span class="err">.</span><span class="m">6</span> <span class="m">92234</span> <span class="m">1</span><span class="err">.</span><span class="m">0</span> <span class="m">92236</span> <span class="m">0</span><span class="err">.</span><span class="m">2</span> <span class="n">end</span>
  <span class="err">uranium</span>  <span class="m">13</span> <span class="err">den=</span><span class="m">18</span><span class="err">.</span><span class="m">76</span> <span class="m">1</span> <span class="m">293</span> <span class="m">92235</span> <span class="m">93</span><span class="err">.</span><span class="m">2</span> <span class="m">92238</span> <span class="m">5</span><span class="err">.</span><span class="m">6</span> <span class="m">92234</span> <span class="m">1</span><span class="err">.</span><span class="m">0</span> <span class="m">92236</span> <span class="m">0</span><span class="err">.</span><span class="m">2</span> <span class="n">end</span>
  <span class="err">uranium</span>  <span class="m">14</span> <span class="err">den=</span><span class="m">18</span><span class="err">.</span><span class="m">76</span> <span class="m">1</span> <span class="m">293</span> <span class="m">92235</span> <span class="m">93</span><span class="err">.</span><span class="m">2</span> <span class="m">92238</span> <span class="m">5</span><span class="err">.</span><span class="m">6</span> <span class="m">92234</span> <span class="m">1</span><span class="err">.</span><span class="m">0</span> <span class="m">92236</span> <span class="m">0</span><span class="err">.</span><span class="m">2</span> <span class="n">end</span>
<span class="n">end</span><span class="err"> </span><span class="n">comp</span>
<span class="n">read</span><span class="err"> </span><span class="n">param</span>
  <span class="err">pgm=yes</span> <span class="err">plt=yes</span>
<span class="n">end</span><span class="err"> </span><span class="n">param</span>
<span class="n">read</span><span class="err"> </span><span class="n">geom</span>
  <span class="err">global</span> <span class="err">unit</span> <span class="m">1</span>
<span class="c">&#39;*** one through three is item 1 in drawing 84-10649 ornl/csd/tm-220 ***</span>
<span class="c">&#39;one    top piece of item 1</span>
  <span class="err">cuboid</span> <span class="m">10</span>  <span class="m">2</span><span class="err">p</span><span class="m">6</span><span class="err">.</span><span class="m">3515</span> <span class="m">1</span><span class="err">.</span><span class="m">2685</span> <span class="err">-</span><span class="m">3</span><span class="err">.</span><span class="m">8115</span> <span class="m">13</span><span class="err">.</span><span class="m">377</span> <span class="m">13</span><span class="err">.</span><span class="m">058</span> <span class="err">origin</span> <span class="err">y=-</span><span class="m">17</span><span class="err">.</span><span class="m">464</span> <span class="err">z=</span><span class="m">0</span><span class="err">.</span><span class="m">15</span> <span class="err">rotate</span> <span class="err">a</span><span class="m">2</span><span class="err">=-</span><span class="m">1</span><span class="err">.</span><span class="m">35</span>
<span class="c">&#39;two    middle piece of item 1</span>
  <span class="err">cuboid</span> <span class="m">20</span>  <span class="m">2</span><span class="err">p</span><span class="m">6</span><span class="err">.</span><span class="m">3515</span> <span class="m">6</span><span class="err">.</span><span class="m">3515</span> <span class="err">-</span><span class="m">3</span><span class="err">.</span><span class="m">8115</span> <span class="m">13</span><span class="err">.</span><span class="m">058</span> <span class="m">11</span><span class="err">.</span><span class="m">155</span> <span class="err">origin</span> <span class="err">y=-</span><span class="m">17</span><span class="err">.</span><span class="m">464</span> <span class="err">z=</span><span class="m">0</span><span class="err">.</span><span class="m">15</span> <span class="err">rotate</span> <span class="err">a</span><span class="m">2</span><span class="err">=-</span><span class="m">1</span><span class="err">.</span><span class="m">35</span>
<span class="c">&#39;three  bottom piece of item 1</span>
  <span class="err">cuboid</span> <span class="m">30</span>  <span class="m">4</span><span class="err">p</span><span class="m">6</span><span class="err">.</span><span class="m">3515</span> <span class="m">11</span><span class="err">.</span><span class="m">155</span> <span class="m">0</span><span class="err">.</span> <span class="err">origin</span> <span class="err">y=-</span><span class="m">17</span><span class="err">.</span><span class="m">464</span> <span class="err">z=</span><span class="m">0</span><span class="err">.</span><span class="m">15</span> <span class="err">rotate</span> <span class="err">a</span><span class="m">2</span><span class="err">=-</span><span class="m">1</span><span class="err">.</span><span class="m">35</span>
<span class="c">&#39;*** four is item 2 in drawing 84-10649 ornl/csd/tm-220 ***</span>
  <span class="err">cylinder</span> <span class="m">40</span> <span class="m">4</span><span class="err">.</span><span class="m">555</span> <span class="m">12</span><span class="err">.</span><span class="m">918</span> <span class="m">0</span><span class="err">.</span>        <span class="err">origin</span> <span class="err">x=-</span><span class="m">12</span><span class="err">.</span><span class="m">176</span> <span class="err">y=-</span><span class="m">9</span><span class="err">.</span><span class="m">343</span> <span class="err">z=</span><span class="m">0</span><span class="err">.</span><span class="m">111</span> <span class="err">rotate</span> <span class="err">a</span><span class="m">1</span><span class="err">=-</span><span class="m">52</span><span class="err">.</span><span class="m">5</span> <span class="err">a</span><span class="m">2</span><span class="err">=-</span><span class="m">1</span><span class="err">.</span><span class="m">400</span>
<span class="c">&#39;*** five is item 3 in drawing 84-10649 ornl/csd/tm-220 ***</span>
  <span class="err">cylinder</span> <span class="m">50</span> <span class="m">5</span><span class="err">.</span><span class="m">761</span> <span class="m">13</span><span class="err">.</span><span class="m">475</span> <span class="m">0</span><span class="err">.</span>        <span class="err">origin</span> <span class="err">x=-</span><span class="m">16</span><span class="err">.</span><span class="m">333</span> <span class="err">y=</span><span class="m">1</span><span class="err">.</span><span class="m">681</span> <span class="err">z=</span><span class="m">0</span><span class="err">.</span><span class="m">174</span> <span class="err">rotate</span> <span class="err">a</span><span class="m">1</span><span class="err">=</span><span class="m">83</span><span class="err">.</span><span class="m">5</span> <span class="err">a</span><span class="m">2</span><span class="err">=+</span><span class="m">1</span><span class="err">.</span><span class="m">173</span>
<span class="c">&#39;*** six is item 4 in drawing 84-10649 ornl/csd/tm-220 ***</span>
  <span class="err">cylinder</span> <span class="m">60</span> <span class="m">4</span><span class="err">.</span><span class="m">5525</span> <span class="m">12</span><span class="err">.</span><span class="m">969</span> <span class="m">0</span><span class="err">.</span>    <span class="err">origin</span> <span class="err">x=-</span><span class="m">9</span><span class="err">.</span><span class="m">539</span> <span class="err">y=</span><span class="m">11</span><span class="err">.</span><span class="m">168</span> <span class="err">z=</span><span class="m">0</span><span class="err">.</span><span class="m">156</span> <span class="err">rotate</span> <span class="err">a</span><span class="m">1</span><span class="err">=</span><span class="m">40</span><span class="err">.</span><span class="m">5</span> <span class="err">a</span><span class="m">2</span><span class="err">=+</span><span class="m">1</span><span class="err">.</span><span class="m">970</span>
<span class="c">&#39;*** seven and eight are item 5 in drawing 84-10649 ornl/csd/tm-220 ***</span>
<span class="c">&#39;seven</span>
  <span class="err">cuboid</span>  <span class="m">70</span>  <span class="m">2</span><span class="err">p</span><span class="m">3</span><span class="err">.</span><span class="m">81</span> <span class="m">8</span><span class="err">.</span><span class="m">13</span> <span class="err">-</span><span class="m">4</span><span class="err">.</span><span class="m">573</span> <span class="m">8</span><span class="err">.</span><span class="m">91</span> <span class="m">0</span><span class="err">.</span> <span class="err">origin</span> <span class="err">y=</span><span class="m">15</span><span class="err">.</span><span class="m">698</span> <span class="err">z=</span><span class="m">0</span><span class="err">.</span><span class="m">290</span> <span class="err">rotate</span> <span class="err">a</span><span class="m">2</span><span class="err">=+</span><span class="m">2</span><span class="err">.</span><span class="m">58</span>
<span class="c">&#39;eight</span>
  <span class="err">cylinder</span> <span class="m">80</span> <span class="m">4</span><span class="err">.</span><span class="m">573</span> <span class="m">13</span><span class="err">.</span><span class="m">229</span> <span class="m">8</span><span class="err">.</span><span class="m">91</span>   <span class="err">origin</span> <span class="err">y=</span><span class="m">15</span><span class="err">.</span><span class="m">698</span> <span class="err">z=</span><span class="m">0</span><span class="err">.</span><span class="m">290</span> <span class="err">rotate</span> <span class="err">a</span><span class="m">2</span><span class="err">=+</span><span class="m">2</span><span class="err">.</span><span class="m">58</span>
<span class="c">&#39;*** nine is item 6 in drawing 84-10649 ornl/csd/tm-220 ***</span>
  <span class="err">cylinder</span> <span class="m">90</span> <span class="m">4</span><span class="err">.</span><span class="m">5545</span> <span class="m">12</span><span class="err">.</span><span class="m">974</span> <span class="m">0</span><span class="err">.</span>    <span class="err">origin</span> <span class="err">x=</span><span class="m">9</span><span class="err">.</span><span class="m">854</span> <span class="err">y=</span><span class="m">10</span><span class="err">.</span><span class="m">964</span> <span class="err">z=</span><span class="m">0</span><span class="err">.</span><span class="m">134</span> <span class="err">rotate</span> <span class="err">a</span><span class="m">1</span><span class="err">=-</span><span class="m">42</span><span class="err">.</span><span class="m">0</span> <span class="err">a</span><span class="m">2</span><span class="err">=+</span><span class="m">1</span><span class="err">.</span><span class="m">680</span>
<span class="c">&#39;*** ten is item 7 in drawing 84-10649 ornl/csd/tm-220 ***</span>
  <span class="err">cylinder</span> <span class="m">100</span> <span class="m">5</span><span class="err">.</span><span class="m">7495</span> <span class="m">13</span><span class="err">.</span><span class="m">475</span> <span class="m">0</span><span class="err">.</span>    <span class="err">origin</span> <span class="err">x=</span><span class="m">16</span><span class="err">.</span><span class="m">388</span> <span class="err">y=</span><span class="m">1</span><span class="err">.</span><span class="m">434</span> <span class="err">z=</span><span class="m">0</span><span class="err">.</span><span class="m">140</span> <span class="err">rotate</span> <span class="err">a</span><span class="m">1</span><span class="err">=-</span><span class="m">86</span><span class="err">.</span><span class="m">0</span> <span class="err">a</span><span class="m">2</span><span class="err">=+</span><span class="m">1</span><span class="err">.</span><span class="m">400</span>
<span class="c">&#39;*** eleven is item 8 in drawing 84-10649 ornl/csd/tm-220 ***</span>
  <span class="err">cylinder</span> <span class="m">110</span> <span class="m">4</span><span class="err">.</span><span class="m">5565</span> <span class="m">12</span><span class="err">.</span><span class="m">954</span> <span class="m">0</span><span class="err">.</span>    <span class="err">origin</span> <span class="err">x=</span><span class="m">12</span><span class="err">.</span><span class="m">029</span> <span class="err">y=-</span><span class="m">9</span><span class="err">.</span><span class="m">398</span> <span class="err">z=</span><span class="m">0</span><span class="err">.</span><span class="m">087</span> <span class="err">rotate</span> <span class="err">a</span><span class="m">1</span><span class="err">=</span><span class="m">38</span><span class="err">.</span><span class="m">0</span> <span class="err">a</span><span class="m">2</span><span class="err">=-</span><span class="m">1</span><span class="err">.</span><span class="m">100</span>
<span class="c">&#39;*12 through 14 is the centerpiece in drawing 84-10649 ornl/csd/tm-220</span>
<span class="c">&#39;twelve</span>
  <span class="err">cylinder</span> <span class="m">120</span> <span class="m">5</span><span class="err">.</span><span class="m">757</span> <span class="m">2</span><span class="err">.</span><span class="m">690</span> <span class="m">0</span><span class="err">.</span>      <span class="err">origin</span> <span class="err">x=-</span><span class="m">0</span><span class="err">.</span><span class="m">593</span> <span class="err">y=-</span><span class="m">0</span><span class="err">.</span><span class="m">593</span> <span class="err">z=-</span><span class="m">1</span><span class="err">.</span><span class="m">753</span>
<span class="c">&#39;thirteen</span>
  <span class="err">cuboid</span> <span class="m">130</span> <span class="m">4</span><span class="err">p</span><span class="m">6</span><span class="err">.</span><span class="m">35</span> <span class="m">5</span><span class="err">.</span><span class="m">718</span> <span class="m">0</span><span class="err">.</span>       <span class="err">origin</span> <span class="err">z=</span><span class="m">0</span><span class="err">.</span><span class="m">937</span>
<span class="c">&#39;fourteen</span>
  <span class="err">sphere</span>    <span class="m">140</span> <span class="m">6</span><span class="err">.</span><span class="m">082</span> <span class="err">chord</span> <span class="err">+z=</span><span class="m">0</span><span class="err">.</span>   <span class="err">origin</span> <span class="err">x=-</span><span class="m">0</span><span class="err">.</span><span class="m">268</span> <span class="err">y=</span><span class="m">0</span><span class="err">.</span><span class="m">268</span> <span class="err">z=</span><span class="m">6</span><span class="err">.</span><span class="m">655</span>
<span class="c">&#39;*** fifteen is the system boundary ***</span>
<span class="c">&#39;fifteen</span>
  <span class="err">cuboid</span>   <span class="m">150</span> <span class="m">4</span><span class="err">p</span><span class="m">25</span><span class="err">.</span><span class="m">0</span> <span class="m">15</span><span class="err">.</span><span class="m">0</span> <span class="err">-</span><span class="m">2</span><span class="err">.</span><span class="m">0</span>
  <span class="err">media</span>  <span class="m">1</span> <span class="m">1</span> <span class="err">+</span><span class="m">10</span>       <span class="err">vol=</span><span class="m">20</span><span class="err">.</span><span class="m">58546556</span>
  <span class="err">media</span>  <span class="m">2</span> <span class="m">1</span> <span class="err">+</span><span class="m">20</span> <span class="err">-</span><span class="m">10</span>   <span class="err">vol=</span><span class="m">245</span><span class="err">.</span><span class="m">678420867</span>
  <span class="err">media</span>  <span class="m">3</span> <span class="m">1</span> <span class="err">+</span><span class="m">30</span> <span class="err">-</span><span class="m">20</span>   <span class="err">vol=</span><span class="m">1800</span><span class="err">.</span><span class="m">040061395</span>
  <span class="err">media</span>  <span class="m">4</span> <span class="m">1</span> <span class="err">+</span><span class="m">40</span>       <span class="err">vol=</span><span class="m">842</span><span class="err">.</span><span class="m">019046637</span>
  <span class="err">media</span>  <span class="m">5</span> <span class="m">1</span> <span class="err">+</span><span class="m">50</span>       <span class="err">vol=</span><span class="m">1404</span><span class="err">.</span><span class="m">99376489</span>
  <span class="err">media</span>  <span class="m">6</span> <span class="m">1</span> <span class="err">+</span><span class="m">60</span>       <span class="err">vol=</span><span class="m">844</span><span class="err">.</span><span class="m">415646269</span>
  <span class="err">media</span>  <span class="m">7</span> <span class="m">1</span> <span class="err">+</span><span class="m">70</span>       <span class="err">vol=</span><span class="m">862</span><span class="err">.</span><span class="m">4600226</span>
  <span class="err">media</span>  <span class="m">8</span> <span class="m">1</span> <span class="err">+</span><span class="m">80</span> <span class="err">-</span><span class="m">70</span>   <span class="err">vol=</span><span class="m">283</span><span class="err">.</span><span class="m">749744681</span>
  <span class="err">media</span>  <span class="m">9</span> <span class="m">1</span> <span class="err">+</span><span class="m">90</span>       <span class="err">vol=</span><span class="m">845</span><span class="err">.</span><span class="m">483582679</span>
  <span class="err">media</span> <span class="m">10</span> <span class="m">1</span> <span class="err">+</span><span class="m">100</span>      <span class="err">vol=</span><span class="m">1399</span><span class="err">.</span><span class="m">390119093</span>
  <span class="err">media</span> <span class="m">11</span> <span class="m">1</span> <span class="err">+</span><span class="m">110</span>      <span class="err">vol=</span><span class="m">844</span><span class="err">.</span><span class="m">921798001</span>
  <span class="err">media</span> <span class="m">12</span> <span class="m">1</span> <span class="err">+</span><span class="m">120</span> <span class="err">-</span><span class="m">130</span> <span class="err">vol=</span><span class="m">280</span><span class="err">.</span><span class="m">088070346</span>
  <span class="err">media</span> <span class="m">13</span> <span class="m">1</span> <span class="err">+</span><span class="m">130</span>      <span class="err">vol=</span><span class="m">922</span><span class="err">.</span><span class="m">25622</span>
  <span class="err">media</span> <span class="m">14</span> <span class="m">1</span> <span class="err">+</span><span class="m">140</span> <span class="err">-</span><span class="m">130</span> <span class="err">vol=</span><span class="m">471</span><span class="err">.</span><span class="m">191948666</span>
  <span class="err">media</span> <span class="m">0</span> <span class="m">1</span> <span class="m">150</span> <span class="err">-</span><span class="m">10</span> <span class="err">-</span><span class="m">20</span> <span class="err">-</span><span class="m">30</span> <span class="err">-</span><span class="m">40</span> <span class="err">-</span><span class="m">50</span> <span class="err">-</span><span class="m">60</span> <span class="err">-</span><span class="m">70</span> <span class="err">-</span><span class="m">80</span> <span class="err">-</span><span class="m">90</span> <span class="err">-</span><span class="m">100</span>
            <span class="err">-</span><span class="m">110</span> <span class="err">-</span><span class="m">120</span> <span class="err">-</span><span class="m">130</span> <span class="err">-</span><span class="m">140</span>  <span class="err">vol=</span><span class="m">31432</span><span class="err">.</span><span class="m">726088316</span>
  <span class="err">boundary</span> <span class="m">150</span>
<span class="n">end</span><span class="err"> </span><span class="n">geom</span>
<span class="n">read</span><span class="err"> </span><span class="n">plot</span>
  <span class="err">scr=yes</span>  <span class="err">lpi=</span><span class="m">10</span>
           <span class="err">clr=</span> <span class="m">1</span> <span class="m">255</span>   <span class="m">0</span>   <span class="m">0</span>
                <span class="m">2</span>   <span class="m">0</span>   <span class="m">0</span> <span class="m">205</span>
                <span class="m">3</span>   <span class="m">0</span> <span class="m">229</span> <span class="m">238</span>
                <span class="m">4</span>   <span class="m">0</span> <span class="m">238</span>   <span class="m">0</span>
                <span class="m">5</span> <span class="m">205</span> <span class="m">205</span>   <span class="m">0</span>
                <span class="m">6</span> <span class="m">255</span> <span class="m">121</span> <span class="m">121</span>
                <span class="m">7</span> <span class="m">145</span>  <span class="m">44</span> <span class="m">238</span>
                <span class="m">8</span> <span class="m">150</span> <span class="m">150</span> <span class="m">150</span>
                <span class="m">9</span> <span class="m">240</span> <span class="m">200</span> <span class="m">220</span>
               <span class="m">10</span>   <span class="m">0</span> <span class="m">191</span> <span class="m">255</span>
               <span class="m">11</span> <span class="m">224</span> <span class="m">255</span> <span class="m">255</span>
               <span class="m">12</span>   <span class="m">0</span> <span class="m">128</span>  <span class="m">64</span>
               <span class="m">13</span> <span class="m">255</span> <span class="m">202</span> <span class="m">149</span>
               <span class="m">14</span> <span class="m">255</span>   <span class="m">0</span> <span class="m">128</span>
            <span class="n">end</span><span class="err"> </span><span class="n">color</span>
  <span class="err">ttl=</span><span class="s">&#39;grotesque x-y slice at z=0.5&#39;</span>
  <span class="err">xul=-</span><span class="m">25</span><span class="err">.</span><span class="m">5</span> <span class="err">yul=</span> <span class="m">25</span><span class="err">.</span><span class="m">5</span> <span class="err">zul=</span><span class="m">0</span><span class="err">.</span><span class="m">5</span>
  <span class="err">xlr=</span> <span class="m">25</span><span class="err">.</span><span class="m">5</span> <span class="err">ylr=-</span><span class="m">25</span><span class="err">.</span><span class="m">5</span> <span class="err">zlr=.</span><span class="m">5</span>
  <span class="err">uax=</span><span class="m">1</span>     <span class="err">vdn=-</span><span class="m">1</span>  <span class="err">nax=</span><span class="m">800</span> <span class="n">end</span>
  <span class="err">ttl=</span><span class="s">&#39;grotesque x-y slice at z=2.0&#39;</span>
  <span class="err">xul=-</span><span class="m">25</span><span class="err">.</span><span class="m">5</span> <span class="err">yul=</span> <span class="m">25</span><span class="err">.</span><span class="m">5</span> <span class="err">zul=</span><span class="m">2</span>
  <span class="err">xlr=</span> <span class="m">25</span><span class="err">.</span><span class="m">5</span> <span class="err">ylr=-</span><span class="m">25</span><span class="err">.</span><span class="m">5</span> <span class="err">zlr=</span><span class="m">2</span> <span class="n">end</span>
  <span class="err">ttl=</span><span class="s">&#39;grotesque x-y slice at z=9.5&#39;</span>
  <span class="err">xul=-</span><span class="m">25</span><span class="err">.</span><span class="m">5</span> <span class="err">yul=</span> <span class="m">25</span><span class="err">.</span><span class="m">5</span> <span class="err">zul=</span><span class="m">9</span><span class="err">.</span><span class="m">5</span>
  <span class="err">xlr=</span> <span class="m">25</span><span class="err">.</span><span class="m">5</span> <span class="err">ylr=-</span><span class="m">25</span><span class="err">.</span><span class="m">5</span> <span class="err">zlr=</span><span class="m">9</span><span class="err">.</span><span class="m">5</span> <span class="n">end</span>
  <span class="err">ttl=</span><span class="s">&#39;grotesque y-z slice at x=-0.593&#39;</span>
  <span class="err">xul=-.</span><span class="m">593</span> <span class="err">yul=-</span><span class="m">25</span><span class="err">.</span><span class="m">5</span> <span class="err">zul=</span><span class="m">15</span><span class="err">.</span><span class="m">5</span>
  <span class="err">xlr=-.</span><span class="m">593</span> <span class="err">ylr=</span> <span class="m">25</span><span class="err">.</span><span class="m">5</span> <span class="err">zlr=-</span><span class="m">3</span><span class="err">.</span><span class="m">5</span>
  <span class="err">uax=</span><span class="m">0</span>     <span class="err">vax=</span><span class="m">1</span>
  <span class="err">vdn=</span><span class="m">0</span>     <span class="err">wdn=-</span><span class="m">1</span> <span class="err">nax=</span><span class="m">800</span>  <span class="n">end</span>
  <span class="err">ttl=</span><span class="s">&#39;grotesque x-z slice at y=0.0&#39;</span>
  <span class="err">xul=-</span><span class="m">25</span><span class="err">.</span><span class="m">5</span> <span class="err">yul=</span><span class="m">0</span><span class="err">.</span><span class="m">0</span>  <span class="err">zul=</span><span class="m">15</span><span class="err">.</span><span class="m">5</span>
  <span class="err">xlr=</span> <span class="m">25</span><span class="err">.</span><span class="m">5</span> <span class="err">ylr=</span><span class="m">0</span><span class="err">.</span><span class="m">0</span>  <span class="err">zlr=-</span><span class="m">3</span><span class="err">.</span><span class="m">5</span>
  <span class="err">uax=</span><span class="m">1</span>     <span class="err">vax=</span><span class="m">0</span>    <span class="err">wax=</span><span class="m">0</span>
  <span class="err">udn=</span><span class="m">0</span>     <span class="err">vdn=</span><span class="m">0</span>    <span class="err">wdn=-</span><span class="m">1</span> <span class="err">nax=</span><span class="m">800</span>  <span class="n">end</span>
  <span class="err">ttl=</span><span class="s">&#39;grotesque x-z slice at y=12.125&#39;</span>
  <span class="err">xul=-</span><span class="m">25</span><span class="err">.</span><span class="m">5</span> <span class="err">yul=</span><span class="m">12</span><span class="err">.</span><span class="m">125</span>  <span class="err">zul=</span><span class="m">15</span><span class="err">.</span><span class="m">5</span>
  <span class="err">xlr=</span> <span class="m">25</span><span class="err">.</span><span class="m">5</span> <span class="err">ylr=</span><span class="m">12</span><span class="err">.</span><span class="m">125</span>  <span class="err">zlr=-</span><span class="m">3</span><span class="err">.</span><span class="m">5</span>
  <span class="err">uax=</span><span class="m">1</span>     <span class="err">vax=</span><span class="m">0</span>       <span class="err">wax=</span><span class="m">0</span>
  <span class="err">udn=</span><span class="m">0</span>     <span class="err">vdn=</span><span class="m">0</span>       <span class="err">wdn=-</span><span class="m">1</span> <span class="err">nax=</span><span class="m">800</span>  <span class="n">end</span>
  <span class="err">ttl=</span><span class="s">&#39;grotesque x-z slice at y=-12.000&#39;</span>
  <span class="err">xul=-</span><span class="m">25</span><span class="err">.</span><span class="m">5</span> <span class="err">yul=-</span><span class="m">12</span><span class="err">.</span><span class="m">000</span>  <span class="err">zul=</span><span class="m">15</span><span class="err">.</span><span class="m">5</span>
  <span class="err">xlr=</span> <span class="m">25</span><span class="err">.</span><span class="m">5</span> <span class="err">ylr=-</span><span class="m">12</span><span class="err">.</span><span class="m">000</span>  <span class="err">zlr=-</span><span class="m">3</span><span class="err">.</span><span class="m">5</span>
  <span class="err">uax=</span><span class="m">1</span>     <span class="err">vax=</span><span class="m">0</span>        <span class="err">wax=</span><span class="m">0</span>
  <span class="err">udn=</span><span class="m">0</span>     <span class="err">vdn=</span><span class="m">0</span>        <span class="err">wdn=-</span><span class="m">1</span> <span class="err">nax=</span><span class="m">800</span>  <span class="n">end</span>
<span class="n">end</span><span class="err"> </span><span class="n">plot</span>
<span class="n">end</span><span class="err"> </span><span class="n">data</span>
<span class="n">end</span>
</pre></div>
</div>
</div>
<div class="section" id="sample-problem-8-infinite-array-of-mox-and-uo2-assemblies">
<h3>Sample Problem 8 Infinite Array of MOX and UO2 Assemblies<a class="headerlink" href="#sample-problem-8-infinite-array-of-mox-and-uo2-assemblies" title="Permalink to this headline"></a></h3>
<p>The purpose of this problem is to calculate the <em>keff</em> of a system
composed of an infinite array of MOX assemblies interspersed between
UO<sub>2</sub> assemblies. Both assembly types contain 331 pins in a
hexagonal lattice with a pin pitch of 1.275 cm and an assembly pitch
of 23.60 cm as shown in
<a class="reference internal" href="#fig2-2-4"><span class="std std-numref">Fig. 15</span></a>. The moderator is borated water at 306°C having a density
of 0.71533 gm/cc and composed of 99.94 wt % H<sub>2</sub>O and
0.06 wt % natural boron. Each fuel rod is 355 cm in length, has a
radius of 0.3860 cm, 0.722-cm-thick Zr cladding with no gap, and is at
a temperature of 754°C.</p>
<p>The UO<sub>2</sub> fuel consists of 4.4 wt % <sup>235</sup>U and 95.6 wt %
<sup>238</sup>U at a density of 8.7922 gm/cc. The UO<sub>2</sub> fuel also
contains 9.4581E–9 atoms/b-cm of <sup>135</sup>Xe and 7.3667E–8 atoms/b-cm
of <sup>149</sup>Sm.</p>
<p>The MOX fuel consists of 96.38 wt % UO<sub>2</sub> and
3.62 wt % PuO<sub>2</sub> at a density of 8.8182 gm/cc. The UO<sub>2</sub>
fuel is composed of 2.0 wt % <sup>235</sup>U and
98.0 wt % <sup>238</sup>U. The PuO<sub>2</sub> fuel is composed of
93.0 wt % <sup>239</sup>Pu, 6.0 wt % <sup>240</sup>Pu- and
1.0 wt % <sup>241</sup>Pu. The MOX fuel also contains 9.4581E–9
atoms/b-cm of <sup>135</sup>Xe and 7.3667E–8 atoms/b-cm of <sup>149</sup>Sm.</p>
<p>These two assemblies are placed so they represent an infinite array in
the X and Y dimensions as shown in <a class="reference internal" href="#fig2-2-5"><span class="std std-numref">Fig. 16</span></a>. There is 20 cm of water
above and below fuel assemblies. This problem uses CENTRM/PMC as the
resolved resonance processor cross section. Since an infinite array
cannot be explicitly modeled, a section of the array is modeled and the
X and Y sides have mirror reflection.</p>
<div class="highlight-scale notranslate"><div class="highlight"><pre><span></span><span class="nf">=csas6        parm=(centrm)</span>
<span class="err">sample</span> <span class="err">problem</span> <span class="m">8</span> <span class="err">-</span> <span class="err">VVER</span> <span class="err">inf.</span> <span class="err">array</span> <span class="err">-</span> <span class="err">MOX</span> <span class="err">&amp;</span> <span class="err">UO</span><span class="m">2</span> <span class="err">Assemblies</span>
<span class="err">v</span><span class="m">7</span><span class="err">-</span><span class="m">238</span>
<span class="n">read</span><span class="err"> </span><span class="n">comp</span>
<span class="c">&#39;  UO2 Fuel</span>