RBMK.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# A RBMK geometry \n",
    "This notebook can be used as a template for modeling RBMK reactors.\n",
    "\n",
    "SOURCES\n",
    "\"Fuel Assembly\": https://web.archive.org/web/20090908013425/http:/www.insc.anl.gov:80/rbmk/reactor/assembly.html\n",
    "\"Fuel with burnable absorber for RBMK-1500\", pg250-252: https://www.osti.gov/etdeweb/servlets/purl/20269236\n",
    "\"DESIGN AND FABRICATION OF NUCLEAR FUEL FOR WWER AND RBMK REACTORS\"\n",
    "Slides 8-11: indico.ictp.it/event/a04215/session/26/contribution/16/material/0/1.pdf"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "from math import pi, sin, cos, sqrt\n",
    "import numpy as np\n",
    "import openmc"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Materials definitions\n",
    "\n",
    "fuel = openmc.Material(name='2.0% Fuel')\n",
    "fuel.add_element('U', 0.995, enrichment=2.0)\n",
    "fuel.add_element('Er', 0.005)\n",
    "fuel.add_nuclide('O16', 2.0)\n",
    "fuel.set_density('g/cm3', 10.400)\n",
    "\n",
    "zircaloy = openmc.Material(name='Zircaloy')\n",
    "zircaloy.add_element('Zr', 0.99)\n",
    "zircaloy.add_element('Nb', 0.01)\n",
    "zircaloy.set_density('g/cm3', 8.59)\n",
    "\n",
    "wall = openmc.Material(name='Carrier Rod Material')\n",
    "wall.add_element('Zr', 0.975)\n",
    "wall.add_element('Nb', 0.025)\n",
    "wall.set_density('g/cm3', 8.59)\n",
    "\n",
    "helium = openmc.Material(name='Helium')\n",
    "helium.add_element('He', 1)\n",
    "helium.set_density('g/cm3', 0.178)\n",
    "\n",
    "# Instantiate a Materials collection and export to xml\n",
    "materials_file = openmc.Materials([fuel, zircaloy, helium, wall])\n",
    "materials_file.export_to_xml()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Geometry definitions for the fuel rod\n",
    "\n",
    "#Making the boundary planes for the sleeves\n",
    "sleeve_min_z = openmc.ZPlane(z0=-12)\n",
    "sleeve_max_z = openmc.ZPlane(z0=+12)\n",
    "\n",
    "# Create cylinders for the fuel and clad\n",
    "fuel_inner_radius = openmc.ZCylinder(r=0.55)\n",
    "fuel_outer_radius = openmc.ZCylinder(r=0.566)\n",
    "clad_inner_radius = openmc.ZCylinder(r=0.596)\n",
    "clad_outer_radius = openmc.ZCylinder(r=0.68)\n",
    "\n",
    "# Create a universe to encapsulate a fuel rod\n",
    "pin_cell_universe = openmc.Universe(name='2.0% Fuel Pin') \n",
    "\n",
    "# Create fuel cell\n",
    "fuel_cell_top = openmc.Cell(name='2.0% Fuel')\n",
    "fuel_cell_top.fill = fuel\n",
    "fuel_cell_top.region = -fuel_inner_radius & +sleeve_max_z\n",
    "pin_cell_universe.add_cell(fuel_cell_top)\n",
    "\n",
    "# Create void space\n",
    "void_space_top = openmc.Cell(name='empty_space')\n",
    "void_space_top.fill = helium\n",
    "void_space_top.region = +fuel_inner_radius & -clad_inner_radius & +sleeve_max_z\n",
    "pin_cell_universe.add_cell(void_space_top)\n",
    "\n",
    "# Create a clad cell\n",
    "clad_cell_top = openmc.Cell(name='2.0% Clad')\n",
    "clad_cell_top.fill = zircaloy\n",
    "clad_cell_top.region = +clad_inner_radius & -clad_outer_radius & +sleeve_max_z\n",
    "pin_cell_universe.add_cell(clad_cell_top)\n",
    "\n",
    "# Create fuel cell\n",
    "fuel_cell_bot = openmc.Cell(name='2.0% Fuel')\n",
    "fuel_cell_bot.fill = fuel\n",
    "fuel_cell_bot.region = -fuel_inner_radius & -sleeve_min_z\n",
    "pin_cell_universe.add_cell(fuel_cell_bot)\n",
    "\n",
    "# Create void space\n",
    "void_space_bot = openmc.Cell(name='empty_space')\n",
    "void_space_bot.fill = helium\n",
    "void_space_bot.region = +fuel_inner_radius & -clad_inner_radius & -sleeve_min_z\n",
    "pin_cell_universe.add_cell(void_space_bot)\n",
    "\n",
    "# Create a clad cell\n",
    "clad_cell_bot = openmc.Cell(name='2.0% Clad')\n",
    "clad_cell_bot.fill = zircaloy\n",
    "clad_cell_bot.region = +clad_inner_radius & -clad_outer_radius & -sleeve_min_z\n",
    "pin_cell_universe.add_cell(clad_cell_bot)\n",
    "\n",
    "# Create an outside of pin cell\n",
    "moderator = openmc.Cell(name='Moderator')\n",
    "moderator.fill = helium\n",
    "moderator.region = +clad_outer_radius\n",
    "pin_cell_universe.add_cell(moderator)\n",
    "\n",
    "# Create a 'sleeve' cell that slices through the middle of the elongated element\n",
    "sleeve_cell = openmc.Cell(name='Sleeve')\n",
    "sleeve_cell.fill = helium\n",
    "sleeve_cell.region = -clad_outer_radius & +sleeve_min_z & -sleeve_max_z\n",
    "pin_cell_universe.add_cell(sleeve_cell)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Geometry definitions for the fuel rod\n",
    "\n",
    "rod_inner_radius = openmc.ZCylinder(r=0.75) \n",
    "rod_outer_radius = openmc.ZCylinder(r=0.7625)\n",
    "\n",
    "rod_cell_universe = openmc.Universe(name='Carrier Rod')\n",
    "\n",
    "inner_rod = openmc.Cell(name='Inner Rod')\n",
    "inner_rod.fill = helium\n",
    "inner_rod.region = -rod_inner_radius\n",
    "rod_cell_universe.add_cell(inner_rod)\n",
    "\n",
    "outer_rod = openmc.Cell(name='Outer Rod')\n",
    "outer_rod.fill = wall\n",
    "outer_rod.region = +rod_inner_radius & -rod_outer_radius\n",
    "rod_cell_universe.add_cell(outer_rod)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Create a circular Lattice for the bundle\n",
    "\n",
    "element_min_z = openmc.ZPlane(z0=-364, boundary_type='reflective')\n",
    "element_max_z = openmc.ZPlane(z0=+364, boundary_type='reflective')\n",
    "carrier_min_z = openmc.ZPlane(z0=-400, boundary_type='reflective')\n",
    "carrier_max_z = openmc.ZPlane(z0=+600, boundary_type='reflective')\n",
    "\n",
    "circlat = openmc.Universe(name='Circular Lattice')\n",
    "\n",
    "channel = openmc.ZCylinder(r=4.5, boundary_type='reflective')\n",
    "\n",
    "channel_cell = openmc.Cell(name='Channel')\n",
    "channel_cell.fill = helium\n",
    "channel_cell.region = -channel & +rod_outer_radius & +element_min_z & -element_max_z\n",
    "\n",
    "#Calculations before finding pin placement\n",
    "r_channel = 4.0\n",
    "r_element = 0.68\n",
    "r_outer = r_channel-r_element\n",
    "\n",
    "padding = pi*r_outer/6-2*r_element\n",
    "r_inner = r_outer*cos(pi/12)-sqrt((r_element*2+padding)**2-(r_element+padding)**2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Add the fuel pins into the channel\n",
    "\n",
    "num_in_rings = [6, 12]\n",
    "radii = [r_inner, r_outer]\n",
    "angles = [0, pi/12]\n",
    "\n",
    "for index in range(len(num_in_rings)):\n",
    "  elem_num = num_in_rings[index]\n",
    "  ring_radius = radii[index]\n",
    "  theta_0 = angles[index]\n",
    "  theta = 2*pi/elem_num\n",
    "  for element in range(elem_num):\n",
    "    x = ring_radius*cos(element*theta + theta_0)\n",
    "    y = ring_radius*sin(element*theta + theta_0)\n",
    "\n",
    "    pin_boundary = openmc.ZCylinder(x0=x, y0=y, r=r_element)\n",
    "    pin = openmc.Cell(fill=pin_cell_universe, region=-pin_boundary & +element_min_z & -element_max_z)\n",
    "    pin.translation = (x,y,0)\n",
    "    pin.id = (index+1)*100 + element\n",
    "    channel_cell.region &= ~pin.region\n",
    "    circlat.add_cell(pin)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Add the 3 Components of the Carrier Rod\n",
    "rod_bound = openmc.ZCylinder(r=0.7625, boundary_type='reflective')\n",
    "rod_channel_min = openmc.ZPlane(z0=-364)\n",
    "rod_channel_max = openmc.ZPlane(z0=+364)\n",
    "\n",
    "rod_above = openmc.Cell(fill=rod_cell_universe, region=-rod_bound & -carrier_max_z & +rod_channel_max)\n",
    "rod_mid = openmc.Cell(fill=rod_cell_universe, region=-rod_outer_radius & +rod_channel_min & -rod_channel_max)\n",
    "rod_below = openmc.Cell(fill=rod_cell_universe, region=-rod_bound & -rod_channel_min & +carrier_min_z)\n",
    "\n",
    "circlat.add_cell(rod_above)\n",
    "circlat.add_cell(rod_mid)\n",
    "circlat.add_cell(rod_below)\n",
    "circlat.add_cell(channel_cell)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "geometry = openmc.Geometry(circlat)\n",
    "geometry.export_to_xml()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.image.AxesImage at 0x12cf0b8b0>"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "circlat.plot(width=(12, 12), basis='xy')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [],
   "source": [
    "# OpenMC simulation parameters\n",
    "\n",
    "batches = 100\n",
    "inactive = 10\n",
    "particles = 5000\n",
    "\n",
    "settings_file = openmc.Settings()\n",
    "settings_file.batches = batches\n",
    "settings_file.inactive = inactive\n",
    "settings_file.particles = particles\n",
    "\n",
    "bounds = [-3, -3, -2.4, 3, 3, 2.4]\n",
    "uniform_dist = openmc.stats.Box(bounds[:3], bounds[3:], only_fissionable=False) #only_fissionable != True due to lots of helium\n",
    "settings_file.source = openmc.Source(space=uniform_dist)\n",
    "\n",
    "settings_file.export_to_xml()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "                                %%%%%%%%%%%%%%%\n",
      "                           %%%%%%%%%%%%%%%%%%%%%%%%\n",
      "                        %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n",
      "                      %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n",
      "                    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n",
      "                   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n",
      "                                    %%%%%%%%%%%%%%%%%%%%%%%%\n",
      "                                     %%%%%%%%%%%%%%%%%%%%%%%%\n",
      "                 ###############      %%%%%%%%%%%%%%%%%%%%%%%%\n",
      "                ##################     %%%%%%%%%%%%%%%%%%%%%%%\n",
      "                ###################     %%%%%%%%%%%%%%%%%%%%%%%\n",
      "                ####################     %%%%%%%%%%%%%%%%%%%%%%\n",
      "                #####################     %%%%%%%%%%%%%%%%%%%%%\n",
      "                ######################     %%%%%%%%%%%%%%%%%%%%\n",
      "                #######################     %%%%%%%%%%%%%%%%%%\n",
      "                 #######################     %%%%%%%%%%%%%%%%%\n",
      "                 ######################     %%%%%%%%%%%%%%%%%\n",
      "                  ####################     %%%%%%%%%%%%%%%%%\n",
      "                    #################     %%%%%%%%%%%%%%%%%\n",
      "                     ###############     %%%%%%%%%%%%%%%%\n",
      "                       ############     %%%%%%%%%%%%%%%\n",
      "                          ########     %%%%%%%%%%%%%%\n",
      "                                      %%%%%%%%%%%\n",
      "\n",
      "                   | The OpenMC Monte Carlo Code\n",
      "         Copyright | 2011-2020 MIT and OpenMC contributors\n",
      "           License | https://docs.openmc.org/en/latest/license.html\n",
      "           Version | 0.12.1-dev\n",
      "          Git SHA1 | 0228ddba1d6399e654a295e51539590f076b2a3c\n",
      "         Date/Time | 2021-04-12 12:40:32\n",
      "    OpenMP Threads | 8\n",
      "\n",
      " Reading settings XML file...\n",
      " Reading cross sections XML file...\n",
      " Reading materials XML file...\n",
      " Reading geometry XML file...\n",
      " Reading U234 from\n",
      " /Users/miriamrathbun/codes/openmc/nucleardata/endfb71_hdf5/U234.h5\n",
      " Reading U235 from\n",
      " /Users/miriamrathbun/codes/openmc/nucleardata/endfb71_hdf5/U235.h5\n",
      " Reading U238 from\n",
      " /Users/miriamrathbun/codes/openmc/nucleardata/endfb71_hdf5/U238.h5\n",
      " Reading U236 from\n",
      " /Users/miriamrathbun/codes/openmc/nucleardata/endfb71_hdf5/U236.h5\n",
      " Reading Er162 from\n",
      " /Users/miriamrathbun/codes/openmc/nucleardata/endfb71_hdf5/Er162.h5\n",
      " Reading Er164 from\n",
      " /Users/miriamrathbun/codes/openmc/nucleardata/endfb71_hdf5/Er164.h5\n",
      " Reading Er166 from\n",
      " /Users/miriamrathbun/codes/openmc/nucleardata/endfb71_hdf5/Er166.h5\n",
      " Reading Er167 from\n",
      " /Users/miriamrathbun/codes/openmc/nucleardata/endfb71_hdf5/Er167.h5\n",
      " Reading Er168 from\n",
      " /Users/miriamrathbun/codes/openmc/nucleardata/endfb71_hdf5/Er168.h5\n",
      " Reading Er170 from\n",
      " /Users/miriamrathbun/codes/openmc/nucleardata/endfb71_hdf5/Er170.h5\n",
      " Reading O16 from\n",
      " /Users/miriamrathbun/codes/openmc/nucleardata/endfb71_hdf5/O16.h5\n",
      " Reading Zr90 from\n",
      " /Users/miriamrathbun/codes/openmc/nucleardata/endfb71_hdf5/Zr90.h5\n",
      " Reading Zr91 from\n",
      " /Users/miriamrathbun/codes/openmc/nucleardata/endfb71_hdf5/Zr91.h5\n",
      " Reading Zr92 from\n",
      " /Users/miriamrathbun/codes/openmc/nucleardata/endfb71_hdf5/Zr92.h5\n",
      " Reading Zr94 from\n",
      " /Users/miriamrathbun/codes/openmc/nucleardata/endfb71_hdf5/Zr94.h5\n",
      " Reading Zr96 from\n",
      " /Users/miriamrathbun/codes/openmc/nucleardata/endfb71_hdf5/Zr96.h5\n",
      " Reading Nb93 from\n",
      " /Users/miriamrathbun/codes/openmc/nucleardata/endfb71_hdf5/Nb93.h5\n",
      " Reading He3 from\n",
      " /Users/miriamrathbun/codes/openmc/nucleardata/endfb71_hdf5/He3.h5\n",
      " Reading He4 from\n",
      " /Users/miriamrathbun/codes/openmc/nucleardata/endfb71_hdf5/He4.h5\n",
      " Minimum neutron data temperature: 294.0 K\n",
      " Maximum neutron data temperature: 294.0 K\n",
      " Preparing distributed cell instances...\n",
      " Writing summary.h5 file...\n",
      " Maximum neutron transport energy: 20000000.0 eV for U235\n",
      " Initializing source particles...\n",
      "\n",
      " ====================>     K EIGENVALUE SIMULATION     <====================\n",
      "\n",
      "  Bat./Gen.      k            Average k\n",
      "  =========   ========   ====================\n",
      "        1/1    0.36726\n",
      "        2/1    0.41917\n",
      "        3/1    0.41996\n",
      "        4/1    0.42104\n",
      "        5/1    0.42013\n",
      "        6/1    0.41539\n",
      "        7/1    0.41936\n",
      "        8/1    0.41825\n",
      "        9/1    0.43324\n",
      "       10/1    0.43023\n",
      "       11/1    0.43863\n",
      "       12/1    0.41949    0.42906 +/- 0.00957\n",
      "       13/1    0.41553    0.42455 +/- 0.00713\n",
      "       14/1    0.43122    0.42622 +/- 0.00531\n",
      "       15/1    0.43523    0.42802 +/- 0.00449\n",
      "       16/1    0.42006    0.42670 +/- 0.00390\n",
      "       17/1    0.42561    0.42654 +/- 0.00330\n",
      "       18/1    0.41654    0.42529 +/- 0.00312\n",
      "       19/1    0.43361    0.42621 +/- 0.00290\n",
      "       20/1    0.41500    0.42509 +/- 0.00283\n",
      "       21/1    0.42439    0.42503 +/- 0.00256\n",
      "       22/1    0.41977    0.42459 +/- 0.00238\n",
      "       23/1    0.41752    0.42405 +/- 0.00225\n",
      "       24/1    0.43619    0.42491 +/- 0.00226\n",
      "       25/1    0.41760    0.42443 +/- 0.00216\n",
      "       26/1    0.42986    0.42477 +/- 0.00205\n",
      "       27/1    0.41637    0.42427 +/- 0.00199\n",
      "       28/1    0.41170    0.42357 +/- 0.00200\n",
      "       29/1    0.41593    0.42317 +/- 0.00193\n",
      "       30/1    0.41912    0.42297 +/- 0.00184\n",
      "       31/1    0.41365    0.42253 +/- 0.00181\n",
      "       32/1    0.41400    0.42214 +/- 0.00177\n",
      "       33/1    0.41753    0.42194 +/- 0.00170\n",
      "       34/1    0.41432    0.42162 +/- 0.00166\n",
      "       35/1    0.41323    0.42128 +/- 0.00163\n",
      "       36/1    0.43010    0.42162 +/- 0.00160\n",
      "       37/1    0.41552    0.42140 +/- 0.00156\n",
      "       38/1    0.41385    0.42113 +/- 0.00152\n",
      "       39/1    0.42110    0.42113 +/- 0.00147\n",
      "       40/1    0.41254    0.42084 +/- 0.00145\n",
      "       41/1    0.41489    0.42065 +/- 0.00141\n",
      "       42/1    0.42521    0.42079 +/- 0.00138\n",
      "       43/1    0.42345    0.42087 +/- 0.00134\n",
      "       44/1    0.42805    0.42108 +/- 0.00131\n",
      "       45/1    0.42292    0.42114 +/- 0.00128\n",
      "       46/1    0.42032    0.42111 +/- 0.00124\n",
      "       47/1    0.42226    0.42114 +/- 0.00121\n",
      "       48/1    0.42795    0.42132 +/- 0.00119\n",
      "       49/1    0.41490    0.42116 +/- 0.00117\n",
      "       50/1    0.42862    0.42134 +/- 0.00115\n",
      "       51/1    0.42480    0.42143 +/- 0.00113\n",
      "       52/1    0.41355    0.42124 +/- 0.00112\n",
      "       53/1    0.42440    0.42131 +/- 0.00109\n",
      "       54/1    0.41865    0.42125 +/- 0.00107\n",
      "       55/1    0.42069    0.42124 +/- 0.00105\n",
      "       56/1    0.41623    0.42113 +/- 0.00103\n",
      "       57/1    0.41685    0.42104 +/- 0.00101\n",
      "       58/1    0.41834    0.42099 +/- 0.00099\n",
      "       59/1    0.43448    0.42126 +/- 0.00101\n",
      "       60/1    0.40917    0.42102 +/- 0.00102\n",
      "       61/1    0.42187    0.42104 +/- 0.00100\n",
      "       62/1    0.42270    0.42107 +/- 0.00098\n",
      "       63/1    0.42030    0.42105 +/- 0.00096\n",
      "       64/1    0.43457    0.42130 +/- 0.00098\n",
      "       65/1    0.41390    0.42117 +/- 0.00097\n",
      "       66/1    0.42198    0.42118 +/- 0.00095\n",
      "       67/1    0.42026    0.42117 +/- 0.00093\n",
      "       68/1    0.41652    0.42109 +/- 0.00092\n",
      "       69/1    0.41120    0.42092 +/- 0.00092\n",
      "       70/1    0.42104    0.42092 +/- 0.00090\n",
      "       71/1    0.40883    0.42072 +/- 0.00091\n",
      "       72/1    0.42478    0.42079 +/- 0.00090\n",
      "       73/1    0.42060    0.42079 +/- 0.00088\n",
      "       74/1    0.41950    0.42077 +/- 0.00087\n",
      "       75/1    0.41800    0.42072 +/- 0.00086\n",
      "       76/1    0.42313    0.42076 +/- 0.00085\n",
      "       77/1    0.42090    0.42076 +/- 0.00083\n",
      "       78/1    0.41947    0.42074 +/- 0.00082\n",
      "       79/1    0.42478    0.42080 +/- 0.00081\n",
      "       80/1    0.43185    0.42096 +/- 0.00082\n",
      "       81/1    0.41999    0.42094 +/- 0.00080\n",
      "       82/1    0.41669    0.42089 +/- 0.00079\n",
      "       83/1    0.42688    0.42097 +/- 0.00079\n",
      "       84/1    0.42583    0.42103 +/- 0.00078\n",
      "       85/1    0.41937    0.42101 +/- 0.00077\n",
      "       86/1    0.42302    0.42104 +/- 0.00076\n",
      "       87/1    0.41306    0.42093 +/- 0.00076\n",
      "       88/1    0.42048    0.42093 +/- 0.00075\n",
      "       89/1    0.41601    0.42087 +/- 0.00074\n",
      "       90/1    0.42380    0.42090 +/- 0.00073\n",
      "       91/1    0.42252    0.42092 +/- 0.00072\n",
      "       92/1    0.42163    0.42093 +/- 0.00071\n",
      "       93/1    0.42060    0.42093 +/- 0.00071\n",
      "       94/1    0.41860    0.42090 +/- 0.00070\n",
      "       95/1    0.41718    0.42086 +/- 0.00069\n",
      "       96/1    0.42078    0.42085 +/- 0.00068\n",
      "       97/1    0.41894    0.42083 +/- 0.00068\n",
      "       98/1    0.40850    0.42069 +/- 0.00068\n",
      "       99/1    0.41439    0.42062 +/- 0.00068\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "      100/1    0.41280    0.42054 +/- 0.00068\n",
      " Creating state point statepoint.100.h5...\n",
      "\n",
      " =======================>     TIMING STATISTICS     <=======================\n",
      "\n",
      " Total time for initialization     = 3.5405e+00 seconds\n",
      "   Reading cross sections          = 3.4979e+00 seconds\n",
      " Total time in simulation          = 7.6338e+01 seconds\n",
      "   Time in transport only          = 7.6297e+01 seconds\n",
      "   Time in inactive batches        = 7.5955e+00 seconds\n",
      "   Time in active batches          = 6.8742e+01 seconds\n",
      "   Time synchronizing fission bank = 1.7017e-02 seconds\n",
      "     Sampling source sites         = 1.3410e-02 seconds\n",
      "     SEND/RECV source sites        = 3.5210e-03 seconds\n",
      "   Time accumulating tallies       = 3.9000e-05 seconds\n",
      "   Time writing statepoints        = 5.7760e-03 seconds\n",
      " Total time for finalization       = 4.0000e-06 seconds\n",
      " Total time elapsed                = 7.9891e+01 seconds\n",
      " Calculation Rate (inactive)       = 6582.89 particles/second\n",
      " Calculation Rate (active)         = 6546.18 particles/second\n",
      "\n",
      " ============================>     RESULTS     <============================\n",
      "\n",
      " k-effective (Collision)     = 0.42078 +/- 0.00067\n",
      " k-effective (Track-length)  = 0.42054 +/- 0.00068\n",
      " k-effective (Absorption)    = 0.42137 +/- 0.00110\n",
      " Combined k-effective        = 0.42081 +/- 0.00058\n",
      " Leakage Fraction            = 0.00000 +/- 0.00000\n",
      "\n"
     ]
    }
   ],
   "source": [
    "openmc.run()"
   ]
  }
 ],
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