7.1.4.3. Examples of Complete XSProc Input Data
7.1.4.3.1. Infinite homogeneous medium input data
Examples of XSProc input data for infinite homogeneous media problems are given below. In these cases the cross section library name “fine_n” indicates that the latest recommended finegroup SCALE library will used in the calculations.
EXAMPLE 1. Default cell definition.
Consider a cylindrical billet of 20 wt % enriched UO_{2}, having a density of 10.85 g/cm^{3} that is 26 cm in diameter and 26 cm tall.
The average meanfree path in the uranium dioxide is on the order of 2.5 cm. Because only a small fraction of the billet is within a meanfree path of the surface, the material can be treated as an infinite homogeneous medium; therefore the CELL DATA block can be omitted. The XSProc data follows:
20% ENRICHED UO2 BILLET
fine_n
READ COMP
UO2 1 0.99 293 92235 20 92238 80 END
END COMP
The volume fraction used for the UO_{2}, 0.99, is calculated by dividing the actual density by the theoretical density obtained from the Isotopes in standard composition library table in the STDCMP chapter, (10.85/10.96). Since the enrichment was specified as 20%, it is assumed that the remainder is ^{238}U.
An alternative input data description follows:
20% ENRICHED UO2 BILLET
fine_n
READ COMP
UO2 1 DEN=10.85 1 293 92235 20 92238 80 END
END COMP
EXAMPLE 2. Specify the cell definition.
Consider a 5liter Plexiglas bottle with an inner radius of 9.525 cm and inner height of 17.78 cm that is filled with highly enriched uranyl nitrate solution at 415 g/L and 0.39 mg of excess nitrate per gram of solution. The uranium isotopic content of the nitrate solution is 92.6 wt % ^{235}U, 5.9 wt % ^{238}U, 1.0 wt % ^{234}U, and 0.5 wt % ^{236}U. Solution density will be calculated from the given data.
The size of the nitrate solution is on the order of 16 to 20 cm in diameter and height. The average meanfree path in the nitrate solution is on the order of 0.5 cm. Therefore, infinite homogeneous medium is an appropriate choice for this problem. By default BONAMI is used for selfshielding the infinite medium of Plexiglas, while CENTRM is used to shield the infinite medium fissile solution.
SET UP 5 LITER URANYL NITRATE SOLUTION IN A PLEXIGLAS CONTAINER
fine_n
READ COMP
PLEXIGLAS 1 END
SOLUTION MIX=2 RHO[UO2(NO3)2]=415
92235 92.6 92238 5.9 92236 0.5
END SOLUTION
END COMP
READ CELLDATA
INFHOMMEDIUM 2 END
END CELLDATA
7.1.4.3.2. LATTICECELL input data
Examples of XSProc input data for LATTICECELL problems are given below.
EXAMPLE 1. SQUAREPITCH ARRAY.
Consider an infinite planar array (infinite in X and Y and one layer in Z) of 20 wt % enriched U metal rods with a 1cm pitch. Each fuel rod is bare uranium metal, 0.75 cm OD \(\times\) 30.0 cm long. The rods are submerged in water.
Because the diameter of the fuel rod, 0.75 cm, is only slightly larger than the average meanfree path in the uranium metal, approximately 0.5, and because the configuration is a regular array, LATTICECELL is the appropriate choice for proper crosssection processing. The parm field is not provided, so the default CENTRM/PMC selfshielding method is used. XSProc data follows:
INFINITE PLANAR ARRAY OF 20% U METAL RODS
fine_n
READ COMP
URANIUM 1 1 293 92235 20 92238 80 END
H2O 2 END
END COMP
READ CELLDATA
LATTICECELL SQUAREPITCH PITCH=1.0 2 FUELD=0.75 1 END
END CELLDATA
Since the MORE DATA and CENTRM DATA blocks were omitted, default options will be used in the selfshielding calculations. The default CENTRM/PMC computation options for a square pitch lattice cell are the methodofcharacteristics (MoC) method with P0 scatter in CENTRM calculations.
EXAMPLE 2. SQUAREPITCH PWR LATTICE.
Consider an infinite, uniform planar array (infinite in X and Y and one layer in Z) of PWRlike fuel pins of 2.35% enriched UO_{2} clad with zirconium. The density of the UO_{2} is 9.21 g/cm^{3}. The fuel in each pin is 0.823 cm in diameter, the clad is 0.9627 cm in diameter, and the length of each pin is 366 cm. The fuel pins are separated by 0.3124 cm of water in the horizontal plane.
LATTICECELL is the appropriate choice for crosssection processing. Assume that all defaults are appropriate; thus the CENTRM/PMC methodology is used, and the MORE DATA and CELL DATA blocks are not entered. The input cross section library named “broad_n” indicates that the recommended broad group SCALE library will be used. In this case CENTRM uses the 2D MoC transport solver. The XSProc data follows:
PWRLIKE FUEL BUNDLE; uniform infinite array model.
broad_n
READ COMP
UO2 1 .84 293. 92235 2.35 92238 97.65 END
ZR 2 1 END
H2O 3 1 END
END COMP
READ CELLDATA
LATTICECELL SQUAREPITCH PITCH=1.2751 3 FUELD=0.823 1 CLADD=0.9627 2 END
END CELLDATA
EXAMPLE 3. SQUAREPITCH PWR LATTICE, with nonuniform Dancoff.
This example is a single PWR assembly of fuel pins of the type described above, contained in a water pool. The interior pins in the assembly can be selfshielded using the same uniform, infinite lattice model in previous example. However selfshielding of the outer boundaryedge pins will be modified to account for being adjacent to a water reflector, rather than surrounded on all sides by similar pins. This requires that the MCDancoff module be executed previously to obtain nonuniform Dancoff factors for the edge pins. The average edgepin value of 0.61 is used to represent Dancoff factors of all boundary pins. The default CENTRM MoC transport solver is used for both cells, but the original pitch of 1.2751 cm for the second cell (i.e., boundary pin) is modified to a new pitch corresponding to a Dancoff value of 0.61.
PWRLIKE FUEL BUNDLE, with boundarypin corrections
broad_n
READ COMP
' mixtures for interior pins
UO2 1 .84 293. 92235 2.35 92238 97.65 END
ZR 2 1 END
H2O 3 1 END
' mixtures for boundary pins
UO2 4 .84 293. 92235 2.35 92238 97.65 END
ZR 5 1 END
H2O 6 1 END
END COMP
READ CELLDATA
LATTICECELL SQUAREPITCH PITCH=1.2751 3 FUELD=0.823 1 CLADD=0.9627 2 END
LATTICECELL SQUAREPITCH PITCH=1.2751 6 FUELD=0.823 4 CLADD=0.9627 5 END
CENTRM DATA DAN2PITCH=0.61 END CENTRM
END CELLDATA
EXAMPLE 6. SPHTRIANGP ARRAY.
Consider an infinite array of spherical pellets of 2.67% enriched UO_{2} with a density of 10.3 g/cm^{3} and a diameter of 1.0724 cm arranged in a “triangular” pitch, flooded with borated water at 4350 ppm. The boron is natural boron; the borated water is created by adding boric acid, H_{3}BO_{3}, and has a density of 1.0078 g/cm^{3}. The temperature is 15\(^{\circ}\).
Because the diameter of the fuel pellet, 1.0724 cm, is smaller than the average meanfree path in the UO_{2}, approximately 1.5 cm, and because the configuration is a regular array, LATTICECELL is the appropriate choice for proper crosssection processing.
The density fraction for the UO_{2} is the ratio of actual to theoretical density (10.3/10.96 = 0.9398). Assume that the U is all ^{235}U and ^{238}U. See Sect. 7.1.4.1.9 for how to define borated water.
The XSProc data follows:
SPHERICAL PELLETS IN BORATED WATER
fine_n
READ COMP
UO2 1 .9398 288 92235 2.67 92238 97.33 END
ATOMH3BO3 2 0.025066 3 5000 1 1001 3 8016 3
1.0 288 END
H2O 2 0.984507 288 END
END COMP
READ CELLDATA
LATTICECELL SPHTRIANGP PITCH=1.1440 2 FUELD=1.0724 1 END
END CELLDATA
7.1.4.3.3. MULTIREGION input data
Examples of XSProc input data for MULTIREGION problems are given below.
EXAMPLE 1. SPHERICAL.
Consider a small highly enriched uranium sphere supported by a Plexiglas collar in a tank of water. The uranium metal sphere has a diameter of 13.1075 cm, is 97.67% enriched, and has a density of 18.794 g/cm^{3}. The cylindrical Plexiglas collar has a 4.1275cmradius central hole, extends to a radius of 12.7 cm and is 2.54 cm thick. The water filled tank is 60 cm in diameter.
The density fraction of the uranium metal is the ratio of actual to theoretical density, where the theoretical density is obtained from the Isotopes in standard composition library table in section 7.2.1. Thus, the density multiplier is 18.794/19.05 = 0.9866. The abundance of uranium is not stated beyond 97.67% enriched, so it is reasonable to assume the remainder is ^{238}U. The Plexiglas collar is not significantly different from water and does not surround the fuel, so it can be ignored. If it is ignored, the problem becomes a 1D geometry that can be defined using the MULTIREGION type of calculation, and the eigenvalue of the system can be obtained without additional data by executing CSAS1. However, the Plexiglas has been included in this data so it can be passed to a code such as KENO V.a which can describe the geometry rigorously. The XSProc data follow:
SMALL WATER REFLECTED SPHERE ON PLEXIGLAS COLLAR
fine_n
READ COMP
URANIUM 1 .9866 293. 92235 97.67 92238 2.33 END
PLEXIGLAS 2 END
H2O 3 END
END COMP
READ CELLDATA
MULTIREGION SPHERICAL RIGHT_BDY=VACUUM END 1 6.55375 3 30.0 END ZONE
END CELLDATA
EXAMPLE 2. BUCKLEDSLAB.
This example features a 93.2% enriched uranylfluoride solution inside a rectangular Plexiglas container immersed in water. The fissile solution contains 578.7 g of UO_{2}F_{2} per liter and has no excess acid. The critical thickness of the fuel is 5.384 cm. The finite height of the fuel slab is 147.32 cm, and the depth is 71.58 cm. The Plexiglas container is 1.905 cm thick and is reflected by 20.32 cm of water.
The half thickness of the fuel (2.692) will be used with a reflected left boundary and a vacuum right boundary (default). The XSProc data follow:
CRITICAL SLAB EXPERIMENT USING URANYLFLUORIDE SOLUTION
fine_n
READ COMP
SOLUTION MIX=1 RHO[UO2F2]=578.7
92235 93.2 92238 6.8 TEMP=300
END SOLUTION
PLEXIGLAS 2 END
H2O 3 END
END COMP
READ CELLDATA
MULTIREGION BUCKLEDSLAB LEFT_BDY=REFLECTED
DY=71.58 DZ=147.32 END 1 2.692 2 4.597 3 24.917 END ZONE
END CELLDATA
7.1.4.3.4. DOUBLEHET input data
EXAMPLE 1: A doublyheterogeneous spherical fuel element with 15,000 UO_{2} particles in a graphite matrix.
Grain fuel radius is 0.025 cm. Grain contains one coating layer that is 0.009cmthick. Pebbles are in a triangular pitch on a 6.4cmpitch. Fuel pebble fuel zone is 2.5 cm in radius and contains a 0.5cmthick graphite clad that contains small amounts of ^{10}B. Pebbles are surrounded by ^{4}He. In this case we designated the homogenized mixture as mixture 10. If we have a KENO V.a or KENOVI input section, we would use mixture 10 in that section. Note that the keyword “FUELR=” is followed by the fuel dimension only, i.e., no mixture number. That is because the fuel mixture number is specified with “FUELMIX=” and therefore need not be repeated.
INFINITE ARRAY OF UO2FUELLED PEBBLES
fine_n
READ COMP
' UO2 FUEL KERNEL
U235 1 0 1.92585E3 293.6 END
O 1 0 4.64272E2 293.6 END
' FIRST COATING
C 2 0 5.26449E2 293.6 END
' GRAPHITE MATRIX
C 6 0 8.77414E2 293.6 END
' CARBON PEBBLE OUTER COATING
C 7 0 8.77414E2 293.6 END
B10 7 0 9.64977E9 293.6 END
HE4 8 0 2.65156E5 293.6 END
END COMP
READ CELLDATA
DOUBLEHET RIGHT_BDY=WHITE FUELMIX=10 END
GFR=0.025 1 COATT=0.009 2 MATRIX=6 NUMPAR=15000 END GRAIN
PEBBLE SPHTRIANGP RIGHT_BDY=WHITE HPITCH=3.2 8 FUELR=2.5 CLADR=3.0 7 END
END CELLDATA
EXAMPLE 2: A doublyheterogeneous spherical fuel element with 10,000 UO_{2} particles and 5,000 PuO_{2} particles in a graphite matrix.
Grain fuel radii for UO_{2} and PuO_{2} particles are 0.025 cm and 0.012 cm, respectively. UO_{2} grains contain one coating layer that is 0.009 cm thick. PuO_{2} grains contain one coating layer that is 0.0095cmthick. Pebbles are in a triangular pitch on a 6.4cmpitch. Fuel pebble fuel zone is 2.5cm in radius and contains a 0.5cmthick graphite clad that contains small amounts of ^{10}B. Pebbles are surrounded by ^{4}He. Since number of particles is entered, the total volume fraction and the pitch can be calculated by the code.
INFINITE ARRAY OF UO2 AND PUO2FUELLED PEBBLES
fine_n
READ COMP
' UO2 FUEL KERNEL
U235 1 0 1.92585E3 293.6 END
O 1 0 4.64272E2 293.6 END
' FIRST COATING
C 2 0 5.26449E2 293.6 END
' GRAPHITE MATRIX
C 6 0 8.77414E2 293.6 END
' CARBON PEBBLE OUTER COATING
C 7 0 8.77414E2 293.6 END
B10 7 0 9.64977E9 293.6 END
HE4 8 0 2.65156E5 293.6 END
' PUO2 FUEL KERNEL
PU239 11 0 1.24470E02 293.6 END
O 11 0 4.60983E02 293.6 END
' FIRST COATING
C 12 0 5.26449E2 293.6 END
' GRAPHITE MATRIX
C 16 0 8.77414E2 293.6 END
END COMP
READ CELLDATA
DOUBLEHET RIGHT_BDY=WHITE FUELMIX=10 END
GFR=0.025 1 COATT=0.009 2 MATRIX=6 NUMPAR=10000 END GRAIN
GFR=0.012 11 COATT=0.0095 12 MATRIX=16 NUMPAR=5000 END GRAIN
PEBBLE SPHTRIANGP RIGHT_BDY=WHITE HPITCH=3.2 8 FUELR=2.5 CLADR=3.0 7 END
END CELLDATA
EXAMPLE 3: A doublyheterogeneous slab fuel element with flibe salt coolant
Grain fuel radii for UO_{2} particles are 0.025 cm. The UO_{2} grains contain four coating layers with thicknesses of 0.01, 0.0035, 0.003, and 0.004 cm, respectively. The fuel grains are embedded in a carbon matrix material to form the fuel compact. The xdimension of fuel plate consists of a 0.5 cm (halfthickness) fuel compact region, a carbon clad with outer dimension of 1.27, followed by the flibe coolant with an outer reflected dimension of 1.62 cm. The width (ydimension) of the slab plate is 22.5 cm and the height (zdimension) is 500 cm. The y and z dimensions are only used to define volumes for the fuel plate.
slab doublehet sample problem: doublehet for slab
v7.1252n
read comp
' fuel kernel
u238 1 0 2.12877e2 293.6 end
u235 1 0 1.92585e3 293.6 end
o 1 0 4.64272e2 293.6 end
b10 1 0 1.14694e7 293.6 end
b11 1 0 4.64570e7 293.6 end
' first coating
c 2 0 5.26449e2 293.6 end
' inner pyro carbon
c 3 0 9.52621e2 293.6 end
' silicon carbide
c 4 0 4.77240e2 293.6 end
si 4 0 4.77240e2 293.6 end
' outer pyro carbon
c 5 0 9.52621e2 293.6 end
' graphite matrix
c 6 0 8.77414e2 293.6 end
b10 6 0 9.64977e9 293.6 end
b11 6 0 3.90864e8 293.6 end
' carbon slab outer coating
c 7 0 8.77414e2 293.6 end
b10 7 0 9.64977e9 293.6 end
b11 7 0 3.90864e8 293.6 end
Li6 8 0 1.38344E06 948.15 end
Li7 8 0 2.37205E02 948.15 end
Be 8 0 1.18609E02 948.15 end
F 8 0 4.74437E02 948.15 end
end comp
read celldata
doublehet fuelmix=10 end
gfr=0.02135 1
coatt=0.01 2
coatt=0.0035 3
coatt=0.003 4
coatt=0.004 5
vf=0.4
matrix=6
end grain
slab symmslabcell
hpitch=1.62 8
cladr=1.27 7
fuelr=0.5
fuelh=500
fuelw=22.500
end
centrm data ixprt=1 isn=8 end centrm
end celldata
EXAMPLE 4: A doublyheterogeneous triangularpitch fuel element with 1,302 UO_{2} particles in a graphite matrix with the DAN2PITCH option for grain.
Grain fuel radius for UO2 particles are 0.02125 cm. The UO2 grains contain four coating layers with radii of 0.03125, 0.03525, 0.03875, and 0.04275 cm, respectively. The fuel grains are embedded in a carbon matrix material to form the fuel compact. Fuel compact is in a triangular pitch on a 1.8796cmpitch. Fuel zone is 0.6225cm in radius and there is a 0.0125 cm gap between fuel and graphite moderator. Since number of particles is entered, the total volume fraction and the pitch can be calculated by the code. Dancoff factor of 0.6552 is inputted for a grain to consider neutron leakage effect.
DH_dan2pitch_nonuniform
v7.1252
read composition
u235 1 0 3.6676E03 600.0 end
u238 1 0 1.9742E02 600.0 end
o16 1 0 3.5114E02 600.0 end
c 1 0 1.1705E02 600.0 end
c 2 0 5.2646E02 600.0 end
c 3 0 9.5263E02 600.0 end
si28 4 0 4.4159E02 600.0 end
si29 4 0 2.2433E03 600.0 end
si30 4 0 1.4805E03 600.0 end
c 4 0 4.7883E02 600.0 end
c 5 0 9.5263E02 600.0 end
cgraphite 6 0 7.2701E02 600.0 end
he 7 0 2.4006E05 600.0 end
cgraphite 8 0 9.2756E02 600.0 end
end composition
read celldata
doublehet fuelmix=9 end
gfr=0.02125 1
coatr=0.03125 2
coatr=0.03525 3
coatr=0.03875 4
coatr=0.04275 5
numpar=1302
matrix=6 end grain
centrm data alump=0.0 dan2pitch=0.6562 end centrm
rod triangpitch
fuelr=0.6225
gapr=0.635 7
hpitch=0.9398 8
fuelh=1.000
right_bdy=white left_bdy=reflected end
centrm data iup=12 isn=16 alump=0.0 end centrm
end celldata
7.1.4.3.5. Two methods of specifying a fissile solution
The standard composition specification data offer flexibility in the choice of input data. This section illustrates two methods of specifying the same fissile solution.
Create a mixture 3 that is aqueous uranyl nitrate solution:
UO_{2}(NO_{3})_{2}, solution density = 1.555 g cm^{3}/
0.2669 g U/gsoln., 0.415 g U/ cm^{3}; excess nitrate = 0.39 mg/gsoln
Uranium isotopic content: 92.6 wt % U235 5.9 wt % U238
1.0 wt % U234 and 0.5 wt % U236
The SCALE atomic weights used in this problem are listed as follows:
H 1.0078
O 15.999
N 14.0067
U234 234.041
U235 235.0439
U236 236.0456
U238 238.0508
Two methods of describing the uranyl nitrate solution will be demonstrated. Method 1 is more rigorous, and method 2 is easier and as accurate.
METHOD 1:
This method involves breaking the solution into its component parts [(HNO_{3}, UO_{2}(NO_{3})_{2}, and H_{2}O)] and entering the basic standard composition specifications for each.
Calculate the density of the HNO_{3} 0.39 \(\times\) 10^{3} g NO_{3}/g soln \(\times\) [(62.997 g HNO_{3}/mole HNO_{3})/(61.990 g NO_{3}/mole NO_{3})] \(\times\) 1.555 g soln/ cm^{3}soln = 6.16 \(\times\) 10^{4} g HNO_{3}/cc soln.
Calculate the density fraction of HNO_{3} (actual density/theoretical density). In the Standard Composition Library the theoretical density of HNO_{3} is 1.0. 6.16 \(\times\) 10^{4}/1.0 = 6.16 \(\times\) 10^{4}.
Calculate the molecular weight of the uranium
The number of atoms in a mole of uranium is the sum of the number of atoms of each isotope in the mole of uranium.
Let AU = the average molecular weight of uranium, g U/mole U
GU = the density of uranium in g/cm^{3}.
Then the number of atoms in a mol of uranium =
(6.023 \(\times\) 10^{+23} * 10^{24} * GU)/AU
or 0.6023 * GU/AU.
The weight fraction of each isotope is the weight % * 100.
Therefore, F235 = 0.926, the weight fraction of U235 in the U
F238 = 0.059, the weight fraction of U238 in the U
F236 = 0.005, the weight fraction of U236 in the U
F234 = 0.010, the weight fraction of U234 in the U
A235 = 235.0442, the molecular weight of U235
A238 = 238.0510, the molecular weight of U238
A236 = 236.0458, the molecular weight of U236
A234 = 234.0406, the molecular weight of U234.
Then the number of atoms of isotopes in a mol of uranium =
6.023 \(\times\) 10^{+23} * 10^{24} * ( (GU*F235/A235) + (GU*F238/A238) +
GU*F236/A236) + (GU*F234/A234) )
or
0.6023*GU * ( 0.926/235.0442 + 0.059/238.0510 +
0.005/236.0458 + 0.010/234.0406 ).
Because the number of atoms of uranium equals the sum of the atoms of isotopes,
0.6023 * GU/AU = 0.6023 * GU *( 0.926/235.0442 + 0.059/238.0510 +
0.005/236.0458 + 0.010/234.0406 )
1/AU = 0.926/235.0442 + 0.059/238.0510 + 0.005/236.0458 + 0.010/234.0406
AU = 235.2144.
Calculate the molecular weight of the UO_{2}(NO_{3})_{2}.
235.2144 + (8 \(\times\) 15.9954) + (2 \(\times\) 14.0033) = 391.184 g UO_{2}(NO_{3})_{2}/mole
Calculate the density of UO_{2}(NO_{3})_{2}
0.415 g U/cc \(\times\) [(391.184 g UO_{2}(NO_{3})_{2}/mol)/(235.2144 g U/mole)] =
0.69018 g UO_{2}(NO_{3})_{2}/ cm^{3}.soln.
Calculate the density fraction (actual density/theoretical density) of UO_{2}(NO_{3})_{2}.
[In the Standard Composition Library the theoretical density of UO_{2}(NO_{3})_{2} is given as 2.2030 g/cm^{3}.]
The density fraction is 0.69018/2.2030 = 0.31329.
Calculate the amount of water in the solution
1.555 g soln/ cm^{3}. soln  6.16 \(\times\) 10^{4} g HNO_{3}/cm^{3} soln  0.69018 g UO_{2}(NO_{3})_{2}LL/ cm^{3}. soln = 0.8642 g H_{2}O/cc soln.
Calculate the density fraction (actual density/theoretical density) of water.
HNO3 3 6.164 293 END
UO2(NO3)2 3 .31329 293 92235 92.6 92238 5.9 92234 1.0
92236 0.5 END
H2O 3 .86575 293 END
METHOD 2:
This method utilizes the solution option available in the standard composition specification data. Because the density is specified in the input data, this method should yield correct number densities that should agree with method 1 except for calculational roundoff.
Calculate the fuel density
0.415 g U/cc is 415 g U/L.
The molecular weight of nitrate NO_{3} is 61.9895.
Calculate the molarity of the solution.
0.39 mg nitrate/g soln \(\times\) 1000 cm^{3}soln/L soln \(\times\) 1 g/1000 mg \(\times\) 1.555 g soln/ cm^{3}soln = 0.60645 g excess nitrate/L soln.
A 1molar solution is 1 mole of acid/L of solution:
(For nitric acid 1 molar is 1 normal because there is only one atom of hydrogen per molecule of acid in HNO_{3}.)
(0.60645 g nitrate/L soln)/(61.9895 g NO_{3}/mole NO_{3}) = 9.783 \(\times\) 10^{3} mole nitrate/L is identical to mole of acid/L, which is identical to molarity.
The density fraction of the solution is 1.0. Do not try to use the density of the solution divided by the theoretical density of UO_{2}(NO_{3})_{2} from the Standard Composition Library for your density multiplier. The UO_{2}(NO_{3})_{2} listed there is the solid, not the solution.
The solution specification data follow:
SOLUTION MIX=1 RHO[UO2(NO3)2] = 415 92235 92.6 92238 5.9
92234 1.0 92236 0.5
MOLAR [HNO3] = 9.7833
TEMP = 293 DENSITY = 1.555 END SOLUTION
Comparison of number densities from the two methods
The number densities of methods 1 and 2 should agree within the limits of the input data. The density multipliers in method 1 are 5 digits and the density multipliers in method 2 are 4 digits. Therefore, the number densities calculated by the two methods should agree to 4 or 5 digits.
Method 1 
Method 2 

Nuclide number 
Atom density 
Atom density 
92235 
9.84603E04 
9.84603E04 
92238 
6.19415E05 
6.19415E05 
92234 
1.06784E05 
1.06784E05 
92236 
5.29387E06 
5.29387E06 
07014 
2.13092E03 
2.13092E03 
08016 
3.74135E02 
3.7410E02 
01001 
5.77973E02 
5.77983E02 
7.1.4.3.6. Multiple unit cells in a single problem
Consider a problem that involves three different UO_{2} fuel assemblies: a 1.98%enriched assembly, a 2.64%enriched assembly, and a 2.96%enriched assembly. All fuel rods are UO_{2} at 10.138 g/cm^{3} and are 0.94 cm in diameter. The Zircaloy4 clad has an inside radius of 0.4875 cm and an outside radius of 0.545 cm. The rod pitch is 1.44 cm. Each fuel assembly is a 15 \(\times\) 15 array of fuel pins with water holes, instrumentation holes, and burnable poison rods. For crosssection processing, the presence of the water holes, instrumentation holes, and burnable poison rods in the assemblies are ignored.
The following XSProc input use the CENTRM/PMC method for selfshielding three latticecells with different fuel enrichments. The remaining mixture (SS304), not specified in a unit cell, is processed as an infinite homogeneous medium using the BONAMI method. Each mixture can appear only in a single zone of one unit cell. For square pitch latticecells the default CENTRM transport solver is MoC with P0 scatter; however in this input, the solver for the 3^{rd} cell is modified through CENTRM DATA to use the tworegion approximation for the CE calculation [npxs=5], and discrete S_{N} transport calculation with P1 anisotropic scatteringfor the MG solutions in the fast and thermal energy ranges [nfst=0, nthr=0].
DEMONSTRATION PROBLEM WITH MULTIPLE RESONANCE CORRECTIONS REQUIRED
broad_n
READ COMP
UO2 1 .925 300 92235 1.98 92238 98.02 END
UO2 2 .925 300 92235 2.64 92238 97.36 END
UO2 3 .925 300 92235 2.96 92238 97.04 END
ZIRC4 4 1.0 300 END
H2O 5 1.0 300 END
ZIRC4 6 1.0 300 END
H2O 7 1.0 300 END
ZIRC4 8 1.0 300 END
H2O 9 1.0 300 END
SS304 10 1.0 300 END
END COMP
READ CELLDATA
LATTICECELL SQUAREPITCH PITCH=1.44 5 FUELD=0.94 1 CLADD=1.09 4 GAPD=0.975 0 END
LATTICECELL SQUAREPITCH PITCH=1.44 7 FUELD=0.94 2 CLADD=1.09 6 GAPD=0.975 0 END
LATTICECELL SQUAREPITCH PITCH=1.44 9 FUELD=0.94 3 CLADD=1.09 8 GAPD=0.975 0 END
CENTRM DATA npxs=5 nthr=0 nfst=0 isct=1 END CENTRM DATA
END CELLDATA
7.1.4.3.7. Multiple fissile mixtures in a single unit cell
The following problem involves large units having the bulk of their fissile material more than one meanfree path away from the surface of the unit. The interaction between the units that occurs in the resonance range is a very small fraction of the total interaction because an overwhelming percentage of the interaction occurs deep within each unit. Therefore, the resonance range interaction between the units can be ignored, and the default infinite homogeneous medium crosssection processing in the resonance range can be considered adequate for this particular application.
Consider a problem that consists of four 20.96kg 93.2%enriched uranium metal cylinders, density 18.76 g/cm^{3}, and four 5liters Plexiglas bottles filled with highly enriched uranyl nitrate solution at 415 g/L, a specific gravity of 1.555, and 0.39 mg of excess nitrate per gram of solution. The isotopic content of the uranium metal is 93.2 wt % ^{235}U, 5.6 wt % ^{238}U, 1.0 wt % ^{234}U, and 0.2 wt % ^{236}U. The uranium isotopic content of the nitrate solution is 92.6 wt % ^{235}U, 5.9 wt % ^{238}U, 1.0 wt % ^{234}U and 0.5 wt % ^{236}U. The size of the metal cylinders is between 10 and 12 cm in diameter and height, and the size of the nitrate solution is on the order of 16 and 20 cm in diameter and height. The average meanfree path in the uranium metal is on the order of 1.5 cm, and the average mean free path in the nitrate solution is on the order of 0.5 cm. Therefore, infinite homogeneous medium is an appropriate choice for this problem and the use of CENTRM/PMC is valid.
See Examples 1–4 of Sect. 7.1.4.1.2 for data input details for the Plexiglas and uranium metal. See Example 1 of Sect. 7.1.4.1.5 for data input details for the uranyl nitrate solution. The XSProc data for this problem follow:
SET UP 4 AQUEOUS 4 METAL
fine_n
READ COMP
URANIUM 1 0.985 293 92235 93.2 92238 5.6 92234 1.0 92236 0.2 END
SOLUTION 2 RHO[UO2(NO3)2]=415 92235 92.6 92238 5.9 92234 1.0 92236 0.5
MOLAR[HNO3]=9.7833 DENSITY=1.555 TEMPERATURE=293 END SOLUTION
PLEXIGLAS 3 END
END COMP
Consider the same materials above except rearrange them so that a 10 cm diameter uranium metal sphere sits inside a 50 cm diameter spherical tank of uranyl nitrate solution having a 1cm thick Plexiglas wall. This problem can be modeled in SCALE but only CENTRM/PMC will treat the resonance processing correctly. This problem is modeled below.
SET UP 4 AQUEOUS 4 METAL
fine_n
READ COMP
URANIUM 1 0.985 293 92235 93.2 92238 5.6 92234 1.0 92236 0.2 END
SOLUTION 2 RHO[UO2(NO3)2]=415 92235 92.6 92238 5.9 92234 1.0 92236 0.5
MOLAR[HNO3]=9.7833 DENSITY=1.555 TEMPERATURE=293 END SOLUTION
PLEXIGLAS 3 END
END COMP
READ CELLDATA
MULTIREGION SPHERICAL END 1 5.0 2 25.0 3 26.0 END ZONE
END CELLDATA
7.1.4.3.8. Cell weighting an infinite homogeneous problem
Cell weighting an infinite homogeneous medium has no effect on the cross sections because there is only one zone and one set of cross sections. However, a cellweighted mixture number can still be supplied using the keyword CELLMIX= followed by an unique mixture number. This cellweighted mixture number can be used in subsequent codes and will produce results similar to the cross sections of the original mixture.
EXAMPLE 1
This problem would probably be run with CSAS1 to provide the kinfinity of 20%enriched UO_{2}.
20% ENRICHED UO2 BILLET
fine_n
READ COMP
UO2 1 0.99 293 92235 20 92238 80 END
END COMP
READ CELLDATA
INFHOMMEDIUM 1 CELLMIX=100 END
END CELLDATA
7.1.4.3.9. Cell weighting a LATTICECELL problem
Cell weighting used with a LATTICECELL problem creates cellweighted homogeneous cross sections that represent the characteristics of the heterogeneous unit cell. This cellweighted mixture can then be used in a subsequent code for the overall volume where the cells are located without having to mock up the actual 3D heterogeneous array of cells. This cellweighted homogeneous mixture is designated by the user with the keyword CELLMIX= immediately followed by an unused mixture number. This needs to follow immediately after the cell description. Note that the mixtures used in the unit cell data cannot be used in a subsequent code because they have been flux weighted to create the user specified mixture. Therefore, if a mixture used in the unit cell description is also to be used in a subsequent code, another mixture must be created that is identical except for the mixture number. Every mixture that is to be used in a subsequent code except zero (i.e., void) must be defined in the standard composition data.
A byproduct of the cellweighting calculation is the eigenvalue (keffective) of an infinite array of the cell described as the unit cell.
EXAMPLE 1
Consider a cylindrical stainless steel tank filled with spherical pellets of 2.67%enriched UO_{2} arranged in a closepacked “triangular” pitch, flooded with borated water at 4350 ppm. The cylindrical stainless tank is sitting in a larger tank filled with borated water at 4350 ppm.
The data for the UO_{2} and borated water were developed in detail in Example 3 of Sect. 7.1.4.3.2. The stainless steel must be defined, and mixture 3 was chosen because mixture 1 was the UO_{2} and mixture 2 was the borated water. Because the borated water will be used as a reflector for the stainless steel tank and has been used in the unit cell data, it must be repeated with a different mixture number (in this case, as mixture 4).
In the subsequent calculation, user specified cell mixture 100 will be used to represent the UO_{2} pellets in the borated water, mixture 3 will represent the stainless steel tank, and mixture 4 will represent the borated water reflector around the stainlesssteel tank.
The XSProc data for creating the cellweighted cross sections on mixture 100 follow:
SPHERICAL PELLETS IN BORATED WATER
fine_n
READ COMP
UO2 1 .9398 293. 92235 2.67 92238 97.33 END
ATOMH3BO3 2 0.025066 3 5000 1 1001 3 8016 3 1.0 293 END
H2O 2 0.984507 293 END
SS304 3 1.0 293 END
ATOMH3BO3 4 0.025066 3 5000 1 1001 3 8016 3 1.0 293 END
H2O 4 0.984507 293 END
END COMP
READ CELLDATA
LATTICECELL SPHTRIANGP PITCH 1.0724 2 FUELD 1.0724 1 CELLMIX=100 END
END CELLDATA
7.1.4.3.10. Cell weighting a MULTIREGION problem
A MULTIREGION problem is cell weighted primarily to obtain a cellweighted homogeneous cross section that represents the characteristics of the heterogeneous unit cell. The eigenvalue obtained for a MULTIREGION problem with cylindrical or spherical geometry having a white boundary condition specified on the right boundary approximates an infinite array of the cells. A vacuum boundary condition would represent a single cell. A slab with reflected boundary conditions for both boundaries represents an infinite array of slab cells. The cellweighted cross sections for spherical or cylindrical geometries with a white right boundary condition do not use a Dancoff correction and thus may not be accurate for representing a large array of the specified units.
EXAMPLE 1
Consider a small, highly enriched uranium sphere supported by a Plexiglas collar in a tank of water. The uranium metal sphere has a diameter of 13.1075 cm, is 97.67% enriched, and has a density of 18.794 g/cm^{3}. The cylindrical Plexiglas collar has a 4.1275cm radius central hole, extends to a radius of 12.7 cm and is 2.54 cm thick. The waterfilled tank is 60 cm in diameter.
The Plexiglas collar is not significantly different from water and does not surround the fuel, so it will be ignored. Because this makes the problem a 1D geometry, it can be defined using the MULTIREGION type of calculation and the eigenvalue of the system can be obtained without additional data by executing CSAS1 with CENTRM/PMC, if PARM=CENTRM is specified on the command line. The abundance of uranium is not stated beyond 97.67% enriched, so assume the remainder is ^{238}U. The XSProc data follow:
=CSAS5
SMALL WATER REFLECTED SPHERE ON PLEXIGLAS COLLAR
fine_n
READ COMP
URANIUM 1 DEN=18.794 1 293. 92235 97.67 92238 2.33 END
H2O 2 END
END COMP
READ CELLDATA
MULTI SPHERICAL CELLMIX=100 END 1 6.5537 2 30.0 END ZONE
END CELLDATA
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KENO DATA THAT USES MIX=100 FOR A HOMOGENEOUS SPHERE OF 30CM RADIUS GOES HERE.
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END