.. _CSAS5App: Additional Example Applications of CSAS5 ======================================== Several example uses of CSAS5 are shown in this section for a variety of applications. Note that many of these examples have been provided since the earliest versions of the CSAS sequences and output data shown may not represent the most current version. Typical search -------------- EXAMPLE 1. CSAS5S — Determine the optimum pitch for an array of Example 4 fuel assemblies Consider a 4 × 4 × 1 array of fuel assemblies in a square aluminum cask. Each assembly consists of a 17 × 17 × 1 array of zirconium-clad, 2.35%-enriched UO\ :sub:2 fuel pins in a square pitched array. The UO\ :sub:2 has a density of 9.21 g/cc. The pin diameter is 0.8 cm and is 366 cm long. The clad is 0.07 cm thick, and the pitch is 1.3 cm. Each fuel bundle is contained in a 0.65-cm-thick Boral sheath. The bundles are separated by an edge-to-edge spacing of 1 cm. The array of bundles is centered in a 10-cm-thick aluminum cask whose inside dimensions are 0.5 cm beyond the outer edges of the fuel bundles. Search for the assembly spacing that yields the maximum value of *k*\ :sub:eff. Because the spacing between the assemblies is to be altered, the last region of unit 2 will be altered. In order to do this, the fuel assembly shroud cannot be defined as a replicate, because the code does not know the size of the array until later. In order to perform a pitch or dimension search, the code calculates the distance between the outermost region of the unit and the region interior to it. Therefore, the fuel assembly gap must be defined as a cuboid, and the water gap between assemblies can be entered as either a cuboid or a replicate. This search has been defined as an optimum dimension search. The input data for this problem follow. .. highlight:: scale :: =CSAS5S SAMPLE FUEL CASK EXAMPLE V7-238 READ COMP UO2 1 DEN=9.21 1.0 293. 92235 2.35 92238 97.65 END ZR 2 1 END H2O 3 1 END B4C 4 0.367 END AL 4 0.636 END AL 5 1 END END COMP READ CELLDATA LATTICECELL SQUAREPITCH PITCH=1.3 3 FUELD=0.8 1 CLADD=0.94 2 END END CELLDATA READ PARAM TME=6.0 NUB=YES FAR=YES GEN=103 END PARAM READ GEOM UNIT 1 COM='FUEL PIN' CYLINDER 1 1 0.4 2P183.0 CYLINDER 2 1 0.47 2P183.07 CUBOID 3 1 4P0.65 2P183.07 UNIT 2 COM='FUEL ASSEMBLY' ARRAY 1 2R-11.05 -183.07 CUBOID 4 1 4P11.7 2P183.72 REPLICATE 3 1 6*0.5 1 GLOBAL UNIT 3 COM='FUEL CASK CONTAINING 4X4 ARRAY OF ASSEMBLIES' ARRAY 2 2R-48.8 -184.22 REPLICATE 5 1 6R10.0 1 END GEOM READ ARRAY ARA=1 NUX=17 NUY=17 NUZ=1 FILL F1 END FILL ARA=2 NUX=4 NUY=4 NUZ=1 FILL F2 END FILL END ARRAY END DATA READ SEARCH OPTIMUM DIMENSION MORE ALTER UNIT=2 REG=3 +X=1 -X=1 +Y=1 -Y=1 +CON=0.204918 -CON= -0.0409836 END SEARCH END An alternative method of entering the search data for this problem is to define the search as an optimum pitch search and require that the spacing of the fuel pin cells remain unchanged. Because a pitch search is always conducted at the lowest array level, in this case the spacing between the pins in the fuel assembly (the outer region of unit 1), it is necessary to countermand the automatic alteration of the outer region of unit 1 by entering the KEEP command. Search constants must be entered for the X and Y faces to instruct the code to KEEP those dimensions unchanged. It is easier to use the keyword ALL, which applies the KEEP command to all of the faces since the optimum pitch search would have changed only the X and Y faces leaving the Z faces unchanged. The ALTER command must then be entered to instruct the search to alter the spacing between the fuel assemblies (region 3 of unit 2). Only the X and Y dimensions are to be altered; so the search constants are entered individually for those dimensions. It is acceptable to enter ALL=1 +Z=0 −Z=0 rather than +X=1 −X=1 +Y=1 −Y=1. It is not necessary to enter the constraints for an optimum pitch search (+CON= and −CON=). They were entered in this case to ensure that the alternative data more nearly duplicate the optimum dimension search data from the previous example. These alternative search data are: :: READ SEARCH OPTIMUM PITCH MORE KEEP UNIT=1 REG=3 ALL=1 ALTER UNIT=2 REG=3 +X=1 −X=1 +Y=1 −Y=1 END SEARCH Auxiliary search commands ------------------------- Auxiliary search commands are entered ONLY if the word MORE was entered in the search type specification data. These data are used to define the method the search will use to alter the pitch, geometry or concentration data and to set the constraints for the parameter search. The auxiliary search commands consist of (1) INDIVIDUAL SEARCH COMMANDS and (2) SEARCH PARAMETER CONSTRAINTS. **EXAMPLE 1** Consider an example in which region 2 of unit 1 is a cuboid and all of the dimensions of the cuboid are to be altered. The search data could be entered as: :: ALTER UNIT=1 REG=2 ALL=1.0 or :: ALTER UNIT=1 REG=2 +X=1.0 −X=1.0 +Y=1.0 −Y=1.0 +Z=1.0 −Z=1.0 Because all of the search constants are nonzero, all of the dimensions will be changed. Because the search constants are identical, the original relationship between the dimensions will be preserved as they are altered. For example, if region 2 of unit 1 is a cylinder and if all the dimensions of the cylinder are to be altered, and if unit 1 is tied to unit cell 1 the search data could be entered as: :: ALTER UNIT=1 REG=2 ALL=1.0 CELL=1 or :: ALTER UNIT=1 REG=2 RADIUS=1.0 +H=1.0 ?H=1.0 CELL=1 Because all of the search constants are nonzero, all of the dimensions will be changed. Because the search constants are identical, the original relationship between the dimensions will be preserved as they are altered. If the original height-to-diameter ratio is 1.5, that ratio will be preserved throughout the search only if the search constants for the radius and + and − height are identical. Search constants can be entered sequentially with each new entry overriding only identical previous entries. For example, if region 2 of unit 1 is a cuboid and if all of the dimensions except the −Z dimension are to be altered, the search data could be entered as: :: ALTER UNIT=1 REG=2 +X=1.0 −X=1.0 +Y=1.0 −Y=1.0 +Z=1.0 or :: ALTER UNIT=1 REG=2 ALL=1.0 −Z=0.0 In the second example, all of the search constants corresponding to the cuboid’s dimensions are set to 1.0 by using the ALL= command. This includes the −Z dimension. To reset the search constant for the −Z dimension to zero, −Z=0.0 is added following the ALL=1.0 command. If the search commands are reversed, −Z=0.0 ALL=1.0, the −Z dimension will also be altered because the −Z portion of the ALL= command will override the previously entered −Z=0.0 command. **EXAMPLE 2** A search command must be entered for each unit and region specification. Consider a problem having units 1, 2, and 3. Unit 1 consists of three concentric spheres in a cuboid. Unit 2 consists of a single sphere in a cuboid, and unit 3 contains three concentric cuboids. A search is to be made that changes the inner sphere and outer cuboid of unit 1, and the sphere and cuboid of unit 2, and the exterior cuboid of unit 3. The thicknesses of the outer spheres of unit 1 are to be maintained, and the two inner cuboids of unit 3 are to remain unchanged. The search data for this problem can be entered as follows: :: ALTER UNIT=1 REG=1 ALL=1.0 ALTER UNIT=1 REG=4 ALL=1.0 MAINTAIN UNIT=1 REG=2 TO 3 ALL=1.0 ALTER UNIT=2 REG=1 TO 2 ALL=1.0 ALTER UNIT=3 REG=3 ALL=1.0 Search constraints for dimension or pitch searches -------------------------------------------------- SEARCH PARAMETER CONSTRAINTS set the parameter limits for the search. The minimum constraint is the minimum value of the parameter allowed in the search. The maximum constraint is the maximum value of the parameter allowed in the search. The initial geometry configuration corresponds to a parameter value of 0.0. A physical limit occurs when the value of the parameter causes geometry intersections. Constraints should be entered for a DIMENSION search. Only one set of constraints (i.e., min and max) are allowed per problem. These constraints apply to all the dimensions that are being altered. For a DIMENSION search, the constraints are given by :eq:CSASapp-1 and :eq:CSASapp-2 .. math:: :label: CSASapp-1 C_{min} = ((D_{min}/D_i) - 1.0)/SC , .. math:: :label: CSASapp-2 C_{max} = ((D_{min}/D_i) - 1.0)/SC , Where C\ :sub:min is the minimum constraint for the search C\ :sub:max is the maximum constraint for the search | D\ :sub:min is the minimum allowed dimension for the search [For a chord, D\ :sub:min = (Radius:sub:min + | Chord\ :sub:min )/ 2 Radius ] min | D\ :sub:max is the maximum allowed dimension for the search [For a chord, D\ :sub:max = (Radius:sub:max + | Chord\ :sub:max )/ 2 Radius\ :sub:max] D\ :sub:i is the initial dimension [For a chord, D\ :sub:i = (Radius:sub:initial + Chord\ :sub:initial )/2 Radius\ :sub:initial] SC is the search constant for that dimension (i.e., +X, −X, RADIUS, etc.) For example, the initial radius of a sphere is 6 cm, and a search is to be conducted to determine the radius at which the sphere is critical. The minimum radius the user wishes to allow is 3 cm, and the maximum radius to be allowed is 9 cm. A nonzero search constant must be entered to cause the radius to be changed. A search constant of 1.0, will be used for the radius (RADIUS=1.0). The constraints calculated from :eq:CSASapp-1 and :eq:CSASapp-2 are: C\ :sub:min = ((3.0/6.0)−1.0)/1.0 = −0.5 C\ :sub:max = ((9.0/6.0)−1.0)/1.0 = 0.5 The constraints would be entered in the problem by entering the following data: :: -CON=-0.5 +CON=0.5 For a PITCH search, the minimum constraint defines the limit for shrinking the system, and the maximum constraint defines the limit for expanding the system: .. math:: & \ \ \ \ \ \ \ \ \ \text{i = number of faces} & C_{min} - \text{MAX} \left[D_{+},D_{\_}\right] \text{for shrinking} \ , & \ \ \ \ \ \ \ \ \ \text{i=1} .. math:: & \ \ \ \ \ \ \ \ \ \text{i = number of faces} & C_{max} - \text{MIN} \left[D_{+},D_{\_}\right] \text{for expanding} \ , & \ \ \ \ \ \ \ \ \ \text{i=1} where C\ :sub:min is the minimum constraint, C\ :sub:max is the maximum constraint, D\ :sub:+ are face constraints for the positive dimensions, D\ :sub:− are face constraints for the negative dimensions. Face constraints must be calculated for each face using :eq:CSASapp-3  and :eq:CSASapp-4 : .. math:: :label: CSASapp-3 D_{+} = \frac{(X_{d+} - X_{i+})(SC_{+} \times \text{del}_{+}\text{SC}_{\_} \times \text{del}_{\_})}{(X_{i+} - X_{i-})(\text{SC}_{+}\text{**}2 \times \text{del}_{\_})} .. math:: :label: CSASapp-4 D_{\_} = \frac{(X_{d-} - X_{i-})(SC_{+} \times \text{del}_{+}\text{SC}_{\_} \times \text{del}_{\_})}{(X_{i+} - X_{i-})(\text{SC}_{+}\text{**}2 \times \text{del}_{\_})} where X :sub:d+ is the desired limit of the positive dimension of the spacing cuboid in that direction (positive dimension of X, Y, or Z, whichever dimension is under consideration) X\ :sub:i+ is the initial positive dimension of the spacing cuboid in that direction (positive dimension of X, Y, or Z, whichever dimension is under consideration) X\ :sub:d− is the desired limit of the negative dimension of the spacing cuboid in that direction (negative dimension of X, Y, or Z, whichever dimension is under consideration) X\ :sub:i− is the initial negative dimension of the spacing cuboid in that direction (negative dimension of X, Y, or Z, whichever dimension is under consideration) SC\ :sub:+ is the search constant for the positive dimension of the spacing cuboid in that direction (positive dimension of X, Y, or Z, whichever dimension is under consideration) SC\ :sub:− is the search constant for the negative dimension of the spacing cuboid in that direction (negative dimension of X, Y, or Z, whichever dimension is under consideration) .. note:: Using a search constant of 1.0 simplifies the determination of C\ :sub:max and C\ :sub:min when the dimensions are to change proportionately. \ del\ :sub:+ is the initial distance from the spacing cuboid to the closest interior region in the positive direction (positive dimension of X, Y, or Z, whichever dimension is under consideration) del\ :sub:− is the initial distance from the spacing cuboid to the closest interior region in the negative direction (negative dimension of X, Y, or Z, whichever dimension is under consideration) The search parameter constraints are entered using the following keywords: −CON=pp is used to set the minimum constraint for the current parameter. The value of pp is defaulted to −10E10 for a dimension search. The value of pp is defaulted to a value that allows geometry regions to touch for a pitch search unless a value was entered for MINPITCH, in which case the parameter corresponding to that pitch is calculated and used. +CON=rr is used to set the maximum constraint for the current parameter. The value of rr is defaulted to +10E10 for a dimension search and to −5 × pp for a pitch search. .. note:: A search will reset a constraint (entered using the keyword +CON= or −CON=) that falls outside the default range to the default value. If a PITCH search is specified and if a value has been entered for MAXPITCH and/or MINPITCH, values should not be entered for the constraints. If values are entered for +CON= and/or −CON= for a PITCH search and if MAXPITCH and/or MINPITCH were specified in the optional search parameters, the maximum and minimum constraints will be set to the values corresponding to MAXPITCH and MINPITCH, even though the value of MINPITCH may result in an intersection. Individual concentration search commands ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ An INDIVIDUAL SEARCH COMMAND for a concentration search consists of (1) a command definition, (2) the mixture number of the mixture to be altered, (3) the name of the standard composition to be altered, (4) a search constant, and (5) a unit cell containing the mixture. A series of individual search commands can be entered to govern the search process. A new search command is initiated whenever a command definition, item (1) above, is encountered. The COMMAND DEFINITION defines the action to be taken for the specified mixture and standard composition component. ALTER, CHANGE, or MODIFY are used to cause the concentration (number densities) of the specified standard composition in the specified mixture to be modified. ALTER CHANGE MODIFY The commands to the left are command definitions, item (1) above. These commands instruct the code to modify the specified concentration data. The MIXTURE NUMBER, item (2) above, defines the mixture that contains the standard composition whose concentration is to be varied during the search. The keyword **MIX=** is entered, followed by the mixture number, nn. MIX=nn is used to define the mixture number associated with the component that is to be changed. The keyword MIX= is entered, followed by the mixture number, nn. There is no default value of nn. The STANDARD COMPOSITION NAME, item (3) above, defines the standard composition whose concentration will be changed in the defined mixture. Only standard compositions listed in the Standard Composition Library chapter can be entered. **SCNAME=**\ mm is used to specify the standard composition name of the component that is to be altered. The keyword SCNAME= is entered, followed by the mixture number, mm. There is no default value for mm. .. note:: If the standard composition component name specified in the Material Information Data (item 1 of the Standard Composition Specification Data) is a solution, for example, SOLNUO2(NO3)2, and the Concentration Search Data specifies SCNAME=UO2(NO3)2, the amount of UO2(NO3)2 in the solution will be altered, but the amount of water and nitric acid will not be changed. Thus, the resultant mixture may no longer meet the criteria associated with the SOLN specification. The SEARCH CONSTANT for a concentration search, item (4) above, is a proportional factor that applies to the standard composition being altered. The keyword FACTOR= followed by a proportionality search constant is used to specify the search constant for a concentration search. **FACTOR=**\ pc is used to specify the search constant. The keyword FACTOR= is entered, followed by the value of the search constant or proportionality factor, pc. The UNIT CELL NUMBER, item (5) above, defines the unit cell containing the mixture to be varied during the search. The keyword **CELL=** is entered, followed by the unit cell number, mm. **CELL=**\ mm is used to link mixture nn to unit cell mm. The keyword CELL= is entered, followed by the unit cell number, mm. The code defaults the cell number to the unit cell containing the mixture specified using MIX=. ENTERING AUXILIARY SEARCH COMMANDS IN THE CONCENTRATION SEARCH DATA ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ EXAMPLE 1 Consider an example in which the density of water is to be varied and is contained in unit cell 2. Full density water, mixture 1, was specified in the standard composition specification data as: H2O 1 END The auxiliary search data could be entered as follows: :: ALTER MIX=1 SCNAME=H2O FACTOR=1.0 CELL=2 or ALTER MIX=1 SCNA=H2O FAC=1.0 CE=2 or ALTER M=1 S=H2O F=1.0 C=2 Note that terse input allows truncation of the keywords. EXAMPLE 2 Consider an example in which the density of UO\ :sub:2\ F\ :sub:2, is to be varied in mixture 2 contained in unit cell 4, a uranyl fluoride solution. The uranyl fluoride solution was specified as: :: SOLNUO2F2 2 300 0 1 293 92235 5 92238 95 END The auxiliary search data could be entered as follows: :: ALTER MIX=2 SCNAME=UO2F2 FACTOR=1 CELL=4 or ALTER M=2 S=UO2F2 F=1 C=4 The terse input option allows truncation of the keywords. Search constraints for concentration searches --------------------------------------------- A SEARCH PARAMETER CONSTRAINT for concentration searches sets the parameter limits for the search. The minimum constraint is the minimum value of the parameter allowed in the search. The maximum constraint is the maximum value of the parameter allowed in the search. The initial concentration corresponds to a parameter value of 0.0. A physical limit occurs when the value of the parameter causes the density of the specified standard composition to become negative. The search can produce an unrealistically high density. Users should manually eliminate those results or set constraints to avoid them. Only one set of constraints (i.e., min and max) are allowed per problem. These constraints apply to all the standard compositions that are being altered. For a CONCENTRATION search, the constraints are given by :eq:CSASApp-5 and :eq:CSASApp-6. The maximum constraint must be larger than the minimum constraint. .. math:: :label: CSASApp-5 C_{min} = \frac{((\frac{D_{min}}{D_{i}})-1)}{\text{FACTOR}}, .. math:: :label: CSASApp-6 C_{max} = \frac{((\frac{D_{max}}{D_{i}})-1)}{\text{FACTOR}}, where C\ :sub:min is the minimum constraint for the search, C\ :sub:max is the maximum constraint for the search, D\ :sub:min is the minimum allowed density for the specified standard composition, D\ :sub:max is the maximum allowed density for the specified standard composition, D\ :sub:i is the initial density of the specified standard composition, FACTOR is the search constant for the standard composition that is being varied. Default search constraints are calculated if +CON and −CON are not entered. The default concentration search constraints are calculated as follows: .. math:: +\text{CON} = \begin{Bmatrix} \text{min}(\frac{1}{FACTOR}) , \text{if any FACTOR} < 0 \text{-S} \times (\text{-CON}), \text{if all FACTOR} > 0 \end{Bmatrix} .. math:: -\text{CON} = \begin{Bmatrix} \text{-S}(+\text{CON}) , \text{if all FACTOR} < 0 \text{max}(\frac{-1}{FACTOR}), \text{if any FACTOR} > 0 \end{Bmatrix} Search considerations --------------------- DIMENSION, PITCH, and CONCENTRATION searches can be performed using CSAS5S. A DIMENSION search alters only those regions specified in the search data. A PITCH search alters the center-to-center spacing of units in an array and any other dimensions specified in the auxiliary data. A CONCENTRATION search alters the density of the standard compositions in the mixtures specified. By default a pitch search is performed at the lowest array level and changes the spacing in either the X dimension for a slab, the X and Y dimensions for an array of cylinders, or the X, Y and Z dimensions for an array of spheres. For example, if an array of fuel assemblies is described in the geometry, the lowest array level is the array of fuel pins comprising an assembly. Therefore, an optimum pitch search would alter the spacing between the fuel pins within the assembly. The spacing can be expanded until the array intersects the first region external to it. The exterior size of the fuel pin array would grow or shrink within the confines of the exterior region (fuel assembly shroud). If the external regions are described using replicate regions, the array can grow or shrink within the confines of the maximum and minimum constraints. If replicate regions are used outside an altered region or an array whose spacing units are altered, the dimensions of the replicate regions are recalculated (maintaining the thickness) at each search pass without having to enter search data for those regions. However, geometry regions specified by a geometry shape (sphere, cuboid, cylinder, etc.) that exist outside an altered region or an array whose spacing units are being altered will remain unchanged unless search data are provided for them. In other words, REPLICATE regions will grow and shrink in response to changes in the dimensions of the interior region, but other geometry shapes will not. Some of the limitations applicable to a pitch or dimension search are: 1. A pitch search is performed only at the lowest array level unless search commands are entered to keep the lowest array level unit unchanged and other commands are entered to cause other units to be altered. 2. A pitch search alters the spacing in either the X dimension for a slab, the X and Y dimensions for an array of cylinders, or the X Y and Z dimensions for an array of spheres. Entering a search constant of zero for +X, −X, +Y, −Y, +Z, and/or −Z will keep the corresponding dimension from being altered. 3. A pitch search alters only the outer region of the unit(s) used in the array at the lowest array level unless the search data specifying otherwise is input. 4. A search cannot alter a region whose boundaries are set by the code (i.e., an ARRAY, CORE BOUNDARY, or REPLICATE following an ARRAY). If the dimensions of a replicate region are to be altered, the dimensions of the region interior to it must be explicitly defined. For example, the interior region can be a standard geometry shape (sphere, cylinder, cuboid, etc.), but cannot be a replicate following an array or core boundary. 5. All searches allow auxiliary search data. Concentration searches can also be performed using CSAS5S. A concentration search alters only those standard compositions specified in the search data. Care must be taken when searching on standard compositions beginning with SOLN, ATOM, WTPT, or ARBM (predecessor to ATOM and WTPT in earlier SCALE versions). One or more of their components can be altered but this will not directly affect the other components. For example, if the standard composition component name specified in the material information data is SOLN (SOLNUO2F2) and the concentration search data specifies SCNAME=UO2F2, the amount of uranyl fluoride salt, UO2F2, in the solution will be altered, but the amount of water and hydrofluoric acid in the solution will remain unchanged. Therefore, the resultant mixture may no longer meet the criteria for a solution (SOLN) specification. Physically, the concentration can vary from zero to some upper limit. The code will prevent the concentration from falling below zero, but the user is responsible for setting constraints that prevent the concentration from exceeding reasonable values. The theoretical density is a reasonable upper limit. Optimum pitch search -------------------- An optimum pitch search searches for the pitch that yields the highest value of *k*\ :sub:eff An optimum pitch search is activated by entering “OPTIMUM PITCH” in the search data. By default, the search is performed at the lowest array level and only the spacing in the X-direction for slabs, the X and Y-directions for cylindrical arrays, and the X, Y, and Z-directions for spherical arrays. The search constants are defaulted to 1.0 for the applicable +X, −X, +Y, −Y, +Z, and −Z dimensions of the outermost region. The dimensions of other geometry regions will not be changed (their search constants are defaulted to 0.0) unless additional search data containing appropriate instructions are supplied. The limits for an optimum pitch search can be set using either the MAXPITCH= and MINPITCH= options in the optional search parameters or the +CON= and −CON= options in the search parameter constraints of the auxiliary search commands. MAXPITCH= and MINPITCH= are used to enter a value of the maximum allowed pitch and minimum allowed pitch respectively. +CON= and −CON= are used to enter values for the parameter constraints (i.e., the maximum and minimum allowed value of the search parameter). Typically, it is easier to set the maximum allowed pitch and minimum allowed pitch using MAXPITCH= and MINPITCH= than it is to calculate the value of the parameter corresponding to those pitches. The default minimum constraint corresponds to MINPITCH, the pitch at which the largest interior region of a unit used in the array is in contact with the spacing cuboid of that unit. The maximum constraint is defaulted to −5 times the minimum constraint. In the first search pass, the code calculates *k*\ :sub:eff for the initial problem geometry. The next two passes calculate the *k*\ :sub:eff for the dimensions corresponding to the minimum constraint and maximum constraint. Subsequent passes fit the previous results to a cubic equation to select new dimensions. Consider a 10 × 10 × 10 array of uranium spheres surrounded by water having a “square” pitch. The uranium spheres are 2 cm in radius, and the center-to-center spacing is 8 cm. The uranium spheres and their associated spacing are defined to be unit 1, and the 10 × 10 × 10 array is defined to be array 1. Search data and results for some optimum pitch searches using this example are given in inputs 1 through 6 below. Default Search — Array of Centered Spheres ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ This problem contains search data for a default optimum pitch search. The problem consists of a 10 × 10 × 10 array of uranium spheres surrounded by low density water. The spheres are 90% \ :sup:235\ U and 10% \ :sup:238\ U and have a radius of 2.0 cm. The spheres are originally centered on an 8.0 cm X, Y, and Z pitch with interstitial low density (0.01 gm/cc) water. The problem searches for the pitch that will produce the maximum *k*\ :sub:eff for the system. :: =CSAS5S 10x10x10 ARRAY DEFAULT PITCH SEARCH V7-238 READ COMP URANIUM 1 1.0 300.0 92235 90.0 92238 10.0 END H2O 2 0.01 300.0 END END COMP READ CELLDATA LATTICECELL SPHSQUAREP PITCH=8.0 2 FUELD=4.0 1 END END CELLDATA READ GEOMETRY UNIT 1 SPHERE 1 1 2.0 CUBOID 2 1 6P4.0 END GEOMETRY READ ARRAY ARA=1 NUX=10 NUY=10 NUZ=10 FILL F1 END FILL END ARRAY END DATA READ SEARCH OPTIMUM PITCH END SEARCH END This data will cause the code to alter the +X, −X, +Y, −Y, +Z, and −Z dimensions of the outer region of unit 1 and search for the dimensions that give the maximum *k*\ :sub:eff. Because the next to last outer dimension of the unit contained in the array are spheres, all six dimensions are altered. The original relationship between the dimensions is preserved (i.e., the original ratio of the X to Y to Z dimensions of the cuboid is preserved throughout the search). Because this is an optimum pitch search, the sphere dimensions will not be changed. For this particular example, results for six passes are given. A final search results for the problem follow: :: ******************* search pass 1 keff= 5.05769E-01 + or - 1.10681E-03 ****************** the parameter was 0.00000E+00 ***** modified keno v data has been rewritten on unit 95 ***** ******************* search pass 2 keff= 1.28382E+00 + or - 1.70155E-03 ****************** the parameter was -5.00000E-01 ***** modified keno v data has been rewritten on unit 95 ***** ******************* search pass 3 keff= 2.58700E-01 + or - 8.16664E-04 ****************** the parameter was 2.50000E+00 ***** modified keno v data has been rewritten on unit 95 ***** ******************* search pass 4 keff= 2.86712E-01 + or - 8.92672E-04 ****************** the parameter was 1.25000E+00 ***** modified keno v data has been rewritten on unit 95 ***** ******************* search pass 5 keff= 2.66952E-01 + or - 8.42503E-04 ****************** the parameter was 1.96018E+00 ***** modified keno v data has been rewritten on unit 95 ***** ******************* search pass 6 keff= 2.63611E-01 + or - 7.74438E-04 ****************** the parameter was 2.16471E+00 1 10x10x10 array default pitch search **************************************************************************************************** **************************************************************************************************** convergence was achieved on pass 2 the parameter was -5.00000E-0 the equation used in the search was: k-eff = +6.69365E-01 -7.39837E-01*p +4.33577E-01*p**2 -8.22064E-02*p**3 k-effective= 1.28382E+00 + or - 1.70155E-03 the corresponding geometry follows; media bias geometry description for those units utilized in this problem region num id ----- unit 1 ----- 1 sphere 1 1 radius = 2.0000 the center is located at ( 0.0000 , 0.0000 , 0.0000 ). 2 cuboid 2 1 +x = 2.0000 -x = -2.0000 +y = 2.0000 -y = -2.0000 +z = 2.0000 -z = -2.0000 **************************************************************************************************** **************************************************************************************************** Default Search — Array of Off-Centered Spheres ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ This is the same problem as described above except the spheres are now off-centered in the X and Y dimensions. The input data and the final search results for this problem follow: :: =CSAS5S 10x10x10 ARRAY DEFAULT PITCH SEARCH- OFFSET V7-238 READ COMP URANIUM 1 1.0 300.0 92235 90.0 92238 10.0 END H2O 2 0.01 300.0 END END COMP READ CELLDATA LATTICECELL SPHSQUAREP PITCH=8.0 2 FUELD=4.0 1 END END CELLDATA READ GEOMETRY UNIT 1 SPHERE 1 1 2.0 CUBOID 2 1 6.0 -2.0 3.0 -5.0 4.0 -4.0 END GEOMETRY READ ARRAY ARA=1 NUX=10 NUY=10 NUZ=10 FILL F1 END FILL END ARRAY END DATA READ SEARCH OPTIMUM PITCH END SEARCH END :: ******************* search pass 1 keff= 5.06585E-01 + or - 1.12636E-03 ****************** the parameter was 0.00000E+00 ***** modified keno v data has been rewritten on unit 95 ***** ******************* search pass 2 keff= 1.28382E+00 + or - 1.70155E-03 ****************** the parameter was -5.00000E-01 ***** modified keno v data has been rewritten on unit 95 ***** ******************* search pass 3 keff= 2.58916E-01 + or - 8.16022E-04 ****************** the parameter was 2.50000E+00 ***** modified keno v data has been rewritten on unit 95 ***** ******************* search pass 4 keff= 2.87506E-01 + or - 8.38203E-04 ****************** the parameter was 1.25000E+00 ***** modified keno v data has been rewritten on unit 95 ***** ******************* search pass 5 keff= 2.67987E-01 + or - 8.48127E-04 ****************** the parameter was 1.95962E+00 ***** modified keno v data has been rewritten on unit 95 ***** ******************* search pass 6 keff= 2.64640E-01 + or - 7.99055E-04 ****************** the parameter was 2.15903E+00 10x10x10 array default pitch search - offset **************************************************************************************************** **************************************************************************************************** convergence was achieved on pass 2 the parameter was -5.00000E-01 the equation used in the search was: k-eff = +6.69265E-01 -7.40343E-01*p +4.35409E-01*p**2 -8.28404E-02*p**3 k-effective= 1.28382E+00 + or - 1.70155E-03 the corresponding geometry follows; media bias geometry description for those units utilized in this problem region num id ----- unit 1 ----- 1 sphere 1 1 radius = 2.0000 the center is located at ( 0.0000 , 0.0000 , 0.0000 ). 2 cuboid 2 1 +x = 2.0000 -x = -2.0000 +y = 2.0000 -y = -2.0000 +z = 2.000 -z = -2.0000 **************************************************************************************************** **************************************************************************************************** Array of Centered Spheres — Constant :math:Z Spacing ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ This is the same problem as the first problem described above except the array does not change in the Z dimensions. This is done by using MORE and specifying a KEEP command. The input data and the final search results for this problem follow: :: =CSAS5S 10x10x10 ARRAY DEFAULT PITCH SEARCH - CHANGE IN X & Y ONLY V7-238 READ COMP URANIUM 1 1.0 300.0 92235 90.0 92238 10.0 END H2O 2 0.01 300.0 END END COMP READ CELLDATA LATTICECELL SPHSQUAREP PITCH=8.0 2 FUELD=4.0 1 END END CELLDATA READ GEOMETRY UNIT 1 SPHERE 1 1 2.0 CUBOID 2 1 6p4.0 END GEOMETRY READ ARRAY ARA=1 NUX=10 NUY=10 NUZ=10 FILL F1 END FILL END ARRAY END DATA READ SEARCH OPTIMUM PITCH MORE KEEP UNIT=1 REG=2 +Z=1.0 -Z=1.0 END SEARCH END :: ******************* search pass 1 keff= 5.05769E-01 + or - 1.10681E-03 ****************** the parameter was 0.00000E+00 ***** modified keno v data has been rewritten on unit 95 ***** ******************* search pass 2 keff= 9.06304E-01 + or - 1.57239E-03 ****************** the parameter was -5.00000E-01 ***** modified keno v data has been rewritten on unit 95 ***** ******************* search pass 3 keff= 2.82956E-01 + or - 8.20501E-04 ****************** the parameter was 2.50000E+00 ***** modified keno v data has been rewritten on unit 95 ***** ******************* search pass 4 keff= 3.19086E-01 + or - 9.27655E-04 ****************** the parameter was 1.25000E+00 ***** modified keno v data has been rewritten on unit 95 ***** ******************* search pass 5 keff= 2.94889E-01 + or - 8.49552E-04 ****************** the parameter was 1.91631E+00 10x10x10 array default pitch search - change in x & y only **************************************************************************************************** **************************************************************************************************** convergence was achieved on pass 2 the parameter was -5.00000E-01 the equation used in the search was: k-eff = +5.70750E-01 -4.42912E-01*p +2.39464E-01*p**2 -4.35513E-02*p**3 k-effective= 9.06304E-01 + or - 1.57239E-03 the corresponding geometry follows media bias geometry description for those units utilized in this problem region num id ----- unit 1 ----- 1 sphere 1 1 radius = 2.0000 the center is located at ( 0.0000 , 0.0000 , 0.0000 ). 2 cuboid 2 1 +x = 2.0000 -x = -2.000 +y = 2.0000 -y = -2.0000 +z = 4.0000 -z = -4.0000 **************************************************************************************************** **************************************************************************************************** Default Search — Array of Centered Cylinders ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ This problem contains search data for a default optimum pitch search. The problem consists of a 10 × 10 × 10 array of uranium cylinders surrounded by low density water. The cylinders are 90% \ :sup:235\ U and 10% :sup:238\ U and have a radius of 2.0 cm and a length of 20 cm. The cylinders are originally centered on an 8.0 cm X and Y pitch and a 24 cm Z pitch with interstitial low density (0.01 gm/cc) water. The problem searches for the pitch that will produce the maximum *k*\ :sub:eff for the system. Because this is an array of cylinders, by default only the X and Y dimensions are modified during the search. The input data and the final search results for this problem follow: :: =CSAS5S 10x10x10 ARRAY DEFAULT PITCH SEARCH - CHANGE IN X & Y ONLY V7-238 READ COMP URANIUM 1 1.0 300.0 92235 90.0 92238 10.0 END H2O 2 0.01 300.0 END END COMP READ CELLDATA LATTICECELL SQUAREPITCH PITCH=8.0 2 FUELD=4.0 1 END END CELLDATA READ GEOMETRY UNIT 1 CYLINDER 1 1 2.0 10.0 -10.0 CUBOID 2 1 4p4.0 2p12.0 END GEOMETRY READ ARRAY ARA=1 NUX=10 NUY=10 NUZ=10 FILL F1 END FILL END ARRAY END DATA READ SEARCH OPTIMUM PITCH END SEARCH END :: ******************* search pass 1 keff= 1.09882E+00 + or - 1.75627E-03 ****************** the parameter was 0.00000E+00 ***** modified keno v data has been rewritten on unit 95 ***** ******************* search pass 2 keff= 1.66591E+00 + or - 1.93770E-03 ****************** the parameter was -5.00000E-01 ***** modified keno v data has been rewritten on unit 95 ***** ******************* search pass 3 keff= 5.08235E-01 + or - 1.28586E-03 ****************** the parameter was 2.50000E+00 ***** modified keno v data has been rewritten on unit 95 ***** ******************* search pass 4 keff= 6.33884E-01 + or - 1.28582E-03 ****************** the parameter was 1.25000E+00 10x10x10 array default pitch search - change in x & y only **************************************************************************************************** **************************************************************************************************** convergence was achieved on pass 2 the parameter was -5.00000E-01 the equation used in the search was: k-eff = +1.09882E+00 -8.48276E-01*p +5.17313E-01*p**2 -1.08998E-01*p**3 k-effective= 1.66591E+00 + or - 1.93770E-03 the corresponding geometry follows; media bias geometry description for those units utilized in this problem regio num id ----- unit 1 ----- 1 cylinder 1 1 radius = 2.0000 +z = 10.000 -z = -10.000 centerline is at x = 0.000 y = 0.000 2 cuboid 2 1 +x = 2.0000 -x = -2.0000 +y = 2.0000 -y = -2.0000 +z = 12.000 -z = -12.000 **************************************************************************************************** **************************************************************************************************** Array of Centered Cylinders — Search Extended to :math:Z Dimension ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ This is the same problem as given above except the search has been extended to include the Z dimension. The input data causes the code to alter the +X, −X, +Y, −Y, +Z, and −Z dimensions of the outer region of unit 1 and search for the dimensions that give the maximum *k\ eff.* The +Z and −Z search constants were chosen to maintain the same spacing between cylinder surfaces in the X, Y and Z dimensions throughout the search. Because this is an optimum pitch search, the cylinder dimensions will not be changed. The input data and the final search results for this problem follow: :: =CSAS5S 10x10x10 ARRAY DEFAULT PITCH SEARCH - CHANGE IN X, Y, Z V7-238 READ COMP URANIUM 1 1.0 300.0 92235 90.0 92238 10.0 END H2O 2 0.01 300.0 END END COMP READ CELLDATA LATTICECELL SQUAREPITCH PITCH=8.0 2 FUELD=4.0 1 END END CELLDATA READ GEOMETRY UNIT 1 CYLINDER 1 1 2.0 10.0 -10.0 CUBOID 2 1 4p4.0 2p12.0 END GEOMETRY READ ARRAY ARA=1 NUX=10 NUY=10 NUZ=10 FILL F1 END FILL END ARRAY END DATA READ SEARCH OPTIMUM PITCH MORE ALTER UNIT=1 REG=2 +Z=0.33333 -Z=0.33333 END SEARCH END :: ******************* search pass 1 keff= 1.09882E+00 + or - 1.75627E-03 ****************** the parameter was 0.00000E+00 ***** modified keno v data has been rewritten on unit 95 ***** ******************* search pass 2 keff= 1.77881E+00 + or - 1.79505E-03 ****************** the parameter was -5.00000E-01 ***** modified keno v data has been rewritten on unit 95 ***** ******************* search pass 3 keff= 4.54894E-01 + or - 1.09005E-03 ****************** the parameter was 2.50000E+00 ***** modified keno v data has been rewritten on unit 95 ***** ******************* search pass 4 keff= 5.70747E-01 + or - 1.32088E-03 ****************** the parameter was 1.25000E+00 10x10x10 array default pitch search - change in x, y, z **************************************************************************************************** **************************************************************************************************** convergence was achieved on pass 2 the parameter was -5.00000E-01 the equation used in the search was: k-eff = +1.09882E+00 -1.00799E+00*p +6.36691E-01*p**2 -1.34609E-01*p**3 k-effective= 1.77881E+00 + or - 1.79505E-03 the corresponding geometry follows; media bias geometry description for those units utilized in this problem region num id ----- unit 1 ----- 1 cylinder 1 1 radius = 2.0000 +z = 10.000 -z = -10.000 centerline is at x = 0.000 y = 0.000 2 cuboid 2 1 +x = 2.0000 -x = -2.0000 +y = 2.0000 -y = -2.0000 +z = 10.000 -z = -10.000 **************************************************************************************************** **************************************************************************************************** Minimum pitch search -------------------- A minimum pitch search searches for the pitch that yields the lowest value of *k\ eff.* A minimum pitch search is activated by entering “MINIMUM PITCH” in the search data. By default, the search is performed at the lowest array level and only the spacing in the X-direction for slabs, the X and Y directions for cylindrical arrays, and the X, Y, and Z directions for spherical arrays. The search constants are defaulted to 1.0 for the applicable +X, −X, +Y, −Y, +Z, and −Z dimensions of the outermost region. The dimensions of other geometry regions will not be changed (their search constants are defaulted to 0.0) unless additional search data containing appropriate instructions are supplied. The limits for a minimum pitch search are the same as an optimum pitch search as described in the previous section. Array of Centered Uranium Slabs ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Consider a 10 × 1 × 1 array of uranium metal slabs. The uranium slabs are 2 cm in thick in the X dimension and 200 cm thick in the Y and Z dimensions. On each side of the uranium slab in the X dimension is 1 cm of H\ :sub:2\ O, then 1 cm of Boral, then 1 cm of H\ :sub:2\ O resulting in an initial center-to-center spacing is 8.0 cm. The slabs are 90% :sup:235\ U and 10% :sup:238\ U, the water is full density, and the Boral is 36.7% B\ :sub:4\ C. The uranium slabs and their associated materials are defined to be unit 1, and the 10 × 1 × 1 array is defined to be array 1. A minimum pitch of 6.01 cm is specified, MINPITCH=6.01, and a maximum pitch of 14.0 cm is specified, MAXPITCH=14.0. The input data and the final search results for this problem follow: :: =CSAS5S 10x1x1 ARRAY DEFAULT PITCH SEARCH - SLAB V7-238 READ COMP URANIUM 1 1.0 300.0 92235 90.0 92238 10.0 END H2O 2 1.0 300.0 END B4C 3 0.367 300.0 END Al 3 0.633 300.0 END H2O 4 1.0 300.0 END END COMP READ CELLDATA LATTICECELL SYMMSLABCELL PITCH=8.0 4 FUELD=2.0 1 CLADD=6.01 3 GAPD=4.0 2 END END CELLDATA READ GEOMETRY UNIT 1 CUBOID 1 1 1.0 -1.0 4P100.0 CUBOID 2 1 2.0 -2.0 4p100.0 CUBOID 3 1 3.0 -3.0 4P100.0 CUBOID 4 1 4.0 -4.0 4P100.0 END GEOMETRY READ ARRAY ARA=1 NUX=10 NUY=1 NUZ=1 FILL F1 END FILL END ARRAY END DATA READ SEARCH MINIMUM PITCH MINPITCH=6.0 MAXPITCH=14.0 END SEARCH END :: ******************* search pass 1 keff= 1.13755E+00 + or - 1.50108E-03 ****************** the parameter was 0.00000E+00 ***** modified keno v data has been rewritten on unit 95 ***** ******************* search pass 2 keff= 1.43116E+00 + or - 1.57018E-03 ****************** the parameter was -2.50000E-01 ***** modified keno v data has been rewritten on unit 95 ***** ******************* search pass 3 keff= 8.04815E-01 + or - 1.47600E-03 ****************** the parameter was 7.50000E-01 ***** modified keno v data has been rewritten on unit 95 ***** ******************* search pass 4 keff= 9.05346E-01 + or - 1.47107E-03 ****************** the parameter was 3.75000E-01 10x1x1 array default pitch search - slab **************************************************************************************************** **************************************************************************************************** convergence was achieved on pass 3 the parameter was 7.50000E-01 the equation used in the search was: k-eff = +1.13755E+00 -9.12964E-01*p +9.40907E-01*p**2 -4.20205E-01*p**3 k-effective= 8.04815E-01 + or - 1.47600E-03 the corresponding geometry follows; media bias geometry description for those units utilized in this problem region num id ----- unit 1 ----- 1 cuboid 1 1 +x = 1.0000 -x = -1.0000 +y = 100.00 -y = -100.00 +z = 100.00 -z = -100.00 2 cuboid 2 1 +x = 2.0000 -x = -2.0000 +y = 100.00 -y = -100.00 +z = 100.00 -z = -100.00 3 cuboid 3 1 +x = 3.0000 -x = -3.0000 +y = 100.00 -y = -100.00 +z = 100.00 -z = -100.00 4 cuboid 4 1 +x = 7.0000 -x = -7.0000 +y = 100.00 -y = -100.00 +z = 100.00 -z = -100.00 **************************************************************************************************** **************************************************************************************************** Critical pitch search --------------------- A critical pitch search alters the outer region of the unit or units at the lowest array level in search of a specified value of *k*\ :sub:eff. A critical pitch search is activated by entering “CRITICAL PITCH” in the search data. By default, the search is performed at the lowest array level, and only the spacing in the X and Y directions (the X and Y dimensions of the outermost region of the unit(s) used in the array at the lowest array level) will be changed. The search constants are defaulted to 1.0 for the +X, −X, +Y, and −Y dimensions of the region. The dimensions of other geometry regions will not be changed (their search constants are defaulted to 0.0) unless additional search data containing appropriate instructions are supplied. The limits for a critical pitch search can be set using either MAXPITCH= and MINPITCH= in the optional search parameters or +CON= and −CON= in the search parameter constraints of the auxiliary search commands. MAXPITCH= and MINPITCH= are used to enter a value of the maximum allowed pitch and minimum allowed pitch respectively. Parameters +CON= and −CON= are used to enter values for the parameter constraints (i.e., the maximum and minimum allowed value of the search parameter). Typically, it is easier to set the maximum allowed pitch and minimum allowed pitch using MAXPITCH= and MINPITCH= than it is to calculate the value of the parameter corresponding to those pitches. The default minimum constraint corresponds to MINPITCH, the pitch at which the largest interior region of a unit used in the array is in contact with the spacing cuboid of that unit. The maximum constraint is defaulted to −5 times the minimum constraint. The code calculates *k*\ :sub:eff for the initial geometry first. Then the dimensions corresponding to the minimum constraint and maximum constraint are calculated. Default Search — Array of Centered Spheres ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ This problem contains search data for a default critical pitch search with *k*\ :sub:eff specified to be 0.95. The problem consists of a 10 × 10 × 10 array of uranium spheres surrounded by low density water. The spheres are 90% :sup:235\ U and 10% :sup:238\ U and have a radius of 2.0 cm. The spheres are originally centered on an 8.0 cm X, Y, and Z pitch with interstitial low density (0.01 gm/cc) water. The problem searches for the pitch that will produce a *k*\ :sub:eff = 0.95 for the system. :: =CSAS5S 10x10x10 ARRAY DEFAULT PITCH SEARCH V7-238 READ COMP URANIUM 1 1.0 300.0 92235 90.0 92238 10.0 END H2O 2 0.01 300.0 END END COMP READ CELLDATA LATTICECELL SPHSQUAREP PITCH=8.0 2 FUELD=4.0 1 END END CELLDATA READ GEOMETRY UNIT 1 SPHERE 1 1 2.0 CUBOID 2 1 6P4.0 END GEOMETRY READ ARRAY ARA=1 NUX=10 NUY=10 NUZ=10 FILL F1 END FILL END ARRAY END DATA READ SEARCH CRITICAL PITCH KEF=0.95 END SEARCH END :: ******************* search pass 1 keff= 5.05769E-01 + or - 1.10681E-03 ****************** the parameter was 0.00000E+00 ***** modified keno v data has been rewritten on unit 95 ***** ******************* search pass 2 keff= 2.58700E-01 + or - 8.16664E-04 ****************** the parameter was 2.50000E+00 ***** modified keno v data has been rewritten on unit 95 ***** ******************* search pass 3 keff= 1.28382E+00 + or - 1.70155E-03 ****************** the parameter was -5.00000E-01 ***** modified keno v data has been rewritten on unit 95 ***** ******************* search pass 4 keff= 8.08580E-01 + or - 1.54692E-03 ****************** the parameter was -3.04076E-01 ***** modified keno v data has been rewritten on unit 95 ***** ******************* search pass 5 keff= 9.37897E-01 + or - 1.63624E-03 ****************** the parameter was -3.73908E-01 ***** modified keno v data has been rewritten on unit 95 ***** ******************* search pass 6 keff= 9.49167E-01 + or - 1.64684E-03 ****************** the parameter was -3.79251E-01 10x10x10 array default pitch search **************************************************************************************************** **************************************************************************************************** convergence was achieved on pass 6 the parameter was -3.79251E-01 the equation used in the search was: k-eff = +5.33889E-01 -9.31824E-02*p +2.29294E+00*p**2 -9.20141E-01*p**3 k-effective= 9.49167E-01 + or - 1.64684E-03 the corresponding geometry follows; media bias geometry description for those units utilized in this problem region num id ----- unit 1 ----- 1 sphere 1 1 radius = 2.0000 the center is located at ( 0.0000 , 0.0000 , 0.0000 ). 2 cuboid 2 1 +x = 2.4830 -x = -2.4830 +y = 2.4830 -y = -2.4830 +z = 2.4830 -z = -2.4830 **************************************************************************************************** **************************************************************************************************** 10x10x10 array default pitch search **************************************************************************************************** **************************************************************************************************** based on the preceding data, the best estimate of the parameter is -3.79602E-01 the geometry corresponding to this parameter follows: media bias geometry description for those units utilized in this problem region num id ----- unit 1 ----- 1 sphere 1 1 radius = 2.0000 the center is located at ( 0.0000 , 0.0000 , 0.0000 ). 2 cuboid 2 1 +x = 2.4816 -x = -2.4816 +y = 2.4816 -y = -2.4816 +z = 2.4816 -z = -2.4816 **************************************************************************************************** **************************************************************************************************** Array of Centered Cylinders — Search Extended to :math:Z Dimension ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ This problem contains search data for a default critical pitch search with *k*\ :sub:eff specified to be 0.95. The problem consists of a 10 × 10 × 10 array of uranium cylinders surrounded by low density water. The cylinders are 90% :sup:235\ U and 10% :sup:238\ U and have a radius of 2.0 cm and a length of 20 cm. The cylinders are originally centered on an 8.0 cm X and Y pitch and a 24 cm Z pitch with interstitial low density (0.01 gm/cc) water. The problem searches for the pitch that will produce a *k*\ :sub:eff = 0.95 for the system. Because this is an array of cylinders, by default only the X and Y dimensions are modified during the search. The +Z and −Z search constants were included and chosen such that the same spacing is maintained between cylinder surfaces in the X, Y and Z dimensions throughout the search. The input data and the final search results for this problem follow: :: =CSAS5S 10x10x10 ARRAY DEFAULT PITCH SEARCH - CHANGE IN X, Y, Z V7-238 READ COMP URANIUM 1 1.0 300.0 92235 90.0 92238 10.0 END H2O 2 0.01 300.0 END END COMP READ CELLDATA LATTICECELL SQUAREPITCH PITCH=8.0 2 FUELD=4.0 1 END END CELLDATA READ GEOMETRY UNIT 1 CYLINDER 1 1 2.0 10.0 -10.0 CUBOID 2 1 4p4.0 2p12.0 END GEOMETRY READ ARRAY ARA=1 NUX=10 NUY=10 NUZ=10 FILL F1 END FILL END ARRAY END DATA READ SEARCH CRITICAL PITCH KEFF=0.95 MORE ALTER UNIT=1 REG=2 +Z=0.33333 -Z=0.33333 END SEARCH END :: ******************* search pass 1 keff= 1.09882E+00 + or - 1.75627E-03 ****************** the parameter was 0.00000E+00 ***** modified keno v data has been rewritten on unit 95 ***** ******************* search pass 2 keff= 1.77881E+00 + or - 1.79505E-03 ****************** the parameter was -5.00000E-01 ***** modified keno v data has been rewritten on unit 95 ***** ******************* search pass 3 keff= 1.00616E+00 + or - 1.61956E-03 ****************** the parameter was 1.09424E-01 ***** modified keno v data has been rewritten on unit 95 ***** ******************* search pass 4 keff= 9.48455E-01 + or - 1.66901E-03 ****************** the parameter was 1.91336E-01 10x10x10 array default pitch search - change in x, y, z **************************************************************************************************** **************************************************************************************************** convergence was achieved on pass 4 the parameter was 1.91336E-01 the equation used in the search was: k-eff = +1.09882E+00 -9.31106E-01*p +7.86483E-01*p**2 -1.42575E-01*p**3 k-effective= 9.48455E-01 + or - 1.66901E-03 the corresponding geometry follows; media bias geometry description for those units utilized in this problem region num id ----- unit 1 ----- 1 cylinder 1 1 radius = 2.0000 +z = 10.000 -z = -10.000 centerline is at x = 0.000 y = 0.000 2 cuboid 2 1 +x = 4.7653 -x = -4.7653 +y = 4.7653 -y = -4.7653 +z = 12.765 -z = -12.765 **************************************************************************************************** **************************************************************************************************** 10x10x10 array default pitch search - change in x, y, z **************************************************************************************************** ****************************************************************************************************0 based on the preceding data, the best estimate of the parameter is 1.88950E-01 the geometry corresponding to this parameter follows: media bias geometry description for those units utilized in this problem region num id ----- unit 1 ----- 1 cylinder 1 1 radius = 2.0000 +z = 10.000 -z = -10.000 centerline is at x = 0.000 y = 0.000 2 cuboid 2 1 +x = 4.7558 -x = -4.7558 +y = 4.7558 -y = -4.7558 +z = 12.756 -z = -12.756 **************************************************************************************************** **************************************************************************************************** Array of Centered Uranium Slabs ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Consider a 10 × 1 × 1 array of uranium metal slabs. The uranium slabs are 2 cm thick in the X dimension and 200 cm thick in the Y and Z dimensions. On each side of the uranium slab in the X dimension is 1 cm of H\ :sub:2\ O, then 1 cm of Boral, then 1 cm of H\ :sub:2\ O resulting in an initial center-to-center spacing of 8.0 cm. The slabs are 90% :sup:235\ U and 10% :sup:238\ U, the water is full density, and the Boral is 36.7% B\ :sub:4\ C. The uranium slabs and their associated materials are defined to be unit 1, and the 10 × 1 × 1 array is defined to be array 1. A minimum pitch of 6.0 cm is specified, MINPITCH=6.0, and a maximum pitch of 14.0 cm is specified, MAXPITCH=14.0. The code searches for the default value of *k*\ :sub:eff. The input data and the final search results for this problem follow: :: =CSAS5S 10x1x1 ARRAY DEFAULT PITCH SEARCH - SLAB V7-238 READ COMP URANIUM 1 1.0 300.0 92235 90.0 92238 10.0 END H2O 2 1.0 300.0 END B4C 3 0.367 300.0 END Al 3 0.633 300.0 END H2O 4 1.0 300.0 END END COMP READ CELLDATA LATTICECELL SYMMSLABCELL PITCH=8.0 4 FUELD=2.0 1 CLADD=6.0 3 GAPD=4.0 2 END END CELLDATA READ GEOMETRY UNIT 1 CUBOID 1 1 1.0 -1.0 4P100.0 CUBOID 2 1 2.0 -2.0 4p100.0 CUBOID 3 1 3.0 -3.0 4P100.0 CUBOID 4 1 4.0 -4.0 4P100.0 END GEOMETRY READ ARRAY ARA=1 GBL=1 NUX=10 NUY=1 NUZ=1 FILL F1 END FILL END ARRAY END DATA READ SEARCH CRITICAL PITCH MINPITCH=6.0 MAXPITCH=14.0 END SEARCH END :: ******************* search pass 1 keff= 1.13755E+00 + or - 1.50108E-03 ****************** the parameter was 0.00000E+00 ***** modified keno v data has been rewritten on unit 95 ***** ******************* search pass 2 keff= 1.43116E+00 + or - 1.57018E-03 ****************** the parameter was -2.50000E-01 ***** modified keno v data has been rewritten on unit 95 ***** ******************* search pass 3 keff= 1.04384E+00 + or - 1.47394E-03 ****************** the parameter was 1.17120E-01 ***** modified keno v data has been rewritten on unit 95 ***** ******************* search pass 4 keff= 9.97707E-01 + or - 1.60798E-03 ****************** the parameter was 1.89330E-01 10x1x1 array default pitch search - slab **************************************************************************************************** **************************************************************************************************** convergence was achieved on pass 4 the parameter was 1.89330E-01 the equation used in the search was: k-eff = +1.13755E+00 -9.08380E-01*p +9.68867E-01*p**2 -3.81717E-01*p**3 k-effective= 9.97707E-01 + or - 1.60798E-03 the corresponding geometry follows; media bias geometry description for those units utilized in this problem region num id ----- unit 1 ----- 1 cuboid 1 1 +x = 1.0000 -x = -1.0000 +y = 100.00 -y = -100.00 +z = 100.00 -z = -100.00 2 cuboid 2 1 +x = 2.0000 -x = -2.0000 +y = 100.00 -y = -100.00 +z = 100.00 -z = -100.00 3 cuboid 3 1 +x = 3.0000 -x = -3.0000 +y = 100.00 -y = -100.00 +z = 100.00 -z = -100.00 4 cuboid 4 1 +x = 4.7573 -x = -4.7573 +y = 100.00 -y = -100.00 +z = 100.00 -z = -100.00 **************************************************************************************************** **************************************************************************************************** 10x1x1 array default pitch search - slab **************************************************************************************************** **************************************************************************************************** based on the preceding data, the best estimate of the parameter is 1.85414E-01 the geometry corresponding to this parameter follows: media bias geometry description for those units utilized in this problem region num id ----- unit 1 ----- 1 cuboid 1 1 +x = 1.0000 -x = -1.0000 +y = 100.00 -y = -100.00 +z = 100.00 -z = -100.00 2 cuboid 2 1 +x = 2.0000 -x = -2.0000 +y = 100.00 -y = -100.00 +z = 100.00 -z = -100.00 3 cuboid 3 1 +x = 3.0000 -x = -3.0000 +y = 100.00 -y = -100.00 +z = 100.00 -z = -100.00 4 cuboid 4 1 +x = 4.7417 -x = -4.7417 +y = 100.00 -y = -100.00 +z = 100.00 -z = -100.00 **************************************************************************************************** **************************************************************************************************** Optimum dimension search ------------------------ An optimum dimension search searches for the geometry dimensions that yield the highest value of *k\ eff.* An optimum dimension search is activated by entering “OPTIMUM DIMENSION” in the search data. There are no defaulted search data in a dimension search. The user must specify the dimensions to be changed and the manner in which they will be changed as described in the auxiliary search commands. A dimension search is performed by altering the regions having nonzero search constants specified in the auxiliary search commands portion of the search data. By default, the search constants for every dimension in the problem are zero. Only those dimensions having a nonzero search constant are altered by the code. By default, a dimension search sets the minimum constraint to −10E10 and the maximum constraint to +10E10. The relationship between the constraints and the search constants are given in :eq:CSASapp-1  and :eq:CSASapp-2. If the default values of the constraints are used, appropriate search constants must be calculated using these equations. It may be simpler to set the search constants to 1.0 and calculate the corresponding maximum and minimum constraints. Several dimension search examples are shown below. Array of Centered Spheres — Search in :math:X, :math:Y, and :math:Z Dimensions ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Consider a 10 × 10 × 10 array of uranium spheres arranged in an array having a “square” pitch. The uranium spheres are 2 cm in radius, and the center-to-center spacing is 8 cm. The uranium spheres and their associated spacing are defined to be unit 1, and the 10 × 10 × 10 array is defined to be array 1. The constraints are set to be consistent with those of this optimum pitch search case. The search data and results for an optimum dimension search that alters the X, Y, and Z dimensions of region 2 of unit 1 are given as follows. :: =CSAS5S 10x10x10 ARRAY - DIMENSION SEARCH V7-238 READ COMP URANIUM 1 1.0 300.0 92235 90.0 92238 10.0 END H2O 2 0.01 300.0 END END COMP READ CELLDATA LATTICECELL SPHSQUAREP PITCH=8.0 2 FUELD=4.0 1 END END CELLDATA READ GEOMETRY UNIT 1 SPHERE 1 1 2.0 CUBOID 2 1 6P4.0 END GEOMETRY READ ARRAY ARA=1 NUX=10 NUY=10 NUZ=10 FILL F1 END FILL END ARRAY END DATA READ SEARCH OPTIMUM DIMENSION MORE ALTER UNIT=1 REG=2 ALL=1.0 -CON=-0.5 +CON=2.5 END SEARCH END :: ******************* search pass 1 keff= 5.05769E-01 + or - 1.10681E-03 ****************** the parameter was 0.00000E+00 ***** modified keno v data has been rewritten on unit 95 ***** ******************* search pass 2 keff= 1.28075E+00 + or - 1.89662E-03 ****************** the parameter was -5.00000E-01 ***** modified keno v data has been rewritten on unit 95 ***** ******************* search pass 3 keff= 2.59110E-01 + or - 8.15306E-04 ****************** the parameter was 2.50000E+00 ***** modified keno v data has been rewritten on unit 95 ***** ******************* search pass 4 keff= 2.87025E-01 + or - 8.24079E-04 ****************** the parameter was 1.25000E+00 ***** modified keno v data has been rewritten on unit 95 ***** ******************* search pass 5 keff= 2.65737E-01 + or - 7.89427E-04 ****************** the parameter was 1.96019E+00 ***** modified keno v data has been rewritten on unit 95 ***** ******************* search pass 6 keff= 2.63297E-01 + or - 8.48658E-04 ****************** the parameter was 2.17436E+00 10x10x10 array - dimension search **************************************************************************************************** **************************************************************************************************** convergence was achieved on pass 2 the parameter was -5.00000E-01 the equation used in the search was: k-eff = +6.70283E-01 -7.35957E-01*p +4.26318E-01*p**2 -7.98562E-02*p**3 k-effective= 1.28075E+00 + or - 1.89662E-03 the corresponding geometry follows; media bias geometry description for those units utilized in this problem region num id ----- unit 1 ----- 1 sphere 1 1 radius = 2.0000 the center is located at ( 0.0000 , 0.0000 , 0.0000 ). 2 cuboid 2 1 +x = 2.0000 -x = -2.0000 +y = 2.0000 -y = -2.0000 +z = 2.0000 -z = -2.0000 **************************************************************************************************** **************************************************************************************************** These MORE search data will cause the code to alter the +X, −X, +Y, −Y, −Z, and +Z dimensions of the outer region (region 2) of unit 1 and search for the dimensions that give the maximum *k*\ :sub:eff. Since all the UNIT 1 outer dimensions are the same (4 or −4) and all the search constants are the same (ALL=1.0) the code will alter each dimension the same amount for a search pass thus preserving the original relationship between dimensions. Because this is an optimum pitch search, the sphere dimensions will not be changed. The optimum dimension for this problem is when the spheres touch as shown in the output data above. Array of Centered Spheres — Search in :math:X and :math:Y Dimensions ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ This is the same problem as described above except the spacing is only varied in the X and Y dimensions. Since only changes in the X and Y dimensions of the outer region are desired, the +X, −X, +Y, and −Y search constants must be set. Since all dimensions are to change at the same rate and have the same initial value, the search constants for the changing surfaces must be the same. The +Z and −Z search constants by default are zero so these dimensions will not change. The input data and the final search results for this problem follow: :: =CSAS5S 10x10x10 ARRAY - X & Y DIMENSION OPTIMUM SEARCH V7-238 READ COMP URANIUM 1 1.0 300.0 92235 90.0 92238 10.0 END H2O 2 0.01 300.0 END END COMP READ CELLDATA LATTICECELL SPHSQUAREP PITCH=8.0 2 FUELD=4.0 1 END END CELLDATA READ GEOMETRY UNIT 1 SPHERE 1 1 2.0 CUBOID 2 1 6P4.0 END GEOMETRY READ ARRAY ARA=1 NUX=10 NUY=10 NUZ=10 FILL F1 END FILL END ARRAY END DATA READ SEARCH OPTIMUM DIMENSION MORE ALTER UNIT=1 REG=2 +X=1.0 -X=1.0 +Y=1.0 -Y=1.0 -CON=-0.5 +CON=2.5 END SEARCH END :: ******************* search pass 1 keff= 5.05769E-01 + or - 1.10681E-03 ****************** the parameter was 0.00000E+00 ***** modified keno v data has been rewritten on unit 95 ***** ******************* search pass 2 keff= 9.06304E-01 + or - 1.57239E-03 ****************** the parameter was -5.00000E-01 ***** modified keno v data has been rewritten on unit 95 ***** ******************* search pass 3 keff= 2.82956E-01 + or - 8.20501E-04 ****************** the parameter was 2.50000E+00 ***** modified keno v data has been rewritten on unit 95 ***** ******************* search pass 4 keff= 3.19086E-01 + or - 9.27655E-04 ****************** the parameter was 1.25000E+00 ***** modified keno v data has been rewritten on unit 95 ***** ******************* search pass 5 keff= 2.94889E-01 + or - 8.49552E-04 ****************** the parameter was 1.91631E+00 10x10x10 array - x & y dimension optimum search **************************************************************************************************** **************************************************************************************************** convergence was achieved on pass 2 the parameter was -5.00000E-01 the equation used in the search was: k-eff = +5.70750E-01 -4.42912E-01*p +2.39464E-01*p**2 -4.35513E-02*p**3 k-effective= 9.06304E-01 + or - 1.57239E-03 the corresponding geometry follows; media bias geometry description for those units utilized in this problem region num id ----- unit 1 ----- 1 sphere 1 1 radius = 2.0000 the center is located at ( 0.0000 , 0.0000 , 0.000 ). 2 cuboid 2 1 +x = 2.0000 -x = -2.0000 +y = 2.0000 -y = -2.0000 +z = 4.0000 -z = -4.0000 **************************************************************************************************** **************************************************************************************************** Array of Offset Spheres — Search in :math:X and :math:Y Dimensions ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ This is the same problem as described above except the spheres are now off-centered in the X and Y dimensions. Since only changes in the X and Y dimensions of the outer region are desired, the +X, −X, +Y, and −Y search constants must be set. All dimensions are to change at the same rate but they have different initial values. The search constants specify how surfaces change relative to each other. For this case the search constants were chosen so that the moderator spacing for the X and Y dimensions will change at the same rate. The +Z and −Z search constants by default are zero so these dimensions will not change. The input data and the final search results for this problem follow: :: =CSAS5S 10x10x10 ARRAY - X & Y DIMENSION OPTIMUM SEARCH - OFFSET V7-238 READ COMP URANIUM 1 1.0 300.0 92235 90.0 92238 10.0 END H2O 2 0.01 300.0 END END COMP READ CELLDATA LATTICECELL SPHSQUAREP PITCH=8.0 2 FUELD=4.0 1 END END CELLDATA READ GEOMETRY UNIT 1 SPHERE 1 1 2.0 ORIGIN 2.0 1.0 0.0 CUBOID 2 1 6.0 -2.0 5.0 -3.0 2P4.0 END GEOMETRY READ ARRAY ARA=1 NUX=10 NUY=10 NUZ=10 FILL F1 END FILL END ARRAY END DATA READ SEARCH OPTIMUM DIMENSION MORE ALTER UNIT=1 REG=2 +X=0.3333333 -X=1.0 +Y=0.4 -Y=0.6666666 -CON=-1.0 +CON=5.0 END SEARCH END :: ******************* search pass 1 keff= 5.07412E-01 + or - 1.16598E-03 ****************** the parameter was 0.00000E+00 ***** modified keno v data has been rewritten on unit 95 *** ******************* search pass 2 keff= 9.05152E-01 + or - 1.53983E-03 ****************** the parameter was -1.00000E+00 ***** modified keno v data has been rewritten on unit 95 ***** ******************* search pass 3 keff= 2.83216E-01 + or - 8.07313E-04 ****************** the parameter was 5.00000E+00 ***** modified keno v data has been rewritten on unit 95 ***** ******************* search pass 4 keff= 3.20108E-01 + or - 7.96799E-04 ****************** the parameter was 2.50000E+00 ***** modified keno v data has been rewritten on unit 95 ***** ******************* search pass 5 keff= 2.93841E-01 + or - 8.39609E-04 ****************** the parameter was 3.82730E+00 10x10x10 array - x & y dimension optimum search - offset **************************************************************************************************** **************************************************************************************************** convergence was achieved on pass 2 the parameter was -1.00000E+00 the equation used in the search was: k-eff = +5.73236E-01 -2.19195E-01*p +5.80895E-02*p**2 -5.19734E-03*p**3 k-effective= 9.05152E-01 + or - 1.53983E-03 the corresponding geometry follows; media bias geometry description for those units utilized in this problem region num id ----- unit 1 ----- 1 sphere 1 1 radius = 2.0000 the center is located at ( 2.0000 , 1.0000 , 0.0000 ). 2 cuboid 2 1 +x = 4.0000 -x = 0.0000 +y = 3.0000 -y = -1.0000 +z = 4.0000 -z = -4.0000 **************************************************************************************************** **************************************************************************************************** Array of Cylinders — Search in :math:X, :math:Y, and :math:Z Dimensions ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ This is the same array of cylinder problem as described in the previous section, except the search is an optimum dimension search. Since changes in the X, Y, and Z dimensions of the outer region are desired, the +X, −X, +Y, −Y, +Z, and −Z search constants must be set. The search data ALL=1.0 sets all the outer region search constants to 1.0. However, a constant ratio for the center-to-center spacing of the units in desired so the Z dimensions will need to be reset to a lower value of 0.333333. This will ensure that the Z dimensions change at 1/3 the rate of the X and Y dimensions relative to their initial values. The input data and the final search results for this problem follow: :: =CSAS5S 10x10x10 ARRAY, OPTIMUM DIMENSION SEARCH - MAINTAIN C-TO-C SPACING RATIO V7-238 READ COMP URANIUM 1 1.0 300.0 92235 90.0 92238 10.0 END H2O 2 0.01 300.0 END END COMP READ CELLDATA LATTICECELL SQUAREPITCH PITCH=8.0 2 FUELD=4.0 1 END END CELLDATA READ GEOMETRY UNIT 1 CYLINDER 1 1 2.0 10.0 -10.0 CUBOID 2 1 4p4.0 2p12.0 END GEOMETRY READ ARRAY ARA=1 NUX=10 NUY=10 NUZ=10 FILL F1 END FILL END ARRAY END DATA READ SEARCH OPTIMUM DIMENSION MORE ALTER UNIT=1 REG=2 ALL=1.0 +Z=0.3333333 -Z=0.3333333 -CON= -0.5 +CON=2.5 END SEARCH END :: ******************* search pass 1 keff= 1.09882E+00 + or - 1.75627E-03 ****************** the parameter was 0.00000E+00 ***** modified keno v data has been rewritten on unit 95 ***** ******************* search pass 2 keff= 1.78193E+00 + or - 2.12022E-03 ****************** the parameter was -5.00000E-01 ***** modified keno v data has been rewritten on unit 95 ***** ******************* search pass 3 keff= 4.54369E-01 + or - 1.10229E-03 ****************** the parameter was 2.50000E+00 ***** modified keno v data has been rewritten on unit 95 ***** ******************* search pass 4 keff= 5.67573E-01 + or - 1.26038E-03 ****************** the parameter was 1.25000E+00 10x10x10 array, optimum dimension search - maintain c-to-c spacing ratio **************************************************************************************************** **************************************************************************************************** convergence was achieved on pass 2 the parameter was -5.00000E-01 the equation used in the search was: k-eff = +1.09882E+00 -1.01312E+00*p +6.38864E-01*p**2 -1.34691E-01*p**3 k-effective= 1.78193E+00 + or - 2.12022E-03 the corresponding geometry follows; media bias geometry description for those units utilized in this problem region num id ----- unit 1 ----- 1 cylinder 1 1 radius = 2.0000 +z = 10.000 -z = -10.000 centerline is at x = 0.000 y = 0.000 2 cuboid 2 1 +x = 2.0000 -x = -2.0000 +y = 2.0000 -y = -2.0000 +z = 10.000 -z = -10.000 **************************************************************************************************** **************************************************************************************************** Minimum dimension search ------------------------ A minimum dimension search searches for the geometry dimensions that yield the lowest value of *k*\ :sub:eff. A minimum dimension search is activated by entering “MINIMUM DIMENSION” in the search data. There are no defaulted search data in a dimension search. The user must specify the dimensions to be changed and the manner in which they will be changed as described in the auxiliary search commands. A dimension search is performed by altering the regions having nonzero search constants specified in the auxiliary search commands portion of the search data. By default, the search constants for every dimension in the problem are zero. Only those dimensions having a nonzero search constant are altered by the code. By default, a dimension search sets the minimum constraint to −10E10 and the maximum constraint to +10E10. The relationship between the constraints and the search constants are given in :eq:CSASapp-1. and :eq:CSASapp-2. If the default values of the constraints are used, appropriate search constants must be calculated using these equations. In some cases it may be simpler to set the search constants to 1.0 and calculate the corresponding maximum and minimum constraints. Several dimension search examples are shown below. Infinite Array of Fuel Bundles Separated by Flux Traps — Search in :math:X and :math:Y-Dimensions ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ The fuel bundles in this problem represent 17 × 17 PWR fuel assemblies. The fuel pins are smeared together, making a mixture 100. The fuel pins consist of 2.35 wt % :sup:235\ U having a diameter of 0.823 cm, zirconium cladding having an outer diameter of 0.9627 cm, and a pitch of 1.275 cm. The fuel bundle is represented as a 10.8375 cm × 10.8375 cm × 366 cm cuboid of mixture 100 surrounded by Boral and then water. The Boral has a density of 2.65 g/cm\ :sup:3 and is composed of 35.17 wt % B\ :sub:4\ C and 64.83 wt % Al. The fuel bundles are at a fixed pitch of 13.0 cm. The problem searches for the thickness of Boral that will produce the lowest system *k*\ :sub:eff. The input data and the final search results for this problem follow: :: =csas5s array of fuel bundles with flux trap v7-238 read comp uo2 1 .84 300. 92235 2.35 92238 97.65 end zr 2 1 end h2o 3 1 end b4c 4 den=2.65 0.3517 end al 4 den=2.65 0.6483 end h2o 5 1 end end comp read celldata latticecell squarepitch pitch=1.275 2 fueld=0.823 1 gapd=0.9627 2 cellmix=100 end end celldata read param far=yes gen=203 npg=1000 end param read geom global unit 1 cuboid 100 1 4p10.8375 2p183.0 cuboid 4 1 4p11.0 2p183.0 cuboid 5 1 4p13.0 2p183.0 end geom read bounds xfc=mirror yfc=mirror end bounds end data read search minimum dimension more alter unit=1 reg=2 +x=1.0 -x=1.0 +y=1.0 -y=1.0 -con=-0.009 +con=0.181818 end search end :: ******************* search pass 1 keff= 7.18665E-01 + or - 1.69354E-03 ****************** the parameter was 0.00000E+00 **** xsdrnpm mesh intervals **** 4 mesh intervals in zone 1 4 mesh intervals in zone 2 14 mesh intervals in zone 3 ***** modified keno v data has been rewritten on unit 95 ***** ******************* search pass 2 keff= 7.95965E-01 + or - 1.50272E-03 ****************** the parameter was 1.81818E-01 **** xsdrnpm mesh intervals **** 4 mesh intervals in zone 1 4 mesh intervals in zone 2 14 mesh intervals in zone 3 ***** modified keno v data has been rewritten on unit 95 ***** ******************* search pass 3 keff= 7.12247E-01 + or - 1.42902E-03 ****************** the parameter was 6.06060E-02 **** xsdrnpm mesh intervals **** 4 mesh intervals in zone 1 4 mesh intervals in zone 2 14 mesh intervals in zone 3 ***** modified keno v data has been rewritten on unit 95 ***** ******************* search pass 4 keff= 7.40472E-01 + or - 1.45576E-03 ****************** the parameter was 1.21212E-01 **** xsdrnpm mesh intervals **** 4 mesh intervals in zone 1 4 mesh intervals in zone 2 14 mesh intervals in zone 3 ***** modified keno v data has been rewritten on unit 95 ***** ******************* search pass 5 keff= 7.08528E-01 + or - 1.56618E-03 ****************** the parameter was 4.01175E-02 array of fuel bundles with flux trap **************************************************************************************************** **************************************************************************************************** convergence was achieved on pass 5 the parameter was 4.01175E-02 the equation used in the search was: k-eff = +7.18297E-01 -5.33084E-01*p +7.78791E+00*p**2 -1.40353E+01*p**3 k-effective= 7.08528E-01 + or - 1.56618E-03 the corresponding geometry follows; media bias geometry description for those units utilized in this problem region num id ******************* global ******************* ----- unit 1 ----- 1 cuboid 100 1 +x = 10.837 -x = -10.837 +y = 10.837 -y = -10.837 +z = 183.00 -z = -183.00 2 cuboid 4 1 +x = 11.441 -x = -11.441 +y = 11.441 -y = -11.441 +z = 183.00 -z = -183.00 3 cuboid 5 1 +x = 13.000 -x = -13.000 +y = 13.000 -y = -13.000 +z = 183.00 -z = -183.00 **************************************************************************************************** **************************************************************************************************** array of fuel bundles with flux trap **************************************************************************************************** **************************************************************************************************** based on the preceding data, the best estimate of the parameter is 3.81620E-02 the geometry corresponding to this parameter follows: media bias geometry description for those units utilized in this problem region num id ******************* global ******************* ----- unit 1 ----- 1 cuboid 100 1 +x = 10.837 -x = -10.837 +y = 10.837 -y = -10.837 +z = 183.00 -z = -183.00 2 cuboid 4 1 +x = 11.420 -x = -11.420 +y = 11.420 -y = -11.420 +z = 183.00 -z = -183.00 3 cuboid 5 1 +x = 13.000 -x = -13.000 +y = 13.000 -y = -13.000 +z = 183.00 -z = -183.00 **************************************************************************************************** **************************************************************************************************** Two Uranium Slabs Separated by a Flux Trap — Search in +\ :math:X Dimensions ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ This problem consists of two 90 wt % enriched uranium metal slabs separated by a flux trap. Each uranium slab is 3.0 cm thick. The spacing between the slabs is 4 cm. Between the slabs is a Boral plate at each slab surface and water in between the two Boral plates. This problem searches for the thickness of the Boral plates and separating water that produces the lowest system *k*\ :sub:eff. The input data and the final search results for this problem follow: :: =CSAS5S MINIMUM DIMENSION SEARCH - SLAB V7-238 READ COMP URANIUM 1 1.0 300.0 92235 90.0 92238 10.0 END B4C 2 0.367 300.0 END AL 2 0.633 300.0 END H2O 3 1.0 300.0 END END COMP READ CELLDATA MULTIREGION SLAB RIGHT_BDY=VACUUM LEFT_BDY=REFLECTED END 3 1 2 2 1 5 END ZONE END CELLDATA READ GEOMETRY GLOBAL UNIT 1 CUBOID 3 1 1.0 0.0 4P100.0 CUBOID 2 1 2.0 0.0 4P100.0 CUBOID 1 1 5.0 0.0 4P100.0 END GEOMETRY READ BOUNDS -XB=MIRROR END BOUNDS END DATA READ SEARCH MINIMUM DIMENSION MORE ALTER UNIT=1 REG=1 +X=1.0 CELL=1 -CON=-1.0 +CON=1.0 END SEARCH END :: ******************* search pass 1 keff= 8.60439E-01 + or - 1.55744E-03 ****************** the parameter was 0.00000E+00 ***** modified keno v data has been rewritten on unit 95 ***** ******************* search pass 2 keff= 8.72016E-01 + or - 1.60673E-03 ****************** the parameter was -1.00000E+00 ***** modified keno v data has been rewritten on unit 95 ***** ******************* search pass 3 keff= 1.08607E+00 + or - 1.73386E-03 ****************** the parameter was 1.00000E+00 ***** modified keno v data has been rewritten on unit 95 ***** ******************* search pass 4 keff= 8.59819E-01 + or - 1.41625E-03 ****************** the parameter was 5.00000E-01 ***** modified keno v data has been rewritten on unit 95 ***** ******************* search pass 5 keff= 8.56856E-01 + or - 1.53569E-03 ****************** the parameter was 2.75785E-01 minimum dimension search - slab **************************************************************************************************** **************************************************************************************************** convergence was achieved on pass 5 the parameter was 2.75785E-01 the equation used in the search was: k-eff = +8.71321E-01 -9.39460E-02*p +9.57041E-02*p**2 +1.97682E-01*p**3 k-effective= 8.56856E-01 + or - 1.53569E-03 the corresponding geometry follows; media bias geometry description for those units utilized in this problem region num id ******************* global ******************* ----- unit 1 ----- 1 cuboid 3 1 +x = 1.2758 -x = 0.0000 +y = 100.00 -y = -100.00 +z = 100.00 -z = -100.00 2 cuboid 2 1 +x = 2.0000 -x = 0.0000 +y = 100.00 -y = -100.00 +z = 100.00 -z = -100.00 3 cuboid 1 1 +x = 5.0000 -x = 0.0000 +y = 100.00 -y = -100.00 +z = 100.00 -z = -100.00 **************************************************************************************************** **************************************************************************************************** minimum dimension search - slab **************************************************************************************************** **************************************************************************************************** based on the preceding data, the best estimate of the parameter is 2.68105E-01 the geometry corresponding to this parameter follows: media bias geometry description for those units utilized in this problem region num id ******************* global ******************* ----- unit 1 ----- 1 cuboid 3 1 +x = 1.2681 -x = 0.0000 +y = 100.00 -y = -100.00 +z = 100.00 -z = -100.00 2 cuboid 2 1 +x = 2.0000 -x = 0.0000 +y = 100.00 -y = -100.00 +z = 100.00 -z = -100.00 3 cuboid 1 1 +x = 5.0000 -x = 0.0000 +y = 100.00 -y = -100.00 +z = 100.00 -z = -100.00 **************************************************************************************************** **************************************************************************************************** Critical dimension search ------------------------- A critical dimension search searches for the geometry dimensions that yield a specified value of *k*\ :sub:eff. A critical dimension search is activated by entering “CRITICAL DIMENSION” in the search data. No defaulted search data are in a dimension search. The user must specify the dimensions to be changed and the manner in which they will be changed. A dimension search is performed by altering the regions having nonzero search constants specified in the individual search commands portion of the search data. By default, the search constants for every dimension in the problem are zero. Only those dimensions having a nonzero search constant are altered by the code. By default, a dimension search sets the minimum constraint to −10E10 and the maximum constraint to +10E10. The relationship between the constraints and the search constants is given in :eq:CSASapp-1 and :eq:CSASapp-2. If the default values of the constraints are used, appropriate search constants must be calculated using these equations. It may be simpler to set the search constants to 1.0 and calculate the corresponding maximum and minimum constraints. Several critical dimension search examples are shown below. Array of Centered Spheres — Search in :math:X, :math:Y, and :math:Z Dimensions ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Consider a 10 × 10 × 10 array of uranium spheres arranged in an array having a “square” pitch. The uranium spheres are 2 cm in radius, and the center-to-center spacing is 8 cm. The uranium spheres and their associated spacing are defined to be unit 1, and the 10 × 10 × 10 array is defined to be array 1. A critical dimension search is performed on the outer dimension of the cuboid that will produce a system *k*\ :sub:eff of 1.0. In the MORE search data, All=1.0 on region 2 of unit 1 specifies all six surfaces of the cuboid change identically. The search constraints are set to search from a sphere center-to-center spacing of 4 cm to 20 cm. The search data and results are given below. :: =CSAS5S 10x10x10 ARRAY - DIMENSION SEARCH V7-238 READ COMP URANIUM 1 1.0 300.0 92235 90.0 92238 10.0 END H2O 2 0.01 300.0 END END COMP READ CELLDATA LATTICECELL SPHSQUAREP PITCH=8.0 2 FUELD=4.0 1 END END CELLDATA READ GEOMETRY UNIT 1 SPHERE 1 1 2.0 CUBOID 2 1 6P4.0 END GEOMETRY READ ARRAY ARA=1 NUX=10 NUY=10 NUZ=10 FILL F1 END FILL END ARRAY END DATA READ SEARCH CRITICAL DIMENSION MORE ALTER UNIT=1 REG=2 ALL=1.0 CELL=1 -CON=-0.5 +CON=2.5 END SEARCH :: ******************* search pass 1 keff= 5.05364E-01 + or - 1.20323E-03 ****************** the parameter was 0.00000E+00 ***** modified keno v data has been rewritten on unit 95 ***** ******************* search pass 2 keff= 2.59843E-01 + or - 8.30139E-04 ****************** the parameter was 2.50000E+00 ***** modified keno v data has been rewritten on unit 95 ***** ******************* search pass 3 keff= 1.27927E+00 + or - 1.51758E-03 ****************** the parameter was -5.00000E-01 ***** modified keno v data has been rewritten on unit 95 ***** ******************* search pass 4 keff= 8.62177E-01 + or - 1.51303E-03 ****************** the parameter was -3.36734E-01 ***** modified keno v data has been rewritten on unit 95 ***** ******************* search pass 5 keff= 9.92494E-01 + or - 1.58120E-03 ****************** the parameter was -3.98906E-01 ***** modified keno v data has been rewritten on unit 95 ***** ******************* search pass 6 keff= 1.00529E+00 + or - 1.78980E-03 ****************** the parameter was -4.01988E-01 ***** modified keno v data has been rewritten on unit 95 ***** ******************* search pass 7 keff= 1.00145E+00 + or - 1.59308E-03 ****************** the parameter was -4.00885E-01 10x10x10 array - dimension search **************************************************************************************************** **************************************************************************************************** convergence was achieved on pass 7 the parameter was -4.00885E-0 the equation used in the search was: k-eff = +4.71062E-01 +7.54530E-02*p +2.99529E+00*p**2 -1.22324E+00*p**3 k-effective= 1.00145E+00 + or - 1.59308E-03 the corresponding geometry follows; media bias geometry description for those units utilized in this problem region num id ----- unit 1 ----- 1 sphere 1 1 radius = 2.0000 the center is located at ( 0.0000 , 0.0000 , 0.0000 ). 2 cuboid 2 1 +x = 2.3965 -x = -2.3965 +y = 2.3965 -y = -2.3965 +z = 2.3965 -z = -2.3965 **************************************************************************************************** **************************************************************************************************** 10x10x10 array - dimension search **************************************************************************************************** **************************************************************************************************** based on the preceding data, the best estimate of the parameter is -4.00545E-01 the geometry corresponding to this parameter follows: media bias geometry description for those units utilized in this problem region num id ----- unit 1 ----- 1 sphere 1 1 radius = 2.0000 the center is located at ( 0.0000 , 0.0000 , 0.0000 ). 2 cuboid 2 1 +x = 2.3978 -x = -2.3978 +y = 2.3978 -y = -2.3978 +z = 2.3978 -z = -2.3978 **************************************************************************************************** **************************************************************************************************** Array of Centered Spheres — Search in :math:X, :math:Y, and :math:Z Dimensions ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Consider a 10 × 10 × 10 array of uranium spheres arranged in an array having a “square” pitch. The uranium spheres are 2 cm in radius, and the center-to-center spacing is 8 cm. The uranium spheres and their associated spacing are defined to be unit 1, and the 10 × 10 × 10 array is defined to be array 1. A critical dimension search is performed on the radius of the uranium sphere that will produce a system *k*\ :sub:eff of 1.0. In the MORE search data, RADIUS=1.0 on region 1 of unit 1 specifies the sphere’s radius is to be altered. The search constraints are set to search from a radius of 0.5 cm to 4.0 cm. The search data and results are given below. :: =CSAS5S 10x10x10 ARRAY - DIMENSION SEARCH V7-238 READ COMP URANIUM 1 1.0 300.0 92235 90.0 92238 10.0 END H2O 2 0.01 300.0 END END COMP READ CELLDATA LATTICECELL SPHSQUAREP PITCH=8.0 2 FUELD=4.0 1 END END CELLDATA READ GEOMETRY UNIT 1 SPHERE 1 1 2.0 CUBOID 2 1 6P4.0 END GEOMETRY READ ARRAY ARA=1 NUX=10 NUY=10 NUZ=10 FILL F1 END FILL END ARRAY END DATA READ SEARCH CRITICAL DIMENSION MORE ALTER UNIT=1 CELL=1 REG=1 RADIUS=1.0 -CON=-0.75 +CON=1.0 END SEARCH END :: ******************* search pass 1 keff= 5.05364E-01 + or - 1.20323E-03 ****************** the parameter was 0.00000E+00 ***** modified keno v data has been rewritten on unit 95 ***** ******************* search pass 2 keff= 1.79497E+00 + or - 1.93055E-03 ****************** the parameter was 1.00000E+00 ***** modified keno v data has been rewritten on unit 95 ***** ******************* search pass 3 keff= 9.91353E-01 + or - 1.56062E-03 ****************** the parameter was 3.83555E-01 ***** modified keno v data has been rewritten on unit 95 ***** ******************* search pass 4 keff= 9.97513E-01 + or - 1.61171E-03 ****************** the parameter was 3.90304E-01 10x10x10 array - dimension search **************************************************************************************************** **************************************************************************************************** convergence was achieved on pass 4 the parameter was 3.90304E-01 the equation used in the search was: k-eff = +5.05364E-01 +1.84705E+00*p -2.10615E+00*p**2 +1.54871E+00*p**3 k-effective= 9.97513E-01 + or - 1.61171E-03 the corresponding geometry follows; media bias geometry description for those units utilized in this problem region num id ----- unit 1 ----- 1 sphere 1 1 radius = 2.7806 the center is located at ( 0.0000 , 0.0000 , 0.0000 ). 2 cuboid 2 1 +x = 4.0000 -x = -4.0000 +y = 4.0000 -y = -4.0000 +z = 4.0000 -z = -4.0000 **************************************************************************************************** **************************************************************************************************** 10x10x10 array - dimension search **************************************************************************************************** based on the preceding data, the best estimate of the parameter is 3.93037E-01 the geometry corresponding to this parameter follows: media bias geometry description for those units utilized in this problem region num id ----- unit 1 ----- 1 sphere 1 1 radius = 2.7861 the center is located at ( 0.0000 , 0.0000 , 0.0000 ). 2 cuboid 2 1 +x = 4.0000 -x = -4.0000 +y = 4.0000 -y = -4.0000 +z = 4.0000 -z = -4.0000 **************************************************************************************************** Array of Centered Cylinders — Search in :math:X and :math:Y Dimensions ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ This problem contains search data for a critical dimension search with *k*\ :sub:eff specified to be 0.95. The problem consists of a 10 × 10 × 2 array of uranium cylinders surrounded by low density water. The cylinders are 90% :sup:235\ U and 10% :sup:238\ U and have a radius of 2.0 cm and a length of 20 cm. The cylinders are originally centered on an 8.0 cm X and Y pitch and a 24 cm Z pitch with interstitial low density (0.01 gm/cc) water. The problem searches for the cylinder center-to-center spacing that will produce a *k*\ :sub:eff = 0.95 for the system. Because this is a dimension search, all search data must be specified. The +X, −X, +Y, and −Y search constants are specified as well as the −CON and +CON search constraints. The search data is set up to alter the cylinder center-to-center spacing from 4.0 cm to 16.0 cm during the search. The MORE search data CELL=1 ties the search data to the first unit cell thus ensuring they change in unison properly modifying the cross-section processing. The input data and the final search results for this problem follow: :: =CSAS5S 10x10x2 ARRAY DIMENSION SEARCH - CHANGE IN X, Y V7-238 READ COMP URANIUM 1 1.0 300.0 92235 90.0 92238 10.0 END H2O 2 0.01 300.0 END END COMP READ CELLDATA LATTICECELL SQUAREPITCH PITCH=8.0 2 FUELD=4.0 1 END END CELLDATA READ GEOMETRY UNIT 1 CYLINDER 1 1 2.0 10.0 -10.0 CUBOID 2 1 4p4.0 2p12.0 END GEOMETRY READ ARRAY ARA=1 NUX=10 NUY=10 NUZ=2 FILL F1 END FILL END ARRAY END DATA READ SEARCH CRITICAL DIMENSION KEFF=0.95 MORE ALTER UNIT=1 REG=2 +X=1.0 -X=1.0 +Y=1.0 -Y=1.0 CELL=1 -CON=-0.5 +CON=2.0 END SEARCH END :: ******************* search pass 1 keff= 8.48434E-01 + or - 1.50137E-03 ****************** the parameter was 0.00000E+00 ***** modified keno v data has been rewritten on unit 95 ***** ******************* search pass 2 keff= 4.22734E-01 + or - 9.98609E-04 ****************** the parameter was 2.00000E+00 ***** modified keno v data has been rewritten on unit 95 ***** ******************* search pass 3 keff= 1.49380E+00 + or - 1.81532E-03 ****************** the parameter was -4.77172E-01 ***** modified keno v data has been rewritten on unit 95 ***** ******************* search pass 4 keff= 9.23415E-01 + or - 1.68810E-03 ****************** the parameter was -8.66017E-02 ***** modified keno v data has been rewritten on unit 95 ***** ******************* search pass 5 keff= 9.47041E-01 + or - 1.57053E-03 ****************** the parameter was -1.13412E-01 10x10x2 array dimension search - change in x, y **************************************************************************************************** **************************************************************************************************** convergence was achieved on pass 5 the parameter was -1.13412E-01 the equation used in the search was: k-eff = +8.52892E-01 -6.96099E-01*p +1.13262E+00*p**2 -4.46134E-01*p**3 k-effective= 9.47041E-01 + or - 1.57053E-03 the corresponding geometry follows; media bias geometry description for those units utilized in this problem region num id ----- unit 1 ----- 1 cylinder 1 1 radius = 2.0000 +z = 10.000 -z = -10.000 centerline is at x = 0.000 y = 0.000 2 cuboid 2 1 +x = 3.5464 -x = -3.5464 +y = 3.5464 -y = -3.5464 +z = 12.000 -z = -12.000 **************************************************************************************************** **************************************************************************************************** 10x10x2 array dimension search - change in x, y **************************************************************************************************** **************************************************************************************************** based on the preceding data, the best estimate of the parameter is -1.16433E-01 the geometry corresponding to this parameter follows: media bias geometry description for those units utilized in this problem region num id ----- unit 1 ----- 1 cylinder 1 1 radius = 2.0000 +z = 10.000 -z = -10.000 centerline is at x = 0.000 y = 0.000 2 cuboid 2 1 +x = 3.5343 -x = -3.5343 +y = 3.5343 -y = -3.5343 +z = 12.000 -z = -12.000 **************************************************************************************************** ****************************************************************************************************z Array of Centered Uranium Slabs ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Consider a 10 × 1 × 1 array of uranium metal slabs. The uranium slabs are 2 cm thick in the X dimension and 200 cm thick in the Y and Z dimensions. On each side of the uranium slab in the X dimension is 1 cm of H\ :sub:2\ O, then 1 cm of Boral, then 1 cm of H\ :sub:2\ O resulting in an initial center-to-center spacing of 8.0 cm. The slabs are 90% :sup:235\ U and 10% :sup:238\ U, the water is full density, and the Boral is 36.7% B\ :sub:4\ C. The uranium slabs and their associated materials are defined to be Unit 1, and the 10 × 1 × 1 array is defined to be array 1. The problem searches for the slab center-to-center spacing that will produce a *k*\ :sub:eff = 1.0 for the system, which is the default search value. Because this is a dimension search all search data must be specified. The +X and −X search constants are specified as well as the −CON and +CON search constraints. The search data is set up to alter the cylinder change the water slab thickness between the Boral plates from 0.8 cm to 10.0 cm thick during the search. The MORE search data CELL=1 ties the search data to the first unit cell thus ensuring they change in unison properly modifying the cross-section processing. The input data and the final search results for this problem follow: :: =CSAS5S 10x1x1 ARRAY DIMENSION SEARCH - SLAB V7-238 READ COMP URANIUM 1 1.0 300.0 92235 90.0 92238 10.0 END H2O 2 1.0 300.0 END B4C 3 0.367 300.0 END AL 3 0.633 300.0 END H2O 4 1.0 300.0 END END COMP READ CELLDATA LATTICECELL SYMMSLABCELL PITCH=8.0 4 FUELD=2.0 1 GAPD=6.0 3 CLADD=4.0 2 END END CELLDATA READ GEOMETRY UNIT 1 CUBOID 1 1 1.0 -1.0 4P100.0 CUBOID 2 1 2.0 -2.0 4p100.0 CUBOID 3 1 3.0 -3.0 4P100.0 CUBOID 4 1 4.0 -4.0 4P100.0 END GEOMETRY READ ARRAY ARA=1 GBL=1 NUX=10 NUY=1 NUZ=1 FILL F1 END FILL END ARRAY END DATA READ SEARCH CRITICAL DIMENSION MORE ALTER UNIT=1 REG=4 +X=1.0 -X=1.0 CELL=1 -CON=-0.24 +CON=1.0 END SEARCH END :: ******************* search pass 1 keff= 1.13914E+00 + or - 1.53132E-03 ****************** the parameter was 0.00000E+00 ***** modified keno v data has been rewritten on unit 95 ***** ******************* search pass 2 keff= 1.41849E+00 + or - 1.58152E-03 ****************** the parameter was -2.40000E-01 ***** modified keno v data has been rewritten on unit 95 ***** ******************* search pass 3 keff= 1.04012E+00 + or - 1.79880E-03 ****************** the parameter was 1.19536E-01 ***** modified keno v data has been rewritten on unit 95 ***** ******************* search pass 4 keff= 9.99495E-01 + or - 1.65714E-03 ****************** the parameter was 1.80331E-01 10x1x1 array dimension search - slab **************************************************************************************************** **************************************************************************************************** convergence was achieved on pass 4 the parameter was 1.80331E-01 the equation used in the search was: k-eff = +1.13914E+00 -9.36766E-01*p +9.20374E-01*p**2 -1.09475E-01*p**3 k-effective= 9.99495E-01 + or - 1.65714E-03 the corresponding geometry follows; media bias geometry description for those units utilized in this problem region num id ----- unit 1 ----- 1 cuboid 1 1 +x = 1.0000 -x = -1.0000 +y = 100.00 -y = -100.00 +z = 100.00 -z = -100.00 2 cuboid 2 1 +x = 2.0000 -x = -2.0000 +y = 100.00 -y = -100.00 +z = 100.00 -z = -100.00 3 cuboid 3 1 +x = 3.0000 -x = -3.0000 +y = 100.00 -y = -100.00 +z = 100.00 -z = -100.00 4 cuboid 4 1 +x = 4.7213 -x = -4.7213 +y = 100.00 -y = -100.00 +z = 100.00 -z = -100.00 **************************************************************************************************** **************************************************************************************************** 10x1x1 array dimension search - slab **************************************************************************************************** **************************************************************************************************** based on the preceding data, the best estimate of the parameter is 1.79511E-01 the geometry corresponding to this parameter follows: media bias geometry description for those units utilized in this problem region num id ----- unit 1 ----- 1 cuboid 1 1 +x = 1.0000 -x = -1.0000 +y = 100.00 -y = -100.00 +z = 100.00 -z = -100.00 2 cuboid 2 1 +x = 2.0000 -x = -2.0000 +y = 100.00 -y = -100.00 +z = 100.00 -z = -100.00 3 cuboid 3 1 +x = 3.0000 -x = -3.0000 +y = 100.00 -y = -100.00 +z = 100.00 -z = -100.00 4 cuboid 4 1 +x = 4.7180 -x = -4.7180 +y = 100.00 -y = -100.00 +z = 100.00 -z = -100.00 **************************************************************************************************** **************************************************************************************************** Search on Multiple Units and Regions ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Consider a problem having Units 1, 2, and 3. Unit 1 consists of three concentric spheres in a cuboid. Unit 2 consists of a single sphere in a cuboid, and Unit 3 contains three concentric cuboids. A search is to be made that changes the exterior dimensions of the three units, the inner sphere of Unit 1, and the sphere of Unit 2. The thicknesses of the outer spheres of Unit 1 are to be maintained, and the two inner cuboids of Unit 3 are to remain unchanged. The search data and results for this problem are the following: :: =CSAS5S DEMONSTRATION OF CRITICAL DIMENSION SEARCH V7-252 READ COMP URANIUM 1 0.985 293 92235 93.2 92238 5.6 92234 1.0 92236 0.2 END PLEXIGLAS 2 END SS304 3 END END COMP READ PARAMETERS GEN=203 NPG=1500 END PARAMETERS READ GEOMETRY UNIT 1 SPHERE 1 1 3.0 SPHERE 0 1 3.25 SPHERE 2 1 3.5 CUBE 0 1 2P5.0 UNIT 2 SPHERE 1 1 3.0 CUBE 0 1 2P5.0 UNIT 3 CUBE 0 1 2P4.9 CUBE 3 1 2P5.0 CUBE 0 1 2P5.0 END GEOMETRY READ ARRAY ARA=1 NUX=3 NUY=3 NUZ=3 FILL 1 3 2 3 2 3 2 3 1 3 2 3 1 3 2 3 1 3 2 3 1 3 1 3 1 3 2 END FILL END ARRAY END DATA READ SEARCH CRITICAL DIMENSION KEF=1.000 EPS=0.005 MORE ALTER UNIT=1 REG=1 ALL=1.0 ALTER UNIT=1 REG=4 ALL=1.0 ALTER UNIT=2 REG=1 TO 2 ALL=1.0 ALTER UNIT=3 REG=3 ALL=1.0 MAINTAIN UNIT=1 REG=2 TO 3 ALL=1.0 +CON=2.0 -CON=-0.5 END SEARCH END :: ******************* search pass 1 keff= 4.04885E-01 + or - 8.44823E-04 ****************** the parameter was 0.00000E+00 ***** modified keno v data has been rewritten on unit 95 ***** ******************* search pass 2 keff= 1.07404E+00 + or - 1.44587E-03 ****************** the parameter was 2.00000E+00 ***** modified keno v data has been rewritten on unit 95 ***** ******************* search pass 3 keff= 1.01654E+00 + or - 1.38559E-03 ****************** the parameter was 1.77869E+00 ***** modified keno v data has been rewritten on unit 95 ***** ******************* search pass 4 keff= 9.94811E-01 + or - 1.34462E-03 ****************** the parameter was 1.71784E+00 ***** modified keno v data has been rewritten on unit 95 ***** ******************* search pass 5 keff= 9.99482E-01 + or - 1.35858E-03 ****************** the parameter was 1.73183E+00 search demonstration of critical dimension search **************************************************************************************************** **************************************************************************************************** convergence was achieved on pass 5 the parameter was 1.73183E+00 the equation used in the search was: k-eff = +4.05037E-01 -7.56139E-02*p +4.77691E-01*p**2 -1.36173E-01*p**3 k-effective= 9.99482E-01 + or - 1.35858E-03 the corresponding geometry follows; media bias geometry description for those units utilized in this problem region num id ----- unit 1 ----- 1 sphere 1 1 radius = 8.1955 the center is located at ( 0.0000 , 0.0000 , 0.0000 ). 2 sphere 0 1 radius = 8.4455 the center is located at ( 0.0000 , 0.0000 , 0.0000 ). 3 sphere 2 1 radius = 8.6955 the center is located at ( 0.0000 , 0.0000 , 0.0000 ). 4 cuboid 0 1 +x = 13.659 -x = -13.659 +y = 13.659 -y = -13.659 +z = 13.659 -z = -13.659 ----- unit 2 ----- 1 sphere 1 1 radius = 8.1955 the center is located at ( 0.0000 , 0.0000 , 0.0000 ). 2 cuboid 0 1 +x = 13.65 -x = -13.659 +y = 13.659 -y = -13.659 +z = 13.659 -z = -13.659 ----- unit 3 ----- 1 cuboid 0 1 +x = 4.9000 -x = -4.9000 +y = 4.9000 -y = -4.9000 +z = 4.9000 -z = -4.9000 2 cuboid 3 1 +x = 5.0000 -x = -5.0000 +y = 5.0000 -y = -5.0000 +z = 5.0000 -z = -5.0000 3 cuboid 0 1 +x = 13.659 -x = -13.659 +y = 13.659 -y = -13.659 +z = 13.659 -z = -13.659 **************************************************************************************************** **************************************************************************************************** search demonstration of critical dimension search **************************************************************************************************** **************************************************************************************************** based on the preceding data, the best estimate of the parameter is 1.73328E+00 the geometry corresponding to this parameter follows: media bias geometry description for those units utilized in this problem region num id ----- unit 1 ----- 1 sphere 1 1 radius = 8.1998 the center is located at ( 0.0000 , 0.0000 , 0.0000 ). 2 sphere 0 1 radius = 8.4498 the center is located at ( 0.0000 , 0.0000 , 0.0000 ). 3 sphere 2 1 radius = 8.6998 the center is located at ( 0.0000 , 0.0000 , 0.0000 ). 4 cuboid 0 1 +x = 13.666 -x = -13.666 +y = 13.666 -y = -13.666 +z = 13.666 -z = -13.666 ----- unit 2 ----- 1 sphere 1 1 radius = 8.1998 the center is located at ( 0.0000 , 0.0000 , 0.000 ). 2 cuboid 0 1 +x = 13.666 -x = -13.666 +y = 13.666 -y = -13.666 +z = 13.666 -z = -13.666 ----- unit 3 ----- 1 cuboid 0 1 +x = 4.9000 -x = -4.9000 +y = 4.9000 -y = -4.9000 +z = 4.9000 -z = -4.9000 2 cuboid 3 1 +x = 5.0000 -x = -5.0000 +y = 5.0000 -y = -5.0000 +z = 5.0000 -z = -5.0000 3 cuboid 0 1 +x = 13.666 -x = -13.666 +y = 13.666 -y = -13.666 +z = 13.666 -z = -13.666 **************************************************************************************************** **************************************************************************************************** UO\ :sub:2\ F\ :sub:2 Solution Tank — Critical Search on Chord Length ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Consider a large spherical tank partially filled with UO\ :sub:2\ F\ :sub:2 solution. The tank has a radius of 34.6 cm and is initially filled with solution to a height of 10.0 cm above the midpoint. The tank is composed of a 0.759-cm-thick Al shell. The UO\ :sub:2\ F\ :sub:2 solution is composed of three standard compositions: UO\ :sub:2\ F\ :sub:2, HF acid, and H\ :sub:2\ O. The code combines these using a set algorithm. This may or may not produce a solution at the desired density. If the density of the solution is known it should be entered. Also, extra acid can be added to the solution by specifying a non-zero acid molarity. A critical dimension search is performed on the length of the chord yielding system *k*\ :sub:eff = 0.98. In the MORE search data, UNIT=1 REG=1 CHORD=1 specifies that the chord length of region 1 in Unit 1 is to be altered with the limits defined by the search constraints. The minimum search constraint is set so region 1 consists of the part of the sphere below an X-Y plane 10.0 cm below the sphere’s midpoint. The maximum search constraint is set so region 1 consists of almost the entire sphere. The constraints are calculated as follows: D\ :sub:i = ( Radius\ :sub:i + Chord\ :sub:i ) / ( 2*Radius\ :sub:i ) = ( 34.6 + 10.0 ) / 2*34.6 ) = 0.6445 D\ :sub:min = ( Radius\ :sub:min + Chord\ :sub:min ) / ( 2*Radius\ :sub:min ) = ( 34.6 + − 10.0 ) / 2*34.6 ) = 0.3555 D\ :sub:max = ( Radius\ :sub:max + Chord\ :sub:max ) / ( 2*Radius\ :sub:max ) = ( 34.6 + 34.6 ) / 2*34.6 ) = 1.0 The maximum constraint is calculated according to: +CON = [ ( D\ :sub:max / D\ :sub:i ) −1.0 ] / CHORD = [ ( 1.0 / 0.6445 ) −1.0 ] / 1.0 = 0.5515 The minimum constraint is calculated according to: −CON = [ ( D\ :sub:min / D\ :sub:i ) − 1.0 ] / CHORD = [ ( 0.3555 / 0.6445 ) − 1.0 ] / 1.0 = −0.4484 The search data and final search results for this problem follow: :: =csas5s sample problem 6 soln tank - crit. dim. search on chord v7-238 read comp solnuo2f2 1 500 0 1 300.0 92235 4.89 92238 95.09 92234 0.02 end al 2 1.0 300.0 end end comp sample problem 6 soln tank - crit. dim. search on chord read geom unit 1 hemisphe-z 1 1 34.6 chord 10. sphere 0 1 34.6 sphere 2 1 34.759 end geom end data READ SEARCH CRITICAL DIMENSION KEF=0.98 MORE ALTER UNIT=1 REG=1 CHORD=1.0 -CON=-0.4484 +CON=0.5515 END SEARCH end :: ******************* search pass 1 keff= 9.32087E-01 + or - 1.25932E-03 ****************** the parameter was 0.00000E+00 ***** modified keno v data has been rewritten on unit 95 ***** ******************* search pass 2 keff= 1.00306E+00 + or - 1.54366E-03 ****************** the parameter was 5.51500E-01 ***** modified keno v data has been rewritten on unit 95 ***** ******************* search pass 3 keff= 9.95554E-01 + or - 1.33223E-03 ****************** the parameter was 3.72325E-01 ***** modified keno v data has been rewritten on unit 95 ***** ******************* search pass 4 keff= 9.80534E-01 + or - 1.47774E-03 ****************** the parameter was 2.37245E-01 solution tank - critical dimension search on chord length **************************************************************************************************** **************************************************************************************************** convergence was achieved on pass 4 the parameter was 2.37245E-01 the equation used in the search was: k-eff = +9.32087E-01 +2.68165E-01*p -2.82174E-01*p**2 +5.30631E-02*p**3 k-effective= 9.80534E-01 + or - 1.47774E-03 the corresponding geometry follows; media bias geometry description for those units utilized in this problem region num id ******************* global ******************* ----- unit 1 ----- 1 -zhemisphere 1 1 radius = 34.600 existing in the -z direction center=(0.00,0.00,0.00) chord=20.581 2 sphere 0 1 radius = 34.600 the center is located at ( 0.0000 , 0.0000 , 0.0000 ). 3 sphere 2 1 radius = 34.759 the center is located at ( 0.0000 , 0.0000 , 0.0000 ). **************************************************************************************************** **************************************************************************************************** solution tank - critical dimension search on chord length **************************************************************************************************** **************************************************************************************************** based on the preceding data, the best estimate of the parameter is 2.33538E-01 the geometry corresponding to this parameter follows: media bias geometry description for those units utilized in this problem region num id ******************* global ******************* ----- unit 1 ----- 1 -zhemisphere 1 1 radius = 34.600 existing in the -z direction center=(0.00,0.00,0.00) chord=20.416 2 sphere 0 1 radius = 34.600 the center is located at ( 0.0000 , 0.0000 , 0.0000 ). 3 sphere 2 1 radius = 34.759 the center is located at ( 0.0000 , 0.0000 , 0.0000 ). **************************************************************************************************** **************************************************************************************************** Optimum concentration search ---------------------------- An optimum concentration search alters the concentration of the specified standard composition in the specified mixture to determine the highest value of *k*\ :sub:eff. The limits for an optimum concentration search are governed by the values entered for the parameter constraints. An optimum concentration search is activated by entering “OPTIMUM CONCENTRATION” in the search data. No defaulted search data are in a concentration search. The user must specify the material and standard composition name to be changed and the manner in which they will be changed. A concentration search is performed by altering the atom densities of the specified standard compositions in the specified materials. The ratio of how the standard compositions change relative to each other is controlled using FACTOR. If a material and standard composition is not listed in the search data it remains unchanged. The concentration search can vary from zero to some upper limit. The code will prevent the concentration from falling below zero, but the user is responsible for setting constraints that prevent the concentration from exceeding reasonable values. In most cases the theoretical density is a reasonable upper limit. Array of Spheres in H\ :sub:2\ O — Search on H\ :sub:2\ O Density ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Consider a 10 × 10 × 10 array of uranium spheres arranged in an array having a “square” pitch. The uranium spheres are 2 cm in radius, and the center-to-center spacing is 8 cm. The uranium spheres and their associated spacing are defined to be unit 1, and the 10 × 10 × 10 array is defined to be array 1. The spheres are initially moderated by ½ density water. An optimum concentration search is performed on the water yielding the maximum system *k*\ :sub:eff for various densities of water. In the MORE search data, MIX=2 and SCNAME=H2O specify that the water component of mixture 2 is to be altered during the search. The maximum allowed density is full density water. The minimum allowed density is 0.05 density water. The maximum constraint is calculated according to: .. math:: +\text{CON}= \frac {\left(\frac{\text{D}_{max}}{\text{D}_{initial}} - 1 \right)}{\text{FACTOR}} \ \ \ \ \ +\text{CON}= \frac{\left(\frac{1.0}{0.5} - 1 \right)}{1} = \frac{1}{0} The minimum constraint is calculated according to: .. math:: -\text{CON}= \frac {\left(\frac{\text{D}_{min}}{\text{D}_{initial}} - 1 \right)}{\text{FACTOR}} \ \ \ \ \ -\text{CON}= \frac{\left(\frac{0.05}{0.5} - 1 \right)}{1} = -0.9 The search data and final search results for this problem follow: :: =CSAS5S 10x10x10 ARRAY - CONCENTRATION SEARCH V7-238 READ COMP URANIUM 1 1.0 300.0 92235 90.0 92238 10.0 END H2O 2 0.5 300.0 END END COMP READ CELLDATA LATTICECELL SPHSQUAREP PITCH=8.0 2 FUELD=4.0 1 END END CELLDATA READ GEOMETRY UNIT 1 SPHERE 1 1 2.0 CUBOID 2 1 6P4.0 END GEOMETRY READ ARRAY ARA=1 NUX=10 NUY=10 NUZ=10 FILL F1 END FILL END ARRAY END DATA READ SEARCH OPTIMUM CONCENTRATION MORE ALTER MIX=2 SCNAME=H2O FACTOR=1.0 -CON=-0.9 +CON=1.0 END SEARCH END :: ******************* search pass 1 keff= 1.22228E+00 + or - 1.75549E-03 ****************** the parameter was 0.00000E+00 ***** modified keno v data has been rewritten on unit 95 ***** ******************* search pass 2 keff= 5.58399E-01 + or - 1.29164E-03 ****************** the parameter was -9.00000E-01 ***** modified keno v data has been rewritten on unit 95 ***** ******************* search pass 3 keff= 1.01673E+00 + or - 1.58517E-03 ****************** the parameter was 1.00000E+00 ***** modified keno v data has been rewritten on unit 95 ***** ******************* search pass 4 keff= 1.14752E+00 + or - 1.59590E-03 ****************** the parameter was 5.00000E-01 ***** modified keno v data has been rewritten on unit 95 ***** ******************* search pass 5 keff= 1.22423E+00 + or - 1.82022E-03 ****************** the parameter was 4.32687E-02 10x10x10 array - concentration search **************************************************************************************************** **************************************************************************************************** convergence was achieved on pass 5 the parameter was 4.32687E-02 the equation used in the search was: k-eff = +1.22278E+00 +5.50106E-02*p -5.25565E-01*p**2 +2.61295E-01*p**3 k-effective= 1.22423E+00 + or - 1.82022E-03 the corresponding geometry follows; 10x10x10 array - concentration search mixing table number of scattering angles = 2 cross section message threshold =5.7E-02 mixture = 1 density(g/cc) = 19.050 nuclide atom-dens. wgt. frac. za awt nuclide title 1092235 4.39277E-02 9.00000E-01 92235 235.0441 92u 235 bnl evalapr77 m.r.bhat mod3 02/28/89 1092238 4.81921E-03 1.00000E-01 92238 238.0510 92U 238 ANL+ EVALJUN77 E.PENNINGTONMOD3 02/13/92 mixture = 2 density(g/cc) = 0.52068 nuclide atom-dens. wgt. frac. za awt nuclide title 2001001 3.48291E-02 1.11926E-01 1001 1.0077 hydrogen in water 1301/1002 mod1 11/23/92 2008016 1.74146E-02 8.88073E-01 8016 15.9904 8O 16 from version 6 evaluation based on the preceding data, the best estimate of the parameter is 5.45542E-02 the mixing table corresponding to this parameter follows: 10x10x10 array - concentration search mixing table number of scattering angles = 2 cross section message threshold =5.7E-02 mixture = 1 density(g/cc) = 19.050 nuclide atom-dens. wgt. frac. za awt nuclide title 1092235 4.39277E-02 9.00000E-01 92235 235.0441 92u 235 bnl evalapr77 m.r.bhat mod3 02/28/89 1092238 4.81921E-03 1.00000E-01 92238 238.0510 92U 238 ANL+ EVALJUN77 E.PENNINGTON MOD3 02/13/92 mixture = 2 density(g/cc) = 0.52631 nuclide atom-dens. wgt. frac. za awt nuclide title 2001001 3.52059E-02 1.11927E-01 1001 1.0077 hydrogen in water 1301/1002 mod1 11/23/92 2008016 1.76029E-02 8.88073E-01 8016 15.9904 8O 16 from version 6 evaluation **************************************************************************************************** **************************************************************************************************** Array of Cylinders in H\ :sub:2\ O — Search on H\ :sub:2\ O Density ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ This problem contains search data for an optimum concentration search. The problem consists of a 10 × 10 × 1 array of uranium cylinders surrounded initially by 0.5 gm/cc density water. The cylinders are 90% :sup:235\ U and 10% :sup:238\ U and have a radius of 2.0 cm and a length of 200 cm. The cylinders are originally centered on an 8.0 cm X and Y pitch with interstitial low density (0.5 gm/cc) water. The problem searches for the H\ :sub:2\ O density that will produce the maximum system *k*\ :sub:eff. In the MORE search data, MIX=2 and SCNAME=H2O specify that the water component of mixture 2 is to be altered during the search. The maximum allowed density is full density water. The minimum allowed density is 0.05 density water. The maximum constraint is calculated according to: .. math:: +\text{CON}= \frac {\left(\frac{\text{D}_{max}}{\text{D}_{initial}} - 1 \right)}{\text{FACTOR}} \ \ \ \ \ +\text{CON}= \frac{\left(\frac{1.0}{0.5} - 1 \right)}{1} = -1.0 The minimum constraint is calculated according to: .. math:: -\text{CON}= \frac {\left(\frac{\text{D}_{min}}{\text{D}_{initial}} - 1 \right)}{\text{FACTOR}} \ \ \ \ \ -\text{CON}= \frac{\left(\frac{0.05}{0.5} - 1 \right)}{1} = -0.9 The search data and final search results for this problem follow: :: =CSAS5S 10x10x1 ARRAY CONCENTRATION SEARCH - CHANGE H2O V7-238 READ COMP URANIUM 1 1.0 300.0 92235 90.0 92238 10.0 END H2O 2 0.5 300.0 END END COMP READ CELLDATA LATTICECELL SQUAREPITCH PITCH=8.0 2 FUELD=4.0 1 END END CELLDATA READ GEOMETRY UNIT 1 CYLINDER 1 1 2.0 100.0 -100.0 CUBOID 2 1 4p4.0 2p100.0 END GEOMETRY READ ARRAY ARA=1 NUX=10 NUY=10 NUZ=1 FILL F1 END FILL END ARRAY END DATA READ SEARCH OPTIMUM CONCENTRATION MORE ALTER MIX=2 SCNAME=H2O FACTOR=1.0 -CON=-0.9 +CON=1.0 END SEARCH END :: ******************* search pass 1 keff= 1.63537E+00 + or - 1.70164E-03 ****************** the parameter was 0.00000E+00 ***** modified keno v data has been rewritten on unit 95 ***** ******************* search pass 2 keff= 1.26936E+00 + or - 1.79640E-03 ****************** the parameter was -9.00000E-01 ***** modified keno v data has been rewritten on unit 95 ***** ******************* search pass 3 keff= 1.51475E+00 + or - 1.50298E-03 ****************** the parameter was 1.00000E+00 ***** modified keno v data has been rewritten on unit 95 ***** ******************* search pass 4 keff= 1.61045E+00 + or - 1.48777E-03 ****************** the parameter was 5.00000E-01 ***** modified keno v data has been rewritten on unit 95 ***** ******************* search pass 5 keff= 1.63831E+00 + or - 1.43843E-03 ****************** the parameter was 1.29491E-01 10x10x10 array concentration search - change h2o **************************************************************************************************** **************************************************************************************************** convergence was achieved on pass 5 the parameter was 1.29491E-01 the equation used in the search was: k-eff = +1.63481E+00 +6.23193E-02*p -2.85686E-01*p**2 +1.05562E-01*p**3 k-effective= 1.63831E+00 + or - 1.43843E-03 the corresponding geometry follows; 10x10x10 array concentration search - change h2o mixing table number of scattering angles = 2 cross section message threshold =5.7E-02 mixture = 1 density(g/cc) = 19.050 nuclide atom-dens. wgt. frac. za awt nuclide title 1092235 4.39277E-02 9.00000E-01 92235 235.0441 92u 235 bnl evalapr77 m.r.bhat mod3 02/28/89 1092238 4.81921E-03 1.00000E-01 92238 238.0510 92U 238 ANL+ EVALJUN77 E.PENNINGTON MOD3 02/13/92 mixture = 2 density(g/cc) = 0.56371 nuclide atom-dens. wgt. frac. za awt nuclide title 2001001 3.77076E-02 1.11926E-01 1001 1.0077 hydrogen in water 1301/1002 mod1 11/23/92 2008016 1.88538E-02 8.88073E-01 8016 15.9904 8O 16 from version 6 evaluation based on the preceding data, the best estimate of the parameter is 1.16606E-01 the mixing table corresponding to this parameter follows: 10x10x10 array concentration search - change h2o mixing table number of scattering angles = 2 cross section message threshold =5.7E-02 mixture = 1 density(g/cc) = 19.050 nuclide atom-dens. wgt. frac. za awt nuclide title 1092235 4.39277E-02 9.00000E-01 92235 235.0441 92u 235 bnl evalapr77 m.r.bhat mod3 02/28/89 1092238 4.81921E-03 1.00000E-01 92238 238.0510 92U 238 ANL+ EVALJUN77 E.PENNINGTON MOD3 02/13/92 mixture = 2 density(g/cc) = 0.55728 nuclide atom-dens. wgt. frac. za awt nuclide title 2001001 3.72774E-02 1.11927E-01 1001 1.0077 hydrogen in water 1301/1002 mod1 11/23/92 2008016 1.86387E-02 8.88074E-01 8016 15.9904 8O 16 from version 6 evaluation **************************************************************************************************** **************************************************************************************************** Cell-Weighted Array of Uranium Cylinders in H\ :sub:2\ O — Search on H\ :sub:2\ O Density ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ This problem contains search data for an optimum concentration search. The problem consists of a single material representing a 10 × 10 × 1 array of uranium cylinders surrounded initially by 0.5 gm/cc density water. The cylinders are 90% \ :sup:235\ U and 10% \ :sup:238\ U and have a radius of 2.0 cm and a length of 200 cm. The cylinders are originally centered on an 8.0 cm X and Y-pitch with interstitial low density (0.5 gm/cc) water. The keyword CELL=100 in the unit cell data indicates this will be a cell-weighted problem with mixture 100 representing the array of uranium rods in water. The problem searches for the H\ :sub:2\ O density that will produce the maximum system *k*\ :sub:eff. In the MORE search data, MIX=2 and SCNAME=H\ :sub:2\ O specify that the water component of mixture 2 is to be altered during the search. On each pass the water density is altered, XSDRNPM is run to produce a new mixture 100, and mixture 100 is used in KENO V.a to calculate a new system *k*\ :sub:eff. The cylinder and cuboid in unit 1 and the array contained in the above problem are replaced by a cuboid of the same size as the array containing mixture 100. The maximum allowed density is full density water. The minimum allowed density is 0.05 density water. The maximum constraint is calculated according to: .. math:: +\text{CON}= \frac {\left(\frac{\text{D}_{max}}{\text{D}_{initial}} - 1 \right)}{\text{FACTOR}} \ \ \ \ \ +\text{CON}= \frac{\left(\frac{1.0}{0.5} - 1 \right)}{1} = -1.0 The minimum constraint is calculated according to: .. math:: -\text{CON}= \frac {\left(\frac{\text{D}_{min}}{\text{D}_{initial}} - 1 \right)}{\text{FACTOR}} \ \ \ \ \ -\text{CON}= \frac{\left(\frac{0.05}{0.5} - 1 \right)}{1} = -0.9 The search data and final search results for this problem follow: :: =CSAS5S 10x10x1 ARRAY CONCENTRATION SEARCH - CHANGE H2O V7-238 READ COMP URANIUM 1 1.0 300.0 92235 90.0 92238 10.0 END H2O 2 0.5 300.0 END END COMP READ CELLDATA LATTICECELL CELL=100 SQUAREPITCH PITCH=8.0 2 FUELD=4.0 1 END END CELLDATA READ GEOMETRY UNIT 1 CUBOID 100 1 4p40.0 2p100.0 END GEOMETRY END DATA READ SEARCH OPTIMUM CONCENTRATION MORE ALTER MIX=2 SCNAME=H2O FACTOR=1.0 -CON=-0.9 +CON=1.0 END SEARCH END :: ******************* search pass 1 keff= 1.63905E+00 + or - 1.69932E-03 ****************** the parameter was 0.00000E+00 **** xsdrnpm mesh intervals **** 20 mesh intervals in zone 1 4 mesh intervals in zone 2 ***** modified keno v data has been rewritten on unit 95 ***** ******************* search pass 2 keff= 1.28883E+00 + or - 1.67697E-03 ****************** the parameter was -9.00000E-01 **** xsdrnpm mesh intervals **** 20 mesh intervals in zone 1 20 mesh intervals in zone 2 ***** modified keno v data has been rewritten on unit 95 ***** ******************* search pass 3 keff= 1.51930E+00 + or - 1.21366E-03 ****************** the parameter was 1.00000E+00 **** xsdrnpm mesh intervals **** 20 mesh intervals in zone 1 20 mesh intervals in zone 2 ***** modified keno v data has been rewritten on unit 95 ***** ******************* search pass 4 keff= 1.61577E+00 + or - 1.47397E-03 ****************** the parameter was 5.00000E-01 0 **** xsdrnpm mesh intervals **** 20 mesh intervals in zone 1 20 mesh intervals in zone 2 ***** modified keno v data has been rewritten on unit 95 ***** ******************* search pass 5 keff= 1.64316E+00 + or - 1.70752E-03 ****************** the parameter was 1.35197E-01 10x10x10 array concentration search - change h2o **************************************************************************************************** **************************************************************************************************** convergence was achieved on pass 5 the parameter was 1.35197E-01 the equation used in the search was: k-eff = +1.63882E+00 +6.77742E-02*p -2.75913E-01*p**2 +8.93705E-02*p**3 k-effective= 1.64316E+00 + or - 1.70752E-03 the corresponding geometry follows; 10x10x10 array concentration search - change h2o mixing table number of scattering angles = 2 cross section message threshold =5.7E-02 mixture = 1 density(g/cc) = 19.050 nuclide atom-dens. wgt. frac. za awt nuclide title 1092235 4.39277E-02 9.00000E-01 92235 235.0441 92u 235 bnl evalapr77 m.r.bhat mod3 02/28/89 1092238 4.81921E-03 1.00000E-01 92238 238.0510 92U 238 ANL+ EVALJUN77 E.PENNINGTON MOD3 02/13/92 mixture = 2 density(g/cc) = 0.56656 nuclide atom-dens. wgt. frac. za awt nuclide title 2001001 3.78981E-02 1.11926E-01 1001 1.0077 hydrogen in water 1301/1002 mod1 11/23/92 2008016 1.89490E-02 8.88073E-01 8016 15.9904 8O 16 from version 6 evaluation mixture = 100 density(g/cc) = 0.0000 nuclide atom-dens. wgt. frac. za awt nuclide title 2001001 0.00000E+00 0.00000E+00 1001 1.0077 hydrogen in water 1301/1002 mod1 11/23/92 2008016 0.00000E+00 0.00000E+00 8016 15.9904 8O 16 from version 6 evaluation 1092235 0.00000E+00 0.00000E+00 92235 235.0441 92u 235 bnl evalapr77 m.r.bhat mod3 02/28/89 1092238 0.00000E+00 0.00000E+00 92238 238.0510 92U 238 ANL+ EVALJUN77 E.PENNINGTON MOD3 02/13/92 based on the preceding data, the best estimate of the parameter is 1.31179E-01 the mixing table corresponding to this parameter follows: 10x10x10 array concentration search - change h2o mixing table number of scattering angles = 2 cross section message threshold =5.7E-02 mixture = 1 density(g/cc) = 19.050 nuclide atom-dens. wgt. frac. za awt nuclide title 1092235 4.39277E-02 9.00000E-01 92235 235.0441 92u 235 bnl evalapr77 m.r.bhat mod3 02/28/89 1092238 4.81921E-03 1.00000E-01 92238 238.0510 92U 238 ANL+ EVALJUN77 E.PENNINGTON MOD3 02/13/92 mixture = 2 density(g/cc) = 0.56455 nuclide atom-dens. wgt. frac. za awt nuclide title 2001001 3.77639E-02 1.11927E-01 1001 1.0077 hydrogen in water 1301/1002 mod1 11/23/92 2008016 1.88820E-02 8.88074E-01 8016 15.9904 8O 16 from version 6 evaluation mixture = 100 density(g/cc) = 0.0000 nuclide atom-dens. wgt. frac. za awt nuclide title 2001001 0.00000E+00 0.00000E+00 1001 1.0077 hydrogen in water 1301/1002 mod 11/23/92 2008016 0.00000E+00 0.00000E+00 8016 15.9904 8O 16 from version 6 evaluation 1092235 0.00000E+00 0.00000E+00 92235 235.0441 92u 235 bnl evalapr77 m.r.bhat mod3 02/28/89 1092238 0.00000E+00 0.00000E+00 92238 238.0510 92U 238 ANL+ EVALJUN77 E.PENNINGTON A. MOD3 02/13/92 **************************************************************************************************** **************************************************************************************************** Minimum concentration search ---------------------------- A minimum concentration search searches for the standard composition density that yields the lowest value of *k*\ :sub:eff. A minimum concentration search is activated by entering “MINIMUM CONCENTRATION” in the search data. There are no defaulted search data in a concentration search. The user must specify the material and standard composition name to be changed and the manner in which they will be changed as described in the auxiliary search commands. A concentration search is performed by altering the atom densities of the specified standard compositions in the specified materials. The ratio of how the standard compositions change relative to each other is controlled using FACTOR. If a material and standard composition is not listed in the search data it remains unchanged. The concentration search can vary from zero to some upper limit. The code will prevent the concentration from falling below zero, but the user is responsible for setting constraints that prevent the concentration from exceeding reasonable values. In most cases the theoretical density is a reasonable upper limit. Infinite Array of Fuel Bundles Separated by Flux Traps — Search on H\ :sub:2\ O Density in Trap ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ The fuel bundles in this problem represent 17 × 17 PWR fuel assemblies. The fuel pins are smeared together, making a mixture 100. The fuel pins consist of 2.35 wt % :sup:235\ U having a diameter of 0.823 cm, zirconium cladding having an outer diameter of 0.9627 cm, and a pitch of 1.275 cm. The fuel bundle is represented as a 10.8375 cm × 10.8375 cm × 366 cm cuboid of mixture 100 surrounded by Boral and then water. The Boral has a density of 2.65 g/cm\ :sup:3 and is composed of 35.17 wt % B\ :sub:4\ C and 64.83 wt % Al. The fuel bundles are at a fixed pitch of 13.0 cm. This is the same problem as described for a minimum dimension search above with the water component of the trap size optimized, thus producing a minimum system *k*\ :sub:eff. The input data and the final search results for this problem follow: :: =csas5s array of fuel bundles with flux trap v7-238 read comp uo2 1 .84 300. 92235 2.35 92238 97.65 end zr 2 1 end h2o 3 1 end b4c 4 den=2.65 0.3517 end al 4 den=2.65 0.6483 end h2o 5 0.5 end end comp read celldata latticecell squarepitch pitch=1.275 3 fueld=0.823 1 cladd=0.9627 2 cellmix=100 end end celldata read param far=yes gen=203 npg=1000 end param read geom cuboid 100 1 4p10.8375 2p183.0 cuboid 4 1 4p11.0 2p183.0 cuboid 5 1 4p13.0 2p183.0 end geom read bounds xfc=mirror yfc=mirror end bounds end data read search minimum concentration more alter mix=5 scname=h2o factor=1.0 -con=-0.9 +con=1.0 end search end :: ******************* search pass 1 keff= 7.70310E-01 + or - 1.31009E-03 ****************** the parameter was 0.00000E+00 **** xsdrnpm mesh intervals **** 4 mesh intervals in zone 1 4 mesh intervals in zone 2 14 mesh intervals in zone 3 ***** modified keno v data has been rewritten on unit 95 ***** ******************* search pass 2 keff= 8.62695E-01 + or - 1.43366E-03 ****************** the parameter was -9.00000E-01 **** xsdrnpm mesh intervals **** 4 mesh intervals in zone 1 4 mesh intervals in zone 2 14 mesh intervals in zone 3 ***** modified keno v data has been rewritten on unit 95 ***** ******************* search pass 3 keff= 7.04284E-01 + or - 1.44875E-03 ****************** the parameter was 1.00000E+00 **** xsdrnpm mesh intervals **** 4 mesh intervals in zone 1 4 mesh intervals in zone 2 14 mesh intervals in zone 3 ***** modified keno v data has been rewritten on unit 95 ***** ******************* search pass 4 keff= 7.36291E-01 + or - 1.54213E-03 ****************** the parameter was 5.00000E-01 array of fuel bundles with flux trap **************************************************************************************************** **************************************************************************************************** convergence was achieved on pass 3 the parameter was 1.00000E+00 the equation used in the search was: k-eff = +7.70310E-01 -7.54970E-02*p +2.03650E-02*p**2 -1.08939E-02*p**3 k-effective= 7.04284E-01 + or - 1.44875E-03 the corresponding geometry follows; array of fuel bundles with flux trap mixing table number of scattering angles = 2 cross section message threshold =5.7E-02 mixture = 1 density(g/cc) = 9.2064 nuclide atom-dens. wgt. frac. za awt nuclide title 1008016 4.10743E-02 1.18465E-01 8016 15.9904 8O 16 from version 6 evaluation 1092235 4.88650E-04 2.07161E-02 92235 235.0441 92u 235 bnl evalapr77 m.r.bhat mod3 02/28/89 1092238 2.00485E-02 8.60819E-01 92238 238.0510 92U 238 ANL+ EVALJUN77 E.PENNINGTON MOD3 02/13/92 mixture = 2 density(g/cc) = 6.4900 nuclide atom-dens. wgt. frac. za awt nuclide title 2040000 4.28457E-02 1.00000E+00 40000 91.2196 40zr sai evalapr76 m.drake d.sa mod2 01/03/89 mixture = 3 density(g/cc) = 0.99817 nuclide atom-dens. wgt. frac. za awt nuclide title 3001001 6.67692E-02 1.11927E-01 1001 1.0077 hydrogen in water 1301/1002 mod1 11/23/92 3008016 3.33846E-02 8.88074E-01 8016 15.9904 8O 16 from version 6 evaluation mixture = 4 density(g/cc) = 2.6500 nuclide atom-dens. wgt. frac. za awt nuclide title 4005010 8.08716E-03 5.07413E-02 5010 10.0130 5b 100lasl evaldec76 g.hale mod1 12/11/92 free gas 4005011 3.25519E-02 2.24568E-01 5011 11.0096 5b11 gebnl evalsep71 cowan mod1 12/11/92 free gas 4006012 1.01598E-02 7.63956E-02 6000 12.0001 6c ornl evaldec73 c.y.fu mod2 12/11/92 free gas 4013027 3.83444E-02 6.48295E-01 13027 26.9818 13al 270lasl evaldec73 p.g. young mod1 11/29/88 mixture = 5 density(g/cc) = 0.99817 nuclide atom-dens. wgt. frac. za awt nuclide title 5001001 6.67692E-02 1.11927E-01 1001 1.0077 hydrogen in water 1301/1002 mod1 11/23/92 5008016 3.33846E-02 8.88074E-01 8016 15.9904 8O 16 from version 6 evaluation mixture = 100 density(g/cc) = 0.0000 nuclide atom-dens. wgt. frac. za awt nuclide title 3001001 0.00000E+00 0.00000E+00 1001 1.0077 hydrogen in water 1301/1002 mod1 11/23/92 3008016 0.00000E+00 0.00000E+00 8016 15.9904 8O 16 from version 6 evaluation 1008016 0.00000E+00 0.00000E+00 8016 15.9904 8O 16 from version 6 evaluation 2040000 0.00000E+00 0.00000E+00 40000 91.2196 40zr sai evalapr76 m.drake d.sa mod2 01/03/89 1092235 0.00000E+00 0.00000E+00 92235 235.0441 92u 235 bnl evalapr77 m.r.bhat mod3 02/28/89 1092238 0.00000E+00 0.00000E+00 92238 238.0510 92U 238 ANL+ EVALJUN77 E.PENNINGTON MOD3 02/13/92 Critical concentration search ----------------------------- A critical concentration search alters the concentration of the specified standard composition in the specified mixture to obtain a specified value of *k*\ :sub:eff. A critical concentration search is activated by entering “CRITICAL CONCENTRATION” in the search data. There are no defaulted search data in a critical concentration search except for the value of *k*\ :sub:eff. If something other than *k*\ :sub:eff = 1.0 is desired the user must specify KEF=**, where \*\* is the desired value of *k*\ :sub:eff. The remaining data is entered after the keyword MORE in the search data block. The user must specify the material and standard composition name to be changed and the manner in which they will be changed as described in the auxiliary search commands. A concentration search is performed by altering the atom densities of the specified standard compositions in the specified materials. The ratio of how the standard compositions change relative to each other is controlled using FACTOR. If a material and standard composition is not listed in the search data it remains unchanged. The concentration search can vary from zero to some upper limit. The code will prevent the concentration from falling below zero, but the user is responsible for setting constraints that prevent the concentration from exceeding reasonable values. In most cases the theoretical density is a reasonable upper limit. Array of Spheres in H\ :sub:2\ O — Search on H\ :sub:2\ O Density ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Consider a 10 × 10 × 10 array of uranium spheres arranged in an array having a “square” pitch. The uranium spheres are 2 cm in radius, and the center-to-center spacing is 8 cm. The uranium spheres and their associated spacing are defined to be unit 1, and the 10 × 10 × 10 array is defined to be array 1.0 cm. The spheres are initially moderated by ½ density water. A critical concentration search is performed on the water yielding system *k*\ :sub:eff = 1.0 for various densities of water. In the MORE search data, MIX=2 and SCNAME=H2O specify that the water component of mixture 2 is to be altered during the search from an initial density of 0.5 gm/cc. The maximum allowed density is full density water (1.0 gm/cc). The minimum allowed water density is 0.0005 gm/cc. The maximum constraint is calculated according to: +CON= (1.0/0.5 − 1)/1.0 = 1.0 The minimum constraint is calculated according to: −CON= (0.0005/0.5 − 1)/1.0 = −0.999 The search data and final search results for this problem follow: :: =CSAS5S 10x10x10 ARRAY - CONCENTRATION SEARCH V7-238 READ COMP URANIUM 1 1.0 300.0 92235 90.0 92238 10.0 END H2O 2 0.5 300.0 END END COMP READ CELLDATA LATTICECELL SPHSQUAREP PITCH=8.0 2 FUELD=4.0 1 END END CELLDATA READ GEOMETRY UNIT 1 SPHERE 1 1 2.0 CUBOID 2 1 6P4.0 END GEOMETRY READ ARRAY ARA=1 NUX=10 NUY=10 NUZ=10 FILL F1 END FILL END ARRAY END DATA READ SEARCH CRITICAL CONCENTRATION KEF=1.0 MORE ALTER MIX=2 SCNAME=H2O factor=1.0 -CON=-0.999 +CON=1.0 END SEARCH END :: ******************* search pass 1 keff= 1.22228E+00 + or - 1.75549E-03 ****************** the parameter was 0.00000E+00 ***** modified keno v data has been rewritten on unit 95 ***** ******************* search pass 2 keff= 4.97499E-01 + or - 1.19161E-03 ****************** the parameter was -9.99000E-01 ***** modified keno v data has been rewritten on unit 95 ***** ******************* search pass 3 keff= 1.15356E+00 + or - 1.85214E-03 ****************** the parameter was -3.06377E-01 ***** modified keno v data has been rewritten on unit 95 ***** ******************* search pass 4 keff= 9.72210E-01 + or - 1.89881E-03 ****************** the parameter was -5.52436E-01 ***** modified keno v data has been rewritten on unit 95 ***** ******************* search pass 5 keff= 1.00805E+00 + or - 1.91023E-03 ****************** the parameter was -5.21036E-01 ***** modified keno v data has been rewritten on unit 95 ***** ******************* search pass 6 keff= 1.00056E+00 + or - 2.05463E-03 ****************** the parameter was -5.28450E-01 10x10x10 array - concentration search **************************************************************************************************** **************************************************************************************************** convergence was achieved on pass 6 the parameter was -5.28450E-01 the equation used in the search was: k-eff = +1.19891E+00 -5.90957E-01*p -2.41575E+00*p**2 -1.11098E+00*p**3 k-effective= 1.00056E+00 + or - 2.05463E-03 the corresponding geometry follows; 10x10x10 array - concentration search mixing table number of scattering angles = 2 cross section message threshold =5.7E-02 mixture = 1 density(g/cc) = 19.050 nuclide atom-dens. wgt. frac. za awt nuclide title 1092235 4.39277E-02 9.00000E-01 92235 235.0441 92u 235 bnl evalapr77 m.r.bhat mod3 02/28/89 1092238 4.81921E-03 1.00000E-01 92238 238.0510 92U 238 ANL+ EVALJUN77 E.PENNINGTON MOD3 02/13/92 mixture = 2 density(g/cc) = 0.23534 nuclide atom-dens. wgt. frac. za awt nuclide title 2001001 1.57425E-02 1.11926E-01 1001 1.0077 hydrogen in water 1301/1002 mod1 11/23/92 2008016 7.87126E-03 8.88074E-01 8016 15.9904 8O 16 from version 6 evaluation based on the preceding data, the best estimate of the parameter is -5.28963E-01 the mixing table corresponding to this parameter follows: 10x10x10 array - concentration search mixing table number of scattering angles = 2 cross section message threshold =5.7E-02 mixture = 1 density(g/cc) = 19.050 nuclide atom-dens. wgt. frac. za awt nuclide title 1092235 4.39277E-02 9.00000E-01 92235 235.0441 92u 235 bnl evalapr77 m.r.bhat mod3 02/28/89 1092238 4.81921E-03 1.00000E-01 92238 238.0510 92U 238 ANL+ EVALJUN77 E.PENNINGTON MOD3 02/13/92 mixture = 2 density(g/cc) = 0.23509 nuclide atom-dens. wgt. frac. za awt nuclide title 2001001 1.57254E-02 1.11927E-01 1001 1.0077 hydrogen in water 1301/1002 mod1 11/23/92 2008016 7.86270E-03 8.88074E-01 8016 15.9904 8O 16 from version 6 evaluation **************************************************************************************************** **************************************************************************************************** Array of Spheres in H\ :sub:2\ O — Search on H\ :sub:2\ O Density ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ This is the same problem as described above with the exception of the initial water density, which is now 0.25 gm/cc. A critical concentration search is performed on the water yielding a system *k*\ :sub:eff = 1.0 for various densities of water. In the MORE search data, MIX=2 and SCNAME=H2O specify that the water component of mixture 2 is to be altered during the search from an initial value of 0.25 gm/cc. The maximum allowed density is ½ density water (0.5 gm/cc). The minimum allowed water density is 0.1 gm/cc. The maximum constraint is calculated according to: +CON= (5.0/0.25 − 1)/1.0 = 1.0 The minimum constraint is calculated according to: −CON= (0.1/0.25 − 1)/1.0 = − 0.6 The search data and final search results for this problem follow: :: =CSAS5S 10x10x10 ARRAY - CONCENTRATION SEARCH V7-238 READ COMP URANIUM 1 1.0 300.0 92235 90.0 92238 10.0 END H2O 2 0.25 300.0 END END COMP READ CELLDATA LATTICECELL SPHSQUAREP PITCH=8.0 2 FUELD=4.0 1 END END CELLDATA READ GEOMETRY UNIT 1 SPHERE 1 1 2.0 CUBOID 2 1 6P4.0 END GEOMETRY READ ARRAY ARA=1 NUX=10 NUY=10 NUZ=10 FILL F1 END FILL END ARRAY END DATA READ SEARCH CRITICAL CONCENTRATION KEF=1.0 MORE ALTER MIX=2 SCNAME=H2O factor=1.0 -CON=-0.6 +CON=1.0 END SEARCH END :: ******************* search pass 1 keff= 1.02599E+00 + or - 1.98346E-03 ****************** the parameter was 0.00000E+00 ***** modified keno v data has been rewritten on unit 95 ***** ******************* search pass 2 keff= 6.69203E-01 + or - 1.55844E-03 ****************** the parameter was -6.00000E-01 ***** modified keno v data has been rewritten on unit 95 ***** ******************* search pass 3 keff= 1.00197E+00 + or - 1.73659E-03 ****************** the parameter was -4.37073E-02 10x10x10 array - concentration search **************************************************************************************************** **************************************************************************************************** convergence was achieved on pass 3 the parameter was -4.37073E-02 the equation used in the search was: k-eff = +0.00000E+00 +0.00000E+00*p +0.00000E+00*p**2 +0.00000E+00*p**3 k-effective= 1.00197E+00 + or - 1.73659E-03 the corresponding geometry follows; 10x10x10 array - concentration search mixing table number of scattering angles = 2 cross section message threshold =5.7E-02 mixture = 1 density(g/cc) = 19.050 nuclide atom-dens. wgt. frac. za awt nuclide title 1092235 4.39277E-02 9.00000E-01 92235 235.0441 92u 235 bnl evalapr77 m.r.bhat mod3 02/28/89 1092238 4.81921E-03 1.00000E-01 92238 238.0510 92U 238 ANL+ EVALJUN77 E.PENNINGTON MOD3 02/13/92 mixture = 2 density(g/cc) = 0.23864 nuclide atom-dens. wgt. frac. za awt nuclide title 2001001 1.59627E-02 1.11926E-01 1001 1.0077 hydrogen in water 1301/1002 mod1 11/23/92 2008016 7.98136E-03 8.88074E-01 8016 15.9904 8O 16 from version 6 evaluation based on the preceding data, the best estimate of the parameter is -4.37073E-02 the mixing table corresponding to this parameter follows: 10x10x10 array - concentration search mixing table number of scattering angles = 2 cross section message threshold =5.7E-02 mixture = 1 density(g/cc) = 19.050 nuclide atom-dens. wgt. frac. za awt nuclide title 1092235 4.39277E-02 9.00000E-01 92235 235.0441 92u 235 bnl evalapr77 m.r.bhat mod3 02/28/89 1092238 4.81921E-03 1.00000E-01 92238 238.0510 92U 238 ANL+ EVALJUN77 E.PENNINGTON MOD3 02/13/92 mixture = 2 density(g/cc) = 0.23864 nuclide atom-dens. wgt. frac. za awt nuclide title 2001001 1.59627E-02 1.11926E-01 1001 1.0077 hydrogen in water 1301/1002 mod1 11/23/92 2008016 7.98136E-03 8.88074E-01 8016 15.9904 8O 16 from version 6 evaluation **************************************************************************************************** **************************************************************************************************** UO\ :sub:2\ F\ :sub:2 Solution Tank — Critical Search on UO\ :sub:2\ F\ :sub:2 Density ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Consider a large spherical tank partially filled with UO\ :sub:2\ F\ :sub:2 solution. The tank has a radius of 34.6 cm and is filled with solution to a height of 30.0 cm above the midpoint. The tank is composed of a 0.759 cm thick Al shell. The UO\ :sub:2\ F\ :sub:2 solution is composed of three standard compositions: UO2F2, HF acid, and H2O. The code combines these using a set algorithm. This may or may not produce a solution at the desired density. If the density of the solution is known it should be entered. Also, extra acid can be added to the solution by specifying a non-zero acid molarity. A critical concentration search is performed on the water yielding system *k*\ :sub:eff = 1.0 for various densities of UO\ :sub:2\ F\ :sub:2 in the solution. In the MORE search data, MIX=1 and SCNAME=UO2F2 specify that the UO\ :sub:2\ F\ :sub:2 component of the mixture 1 solution is to be altered during the search. The code calculates the density of the solution. The initial uranium fuel density is 300 gm/liter. The maximum allowed uranium density is 600 gm/liter. The minimum allowed uranium density is 150 gm/liter. The maximum constraint is calculated according to: +CON= (600/300 − 1)/1.0 = 1.0 The minimum constraint is calculated according to: −CON= (150/300 − 1)/1.0 = −0.5 The search data and final search results for this problem follow: :: =CSAS5S SOLUTION TANK - CRITICAL CONCENTRATION SEARCH V7-238 READ COMP SOLNUO2F2 1 300 0 1 300.0 92235 4.89 92238 95.09 92234 0.02 END AL 2 1.0 300.0 END END COMP READ GEOM HEMISPHE-Z 1 1 34.6 CHORD 30. SPHERE 0 1 34.6 SPHERE 2 1 34.759 END GEOM END DATA READ SEARCH CRITICAL CONCENTRATION KEF=1.0 MORE ALTER MIX=1 SCNAME=UO2F2 FACTOR=1.0 -CON=-0.5 +CON=1.0 END SEARCH END :: ******************* search pass 1 keff= 8.31461E-01 + or - 1.14626E-03 ****************** the parameter was 0.00000E+00 ***** modified keno v data has been rewritten on unit 95 ***** ******************* search pass 2 keff= 1.05426E+00 + or - 1.63141E-03 ****************** the parameter was 1.00000E+00 ***** modified keno v data has been rewritten on unit 95 ***** ******************* search pass 3 keff= 1.01773E+00 + or - 1.42881E-03 ****************** the parameter was 7.56451E-01 ***** modified keno v data has been rewritten on unit 95 ***** ******************* search pass 4 keff= 9.97887E-01 + or - 1.42235E-03 ****************** the parameter was 6.59454E-01 solution tank - critical concentration search **************************************************************************************************** **************************************************************************************************** convergence was achieved on pass 4 the parameter was 6.59454E-01 the equation used in the search was: k-eff = +8.31461E-01 +2.45656E-01*p +7.41484E-02*p**2 -9.70021E-02*p**3 k-effective= 9.97887E-01 + or - 1.42235E-03 the corresponding geometry follows; solution tank - critical concentration search mixing table number of scattering angles = 2 cross section message threshold =5.7E-02 mixture = 1 density(g/cc) = 1.5960 nuclide atom-dens. wgt. frac. za awt nuclide title 1001001 6.36618E-02 6.67438E-02 1001 1.0077 hydrogen in water 1301/1002 mod1 11/23/92 1008016 3.43513E-02 5.71507E-01 8016 15.9904 8O 16 from version 6 evaluation 1009019 2.52040E-03 4.98199E-02 9019 18.9982 9f 19 ornl evaljul74 c.y.fu d.c.lars mod3 12/16/88 1092234 2.56198E-07 6.23860E-05 92234 234.0405 92U234 BNL HEDL+EVALJUL78 DIVADEENAM MOD3 01/10/91 1092235 6.23730E-05 1.52534E-02 92235 235.0441 92u 235 bnl evalapr77 m.r.bhat mod3 02/28/89 1092238 1.19757E-03 2.96614E-01 92238 238.0510 92U 238 ANL+ EVALJUN77 E.PENNINGTON MOD3 02/13/92 mixture = 2 density(g/cc) = 2.7020 nuclide atom-dens. wgt. frac. za awt nuclide title 2013027 6.03066E-02 1.00000E+00 13027 26.9818 13al 270lasl evaldec73 p.g. young d.g mod1 11/29/88 based on the preceding data, the best estimate of the parameter is 6.69249E-01 the mixing table corresponding to this parameter follows: solution tank - critical concentration search mixing table number of scattering angles = 2 cross section message threshold =5.7E-02 mixture = 1 density(g/cc) = 1.5998 nuclide atom-dens. wgt. frac. za awt nuclide title 1001001 6.36618E-02 6.65852E-02 1001 1.0077 hydrogen in water 1301/1002 mod1 11/23/92 1008016 3.43662E-02 5.70395E-01 8016 15.9904 8O 16 from version 6 evaluation 1009019 2.53528E-03 4.99948E-02 9019 18.9982 9f 19 ornl evaljul74 c.y.fu d.c.lars mod3 12/16/88 1092234 2.57710E-07 6.26050E-05 92234 234.0405 92U 234 BNL HEDL+EVALJUL78 DIVADEENAM MOD3 01/10/91 1092235 6.27412E-05 1.53069E-02 92235 235.0441 92u 235 bnl evalapr77 m.r.bhat mod3 02/28/89 1092238 1.20464E-03 2.97655E-01 92238 238.0510 92U 238 ANL+ EVALJUN77 E.PENNINGTON MOD3 02/13/92 mixture = 2 density(g/cc) = 2.7020 nuclide atom-dens. wgt. frac. za awt nuclide title 2013027 6.03066E-02 1.00000E+00 13027 26.9818 13al 270lasl evaldec73 p.g. young d.g mod1 11/29/88 **************************************************************************************************** **************************************************************************************************** The final solution for this search contains 500 gm/liter of uranium. The search, however, did not change the amount of acid or water in the solution. To get a better estimation of the solution density and *k*\ :sub:eff for this concentration of uranium the problem should be run again with an initial uranium density of 500 gm/liter as shown below. UO\ :sub:2\ F\ :sub:2 Solution Tank — Critical Search on UO\ :sub:2\ F\ :sub:2 Density, Check ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ This is the same UO\ :sub:2\ F\ :sub:2 solution tank problem examined above except the initial uranium density is 500 gm/liter. This problem is run again so the code will calculate water and acid densities associated with this density of uranium. The search data has also been modified to account for the new density of uranium. A critical concentration search is performed on the water yielding system *k*\ :sub:eff = 1.0 for various densities of UO\ :sub:2\ F\ :sub:2 in the solution. In the MORE search data, MIX=1 and SCNAME=UO2F2 specify that the UO\ :sub:2\ F\ :sub:2 component of the mixture 1 solution is to be altered during the search. The code calculates the density of the solution. The initial uranium fuel density is 500 gm/liter. The maximum allowed uranium density is 600 gm/liter. The minimum allowed uranium density is 400 gm/liter. The maximum constraint is calculated according to: +CON= (600/500 − 1)/1.0 = 0.2 The minimum constraint is calculated according to: −CON= (400/500 − 1)/1.0 = −0.2 The search data and final search results for this problem follow: :: =csas5s solution tank - critical concentration search v7-238 read comp solnuo2f2 1 500 0 1 300.0 92235 4.89 92238 95.09 92234 0.02 end al 2 1.0 300.0 end end comp read geom hemisphe-z 1 1 34.6 chord 30. sphere 0 1 34.6 sphere 2 1 34.759 end geom end data READ SEARCH CRITICAL CONCENTRATION KEF=1.0 MORE ALTER MIX=1 SCNAME=uo2f2 factor=1.0 -CON=-0.2 +CON=0.2 END SEARCH end :: ******************* search pass 1 keff= 9.99662E-01 + or - 1.40567E-03 ****************** the parameter was 0.00000E+00 solution tank - critical concentration search **************************************************************************************************** convergence was achieved on pass 1 the parameter was 0.00000E+00 the equation used in the search was: k-eff = +0.00000E+00 +0.00000E+00*p +0.00000E+00*p**2 +0.00000E+00*p**3 k-effective= 9.99662E-01 + or - 1.40567E-03 the corresponding geometry follows; solution tank - critical concentration search mixing table number of scattering angles = 2 cross section message threshold =5.7E-02 mixture = 1 density(g/cc) = 1.5656 nuclide atom-dens. wgt. frac. za awt nuclide title 1001001 6.14448E-02 6.56680E-02 1001 1.0077 hydrogen in water 1301/1002 mod1 11/23/92 1008016 3.32538E-02 5.63969E-01 8016 15.9904 8O 16 from version 6 evaluation 1009019 2.53136E-03 5.10061E-02 9019 18.9982 9f 19 ornl evaljul74 c.y.fu d.c.lars mod312/16/88 1092234 2.57312E-07 6.38714E-05 92234 234.0405 92U 234 BNL HEDL+EVALJUL78 DIVADEENAMMOD3 01/10/91 1092235 6.26441E-05 1.56166E-02 92235 235.0441 92u 235 bnl evalapr77 m.r.bhat mod3 02/28/89 1092238 1.20278E-03 3.03676E-01 92238 238.0510 92U 238 ANL+ EVALJUN77 E.PENNINGTON MOD3 02/13/92 mixture = 2 density(g/cc) = 2.7020 nuclide atom-dens. wgt. frac. za awt nuclide title 2013027 6.03066E-02 1.00000E+00 13027 26.9818 13al 270lasl evaldec73 p.g. young d.g mod1 11/29/88 based on the preceding data, the best estimate of the parameter is 0.00000E+00 the mixing table corresponding to this parameter follows: solution tank - critical concentration search mixing table number of scattering angles = 2 cross section message threshold =5.7E-02 mixture = 1 density(g/cc) = 1.5656 nuclide atom-dens. wgt. frac. za awt nuclide title 1001001 6.14448E-02 6.56680E-02 1001 1.0077 hydrogen in water 1301/1002 mod1 11/23/92 1008016 3.32538E-02 5.63969E-01 8016 15.9904 8O 16 from version 6 evaluation 1009019 2.53136E-03 5.10061E-02 9019 18.9982 9f 19 ornl evaljul74 c.y.fu d.c.lars mod3 12/16/88 1092234 2.57312E-07 6.38714E-05 92234 234.0405 92U 234 BNL HEDL+EVALJUL78 DIVADEENAM MOD3 01/10/91 1092235 6.26441E-05 1.56166E-02 92235 235.0441 92u 235 bnl evalapr77 m.r.bhat mod3 02/28/89 1092238 1.20278E-03 3.03676E-01 92238 238.0510 92U 238 ANL+ EVALJUN77 E.PENNINGTON MOD3 02/13/92 mixture = 2 density(g/cc) = 2.7020 nuclide atom-dens. wgt. frac. za awt nuclide title 2013027 6.03066E-02 1.00000E+00 13027 26.9818 13al 270lasl evaldec73 p.g. young d.g mod1 11/29/88 **************************************************************************************************** This problem converged on the first pass, so the amount of uranium, HF acid, and water in the solution were reasonably good estimates. For this problem the HF acid and H\ :sub:2\ O only changed marginally and, therefore, had very little effect on the system *k*\ :sub:eff. This is not always the case; therefore if a search is being done on a solution, the problem should always be rerun with the final search densities. Fuel Bundles Separated by Flux Traps — Critical Search Boron and Al Densities ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ The fuel bundles in this problem represent 17 × 17 PWR fuel assemblies. The fuel pins are smeared together, making a mixture 100. The fuel pins consist of 4.35 wt % :sup:235\ U having a diameter of 0.823 cm, zirconium cladding having an outer diameter of 0.9627 cm, and a pitch of 1.275 cm. The fuel bundle is represented as a 10.8375 cm × 10.8375 cm × 366 cm cuboid of mixture 100 surrounded by Boral and then water. The Boral has a density of 2.61 g/cm\ :sup:3 and has an initial composed of 50.0 wt % B\ :sub:4\ C and 50.0 wt % Al. The fuel bundles are at a fixed pitch of 13.0 cm. Boral plates surrounding the X and Y sides of each fuel assembly are 0.1625 cm thick. Full density water is between the Boral plates. A critical concentration search is performed on the Boral plates searching for a system *k*\ :sub:eff = 0.95. The Boral plates are at a fixed density of 2.61 gm/cc. As the density of the B\ :sub:4\ C changes, the density of the Al changes in the opposite direction maintaining a constant Boral density. There are two entries in the MORE search data. The first entry, MIX=4 SCNAME=b4c factor=1.0 specifies that the B\ :sub:4\ C of mixture 4 is to be altered during the search. The second entry, MIX=4 SCNAME=al factor = −1.0 specifies that aluminum is to be changed in the opposite direction and proportionally to B\ :sub:4\ C during the search. Both B\ :sub:4\ C and Al have the same initial density of 0.5 \* 2.61 = 1.305 gm/cc. The maximum constraint is calculated according to: +CON= (2.59695/1.305 − 1)/1.0 = 0.99 The minimum constraint is calculated according to: −CON= (0.1305/1.305 − 1)/1.0 = −0.9 The search data and final search results for this problem follow: :: =csas5s array of fuel bundles with flux trap v7-238 read comp uo2 1 .84 300. 92235 4.35 92238 95.65 end zr 2 1 end h2o 3 1 end b4c 4 den=2.61 0.5 end al 4 den=2.61 0.5 end h2o 5 1.0 end end comp read celldata latticecell squarepitch pitch=1.275 3 fueld=0.823 1 cladd=0.9627 2 cellmix=100 end end celldata read param far=yes gen=203 npg=1000 end param read geom global unit 1 cuboid 100 1 4p10.8375 2p183.0 cuboid 4 1 4p11.0 2p183.0 cuboid 5 1 4p13.0 2p183.0 end geom read bounds xfc=mirror yfc=mirror end bounds end data read search critical concentration kef=0.95 more alter mix=4 scname=arbmb4c factor=1.0 alter mix=4 scname=al factor=-1.0 -con=-0.9 +con=0.99 end search end :: =csas5s array of fuel bundles with flux trap v7-238 read comp uo2 1 .84 300. 92235 4.35 92238 95.65 end zr 2 1 end h2o 3 1 end b4c 4 den=2.61 0.5 end al 4 den=2.61 0.5 end h2o 5 1.0 end end comp read celldata latticecell squarepitch pitch=1.275 3 fueld=0.823 1 cladd=0.9627 2 cellmix=100 end end celldata read param far=yes gen=203 npg=1000 end param read geom global unit 1 cuboid 100 1 4p10.8375 2p183.0 cuboid 4 1 4p11.0 2p183.0 cuboid 5 1 4p13.0 2p183.0 end geom read bounds xfc=mirror yfc=mirror end bounds end data read search critical concentration kef=0.95 more alter mix=4 scname=arbmb4c factor=1.0 alter mix=4 scname=al factor=-1.0 -con=-0.9 +con=0.99 end search end ******************* search pass 1 keff= 8.22991E-01 + or - 1.79474E-03 ****************** the parameter was 0.00000E+00 **** xsdrnpm mesh intervals **** 4 mesh intervals in zone 1 4 mesh intervals in zone 2 14 mesh intervals in zone 3 ***** modified keno v data has been rewritten on unit 95 ***** ******************* search pass 2 keff= 8.01629E-01 + or - 1.78327E-03 ****************** the parameter was 9.90000E-01 **** xsdrnpm mesh intervals **** 4 mesh intervals in zone 1 4 mesh intervals in zone 2 14 mesh intervals in zone 3 ***** modified keno v data has been rewritten on unit 95 ***** ******************* search pass 3 keff= 9.82092E-01 + or - 1.63305E-03 ****************** the parameter was -9.00000E-01 **** xsdrnpm mesh intervals **** 4 mesh intervals in zone 1 4 mesh intervals in zone 2 14 mesh intervals in zone 3 ***** modified keno v data has been rewritten on unit 95 ***** ******************* search pass 4 keff= 9.01534E-01 + or - 1.74845E-03 ****************** the parameter was -7.66111E-01 **** xsdrnpm mesh intervals **** 4 mesh intervals in zone 1 4 mesh intervals in zone 2 14 mesh intervals in zone 3 ***** modified keno v data has been rewritten on unit 95 ***** ******************* search pass 5 keff= 9.38557E-01 + or - 1.59285E-03 ****************** the parameter was -8.52032E-01 **** xsdrnpm mesh intervals **** 4 mesh intervals in zone 1 4 mesh intervals in zone 2 14 mesh intervals in zone 3 ***** modified keno v data has been rewritten on unit 95 ***** ******************* search pass 6 keff= 9.48930E-01 + or - 1.69920E-03 ****************** the parameter was -8.65663E-01 array of fuel bundles with flux trap **************************************************************************************************** **************************************************************************************************** convergence was achieved on pass 6 the parameter was -8.65663E-01 the equation used in the search was: k-eff = +8.66738E-01 +2.74121E-01*p +5.80563E-02*p**2 -4.25484E-01*p**3 k-effective= 9.48930E-01 + or - 1.69920E-03 the corresponding geometry follows; array of fuel bundles with flux trap mixing table number of scattering angles = 2 cross section message threshold =5.7E-02 mixture = 1 density(g/cc) = 9.2064 nuclide atom-dens. wgt. frac. za awt nuclide title 1008016 4.10835E-02 1.18491E-01 8016 15.9904 8O 16 from version 6 evaluation 1092235 9.04496E-04 3.83457E-02 92235 235.0441 92u 235 bnl evalapr77 m.r.bhat mod3 02/28/89 1092238 1.96373E-02 8.43163E-01 92238 238.0510 92U 238 ANL+EVALJUN77 E.PENNINGTON MOD3 02/13/92 mixture = 2 density(g/cc) = 6.4900 nuclide atom-dens. wgt. frac. za awt nuclide title 2040000 4.28457E-02 1.00000E+00 40000 91.2196 40zr sai evalapr76 m.drake d.sa mod2 01/03/89 mixture = 3 density(g/cc) = 0.99817 nuclide atom-dens. wgt. frac. za awt nuclide title 3001001 6.67692E-02 1.11927E-01 1001 1.0077 hydrogen in water 1301/1002 mod1 11/23/92 3008016 3.33846E-02 8.88074E-01 8016 15.9904 8O 16 from version 6 evaluation mixture = 4 density(g/cc) = 2.6100 nuclide atom-dens. wgt. frac. za awt nuclide title 4005010 1.52119E-03 9.69077E-03 5010 10.0130 5b100lasl evaldec76 g.hale l.stewart mod1 12/11/92 4005011 6.12300E-03 4.28888E-02 5011 11.0096 5b11 gebnl evalsep71 c.cowan mod1 12/11/92 free gas 4006012 1.91105E-03 1.45903E-02 6000 12.0001 6cornl evaldec73 c.y.fu and f.g. perey mod2 12/11/92 4013027 5.43405E-02 9.32830E-01 13027 26.9818 13al270lasl evaldec73 p.g. young d.g mod1 11/29/88 mixture = 5 density(g/cc) = 0.99817 nuclide atom-dens. wgt. frac. za awt nuclide title 5001001 6.67692E-02 1.11927E-01 1001 1.0077 hydrogen in water 1301/1002 mod1 11/23/92 5008016 3.33846E-02 8.88074E-01 8016 15.9904 8O 16 from version 6 evaluation mixture = 100 density(g/cc) = 0.0000 nuclide atom-dens. wgt. frac. za awt nuclide title 3001001 0.00000E+00 0.00000E+00 1001 1.0077 hydrogen in water 1301/1002 mod1 11/23/92 3008016 0.00000E+00 0.00000E+00 8016 15.9904 8O 16 from version 6 evaluation 1008016 0.00000E+00 0.00000E+00 8016 15.9904 8O 16 from version 6 evaluation 2040000 0.00000E+00 0.00000E+00 40000 91.2196 40zr sai evalapr76 m.drake d.sa mod2 01/03/89 1092235 0.00000E+00 0.00000E+00 92235 235.0441 92u 235 bnl evalapr77 m.r.bhat mod3 02/28/89 1092238 0.00000E+00 0.00000E+00 92238 238.0510 92U 238 ANL+ EVALJUN77 E.PENNINGTON MOD3 02/13/92 based on the preceding data, the best estimate of the parameter is -8.66988E-01 the mixing table corresponding to this parameter follows: array of fuel bundles with flux trap mixing table number of scattering angles = 2 cross section message threshold =5.7E-02 mixture = 1 density(g/cc) = 9.2064 nuclide atom-dens. wgt. frac. za awt nuclide title 1008016 4.10835E-02 1.18491E-01 8016 15.9904 8O 16 from version 6 evaluation 1092235 9.04496E-04 3.83457E-02 92235 235.0441 92u 235 bnl evalapr77 m.r.bhat mod3 02/28/89 1092238 1.96373E-02 8.43163E-01 92238 238.0510 92U 238 ANL+EVALJUN77 E.PENNINGTON MOD3 02/13/92 mixture = 2 density(g/cc) = 6.4900 nuclide atom-dens. wgt. frac. za awt nuclide title 2040000 4.28457E-02 1.00000E+00 40000 91.2196 40zr sai evalapr76 m.drake d.sa mod2 01/03/89 mixture = 3 density(g/cc) = 0.99817 nuclide atom-dens. wgt. frac. za awt nuclide title 3001001 6.67692E-02 1.11927E-01 1001 1.0077 hydrogen in water 1301/1002 mod1 11/23/92 3008016 3.33846E-02 8.88074E-01 8016 15.9904 8O 16 from version 6 evaluation mixture = 4 density(g/cc) = 2.6100 nuclide atom-dens. wgt. frac. za awt nuclide title 4005010 1.50619E-03 9.59518E-03 5010 10.0130 5b 100lasl evaldec76 g.hale l.stewart mod1 12/11/92 4005011 6.06261E-03 4.24657E-02 5011 11.0096 5b11 gebnl evalsep71 c.cowan mod1 12/11/92 free gas 4006012 1.89220E-03 1.44464E-02 6000 12.0001 6c ornl evaldec73 c.y.fu and f.g. perey mod2 12/11/92 4013027 5.43791E-02 9.33493E-01 13027 26.9818 13al 270lasl evaldec73 p.g. young d.g mod1 11/29/88 mixture = 5 density(g/cc) = 0.99817 nuclide atom-dens. wgt. frac. za awt nuclide title 5001001 6.67692E-02 1.11927E-01 1001 1.0077 hydrogen in water 1301/1002 mod1 11/23/92 5008016 3.33846E-02 8.88074E-01 8016 15.9904 8O 16 from version 6 evaluation mixture = 100 density(g/cc) = 0.0000 nuclide atom-dens. wgt. frac. za awt nuclide title 3001001 0.00000E+00 0.00000E+00 1001 1.0077 hydrogen in water 1301/1002 mod1 11/23/92 3008016 0.00000E+00 0.00000E+00 8016 15.9904 8O 16 from version 6 evaluation 1008016 0.00000E+00 0.00000E+00 8016 15.9904 8O 16 from version 6 evaluation 2040000 0.00000E+00 0.00000E+00 40000 91.2196 40zr sai evalapr76 m.drake d.sa mod2 01/03/89 1092235 0.00000E+00 0.00000E+00 92235 235.0441 92u 235 bnl evalapr77 m.r.bhat mod3 02/28/89 1092238 0.00000E+00 0.00000E+00 92238 238.0510 92U 238 ANL+ EVALJUN77 E.PENNINGTON MOD3 02/13/92 ******************************************************************************************************* *******************************************************************************************************