Users manual of CBZ/FRBurnerRZ: A module for fast reactor core design

Transcription

1 Users manual of CBZ/FRBurnerRZ: A module for fast reactor core design Go CHIBA May 25, 2018 Contents 1 Brief summary of FRBurnerRZ 2 2 Preparation of input deck Preparation of fuel pellet composition data Preparation of cladding composition data Preparation of fuel assembly geometry data Preparation of fuel assembly data Preparation of control rod data Medium data preparation except fuel, blanket and control rod assemblies Data preparation of geometrical configuration of reactor core Running of reactor core burnup calculations and how to obtain output data Burnup calculation condition Information extraction before burnup calculations running Running of burnup calculations and standard output Calculation option Extraction of results after running Calculations at specific burnup point after running Kinetics parameters calculations Time-dependent line power spatial distribution Exercise 14 A Detail of model simplification to two-dimensional cylinder 17 1

2 1 BRIEF SUMMARY OF FRBURNERRZ 2 1 Brief summary of FRBurnerRZ A module FRBurnerRZ is to perform burnup calculations of fast neutron reactor core which is modeled to a twodimensional cylinder. All the assemblies such as fuel assemblies, blanket assemblies and control rod assemblies are treated as homogeneous media, in which internal heterogeneous structure is ignored. Thus nuclide number densities homogenized over a whole assembly are used. 1 A reactor core in three dimension is modeled to a two-dimensional cylinder. A concept of this simplification is shown in Fig. 1. Its detail will be described in the appendix(, but not yet prepared...) Fig. 1: Simplification of a reactor core from three-dimension to two-dimension In typical fast breeder reactor operations, plant is in operation for a specific period, it stops to operate, some fuel assemblies and blanket assemblies are discharged, new assemblies are loaded, and the plant restarts its operation. Length of the operation period is called cycle length. Fuel assemblies and blanket assemblies are loaded for a certain number of cycles, and after the use they are discharged from a reactor core. For example, if all the assemblies are loaded for five cycles, at the end of each cycle, 1/5 of total assemblies are discharged. In such a case, we can say that the number of fuel exchange batches is five. 2 Effective (energy-averaged, or multi-group) cross section of each medium is calculated with 70-group CBZLIB, and calculated 70-group macroscopic cross sections are used in neutron flux calculations of a whole reactor core. Neutron flux distributions are obtained by solving the neutron diffusion equation. In FRBurnerRZ, a diffusion solver PLOS of CBZ is used. Nuclear fuel burnup calculations are performed by a module Burnup. Nuclides burnup chain used in burnup calculations can be chosen by users, but in this manual a default burnup chain implemented in Burnup is used. 3 2 Preparation of input deck This manual is based on a sample input main.rz.halfcore.cxx stored in a directory CBGCAL/monju of the CBZ package. This sample is to calculate simplified model of a proto-type fast breeder reactor. 2.1 Preparation of fuel pellet composition data In fuel pellet of fast reactors, uranium and plutonium mixed-oxide fuel (MOX fuel) is generally used. Pellet composition data, which are kinds of nuclides and their number densities, are defined in a class FRDTFuelComposition. In the following, an example of instance generation of FRDTFuelComposition is shown. 1 // +++ Fuel composition MATIDTranslator midt ; Listing 1: Example of instance generation of FRDTFuelComposition 1 If volume of region i is V i and number density of concerned nuclide in region i is N i, homogenized number density of this nuclide N is calculated as N = N i V i / V i. 2 i i Actually we cannot model this fuel exchange rigorously in two-dimensional core model. In FRBurnerRZ, fictitious batch-wise nuclide number densities are calculated for each medium in a two-dimensional reactor core, and burnup calculations are performed for each batchwise number density. In neutron flux calculations in a reactor core, medium-wise macroscopic cross sections are calculated from number densities averaged over all the batches. 3 As heavy nuclides, 17 nuclides such as U-235, -236, -238, Np-237, Pu-238, -239, -240, -241, Am-241, -242m, -243, Cm-242, -243, -244, -245 and -246 are treated, and fission product nuclides are considered as a pseudo unique nuclide. An example of heavy nuclides burnup chain will be shown in the exercise section.

3 2 PREPARATION OF INPUT DECK // [ Inner core ] 5 FRDTFuelComposition fc ic ; 6 fc ic. PutPuFissileEnrichment (16.1); // [wt%] 7 //fc ic.putpuenrichment(20.); // [wt%] 8 fc ic. PutU5Enrichment (0.2); // [wt%] 9 fc ic.putom(1.97); 10 fc ic. PutPelletTheoreticalDensity (85.); // % 11 string matnum[]={ Pu238, Pu239, Pu240, Pu241, Pu242, Am241 }; 12 real fc ic inp []={0., 58., 24., 14., 4., 0.}; 13 fc ic.puttrucomposition(6,matnum, fc ic inp, midt ); 14 / 15 i n t matid[]={942380,942390,942400,942410,942420,952410}; 16 fc ic.puttrucomposition(6,matid, fc ic inp ); 17 / In FRBurnerRZ, mixed fuel of UO 2 and PuO 2 is expected, but it is possible to add minor actinoid (MA) nuclides such as Np and Am to Pu. A method PutPuFissileEnrichment in line 6 determines weight percent of fissile plutonium isotopes, a sum of Pu-239 and Pu-241, to total actinoid nuclides in fresh fuel. If users want to determine weight percent of all plutonium isotopes to total actinoid nuclides, a method PutPuEnrichment can be used. A method PutU5Enrichment in line 8 determines weight percent of uranium-235 to total uranium isotopes in UO 2. Note that in UO 2, uranium-235 and -238 are included. A method PutOM in line 9 determines number density ratio of oxygen to total actinoid nuclides. 4 A method PutPelletTheoreticalDensity in line 10 determines volume theoretical density of oxide fuel within pellet. This theoretical density is a ratio of fuel material volume to whole pellet volume. In this example, 85% of pellet is occupied by oxide fuel and other 15% is by void. Finally, a method PutTRUComposition in line 13 determines a weight percent of transuranic (TRU) element included in PuO 2. The second argument corresponds to an array for nuclide names (matnum) and weight percent of these nuclides is assigned in the third argument. If a list of nuclide names is provided in character style like this example, we have to assign an instance of the MATIDTranslator class, which is to translate nuclide name to nuclide ID, as the forth argument. If nuclide IDs are assigned in the second argument, it is unnecessary to assign the forth argument. Please see line 16. In a fast reactor core, blanket assemblies, which reduce neutron leakage from a reactor core and promote conversion from U-238 to Pu-239, are as important as fuel assemblies. An example of data preparation of pellet of blanket assemblies with the FRDTFuelComposition class is shown below: Listing 2: Example of instance generation of FRDTFuelComposition (blanket assemblies) 1 // [ Blanket ] 2 FRDTFuelComposition f c b ; 3 fc b. PutPuFissileEnrichment (0); 4 fc b. PutU5Enrichment (0.2); 5 fc b.putom(2.01); 6 fc b. PutPelletTheoreticalDensity (92.79); 7 fc b.puttrucomposition(6,matnum, fc ic inp, midt ); Since pellet of blanket assemblies is composed of only UO 2, fissile plutonium isotopes weight percent is zero as shown in line 3. Note that even if pellet does not include PuO 2, users have to do the PutTRUComposition method like this example. Number densities of pellet regions can be printed out on a screen by a method ShowSelf of FRDTFuelComposition. 2.2 Preparation of cladding composition data A geometrical configuration of a fuel assembly of middle-size fast breeder reactor is shown in Fig. 2. Fuel pellet is contained in a cladding, and more than 100 fuel pins consisting of pellet and cladding are contained by a hexagonal tube called wrapper tube or duct. In this figure, the outermost hexagonal is just an external boundary of an assembly, and is not structure. FRBurnerRZ assumes that cladding and wrapper tube are made of the same unique composition. These composition data are defined by the FRDTSUSComposition class. An example of instance generation of this class is shown below: 1 // +++ SUS composition +++ Listing 3: Example of instance generation of the FRDTSUSComposition class 4 A number density ratio of oxygen to actinoid is referred to as O/M ratio. When this value is over 2.0, oxidization of cladding begins, so a value smaller than 2.0 is generally used. When nuclear fuel is depleted, number densities of actinoid nuclides decrease, so the O/M ratio should increase with operation.

4 2 PREPARATION OF INPUT DECK Y [cm] X [cm] Fig. 2: Geometrical configuration of a fuel assembly of a middle-size fast breeder reactor 2 FRDTSUSComposition sus316 ; 3 real sus inp2 []={1.7, 13.5, 17., 2.5, 65.3, 0}; // Mn Ni Cr Mo Fe W 4 sus316. PutDensityAndRatio (7.98, sus inp2 ); In FRDTSUSComposition, number density data are generated by determining weight ratio of materials such as Mn, Ni, Cr, Mo, Fe and W. The first argument of a method PutDensityAndRatio is density of a material in [g/cm 3 ]. In this example, SUS316 is assumed. 2.3 Preparation of fuel assembly geometry data An example of instance generation of the class FRDTSubAssemblyGeometry to define geometry of a fuel assembly is shown below: Listing 4: Example of instance generation of the FRDTSubAssemblyGeometry class 1 // +++ Fuel subassembly geometry FRDTSubAssemblyGeometry sa geom ; 3 sa geom. PutAssemblyPitch (115.6); 4 sa geom. PutDuctOutersize (110.6); 5 sa geom. PutDuctThickness (3.); 6 sa geom. PutPinOuterDiameter (6.5); 7 sa geom. PutPinThickness (0.47); 8 sa geom.putnumberofpin(169); 9 sa geom. PutSpacerwireDiameter (1.32); 10 sa geom. PutSpacerwirePitch (307.); 11 sa geom. PutPelletOuterDiameter (5.4); 12 sa geom. PutPelletInvoidDiameter (0.); 13 sa geom. PutPinPitch (7.9); //sa geom.showvolumeinformation (); In this example, various parameters related to length is assigned in a unit [mm]. Parameter definition is shown in Fig. 3. Pin-pitch is a distance between center positions of two neighboring fuel pins, and it is about 7.9[mm] in a prototype fast reactor. In a peripheral region, a wrapper tube (or duct) is located, but coolant can flow outside of a wrapper tube, so there is no agreement between an assembly pitch and duct outer size. It is possible to consider a void region inside of fuel pellet: a hollow fuel pellet. In lines 9 and 10, data about spacer wire are defined. Spacer wire is used to avoid direct contact between neighboring fuel pins and this is rounded outside each fuel pin. Wire pitch defined in line 10 is axial length within which spacer wire is fully rotated around fuel pin. The number of fuel pins is given as 3N(N 1)+1 where N is the number of fuel ring layers. The number of fuel ring layers is 8 in a prototype fast reactor as shown in Fig. 2, so the number of pins is 169. The number of pins can be assigned by the method PutNumberOfPin. When the duct specification, the number of fuel pins and pin pitch are physically inconsistent with each other, an error message will be provided.

5 2 PREPARATION OF INPUT DECK 5 Fig. 3: Parameters related to geometry defined in the FRDTSubAssemblyGeometory class In line 15 of this example, a method ShowVolumeInformation is commented out. This method is to print out a volume of each region of a fuel assembly. Results of this method for a prototype fast reactor are shown below: Listing 5: Example of the ShowVolumeInformation method of the FRDTSubAssemblyGeometry class 1 ##################################################### 2 # Region wise volume of subassembly [mmˆ2]. 3 # 4 # Total : e+04 5 # Fuel pellet : e+03 6 # Void in pellet : e+00 7 # Void at pellet clad gap : e+02 8 # Cladding : e+03 9 # Spacer wire : e # Wrapper tube : e # Coolant : e # ( inside of wrapper tube) : e ##################################################### 2.4 Preparation of fuel assembly data Using fuel assembly geometry data, fuel pellet composition data and cladding composition data, an instance of the class FRDTSubAssembly which defines a fuel assembly can be generated. An example of instance generation is shown below: 1 // +++ Fuel subassembly FRDTSubAssembly sa ic ; 3 sa ic. PutSubAssemblyGeometry(&sa geom ); 4 sa ic. PutFuelComposition(&fc ic ); 5 sa ic. PutSUSComposition(&sus316 ); 6 sa ic. PutSodiumDensity ( ); Listing 6: Example of instance generation of the FRDTSubAssembly class From lines 3 to 5, instances of fuel geometry data, pellet composition data and cladding composition data are transferred. In line 6, density of sodium flowing through coolant path is defined in unit of [g/cm 3 ]. This instance of the FRDTSubAssembly class contains information about fuel assembly, so it can generate assembly-averaged nuclides number densities for burnup calculations. In the following example, average number density data calculated by the instance of FRDTSubAssembly is transferred to an instance of the Medium class in line 5. Listing 7: Example of number density data transfer from instance of FRDTSubAssembly to that of the medium class 1 Medium min ic ; 2 min ic.putimax(group ); 3 min ic.putpl(1); 4 min ic. PutNuclide(fuel nuc num,matno); 5 sa ic. PutHomogenizedNumberDensity( min ic ); 6 min ic. PutTemperatureForAllNuclide ( ); 7 min ic. GetNuclide (110230).PutTemperature( ); In this example, an instance sa ic of the FRDTSubAssembly class for inner core fuel assembly provides number density data to an instance min ic of the Medium class, and in line 6, temperature of all the nuclides contained in this medium

6 2 PREPARATION OF INPUT DECK 6 is assigned. Note that temperature of sodium nuclide (ID is ) which is included in coolant is redefined in line Preparation of control rod data Data of control rod assembly can be generated with the same manner as that of fuel assembly basically. A file main.cr.cxx is for control rod data generation in the CBZ package whereas control data preparation is not required in fast reactor core burnup calculations with a file main.rz.halfcore.cxx. Example of composition data generation of neutron absorption pellet, B 4 C pellet, is shown in the following. The first and second arguments correspond to the B-10 enrichment and pellet theoretical density. In this example, data for main regulation rods and backup rods are prepared in lines 2 and 3. Listing 8: Example of instance generation of FRDTB4CComposition class 1 // +++ B4C composition FRDTB4CComposition b4c cr (0.39, ); // B 10 ratio, theoretical density 3 FRDTB4CComposition b4c bcr (0.90, ); // B 10 ratio, theoretical density Next example of data preparation for geometrical information of control rod is shown in the following. In addition to geometrical parameters as fuel assembly, specific parameters such as guide tube and protect tube should be assigned. Listing 9: Example of instance generation of the FRDTControlRodGeometry class 1 FRDTControlRodGeometry cr geom ; 2 cr geom. PutAssemblyPitch (115.6); 3 cr geom. PutPinOuterDiameter (16.9); 4 cr geom. PutPinThickness (2.0); 5 cr geom.putnumberofpin(19); 6 cr geom. PutSpacerwireDiameter (1.2); 7 cr geom. PutSpacerwirePitch (239.); 8 cr geom. PutPelletOuterDiameter (12.2); 9 cr geom. PutPelletInvoidDiameter (0.); 10 // The following are specific for control rod 11 cr geom. PutGuidetubeOuterDiameter (110.6); 12 cr geom. PutGuidetubeThickness (3.); 13 cr geom. PutShieldtubeOuterDiameter (94.); 14 cr geom. PutShieldtubeThickness (2.); Geometrical configuration of a control rod of a prototype fast reactor is shown in Fig. 4. Nineteen neutron absorption pins are contained in a protect tube, and these absorption pins and the protect tube are simultaneously moved up and down in a guide tube. When a control rod is fully withdrawn, only a guide tube exists with coolant in a control rod. Geometrical structure of neutron absorption pins of a control rod is shown also in Fig. 4. Note that structures of main regulation rods and backup rods differ from each other. Fig. 4: Geometrical configuration of control rod of a prototype fast reactor Finally an example of instance generation of the class FRDTControlRod which defines a control rod, and an example of number density data transfer from this instance to an instance of the Medium class are shown below: 5 Rigorously speaking, temperature of fuel pellet is the highest, and that of cladding is lower than that, and that of coolant should be the lowest. This is not considered in this example, and only sodium temperature is differently treated from others.

7 2 PREPARATION OF INPUT DECK 7 Listing 10: Example of instance generation of the FRDTControlRod class 1 FRDTControlRod cr ; 2 cr. PutControlRodGeometry(&cr geom ); 3 cr. PutB4CComposition(&b4c cr ); 4 cr. PutSUSComposition(& sus316 ); 5 cr. PutSodiumDensity ( ); // 400oC 6 7 Medium med cr ; 8 9 med cr.putimax(group ); 10 med cr.putpl(1); cr. PutHomogenizedNumberDensity( med cr ); med cr. PutTemperatureForAllNuclide ( ); 2.6 Medium data preparation except fuel, blanket and control rod assemblies Medium data generation procedures for inner/outer cores, radial/axial blankets and control rod have been already described, so data preparation for other media such as neutron reflectors are described here. In the following, medium data preparation for a reflector is shown. A method PutNuclide in line 10 supplies nuclides information contained in a medium. The first, second and third arguments are the number of nuclides, a list of nuclide ID and a list of number densities. 6 In line 11, temperature of nuclides in a medium is determined. Listing 11: Example of medium data generation except fuel, blanket and control rod assemblies 1 // ( axial shielding (core)) 2 Medium min as ; 3 min as.putimax(group ); 4 min as.putpl(1); 5 int mat3[]={110230,240000,280000,420000,260000,250550}; 6 real den3[]={ e 3, e 3, e 3, e 4, e 2, e 4 9 }; 10 min as. PutNuclide (6,mat3, den3 ); 11 min as. PutTemperatureForAllNuclide ( ); 2.7 Data preparation of geometrical configuration of reactor core Geometrical information of two-dimensional cylindrical reactor core, scheme of spatial mesh division and assignment of material (or medium) into a reactor core are defined by an instance of the CartMeshInfo class. An example of this is shown below: Listing 12: Example of data generation of reactor core geometrical configuration by the CartMeshInfo class 1 // +++ CartMeshInfo 2 CartMeshInfo cmi ; 3 real xl []={ , , , , , , , , , , , , , , , }; 9 real yl []={ , 15., 16.5, 15., 20., }; 12 int xm[]={1,2,2,1,1, 1,2,2,1,2, 2,2,2,2,2, 6}; 13 int ym[]={3,3,3,3,4, 3}; 14 int mat[]={ 15 40, 0, 3, 6,40, 6, 9,12,40,15,18,21,24,29,31,39, 16 40, 1, 4, 7,40, 7,10,13,40,16,19,22,25,29,31,39, 17 40, 2, 5, 8,40, 8,11,14,40,17,20,23,26,29,31,39, 18 40,32,32,32,40,32,33,33,40,33,35,35,27,30,31,39, 19 40,34,34,34,40,34,34,34,40,34,36,36,28,30,31,39, 20 40,37,37,37,40,37,37,37,40,37,37,37,38,38,38,39, 21 }; 22 cmi. PutMeshInfo(16,6,xm,ym, xl, yl,mat); 23 cmi. PutBoundaryCondition 24 ( Reflective, Vacuum, Reflective, Vacuum ); 6 This is a definition of nuclide ID: when atomic number is N, mass number is A and excitation level is L, nuclide ID is defined as an integer, N A 10 + L. When a nuclide is in a ground level, L should be zero. Note that ID should be an integer, so the initial character should not be zero.

8 3 RUNNING OF REACTOR CORE BURNUP CALCULATIONS AND HOW TO OBTAIN OUTPUT DATA 8 Material assignment is defined by an array mat, and in this example totally 41 medium data are defined. In FRBurnerRZ, two different core regions, an inner core and an outer core, can be specified; different media (or materials) can be assigned to inner and outer cores. Also axial blanket which is attached at the top and bottom of a fuel assembly and radial blanket are differently defined. An example of medium data assignment to an instance of FRBurnerRZ is shown below: Listing 13: Example of assignment of medium data to an instance of the FRBurnerRZ class 1 FRBurnerRZ frbn ; 2 frbn.putnumberofmedium(41); 3 // ( core geometry information) 4 frbn. PutCartMeshInfo(cmi ); 5 frbn.putmediumidforic(0,17); 6 frbn.putmediumidforoc(18,23); 7 frbn.putmediumidforrb(24,31); 8 frbn.putmediumidforabic(32,34); 9 frbn.putmediumidforaboc(35,36); 10 // (medium information) 11 frbn.putmediumic( min ic ); 12 frbn.putmediumoc(min oc ); 13 frbn.putmediumrb(min rb ); 14 frbn.putmediumab(min ab ); 15 frbn.putmedium(min as,37); 16 frbn.putmedium(min asb,38); 17 frbn.putmedium(min rs,39); 18 frbn.putmedium( min fol,40); 19 // ( subassembly geometry information) 20 frbn.putsadatafuel(sa geom ); 21 frbn. PutSADataRadialBlanket( sarb geom ); In line 2, the number of media in a reactor core is defined. From lines 5 to 9, medium (or material) ID, which has been already defined above as an array mat, is assigned to each of media. Users have to assign a medium ID of zero to the initial medium of an inner core, and assign medium IDs from inner core, outer core, radial blanket, axial blanket and the others in order. From lines 11 to 14, instances of the Medium class about inner/outer cores and radial/axial blankets are transferred to an instance of FRBurnerRZ. From lines 15 to 18, instance of the Medium class about other media are transferred also. 3 Running of reactor core burnup calculations and how to obtain output data When ones operate a nuclear reactor for a specific period under an certain thermal output, nuclide number densities in fuel pellets change with time, so nuclear (reactor physics) properties of a reactor core also should change with time. Numerical calculations for this behavior due to reactor operation are referred to as reactor core burnup calculations. The FRBurnerRZ class performs burnup calculations for a simplified two-dimensional cylindrical reactor core. In this section, an example of burnup calculations with an instance frbn of RRBurnerRZ is described. 3.1 Burnup calculation condition An example of burnup calculation condition determination is shown below: Listing 14: Example of burnup calculation condition 1 frbn. PutReactorPower( ); // [Wth] 2 // A factor of 0.5 is used because we are treating a half core model. 3 frbn. PutTraceCycle (6); 4 frbn. PutCycleDiv (2); 5 frbn. PutCycleLength (148); // day 6 frbn.putbatchnumber(5,5,5);// Inner core/outer core/radial blanket 7 frbn. PutRefuelDay (0.); // days required for refueling A reactor thermal power in unit of [W] is determined by the method PutReactorPower. Note that a half of a total reactor power is assigned in this example since a half core is treated here by assuming axial symmetry. The number of total cycles for which burnup calculations are performed is determined by the method PutTraceCycle. In the method PutCycleDiv, the number of step divisions for each operation cycle is determined. In this example, each of cycles is divided into two, so neutron flux distribution and eigenvalue calculations are conducted at three burnup points, beginning, middle and end of cycle. Length of each operation cycle is determined in unit of day by the method PutCycleLength. Note that unique cycle length is assigned to all the operation cycles in FRBurnerRZ.

9 3 RUNNING OF REACTOR CORE BURNUP CALCULATIONS AND HOW TO OBTAIN OUTPUT DATA 9 The number of fuel exchange batches, which is the number of operation cycles for which fuel or blanket assembly is loaded in a reactor core, is determined by the method PutBatchNumber for inner/outer core and radial blanket. In multi-cycle operations, certain length of period should be considered after the end of each operation cycle to restart the new cycle. A length of this period, refueling time, is determined by the method PutRefuelDay. 3.2 Information extraction before burnup calculations running Before running burnup calculations, several information can be extracted from an instance of FRBurnerRZ. A list of methods to do that is shown below: 1 frbn. PrintMacroXS (); 2 frbn. PrintInitialND (); Listing 15: Methods to extract information before running PrintMacroXS: To print out macroscopic cross sections of inner/outer cores and radial/axial blankets at the beginning of operation PrintInitialND: To print out nuclides number densities of inner/outer cores and radial/axial blankets at the beginning of operation 3.3 Running of burnup calculations and standard output To run burnup calculations, a method Run of FRBurnerRZ should be used. Two arguments are required in this method, and the first one is a nuclear data library used for burnup calculations, that is an instance of the XSLibrary class, and the second one is a burnup calculation module, that is an instance of the Burnup class. When ones perform the Run method, a kind of the following output might be printed on a screen: Listing 16: Example of output of the Run method 1 ################################################################ 2 #C Step Day Keff Max. line C.R. Power Dist. [%] 3 #y power [W/cm] IC OC RB AB 4 # (pos : r, z) 5 # (1,0) (1,0) (1,0) (1,0) (1,0) (1,0) (1,0) (1,0) (1,0) (1,0) (1,0) (1,0) (1,0) (1,0) (1,0) (1,0) (1,0) (1,0) On each line, from the left side burnup cycle, step during cycle, elapsed days, effective neutron multiplication factor, maximum liner power, conversion ratio and thermal power fractions are printed. In this example, each burnup cycle is divided to 2, so parameters are provided at 3 different steps. Length of operation cycle is 148 and no cooling (refueling) time is considered. A pair of integers printed with maximum line power corresponds to a spatial mesh position in which maximum liner power is detected. Spatial mesh ID is defined from zero, so if this position is (1,0), the second radial and the first axial mesh gives the maximum line power. A conversion ratio is defined as a ratio of macroscopic neutron capture reaction rates of fertile nuclides to macroscopic neutron absorption reaction rates of fissile nuclides. Fertile nuclides are Th-232, U-234, U-238 and Pu-240, and fissile nuclides are U-233, U-235, Pu-239 and Pu-241 in FRBurnerRZ. 7 7 A conversion ratio can be presented in the following equation: Note that σ a = σ c +σ f. C.R. = (NTh 232 σc Th 232 (N U 233 σ U 233 a +N U 234 σ U 234 c +N U 235 σ U 235 a +N U 238 σ U 238 c +N Pu 240 σc Pu 240 N Pa 233 σc Pa 233 )φ +N Pu 239 σa Pu 239 +N Pu 241 σa Pu 241 )φ. (1)

10 3 RUNNING OF REACTOR CORE BURNUP CALCULATIONS AND HOW TO OBTAIN OUTPUT DATA 10 Four values are printed as thermal power fractions. These show contributions of region-wise power to total power in percent. IC, OC, RB and AB stand for inner core, outer core, radial blanket and axial blanket. 3.4 Calculation option Before conducting the Run method, calculation conditions can be determined with the following methods: 1 frbn. PrintLinepowerMap (); 2 frbn. VoidCalculationOn (); 3 frbn. DopplerCalculationOn (); 4 frbn. ShowFissionInfoOn (); 5 frbn.cmfdoff(); Listing 17: List of methods to assign calculation options PrintLinePowerMap: During burnup calculations with the Run method, spatial distribution of line power is printed on a screen. Furthermore, data of time-dependent line power spatial distribution is stored in an external file named as linepower map. This can be efficiently utilized to make animation of power distribution change with time. VoidCalculationOn: Reactivity which is induced by changing sodium density in all the media is calculated at every time points in which neutron flux distribution and eigenvalue calculations are performed. As default, coolant number density is changed to zero, but the degree of the change can be assigned through argument of this method. In this case, a value provided as an argument is a factor which is multiplied to original coolant number density. DopplerCalculationOn: Reactivity which is induced by changing heavy nuclides temperature in all the media is calculated at every time points in which neutron flux distribution and eigenvalue calculations are performed. As default, temperature change is +10K, but users can assign this by using arguments of this method. ShowFissionInfoOn: Detail information on neutron multiplication is printed out when conducting the Run method. CMFDOff: To put off the coarse-mesh finite-difference acceleration, which is adopted as default. When systems are not near critical, this acceleration sometimes does not work properly. In such cases, this method should be used. This should be done before conducting the PreCalculation method of FRBurnerRZ. 3.5 Extraction of results after running By conducting the Run method, several information such as neutron multiplication factors and maximum line power are printed on a screen, but these are a fraction of obtained results. The other information can be extracted by methods of FRBurnerRZ below: Listing 18: List of methods to extract calculation results after running 1 frbn.printnd(bu, nd ); 2 frbn. PrintBurnup (); 3 frbn. PrintFluxLevelHistory (); 4 frbn. Print1groupXS( Pu239,0); 5 frbn. PrintNuclideWeightPerBatch( bu); // Discharged HM weights are printed. 6 frbn. PrintNuclideWeightPerBatch(bu,true ); // Loaded HM weights are printed. PrintND: Batch-wise data of every media after burnup calculations are printed. The first argument is an instance of the Burnup class, and users can chose a type of extracted information by the second argument. In this example, nd is used as the second argument, so number density data are printed. 8 Note that macroregion-wise data such as inner/outer cores and radial/axial blankets are also printed out. PrintBurnup: To print out average and maximum burnup for every media. If users want to know batch-wise burnup, they can know by using the method PrintND. PrintFluxLevelHistory: To print out average neutron flux level for every media at every burnup steps. 8 In FRBurnerRZ, fission product nuclides are modeled to a pseudo unique nuclide. This nuclide is referred to as (pseudo) lumped fission product nuclide. In FRBurnerRZ, there four types of pseudo lumped fission product; FP.U235, FP.U238, FP.Pu239 and FP.Pu241. When U-235 and -236 fission, a nuclide FP.U235 is generated, when U-238, Np-237 and Pu-238 fission, FP.U238 is generated, when Pu-239 and -240 fission, FP.Pu239 is generated and when Pu-241 and -242, Am and Cm fission, FP.Pu241 is generated.

11 3 RUNNING OF REACTOR CORE BURNUP CALCULATIONS AND HOW TO OBTAIN OUTPUT DATA 11 Print1groupXS: To print out microscopic one-group cross section. Nuclide is specified by the first argument and the medium is specified by the second argument. 9 PrintNuclideWeightPerBatch: To print out nuclide-wise and region-wise weights of discharged fuel per one patch. When users want to print out weights of loaded fuel per patch, the second argument should be true. Users can calculate material balance in an equilibrium state of a reactor core. 3.6 Calculations at specific burnup point after running After performing burnup calculations, various calculations can be done at arbitrary operation cycle (burnup point). To do that, number density data after burnup calculations are overwritten by those at specific burnup point which are chosen by users by using a method PutNumberDensity as used in the following example: Listing 19: Example of a method to overwrite nuclide number density data at specific burnup point 1 frbn. PutNumberDensity(0,0); In this method, the first and second arguments are used to determine concerned operation cycle and step. After overwriting number density data, the following methods can be used to perform various calculations and to get useful information. If users do not perform PutNumberDensity before doing the following methods, all the calculations are performed for a state at the end of the reactor operation. Listing 20: Example of methods used after overwriting number density data 1 frbn. CalNeutronFluxEnergySpectrum (); // forward fluxes for all mediums 2 frbn. CalNeutronFluxEnergySpectrum(true ); // adjoint fluxes for all mediums 3 frbn. CalNeutronFluxEnergySpectrum(false,8); // forward flux for medium 8 4 frbn. CalNeutronFluxEnergySpectrum(false,0,15); // forward fluxes for mediums 0 to 15 5 frbn. CalVoidReactivity (0.25); 6 frbn. CalDopplerReactivity( xslib ); 7 frbn. Cal1groupXS (); 8 frbn. ShowNeutronMultiplicationInfo (); CalNeutronFluxEnergySpectrum: To print out neutron flux energy spectra in every media. If the first argument, whose default is false, is true, adjoint neutron flux energy spectra are printed out. The second argument is used to assign the medium ID for which neutron flux energy spectrum is printed out. If this is not assigned, neutron flux energy spectra in all the media are printed out. If users assign the third argument, neutron flux energy spectra in several media are printed out; the initial and last medium IDs are the second and third arguments. In line 4 of the above example, neutron flux energy spectra from medium 0 to medium 15 are printed out. CalVoidReactivity: To calculate reactivity induced by changing coolant density by the exact perturbation theory. The first argument is used to assign a factor multiplied to the original density. Its default value is 0.0, so if this is omitted, coolant void reactivity is calculated. By using the second and third arguments, users can assign media for which coolant density is changed like the CalNeutronFluxEnergySpectrum method. For example, if users do CalVoidReactivity(0.25, 0, 16), reactivity induced by reducing coolant density by 75% in media 0 to 16 is calculated. CalDopplerReactivity: To calculate reactivity induced by changing heavy nuclides temperature by the exact perturbation theory. The first argument is used to assign an instance of XSLibrary class, and the second argument is used to determine a degree of temperature change, whose default value is 10. The third and forth arguments are used to assign media for which heavy nuclides temperature are changed like the CalNeutronFluxEnergySpectrum method. Cal1groupXS: To print out one-group cross section of all the media including non-fuel media. 9 Neutron-nuclide reaction cross sections are physical quantities dependent on incident neutron energy, but if we average over a whole energy range by weighing neutron flux energy spectra, we can derive one-group cross section. Actually one-group cross section σ is calculated as G where σ(e) is cross section and φ(e) is neutron flux. σ = σ(e)φ(e)de φ(e)de σ gφ g g=1 = G φ g g=1, (2)

12 3 RUNNING OF REACTOR CORE BURNUP CALCULATIONS AND HOW TO OBTAIN OUTPUT DATA 12 ShowNeutronMultiplicationInfo: To print out detail information on fission chain reactions. For example, users can know nuclide-wise contribution to total fission reactions in output of this method. An example of a reactivity calculation by the CalVoidReactivity method is shown below: Listing 21: Example of reactivity calculation by the perturbation theory 1 # 2 # System CBG 3 # Perturbation calculation 4 # Solver : PLOS 5 # 6 # Energy Yield Absorption Scattering Leakage (n,2n) Total e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e # 16 # +++ Summary (energy integrated value ) # 18 # Yield : 0.000e # Absorption : 1.698e # Scattering : 2.242e # N2N : 0.000e # (Non Leak) : 2.412e # Leakage r : 1.045e # Leakage z : 3.055e # ( Leakage) : 4.100e # 27 # Perturbation Cal. : 1.688e # Direct Cal. : 1.687e 02 Reactivity can be found in line 27 (the second line from the bottom) as Perturbation Cal. Component-wise reactivity is also shown in lines 18 to 25. Absorption in line 19 and Scattering in line 20 are reactivity components due to absorption cross section change and scattering cross section change, respectively. (Leakage) in line 25 is a component related to neutron leakage from a reactor core. Above the summary section, energy group-wise and component-wise reactivity data are printed out. Furthermore, all the data about all the media included in a reactor core can be outputted to external files, and this is done by the method WriteFileMediumData of FRBurnerRZ. The following is an example: Listing 22: Example of output of all the medium data written in an external file 1 frbn. WriteFileMediumData(./MEDDATA/, med ); The first and second arguments are used to determine a directory in which external files are stored and the name of the external files. Data of each medium are stored in a file named as this name with the medium ID. In this example, files whose names are med0 and med1 are stored in the directory MED DATA. 3.7 Kinetics parameters calculations An example of input data to calculate effective delayed neutron fraction β eff and prompt neutron life time l is shown below. Note that this part is commented out in a file main.rz.halfcore.cxx in the CBZ package. 1 DelayedNeutronData Del ( 9); 2 Del.SetThUPu(); 3 string mdir2(../../cbglib/delayed/j4.70g/ ); 4 Del. PutDataFromFile(mdir2, Th232 ); 5 Del. PutDataFromFile(mdir2, U233 ); 6 Del. PutDataFromFile(mdir2, U234 ); 7 Del. PutDataFromFile(mdir2, U235 ); 8 Del. PutDataFromFile(mdir2, U238 ); 9 Del. PutDataFromFile(mdir2, Pu239 ); 10 Del. PutDataFromFile(mdir2, Pu240 ); 11 Del. PutDataFromFile(mdir2, Pu241 ); 12 Del. PutDataFromFile(mdir2, Pu242 ); 13 frbn. CalDelayedNeutronParameters( Del ); 14 //Del. ShowSelf (); Listing 23: Example of an input for kinetics parameters calculations Details of heavy nuclides delayed neutron data can be printed out by the method ShowSelf of the DelayedNeutronData class. An example of output is shown below:

13 3 RUNNING OF REACTOR CORE BURNUP CALCULATIONS AND HOW TO OBTAIN OUTPUT DATA 13 Listing 24: Example of output of delayed neutron data 1 # Nu d 2 # e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e # 19 # Family wise abundance 20 # e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e # 31 # Family wise decay constant 32 # e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e+00 At the beginning, the numbers of delayed neutrons per one fission reaction ν d dependent on incident neutron energy and nuclide are shown. In quite a high energy range ν d takes large values, but they become almost constant in lower energy range. The ν d values are in Th-232 whose ID is and in U-238 whose IS is Time-dependent line power spatial distribution As described in the preceding section, if a method PrintLinePowerMap is done before running, time-dependent data of line power spatial distribution is printed out in a file linepower map, and users can make a simple animation with this. Line power spatial distribution is printed out at each step, so if the number of burnup cycles is n 1 and the number of step divisions is n 2, totally spatial distributions at n 1 (n 2 + 1) burnup points are printed out. This number is printed out in the first line of a file linepower map. An example of shell script to make a simple animation with GNUPLOT and this linepower map file is shown below: Listing 25: Example of shell script to make a simple animation of time-dependent line power spatial distribution 1 #!/ bin/sh 2 3 for loop in seq do 5 6 loop10= expr $loop if test $loop lt 10 ; then 9 output=pmap00$loop. png 10 elif test $loop lt 100 ; then 11 output=pmap0$loop. png 12 else 13 output=pmap$loop. png 14 fi 15 echo writing file : $output gnuplot persist <<EOF 18 #set size 0.7, set pm3d map 20 #set xrange [50:100] 21 #set yrange [30:60] 22 set xlabel R [cm]

14 4 EXERCISE set ylabel Z [cm] 24 #set logscale cb 25 set cbrange [:350] 26 #set format cb 10ˆ{%L} 27 set terminal png 28 set output $output 29 set title $loop 30 splot linepower map i $loop : $loop ti 31 EOF done convert loop 0 delay 20 pmap.png pmap. gif 36 rm pmap.png In line 3, an iteration loop is defined, and a loop from 0 to 17 is done in this example because there are data at 18 points. Ranges of plot in x-axis and y-axis are defined in lines 20 to 21, which are commented out in this example. Range of data (line power) is defined in line 25. A speed of changing pages can be adjusted by changing a value after -delay in line 35. Finally, a file pmap.gif is generated. 4 Exercise Figure 5 shows assemblies arrangement (top-view) of a middle-size fast breeder reactor. Assemblies without any symbols are inner core fuel assemblies and assemblies with mark # are outer core fuel assemblies. Difference between inner and outer cores is in plutonium enrichment. Assemblies with C, F and B in an inner core region are coarse regulation, fine regulation and back up control rod assemblies. Three-layer radial blanket assemblies surround a outer core region, and are surrounded by neutron reflector assemblies. As already described, all the assemblies are treated as homogeneous media and inner structure of each assembly is ignored in FRBurnerRZ. Fig. 5: Assemblies arrangement in middle-size fast breeder reactors This three-dimensional reactor core is simplified into two-dimensional cylinder as shown in Fig. 6. Rigorously speaking, there is no symmetry in the axial direction, but symmetry is assumed for simplification in the example provided in the CBZ package. In burnup calculations with FRBurnerRZ, the same fresh fuels are loaded at the beginning of operation. This kind of a reactor core is referred to as initially-loaded core, and it should possess high reactivity. As operation cycle proceeds, one fifth of fuel assemblies are discharged after the end of each cycle when the number of batches is 5. After five-cycle operation, all the assemblies had experienced fuel exchange, so a reactor core reach an equilibrium state. Such a reactor core is referred to as an equilibrium core. In this exercise, reactor physics parameters such as neutron effective multiplication factors and maximum line powers are calculated at the beginning, middle and end of equilibrium cycles, those are denoted as BOEC, MOEC and EOEC.

15 4 EXERCISE 15 Fig. 6: Cylindrical model of middle-size fast breeder reactors Please perform burnup calculations for this middle-size fast reactor core with FRBurnerRZ, prepare answers to the following questions and summarize them in a ppt file. You do not have to prepare a high-quality ppt file; you have only to show your results to me. If you want to see multi-group cross section data of nuclides, please ask me directly. To do this exercise, you have to understand nuclides transmutation process during nuclear fuel burnup. Figure 7 shows nuclides burnup-chain for major heavy nuclides used in the SRAC-2006 code. A burnup-chain used in the present exercise is a bit different from this, but this might be helpful for you. Fig. 7: Nuclides burnup-chain for major heavy nuclides of the SRAC-2006 code 1. Please plot average macroscopic fission production cross sections of each of inner/outer cores and radial/axial blankets. X-axis and y-axis should be neutron energy and cross section in log-scale. These data can be obtained by a method PrintMacroXS. Also please mention a difference of this cross section between inner and outer cores. This cross section of inner core should be smaller than that of outer core. Please describe this reason from a viewpoint of reactor core design. Furthermore, please compare this cross section between inner (or outer) core and blanket. You might see a significant difference between these two, so please describe this reason. 2. Please plot effective neutron multiplication factors and maximum line power with operation days. Note that the number of exchange batches is five for both fuel assemblies and radial blanket assemblies. Also please describe time-dependence of maximum line power. You can see this by using the PrintLinepowerMap method.

16 4 EXERCISE Plot effective neutron multiplication factors with operation days when the number of exchange batches is changed to four and three. Also please describe what happens when you change the number of batches. You can change the number of batches by using a method PutBatchNumber. 4. Please adjust plutonium enrichment of inner and outer cores so as to make effective neutron multiplication factors slightly larger unity (1.000 to 1.003) at EOEC, and to minimize maximum line power at BOEC, MOEC and EOEC. The number of fuel exchange batches is five. Note that a reactor core prepared here is referred to as a reference core. You can adjust plutonium enrichment with a method PutPuFissileEnrichment. 5. Please compare neutron flux energy spectra at EOEC of the reference core in inner/outer cores and radial/axial blankets. You can choose representative medium for each of these assemblies. Also please describe differences in neutron flux energy spectra among these four regions, and reason of these differences. 6. On the reference core, please plot number densities change of U-238, Pu-239 and Pu-241 with operation cycle in inner/outer cores and blanket. In figures, x-axis and y-axis should be operation cycle and number density. 7. Please adjust plutonium enrichment of inner/outer cores for the reference core, but U-235 enrichment should be 5.0wt% and 10.0wt% for inner/outer cores, respectively. As the reference core calculation, effective neutron multiplication factor at EOEC should be slightly larger than unity, and maximum line power should be minimized. Also please compare effective neutron multiplication factor and maximum line power of this core with those of the reference core, and describe reason of difference. 8. Please adjust plutonium enrichment of inner/outer cores for the reference core, but you have to consider longcooling fuel, in which Am-241 is generated from Pu-241 decay; you have to make weight percents of Pu-241 and Am-241 (TRU weight percent) 7%. Core design conditions are same as the above questions. Also please compare effective neutron multiplication factor and maximum line power of this core with those of the reference core, and describe reason of difference. 9. Please attempt to design a reactor core which does not use any plutonium isotopes based on the reference core. Different U-235 enrichments can be assigned to inner/outer cores. If you can design such a Pu-free core, please compare effective neutron multiplication factor and maximum line power of this core with those of the reference core, and describe reason of difference. Note that blanket fuel should be the same as that of the reference core. 10. Please attempt to design a reactor core to maximize operation cycle length by changing U-235 enrichment, plutonium enrichment and TRU composition. Note that blanket fuel should be the same as that of the reference core, and maximum line power in an equilibrium state should be smaller than 360W/cm. If you can design such a long-lived core, please describe why you can attain this in comparison with the reference core. Furthermore, please do the same thing under an additional design condition that burnup reactivity loss (reactivity difference between BOEC and EOEC) is less than 0.04 k. 11. Please calculate reactivity induced when coolant in each of inner core, outer core, radial blanket and axial blanket is voided; you have to calculate four reactivity. Some of results should be positive and the others are negative. Please describe this difference.

17 A DETAIL OF MODEL SIMPLIFICATION TO TWO-DIMENSIONAL CYLINDER 17 A Detail of model simplification to two-dimensional cylinder In the following, a method to simplify a fast reactor core shown in Fig. 8 to cylindrical geometry is described in detail. In this figure, a coarse regulation rod depicted as C is located at the first layer position, and the second, third, forth and fifth layers surround it. Description of the sixth layers and others are omitted here. Fig. 8: Fast reactor core which is simplified to cylindrical geometry When one simplifies this reactor core to two-dimensional cylinder, he has to prepare a cylindrical ring to preserve an area of each layer. This concept is briefly described in Fig. 9. In this figure, a fraction (the first layer to the fifth layer) of cylindrical reactor core model is shown. On the fourth layer, control rod assemblies are loaded in addition to fuel assemblies. For this kind of layer, it is decomposed to three sub-layers, such as fuel, control rod and fuel sub-layers. A black ring in this figure corresponds to the control rod sub-layer. On the fourth layer, fuel sub-layers are located inside and outside of the control sub-layers. Ratio of in- to out-fuel sub layers areas is arbitrary, so the same areas are assigned to the in- and out-fuel sub-layers. Fig. 9: Fast reactor core in cylindrical model

18 A DETAIL OF MODEL SIMPLIFICATION TO TWO-DIMENSIONAL CYLINDER 18 Finally, a material map, which is an array mat in the program, in two-dimensional cylinder model is shown in Fig. 10. An array xl defines width of each layer. Colored frames in this figure corresponds to fuel layer in Figs. 8 and 9. On the fourth layer in which fuel assemblies and control rod assemblies are located, the same medium indices (6, 7 and 8) are assigned to in- and out-fuel sub-array. Fig. 10: Material map of a reactor core simplified to cylindrical model