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1. O lt 2 x ES ae o Aluminum 0 UA AT Y yo y o i 0 0 5 1 0 15 20 25 30 3 u thickness of platelet on each side of foil mm Fig 2 Change in 08 g as function of the thickness of material surrounding foil SNEAK 7B 59 1 24 R 8 1 20 x 1 16 1 12 1 08 1 04 1 00 0 96 Unat Unat U nat U35 Unat U Unat depl x Experiment KAPER Calculation Fig 3 Fine structure of fission rate Ryg in SNEAK 8Z 60 measurement position nS ee ee measurement position 1 graphite Fig 4 Principal cell structure of SNEAK 5C 61 Ap p g 90 80 70 60 40 N _ bees 5 zos E SEE 90 50 10 Thickness of U238 sample in g cm 0 y measurement with sample in graphite I measurement with sample in U nat sample in graphite heterogeneous perturbation Sample in U nat calculation KAPER Fig 5 Central Reactivity Worth of U238 in SNEAK 5C Fig 6 Exp Platelets in horizontal position b E a vertical D Calc Diffusion perpendicular to platelets cross sections e Dy parallel u a En KAPER N Cross sections from a acia ig cas 10 20 zi a 50 60 2 480 80 100 void height Zicm Integrated axial sodium void reactivity effect for different platelet orientations SNEAK 9B Z9 63 VII C
2. IHI MORE NUNPER ENORM 22 1 Flux and adjoint distributions are to be superimposed on a cosine group depend ent curve This curve is defined by the bucklings given on card K4 or K5 of phase 99998 as f x cos Bx where the center point x 0 is located at the center of the cell containing the sample 0 Option not used 1 Perturbed flux calculated and used in perturbation calculation O Unperturbed flux as calculated in phase 99998 used in perturbation calculation 1 Cross sections and or region dimensions are to be changed for the perturbed cell as they are transfered from phase 99998 0 Option not used 1 A void is specified in one or more regions of the perturbed cell with card K5 When MORE 1 external unit number with new cross sections and or region dimensions for the perturbed cell as prepared from phase 99999 For MORE 1 the parameter NUNPER is meaningless This variable is used to normalize the reactivity values to the results of a multi dimensional calculation Normally ENORM is equal to the following 23 S 0 s o A ee ee d r S r S 1 where S 1 and S r are the fission and adjoint sources respectively and the integration is over the reactor core and reflector SI If MORE 1 go to card K6 otherwise input card K5 K5 VOID I I 1 WW VOID I 0 0 for the I region in which a void is desired in the unperturbed state ot
3. Of 3 FREEFO called from MAIN This subprogram reads all card input without control of format and transfers the data to a temporary unit assigned the number 8 All subsequent read statements for card input refer to this temporary unit LILFIN called by DUMMY3 DUMMY1 DUMMY2 LILFIN determines the next segment of the overlay program re quired to continue the calculation LILHOL called from GRREAP Used to place a literal constant in a particular variable address GROOCP called from GRREAP Reading of group microscopic cross section data from the cross section library REACT called from HETERO and HOMOCA In this subroutine space dependent reaction rates in the cell are calculated and printed In addition the integral flux and cell averaged reaction rates are calculated YZ3 called from SLAB ASLAB and SLOOI A routine for the calculation of the difference of the infinite sums of exponential integrals of order 3 32 E3 called from SLAB ASLAB PE3 SLABP DEMO SLOOI SLOO2 COLL Calculation of exponential integrals of order 3 Ey 0 23 called from YZ3 Infin te summation of exponential integrals of order 3 PE3 called from EZ3 Evaluation of exponential integral of order 3 for small arguments E2 called from 23 YZ3 EZ5 Calculation of exponential integral of order 2 E x PFUNC called from YZ3 The probability of escape without a collision from a slab DU
4. where De and D are defined as cell flux averaged quantities Since the fluxes needed to calculate E and D are not known a priori the calculation of the 45 scaling factor is included in the outer iteration procedure in the solution of the flux equation In some experiments performed in critical assemblies the anisotropic effect of diffusion resulting from the orientation of the platelets composing the assembly plays an important role To account for this effect the DB term in the scaling factor is calculated as DB D where D is the component of the diffusion coefficient perpendicular to the platelets and D is the component parallel to the platelets The diffusion coefficient components are calculated with the equation Lo aaa een n n nm k m er D SS SS SES SES Sr 2 12 V 0 non w us where Egy is the transport cross section in the mean and Pun k are directional collision probabilities These probabilities are related to the normal collision probabilities by 3 P 2 3 P P nm 1 nm 11 nm in infinite slab geometry The diffusion coefficient Eq 2 12 was derived from integral transport theory by utilizing the mean square distance relation for the diffusion area This procedure is equivalent to the results of Benoist 7 if one neglects the angular correlation terms which are considered to be small in thin plate type fast critical assemblies For the collision probabilities in Eq 2 12 the r
5. Argonne National Laboratory ANL 7645 April 1970 7 P Benoist Streaming Effects and Collision Probabilities in Lattices Nucl Sci and Eng 34 285 307 1968 8 K B hnel and H Meister A Fast Reactor Lattice Experiment for Inves tigation of ko and Reaction Rate Ratios in SNEAK Assembly 5 Kern forschungszentrum Karlsruhe KFK 1176 1970 9 E Kiefhaber Unpublished note Kernforschungszentrum Karlsruhe 1969 10 R B hme et al Experimental Results from Pu Fueled Fast Critical Assemblies in National Topical Meeting on New Developments in Reactor Physics and Shielding Sept 12 15 1972 Kiamesha Lake N Y CONF 720901 53 Table I ko for the SNEAK 5C Simplified Cell Relative Thickness of the cell 0 quasi homogeneous aie Atami trary 0 9627 0 9628 0 0298 0 9828 0 9825 0 9849 0 0507 1 010 1 0094 0 0814 0 0758 k values in parentheses are values before iteration on the source densities Table II Dependence of k on the background cross section of 384 en SNEAK 5C ZERA 4 Simplified SNEAK 5C KAPER Soe aa o 0 9404 Self shielded with 08 Self shielded 0 9504 0 0100 0 9414 1 0211 0 0797 with y t8 KOM is defined as k of the quasi homogeneous cell wherein the cell thickness is multiplied by 10 and oer is defined as k
6. 4 ee l L Xj n Pn D E s ES NOR DAR n and 3 g g k tk paa l VEE oV I E o O E E E SS In these equations it is assumed that the effect of the reactivity sample in the calculation of the normalization integral is negligible Therefore the perturbed fission source is replaced by its unperturbed value For small sample reactivity worth for which the program is designed to handle this approximate is quite valid The factor Dyor is calculated for a normal assembly cell F o is the normalization integral normalized by the neutron and importance source at the center of the assembly The integration in F o is over the entire assembly core and reflector This factor is obtained in an independent calculation such as a two dimensional diffusion calculation and is used as input data to KAPER 5 Numerical Results In this section results of a number of various calculations are given which illustrate the versatility and validity of the methods utilized in the KAPER program In some instances the results are compared to those of other computer programs and in other instances to direct experimental results The results selected as illustrative examples depend primarily on the calculational model and only to a second order on the cross section data used 5 1 Heterogeneity Calculations To demonstrate the improvement of the methods used in KAPER over those in ZERA see Section 2 a series of k calculations were performed for a cell simi
7. Multigruppen Zellcode Dieses Programm erm glicht die Analyse von Messungen an heterogenen kri tischen Anordnungen in Plattengeometrie Der Bericht ent h lt eine genaue Beschreibung der f r das Programm erfor derlichen Eingabedaten der Programmstruktur sowie der im Programm verwendeten theoretischen Methoden Table of Contents II III IV VI VII VIII Introduction Input Card Description Computer Core Space Requirements Control Cards and Deck Structure Program Structure and Operation Computational Methods Utilized in KAPER Cross Section Library and Isotope Name Convention References Page 24 26 29 40 63 69 KAPER Lattice Program for Heterogeneous Critical Facilities User s Guide This report is to serve as a documentation of the KAPER pro gram as it exists since January 1973 Included in this docu mentation is a detailed description of the input data required for the program the structure of the program and an outline of the theoretical methods utilized in the program The program was developed to function on the IBM 370 within the Karlsruhe NUSYS program system However the program can be easily used outside of the system without any internal changes in the code In addition sufficient details are given so that for example interfacing with other codes as multi dimensional diffusion theory programs can be performed The theoretical methods utilized ar
8. Number of isotopes for which reaction rates are to be calculated NOR lt NIS of card K5 The cross section types SFISS fission and SCAPT capture are calculated along with the cell cross sections and are stored on unit NAP see card K9 for the isotopes spe cified by IS I see also card K9 NCHI a If NCHI 0 the internal fission neutron spectra are used in all calculations b For NCHI gt O specifies the number of fissionable isotopes for which a fission spectrum is read in as input cards KIO and Kll c For NCHI lt O new fission neutron spectra having a Maxwellian shape are to be calculated for one of the following reasons 1 The energy group structure of the cross section set used is different than that of the Karlsruhe sets 2 The temperature of the fissionable isotopes is to be changed from that given internally in the program 3 A combination of 1 and 2 Within KAPER are fission neutron spectra for the fissionable iso topes These spectra have a Maxwellian shape VE exp E T AE TVaT where g is the energy group index The temperatures T for the various isotopes as given in KAPER are OOD al A 2335 1 31 MeV 7 A 2350 1 30 MeV 23607 1 31 MeV 2383 1 35 MeV dp 1 41 MeV pu 1 39 Mev pn 1 34 MeV 242 1 39 MeV The energy group structure of the spectra is that of the 26 groups in the Karlsruhe cross section sets see Ref 2 and Section VII K4 GAMI
9. of exponential integral of order 5 E5 x EZ5 called from YZ5 Infinite summation of exponential integrals of order 5 PES called from EZ5 Evaluation of exponential integral of order 5 for small arguments E4 called from 25 Calculation of exponential integral of order 4 E 0 YZ5 called from BENOIS A routine for the calculation of the difference of the infinite sums of exponential integrals of order 5 DUMMY called from MAIN This subprogram provides the control for this segment of the overlay program The input cards for this phase of the calcu 38 lation are sampled and the core storage needed for the problem is determined The relative addresses of the variables are cal culated and passed to the subprogram of this segment CELLO called from DUMMY1 This subroutine calculates the space requirements for the collision probabilities PERT called from DUMMY 1 The input cards are read for the calculations as well as the data cross sections and fluxes from the phases 99999 and 99998 Printing of initial variables is performed DEMO called from PERT The normalization integral perturbation denominator is calculated for the cell under the assumption that the flux and adjoint have a curvature cosine away from the cell as given by the Buckling PEFLUX called from PERT The perturbed flux in the cell containing the sample is calculated The environment of the cell containing the sample i
10. of the full cell y6 Table III Dependence of the effective cross sections of I on the background U KAPER SNEAK 5C simplified cell region 5 238 cross section of Self shielded cross sections of 23 Background cross section of 2387 b difference Z difference Group 14 1 00 2 15 keV Infinite Dilution 3 929 2 324 selirehteldeg 10 6 3 778 3 8 DATE E 5 7 with 8 i Self shielded 19 9 3 794 0 4 2 206 0 6 with Es Group 16 215 465 eV Infinite Dilution Self shielded with 0 p8 Self shielded i o with t8 Group 18 46 5 100 eV Infinite Dilution Self shielded 1 O with p8 Self shielded ej with t8 GG 56 Table IV Reactivity Worth Ratios in SNEAK 5C Position I Sample Weight Position 2 e 5 1 135 0 019 Experiment Calculation KAPER 1 146 5 3 440 0 430 60 1 545 0 251 3 1 635 0 092 Fe 0 3 0 440 0 109 3 127 1 310 430 57 5 or 344mm Aluminum Uranium foil 5 Cells Uranium foil reference 5 Cells SA 1 6 or 3 2mm Stainless steel Uranium foil Fig 1 Spectral indices measurements in SNEAK 7B 10 Calculation with KAPER o 2 7 Measurement v i i 2 o 84 Le e lt 3 9 i 2 Stainless steel Ww zi p 6 S f I z un 5 x 2 Ll dl dh o
11. one calculates with less than 26 groups 64 In assigning fission neutron spectra to the various fissionable isotopes encountered within a given problem the program searches the isotope lists for recognizable fissionable isotopes For this purpose the isotopes recognized by the program and assigned the correct fission neutron spectrum are the following Th TH320 233y 12330 SE 2340 2357 2350 U SAO U 5B0 236y y2360 238 2380 U 840 U 880 239 Pu PU390 PUOBO PU2BO PU9AO PU9BO 24054 PU400 ale PU410 242 PU420 Faas 8 239 The several possibilities for the isotopes y 23 U and 3 Pu allow for the inclusion of several cross section data sets for the same isotope but differing in their respective temperatures Doppler broading of the cross sections However when assigning a fission neutron spectrum to a particular isotope using the in put option on cards K10 and KI of phase 99999 the program only 65 recognizes the first identification of the isotope as given in the list above or as spelled out on card KIO of phase 99999 Therefore on the input cards K5 and K6 all possible combina tions above may be used but on input cards KIO only the first given isotope identification name given above may be used The energy group limits for the 26 group structure used to gen erate the built in fission neutron spectra from the Maxwellian distribution given on page 6 are shown in the following tabl
12. procedure outlined above as used in KAPER is the first step of a rapidly converging iterative method which consists of using the self shielded total cross sections Eq 2 9 to obtain the background cross sections in Eqs 2 7 and 2 8 and then repeating the entire proce dure In this manner one could derive a consistent set of self shielded cross sections the KAPER cross sections are very close approximations to this consistent set The source densities Spy which appear in Eq 2 9 are first approximated in KAPER by the fully self shielded total cross sections Sa y N 0 0 0 2 10 v m 44 An iteration on the source densities can be carried out if required How ever in most cases the first approximation Eq 2 10 is sufficiently good The heterogeneous flux distribution within a unit cell is calculated by solving Eq 2 4 with the cross sections defined by Eq 2 9 The equation is solved by the power iteration method Convergence is assumed when the following condition is fulfilled in an outer iteration 2 1 1 ne De lt e a gern i as ERE th 3 where a is the total cell fission source for the i iteration The equation adjoint to Eq 2 4 z8 o8 SP PK Be otk 2 11 t n nm s f m n m k n n is solved in an analogous manner Because the collision probabilities in all the equations are calculated exactly there is no assumption made about the flux or neutron current at the boun
13. 0 0 GAM2 30 0 GAM3 0 0 Recommended values EBS 0 02 EBS2 0 005 These variables of card K4 are utilized in the calculation of the resonance self shielding of the cross sections The first three variables are used to define the fictitious background cross sections bj in the approximation P mn see Eq 2 8 in Section VI NB is defined on card K3 The background cross sections are defined as b Lo ap 71 i 2 where ALP GAMI came GAM3 lt o gt CES NB 1 t The last two variables on card K4 are used to test the degree of resonance self shielding of the isotopes In the energy groups in which the condition 0 0 1 0 lt EBSI is satisfied where 0 0 is the total self shielding factor for the fully self shielded condition the resonance self shield ing is handled as in the homogenized cell For the case that the condition 0 750 1 0 lt EBS2 is satisfied the resonance self shielding is completely neglected K5 NIS Total number of isotopes NIS lt 20 ISOT I I 1 NIS Isotope names as specified by the con vention of the cross section set e g aPU9A0a 5 character name See Sec tion VII for clarifications K6 NISS Number of isotopes for which the influence of the cell structure on the resonance self shielding of the cross sections is to be considered The resonance self shielding of all isotopes of card K5 which are not given again on card K6 i
14. 0 0 0 0 0 0 0 1 39 0 025134 0 09918 0 1842 0 2610 0 1953 0 1372 0 0603 0 02378 0 009394 0 003078 0 001434 0 0 1 41 0 02656 0 1021 0 1862 0 2605 0 1933 0 1352 0 05921 0 0233 0 009201 0 003014 0 001404 0 0 67 SUERPUTINE SPREAF ORSI SNOG Qa STE Tet SZy4NOG TOS4FSTE C Cka PEOCRAM TC READ CPOSS SECTIONS FECM LIRPARY FOR KAPEF Cee FNR JEE NUTSIDE OF TRE KAPLSRUHE NUSYS FROGRAM SYSTEM C PELAR ISOT 20 GP SNC 2 TYP ATEXT SC BTEXT USO ISFTISChNC 20 ISC CIEMS ION SNOGO7sT SZ 4 NOG 4QUIEZ 49 10 y SMTCTCISZ NOG NCG X JUS 20 FSTFUNOG 20 COMMON ATEXT ISOT y ISOM BTEXT K3 C1 5C COMMCN M 07 0 NE2 NG M E GNIS INf C NS f 4 RFNIND NSF PRINT 2002 2002 FOFMAT 7 XCRLOSS SECTION CATA FROM SPECIAL LIPPADYA at CM 160 15 1 157 100 JTS IS 0 16 2 NNG C Caro EXPLANATION CF CROSS SECTICN DATA CEQUISED FROM LIBRARY lack C FOUR REFOBDS PEQUIPED FOR EACH TSNTNPE c fas it si a c ISO IS A PEAL B IDENTIFICATICN CF ISCTOPE FER EXAMPLE IJ2280 GGG EGO ISO SES DEDE PHO RECORD DARA A lee ale a je fe ak akc 3K ME Mei mmn mn Mm CD EY I A Ar Am er REA And To fm MM u A ME An ya ME nD Mn ey AA A A A A A ry SD ER a oe m m mm ns et QUES yJebl MATRIX IS TSPIPPE INDEX t 1 TO NOG TRE EMER CY GPOUP DEPENDENT VALVES CF THE FFLLOwIAG CRESS SECTIONS WHERE NPG IS THE NUMBER OF ENERGY GOrues J l Q NU STGMA FISSTION INFINITE DILUTION J 2 Q STGMA ABSOPPTICN CINFINT TE D
15. 1 This is a routine for the determination of the coefficients in the fitting of a series of partial factions to the collision probabilities PMC called from PPM761 In PMC collision probabilities for use in the determination of effective cross sections are calculated cu This is a BLOCK DATA routine used to initialize a common block having fission neutron spectra with corresponding Maxwellian temperatures and the energy group limits of the 26 group struc ture of the Karlsruhe cross section set M10760 Common block used in the segment controlled by the subprogram DUMMY 3 35 DUMMY2 called from MAIN This subprogram provides the control for this segment of the overlay program The input cards for this phase of the calcu lation are sampled and the core storage needed for the problem is determined The relative addresses of the variables are cal culated and passed to the subprograms of this segment HET called from DUMMY2 All input data cards and cross sections from PPM761 are read in by HET Printing of initial variables for the problem is performed HOMOCA called from DUMMY2 This routine performs a zero dimensional homogeneous calcula tion of flux and adjo nt QUERP called from DUMMY2 Homogenized cell cross sections are calculated and stored on an external unit for later use in multi dimensional computer codes The homogenized cross sections include macroscopic cell cross sections avera
16. APER was set equal to the total unshielded cross section o g Whereas the k obtained by Kiefhaber with ZERA Table II depends strongly on the background cross section this dependences is very weak in the KAPER results In fact the KAPER kygr depends on the background cross section in much the same manner as kyom which is addi tional evidence for the conclusion that the approximations in the KAPER program are similar to the approximations used in homogeneous calculations Further more as the resonances of 238y are strongly self shielded one would expect better results from ZERA by using 0 g rather than O g as a background cross section This expectation is borne out by the results shown in Table I where the ZERA k is fairly close to the true KAPER k The influence of the 2a background cross section on the self shielded cross section of 239Pu is shown in Table III Though the change in the back ground cross section is very large ogg is certainly an extreme overestimate the changes in the self shielded cross sections are small except in group 18 Thus it is demonstrated that the cross sections as defined in this paper are insensitive to the background cross sections 5 2 Calculation of Cell Reaction Rates In the hard spectrum core of SNEAK 7B 10 studies were made on the effect of structural materials stainless steel and aluminium on spectral indices measured with foils In addition the measurement was designed to provide a v
17. Gruppendarstellung KFK 770 EUR 3953d April 1968 R B Kidman R E Schenter Group Constants for Fast Reactor Calculations Hanford Engineering Development Laboratory HEDL TME 71 36 March 1971
18. ICES C AND IS ISOTOPE INDE X C ee ae ee ees SMTMTITS JG TGI STEMA TNELASTTC SCATTESING MATRIX PLUS C ZoC SIGMAW i92 MATRIX FROM GROUP TG IT JG C nce ey ane e A un A A A mn nn A nn A ee eee C 99 CONTINLE RELCINSF END 1010 JISN CO 110 fS 1 782 TFITSN EQ TSOTETS I GN TN 12C 110 CONTINUE RELTANEF PEAGINSF REACINSF GO TO 99 120 INC INC JISCIND I TS PELTINSF COCTSsJelh 19 JS 194 9L 14 IGZ LEAD INSE SMNOG ly 1S 16 1 21 7 1G 1 N0G FSTF I1G 1S 1G 1 NCG PEACINSE SMTOT TIS 36 16 1 JG 19 NCG pI 6 1 41106 CO 200 IC 1 NCG SUM C e C9 170 JG 1 NAG 170 SUM SUM SMTPT IS 1G 416 SMTOT IS IG IGI SNOG 7 IS IE I SUM O01 Ss 2 TGI SNCG 4 IS IG X SMTOT TS 16 16 200 CONTINUE GO TO 59 1010 IF IND EQ ISZ IGN TO 289 PFINT 2004 2004 FOFMAT xeNOT ALL OF THE ISOTOPES PEQLESTFO AFE FFJACL IN THE L XT ER AR Yao CO 266 J 1 ISZ CO 265 K 14 IND TF ISTS K 19 FQ TIGP TO 27 265 CONTINUE PRINT 20059 150T1 1 2005 FOFMAT ISOTOPE NOT FOUND TSty2x A8 270 CONTINUE 260 CORTINUE PF INT 20C6 2006 FORMATS 9 K PROGRAM STAPXX ET STCP 250 PETURN EME VIII 11 2 3 69 References P E Me Grath and E A Fischer Calculation of Heterogeneous Fluxes Reaction Rates and Reactivity Worths in the Plate Structure of Zero Power Fast Critical Assemblies KFK 1557 Marz 1972 H Huschke Gruppenkonstanten fiir dampf und natriumgekiihlte schnelle Reaktoren in einer 26
19. ILUTION J 3 N SICMA ELASTIC SCATTERING INFINITE DILUTICAN J 45 Q STGMA TRANSPOPT INFINITE DILLTIEN Tepe A a D aaa pm m A A ee es eA EN Mr Mm Mr SH Ne AT nk Mc I a An Ey m ME A ME me m ec a Ae ey m ER m m m SELF SHIELDING FACTOPS FOR THE RESPECTIVE CPASS SECTTFNS J l TC 4 FOF VAR INUS BACKGPNUND CROSS SECTIONS L NNCaL TO 2 NOG Q F SIGMA FACKGRCUNND 0 0 L 2 NO0G 1 TO 3XNNG Q F SIGMA BACKGROUAD 1 C0EF 1 L 3 NNG 1l TO 4 N0Gy N F SIGMA BACKGROIUND 1 0F 2 L 4 NOG 1 TM EXNOG N Ft SIGMA RACKGREIUND 1 0E 3 L 5 NOG TO ERNNG Q F STGMA RACKGROUND 1 0E 4 L 6 NM0G 41 TO 7 amp NNG Q F SIGMA BACKG CUND AL OFF5 L 7 NOG L TO BXNNG Q F SIGAA BACKGRPDIUNO 1 CE 6 CO lata aaa se oka REEMRD Boalo DIGI IGRI RoR tok ak x Ok aC HOI K OK X BAMAMDANAARAADANRMAANNAANBHAVBAAHDA c I 1y SMHOG SIGMA TOTAL INFINITE DILUTION c I 2 SNNG SIGMA TRANSPOPT CINTFINT TE DILUTTON 68 I 3 SNOG SIGMA INELASTIC SCATTERING FLUS STGME My2N 4 SHOG STGMA ELASTIC FEMMVAL T 5 SUOG MU AVERAGE COSINE CF THE ELASTIC SCA amp TTESTNE ANCLF T 6 SMOGSNU NUMBER OF MEUTFMNS PER FISSIEN T 7 SNNG STGMA TOTAL GROLP REMCVAL INFINITE DTUITICA FSTFCIG IS MATRIX IG ENERGY GROLD INDEX IS ISCTLPE INDEX FSTF TMTAL CROSS SFCTICN SELF SHIFLDING FACTOR Fre RECK GROUND CROSS SER TICN EQUAL ZERS CORO aaa lalola IOI Sk ROR la CNA OOOO DE IO SHORE Rp IOI ICR HK OA ASOMO C SMTOT TSJC TG MATRIX IG AND JG ARE ENERGY COCUP TNC
20. KFK 1893 les A S 2 5 S 9 2 x 5 D Q 7 um CE 5 O z 3 i D s 5 lt o 2 x un JE 5 I 2 T z E U S N 25 z 4 0 oH e 2 D o em mo Pr E 33 y gt 4 i s 8 13 pe La j a X o Y E 35 x wu o 0 r N DD a N A LL e IS 3 a as a Se an Als Manuskript vervielf ltigt Fur diesen Bericht behalten wir uns alle Rechte vor GESELLSCHAFT FUR KERNFORSCHUNG M B H KARLSRUHE KERNFORSCHUNGSZENTRUM KARLSRUHE 1973 KFK 1893 Institut fiir Angewandte Systemtechnik und Reaktorphysik Projekt Schneller Briiter KAPER Lattice Program for Heterogeneous Critical Facilities User s Guide by P E Mc Grath Gesellschaft fiir Kernforschung mbH Karlsruhe KAPER Lattice Program for Heterogeneous Critical Facilities User s Guide Abstract This report is a documentation of the KAPER program which is a multigroup lattice code developed to analyze experi ments performed in plate type heterogeneous critical fa cilities Included in this documentation is a detailed description of the input data required for the program the structure of the program and an outline of the theo retical methods utilized in the program 20 11 1973 KAPER Zellprogramm fiir heterogene kritische Anordnungen Benutzer Anleitung Zusammenfassung Der vorliegende Bericht ist eine Dokumentation des KAPER Programms einem
21. MMY3 called from MAIN This subprogram provides the control for this segment of the over lay program The input cards for this phase of the calculation are sampled and the core storage needed for the problem is determined The relative addresses of the variables are calculated and passed to the subprograms of this segment 33 PPM760 called from DUMMY3 Reading of all input data on cards for problem in phase 99999 is performed in this subprogram Some checks are performed on input data GRREAP called from PPM760 Assignment of locatiom for cross section data from the cross sec tion library is done as well as the calculation of certain cross section types not given directly in the library PPM761 called from DUMMY3 This large subroutine calculations the heterogeneous resonance self shielded cross sections and stores them on an external unit for use in subsequent segments DUMMY or DUMMY2 SPEK called from PPM761 Determination of location im COMMON CH of fission neutron spectra for the individual fissionable isotopes in a calculation NSPEK called from PPM761 NSPEK calculates fission neutron spectra from a Maxwellian distribution INPOO called from PPM761 A routine for the interpolation of resonance self shielding factors 34 INDX called from PPM761 This routine is for indexing of reaction coefficients used in the calculation of effective cross sections BRB called from PPM76
22. asurements The fine structure of the cell is slightly overestimated by KAPER These results are fairly representative of the agreement in a wide range of measurements 5 3 Analysis of Reactivity Worth Measurements A series of small sample reactivity worth measurements performed in SNEAK have been analyzed with the methods described in this paper These include measurements performed in SNEAK 5C 8 The assembly 5C of SNEAK was a null reactivity core with a soft spectrum and strong heterogeneity effects as ex plained previously The sample reactivity measurements were performed in two positions within the unit cell of the core as shown in Fig 4 For comparison with experiment the ratio of the sample worth in position 1 to that in position 2 is used This eliminates the uncertainty associated with Beff and the absolute magnitude of the normalization integral The results of these calculations are given in Table IV This is quite clearly an example of where first order homogeneous perturbation as normally used to interpret these types of measurements fails The measurements in this assembly are extremely position dependent and must be therefore interpreted with a pro gram having the capabilities of KAPER Except for the 240Pu sample the cal culated and measured reactivity worth ratios agree within the experimental errors The calculation of the sample size effect of the 238y samples for the two measurement positions is shown in Fig 5 It
23. ceeding calculation 2 2 in a one dimensional dif fusion theory program Input card K7 and K8 O Otherwise Minimum number of outer iterations for convergence on the eigenvalue IMIN gt 3 Maximum number of outer iterations Convergence criterion for the fluxes Convergence criterion for the eigenvalue koe S1 K4 S2 K5 53 K6 15 If NB 0 go to S3 otherwise go to K4 if IB or to K5 if IB gt 1 If IAN 1 cards K4 and K5 consist of two cards each The first and second cards are respectively the Buckling B2 parallel and perpendicular to the plate structure Normally as in the SNEAK assembly they are the radial and axial B respectively BSQ Go to S3 BSQ K K 1 NOG Universal Buckling Group dependent Buckling where NOG is the number of energy groups If KBSQ 0 go to S4 otherwise input card K6 MAXB AKE EBK FAB NBQ Maximum number of iterations on the Buckling to the specified keff Desired koff Convergence criterion for k aff Factor used to multiply the initial Bucklings for a second guess of the Bucklings recommended FAB 1 1 to 1 2 Group number arbitrary in which the Buckling is positive S4 If IHCS 0 end of card input for phase 99998 otherwise input card K7 K7 NPT IRR IM MN T MD T I 1 IM External unit number on which the homo genized cross sections are to be written NPT may be the same unit number as NXST i
24. ctions in a multiregion cell B phase for the calculation of the flux and adjoint distribu tions in the cell with associated reaction rates C phase for the calculation of heterogeneous small sample reactivity worths These segments overlay each other during the sequence of the calculation as given by the input cards Information is passed to each segment by means of external units which are declared by job control cards At present the program considers only infinite plate geometry No symmetry is assumed for the cell so that a cell specified in a calculation is assumed to be repeated infinitly in both directions II Input Card Description All data required on the following cards is entered format free The convention is that the first parameter of an input card labeled here as K4 for example must begin in the first col umn of the card The following parameters of the input card can be entired anywhere on the card up to and including col umn 80 with a minimum of one blank space between parameters Two or more cards may be used for the data of one input card K In the second and succeeding cards the first column of the card must be blank All literal data required must be entered between alphas a or apostraples 1 Initiation of a series of KAPER calculations The following three cards can be identical to those used in a NUSYS cal culation of the Karlsruhe program system Kl AKAPERO The field
25. dary of the cell The procedures used to evaluate the collision probabilities are based on the methods developed by A P Olson 6 for the RABID code In KAPER Eqs 2 4 and 2 11 are used in a slightly more refined form for the fission source The fission spectrum y8 used in multigroup equations should be a properly weighted spectrum derived from the contributions of the various fissionable isotopes present in the unit cell Generally the correct spectrum is not known a priori and must therefore be approximated To circum vent this problem the KAPER program uses a fission spectrum for each particular fissionable isotope present in the unit cell and calculates the fission source in energy group g and region m as g k k X Wr Vin En eK a where j is the fissionable isotope index This representation of the fission source is particularly important in the calculation of reactivity worths Section 4 For example the use of a universal fission spectrum for the calculation of uranium sample worths in an assembly whose predominant fission source is from the plutonium isotopes can result in a 2 error even if the universal fission spectrum is correctly weighted The above representation of the fission source is used throughout the KAPER program For simplicity however the fission sources in this paper are written with a universal fission spectrum To account for leakage the collision probabilities are scaled in the following manner
26. e Group Upper Energy Limit ev Au 10 5 x 10 0 48 2 6 5 x 10 0 48 3 4 0 x 10 0 48 4 2 5 210 0 57 5 1 4 x 10 0 57 6 8 0 x 10 gt 0 69 7 4 0 x 10 0 69 8 2 0 x 10 0 69 9 1 0 x 10 0 77 10 4 65 x 10 0 77 11 2 15 x 10 0 77 12 1 6 40 0 77 13 4 65 x 10 0 77 14 2 15 x 10 0 77 15 1 0 x 10 0 77 16 4 65 x 10 0 77 17 2 15 x 10 0 77 18 1 0 x 10 0 77 19 46 5 0 77 20 21 5 0 77 21 10 0 0 77 22 4 65 0 77 23 2 15 0 77 24 1 0 0 77 25 0 465 0377 0 215 0 77 N On Group l 2 26 66 In the generation of the fission neutron spectra the assumption used in the KAPER program are that neutrons emitted above the upper energy limit of the first group are included in the first group and all neutrons emitted below 10 keV are included in the group containing that energy point These spectra as included in KAPER for the different temperatures which characterize the various fissionable isotopes are shown below Temperatures MeV 1 30 PEL 1 32 1 34 1 35 0 018609 0 019285 0 01995 0 021233 0 02194 0 08579 0 0873 0 0888 0 09179 0 09328 0 1742 0 1754 0 1766 0 1789 0 1800 0 2625 0 2624 0 2623 0 2621 0 2619 0 2045 0 2035 0 2025 0 2004 0 1994 0 1473 0 1461 0 1449 0 1427 0 1416 0 06567 0 06504 0 06441 0 06319 0 06260 0 0261 0 02582 0 02555 0 02502 0 02477 0 01035 0 01023 0 01012 0 009904 0 009799 0 003397 0 003359 0 003321 0 003249 0 003213 0 001584 0 001566 0 001549 0 001514 0 001498 0 0 0
27. e described in a reprint of a paper published in the proceedings of the conference on Mathematical Models and Computational Techniques for the Analysis of Nuclear Systems This paper is included in a section of this report and describes among other things a number of improvements which have been made in the program since the release of Ref 1 The most significant changes are the following 1 A procedure for the calculation of anisotropic diffusion coefficient in the cell 2 Fission source representation generalized to include indi vidual fission spectra for all fissionable isotopes of the cell This relaxes the assumption associated with a uni versal fission spectrum The user of this program is requested to communicate to the authors any difficulties or errors associated with the use of the program and errors or points needing greater clarifica tion in this documentation I Introduction The KAPER program is a dynamically dimensioned program in an overlay structure The computer core space needed for the pro gram is adjusted at the beginning of each calculation by spe cifing on the job control cards the amount of core space de sired The calculation of the space required for a problem is given in a section following the description of the input cards The three main segments of the KAPER program consist of the following A phase for the calculation of heterogeneous resonance self shielded cross se
28. ecord is written as shown below The form of the records are Record type Ry NAI SIGMA format free NA2 cross which SCAPT NUSF STR lt SREM SFISS CHI SMTOT constant 8 byte word section type name 8 byte word include the following total absorption capture plus fission nu sigma fission transport cross section total group removal fission always set equal to zero fission neutron spectrum scattering matrix IS column index for the scattering matrix for other NA3 MAGRO cross section types IS 0 constant 8 byte word MN 1 composition number assigned to this cross section set see card K7 NOG IG SX IG IG 1 N0G Record type Ry NAI format free NA2 IS NARR I MN 1 NOG 16 SX IG IG 1 NOG number of energy groups group index respective cross section value SRATE constant 8 byte word reaction rate cross section type name 8 byte word which include the following SCAPT capture cross section SFISS fission cross section O constant constant 8 byte word see card K8 composition number assigned to this cross section set see card K7 number of energy groups group index respective cross section value The order in which the records are written on unit NPT is the following 19 1 1 record containing two words IX 6 NO0G IM IRRe2 IM total number of records on unit NPT excluding the fir
29. egion optical thicknesses of the cell are calculated with the transport cross section rather than the total cross sec tion In this case one obtains the correct homogeneous limit of the diffusion coefficient in Eq 2 12 With the solution of the unit cell flux and self shielded cross sections the KAPER program is able to generate homogenized cross sections for use in a multidimensional homogeneous flux program to solve for the global parameters of the assembly 3 Calculations within a Local Perturbation of the Unit Cell In many instances measurements in a critical assembly involve a disturbance of the repeating unit cell of the assembly For example a portion of a cell may be removed for the insertion of a channel in which reaction rates are to E be measured with chambers or in a reactivity worth measurement a low density plate of inert material may be inserted between two plates of the cell at the position into which a sample is to be placed In both of these cases the period icity of the unit cell is disturbed We shall call th s cell containing the local perturbation including the surrounding unit cells in which the flux is significantly disturbed by the perturbation a perturbed cell As this is a very practical problem of interest to the evaluators of experiments performed in critical assemblies a capability of solving for the flux and therefore reaction rates in such a situation was built into the KAPER program To find
30. erification check for the method used in KAPER to calculate the flux and reaction rate distribution in a local perturbation of the assembly core The core of SNEAK 7B consisted of a simple unit cell of one mixed oxide platelet uranium and plutonium and one uranium oxide platelet For the measurements uranium foils were placed between the normal platelets of the cell to serve as a reference Additional foils were placed between stainless steel platelets and aluminium platelets of two thicknesses This configuration is shown in Fig 1 The addition of the stainless steel and aluminium represents a local perturbation in the normal repeating unit cell The results of the KAPER cal culations for this experiment are shown in Fig 2 These results are given as the percent change in the spectral index o g ogs5 with respect to the reference measurement The effect of the cross section set used in the calculation of the change of the spectral index was small 10 although not negligible In general the agreement with the experimental results is quite good As an illustration of the calculation of cell reaction rate fine structures the results of a measurement and calculation are shown in Fig 3 In this figure the cell composition is shown along the ordinate axis The assembly for this 51 measurement was a uranium null reactivity core Shown in Fig 3 is the calcu lated cell fine structure of 238U fission along with results of uranium foil me
31. f the data presently on it is no longer needed at this point Number of isotopes for which homogenized reaction rate cross sections are to be calculated The first IRR isotopes of the NOR isotopes specified on card K9 of phase 99999 will be treated IRR lt N R Number of homogenized cross section sets produced IM lt 3 It is possible to pro duce up to three sets in which all the cross sections are identical except for the diffusion coefficient as determined by MD I Number assigned to this set of cross sections Diffusion coef ficient for this set EnA 1 a a 1 D D perpendicular en 2 D D parallel The diffusion coefficient is not written on the unit NPT but rather the transport cross section is calculated from the diffusion coefficient specified and is written on TO S5 If IRR 0 end of card input for phase 99998 otherwise input K8 17 K8 NARR T I 1 IRR Material combination name 8 byte word assigned to the homogenized reaction rate cross sections The homogenized cross sections as specified on cards K7 and K8 are written on unit NPT in a form that can be read directly by phase 2004510 of the Karlsruhe NUSYS program system The variable MN I is the composition number of the cross sections in the SIGMA block of NUSYS which will be replaced by the KAPER cross sections For clarity the form of the cross sections on unit NPT is given For each cross section type one r
32. for the program IV Control Cards and Deck Structure The following is a list of the necessary control cards needed for a normal calculation In these cards a blank is denoted with _ In this example the compiled program is found on the tape DVO377 27 Job Card SETUP_DEVICE TAPE9 ID DV0377 NGRING SAVE SL EXEC _FHLG PARM L VLY PARM G XXXXXX TIME G XX L SYSUTI_DD_SPACE 3303 700 l SYSLMAD_DD_SPACE 3303 700 1 l KAPER DD_DSN KAPER UNIT TAPE9 V L SER DVO377 DISP GLD PASS L SYSIN_DD_x _ENTRY_MAIN _ INCLUDE _KAPER _OVERLAY_A _INSERT_DUMMY3 PPM760 PPM761 SPEK GRREAP INP _INSERT_INDX PMC BRB CH M10760 NSPEK _OVERLAY_A _INSERT_DUMMY2 _PVERLAY_B _INSERT_H MOCA _ VERLAY_B _INSERT_HET _OVERLAY_B _INSERT_QUERP _OVERLAY B INSERT HETERG TABLE ASLAB BENQIS SLAB SLABP E5 EZ5 _INSERT_PES E4 YZ5 _OVERLAY A _INSERT_DUMMY I 28 _OVERLAY_B _INSERT CELLO _QVERLAY_B _INSERT_DEM PEFLUX SLOO1 SLOO2 CPERT CQLL PERT _ OVERLAY B _INSERT_HOMPET G FTO8FOO1_DD_UNIT SYSDA SPACE TRK 9 G FTO4FOO1_DD_DSN GROUCO VOL SER NUSICE UNIT 3330 _DISP SHR plus needed units for data transfer G SYSIN DD Input cards o The declaration of unit 8 is always necessary since the input cards are transfer to it for reading by the program without format control The unit number 4 must always be declared as it is the cross section lib
33. g up the collision density balance within the imaginary boundaries we have the contribution of those neutrons which always remain within the bound aries and those that come from outside Due to the particular location of the imaginary boundaries there is no contribution to the collision density of neu trons that are leaving the perturbed cell and returning after one or more collisions as these are already included in the source coming from outside our boundaries Our equation would then read as follows v 8 o J pte E v ok 4 s 3 1 n t n mn s m m n n m k m m where Pa is the probability a neutron from region m suffers its first collision in region n while remaining within our imaginary boundaries The fluxes and cross sections explicitly written in Eq 3 1 are defined for the regions that compose the perturbed cell Therefore the first term on the right hand side of Eq 3 1 represents the contributions from within our imaginary boundaries and the second term s the contribution from outside the boundaries The source term SB has an appearance similar to the first term in Eq 3 1 except that the collision probabilities have a different definition We may write the source equation as B 8 k g g sk k S en alle ge ay Ses 3 2 m k m m 47 where Emp is the probability that a neutron in a region m outside our imaginary boundaries suffers its first collision in a region n inside the boundaries The cross section
34. ge fission neutron spectrum for the cell and reaction rate cross sections HETERO called from DUMMY2 The flux and adjoint in the heterogeneous cell are calculated in HETERO A neutron balance in the cell is also calculated 36 For the case that reactivity worths are desired the cell nor malization integral perturbation denominator is calculated TABLE called from HETERO A table of exponential integrals of order 3 4 and 5 are cal culated from two polynomials over a discrete mesh spacing and are stored in a common block for use by the collision probab ility routines In addition the weights and modes of the Gaussian Quadrature approximation to the infinite summation of the exponential integrals are initialized ASLAB called from HETERO The collision probabilities for a cell in an infinite repeating lattice are calculated BENOIS called from HETERO The components of the diffusion coefficient parallel and per pendicular to the plate structure are calculated using the first term of the Benoist formulation see Eq 2 12 in Sec tion VI SLAB called from HETERO The collision probabilities of a perturbed cell located in an infinitely repeating lattice of normal core cells are calculated 37 SLABP called from HETERO The collision probabilities within a single unit cell are calcu lated for neutronsoriginating only within the cell E5 called from PE5 and BENOTS Calculation
35. herwise set VOID I 1 0 where I 1 NW are the region indices of the perturbed cell It is possible to use VOID I to adjust the density of the ER region by simply putting VOID I equal to an appro priate value e g to reduce the density in the th region by a factor of 2 VOID I 0 5 Card K6 is repeated NSAM times with a descriptive title for each sample in the order in which they are calculated K6 NT Number of four character words in the description title lt 15 TITLE I I 1 NT The title must be placed between apostraples and can consist of any character except apostraples An example of this is shown 24 e g 7 SNEAK 900A WORTH FOR U 238 End of card input for phase 99997 III Computer Core Space Requirements The KAPER program requires 130K bytes of core storage for the overlay structure suggested in this guide plus N Ny or N whichever is the largest The values of the N s are defined by the following equations for each phase of the program 1 Phase 99999 cross section phase N ve ISENDOS 39 06 1081mm Jamie 2015 mmoc where NIS number of isotopes NOG NR fi number of energy groups number of regions 2 Phase 99998 flux adjoint and reaction rate phase N 4a NOGKNRE NOGHNRA 13 NFI 2 NRS NOG 12 NFI 8 NRe 5 NFI NRS IAR IAR where NOG number of energy groups 25 NRS number of isotopes for reaction rate calculations NRS NOR on card K3 in pha
36. ical interpreta tion A source neutron from a fission or scattering reaction is weighted by the source importance function while a colliding neutron is weighted by the colliding neutron importance function The last term in Eq 4 3 accounts for diffusion effects The form utilized in the KAPER program is however slightly different than Eq 4 3 We can rearrange the equation to obtain the following results af Blech FEM ceo tk tg P g D Ya SK SE Yo 824 ba Va gm m k m g 1 N SOUR Dun 88 5 0 4 4 m The only advantage to Eq 4 4 is that the first three terms can be identified as the normal absorption scattering and fission perturbation terms The perturbed flux in Eq 4 4 is obtained with the procedure explained in Section 3 In this case the disturbance in the unit cell is the inserted reactivity sample By utilizing Eq 3 1 to find the flux in and around the sample one accounts also for the perturbation due to the insertion of the sample in the sample environment Since the KAPER program is a lattice program the denominator of Eq 4 4 can not be calculated for the entire assembly core and reflector The calcula tion of the denominator or normalization integral as it is commonly called 49 is best accomplished with a multidimensional flux program Therefore the proce dure selected for the calculation of the denominator is as follows Do F DvoR where g 6 kok k A V_ re m m
37. ients A depend fairly strongly on the background cross section which is not well defined Presently the unshielded total cross sections are generally used as background cross sections Therefore an improvement in Wintzer s method was proposed and is used in the KAPER program The coefficients Axvmn are calculated as described above Then the effective self shielded cross sections for each reaction region and isotope are obtained from the equation 2 9 These cross sections are used in Eq 2 4 The collision probabilities Pan occurring in Eq 2 4 are calculated with the self shielded total cross sections The approximation of Eq 2 9 is good since the effective cross sections depend only very weakly on the background cross sections This was observed for a similar homogeneous case in 5 and is demonstrated for the heterogeneous cases in the results of Section 5 of this paper Thus it is possible to say that the self shielded cross sections are fairly well defined by Eq 2 9 It is for this reason that Eq 2 4 is used in KAPER rather than Eq 2 3 with its coefficients which are sensitive to the background cross sections The use of the self shielded total cross section Eq 2 9 to calculate the collision probabilities is an approximation which we could not derive from mathematical principles however we feel that the approximation is adequate for the purposes of practical calculations It should be noted that the
38. is free and can be used for identication of the job as for example oNORMAL CORE CELL OF SNEAK 900Aa K2 NFPH Number of first phase for KAPER calcula tion normally 99999 K3 QENDE a Constant 2 Phase 99999 Cross Section Phase In this phase the resonance self shielded cross sections for the heterogeneous cell are calculated and stored on external units for use in subsequent phases While most calculations will be for a true heterogeneous cell it is possible in this phase to prepare cross sections for a homogeneous medium This is accomplished by specifying a two region cell with the same isotope mixture in each region In addition card K6 would contain NISS 1 and ISON 1 aKEINEo The resonance self shielding is then calculated for the homogeneous composition as it is customarily performed in the program 0004460 of the NUSYS system of Karlsruhe To identify these cross sec tions as homogeneous for other phases of the KAPER program card K9 should contain NREG gt 1 K3 0999990 GRSN I I 1 3 NOG NB KGEO NFPH IH NOR Constant Group cross section set name 15 character name g a26 GR KFKINROO 1a Number of energy groups lt 26 see Section VII Number of terms recommended YB 6 in the approximation of the reaction coefficients by a series of rational functions See explanation on card K4 Free parameter no meaning at present Number of the following phase 5 Constant
39. is seen that the pro gram slightly underpredicts the sample size effect 5 4 Utilization of Cell Homogenized Cross Sections The importance of correctly calculating cross sections for the heteroge neous assembly is illustrated in the following example In an experiment to study the effect of leakage in an axial sodium void traverse the platelets of the assembly in the voided region were oriented first parallel to the direction of the traverse and second perpendicular to the traverse direction To analyze the experiment one dimensional diffusion theory perturbation theory was used The cross sections for the one dimensional calculation were generated by a routine at Karlsruhe which calculates self shielded cross sec tions using the f factor concept for homogeneous media In addition the KAPER program was used to calculate homogenized cross sections for the heterogeneous cell of the assembly For the two different platelet orientations the diffu sion coefficient was set equal to its component parallel to the platelets and to its component perpendicular to the platelets for the respective cases The results of using these cross sections in the one dimensional diffusion calcu lation are shown in Fig 6 The use of the homogenized cross sections signifi cantly lowers the calculated curve The effect of the leakage in the two dif ferent platelet orientations is quite nicely described by the diffusion coef ficients calculated by KAPER 6 Concl
40. lained later an alternative procedure to that above is used in KAPER it consists of eliminating the source densities S bet ween Eqs 2 1 and 2 2 One then obtains the following equations for the neutron flux gt vie oS J pE J rks nt n mn s n m k m m ry8ves y ek Ka 2 4 42 8 RYD where Pon Pin E gt E and the effective cross sections re are defined by n E x Y z g 4 z E Bon E g Yon re t n CET x 1 g n ea m e nm m E It must be emphasized that the two systems of equations Eqs 2 3 and 2 4 are completely equivalent within the narrow resonance approximation Both Eq 2 3 and Eq 2 4 can be readily solved if the coefficients are available However for a typical cell of a fast critical assembly where resonance ab sorbers may be present with different concentrations in several regions of the cell the calculation of these coefficients can be a difficult problem In this type of problem the usual equivalence theorem of resonance absorption is no longer valid and one has to search for more elaborate methods to calculate the self shielding in the different regions 2 5 For the calculation of the coefficients in the KAPER program we have modi fied a method which was originally proposed by D Wintzer 4 and which is used in the Karlsruhe code ZERA In Wintzer s method the coefficients of Eq 2 3 here written as Axmn for a particular reaction rate of type x are constructed f
41. lar to that of the SNEAK 5C assembly 8 This was a null reactivity assembly with a soft spectrum and strong heterogeneity effects The core contained mainly mixed oxide and graphite The k values obtained for cells of different thicknesses are given in Table I The following comments concerning the results can be made a As expected from the theory the results for the quasi homogeneous case agree well 50 b The two codes ZERA and KAPER using the same approximation for the colli sion probabilities give ko values which differ by 0 6 for the full cell and less than that for the smaller cells Thus the methods in the ZERA code may be used unless large heterogeneities are involved c The ko values before iteration on the source densities are given in brack ets The figures indicate that the changes due to the iteration are by one order of magnitude smaller than the difference in values given by the two codes Therefore the iteration on the source is necessary only in cases of large heterogeneity The dependence of the heterogeneity effect on the background cross section as used in the formulation of the effective resonance self shielded cross sec tion is shown in Table IL Routinely the KAPER program uses the background cross sections of 238y equal to its potential cross section 998 10 6 barns in the resonance groups For comparison with ZERA calculations performed by Kiefhaber 9 the background cross section of 2384 in K
42. ods employed to calculate effective multigroup self shielded cross sections flux and reaction rate distributions and reactivity worths in a heterogeneous cell are presented The methods are based on a heterogeneous formulation of the self shielding factor f factor concept in the integral transport theory equation Representative numerical results are included and compared to experiments I Introduction The KAPER program 1 is a multigroup lattice code developed to analyze experiments performed in plate type heterogeneous critical facilities These experiments include those in which the flux fine structure in the lattice must be taken into account as for example in reaction rate and small sample reac tivity worth measurements The program may also be used to provide homogenized heterogeneous resonance self shielded cross sections for multidimensional dif fusion or transport codes This paper provides a discussion of the methods employed in the program Included also is a comparison of results from KAPER with results of experiments and other computer codes Some typical results of experiments analyzed with KAPER are given The program is a dynamically dimensioned code in an overlay structure The three main segments consist of a procedure for the calculation of resonance self shielded cross sections in the multiregion cell a procedure for the cal culation of the cell fluxes real and adjoint including reaction rates and a procedure for
43. on in the cell is not cal culated External unit number which contains cross sections for the perturbed cell This is an option which can be used to calculate flux and reaction rate distributions in a cell inserted between the normal repeating cells of the assembly but which is different in some manner than the normal cell When the perturbed cell option is not used NPERT must be Zero O No Bucklings are to be used therefore leakage is set equal to zero l Universal Buckling to be used gt Group dependent Bucklings to be used In the case IB gt 1 input card K4 or K5 gt 0 Iteration on Buckling B2 to a desired k as specified on card K6 otherwise 0 eff 1 If the perturbed cell as defined on unit NPERT is to be extended by a normal cell on each side from unit NXST before K3 IAN THCS IMIN IMAX EPSCON EPS calculation of the flux distribution in the perturbed cell This option can be used to investigate the influence of the perturbed cell boundary selection without redefining the cell in phase 99999 O For all other cases 1 Anisotropic leakage considered and di rectional diffusion coefficients calculated If this option is selected and Bucklings are input be sure to see comment Sl The diffusion coefficient is calculated par allel and perpendicular to the plate structure O Otherwise gt 0 The cross sections for the heterogeneous cell are homogenized for use in a suc
44. ous Cross Sections and Fluxes The equations used for cell calculations in the KAPER program are derived from the energy dependent integral Boltzmann equation which is in the case of isotropic scattering E EDO far S E r P e gt r E 2 1 where is the flux S is the neutron source density 2 is the total cross section and P is the first flight collision probability The source density is composed of the fission source and the slowing down source in the following manner S E x faz Ba E r Ax E vE E r o E r 2 2 where X is the eigenvalue The remaining notation is standard To obtain the multigroup equations for a fast reactor system where reso nance effects are important one usually postulates the narrow resonance ap proximation and observes that the source density S shows no resonance struc ture within this approximation Then the straightforward procedure is to eliminate the flux between Eqs 2 1 and 2 2 and to write the m ltigroup equations with the source density S as a variable If the unit cell is sub divided into N regions with index n the resulting equations are po k En mm z vie n E PG y tax gO Pm 4 2 3 n t n n where Vp is the volume of the nt region and g is the energy group index The brackets gt indicate averages over energy The problem of the weighting spec trum within the energy group will not be discussed here For reasons which will be exp
45. rary 29 V Program Structure and Operation The KAPER consists of the main program and many subprograms as shown on the overlay map Fig l In this section a brief des eription of the function of the various subprograms is given MAIN This program controls the logical sequence of the code Here the maximum amount of core storage is allocated for the problem based on the information supplied by the user Calling of the three segments phases 99999 99998 and 99997 is performed as specified by the card input PARM called from MAIN PARM is an assembler program which reads the contents of the PARM G field on the EXEC card for use within a FORTRAN program CONVY called from MAIN The subprogram performs a conversion from machine internal fixed or floating point to alphanumerical or the reserve ALLOCX called from MAIN Dynamic assignment of main core storage at execution time FREEX called from MAIN Release of dynamically assigned main core storage Phase 99999 DUMMY 3 PPM760 PPM761 SPEK GRREAP INPOO INDX PMC BRB NSPEK CH M10760 Fig 1 MAIN FREEFO CONVY LILFIN REACT ALLOCX LILHOL YZ3 FREEX GROOCP E3 PE3 PARM EZ3 E2 PFUNC Phase 99998 DUMMY 2 HOMOCA HET HETERO QUERP TABLE ASLAB BENOIS SLAB SLABP E5 EZ5 PE5 E4 YZ5 Recommended Overlay Structure for KAPER CELLO Phase 99997 DUMMY 1 PERT HOMPET DEMO PEFLUX SLOO 1 SLOO2 CPERT COLL
46. regions in the cell NOG number of energy groups NFI number of fissionable isotopes included in the isotope list of the problem NOR number of isotopes for reaction rate calculation NWY O if NREG 0 o i 3 NR NR NOG if NREG gt O Number of the isotope in the IS T I array for which reaction rate cross sections are desired If NOR 0 then the IS I s are omitted If NCHI 0 end of card input for phase 99999 otherwise go to KIO if NCHI gt O or to Kl2 if NCHI lt O Cards KIO and Kll are repeated NCHI times NAME Name of fissionable isotope for which a fission neutron spectrum is to be read in as input The fissionable isotopes K11 CHI K K 1 NOG S2 End of card input for K12 NI NE 1 in KAPER include aTH320a QU2330a aU23400 aU23500 aU2360a aU238004 aPU3900 aPU400a oPU4100 and aPU420a See Section VII for clarifications Fission neutron spectrum by energy group for isotope NAME phase 99999 Number of isotopes for which a new tempera ture for the Maxwellian shape of the fis sion neutron spectrum is to be read as input gt 0 Read in new energy group boundaries for the group structure of the cross sec tion set used S3 If NE gt O input card K13 otherwise go to card Kl4 KI3 N E K K 1 N Number of new energy limit values to be read in Upper energy limit of the geh energy group MeV S4 If NI gt O input card Kl4 otherwise end of card inp
47. rom the contri butions of the individual isotopes Thus one has E xn Am E m il o Ace tn v where the summation is over isotopes Each isotopic DS its can be written as Sy E oe i o o E Fan u 2 7 on tv where o is the total cross section of the isotope v and o is the back ground cross section per atom of isotope v due to the other isotopes pre sent in region n The background cross section 69 is assumed constant in energy Note that the collision probability Pn depends on the background cross sections of all other isotopes present between the regions m and n this dependence is indicated by Toe The main point in Wintzer s method is that the functions involving the collision probabilities are fitted to a sum of partial fractions ai I Pan Fe E EEE RR Tey E m tv o i b dey 8 2 8 With this approximation the averaging over resonances can be easily carried out The resulting expressions are related to the self shielding factors by the equation Te o x f ba xv EN v br FO by f b where f by is the tabulated self shielding factor for a background cross section having the value b and oyis the infinitely dilute cross section i e no resonance self shielding This method of Wintzer s works well for lattices with weak heterogeneity but it fails if the heterogeneity is large that is if the optical thickness of at least one region is large The reason is that the coeffic
48. ross Section Library and Isotope Name Convention Use outside of Karlsruhe NUSYS system mn rm a a m EG a a a Fe mn er rm mm Am re a an An An m Aa Mn a o ma o ee Hs ann ts o The KAPER program was designed to function within the NUSYS program system of Karlsruhe Therefore the program reads the required cross section data directly from the prepared cross section data units of NUSYS with the two subroutines GRREAP and GROOCP Therefore built into the program was a certain isotope identification convention and the established energy group structure containing 26 groups To use the program out side of the Karlsruhe NUSYS program system a new subroutine GRREAP is provided A listing of this short subprogram is in cluded in the following The data and format required by the program are explained in the FORTRAN listing The cross sec tions and their respective self shielding factors can be gen erated from ENDF data or similar data files by a number of available programs for example ETOX or 1DX from the Argonne Computer Code Center see Ref 3 To calculate with more than 26 energy groups it is only neces sary to change the common block CH which contains the prepared fission neutron spectra for the fissionable isotopes This common block is contained in subroutines PPM760 NSPEK SPEK PPM761 and in BLOCK DATA The rest of the program is variably dimensioned for the number of energy groups No change in common block CH is necessary if
49. s and fluxes in Eq 3 2 are defined for the normal unit cell and are available from a previous calculation Therefore Bn can be calculated di rectly and used in Eq 3 1 to solve for the flux within the perturbed cell inside the imaginary boundaries The solution of both Eq 3 1 and the corresponding adjoint equation are carried out by the power iteration method as briefly outlined in the previous section 4 Heterogeneous Perturbation Calculation For the calculation of heterogeneous reactivity worths perturbation theory is used Perturbation theory offers an advantage for the calculation of small changes in a system this being that the change in the system is expressed di rectly rather than being the difference of two nearly equal quantities as one would have by calculating the eigenvalue separately for the perturbed and un perturbed systems Therefore the heterogeneous fluxes and cross sections ob tained as described in the previous sections are used in a perturbation theory formulation of the integral transport theory equation to obtain reactivity worths of small changes introduced into the assembly core However since the flux depression or peaking in the sample can be as important an effect as the self shielding of the sample cross sections the exact form of the perturbation equation is utilized in the KAPER program rather than a first order form as is commonly employed in perturbation programs Therefore formulating the per
50. s calculated as for a homo genized cell NISS lt NIS K7 K8 K9 ISON I I 1 NISS Isotope names e g aPU9A00 NR D I I 1 NR MIS T I 1 NR NMIS Number of regions into which the cell is divided NR gt 2 th i Thickness of the i region in cm Mixture number of the geh ton as determined by the order in card K8 Number of different mixtures CON J 1 I 1 NIS J 1 NMIS Isotope concentration as NREG NAP atoms fn 10724 1 If the cross sections are being prepared for the calculation of fluxes of the normal core cell in phase 99998 O If the cross sections are being prepared for reactivity worth calcula tions or for the perturbed cell option in phase 99998 gt 1 If the cross sections are being prepared for a homogeneous flux calcu lation in phase 99998 External unit number on which cross sec tions are to be written for use in a suc ceeding phase If NAP lt O a rewind is exe cuted before writing otherwise the unit S1 K10 IS I I 1 NOR 10 is held at the end of the last record written in a preceding phase 99999 cal culation In this manner the cross sec tions for reactivity worth samples can be stacked one after the other on the same unit The space needed on this unit for each calculation can be de termined from the equation for SPACE SPACE words 5 NR HNOGHNOR 2 NOG NF NOG NOG NR NR NOGX 2 NFT 4 NR NWY where NR number of
51. s taken as an infinite repetition of normal core cells SLOOI called from PEFLUX Collision probabilities of the cell containing the sample located in an infinitely repeating lattice of normal core cells are calculated 39 SLOO2 called from PEFLUX The collision probabilities within the sample cell are calcu lated for neutrons originating only within the cell COLL called from PERT The collision probabilities for the two cases sample in and sample out are calculated for use in the perturbation theory calculation They are written on an external unit for use in subroutine CPERT CPERT called from PERT The reactivity worth of a sample is calculated with integral transport perturbation theory HOMPET called from DUMMY1 This routine is a homogeneous version of the CPERT routine for the calculations of reactivity worths A homogeneous formulation of perturbation theory is utilized 40 VI Computational Methods Utilized in KAPER Paper presented at ANS National Topical Meeting on Mathematical Models and Computational Techniques for Analysis of Nuclear Systems April 9 11 1973 in Ann Arbor Michigan USA KAPER A COMPUTER PROGRAM FOR THE ANALYSIS OF EXPERIMENTS PERFORM D IN HETEROGENEOUS CRITICAL FACILITIES P E Me Grath and E A Fischer institut f r Angewandte Systemtechnik und Reaktorphysik Kernforschungszentrum Karlsruhe Germany Abstract The essential features of the meth
52. se 99999 if NOR gt O or NRS 1 if NOR 0 NR number of regions NR MAX NY NW where NY is the number of regions in the normal cell and NW is the number of regions in the perturbed cell If NGC 1 then NR 2 NY NW see card K2 of phase 99998 IAR NY if IAN 1 otherwise IAR 1 see card K2 of phase 99998 NFI number of fissionable isotopes in the cells 3 Phase 99997 reactivity worth phase N3 4 NOCANR 4 NE NOGENW 4 NFP NOG NOGH NW NR IXO 2wIXO NOG 2 NR 2 NOG 14 NF 28 NEP NReMAF NOGeL1WT1 1 where NY number of regions in the normal cell NW number of regions in the perturbed cell if the perturbed cell option in the phase 99998 is not used then NW NY NR NY NW IXO 2 NCELL NY NW where NCELL is defined on card K2 of phase 99997 NOG number of energy groups NF number of fissionable isotopes in the normal cell 26 NFP maximum number of different fissionable isotopes in any one set of data for sample reactivity worths see card K2 of phase 99997 z MAX NF NFP MAX NY NW N I In the above equations for the N s the formulation X MAX Y Yo Yg ro means that the maximum value of the Y variables is assigned to X The N s are computed in bytes The PARM G in the EXEC card see the section IV on control cards is set equal to or greater than the largest of NING or N In other words PARM G MAX Nj No Na The REGION size is then PARM G plus 130K
53. st NOG number of energy groups 2 IM records of type Ry for NA2 CHI 3 5 records of type Ry for NA2 SCAPT NA2 NUSF NA2 STR NA2 SREM NA2 SFISS respectively 4 NOG records of type Ry for NA2 SMTOT with the index IS running from to NOG Note 3 and 4 are repeated then IM 1 times once for each composition number MN I selected The cross sections are the same except for the transport cross section which is defined by MD I of card K7 5 2 records of the type R for NA2 SCAPT and NA2 SFISS respectively 20 Note 5 is repeated for each of the IRR isotopes for which reac tion rate cross sections are produced 4 Phase 99997 Reactivity Worth Phase In this phase heterogeneous reactivity worths are calculated If homogeneous cross sections were prepared in phase 99999 the program will perform a homogeneous first order perturbation calculation automatically In this case a number of the variables as input on the following cards have no mean ing as is noted Kl 0999970 Constant K2 NFPH Number of following phase NFLUX External unit number containing cross sec tions and fluxes for normal cell Normally NFLUX NXST on card K2 of phase 99998 NPERT OM External unit number containing cross sec tions and fluxes for the perturbed cell Normally NPERT NPERT on card K2 of phase 99998 When option is not used NPERT 0 In this case the program substitutes a normal cell for the pert
54. the calculation of small sample reactivity worths The fundamental basis of the program is integral transport theory in the collision probability formulation The multigroup resonance self shielded cross sections for the multiregion cell are defined by a procedure utilizing the f factor concept The concept of the composition dependent self shield ing factor factor for homogeneous media was first introduced by Abagjan et al 2 and recently tested against a more exact model by Kidman et al 3 A consistent formulation for the heterogeneous medium was developed through 4l the integration of the space and energy dependent integral transport equation and represents an improvement in a method originally developed by Wintzer 4 The program has a particular feature which allows one to calculate the flux distributions real and adjoint and reaction rates in a cell differing from the normal unit cell of the core This feature has great utility for analyzing experiments that disturb the properties of the cell in the measurement proce dure In this problem the flux is found by solving the integral transport equa tion as a fixed source equation in which the normal unit cell serves as an ex ternal source Reactivity worths are calculated with an exact perturbation formulation of the integral transport equation By exact is meant that the perturbed flux and unperturbed adjoint are used in the formulation 2 Calculation of Heterogene
55. the flux and adjoint distribution in the perturbed cell it is assumed that the change in the assembly introduction of the perturbed cell is sufficiently small as to not affect the criticality of the assembly nor the spectrum several mean free paths from the perturbed cell position With this assumption the flux and adjoint distribution in the perturbed cell can be obtained by solving the integral transport equation Eq 2 4 as a fixed source equation The source is the first flight leakage uncollided neutrons from the surrounding normal unit cells several mean free paths removed from the perturbation or in the case of the adjoint equation the importance a perturbed cell neutron has upon escaping from the perturbed cell To write the equation for this case it is sufficient to formulate the equation from physical processes For example Eq 2 4 can be simply derived by equating the total collision density in a particular energy group and region to the sum of the contributions from all energy groups and regions from which it is possible for neutrons to come Let us draw imaginary boundaries around the perturbed cell of the assembly We have located these imaginary boundaries as a result of the definition of the perturbed cell given above at a point where the equilibrium spectrum of the assembly is reestablished From a previous calculation of the normal unit cell we have therefore the flux solution outside these imaginary boundaries Settin
56. turbation equation with the integral transport theory flux equation representing the perturbed state and the adjoint equation representing the unperturbed state one obtains gt A 8 u ee se pk y tk g m n k mn p E m 4 1 g k k P A 85 ve Re nk j de where p 81 A and the region index summations are over all regions where the perturbation operators are non zero The perturbation operator in general N is defined as N N where the prime denotes the quantity defined in the perturbed state In the perturbation equations we will represent the fission source as a sum of contributions index j from each fissionable isotope pres ent The denominator of Eq 4 1 is o t tk k pk De NR vi Sn yy X MN a 4 2 gm J kn We may rewrite Eq 4 1 if we use the following form of the perturbation operator U IT ER A 48 In addition we may also use the relationship between the source importance function u and the colliding neutron importance function a H k pk gtk m mn n Y Introducing these relationships into Eq 4 1 we can write the results as after some rearranging I g l ep8 gtg g gt k tk ts P D 3 en Bea ea Oe MER RE 8559 g m k m he k gtk a k k York tk tg Ex i VE wo I E x VE RM k des di k m j A where 2 is the total absorption cross section in region m and energy group g m This equation has a form that renders itself to easy phys
57. urbed cell and therefore all remarks pertaining to the perturbed cell in the following input data apply to the substituted normal cell does not apply in homogeneous case NXECT External unit number containing cross sec tions for the various samples to be calcu lated NTAPE NTAPE 1 NSAM NFP NCELL 21 Reserve external unit for collision probabilities The space needed on this unit is SPACE bytes 4x NOGXIOX IOX where NOG number of energy groups IOX 2 NCELL NY NW NCELL input data NY number of regions into which the normal cell is divided NW number of regions into which the perturbed cell is divided Reserve external unit for collision probabilities The space needed on this unit is the same as computed for NIAPE Number of samples to be calculated This is the number of cross section sets on unit NXECT Maximum number of fissionable isotopes in any one cross section set on unit NXECT Number of normal cells gt 1 to be placed on either side of the perturbed cell for the calculation of the reactivity effects It is for these cells that the range of the terms ose are defined NCELL should be large enough so that the probability a neutron from the perturbed cell suffers a collision before reaching the outer boundary of the last normal cell is at least 0 98 If the parameter is given as NCELL lt O the program will select an appro priate value for it K3 K4 IH
58. usions It has been demonstrated that the methods employed in the KAPER program are extremely useful for the analysis of measurements performed in a hetero geneous environment of fast critical assemblies The application of the 52 f factor concept is not as accurate as the methods used in ultrafine group slowing down codes nevertheless the procedure certainly yields sufficient accuracy with a tremendous savings in computer time for routine calcula tions 1 P E Mc Grath and E A Fischer Calculation of Heterogeneous Fluxes Reaction Rates and Reactivity Worths in the Plate Structure of Zero Power Fast Critical Assemblies Kernforschungszentrum Karlsruhe KFK 1557 March 1972 2 I I Bondarenko et al Group Constants for Nuclear Reactor Calcul tions Consultants Bureau New York 1964 3 R B Kidman R E Schenter R W Hardie and W W Little The Shielding Factor Method of Generating Multigroup Cross Sections for Fast Reactor Analysis Nucl Sci and Eng 48 189 201 1972 4 D Wintzer Heterogeneity Calculations including Space Dependent Reso nance Self Shielding in Proceedings of an IAEA Symposium Fast Reactor Physics Karlsruhe Nov 1967 15 B A Fischer Ihe Overlap Effect of Resonances of Different Fuel Iso topes in the Doppler Coefficient Calculations For Fast Reactors Nukleonik 8 146 1966 6 A P Olson RABID An Integral Transport theory Code for Neutron Slowing Down in Slab Cells
59. ut for phase 99999 Card K14 is repeated NI times K14 NAME TEMP pps Name of fissionable isotope for which a new Maxwellian temperature is to be read in The name convention is the same as on card Kl0 Maxwellian temperature MeV for isotope NAME S5 End of card input for phase 99999 3 Phase 99998 flux adjoint_and reaction rate phase In this portion of the program the flux and adjoint distributions in the heterogeneous cell are calculated as well as heterogeneous reaction rates if they were specified during the cross section preparation in phase 99999 If the cross section prepared in phase 99999 are for a homogeneous mixture the program will auto matically select the correct computational path In either case the card input in this phase remains the same Kl 999980 K2 NFPH NHOM Constant Number of the following phase gt 0 for quasi homogeneous calculations in which the optical thickness of the cell is less than or equal to 0 otherwise 0 NXST NPERT IB KBSQ NGC 13 External unit number which contains cross sections for the normal cell When fluxes from this calculation are to be used in the reactivity worth phase NXST must be a positive number In this case the cross sections are saved and the fluxes are written on NXST for transfer to phase 99997 For all other calculations enter NXST as a negative number When NXST is negative the adjoint distributi

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