Calculation of Power Density with MCNP in TRIGA Reactor

Transcription

1 Internatonal Conerence Nuclear Energy or New Europe 26 Portorož, Slovena, September 18-21, 26 Calculaton o Power Densty wth MCNP n TRIGA Reactor ABSTRACT Luka Snoj, Matjaž Ravnk Jože Stean Insttute Jamova 39, SI-1 Ljubljana, Slovena Luka.Snoj@js.s, Matjaz.Ravnk@js.s Modern Monte Carlo codes (e.g. MCNP allow calculaton o power densty dstrbuton n 3-D geometry assumng detaled geometry wthout unt-cell homogenzaton. To normalze MCNP calculaton by the steady-state thermal power o a reactor, one must use approprate scalng actors. The descrpton o the scalng actors s not adequately descrbed n the MCNP manual and requres detaled knowledge o the code model. As the applcaton o MCNP or power densty calculaton n TRIGA reactors has not been reported n open lterature, the procedure o calculatng power densty wth MCNP and ts normalzaton to the power level o a reactor s descrbed n the paper. 1 INTRODUCTION Monte Carlo computer code, MCNP, s a very powerul and versatle tool or partcle transport calculatons. It can be used or transport o neutrons, photons and electrons. Transport o neutrons s o specal nterest or a reactor physcst. MCNP code can be used or calculatons o multplcaton actor, reacton rates, saturated actvtes, neutron luxes and spectra, power peakng actors, reacton rate dstrbutons, sheldng etc. Its man advantage s the ablty to handle complcated geometres. MCNP also provdes seven standard tally types [1]. All talles are normalzed to one "startng" partcle except n KCODE crtcalty problems, whch are normalzed to one sson neutron. In order to normalze the result by the thermal power o a system, one must use approprate scalng actors. As the procedure o calculatng the scalng actors or KCODE calculaton s not adequately descrbed n the MCNP manual, t wll be descrbed n the paper. The present paper wll ocus on standard cell lux (track length estmate o cell lux F4 tally [1] whch s o greatest nterest or reactor physcs calculatons. However the results can be appled also to other neutron talles. 2 NORMALISATION OF F4 TALLIES IN A CRITICALITY CALCULATION The multplcaton actor s one o the most mportant propertes o a reactor or other system made o ssle materal. In MCNP the most common way to calculate the multplcaton actor s through the use o KCODE card. It s mportant to note that all the standard MCNP talles can be made durng a crtcalty calculaton. Snce the MCNP results are normalsed to one source neutron, the result has to be properly scaled n order to get absolute comparson to the measured quanttes (lux, reacton rate, sson densty, etc.. The F4 tally results can be scaled to a desred sson neutron 19.1

2 19.2 source (power level or total neutron pulse strength. The scalng actor can be entered on the FM (tally multpler card or can be appled later n data processng. The neutron brth rate n a ssle system can be calculated rom the released energy per P unt tme.e. the power o the system. The system producng power P needs ssons per w second, where w denotes eectve energy released per sson event. Although the value o w wll vary somewhat wth the type o reactor and the detaled core composton, t s typcally Pν o the order o 198 MeV or steady state condton. Ths sson rate produces neutrons w per second, where ν denotes the average number o neutrons released per sson. (The value o ν s lsted n the MCNP output le n the box contanng the nal k e result and represents the value averaged over ssle sotopes and neutron energes. Thereore to normalze an F4 tally by the steady-state thermal power o a crtcal system, the ollowng scalng actor n unts o sson neutrons per unt tme should be used neutron P[ W] ν sson S = 13 J MeV w MeV sson (1. The upper scalng actor s approprate or crtcal.e. steady-state power level systems only (k = 1. KCODE talles or subcrtcal and supercrtcal systems do not nclude any multplcaton eects because sson s treated as absorpton. Thereore one must multply the equaton (1 by 1 on the rght-hand sde or subcrtcal and supercrtcal systems, ke respectvely. It s mportant to note that the scalng actor or subcrtcal and supercrtcal systems s vald only when one uses neutron source dstrbuton dentcal n space and energy to the source dstrbuton obtaned rom the soluton o an egenvalue problem wth k e 1. To conclude, when one wants to scale the calculated F4 tally "lux", Φ F4, one must use the ollowng equaton neutron P[ W] ν neutron sson 1 1 Φ = Φ cm s J MeV MeV sson 2 F ke cm w where Φ denotes the actual total neutron lux n the system., (2 3 CALCULATION OF POWER DENSITY DISTRIBUTION 3.1 Calculaton o power densty The energy released n a nuclear sson reacton s dstrbuted among a varety o reacton products. The majorty o the sson energy appears as the knetc energy o the sson ragments and s deposted essentally at the pont o sson. About 97 % o the Proceedngs o the Internatonal Conerence Nuclear Energy or New Europe, 26

3 19.3 recoverable sson energy s deposted drectly n the ssle materal [2]. In our calculaton o the power densty dstrbuton we wll assume that power densty s proportonal to sson densty. In other word that means that we assume that all o the recoverable sson energy s deposted at the pont o sson. We wll also assume that there are no temperature eedback eects and that there s only one ssle sotope n the system. Power densty, dened as the energy deposted n the ssle materal per unt volume per unt tme, can be wrtten as p( r = w ( E, r Σ ( E, r ϕ( E, r de, (3 where Σ and φ denote macroscopc sson cross secton and neutron spectrum, respectvely. Neutron spectrum φ(e,r s normalzed such as ϕ( E, r de =Φ( r, (4 where Φ(r s total neutron lux n cm -2 s -1. The same s vald also or MCNP F4 tally ϕ F ( E, r de =Φ ( r. (5 4 F4 Equaton (3 represents thermal power densty at poston r n the sson system. Hence the total power generated by the sson system s just the ntegral o the power densty over the total volume where Σ. 3 P = d r w ( E, r Σ ( E, r ϕ( E, r de V, Σ (6 Assumng that w and number densty o the ssle materal do not depend sgncantly on the energy and the poston n the ssle system, equaton (3 can be wrtten as p( r = wn σ (, (, r E, (7 E r ϕ E d where N denotes ssle materal atom densty and σ denotes mcroscopc total sson cross secton. In MCNP the F4 cell lux tally, Φ F4 (r, s averaged over the volume o the samplng cell. Thereore, when we calculate sson densty n a certan cell, Δ V ( runs over all spatal cells o the reactor, we can omt the spatal dependence rom the ntegral n equaton (7 and obtan p( r = w N σ ( (, E ϕ E de r V (8 Δ The ntegral can be qute easly calculated by usng the tally multpler (FM card [1], whch s used to calculate, or Δ V, any quantty o the orm Proceedngs o the Internatonal Conerence Nuclear Energy or New Europe, 26

4 Emax 19.4 FR = C σr( E ϕf4( E de, (9 Emn where the constant C s any arbtrary scalar quantty that can be used or normalsaton, σ R (E s mcroscopc cross secton or reacton R taken rom MCNP cross-secton lbrares and E mn and E max denote mnmum and maxmum neutron energy n the system (usually and 2 MeV, respectvely. F R s the quantty or reacton R that s calculated by MCNP and ts value can be ound n the output le. Any number o ENDF or specal reactons can be used as a multpler as long as they are present n the MCNP cross-secton lbrares. I the C entry s negatve (or type 4 tally only, C s replaced by C tmes the atom densty o the cell where the tally s made. Thus or calculatng power densty we use F4 tally wth tally multpler -6, whch s the mcroscopc total sson cross secton, and normalse the result to a desred sson neutron source (power level or total neutron pulse strength. Usng equatons (8, (9 and the scalng actor S we obtan p( r = Pν M w w Emax σ ( ϕf4(, r E Δ V (1 mn N E E de It s nterestng that the power densty s ndependent on w. Fnally we can wrte power densty as p( r = Pν NMF, r V. (11 Note that p(r s a uncton o r, as t s constant nsde each Δ V. I we want to obtan "smooth" p(r dstrbuton, all Δ V must be small. It s mportant to note that σ n equaton (1 s actually macroscopc total sson cross secton normalzed to one atom o the sampled materal. Thereore n case the sampled materal contans ssle and non-ssle components (e.g. UO 2, UCO, UZrH, UO 2 + PuO 2, etc. the ntegral n equaton (1 can be smply multpled by the sampled materal atom densty,.e. a mxture o several nucldes, and not the atom densty o the ssle materal only. I the sampled materal contans several ssle sotopes, mcroscopc total sson cross secton, σ, s calculated as the weghted average over all ssle sotopes, wth regard to ther atom racton. The same s true orν. Δ 3.2 Alternatve opton or calculaton o power densty The alternatve opton or calculaton o power densty s much smpler, but t has some lmtatons. Ths opton s partcularly useul when we want to calculate the power densty n ndvdual cells or uel elements. Frst we calculate sson densty (normalsed "per atom".e. calculate F n each cell contanng ssle materal and obtan E max F = Φ( E σ ( E de E mn (, (12 Proceedngs o the Internatonal Conerence Nuclear Energy or New Europe, 26

5 19.5 where denotes the cell ndex. Snce the power produced n one cell s lnearly proportonal to the number o ssons n that cell, we can calculate the power produced n cell, P, by multplyng the ssle system thermal power, P, by the relatve number o ssons n cell. As the relatve number o ssons n cell s proportonal to the product o sson densty n cell and the volume o cell, we obtan P ( F ( F V = P. (13 V Usng the denton o power densty as beng the power produced per unt volume we obtan the power densty n cell as p ( F ( F = P. (14 V The man lmtatons o ths procedure o sson densty calculaton are that we have to know the exact volume o each cell and that we have to sample all cells contanng ssle materal, whch s not necessary or the prevously descrbed opton. Note that V can be the volume o the uel nsde the uel element or very small volume o arbtrary materal composton and needs not to contan only ssle materal. In both cases p corresponds to the average power densty n that volume. 3.3 Calculaton o power densty dstrbuton The easest way or calculatng the power densty dstrbuton can be by usng the supermposed mesh tally card, FMESHn (or the present only type 4 talles are permtted. FMESH card allows the user to dene a mesh tally supermposed over the problem geometry. By deault, the mesh tally calculates the track length estmate o partcle lux, averaged over a mesh cell. I we use the tally multpler card or sson together wth the mesh tally and scale the results to approprate power level, we can obtan power densty dstrbuton. When usng tally multpler card together wth supermposed mesh tally card t s recommended to set materal number to, whch causes that the reacton cross sectons or the materal n whch the partcle s travellng are used. Thus we do not have to worry n whch materal partcle s travellng. The FMESH card s extremely powerul and useul method also or calculatng lux dstrbutons, power peakng actors, etc.. By usng the very ne mesh we can calculate local power peakngs and power densty dstrbutons near the water channels, whch s very dcult not mpossble to do wth determnstc methods. 4 POWER DENSITY DISTRIBUTION IN TRIGA MARK II CORE The supermposed mesh tally eature o the MCNP code was used to calculate detaled power densty dstrbuton and power peakng actors o varous hypothetcal mxed cores o TRIGA Mark II reactor located at Joze Stean Insttute. The mesh o the mesh tally was so ne that power densty dstrbuton was calculated wth a resoluton o 1mm 1mm. The dsadvantage o such hgh resoluton s the very large number o samplng cells (5 5 Proceedngs o the Internatonal Conerence Nuclear Energy or New Europe, 26

6 19.6 and consequently relatvely hgh relatve error o the tally. That s the man reason why we calculated the power densty dstrbuton averaged over the uel heght. The results are presented n Fgure 1. Fgure 1: Power densty dstrbuton (rel. unts n mxed core wth 8.5 w/o standard uel and 6 LEU uel elements n the D rng. The numbers on x n y axs are dstances rom the centre o the core n cm. 5 CONCLUSIONS We have shown how to scale MCNP tally results to a desred power level o the system. The scalng actors can be derved rom the system power, multplcaton actor, average number o neutrons released per sson and average energy produced per sson event. When one s amlar wth normalzng the results to a certan power level, the MCNP becomes a versatle calculaton tool or varous reactor physcs calculatons. One can calculate reacton rates, saturated actvtes, neutron luxes and spectra, power peakng actors etc.. The MCNP ablty to handle complcated geometres and use o the FMESH card eature o the MCNP code enables one to calculate detaled neutron lux and varous reacton rate dstrbutons wth a resoluton o 1mm 1mm or less (depends on the sze o the system, the computer hardware and sotware. The later eature s especally useul or calculaton o power densty dstrbuton and local power peakng actors calculaton snce t can reproduce properly the eects o local power gradents due to small heterogenetes n the core. REFERENCES [1] J.J. Bresmester, "MCNP5- A General Monte Carlo N-Partcle Transport Code, Verson 5 Los Alamos Natonal Laboratory", March, 25 [2] J. J. Duderstadt, L. J. Hamlton, Nuclear Reactor Analyss, John Wley & Sons, 1976 Proceedngs o the Internatonal Conerence Nuclear Energy or New Europe, 26