MONTE CARLO ALGORITHM FOR CLASPING SEARCH AND NEUTRON LEAKAGE

Transcription

1 Sep. 5. Vol. 7. No. 3 Inernaonal Journal of Engneerng and Appled Scences - 5 EAAS & ARF. All rghs reserved MONTE CARLO ALGORITHM FOR CLASPING SEARCH AND NEUTRON LEAKAGE PEYMAN MAJNOUN Faculy of Mechancal Engneerng, Unversy of Tabrz-IRAN Correspondng Emal: p.manoon@gmal.com ASTRACT A Mone Carlo algorhm has been developed for akng no accoun neuron leakage effec by a claspng. The algorhm allows one o perform claspng search mode proper value calculaons where he claspng s reaed as a proper value. For nroducng neuron leakage effec, he spaal dependence of neuron nsably s come near a sngle Fourer mode. The parcle wegh s a complex number.the magnary par, however, dsappears and only he real par needs o be reaed f symmery exss wh respec o he drecon represened by he claspng and he dsrbung and splng neuron emsson are soropous. The algorhm has been verfed by comparng buck lngs acqure by claspng search mode proper value calculaons wh hose by he dffuson approxmaon and he mehod.when a horzonal calculave feld has wo orhogonal symmery planes and he dsrbung and splng neuron emsson are soropous he magnary pars of complex weghs dsappear. If does no, complex weghs need o be reaed. The mehodology for reang complex weghs has no ye been esablshed and would be a fuure work. The newly developed algorhm can also be appled o leakage-correced Mone Carlo calculaons, generang leakage correced neuron specra and nsably dsrbuons. Alhough hs paper has presened ha he algorhm can conrbue o he generaon of leakage-correced group connuous s based on Mone Carlo calculaon echnques. Keywords: Mone Carlo algorhm, Claspng, Neuron leakage INTRODUCTION Three-dmensonal of core calculaons by Mone Carlo echnques are gradually becomng a calculaon ool for reacor core desgns. Meanwhle, connuous energy Mone Carlo calculaons are now beleved an effcen ool for calculang some neuronc parameers such as dffuson coeffcen, whch wll be used for deermnsc core desgn codes. Yoshoka and Ando [], and Leppänen [], have been performed for generang deermnsc group connuous by usng Mone Carlo echnques. Calculaons are usually performed for un fuel pn cells or fuel assembles havng nfne lengh n he vercal drecon. To oban accurae group connuous, he effec of neuron leakage o he radal and axal drecons has o be aken no accoun n he Mone Carlo calculaons. Neuron leakage correcons are ncorporaed no deermnsc fuel pn cell or fuel assembly calculaon codes by usng some leakage correcon echnques, e.g., he mehod (Dudersad and Hamlon, [3]). In hs mehod, he neuron leakage effec s aken no accoun by specfyng he claspng of a reacor core. In he feld of Mone Carlo, however, some echnques have been developed o presen he neuron leakage effec no fuel pn cell or fuel assembly calculaons (Yoshoka and Ando []; Frdman and Leppänen [4]; Shm e al. [5]). Leppänen [], presen an addonal cross secon-lke parameer for ncorporang he neuron leakage effec. Ths echnque s smlar o he calculaons of he proper value mode. A erm Dgg s arfcally added as smulaed absorpon where Dg s a dffuson coeffcen n he gh energy group and g s a geomerc claspng. Ths mehodology s based on he assumpon ha he dffuson holds whn reasonable accuracy. A dffculy n esmang he dffuson coeffcens for connuous energy Mone Carlo calculaons sll remans unresolved. The mehods are beleved unsubsanaed and no recommended o be used for group connuous generaon. Frdman and Leppänen [4] and Shm e al. [5] presen a mehod based on he mehod for 6

2 Sep. 5. Vol. 7. No. 3 Inernaonal Journal of Engneerng and Appled Scences - 5 EAAS & ARF. All rghs reserved few-group connuous generaon. A Mone Carlo calculaon s used o produce fne group cross secons used for solvng he equaons. Then, he equaons are solved o oban he crcal neuron specrum. Ths mehod s sem-deermnsc, and leakage-correced effec s no explcly ncluded n he Mone Carlo calculaon. The change of keff caused by neuron leakage s calculaed by he perurbaon heory. Ths echnque requres he ad on source dsrbuon ha s usually dffcul o be esmaed by a connuous energy Mone Carlo calculaon. Anoher relaed work usng Mone Carlo echnque was performed by Maorov [6]. To ake no accoun he effec of neuron leakage, parcle weghs are gven by a formula dependng on a parcle s random walks. Recenly, a leakagecorreced Mone Carlo calculaon echnque was proposed by Yun and Cho [7]. Ths mehod presens he effec of neuron leakage by specfyng on he ouer surfaces of a calculave feld nsead of nroducng a geomerc claspng. Durng he course of a Mone Carlo crcaly calculaon, an albedo s updaed a each generaon n such a way ha keff s equal o uny. The presen paper focuses on deermnaon of he geomerc claspng or spaal decompose connuous c n he vercal drecon of a fuel pn cell or a fuel assembly. If he geomerc claspng n he horzonal drecon s larger han he maeral claspng, he neuron nsably decomposes n he vercal drecon. The spaal decompose connuous s nmaely relaed o he sub crcaly of he fuel pn cell or fuel assembly and can be used as an ndcaor of nuclear crcaly safey. In such a case, a proper value mode neuron ranspor equaon can be defned. Yamamoo and Myosh [8] proposed a mehod o solve he proper value mode equaon by Mone Carlo. The mehod of he proper value mode calculaon has been mplemened no a connuous energy Mone Carlo code MCNP4C (resmeser, [9]). On he oher hand, f he geomerc claspng n he horzonal drecon s smaller han he maeral claspng, he vercal neuron nsably dsrbuon s expressed by a. ( x, y,, E ) ( x, y,, E ) ( x, y,, E ) d de ( x, y,, E ) ( x, y,, E E ) S ( E ) d de ( x, y,, E ) ( x, y, E ) ( x, y,, E ) f 4 Where s a drecon cosne wh he z-axs, and oher noaons are sandard whn he nuclear engneerng communy. Ths equaon s a proper value equaon whose proper value s he spaal 7 cosne funcon where z s he geomerc claspng n he z-drecon. The vercal claspng z s defned by where H s he acve hegh of a core and s he exrapolaed lengh. Some deermnsc reacor calculaon codes (Fowler e al. []) have a funcon of fndng a crcal geomerc claspng. The presen paper s o develop a new algorhm of Mone Carlo calculaon mehod for fndng a crcal geomerc claspng. If ceran condons dscussed below are me, he algorhm s also avalable for fuel pn cell or fuel assembly calculaons n whch he effec of neuron leakage specfed by a claspng s aken no accoun n he same manner as he mehod. The mehod adops ha he leakage curren s a funcon of neuron energy only and ha he leakage occurs whou dreconal dependence. On he oher hand, he mehod of hs paper adops ha he leakage curren depends only on drecon. Crcal claspng's acqure by he presen mehod are o be compared wh he mehod n one-energy group n a laer secon.. Revs of spaal decompose connuous search by Mone Carlo mehod.. Theory of proper value mode calculaon by Mone Carlo mehod When he horzonal claspng s smaller han he maeral claspng, spaal decompose connuous search mode calculaons by Mone Carlo (Yamamoo and Myosh, [8]) are revsed. Suppose ha he horzonal claspng n he x and y-drecons s larger han he maeral claspng and ha an exernal neuron source s locaed n he z-drecon. If he subcrcal sysem s homogeneous n he z- drecon, he asympoc neuron nsably dsrbuon n he z-drecon s gven by where s a spaal decompose connuous n he z-drecon. The hree-dmensonal neuron ranspor equaon s rewren o a wo-dmensonal form as: () decompose connuous. Ths equaon s o be solved by he repeon mehod as crcaly calculaons. To nclude he las erm n Eq. () durng he random processes n a Mone Carlo calculaon, he dfferenal of he wegh of a neuron

3 Sep. 5. Vol. 7. No. 3 Inernaonal Journal of Engneerng and Appled Scences - 5 EAAS & ARF. All rghs reserved parcle ha fles over an nfnesmal dsance ds s gven by (Yamamoo, []): dw Wds () Thus, afer ha S n he h pah wh a drecon cosne, he nal wegh W changes o: W W exp( s ) (3) a hs sage s an unknown proper value and s gven from he prevous generaon. Unless he wegh changes durng s pah, he produc of he rack lengh lengh S and s wegh W s gven by W S. The wegh, however, changes durng s pah. Afer he h pah wh he dsance S, he, he produc of he rack lengh and he parcle s wegh s gven by: exp( s TL W s ds W (4) s ) exp( ) TL n Eq. (4) s he rack lengh esmaor of neuron nsably when neuron wegh s a connuously changng funcon across he pah. Ths s no only o he calculaon of k eff bu o all alles as well. Maor modfcaons o a Mone Carlo crcaly calculaon code needed for proper value mode calculaons are mplemenaon of Eqs. (3) and (4). All oher procedures for crcaly calculaons can be ulzed for proper value mode calculaons whou any modfcaons. Afer all random processes whn one generaon are compleed, he proper value s esmaed as: : N. W W v. TL TL. where s summarze over all pahs whn one generaon, N s he number of source per generaon, D W = W (exp( zgs) - ) and W s s he wegh of S f parcles ha sar from splng source ses. The sarng wegh W s s gven by M/N where M s a nomnal source sze. N pons come from he splng source pons of he prevous generaon. Thus, he oal source wegh per generaon s a connuous M. An ndcaor of, k, s defned as: k f M. TL (6) Ths ndcaor has he same defnon as k eff and s supposed o be uny f converged. The s used for he nex generaon calculaon s deermned n such a way ha k approaches uny as: c( k ) (6) where he sands for he generaon number and s an unreasonable posve value. The are averaged over all generaons afer generaons o deermne a fnal. The mode calculaon s exends o nfny lengh along he z-axs.... Confrmaon of mode calculaon by Mone Carlo mehod The algorhm of he mode calculaon by Mone Carlo mehod s nsalled no he connuous energy Mone Carlo code MCNP4C. The mode calculaon was numercally verfed by comparng decompose connuous calculaed by he mode calculaons wh hose by fxed source calculaons for uranyl nrae aqueous fuel soluons (Yamamoo and Myosh, [8]). In he fxed source calculaons, neuron nsably dsrbuons n he z-drecon were calculaed and he decompose connuous were acqure by fng he nsably dsrbuons. Ths paper presens a numercal confrmaon usng a (5) homogeneous, one-energy group problem wh soropous dsrbung n he laboraory sysem snce n hs suaon an accurae reference soluon can be acqure by he dffuson heory. A es Mone Carlo code has been prepared for he numercal es. Suppose ha a recangular parallelepped wh he sde lengh of 4.3 cm s composed of a homogeneous mulplyng medum. The one-group connuous s shown n Table. The spaal 8

4 Sep. 5. Vol. 7. No. 3 Inernaonal Journal of Engneerng and Appled Scences - 5 EAAS & ARF. All rghs reserved decompose connuous based on he dffuson heory s gven by: where, X y (7) ( / ( H )), (H) s sde lengh, ( ) / D, m f a D / (3 ),.74 The spaal decay consan s based on he dffuson n he horzonal drecon as well as n he vercal drecon. To mprove he accuracy of s as he reference soluon, anoher spaal decompose connuous s defned by: where X y m h f n a (8) ( k ) D, K n s neuron mulplcaon facor of he recangular parallelepped wh nfne lengh n he vercal drecon. K n was calculaed by a Mone Carlo crcaly calculaon. Thus, he spaal decompose connuous s more accurae han snce he horzonal drecon s reaed by he neuron ranspor heory. Anoher spaal decompose connuous 3 s acqure by nsably dsrbuon n he vercal drecon o. The nsably dsrbuon was exp( z) 3 h m calculaed by a fxed source calculaon for he recangular parallelepped wh nfne lengh n he vercal drecon. In he calculaon for he nsably dsrbuon, an exernal neuron source was placed a z =. The calculaed vercal nsably dsrbuon s shown n Fg.. The nsably s fed n he regon far enough away from he source o exclude he effec of hgher harmoncs. Table - Group consans and calculaon resuls for exponenal flux dsrbuon Parameers Values (cm - ).8 (cm - ).9 f (cm - ).4 3 k n ±.3 (cm - ) a (cm - ) b.9 ±. 3 (cm - ) c.9345 ±. e (cm - ) d.9489 ±.5 k.999 ±.6 a) Spaal decay consan defned by Eq. (8). b) Spaal decay consan defned by Eq. (9). c) Spaal decay consan fed o exponenal funcon. d) Spaal decay consan by c-egenvalue mode calculaon. Vercal flux (arbrary) The e was acqure by he mode calculaon wh 55, neurons per generaon, skppng 5 generaons and runnng 85, acve generaons. A he same me, he ndcaor k defned by Eq. (6) was calculaed. The calculaon resuls of he spaal decompose connuous are shown n Table. Four values n Table relavely agree wh each oher. and slghly dffer from 3 and e due o he naccuracy nvolved n usng he dffuson approxmaon. The shows excellen agreemen e wh he decompose connuous 3 ha s rgorously 9

5 Sep. 5. Vol. 7. No. 3 Inernaonal Journal of Engneerng and Appled Scences - 5 EAAS & ARF. All rghs reserved based on he neuron ranspor heory. Therefore, one may conclude ha he algorhm shown n Secon. provdes an accurae Eq. () and he algorhm s verfed numercally.. ( x, y,, E ) ( x, y, E ) ( x, y,, E ) 3. Claspng search mode calculaon by Mone Carlo mehod 3.. Theory of claspng search mode calculaon Ths secon deals wh cases where he horzonal claspng s smaller han he maeral claspng. Accordng o ha he energy-dependen horzonal nsably and he energy-ndependen vercal nsably are separable and he vercal nsably dsrbuon s characerzed by a claspng mode as: f (x,y,z, WE, ) = f ( x, y, WE, )exp( z) (9) where s he magnary un, s geomerc claspng n he z-drecon. Here, one of he spaal dependence n he z-drecon by a sngle Fourer mode, exp( z). I s assumed ha he vercal nsably s ndependen of neuron energy. Ths assumpon s no correc bu s forced o be presened for represenng he leakage effec by he sngle facor as n he wdely-used mehod. Subsue d de ( x, y,, E ) ( x, y,, E E ) S ( E ) d de ( x, y,, E ) ( x, y, E ) ( x, y,, E ) f 4 Eq. () no he hree-dmensonal neuron ranspor equaon. Then, afer a algebrac manpulaon, one obans a wo-dmensonal ranspor equaon ncludng he vercal claspng : Where s a drecon cosne wh he z-axs. Eq. () s very smlar o Eq. ().To ncorporae he las erm n Eq. () no random processes of a Mone Carlo calculaon, we follow he same procedure as n Secon.. The dfferenal of he wegh of a neuron parcle caused by flyng over nfnesmal dsance ds s gven by: dw Wds () Thus, afer a parcle fles over a dsance S n he h pah wh a drecon cosne, he nal wegh W () changes o: W W exp( S ) W (cos( S ) sn( S )) Dsp The produc of he rack lengh and he parcle s wegh afer he h pah wh a dsance S s gven by: S exp( S ) sn( S ) cos( S ) TL W exp( ) S ds W W (3) Whle he nal wegh of a parcle sarng from a splng source s a real number, he subsequen weghs can be complex numbers. If he splng neuron emsson s soropous n he laboraory sysem, he magnary par of W + n Eq. (3), W sn( z s ), appears wh he same frequency as W sn( z s ). Furhermore, f he dsrbung s soropous, he magnary pars of weghs dsappear for he same reason. Snce he calculave feld s homogeneous n he z-drecon, he weghs are symmerc wh respec o he z-drecon. In he same way, he magnary par of TL n Eq. (4), (cos( x S )- ) x, whch s an odd funcon of x, s supposed o dsappear. Therefore by magnary pars n Eqs. (3) and (4) can be omed only f he splng neuron emsson and dsrbung are soropous. Then, under such condons, Eqs. (3) and (4) are rewren as follows: W W cos( S ) (4) TL sn( S ) W (5) W and TL can be negave when z > p. When s large (.e., large neuron leakage n he z- drecon) and a mean free pah s long, a negave wegh ends o occur n Eq. (5). Even afer a wegh s

6 Sep. 5. Vol. 7. No. 3 Inernaonal Journal of Engneerng and Appled Scences - 5 EAAS & ARF. All rghs reserved The claspng used for he nex generaon calculaon s deermned ha k eff approaches uny as: Where l sands for he generaon number and d s an unreasonable posve value. Splng sources wh negave weghs mus be cancelled a he end of each generaon. Ths presen paper uses he procedure s for he wegh cancellaon (Yamamoo, []). A whole regon s dvded no a large number of small regons. Splng sources wh posve and negave weghs are accumulaed n he bns. The number of splng source parcles n he h bn s gven by n = n p - n m where n p and n m are he number of splng sources wh posve and negave weghs, s respecvely. n splng sources ha are used for he nex generaon s sarers are dsrbued unformly whn he h bn. 3.. Confrmaon of claspng search mode calculaon As shown n Secon 3., homogeneous one-energy group problems were used for numercal confrmaon of he claspng search mode calculaon. The one-energy group does no lm he generaly of n = In W + R (6) = + l ( - l + l d K eff ) () urns negave, he parcle wh a negave wegh keeps beng followed. The negave parcle wegh s reaed he same way as f were posve. Dscardng he negave weghs s equvalen o shorenng he pah lengh ha z < p. The proposed mehod, however, adops an nfne dmenson n he z-drecon. The wegh gven by Eq. (5) s a perodc funcon of he pah lengh, and he negave wegh becomes posve agan. If he pah lengh s very long n such a way ha z > 3p, he wegh may become larger han wha was before he pah sared. Ths problem may be emphaszed n reacor sysems wh gas-flled regons. Such heerogeneous sysems ncludng vod regons are problemac and should be reaed as a fuure work. If a parcle colldes, (n ) splng source ses are sored for use as splng sources n he nex generaon where: ( ) n å å f In(.) s he neger par, R s unform smulaed random number n [, ). If W <, a negave wegh s a sgned o he n splng sources. Afer all random processes whn one generaon are compleed, he proper value s esmaed as: é S ù W = Re - x exp( - x ) = cos( x )- ê ò A W S ds S ë úû generaon, N s he number of source per generaon, D W = W (cos( z S )- ), and: Here, Re[.] s a real par, and N = N p - N m where N p and N m are he numbers of splng source ses wh posve and negave weghs, respecvely. W s s å n å. TL k = f (9) eff M gven by W s = M/N where M s a nomnal source sze. k eff of Eq. () s supposed o be uny and s gven by: hs developed mehod. I s sraghforward o exend hs mehod o general connuous energy problems. Frs, one-energy group problems ha have nfne dmenson n he horzonal drecon were beleved. In hs case, he neuron nsably n he neuron ranspor equaon, Eq. (), becomes a funcon of drecon only as follows: N. + å f ( W) - S ( å D - n. + Wn å å) åf ( TL f ) xf ( ) = 4p ò W d W + W = () s f 4p å A (7) Ths problem s suable for confrmaon of he claspng search mode calculaon model snce no neuron ranspor n he horzonal drecon s presened. The proper values s of Eq. () were acqure by search mode calculaons wh 55, Where s summarze over all pahs whn one neurons per generaon, skppng 5 generaons ( ) (8) and runnng 8, acve generaons. Snce hs problem adops ha he splng neuron emsson and dsrbung are soropous, he magnary pars of wegh cancel ou and he weghs are expressed by usng Eqs. (5) and (6). The calculaons were performed for several cases. The group connuous and calculaon resuls are shown n Table. s he claspng based on he dffuson and s gven by, m e n Table s he claspng acqure by he claspng search calculaon. In cases,, and 3 n Table, we have a negave wegh n each pah s

7 Sep. 5. Vol. 7. No. 3 Inernaonal Journal of Engneerng and Appled Scences - 5 EAAS & ARF. All rghs reserved u J( u) + å ( u) f ( u) = du å ( u u) f ( u ) + c ( u) du nå ( u ) f ( u ) k ò ò () S f eff ( ) (, ) ( ) ( ) ( ) ( ) 3 f u + g u å = ò u å u J u du u u J u (3) S æ ö æ ö - an ç ( ) ( ) èå ç u ø èå u ø g(, u ) = æ - æ ö 3 an - ç çå ( ) ç å ( ) è è u u ø (4) less han 3-7 even when n = or - Thus, he effec of negave weghs s almos neglgble. In case 4, he probably s 3% when n = or - and he fracon of splng sources wh negave weghs s.3%. Thus, he effec of negave weghs s sgnfcan n case 4. In cases,, and 3, he resuls of he claspng search calculaons agree wh hose by he dffuson whn.3%. On he oher hand, n case 4 a noable dfference beween and e s found. Fg. shows he dependences of neuron nsably for cases and 4 ha were acqure durng he claspng search mode calculaons. The dependence for case s almos fla and, he slgh dependence can be accuraely come near by a lnear funcon of he drecon cosne, whch s one of he requses of he dffuson approxmaon. Ths fac means ha he dffuson gves a que accurae claspng. For case 4, however, he dependence clearly devaes from a lnear funcon of he drecon cosne. Thus, n case 4 s beleved o be based, and he dfference beween and e may be due o he naccuracy of. To oban a more accurae claspng, an nsably dsrbuon n he z-drecon was calculaed for a horzonally nfne bu vercally fne slab ha s nearly crcal (k eff =.8 ±.3). The hckness of he crcal slab was 5.75 cm. The vercal nsably dsrbuon was fed o a cosne funcon, he vercal claspng. The vercal nsably dsrbuon, however, s no exacly cosne shape due o neuron ranspor ransen effecs near he ouer boundares. The nsably dsrbuon only near he cener of he slab was used for he fng. The nsably dsrbuon and fed cosne curve are shown n Fg. 3. The area used for he fng was.35 cm. The claspng acqure from he nsably dsrbuon, c, s gven n Table. The claspng c agrees well wh e, whch may show confrmaon of he claspng search mode calculaon wh large neuron leakage. Anoher confrmaon can be done by comparng wh he mehod. The equaons are gven n (Dudersad and Hamlon, [3]) as: Table - Group consans and calculaon resuls for he bucklng search mode calculaons wh nfne dmenson n he horzonal drecon. Case Case Case 3 Case 4 (cm - ) a (cm - ) f (cm - ) (cm - ) a e (cm - ) b.3368 ± ±..74 ±..59e ±. c (cm - ) c ±.5 (cm - ) d k eff. ±.3. ±.3. ±.3. ±.5 a) ucklng by dffuson approxmaon. c) ucklng by fng o cosne funcon. b) ucklng by bucklng search mode calculaon. d) ucklng by he mehod.

8 Sep. 5. Vol. 7. No. 3 Inernaonal Journal of Engneerng and Appled Scences - 5 EAAS & ARF. All rghs reserved e) If he negave weghs are dscarded, c =.6378 ±. cm-. very accuraely he mehod. For numercal confrmaon of he mehod for solvng Eq. (), a recangular parallelepped wh a fne sde lengh n he horzonal drecon was beleved for each of cases 4. The group connuous, dmensons, and calculaon resuls are shown n Table 3. n Table 3 s a vercal claspng and s defned by: = - - (6) m x y w where u s he neuron lehargy, J(u) s neuron curren, s soropous componen of he dsrbung cross secon, and s he lnearly an soropous dsrbung componen of he dsrbung cross secon. The equaons are modfed from he orgnal ones n he leraure. Assumng ha he dsrbung s soropous (.e,. = ) and he neuron s monoenergec, Eqs. (3) and (4) are smplfed o: ( å - å ) g( ) + = n å g( ) å 3 k Dspl S f eff where he varable u and he subscrps for he dsrbung orders are omed. Usng Eq.(6), he crcal buck lngs for k eff = can be easly acqure by erave calculaons. The crcal buck lngs acqure by Eq.(6) are lsed n Table. The crcal bucks lngs acqure by he mehod agree very well wh he Mone Carlo calculaons for all cases. If he negave eghs are mmedaely dscarded, he calculaed crcal claspng s cm whch s defnely wrong. The proposed algorhm for he claspng search calculaon by Mone Carlo mehod can reproduce Anoher vercal claspng s gven n Table 3, whch s defned by: where = - (7) m h = ( n å k - å ) / D, k = k of h f n a m eff he recangular parallelepped wh nfne lengh n he vercal drecon. k n was calculaed by a Mone Carlo crcaly calculaon. s more accurae snce he horzonal drecon s reaed by he neuron ranspor heory. Excellen agreemen s found beween and e excep for case 4f where ansoropy of neuron nsably s sgnfcan due o he large vercal claspng and long mean free pah. 4.Leakage-correced calculaon by Mone Carlo mehod Suppose ha a claspng n he vercal drecon s known a pror and he horzonal drecon s reaed, ncludng he horzonal neuron leakage, by he neuron ranspor heory. The neuron ranspor equaon for hs case s gven by: 3

9 Sep. 5. Vol. 7. No. 3 Inernaonal Journal of Engneerng and Appled Scences - 5 EAAS & ARF. All rghs reserved WÑ. f ( x, y, W, E ) + å ( x, y, E ) f ( x, y, W, E ) ò = d W de f ( x, y, W, E ) å ( x, y, W W, E E ) c ( E ) k 4p ò ò eff where s no an proper value bu a user-specfed claspng n he z-drecon. The calculaon algorhm for solvng Eq.(9) s almos he same as convenonal Mone Carlo crcaly calculaons for seekng k eff. The excepons are ha a parcle wegh and he produc of a rack lengh and parcle wegh are gven by Eqs. (5) and )6 ( respecvely. k eff appears explcly as an proper value of Eq.(9). The claspng s no updaed and s fxed a he userspecfed value hroughou he calculaon. The calculaon n whch neuron leakage n he vercal drecon s correced by a user-specfed vercal claspng s easer han he claspng search mode calculaon snce he claspng need no be updaed. y solvng Eq. (9), one can oban a leakagecorreced neuron nsably dsrbuon and neuron specrum, whch wll subsequenly be used for generang group connuous. Nex, consder a consuen un of a whole core confguraon such as a un fuel, a sngle fuel assembly, or mul-fuel assembles. The reflecve condons are requre on he boundary surfaces of he consuen un. Alhough he neuron ranspor phenomenon s W = + W exp( - ( m + h + x ) S ) = + d W de f ( x, y, W, E ) n å ( x, y, E )- xf ( x, y, W, E ) x y z 4 reaed by Mone Carlo mehod whn he range of he un, he effecs of vercal and horzonal neuron leakage are presened by vercal and horzonal buck lngs, respecvely. Agan, he un s homogeneous and nfne n he z-drecon. Assume ha he hreedmensonal nsably dsrbuon n an x y z Caresan coordnae sysem s characerzed by a smple claspng mode as: f ( x, y, z, W, E ) = f ( x, y, W, E )exp( ( x x + y y + z z )) Dsp where x, y and z are geomerc drecon buck lngs n x, y, and z-drecons, respecvely. Subsung Eq. (3) no he hree dmensonal neuron ranspor equaon, one obans a wodmensonal ranspor equaon ha s smlar o Eq. (). However, he las erm s dfferen and s gven by: - ( m+ h + x) f ( x, y, W, E ) (3) x y z where u, n are drecon cosnes wh he x- and y- axes, respecvely. Eqs. (3) and (4) are, respecvely, rewren as: = W (cos(( m + h + x ) S )- sn(( m + h + x ) S )) ò x y z x y z S ò TL = W exp( - ( m + h + x ) S ) ds x y z æsn(( xm + yh + zx ) S ) cos(( xm + yh + zx ) S )- ö = W + çè xm + yh + zx xm + yh + zx ø Consder ha a horzonal calculave feld has wo orhogonal symmery planes and he dsrbung and splng neuron emsson are soropous. For such a case, Gelbard and Lell [3] menoned a claspng canno exs anywhere n he lace one canno f a cosne very well o any segmen of hs curve. However, suppose ha one connues o presume, roughly, a cosne shape even for an asymmerc lace cell and he horzonal bucklng x and y can be defned whn a ceran range of accuracy. Then, one opon s o neglec he magnary pars of Eqs. (3) and (33 (whch however may presen errors ha are dffcul o be quanavely esmaed. Anoher opon s o rea complex weghs defned by Eq. (3). A deermnsc core calculaon code equpped S f (8) (3) (3) wh he mehod could be used for confrmaon of he new algorhm ha handles complex weghs. When one s neresed n a crcal claspng search calculaon n a hree-dmensonal confguraon, hree drecon bucklng, x, y and z, need o be deermned. However, he combnaon of hree drecon bucklng s unreasonable. Thus, some resran condons are requred on he combnaon of x, y and z. For nsance, x and y are fxed a connuous and hen only z s searched. Or, x s equal o y and he rao of he horzonal claspng o he vercal claspng s fxed and so on.

10 Sep. 5. Vol. 7. No. 3 Inernaonal Journal of Engneerng and Appled Scences - 5 EAAS & ARF. All rghs reserved 5. CONCLUSIONS Ths paper has used a Mone Carlo calculaon algorhm for claspng search mode calculaons where he horzonal drecon s reaed by neuron ranspor calculaon and neuron leakage effec n he remanng vercal drecon whch s aken no accoun by he vercal claspng. When he horzonal claspng s larger han he maeral claspng, he vercal neuron nsably dsrbuon akes he form of an exponenal funcon. The spaal decompose connuous of he vercal neuron nsably dsrbuon can be acqure as a proper value by solvng a claspng search mode proper value equaon. The proper value can be verfed by comparng wh he spaal decompose connuous calculaed by a fxed source problem. On he oher hand, when he horzonal claspng s smaller han he maeral claspng, a proper value equaon s a claspng n he vercal drecon can be defned by he vercal drecon by a sngle Fourer mode, exp(z). The proper value equaon ncludes a complex erm, whch produces complex parcle weghs durng he Mone Carlo random processes. However, f he calculave feld s unform n he vercal drecon and he dsrbung and splng neuron emsson are soropous, he weghs urn ou o be symmerc wh respec o he vercal drecon, whch elmnaes he magnary pars from he complex weghs. Thus, only he real pars reman n he random processes. When a claspng s large and a neuron s mean free pah s long, a negave wegh may appear durng he random processes. The negave wegh s reaed he same way as he posve wegh. When appearance of negave weghs s sgnfcan, he dffuson becomes worse due o noable ansoropy of neuron nsably. In such a case, confrmaon by comparng wh he dffuson s naccurae. The claspng acqure by he claspng search mode proper value calculaon was compared wh he one acqure by he nsably dsrbuon or by he mehod. oh bucklng agree well, and he claspng search mode calculaon has been verfed for large neuron leakage sysems. Ths Mone Carlo algorhm for nroducng neuron leakage effec by a claspng can also be appled o leakage-correced Mone Carlo calculaons. The bucklng n he x, y, and z-drecons are fxed a user-specfed values, and hen Keff, neuron specrum, splng source dsrbuon and so on are calculaed for generang group connuous used for a subsequen reacor core desgn. When a horzonal calculave feld has wo orhogonal symmery planes and he dsrbung and splng neuron emsson are soropous he magnary pars of complex weghs dsappear. If does no, complex weghs need o be reaed. The mehodology for reang complex weghs has no ye been esablshed and would be a fuure work. The newly developed algorhm for solvng he claspng search mode of proper value equaon was verfed hrough comparson wh he claspng does acqure by he dffuson and he mehod. As long as he dffuson s reasonable, boh claspng's acqure by he claspng search mode of proper value and he dffuson agree well, whch exhbs confrmaon of he new algorhm. REFERENCES. Yoshoka, K., Ando, Y.,. Mulgroup scaerng marx generaon mehod usng wegh-o-flux rao based on a connuous energy Mone Carlo echnque. Journal of Nuclear Scence and Technology 47, Leppänen, J., 7. Developmen of a New Mone Carlo Reacor Physcs Code. VTT Publcaons Dudersad, J.J., Hamlon, L.J., 976. Nuclear Reacor Analyss. John Wley & Sons, New York. 4. Frdman, E., Leppänen, J.,. On he use of he Serpen Mone Carlo code for few group cross secon generaon. Annals of Nuclear Energy 38, Shm, H.J., Han,.S., Jung, J.S., Park, H.J., Km, C.H.,. McCARD: Mone Carlo code for advanced reacor desgn and analyss. Nuclear Engneerng and Technology 44, Maorov, L.V., 985. Calculang neuronflux funconals by he Mone Carlo mehod n breeder sysems wh leakage specfed by a geomerc parameer. Translaed from Aomnaya Énergya 58, Yun, S., Cho, N.Z.,. Mone Carlo depleon under leakage-correced crcal specrum va albedo search. Nuclear Engneerng and Technology 4, Yamamoo, T., Myosh, Y., 3. An algorhm of α and c-mode egenvalue calculaons by Mone Carlo mehod. In: Proc. 7h In. Conf. on Nuclear 9. resmeser, J.F., (Ed.). MCNP A General Mone Carlo N-parcle ranspor code, Verson 4C. LA-379-M. 5

11 Sep. 5. Vol. 7. No. 3 Inernaonal Journal of Engneerng and Appled Scences - 5 EAAS & ARF. All rghs reserved Fowler, T.., Vondy, D.R., Cunnngham, G.W., 969. Nuclear Reacor Core Analyss Code: CITATION, ORNL-TM-496, Rev... Yamamoo, T., a. Hgher order a mode egenvalue calculaon by Mone Carlo power eraon. Progress n Nuclear Scence and Technology, Yamamoo, T., b. Non-regon wse wegh cancellaon for Mone Carlo hgher order crcaly calculaons usng kernel densy esmaor. Annals of Nuclear Energy 38, Gelbard, E.M., Lell, R., 977. Mone Carlo reamen of fundamenal-mode neuron leakage n he presence of vods. Nuclear Scence and Engneerng 63,