TREATMENT OF HETEROGENEOUS REACTOR SYSTEMS

Transcription

1 TEATMENT OF HETEOGENEOUS EACTO SYSTEMS M. gheb 7/9/4 NTODUCTON Mo recor yem nowdy re o he heerogeneou ype, where he uel nd moderor/cooln re lumped nd do no orm homogeneou mixure. The criicliy prmeer o inere uch he regenerion cor η uel uilizion cor nd he reonnce ecpe probbiliy p ued in he our cor ormul or he ininie medium muliplicion cor will hve o ccoun in heir clculion o hi c. COSS-SECTONS HOMOGENZATON N HETEOGENEOUS SYSTEMS We cn mix he dieren meril region in heerogeneou yem nd re i homogeneou yem only i he diuion lengh L or he homogenized region i lrger hn ny chrceriic dimenion d o he yem, or herml neuron: L >> d Similrly or neuron, he ollowing condiion mu be iied: τ>> d, where: τ i he lowing down re, or ge.

2 n ny homogenizion, we require he preervion o recion re in he heerogeneou nd homogenized yem. Thi cn be expreed : r r d where: i he ol volume o he recor, i he verge lux in he yem: d r i homogenized cro ecion. For everl m epre region, Eqn. cn be wrien : m i i i m i i i i i which i requiremen h he cro ecion re o be lux weighed nd volume weighed in obining homogeneou cro ecion. he luxe re reonbly conn, nd we cn ue only volume weighing: i m i i m i i i i 3 The cor / i oen reerred o he el-hielding cor. i EFFECT OF HETEOGENETY ON THE EGENEATON FACTO The quniy η i he number o iion neuron produced per neuron borbed in he uel. n homogeneou recor, uel men he iionble ioope. n lumped

3 recor, uel now reer o he iionble ioope nd ny ioope which re mixed wih he iionble ioope. he uel i lumped, i my coni o number o ioope, ome o which re iionble, nd ome re no. The verge lux my no be he me in ll ioope conidered pr o he uel. n hi ce, he regenerion cor i given by: η i i vnσ i i i i N σ i i i 4 where i i he verge lux over he i-h ioope. he vriou ioope re inimely mixed, i conn, nd Eqn. 4 become: η i i vnσ i i i N σ i i 5 A n exmple, or nurl urnium: vn U σ U σ σ c σc η NU U NU U NU U 35 vσ U σ σ σ U c U 38 c U r 35 NU where: r NU 4 Since cro ecion d be or ue in herml recor li he [m/ec] vlue, hee hve o be modiied by he non-/v cor g, nd he emperure nd Mxwell diribuion cor:

4 / 93 π σ g σ 73 6 where: v p v kt π m 8kT i he rio o he mo probble o he verge energy π m or velociy Mxwellin diribuion: mv 3/ kt m nv 4 π n ve π kt g i he non-/v cor, uming well-modered yem. i uncion o emperure. Noe h cering cro ecion re no ubjec o hi correcion. i he medium emperure in degree Celciu room emperure. Since g U., nd g U.975, room emperure, we ge: o C η The cro ecion d were ken rom Tble, he non-/v cor rom Tble nd he vlue o he verge number o neuron rom iion ν rom Tble.

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6 Tble : Neuron egenerion D. U 33 U 35 u 39 Nurl Urnium ν η σ c α σ EFFECT OF HETEOGENETY ON THE FUEL UTLZATON FACTO The uel uilizion cor i deined he rio o he number o neuron borbed per uni ime in he uel, o he ol number o neuron borbed per uni ime in he whole recor, including uel, moderor, rucure nd conrol elemen. For lumped yem coniing o uel region nd moderor region:

7 uel rd rd rd uel mod eror 7 Le: uel rd moderor rd where:, re he volume o he uel nd moderor region, repecively. Then we cn wrie: 8 or: 9 The cor / i clled he didvnge cor nd i given by rerrnging Eqn. 9 :

8 i didvnge ince normlly, or lumped yem, >, conequenly / >, nd or heerogeneou yem will be le hn or he equivlen homogeneou one hown in Fig.. For lice conigurion, he ir ep or he clculion o i o chooe uni cell hown in Fig..

9 The nex ep i o pproxime he rue hpe o he cell by hpe which cn be decribed by ingle dimenion, in uch wy o minin he rue volume o he cell. For he cylinder nd he qure prllelepiped o hve he me volume: π L L rom which: π Wih he problem reduced o one dimenion, diuion heory my now be pplied. The lowing down deniy q i umed uniorm over he region o he cell occupied by he moderor, nd o be zero in he uel. The diuion blnce equion or hi ce will be in he uel nd in he moderor: D D q

10 where he ubcrip denoe he uel, nd ubcrip denoe he moderor. Dividing by he diuion coeicien, wriing: coordine yem or he Lplcin operor, Eqn. become: / D, nd uing cylindricl d d dr r dr d d q dr r dr D The oluion in he uel region in erm o Beel uncion i: A r DK r A r 3 ince: D, K goe o ininiy r, which would led o n unphyicl oluion.. The oluion in he moderor region i obined by dding complemenry uncion nd priculr oluion: q B r CK r 4 Since ll he cell re idenicl nd re umed o coniue n ininie rry, here mu be no neuron curren rom cell o cell; hu: d dr 5 C: Applying hi condiion o Eqn. 4, we ge he relion beween he conn B nd

11 d B C K, dr where we ued he Beel uncion he relionhip: We ge or he vlue o C: z z, K z K z, C K B 6 Thu Eqn. 4 cn be rewrien : where: q GK [ r K r] 7 B G K Applying he condiion o he coninuiy o he lux nd curren he inerce beween he uel: nd moderor, or he lux:, nd or he curren D D, 8 yield wo equion or he vlue o he conn A nd G lhough or he evluion o, he vlue o G i no required:

12 [ ] [ ] q A GK K DA DG K K 9 From he econd equion in 9, we ge he vlue o G: K K D D A G Upon ubiuing he vlue o G in he ir equion in 9, we ge he invere o A : [ ] K K K K K D D q A We recll now he deiniion o : A A nd eime he recion re per uni lengh o he cell. The borpion re in uel i ju he neuron curren ino region, or: D A dr d D J A π π π The ol borpion per uni lengh o cell i ju he ol ource rengh per uni lengh: q A π, hu:

13 A q A D q π π nering he vlue o /A rom Eqn. ino Eqn. : K K K K Thu / coni o wo erm one which involve o he uel only, nd he oher o he moderor only. The generl orm o Eqn. i: E F 3 The uncion E nd F re lied or dieren geomerie in Tble. he uel i urrounded by hin lyer o noniionble meril o volume nd borpion cro ecion i eec cn be included in he expreion or / : E F 4 we imply here h he eec o he lyer on he lux diribuion i negligible.

14 EFFECT OF HETEOGENETY ON THE ESONANCE ESCAE OBABLTY The expreion or he reonnce ecpe probbiliy or homogeneou yem h been derived : N e p exp 5 ξ where: e i he eecive reonnce inegrl. n ce o heerogeneou yem, he reonnce borpion inide he uel i much le hn he borpion he urce. When he uel meril h low modering power, he energy region depleed by reonnce borpion he urce re no replenihed by moderion, nd hi el-hielding o he uel core gin reonnce borpion i igniicn. Thi eec mke poible he conrucion o heerogeneou recor uing urnium moderor. homogeneou yem i ued, criicliy cnno be chieved.

15 A emi-empiricl mehod i ued or lumped yem ince well-known reonnce rucure doe no exi. On heoreicl nd experimenl ground he re o reonnce borpion per uni ime per uni energy inervl or urnium lump o imple hpe i given by: S N Nb M where: cor: i he verge lux per uni energy inervl he urce o he lump, S i he urce re o he lump,, b re experimenlly deermined conn, i he volume o he lump. Thee borpion would led o decree in he lowing-down deniy given by dq 6 de From which we cn wrie: dq S N Nb 7 de M eclling he expreion o lowing-down deniy: q ξ E 8 we cn wrie:

16 dq q N Nb ξ E S M de eclling he deiniion o reonnce ecpe probbiliy, nd inegring beween he limi E nd E, we ge: h E E q h N de S de p exp b q ξ E M E E E h h 9 i i umed h: con n, we ge: where: E N p exp σ ξ Eh e de E 3 E de de S de S e σ e b A µ E E M E M Eh E Eh E Eh The vlue o A nd µ re given here or Nurl Urnium nd ome o i compound:

17 ill remin o eime he rio: /. n nlogy o he previouly derived didvnge cor or he herml uilizion cor: Didvnge cor, We deine reonnce uilizion cor, nd cn wrie: r E F r, Thu: E F 3 which cn be ubiued ino Eq. 3 o yield: e e N E N F p exp ξ ξ 3

18 n Eqn. 3, denoe iciiou borpion cro ecion which i relly lowing-down cro ecion expreing he probbiliy h neuron will be lowed down rom he reonnce energy rnge. A previouly derived in he conex o Fermi Age Theory where i w reerenced removl cro ecion, i i given by: ξ ln E E h 33 The verge borpion cro ecion in he borber i: N σ e E N e 34 de E de E ln E h From Eqn. 33 nd Eqn. 34: N ξ e 35 Subiuing rom Eqn. 35 ino Eqn. 3, we inlly ge: p exp ξ N e F E 36 THE THEMAL DFFUSON COEFFCENT

19 For homogeneou medium he ollowing relionhip or he diuion coeicien, bed on rnpor heory, cn be ued: µ µ µ D 37 which reduce, or wekly borbing medium o: 3 3 r D λ µ 38 For heerogeneou recor, he rio o moderor volume o uel volume i lrge, o h he diuion coeicien o he moderor cn be ued in he clculion. hi i no rue, he ollowing expreion which ccoun or he didvnge cor cn be ued: 3 3 r r r D 39 eclling he expreion or he didvnge cor: ξ, where i he herml uilizion cor, we cn wrie i : 3 r r D ξ ξ 4

20 Tble li vlue o he rnpor men ree ph or ome meril o inere. THE THEMAL DFFUSON AEA The herml diuion re i given by: D L 4 We hve lredy derived n expreion or, we cn wrie imilr expreion or D h : ξ ξ ξ

21 >> ξ, hen we cn wrie: 4 he diuion coeicien i ken being he one or he moderor, hen we ge: L D L 43 where: L given in Tble. i he diuion re o he moderor. Some experimenl vlue or L re

22 THE FAST FSSON EFFECT Non-herml iion cn occur in he reonnce region e.g. in U 33, U 35 nd u 39, or in he region in meril which hve iion hrehold in he Me rnge e.g. in U 38 nd Th 3. For heerogeneou yem he clculion o he iion cor mu ccoun or wo phenomen: The probbiliy h neuron born in he uel lump mke colliion beore ecping. The conribuion hi neuron mke o he eec, well he conribuion o ubequen generion by ccde eec. Le u inroduce: [probbiliy h neuron produced in he n-h generion n iion mke colliion beore ecping rom he uel lump] Thu, n i he probbiliy o ecping rom he uel lump, nd i n i he probbiliy o recion i, where: γ in For he neuron produced in he ir generion o iion, we ge he ollowing number o neuron or ech ir generion neuron: ν r

23 where we imply h n elic-cering recion h negligible eec in reducing he neuron energy. From h ir generion, ome neuron re lo by ecpe, inelic colliion, nd rdiive cpure nd: [rcion o ir generion neuron enering lowing-down proce] in For he ollowing generion he ollowing ccding proce occur: By dding up he erm in he lo column, we ge: in in n in ε r... r... n n n we ume h n or ll generion, we ge:

24 < in in n n in r r r r r ν ε, Subrcing he denominor rom he numeror on boh ide o Eqn. 45, we ge: in ν ν ε in ν ν ε Subiuing or rom Eqn. 43, we ge: in in γ ν ν ε And inlly: γ ν ν ε 46

25 We now need o clcule he probbiliy o colliion. Conider uel rod o rdiu hown in Fig. 4. We hve lredy deduced: -Number o neuron ecping rom lump / Number o neuron produced in he uel lump The denominor i imply: ν r d Wriing down he numeror, we ge: S ν r e ρ ν r d coθ d ds 4πρ 47

26 where we re conidering he number o neuron produced in volume elemen d, enued by he cor ρ e, nd by he invere qure dince cor 4πρ. We re correcing or he coine o he ngle beween he norml o he uni re ds nd he direcion o he ougoing neuron. An inegrion over he urce nd over he volume i hen crried ou. The inegrion o Eqn. 47 cn be ediou nd depend on he reed geomery. lue o or dieren geomerie re hown in Fig. 5 where x i he invere o he diuion lengh, nd correpond o he ol cro ecion. we ume h he lux i conn in he uel rod, nd h he rdiu o he rod i mll compred o he men ree ph, hen we cn wrie: OTMZATON OF LATTCE CONFGUATON

27 For given uni cell pich, he reonnce ecpe probbiliy decree he rod rdiu i increed, where he herml uilizion cor incree o urion. For given rod rdiu, he reonnce ecpe probbiliy incree wih lrger cell rdiu, where he herml uilizion cor decree. Thu, i i eviden h condiion which vor n incree in will cue decree in p, nd vie ver. n he deign o heerogeneou lice we mu hen ind priculr rrngemen o uel nd moderor which give mximum vlue o he our cor ormul or he ininie medium muliplicion cor: k ηε p 49 The procedure o be ued i hu o conider number o uel-moderor lice, wih vriou rod rdii nd lice pcing, nd conruc urce or k uncion o hee wo prmeer. Then chooe he lice h will mximize he vlue o. Thi opiml vlue occur normlly when he vlue o p nd re decreing or increing he me re. k

28 EFEENCES. M. gheb, Lecure Noe on Fiion ecor Deign Theory, FSL-33, Univeriy o llinoi, 98.. J.. Lmrh, nroducion o Nucler Engineering, Addion-Weley ublihing Compny, 983.