22.05 Reactor Physics - Part Twenty. Extension of Group Theory to Reactors of Multiple Regions One Energy Group *

Transcription

1 22.5 eaor Phyi - Par Tweny Exenion o Group Theory o eaor o Muliple egion One Energy Group *. Baground: The objeive reain o deerine Φ ( or reaor o inie ize. The ir uh ae ha we exained wa a bare hoogeneou ore. The ajor ep in he oluion were o: Modiy he oninuou energy diuion equaion, in whih ro-eion and lux were union o r and E or a hoogeneou ediu. Hene, ro-eion beoe union o E alone. Flux reain a union o r and E. Aue eparabiliy o he energy and paial union o he lux. Tha i Φ( = ( E). Subiue he eparabiliy aupion ino he hoogenized diuion equaion and eparae er o ha a union o poiion equal a union o energy. Thi i only poible i boh union equal a onan. Thi 2 onan i alled he buling,. Solve or he paial oluion. B r Subiue he paial oluion ino he hoogenized diuion equaion. Deine group onan wihin eah energy group. Solve or he energy dependene o he lux. Thi equene i unique o a bare (unreleed) ore and i NOT he general ae. Speiially, or ore o uliple region (uel/oderaor and releo, he eparabiliy aupion doe no hold. A a reul, he equene or obaining a oluion or Φ ( hange. In pariular, he energy dependene i obained beore he paial dependene. * Maerial in hi eion ollow ha o Henry pp Porion ha are verbai are indiaed by quoaion.

2 The proble i onidered here or a wo region reaor wih one energy group being aued. In he nex eion o hee noe, we onider a wo region reaor wih wo energy group. Finally, we exend he idea o uliple region reaor wih uliple energy group. 2. One Energy Group: The er one-group heory i ued o deribe everal dieren ehod o oluion in reaor phyi. Earlier (Seion o hee noe) we derived a oneveloiy odel and ued i o olve or he paial lux in bare ore. Tha approah i oen alled one-group heory beaue he ingle veloiy orrepond o a ingle energy group. Bu, a diinion hould be ade beween ha derivaion and wha we will do nex beaue, even hough he reul i he ae, he ehod o derivaion eployed here i ar ore rigorou. The aring poin i he oninuou energy diuion equaion whih we reae below: D( r, E) Φ( r, E) + Σ = χ j j j (E) ν Σ j ( r, E ) + Σ ( r, E) Φ( r, E) For a hoogeneou ediu, hi beoe: ( r, E E) Φ ( r, E )de 2 D(E) Φ( + Σ (E) Φ( = (E) (E ) (E E) (,E )de χ νσ + Σ λ Φ r We wih o apply hi equaion o a wo region reaor ubje o he aupion ha eah region i hoogeneou. Thu, we odiy he above equaion by adding a uperrip o eah paraeer (D, Σ, Σ, Σ,e.). A given value o hi uperrip orrepond o a pariular region. For exaple, = igh be ore/oderaor; =2 igh be releor, =3 igh be hielding, e. The equaion beoe: D (E) Φ( + Σ (E) Φ( = (E) (E ) (E E) (,E )de χ νσ + Σ Φ r λ ( =, 2, 3,..., K) For he ae a hand, = 2 beaue here are wo phyial region. (Noe: The one in one-group heory reer o he nuber o energy group and hene here i one energy union (E) ha will apply o eah region.) We an now ee why he eparabiliy aupion ( Φ (=(ψ(e)) ha wa ued or he (A) 2

3 hoogeneou bare ore (=) will no wor or a uli-region reaor ( 2). Eah region ha i own e o aerial properie (i.e., D, Σ, e.) A ingle union ψ(e) anno aiy dieren e o properie. A preerred approah, and one ha i viable, i o reognize ha aer a ew olliion, neuron behavior i independen o neuron origin. Thu, he neuron energy peru in a given region hould be haraerii o only he hoogenized aerial ha are preen in ha region. Thu, we aue eparabiliy wihin eah region. Here he word wihin iplie a ew ean pah inide a given region. Thu, we aue: Φ( (E) (B) Where (E) i an energy peru union or region. Thi i he eenial aupion o one-group diuion heory. A a reul o aing i, we an urher aue ha, wihin a given region, he ype o analyi done or he bare hoogeneou reaor reain relevan. Thu, he eenial raegy o one-group diuion heory i: ) aue ha eparabily (equaion B above) i valid wihin eah region ; 2) ind an appropriae peru union ψ (E) or eah region; and 3) onne he paial union ( or eah region o he orreponding paial union o neighboring region. (Henry, p. 46). Given he above raegy, we expe ha he leaage er an be approxiaed a: ( B ) 2 2 ( B ) (,E) D (E) Φ( = D (E) Φ r Where i a nuber and i alled he aerial buling. For he ingle 2 region reaor, we enounered a iilar quaniy, B r, whih depended on he geoery o he reaor and wa ered he geoeri buling. The aerial buling i dieren. Speiially, ( B ) 2 ha nohing o do wih eiher he geoery o he reaor or o an individual region. I i ound by reognizing ha eah region i par o a riial reaor or whih λ i uniy. Hene, ( B ) 2 i ound by wriing he hoogeneou diuion equaion or eah region a: D (E) 2 ( B ) + Σ (E) Φ( r,e) = χ (E) νσ (E ) + Σ (E E) Φ( r,e )de λ or r in he inerior o region, 3

4 ) 2 and eleing ( B a he eigenvalue ha yield an everywhere poiive oluion or he lux. I i iporan o underand he phyial eaning o hi nuber and we quoe ro Henry (p. 46): ( ) 2 B The nuber i haraerii o he ixure o aerial aing up region and, or ha reaon, i alled he aerial buling. Maheaially i i an eigenvalue o he hoogeneou equaion. A an be een ro ha equaion, i ee on neuron balane i o regulae he leaage rae o ha he porion o he reaor in he inerior o region behave in a uained, riial ahion. I he o ha region exeed uniy, he nuber o neuron reoved ro de dv per eond by ou-leaage, aering, and aborpion will equal he nuber inrodued ino de dv by iioning and aering ro oher energie; while, i he i le han uniy, he nuber reoved by aering and aborpion will equal he nuber inrodued by iioning, aering, and leaage ino he region. Thu, i he or region- aerial i le han uniy, ( ) 2 B will be a negaive nuber o ha he exra neuron needed or overall balane will be upplied by negaive leaage, i.e., by ne leaage ino dv. In releor aerial, or whih ( ) 2 νσ ( E) =, will alway be negaive. (Henry, pp ) B We now ubiue Φ( = (E) ino he odiied hoogeneou diuion equaion or region, eliinae he paial union, and obain: D (E) 2 ( B ) (E) + Σ (E) (E) = [ χ (E) νσ (E ) + Σ (E E) ] (E )de (C) Thi relaion an be olved uing he ehod developed earlier (Seion 6 o hee noe) o ind (E). I will be realled ha he approah i o divide he energy range (- MeV) ino G group where G igh be a large a 3. The widh o eah group, ΔE g, i eleed o ha ro-eion properie will be onan over ha group. Thi allow a urher aupion ha he lux (E) will be onan over eah group. Group onan an now be deined. Thee are o he or: g E g Eg Σg Eg Σ E (E) g g Noe ha hee group onan are deined: ) or an aued lux hape, (E) and 2) a narrow energy range, Δ E g. Thi will be in onra o a dieren e o 4

5 group onan ha we will deine laer when we olve or paial lux hape. Ue o he above group onan allow u o rewrie Equaion (C) a a e o algebrai relaion, whih are readily olvable nuerially. So, a hi poin we an obain (E). The objeive hereore beoe o obain he paial lux hape. Thu ar he derivaion ha been rigorou. Bu a proble now arie in ha we have o rea he inerae beween region. The preerred boundary ondiion are he andard one ha lux and he noral oponen o he urren be oninuou a all energie. However, hi in poible beaue he (E) dier or eah region. To quoe Henry (p. 47), The bai proble i ha he eparable or (E) (=,2, K,), whih are good approxiaion o Φ(r, E) in he inerior porion o he region, anno be valid near inerae beween region, ine, i uperrip and l deignae wo adjaen region o dieren opoiion o ha (E) and l (E) are dieren union o E, hen, i r i any poin on he inerae beween hee region, ( r ) (E) anno equal ( r ) (E) or all energie. (I i did, hen (E) would have o have he ae energy hape (E). Thu, auing ha he eparable or (E) i valid hroughou eah region ae i ipoible o ee he oninuiy-o-lux boundary ondiion a he inerae beween region. The be one an do i o adop he uh weaer boundary ondiion ha he inegral over all energie o Φ ( and n Φ(r, E) be oninuou. Thi boundary ondiion i enirely onien wih he oninuiy o Φ ( and n D Φ( a all energie ine, i here were oninuiy a eah energy, he inegral over all energie would erainly be oninuou. (The revere ipliaion anno, o oure, be drawn.) To be peii, one-group heory i baed on he ollowing aupion:. Φ( r,e) = ( (E) hroughou eah opoiion, (E) being ound by olving Equaion C a deribed above. 2. I and l indiae any wo adjaen opoiion and r i a poin on he inerae, we require ha and de de n ( r ) D (E) = (E) [ ( r ) (E)] = den D (E) [ de ( r ) (E) ( r ) (E) Noe: I (E) i noralized o ha =, hen hee ondiion beoe: ] 5

6 and ( r ) = ( r ) D (E) n ( r ) = D (E) n ( r ) We now ubiue he eparabiliy relaion (equaion labeled a B above) ino he hoogenized diuion equaion or eah region (equaion labeled a (A) above) and inegrae over all energie o a o oply wih he boundary ondiion. The reul i: = χ λ D (E) (E) (E ) (E )de νσ (E ) (E )de = νσ λ + + de de (E E) (E ) Σ (E ) (E )de + Σ Σ (E) Nex, we deine a e o one-group ro-eion: (D) D Σ νσ Σ Σ Σ D (E ) Σ (E ) Σ = Σ Σ (E ) νσ (E ) a Thee nuber are independen o boh pae and energy. They ay be opued or any region one we have deerined he nulear onenraion n j and he iroopi ro-eion σ j (E) o he aerial in region and have obained he peru union (E). (Henry, pp ) I i iporan o reognize he dierene beween hee group onan ha are ued o obain he paial lux hape ro he one deined earlier o obain he energy lux hape. Fir, he urren group onan (paial alulaion) are 6

7 inegraed over a very wide energy range. Here, ha range i o. The ore oon range and he one ha we ue laer on are heral, epiheral, and a. Seond, he hape o (E) i nown. Thi i wha allow evaluaion o he inegraion over uh wide range. One oher lariiaion i appropriae. The word group a ued here reer o he nuber o energy range over whih he above inegraion are perored. For he preen e o leure noe, here i one range ( o ) and hene one group. The ore oon approah, a i already noed, i or hree range (heral, epiheral, and a) and hene hree group. Upon ubiuion o hoe one-group ro-eion, we obain: D + Σ Wih boundary ondiion: n ( r ) = ( r ) D ( r ) = n = νσ λ D ( r ) The ondiion ha he lux i o vanih on he exernal urae o he reaor i e by requiring ha he appropriae ( vanih or any r on ha urae. Finally, ine ( i everywhere oninuou hroughou he reaor, he uperrip i redundan. The poin r auoaially peiie he region. Aordingly, we hall replae all he ( by he ingle union Φ(, he onegroup alar lux. (Noe ha Φ( i no a lux per uni energy; i uni are (peed) x (neuron/) = neuron per quare per e.) Wih hi hange in noaion he one-group diuion equaion or a reaor opoed o everal hoogeneou region beoe D Φ( + Σ Φ( = νσ λ Wih he ollowing boundary ondiion: Φ(. Φ ( and he noral oponen n D Φ( are oninuou aro inerae beween dieren opoiion; 2. Φ = on he ouer boundary o he reaor. 7

8 The above equaion i oeie alled he one-veloiy diuion equaion ine i an alo be derived by auing ha all he neuron are raveling a a ingle peed. The rouble wih ha approah i ha i provide no yeai proedure or inding he one-group onan D, Σ, and νσ, and ae i hard o aoun or uh ee a a iion, reonane apure, and a-neuron leaage. I alo ae one-group heory appear uh ruder han i aually i. Aording o he view we have adoped, he one-group heory aually provide a very deailed energy-dependen lux, naely, (E) Φ(, in eah region. The eenial approxiaion oni o a very rude reaen o he raniion zone beween he dieren aerial opoiion. In hee zone he rue alar lux deniy Φ ( aually hange gradually ro he (E) appropriae o he inerior par o one region o ha appropriae o he inerior par o he nex. We replae hi gradual hange by an abrup one and in o doing ae an error in he ne leaage rae ro one region o he nex. (Henry, pp. 49-5) I i iporan o reognize ha or hi uli-region reaor exaple, he peral (or energy) union are ideniied beore he paial oluion i obained. The laer i done nuerially uing inie dierene equaion. 3. Suary. Conider he reaor ha i o be analyzed and he deired auray o he oluion. Deerine: a) The nuber o diin region ino whih he reaor will be divided (i.e., uel/oderaor; releor; hield) b) The nuber o energy group ha will be ued when obaining he hape o he paial lux. (Noe: The nuber ued o deerine he hape o he energy lux i norally everal houand.) 2. Wrie ou he oninuou energy diuion equaion or eah region. Aue hoogeneiy wihin eah region. 3. Aue eparabiliy o he lux (i.e., Φ( (E)) wihin eah region. 4. Approxiae he leaage er via he relaion: 2 ( B ) (,E) D (E) Φ( = D (E) Φ r 8

9 Thi i obained by analogy wih he bare hoogeneou ae where we were able o olve direly or he paial par. 5. Subiue eparabiliy (ep 3) ino he equaion ep (4) and eliinae (. Solve he reuling equaion or (E) by dividing he ull energy range ino everal houand group, aing he lux hape a onan over eah group, and deining group onan over eah Δ E. 6. Sele boundary ondiion in an inegral ene. The range() o inegraion on he boundary ondiion i he ae a he range() hoen or he nuber o energy group ued o opue he paial lux. 7. Cobine he eparabiliy relaion, he oluion or (E), and he deiniion o he boundary ondiion o obain he equaion labeled a D in he noe. 8. Deine a e o group ro-eion ha are o be inegraed over he energy range() eleed or he paial lux deerinaion. Evaluae hee group onan hi i poible beaue (E) i now nown. 9. ewrie he inegral equaion labeled a D a a e o algebrai one. Solve, nuerially.. eadju uel onenraion, e., and repea he alulaion unil a deign or whih λ= i obained. When deigning an aual reaor, i i oon praie o do ep () (5) o he above equene only one. I i hen aued ha (E) i ineniive o all hange in aerial onenraion and/or geoery. Sep (6) () are hen done any ie unil a riial oniguraion i ideniied. Thi approah redue opuer ie beaue Sep 5 whih involve houand o group i only done one. g 9