NEUTRON DIFFUSION THEORY

Transcription

1 NEUTRON DIFFUSION THEORY M. Ragheb 4//7. INTRODUCTION The diffuion heory model of neuron ranpor play a crucial role in reacor heory ince i i imple enough o allow cienific inigh, and i i ufficienly realiic o udy many imporan deign problem. The neuron are here characeried by a ingle energy or peed, and he model allow preliminary deign eimae. The mahemaical mehod ued o analye uch a model are he ame a hoe applied in more ophiicaed mehod uch a muli-group diffuion heory, and ranpor heory. The derivaion of he diffuion equaion will depend on Fick law, even hough a direc derivaion from he ranpor equaion i alo poible. The Helmhol equaion i derived, and he limiaion on diffuion equaion a well a he boundary condiion ued in i applicaion o realiic engineering and phyic problem are dicued.. TRANSPORT CROSS-SECTION The effec of he caering angular diribuion on he moion of a neuron i aken ino accoun by ue of he ranpor cro ecion. Le u aume:. An infinie medium,. A purely caering medium wihou aborpion, 3. The energy of he neuron doe no change a a reul of a colliion wih he nuclei of he medium, 4. Afer each colliion he paricle ravel a caering mean free pah and i defleced by he caering angle. Afer injecion o he yem, he neuron move a diance: Z () he projeced diance raveled afer he fir colliion along he -axi i, a hown in Fig. : Z co( ) The average value of Z i: Z co( ) ()

2 where: and:, 3A A i he ma number of he nuclei in he caering medium. The diance raveled afer he econd colliion i: Z co( ).co( ) where he aimuhal angle ( ) caering i ioropic. Figure. Geomery for he ranpor of a paricle over hree colliion. The average value of Z i: Z co( ).co( ) co( ) (3) In general, for (n-) colliion: Z n n, for n=,,, (4)

3 Since Z n a n, hi implie ha he neuron could be caered in eiher direcion wih equal probabiliy, a i nex colliion. Thi mean he neuron forge i original direcion of moion afer a ucceion of colliion which i characeriic of Markov chain, and having been carried a diance of a ranpor mean free pah: r n ( ), n (5) The ranpor cro ecion i defined a: r ( ) (6) r If aborpion i preen, we generalie he definiion of ranpor cro ecion o be: ( ) r a (7) where i he oal macrocopic cro ecion. 3. FICK S LAW AND THE DIFFUSION APPROXIMATION The neuron flux ( ) and neuron curren () are relaed in a imple way under cerain condiion. Thi relaionhip beween and i idenical in form o a law ued in he udy of diffuion phenomena in liquid and gae: Fick law. In Phyical Chemiry, Fick law ae ha: If he concenraion of a olue in one region i greaer han in anoher of a oluion, he olue diffue from he region of higher concenraion o he region of lower concenraion. The ue of hi law in Reacor Theory lead o he Diffuion Approximaion. Le u make he following aumpion:. We conider an infinie medium,. The cro ecion are conan, independen of poiion, implying a uniform medium, 3. Scaering i ioropic in he Laboraory (LAB) yem, 4. The neuron flux i a lowly varying funcion of he poiion,

4 5. We ue a one peed yem where he neuron deniy i no a funcion of ime, 6. A eady ae yem where he neuron deniy i no a funcion of ime, 7. No fiion ource in he yem. Some of hee aumpion will be laer relaxed. For inance, he diffuing medium will be aken a finie in ie raher han infinie. We hall aemp o calculae he curren deniy a he cener of he coordinae yem of Fig.. The vecor i given by: iˆ x y ˆj k ˆ, (8) o ha we mu evaluae he componen x, y,. Figure. Spherical geomery for he derivaion of he neuron curren and Fick Law. Thee ne curren componen can be wrien in erm of he parial axial curren a: x y x y x y (9)

5 Le u concenrae on he eimaion of one ingle componen: croing he elemen of area ds a he origin of he coordinae yem in he negaive direcion, a hown in Fig.. Every neuron paing hrough ds in he x-y plane come from a caering colliion. Neuron caering above he x-y plane will hu flow downward hrough he elemen of area ds. Conider he volume elemen: dv r in( ) drd d () The number of caering colliion occurring per uni ime in he volume elemen dv i: r dv r r drdd () ( ) ( ) in( ) where: () r i he paricle flux, i he macrocopic caering cro ecion. Since caering i ioropic in he LAB yem, he fracion arriving o ubended by he olid angle d, and i given by: ds i ha d ds r ds co( ) 4 4r () Thu he number of neuron caered per uni ime in dv reaching aenuaed in he medium by he exponenial facor r e i: ds afer being ds co( ) r dn e ( r) r in( ) drd d 4 r The parial curren can now be wrien a: dn / r e ( r)in( )co( ) dr d d ds 4 (3) Since (r) i an unknown funcion, we expand i in a Taylor erie auming i varie lowly wih poiion: () r x y x y

6 Wriing x, y, and in pherical coordinae: ( r) rin( )co( ) rin( )in( ) rco( ) x y (4) Subiuing Eq. 4 ino Eq. 3, we ge, r x y e rin( )co( ) rin( )in( ) r co( ) in( )co( ) drd d 4 (5) The erm conaining co( ) and ) inerval, hu: in( inegrae o ero over he angle, / r 4 (6) e rco( ) in( )co( ) drd d The fir erm can be evaluaed a: / r 4 I e in( )co( ) drd d 4 4 The econd erm alo become: / I r re in( )co ( ) drd d Thu Eq. 6 become:

7 4 6 (7) Similarly: 4 6 (8) Subiuing Eq. 7 and 8 ino Eq. 9, we can wrie: x y 3 x 3 y 3 (9) Subiuing ino Eq. 8, we ge he expreion for he curren deniy afer dropping he evaluaion a he origin noaion, ince he origin of he coordinae i arbirary: ˆ i ˆ j k ˆ 3 x y grad 3 3 () We define he diffuion coefficien : D () 3 Thu Fick law for neuron diffuion i given by: D () I ae ha he curren deniy vecor i proporional o he negaive gradien of he flux, and eablihe a relaionhip beween hem under he enunciaed aumpion. Noice ha he gradien operaor urn he neuron flux, which i a calar quaniy, analogou o he volage and he emperaure in oher field, ino he neuron curren, which i a vecor quaniy. 4. LIMITATIONS OF DIFFUSION THEORY

8 Fick law expree he fac ha if he gradien of he flux i negaive, hen he curren deniy i poiive. Thi mean ha he paricle will diffue from he region of higher flux o he region of lower flux hrough colliion in he medium. Fick law impoe ome limiaion on he problem olved, becaue of he inheren aumpion, and i become invalid, needing correcion, under he following condiion:. Cloene o boundarie: The derivaion aumed an infinie medium. For a finie medium, Fick law i valid only a poin which are more han a few mean free pah from he edge of he medium. Thi i o, ince he exponenial erm die off quickly wih diance, and only poin of a few mean free pah from he poin where he flux i compued make a ignifican conribuion o he inegral.. Proximiy o ource or ink: I wa aumed ha he conribuion o he flux i moly from caering colliion. Source can be preen. Becaue of he aenuaion facor, however, a mall number of he ource neuron will conribue o he flux if hey are more han a few mean free pah from ource. 3. Anioropic LAB yem caering: Ioropic caering in he LAB yem occur a low energie, bu i no rue in general. However, Fick law i ill valid wih moderae anioropy in caering, if a modified form of he diffuion coefficien i ued, baed on ranpor heory. Such an expreion for D i given by: D 4 3 ( ) 5 a (3) If Eq, 3 reduce o: a r D, (4) 3 ( ) 3 3 r ince: where: ( ) r, r r i he ranpor mean free pah defined earlier. 4. Highly aborbing media:

9 The flux wa expanded in a Taylor erie and wa aumed a lowly varying. The flux however change rapidly in rongly aborbing media. Thu Fick law applie o yem in which: a When aborpion i preen, D hould be compued uing Eq. 3 and no Eq. 4 or. 5. Proximiy o inerface: The aumpion of a uniform medium wa ued in he derivaion of Fick law. A he boundary beween wo media of differen caering properie, Fick law i ill valid, provided ha he harp change doe no lead o a rapidly varying flux, which invalidae he Taylor erie expanion of he flux in he derivaion. In fac, he econd order derivaive, eiher inegrae o ero, or cancel ou in heir conribuion o. The hird derivaive do in fac conribue o he inegral of. Thu Fick law would be valid if he econd derivaive of he flux doe no vary appreciably. 6. Rapidly ime-varying flux I wa aumed ha he flux i independen of ime. I i poible o relax hi requiremen if he change in i mall during he ime for a neuron o ravel a few mean free pah. If he 5 low neuron in a reacor have velociy of v cm/ec, and heir ravel over hree caering mean free pah, hen we can ae ha he ime variaion of he flux mu aify he condiion: 3 3 d 5 v d [ec] which ae ha he ime needed for he neuron o ravel hree mean free pah i maller han he characeriic ime for he neuron flux variaion. 5. DERIVATION OF THE NEUTRON DIFFUSION EQUATION To derive he neuron diffuion equaion we adop he following aumpion:. We ue a one-peed or one-group approximaion where he neuron can be characeried by a ingle average kineic energy,. We characerie he neuron diribuion in he reacor by he paricle deniy n(r,) which i he number of neuron per uni volume a a poiion r a ime. I relaionhip o he flux i: r, vn( r, )

10 We conider an arbirary volume V and wrie he balance equaion: Time rae of change Producion rae Aborpion Ne leakage from of he number of = - - in V in V he urface of V neuron in V The fir erm i expreed mahemaically a: d d n( r, ) dv ( r, ) dv ( r, ) dv d d v v V V V The producion rae can be wrien a: V S ( r, ) dv The aborpion erm i: and he leakage erm i: V ( ) (, ) a r r dv ( r, ) n ˆd ( r, ) dv V where we convered he urface inegral o a volume inegral by ue of Gau Theorem or he divergence heorem. Subiuing for he differen erm in he balance equaion we ge: or: ( r, ) dv S( r, ) dv a( r) ( r, ) dv ( r, ) dv v V V V V ( r, ) S a dv v (5) V Since he volume V i arbirary we can wrie:

11 ( r, ) a S v (6) We now ue he relaionhip beween and (Fick law) o wrie he diffuion equaion: ( r, ) D( r) ( r, ) a( r) ( r, ) S( r, ) v (7) Thi equaion i he bai of much of he developmen in reacor heory uing diffuion heory. 6. THE HELMHOLTZ EQUATION The diffuion equaion (7) i a parial differenial equaion of he parabolic ype. I alo decribe phyical phenomena in hea conducion, ga diffuion, and maerial diffuion. Thi equaion implifie in cae he medium i uniform or homogeneou uch ha D and do no depend on he poiion r a: a ( r, ) D ( r, ) a( r, ) S( r, ) v (8) where we ued he fac ha he divergence of he gradien lead o he Laplacian operaor:. The Laplacian Operaor depend on he coordinae yem ued: Careian: (9) x y Cylindrical: r r r r r Spherical: r in( ) r r r r in( ) r in ( ) (3) (3) In cae of ymmery around he angular variable and, hee equaion implify in he one dimenional cae ino:

12 r d dr r d dr where: =, for he careian coordinae, =, for he cylindrical coordinae, =, for he pherical coordinae, and he parial derivaive ha been replaced by he oal derivaive. When he flux i no a funcion of ime, we ue he eady ae diffuion equaion, or he calar Helmhol equaion; D a S (3) which i a parial differenial equaion of he ellipic ype. The Helmhol equaion can be wrien a: where D Sr ( ) L D ( r) ( r) (33) L, L i he diffuion lengh. a If on he oher hand, he neuron deniy and flux are independen of poiion, we can wrie from Eq. 8; conidering div : d( ) S( ) a( ) (34) v d which i a ime dependen equaion in he flux. 7. BOUNDARY CONDITIONS FOR THE STEADY-STATE DIFFUSION EQUATION Mahemaically, for he Helmhol equaion he following boundary condiion are needed: eiher or he normal derivaive, or a linear combinaion of he wo mu be n pecified. Boh and canno be pecified independenly. Normally, boundary condiion are n pecified baed on phyical argumen ha do no violae hi condiion.. Vacuum Boundary Condiion:

13 The mean free pah of neuron in air i much larger han in he reacor, o ha i i poible o rea i a a vacuum in reacor calculaion. If we conider no neuron refleced from he vacuum back o he reacor core (Fig. 3), hen we can wrie he equivalen o Eq. 7: ( x) 4 6 x (35) Recalling ha from he Fick law derivaion, D 3 we can wrie: D ( x) 4 x (36) From which: x D (37) If a he boundary he maerial i moly a caerer a, hen, and we can ubiue for D r : 3 3 x D r (38) However from he geomery of Fig. 3, if he diffuion heory i linearly exrapolaed, we can wrie: an( ) (39) d x where d i he exrapolaed lengh. Comparing Eq. 38 and 39, we ge: 3 x d r

14 from which: d r (4) 3 More deailed analye give more complicaed expreion for d in he iuaion of curved boundarie. For inance, for plane geomery, i i given by: d.74 r (4) The boundary condiion i now enunciaed a: The flux vanihe a he exrapolaed diance beyond he edge of he reacor urface, or: ( Rd) (4) where R i he reacor radiu. Figure 3. Geomery for he eimaion of he exrapolaion lengh in diffuion heory.. Inerface boundary condiion:

15 The flux i coninuou acro he boundary beween wo differen media A and B: A in erface (43) B in erface and he normal componen of he curren deniy a he boundary hould be equal: A (44) n in erface B n in erface 3. Phyical requiremen boundary condiion: The oluion o he diffuion equaion mu be real, non-negaive, and ingle valued: (45) Alo, he oluion o he diffuion equaion mu be finie in hoe region where he equaion i valid excep, bu no necearily, a ingular poin of he ource diribuion: (46) Thee condiion can be ued o eliminae unneceary funcion from he oluion. Thi pave he way for he calculaion of flux and power diribuion in pracical reacor deign iuaion. REFERENCES. M. Ragheb, Lecure Noe on Fiion Reacor Deign Theory, FSL-33, Univeriy of Illinoi, R. Lamarh, Inroducion o Nuclear Engineering, Addion-Weley Publihing Company, 983.