Dresner, L.

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SEPTEMBER 1961 KFK 85 INSTITUT F Ü R  NEUTRONENPHYSIK U N D  REAKTORTECHNIK 0 SOME REMARKS O N  THE EFFECT OF A N O N U N I F O R M  TEMPERATURE DISTRIBUTION O N  THE TEMPERATURE DEPENDENCE OF RESONANCE ABSORPTION LAWRENCE DRESNER Reprinted frortl NUCLEAR SCIENCE AN[) ENGINEERIXG, Voilline 11, NO. 1, Septenikr 1961 Acudeiiiic Press Inc. l'rinted in U S A .  NUCLEAR CCIEIVCE E N < : I N E E I ~ ~ N G :  I r  3<332 (19ö1) der ZentrabUchersl Some Remarks on the Effect of a Nonuniform Temperature Distribution on the Temperature Dependence of Resonance Absorption Institut für ATeutronenphysik und Reaktortechnik, Kernreaktor Bau- und Betriebs-Gesellscllaft m.b.ZZ., Karlsruhe, Germany Received Nooember 28, 1960 The inffuence of a non-uiiiform ternperature distribution on the absorption of neutrons in n purely absorbing resonance of an isolated lump is studied. I t  is sholvn that in practical situations the absorption in the lump depends only on its average teniperature and not on 11-liether its temperature distribution is uniform or not. Tlie criticulity, on oiie hand, aiid the initial time behavior of heterogcrieous reactors following a large instantarieoiis reactivity additiori, on the other, de- pend to a great exteiit on the increase in resoiiance absorption with iiicreasing temperature. Therefore, tlie temperature deperidence of resonance absorption is oiie of the important quantities determining both the criticality arid safety of heterogeiieous reactors. This tenlperature dependence has beeri the subject of niiich stiidy in the past, but iii all previous work the temperature distributiori was uniform in tlie ab- sorber liimp, quite in coiitrast to the state of affairs in an actual reactor. The effects of non-uiiiformitg iii the temperature distribution may be important, niid it is the purposc of this paper to discuss them. IVe shall only consider the case of absorption in an isolated lump (i.e., no Daiicoff effect) by a purely ab- sorbing resonance (i.e., I'r N I'). Furthermore, we shall ignore potential scattering altogether. Thus we shall not corisider either monoeiiergetic scattering or scattering with moderation in the absorber lump. Under these conditions the effective resonance iiitegral of thc absorber is givcii by the eyuation cos d 4 7r 4 where = the atomic deilsity of nbsorber atoms iii the lump = the liimp voliime = the effective resonaiicc integral = the fliix in the moderator, taken as iini- form in space and lethargy = the lethargy = thc differential element of the Iiimp surface, S = the posit,ioil of a poiiit oii the lump surface = the inward normal to S at r8 = a unit. vector a t  r, d = the angle bet~veeii o aiid n uo = the peak resoiiance cross sectioii in t'he abseiice of Doppler broadening l(rs, W )  = the leiigth of the chord a t  r, in di- rection o $(6, .V) = the Doppler broadeiied line shape giren below in (2) 6 = the ratio of the natural width of the resoiiance, I', to  the Doppler ~vidth, ( A E , , ~ T / A ) ~ ~  2 (r8.o) E,, = the resonailce eiiergy -(L-erp{- NU,^ $[6(so),x]ds)) Ii?' = the temperature of the absorber niiclei iii energy units * On leave from Oak Ridpe Xational Laboratory, Oak Itidge, Tennessee operated by Union Carbide Corporation A = the mass iiumber of the absorber iiuclci for the U. S. Atomic Energy Commission. X = the ciiergy measiired from exact Ais WC s11:~ll SCC l : ~ t ~ r ,  t l i ~  difY(~r(~lr(.(~ 1)vt \\-(vII ill(% 1:IIS of ( l )  : I I I ~  the I<IlS of (4) is : tc t~~:~l ly  I.:L~II(>I~ s111:111. IIc11c~>, ill ~vh:lt follo\vs \\-c shall drop thc. < --sigu. Thr  rxpollcwt ill (1) ('a11 rc\vrit tc~r  1)y i ~ r t  crc,h:u~g- itlg tllc ordtbr of the. S a~rtl  w ilrtcgl.atio~rs: \\-llcrcl 7 - ( ! J )  = 0 if y < 0 :lnd l *(!l) = 1 i f  ! j  > 0. 1'111. 1:lst two i~~trgr:lls over S :r~rd s givtl csac*tIy the volmmc intcgrul of + over 1'. Thus fin:llly \v(. have. 1 (rg.wI (1s cos 7.9 I .( n . o ) . .~- . ~ - ( 6 )  .*[%(SO ),  .l .] = l*( B, .c) \vhcrc. th t  Imr o v t ~  + cIcnotc+ :I sl):lti:ll :tvc,~.:tgc' ovllr I -  :uld 7 = 11- S. Sr~l)..;tituti~lg ((i) llto (4) ~lo\ \ -  givw a l ~ d  \vhcl.cl 7 still also c3clr~uls 4lr,,S. Tl1(> f:lc,tol, - pif,,i"' is rcspc~c.tivcly (~l t1:~1 to 0.!)80, ().!)75, :Lll(l o.!)f:< for spl~c>~.cs, .yli~~tl(.rs, : ~ ~ l d  SI:LIIS. '171111s O l l ~ y  ;L sniall c ~ r o r  is introtlr~c+c'd illto (4) \vhr11 it is t:ll;llll \\it11 tlic. c~clualitg sigu. l'h:~t tllis l:\st c.o~rc,lr~sio~~ tiocls 11ot t lcpc~~d v(,rv st~.o~igIy 011 t l i ~  pr(viso sli:lp(> of' 1i11p ~.:LII \)cl bcx(~l l  ~.clpt*atilrg tht' a1)ol.c (-:LI(YII:L~ iorr for :l l i ~ r ( *  \vitll fll(x sh:~pc' ( 1  + .c")-', 11 > 1 . 111 this c.ascb, tllt~ ratio of tllcx IUIS of ( l )  to that of' (4) ih g i v o ~ ~  1)y F/ll ". spli(.r(.s : L I I ~  s l i i l ~  illis t.:lt io 11:~s :l sll:~llo\v ~ ~ l i ~ l i ~ ~ ~ ~ ~ ~ ~ ~  4 11(.:1r 11 = L', rising to rlnity for U = l :urd 11 = z . , lhp  . v : ~ l r ~ c ~  : ~ tt11(1 ~~l i l~i l i l t tm arc. cssc.~rti:~Ily t ] 1 0 ~ ( .  g i \ ~ ( ~ ~ r  i l l  the. pr(3viorls I):II.:L~~:LI)~I. Tlir~s TV(, cisp(b(-t 111:~t for sp11c~rc.s  id (.ylilrd(lrs clspc~c.i:111y, t11(. c1l,r()r arisi~rg fro111 thtl tlscX of ( : 2 )  i l l  (1) is 1rclitht1r v1.l.y grc:lt 11or vn~.ic.s rapidly wit h tlio sli:~pcx of thc I i l ~ o .  rlccortii~lg to ( 2 )  \vli(>r(l j (B1j  (10' is t1i:it f~.:trtio~r of tlic 1111rq) vo1111nc i~r  \vhic.h B lit.:: I)c.t\vc~o~~ B' :urd B' + dB'. So\v j(8) 8 = "j(B2), \\-lir~,c\ t llc l:lt t c ~  fulrc.t ion, g, is t11r11 :l l,os>il)l(. (1 /~or,rlalizcrl tlistril)utio~~ of R .  111 this (.:is(,, it follo\vh from (!l) :~lrd (:i) t11:~t 111 tllc last s t ~ p  I\-(' h:~ve I I S ( ~  t l ~ :  r(>l;ltio~~ - ( / l 1  = d.r(F, '2rq. The difft~rrncac I)ct~vclc~~ thc. 1jIIS of (4) : L I I ~  tllr \vllor(, tllr I ) : , ~ ~ ,  as usrlal, dl.llotc spaw :Ivcrugcs. ItHS of ( l )  tfrlc to the us(. of illc~clui~lity ( 5 )  (-:l11 l)(, I:roln (10) :Llltl (11) it f()l l~n.s that cal(.ulatt~l ill tllo case of I I ~  Ilopplcr cf'ft1c.t at all, i.c., +(e, .C) >= c-'+(cO, .c) (1%) fi = (1 + , by the mrthod of Gtucvic.11 : I I I ~  I'omera~~chouk (2). Tfiost~ authors filrtl that llc r:~tio 2 = S!:$' (121)) \vll(>l.l> i l , i i l  is ~ h f '  r(:so~iaii(~c~ iiitcgral of a 1111np of i h ~  saiil(1 lii:kt(>rial aiid sliapo, witli :I sizt. gircii hy c-'7, :Lli(I : ~ t  a ~c~iijorm tciiripctruture c8. 1'li(\ - sigii ocsch\irs r:ltlici. th:~ii :III  ~(l i iali ty sigii 10 i i ~ d i i ~ a t ~  that (13) liolds :LS :lii ii~c~c~iiality orily witkiiii llic. (sinall) (>ri.or urisiiig froiii tli.oppiiig the < sigii ii i  (1). f-1listril)tit ioiis arc (.oiie(~i~~:~t)It~ i i i  \\-liic.li c  docbs iiot. (1ifft.r vcry iiiiic.h from iiriity, alt1ioiigIi 0 varics o v c ~  :I rcasoiiat)Iy ividc raiige of v:lliies. (\Y(> s1i:~ll s i ~  : ~ i i  c~sniiipI(~ of siic~li a distri1)iitioii stiortly.) If c - 1 is sni:~Il, lio\vchvc~, tIi(> riglit- :~ird lclft-hniitl sidcs of (12:~) c~:riiiiot I)(. vcbry dificrcliit. I:or, iiidcwl, tlic3ii. :iiid siiicgc. tlic. iiit('graiid p~ i h  positiv(> (av(~ryivh~r(3, it is calcar that if C - 1 << 1 , #(o, .T) aiid c-'#(c8, .T) (*aiiiiot t)r \-(>ry diff(ki.t~iit : ~ t  311. Iii siii.11 :I Vase \rc ivoiild i.sp(~at tli:it :I ~.alid approxirnatioii to (13) ivo~ild 1~ tlic. cyiialii!~ I = cp'l„,,,f (1 -c-I(, 0  - c6') 2 I„, t  ( 8 )  (1.5) \\-tii.rc1 I„„,l (8) is thc. r(~soii:~iic.c~ iiitcgr:il of thi, :ic>tii:il liiriip t)(biiig c~oiisiiic~rc~d at :L iiiiiforiii tcliilpi~ratiircx 8.111 i ) r d ( ~  to piirsii~ tliis stiidy fiirtliei. aiid, iii par- tic*iil:ii., tcst tkir :rcac.iirac.y of (I;i), sorncb rc.:~soiint)lc. ~f-tIi,.tril)iitio~i ~ i i i i s t  :issiiriicd. 1,c.t 11s tlicrt.forc. iiii:\giiii> tti:it iii a c.yliiidric.nl fiicl i.od i i i  :I poivtlr i(hncatoi. tlic fihsioii hc.at soiii.c.~ is cffcc.tivc.ly sputi:illy iiiiif'oiiii. Iii tliis tllc t c ~ i i p ~ ~ i ~ a i i i i ~ i ~  d striI)iitioii iii i,od 11-ill 1 ~ 3  par:il)olic., i.r., Y'(? ) = Y',, + (T1 - Y'lli(r I?)? ( 1  (;I I\-1ici.c. r i:: tlicx radi:~l c~oordiii:itt~ iii tli(~ rod, X is tlicl r:i(liiis, :liitl T, , :liid T l  :~rc, rcspc~i~tii.t~ly, tlit. tciiip(~r:~triri~s :it aiiy poiiit , :it tlic c(.iitcr, :~iid :it tlic 1)oiiiid:li.y. IS is iiot v ( ~ y  diffci.c~ii1 froiil T „  , tlicl c*orrc.spoiitliiig distri\)iitioii of 0 is also p:ii:il)olic*, i.c.., I'roiii ( l i )  it followh t1i:it .fiel = ie, - e,)vi e,, 5 e _I e, (1 8:~)  {](B2) = ( O I 2  - B;"-' ol;> j ,.I tdic~si. f- aiid {I-distril)iitio~is. \ITitti rt.spcc+t to gr.iitrraI foriri ( 1  8) is, of c.oiirsc3, rcprctscritat ive. Fro~ri (1 8a) it folloivs that 1:or (Y = d?, \vlii(.li c.orrc>spoiids to ü iatio 7',/7', = 2, c difitrrs from iiiiity t)y lrsb t h:lii 1 ..? ' ; ; for (Y = f l 2 ,  C = 1.003. IIriic.(1, iiiid(~r thil aiitic5i- patc'd ~iri~~iiiistaiic'( th:~t iii prachtic.al cascs oc is iiot ioo largc., iv1. rspcc.1 (13) to hc valid. \\T(> ( o : l ~ i  fest this c~oiic.lusioii for the iiitcrestiiig c8asc of estrc.rri(. I>oppl(.r t)road(.iiiiig (i.c., 0  << 1 ) for \\-liic.11 \iTigiic>r CI a1. (,?) liav(1 siiggtlstcd tk ic .  i i s ~  of ttic :~pprosirii:it(> liiic. sli:lpc. I<iliiatioii (20) is partic.iilarly appropriatc for thosc. rc~soiiaiic.(~s n-liic.ti c.oiitribiite stroiigly to tlic. tc3rii- pcratiirc c.ocfficaii>iit of thcx r<'hOllall(.<' iiitc>gral. 1:rom (20) niid (18) it follo\vs that 2 J (I - esp{ -ßcxp( - 2 ' )  --<, ß o ( 2 2  1 I - (>SI1 [- ( (Y2 - 1 1 . ~- P-- ((Y? - 1 )  z? ~\-1i(~i.(. = .Yo„T(-\/a 216' :iiid cu = (01 t ~ ~ , ) .  I\-Iit'ii cu = i , 1  = I ,,,,, f ( e ) .  l'ypi(+al v:iliir,q foi tlic par:tnictci.h :1i.i1 = 0.0.j I ) : I ~ I I s - ~  ~ i i i - ~  f ~ r  iiict:illic. l - ,  7 = 2 c'iii, e = 0.05, nl, = .5000 1):iriih. 1:or tli(>s(> v:~liics P 2 22. Siiii~(~ii(-:il iiitegr:~tioii of (22) for tlicl f'oiir c.ahcx3. cu = 1 ,  d-? ,3 = 10, :NI givils tlic follo\\-iiig rc~siiltq: I'or 9 = 10, I :iiid I u r , ,  1 (G) ,  i,(,., I ( @  = Qj) aii(1 I [ a  = I ) ,  :IIX\ ~vitliiii :ihoiit 1 .;> ' ; of oiic. :iiiotlicr. thc. foriiior 1)c.iiig l:~rgt.r. I'or i j l = :X), tlio c.orrc.spoiiding diffcrciirt. is :it)oiit 2.3 ';. 'rliiis f'oi thes(> coiiditioiis otir c.spectr~- tioii t 1i:lt (1 5 )  \voril(l I)(. :ic.ciirat c, is vriificd. I3:iscd oii tli(. forcgoiiig aiinlysis I\-c art. iirc'liiic~d to c*oiisidrr tIic :~pprosini:~tioi~ I = I,l,,,f ( 8 )  :I% n f:iirIy accaiir:itc oiic'. I'or pi.:ii.tic.nl piirposcs it is \-irtii:illy 'I'liis :11)1)rosi1ll:itio11 is origiii:iIly (Ilit, to  G .  11. Iioc; "1irsoii:iiicr hl)horl)tioii of Seiitroiis ii i  1)opl)lrr I3roa(leiied I~esoii:r~ic.rs," 1<a11'12-1241 (0ctol)er. 1054) Ilor :ii;riiiiie<i t 11:tt i . t l . ,  .f :~iitl C/ : r i ~  tiotli iiiiifoisiii clisti~il)iitioiis. \I-c will for sll,~,s, 1 gi 1 ) ,  ,,.jierC. o ,  (~c,t(,rlllillr~~ ~p~ :issiiiiic Iiciic*c~fort 11 for t hc s:~lie of argiiriit~iit tliat the t l l r  i.eil,iii.c.Iliciit li:it t l i r  iritcgritls J,, r,- ,lx :lii(i distri1)iitioiis (18) Iiold csai+tIy; foi.tiiii:ittly, oiii. fiii:il J u G ? ( B „ ~ )  I ) r  tlir s:liilr, ~ l ~ i ~  iolltlitiolr g:ivc 0, :llll,rosi- ! i~oii(~~iisioiis (10 iiot dcpciid c*ritii*:illy oii tli(. d(7ioiis of i~i;itrly riiii:il t o  8 - (3;?1[~$ - @ i i I  + 1811. Tlir foiiii  identical with the oft-repeated prescrjption that the ACKNOWLEDQ~IENT actual resonaiice integral is equal to that of the Same Thc author wishes to  thank Dr. W. E~~~~ for ar, lump uniformly held a t  the average temperature, interesting correspondence regarding this Problem. since the effect,ive temperat,ure determined from 6 is REFERENCES usually not very different from the nverage tem- perature. The virtue of the present note is that it !. G .  H .  H A ~ ~ Y ,  J .  E. J A I ~ L E ~ V O O D ,  A N D  G, poLYA, indicates that this prcscription is, in fact, quite M- equalities," 11. 137, theorem 187. Cambfidge univ,  Press, London and New York, 1934, curate, a t  least in the special casc treated. 8. I. I. GUHEVICIX A N D  I. Y. P O M E R A ~ C H ~ ~ ~ ,  pTOC. of the second term depends on ho\1*8. is determined and is to Intern. C0n.f. Peaceful Uses rltomic EnergZl, G~~~~~ some extent arbitrary. Thus the real meaning of Roe's 6, 4GG (1955). result is that if the second term is not too largc, 8 ,  = 8 to  3. E.  P. WIGNER, E- CREUTZ, H. JUPNIK, AND T.  SN^^^^, a good order of approximation. J .  r lppl .  Phgs. 26, 200 (1955). 

Some remarks on the effect of a nonuniform temperature distribution on the temperature dependence of resonance absorption

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Some remarks on the effect of a nonuniform temperature distribution on the temperature dependence of resonance absorption

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