REACTOR HEAT GENERATION

Transcription

1 REACTOR HEAT GENERATION Th nrgy rlasd rom ission appars as kintic nrgy o ission ragmnts, ission nutrons and gamma and bta radiation. Approximatly 193 Mv is rlasd dirctly as a rsult o ission, and consists o that nrgy producd promptly at th tim o ission, and a latr dlayd componnt rsulting rom dlayd nutrons and dcay o radioactiv ission products. Th prompt componnt is associatd with th kintic nrgy o ission products, prompt ission nutrons and prompt ission gammas. An additional 7 Mv pr ission is producd by th captur o xcss ission nutrons and th subsqunt gamma and bta dcay o activation products. Thrmal nrgy (hat) is producd as ths particls intract with, and transr thir kintic nrgy to, th lattic atoms o th ul and othr ractor matrials. O th 2 Mv availabl rom a ission raction, approximatly 1 Mv is du to th kintic nrgy o nutrinos associatd with bta dcay and is unrcovrabl. Fission ragmnts and bta particls hav vry short rangs in ractor matrials and thir kintic nrgy can b considrd absorbd at thir point o origin. Approximatly 16 Mv is carrid as kintic nrgy o th ission ragmnts alon, and as such all o this nrgy would b dpositd locally in th ul. Gamma radiation on th othr hand has a rlativly long rang as compard to th dimnsions o th ractor ul and thror dposits its nrgy both in th ul, th modrator and ractor structur. Du to th larg mass o ul in th cor howvr, th maority o th gamma nrgy is dpositd in th ul, though th point o intraction may b ar rom th point o origin. In thrmal ractors, th nrgy rlasd by th ission nutrons is dpositd primarily in th modrator as a rsult o th thrmalization procss. In Light Watr Ractors whr th modrator and th coolant ar th sam, this rsults in approximatly 5 Mv pr ission dpositd dirctly into th coolant by nutron thrmalization alon. Th amount o hat producd in various ractor componnts is a unction o th ractor matrials and coniguration. This is particularly tru o th activation componnt. Th ollowing valus howvr provid guidlins whn mor prcis inormation is unavailabl. ENERGY DISTRIBUTION (a) Ful (b) Modrator (c) Ractor structur (d) Nutrinos 18 Mv/ission 8 Mv/ission 2 Mv/ission 1 Mv/ission It is o intrst to not, that o th approximatly 19 Mv pr ission that is rcovrabl, only 18 Mv or approximatly 95 % is producd in th ul itsl, with th rmaindr coming rom outsid th ul. O additional intrst is th hat producd by th dcay o th radioactiv ission ragmnts, thir daughtrs and activation products producd by xcss ission nutrons. At th bginning o cor li this nrgy is unavailabl. Howvr, ths radioactiv products rach quilibrium atr a short priod o tim, and at quilibrium constitut approximatly 7% o th ractor powr. Exampl: Dtrmin th nrgy rlasd in th ission raction U n Ba + Kr +2 n A mass balanc on th ractants givs

2 Th nrgy associatd with th mass dicit is thn Δm = =. 28 amu Mv. 28 amu 931 = amu Mv ission As th ovrwhlming maority o th nrgy dpositd in th ul is du to th short rangd ission ragmnts and bta particls, w approximat th hat gnration rat in th ractor ul as a constant tims th ission rat whr: q ( r v ) G v v Mv q ( r) = GΣ φ( r) 3 cm sc = Volumtric hat gnration rat (Enrgy/vol-tim) = Fission nrgy absorbd by th ul pr ission ( 18 Mv/ission) Both th cross sction and th lux ar in gnral unctions o position, nrgy and tim, i.. v v v q ( r, E,) t = GΣ ( r, E,) t φ ( r, E,) t (1) whr th lux and volumtric hat gnration rat ar now dirntial quantitis with rspct to nrgy. I w assum stady-stat opration, th hat gnration rat is indpndnt o tim and th volumtric hat gnration rat as a unction o position is q ( r v ) = G Σ ( r v, E) φ( r v, E) de (2) Th nrgy dpndnc o th lux is usually tratd by rwriting th intgral as th sum o intgrals ovr nrgy bands or groups v q ( r ) = G N G g = 1 E g 1 Σ E g v v ( r, E) φ ( r, E) de (3) and dining a group avragd lux and cross sction such that th ission rat ovr th group is prsrvd, i.. Σ E E g 1 g ( r v, E) φ( r v, E) de Σ ( r v ) φ ( r v ) g g (4) whr E g 1 v v φ ( r ) φ( r, E) de (5) g E g Not, this implis th appropriat group avragd cross sction is givn by 43

3 Σ g Eg 1 v v Σ ( r, E) φ( r, E) de v Eg ( r ) = Eg 1 v φ( r, E) de Eg Eg 1 Σ Eg v v ( r, E) φ( r, E) de v φ ( r ) g (6) Th volumtric hat gnration rat may thn b writtn in trms o th group luxs and group avragd cross sctions as N G v v v q ( r ) = G Σ ( r ) φ ( r ) (7) = g 1 From Equation 7, w can inr that to a vry good approximation th spatial distribution o th hat gnratd in a nuclar ractor is proportional to th spatial distribution o th ission rat. In thrmal ractors, th ission rat is dominatd by nutrons in th thrmal nrgy rang, primarily du to th larg macroscopic cross sction associatd with thrmal ission rlativ to th ission cross sction at highr nrgis. Th volumtric hat gnration rat can thn b approximatd as g g v q ( r ) = G N G = g 1 Σ g v v ( r ) φ ( r) GΣ g th v v ( r ) φ ( r ) th (8) which implis th hat gnration rat is proportional to th thrmal nutron lux distribution in th ul. 44

4 HEAT GENERATED IN A FUEL ROD Considr a long, thin ul rod orintd vrtically at som arbitrary point within a cylindrical cor. As th nutron lux is in gnral a unction o spac, th lux and thror hat gnration rat in any particular rod will b a unction o spac. Figur 1: Cor Flux Distribution in th Vicinity o a Ful Rod Du to th rlativly small cross sctional ara o a ul rod compard to that o th cor, th lux in th vicinity o th ul rod may b considrd constant in trms o th ovrall cor radial bhavior with th magnitud o th lux govrnd by th rod position. Th axial distribution in th rod howvr ollows th axial distribution in th cor. At any location within th cor, th thrmal lux within an individual ul rod is dprssd radially du to th strong nutron absorption in th ul. Thrmal Flux Modrator Fast Flux Ful Figur 2: Fast and Thrmal Flux Distributions in a Ful Rod I w assum th lux within th ul rod to b sparabl in th radial and axial dirctions thn φ ( r, z) = φ Φ( r) Ψ( z) (1) 45

5 with th amplitud φ a unction o cor position. This implis th volumtric hat gnration rat is also sparabl in r and z such that q ( r, z) = q Φ( r) Ψ( z) (2) Th total hat gnratd in th rod is th intgral o th volumtric hat gnration rat ovr th rod volum q = q (,) r z dv (3) V or or a cylindrical ul lmnt R H q = q Φ ( r)2π rdr Ψ ( z) dz (4) Assum or sak o illustration that th lux in th ul rod is uniorm radially and cosin shapd axially with z = th cor mid plan, i.. πz q (,) r z = q cos (5) whr H is th xtrapolatd cor hight. Th total hat gnratd in th ul rod would b q = q πr 2 q = q πr H 2 H or H πz cos H dz (6) H πh sin (7) π 2H whr R is th ul (pllt) radius. I th xtrapolation distancs ar small rlativ to th dimnsions o th ractor, thn 2 q q 2 R H (8) Rlationships which ar usul in dscribing th hat gnratd in ractor ul lmnts includ: Linar Hat Rat: Th linar hat rat q is dind to b th hat gnratd pr unit lngth in a ul lmnt. Local linar hat rat can b rlatd to th hat producd at a spciic location within a ul lmnt and th volumtric hat gnration rat at that location through th ollowing rlationships. Th total hat producd within a dirntial lngth dz about z is = v q ( z) dz q ( r, z) dadz A x (9) such that th local linar hat rat is = v q ( z) q ( r, z) da A x (1) 46

6 Th total hat gnratd within th ul lmnt is thn H q = q z dz (). (11) Th avrag linar hat rat in a particular lmnt is obtaind by avraging th local linar hat rat ovr th hight. H 1 q = = H q z dz q (). (12) H Th cor avragd linar hat rat can b obtaind rom th cor thrmal output, and th numbr o ul lmnts in th cor by rcognizing that th total hat gnratd in th ractor ul is simply th sum o th hat gnratd in th individual ul lmnts. I q i is th hat gnratd in an arbitrary ul lmnt, thn n n i i i= 1 i= 1 γ Q& = q = q H (13) whr : Q & = Cor thrmal output rom all sourcs γ = Fraction o cor thrmal nrgy gnratd in th ul n = Total numbr o ul lmnts I w din th cor avragd linar hat rat as 1 q n c q i n = i 1 (14) thn γ Q& = q nh (15) c or Q & γ q c = nh (16) It will b shown latr, that linar hat rat can b rlatd to ul tmpratur. As a rsult, maximum linar hat rat is usually st by ul mlt or othr maximum tmpratur considrations. For a givn ul hight, th maximum linar hat rat dictats th numbr o ul lmnts in a cor. Hat Flux: Th hat lux is th hat transr rat pr unit surac ara. Whil th hat lux can b rrncd to any surac, in ractors th hat lux is most otn rrncd to th outr clad surac, i.. th clad/coolant intrac. Assuming all hat transr in a ul lmnt is in th 47

7 radial dirction, at stady-stat th local hat lux can b rlatd to local linar hat rat and volumtric hat gnration rat by th simpl nrgy balanc q ( z) P dz = q ( r, z) dadz = q ( z dz (17) w ) A x or whr P w is th hatd prmitr. q ( z) P = q ( r, z) da = q ( z) (18) w A x W can thn rlat th hat lux and total hat gnratd in a ul lmnt by q q ( z) P w dz. (19) = H Th avrag hat lux in a particular ul lmnt is thn H 1 q q = q ( z) dz = = H P H w q A s. (2) As with th cor avragd linar hat rat, th cor avragd hat lux can b obtaind rom th cor thrmal output, and th numbr o ul lmnts in th cor by Q & γ Q & γ qc = = (21) na np H s w As will b shown latr, maximum hat lux is usually st by critical hat lux (DNB, Dryout) considrations. For a givn ul hight and numbr o ul lmnts, th maximum hat lux dictats th cross sctional dimnsions (radius, thicknss, tc.) o th ul. 48

8 HEAT GENERATED IN A REACTOR CORE W hav sn, that th total hat gnratd in th ractor ul is simply th sum o th hat gnratd in th individual ul lmnts γ n & = i= 1 Q q i (1) whr again th magnitud o q i is a unction o th ul lmnt s location in th cor. This is quivalnt to intgrating th ission rat ovr th ntir cor, i.. γ Q& = G ( r) φ( r) dv Σ v v (2) Vcor Du to th complx spatial distribution o th lux and th cross sction, this intgral can only b valuatd undr spcial conditions. Lt s considr on such spcial cas, whr w assum th ission cross sction is a constant in th ul, and zro i outsid th ul. Σ v v Σ o r r ( r v ) = (3) v v r r whr r v dnots locations within th ul. W din an quivalnt homognous cross sction or th ntir cor ( Σ = σ N ) such that th total numbr o ul atoms is consrvd. Th total hat gnratd in th ul can thn b writtn as γ Q & = G φ( r) dv Σ v Vcor (4) I N o is th ul numbr dnsity, thn th total numbr o ul atoms is NV = NV o ul cor N = N V o V ul cor (5) Th quivalnt homognous macroscopic cross sction can thn b writtn in trms o our original ul macroscopic cross sction as such that th total hat gnratd in th ul is ul ul Σ = N = N V V σ σ o = Σ o (6) V V cor cor 49

9 V ul γ Q & = G o φ( r) dv Σ v Vcor V cor (7) I w urthr assum, that th hat gnration rat in th modrator and structural matrials is proportional to th cor wid ission rat, thn & V ul Q = G o ( r) dv Σ Vcor φ v (8) V cor whr G contains contributions to th hat gnration rat rom all sourcs. Evaluation o Equation 8 still rquirs spciication o th cor wid lux distribution. I w assum th local variations in th lux du to ul lmnts is small compard to th total lux, thn w can trat th cor as approximatly homognous and us Diusion v Thory or som othr suitabl nutron lux modl to gnrat φ( r ). On group Diusion Thory givs th ollowing simpl lux shaps or idalizd ractor gomtris. Flux Distributions in Idal Gomtris Ininit Slab Paralllpipd Sphr Finit Cylindr πx φ cos a πx πy πz φ cos cos cos a b c φ πrr φ J πr sin R 2. 45r πz cos R H Tabl 1 (All dimnsions ar xtrapolatd) Flux shaps in actual ractor systms almost nvr ollow th simpl unctional orms givn in Tabl 1. To account or powr variations du to non idal gomtris and/or uncrtaintis du to manuacturing tolrancs and physical changs during opration, th concpt o a Hot Spot or Powr Paking Factor is introducd. Th Powr Paking Factor is dind such that F q = This implis that th maximum local hat lux at any point in th cor is and sinc th cor avragd hat lux is proportional to th linar hat rat Maximum Cor Hat Flux Cor Avragd Hat Flux. (9) q max = F q q c (1) q max = F q q c. (11) 5

10 Th simpl lux shaps givn in Tabl 1 can b usd to stimat th powr paking actor as illustratd in th ollowing xampl. Exampl: Comput th powr paking actor or a cylindrical ractor having a nutron lux distribution whr z = is at th cor midplan. φ J 245. r πz cos R H SOLUTION Th local hat lux in a powr ractor is proportional to th local ission rat and thror th local lux, i r πz q (,) r z φj cos R H 2. 45r πz q (,) r z = C1J cos R H For this lux distribution, th maximum hat lux occurs at r =, and z =, such that qmax = C1. Th cor avragd hat lux is obtaind by avraging th hat lux ovr th cor volum q = r z q = C1J cos π V R H dv cor V cor H/ 2 R cor r πz 2 C1J cos 2πrdrdz πr H R H cor H/ 2 q = 1 R H C 4R R H 245. R 2 1 J 1 π 245. R cor cor cor I th ractor dimnsions ar larg compard to th xtrapolation distancs R rducs to Th powr paking actor is thn q = 1 C 4 1 π 245. J 1 ( 245. ) πh sin 2H Rcor, H H and th abov q Fq = q max C1 = 1 C 4 1 J π (. 245) π = = J (. ) ( 245. ) 51

11 Th ul in powr ractors is normally loadd in such a way as to rduc th powr paking actor, with typical valus around 2.3. Exampl: A nuclar ractor is to b constructd to produc 3411 Mwt, 97.4 % o which is producd in th ul. Th ul lmnts ar in th orm o cylindrical rods. Th powr distribution prdictd by a nutronics analysis indicats a cor powr pak-to-avrag ratio o 2.5. Accidnt analyss plac th maximum allowabl linar hat rat at kw/t at any point in th cor and rquirs th maximum cor hat lux not xcd 474,5 Btu/hr-t 2 at any point. a) I th ul rods ar to b 12 t long, how many ul rods ar rquird? b) What diamtr ar th rods? c) In light watr ractors, th ul rods ar typically arrangd in a squar lattic. Assuming a rod pitch (cntr-tocntr spacing) o.496 inchs, what is th ctiv diamtr o th cor? SOLUTION a) Th numbr o rods in th cor ar rlatd to th total cor powr and th linar hat rat through th rlationship Q& nq H n q max = = F H γ c q Th numbr o rods in th cor is thn n = Fqγ Q & 3 ( 2. 5)(. 974)( ) = = 5, 968 Hq ( 12)( ) max b) Th rod diamtr is dictatd by th surac hat lux. Th rlationship btwn th surac hat lux and th linar hat rat is q = q Pw = q πd such that th rod diamtr is max max max D q = πq max max ( )( 3413) = =. 319 t =. 373 inchs π( 474, 5) c) For a rod pitch o S =.496 inchs, th total cor ara is Th ctiv cor diamtr is thn Acor = ns = ( 5, 968)(. 496 / 12) = t 2 D π 4 2 ( 4)( ) = t D = =1. 53 t π 52

12 HEAT GENERATION DURING SHUTDOWN In a ractor shutdown, th ractor powr dos not immdiatly go to zro, but alls o rapidly with a rat govrnd by th longst livd dlayd nutron prcursor. This may b asily shown i w assum that ractor powr can b dscribd by th point kintics quations dp ( ρ β ) Pk = + dt λ l i i C i (1) dci dt βipk = λic l i (2) or taking advantag o th dinition o ractivity dp ( k 1 βk) P = + dt λ l i i C i (3) dci dt βipk = λic l i (4) I w urthr assum that ollowing control rod insrtion, nutron multiplication is vry small ( k ) thn th solution or th powr may b approximatd by dp P + = dt λ l i i C i (5) dci dt = λ C (6) i i which has solution λ C () λ t t [ ] t / l i i i / l P( t) = P() + ( 1 l λi ) i (7) Not, l is th prompt nutron litim and is xtrmly short ( sconds) such that xponntial trms containing l di out quickly laving only thos asssociatd with th dlayd nutron prcursors. As tim progrsss, th short livd dlayd nutron prcursors also di out such that vntually only trms associatd with th longst livd dlayd nutron prcursor rmain and control th rat at which th ractor powr dcays. In addition to this rsidual ission nrgy, th ractor also continus to gnrat hat du to th dcay o ission ragmnts and activation products built up during opration. Th magnitud o this hat sourc and th rat at which it dcays dpnds partly on th oprating history o th cor. In particular, shutdown powr (P s ) dpnds on th oprating powr lvl (P o ), th oprating tim (t o ) at which th ractor opratd at powr lvl P o, and th shutdown tim (t s ). In rality, th dcay hat sourc is th many bta and gamma transitions o th xcitd nucli ormd as ission ragmnts or nutron captur products. To account or all, or vn most o ths dcay chains is impractical at bst or routin stimats o th dcay hat rat. As a rsult, mpirical its hav bn dvlopd which rlat th ratio o th dcay powr or shutdown powr o th ractor to th oprating powr in trms o th oprating and shutdown tims. Th spatial distribution o dcay hat can b assumd to ollow th oprating powr distribution. To obtain an ida o th rat at which dcay hat is built up in a ractor cor th ollowing tabl is providd. 1 4 P s /P o Vrsus Oprating Tim 53

13 Oprating Tim (sc) P s /P o , , ,.665 3,6,.682 ininit.699 Tabl 1 Th rat o dcay o shutdown powr is sn by th ollowing tabl or an ininit oprating tim. Ths tabls contain inormation rom th ANS Standard or computing dcay hat. P s /P o Vrsus Shutdown Tim Shutdown Tim (sc) P s /P o ,.185 1,.97 1,.48 Tabl 2 Not: Th dcay hat rat approachs its quilibrium valu vry quickly, and as a rsult th dcay hat rat is a strong unction o th ractor oprating history with th most rcnt oprating conditions bing th most inluntial. 54

14 Exampl: An mpirical rlationship or dtrmining dcay hat rats is givn by P P s o s o s s =. 1 ( t + 1) ( t + t + 1) +. 87( t + t ). 87( t ) whr t o is th tim (sconds) th ractor opratd at powr P and t s is th tim (sconds) sinc ractor shutdown. a) Comput and plot th dcay hat rat as a unction o oprating and shutdown tims or oprating tims o on day, on wk, on month and on yar. b) Dtrmin th oprating tim rquird or th dcay hat rat to rach 95% o its quilibrium valu. c) A ractor oprats or th irst 6 months o a on yar cycl at 5% powr, and th rmaining six months at 1 % powr. Compar th dcay hat rat at th nd o th yar to that which would b obtaind had th ractor opratd or th ntir yar at 1 % powr. SOLUTION a) Th dcay hat rat as a unction o oprating and shutdown tim is givn in th ollowing graphs Dcay Hat Rat Ptot k t k Tim (hours) Dcay hat rat or opration and shut down tims o on day 55

15 Dcay Hat Rat Ptot k t k Tim (days) Dcay hat rat or opration and shut down tims o on wk Dcay Hat Rat Ptot k t k Tim (days) Dcay hat rat or opration and shut down tims o on month 56

16 Dcay Hat Rat Ptot k t k Tim (days) Dcay hat rat or opration and shut down tims o on yar b) Th quilibrium dcay hat rat (P ) is 6 % o th oprating powr lvl or this corrlation. Th tim to rach 95 % o this valu (P = 5.7 %) is ound by itrativly solving th abov dcay hat quation or t o, with t s =. Th rsulting oprating tim is days. c) For th irst six months, th dcay hat rat is du only to continuous opration at 5 % powr. For th scond six months, th dcay hat rat is th sum o th dcay hat rats rom opration at 5 % powr or th total oprating tim and th dcay hat rat rom oprating at 5 % powr or th scond six month priod. For th dcay hat rat writtn as to < PP ( o, to, ts) = Po 1. [( ts + 1) ( to + ts + 1) +. 87( to + ts ) 87. ( ts ) ] to > this may b xprssd as Th rrnc dcay hat rat is P ( t ) = P(. 5, t, ) + P(. 5, t 6months, ) op o o o P ( t ) = P( 1, t, ). r o o Th rsulting dcay hat rats or on yar o opration ar illustratd blow. 57

17 Dcay Hat Rat Pop i.3 Pr i t i Tim (days) Th initial stp ris in th dcay hat rat upon initiation o ractor opration, and corrsponding stp dclin in th dcay hat rat upon ractor shutdown is du to th buildup and dcay o th short livd ission products. It is obvious rom ths graphs, that th short tim bhavior o th dcay hat is dominatd by th most rcnt oprating conditions. Exampl: A powr ractor oprats at 34 Mwt or on yar. Dtrmin th dcay hat rat 5 minuts ollowing ractor shutdown. SOLUTION W again us th mpirical quation or dcay hat P P s o s o s s =. 1 ( t + 1) ( t + t + 1) +. 87( t + t ). 87( t ) t o = 1yar = sconds t s = 5 minuts = 3 sconds From th givn quation or dcay hat Ps P o =. 28 P = ( 34)(. 28) = 95 Mwt s 58

18 It should b obvious rom this lvl o dcay hat, that cor cooling is ncssary vn whn th ractor is shutdown. In powr ractors, th lack o sustaind cooling can rsult in svr structural damag, including cor mlt. Altrnat Approach An altrnat approach to th purly mpirical corrlations givn abov, is to assum th numrous componnts o th dcay hat sourc can b lumpd into a rlativly small numbr o groups, similar to th approach takn with dlayd nutrons. I q is th concntration o dcay hat group, thn q is assumd to satisy th simpl balanc quation dq V dt = E Σ φ( t) V λ q V (8) ission rat 4 loss rat production rat whr E is th yild raction and λ th dcay constant or dcay hat group. As was shown prviously, th hat production rat is proportional to th ission rat, such that dgq V dt = E GΣ φ( t) V λ Gq V (9) P ( t) o γ whr G is nrgy pr ission and P o (t) th total ractor powr. Th balanc quation or dcay hat group can thn b writtn dγ dt = E P ( t) λ γ (1) o and th total dcay hat sourc is P ( t) = λ γ ( t) (11) In principl, Equation 1 can b solvd or any oprating history. Considr th spcial cas o an ininit oprating tim, such that th systm has rachd quilibrium and dγ dt = Thn rom Equation 1, E Po γ ( ) = (12) λ I th systm is thn shut down, thn or any shutdown tim t s, γ is th solution o subct to th initial condition dγ dt = λ γ (13) 59

19 Solution o Equation 13 givs E Po γ ( ts = ) = γ ( ) = (14) λ γ E Po ) = xp( λ t ) (15) λ ( ts s and th dcay hat sourc is P( ts ) Γ d ( ts ) = E xp( λ ts ) P Typical valus or th yilds and dcay constants ar givn in Tabl 3 blow. Group o E λ (sc -1 ) x x x x x x x x x 1-1 Tabl 3 Dcay Hat Group Constants (From RETRAN cod manual) 6

20 HEAT GENERATION IN REACTOR STRUCTURE Gamma and nutron radiation manating rom th ractor cor intracts with and is absorbd by structural matrials such as cor barrls, prssur vssls, tc. Th absorbd radiation is convrtd into hat which must b rmovd. As cor barrls and ractor prssur vssls ar rlativly thin compard to thir diamtr, thy can b accuratly approximatd as slabs. Considr a monodirctional, mononrgtic gamma lux incidnt upon a slab wall as illustratd blow. x= x=l Figur 1: Gamma Radiation Incidnt on a Slab Wall Th intraction rat within th slab is givn by μφ( x ) whr μ is th total attnuation coicint. Photon intractions invitably lad to th production o short rang lctrons through Compton scattring, pair production, and th photolctric ct. W can again assum th dposition o th lctron nrgy occurs locally such that th hat gnratd by photons is proportional to th photon intraction rat. W thror writ th hat gnration rat as q ( x) = Eμ φ ( x) (1) whr μ a is th nrgy absorption coicint or th slab matrial at photon nrgy E. Valus o attnuation coicint and nrgy absorption coicint ar givn in Tabl 1. Th nrgy absorption coicint accounts or th raction o th incidnt photon nrgy carrid by th lctrons atr an intraction. For th simpl xampl o a monodirctional bam, th uncollidd gamma lux within th slab is o x a φ ( ) = φ xp( μx) (2) To account or th contribution o scattrd photons w introduc th concpt o a Buildup Factor B( E, μ x) whr th Buildup Factor is dind as Total Enrgy Absorbd at x rom Scattrd and Unscattrd Photons o Incidnt Enrgy E BE (, μx) Enrgy Absorbd at x rom Unscattrd Photons o Incidnt Enrgy E (3) Th volumtric hat gnration rat or this xampl would thn b q ( x) = Eμa φ xp( μx) B( E, μx). (4) Th Buildup Factor is an mpirical it to data obtaind rom dtaild radiation transport calculations and is availabl in most standard shilding txts. 61

21 Attnuation Coicints (cm -1 ) Photon Enrgy Watr Iron Lad Concrt (Mv).5 μ =.966 μ a =.33 μ =.651 μ a =.231 μ = 1.64 μ a =.924 μ =.24 μ a =.7 1. μ =.76 μ a =.311 μ =.468 μ a =.25 μ =.776 μ a =.375 μ =.149 μ a = μ =.574 μ a =.285 μ =.381 μ a =.19 μ =.581 μ a =.285 μ =.121 μ a =.6 2. μ =.493 μ a =.264 μ =.333 μ a =.182 μ =.518 μ a =.273 μ =.15 μ a = μ =.396 μ a =.233 μ =.284 μ a =.176 μ =.477 μ a =.284 μ =.853 μ a = μ =.31 μ a =.198 μ =.246 μ a =.178 μ =.483 μ a =.328 μ =.674 μ a = μ =.219 μ a =.165 μ =.231 μ a =.197 μ =.554 μ a =.419 μ =.538 μ a =.416 Tabl 1: (Photon Attnuation and Enrgy Absorption Coicints, rom Todras and Kazimi) For nrgy distributions othr than mononrgtic, th volumtric hat gnration rat is somwhat mor complicatd. W considr two cass: on whr th incidnt photon lux consists o a init numbr o discrt photon nrgis, and th scond whr th incidnt photon lux is a continuous spctrum o nrgis. Multipl Discrt Photon Enrgis For th cas o multipl discrt photon nrgis, th volumtric hat gnration rat is th sum o th hat producd by ach incidnt photon, i.. whr th subscript i dnots th individual nrgis. Continuous Spctrum o Photon Enrgis q ( x ) = E μ φ xp( μ x ) B ( E, μ x ) (5) i i ai i i i i I th incidnt photon lux is a continuous spctrum o nrgis, th volumtric hat gnration rat is obtaind by intgrating ovr all incidnt nrgis, i.. q ( x) = Eμ φ ( E)xp( μx) B( E, μx) de a (6) whr φ ( E ) contains th incidnt photon spctrum. 62

22 HEAT GENERATION FROM RADIOISOTOPE SOURCES Radioisotop powr sourcs produc hat as a rsult o xothrmic dcay ractions. This hat is thn convrtd to lctricity through thrmolctric dvics or othr similar mans. Chargd particls mittd during th dcay procss hav kintic nrgis corrsponding to th mass dct o th dcay raction. W again assum that th kintic nrgy o ths particls is dpositd locally at th point o dcay, such that th hat gnration rat is proportional to th spatial distribution o th radioisotop within th powr sourc. This is gnrally uniorm. I ΔE is th nrgy associatd with th mass dct o th dcay raction, thn th volumtric hat gnration rat du to radioactiv dcay is v v q ( r) = ΔEλ N( r) (1) whr: λ v Nr ( ) = Dcay constant o th radioisotop = Radioisotop numbr dnsity as a unction o position Exampl: (Adaptd rom Exampl 4-6, El-Wakil) A radioisotop powr sourc is uld with 475 gm o Pu 238 C, 1 % nrichd in Pu 238. I th dnsity o PuC is 12.5 gm/cm 3, calculat th volumtric hat gnration rat and total thrmal output o th radioisotop sourc. SOLUTION Pu 238 dcays to U 234 via alpha mission with an 86 yar hal li, i Pu U + H. U 234 has a hal li o 2.47 x 1 5 yars and rlativ to Pu 238 can b considrd stabl. Th mass dct is givn by Th dcay constant is rlatd to th hal li through Th Pu 238 numbr dnsity is givn by Δm = =. 6 amu or in trms o nrgy Mv ΔE = 931 Δm amu = ( 931)(. 6) = Mv/raction λ = = = = sc 9 t yr sc ρ N = PuC Av ( 12. 5)( ) = = M PuC such that th volumtric hat gnration rat is nucli/cm 3. 63

23 q = ΔEλ N = ( )( )( ) = Mv/cm 3 sc Th total thrmal output o th dvic is Q= q V = q m ρ 13 = ( )( ) = Mv sc = 261 W 64