ANALYSIS OF FAST REACTORS SYSTEMS M. Rghe 4/7/006 INTRODUCTION Fst rectors differ from therml rectors in severl spects nd require specil tretment. The prsitic cpture cross sections in the fuel, coolnt nd structure fll with incresing neutron energy fster thn the fission cross section. Thus the neutron economy is improved in the fst region of the neutron spectrum. This is chieved y voiding the use of modertor mterils s coolnts. Since the fission cross section is less in fst rector thn in therml rector, fst rector would contin much more fissile fuel thn therml rector for the ttinment of criticl mss. Fst rector cores, hving no modertor, will e very compct in size. This leds to higher power density necessitting the use of efficient coolnts such s liquid metls. Sodium, led, nd sodium-potssium eutectic tht is liquid t room temperture re prime cndidtes. A smll core would imply reltively lrge mount of neutron lekge from its surfce. A reflector is voided nd is replced y lnket to intercept the leking neutrons into reeding mteril. The multipliction nd energy production in the lnket must e ccounted for from the perspective of power production. We proceed to tret such system using sudivision into fst group for neutrons of energy ove 1.35 Me, the fission threshold of U 38, nd therml group for energies elow it. THE CORE AND BLANKET TWO GROUP EQUATIONS The fst nd therml group core equtions cn e written s: Core fst group: -[Lekge from fst group]-[asorptions in fst group] -[Downscttering from fst to therml group] +[Fissions in fst group]+[fissions in therml group]=0 + D - - + χν c + χν c fc c = 0 1c 1c 1c 1c 1sc 1c 1 1 1fc 1c 1 (1) Core therml group:
-[Lekge from therml group]-[asorptions in therml group] +[Downscttering neutrons from fst group] +[Fissions in fst group]+[fissions in therml group]=0 D - + + χν c + χν fc c = 0 c c c c 1sc 1c 1 1fc 1c c () We cn notice tht oth fst nd therml neutrons re cusing fissions in either group such s: χ 1+ χ = 1 The verge numers of neutrons per fission in different mterils re listed in Tle 1. Tle 1: Two group fst rector constnts. (From ANL-5800). Group 1: Energy rnge 1.35 Me to, χ 1 = 0.575. Group : Energy rnge 0 to 1.35 Me, χ = 0.45. Element ν σ f σ c σ tr σ 1s 1 1 1 1 Pu 39 3.10.93 1.95 1.78 0.10 0.30 4.6 7.0 0.90 U 35.70.50 1.9 1.44 0.08 0.8 4.5 7. 1.50 U 38.60.47 0.54 0.005 0.036 0.19 4.6 7.1.05 Fe - - - - 0.005 0.006.0.8 0.70 N - - - - 0.0005 0.0008.0 3.5 0.30 Al - - - - 0.004 0.00 1.8 3.5 0.38 As in the cse of therml rectors: = f + c. In the sence of modertor, the role of inelstic scttering is s importnt s the elstic scttering, so tht: 1s = 1in+ 1e. The diffusion coefficients re defined s: 1 1 D =, D =, 1 3 1tr 3 tr where: = + + + + (1 ) 1tr 1 1in 1e 1in1 1e1 = + + (1 µ ) tr in e 0 µ 0
σ σ jej jinj 1tr tr is the elstic scttering cross section within group j is the inelstic scttering cross section within group j, re the mcroscopic trnsport cross sections, which re the reciprocls of the men free pths for ll interctions in groups 1 nd. In the sme mnner, we cn now write the U 38 lnket equtions, neglecting the therml fission of U 38. Blnket fst group: + D - - + χν =0 1 1s 1 1 1f (3) Blnket therml group: +D - + + χν = 0 1s 1f (4) CORE AND BLANKET COUPLING COEFFICIENTS AND CRITICALITY DETERMINANT If we let: = B, we cn rewrite Eqns. 1 nd in the form: D B +( + χν ) = χν (5) 1c 1c 1c 1sc 1 1c 1fc 1c 1 c fc c D B +( χν ) = ( + χν ) (6) c c c c fc c 1sc 1c 1fc 1c Let us denote: ( + χ ν ) = 1c 1sc 1 1c 1fc 1nc ( χν ) = c c fc 1pc s the net core removl cross sections, nd: ( + χ ν ) = 1sc 1c 1fc 1pc s the totl production cross section of group neutrons from group 1 neutrons, nd define:
L D D =, L = 1c c 1nc nc 1 nc nc then, Eqns. 5 nd 6 ecome: χ ν c (1+L B ) = 1 fc 1nc 1c c 1 nc (7) (1+L B ) = (8) ipc nc c 1c nc Eliminting 1c nd c from Eqns. 7 nd 8, we get: 1 1 1 = k n (1+L B ) (1+L B ) (9) 1nc nc where: k n χ ν = 1 c fc 1pc 1nc nc (10) hving estimted the quntities L1 nc, Lnc nd kn, nd similrly to the cse of therml rectors, we cn estimte the principl nd lternte ucklings s: + + 4c µ = (11) ν = µ + (1) where: 1 1 = + L L k c = L 1nc nc 1 n 1nc. Lnc For the principl uckling, B = µ, + µ = 0, = AX 1c 1c 1c + µ = 0, = A X ' c c c
By sustituting in Eqn. : -D µ AX- AX + AX = 0, ' ' c nc 1pc from which we cn get the principl coupling coefficient s: S A = = (13) ( + ) ' 1pc 1 A nc D cµ For the lternte uckling, B = ν, ν = 0, = CY 1c 1c 1c ν = 0, = CY ' c c c By sustituting in Eqn., S C = = (14) ( + ) ' 1pc C nc Dcν To estimte the coupling coefficients in the reflector, we define the removl cross section in the lnket s: = + χ ν 1n 1 1s 1 1f nd the totl production cross section of therml neutrons from fst group neutrons: = + χ ν 1p 1s 1f We cn thus write Eqns. 3 nd 4 s: = 0 (15) 1n D1 + = 0 (16) 1p D D Defining the diffusion res: L D D =, L =, 1n
the lst equtions ecome: 1 0 = (17) L1 1 + = 0 (18) 1p L D In the sme wy s in the cse of therml rectors, we get: S 3 = D 1p 1 1 1 ( + ) L L (19) If the currents nd fluxes continuity conditions re pplied t the core nd lnket oundry we otin 4 y 4 criticlity determinnt s in the cse of therml rectors, ut with different definition of the coupling coefficients nd constnts. If the rector composition is vried, then single vlue of the criticl rdius for the core cn e otined. ESTIMATION OF THE FLUX DISTRIBUTIONS We seek to otin expressions for the flux distriutions normlized to 1 Wtt of rector power production. The difference etween fst rectors nd therml rectors is tht fst fissions re generting power in the lnket. Thus the totl fst rector power cn e written s: fissions 3 Me 13 Joule P[Wtts] = ( fc cc + 1fc 1cc +1f1 ). cm.00 1.6x10 3 cm.sec fission Me (0) = + + 11 3.x10 ( fc cc 1fc 1cc 1f 1 ) We define the fst fission fctor s: ε = Totl fission rte in core nd lnket Therml group energy fissions in core = 1+ + 1fc 1c 1f fc c fc c c (1) From Eqns. 0 nd 1, we get: P = x ε () 11 [Wtts] 3. 10 fc c c
To compute the power P, one needs to estimte the fst fission fctor ε. This in turn needs the knowledge of:,,. 1c c The first two re the sme s the fst nd therml core fluxes 1 c, c in the cse of therml rectors ssocited with the pproprite constnts. The quntity cn e written s: d = = d R+ T R F ( R+ T + d r) sinh 4π r L R+ T R+ T 4π rdr 3 F ( R+ T + d r) = rsinh d ( ) r 3 3 R+ T R L R R rdr where the integrtion is crried out over the lnket thickness. Now: d d r r I = rsinh c rd[cosh ] c c r r = cr [ cosh c cosh dr] c c d r r d = cr [ cosh + csinh ] c c d d = c( d) cosh + cosh c sinh + c sinh c c c c Thus: d 3F T + d T + d 1 = [ RL 3 3 cosh + L1 sinh ( R+ T) R L1 L d d L1 ( R+ d)cosh Lsinh ] L L (3) Thus the fst fission fctor ε cn e evluted from Eqn. 1. The quntity A cn e clculted for P = 1 Wtt, nd the fluxes cn e plotted.
ESTIMATION OF THE BREEDER RATIO In fst rector systems sed upon the U 35 nd Pu 39 fuel cycle, it is possile to produce more fissile fuel toms thn those tht re consumed. As mesure of this cpility we define the reeding rtio BR s: Rte of fissile toms production BR= Rte of fissile toms consumption (4) When BR<1, it is denoted s the conversion rtio insted. If α is the cpture to fission rtio, then the numer of fissile toms consumed per second in the core: (1 + α ) + (1 + α ) 1 1fc 1c c fc c c Writing down the numer of fissile toms production in the core nd lnket, Eqn. 4 cn e written s: + + + BR = (1 + α ) + (1 + α ) 1cc 1c c cc c c 1c c 1 1fc 1c c fc c + + ( + ) = (1 + α ) + (1 + α ) 1c 1cc cc 1c c c cc 1c 1 1fc fc c (5) where re the cpture cross sections in the core for groups 1 nd. wy:, 1c c In Eqn. 5 we need to find n expression for. This cn e done in the following d ( S + GZ ) d 3 1 = = d R+ T 4 πg ( R+ T + d r) = S31 + rsinh dr L R Mking use of the result in Eqn. 3, we cn write:
3G T + d T + d = S31 + [ RL 3 3 cosh + Lsinh ( R+ T) R L L d d L( R+ d)cosh Lsinh ] L L (6) Knowing from eqn. 6 nd from Eqn. 3, the vlue of the reeding rtion cn e determined from Eqn. 5. REFERENCES 1. M. Rghe, Lecture Notes on Fission Rectors Design Theory, FSL-33, University of Illinois, 198.. J. R. Lmrsh, Introduction to Nucler Engineering, Addison-Wesley Pulishing Compny, 1983. 3. K. Wirtz, Lectures on Fst Rectors, 1979. 4. ----------, Rector Physics Constnts, ANL-5800, Argonne Ntionl Lortory, 1979.