PHGN590. Introduction to Nuclear Reactor Physics. Two Group Flux Profile
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1 PHGN590 Introduction to Nuclear Reactor Physics Two Group Flux Profile J. A. McNeil Physics Department Colorado School of Mines 4/2009
2 2 Flux_Profile_2group.nb Two velocity groups ü Parameters based on USGS Triga (group 1-> fast, group 2 -> thermal) H* Geometry is given in inches and converted to cm *L geom = 8RC Ø 2.54 H ê 16L, HC Ø 2.54 µ 15, RRef Ø 2.54 H ê 8L, HRef Ø 2.54 H µ 3.47L<; Print@" Reactor geometry: Core Radius = ", RC ê. geom, " cm Core Height = ", HC ê. geom, " cm"d Print@" Reflector radius = ", RRef ê. geom, " cm Reflector Height = ", HRef ê. geom, " cm"d params1 = 8n Ø 2.07, Sf Ø.00258, SCa Ø H L, SRefa Ø , SCs Ø 3.29, SRefs Ø.3811, mc Ø.025, mref Ø 1 ê 18.<; params2 = 8n Ø 2.07, Sf Ø.0712, SCa Ø H L, SRefa Ø , SCs Ø 3.29, SRefs Ø.3811, mc Ø.025, mref Ø 1 ê 18.<; H* Diffusion constants for core *L StrC = SCa + Sf + H1 - mcl SCs; D C = 1 ê H3 StrCL; L C = D C SCa ;
3 Flux_Profile_2group.nb 3 Print@" Core parameters "D Print@" Fast group: Str1 = ", StrC ê. params1, " cm^-1 D C1 = ", D C ê. params1, " cm L C1 = ", L C ê. params1, " cm"d Print@" Thermal group: Str2 = ", StrC ê. params2, " cm^-1 D C2 = ", D C ê. params2, " cm L C1 = ", L C ê. params2, " cm"d H* Diffusion constants for reflector *L StrRef = SRefa + H1 - mrefl SRefs; lex = 1 ê H3 StrRefL; D R = 1 ê H3 StrRefL; L R = D R SRefa ; Print@" Reflector parameters "D Print@" Fast group: Str1 = ", StrRef ê. params1, " cm^-1 D R1 = ", D R ê. params1, " cm L R1 = ", L R ê. params1, " cm"d Print@" Thermal group: Str2 = ", StrRef ê. params2, " cm^-1 D R2 = ", D R ê. params2, " cm L R1 = ", L R ê. params2, " cm"d Reactor geometry: Core Radius = cm Core Height = 38.1 cm Reflector radius = cm Reflector Height = cm Core parameters Fast group: Str1 = cm^-1 D C1 = cm L C1 = cm Thermal group: Str2 = cm^-1 D C2 = cm L C1 = cm Reflector parameters Fast group: Str1 = cm^-1 D R1 = cm L R1 = cm Thermal group: Str2 = cm^-1 D R2 = cm L R1 = cm
4 4 Flux_Profile_2group.nb Rex = HRRef + lexl ê. params1 ê. geom; Hex = HHRef + 2 lexl ê. params1 ê. geom; Print@" Extrapolation distance: lex = ", lex ê. params1, " cm"d Extrapolation distance: lex = cm ü Bare Cylindrical Reactor The diffusion equations for the two velocity group fluxes are: D 1 2 F 1 == H-Sa 1 - Ss 1->2 L F 1 + n Sf 2 F k 2 D 2 2 F 2 == -Sa 2 F 2 + Ss 1->2 F 1 For cylindrical geometry these factor into r and z components: F = R[r] Z[z]. Reqn = 1 r D@Hr R'@rDL, rd == -Br2 R@rD; Zeqn = Z''@zD ã -Bz 2 Z@zD; For both groups the boundary conditions are R'[0]=Z'[0]==0 and R[Rex]=Z[- Hex/2]==0 which gives: Z1soln@z_D = Az1 Cos@Bz zd R1soln@r_D = Ar1 BesselJ@0, Br rd Z2soln@z_D = Az2 Cos@Bz zd R2soln@r_D = Ar2 BesselJ@0, Br rd Az1 Cos@Bz zd Ar1 BesselJ@0, Br rd Az2 Cos@Bz zd Ar2 BesselJ@0, Br rd where Bz = p / Hex and Br = z0 / Rex, where z0 is the first zero of J0:
5 Flux_Profile_2group.nb 5 z0soln = z0 ê. FindRoot@BesselJ@0, z0d ã 0, 8z0, 2.5<D; Bconstants = 8Br -> z0soln ê Rex, Bz -> p ê Hex<; Bnet = Br 2 + Bz 2 ê. Bconstants ê. geom; PNL = 1 ê H1 + Bnet^2 L C ^2L ê. params1 ê. geom; r21 = SCs ê HSCa + D C Bnet^2L ê. params2 ê. Bconstants; kmax = HHHn SfL ê. params1l + r21 HHn SfL ê. params2ll ê HBnet ^ 2 + HHSCa + SCsL ê. params2ll ê. Bconstants; Print@" Buckling constants Hcm^-1L: Br = ", Br ê. Bconstants, " Bz = ", Bz ê. Bconstants, " BHtotalL = ", BnetD Print@" Nonleakage probability = ", PNLD Print@" Ratio of thermal to fast flux = ", r21d Print@ " Maximum neutron multiplication: k = ", kmaxd Buckling constants Hcm^-1L: Br = Bz = BHtotalL = Nonleakage probability = Ratio of thermal to fast flux = Maximum neutron multiplication: k = soln = 8Br -> z0soln ê Rex, Bz -> p ê Hex, Ar1 Ø 1, Az1 Ø 1, Ar2 Ø r21, Az2 Ø r21<; Plot@8 R1soln@rD ê. soln, R2soln@rD ê. soln<, 8r, 0, Rex<D Plot@8 Z1soln@zD ê. soln, Z2soln@zD ê. soln<, 8z, 0, Hex ê 2<D
6 6 Flux_Profile_2group.nb
7 Flux_Profile_2group.nb
8 8 Flux_Profile_2group.nb ü Reflected Slab Reactor Consider a slsb reactor having a core of width, a, which is surrounded by an infinite reflector. There are two velocity groups and two regions (core and reflector) giving the following four diffusion equations for the neutron flux: [1a] D C1 2 F C1 ã HS ac1 + S 12 C L F C1-1 k Hn 1 S f1 F C1 + n 2 S f2 F C2 L [1b] D R1 2 F R1 == HS ar1 + S 12 R L F R1 [1c] D C2 2 F C2 == S ac2 F C2 - S sc1 F C1 D R2 2 F R2 == S ar2 F R2 - S sr1 F R1 [1d] In the following we will solve for the conditions that give k= 1. For slab symmetry 2 F -> d2 F. The (8) boundary conditions are: dx 2 F C1 '[0] = F C2 '[0] = 0 F C1 A a 2 E = F R1A a 2 E J C1 A a 2 E = J R1A a 2 E F C2 A a 2 E = F R2A a 2 E J C2 A a 2 E = J R2A a 2 E F D = F D = 0 [2a,b] [2c] [2d] [2e] [2f] [2g,h] The general solutions, satisfying [2a,b] and [2g,h] and normalized by the first sineterm, are: F = Cos@m xd + r C Cosh@n xd; F = S 1 Cos@m xd + S 2 r C Cosh@n xd; F = r F Exp@-k 1 xd; F = r G Exp@-k 2 xd + S 3 r F Exp@-k 1 xd;
9 Flux_Profile_2group.nb 9 We will work only in the positive x region and use reflection symmetry to obtain the solution in the negative x region. There remain 10 parameters, {m, n, k 1, k 2, r C, r F, r G, S 1, S 2, S 3 <, to be determined. Substitution in to Eq.[1c,d] gives: k 1 soln = S ar1 + S 12 R D R1 ; k 2 soln = S ar2 D R2 ; Substitution into [1a,b] and equating sine and sinh-terms gives: S 1 soln = S 2 soln = S 3 soln = S 12 C S ac2 + D C2 m 2 ; S 12 C S ac2 - D C2 n ; 2 D R1 S 12 R ; D R1 S ar2 - D R2 S ar1 - D R2 S 12 R The joining conditions for the currents at r = R gives the flux ratios, { r C, r F, r G <:
10 10 Flux_Profile_2group.nb eq1 = F C1 B a 2 F ã F R1B a 2 F; eq2 = D C1 D@F xd == D R1 D@F xd ê. x Ø a 2 ; rcfsoln = Flatten@Solve@8eq1, eq2<, 8r C, r F <DD; r Csoln = r C ê. rcfsoln; r Fsoln = r F ê. rcfsoln; eq3 = F C2 B a 2 F ã F R2B a 2 F; eq4 = D C2 D@F xd == D R2 D@F xd ê. x Ø a 2 ; rfgsoln = Flatten@Solve@8eq3, eq4<, 8r G, r F <DD; r Gsoln = Simplify@r G ê. rfgsoln ê. r C -> r Csoln D; r Fsoln2 = Simplify@r F ê. rfgsoln ê. r C -> r Csoln D; Print@" r C Print@" r F Print@" r G = ", r Csoln D = ", r Fsoln D = ", r Gsoln D r C = - -m SinA a m 2 E D C1 + CosA a m 2 E D R1 k 1 n SinhA a n 2 E D C1 + CoshA a n 2 E D R1 k 1 r F = r G = a k 1 2 Im CoshA a n 2 E SinA a m 2 E + n CosA a m 2 E SinhA a n 2 EM D C1 n SinhA a n 2 E D C1 + CoshA a n 2 E D R1 k 1 1 D R2 Hk 1 - k 2 L a k2 2 -m SinB a m 2 F D C2 S 1 + CosB a m 2 F D R2 S 1 k 1 + n SinhA a n 2 E D C2 S 2 Im SinA a m 2 E D C1 - CosA a m 2 E D R1 k 1 M n SinhA a n 2 E D C1 + CoshA a n 2 E D R1 k 1 - CoshA a n 2 E D R2 S 2 k 1 I-m SinA a m 2 E D C1 + CosA a m 2 E D R1 k 1 M n SinhA a n 2 E D C1 + CoshA a n 2 E D R1 k 1 The second solution for r F will be used to obtain the reactor width, a, self-consistently. We start with the value for a bare reactor based on the reactor parameters are taken from Meem, Two Group Reactor Theory,
11 Flux_Profile_2group.nb 11 H* Meems parameters *L params1 = 8n 1 Ø 2.47, S f1 Ø , S ac1 Ø , S ar1 Ø 0.0, S sc1 Ø , S sr1 Ø.6871, z C Ø , z R Ø.8203, E 0 Ø 1.62 µ 10^6, E th Ø 0.1, mc Ø.5078, mref Ø.5391<; params2 = 8n 2 Ø 2.47, S f2 Ø.06147, S ac2 Ø , S ar2 Ø , S sc2 Ø 4.715, S sr2 Ø 6.529, mc Ø , mref Ø.6533<; Calculate the various parameters appropriate to this reactor: S 12 Cvalue = z C S sc1 ê Log@E 0 ê E th D ê. params1; S 12 Rvalue = z R S sr1 ê Log@E 0 ê E th D ê. params1; PrintA" S 12 C = ", S 12 Cvalue, " cm -1 S 12 R = ", S 12 Rvalue, " cm -1 "E StrC1 = H1 - mcl S sc1 ê. params1; StrC2 = H1 - mcl S sc2 ê. params2; StrR1 = H1 - mrefl S sr1 ê. params1; StrR2 = H1 - mrefl S sr2 ê. params2; D C1value = 1 ê H3 HS ac1 + StrC1LL ê. params1; D C2value = 1 ê H3 HS ac2 + StrC2LL ê. params2; D R1value = 1 ê H3 HS ar1 + StrR1LL ê. params1; D R2value = 1 ê H3 HS ar2 + StrR2LL ê. params2; Print@" D C1 = ", D C1value, " cm D C2 = ", D C2value, " cm D R1 = ", D R1value, " cm D R2 = ", D R2value, " cm"d Dparams = 8D C1 -> D C1value, D C2 -> D C2value, D R1 -> D R1value, D R2 -> D R2value, S 12 C -> S 12 Cvalue, S 12 R -> S 12 Rvalue <; k 1 value = k 1 soln ê. Dparams ê. params1; k 2 value = k 2 soln ê. Dparams ê. params2;
12 12 Flux_Profile_2group.nb PrintA" k 1 = ", k 1 value, " cm -1 k 2 = ", k 2 value, " cm -1 "E L nsol = D C1 S 12 C + S ac1 - n 1 S f1 ; L nvalue = L nsol ê. Dparams ê. params1; Print@" L n = ", L nvalue, " cm"d slabparams = 8D C1 -> D C1value, D C2 -> D C2value, D R1 -> D R1value, D R2 -> D R2value, S 12 C -> S 12 Cvalue, S 12 R -> S 12 Rvalue, L n -> L nvalue, R Ø R value <; S 12 C = cm -1 S 12 R = cm -1 D C1 = cm D C2 = cm D R1 = cm D R2 = cm k 1 = cm -1 k 2 = cm -1 L n = cm The initial guess for the slab width is estimated from the buckling constant using the geometric mean of the fast and thermal buckling:
13 Flux_Profile_2group.nb 13 a new = 0; k = n 1 S f1 ê S ac1 ê. params1; L1 = D C1value S ac1 ê. params1; B1 = Hk - 1L ë IL1 2 M ; k = n 2 S f2 ê S ac2 ê. params2; L2 = D C2value S ac2 ê. params2; B2 = Hk - 1L ë L2 2 ; a 0 = p í B1 B2 ; Print@" a HguessL = ", a 0, " cm"d a HguessL = cm The parameters list needs to be re-run once the self consistent radius is determined. Initially, however, set it to a 0. RUN SOLUTION HERE --> a value = If@a new < 1, a 0, a new D; slabparams = 8D C1 -> D C1value, D C2 -> D C2value, D R1 -> D R1value, D R2 -> D R2value, S 12 C -> S 12 Cvalue, S 12 R -> S 12 Rvalue, L n -> L nvalue, a Ø a value < 8D C1 Ø , D C2 Ø , D R1 Ø , D R2 Ø , S 12 C Ø , S 12 R Ø , L n Ø , a Ø < Find the remaining parameters needed to specify the solutions. b m = b n = 1 L + S ac2 2 n D C2 ê. slabparams ê. params1 ê. params2;
14 14 Flux_Profile_2group.nb - 1 L n 2 + S ac2 D C2 ê. slabparams ê. params1 ê. params2; c = - n 2 S f2 S 12 C D C1 D C2 + S ac2 L n 2 D C2 ê. slabparams ê. params1 ê. params2; m value = I-b m + SqrtAb m 2-4 cem ë 2 ; n value = I-b n + SqrtAb n 2-4 cem ë 2 ; S 1 value = S 1 soln ê. 8m -> m value < ê. slabparams ê. params2; S 2 value = S 2 soln ê. 8n Ø n value < ê. slabparams ê. params2; S 3 value = S 3 soln ê. slabparams ê. params2 ê. params1; musparams = 8m -> m value, n -> n value, S 1 -> S 1 value, S 2 -> S 2 value, S 3 -> S 3 value, k 1 -> k 1 value, k 2 -> k 2 value <; r Cvalue = r Csoln ê. slabparams ê. params2 ê. params1 ê. musparams; r Fvalue = r Fsoln ê. slabparams ê. params2 ê. params1 ê. musparams; r Gvalue = r Gsoln ê. slabparams ê. params2 ê. params1 ê. musparams; solnparams = 8m -> m value, n -> n value, S 1 -> S 1 value, S 2 -> S 2 value, S 3 -> S 3 value, k 1 -> k 1 value, k 2 -> k 2 value <; rparams = 8r C -> r Cvalue, r F -> r Fvalue, r G -> r Gvalue < ê. solnparams; PrintA" m = ", m value, " cm -1 n = ", n value, " cm -1 "E Print@" S 1 = ", S 1 value, " S 2 = ", S 2 value, " S 3 = ", S 3 value D
15 Flux_Profile_2group.nb 15 PrintA" k 1 = ", k 1 value, " cm -1 k 2 = ", k 2 value, " cm -1 "E Print@" r C = ", r Cvalue, " r F = ", r Fvalue, " r G = ", r Gvalue D m = cm -1 n = cm -1 S 1 = S 2 = S 3 = k 1 = cm -1 k 2 = cm -1 r C = r F = r G = Test the solution by examining the matching conditions at the reactor boundary, a/2: FC1 = F C1 B a F ê. params1 ê. slabparams ê. Dparams ê. 2 solnparams ê. rparams; FR1 = F R1 B a F ê. params1 ê. slabparams ê. Dparams ê. 2 solnparams ê. rparams; PrintB" F a D = ", FC1, " 2 F R1@ a D = ", FR1F 2 FC2 = F C2 B a F ê. params1 ê. params2 ê. slabparams ê. 2 Dparams ê. solnparams ê. rparams; FR2 = F R2 B a F ê. params1 ê. params2 ê. slabparams ê. 2 Dparams ê. solnparams ê. rparams; PrintB" F a D = ", FC2, " 2 F R2@ a D = ", FR2F 2
16 16 Flux_Profile_2group.nb jc1 = D C1 D@F xd ê. x Ø a 2 ê. params1 ê. slabparams ê. Dparams ê. solnparams ê. rparams; jr1 = D R1 D@F xd ê. x Ø a 2 ê. params1 ê. slabparams ê. Dparams ê. solnparams ê. rparams; PrintB" J a D = ", jc1, " 2 J R1@ a D = ", jr1f 2 jc2 = D C2 D@F xd ê. x Ø a 2 ê. params1 ê. params2 ê. slabparams ê. Dparams ê. solnparams ê. rparams; jr2 = D R2 D@F xd ê. x Ø a 2 ê. params1 ê. params2 ê. slabparams ê. Dparams ê. solnparams ê. rparams; PrintB" J a D = ", jc2, " 2 J R2@ a D = ", jr2f 2 F a 2 D = F R1@ a 2 D = F a 2 D = F R2@ a 2 D = J a 2 D = J R1@ a 2 D = J a 2 D = J R2@ a 2 D = The first attempt using the initial guess for a does not quite satisfy the matching conditions. To find the correcd slab width, find the value of a which gives the same flux ratio, r F, for the two cases:
17 Flux_Profile_2group.nb 17 a new = a ê. FindRoot@Hr Fsoln - r Fsoln2 L ê. params1 ê. params2 ê. Dparams ê. solnparams, 8a, a value <D; Print@" Self-consistent a = ", a new, " cm"d Self-consistent a = cm --> Now re-run the solutions to show that the matching conditions are now exactly satisfied (go to "RUN SOLUTION HERE" and re-run parameter list ). F1plot@x_D = If@x < a value ê 2, HF ê. solnparams ê. rparams, HF ê. solnparams ê. rparamsd; F2plot@x_D = If@x < a value ê 2, HF ê. solnparams ê. rparams, HF ê. solnparams ê. rparamsd; Plot@8F1plot@xD, F2plot@xD<, 8x,.01, 2 a value <, PlotRange Ø AllD
18 18 Flux_Profile_2group.nb Reflected Spherical Reactor Consider a spherical reactor having a core of radius, R, which is surrounded by an infinite reflector. There are two velocity groups and two regions (core and reflector) giving the following four diffusion equations for the neutron flux: [1a] D C1 2 F C1 ã HS ac1 + S 12 C L F C1-1 k Hn 1 S f1 F C1 + n 2 S f2 F C2 L [1b] D R1 2 F R1 == HS ar1 + S 12 R L F R1 [1c] D C2 2 F C2 == S ac2 F C2 - S sc1 F C1 [1d] D R2 2 F R2 == S ar2 F R2 - S sr1 F R1 For spherical symmetry we define F = u/r and, 2 F -> 1 d2 r dr 2 u. The (8) boundary conditions are: u C1 [0] = u C2 [0] = 0 [2a,b] u = u [2c] J = J [2d] u = u [2e] J = J [2f] u D = u D = 0 [2g,h] The general solutions, satisfying [2a,b] and [2g,h] and normalized by the first sineterm, are: u = Sin@m rd + r C Sinh@n rd; u = S 1 Sin@m rd + S 2 r C Sinh@n rd; u = r F Exp@-k 1 rd; u = r G Exp@-k 2 rd + S 3 r F Exp@-k 1 rd; There remain 10 parameters, {m, n, k 1, k 2, r C, r F, r G, S 1, S 2, S 3 <, to be deter-
19 Flux_Profile_2group.nb C F G 1 2 3< mined. Substitution in to Eq.[1c,d] gives: k 1 soln = S ar1 + S 12 R D R1 ; k 2 soln = S ar2 D R2 ; Substitution into [1a,b] and equating sine and sinh-terms gives: S 1 soln = S 2 soln = S 3 soln = S 12 C S ac2 + D C2 m 2 ; S 12 C S ac2 - D C2 n ; 2 D R1 S 12 R ; D R1 S ar2 - D R2 S ar1 - D R2 S 12 R The joining conditions for the currents at r = R gives the flux ratios, { r C, r F, r G <:
20 20 Flux_Profile_2group.nb eq1 = u == u eq2 = D C1 D@u ê r, rd == D R1 D@u ê r, rd ê. r Ø R; rcfsoln = Flatten@Solve@8eq1, eq2<, 8r C, r F <DD; r Csoln = r C ê. rcfsoln; r Fsoln = r F ê. rcfsoln; eq3 = u == u eq4 = D C2 D@u ê r, rd == D R2 D@u ê r, rd ê. r Ø R; rfgsoln = Flatten@Solve@8eq3, eq4<, 8r G, r F <DD; r Gsoln = Simplify@r G ê. rfgsoln ê. r C -> r Csoln D; r Fsoln2 = Simplify@r F ê. rfgsoln ê. r C -> r Csoln D; Print@" r C Print@" r F Print@" r G = ", r Csoln D = ", r Fsoln D = ", r Gsoln D R m Cos@R md D C1 - Sin@R md D C1 + Sin@R md D R1 + R Sin@R md D R1 k 1 r C = - R n Cosh@R nd D C1 - Sinh@R nd D C1 + Sinh@R nd D R1 + R Sinh@R nd D R1 k 1 r F = r G = R k 1 R Hn Cosh@R nd Sin@R md - m Cos@R md Sinh@R ndl DC1 R n Cosh@R nd D C1 - Sinh@R nd D C1 + Sinh@R nd D R1 + R Sinh@R nd D R1 k 1 -I R k 2 HD R1 H1 + R k 1 L HD C2 HHR m Cos@R md - Sin@R mdl Sinh@R nd S 1 + Sin@R md H-R n Cosh@R nd + Sinh@R ndl S 2 L + Sin@R md Sinh@R nd D R2 HS 1 - S 2 L H1 + R k 1 LL + D C1 HHR m Cos@R md - Sin@R mdl HR n Cosh@R nd - Sinh@R ndl D C2 HS 1 - S 2 L + D R2 HSin@R md HR n Cosh@R nd - Sinh@R ndl S 1 + H-R m Cos@R md + Sin@R mdl Sinh@R nd S 2 L H1 + R k 1 LLLM ë HR D R2 HHR n Cosh@R nd - Sinh@R ndl D C1 + Sinh@R nd D R1 H1 + R k 1 LL H-k 1 + k 2 LL The second solution for r F will be used to obtain the reactor radius, R, self-consistently. We start with the value for a bare reactor based on the reactor parameters are taken from Meem, Two Group Reactor Theory,
21 Flux_Profile_2group.nb 21 H* Meems parameters *L params1 = 8n 1 Ø 2.47, S f1 Ø , S ac1 Ø , S ar1 Ø 0.0, S sc1 Ø , S sr1 Ø.6871, z C Ø , z R Ø.8203, E 0 Ø 1.62 µ 10^6, E th Ø 0.1, mc Ø.5078, mref Ø.5391<; params2 = 8n 2 Ø 2.47, S f2 Ø.06147, S ac2 Ø , S ar2 Ø , S sc2 Ø 4.715, S sr2 Ø 6.529, mc Ø , mref Ø.6533<; Calculate the various parameters appropriate to this reactor: S 12 Cvalue = z C S sc1 ê Log@E 0 ê E th D ê. params1; S 12 Rvalue = z R S sr1 ê Log@E 0 ê E th D ê. params1; PrintA" S 12 C = ", S 12 Cvalue, " cm -1 S 12 R = ", S 12 Rvalue, " cm -1 "E StrC1 = H1 - mcl S sc1 ê. params1; StrC2 = H1 - mcl S sc2 ê. params2; StrR1 = H1 - mrefl S sr1 ê. params1; StrR2 = H1 - mrefl S sr2 ê. params2; D C1value = 1 ê H3 HS ac1 + StrC1LL ê. params1; D C2value = 1 ê H3 HS ac2 + StrC2LL ê. params2; D R1value = 1 ê H3 HS ar1 + StrR1LL ê. params1; D R2value = 1 ê H3 HS ar2 + StrR2LL ê. params2; Print@" D C1 = ", D C1value, " cm D C2 = ", D C2value, " cm D R1 = ", D R1value, " cm D R2 = ", D R2value, " cm"d Dparams = 8D C1 -> D C1value, D C2 -> D C2value, D R1 -> D R1value, D R2 -> D R2value, S 12 C -> S 12 Cvalue, S 12 R -> S 12 Rvalue <; k 1 value = k 1 soln ê. Dparams ê. params1; k 2 value = k 2 soln ê. Dparams ê. params2;
22 22 Flux_Profile_2group.nb PrintA" k 1 = ", k 1 value, " cm -1 k 2 = ", k 2 value, " cm -1 "E L nsol = D C1 S 12 C + S ac1 - n 1 S f1 ; L nvalue = L nsol ê. Dparams ê. params1; Print@" L n = ", L nvalue, " cm"d sphereparams = 8D C1 -> D C1value, D C2 -> D C2value, D R1 -> D R1value, D R2 -> D R2value, S 12 C -> S 12 Cvalue, S 12 R -> S 12 Rvalue, L n -> L nvalue, R Ø R value <; S 12 C = cm -1 S 12 R = cm -1 D C1 = cm D C2 = cm D R1 = cm D R2 = cm k 1 = cm -1 k 2 = cm -1 L n = cm The initial guess for the radius is estimated from the buckling constant using the geometric mean of the fast and thermal buckling:
23 Flux_Profile_2group.nb 23 R new = 0; k = n 1 S f1 ê S ac1 ê. params1; L1 = D C1value S ac1 ê. params1; B1 = Hk - 1L ë IL1 2 M ; k = n 2 S f2 ê S ac2 ê. params2; L2 = D C2value S ac2 ê. params2; B2 = Hk - 1L ë L2 2 ; R 0 = p í B1 B2 ; Print@" R HguessL = ", R 0, " cm"d R HguessL = cm The parameters list needs to be re-run once the self consistent radius is determined. Initially, however, set it to R 0. RUN SOLUTION HERE --> R value = If@R new < 1, R 0, R new D; sphereparams = 8D C1 -> D C1value, D C2 -> D C2value, D R1 -> D R1value, D R2 -> D R2value, S 12 C -> S 12 Cvalue, S 12 R -> S 12 Rvalue, L n -> L nvalue, R Ø R value < 8D C1 Ø , D C2 Ø , D R1 Ø , D R2 Ø , S 12 C Ø , S 12 R Ø , L n Ø , R Ø < Find the remaining parameters needed to specify the solutions. b m = 1 L + S ac2 2 n D C2 ê. sphereparams ê. params1 ê. params2;
24 24 Flux_Profile_2group.nb b n = - 1 L n 2 + S ac2 D C2 ê. sphereparams ê. params1 ê. params2; c = - n 2 S f2 S 12 C D C1 D C2 + S ac2 L n 2 D C2 ê. sphereparams ê. params1 ê. params2; m value = I-b m + SqrtAb m 2-4 cem ë 2 ; n value = I-b n + SqrtAb n 2-4 cem ë 2 ; S 1 value = S 1 soln ê. 8m -> m value < ê. sphereparams ê. params2; S 2 value = S 2 soln ê. 8n Ø n value < ê. sphereparams ê. params2; S 3 value = S 3 soln ê. sphereparams ê. params2 ê. params1; musparams = 8m -> m value, n -> n value, S 1 -> S 1 value, S 2 -> S 2 value, S 3 -> S 3 value, k 1 -> k 1 value, k 2 -> k 2 value <; r Cvalue = r Csoln ê. sphereparams ê. params2 ê. params1 ê. musparams; r Fvalue = r Fsoln ê. sphereparams ê. params2 ê. params1 ê. musparams; r Gvalue = r Gsoln ê. sphereparams ê. params2 ê. params1 ê. musparams; solnparams = 8m -> m value, n -> n value, S 1 -> S 1 value, S 2 -> S 2 value, S 3 -> S 3 value, k 1 -> k 1 value, k 2 -> k 2 value <; rparams = 8r C -> r Cvalue, r F -> r Fvalue, r G -> r Gvalue < ê. solnparams; Print@" m = ", m value, " n = ", n value D Print@" S 1 = ", S 1 value, " S 2 = ", S 2 value, " S 3 = ", S 3 value D
25 Flux_Profile_2group.nb 25 PrintA" k 1 = ", k 1 value, " cm -1 k 2 = ", k 2 value, " cm -1 "E Print@" r C = ", r Cvalue, " r F = ", r Fvalue, " r G = ", r Gvalue D m = n = S 1 = S 2 = S 3 = k 1 = cm -1 k 2 = cm -1 r C = µ 10-8 r F = r G = Test the solution by examining the matching conditions at the reactor radius, R:
26 26 Flux_Profile_2group.nb uc1 = u ê. params1 ê. sphereparams ê. Dparams ê. solnparams ê. rparams; ur1 = u ê. params1 ê. sphereparams ê. Dparams ê. solnparams ê. rparams; Print@" u = ", uc1, " u = ", ur1d uc2 = u ê. params1 ê. params2 ê. sphereparams ê. Dparams ê. solnparams ê. rparams; ur2 = u ê. params1 ê. params2 ê. sphereparams ê. Dparams ê. solnparams ê. rparams; Print@" u = ", uc2, " u = ", ur2d jc1 = D C1 D@u ê r, rd ê. r Ø R ê. params1 ê. sphereparams ê. Dparams ê. solnparams ê. rparams; jr1 = D R1 D@u ê r, rd ê. r Ø R ê. params1 ê. sphereparams ê. Dparams ê. solnparams ê. rparams; Print@" J = ", jc1, " J = ", jr1d jc2 = D C2 D@u ê r, rd ê. r Ø R ê. params1 ê. params2 ê. sphereparams ê. Dparams ê. solnparams ê. rparams; jr2 = D R2 D@u ê r, rd ê. r Ø R ê. params1 ê. params2 ê. sphereparams ê. Dparams ê. solnparams ê. rparams; Print@" J = ", jc2, " J = ", jr2d u = u = u = u = J = J = J = J =
27 Flux_Profile_2group.nb 27 The first attempt using the initial guess for R does not quite satisfy the matching conditions. To find the correcd radius, find the value of R which gives the same flux ratio, r F, for the two cases: R new = R ê. FindRoot@Hr Fsoln - r Fsoln2 L ê. params1 ê. params2 ê. Dparams ê. solnparams, 8R, R value <D; Print@" Self-consistent R = ", R new, " cm"d Self-consistent R = cm Now re-run the solutions to show that the matching conditions are now exactly satisfied (go to "RUN SOLUTION HERE").
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