Task: Role of moderation

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1 PHGN590 Introduction to Nuclear Reactor Physics Modeling Neutron Slowing in Reactors J. A. McNeil Physics Department Colorado School of Mines 2/2009 Task: Role of moderation In this task we explore the role of moderating (slowing) the neutrons. As can be seen from the data lists, for 235 U the macroscopic cross section for fission by a fast neutron is cm -1, while that for a thermal (slow) neutron is 28.4 cm -1. A moderator is a material that slows the neutrons down without absorbing them. Graphite ( 12 C) is an excellent moderator. It has a macroscopic scattering cross section of 0.381cm -1 and an absorption cross section of cm -1. The addition of a moderating material, like graphite, can alter the critical reactivity. Enrico Fermi used this to construct the first sustained chain reaction using natural uranium as the fuel. ü Analytic tasks (a) To illustrate this calculate k, Eq.(8), for natural Uranium ( 235 U (.72% ) and 238 U (99.28%)). Since Uranium is so much heavier than a neutron, elastic scattering does not slow the neutron down and the fast cross sections must be used.) H* Constants *L Cons = 8k B Ø µ 10^-23, T room Ø , e -> µ 10^-19, m n Ø µ 10^-27<; H* Thermal neutron values *L U235Th = 8r Ø.01886, n d Ø.04833, S s Ø.01588, S g Ø 4.833, S f -> 28.37, n Ø 2.42<; U238Th = 8r Ø.0191, n d Ø.04833, S s Ø.4301, S g Ø.13194, S f -> 0, n Ø 0<; U235frac =.0072; SigSNatUTh =

2 2 Moderation.nb HU235frac S s ê. U235ThL + HH1 - U235fracL S s ê. U238ThL; SigGNatUTh = HU235frac S g ê. U235ThL + HH1 - U235fracL S g ê. U238ThL; SigFNatUTh = HU235frac S f ê. U235ThL + HH1 - U235fracL S f ê. U238ThL; kinfnatuth = HHn U235frac S f ê. U235ThL + Hn H1 - U235fracL S f ê. U238ThLL ê HSigGNatUTh + SigFNatUThL; Print@" The thermal k-factor for an infinite body of natural uranium is ", kinfnatuthd; H* Fast neutron values *L C12data = 8r Ø.00160, n d Ø.08023, S s ->.3811, S g Ø , S f Ø 0, n Ø 0<; U235Fast = 8r Ø.01886, n d Ø.04833, S s Ø , S g Ø , S f ->.06766, n Ø 2.6<; U238Fast = 8r Ø.0191, n d Ø.04833, S s Ø.33347, S g Ø , S f -> , n Ø 2.6<; SigSNatUFast = HU235frac S s ê. U235FastL + HH1 - U235fracL S s ê. U238FastL; SigGNatUFast = HU235frac S g ê. U235FastL + HH1 - U235fracL S g ê. U238FastL; SigFNatUFast = HU235frac S f ê. U235FastL + HH1 - U235fracL S f ê. U238FastL; kinfnatufast = HHn U235frac S f ê. U235FastL + Hn H1 - U235fracL S f ê. U238FastLL ê HSigGNatUFast + SigFNatUFastL; Print@" The fast k-factor for an infinite body of natural uranium is ", kinfnatufastd; The thermal k-factor for an infinite body of natural uranium is The fast k-factor for an infinite body of natural uranium is (b) The average energy of a neutron created in a fission event is about 2 MeV. What fraction of the neutron's energy is lost in an elastic collision with a 12 C nucleus where the average of the cosine of the scattering angle is 2 3 A?

3 Moderation.nb 3 (c) How many collisions does it take to slow the neutron down to thermal energy (~1 ev)? H* ModFac is the moderation factor = fractional energy loss in an elastic scattering collision EF = Energy of the fast neutrons H2.5 MeVL ETh = Energy when thermal cross sections are applicable H1 evl NumScatt = number of elastic scatterings EThêEF = ModFac^NumScatt --Ø NumScatt = Log@EThêEFDêLog@1-ModFacD *L 3 A Avalue = 12; EF = 2. µ 10^6; ETh = 1.; ModFac = N@2 Avalue ê H1 + AvalueL^2D; NumScatt = Floor@Log@ETh ê EFD ê Log@1 - ModFacDD; Print@" For scattering from Carbon the fraction of neutron energy lost is ", ModFacD; Print@" Thus, ", NumScatt, " scatterings are required to bring the neutron to thermal energies"d; For scattering from Carbon the fraction of neutron energy lost is Thus, 94 scatterings are required to bring the neutron to thermal energies (d) Now mix in graphite so that the number densities fractions are x of 12 C and (1-x) uranium (natural). Calculate the aggregate macroscopic cross sections for this mixture using the fast neutron values. From these, calculate the probability, p(x), that a neutron survives to reach thermal energies. Plot the survival probability as a function of x. (e) Now, that the neutrons are slowed down (with probability, p), recalculate the macroscopic cross sections for the x mixture using the thermal values and calculate the critical factor, k(x), given by k HxL = p n S f S a Thermal + H1 - pl n S f S a. Fast

4 4 Moderation.nb What is the optimal mixing fraction yielding the greatest value for k? SigSMixFast = HCmix S s ê. C12dataL + H1 - CmixL SigSNatUFast; SigGMixFast = HCmix S g ê. C12dataL + H1 - CmixL SigGNatUFast; SigFMixFast = H1 - CmixL SigFNatUFast; Prob = SigSMixFast SigGMixFast + SigSMixFast + SigFMixFast NumScatt ; H1 - ProbL n SigFMixFast kmixfast = ê. U235Fast; SigGMixFast + SigFMixFast SigSMixTh = HCmix S s ê. C12dataL + H1 - CmixL SigSNatUTh; SigGMixTh = HCmix S g ê. C12dataL + H1 - CmixL SigGNatUTh; SigFMixTh = H1 - CmixL SigFNatUTh; Prob n SigFMixTh kmixth = ê. U235Th; SigGMixTh + SigFMixTh kinfnatu = kmixth + kmixfast; Plot@kinfNatU ê. Cmix Ø x, 8x, 0, 1<D CmixMax = Cmix ê. Flatten@FindRoot@ Cmix kinfnatu, 8Cmix, 0.98`<DD; kcmix = kinfnatu ê. Cmix Ø CmixMax; kmixfast ê. Cmix Ø CmixMax; kmixth ê. Cmix Ø CmixMax; 1 MFPFast = SigSMixFast + SigGMixFast + SigFMixFast ê. Cmix Ø CmixMax; 1 MFPTh = SigSMixTh + SigGMixTh + SigFMixTh ê. Cmix Ø CmixMax; AbsLenFast = 1 SigGMixFast + SigFMixFast ê. Cmix Ø CmixMax;

5 Moderation.nb 5 AbsLenTh = 1 SigGMixTh + SigFMixTh ê. Cmix Ø CmixMax; LFast = AbsLenFast MFPFast 3 ê. Cmix Ø CmixMax; LTh = AbsLenTh MFPTh 3 ê. Cmix Ø CmixMax; LMix = Prob LTh + H1 - ProbL LFast ê. Cmix Ø CmixMax; Print@" The probability of surviving to thermal speeds is ", Prob ê. Cmix Ø CmixMaxD Print@" The optimal mix of carbon is ", CmixMax, " yielding k = ", kcmixd Print@" Absorption lengths: AbsHFastL = ", AbsLenFast, " AbsHThermalL = ", AbsLenTh, " cm"d Print@" Mean Free Path lengths: MFPHFastL = ", MFPFast, " MFPHThermalL = ", MFPTh, " cm"d Print@" Diffusion lengths: LHFastL = ", LFast, " LHThermalL = ", LTh, " cm \!\H\*OverscriptBox@\HL\L, \H_\LD\L = ", LMixD

6 6 Moderation.nb The probability of surviving to thermal speeds is The optimal mix of carbon is yielding k = Absorption lengths: AbsHFastL = AbsHThermalL = cm Mean Free Path lengths: MFPHFastL = MFPHThermalL = cm Diffusion lengths: LHFastL = LHThermalL = cm L ê = ü Monte Carlo simulation (f) Develop a Monte Carlo simulation for this reactor configuration and calculate the neutron multiplication factor, k, for your optimal mix.

7 Moderation.nb 7 ü Set up the simulation by defining the time step in terms of the velocity and group cross sections Clear@vcm, vboost, dps, dpg, dpf, dpa, dp, dt, vmagd H* This block can only be executed after the previous section *L Fastdata = 8r Ø , n d Ø , S s Ø SigSMixFast, S g Ø SigGMixFast, S f Ø SigFMixFast, n Ø 2.6< ê. Cmix Ø CmixMax Thermaldata = 8r Ø , n d Ø , S s Ø SigSMixTh, S g Ø SigGMixTh, S f Ø SigFMixTh, n Ø 2.42< ê. Cmix Ø CmixMax data = 8Fastdata, Thermaldata<; muavg@a_d = 2 ê H3 AL; Avalue = 12; vboost@vmag_d = vmag ê H1 + AvalueL ; H* boost velocity connecting CM and Lab frames *L vcm@vmag_d = Avalue vmag ê H1 + AvalueL ; H* velocity of neutron in CM frame *L EFast = 2.5 µ 10^6; vf = 100 Sqrt@2 He EFastL ê m n D ê. Cons; ETh = 1.0; vth = 100 Sqrt@2 He EThL ê m n D ê. Cons; vroom = 100 vavg@t room D ê. Cons; dtofv@v_, ig_d := 1 ê H10 v HS s + S g + S f LL ê. data@@igdd; dtth = dtofv@vth, 2D; dtf = dtofv@vf, 1D; dsf = vf dtf; dsth = vth dtth; dpsf = S s dsf ê. Fastdata; dpsth = S s dsth ê. Thermaldata; dpff = S f dsf ê. Fastdata; dpfth = S f dsth ê. Thermaldata; dpgf = S g dsf ê. Fastdata; dpgth = S g dsth ê. Thermaldata; dpaf = dpff + dpgf; dpath = dpfth + dpgth; dpf = dpsf + dpaf; dpth = dpsth + dpath; Print@" Fast time step = ", dtf, " sec, and average speed of, v = ", vf, " cmês, distance per step = ", dsf, " cm"d Print@" Thermal time step = ", dtth,

8 8 Moderation.nb " sec, and average speed of, v = ", vth, " cmês, distance per step = ", dsth, " cm"d Print@" The probabilities of each possible event in one time step:"d Print@" Fast values: scatter = ", dpsf, " fission = ", dpff, " capture =", dpgfd; Print@" Thermal values: scatter = ", dpsth, " fission = ", dpfth, " capture =", dpgthd 8r Ø , n d Ø , S s Ø , S g Ø , S f Ø , n Ø 2.6< 8r Ø , n d Ø , S s Ø , S g Ø , S f Ø , n Ø 2.42< Fast time step = µ sec, and average speed of, v = µ 10 9 cmês, distance per step = cm Thermal time step = µ 10-7 sec, and average speed of, v = µ 10 6 cmês, distance per step = cm The probabilities of each possible event in one time step: Fast values: scatter = fission = capture = Thermal values: scatter = fission = capture =

9 Moderation.nb 9 ü Run the simulation: Nexp = number of "experiments" Nneutron= number of neutrons per experiment TimingBRAbsAvgTable = 8<; ktable = 8<; NScattTable = 8<; ElossTable = 8<; Rmax = 4 LMix; NRbins = 40; dr = Rmax ; NumDen = Table@0, 8i, 1, NRbins + 1<D; NRbins SsGroup = 8S s ê. datap1t, S s ê. datap2t<; SgGroup = 8S g ê. datap1t, S g ê. datap2t<; SfGroup = 8S f ê. datap1t, S f ê. datap2t<; ngroup = 8n ê. datap1t, n ê. datap2t<; Nexp = 10; Nneutrons = 200; NThTotal = 0; 0. DoBNFiss = 0; RAbsTable = 8<; NScattToTh = 0; NTh = 0; NFastScattTotal = 0; Eloss = 0; DoBForB8ig = 1, v = 80, 0, vf<; vmag = vf; dt = dtofv@vmag, igd; r = 80, 0, 0<; istep = 1; istop = -1; NFastScatt = 0<, istop < 0 && istep < 10 5, istep++, :ran = RandomReal@D; ds = vmag dtofv@vmag, igd; dps = ds SsGroupPigT; dpg = ds SgGroupPigT; dpf = ds SfGroupPigT; r = r + v dt; IfBran < dps + dpg + dpfh* does it interact? *L, H*YES, it interacts*l:ifbran < dps H* Is the interaction elastic? *L, H*YES, elastic *L:vmagsave = vmag; thcm = p RandomReal@D; phicm = 2 p RandomReal@D; v = vcm@vmagd 8Sin@thcmD Cos@phicmD, Sin@thcmD Sin@phicmD, Cos@thcmD< + vboost@vd; vmag = v.v ; IfBvmag vth H* Is the new speed thermal? *L,

10 10 Moderation.nb H* Is the new speed thermal? *L, H* yes, change to group 2, and keep speed constant from now on *L :ig = 2; v = vth v >, H* no, accumulate vmag fractional energy loss *L:Eloss = Eloss vmag 2 vmagsave 2 ; NFastScatt++>F;>, H* NO, inelastic *L8iStop = 1; If@ran < dps + dpfh* Was the inelastic event fission? *L, H* Yes, accumulate fission neutrons generated *LNFiss = NFiss + ngrouppigth* no, do nothing *LD< FH* End elastic scattering IF *L >FH* End interaction IF *L; >FH* End For *L; If@ig ã 2, NScattToTh = NScattToTh + NFastScatt; NTh ++D; NFastScattTotal = NFastScattTotal + NFastScatt; rmag = r.r ; AppendTo@RAbsTable, rmagd; rindex = FloorB rmag F + 1; If@rindex > NRbins, dr NumDenPNRbins + 1T ++, NumDenPrindexT ++D;, NFiss 8j, 1, Nneutrons<F; AppendToBkTable, Nneutrons F; LenRabs = Length@RAbsTableD; AppendToBRAbsAvgTable, i=1 Nneutrons RAbsTablePiT F; Nneutrons AppendToBNScattTable, NB NScattToTh FF; NTh

11 Moderation.nb 11 Eloss AppendToBElossTable, NB NFastScattTotal FF; NThTotal = NThTotal + NTh;, 8iexp, 1, Nexp<F; ElossAvg = Nexp iexp= , < ElossTablePiexpT F Nexp

12 12 Moderation.nb ü Calculate averages and deviations and plot the flux density H* k Avergage *L Lenk = Length@kTableD; kavg = N@Sum@kTable@@iDD, 8i, 1, Lenk<D ê LenkD; Delk = Sqrt@Sum@HkTable@@iDD - kavgl ^ 2, 8i, 1, Lenk<D ê LenkD; H* Average absorption radius *L LenR = Length@RAbsAvgTableD; RAbsAvg = N@Sum@RAbsAvgTable@@iDD, 8i, 1, LenR<D ê LenRD; DRAbs = Sqrt@N@Sum@HRAbsAvgTable@@iDD ^ 2 - RAbsAvg ^ 2L, 8i, 1, LenR<D ê LenRDD; H* Average number of scatterings to reach thermal speed *L LenNScatt = Length@NScattTableD; NScattAvg = Sum@NScattTable@@iDD, 8i, 1, LenNScatt<D ê LenNScatt; ProbSurviveToThermal = N@NThTotal ê Nneutrons ê NexpD; H* Print Monte Carlo results *L Print@" Neutron multiplication constant khmonte CarloL = ", kavg, " +ê- ", Delk, " khanalyticl = ", kcmixd Print@" Average radius value at absorption = ", RAbsAvg, " +ê_ ", DRAbs, " cm"d Print@" Average number of scatterings needed to reach thermal speeds = ", NScattAvgD Print@" Survival probability HMCL = ", ProbSurviveToThermal, " HanalyticL =", Prob ê. Cmix -> CmixMaxD

13 Moderation.nb 13 Neutron multiplication constant khmonte CarloL = ê khanalyticl = Average radius value at absorption = ê_ cm Average number of scatterings needed to reach thermal speeds = Survival probability HMCL = HanalyticL = NRbins norm = i=1 NumDenPiT; DenNormed = TableBNB NumDenPiT norm IShellVol@iR_D = dr 3 4 p ir 2 ;M PhiMCTable = TableB DenNormedPiT ShellVol@iD Phi@x_D = - x LD F, 8i, 1, NRbins<F;, 8i, 1, NRbins<F; x ; Inorm = 4 p IntegrateAPhi@xD x2, 8x, 0, <, Assumptions Ø Re@LDD > 0E;M Phi0@x_D = Phi@xD ê. LD Ø LMix; ; PhiMax = norm Max@PhiMCTableD; PhiAnalyticTable = Table@Phi0@Hi - 0.5`L drd, 8i, 1, NRbins<D; horizaxis = Style@"r HcmL", FontFamily Ø "Tahoma", FontColor Ø Blue, FontWeight Ø Bold, FontSize Ø 12D; vertaxis = Style@"j", FontFamily Ø "Tahoma", FontColor Ø Blue, FontWeight Ø Bold, FontSize Ø 12D; plotname = Style@"Neutron Flux", FontFamily Ø "Tahoma", FontColor Ø Black, FontWeight Ø Bold, FontSize Ø 14D; p1 = ListPlot@PhiMCTable, DisplayFunction Ø Identity, PlotStyle Ø 8RGBColor@1, 0, 0D, PointSize@0.015`D<, Frame Ø True, GridLines Ø Automatic, PlotLabel Ø plotname, FrameLabel Ø 8horizaxis, vertaxis<,

14 14 Moderation.nb plotname, FrameLabel Ø 8horizaxis, vertaxis<, ImageSize Ø 400, Background Ø LightOrange, PlotRange Ø PhiMax, 1.2 PhiMax<D; p2 = ListPlot@PhiAnalyticTable, Joined Ø True, DisplayFunction Ø Identity, PlotStyle Ø 8Black, Thickness@0.005`D<, Frame Ø True, GridLines Ø Automatic, PlotLabel Ø plotname, FrameLabel Ø 8horizaxis, vertaxis<, ImageSize Ø 400, Background Ø LightOrange, PlotRange Ø PhiMax, 1.2 PhiMax<D; Show@p1, p2, DisplayFunction Ø $DisplayFunctionD Neutron Flux j r HcmL

Elastic scattering. Elastic scattering

Elastic scattering. Elastic scattering Elastic scattering Now we have worked out how much energy is lost when a neutron is scattered through an angle, θ We would like to know how much energy, on average, is lost per collision In order to do