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Introduction to Reactor Physics and Analysis

by: Donato Hoeger Jr.

Introduction to Reactor Physics and Analysis PHGN 590

Marketplace > Colorado School of Mines > Engineering Physics > PHGN 590 > Introduction to Reactor Physics and Analysis
Donato Hoeger Jr.
Colorado School of Mines
GPA 3.94


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This 47 page Class Notes was uploaded by Donato Hoeger Jr. on Monday October 5, 2015. The Class Notes belongs to PHGN 590 at Colorado School of Mines taught by Staff in Fall. Since its upload, it has received 60 views. For similar materials see /class/219611/phgn-590-colorado-school-of-mines in Engineering Physics at Colorado School of Mines.

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Date Created: 10/05/15
PHGN590 Introduction to Nuclear Reactor Physics Modeling Neutron Slowing in Reactors J A McNeil Physics Department Colorado School of Mines 22009 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 macro scopic cross section for fission by a fast neutron is 0068 cm l while that for a thermal slow neutron is 284 cm l A moderator is a material that slows the neutrons down without absorbing them Graphite 12C is an excellent moderator It has a macro scopic scattering cross section of 038lcm 1and an absorption cross section of 000027 cm l 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 Analytic tasks a To illustrate this calculate km Eq8 for natural Uranium 235U 72 and 238U 9928 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 Constants Cons k3 gt 138066 gtr 10quot 23 Troom gt 29315 e gt 160219 gtlt10quot 19 mn gt 1674929gt10quot 27 Thermal neutron values U235Th p gt 01886 nd gt 04833 ZS gt 01588 27 4833 2f gt 2837 v gt242 U238Th p gt 0191 nd gt 04833 ZS gt 4301 21 13194 2f gt 0 v gt0 U235frac 0072 SigSNatUTh 2 Moderation nb U235frac ZS U235Th 1 U235frac ZS U238Th SigGNatUTh U235frac 239Y U235Th 1 U235frac 2 U238Th SigFNatUTh U235frac 2f U235Th 1 U235frac 2f U238Th kinfNatUTh v U235frac 2f U235Th v 1 U235frac 2f U238Th SigGNatUTh SigFNatUTh Printquot The thermal k factor for an infinite body of natural uranium is quot kinfNatUTh Fast neutron values C12data p gt 00160 nd gt 08023 ZS gt 3811 27 0002728 2f gt 0 v gt 0 U235Fast p gt 01886 nd gt 04833 ZS gt 328644 2 gt 0120825 2f gt 06766 v gt 26 U238Fast p gt 0191 nd gt 04833 ZS gt 33347 239Y gt 007732 2f gt 004591 v gt 26 SigSNatUFast U235frac ZS U235Fast 1 U235frac ZS U238Fast SigGNatUFast U235frac 239Y U235Fast 1 U235frac 239Y U238Fast SigFNatUFast U235frac 2f U235Fast 1 U235frac 2f U238Fast kinfNatUFast v U235frac 2f U235Fast v 1 U235frac 2f U238Fast SigGNatUFast SigFNatUFast Printquot The fast k factor for an infinite body of natural uranium is quot kinfNatUFast The thermal k factor for an infinite body of natural uranium is 133581 The fast k factor for an infinite body of natural uranium is 102411 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 12C nucleus Where the average of the cosine of the scattering angle is c How many collisions does it take to slow the neutron down to thermal energy 1 eV ModFac is the moderation factor fractional energy loss in an elastic scattering collision EF Energy of the fast neutrons 25 MeV ETh Energy when thermal cross sections are applicable 1 eV NumScatt number of elastic scatterings EThEF ModFacANumScatt NumScatt LogEThEFLog1 ModFac Avalue 12 EF 2 x10quot6 ETh 1 ModFac N2 Avalue 1 Avalue A2 NumScatt Floor Log ETh EF Log 1 ModFac Printquot For scattering from Carbon the fraction of neutron energy lost is quot ModFac Printquot Thus quot NumScatt quot scatterings are required to bring the neutron to thermal energiesquot For scattering from Carbon the fraction of neutron energy lost is 0142012 Thus 94 scatterings are required to bring the neutron to thermal energies 1 Now mix in graphite so that the number densities fractions are x of 12C and l x uranium natural Calculate the aggregate macroscopic cross sections for this mixture using the fast neutron values From these calculate the probability px that a neu tron 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 mac roscopic cross sections for the x mixture using the thermal values and calculate the critical factor kx given by V 2f V 2f 23961 Fast kxp lp 23961 Thermal 4 Moderation nb What is the Optimal mixing fraction yielding the greatest value for k C1earCmix SigSMixFast Cmix ZS C12data 1 Cmix SigSNatUFast SigGMixFast Cmix Z C12data 1 Cmix SigGNatUFast SigFMixFast 1 Cmix SigFNatUFast SigSMixFast Numscatt SigGMixFast SigSMixFast SigFMixFast 1 Prob v SigFMixFast lexFast U235Fast SlgGMleast SlgFMleast SigSMixTh Cmix ZS C12data 1 Cmix SigSNatUTh SigGMixTh Cmix 239Y C12data 1 Cmix SigGNatUTh SigFMixTh 1 Cmix SigFNatUTh Prob v SigFMixTh lexTh U235Th SigGMixTh SigFMixTh kinfNatU kMixTh kMixFast o I Prob PlotkinfNatU Cmix gt x x 0 1 CmixMax Cmix Flatten FindRoot amixkinfNatU Cmix 0 98 kaix kinfNatU Cmix gt CmixMax kMixFast Cmix gt CmixMax kMixTh Cmix gt CmixMax 1 SigSMixFast SigGMixFast SigFMixFast MFPFast Cmix gt CmixMax 1 SigSMixTh SigGMixTh SigFMixTh Cmix gt CmixMax MFPTh 1 AbsLenFast Cmix gt CmixMax SlgGMleast SlgFMleast Moderation nb 5 1 AbsLenTh Cmix gt CmixMax SigGMixTh SigFMixTh AbsLenFastMFPFast LFast f leX gt CmixMax AbsLenThMFPTh LTh f leX gt CmixMax LMix Prob LTh 1 Prob LFast Cmix gt CmixMax Printquot The probability of surviving to thermal speeds is quot Prob Cmix gt CmixMax Print quot The optimal mix of carbon is quot CmixMax quot yielding k quot kaix Printquot Absorption lengths AbsFast quot AbsLenFast quot AbsThermal quot AbsLenTh quot cmquot Printquot Mean Free Path lengths MFPFast quot MFPFast quot MFPThermal quot MFPTh quot cmquot Printquot Diffusion lengths LFast quot LFast quot LThermal quot LTh quot cm 0verscriptBoxL quot LMix 110 120 115 110 105 100 095 L 0 2 0 4 0 5 0 8 6 Moderation nb The probability of surviving to thermal speeds is 0834869 The optimal mix of carbon is 0963601 yielding k 120202 Absorption lengths AbsFast 137158 AbsThermal 728205 cm Mean Free Path lengths MFPFast 263093 MFPThermal 252202 cm Diffusion lengths LFast 346821 LThermal 782421 cm E 122593 Monte Carlo simulation f Develop a Monte Carlo simulation for this reactor configuration and calculate the neutron multiplication factor k for your optimal mix Moderation nb 7 I Set up the simulation by defining the time step in terms of the velocity and group cross sections C1earvcm vboost dPs dPg de dPa dP dt vmag This block can only be executed after the previous section Fastdata p gt 001886 nd gt 004833 ZS gt SigSMixFast 239Y gt SigGMixFast 2f gt SigFMixFast v gt 2 6 Cmix gt CmixMax Thermaldata p gt 001886 nd gt 004833 ZS gt SigSMixTh 239Y gt SigGMixTh 2f gt SigFMixTh v gt 2 42 Cmix gt CmixMax data Fastdata Thermaldata muavgA 2 3 A Avalue 12 vboost vmag vmag 1 Avalue boost velocity connecting CM and Lab frames vcmvmag Avalue vmag 1 Avalue velocity of neutron in CM frame EFast 25 10quot6 vF 100 Sqrt2 e EFast mn Cons ETh 10 vTh 100 Sqrt2 e ETh mn Cons vroom 100 vavg Troom Cons dtofvv ig 1 10 v ZS 239Y 2f dataig dtTh dtofvvTh 2 th dtofvvF 1 dsF vF th dsTh vTh dtTh dPsF ZS dsF Fastdata dPsTh ZS dsTh Thermaldata deF 2f dsF Fastdata deTh 2f dsTh Thermaldata dPgF 239Y dsF Fastdata dPgTh 239Y dsTh Thermaldata dPaF deF dPgF dPaTh deTh dPgTh dPF dPsF dPaF dPTh dPsTh dPaTh Printquot Fast time step quot th quot sec and average speed of v quot vF quot cms distance per step quot dsF quot cmquot Printquot Thermal time step quot dtTh 8 Moderation nb quot sec and average speed of v quot vTh quot cms distance per step quot dsTh quot cmquot Printquot The probabilities of each possible event in one time stepquot Printquot Fast values scatter quot dPsF quot fission quot deF quot capture quot dPgF Printquot Thermal values scatter quot dPsTh quot fission quot deTh quot capture quot dPgTh p9001886 nd 9004833 25 90379365 ZY 90000545448 Zf 90000183637 v 926 p9001886 nd 9004833 25 90382775 ZY 9000629738 Zf 9000743502 v9242 Fast time step l203gtlt1039M sec and average speed of v 218697gtlt109 cms distance per step 0263093 cm Thermal time step l82337gtlt10397 sec and average speed of v l383l6gtlt106 cms distance per step 0252202 cm The probabilities of each possible event in one time step Fast values scatter 00998082 fission 00000483135 capture 0000143503 Thermal values scatter 00965367 fission 000187513 capture 000158821 Moderation nb 9 Run the simulation Nexp number of quotexperimentsquot Nneutron number of neutrons per experiment TimingRAbsAngable kTable NScattTable ElossTable Rmax 4 LMix NRbins 40 Rmax dR NumDen Table0 1 1 NRbins 1 NRbins ZsGroup ZS data 1 ZS data 2 ZyGroup 2Y data 1 239Y data 2 ZfGroup 2f data 1 2f data 2 vGroup v data 1 v data 2 Nexp 10 Nneutrons 200 NThTotal 0 0 DoNFiss 0 RAbsTable NScattToTh 0 NTh 0 NFastScattTotal 0 Eloss 0 DoForig 1 v 0 0 VF vmag vF dt dtofvvmag ig r 0 0 0 iStep 1 iStop 1 NFastScatt 0 iStop lt o ampamp iStep lt 105 iStep ran RandomReal ds vmag dtofvvmag ig dPs ds ZsGroup ig dP ds ZyGroup ig de ds ZfGroup ig r r v dt If ran lt dPs dP de does it interact YES it interacts If ran lt dPs Is the interaction elastic YES elastic vmagsave vmag thcm 7r RandomReal phicm 2 7r RandomReal v vcmvmag Sin thcm Cos phicm Sinthcm Sin phicm Cos thcm vboostv vmag Vvv Ifvmag s vTh Is the new speed thermal 10 Moderation nb yes change to group 2 and keep speed constant from now on vThv ig 2 v vmag no accumulate fractional energy loss Eloss vmag2 Eloss 1 NFastScatt vmagsave2 N0 inelastic iStop 1 If ran lt dPs de Was the inelastic event fission Yes accumulate fission neutrons generated NFiss NFiss vGroupig no do nothing End elastic scattering IF End interaction IF End For If I NScattToTh NScattToTh NFastScatt NTh NFastScattTotal NFastScattTotal NFastScatt rmag rr AppendToRAbsTable rmag rmag rindex Floor 1 If rindex gt NRbins NumDenNRbins 1 NumDenrindex NFiss J 1 Nneutrons AppendTokTable Nneutrons LenRabs Length RAbsTable E futmns RAbsTable i AppendTo RAbsAngable Nneutrons NScattToTh 7 AppendTo NScattTable N NTh Moderation nb 11 Eloss AppendTo ElossTable N NFastScattTotal NThTotal NThTotal NTh iexp 1 Nexp Z i39Z ElossTable iexp ElossAvg Nexp 13741 0141855 1 2 Moderation nb Calculate averages and deviations and plot the flux density k Avergage LenkLengthkTable kAvg NSumkTablei i 1 LenR Lenk Delk SqrtSumkTablei kAvgA2 i1 LenkLenk Average absorption radius LenRLengthRAbsAngable RAbsAvg NSumRAbsAngablei i 1 LenR LenR ARAbs SqrtNSumRAbsAngableiA2 RAbsAng2 i 1LenRLenR Average number of scatterings to reach thermal speed LenNScattLengthNScattTable NScattAvg SumNScattTablei i 1 LenNScatt LenNScatt ProbSurviveToThermalNNThTotalNneutronsNexp Print Monte Carlo results Printquot Neutron multiplication constant kMonte Carlo quot kAvg quot quot Delk quot kanalytic quot kaix Printquot Average radius value at absorption quot RAbsAvg quot quot ARAbs quot cmquot Printquot Average number of scatterings needed to reach thermal speeds quot NScattAvg Printquot Survival probability MC quot ProbSurviveToThermal quot analytic Prob Cmix gt CmixMax Moderation nb 13 Neutron multiplication constant kMonte Carlo 118256 00921539 kanalytic 120202 Average radius value at absorption 888485 0310202 cm Average number of scatterings needed to reach thermal speeds 923824 Survival probability MC 08345 analytic 0834869 NRbins norm 2 NumDen i i1 NumDen i DenNormedTab1eN i 1 NRbins norun ShellVol iR dR3 4 7r 1112 DenNormed i i 1 NRbins ShellVoli 39 39 39 PhiMCTable Tab1e e39f Phix norm 47rIntegratePhix x2 x x 0 co Assumptions gt Re LD gt 0 Phix norm MaxPhiMCTab1e PhiAnalyticTable TablePhi0i 05 dR i 1 NRbins horizaxis Sty1equotr cmquot FontFamilyquotTahomaquot FontColoraBlue FontWeight aBold FontSize912 vertaxis Sty1equot quot FontFamily9quotTahomaquot FontColoraBlue FontWeight aBold FontSize912 plotname Sty1equotNeutron Fluxquot FontFamilyquotTahomaquot FontColoraBlack FontWeightaBold FontSize914 p1 ListPlotPhiMCTable DisplayFunctionaIdentity PlotStylea RGBColor1 0 0 PointSize0015 FrameaTrue GridLines aAutomatic PlotLabele plotname FrameLabelhorizaxis vertaxis Phi0x LDLMix PhiMax 14 Moderation nb ImageSize gt 400 Background gt LightOrange PlotRange gt 01 PhiMax 1 2 PhiMax p2 ListPlotPhiAnalyticTab1e Joined gt True DisplayFunction gt Identity PlotStyle gt Black Thickness 0 005 Frame gt True GridLines gt Automatic PlotLabel gt plotname FrameLabel gt horizaxis vertaxis ImageSize gt 400 Background gt LightOrange PlotRange gt 01 PhiMax 1 2 PhiMax Showp1 p2 DisplayFunction gt Disp1ayFunction Neutron Flux r cm Interaction of Radiation with Matter I Energetic charged particles BetheBloch stopping formula H Gamma radiation photoelectric compton pair production 11 neutrons elastic scattering photocapture ssion action of energetic charged particles Inter with matter 3 j e 3 Atoms of target 6 material Dominant energyloss mechanism IONiZATION Stopping Power average kinetic energy lost per unit distance in the target material dEdx Systematics of Stopping Power V 096 Mug Bowler average kinetic energy lost per unit distance in the target material dEdx BetheBloch Quantum calculation dE 41thntZpe dX mecz I 1 2m 02 2 x 21n eB391 3 IOU3932 Ztnt number of electrons per unit volume a Zp 2 charge number Z of projectile 32 v2c2 1 11KEmpc22 mac2 electron rest energy 511 MeV O O mpc2 projectile rest energy IO average energy of the knockedout electron In a 6 E cV Stopping power example Use the BetheBloch formula to calculate the stopping power of air for 5 MeV alphas air IO 16 2t 09 eV dE39 J lMoVm hplhohln I win BetheBloch Energy Loss Formula The BetheBloch formula for the magnitude of the ionization energy loss dded P1 2 n 2 62 hc2 Mec2 LOQEZ H862 beta ZIon 1beta 2beta2 1 2 beta2 Mec2 4 922 hc2 n Pi zpz u H w beta Mecz where n is the atomic number density of the target Ztnne the electron number density in units of number per cubic angstrom Mec2 is the rest energy of the electron ZtZp is the target and projectile Z betavc hc 19733 evAngstrcrus and on is the mean electron excitation energy which is well approximated by 6 Zquot0t9 eV This gives the stopping power in units of eVAngstrom dividing by 100 gives MeVrnicrom The formula does not apply for very low speeds betaltltli beMwbkwhnuz 2 Convert to a function of kinetic energy with the following relativistic substitution rule The constants are taken from the Review of Particle Properties dsdxxldzdxlbetagtSqrt111xERE2 coastloc2gt51110 692gt1137hcgt19733 const81Ztgt14ngt04978ongt16149 conItAnztgt79ngt05907Iongt16799 conItAqZtgt47ngt05857Iongt16479 consthirCZtgt72ngt5377105Iongt1672 9 protonRlgt938327Zpgt1 ddeprotonddeKEconstlprotonconstsiz plotprotonPlotdzdxprotonlooKE11O1 electronRE gt511Zpgt1z dxdxaloctrondzdxxEconsteloctronconst81 plotoloctronPlotdzdxelectronlooK355 P10tLabo1gt39Stopping Power of electrons in 8139 AxesLabelgt39Energy Mev3939dzdx MeVmicron39 alphaREgt49315zngt2 dldxnlphaddeKxconstalphalconst81 plotalphnPlotddealphalooKE310 1de MeV 39 cron Atoppfgb Pager of electrons in Si 000044 000043 000042 000041 5 Energy Kev batheme 3 show I 9101319111 plotproton Plotubolo IIStopping Pm in 81 Howlmicron quot I AxeLabel In Its quot 39dxdx luvmicron 391 1de MeVmicron 0 2 r Stopping Power in Si MeVmicron 4 39 Energ Graphics Mean ion depth Max100 Max4 1 max40 3 damm lnllnlx Do hulaNI 01dx39dldxproton ngtln 1 Maud In 3 29629 N 01dldxproton ltlgt1 INI 01dldauloctron ngt1 NI 01dldxalphl JCSgt1 Int Olidlduloctron Klgt1 107181 776 932 bedwbkmhmuz 4 multiplying by 100 converts dVAnq gt IbVcn lthO39dldxxlconntlalphaconstairKEgt5 0992732 NtdldxxlIOOIconstprotonlconltAgxEgt1 00954981 hb8qrt1111938327 2 compareal0001 dxdxlZtgt47ngt585710 222pgt102gt113 lac2gt511betagtbbIongt1610 6479 00461308 00954981 S 14 STOPPING MeV mgcmz 05 TOTAL 0 2 1 0 E E l 1 Alum Dmuz ION 1 97910 Mun0 y 149 TARGET 9 rump 11x by 2321 2 3 4 ENERGY MeVamu ldbbr su14 Nullm 2 SNOI v mum n91 um m7 m 315 um Interaction of gamma radiation with matter Basically an electromagnetic interaction with the atomic electrons Three possibilities 6 39Y Photoelectrlc Y Y Compton Scattering e y 6 Pair Production Attenuation coef cient initial intensity surviving intensity IO It W M W W m v M M W It 10 5M 10 e up 9 c u linear attenuation coef cient cm1 p mass density gcm3 ulp mass attenuation coef cient cm2g Radiation shielding example What thickness of lead shielding is necessary to reduce an initial ux of 06 MeV 1 s by a factor of 1 million Use Figure 316 of Meyerhof 39 393 55 F 1 l I 4 at 047 18 1quot 036 a Z Wquot 22 am 7 g I Fig3 104 Interaction of neutrons with matter Since neutrons do not interact electromagnetically they do not interact with the atomic electrons and therefore will interact with the nuclei of the target material Possibilities 1 Elastic scattering bounces off A n n A nucleus with no excitation A J J Iquot q 2 Bounces off but leaves the target nucleus in an excited state A Inm t 1quot 3 Be absorbed by the nucleus emitting either a gamma or an alpha A 1364 4 Be absorbed by the nucleus causing it to undergo ssion A I L Elastic scattering of neutrons by target nuclei Billiardball kinematics f n n pn 6 n A O A 9 Initial Final pA Conservation of energy and momentum gt 2 9 2 2 PH Pn P 1 2mn 2mn 2m A gt a gt pn p 1 1 pA Energymomentum conservation gives Maximum recoil energy L L I max 1 21 s KEreco mg KEneutron For heavy target nuclei the neutron loses almost no energy For light target nuclei the neutron can lose a signi cant fraction of its energy for a proton target the neutron can loose all of its energy through an elastic collision quot m dim quot PHGN590 Introduction to Nuclear Reactor Physics One Group Flux Profile J A McNeil Physics Department Colorado School of Mines 22009 2 Flux Pro le lgroup nb One velocity group I Parameters based on USGS Triga Geometry is given in inches and converted to cm geom RC gt 254 10 5 16 HC gt254lt15 RRef gt254 2158 HRef gt 254 15 2gtlt347 Print quot Reactor geometry Core Radius RC geom quot cm Core Height quot HC geom quot cmquot Print Reflector radius RRef geom quot cm Reflector Height quot HRef geom quot cmquot params v gt243 zf gt 0712 ZCa gt 0894 0712 ZRefa gt 00002728 ZCs gt 329 ZRefs gt 3811 uC gt 025 uRef gt 118 Ztr 2Ca 1 uC ZCs Aex 1 3 Ztr Rex RC Aex params geom Hex HC ZAex params geom D0 1 3 Ztr params geom Dc ZCa L0 params Flux Pro le lgroup nb 3 Reactor geometry Core Radius 261938 cm Core Height 381 cm Reflector radius 549275 cm Reflector Height 557276 cm v gt243 2f gt00712 2Ca gt01606 2Refa gt00002728 2Cs gt329 2Refs gt03811 HC 0025 pLRef gt00555556 I Bare Cylindrical Reactor The diffusion equation for the ux is DV2 4 2 a e For cylindrical geometry this factors into r and 2 components 5 Rr Zz Reqn iDrR39r Br2Rr r Zeqn Z 39 39 2 Bz2 Zz The boundary conditions are R390Z3900 and RReXZ Hex20 Which gives ZOsolnz A2 Cos Bz z ROSoln r Ar Besse1J0 Br r A2 Cos B2 2 Ar BesselJ 0 Br r Where Bz 7r Hex and Br 0 Rex Where 0 is the first zero of J0 4 Flux Pro le lgroup nb Ztr ZCa 1 uC ZCS Aex 1 3 Ztr Rex RC Aex params geom Hex HC 2 Aex params geom D0 1 3 Ztr params Dc L0 params ZCa Osoln 0 FindRootBesse1J0 0 0 0 25 Bconstants Br gt Osoln Rex Bz gt7tHex Ar gt 1 Az gt 1 Bnet Br2 322 Bconstants PNL 1 1BnetA2LcA2 012286 0990785 FuxProfie1 group nb 5 isolnr z ROsolnr ZOsolnz Bconstants Plot3D solnr z z 11Hex2 11Hex2 r O 11Rex 6 Flux Pro le lgroup nb I Reflected Cylindrical Reactor Ztr ZRefa 1 uRef ZRefs Aex 1 3 Ztr Rex RRef Aex params geom Hex HRef 2 Aex params geom DR 1 3 Ztr params L DR aramS R ZRefa p I Printquot Extrapolated radius quot Rex quot cmquot Printquot Extrapolated height quot Hex quot cmquot Print quot Reflector parameters DR quot DR quot cmquot Printquot LR quot ll 1 ll LR cm Extrapolated radius 558529 cm Extrapolated height 575784 cm Reflector parameters DR 092541 cm LR 582432 cm 1 The diffusion equations for the ux are DC V2 C vf Zac C DR V2 R ZaR R For cylindrical geometry this factors into r and 2 components I Rr Zz Flux Pro le lgroup nb 7 1 RCoreEqn D r RC 39 r r Br2 Rcr r RRfE 1d dR RL2 e n r r r 1 rdr dr R R RI d 2 ZCoreEqn Zcz 322 Zcz z d 2 2 ZRequn d zRz ZRz LR Z The boundary conditions are R390Z3900 and RReXZ Hex2 20 Which gives ZCorez ACz Cos Bz z ZRef z ARz Sinh z Hex 2 LR RCorer ACr Besse1J0 Br r RRefr ARr Besse1K0 Rex LR BesselI 0 r LR BesselI 0 Rex LR Besse1K0 r LR The solutions must be joined at their mutual boundaries given by the core radius and core height RCHC Joinz Dc ZCore 39 z ZCorez DR ZRef 39 z ZRef z z gtHC params geom PlotJoinZ 32 0 1 stoln Bz FindRootJoinZ 32 05 320 7rHex 8 Flux Pro le lgroup nb JoinR Dc RCore 39 r RCorer DR RRef 39 r RRef r r gtRC params geom PlotJoinR Br 0 2 Brsoln Br FindRootJoinR Br 08 Br0 2404 Rex Printquot Buckling constants B2 quot stoln quot cmquot1 BzO quot BzO quot cm391quot Printquot Br quot Brsoln quot cmquot1 BrO quot BrO quot cm391quot Buckling constants Bz 00423243 our1 320 0054562 cur1 Br 008511 cur1 Bro 00430416 cur1 ARrsoln RCoreRC geom RRef RC geom Br gt Brsoln ACr gt 1 ARr gt 1 Bconstants Br gt Brsoln Bz gt stoln ACr gt 1 ARr gt ARrsoln Flux Pro le lgroup nb 9 Rsolnr If r gt RC geom RRef r Bconstants RCorer Bconstants ROex Aex RC params geom BOr 2 404 R0ex ROSolnr Ifr lt ROex Besse1J0 BOr r 0 PlotROSolnr Rsolnr r 0 1 Rex params 10 FluxiProfie lgroup nb Astoln ZCore5 HC geom ZRef5 HC geom 32 gt stoln ACz gt 1 ARz gt 1 Bconstants 32 gt stoln ACz gt 1 ARz gtAstoln Zsolnz Ifz gt 5 HC geom ZRefz Bconstants ZCorez Bconstants HOex 2 Aex HC geom params 302 7r HOex ZOsolnz Ifz lt HOex 2 CosBOz z 0 P10tZOsolnz Zsolnz z 0 5 Hex geom 10 Flux Pro le lgroup nb 11 Bconstants Br gt Brsoln 32 gt stoln ACr gt 1 ARr gt ARrsoln 32 gt stoln ACz gt 1 ARz gt Astoln solnr z Rsolnr Zsolnz Bconstants params Hlimit 1 05 Hex 2 geom params Rlimit 6 Rex geom params Plot3D solnr z z H1imit Hlimit r 0 R1imit gr0up nb Flux Profile 1 1239 PHG N590 Introduction to Nuclear Reactor Physics Modeling Xe Poisoning J A McNeil Physics Department Colorado School of Mines 2 O9 Governing equation for fission fragment and daughter build up while under power ignore burnup term since we dont39 have the flux profile n1t in 1 Exp ll t1 eql M2 t 11 in t 42 n2 1 yzfis N239t 1 1 len yzfis 12 N2t N2501nt SimplifyN2 t F1attenDSolveeq1 N2 0 0 N2 t t l e tAZ 12 11 12 leAl 1 et1211 e 2 etlt11AZgt12 1 wt y 11 12 Zfis SecHour 3600 71 06 yxe 00237 1 Log2 657 Sealtour Axa Log2 91 Sealtour RxPower 930x10quot6 W Xi 0831 mix 254 9 9 16 mix 15 gt4254V01 pi RRx Z LRx avg RxPower 200 x16 x10quot 13 2f Vol tstop 8 Sealtour Ifinal 71 RxPower 200 x 16 x10quot 13AI Printquot Average flux quot avg quot ncmA25quot Printquot III final quot Ifinal xat 2501nt 1 411 2 AAXe lillf gt lilIfinal y gt yxe Efis gt 2f i gt avg III t I t 1 411 2 AAXe if gt lilIfinal y gt yxe Efis gt 2f i gt avg 2 XePoison nb Average flux 4195275gtlt1012 ncm 2s NI final 595013x1019 Governing equation for Xe after shut down lSDt ulsno Exp ll t eqSD nzsn t 11 lSDt 42 nzsnm ust t 1 NlSDOAl 12 NZSDt NZSDsolnt SimplifyNZSDt F1atten DSolveeqSD NZSD0 NZSDO NZSDt t l e 1 1 et Al12gt NlSDO Al Al 12 et 12 NZSDO Al 12 params lSDO gt Itstop anDt 25Dsolnt pa nzsno gt nxetstop 1 all 12 alxe ra s NISDt lSDt params Aggregate fragment and daughter history xaPlott Ift lt tstop llxat NanDH tstop IPlott Ift lt tstop NIH1 ulsnu tstop 1 Printquot A I quot1 quot sA l A xe quot Axa quot sA l Printquot Number of Xe at shut down I quot Xe tstop Printquot Number of I at shut down quot Itstop III asymptotic quot Ifinal horizaxis StyleFomquott hours quot FontFamily gt quotTahomaquot FontColor gt Blue FontWeight gt Bold Fontsize gt 12 vertaxis StyleFomquot umber 10quot20 quot FontFamilya quotTahomaquot FontColor gt Blue FontWeight gt Bold Fontsize gt 12 plotname StyleFomquot Iodine red and Xe blue Buildup and Decayquot Family gt quotTahomaquot FontColor gt Black FontWeight gt Bold Fontsize gt 14 Plot NIPlotHz Secttour lO quot 20 xaPlotH Secttour lO quot 20 t Plotstyle gt Red Thiekness 005 Blue Thiekness 005 Frame gt It rue GridLines gt Automatic PlotLabel gt plotna FrameLabel gt horizaxis vertaxis Imagesize gt 600 Background gt Light lellow PlotRange gt All AI 00000293061 sA l Number of Xe at shut down Number of I at shut down Number 10quot20 XePoison nb 3 AXe 00000211583 SA l NI asymptotic 595013x1019 132683x1019 339169x1019 Iodine red and Xe blue Buildup and Decay 1 hours 4 XePoison nb Xe reactivity data versus theory scaled Cleart tstart 1308 p0 010 m 010 B 007 XeData 0 0107p0 10 011 p0 20 0217p0 30 0237p0 40 0347p0 50 0467p0 60 0557p0 70 067 p0 24260 1217p0 25 360 114 p0 26460 1os7p0 27660 1097p0 28360 1037p0 29660 100 p0 30 360 0997 3112 60 0947p0 32 089 p0 33 oss7p0 dataLengthXeData scale 64x10quot 19 plot 100 at XeData data 1 plot GXe26x10quot6lt10quot 24 2xet o39XelWIXePlott Vol 2at 2 2m2xet pxet zxem B 2at Xe39I h It ablei dt scale XePlotidt SecHour i 1 plot Xe39I h It ablei dt 22 pXei dt SecHour i 1 plot horizaxis StyleFom IIt hours quot Fontt amily gt quotTahomaquot FontColor gt Blue FontWeight gt Bold Fontsize gt 12 vertaxis StyleFormquot Reactivity S quot Fontt amilya quotTahomaquot Fon Co or gt Blue FontWeight gt Bold Fontsize gt 12 plotname StyleFormquotXe Poisoningquot FontFamilya quotTahomaquot FontColor gt Black FontWeight gt Bold Fontsize gt 14 p39I h ListPlotXe39I h PlotMarkers gt Automatic Frame gt It rue Background gt LightYellow PlotLabel gt plotname FrameLabel gt horizaxis vertaxis Imagesize gt 500 PlotRange gt All pData ListPlot XeData PlotMarkers gt Automatic Plotstyle gt Black Showp391 h pData Xe lquot a 7 W W W W W W W W W W W W W W 12 7 O 7 O 10 7 0 0 7 o A 3 08 7 r E i o i a 7 7 0 057 0 7 N a 7 o o a W W W W W W W W W W W W W W W W W W W W W W W W W W W 7 1o 15 20 25 30 t hours


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