Chemical Engineering 412
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1 Chemical Engineeing 41 Intoductoy Nuclea Engineeing Lectue 16 Nuclea eacto Theoy III Neuton Tanspot 1
2 One-goup eacto Equation Mono-enegetic neutons (Neuton Balance) DD φφ aa φφ + ss 1 vv vv is neuton speed Fo eacto, ss νν ff φφ νν is neutons/fission In eigenfunction fom and at steady state DD φφ aa φφ + νν kk ffφφ 0 φφ aa νν kk f DD φφ φφ + BB φφ 0
3 Mateial Buckling φφ BB φφ φφ + BB φφ 0 νν BB kk ff aa DD kk νν ff φφ DDBB φφ + aa φφ eigenfunction fom of eacto equation one-goup, steady-state, eacto equation mateial buckling (neuton geneation absoption)/diffusion νν ff DDBB + aa eacto multiplication facto multiplication facto neuton geneation ate/(leakage + absoption)
4 Fuel utilization and k ss ηη aaaa φφ ηη aaaa aa aa φφ ηηηη aa φφ ff aaaa aa kk ηηηη aaφφ aa φφ ff fuel utilization facto neutons absobed by fuel / (those absobed by fuel + by othe means coolant, modeato, etc.) ηηηη kk k-value fo infinite (no oveall leakage) eacto a mateial popety η fission neutons geneated pe absobed neuton
5 Opeating Citical eacto Equation kk 1 DDBB φφ aa φφ + kk kk aaφφ 1 vv DDBB φφ + kk 1 aa φφ 0 eacto opeating at steady state BB φφ + kk 1 aa DD φφ BB φφ + kk 1 LL φφ 0 LL DD aa One-goup diffusion aea BB kk 1 LL One-goup buckling
6 Pespective Pevious equations show how to solve fo neuton flux pofile φφ as a function of space How to detemine citical eacto dimensions BB kk 11 LL kk 11 aa, fo a bae eacto (B DD gb mat) Fist find solutions to the eacto equations 1D, D, o 3D Then find dimensions fo a citical eacto Assumptions: Bae, homogeneous eactos Constant (special and tempoal) popeties None ae valid but, but help to develop insight into eacto opeations Because souce tems ae popotional to the flux, the geneally inhomogeneous diffeential equations ae now homogeneous equations.
7 Bae Slab eacto Solution dd φφ ddxx BB φφ eacto equation φφ aa φφ aa φφ 0 0 bounday conditions φφ xx AA cos BBBB + CC sin BBBB CC 0 φφ xx AA cos BBBB φφ aa/ 0 BB nn nnnn aa dd φφ geneal solution fom symmety o by substitution Eigenvalues fom bounday conditions all n impotant in tansient solution, only n1 impotant fo steady solution BB 1 is buckling (pop. to flux cuvatue) BB 1 1 φφ ddxx The constant A is as yet undetemined and elates to the powe. Thee ae diffeent solutions to this poblem fo evey powe level. aa aa Infinite plane indicates no net flux fom sides
8 PP EE ff Bae Slab eacto Powe aa/ aa/ PP AA aaee ff ππ φφ xx φφ xx dddd sin ππππ aa ππππ aaee ff sin ππππ aa cos ππππ aa EE is the ecoveable enegy pe fission Powe Scales with flux! aa aa Infinite plane indicates no net flux fom sides
9 Absobe/emitte vs eacto 1 LL φφ dd φφ ddxx ss DD dd φφ ddxx BB φφ tanspot in an absobe/emitte tanspot in a eacto souce popotional to flux φφ xx ss 1 cosh xx LL aa + dd aa cosh LL ππππ φφ xx aaee ff sin ππππ cos ππππ aa aa flux in an absobe/emitte flux in a eacto
10 1 dd dddd dddd dddd Spheical eacto BB φφ φφ φφ 0 0 φφ AA sin(bbbb) + C cos(bbbb) CC 0 sin BBBB φφ AA BB nn nnnn eacto tanspot equation bounday conditions fom symmety o by substitution Eigen values specific solution geneal solution BB 1 ππ buckling φφ AA sin ππ
11 Spheical eacto Powe Integate ove symmetic dimensions tansfom volume integal to adial integal PP EE ff φφ dddd 4ππEE ff PP 4ππEE ff AA ππ ππ sin ππππ 0 cos ππππ φφ dd again, powe is popotional to flux and highest at cente φφ PP sin ππππ 4 EE ff
12 Infinite Cylindical eacto 1 dd dddd dddd dddd BB φφ dd φφ dd + 1 dddd dddd eacto tanspot equation φφ φφ 0 0; φφ < bounday conditions dd φφ dd + 1 dddd dddd + BB mm φφ 0 φφ AAJJ 0 BBBB + CCYY 0 BBBB φφ AAJJ 0 BBBB BB nn xx nn BB 1 xx 1 φφ AAJJ zeo-ode (m0) Bessel equation geneal solution involves Bessel functions of fist and second kind flux is finite oots of Bessel functions - φφ is zeo at bounday fist oot solution (powe poduction detemines A)
13 Bessel Functions J0 J1 J Y0 Y1 Y Bessel Function x
14 Infinite Cylindical eacto Powe PP EE ff φφ dddd ππee ff.405 PP ππee ff JJ 0 0 xxxjj 0 xx ddxxx xxjj 1 xx 0 PP ππee ff AAJJ φφ 0.738PP EE ff JJ dddd 0 dddd tansfom volume integal to adial integal becomes powe pe unit length again, powe is popotional to powe and highest at cente
15 1 Finite Cylindical eacto + φφ zz BB φφ eacto tanspot equation φφ, zz φφ 0, zz φφ, HH φφ, HH 0 φφ, zz ZZ(zz) φφ zz ZZ + 1 ZZ ZZ BB zz sepaation of vaiables bounday conditions + ZZ zz BB ZZ(zz) since and ZZ vay independently, both potions of the equation must equal (geneally diffeent) constants, designated as BB and BB ZZ, espectively HH HH
16 1 BB.405 AA JJ 0 ZZ zz BB ZZ ZZ Finite Cylinde Solution a poblem we aleady solved, w/ same bcs again a poblem we aleady solved, w/ same bcs ZZ zz AA cos ππππ HH φφ, zz AA JJ BB BB + BB HH cos ππππ HH solution is the poduct of the infinite cylinde and infinite slab solutions Buckling is highe than fo eithe the infinite plane o the infinite cylinde. Buckling geneally inceases with inceasing leakage, and thee ae moe sufaces to leak hee than eithe of the infinite cases. HH HH
17 Neuton Flux Contous Neuton flux in finite cylindical eacto 3D contous of neuton flux at high powe 3D contous w colo scaled to magnitude intemediate powe 3D contous of neuton flux at low powe
18 Neuton Flux Contous Neuton flux in finite paallelepiped (cubical) eacto 3D contous of neuton flux at high powe 3D contous w colo scaled to magnitude intemediate powe 3D contous of neuton flux at low powe
19 Citical Buckling kk νν ff aa + DDBB value of k fo citical eacto BB νν ff aa DD BB cc νν ff aa DD BB ll kk 1 LL value of B when k 1 citical mateial buckling geometic buckling νν ff aa DD kk 1 LL geometic and mateial buckling must be equal fo a citical eacto
20 Citical Equation (One goup) kk 1 + BB LL 1
21 Physical Intepetation k 1+ B k φdv a a L a + DB a φdv + DB φdv k k P L The pobability of non-leakage is invesely popotional to k k P k L a k a φdv φdv P L ηfp L geometic and mateial buckling must be equal fo a citical eacto
22 a f Themal eactos Fou Facto Fomula af a af + am af af + total coss section sum of fuel and modeato am themal fuel utilization facto η T η ( E) σ ( E) φ( E) σ af af ( E) φ( E) de de neutons emitted/themal neuton absobed in the fuel k pεηt f φ a T a φ T pεη f T infinite multiplication facto popotional to pobability of escaping esonance absoption, fast fission contibution,
23 Citicality Calculations T T ( 1 ) ( + B L 1+ B τ ) T T 1/ pob themal neuton1/ pob fast neuton doesn' t leak doesn' t leak while slowing L τ k T D D P P 1 1 F a k 1 ( )( 1 1 ) 1 ( + B L + B τ + B L + τ ) T k k T k T T Two-goup equation fo a bae (themal) eacto Modified one-goup citical equation k 1+ B M T 1
24 Bae eacto Summay / cos 3.64 / 3.63 cos.405 J / J / 3.85 cos cos cos / 1.57 cos f f f f f E P A sphee VE P H z A H z D cylinde E P A D cylinde VE P c z b y a x A c b a D plate ae P a x A a D plate π π π π π π π π π π π geomety Buckling (B ) Flux A φ av Ω φ max
25 eflected eactos L A L C L A B A B C B A L L k B B c c c c c exp ' exp ' exp ' sin cos sin φ φ φ φ φ φ φ φ coe tanspot equation coe mateials popeties eflecto tanspot equation geneal solution fo coe flux must be finite at the cente geneal solution fo the eflecto flux must be finite as inceases
26 φc J c D A D ( ) φ ( ) ( ) n J ( ) n φ c ( ) Dφ ( ) sinb exp( / L) c AD c c A B cosb B cot B 1 D B cot B 1 D 1 B cot B L sinb D c eflected eactos L A' D 1 L fluxes equal at coe-eflecto inteface cuent densities also equal equate fluxes and cuent densities 1 L 1 + exp L divide cuent density by flux equation citical equation fo eflected eacto (tanscendental equation) citical equation when D D c (not tanscendental in )
27 eflected eactos < Bae eactos B cot B D Dc L
28 Detemine emaining Unknown ( ) ( ) B B B E PB A B B B B A E d B A E P dv E P B L A A f f f c f cos sin 4 cos sin 4 sin 4 sin exp ' 0 0 π π π φ
29 Some details eflected eactos lend themselves less easily to analytical solution commonly eactos ae consideed as sphee equivalents athe than tying to solve the equations. easonable epesentation fo fast neutons not fo themal eactos eflecto savings in size is typically about the thickness of the extapolated distance.
30 Flux Compaisons
31 Themal Flux Vaiations
32 Two-enegy, detailed model
33 Position 1 (cente of od) fast themal
34 Position (outside od at :30/7:30) fast themal
35 In a Vacuum fast themal
36 Azimuthal Dependence fast themal
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