13: Diffusion in 2 Energy Groups

Transcription

1 3: Diffusion in Energy Groups B. Rouben McMster University Course EP 4D3/6D3 Nucler Rector Anlysis (Rector Physics) 5 Sept.-Dec. 5 September

2 Contents We study the diffusion eqution in two energy groups of neutrons, which is the methodology most often used for full-core diffusion clcultions (but not for lttice clcultions). We tret first the infinite lttice, then move on to the finite rector. 5 September

3 The Diffusion Eqution in Groups In multigroup theory it is conventionl to number groups from the highest energy rnge to the lowest energy rnge. Following this convention, in the -group model: Group is the fst (or slowing-down ) group Group is the therml group The energy boundry seprting the groups is typiclly ten s.65 ev. Subscripts nd refer to groups nd respectively, e.g., we will hve quntities,,,, D, D, etc. cont d 5 September 3

4 The Diffusion Eqution in Groups For now we will me the following simplifying ssumptions: All fission neutrons re born in the fst group All fissions re lumped s if creted by the therml group, i.e. we hve quntity f, but no f (i.e., f is ssumed to be renormlized to ccount for the few percent of fst fissions) We lso hve downscttering or modertion (i.e., group--to-group-) cross section, which we shll denote, but we me the ssumption tht upscttering is negligible, i.e. we te, nd ignore it. 5 September 4

5 The Eqution for the Infinite Lttice As we did in the -group tretment, we will strt the discussion with the infinite lttice, for which there re no lege terms. With the simplifying ssumptions in the previous slide, then, the time-independent diffusion eqution for the infinite lttice in groups is: f Eq. () [ set of equtions] expresses the neutron blnce: the bsorption (loss) terms re blnced by the production terms, in ech energy group. cont d 5 September 5 ()

6 Infinite-Lttice Eqution (cont d) However, this is set of homogeneous liner equtions in unnowns. Just s in the -group cse, there is not lwys nontrivil solution! In order tht there be non-trivil solution, the determinnt of the eqution set hs to be, i.e. f However, we cnnot expect this to be the cse in generl: physiclly speing, we cnnot throw together just ny ind of mteril nd expect it to constitute sttic rector (except one with neutron flux everywhere)! cont d 5 September 6 ()

7 Infinite-Lttice Eqution (cont d) Therefore, in order to ensure solution, we divide f by n djustble fctor, which will represent the infinite-lttice s multipliction constnt. The eqution set then becomes: f The criticlity criterion then becomes: f (4) cont d (3) 5 September 7

8 i.e., Infinite-Lttice Eqution (cont d) i. e., f f Eq. (5) cn be interpreted ccording to the neutron cycle (in the sme wy s we derived the 4-fctor formul) s follows: Suppose N fst neutrons re born from therml fissions These neutrons cn be bsorbed or be moderted (downscttered to the therml group). The number which succeed in being moderted is given by the rtio N These surviving neutrons will be bsorbed ccording to cross section, but some of these bsorptions (if in fuel) will result in fissions nd new numbers of neutrons, ccording to the yield cross section f. The rtio of number of neutrons in successive genertions is then seen to be exctly, s given by Eq. (5). (5) 5 September 8

9 Infinite-Lttice Eqution (cont d) In summry: In the -group formlism, the infinitelttice multipliction constnt is given by f which is to be compred to the -group result f It is esy to see tht the rtio is relly the resonnce escpe probbility p, since it is the frction of fst neutrons which survive to modertion: p cont d 5 September 9 (6) (5)

10 Infinite-Lttice Eqution (cont d) Then, by compring the rewritten Eq. (5) with the 4-fctor formul: we see tht we must hve f p fp f f This is ctully esy to understnd: Multiply Eq. (8) by nd rewrite it s f = f. This is very similr to the version of this reltionship tht we hd derived for totlly uniform medium: f =. The difference in the current cse is tht we distinguish between fuel nd other mteril, so we need the fctor f to give the therml bsorptions in fuel. And in the nlysis in this lerning module we hve essentilly set =. 5 September (7) (8) (5)'

11 5 September Eqution in Opertor Form Note tht our originl -group eqution for the infinite lttice, Eq. (), cn be written in mtrix form s: This cn be rewritten in opertor form s: (9) f ) ( (3) ) ( () () () opertor yield the opertor nd scttering bsorption the where f F M Φ FΦ MΦ

12 Eqution in Opertor Form (cont d) With the introduction of eff to ensure non-trivil solution, the finl eqution cn be written in more conventionl eigenvlueproblem form s MΦ FΦ where nd is the eff (4) (5) eigenvlue problem. Note tht if we integrte Eq. (4), we get s we hd shown previously eff FΦdr MΦdr of the Neutron production rte Neutron loss rte 5 September

13 Discussion/Exercise If we remove the simplifictions of ssuming tht: ) fst fissions re included in therml fissions, nd ) there is no upscttering, write the eqution for the bsorption nd scttering opertor M nd for the fission-source opertor F for the infinite lttice. Note: The dvntge of the opertor nottion is tht the bsic eqution (4) will be the sme no mtter wht exct model we use, i.e., M nd F my include different things, but the eqution remins the sme. 5 September 3

14 5 September 4 Include Fst Fission & Upscttering ) ( (3) ) ( opertor yield opertor nd scttering bsorption f f F M Φ FΦ MΦ The opertors in the more generl cse:

15 The Finite Rector We now move to the cse of the finite homogeneous rector, nd for now retin the simplifying ssumptions we hd ssumed for the infinite lttice For the finite lttice, we will hve to chnge to eff nd dd the lege terms. The lege out of the rector in ech group is D dv (This must be positive number for ech group why?) cont d 5 September 5

16 5 September 6 Finite-Rector Eqution in Groups The eqution for the finite homogeneous rector, with lege, is: [Note: This hs to be understood s pplying t every point in the rector.] As we did in the one-energy-group formlism, let s write the term s DB for ech group, with the sme bucling (B ) in the groups. [This mens tht the flux hs the sme shpe in the two groups.] (6) eff f D D D (7) eff f D B D B

17 Criticlity Criterion for Finite Rector in Groups The criterion for hving non-trivil solution to this set of homogeneous equtions is (s usul) tht the determinnt hve zero vlue: D B D B eff f (8) This reduces to i. e., D B D B eff D B f D B (9) 5 September 7 f eff

18 Significnce of eff The significnce of Eq. (9) is tht the expression on the right-hnd side of Eq. (9), which is n expression in the vlues of the mteril properties of the rector, must hve the vlue if the rector is to be criticl (nd llowed, therefore, to be time-independent rector). If the right-hnd side of Eq. (9) is not zero, then he properties of the rector do not llow timeindependent solution. However, eff tells us how fr off we re, nd it lso tells us tht if we chnge the rector to hve fission cross section f / eff, then tht rector will be criticl. 5 September 8

19 Criticlity Criterion for Finite Rector in Groups If we me use of the following definitions for the therml diffusion re nd the fst diffusion re [lso clled the neutron ge ] L L L D nd L D we get for the criticlity criterion eff f ( ) L B L B. () 5 September 9

20 Reltionship Between nd eff If we compre this to the criticlity criterion for the infinite lttice, which we obtined previously: f. () we see tht we hve the reltionship eff L B L B (3) Since the physicl difference between nd eff is the lege, the fctor multiplying is obviously relted to the lege question. In fct the next slides show tht this fctor is the totl non-lege probbility. 5 September

21 Fst Non-Lege Probbility Consider the fst group. The rection rtes for neutrons in this group re: Fst bsorption, Downscttering, Lege, D B So the non-lege probbility P is given by P ( 5 September D D B B ) D B L B

22 Therml Non-Lege Probbility Now consider the therml group. The rection rtes for neutrons in this group re: Therml bsorption, Lege, So the non-lege probbility P is given by 5 September B D B L B D B D B D P

23 6-fctor formul for eff Since for neutron not to le from the rector demnds tht it not le s fst neutron nd tht it not le s therml neutron, we see tht P P P represents the full non-lege probbility of neutrons from the rector. So the reltionship between nd eff becomes P P P eff *( Non Lege Probbility )(6) Incidentlly, if we substitute into Eq. (6) the 4-fctor formul for : fp Eq.(7) we get the 6-fctor formul for eff : nother importnt formul fp P P eff 7 5 September 3

24 Summry for -Group Model The criticlity criterion in energy groups for finite homogeneous rector reltes the bucling B to the mteril properties: eff f ( ) L B L B. Eq.() Regrding the flux shpe in the rector, this is given (for both groups) by the sme eigenvlue eqution s before: B (8) except tht in groups the vlue of the geometricl bucling B must be relted to the mteril properties s per Eq. (). (cont d) 5 September 4

25 Discussion/Exercise Derive the reltive mplitudes of the fst nd therml fluxes. 5 September 5

26 5 September 6 Reltive Amplitudes of Fluxes We cn mnipulte the blnce equtions (7) to determine the rtio of fluxes, Eq.(9) below: Either of the equtions in Eq. (7) cn just s esily be used to relte the fluxes s bove. (9).(7) eff f eff f D B or B D Eq B D D B

27 5 September 7 Modified -Group Criticlity Criterion Eq. () is the reltionship between nd eff is. A modified, pproximte reltionship cn sometimes be invoed, for lrge rectors where the bucling is sufficiently smll nd the lege term is usully smll few %. In the cse where B is smll, we cn neglect the term involving B *B, which will be even smller, nd we get: where 3.(). ) ( B M B L L Eq B L B L f eff f eff (3) Ares Diffusion Therml Fst L L M

28 Another Exercise For the finite homogeneous rector in 3 groups: MΦ FΦ Eq.(4) write the equtions for nd the opertors M nd F, if: ll fission neutrons re born in the fst group, but from fissions in ny group, nd there is scttering from ny group to ny other group. 5 September 8

29 Exercise The usul mthemticl intervention (which hs been used bove) for ensuring tht the rector eqution hs solution is to include n eigenvlue, (= / eff ), in front of the yield opertor F. Relte/contrst this intervention with the physicl, concrete steps which re or cn be used to ensure criticlity of rel rectors. 5 September 9

30 END 5 September 3