CH.III : APPROXIMATIONS OF THE TRANSPORT EQUATION

1. CH.III : APPROXIMATIONS OF THE TRANSPORT EQUATION ONE SPEED BOLTZMANN EQUATION • ONE SPEED TRANSPORT EQUATION • INTEGRAL FORM • RECIPROCITY THEOREM AND COROLLARIES DIFFUSION APPROXIMATION • CONTINUITY EQUATION • DIFFUSION EQUATION • BOUNDARY CONDITIONS • VALIDITY CONDITIONS • P1 APPROXIMATION IN ONE SPEED DIFFUSION • ONE SPEED SOLUTION OF THE DIFFUSION EQUATION MULTI-GROUP APPROXIMATION • ENERGY GROUPS • SOLUTION METHOD 1st–FLIGHT COLLISION PROBABILITIES METHODS

2. III.1 ONE SPEED BOLTZMANN EQUATION ONE SPEED TRANSPORT EQUATION  Suppressing the dependence on v in the Boltzmann eq.: Let : expectednb of secundary n/interaction, and : distribution of the scattering angle  (why?) (why?)

3. Development of the scattering angle distribution in Legendre polynomials: with and Weak anisotropy with

4. INTEGRAL FORM Isotropic scattering and source (see chap.II)  In the one speed case: with = transport kernel = solution for a point source in a purely absorbing media (Dimensions !!??)

5. S V RECIPROCITY THEOREM AND COROLLARIES with Proof +BC in vacuum - V dr (BC in vacuum!) 4 d

6. Corollary Isotropic source in Collision probabilities Set of homogeneous zones Vi Ptij : proba that 1 n appearing uniformly and isotropically in Vi will make a next collision in Vj Then Rem: applicable to the absorption (Paij) and 1st-flight collision proba’s (P1tij) Nb of n emitted in dro about ro (dimensions!!) Reaction rate in dr about r per n emittedatro

7. Escape probabilities Homogeneous region V with surface S Po : escape proba for 1 n appearing uniformly and isotropically in V o : absorption proba for 1 n incident uniformly and isotropically on S Rem: applicable to the collision and 1st-flight collision probas

8. III.2 DIFFUSION APPROXIMATION CONTINUITY EQUATION Objective: eliminate the dependence on the angular direction  Boltzmann eq. integrated on (see weak anisotropy): with  Angulardependencestillexplicitlypresent in the expression of the integratedcurrent(i.e. not a self-contained eq.in ) 4 d

9. DIFFUSION EQUATION Continuity eq.: integrated flux everywhere except for • Still 6 var. to consider! Objective of the diffusion approximation: eliminate the two angular variables to simplify the transport problem Postulated Fick’s law: with : diffusion coefficient [dimensions?] (comparison with other physical phenomena!) 

10. BOUNDARY CONDITIONS Reminder: BC in vacuum  angular dependence  not applicable in diffusion Integration of the continuity eq. on a small volume around a discontinuity (without superficial source): • Continuity of the normal comp. of the current: • Discontinuity of the normal derivative of the flux But continuity of the flux because  Continuity of the tangential derivative of the flux

11. External boundary: partial ingoing current vanishes Not directly deductible from Fick’s law (why?) Weak anisotropy 1st-order development of the flux in Expression of the partial currents with

12. Partial ingoing current vanishing at the boundary: Linear extrapolation of the flux outside the reactor • Nullity of the flux in : extrapolationdistance Simplification Use of the BC at the extrapoled boundary VALIDITY CONDITIONS Implicit assumption: D = material coefficient • m.f.p. < dimensions of the media  last collision occurred in the media considered  D : fct of this media only • Diffusion approximation questionable close to the boundaries • BC in vacuum! • Possible improvements (see below)

13. (link between cross sections and diffusion coefficient) P1 APPROXIMATION IN ONE SPEED DIFFUSION Anisotropy at 1st order (P1 approximation): In the one speed transport eq. 0-order angular momentum (one speed continuity eq.) 1st-order momentum Preliminary:

14. Consequently Reminder: Addition theorem for the Legendre polynomials:  Thus:

15. In 3D: with and Homogeneous material + isotropic sources • Fick’s law with Transport cross section:  Approximation of the diffusion coefficient: (without fission)

16. ONE-SPEED SOLUTION OF THE DIFFUSION EQUATION (WITHOUT FISSION) Infinite media Diffusion at cst v,  homogeneous media, point source in O Define Fourier transform:  Green function:  For a general source: Comparison with transport ?

17. Particular cases(see exercises) • Planar source • Spherical source • Cylindrical source As with Kn(u), In(u): modified Bessel fcts

18. Finite media • Allowance to be given to the BC! • Virtual sources method • Virtual superficial sources at the boundary (<0 to embody the leakages)  no modification of the actual problem • Media artificially extended till  • Intensity of the virtual sources s.t. BC satisfied • Physical solution limited to the finite media Examples on an infinite slab Centered planar source (slab of extrapolated thickness 2a) BC at the extrapolated boundary: Virtual sources:

19. Flux induced by the 3 sources: BC  Uniform source (slab of physical thickness 2a) Solution in  media (source of constant intensity): Diffusion BC: Solution in finite media: Accounting for the BC:

20. Diffusion length Let : diffusion length We have Planar source: • L = relaxation length Point source: use of the migration area (mean square distance to absorption)

21. III.3 MULTI-GROUP APPROXIMATION ENERGY GROUPS One speed simplification not realistic (E  [10-2,106] eV) • Discretization of the energy range in Ggroups: EG < … < Eg < … < Eo (Eo: fast n; EG: thermal n) • transport or diffusion eq. integrated on a group Flux in group g: Total cross section of group g: (reaction rate conserved) Diffusion coefficient for group gAND direction x ( possible loss of isotropy!) Isotropic case:

22. Transfer cross section between groups: Fission in group g: External source: • Multi-group diffusion equations Removal cross section:  = proba / u.l. that a n is removed from group g

23. If thermal n only in group G  sg’g = 0 if g’ > g SOLUTION METHOD Characteristic quantities of a group = f() usually • Multi-group equations = reformulation, not solution! • Basis for numerical schemes however (see below)

24. III.4 1st-FLIGHT COLLISION PROBABILITIES METHODS MULTI-GROUP APPROXIMATION Integral form of the transport equation Isotropic case with the energy variable:

25. ( ) Energy discretization Optical distance in group g: • Multi-grouptransport equations (isotropic case) with source: (compare with the integral form of the one speed Boltzmann eq.)

26. Multi-group approximation  Solve in each energy group a one speed Boltzmann equation with sources modified by scatterings coming from the previous groups (see convention in numbering the groups) • Within a group, problem amounts to studying 1st collisions • Iterative process to account for the other groups Remark Characteristics of each group = f() !!!  2nd (external) loop of iterations necessary to evaluate the neutronics parameters in each group

27. IMPLEMENTING THE FIRST-COLLISION PROBABILITIES METHOD Integral form of the one speed, isotropic transport equation where S contains the various sources, and Partition of the reactor in small volumes Vi: • homogeneous • on which the flux is constant (hyp. of flat flux)

28. Multiplying the Boltzmann eq. by t and integrating on Vi: Then, given the homogeneity of the volumes: Uniform source  : proba that 1 n unif. and isotr. emitted in Vi undergoes its 1st collision in Vj avec (+ flat flux)

29. How to apply the method? • Calculation of the 1st-flight collision probas (fct of the chosen partition geometry) • Evaluation of the average fluxes by solving the linear system above Reducing the nb of 1st-flight collision probas to estimate Conservation of probabilities Infinite reactor: Finite reactor in vacuum: with Pio: leakage proba outside the reactor without collision for 1 n appearing in Vi Finite reactor: with PiS: leakage proba through the external surface S of the reactor, without collision, for 1 n appearing in Vi

30. For the ingoing n: with • Sj : proba that 1 n appearing uniformly and isotropically across surface S undergoes its 1st collision in Vj • SS : proba that 1 n appearing uniformly and isotropically across surface S in the reactor escapes it without collision across S Reciprocity 1 Reciprocity 2

31. Partition of a reactor in an infinite and regular network of identical cells • Division of each cell in sub-volumes • 1st–flight collision proba from volume Vi to volume Vj: • Collision in the cell proper • Collision in an adjacent cell • Collision after crossing one cell • Collision after crossing two cells, … Second term: Dancoffeffect (interaction between cells)