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

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

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

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

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

Collision probabilities 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 emitted at ro

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

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

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!) 

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

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

VALIDITY CONDITIONS Partial ingoing current vanishing at the boundary: Linear extrapolation of the flux outside the reactor Nullity of the flux in : extrapolation distance 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)

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: (link between cross sections and diffusion coefficient)

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

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

Comparison with transport ? 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 ?

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

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:

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:

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

III.3 MULTI-GROUP APPROXIMATION ENERGY GROUPS One speed simplification not realistic (E  [10-2,106] eV) Discretization of the energy range in G groups: 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 g AND direction x ( possible loss of isotropy!) Isotropic case:

Multi-group diffusion equations 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

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

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

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

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

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)

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)

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

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

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: Dancoff effect (interaction between cells)

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    