The collision probability method in 1D part 2

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1 The collision probability method in 1D part 2 Alain Hébert alain.hebert@polymtl.ca Institut de génie nucléaire École Polytechnique de Montréal ENE611: Week 9 The collision probability method in 1D part 2 1/25

2 Content (week 9) 1 Calculation of 1D collision probabilities plane 2D geometry cylindrical 1D geometry spherical 1D geometry ENE611: Week 9 The collision probability method in 1D part 2 2/25

3 The collision probability method 1 Consider an infinite lattice of unit cells, each of them represented as i V i V i represents the infinite set of regions V i belonging to all the cells in the lattice The sources of secondary neutrons are uniform and equal to Q i on each region V i. The collision probabilityp ij,g is the probability for a neutron born uniformally and isotropically in any of the regions V i of the lattice to undergo its first collision in region V j of a unit cell or assembly. If the total cross section Σ(r) is constant and equal to Σ j in region V j, reduced CPs are (1) p ij = P ij Σ j = 1 4πV i V i d 3 r d 3 r e τ(s) V j s 2 where the optical path τ(s) is given by (2) τ(s) = s ds Σ(r s Ω). ENE611: Week 9 The collision probability method in 1D part 2 3/25

4 Plane 2D geometry 1 We will now study the case of a 2D geometry defined in the x y plane and homogeneous along the z axis. A cylinder or tube of infinite length is an example of such a geometry. In this case, it is possible to integrate analytically the integral transport equation along the axial angle θ. This angle is use to define the direction cosine of the particle relative to the z axis. z Ω r = (x,y,z) The direction of the particle is Ω = sinθ cosǫi+sinθ sinǫj +cosθk. Integration over θ is possible by taking the projection of each particle free path on the x y plane. We write θ y d 2 Ω = dθdǫ sinθ x ε (x,y) (3) τ(ρ) = τ(s) sinθ dρ = ds sinθ where ρ is the projection of s on the x y plane. ENE611: Week 9 The collision probability method in 1D part 2 4/25

5 Plane 2D geometry 2 We will limit ourself on the integration of the integral transport equation, defined on the domain of a unique unit cell. We use the relations r = r +sω and d 3 r = s 2 d 2 Ωds and obtain (4) p ij = 1 4πV i d 3 r V i 4π d 2 Ω dse τ(s) I j where the quantity I j is the set of points belonging simultaneously to the half straight line of origin r and direction Ω; to volume V j of the unit cell. With the help of figure, Eq. (4) can be rewritten as (5) p ij = 1 4πV i 2π dǫ d 2 r V i π dθ dρe I j τ(ρ) sinθ where V i now represents a surface in the x y plane and I j is the set of points belonging simultaneously to the half straight line of origin r and direction ǫ; to surface V j of the unit cell. ENE611: Week 9 The collision probability method in 1D part 2 5/25

6 Bickley functions 1 Bickley functions are defined as (6) with n 1. Ki n (x) = π/2 dθ sin n 1 θe x sinθ = π/2 dθ cos n 1 θe x cosθ Ki1(x) Ki2(x) Ki3(x) The following equations are relating Bickley functions of different orders: x x du Ki n (u) = Ki n+1 (x) Ki n+1 (x ),.5 Ki4(x) Ki5(x) and d dx Ki n(x) = Ki n 1 (x) x They can be evaluated efficiently using the matlab scriptakin. ENE611: Week 9 The collision probability method in 1D part 2 6/25

7 Plane 2D geometry 1 Equation (5) can be simplified by introducing the Bickley functions. We obtain (7) p ij = 1 2πV i 2π dǫ d 2 r dρki 1 [τ(ρ)]. V i I j Sets of integration lines, referred as tracks, are draw over the complete domain. Each set is characterized by a given angle ǫ and contains parallel tracks covering the domain. Tracks in a set can be separated by a constant distance h or can be placed at optimal locations, as depicted in sub-figures (a) and (b), respectively. Each track is used forward and backward, corresponding to angles ǫ and ǫ. Three-point formula h Three-point formula Three-point formula (a) Rectangular quadrature (b) Gaussian quadrature ENE611: Week 9 The collision probability method in 1D part 2 7/25

8 Plane 2D geometry 2 In the case of convex volumes i and j, Eq. (7) can be rewritten as (8) p ij = 1 2πV i 2π dǫ hmax h min dh lj dl dl Ki 1 (τ ij +Σ i l +Σ j l) if i j where τ ij is the optical path of the materials between regions i and j and (9) p ii = 1 2πV i 2π dǫ hmax h min dh dl l dl Ki 1 [Σ i (l l )]. y V i l l i l ij l j l' r V j r' h ε x ENE611: Week 9 The collision probability method in 1D part 2 8/25

9 Cylindrical 1D geometry 1 In the case of 1D cylindrical geometry, the tracks are identical for any value of the angle ǫ. track 1 track 2 The tubular volumes are concave, making extra terms to appear. The corresponding geometry is depicted in figure. Two tracks are represented, both contributing to collision probability components p ij with i < j and p ii. Track 1 corresponds to the integration domain where region i is concave and track 2 corresponds to the integration domain where it is convex. In this figure, τ ij and τ ii are the optical paths of the materials located between regions i and j or between two instances of region i. Vj Vi l j l j τij l i τii r j-1/2 r l i-1/2 i r i+1/2 l i l j l j r j+1/2 ENE611: Week 9 The collision probability method in 1D part 2 9/25

10 Cylindrical 1D geometry 2 In this case, Eq. (8) can be rewritten as (1) p ij { = 2 ri 1/2 dh V i + lj dl dl ] + Ki 1 (τ ij +Σ i l +Σ j l) ri+1/2 r i 1/2 dh [ Ki 1 [(τ ij +τ ii +Σ i l i )+Σ i l +Σ j l] } lj dl dl Ki 1 (τ ij +Σ i l +Σ j l) if i < j where r i±1/2 are the radii bounding region i and Eq. (9) can be rewritten as { p ii = 2 ri 1/2 dh V i (11) + ri+1/2 r i 1/2 dh dl dl [2 dl Ki 1 [Σ i (l l )]+ l } l dl Ki 1 [Σ i (l l )] dl Ki 1 [τ ii +Σ i (l +l)] where the first and second terms are the contributions from tracks 1 and 2, respectively. ] ENE611: Week 9 The collision probability method in 1D part 2 1/25

11 Cylindrical 1D geometry 3 These equations can be simplified by introducing the following definitions: (12) C ij (τ ) = lj dl dl Ki 1 (τ +Σ i l +Σ j l) and (13) so that (14) D i = dl dl Ki 1 [Σ i (l l )] l { p ij = 2 ri 1/2 dh[c ij (τ ij +τ ii +Σ i l i )+C ij (τ ij )]+ V i if i < j, and ri+1/2 r i 1/2 dhc ij (τ ij ) } (15) { p ii = 2 ri 1/2 dh[2d i +C ii (τ ii )]+ V i ri+1/2 r i 1/2 dhd i }. ENE611: Week 9 The collision probability method in 1D part 2 11/25

12 Cylindrical 1D geometry 4 Integration in l and l is next performed, leading to the following relations: a)σ i andσ j : 1 [ C ij (τ ) = Ki 3 (τ ) Ki 3 (τ +Σ i l i ) Ki 3 (τ +Σ j l j ) Σ i Σ j ] + Ki 3 (τ +Σ i l i +Σ j l j ) b)σ i = andσ j : c)σ i andσ j = : C ij (τ ) = l i Σ j [Ki 2 (τ ) Ki 2 (τ +Σ j l j )] C ij (τ ) = l j Σ i [Ki 2 (τ ) Ki 2 (τ +Σ i l i )] d)σ i = Σ j = : C ij (τ ) = l i l j Ki 1 (τ ) ENE611: Week 9 The collision probability method in 1D part 2 12/25

13 Cylindrical 1D geometry 5 e)σ i : f)σ i = : D i = l i Σ i 1 Σ 2 i [Ki 3 () Ki 3 (Σ i l i )] D i = πl2 i 4. ENE611: Week 9 The collision probability method in 1D part 2 13/25

14 Cylindrical 1D geometry 6 The above relations are implemented in Matlab using the following two scripts: function f=cij_f(tau,sigi,sigj,segmenti,segmentj) if sigi = && sigj = f=(akin(3,tau)-akin(3,tau+sigi*segmenti)-... akin(3,tau+sigj*segmentj)+... akin(3,tau+sigi*segmenti+sigj*segmentj))/(sigi*sigj) ; elseif sigi == && sigj = f=(akin(2,tau)-akin(2,tau+sigj*segmentj))*segmenti/sigj ; elseif sigi = && sigj == f=(akin(2,tau)-akin(2,tau+sigi*segmenti))*segmentj/sigi ; else f=akin(1,tau)*segmenti*segmentj ; end function f=di_f(sig,segment) if sig = f=segment/sig-(akin(3,)-akin(3,sig*segment))/sigˆ2 ; else f=pi*segmentˆ2/4 ; end ENE611: Week 9 The collision probability method in 1D part 2 14/25

15 Tracking 1 The tracking of 1D cylindrical and spherical geometries is similar and can be generated with the same algorithm. A matlab scriptf=sybt1d(rad,lgsph,ngauss) is presented to produce a tracking object in these cases. The figure presents an example with I = 2 regions and K = 2 tracks per regions. Tracks 1 and 2 have two segments, identified as l 1 and l 2. Tracks 3 and 4 have only one segment, identified as l φ l 2 3 l 2 l 2 l 1 l 1 l 2 h 1 h 3 h 2 h 4 h ENE611: Week 9 The collision probability method in 1D part 2 15/25

16 Tracking 2 A K point Gauss-Jacobi quadrature is used The four tracks are obtained using rad=[.75.15] ; track=sybt1d(rad,false,2) ; track(i,j) is a 2D cell array containing I K elements, one per track. The i index refers to the tracks located in r i 1/2 < h < r i+1/2. The (i,j) th cell is also a cell array containing three elements: the weight w k of the track, the product w k cosφ, a 1D array of dimension I i+1 containing the track segments. The l 1 segment length of track 2 is l2=track{1,2}{3}(1) leading tol2 = Any lengthtrack{ik,il}{3}(1) corresponding to a segment that cross the h axis must be multiplied by 2. 2 l 2 l 1 1 l 2 l 1 4 l 2 φ 3 l 2 h 1 h 3 h 2 h 4 h ENE611: Week 9 The collision probability method in 1D part 2 16/25

17 Tracking 3 The following script will perform a numerical integration of p 11 in 1D cylindrical tube of radius rad and macroscopic total cross sectionsigt: function f=p11_cyl(rad,sigt) % compute the p11 component in 1D cylindrical geometry track=sybt1d(rad,false,5) ; val= ; for i=1:5 segment=2*track{1,i}{3}(1) ; if sigt = d=segment/sigt-(akin(3,)-akin(3,sigt*segment))/sigtˆ2 ; else d=pi*segmentˆ2/4 ; end val=val+track{1,i}{1}*d ; end f=val/(pi*radˆ2) ; ENE611: Week 9 The collision probability method in 1D part 2 17/25

18 Boundary conditions 1 The external boundary condition of a 1D cylindrical geometry is introduced by choosing an albedo β + set to zero for representing a voided boundary, or set to one for representing reflection of particles. The voided boundary condition is similar to the vacuum condition used for 1D slab geometry. The reflecting boundary condition is implemented in the context of the Wigner-Seitz approximation for lattice calculations in reactor physics. The Wigner-Seitz approximation consists of replacing the exact boundary with an equivalent cylindrical boundary, taking care to conserve the amount of moderator present in the cell. The correct condition in this case is the white boundary condition in which the particles are reflected with an isotropic angular distribution. Fuel Fuel ENE611: Week 9 The collision probability method in 1D part 2 18/25

19 Boundary conditions 2 The application of a white boundary condition is based on the transformation of the reduced collision probability matrix P = {p ij, i = 1,I and j = 1,I} corresponding to a vacuum boundary condition. We first compute the escape probability vector P is = {P is, i = 1,I}, a column vector whose components are obtained from (16) I P is = 1 p ij Σ j j=1 representing the probability for a neutron born in region i to escape from the unit cell without collision. The probability for a neutron entering in the unit cell by its boundary, with an isotropic angular distribution, to first collide in region i is represented by the component P Si. It is possible to show that (17) where S is the surface of the boundary. p Si = P Si Σ i = 4V i S P is ENE611: Week 9 The collision probability method in 1D part 2 19/25

20 Boundary conditions 3 The transmission probability P SS is therefore given as (18) P SS = 1 I p Si Σ i. i=1 The reduced collision probability matrix P = { p ij, i = 1,I and j = 1,I} corresponding to a white boundary condition can now be obtained using P = P+P is [ β + +(β + ) 2 P SS +(β + ) 3 P 2 SS +(β+ ) 4 P 3 SS +...] p Sj or, using a geometric progression, as (19) P = P+ β + 1 β + P SS P is p Sj. The collision probability matrix P can be scattering-reduced and used in the matrix flux equation of the CP method. ENE611: Week 9 The collision probability method in 1D part 2 2/25

21 Spherical 1D geometry 1 Computing the tracking and collision probabilities in 1D spherical geometry is similar to the cylindrical geometry case. The tracking is obtained by thesybt1d Matlab script The expression of reduced collision probabilities is written (2) p ij = 1 d 3 r V i V i I j dse τ(s) ENE611: Week 9 The collision probability method in 1D part 2 21/25

22 Spherical 1D geometry 2 which can be rewritten as (21) p ij = 2π V i { ri 1/2 dhh ] + e (τ ij+σ i l +Σ j l) + ri+1/2 r i 1/2 dhh lj dl dl } lj dl dl e (τ ij+σ i l +Σ j l) [ e [(τ ij+τ ii +Σ i l i )+Σ i l +Σ j l] if i < j where r i±1/2 are the radii bounding region i and (22) p ii = 2π V i + { ri 1/2 dhh ri+1/2 r i 1/2 dhh dl dl [2 l dl e [Σ i (l l l dl e [Σ i (l l )] } )] + dl e [τ ii+σ i (l +l)] where the first and second terms are the contributions from tracks 1 and 2, respectively. ] ENE611: Week 9 The collision probability method in 1D part 2 22/25

23 Spherical 1D geometry 3 These equations can be simplified by introducing the following definitions: (23) C ij (τ ) = lj dl dl e (τ +Σ i l +Σ j l) and (24) so that p ij = 2π V i (25) D i = dl l dl e Σ i (l l { ri 1/2 dhh[c ij (τ ij +τ ii +Σ i l i )+C ij (τ ij )]+ ) ri+1/2 r i 1/2 dhhc ij (τ ij ) } if i < j and (26) p ii = 2π V i { ri 1/2 dhh[2d i +C ii (τ ii )]+ ri+1/2 r i 1/2 dhhd i }. ENE611: Week 9 The collision probability method in 1D part 2 23/25

24 Spherical 1D geometry 4 Integration in l and l is next performed, leading to the following relations: a)σ i andσ j : C ij (τ ) = 1 Σ i Σ j [ e τ e (τ +Σ i l i ) e (τ +Σ j l j ) +e (τ +Σ i l i +Σ j l j ) ] b)σ i = andσ j : c)σ i andσ j = : d)σ i = Σ j = : C ij (τ ) = l i Σ j [ e τ e (τ +Σ j l j ) ] C ij (τ ) = l j Σ i [ e τ e (τ +Σ i l i ) ] C ij (τ ) = l i l j e τ ENE611: Week 9 The collision probability method in 1D part 2 24/25

25 Spherical 1D geometry 5 e)σ i : f)σ i = : D i = l i Σ i 1 Σ 2 i [1 e Σ i l i ] D i = l2 i 2. ENE611: Week 9 The collision probability method in 1D part 2 25/25

The collision probability method in 1D part 1

The collision probability method in 1D part 1 Alain Hébert alain.hebert@polymtl.ca Institut de génie nucléaire École Polytechnique de Montréal ENE6101: Week 8 The collision probability method in 1D part