The collision probability method in 1D part 1

Transcription

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 1 1/28

2 Content (week 8) 1 The collision probability method Scattering reduction and power iteration The Matlab scriptaleig Calculation of 1D collision probabilities slab geometry ENE6101: Week 8 The collision probability method in 1D part 1 2/28

3 The collision probability method 1 The collision probability (CP) method results from the spatial discretization of the integral transport equation in multigroup form, assuming isotropic particle sources. For a problem with I regions, produces a I I matrix in each energy group. Preferred for treating general unstructured meshes and few-region problems. The integral infinite-domain form of the transport equation is (1) φ g (r,ω) = 0 ds e τ g(s) Q g (r sω,ω) Integrating over the solid angles, we directly obtain the integrated flux φ g (r): (2) φ g (r) = 4π d 2 Ω φ g (r,ω) = 1 4π where the optical path τ g (s) is given by 4π d 2 Ω 0 ds e τ g(s) Q g (r sω) (3) τ g (s) = s 0 ds Σ g (r s Ω). ENE6101: Week 8 The collision probability method in 1D part 1 3/28

4 The collision probability method 2 r' s r Ω We now introduce the change of variable r = r sω with d 3 r = s 2 d 2 Ω ds. We obtain (4) φ g (r) = 1 4π d 3 r e τ g(s) s 2 Q g (r ) with s = r r. 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,g on each region V i. After multiplication by Σ g (r) and integration over each region V i, Eq. (4) can be written (5) V j d 3 rσ g (r)φ g (r) = 1 4π V j d 3 r Σ g (r) i Q i,g V i d 3 r e τ g(s) s 2 ENE6101: Week 8 The collision probability method in 1D part 1 4/28

5 The collision probability method 3 The neutron source in Eq. (5) is defined as (6) Q i,g = h Σ s0,i,g h φ i,h + 1 K eff Q fiss i,g. The fission source in Eq. (6) is defined as (7) fiss J Q fiss i,g = j=1 χ j,g νσ f,j,h φ i,h h where j is a fissionable isotope index, χ j,g is the fission spectrum of isotope j and Σ f,j,h is the macroscopic fission cross section of isotope j for neutrons in group h. Equation (5) simplifies to (8) V j Σ j,g φ j,g = i Q i,g V i P ij,g ENE6101: Week 8 The collision probability method in 1D part 1 5/28

6 The collision probability method 4 where (9) φ j,g = 1 V j V j d 3 rφ g (r), (10) Σ j,g = 1 V j φ j,g V j d 3 rσ g (r)φ g (r) and (11) P ij,g = 1 4πV i V i d 3 r V j d 3 rσ g (r) e τg(s) s 2. 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 g (r) is constant and equal to Σ j,g in region V j, reduced CPs are (12) p ij,g = P ij,g Σ j,g = 1 4πV i V i d 3 r V j d 3 r e τg(s) s 2. ENE6101: Week 8 The collision probability method in 1D part 1 6/28

7 The collision probability method 5 Reduced CPs generally remain finite in the limit where Σ j,g tends to zero. This ensures the correct behaviour of the collision probability theory in cases where some regions of the lattice are voided. Other interesting properties of CPs are the reciprocity and conservation properties, which can be written (13) and (14) p ij,g V i = p ji,g V j p ij,g Σ j,g = 1 ; i. j Using the reciprocity property, Eq. (8) can be further simplified to (15) φ i,g = j Q j,g p ij,g. ENE6101: Week 8 The collision probability method in 1D part 1 7/28

8 The collision probability method 6 The CP method generally proceeds in three steps: 1. A tracking process is applied over the lattice geometry to span a sufficiently large number of neutron trajectories. In a 2D domain, the tracking parameters are the number of azimuthal angles and the number of parallel tracks per centimetre. The tracking is generally independent of the energy group. 2. A numerical integration process is required to compute the CPs, using tracking information and knowledge of the macroscopic total cross sections in each region. This integration should be repeated for each energy group. 3. Once the CPs are known, the integrated flux can be computed from Eqs. (6) and (15). Collision probability techniques can also be applied to the case of a domain D surrounded by a surface D. The free path lengths are restricted to finite lengths defined inside D. Infinite lattices can still be described by incorporating reflective or periodic boundary conditions over D. ENE6101: Week 8 The collision probability method in 1D part 1 8/28

9 Power iteration 1 The within-group scattering term is first included in the left-hand side of Eq. (15) to obtain (16) φ i,g j p ij,g Σ s0,j,g g φ j,g = j Q j,g p ij,g where Q i,g includes the fission sources and the diffusion sources from all groups except group g. It is obtained from Eq. (6) as (17) Q i,g = h g Σ s0,i,g h φ i,h + 1 K eff Q fiss i,g. Equation (16) can be written in matrix form as (18) Φ g = W g Q g where Φ g = {φ i,g ; i} and Q g = {Q i,g ; i}. ENE6101: Week 8 The collision probability method in 1D part 1 9/28

10 Power iteration 2 The W g matrix is the scattering-reduced collision probability matrix. It is defined as (19) W g = [I P g S s0,g g ] 1 P g where I is the identity matrix, P g = {p ij,g ; i and j} and S s0,g g = diag{σ s0,i,g g ; i}. Two iterative processes are generally superimposed on these 1 G flux solution methods: 1. The inner process iterates over the diffusion up-scattering sources until convergence of the thermal flux. This iteration process is accelerated using two different techniques. The rebalancing technique produces a group-dependent factor which restores the exact multigroup flux distribution homogenized over all regions. The variational acceleration technique. 2. The outer (or power) iteration process is over the eigenvalue and is not required for fixed source problems. It is generally not accelerated in a lattice code and consists in computing neutron flux at outer iteration k +1 from source at iteration k, using (20) Φ (k+1) g = W g Q (k) g The critical parameter (K eff ) is then adjusted at the end of each power iteration. ENE6101: Week 8 The collision probability method in 1D part 1 10/28

11 The Matlab scriptaleig 1 The numerical solution of an eigenvalue equation is possible with Matlab scriptaleig. This script is called as [iter,eval,evect]=aleig(a,b,eps) ; This script find the fondamental eigenvalue and corresponding eigenvector of equation using the inverse power method. Dummy variables are defined as follows: ( A 1 ) B Φ = 0 K eff variablesaandbare the input matrices eps is the convergence parameter of the inverse power method. The script returns a list containing the number of iterations the fundamental eigenvalue 1/K eff the fundamental eigenvector Φ. ENE6101: Week 8 The collision probability method in 1D part 1 11/28

12 Examples of 2D geometries insybilt: 1 Only compatible with the collision probability method. 2D Physical" cell examples 2D assembly example (use the interface current method) ENE6101: Week 8 The collision probability method in 1D part 1 12/28

13 Example of 2D geometries innxt: 1 Compatible with the CP, MOC and Monte-Carlo methods in 2D Can generate prismatic 3D geometries ENE6101: Week 8 The collision probability method in 1D part 1 13/28

14 Example of 3D geometries innxt: 1 Compatible with the CP, MOC and Monte-Carlo methods in 3D z x y ENE6101: Week 8 The collision probability method in 1D part 1 14/28

15 Example of 2D geometries in SALT: 1 mesh-splitting for flat-source approximation same geometry without mesh-splitting ENE6101: Week 8 The collision probability method in 1D part 1 15/28

16 Example of 2D geometries in SALT: 1 mesh-splitting for flat-source approximation same geometry without mesh-splitting ENE6101: Week 8 The collision probability method in 1D part 1 16/28

17 Example of 2D tracking 1 tracking with two values of DELX Dir= 0 Plane=0 IU= 0 IV= 0 Dir= 0 Plane=0 IU= 0 IV= y 2 y x x DELX = 0.2 DELX = 0.01 ENE6101: Week 8 The collision probability method in 1D part 1 17/28

18 Slab geometry 1 The CP formulation in one-dimensional (1D) slab geometry makes possible the analytical integration of Eq. (12) with respect to some dependent variables. We will first consider the case of a geometry made of an infinite lattice of identical unit cells. Each unit cell is made of a succession of I regions, each of volume V i with 1 i I. The reference unit cell is defined in x 1/2 x x I+1/2 and the reference region i is defined in x i 1/2 x x i+1/2. C i is the infinite set of all indices i, representing instances of a region repeating itself by translation of the unit cell ρ x i-1/2 r' ε x i+1/2 s x i+3/2 θ r Ω x x i region i x i+1 region i+1 ENE6101: Week 8 The collision probability method in 1D part 1 18/28

19 Slab geometry 2 Equation (12) is the expression of a CP component in the infinite domain case. It can be rewritten in a convenient form. r = r +sω and d 3 r = s 2 d 2 Ωds, so that (21) p ij = 1 4πV i V i d 3 r 4π d 2 Ω dse τ(s) I j where we have omitted the energy group index in order to simplify the equations. The quantity I j is the set of points belonging simultaneously to to the half straight line of origin r and direction Ω; to a single instance of volume V j. We also define the direction Ω of the particle in term of the colatitude θ and azimuth ǫ using Ω = cosθi+sinθ cosǫj +sinθ sinǫk. With this definition and with the help of figure, we can write (22) d 2 Ω = dθdǫ sinθ l(ρ) = τ(s) cosθ dx = ds cosθ where ρ is a positive number representing the projection of s on the x axis. The material properties are independent of ǫ so that the integration in this variable can be performed analytically. ENE6101: Week 8 The collision probability method in 1D part 1 19/28

20 Slab geometry 3 Equation (21) can be simplified as (23) where p ij = 1 2 x i i C i xi +1/2 π/2 xj+1/2 l(x,x) dx dθ tanθ dx e cosθ x i 1/2 0 x j 1/2 (24) x i = V i = x i+1/2 x i 1/2 andl(x,x) = l(x,x ) is the projected optical path l(ρ) defined between pointsx andxas (25) l(x,x) = x x dx Σ(x ) so that l(x,x j+1/2 ) = l(x,x j 1/2 )+ x j Σ j. Summation over Vi includes instances of V i belonging to all unit cells of the lattice. The presence of neighboring cells will be taken into account in a specular way. ENE6101: Week 8 The collision probability method in 1D part 1 20/28

21 Slab geometry 4 Equation (23) can be rewritten so as to explicitly represent all contributions from other cells: (26) p ij = xi+1/2 π/2 1 dx dθ tanθ 2 x i m=0 x i 1/2 0 dx e ml cell +l(x,x) cosθ +e (m+1)l cell l(x,x) cosθ x j 1/2 xj+1/2 where 1 i < j I and where the unit-cell optical path l cell is defined as (27) l cell = xi+1/2 x 1/2 dx Σ(x ). ENE6101: Week 8 The collision probability method in 1D part 1 21/28

22 Slab geometry 5 The self-region collision probability p ii is (28) p ii = x i x m=0 x i 1/2 dx xi+1/2 x dx xi+1/2 dx x i 1/2 π/2 0 dθ tanθ e ml cell +l(x,x ) cosθ +e (m+1)l cell l(x,x ) cosθ e ml cell +l(x,x) cosθ +e (m+1)l cell l(x,x) cosθ where 1 i I. ENE6101: Week 8 The collision probability method in 1D part 1 22/28

23 Exponential functions 1 Exponential functions are defined as (29) E n (x) = 1 0 duu n 2 e x/u = 1 duu n e xu with n The exponential functions satisfy relations 1 E 1 (x) x x due n (u) = E n+1 (x) E n+1 (x ) 0.5 E 2 (x) E 3 (x) and E n (x) = d dx E n+1(u) E 4 (x) x They can be evaluated efficiently using the matlab scripttaben. ENE6101: Week 8 The collision probability method in 1D part 1 23/28

24 Slab geometry cont n 1 We will now present the Kavenoky method for obtaining the numerical values of the CP components in slab geometry. The principle of this method is to keep the sum over n in Eqs. (26) and (28) and to introduce exponential functions E n (x), with n 1. Integration of Eq. (26) in θ leads to (30) p ij = xi+1/2 1 dx 2 x i m=0 x i 1/2 xj+1/2 [ [ ]} dx {E 1 ml cell +l(x,x) ]+E 1 (m+1)l cell l(x,x) x j 1/2 if i j and integration of Eq. (28) in θ leads to (31) p ii = x i { x m=0 x i 1/2 dx xi+1/2 x dx xi+1/2 x i 1/2 dx [ [ ]} {E 1 ml cell +l(x,x ) ]+E 1 (m+1)l cell l(x,x ) [ [ {E 1 ml cell +l(x,x) ]+E 1 (m+1)l cell l(x,x)]} }. ENE6101: Week 8 The collision probability method in 1D part 1 24/28

25 Slab geometry cont n 2 Equations (30) and (31) involve two types on spatial integration which can be written in functional form as (32) and (33) R ij,m {±l} = 1 2 xi+1/2 x i 1/2 dx R ii,m {±l} = 1 dx 2 x i 1/2 { x xi+1/2 x i 1/2 dxe 1 xj+1/2 x j 1/2 dxe 1 [ ] ml cell ±l(x,x) where i < j [ ] xi+1/2 [ ] } ml cell ±l(x,x ) + dxe 1 ml cell ±l(x,x) x so that (34) p ij = 1 x i m=0 R ij,m {l}+r ij,m+1 { l} where i j. ENE6101: Week 8 The collision probability method in 1D part 1 25/28

26 Slab geometry cont n 3 The collision probabilities defined in Eq. (34) are representing an infinite lattice and are normalized as I (35) p ij Σ j = 1, i. j=1 We will now find analytical reductions for functionals R ij,m {±l} and use them to evaluate the CPs. Integration in x and x is performed, leading to the following relations: a)i < j,σ i 0 andσ j 0 : 1 [ ] R ij,m {±l} = {E 3 ml cell ±l(x i+1/2,x j 1/2 ) 2Σ i Σ j ] E 3 [ml cell ±l(x i+1/2,x j+1/2 ) [ ] + E 3 [ml cell ±l(x i 1/2,x j+1/2 ) ] E } 3 ml cell ±l(x i 1/2,x j 1/2 ) b)i < j,σ i = 0 andσ j 0 : R ij,m {±l} = ± x i 2Σ j [ [ ] {E 2 ml cell ±l(x i 1/2,x j 1/2 ) ] E } 2 ml cell ±l(x i 1/2,x j+1/2 ) ENE6101: Week 8 The collision probability method in 1D part 1 26/28

27 Slab geometry cont n 4 c)i < j,σ i 0 andσ j = 0 : R ij,m {±l} = ± x j 2Σ i d)i < j andσ i = Σ j = 0 : [ [ ] {E 2 ml cell ±l(x i+1/2,x j 1/2 ) ] E } 2 ml cell ±l(x i 1/2,x j 1/2 ) R ij,m {±l} = x i x j 2 ] E 1 [ml cell ±l(x i 1/2,x j 1/2 ) e)i = j andσ i 0 : R ii,m {±l} = ± x i E 2 [ ml cell ] Σ i 1 Σ 2 i [ ] {E 3 [ml cell ] E } 3 ml cell ±l(x i 1/2,x i+1/2 ) f)i = j andσ i = 0 : R ii,m {±l} = E 1 [ ml cell ] x 2 i 2. ENE6101: Week 8 The collision probability method in 1D part 1 27/28

28 Matlab scripts 1 function f=rij_f(sg,tau0,sigi,sigj,segi,segj) if sigi = 0 && sigj = 0 f=0.5*(taben(3,tau0)-taben(3,tau0+sg*sigi*segi)-... taben(3,tau0+sg*sigj*segj)+... taben(3,tau0+sg*sigi*segi+sg*sigj*segj))/(sigi*sigj) ; elseif sigi == 0 && sigj = 0 f=sg*0.5*segi*(taben(2,tau0)-taben(2,tau0+sg*sigj*segj))/sigj ; elseif sigi = 0 && sigj == 0 f=sg*0.5*segj*(taben(2,tau0)-taben(2,tau0+sg*sigi*segi))/sigi ; else f=0.5*segi*segj*taben(1,tau0) ; end function f=rii_f(sg,tau0,sigi,segi) if sigi = 0 f=sg*segi*taben(2,tau0)/sigi-(taben(3,tau0)-... taben(3,tau0+sg*sigi*segi))/sigiˆ2 ; else f=0.5*segiˆ2*taben(1,tau0) ; end ENE6101: Week 8 The collision probability method in 1D part 1 28/28