Space-time kinetics. Alain Hébert. Institut de génie nucléaire École Polytechnique de Montréal.

Transcription

1 Space-time kinetics Alain Hébert Institut de génie nucléaire École Polytechnique de Montréal ENE613: Week 1 Space-time kinetics 1/24

2 Content (week 1) 1 Space-time kinetics equations Point kinetics equations Equivalent point kinetics parameters Point kinetics equations (classical form) Inhour equation with precursors Calculation of an exponential matrix The theta finite-difference scheme The theta finite-difference scheme Theta scheme with point kinetics Stability condition Space-time kinetics with Θ = 1 Space-time kinetics with Θ = ENE613: Week 1 Space-time kinetics 2/24

3 Space-time kinetics equations 1 1 V n,g = t φ g(r, t) D g (r) φ g (r, t) + Σ rg (r) φ g (r, t) + X l GX h=1 h g Σ g h (r) φ h (r, t) + χ pr g (r) (1 β) G X χ del l,g (r) λ l c l (r, t) + q g (r, t) h=1 νσ fh (r) φ h (r, t) c l (r, t) t X G = β l νσ fh (r) φ h (r, t) λ l c l (r, t) ; l = 1, N d h=1 β = X l β l. ENE613: Week 1 Space-time kinetics 3/24

4 Point kinetics equations 1 1 V n (E) φ(e, t) + Σ(E) φ(e, t) = t + χ pr (E)(1 β) + X l Z Z χ del l (E) λ l c l (t) + q(e, t) de Σ s, (E E ) φ(e, t) de νσ f (E ) φ(e, t) together with the set of N d precursor equations: c l (t) t = β l Z de νσ f (E) φ(e, t) λ l c l (t) ; l = 1, N d. ENE613: Week 1 Space-time kinetics 4/24

5 Equivalent point kinetics parameters 1 The time-dependent reactivity of the reactor is ρ(t) = 1 R de φ (E)»Σ(E) φ(e, t) R de φ (E) χss (E) R R de Σ s, (E E ) φ(e, t) de νσ f (E ) φ(e, t), the steady-state fission spectrum is χ ss (E) = (1 β) χ pr (E) + X l β l χ del l (E), ENE613: Week 1 Space-time kinetics 5/24

6 Equivalent point kinetics parameters 2 the mean neutron generation time of the reactor is Λ(t) = R R de V 1 n (E) φ (E) φ(e, t) R de φ (E) χss (E) de νσ f (E ) φ(e, t), the average delayed fractions are β l (t) = β l R R de φ (E) χdel l (E) de φ (E) χss (E) ENE613: Week 1 Space-time kinetics 6/24

7 Equivalent point kinetics parameters 3 and β(t) = X l β l (t) = 1 (1 β) R R de φ (E) χpr (E) de φ (E) χss (E) and where the neutron population, precursor concentration and external source are n(t) = Z de 1 V n (E) φ (E) φ(e, t), c l (t) = c l (t) Z de φ (E) χ del l (E) and q(t) = Z de φ (E) q(e, t). ENE613: Week 1 Space-time kinetics 7/24

8 Point kinetics equations (classical form) 1 d dt n(t) = ρ(t) β(t) Λ(t) n(t) + X l λ l c l (t) + q(t) d dt c l(t) = β l (t) Λ(t) n(t) λ l c l (t) ; l = 1, N d ENE613: Week 1 Space-time kinetics 8/24

9 Point kinetics equations (classical form) 2 Point kinetics equations are written in matrix form as x() = x d x(t) = A x(t) + S dt x(t) = [ n(t) c 1 (t) c 2 (t)... c Nd (t)], S = [ q... ] A = ρ β Λ β 1 Λ β 2 Λ. β Nd Λ λ 1 λ 2... λ Nd λ 1... λ λ Nd ENE613: Week 1 Space-time kinetics 9/24

10 Point kinetics equations (classical form) 3 Steady-state equation: A x = c l () c l, = β l Λ λ l n ; l = 1, N d. Each side of the point kinetics equation is multiplied by the integrating factor e At and integrated in time. We obtain the analytical solution of the point-kinetics equations as x(t) = e At x + (e At I) A 1 S where we made use of the identity Z t» d dt e At dt x(t ) A x(t ) = e At x(t) x(). ENE613: Week 1 Space-time kinetics 1/24

11 Calculation of an exponential matrix 1 A straightforward technique consists to obtain the N d + 1 eigenvalues and eigenvectors of linear system (A ω i I) p i = where 1 i N d + 1 The eigenvalues are the root of the characteristic polynomial det (A ω I) = which can be obtained as the roots of the inhour equation : ρ β Λ ω + X l β l λ l λ l + ω =. ENE613: Week 1 Space-time kinetics 11/24

12 Inhour equation with precursors 1 ω = ρ β Λ ρ = β ω = ω = λ 6 ω = β Λ ρ = ENE613: Week 1 Space-time kinetics 12/24

13 Inhour equation with precursors 2 We reproduce here a simple Matlab script for solving the inhour equation ρ β Λ ω + X l β l λ l λ l + ω = function y = inhour(lambda, L, beta, rho) % Find the roots of the inhour equation % function y = inhour(lambda, L, beta, rho) % (c) 27 Alain Hebert, Ecole Polytechnique de Montreal [nn, dd]=residue(lambda.*beta,-lambda,[]) ; y=sort(roots(conv([-l, rho-sum(beta)],dd)+[ nn])) ; The script parameters are defined as lambda= array of dimension N d containing the decay constants λ l for the delayed neutrons L= neutron generation time Λ beta= array of dimension N d containing the delayed fractions β l rho= reactivity ρ y= array of dimension N d + 1 containing the roots ω i of the inhour equation. ENE613: Week 1 Space-time kinetics 13/24

14 Calculation of an exponential matrix 1 The analytical solution also requires the knowledge of the matrix P whose columns are the eigenvectors of square matrix A. This matrix is equal to P = {p i,j ; i = 1, N d + 1 and j = 1, N d + 1} where p 1,j = 1 and p i+1,j = β i Λ (λ i + ω j ). ENE613: Week 1 Space-time kinetics 14/24

15 Calculation of an exponential matrix 1 Similarly, we can show that P 1 = {y i,j ; i = 1, N d + 1 and j = 1, N d + 1} where y i,1 = " 1 + X l β l λ l Λ (λ l + ω i ) 2 # 1 and y i,j+1 = y i,1 λ j λ j + ω i. Theorem If P is the matrix whose columns are the eigenvectors of square matrix A, and if ω i are the corresponding eigenvalues, then e At = P diag(e ω it ) P 1. ENE613: Week 1 Space-time kinetics 15/24

16 The theta finite-difference scheme 1 A finite difference relation is used for time discretization of neutron flux and precursor equations, written as (1 Θ) t φ g(r, t) + Θ tn 1 t φ g(r, t) = φ g(r, t n ) φ g (r, t n 1 ) tn t n (1 Θ) t c l(r, t) + Θ tn 1 t c l(r, t) = c l(r, t n ) c l (r, t n 1 ) tn t n where l = 1, N d. Setting Θ =, Θ = 1 and Θ = 1/2 yield the explicit scheme, the fully implicit scheme and the Crank-Nicholson scheme, respectively. ENE613: Week 1 Space-time kinetics 16/24

17 Theta scheme with point kinetics 1 x (n+1) x (n) t n = Θ A x (n+1) + (1 Θ) A x (n) + S where x (n) x(t n ). The theta scheme can be applied to the point-kinetics equations and used to study its stability characteristics. It can also be applied to the space-time kinetics equations for solving full-core problems. Above equation with S = may be written as x (n+1) = R x (n) where I is the identity matrix and where the iterative matrix is defined as R =» I 1» I Θ A + (1 Θ) A t n t n. ENE613: Week 1 Space-time kinetics 17/24

18 Theta scheme with point kinetics 2 The eigenvalue problem associated to the above iterative scheme is written γ i p i R p i = where γ i is the i th eigenvalue of matrix R and p i is the corresponding eigenvector. We can easily show that these eigenvectors are identical to those of the inhour equation and that the corresponding eigenvalues are related using γ i = 1 + (1 Θ) ω i t n 1 Θ ω i t n. The eigenvectors of matrix R are linearly independent, so that the initial condition x can be expressed as a linear combination: x x () = X i c i p i. ENE613: Week 1 Space-time kinetics 18/24

19 Stability condition 1 After a progression of n time-steps, we have x (n) = R x (n 1) = R X i c i γ n 1 i p i = X i c i γ n i p i. We apply a negative reactivity ρ < to a point kinetics problem and verify that the numerical solution given by previous equation vanishes as t increases. The stability will be guaranteed if γ i < 1 for all values of i. If ρ <, we observe from inhour figure that all eigenvalues ω i are negative. γ i equation leads to condition γ i < 1 if (1 + Θ ω i t n ) < 1 (1 Θ) ω i t n < 1 + Θ ω i t n which is satisfied if 1 2Θ 2 ω i t n < 1 ; i. ENE613: Week 1 Space-time kinetics 19/24

20 Stability condition 2 We conclude that the explicit scheme (Θ = ) is stable if the time-step size is set below a threshold value written as 2 t n < max i ( ω i ). the fully implicit and Crank-Nicholson schemes are unconditionally stable. The practical choice of Θ for production calculations is selected in interval 1/2 Θ 1. ENE613: Week 1 Space-time kinetics 2/24

21 Space-time kinetics with Θ = V n,g t n φ g (r, t n ) D g (r) φ g (r, t n ) + Σ rg (r) φ g (r, t n ) = S imp g (r, t n ) + + ( χ ss g (r) X l GX h=1 h g χ del l,g (r) Σ g h (r) φ h (r, t n ) ) β l X G νσ fh (r) φ h (r, t n ) 1 + λ l t n h=1 c l (r, t n ) = λ l t n " G # X c l (r, t n 1 ) + β l t n νσ fh (r) φ h (r, t n ) h=1 ENE613: Week 1 Space-time kinetics 21/24

22 Space-time kinetics with Θ = 1 2 with the fixed source defined as S imp g (r, t n ) = 1 V n,g t n φ g (r, t n 1 ) + X l χ del l,g (r) λ l 1 + λ l t n c l (r, t n 1 ) + q g (r, t n ) ENE613: Week 1 Space-time kinetics 22/24

23 Space-time kinetics with Θ = 1 1 V n,g t n φ g (r, t n ) = S exp g (r, t n ) X G c l (r, t n ) = (1 λ l t n ) c l (r, t n 1 ) + β l t n νσ fh (r) φ h (r, t n 1 ) h=1 ENE613: Week 1 Space-time kinetics 23/24

24 Space-time kinetics with Θ = 2 with the fixed source defined as S exp g (r, t n ) = 1 V n,g t n φ g (r, t n 1 ) + D g (r) φ g (r, t n 1 ) Σ rg (r) φ g (r, t n 1 ) + + " + X l χ ss g (r) X l GX h=1 h g β l χ del l,g (r) # G X Σ g h (r) φ h (r, t n 1 ) h=1 νσ fh (r) φ h (r, t n 1 ) χ del l,g (r) λ l c l (r, t n 1 ) + q g (r, t n 1 ). ENE613: Week 1 Space-time kinetics 24/24