Lecture 18 Neutron Kinetics Equations

Transcription

1 Lecture 18 Neutron Kinetics Equations Prof. Dean Wang For a nuclear reactor to operate at a constant power level, the rate of neutron production via fission reactions should be exactly balanced by neutron loss via absorption and leakage. Any deviation from the balance condition will result in a time- dependence of the neutron population and hence the power level of the reactor. It is important that one be able to predict the time behavior of the neutron population in a reactor core induced by changes in reactor multiplication. Such a topic is known as nuclear reactor kinetics. Nuclear safety studies require a precise understanding of the reactor core dynamic response under transient or accident conditions. The neutronic behavior of the core during a transient is governed by space- time kinetics equations. 1. Neutron Space- Time Kinetics Equations The continuous- energy transient transport equation is governed by " r,,ω, " + Ω φ r, E, Ω, t + Σ r, E φ r, E, Ω, t = Q r, E, Ω, t (1) Q r, E, Ω, t = dω de Σ r, E E, Ω Ω φ r, E, Ω, t χ E 1 β d E νσ r, E φ r, E, t + together with the set of N precursor equations: r, " + χ E λ C r, t (2) = β d EνΣ r, E φ r, E, t λ C r, t, i = 1,, N (3) β = β (4) χ E = prompt fission neutron spectrum νσ r, E = nu macroscopic fission cross section β = delayed neutron fraction of precursor group i χ E = delayed neutron spectrum of precursor group i λ = radioactive decay constant of precursor group i C r, t = precursor concentration of precursor group i We can write the above time- dependent neutron transport equation can be approximated by the multigroup diffusion equations: χ 1 β r, " D r φ r, t + Σ, r φ r, t = Σ, r φ r, t + νσ, r φ r, t + χ λ C r, t (5a) 1

2 r, " = β νσ, r φ r, t λ C r, t (5b) g = 1,, G. Note that here we assume time- independent nuclear properties, e.g., cross sections. In fact, a change in these nuclear properties is the most probable cause of any transient. 2. Point- Kinetics Equations The point- kinetics equations are those describing the transient behavior of the neutron flux as a point model, the flux is independent of the spatial position r. Here we follow the derivation of the point- kinetics equations given in reference [1]. The main idea in point kinetics is to separate neutron multigroup φ r, t in a part depending only on space, and another depending only on time as φ r, t = S r, t T t (6) φ r, t φ r, t = φ r, t φ r, t S r, t = S r, t S r, t S r, t (7) This equation is still exact, but S r, t depends on both space and time in this approach. There is a degree of arbitrariness in the choice of S r, t and T t. Now we introduce a weight vector W r, t W r = W r, t (8) W r, t Now we define T t = W r v φ r, t (10) and it follows that after (6), S r, t must satisfy the constraint W r v S r, t = 1 (11) the symbol = dr implies spatial integration over the whole reactor core. Remarks: The factor S r, t is the form function or shape function, and T t is the amplitude function. T t represents loosely the total umber of neutrons in the reactor, this number depends somewhat on the chosen weight vector W r. 2

3 The form function S r, t may vary in time, but it s weighted spatial integral is time independent. Thus T t can be considered as the time dependence of the neutron poputation. Now we can obtain the neutron point- kinetics equations by substituting (6) into the space time diffusion equations (5) which have been pre- multiplied by W r and by spatially integrating over the whole reactor core: " " " = T t + λ C t (12a) = T t λ C t (12b) C t = Λ t = (13a) (13b) β t = β (13c) β t = β t (13d) ρ t = (13e) So far no approximations have been made. However, the calculation of the kinetics parameters ρ t, β t, and Λ t depend, by definition on the form function S r, t. In order to know S r, t, it implies in turn knowing the neutron flux φ r, t which necessitates a complete solution of the space time neutron transport or diffusion equation. Therefore, it is very difficult to determine the point kinetics parameter. The way out of this is by a function depending on space alone, S r. This function is usually provided by the static, initial neutron distribution, before perturbations have been applied. For instance, β t, and Λ t become time independent. As for the reactivity ρ t, its value depends on the time variations of nuclear cross sections and diffusion coefficients. But these parameters are applied to a form function which does not correspond to the instantaneous state of the core during the transient. In this case, it can be shown that the best choice for the weight function W r is the adjoint flux of the initial neutron flux, φ r at t = 0. Remarks on β t : We use the adjoint flux φ r for the weight function W r, the effective delayed neutron fraction β t can be written as β t = β (14) We can apply a separation approximation for the initial adjoint flux as φ r φ r φ r φ r = φ r φ φ φ = φ r φ (15) 3

4 Substituting (15) into (14) gives β t = β γ = = β γ = =, (16) (17) Thus, the effective delayed neutron fraction β t is weighted by the ratio of the importance weight of the delayed and total fission neutron spectra, γ. In general, γ < 1 for fast reactors and γ > 1 for thermal reactors. 3. An Analytic Solution of the Point- Kinetics Equations We can cast (12) in matrix form as " " ψ = A = = Aψ (18) T t C t C t C t λ λ λ λ λ (19a) (19b) λ We diagonalize matrix A with matrix V as D = V AV (20) ω ω D = ω (21) ω = the eigenvalues of matrix A, V = N + 1 N + 1 matrix with the columns consisting of the eigenvectors of matrix A. let ψ = Vψ (22) 4

5 ψ = C C (23) Substituting (22) into (18) gives " = " V AVψ = Dψ (24) or ω ω " C C = For the initial condition ψ 0 = ψ = e e ω 0 C 0 C 0 e After the inverse transformation, we have T t C t e ψ = C t C t = Vψ = V C C, we can obtain the solution as 0 C 0 C 0 e e (25) (26) V ψ 0 (27) The eigenvalues ω can be determined by det A ωi = 0, (28) or we expand the determinant as ω + λ ω = 0 (29) Since ω λ, they should be the solution of ω + = 0 (30) This is known as Nordheim equations. The values of this equation can be only obtained approximately. References 1. J. Koclas, Reactor Control and Simulation Lecture Notes. 2. D. Wang, B.J. Ade, A.M. Ward, Cross Section Generation Guidelines for TRACE- PARCS, NUREG/CR- 7164, June