AN ANALYTICAL SOLUTION FOR THE TWO-GROUP KINETIC NEUTRON DIFFUSION EQUATION IN CYLINDRICAL GEOMETRY

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1 2011 International Nuclear Atlantic Conference - INAC 2011 Belo Horizonte, MG, Brazil, October ASSOCIAÇÃO BRASILEIRA DE ENERGIA NUCLEAR - ABEN ISBN: AN ANALYTICAL SOLUTION FOR THE TWO-GROUP KINETIC NEUTRON DIFFUSION EQUATION IN CYLINDRICAL GEOMETRY Julio Cesar L. Fernandes 1, Bardo Ernst Bodmann 2 and Marco Túllio Vilhena 1,2 1 Programa de Pós Graduação em Matemática Aplicada (DMPA / UFRGS) Universidade Federal do Rio Grande do Sul Rua: Av. Bento Gonçalves, Porto Alegre, RS julio.lombaldo@ufrgs.br, vilhena@pq.cnpq.br 2 Programa de Pós-Graduação em Engenharia Mecânica - (PROMEC / UFRGS) Universidade Federal do Rio Grande do Sul Rua: Sarmento Leite, 425/ Porto Alegre, RS bardo.bodmann@ufrgs.br ABSTRACT Recently the two-group Kinetic Neutron Diffusion Equation with six groups of delay neutron precursor in a rectangle was solved by the Laplace Transform Technique. In this work, we report on an analytical solution for this sort of problem but in cylindrical geometry, assuming a homogeneous and infinite height cylinder. The solution is obtained applying the Hankel Transform to the Kinetic Diffusion equation and solving the transformed problem by the same procedure used in the rectangle. We also present numerical simulations and comparisons against results available in literature. 1. INTRODUCTION In the present work we elaborate a methodology to solve analytically the kinetic diffusion equation by integral transform technique. Recently Lemos et al [5] solved the diffusion equation of neutrons in slab geometry for a model of two energy groups by the technique of Laplace transform. In another work, Gocalves et al [3] and Vilhena et al [9] solved the transport equation of neutrons in cylindrical geometry considering isotropic scattering, using the Hankel transform. In view of the promising results in this works, and the fact that the approximation S 2 of Boltzmann s transport equation of neutrons results in the kinetic diffusion equation, in this work we focus on the derivation of an analytical formulation for the neutron flux of fast and thermal groups and contributions from the precursors. These equations can be expressed in cylindrical geometry, relevant in calculus of cells of nuclear reactors. The principal idea is solve the one-dimensional diffusion equation of neutrons, for the model of two energy groups in homogeneous cylindrical geometry by the Hankel transform. To this end, we apply the Hankel transform in the diffusion equation and obtain a generalized system involving two groups of equations, two for flux, fast and thermal, and six of kinetic origin, representing six different groups of delayed neutrons. For simplicity, however without loosing generality, we discuss the one-dimensional, multigroup diffusion kinetic equation with six delayed neutron precursor concentration groups.

2 Note, the equation for the delayed neutron precursor concentration is a first order linear differential equation in the time variable. Upon application of the Hankel transform in our equations results in a matrix differential equation, that may be solved in general even for large matrix order, due to the non-degeneracy of eigenvalues. This is especially an advantage by virtue of the stiff character of this sort of problems, generated by the considerable time scale differences of the prompt and delayed neutrons. Differently than other methods the adopted methodology is robust, that allows us to work out the one-dimensional diffusion kinetic equation, in a straightforward manner also for problems that consider a significant number of energy groups (up to 200), and further may be extended to multidimensional and multi-layered problems. 2 THE PROBLEM FORMULATION Our starting point is the kinetic diffusion equation with two energy groups, together with the kinetic equation for one neutron precursor. The diffusion equation is solved upon applying the zero order Hankel transform in r with cylindrical geometry. Let the diffusion equation for the groups 1 and 2 is given by 1 v g t φ g(r, t) = r D g φ g (r, t) Σ Rg φ g (r, t) + (1 β) t C i(r, t) = λ i C i (r, t) + β i ν g Σ fg φ g (r, t) + 6 λ i C i (r, t) (1) i=1 ν g Σ fg φ g (r, t) (2) where the operator r is the radial part of the Laplace operator in cylinder coordinates r = r 1 r(rr). (3) Here D g is the diffusion coefficient for neutrons of group g, Σ Rg is the macroscopic removal cross section for group g, C is the concentration of neutron precursors, λ i is the decay constant, νσ (R) fg is the product of the medium number of neutrons per fission and the macroscopic fission cross section of group g and νσ (R) gg is the macroscopic scattering cross section of group g into group g. In our simplified case, we will assume only one precursor concentration C(r, t). Moreover, we consider D g a constant independent of r, so that the commutator [ r, D g ] = 0. In this case, our governing equations are 1 v g t φ g(r, t) = D g r φ g (r, t) Σ Rg φ g (r, t) + (1 β) C(r, t) = λc(r, t) + β t ν g Σ fg φ g (r, t) + λc(r, t) (4) ν g Σ fg φ g (r, t) (5) 3 THE FINITE HANKEL TRANSFORM Recalling, that the Finite Hankel transform of order p is defined by H p [f(r); r α n ] = a 0 rf(r)j p (α n r) dr (6)

3 where α n are the values such that J(α n r) = 0. We apply the Finite Hankel Transform, because the flux outside the cylinder limited by R vanishes. In order to establish the boundary conditions, we assume zero flux at the border φ(r, t) = 0 and make use of the distributional character of the flux, i.e. φ(0, t) is limited. The unique initial condition is φ(r, 0) = 0. For the inversion, we apply the Hankel inversion theorem, that says that for φ we can find φ by the following prescription. φ(r, t) = 2 R 2 n=1 φ(α n, t) J 0(α n r) [J 0(α n R)] 2 (7) The operator r has the property H 0 { r φ} = α 2 nh 0 {φ}, so that (1) and (2) are 1 d v dt φ g (α n, t) + D g αn 2 φ g (α n, t) + Σ Rg φg (α n, t) (1 β) and t C(α n, t) = λ C(α n, t) + β i νσ fg φg (α n, t) = C(α n, t) (8) ν g Σ fg φg (α n, t). (9) 4 SOLUTION OF THE MATRIX SYSTEM The afore presented equations may be cast into matrix form. 1 v g t φ g (r, t) = D g α n φg (r, t) Σ Rg φg (r, t) + (1 β) ν g Σ fg φg (r, t) + λ C(r, t) (10) t C(r, t) = λ C(r, t) + β ν g Σ fg φg (r, t) (11) Using the notation t ( ) = ( ), the matrix system is given by φ 1 φ 2 C + [ v1 D 1 αn 2 + v 1 Σ R1 v 1 (1 β)ν 1 Σ f1 v 2 (1 β)ν 2 Σ f2 λ v 1 (1 β)ν 1 Σ f1 v 2 D 2 αn 2 + v 2 Σ R2 v 2 (1 β)ν 2 Σ f2 λ βν 1 Σ f1 βν 2 Σ f2 λ ] φ 1 φ 2 C = 0 Our system in shorthand notation has the form X + AX = 0 (12) with well known solution X(t) = e At X(0) (13)

4 which may be written in diagonalised form for A e At = Y e Kt Y 1. (14) Here, A = Y KY 1, where K is the matrix of eigenvalues of A, Y is the matrix of eigenvectors of A, and Y 1 is the inverse of Y and e Kt is diagonal. e Kt = e k 1t 0 e k 2t 0 e k 3t (15) Here k 1, k 2, k 3 are the eigenvalues of A and the components of diagonal matrix K. Then, the global solution can be written as X(t) = Y e Kt Y 1 X(0) (16) and φ 1 (α n, t) φ 2 (α n, t) C(α n, t) = Y e k 1t 0 e k 2t 0 e k 3t Y 1 φ 1 (α n, 0) φ 2 (α n, 0) C(α n, 0) (17) where the initial conditions for both, fluxes and precursor concentration are taken from is the steady state solution, so that the final solution for both the fluxes and precursor concentration is given by φ g (r, t) = 2 R 2 n=1 φ g (α n, t) J 0(α n r) [J 0(α n R)] 2 (18) C(r, t) = 2 R 2 n=1 C(α n, t) J 0(α n r) [J 0(α n R)] 2 (19) 5 RESULTS In figures 1-3 we show the numerical results for the fast and thermal radial neutron flux as well as the neutron precursor concentration and their time dependence. These findings agree with those found in the literature.

5 Figure 1. Profile of φ 1, for different values of t. Figure 2. Profile of φ 2, for different values of t. Figure 3. Profile of C, for different values of t.

6 6 CONCLUSION In this work, we determined the fast and thermal flux solutions together with a neutron precursor concentration of the kinetic diffusion equation in cylindrical geometry using the technique of zero order Hankel transform. Since existence and uniqueness of the solution is proven by the Cauchy-Kowalewsky theorem, the present work established a new approach to solve the diffusion equation in a cylindrical coordinates. Our procedure allows us to construct an efficient algorithm for solving these equations in an exact fashion. Finally motivated by the promising results attained by this methodology, in a forthcoming paper we extend the present case to a problem with multi regions, each one with its distinct and specific physical parameters. This will open pathways to solve the global calculus of criticality of a nuclear reactor nucleus, applying continuity conditions of the flux and current density of neutrons across the interfaces. ACKNOWLEDGEMENT The authors are gratefully to CNPq (Conselho Nacional de Desenvolvimento Científico e Tecnológico) for the partial financial support of this work. J. C. L. Fernandes was supported by a master degree fellowship of the CAPES(Brazil). A special acknowledgment for the project INCT (Instituo Nacional de Ciência e Tecnologia - Reatores Nucleares Inovadores) for financial support for participation in this congress. REFERENCES [1] B. E.J. Bodmann, M. T. Vilhena, L. S.Ferreira, J. B.Bardaji. An analytical solver for the multi-group tw dimensional neutron-diffusion equation by integral transform techniques,il Nuovo Cimento della Societ Italiana di Fisica. C 33, pp. 1-10, [2] I. A. Snneddon, The use of integral transforms, McGraw-Hill company., New York, [3] G. A. Goncalves, M. T. Vilhena, B. E.J. Bodmann, Heuristic Geometric Eigenvalue universaly in a one-dimensional neutron transport problem with anisotropic scattering, Kertechnik, 75, pp.50-52, [4] G. A. Goncalves, S. B. Leite, M. T. Vilhena, Solutionof the neutron transport equation problem with anysotropic scattering, Annals of Nuclear Energy 36, pp , [5] R. Lemos, M. T. Vilhena, F. C. da Sila, S. Wortmann, Analytic Solution for Two-Group Diffusion equations in a Multilayered Slab using Laplace Transform Technique, Progress in Nuclear Energy, 50, pp , [6] M. T. Vilhena, C. Segatto, H. Velho, G. A. Goncalves, Analytical Solution of the onedimensional discrete ordinates equation by the Laplace and Hnakel integral Transform, Integral Methods in Science and Engineering., pp , [7] J. Lamarsh, Introduction to Nuclear Reactor Theory, McGraw-Hill company., New York, [8] Segatto, C.F., Vilhena, M.T. and Pazos, R.P. (2000) On the convergence of the spherical harmonics approximations, Nuclear Science and Engineering, Vol. 134, No. 1, pp

7 [9] Vilhena, M.T., Segatto, C.F. and Barichello, L.B. (1995) A particular solution for the Sn radiative transfer problems, Journal of Quantitative Spectroscopy and Radiative Transfer, Vol. 53, No. 4, pp [10] R. C. Barros, Um Método Numérico Livre de Erro de Truncamento Espacial para Cálculos Unidimensionais de Multigrupo Difusão, IV Congresso Geral de Energia Nuclear.,vol. 1, pp , [11] G. J. Mitsis, Transport Solutions to the Monoenergetic Critical Problems, PhD Tesis, Argonne, Illinois, USA, 1963.