Numerical Techniques for the Neutron Diffusion Equations in the Nuclear Reactors

Transcription

1 Adv. Studies Theor. Phys., Vol. 6, 01, no. 14, Numerical Techniques for the Neutron Diffusion Equations in the Nuclear Reactors Abdallah A. Nahla Department of Mathematics, Faculty of Science Taif University, Taif 888, Saudi Arabia Faisal A. Al-Malki Department of Mathematics, Faculty of Science Taif University, Taif 888, Saudi Arabia faisal Mahmoud Rokaya Department of Information Technology College of Computing & Information Technology Taif University, Taif 888, Saudi Arabia Abstract The space-time neutron diffusion equations with multi-group of delayed neutrons are a couple of the stiff nonlinear partial differential equations. The finite difference method is used to reduce the partial differential equations into ordinary differential equations. This ordinary differential equations are rewritten in a matrix form. The general solution of the matrix differential equation contains the exponential function of the coefficient matrix. The numerical techniques for processing the exponential function of the coefficient matrix are presented using analytical method and fundamental matrix method. The eigenvalues of the coefficient matrix are calculated numerically using FORTRAN computer code based on Laguerre s method. The fundamental matrix and its inverse are calculated analytically for two energy groups of reactor kinetics and one group of precursor delayed neutrons. The numerical techniques are applied to three-dimensional space-time neutron diffusion equations with average one group of delayed neutrons in the different nuclear reactors. The results of numerical technique codes are compared with the results of traditional codes. Corresponding author: Department of Mathematics, Faculty of Science, Tanta University, Tanta 3157, Egypt.

2 650 A. Nahla, F. Al-Malki and M. Rokaya Keywords: Reactor kinetics equations; Finite difference; Analytical and fundamental matrix methods 1 Introduction The nuclear reactor problems, especially those involving safety considerations, the coefficients of the neutron diffusion equations depend upon parameters such as the neutron power level, precursor concentration of delayed neutrons groups, time, space and temperature feedback. In general, the reactor problem in the presence Newtonian temperature feedback effects comprises a very large and complex system of coupled nonlinear partial differential equations. Numerical methods for solving the space-time neutron diffusion equations in the nuclear reactor have been of interest in the nuclear reactor physics and engineering. The mixed dual nodal method MINOS was used to solve the reactor kinetics equations with improved quasi-static IQS model and the θ-method was used to solve the precursor equations (Dahmani et al., 001)[16]. An efficient solution method was presented to solve the time dependent multi-group diffusion equations for subcritical systems with external sources using a rigorous weight function in the quasi-static method (Kobayashi, 005)[13]. Parallelised Krylov methods were applied to the improved quasi-static approach in addition to the direct, implicit time difference, approach for solving space-time dependent multi-group neutron diffusion equations (Gupta et al., 005)[6]. The nodal diffusion method was developed to solve space-time neutron kinetics using the finite-element, primal and mixed hybrid nodal methods (Grossman and Hennart, 007)[15]. A one-step implicit method and a nodal modal method were studied to solve the time dependent neutron diffusion equations which based on a hexagonal spatial mesh (Ginestar et al., 00[9]; Miró et al., 00[18]; González-Pintor et al., 010[]). An adiabatic kinetics model was developed for a BWR core simulator AETNA (Tamitani et al., 003)[17]. Mathematical modeling of the space-time kinetics phenomena in advanced heavy water reactor, a 90 MW thermal, vertical pressure tube type thorium based nuclear reactor was presented using nodal modal method (Shimjith et al., 010)[3]. Numerical technique, based on the class of Padé and cut-product approximations, was applied to solve the two-energy group space-time nuclear reactor kinetics equations in two dimensions (Aboanber and Nahla, 006)[]. Adaptive Matrix Formation (AMF) method was presented and applied to homogeneous, symmetric heterogeneous and non-symmetric heterogeneous reactors in the case of two- and three-dimensions (Aboanber and Nahla, 007)[3]. The generalized Runge-Kutta method was developed for solving the multi-group, multidimensional, static and transient neutron diffusion kinetics equations (Aboanber and Hamada, 008)[4]. Computation accuracy and efficiency of a power series analytic method were presented for the time- space dependent neutron

3 Numerical techniques for the neutron diffusion equations 651 diffusion equations with adiabatic heat up and doppler feedback (Aboanber and Hamada, 009)[5]. The multi-group integro-differential equations of the neutron diffusion kinetics was presented and solved numerically in multi-slab geometry with the use of the progressive polynomial approximation (Quintero- Leyva, 010)[8]. In this work, numerical techniques for space-time neutron diffusion equations with multi-group of delayed neutrons are developed. The partial differential equations is changed to ordinary differential equations using the finite difference method. The matrix form of ordinary differential equations is obtained. The analytical method and fundamental matrix method for the exponential function of the coefficient matrix are presented. The results of numerical technique are discussed and compared with the results of traditional methods. Neutron Diffusion Equations The multi-energy group nuclear reactor kinetics equations with multi-group delayed precursor neutrons can be written in the following form [10, 14, 1, 4] 1 G v g t Φ g(r, t) = D g (r) Φ g (r, t) Σ ag Φ g (r, t) Σ sg,g Φ g (r, t) g >g G ( ) I + χg νσ fg (1 β) + Σ sg,g Φg (r, t) + χ g,i λ i C i (r, t), g =1 i=1 g = 1,,, G (1) t C i(r, t) = β i G νσ fg Φ g (r, t) λ i C i (r, t), i = 1,,, I () g=1 where r is the position vector (cm), t is the time (s), Φ g (r, t) is the scalar neutron flux (cm s 1 ) in group g, C i (r, t) is the concentration of delayed neutron precursors (cm 3 ) in group i, v g is the mean velocity of the neutron (cm s 1 ) in group g, D g (r) is the diffusion coefficient (cm) in group g, Σ ag (r, t) is the absorption cross-section (cm 1 ) in group g, Σ fg (r) is the fission crosssection (cm 1 ) in group g, Σ sg,g (r) is the scattering cross-section (cm 1 ) from group g to group g such that (Σ sg,g (r) = 0 for g > g), ν is the mean number of fission neutrons, χ g is the spectrum of prompt neutrons in group g, χ g,i is the spectrum of i-group delayed neutrons in group g, λ i is the decay constant (s 1 ) of group i precursors, β i is the fraction of delayed neutrons in group i, and β = I β i is the total fraction of delayed neutrons. i=1

4 65 A. Nahla, F. Al-Malki and M. Rokaya Using the finite difference method, the equations (1) and () lead to the following matrix form [7] where, dψ(t) dt = AΨ(t) + B, (3) Ψ(t) = [ Φ 1 Φ Φ G C 1 C C I ] T, (4) A = σ 1 τ 1, τ 1,G f 1,1 f 1, f 1,I τ,1 σ τ,g f,1 f, f,i τ G,1 τ G, σ G f G,1 f G, f G,I p 1,1 p 1, p 1,G λ p,1 p, p,g 0 λ p I,1 p I, p I,G 0 0 λ I, (5) and B = [ σ 1 σ σ G ] T, (6) Here, σ g, σ g, τ g,g, f g,i and p i,g are defined as σ g = v g χ g νσ fg (1 β) Σ ag v g G Σ sg,g g >g v g ( x) ( Dg,j 1,k,l + D g,j+ 1,k,l ) ( Dg,j,k ( y) 1,l + D ) v ( g g,j,k+ 1,l Dg,j,k,l ( z) 1 ) + D g,j,k,l+ 1, g = 1,,, G (7) σ g = v g ( x) ( Dg,j 1,k,lΦ g,j 1,k,l + D g,j+ 1 v g ),k,lφ g,j 1,k,l ) ( Dg,j,k ( y) 1,lΦ g,j,k 1,l + D g,j,k+ 1,lΦ g,j,k+1,l v ( ) g Dg,j,k,l ( z) 1 Φ g,j,k,l 1 + D g,j,k,l+ 1 Φ g,j,k,l+1, g = 1,,, G (8) τ g,g = v g ( χg νσ fg (1 β) + Σ sg,g ) ; g, g = 1,,, G; g g (9)

5 Numerical techniques for the neutron diffusion equations 653 f g,i = v g χ g,i λ i ; g = 1,,, G; i = 1,,, I (10) p i,g = β i νσ fg ; g = 1,,, G; i = 1,,, I (11) Assume that, the matrices A and B are constant during the interval time t m and t m+1 = t m + h considering the length of the time interval is small. Then, the general analytical solution of equation (3) takes the following form: Ψ(t m+1 ) = exp(ha)ψ(t m ) + (exp(ha) I) A 1 B (1) There are the different numerical techniques for calculating the exponential function of the coefficient matrix A. 3 Analytical Method Let us rewrite the exponential function of the coefficient matrix exp(ha) like analytical method[1] K exp(ha) = α k (ha) k (13) k=0 where α 0, α 1,, α K are coefficients and K = G + I 1 is the number of the rows and columns of the coefficient matrix A. The coefficients α k can be determined by solving the following equation exp(hω 0 ) exp(hω 1 ). exp(hω K ) = 1 (hω 0 ) (hω 0 ) (hω 0 ) K 1 (hω 1 ) (hω 1 ) (hω 1 ) K (hω K ) (hω K ) (hω K ) K α 0 α 1. α K (14) where ω 0, ω 1,, ω K are the eigenvalues of the matrix A. It is difficult to calculate analytically the eigenvalues of the coefficient matrix A in general form. So, the eigenvalues are calculated numerically using FORTRAN computer code based on Laguerre s method (Appendix B). Gauss elimination method is used to solve the equation (14) which yields α k = ( 1) k K j=0 h k exp(hω j )Q K k, j, k = 0, 1,, K (15) K (ω i ω j ) i = 0 i j

6 654 A. Nahla, F. Al-Malki and M. Rokaya where 0 for m < 0 or m > K 1 for m = 0 Q m, j = K K K ω i1 ω i ω im for 0 < m K such that, K i 1 = 0 i 1 j i 1 = 0 i 1 j K i = i i j i = i i j K i m = i m i m j i m = i m i m j ω i1 ω i ω im = 0 for i m > K or i m = j = K. These coefficients, α k, are expanded for a special case at two energy groups and average one group of delayed neutrons (Appendix A). Substituting equation (15) into equations (13) and (1) yields K Ψ(t m+1 ) = ( 1) k k=0 K + ( 1) k k=0 K j=0 K j=0 exp(hω j )Q K k, j K (ω i ω j ) i = 0 i j exp(hω j )Q K k, j K (ω i ω j ) i = 0 i j A k A k Ψ(t m ) I A 1 B (16) This equation represents the numerical technique for solving the space-time neutron diffusion equations with multi-group of delayed neutrons which based on analytical method for the exponential function of matrix A. 4 Fundamental Matrix Method In this section, the fundamental matrix method for calculating the exponential function of the coefficient matrix is introduced. Definition 1:[1] For homogenous differential equation, (3), with A, a real constant matrix whose eigenvalues ω 0, ω 1,, ω K are all different, Θ(t) = [ U 0 e ω 0t U 1 e ω 1t U K e ω Kt ] (17) is a fundamental matrix, where U k is any eigenvector corresponding to ω k. The exponential function exp(at) is taken the following form exp(at) = Θ(t)Θ 1 (t 0 ) (18)

7 Numerical techniques for the neutron diffusion equations 655 Substituting equation (18) into (1) yields the solution of equation (3) during the time interval [t m, t m+1 ] which takes the following form Ψ(t m+1 ) = Θ(t m+1 )Θ 1 (t m )Ψ(t m ) + ( Θ(t m+1 )Θ 1 (t m ) I ) A 1 B (19) where Θ(t m+1 ) is the fundamental matrix at t m+1 for the homogenous differential equation (3) and Θ 1 (t m ) is the inverse of the fundamental matrix at t m. According to Definition 1 the fundamental matrices Θ(t m+1 ) and Θ(t m ) are taken the form and Θ(t m+1 ) = [ U 0 e hω 0 U 1 e hω 1 U K e hω K Θ(t m ) = [ U 0 U 1 U K ] ] (0) (1) where U k is the eigenvectors corresponding to the eigenvalues ω k. The fundamental matrix and its inverse are calculated analytically for two energy groups and average one group of delayed neutron (Appendix B). 5 Numerical Results and Discussions To substantiate the feasibility of the numerical techniques for solving the neutron diffusion equations in the nuclear reactor, we developed the FOR- TRAN computer code that solves three-dimensional nuclear reactor problems. The numerical techniques using the analytical method and fundamental matrix of exponential function are applied to three-dimensional homogenous and TWIGL heterogenous reactors. The parameters of these two different nuclear reactors are shown in the Table Three-dimensional homogenous reactor Three-dimensional homogenous reactor is a bare homogenous cube of side length 00 (cm) on each side with two energy groups and average one precursor group. The boundary conditions were homogenous Dirichlet on all six sides, Φ 1 = Φ = 0 at x = 0, 00; y = 0, 00; and z = 0, 00 (cm). Ten mesh intervals were used in each direction, x = y = z = 0 (cm). Two cases of the perturbation are occurred as step change of the thermal absorption cross section. Two numerical techniques using the fundamental matrix method (NT-FMM) and analytical method (NT-AM) of exponential function are applied to solve these cases. Case I: The fast and thermal neutron flux at the center point of threedimensional homogenous reactor with a positive step reactivity, Σ a = 0.369

8 656 A. Nahla, F. Al-Malki and M. Rokaya Table 1: Data of three-dimensional homogeneous and TWIGL heterogenous reactors. Parameters Homogenous reactor TWIGL heterogenous reactor Material 1, Material 3 Group 1 Group Group 1 Group Group 1 Group D(cm) Σ a (cm 1 ) ν(neutron) Σ f (cm 1 ) Σ sg,g+1 (cm 1 ) v(cm/s) λ(s 1 ) β Table : Fast and Thermal neutron flux at the center point of threedimensional homogeneous reactor with a positive step reactivity Time Fast neutron flux Thermal neutron flux (s) AMF NT-FMM NT-AM 3DKIN AMF NT-FMM NT-AM , are shown in Table. This table shows the results of the NT-FMM and NT-AM compared with the 3DKIN (Ferguson and Hanson, 1973[11]) and adaptive matrix formation, AMF (Aboanber and Nahla, 007[3]). Case II: Table 3 shows the fast and thermal neutron flux at the center point of three-dimensional homogenous reactor with a negative step reactivity, Σ a = This table substantiates that the results of the NT- FMM and NT-AM are an accurate comparing to the results of the AMF. These comparisons substantiate the feasibility of the numerical techniques, NT-FMM and NT-AM, for solving the neutron diffusion equations in threedimensional homogenous reactor with different types of step reactivity. 5. Three-dimensional TWIGL reactor The TWIGL reactor is The three-dimensional heterogenous symmetric reactor. The overall reactor geometry is cubic with a side length of 160 (cm). An axial

9 Numerical techniques for the neutron diffusion equations 657 Table 3: Fast and Thermal neutron flux at the center point of threedimensional homogeneous reactor with a negative step reactivity Time Fast neutron flux Thermal neutron flux (s) AMF NT-FMM NT-AM AMF NT-FMM NT-AM blanket material of 4 (cm) in thickness is presented at the top and the bottom as in the Figure 1. Four regions containing material 1 are perturbed. They are located symmetrically along the diagonal of the four quadrants of the core. In the three-dimensions geometry, this configuration was made of 11 (cm) in thickness to the z direction, and a blanket of 4 (cm) in thickness was added to the top and to the bottom. It has two energy groups and one delayed precursor group. Due to the symmetry, only the right half of the cube was considered with a homogenous Neumann boundary condition imposed at the exposed mid-plane, D Φ = 0 at x = 0; 0 y 160 and 0 z 160. ten mesh intervals were used in x direction, x = 8 (cm), and twenty mesh intervals were used in each y and z directions, y = z = 8 (cm). Two cases of the perturbation are occurred into the regions of material 1. The perturbation is a linear ramp with the above cross section changed effected in 0. (s) and the perturbed cross section is maintained thereafter. Two numerical techniques, NT-FMM and NT-AM, are applied to solve these cases. Case I: Table 4 shows the thermal neutron flux at the different points, namely p(0, 80, 16); p(40, 40, 16); p(0, 80, 80); p(40, 40, 80); p(0, 80, 144) and p(40, 40, 16) (cm) with a positive linear ramp reactivity defined as { t, for 0 t 0. (s); Σ a (material 1) = , for t 0. (s). The thermal neutron flux of the NT-FMM and NT-AM is compared with the thermal neutron flux of 3DKIN and AMF. From this comparison, the thermal flux of NT-FMM and NT-AM are coincident. The relative errors of the thermal flux of numerical Techniques with respect to the thermal flux of 3DKIN at different points are shown in Figure. The maximum relative error is less than So, we can say that the results of numerical techniques are agreeable with the results of reference methods. Case II: Table 5 shows the thermal neutron flux at the different points,

10 658 A. Nahla, F. Al-Malki and M. Rokaya Figure 1: Three-dimensional TWIGL reactor geometry: (a) x y plane for 4 z 136 cm; (b) x y plane for 0 z 4 cm and 136 z 160 cm. Figure : The relative errors of the thermal flux of numerical techniques with respect to 3DKIN.

11 Numerical techniques for the neutron diffusion equations 659 Table 4: Thermal neutron flux at different points of three-dimensional TWIGL reactor with a positive ramp reactivity Method Time Thermal neutron flux at different points (s) (0,80,16) (40,40,16) (0,80,80) (40,40,80) (0,80,144) (40,40,144) 3DKIN AMF NT-FMM NT-AM DKIN AMF NT-FMM NT-AM DKIN AMF NT-FMM NT-AM DKIN AMF NT-FMM NT-AM DKIN AMF NT-FMM NT-AM DKIN AMF NT-FMM NT-AM DKIN AMF NT-FMM NT-AM

12 660 A. Nahla, F. Al-Malki and M. Rokaya Table 5: Thermal neutron flux at different points of three-dimensional TWIGL reactor with a negative ramp reactivity Method Time Thermal neutron flux at different points (s) (0,80,16) (40,40,16) (0,80,80) (40,40,80) (0,80,144) (40,40,144) AMF NT-FMM NT-AM AMF NT-FMM NT-AM AMF NT-FMM NT-AM AMF NT-FMM NT-AM AMF NT-FMM NT-AM AMF NT-FMM NT-AM AMF NT-FMM NT-AM

13 Numerical techniques for the neutron diffusion equations 661 p(0, 80, 16); p(40, 40, 16); p(0, 80, 80); p(40, 40, 80); p(0, 80, 144) and p(40, 40, 16) (cm) with a negative linear { ramp reactivity as t, for 0 t 0. (s); Σ a (material 1) = , for t 0. (s). This table substantiates that the thermal neutron flux of the NT-FMM and NT-AM are accurate comparing to the thermal neutron flux of the AMF. Tables 4 and 5 substantiate the accuracy of the numerical techniques, NT-FMM and NT-AM, for solving the neutron diffusion equations in three-dimensional TWIGL reactor with different types of a linear ramp reactivity. 6 Conclusions The numerical techniques based on finite difference method, fundamental matrix method and analytical method are presented for the numerical solution of the transient two energy groups of neutron diffusion and one group of delayed precursor equations in three-dimensional reactors. These codes are characterized by the expansion of the coefficients matrix for all mesh points of the reactor into two different matrices, specified for each mesh point of the reactor. The codes were tested with two different benchmark problems in three-dimensional geometry, namely the three-dimensional homogenous and TWIGL heterogenous reactors. The fast and thermal neutron flux of the numerical techniques, NT-FMM and NT-AM, were compared to those provided by traditional codes, with an overall agreement and good performance. We can conclude that NT-FMM and NT-AM are accurate codes and they are the feasibility for solving the neutron diffusion equations in three-dimensional homogenous and heterogenous reactors. ACKNOWLEDGEMENTS The authors gratefully acknowledge the Taif University for the financial support (Project No: ). References [1] A.A. Nahla, Analytical solution to solve the point reactor kinetics equations, Nuclear Engineering and Design, 40 (010), [] A.E. Aboanber and A.A. Nahla, Solution of two-dimensional spacetime multigroup reactor kinetics equations by generalized Padé and cutproduct approximations, Annals of Nuclear Energy, 33 (006), 09 -.

14 66 A. Nahla, F. Al-Malki and M. Rokaya [3] A.E. Aboanber and A.A. Nahla, Adaptive matrix formation (AMF) method of spacetime multigroup reactor kinetics equations in multidimensional model, Annals of Nuclear Energy, 34 (007), [4] A.E. Aboanber and Y.M. Hamada, Generalized RungeKutta method for two- and three-dimensional spacetime diffusion equations with a variable time step, Annals of Nuclear Energy, 35 (008), [5] A.E. Aboanber and Y.M. Hamada, Computation accuracy and efficiency of a power series analytic method for two- and three-space-dependent transient problems, Progress in Nuclear Energy, 51 (009), [6] A. Gupta, R.S. Modak, H.P. Gupta, V. Kumar and K. Bhatt, Parallelised Krylov subspace method for reactor kinetics by IQS approach, Annals of Nuclear Energy, 3 (005), [7] A. Quarteroni, R. Sacco and F. Saleri, Numerical Mathematics, Springer- Verlag, New York, Inc. USA, 000. [8] B. Quintero-Leyva, The multi-group integro-differential equations of the neutron diffusion kinetics. Solutions with the progressive polynomial approximation in multi-slab geometry, Annals of Nuclear Energy, 37 (010), [9] D. Ginestar, R. Miró, G. Verdú and D. Hennig, A transient modal analysis of a BWR instability event, Journal of Nuclear Science and Technology, 39 (00), [10] D.L. Hetrick, Dynamics of Nuclear Reactors, American Nuclear Society, La Grange Park, [11] D.R. Ferguson and K.F. Hansen, Solution of the space-dependent reactor kinetics equations in the three dimension, Nuclear Science and Engineering, 51 (1973), [1] D.W. Jordan and P. Smith, Nonlinear Ordinary Differential Equations, 4th edition, Oxford University Press, 007. [13] K. Kobayashi, A rigorous weight function for neutron kinetics equations of the quasi-static method for subcritical systems, Annals of Nuclear Energy, 3 (005), [14] K.S. Smith, An analytic nodal method for solving the two-group, multidimensional, static and transient neutron diffusion equations, M. Sc. Thesis, Nuclear Engineering, Massachusetts Institute of Technology, 1979.

15 Numerical techniques for the neutron diffusion equations 663 [15] L.M. Grossman and J.P. Hennart, Nodal diffusion methods for space-time neutron kinetics, Progress in Nuclear Energy, 49 (007), [16] M. Dahmani, A.M. Baudron, J.J. Lautard and L. Erradi, A 3D nodal mixed dual method for nuclear reactor kinetics with improved quasistatic model and a semi-implicit scheme to solve the precursor equations, Annals of Nuclear Energy, 8 (001), [17] M. Tamitani, T. Iwamoto and B.R. Moore, Development of kinetics model for BWR core simulator AETNA, Journal of Nuclear Science and Technology, 40 (003), [18] R. Miró, D. Ginestar, G. Verdú and D. Hennig, A nodal modal method for the neutron diffusion equation. Application to BWR instabilities analysis, Annals of Nuclear Energy, 9 (00), [19] R.S. Modak and A. Gupta, A scheme for the evaluation of dominant timeeigenvalues of a nuclear reactor, Annals of Nuclear Energy, 34 (007), [0] R.S. Varga, Matrix Iterative Analysis, Prentice Hall, Englewood Cliffs, New Jersey, 196. [1] S. Glasstone and A. Sesonske, Nuclear Reactor Engineering, Chapman & Hall Inc, [] S. González-Pintor, D. Ginestar and G. Verdú, 010. Time integration of the neutron diffusion equation on hexagonal geometries, Mathematical and Computer Modelling, 5 (010), [3] S.R. Shimjith, A.P. Tiwari, M. Naskar and B. Bandyopadhyay, Spacetime kinetics modeling of Advanced Heavy Water Reactor for control studies, Annals of Nuclear Energy, 37 (010), [4] W.M. Stacey, Nuclear Reactor Physics, John Wiley and Sons, Inc. USA, 001. Appendix A: Coefficients α k of analytical method Coefficients α k of analytical method for two energy groups and average one group of delayed neutron are expanded as follows α 0 = ω 1ω exp(hω 0 ) (ω 1 ω 0 )(ω ω 0 ) + ω 0ω exp(hω 1 ) (ω 0 ω 1 )(ω ω 1 ) + ω 0ω 1 exp(hω ) (ω 0 ω )(ω 1 ω ) ()

16 664 A. Nahla, F. Al-Malki and M. Rokaya α 1 = (ω 1 + ω ) exp(hω 0 ) h(ω 1 ω 0 )(ω ω 0 ) + (ω 0 + ω ) exp(hω 1 ) h(ω 0 ω 1 )(ω ω 1 ) + (ω 0 + ω 1 ) exp(hω ) h(ω 0 ω )(ω 1 ω ) (3) α = exp(hω 0 ) h (ω 1 ω 0 )(ω ω 0 ) + exp(hω 1 ) h (ω 0 ω 1 )(ω ω 1 ) + exp(hω ) h (ω 0 ω )(ω 1 ω ) (4) Appendix B: Fundamental Matrix For two energy groups and average one group of delayed neutrons, the matrix A takes the form A = σ 1 τ 1, f 1 τ,1 σ 0 p 1 p λ (5) where, λ is average decay precursors constant. The eigenvalues ω k of matrix A are the roots of the following equations ω 3 (σ 1 + σ λ)ω (τ 1, τ,1 + p 1 f 1 σ 1 σ + λσ 1 + λσ )ω +f 1 p 1 σ f 1 p τ,1 + λσ 1 σ λτ 1, τ,1 = 0 (6) The eigenvectors U k of matrix A are taken the following form U k = 1 τ,1 (ω k σ ) φ k (ω k σ )(ω k +λ) (7) where, φ k = p 1 (ω k σ ) + p τ,1, and k = 0, 1,. The fundamental matrix Θ(t m ), equation (1), takes the following form Θ(t m ) = τ,1 (ω 0 σ ) φ 0 (ω 0 σ )(ω 0 +λ) τ,1 (ω 1 σ ) φ 1 (ω 1 σ )(ω 1 +λ) τ,1 (ω σ ) φ (ω σ )(ω +λ) (8) The inverse of the fundamental matrix Θ(t m ), is taken the following form Θ 1 (t m) = 1 Θ (ω 0 σ ) { φ (ω +λ) φ 1 (ω 1 σ ) { φ 0 (ω 0 +λ) φ (ω σ ) { φ 1 (ω 1 +λ) φ 0 (ω 0 +λ) } (ω0 σ { ) φ1 (ω σ ) φ (ω 1 σ ) (ω 1 +λ) τ,1 (ω 1 +λ) (ω +λ) } (ω1 σ { ) φ (ω 0 σ ) φ 0(ω σ ) (ω +λ) τ,1 (ω +λ) (ω 0 +λ) } (ω σ { ) φ0 (ω 1 σ ) φ 1(ω 0 σ ) τ,1 (ω 0 +λ) (ω 1 +λ) (ω where, Θ = φ 1 ω ) 0 + φ (ω 0 +λ) 1 (ω ω 0 ) + φ (ω 1 +λ) (ω 0 ω 1 ) Received: February, 01 (ω +λ). } (ω0 σ )(ω 1 ω ) } (ω1 σ )(ω ω 0 ) } (ω σ )(ω 0 ω 1 ) (9)