Processing of Basic Nuclear Data for Criticality Coefficients Calculations of Fast Homogeneous Systems

Transcription

1 Processing of Basic Nuclear Data for Criticality Coefficients Calculations of Fast Homogeneous Systems Jefferson Neves Pereira Instituto Militar de Engenharia Sergio de Oliveira Vellozo Instituto Militar de Engenharia Paulo Henrique Pereira da Silva Instituto Militar de Engenharia

2 1. Objectives The objective of this work is to solve the Diffusion Equation for an infinite homogeneous system, using a very large number of energy groups, in order to calculate criticality coefficients.

3 Supposing I groups of energy, one will have to solve a equations system with I variables, where the neutron fluxes are the solutions. Let 1 be the smallest energy group and I, the greatest one. Σ t1 ϕ 1 = χ 1 k ( I Σ t2 ϕ 2 = χ 2 k ( I Σ t3 ϕ 3 = χ 3 k ( I i=i i '=I ν Σ fi ϕ i )+ i '=i i=i i' =I ν Σ fi ϕ i )+ i=2 i' =i i=i i' =I ν Σ fi ϕ i )+ i=3 i '=i... Σ s i' i Σ s i' i Σ s i' i Σ ti 1 ϕ I 1 = χ I 1 k ( I νσ fi ϕ i )+Σ s (I 1)(I 1) +Σ s I (I 1) I Σ ti ϕ I = χ I k ( ν Σ fi ϕ i )+Σ s II

4 2.1. Obtaining and organizing the data 2. Procedures Firstly, one obtained the values of cross sections (inelastic and elastic scattering, radioactive capture, fission and total ones) that are available on << exfor/endf.htm>> for the nucleus of Uranium-235. The data were organized as follows:

5 2.2. Data Manipulation using the Fortran Program Data Interpolation. The Diffusion Equation, for neutrons which the solution is the target of this work, asks the cross sections from the same group of energy However, each range of cross sections provided by the IAEA had its specific number of values of energy, varying from 44,901 to 107,560. To solve this problem, one chose specified values of energy, according to the following equation E i =10 5 e (i 1) ev,(,2,3,...,28325 )

6 Data Integration. The Diffusion equation requires groups of energies, and not isolated values of energy. Then, one can not simply solve it using the values of cross sections for each point of energy Ei, but, instead, using integrated cross sections for an energy range,,in this specific case. ΔE=E i+1 E i According to Henry (1975), the integrated cross sections for a group i can be written as follows: νσ fi 1 ΔE i ν Σ fi (E )de Σ ti 1 ΔE i Σ ti (E )de Σ ii' de i ΔE i' Σ s (E i' E i )de i' χ i χ i (E)dE

7 Taking into account that the energy ranges were very small, the chosen method of numerical integration was the Trapezoidal Rule. Comparisons between exact values and the results provided by that rule showed the method to be coherent. The function to be integrated in equation of the inelastic scattering cross section follows the principle of the evaporation theory. Σ s (E' E)=aΣ si' E E' e a E E 1 (1+ ae' )e ae' The fission spectrum has an algebraic expression dependent on Ei (Duderstadt, 1975) χ= e E i sinh ( E i )

8 Solving the Diffusion Equation. The final part of the program should be able to solve a 28,324-unknown variables homogeneous system of equations, in which the variables were the neutron fluxes. One must remember that, despite of 28,325 values of energy, after integration, one found 28,324 energy ranges. To solve such problem, it was necessary to use an iterative process. In this problem, the up-scattering is considered to be null, i.e., there is no scattering from lower to upper energy ranges. Considering that, the Diffusion equation for the last energy range I = can be written simply as: Where S0 represents all the neutron sources and k is called the criticality coefficient. In this way, giving random guessing values for both S0 and k, the value of the flux of the last energy range is found as: where Σ ti ϕ I =χ I S 0 K +Σ sii ϕ I ϕ I =χ I S 0 K 0 Σ RI Σ RI =Σ ti Σ sii

9 Rearranging the Diffusion Equation for the penultimate energy range, one finds: ϕ I 1 = χ ( I 1 ) S 0 K 0 +Σ (I 1) I ϕ I ϕ R (I 1) However, the value of the last flux was previously found. Substituting this value in the equation above, it is found the value of the flux of the penultimate energy range. Doing so, one can find all the remaining fluxes, until reaching the minimum energy range. After that, all the fluxes will be known and one can, therefore, calculate a real value for S0: I S 1 = ν Σ fi ϕ i

10 Through the relation S 1 K 1 = S 0 K 0 It is also possible to calculate a new value for K. With these new values of S0 and K, one can recalculate all the fluxes, until the value of k converges. In this work, the precision value for the convergence of was After the necessary iterations to reach such precision, the program stops calculating the fluxes and provides the last value for k.

11 3. Results After having all the presented procedures done, the fortran program provided a criticality coefficient k = 2.2 after 2 iterations.

12 4. Conclusions Using the software scale, simulating a system in the same conditions as that on worked with, that means, an infinite medium fulled with only Uranium-238, the criticality coefficient found was k = 2.3. Therefore, using this last result as a pattern, it is possible to infer that the procedures done to get the final result were satisfactorily correct and it will be now possible to improve them in order to obtain other results as accurate as possible. Moreover, continuing in an analogous way, the program can be extended to calculate criticality coefficients for heterogeneous systems, which can contain Uranium-238, Plutonium-239, Plutonium-240, Plutonium-241 and Plutonium-242.

13 5. References Evaluatd Nuclear Data File (ENDF), Pacitti, Tercio. FORTRAN-monitor; princípios. 3ed. Rev. e atual. Rio de Janeiro, Livros Técnicos e Científicos, Henry, Allan F. Nuclear analysis. 2ed. The MIT Press, Duderstadt, James. Nuclear Reactor Analisys, Department of Nuclear Engineering, The University of Michigan Press Nasjleti, E. Diffusion of Neutrons in a Heavy Elements Medium and Application to the Multigroup Diffusion Theory. The Engineering Research Institute, The University of Michigan, Ann Arbor. 1957