A Rapid Method for calculating k_(eff) values for Spent Partitions of radially or axially inhomogeneous Densities, where Surfaces of Fuel Elements are tapered with Porosity

摘要

We see rapid convergence of KK in Tables I through V. In all of these tables we can interpret KK as being buckling in the middle or center of a given 'fuel' element or simple assembly. In other words, in the Tables shown and numerous undisplayed tables, the KK value of the material buckling of the our samples converges rapidly to the correct values. Agreement between the analytical value for critical material buckling and the matrix extracted value of material buckling is to more than 10 significant figures for matrices of size 12 by 12 and over 10 figures for Tables I through n. Table Ⅲ shows similar agreement to reliable independent calculations via the "shooting method" for Ordinary Differential Equations (ODE's). For Table Ⅳ, agreement to 11 significant figures was attained by the 18 × 18 matrix. Table V is especially impressive in that agreement to 11 significant figures is already in the 10 × 10 matrix. Likewise in Tables Ⅳ and V there was verified agreement by selectively using the "shooting" method of ODE's find the properly fitting KK value. In all examples it takes less than 30 seconds of Intel Core-Ⅱ cpu time to extract these roots for Tables I through Ⅲ. In Tables Ⅳ and V, it took less than 40 seconds to extract the lowest root from the 14 × 14 and 16 × 16 matrices of our EBMaG code and formulation package. We took advantage of Fick's 'law' for neutron diffusion in a secondary way by allowing ourselves to transfer the diffusion length 'D' outside to the right-hand term of the n-diffusion equation, so that the 'D' is absorbed within the tapered KK factor, no being subject to the tedious but negligible effects of taking the gradient of D of the fuel element. Although more exhaustive of CPU time and RAM, our EBMaG code and algorithm can be modified to successfully include term of gradient 'D' into the function operator G, which was described within or after (Eq 3) on a prior page.