Multidimensional spacial eigenmode analysis. Page: 4 of 8
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where v, is the sth zero of the mth order Bessel function. As expected, two-dimensional squares (or
rectangles) yield cosines and sines [Ref. 3].
When we examine the eigenvalues derived from a fission matrix analysis for the two-
dimensional cylinder [Eq. (2)], we see a single eigenvalue followed by a series of pairs, followed by a
singlet, followed by a series of pairs, etc. As evident in Eq. (2), when m = 0, the eigenfunction
associated with the sine function is eliminated and we are left with the zeroth order Bessel function
eigenfunctions (these are the resultant eigenfunctions in the one-dimensional cylindrical system).
Thus, the fission matrix analysis yields the expected eigensystems for the two-dimensional cylinder
associated with a simple analytic diffusion analysis.
The importance of this result is seen when we examine reflected cylindrical systems in a
similar fashion to Ref. 3. The goal is to approximate the dominance ratio (the ratio of the first
higher mode eigenvalue to the fundamental eigenvalue, or k,), which can change as we introduce
reflective boundary conditions (e.g., a half- or quarter-core).
We consider a homogeneous infinite cylinder (E,= 1, vEX= 0.24, E, = 0.7). Table 1 contains
the results of the two-dimensional cylinder fission matrix analysis. The asymptotic results are
determined from diffusion theory analysis. A single reflecting plane yields a first higher mode
eigenvalue related to the first zero of the first order Bessel function (3.831...), while a quarter
cylinder yields a higher eigenvalue related to the second zero of the zeroth order Bessel function
(5.520...). From Ref. 3, we note that the DR asymptotically approaches (k /k), where
7 ~ (v,,2 - v"02)/v102, (vI, = 2.4048...). In general, results for the larger cylinder are closer to the
asymptotic values, as expected. We note a decrease in the dominance ratio only if we reflect two
surfaces, i.e., quarter-core. In Cartesian systems, there were also occasions where introducing a
single reflecting surface did not decrease the dominance ratio. Because of the azimuthal symmetry
of the surviving eigenmode in the quarter-core, no further DR reductions are possible.
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Parsons, Donald Kent & Kornreich, D. E. (Drew E.). Multidimensional spacial eigenmode analysis., article, January 1, 2003; United States. (https://digital.library.unt.edu/ark:/67531/metadc933229/m1/4/: accessed April 22, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.