Recently, there is rekindled interest in developing pebble bed reactors (PBRs) as a candidate of very high temperature gas-cooled reactors (VHTRs). A typical pebble bed reactor core takes cylindrical shape and houses a multitude of graphite balls that are cycled continuously through the core. Until now, most existing methods of nuclear design analysis for this type of reactors are based on old finite-difference solvers or on statistical methods. Therefore, there is strong desire of making available high fidelity coarse-mesh nodal codes in cylindrical geometry.
During the last decade, the analytic function expansion nodal (AFEN) method has been developed and successfully applied to the static and kinetic problems in Cartesian (x,y,z) geometry and hexagonal-z geometry. In this thesis, we extended the analytic function expansion nodal (AFEN) method to the treatment of the full three-dimensional cylindrical (r,$\theta$,z) geometry. The AFEN methodology in this geometry is ``robust``, due to the unique feature of the AFEN method that it does not use the transverse integration. The transverse integration in the usual nodal methods, however, leads to an impasse, that is, failure of the azimuthal term to be transverse-integrated over r-z surface.
More specifically, the features of the work in this thesis are as follows. We use thirteen analytic functions in outer nodes and seven analytic functions in innermost nodes for the new general solution in this geometry. Based on these new general solution, we developed the TOPS code for cylindrical (r,$\theta$,z) geometry. The multi-group extension based on matrix function theory and the coarse group rebalance (CGR) acceleration are also described. The typical pebble bed reactors have void regions in the top and side regions of the core. To treat void regions, we introduced a partial current translation (PCT) method which does not use diffusion coefficients and an AFEN-consistent method which is based on the AFEN method.