Bulk Dispersion and Microscale Diffusion in Fibrous Reactors

Direct modeling of time-dependent transport and reactions in realistic heterogeneous
systems, in a manner that considers the evolution of the quantities of interest in both, the macro-scale
(suspending fluid) and the micro-scale (suspended particles), is currently well beyond the capabilities
of modern supercomputing. This is understandable, since even a simple system such as this can
easily contain over 107 particles, whose length and time scales differ from those of the macro-scale
by several orders of magnitude. While much can be gained by applying direct numerical solution
to representative model systems, the direct approach is impractical when the performance of large,
realistic systems is to be modeled. In this study we derive and analyze a “hybrid” model that is
suitable for fibrous reactors. The model considers convection/diffusion in the bulk liquid, as well
as intra-fiber diffusion and reaction. The essence of our approach is that diffusion and (first-order)
reaction in the intra-fiber space are handled semi-analytically, based on well-established theory. As
a result, the problem of intra-fiber transport and reaction is reduced to an easily solvable set of
n0 ODEs, where n0 is the number of terms in the Bessel expansion evaluated without recourse to
approximation; this set is coupled, point-wise, with a numerical model of the macro-scale. When the
latter is discretized using N nodes, the total “hybrid” model for the system consists of a system of
N(2 + n0) ODEs, which is easily solvable on a modest workstation. Parametric analyses are presented
and discussed.