Multi-Scale Modeling of the Dynamics of a Fibrous Reactor: Use of an Analytical Solution at the Micro-Scale to Avoid the Spatial Discretization of the Intra-Fiber Space

• Main Authors: Adam Dobri, Thanasis D. Papathanasiou
• Format: Article
• Language: English
• Published: MDPI AG 2019-12-01
• Series: Fluids
• Subjects: multi-scale analysis; fibrous reactors
• Online Access: https://www.mdpi.com/2311-5521/5/1/3

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 &#8220;hybrid&#8221; 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 <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>n</mi> <mn>0</mn> </msub> </mrow> </semantics> </math> </inline-formula> ODEs, where <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>n</mi> <mn>0</mn> </msub> </mrow> </semantics> </math> </inline-formula> 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 &#8220;hybrid&#8221; model for the system consists of a system of <inline-formula> <math display="inline"> <semantics> <mrow> <mi>N</mi> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mo>+</mo> <msub> <mi>n</mi> <mn>0</mn> </msub> </mrow> <mo>)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> ODEs, which is easily solvable on a modest workstation. Parametric analyses are presented and discussed.