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A Finite-Difference Solution of Solute Transport through a Membrane Bioreactor

机译：通过膜生物反应器的溶质运移的有限差分解决方案

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摘要

The current paper presents a theoretical analysis of the transport of solutes through a fixed-film membrane bioreactor (MBR), immobilised with an active biocatalyst. The dimensionless convection-diffusion equation with variable coefficients was solved analytically and numerically for concentration profiles of the solutes through the MBR. The analytical solution makes use of regular perturbation and accounts for radial convective flow as well as axial diffusion of the substrate species. The Michaelis-Menten (or Monod) rate equation was assumed for the sink term, and the perturbation was extended up to second-order. In the analytical solution only the first-order limit of the Michaelis-Menten equation was considered; hence the linearized equation was solved. In the numerical solution, however, this restriction was lifted. The solution of the nonlinear, elliptic, partial differential equation was based on an implicit finite-difference method (FDM). An upwind scheme was employed for numerical stability. The resulting algebraic equations were solved simultaneously using the multivariate Newton-Raphson iteration method. The solution allows for the evaluation of the effect on the concentration profiles of (i) the radial and axial convective velocity, (ii) the convective mass transfer rates, (iii) the reaction rates, (iv) the fraction retentate, and (v) the aspect ratio.
机译：目前的论文提出了通过固定化膜生物反应器（MBR）的溶质的运输的理论分析，该反应器固定有活性生物催化剂。通过MBR解析和数值求解了变系数的无因次对流扩散方程。该分析解决方案利用规则的扰动，并考虑了径向对流以及基底物质的轴向扩散。对于沉项，假设为Michaelis-Menten（或Monod）速率方程，并且扰动扩展到二阶。在解析解中，仅考虑了Michaelis-Menten方程的一阶极限。因此解决了线性方程。然而，在数值解中，这一限制被解除了。非线性，椭圆形，偏微分方程的求解基于隐式有限差分法（FDM）。为了数值稳定，采用了迎风方案。使用多元牛顿-拉夫森迭代方法同时求解所得的代数方程。该解决方案允许评估以下浓度对浓度分布的影响：（i）径向和轴向对流速度；（ii）对流传质速率；（iii）反应速率；（iv）截留馏分；和（v ）长宽比。

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  • 来源
    《Mathematical Problems in Engineering》 |2015年第5期|810843.1-810843.8|共8页
  • 作者

    Godongwana B.; Solomons D.; Sheldon M. S.;

  • 作者单位

    Cape Peninsula Univ Technol, Dept Chem Engn, ZA-8000 Cape Town, South Africa.;

    Univ Cape Town, Dept Math & Appl Math, ZA-7700 Rondebosch, South Africa.;

    Cape Peninsula Univ Technol, Dept Chem Engn, ZA-8000 Cape Town, South Africa.;

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