Modelling and Simulation of a Packed Bed of Pulp Fibers Using Mixed Collocation Method

A convenient computational approach for solving mathematical model related to diffusion dispersion during flow through packed bed is presented. The algorithm is based on the mixed collocation method. The method is particularly useful for solving stiff system arising in chemical and process engineering. The convergence of the method is found to be of order 2 using the roots of shifted Chebyshev polynomial. Model is verified using the literature data. This method has provided a convenient check on the accuracy of the results for wide range of parameters, namely, Peclet numbers. Breakthrough curves are plotted to check the effect of Peclet number on average and exit solute concentrations. 1. Introduction The removal of the soluble material, occupying the interstitial space between the particles and in the pores of the particles, from a saturated packed bed of particles is carried out by the introduction of a solvent, for example, water or weak wash liquor flowing through the bed. Solute removal is associated with diffusion like dispersion of the solvent in the direction of flow, known as longitudinal dispersion. The mechanics involved are the sum of displacement of the fluid containing solute by movement of water plug controlled by fluid mechanics, dispersion due to back mixing, diffusion due to concentration gradient, and adsorption-desorption due to relative affinity of various solutes towards the particle surface. The core problem is the prediction of the behavior of initially sharp interface between the liquids having identical dynamical and kinematical properties. This problem is of considerable practical importance, for example, in determining the efficiency of solvent utilization or filtrate recovery in washing of filter cakes. Mathematically, such chemical engineering processes can be best described by two point boundary value problems as follows: where , along with the boundary conditions as follows: A great deal of effort has been applied to compute efficiently the solution of transient partial differential equations (1) and (2) analytically [1–7] and numerical algorithms such as Hermite radial basis function interpolation numerical scheme [8], two-stage Lie-group shooting method [9], finite difference method [10–12], spectral collocation method [13], Sinc differential quadrature method [14], orthogonal collocation method [10, 15–17], fitted mesh collocation method [18], a novel numerical scheme [19], Galerkin/Petrov Galerkin method [20–22], orthogonal collocation on finite elements [23–25], factorized diagonal Padé approximation [26], spline