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# Application of Mathematica Software to Solve Pulp Washing Model

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Abstract:

The removal of the bulk liquor surrounding the pulp fibers using less concentrated liquor is known as pulp washing. In the present study, a pulp washing model involving diffusion-dispersion through packed beds of finite length is presented. Separation of variables is applied to solve system of governing partial differential equations and the resulting equations are solved using Mathematica. Results from the present case are compared with those of previous investigators. The present case is giving better results than the previous investigators. 1. Introduction The objective of pulp and paper industry to produce its target production with high efficiency and less environmental load can only be met by initiating a meticulously planned research on mathematical methods. Pulp washing plays an important role in reduction of black liquor solids in the pulp being carried forward for further processing. The efficiency of washing depends on the degree of mixing, rate of desorption, diffusion-dispersion of dissolved solids, and chemicals from the fibrous matrix. Modeling of pulp washing is done mainly using three approaches, namely, (a) process modeling (b) physical modeling, and (c) statistical modeling. A complete review of the various process models used to describe pulp washing has been presented by . Initially researchers like in [2, 3] proposed the models based on axial dispersion. Pellett  introduced a mathematical model combining the effects of particle diffusion and axial dispersion. A detailed model related to mass transfer in fibrous particle was given by ; it was also restricted for axial dispersion only. Comprehensive models involving physical features of the fibers such as fiber porosity and fiber radius were presented by [6, 7]. Extensive study of axial dispersion model has been carried out by [8–26]. The model has been solved using analytic and numerical techniques like Laplace transform technique [2–4, 10, 15, 23, 26], finite difference technique , orthogonal collocation method [5, 7, 12], orthogonal collocation on finite elements [6, 20, 21], Galerkin/Petrov Galerkin method [8, 19], Hermite collocation method by [11, 17, 24] and Spline collocation method . The accuracy of the analytic solution undoubtedly exceeds the limit of applicability of the theory to real situations. Moreover, it is highly desirable to have a simple and consistent model of the transport phenomenon based on essential features of real situation. Keeping this modest goal in mind axial dispersion model is solved along with linear adsorption isotherm. The method

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