# Mathematical Modelling of Sand Filtration to Improve Water Quality

## Amelia Warton

### Abstract

The use of sand filters is investigated to reduce the concentration of pollution particles present in unclean water. This is achieved by constructing, solving and analysing a simple model; then extending the model to incorporate more physical phenomena. The initial model which is derived accounts for dispersion, advection and adsorption, where adsorption is represented by a linear function. It is solved analytically, and then numerically via finite difference methods and the numerical method of lines. The effects of changes in velocity, length and attachment rate were considered, with the ultimate goal being to determine how certain changes in these parameters affect the efficiency of the filter.

The extended model was created to improve upon the limitations of the simple model. In particular, the extended model accounts for the porosity of the sand and incorporates three stages of filtering which have been implied by experimental data [1]. This allows the attachment of pollution to sand particles via advection to become a process whereby pollution adheres to the sand while also continually being washed back into the water. The governing equation is formed by combining an advection-dispersion equation with an equation of nonlinear multistage accumulation kinetics.

Matlab code was formulated to implement the method of lines in solving both the simple and extended model. Additionally, the method of lines solution for the extended model is an improvement upon the numerical solution formed by Gitis et al. [2], which involved fully discretising the problem. Finally, the graphs generated provide the basis for a discussion about the outcomes of improving the simple model.

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Page Created: 20th June 2013.
Last Updated: 20th June 2013.