Mathematical modeling of the undetachable filtration of a water suspension with changing the flow direction

Abstract

A theoretical study of the clarifying effect at a rapid filter when changing the direction of the suspension flow in its medium during the filter run (descending to ascending or vice versa) is performed. As an instrument for research, mostly exact analytical methods have been used. In the technological process of filtration, two stages are conventionally distinguished — before and after changing the place of a suspension supply. The theoretical analysis is based on a mathematical model of undetachable nonlinear filtration. The composition and amount of a dispersed contamination in the initial suspension are stable, and the filter medium is initially clean (first stage) or already contains a large amount of a deposition mainly near the outlet (second stage). The suspension flow in the contaminated medium obeys the linear law with hydraulic conductivity, which is an empirical function of the concentration of deposited particles. The exact solution in implicit form of the corresponding mathematical problem is presented in relation to the first stage of filtration, which allows us to specify the physico-chemical picture in the medium layer as far as the beginning of the second stage. An arbitrary form of the functional filtration coefficient is allowed, which requires the use of numerical methods to solve this problem in the second stage. The initial system of ordinary differential equations in the canonical form and the procedure for calculating the most important filtration characteristics based on the data array thus obtained are presented. A special case of a linear form of the filtration coefficient is analyzed separately by strict analytical methods. In a number of examples with typical initial data, the derived calculation equations and dependences are used to establish the effect due to a sharp change in the direction of the suspension flow during one filter run. It is shown that, in this way, the quality of the filtrate is deteriorated minimally. However, due to the active participation in the suspension clarification of the virtually entire volume of the filter medium, it is possible to achieve a more uniform distribution of the deposition in it and, as a result, a very significant reduction in head losses. Thus, based on the calculations of the technological time (the maximum permissible head losses are achieved), there is a real opportunity to extend the continuous operation time of rapid filters by 25 % or more. The nonlinear problem of transfer and deposition of ferric iron in the layer of a fast filter bed is formulated with regard for the oxidation of ferrous iron and definitely solved. The equations for the calculation of changes over time and over the height of the bed in the concentrations of suspended and deposited particles of iron hydroxide and the increase of a head loss in it are constructed. The forecast of the concentration of iron hydroxide in the filtrate and deposited form is done on examples. The possibility of a reliable substantiation of technological and constructive parameters based on the obtained solutions is shown.