Numerical solution of differential and integral equations
documentclass [12pt, a4paper] {article} itle {One non-stationary mathematical model of a method ultrafiltration} author {Zaharov J.N., Lobasenko R.B., Ragulin V.J.} egin {document} maketitle indent In the modern food-processing industry wide application Have found various membrane methods. In particular, for division and condensation solutions of high-molecular substances The method of a ultrafiltration, is used by prominent feature which formation at a surface of a membrane of a layer is with the increased concentration of the dissolved substances (polarizing layer). Its thickness gradually grows. At excess level of concentration (concentration gel-forming c*), on surfaces of a membrane there is a formation of a layer of gel. Formation less-permeability a layer of gel on a surface of the membrane, arising in result of the phenomenon of concentration polarization, serves one of principal causes of decline of productivity membrane ultrafiltrational devices. A numerical estimation of features this process, it is important as for the analysis of a ultrafiltration in general, so and for development membrane devices, in which basis of work removal of the concentrated solution directly lays from closed-membrane areas. ewline indent One of mathematical models given process and its decision have been shown in ~cite {semenov, zaharov}. However, for the profound research developments of examined process in time for us it is necessary non-stationary spacing mathematical model. We shall consider non-stationary model of process of a filtration in round cylindrical to membrane on area $xin [0, l] $, $yin [0, a] $ in the following kind: egin {equation} label {trivial} left { egin {array} {rcl} displaystyle frac {{partial} vec {v}} {{partial} t} + (vec {v} abla) vec {v} + frac { abla p} { ho} = vec {f} + u Delta vec {f}, div vec {v} =0 end {array} ight. end {equation} egin {equation} label {trivial} displaystyle frac {{partial} vec {v}} {{partial} t} + (vec {v} abla) C=div (D abla C), end {equation} oindent where $D=const $ - factor of diffusion, $ u $-- factor of kinematic viscosity, $ vec v $ - speed of current, $ ho $ - density, $ abla p $ - a gradient of pressure on the ends membranes, $C $ - сoncentration of solvent. ewline indent the boundary condition on a surface of a membrane we shall set as: egin {equation} label {trivial} displaystyle frac {{partial} C (t, x, y)} {{partial} y} mid _ {y=a} =-V (t) cdot (P _ {l}-P _ {atm}) cdot C (t, x, y) mid _ {y=a} end {equation} Here $V (t) $ - permeability of a membrane. ewline indent In practice the offered model can be realized in several ways. Simplest from them consists in consideration of current of a liquid as current Puazeil. Development this variant consideration of current Puazeil is with piece-constant viscosity that will allow to reflect the fact formations of a boundary layer at a surface of a membrane. And, at last, finding of speed of current from the equation (1) for reception the most exact decision. ewline indent The numerical realization, formulated by us mathematical models of non-stationary process of a ultrafiltration in round to cylindrical membrane, on a complex of programs have shown it adequacy and the practical importance. In particular us was it is simulated effect of mechanical clearing membranes. egin {thebibliography} {10} ibitem {semenov} Semenov A.G., Lobasenko B.A. " Mathematical description of process of a ultrafiltration with the account gel-forming on a surface of a membrane ". // Storage and processing, 2001, №8, with. 15-17. ibitem {zaharov} Zaharov J.N., Lobasenko R.B., Semenov A.G. " The analysis of model of process gel-forming at a ultrafiltration on a flat membrane ". // Prospects manufactures of food stuffs of new generation: the collection of materials of the international scientific - practical conference devoted to the 85-anniversary Omsk state agrarian university / OmSU. - Omsk, 2003. - 338с. with ill. end {thebibliography} end {document}
Note. Abstracts are published in author's edition
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