Diffusion with Nonlinear Adsorption
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Diffusion with Nonlinear Adsorption Tomáš Fürst · Rostislav Vodák
Received: 8 April 2008 / Accepted: 5 July 2008 / Published online: 26 July 2008 © Springer Science+Business Media B.V. 2008
Abstract This article deals with the mathematical description of the process of washing a contaminant out of a porous material sample. Important applications of this process in the leather-making industry are discussed. Part of the contaminant is held in the porous material by an active chemical bond due to adsorption which makes the governing equations nonlinear. We prove existence and uniqueness of a weak solution to the proposed model and show its stabilization for time converging to infinity. Keywords Diffusion · Adsorption · Stabilization 1 Introduction and Physical Background During the process of turning raw hide into leather, one meets the problem of washing a certain chemical contaminant out of the treated hide. Part of the contaminant is present in the form of water solution but a certain portion of it is held in the hide by an active chemical bond due to adsorption. It is assumed that the process of adsorption takes place in the entire volume of the treated hide due to high porosity of the material. At each point of the hide and any instant, the adsorbed concentration of the contaminant is in equilibrium with the concentration of the water solution. The equilibrium is described by the so-called Langmuir isotherm [1, 2] which takes the form cA =
Bc , 1 + Lc
where cA stands for the concentration of the adsorbed contaminant, c stands for the concentration of the water solution and B and L are positive constants. T. Fürst () · R. Vodák Dept. of Math. Analysis, Fac. of Science, Palacký University, Tomkova 40, 779 00 Olomouc-Hejˇcín, Czech Republic e-mail: [email protected] R. Vodák e-mail: [email protected]
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T. Fürst, R. Vodák
During the process of washing, one needs to get rid of the contaminant so that its overall concentration cA + c in the treated hide decreases below a certain limit. It is usually very time-consuming (and thus expensive) to reach this limit only by washing the hide in pure water (i.e. only by the mechanism of diffusion). Thus, at certain instant, a chemical agent is added which neutralizes the remaining contaminant. However, the chemical agent is expensive and its application results in an additional need to clean the sewage waters. Consequently, there arises the need to understand the washing process well to be able to optimize the described procedure. This paper deals with the mathematical description of the washing process which precedes the application of the neutralizing agent. Let us consider a hide sample that is represented by a set 0 ⊂ RN . The sample is placed in a basin of liquid represented by a set ⊃ 0 . At time t = 0, the sample contains the contaminant at a given concentration (a portion of which is adsorbed), which is then gradually washed away into the surrounding liquid by the process of diffusion. Diffusion takes place in the whole system, in the sample together wi
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