Abstract
Diffusivity is a strong function of concentration and an important transport property. Diffusion of multiple species is far more frequent than the diffusion of one species. However, there are limited experimental data available on multi-component diffusivity. The objective of this study is to develop an optimal control framework to determine multi-component concentration-dependent diffusivities of two gases in a non-volatile phase such as polymer. In Part 1 of this study, we derived a detailed mass-transfer model of the experimental diffusion process for the non-volatile phase to provide the temporal masses of gases in the polymer. The determination of diffusivities is an inverse problem involving principles of optimal control. Necessary conditions are determined to solve this problem. In Part 2 of this study, we utilized the results of Part 1 to determine the concentration-dependent, multi-component diffusivities of nitrogen and carbon dioxide in polystyrene. To that end, solubility and diffusion experiments are conducted to obtain necessary data. In the ternary system of nitrogen (1), carbon dioxide (2), and polystyrene (3), the diffusivities and D11, D12, D21, and D22 versus the gas mass fractions are two-dimensional surfaces. The diffusivity of carbon dioxide was found to be greater than that of nitrogen. The value of the main diffusion coefficient D11 was found to increase as the concentration of carbon dioxide increased. The highest value of D11 obtained was 2.2 X 10^-8m^2s^-1 for nitrogen mass fraction of 3.14 X10^-4 and for a carbon dioxide mass fraction of 5.67 X 10^-4 . The cross-diffusion coefficient increased as the concentrations of nitrogen and carbon dioxide increased. The diffusivity reached its maximum value when the concentrations of nitrogen and carbon dioxide were at their maximum values. The diffusivity was of the order of 10^-9m^2s^-1. The diffusivity of the cross-diffusion coefficient D21 was found to be increased for the mass The diffusivity of the cross-diffusion coefficient was found to be increased for the mass fractions of carbon dioxide ranging from 0 to 1.70 X 10^-3 . The diffusivity was found to be of the order of . The diffusion coefficient, D22, was found to increase with the concentrations of nitrogen and carbon dioxide, D22 remained high with low concentrations of carbon dioxide. The diffusivity was found to be of the order of 10^-7m^2s^-1
Highlights
Diffusion can be defined in very simple terms as the movement of molecules from one place to another due to a concentration gradient (Philibert, 2005)
The problem is that this assumption is only good for cases in which diffusion is occurring in a two-component system, in a system in which one component is diluted by a large excess of one or more of the other components, or in a system in which all of the components in the mixture have similar quantities and natures
The first component can diffuse against its concentration gradient, i.e., from a region of lower concentration to the region of higher concentration
Summary
Diffusion can be defined in very simple terms as the movement of molecules from one place to another due to a concentration gradient (Philibert, 2005). The question arises concerning whether the three or more components diffusing in a system can be dealt in a same way as the components of a binary system. Answer to this question is “No,” so we must deal with the problem of how to deal with a multi-component system. Concerns such as this have been on the minds of chemical engineers for a long time(Krishnamurthy & Taylor, 1982). The matrix of a multi-component system can be incorporated into powerful computer software, which can be used in equipment design.
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