Critical assessment of the stability and convergence of the equations of multi-component diffusion

The transport phenomenon in many engineering applications is dominated by multi-component diffusion. Examples include low-pressure chemical vapor deposition, mass transport in gas diffusion layers in fuel cells, and transport of chemicals in biomedical devices. With increasing interest in small-scale devices, diffusion dominated (i.e., low Peclet number) transport is likely to become more important in engineering applications of the next generation. Unlike in binary systems (consisting of two species), diffusion of a certain species in a multi-component system is dictated not only by its own concentration gradient but also by the concentration gradient of the other species in the system [1,2]. This results in a system of strongly coupled nonlinear second-order elliptic partial differential equations. In the case of binary systems, or in the case of dilute mixtures wherein a certain ‘‘reference’’ species constitute most of the mixture, these equations become segregated, or are only weakly coupled [1,2]. For concentrated non-binary mixtures, segregation of the governing equations cannot be performed naturally, and is performed artificially only to make the system of equations amenable to numerical solution. Segregated solution of the governing equations, wherein only the self-diffusion operator is treated implicitly, while the diffusion due to the other species is treated explicitly, results in an iterative algorithm whose convergence depends on the strategy used to conserve overall mass. In this short note, it is shown that if the mass fraction summation criterion (i.e., mass fractions of species summing to unity) is imposed explicitly, and if the transport properties are not held constant, the convergence of this semi-implicit system of equations requires under-relaxation even for one-dimensional calculations. For multi-dimensional problems,