Mathematical models variable diffusion coefficient

Scientists developed a two-stage model, which takes into account water-vapor-sorption kinetics of wool fibers and can be used to describe the coupled heat and moisture transfer in wool fabrics. The predictions fi om the model showed good agreement with experimental observations obtained from a sorption-cell experiment. More recently, Scientists further improved the method of mathematical simulation of the coupled diffusion of the moisture and heat in wool fabric by using a direct numerical solution of the moisture-diffusion equation in the fibers with two sets of variable diffusion coefficients. These researeh publieations were focused on fabrics made fi om one type of fiber. The features and differences in the physical mechanisms of coupled moisture and heat diffusion into fabrics made fi om different fibers have not been systematieally investigated. 

In this chapter generalized mathematical models of three dimensional electrodes are developed. The models describe the coupled potential and concentration distributions in porous or packed bed electrodes. Four dimensionless variables that characterize the systems have been derived from modeling a dimensionless conduction modulus ju, a dimensionless diffusion (or lateral dispersion) modulus 5, a dimensionless transfer coefficient a and a dimensionless limiting current density y. The first three are... 

One major shortcoming of these models is that the flow property during dyeing is not defined in a sonnd mathematical form, since, for both the STM and MIM approaches, an exact solntion of the problem of convective diffusion to a sohd surface first requires the solution of the hydrodynamic equations of motion of the fiuid for boundary conditions appropriate to the mainstream velocity of flow and the shape of the package. Another limitation of their work is that those workers did not consider situations like variable boundary conditions and variable dispersion coefficients, which are quite common situations in dyeing practice, in their numerical simulations. 

The applicable mathematical form of the flux equation is model dependent. For example, the first type model consists of differential equations (DEs). They are developed to yield concentration profiles in the sediment layers as well as the flux. These DEs typically use Equation 4.1 as a boundary condition. The solutions to these DEs require one or more of the following boundary condition categories the Dirichlet condition, the Neuman condition, or a third condition. The first two types are the most common these require mathematical functions containing gradients of the dependent variable (i.e., Cw) as well as functions of the dependent variable itself. For these diffusive-type fluxes, the transport parameter is a diffusion coefficient such as Dg. Several other transport parameters are commonly used and represent diffusion in air and the biodiffusion or bioturbation of soil/sediment particles. 

Thus the model based on the diffusion coefficient gives results of more fundamental value than the model based on mass transfer coefficients. In mathematical terms, the diffusion model is said to have distributed parameters, for the dependent variable (the concentration) is allowed to vary with all independent variables (like position and time). In contrast, the mass transfer model is said to have lumped parameters (like the average hydrogen concentration in the metal). 

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