Diffusion flux/effective coefficient

The Chemkin package deals with problems that can be stated in terms of equation of state, thermodynamic properties, and chemical kinetics, but it does not consider the effects of fluid transport. Once fluid transport is introduced it is usually necessary to model diffusive fluxes of mass, momentum, and energy, which requires knowledge of transport coefficients such as viscosity, thermal conductivity, species diffusion coefficients, and thermal diffusion coefficients. Therefore, in a software package analogous to Chemkin, we provide the capabilities for evaluating these coefficients. ... 

In considering axial dispersion as a diffusive flow, we assume that Fick s first law applies, with the diffusion or effective diffusion coefficient (equation 8.5-4) replaced by an axial dispersion coefficient, D,. Thus, for unsteady-state behavior with respect to a species A (e.g., a tracer), the molar flux (NA) of A at an axial position x is... 

It should be pointed out that for a low pressure gas the radial- and axial diffusion coefficients are about the same at low Reynolds numbers (Rediffusion effects may be important at velocities where the dispersion effects are controlled by molecular diffusion. For Re = 1 to 20, however, the axial diffusivity becomes about five times larger than the radial diffusivity [31]. Therefore, the radial diffusion flux could be neglected relative to the longitudinal flux. If these phenomena were also present in a high-pressure gas, it would be true that radial diffusion could be neglected. In dense- gas extraction, packed beds are operated at Re > 10, so it will be supposed that the Peclet number for axial dispersion only is important (Peax Per). 

Unsteady state diffusion in monodisperse porous solids using a Wicke-Kallenbach cell have shown that non-equimolal diffusion fluxes can induce total pressure gradients which require a non-isobaric model to interpret the data. The values obtained from this analysis are then suitable for use in predicting effectiveness factors. There is evidence that adsorption of the non-tracer component can have a considerable influence on the diffusional flux of the tracer and hence on the estimation of the effective diffusion coefficient. For the simple porous structures used in these tests, it is shown that a consistent definition of the effective diffusion coefficient can be obtained which applies to both the steady and unsteady state and so can be used as a basis of examining the more complex bimodal pore size distributions found in many catalysts. 

Diffusion coefficient. Dm. Coefficient characterizing the intensity of the diffusive flux or the rate of diffusion along a concentration gradient. In principle, the diffusion coefficient is a function of the concentration, and the diffusive flux of one component depends on the concentrations of all the components of the system. These effects are neglected here, since only dilute solutions are used in chromatography. For the same reason, diffusion coefficients at infinite dilution are used. 

This allows us to make a homogeneous model of the porous catalyst pellet in which we now have a diffusive flux given as the product of an effective diffusion coefficient and a concentration gradient, and a rate of reaction given by the product of the catalytic area and the reaction rate per unit area. 

However, if the axial dispersion can be described by an effective diffusion coefficient giving a diffusive flux of Eai—dc/dz), then a balance over the clement between z and z + dz will give ... 

Wilke [103] proposed a simpler model for calculating the effective diffusion coefficients for diffusion of a species s into a multicomponent mixture of stagnant gases. For dilute gases the Maxwell-Stefan diffusion equation is reduced to a multicomponent diffusion flux model on the binary Pick s law form in which the binary diffusivity is substituted by an effective multicomponent diffusivity. The Wilke model derivation is examined in the sequel. 

Looking at diffusion processes in the pore water volume of sediments, it must be taken into consideration that diffusion can only take place within the pore water volume (porosity ( )), hence a diffusive flux can only be proportionally effective with regard to this spatial compartment. Beyond this limitation, the diffusion coefficient is distinctly lower in the pore water volume of a sediment (Dsed) than in free solution. The diffusive flux in the sediment (J ed) is calculated as ... 

In this section we examine the flow of a suspension of particles, particularly the apparent viscosity coefficient of the suspension. Our interest is in calculating the convective mass flux of a suspension as distinct from the diffusive flux of Brownian motion. As previously, we shall assume a very dilute suspension in which each particle behaves as if it were in a liquid of infinite extent. To simplify the calculation, we neglect Brownian motion, although, as we discuss later, in the very dilute limit considered and for spherical particles it has no effect on the suspension viscosity. 

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