Total diffusion flux

Diffusion in the bulk crystals may sometimes be short circuited by diffusion down grain boundaries or dislocation cores. The boundary acts as a planar channel, about two atoms wide, with a local diffusion rate which can be as much as 10 times greater than in the bulk (Figs. 18.8 and 10.4). The dislocation core, too, can act as a high conductivity wire of cross-section about (2b), where b is the atom size (Fig. 18.9). Of course, their contribution to the total diffusive flux depends also on how many grain boundaries or dislocations there are when grains are small or dislocations numerous, their contribution becomes important. 

The sorption of a penetrant by a class (a) membrane can be treated most simply, if the processes of sorption in the polymer matrix and into each kind of specific sorption site within it can be treated as mutually independent and hence additive. Under these conditions, each sorbed species may similarly be expected to contribute additively to the total diffusion flux. Hence, if there are N — 1 kinds of specific sorption sites, the overall sorption and permeability or diffusion coefficients can be written as... 

In this expression, 3 is a purely geometric factor generated from the ratio of the surface area to the volume of the sphere the first bracket is the maximum total rate of reaction, achieved when the concentration is everywhere at cf the second bracket is the total diffusive flux across the surface when the... 

When using the dusty gas model we obtain the following gradient equation for the total diffusive flux... 

The boundary condition of zero accumulation on the interface between macropores and solid phase is imposed. The effective diffusivity of the porous sample G1 with bimodal pore size distribution is summarized in Fig. 16, where the sample macro-porosity macro is varied on the horizontal axis. This effective diffusivity is compared with a situation where the diffusion transport in nanopores is omitted. The contribution of the transport through the nano-porous solid phase to the total diffusion flux is significant. The calculated effective... 

The total diffusion flux of any species within the pores of the particles of packing material is obtained by combining these different contributions. 

The total diffusion flux J is the amount of substance that reacts on the entire surface per unit time. 

The dimensionless total diffusion flux interval from 0 to a is equal to... 

Taking into account (3.4.12), we obtain the dimensional total diffusion flux to the part of the film boundary from 0 to in the form... 

The corresponding total diffusion flux to the film boundary is fx... 

The analysis of many technological processes involving dissolution, extraction, vaporization, combustion, chemical transformations in dispersions, sedimentation of colloids, etc. are based on the solution of the problem of mass exchange between particles, drops, or bubbles and the ambient medium. For example, in industry one often deals with processes of extraction from drops or bubbles or with heterogeneous transformations on the surface of catalyst particles suspended in a fluid. The rate of extraction and the intensity of a catalytic process to a large extent are determined by the value of the total diffusion flux of a reactant to the surface of particles of the disperse phase, which, in turn, depends on the character of flow and the particle shape, the influence of neighboring particles, the kinetics of the surface chemical reaction, and some other factors. 

For a particle of arbitrary shape in a translational flow, the first three terms of the asymptotic expansion of the dimensionless total diffusion flux as Pe — 0 have the form [62]... 

Mass transfer between a circular cylinder of radius a and a simple shear flow (G 2 = 1, the other Gkm = 0) was studied in [132]. For the dimensionless total diffusion flux per unit length of the cylinder, the following expression was obtained as Pe 0 ... 

Formulas for calculating diffusion fluxes. The dimensionless total diffusion flux to a part of the surface of a particle (drop or bubble) between neighboring stagnation lines (points) rjk and rjk+i can be calculated by the formula [166]... 

In the expression (4.6.23), the superscript s refers to values on the interface at = s. The value of the constant A corresponds to the conventional practice to determine the plane total diffusion flux per unit length of the cylinder (0 <

solid particles in a viscous fluid. 

Dimensionless total diffusion flux to the surface of an ellipsoidal particle at high Peclet numbers is determined by the formula [433]... 

The dimensionless total diffusion flux corresponding to a potential flow past an ellipsoidal bubble (Re = oo) is calculated at high Peclet numbers by the formula [174]... 

The following limit relation for the total diffusion fluxes on the surface of two identical solid particles arranged on the axis of a translational Stokes flow (Figure 4.8) was derived in [169] ... 

It follows from this expression that the total diffusion flux to the second drop is less than half that for the first drop, and we have h/h —> 0 as /c —> oo. The total diffusion flux for all drops of the chain is equal to... 

The particles located upstream are shielding the subsequent particles. As a result, the total diffusion flux on their surfaces decreases monotonically so that I > h> > Ik> ik+i > , and the ratio h/I tends to zero as k - oo. As follows from (4.14.5), the total diffusion flux on the second sphere is nearly half the total diffusion flux on the first sphere, and that on the seventh sphere is less than one-third of that on the first. 

The total diffusion flux on all particles in the chain is calculated with the help of (4.14.5) ... 

The maximal substrate uptake rate of a cell is reached when the substrate concentration at the cell surface is zero. The total diffusion flux, J, to the cell is then ... 

If we integrate (6.167) over the entire surface of a plate of length h and width h, we will obtain the total diffusion flux to the plate ... 

The model accounts for three different transport mechanisms, molecular diffusion, Knudsen diffusion, and viscous transport. The total diffusive flux in DGM results from molecular diffusion acting in series with Knudsen diffusion. The viscous porous media flow (Darcy flow) acts in parallel with diffusive flux. The DGM can be written as an implicit relationship among molar concentrations, fluxes, concentration gradients and pressure gradient as... 

According to the model of Paul and Koros, the total diffusive flux N is given... 

Mean pore radius in the CL is Vd — 3 10 nm. Thus, the mean free path of molecules, If (Section 1.4.6), is in the order of In this situation Knudsen diffusion contributes to the total diffusion flux of species. Formally, the expression for the ith species flux due to the Knudsen diffusion has the form of Pick s law ... 

Let us also examine Eq. (2.82) for a convection-free system at isothermal conditions, where D = 0, and VT = 0. For the system to have no entropy production (that is, to be at equilibrium state), V/i, == —g. Since in a gravity field gx —gv = 0 and g = g, then djijdz = —g or dpii = —M gdz, which is the same as Eq. (2.13). With the above background, we now switch to the expression for the total diffusion flux, which can be derived from the thermodynamics of irreversible processes. 

The previously described models for the structural parameters Bq, Kq, and T, while straightforward, are oversimplified. More complex equations have been developed to describe diffusion in solids with broad pore size distribution. The model of Johnson and Stewart [37] (henceforth called the parallel-pore model) assumes that each pore makes its contribution to the total diffusive flux independently of the others. For isobaric conditions,... 

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