Local diffusion flux

According to the scheme of the compartments in the liquid membrane system in Figure 13.5b, all local diffusion fluxes of M species from ktok+1 compartment can be defined by a phenomenological Equation 13.41 corresponding with the first Pick s law for diffusion ... 

In a similar way, by substituting the first term of the expansion (3.2.5) into (3.2.9), we obtain the dimensionless local diffusion flux... 

By differentiating (3.4.9), we obtain the local diffusion flux to the film surface [270] ... 

One can see that the local diffusion flux attains its maximum at the front stagnation point on the surface of the sphere (at 0 = 7r) and monotonically decreases with the angular coordinate to the minimum value, which is equal to zero and is attained at 0 = 0. 

Let us calculate the dimensionless local diffusion flux to the surface of the drop ... 

The local diffusion flux for the first-order reaction is calculated by the formula... 

Diffusion flux. To approximate the dimensionless local diffusion flux j = (dc/dy)y=o on the disk surface, it is convenient to use the cubic equation... 

The distribution of the dimensionless local diffusion flux along the plate, j = -(dc/dy)y=Q, can be approximately found by solving the cubic equation... 

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. 

Reid, Sherwood and Prausnitz [11] provide a wide variety of models for calculation of molecular diffusion. Dr is the Knudsen diffusion coefficient. It has been given in several articles as 9700r(T/MW). Once we have both diffusion coefficients we can obtain an expression for the macro-pore diffusion coefficient 1/D = 1/Dk -i-1/Dm- We next obtain the pore diffusivity by inclusion of the tortuosity Dp = D/t, and finally the local molar flux J in the macro-pores is described by the famiUar relationship J = —e D dcjdz. Thus flux in the macro-pores of the adsorbent product is related to the term CpD/r. This last quantity may be thought of as the effective macro-pore diffusivity. The resistance to mass transfer that develops due to macropore diffusion has a length dependence of R]. 

In the vicinity of the formation point, r0, the probability of a pair existing must be continuous, else this diffusive flux would be locally infinite (singular), i.e. 

There is a local (Fickian transport) and a non-local (stress induced) term in this flux equation. In the local term, the stress acts in the same way as an activity coefficient does. It always increases local diffusion since V] is positive and independent of the sign of the partial molar volume of /. 

The diffusion potential of an atom at the surface is proportional to local surface curvature as demonstrated in Section 3.4. The curvature can be determined from Eq. 14.1 and is a function of x. The local diffusion potential produces boundary conditions for diffusion through the bulk or transport via the vapor phase. For surface diffusion, gradients in the diffusion potential produce fluxes along the surface. 

Let us first consider, for simplicity, the case of diffusive transfer without ultrafiltra-tion. The local mass flux through the membrane per unit area Js (x) of a specific toxin is given by... 

Surface tractions or contact forces produce a stress field in the fluid element characterized by a stress tensor T. Its negative is interpreted as the diffusive flux of momentum, and x x (—T) is the diffusive flux of angular momentum or torque distribution. If stresses and torques are presumed to be in local equilibrium, the tensor T is easily shown to be symmetric. 

Extended nonequilibrium thermodynamics is not based on the local equilibrium hypothesis, and uses the conserved variables and nonconserved dissipative fluxes as the independent variables to establish evolution equations for the dissipative fluxes satisfying the second law of thermodynamics. For conservation laws in hydrodynamic systems, the independent variables are the mass density, p, velocity, v, and specific internal energy, u, while the nonconserved variables are the heat flux, shear and bulk viscous pressure, diffusion flux, and electrical flux. For the generalized entropy with the properties of additivity and convex function considered, extended nonequilibrium thermodynamics formulations provide a more complete formulation of transport and rate processes beyond local equilibrium. The formulations can relate microscopic phenomena to a macroscopic thermodynamic interpretation by deriving the generalized transport laws expressed in terms of the generalized frequency and wave-vector-dependent transport coefficients. 

To normalize the governing equations, we introduce a dimensionless position, z = x/a, and two dimensionless dependent variables,/ =/// and u = ua/DD. Note that the normalized velocity m is equivalent to a local Peclet number, indicating the relative magnitudes of the advective and diffusive fluxes of the reactive species. Applying these definitions to the transport equations yields the dimensionless governing equations... 

If local equilibrium is established instantaneously the ionic composition of parallel diffusion fluxes in the bead pores (i.e., free ions) and the gel phase of the bead (i.e., condensed and bound ions) is determined in particular by the selectivity factor, i.e., by the shape of the ex-... 

---