Diffusion-convection layer thickness

Note that this do (cm) is different from that of diffusion layer thickness because within the diffusion layer, there is no solution convection, hut within this diffusion—convection layer, both diffusion and convection coexist. Based on convection kinetic theory, this diffusion—convection layer thickness can be approximately expressed as Eqn (5.1) ... 

For convective crystal dissolution, the dissolution rate is u = (p/p )bD/8. For diffusive crystal dissolution, the dissolution rate is u = diffusive boundary layer thickness as 5 = (Df), the diffusive crystal dissolution rate can be written as u = aD/5, where a is positively related to b through Equation 4-100. Therefore, mass-transfer-controlled crystal dissolution rates (and crystal growth rates, discussed below) are controlled by three parameters the diffusion coefficient D, the boundary layer thickness 5, and the compositional parameter b. The variation and magnitude of these parameters are summarized below. 

Measurements of the rate of deposition of particles, suspended in a moving phase, onto a surface also change dramatically with ionic strength (Marshall and Kitchener, 1966 Hull and Kitchener, 1969 Fitzpatrick and Spiel-man, 1973 Clint et al., 1973). This indicates that repulsive double-layer forces are also of importance to the transport rates of particulate solutes. When the interactions act over distances that are small compared to the diffusion boundary-layer thickness, the rate of transport can be computed (Ruckenstein and Prieve, 1973 Spiel-man and Friedlander, 1974) by lumping the interactions into a boundary condition on the usual convective-diffusion equation. This takes die form of an irreversible, first-order reaction on tlie surface. A similar analysis has also been performed for the case of unsteady deposition from stagnant suspensions (Ruckenstein and Prieve, 1975). 

When van der Waals and double-layer forces are effective over a distance which is short compared to the diffusion boundary-layer thickness, the rate of deposition may be calculated by lumping the effect of the particle-collector interactions into a boundary condition on the usual convective-diffusion equation. This condition takes the form of a first-order irreversible reaction (10, 11). Using this boundary condition to eliminate the solute concentration next to the disk from Levich s (12) boundaiy-kyersolution of the convective-diffusion equation for a rotating disk, one obtains... 

Since D plays the same role as the kinematic viscosity v, we may expect for large Schmidt numbers (v>D) that the viscous boundary layer thickness should be considerably larger than the diffusion boundary layer thickness. A consequence of this is that the velocity seen by the concentration layer at its edge is not the free stream velocity U but something much less, which is more characteristic of the velocity close to the wall (Fig. 4.2.1). We note also that since c is understood to be c, then in a multicomponent solution there may be as many distinct boundary layers as there are species, with the thickness of each defined by the appropriate diffusion coefficient. With this caveat in mind, we may write the convective diffusion equation for a two-dimensional diffusion boundary layer and estimate the diffusion layer thickness. 

Comparing Eqn (5.14a) with (5.3), it can define the thickness of diffusion—convection layer for a rotating electrode ... 

It can be seen that this thickness of the diffusion—convection layer is not a function of the location on the electrode surface, which is different from that of Eqn (5.1), and therefore, the current density over the entire RDE surface is uniformly distributed. 

The good agreement of the experimental data shown in Fig. 56 with the classical RSA model is due to the fact that, in these experiments, the diffusion boundary layer thickness was fixed at a value eomparable with the partiele diameter as a result of convection. Hence, K was... 

Diffusion in a convective flow is called convective diffusion. The layer within which diffnsional transport is effective (the diffnsion iayer) does not coincide with the hydrodynamic bonndary layer. It is an important theoretical problem to calcnlate the diffnsion-layer thickness 5. Since the transition from convection to diffnsion is gradnal, the concept of diffusion-layer thickness is somewhat vagne. In practice, this thickness is defined so that Acjl8 = (dCj/ff) Q. This calcniated distance 5 (or the valne of k ) can then be used to And the relation between cnrrent density and concentration difference. 

It follows that convection of the hqnid has a twofold influence It levels the concentrations in the bnlk liquid, and it influences the diffusional transport by governing the diffusion-layer thickness. Shght convection is sufficient for the first effect, but the second effect is related in a qnantitative way to the convective flow velocity The higher this velocity is, the thinner will be the diffusion layer and the larger the concentration gradients and diffusional fluxes. 

Natural convection depends strongly on cell geometry. No convection can arise in capillaries or in the thin liquid layers found in narrow gaps between electrodes. The rates of natural convective flows and the associated diffusion-layer thicknesses depend on numerous factors and cannot be calculated in a general form. Very rough estimates show that the diffusion-layer thickness under a variety of conditions may be between 100 and 500 pm. 

In free-convection mass transfer at electrodes, as well as in forced convection, the concentration (diffusion) boundary layer (5d extends only over a very small part of the hydrodynamic boundary layer <5h. In laminar free convection, the ratio of the thicknesses is... 

Free convection at vertical cylinders differs from that along vertical plates because of the influence of curvature, but the effect is important only if the diffusion layer thickness is appreciable compared with the diameter. This can be expressed as... 

Tobias and Hickman (T2), the only investigators to date to study combined free and forced convection in horizontal channel flow, found a remarkably sharp separation between forced- and free-convection dominated mass transfer. In forced convection, the critical Grashof number, based on the diffusion layer thickness, is... 

Then an approximate analytical solution of the convective diffusion equation (43), which satisfies the boundary conditions, equation (44), is available under the assumption that the thickness of the diffusion layer <5, is small compared with the body radius r0 (p. 80 in [25]). As in the example of Section 4.1 (see equation (33)), the results of the derivation can be formally written in terms of the diffusion layer thickness, which now is ... 

In a simple model, one assumes that convective mass transport keeps the concentration constant at some fixed distance 5 from the solid wall. Thus, the diffusion layer thickness is constant. 

Lionbashevski et al. (2007) proposed a quantitative model that accounts for the magnetic held effect on electrochemical reactions at planar electrode surfaces, with the uniform or nonuniform held being perpendicular to the surface. The model couples the thickness of the diffusion boundary layer, resulting from the electrochemical process, with the convective hydrodynamic flow of the solution at the electrode interface induced by the magnetic held as a result of the magnetic force action. The model can serve as a background for future development of the problem. 

Dissolution distance in 18,000 s would be 174/im, greater than the diffusive dissolution distance of 48 ixm obtained earlier. There are no experimental data to compare. The convective dissolution rate can be applied only when the diffusion distance (Dt) is greater than the boundary layer thickness. If diffusion distance (Dt) is smaller than the boundary layer thickness (86.4 fim), i.e., if t< 1408 s, the dissolution would be controlled by diffusion even for a falling crystal, and the method in Section 4.2.2.3 should be used. 

One example would be ice melting or methane hydrate dissociation when rising in seawater. Convective melting rate may be obtained by analogy to convective dissolution rate. Heat diffusivity k would play the role of mass diffusivity. The thermal Peclet number (defined as Pet = 2aw/K) would play the role of the compositional Peclet number. The Nusselt number (defined as Nu = 2u/5t, where 8t is the thermal boundary layer thickness) would play the role of Sherwood number. The thermal boundary layer (thickness 8t) would play the role of compositional boundary layer. The melting equation may be written as... 

The boundary layer thickness 5. For convective crystal dissolution, the steady-state boundary layer thickness increases slowly with increasing viscosity and decreasing density difference between the crystal and the fluid. It does not depend strongly on the crystal size. Typical boundary layer thickness is 10 to 100/rm. For diffusive crystal dissolution, the boundary layer thickness is proportional to square root of time. 

Figure 3.6. Schlieren photographs showing the changes in thickness of the diffusion boundary layer and the behavior of buoyancy-driven convection shown in relation to bulk supersaturation [1], [2]. The figure shows the (111) faceofaBa(N03)2 crystal from an aqueous solution. In region I, only the thickness of the diffusion boundary layer increases in region II, we see unstable lateral convection (HA) and intermittently rising plumes (IIB) and in region III we see steady buoyancy-driven convection.

The time variation of 8 before the onset of natural convection depends on how the diffusion process is provoked. If a constant current density is switched on at t = 0, then the time variation of the effective diffusion-layer thickness can be obtained from Eqs. (7.179) and (7.202)... 

Fig. 7.96. When diffusion is not disturbed by convection, it can be shown that the diffusion-layer thickness 5 varies with the square root of time.

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