Thin boundary layers, mass transfer equation

Now the simplified mass transfer equation that accounts for convection normal and parallel to the spherical interface, and radial diffusion into a thin boundary layer, is... 

Hence, the thin boundary layer approximation is justified, and the locally flat description of the equation of continuity and the mass transfer equation is valid. 

Locally Flat Description. Analogous to the discussion on pages 279-280, one invokes the thin boundary layer approximation for either short contact times or small diffusivities and arrives at a locally flat description of the mass transfer equation for Ca(t, f) ... 

Step 9. Linearize the axial velocity profile within a thin mass transfer boundary layer adjacent to the catalytic surface and incorporate this information in the mass transfer equation for the reactant concentration... 

The solution of such an equation for an actual membrane device for ultrafiltration is difficult to obtain (see Zeman and Zydney (1996) for background information). One therefore usually falls back on the stagnant film model for determining the relation between the solvent flux and the concentration profile (see result (6.3.142b)). To use this result, we need to estimate the mass-transfer coefficient kit = Dit/dt), for the protein/macromolecule. One can focus on the entrance region of the concentration boundary layer, assume to be constant for a dilute solution, V = V, Vj, = 0 in the thin boundary layer, v = y ,y (where is the wall shear rate of magnitude AVz/Ay ) and obtain the result known as the Leveque solution at any location z in terms of the Sherwood number ... 

The surface velocities of Haberman and Sayre (HI), when used in the thin concentration boundary layer equation for circulating spheres, Eq. (3-51), yield the mass transfer factors and X d shown in Fig. 9.7 for k <2. For a fluid sphere in creeping flow the relationship between the mass transfer factors is... 

When the concentration boundary layer is sufficiently thin the mass transport problem can be solved under the approximation that the solution velocity within the concentration boundary layer varies linearly with distance away from the surface. This is called the L6v que approximation (8, 9] and is satisfactory under conditions where convection is efficient compared with diffusion. More accurate treatments of mass transfer taking account of the full velocity profile can be obtained numerically [10, 11] but the Ldveque approximation has been shown to be valid for most practical electrodes and solution velocities. Using the L vSque approximation, the local value of the concentration boundary layer thickness, 8k, (determined by equating the calculated flux to the flux that would be obtained according to a Nernstian diffusion layer approximation that is with a linear variation of concentration across the boundary layer) is given by equation (10.6) [12]. 

Boundary layer theory, just like film theory, is also based on the concept that mass transfer takes place in a thin him next to the wall as shown in Fig. 1.48. It differs from the him theory in that the concentration and velocity can vary not only in the y-direction but also along the other coordinate axes. However, as the change in the concentration prohle in this thin him is larger in the y-direction than any of the other coordinates, it is sufficient to just consider diffusion in the direction of the y-axis. This simplihes the differential equations for the concentration signihcantly. The concentration prohle is obtained as a result of this simplihcation, and from this the mass transfer coefficient [3 can be calculated according to the dehnition in (1.179). In practice it is normally enough to use the mean mass transfer coefficient... 

In Chap. 9, we considered the solution of this equation in the limit Re 1, where the velocity distribution could be approximated by means of solutions of the creeping-flow equations. When Pe 1, we found that the fluid was heated (or cooled) significantly in only a very thin thermal boundary layer of 0(Pe l/3) in thickness, immediately adjacent to the surface of a no-slip body, or () Pe l/2) in thickness if the surface were a slip surface with finite interfacial velocities. We may recall that the governing convection di ffusion equation for mass transfer of a single solute in a solvent takes the same form as (111) except that 6 now stands for a dimensionless solute concentration, and the Peclet number is now the product of Reynolds number and Schmidt number,... 

If T terface and Tbuik replace Ca, equilibrium and Ca, bulks respectively, in the definition of the dimensionless profile P, and the thermal diffusiv-ity replaces a. mix. then the preceding equation represents the thermal energy balance from which temperature profiles can be obtained. The tangential velocity component within the mass transfer boundary layer is calculated from the potential flow solution for vg if the interface is characterized by zero shear and the Reynolds number is in the laminar flow regime. Since the concentration and thermal boundary layers are thin for large values of the Schmidt and Prandtl... 

Realize that dvo/dr)r=R and Kdy/dt)re]r=R are nonzero for potential flow around a cylinder and that the first-order term in the polynomial expansion for V(, does not vanish, but this first-order term is small relative to the leading zeroth-order term. Now the locally flat description of the equation of continuity allows one to calculate the radial velocity component. For example, integration from the nondeformable solid-liquid interface at y = 0, where Vy = 0, to any position y within the thin mass transfer boundary layer produces the following result ... 

The tangential component of the dimensionless equation of motion is written explicitly for steady-state two-dimensional flow in rectangular coordinates. This locally flat description is valid for laminar flow around a solid sphere because it is only necessary to consider momentum transport within a thin mass transfer boundary layer at sufficiently large Schmidt numbers. The polar velocity component Vo is written as Vx parallel to the solid-liquid interface, and the x direction accounts for arc length (i.e., x = R9). The radial velocity component Vr is written... 

The underlined term in the preceding equation is negligible when mass transfer boundary layers adjacent to the catalytic surface are very thin at large Schmidt numbers. The locally flat approximation is valid when Sc asymptotically approaches infinity. 

Sub-millimeter inter-electrode gaps (in the case of plate and charmel reactors) or electrode widths (in the case of coplanar interdigitated band electrodes) lead to thin concentration boundary layers with any flow rate [14,23] resulting in enhanced mass transfer rates and thus increasing the attainable space-time-yield [Equation (17.17)]. 

According to Equation 7.4, pure crystals, i.e., keff close to 0, are obtained at low growth rates and high mass transfer, that is, at a thin interfacial boundary layer. This sets the conditions for an optimized process aiming at a highly pure soUd product. 

---