Mass transfer, boundary layer for

The generalized form for the linear tangential velocity profile within the mass transfer boundary layer for a no-slip interface is... 

This second-order ordinary differential equation given by (16-4), which represents the mass balance for one-dimensional diffusion and chemical reaction, is very simple to integrate. The reactant molar density is a quadratic function of the spatial coordinate rj. Conceptual difficulty arises for zeroth-order kinetics because it is necessary to introduce a critical dimensionless spatial coordinate, ilcriticai. which has the following physically realistic definition. When jcriticai which is a function of the intrapellet Damkohler number, takes on values between 0 and 1, regions within the central core of the catalyst are inaccessible to reactants because the rate of chemical reaction is much faster than the rate of intrapellet diffusion. The thickness of the dimensionless mass transfer boundary layer for reactant A, measured inward from the external surface of the catalyst,... 

Equations (17.30) and (17.31) demonstrate that sin e-pass h h-conversion microreactors which are characterized by NTU hitter than unity (see Section 17.3.3.2) imply fitUy developed mass transfer boundary layers for the major part of the electrode surface. The electrode processes are thus coupled. The mass tranter coefficient is independent of the liquid phase flow rate and can be estimated using Equation (17.28). 

For flow parallel to an electrode, a maximum in the value of the mass-transfer rate occurs at the leading edge of the electrode. This is not only the case in flow over a flat plate, but also in pipes, annuli, and channels. In all these cases, the parallel velocity component in the mass-transfer boundary layer is practically a linear function of the distance to the electrode. Even though the parallel velocity profile over the hydrodynamic boundary layer (of thickness h) or over the duct diameter (with equivalent diameter de) is parabolic or more complicated, a linear profile within the diffusion layer (of thickness 8d) may be assumed. This is justified by the extreme thinness of the diffusion layer in liquids of high Schmidt number ... 

In an ideal stagnation flow, a certain amount of the flow that enters through the inlet manifold can leave without entering the thermal or mass-transfer boundary layers above the surface. For an axisymmetric, finite-gap, flow, determine how the bypass fraction depends on the separation distance and the inlet velocity. 

Mass transfer from a surface to the gas-solid suspension is significantly higher than that for particle-free conditions. This increase is due to a possible reduction of the mass transfer boundary layer and an increase in the interstitial gas velocity when particles are present. The effect on heat transfer is even more significant as solid particles also act as heat carriers. 

Fig. 19. Isoconcentration contours for a mass transfer boundary layer thickness equivalent to the notch depth d. (Figure and caption reprinted from Jordan and Tobias [57] by permission of the publisher, The Electrochemical Society, Inc.).

Fig. 20. Increase in flux as a function of distance along the perimeter for a mass transfer boundary layer thickness equivalent to the notch depth d. Note that the perimeter is longer than the distance in the X direction, hence the locations on the perimeter of the left corner, central trench, and right corner are also shown. (Figure and caption reprinted from Jordan and Tobias [57] by permission of the pubUsher, The Electrochemical Society, Inc.).

The influence of a wall on the turbulent transport of scalar (species or enthalpy) at the wall can also be modeled using the wall function approach, similar to that described earlier for modeling momentum transport at the wall. It must be noted that the thermal or mass transfer boundary layer will, in general, be of different thickness than the momentum boundary layer and may change from fluid to fluid. For example, the thermal boundary layer of a high Prandtl number fluid (e.g. oil) is much less than its momentum boundary layer. The wall functions for the enthalpy equations in the form of temperature T can be written as ... 

The Schmidt number for the mass transfer is analogous to the Prandtl number for heat transfer. Its physical implication means the relative thickness of the hydrodynamic layer and mass-transfer boundary layer. The ratio of the velocity boundary layer (S) to concentration boundary layer (Sc) is governed by the Schmidt number. The relationship is given by... 

Problem 11-7. Mass Transfer with Finite Interfacial Velocities. In Section G, we considered the problem of mass transfer at large Reynolds and Schmidt numbers from an arbitrary 2D body with a no-slip boundary condition imposed at the particle surface. We noted that the form of the solution would be different if the tangential velocity at the body surface were nonzero, i.e., us(x) / 0. Determine the form of the mass transfer boundary-layer equation for this case, and solve it by using a similarity transformation. What conditions, if any, are required of us(x) for a similarity solution to exist ... 

Equation (26.69) (or Equation (26.70) for the case of a supporting electrolyte) was originally derived under the assumption of no convective velocities. These same relationships can be utilized within the effective mass transfer boundary layer when fluid stirring exists. A current density/con-centration equation with fluid stirring can now be generated starting with Equation (26.69) or Equation (26.70). The dCldy term in Equation (26.69) is replaced by AC/8, (where 8, is the effective mass transfer boundary-layer thickness) ... 

Consider creeping viscous flow of an incompressible Newtonian fluid past a stationary gas bubble that is located at the origin of a spherical coordinate system. Do not derive, but write an expression for the tangential velocity component (i.e., vg) and then linearize this function with respect to the normal coordinate r within a Ihin mass transfer boundary layer in the liquid phase adjacent to the gas-liquid interface. Hint Consider the r-9 component of the rate-of-strain tensor ... 

The denominator of the second term in parentheses on the right side of (11-17) cannot be evalnated explicitly until one obtains basic information for the concentration profile. However, the following qualitative analysis is based solely on the fact that the mass transfer boundary layer is much thinner than the corresponding momentum boundary layer at very large Schmidt numbers. Under these conditions, the transfer of component A into the liquid phase is limited to a very thin shell that surrounds the sphere. Rates of interphase mass transfer are... 

In other words, if the mass transfer boundary layer is very thin, then there is a tremendous change in the radial concentration gradient as one moves a very short distance radially outward from the spherical interface. Hence, at large Schmidt numbers where Sc(9) R, the following locally flat approximation is valid for spherical interfaces because curvature within the boundary layer is negligible ... 

The dimensionless profile for mobile component A within the mass transfer boundary layer can be expressed in terms of the incomplete gamma function... 

Since dx — Rd9, integration of (11-85), subject to the condition that 5c = 0 at 0 = 0, yields the desired result for the simplified mass transfer boundary layer thickness ... 

If the Reynolds number is based on the sphere diameter, as defined earlier, then the group of terms prior to the integral in (11-93) is proportional to the inverse of the mass transfer Peclet number. The general expression for the mass transfer boundary layer thickness is... 

In dimensionless notation, the generalized expression for the simplified mass transfer boundary layer thickness is... 

The former calculation for 8ci9 = tt/2) yields the maximum value of the dimensionless radial variable (i.e., normal coordinate) at the outer edge of the mass transfer boundary layer ... 

Unlike creeping flow about a solid sphere, the r9 component of the rate-of-strain tensor vanishes at the gas-liquid interface, as expected for zero shear, but the simple velocity gradient (dvg/dr)r R is not zero. The fluid dynamics boundary conditions require that [(Sy/dt)rg]r=R = 0- The leading term in the polynomial expansion for vg, given by (11-126), is most important for flow around a bubble, but this term vanishes for a no-slip interface when the solid sphere is stationary. For creeping flow around a gas bubble, the tangential velocity component within the mass transfer boundary layer is approximated as... 

Notice that vgir = / )/Vapproach within the mass transfer boundary layer is threefold larger for potential flow relative to creeping flow. 

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