Mass transfer equation large Schmidt numbers

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 ... 

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... 

Numerical solution of the mass transfer equation begins at a small nonzero value of z = Zstart, uot at the inlet where Cp, x, y,z = 0) = Ca, miet for all values of x and y. This is achieved by invoking an asymptotically exact analytical solution for the molar density of reactant A from laminar mass transfer boundary layer theory in the limit of very large Schmidt and Peclet numbers. The boundary layer starting profile is valid under the following condition ... 

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. 

A comparison of equations 6.63 and 6.69 shows that similar forms of equations describe the processes of heat and mass transfer. The values of the coefficients are however different in the two cases, largely to the fact that the average value for the Prandtl number, Pr, in the heat transfer work was lower than the value of the Schmidt number, Sc, in the mass transfer tests. 

When the Schmidt number is infinitely large, W,- is reduced to f p) h + jh) and appears as the product of a hydrodynamic transfer fimction f p) and a mass-transport transfer function Zc = fi -I- jt2- The mass-transport transfer function Zc is presented in Figure 15.3. It is eeisily verified that W,- approaches 0.5 when the frequency tends toward zero, in agreement with the exponent of the rotation speed in the Levich equation. This value of 0.5 is also verified if the complete expression of W,- is used. The complex function 2W,- is presented in Figure 15.4 in... 

The equations also hold in a corresponding manner for mass transfer. This merely requires the Nusselt number to be replaced by the Sherwood number and the Prandtl by the Schmidt number. Prerequisite for the validity of the equations is however, a sufficiently large value for the Peclet number Pe = RePr A 500... 

We have already noted that mass transfer in a liquid is almost always characterized by large values of the Peclet number (the Peclet number for mass transfer involves the product of the Schmidt number and Reynolds number instead of the Prandtl number and Reynolds number) and that the dimensionless form of the convection-diffusion equation governing transport of a single solute through a solvent is still (9-7), with 6 now being a dimensionless solute concentration. For transfer of a solute from a bubble or drop into a liquid that previously contained no solute, the concentration 6 at large distances from the bubble or drop will satisfy the condition... 

Order-of-magnitude analysis indicates that diffusion is neghgible relative to convective mass transfer in the primary flow direction within the concentration boundary layer at large values of the Peclet number. Typically, liquid-phase Schmidt numbers are at least 10 because momentum diffusivities (i.e., i/p) are on the order of 10 cm /s and the Stokes-Einstein equation predicts diffusion coefficients on the order of 10 cm /s. Hence, the Peclet number should be large for liquids even under slow-flow conditions. Now, the partial differential mass balance for Cji,(r,0) is simplified for axisynunetric flow (i.e., = 0), angu-... 

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... 

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