Boundary Layer Solution of the Mass Transfer Equation

Integration of (11-52) from the solid-liquid interface where = 0 at y = 0 to any position y within the mass transfer boundary layer prodnces the following result  

The eqnation of continuity has served its purpose for this two-dimensional flow problem. In fact, momentnm bonndary layer theory employs the same methodology by postulating the fnnctional form for the velocity component parallel to the interface, and calculating the velocity component in the normal coordinate direction via the equation of continnity. Now Vr and v are incorporated into the mass transfer equation. 

11-3 BOUNDARY LAYER SOLUTION OF THE MASS TRANSFER EQUATION 

The concentration of mobile component A is expressed in terms of the dimensionless profile  

The boundary layer boundary condition (BLBC) suggests that a combination-of-variables approach should be successful if a new independent variable is defined as 

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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 ... 

This boundary-layer theory applies to mass-transfer controlled systems where the membrane permeation rate is independent of pressure, for there is no pressure term in the model. In such cases it has been proposed that, as the concentration at the membrane increases, the solute eventually precipitates on the membrane surface. This layer of precipitated solute is known as the gel-layer, and the theory has thus become known as the gel-polarisation model proposed by Micii i i.si 0). Under such conditions C, in equation 8.15 becomes replaced by a constant Cq the concentration of solute in the gel-layer, and ... 

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 we assume that the concentration of solute is constant at the surface of the body, then the dimensionless version of the mass transfer problem is formally identical to that previously, solved, namely the governing equation (11-1) with the boundary conditions 6 = 1 at the body surface and 0 —> 0 at infinity, plus the initial condition 9 = 1 at the leading edge of the boundary layer x = 0. The primary difference is that the normal velocity at the surface of the body is now nonzero with a magnitude given in dimensionless form 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 ... 

Uniform Surface Injection. Although a mass transfer distribution yielding a uniform surface temperature is most efficient, it is much easier to construct a porous surface with a uniform mass transfer distribution. Libby and Chen [34] have considered the effects of uniform foreign gas injection on the temperature distribution of a porous flat plate. For these conditions, however, boundary layer similarity does not hold. Libby and Chen extended the work of Iglisch [35] and Lew and Fanucci [36], where direct numerical solutions of the partial differential equations were employed. An example of the nonuniform surface enthalpy and coolant concentrations resulting from these calculations is shown in Fig. 6.16. 

The solution to this laminar boundary layer problem must satisfy conservation of species mass via the mass transfer equation and conservation of overall mass via the equation of continuity. The two equations have been simplified for (1) two-dimensional axisymmetric flow in spherical coordinates, (2) negligible tangential diffusion at high-mass-transfer Peclet numbers, and (3) negligible curvature for mass flux in the radial direction at high Schmidt numbers, where the mass transfer... 

Answer Begin with the equation of continuity and the mass transfer equation in cylindrical coordinates with two-dimensional flow (i.e., Vr and vq) in the mass transfer boundary layer and no dependence of Ca on z because the length of the cylinder exceeds its radius by a factor of 100. Heat transfer results will be generated by analogy with the mass transfer solution. The equations of interest for an incompressible fluid with constant physical properties are... 

Step 11. Write all the boundary conditions that are required to solve this boundary layer problem. It is important to remember that the rate of reactant transport by concentration difhision toward the catalytic surface is balanced by the rate of disappearance of A via first-order irreversible chemical kinetics (i.e., ksCpJ, where is the reaction velocity constant for the heterogeneous surface-catalyzed reaction. At very small distances from the inlet, the concentration of A is not very different from Cao at z = 0. If the mass transfer equation were written in terms of Ca, then the solution is trivial if the boundary conditions state that the molar density of reactant A is Cao at the inlet, the wall, and far from the wall if z is not too large. However, when the mass transfer equation is written in terms of Jas, the boundary condition at the catalytic surface can be characterized by constant flux at = 0 instead of, simply, constant composition. Furthermore, the constant flux boundary condition at the catalytic surface for small z is different from the values of Jas at the reactor inlet, and far from the wall. Hence, it is advantageous to rewrite the mass transfer equation in terms of diffusional flux away from the catalytic surface, Jas. 

The Nernst boundary layer thickness is a simple characteristic of the mass transfer but its definition is formal since no boundary layer is in fact stagnant and least of all boundary layers on gas-evolving electrodes furthermore, the Schmidt number, known to influence mass transfer, is not incorporated in the usual dimensionless form. For this reason, lines representing data from gas evolution in two different solutions can be displaced from one another because of viscosity differences. Nevertheless, the exponent in the equation = aib (32)... 

The calculation of the temperature and species concentrations in the channel is based on the solution of the quasi-steady equations for the gas phase heat and mass transfer. In order to avoid the solution of the complete boundary layer, the so-called film-approach is used which is based on the use of local heat transfer coefficients. 

As soon as the functional relationships between the Nusselt, Reynolds and Prandtl numbers or the Sherwood, Reynolds and Schmidt numbers have been found, be it by measurement or calculation, the heat and mass transfer laws worked out from this hold for all fluids, velocities and length scales. It is also valid for all geometrically similar bodies. This is presuming that the assumptions which lead to the boundary layer equations apply, namely negligible viscous dissipation and body forces and no chemical reactions. As the differential equations (3.123) and (3.124) basically agree with each other, the solutions must also be in agreement, presuming that the boundary conditions are of the same kind. The functions (3.126) and (3.128) as well as (3.127) and (3.129) are therefore of the same type. So, it holds that... 

While the film and surface-renewal theories are based on a simplified physical model of the flow situation at the interface, the boundary layer methods couple the heat and mass transfer equation directly with the momentum balance. These theories thus result in anal3dical solutions that may be considered more accurate in comparison to the film or surface-renewal models. However, to be able to solve the governing equations analytically, only very idealized flow situations can be considered. Alternatively, more realistic functional forms of the local velocity, species concentration and temperature profiles can be postulated while the functions themselves are specified under certain constraints on integral conservation. Prom these integral relationships models for the shear stress (momentum transfer), the conductive heat flux (heat transfer) and the species diffusive flux (mass transfer) can be obtained. 

This is the appropriate correlation to use when there is heat or mass (i.e., substitute Nu by Sh) transfer from a sphere immersed in a stagnant film is studied, Nu = 2. The second term in (5.294) accounts for convective mechanisms, and the relation is derived from the solution of the boundary layer equations. For higher Re3molds numbers the Nusselt number is set equal to the relation resulting from the boundary layer analysis of a flat plate ... 

The model was solved using orthogonal collocation on finite elements (OCFE). Orthogonal collocation on finite elements was developed by Carey and Finlayson (26) for solution of boundary layer problems. Carey and Finlayson used OCFE to solve the simultaneous heat and mass transfer equations describing a catalyst pellet and found the new method to be more efficient than finite difference techniques. They also showed that OCFE was applicable to boundary layer problems that could not be solved by global orthogonal collocation. Jain and Schultz (27)... 

Discrepancies between experimentally obtained and theoretically calculated data for cadmium concentration in the strip phase are 10-150 times at feed or strip flow rate variations. These differences between the experimental and simulated data have the following explanation. According to the model, mass transfer of cadmium from the feed through the carrier to the strip solutions is dependent on the diffusion resistances boundary layer resistances on the feed and strip sides, resistances of the free carrier and cadmium-carrier complex through the carrier solution boundary layers, including those in the pores of the membrane, and resistances due to interfacial reactions at the feed- and strip-side interfaces. In the model equations we took into consideration only mass-transfer relations, motivated by internal driving force (forward... 

In Chap. 26, concentration polarization in reverse osmosis was treated using a simple mass-transfer equation, Eq. (26.48), which is satisfactory where the surface concentration is only moderately higher than the bulk concentration. For UF, the large change in concentration near the surface requires integration to get the concentration profile. The basic equation states that the flux of solute due to convection plus diffusion is constant in the boundary layer and equal to the flux of solute in the permeate ... 

In a few limited situations mass-transfer coefficients can be deduced from theoretical principles. One very important case in which an analytical solution of the equations of momentum transfer, heat transfer, and mass transfer has been achieved is that for the laminar boundary layer on a flat plate in steady flow. 

The mass-transfer conditions prevailing in the boundary layer described so far correspond to diffusion of A through stagnant B in dilute solutions. Therefore, the flux can be written in terms of a -type mass-transfer coefficient, as described by equation (2-7), using the difference in surface and bulk concentrations as the driving force ... 

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... 

Gas flow through the small channels of a honeycomb matrix is nearly always laminar, and analytical solutions are available for heat and mass transfer for fully developed laminar flow in smooth tubes. In the inlet region, where the boundary layers are developing, the coefficients are higher, and numerical solutions were combined with the analytical solution for fully developed flow and fitted to a semitheoretical equation [14] ... 

The correlating equation [67] established here can be used to evaluate the mass transfer coefficient and the thickness of the diffusion boundary layer, S(= d/sh). The thickness of this layer calculated for an organic solvent and aqueous solution were 10 -10 and 10 -10 cm, respectively, for the four types of quaternary salts studied. For a solute crossing a mass transfer resistance film, the transfer time can be approximately estimated by the following equation [131] ... 

The approximations given by Equations 8.35 are the solution to Leveque s problem given in Equation 8.30 with a linear wall reaction. Since the formulation of the problem leads to a linearized velocity profile in a planar boundary layer, laminar flows (parabolic velocity profiles) in curved channels are more susceptible to present higher deviations from these results. For a fully developed flow in a round tube, the error associated with Equation 8.35b is 1.4 and 0.13% for aPe ,lz equal to 100 and 1000, respectively. Lopes et al. [40] observed that these differences are visible mainly for Da — 00 and calculated corrections to account for these effects. It was shown that in the mass transfer-controlled limit. 

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