Boundary Layer Mass Transfer Analysis

As an incompressible fluid of infinite extent approaches and flows past either a spherical solid pellet or a gas bubble, a mobile component undergoes inteiphase mass transfer via convection and diffusion from the sphere to the fluid phase. The overall objective is to calculate the mass transfer coefficient and the Sherwood number at any point along the interface (i.e., the local transfer coefficients), as well as surface-averaged transfer coefficients. The results are applicable in the laminar flow regime (1) when the sphere is stationary and the fluid moves, 

Steady-state analysis in the absence of any chemical reactions produces the following mass balance for mobile component A in an incompressible fluid when the control volume is differentially thick in all coordinate directions  

The only assumption is that the physical properties of the fluid (i.e p and A.mix) are constant. The left-hand side of equation (11-1) represents convective mass transfer in three coordinate directions, and diffusion is accounted for via three terms on the right side. If the mass balance is written in dimensionless form, then the mass transfer Peclet number appears as a coefficient on the left-hand side. Basic information for dimensional molar density Ca will be developed before dimensionless quantities are introduced. In spherical coordinates, the concentration profile CA(r,6,4 ) must satisfy the following partial differential equation (PDE)  

This is a horrendous equation that requires simplification via reasonable engineering approximations before one can derive any meaningful results from an analytical solution. The origin of an xyz Cartesian coordinate system is placed at the center of the sphere, and it remains there throughout the analysis. The fluid approaches the stationary sphere from above and moves downward along the z axis in the negative z direction. Hence, the velocity vector (i.e., approach velocity) of the fluid far from the sphere is 

When the fluid approaches the sphere from above, the fluid initially contacts the sphere at 0 = 0 (i.e., the stagnation point) because polar angle 6 is defined relative to the positive z axis. This is convenient because the mass transfer boundary layer thickness Sc is a function of 6, and 5c = 0 at 0 = 0. In the laminar and creeping flow regimes, the two-dimensional fluid dynamics problem is axisymmetric (i.e., about the z axis) with 

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One should realize that these calculations are based on an expression for Vr which corresponds to potential flow past a stationary nonde-formable bubble, as seen by an observer in a stationary reference frame. However, this analysis rigorously requires the radial velocity profile for potential flow in the Uquid phase as a nondeformable bubble rises through an incompressible liquid that is stationary far from the bubble. When submerged objects are in motion, it is important to use liquid-phase velocity components that are referenced to the motion of the interface for boundary layer mass transfer analysis. This is accomplished best by solving the flow problem in a body-fixed reference frame which translates and, if necessary, rotates with the bubble such that the center of the bubble and the origin of the coordinate system are coincident. Now the problem is equivalent to one where an ideal fluid impinges on a stationary nondeformable gas bubble of radius R. As illustrated above, results for the latter problem have been employed to estimate the maximum error associated with the neglect of curvature in the radial term of the equation of continuity. 

When the boundary layer mass or heat transfer is the rate determining step for the whole drying time even after the liquid recedes into the pores, the following analysis is applicable. By performing a mass balance on the sphere, the rate at which the liquid will recede into the pores (see Figure 14.6), dR/dt, is given by... 

In Equation 6.18, denotes the depth of penetration by an eddy in the neighboring mass transfer boundary layer. Harriott s analysis of his data indicated that both slip velocity and transient effects are important in determining the overall mass transfer process. Although Harriott could not provide actual slip velocity and unsteady-state mass transfer data, his conclusions based on effects of diffusivity and particle density on in stirred tanks are quite logical. In his conclusion, Harriott argued that the slip velocity theory is relatively easy to use for quantitative predictions as compared to the modified penetration theory. 

Tong, L. S., 1968b, Boundary Layer Analysis of the Flow Boiling Crisis, Int. J. Heat Mass Transfer 77 1208—1211.(5)... 

There are some good chemical-vapor-deposition reactors that deliberately starve the rotating disk. However, the similarity is broken by the recirculation, and the one-dimensional analysis techniques described herein lose their validity. If the chemical reaction on the surface is sufficiently slow, compared to mass transfer through the boundary layer, then the deposition uniformity will not be much affected by the boundary-layer similarity. In these... 

In flow situations, empirical analogies between mass and heat transfer are usually employed. For single-particle mass transfer, the boundary layer analysis for mass transfer is similar to that for heat transfer and thus is used for typical applications such as sublimation of a solid (e.g., naphthalene ball) or evaporation of a liquid drop falling in air. For a single sphere of diameter dp moving in a fluid, in terms of a boundary layer analysis analogous... 

Numerous empirical correlations for the prediction of residual NAPL dissolution have been presented in the literature and have been compiled by Khachikian and Harmon [68]. On the other hand, just a few correlations for the rate of interface mass transfer from single-component NAPL pools in saturated, homogeneous porous media have been established, and they are based on numerically determined mass transfer coefficients [69, 70]. These correlations relate a dimensionless mass transfer coefficient, i.e., Sherwood number, to appropriate Peclet numbers, as dictated by dimensional analysis with application of the Buckingham Pi theorem [71,72], and they have been developed under the assumption that the thickness of the concentration boundary layer originating from a dissolving NAPL pool is mainly controlled by the contact time of groundwater with the NAPL-water interface that is directly affected by the interstitial groundwater velocity, hydrodynamic dispersion, and pool size. For uniform... 

Based on the boundary layer analysis for laminar flow past a plate we might suggest that the mass transfer coefficient generally depends on the Reynolds number and the Prandtl number ... 

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 approach developed by Newman for the treatment of both mass-transfer and electric-field effects in boundary-layer flows has had considerable success.L2 6 However, many flows of practical interest have separation and recirculation regions, features not amenable to a boundary-layer analysis. Fortunately, there has been significant progress in the heat-transfer and other communities in computational fluid dynamics (CFD), providing numerical methods applicable to problems important to electrochemistry. The pioneers in using CFD for electrochemical applications are Alkire and co-workers, who have been largely interested in flow effects in localized corrosion. The literature is briefly reviewed in the next section. 

With a few exceptions, the fluid flow must be simulated before the mass-transfer simulations can be rigorously performed. Nevertheless, here are several important situations, such as that at a rotating disk electrode, where the fluid flow is known analytically or from an exact, numerical solution. Thus there exists a body of work that was done before CFD was a readily available tool (for example, see Refs. 34-37). In many of these studies, a boundary-layer analysis, based on a Lighthill transformation (Ref. 1, Chapter 17), is employed. 

There have been fewer studies in electrochemistry where the flow is known but the boundary-layer approach is inapplicable. One example has been recently analyzed and compared with experiment. In this case, mass transfer to a line electrode or an array of line electrodes in the presence of an oscillatory shear flow was treated. A finite-volume approach was used for the numerical analysis and a ferri/ferrocyanide redox couple was used to measure the mass-transfer rate. The studies show that boundary-... 

But before the virtues of the results and the approach are extolled, the method must be described in detail. Let us therefore return to a systematic development of the ideas necessary to solve transport (heat or mass transfer) problems (and ultimately also fluid flow problems) in the strong-convection limit. To do this, we begin again with the already-familiar problem of heat transfer from a solid sphere in a uniform streaming flow at sufficiently low Reynolds number that the velocity field in the domain of interest can be approximated adequately by Stokes solution of the creeping-flow problem. In the present case we consider the limit Pe I. The resulting analysis will introduce us to the main ideas of thermal (or mass transfer) boundary-layer theory. 

We saw in Chapter 10 that the boundary-layer structure, which arises naturally in flows past bodies at large Reynolds numbers, provides a basis for approximate analysis of the flow. In this chapter, we consider heat transfer (or mass transfer for a single solute in a solvent) in the same high-Reynolds-number limit for problems in which the velocity field takes the boundary-layer form. We saw previously that the thermal energy equation in the absence of significant dissipation, and at steady state, takes the dimensionless form... 

The analysis in this chapter largely follows the original developments of J. D. Goddard and A. Acrivos, Asymptotic expansions for laminar forced-convection heat and mass transfer, Part 2, Boundary-layer flows, J. Fluid Mech. 24, 339-66 (1966). 

In the preceding section, it was assumed that the concentration at the surface is known. In other words, the surface concentration can be taken as equal to the concentration in the bulk of the fluid phase. This is true only if there is essentially no mass transfer resistance across the film (or boundary layer) surrounding the pellet. If such a resistance is present, the actual surface concentration will be lower for the reactant and higher for the product than in the fluid bulk. A Sherwood number for mass transfer would then be needed in the analysis. Several empirical correlations have been proposed that relate the Sherwood number (for mass transfer) to the Reynolds number (for flow) and Schmidt... 

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