Thin Boundary Layer Approximation

Following this procedure, and discarding all terms involving concentration gradients, the Nernst-Planck equation (10) reduces in the bulk to the Laplace equation for the potential  

In deriving (25), we also set the transient concentration term on the left in (10) to zero, thus considering only steady-state (or pseudosteady-state) processes. One may still apply the approximation of the thin boundary layer, as stated by (25), to transient problems, allowing the concentration within the thin boundary layer to vary with time. Detailed discussion of this class of problems is, however, outside the scope of this review and can be found in publications focusing on transients (e.g.. Refs. [9-12]). 

Equation (25) is solved for the bulk region by applying, in the absence of concentration gradient in the bulk, simplified boundary conditions. 

On electrodes we apply (23), recognizing that now, in the absence of concentration gradients in the bulk, the current on the solution side is driven only by electric migration  

As before, the right-hand side on (27) is given by either (18) or (20). Alternatively, we can specify a potential balance at the electrode by applying (lumped across the thin boundary layer) either (14) or (24), rearranged in the form of (28)  

---

Two particularly useful equations can be derived by applying the thin concentration boundary layer approximation to steady-state transfer from an axisymmetric particle (L2). The particle and the appropriate boundary layer coordinates are sketched in Fig. 1.1. The x coordinate is parallel to the surface x == 0 at the front stagnation point), while the y coordinate is normal to the surface. The distance from the axis of symmetry to the surface is R. Equation (1-38), subject to the thin boundary layer approximation, then becomes... 

Hence, for Nj > 4, the error of the thin boundary layer approximation is less than 10 percent, while at Nj = 2.5 the error is 16 percent. The first two terms of the above perturbation solution, Eqs. (50) and (51), yield, upon inverting to the real time domain,... 

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. 

Local mass transfer coefficients vary with polar angle 9 because of the inverse relation between fec.iocai and 5c (0). Once again, it is recommended to average fee. local or 8c 9) over the front hemisphere only, where the thin boundary layer approximation is justified, particularly for creeping flow. Hence,... 

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) ... 

When the Peclet number is very large Pe oo), the temperature or the concentration varies only in a very thin layer adjacent to the surface of the sphere. Using the Stokes flow field and the thin boundary layer approximation, Friedlander [8] and Lochiel and Calderbank [9] derive the following expression for rigid sphere ... 

For a fluid sphere with Fe —> oo, the thin boundary layer approximation and the flow field of Hadamard-Rybczynski give the following equation [1] ... 

Figure 1. An electrochemical cell, depicting the thin boundary layer approximation. The bulk of the cell is well mixed and all concentration variations are assigned to a thin boundary layer next to the electrodes. Typically, the boundary layer thickness, 5, is far thinner with respect to the bulk than illustrated here. In the region of uniform concentration, the Laplace equation for the potential holds.

Even when applying the thin boundary layer approximation, the equations required for solving the current and potential distributions in the electrochemical cell yield a nonlinear system requiring iterative solution. The reason is that the boundary conditions incorporate the unknown term (the electrostatic potential or the current density). While this presents no serious hurdle for computer-implemented numerical solutions, analytical solutions of nonlinear systems are difficult and generally require a linearization procedure. To analytically characterize features of the current distribution, some simplifying approximations are frequently applied. These are summarized in Table 1, and are discussed below. 

This is the most general case. Here, none of the dissipative processes are controlhng, and therefore no mechanism is assumed to be negligible. The Nemst Planck equation, (10), or its thin boundary layer approximation (the Laplace equation (25)), is solved subject to the... 

The procedure outlined below describes an algorithm for numerical solution of the current distribution. It is typically not the one implemented in practical software since is not efficient. However, for a conceptual description of the procedure it is the simplest to discuss, as it is not hampered by extraneous mathematical considerations. The thin boundary layer approximation is invoked and rather than a complete solution of the flow field, it is assumed that the equivalent stagnant concentration boimdary layer, %, along the electrode is available (may vary with position). The computational routine consists of the following steps applied to numerous points along the electrodes ... 

The modeling of the current distribution in a general-geometry cell nearly always requires a numerical solution. The following discussion focuses on the thin boundary layer approximation, with the overpotential components lumped within a thin boundary layer which may be of a varying thickness. The Laplace equation for the potential with nonlinear boundary conditions must be solved. Similar considerations typically apply to the more comprehensive solution of the Nernst-Planck equation (10) however, the need to account for the convective fluid flow in the latter case makes the application of the boundary methods more complex. We focus our brief discussion on the most common methods the finite-difference method, the finite-element method, and the boundary-element method, schematically depicted in Fig. 4. Since the finite-difference method is the simplest to implement and the best known technique, it is discussed in somewhat more detail. 

What happens is here that the floes evidently move into the thin boundary layers adjaeent to the hydraulieally smooth surfaees, where mueh of the energy is dissipated, with the result that the partieles are subjeeted to strong stresses beeause of the small volume of the boundary layers. This hypothesis is supported by the good eorrelation of the results for the smooth disc with the results for various impellers in Fig. 6 it was assumed here that in the case of the disc, the majority of the power is dissipated in the boundary-layer volume V5, and the relationship nj/e V/V5 is approximately valid. The volume of the boundary layer (Eq. (21)) was obtained by integration from the theoretical solution [65] for the thickness of the boundary layer (Eq. (21)) of a smooth disc with turbulent flow. 

Fig. 1.1 Coordinates for the thin concentration boundary layer approximation.

For a fluid sphere with Pe oo the thin concentration boundary layer approximation, Eq. (1-63), becomes... 

The thin concentration boundary layer approximation, Eq. (3-51), has also been solved for bubbles k = 0) using surface velocities from the Galerkin method (B3) and from boundary layer theory (El5, W8). The Galerkin method agrees with the numerical calculations only over a small range of Re (L7). Boundary layer theory yields... 

The resistance to mass transfer within a slug in a liquid of low viscosity has been measured by Filla et ai (F5), who found that kA) was approximately proportional to the square root of the diffusivity within the bubble, p, as predicted by the thin concentration boundary layer approximation. In addition, kA JA was independent of slug length for 1 < L/D < 2.5. 

Diffusion boundary layer approximation. For x = 0(1), the concentration mostly varies on the initial interval in a thin diffusion boundary layer near the free boundary of the film. In this region we expand the transverse coordinate according to the rule... 

The thin region near the body surface, which is known as the boundary layer, lends itself to relatively simple analysis by the very fact of its thinness relative to the dimensions of the body. A fundamental assumption of the boundary layer approximation is that the fluid immediately adjacent to the body surface is at rest relative to the body, an assumption that appears to be valid except for very low-pressure gases, when the mean free path of the gas molecules is large relative to the body [6]. Thus the hydrodynamic or velocity boundary layer 8 may be defined as the region in which the fluid velocity changes from its free-stream, or potential flow, value to zero at the body surface (Fig. 1.3). In reality there is no precise thickness to a boundary layer defined in this manner, since the velocity asymptotically approaches the free-stream value. In practice we simply imply that the boundary layer thickness is the distance in which most of the velocity change takes place. 

Since l/r)d(rvr)/dr is replaced by dvr/dr in the equation of continuity when analysis is required only within a thin boundary layer, the relative error based on this approximation is... 

In potential flow, the stream function and the potential function are used to represent the flow in the main body of the fluid. These ideal fluid solutions do not satisfy the condition that = Vy = 0 on the wall surface. Near the wall we have viscous drag and we use boundary-layer theory where we obtain approximate solutions for the velocity profiles in this thin. boundary layer taking into account viscosity. This is discussed in Section 3.10. Then we splice this solution onto the ideal flow solution that describes flow outside the boundary layer. 

The tertiary distribution represents the formal thin boundary layer solution with no farther approximation. 

Besides the resuspension of particles, the perfect sink model also neglects the effect of deposited particles on incoming particles. To overcome these limitations, recent models [72, 97-99] assume that particles accumulate within a thin adsorption layer adjacent to the collector surface, and replace the perfect sink conditions with the boundary condition that particles cannot penetrate the collector. General continuity equations are formulated both for the mobile phase and for the immobilized particles in which the immobilization reaction term is decomposed in an accumulation and a removal term, respectively. Through such equations, one can keep track of the particles which arrive at the primary minimum distance and account for their normal and tangential motion. These equations were solved both approximately, and by numerical integration of the governing non-stationary transport equations. 

Let us recapitulate. We have achieved a solution to boundary-layer-like burning of a steady liquid-like fuel. A thin flame or fast chemistry relative to the mixing of fuel and oxygen is assumed. All effects of radiation have been ignored - a serious omission for flames of any considerable thickness. This radiation issue cannot easily be resolved exactly, but we will return to a way to include its effects approximately. 

Unsteady transfer with Pe oo has been treated using the thin concentration boundary layer assumptions. With this approximation, the last term in Eq. (3-56) is deleted. Hence, for small t where the convection term is negligible, the transfer rate for rigid or circulating spheres is identical to that for diffusion from a plane into a semi-infinite region ... 

It was assumed that the concentration boundary layer was thin relative to the shorter axis of the spheroid. The order of magnitude of the boundary layer thickness can be approximated by the thickness 5 of a fictitious film... 

It is difficult to solve the system of Eqs. (39)—(41) for these boundary conditions. However, certain simplifying assumptions can be made, if the Prandtl number approaches large values. In this case, the thermal boundary layer becomes very thin and, therefore, only the fluid layer near the plate contributes significantly to the heat transfer resistance. The velocity components in Eq. (41) can then be approximated by the first term of their Taylor series expansions in terms of y. In addition, because the nonlinear inertial terms are negligible near the wall, one can further assume that the combined forced and free convection velocity is approximately equal to the sum of the velocities that would exist when these effects act independently. Therefore, for assisting flows at large Prandtl numbers (theoretically for Pr -> oo), Eq. (41) can be rewritten in the form ... 

---