Boundary layers approximation

The full concentration equation for the contaminant may be simplified in the same manner as the Navier-Stokes equations to derive a boundary-layer approximation for the concentration, namely,... 

To describe the velocity profile in laminar flow, let us consider a hemisphere of radius a, which is mounted on a cylindrical support as shown in Fig. 2 and is rotating in an otherwise undisturbed fluid about its symmetric axis. The fluid domain around the hemisphere may be specified by a set of spherical polar coordinates, r, 8, , where r is the radial distance from the center of the hemisphere, 0 is the meridional angle measured from the axis of rotation, and (j> is the azimuthal angle. The velocity components along the r, 8, and (j> directions, are designated by Vr, V9, and V. It is assumed that the fluid is incompressible with constant properties and the Reynolds number is sufficiently high to permit the application of boundary layer approximation [54], Under these conditions, the laminar boundary layer equations describing the steady-state axisymmetric fluid motion near the spherical surface may be written as ... 

This condition can be met by choosing either a RHSE of a sufficiently large radius, or by maintaining a high speed of rotation. From the results of the turning moment measurements shown in Fig. 4, one may take Re = 200 as the lower limit where the boundary layer approximation is valid. Thus the useful flow regime for electrochemical application is ... 

In filtration, the particle-collector interaction is taken as the sum of the London-van der Waals and double layer interactions, i.e. the Deijagin-Landau-Verwey-Overbeek (DLVO) theory. In most cases, the London-van der Waals force is attractive. The double layer interaction, on the other hand, may be repulsive or attractive depending on whether the surface of the particle and the collector bear like or opposite charges. The range and distance dependence is also different. The DLVO theory was later extended with contributions from the Born repulsion, hydration (structural) forces, hydrophobic interactions and steric hindrance originating from adsorbed macromolecules or polymers. Because no analytical solutions exist for the full convective diffusion equation, a number of approximations were devised (e.g., Smoluchowski-Levich approximation, and the surface force boundary layer approximation) to solve the equations in an approximate way, using analytical methods. 

He solved this equation, using three different boundary conditions, two of which are also used in the field of particle deposition on collectors the Perfect Sink (SINK) model, the Surface Force Boundary Layer Approximation (SFBLA) and the Electrode-Ion-Particle-Electron Transfer (EIPET) model. 

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... 

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 dependence of Sh on Pe/(1 + k) at high Pe results because the Hadamard -Rybczynski analysis gives dimensionless velocities iiJU, iio/V) proportional to (1 + k) within and close to the particle (Eqs. (3-7) and (3-8)). Similar dependence is encountered for unsteady external transfer (Section B.2), and for internal transfer at all Pe (Section C.4). These results do not give the rigid sphere values as /c x, because of fundamental diflerences between the boundary layer approximations for the two cases (see Chapter 1), and arc only valid for /c < 2. 

Analytic solutions for flow around and transfer from rigid and fluid spheres are effectively limited to Re < 1 as discussed in Chapter 3. Phenomena occurring at Reynolds numbers beyond this range are discussed in the present chapter. In the absence of analytic results, sources of information include experimental observations, numerical solutions, and boundary-layer approximations. At intermediate Reynolds numbers when flow is steady and axisym-metric, numerical solutions give more information than can be obtained experimentally. Once flow becomes unsteady, complete calculation of the flow field and of the resistance to heat and mass transfer is no longer feasible. Description is then based primarily on experimental results, with additional information from boundary layer theory. 

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. 

CA 59, 6189(1963) (Erosive burning of solid proplnts is considered by the boundary layer approximation in the aerothermochemical point of view. The numerical calcn shows that the burning rate becomes larger with increase of the velocity gradient of hot gas stream on the surface of the propint)... 

From the solution shown in Fig. 1.4, it is evident that a plug-flow representation is not appropriate. However, for these flow conditions (which are typical), a boundary-layer approximation is appropriate. In fact, based on direct comparison between full Navier-... 

Unfortunately, these equations cannot be modeled using the simple parallel-flow assumptions. In the entry region the radial velocity v and the pressure gradient will have an important influence on the axial-velocity profile development. Therefore we defer the detailed discussion and solution of this problem to Chapter 7 on boundary-layer approximations. 

When the boundary-layer approximations are applicable, the characteristics of the steady-state governing equations change from elliptic to parabolic. This is a huge simplification, leading to efficient computational algorithms. After finite-difference or finite-volume discretization, the resulting problem may be solved numerically by the method of lines as a differential-algebraic system. 

For channels that are narrow compared to their length (rs zs) and for Rer > 1, it is apparent from Eq. 7.15 that the only order-one term is the pressure gradient. Therefore we conclude that in the boundary-layer approximation the entire radial-momentum equation reduces to... 

There are numerous applications that depend on chemically reacting flow in a channel, many of which can be represented accurately using boundary-layer approximations. One important set of applications is chemical vapor deposition in a channel reactor (e.g., Figs. 1.5, 5.1, or 5.6), where both gas-phase and surface chemistry are usually important. Fuel cells often have channels that distribute the fuel and air to the electrochemically active surfaces (e.g., Fig. 1.6). While the flow rates and channel dimensions may be sufficiently small to justify plug-flow models, large systems may require boundary-layer models to represent spatial variations across the channel width. A great variety of catalyst systems use... 

The flow conditions are chosen to represent a range of gas-turbine-combustor conditions, covering a range of physical parameters that include inlet velocities from 0.5 to 5 m/s and pressures from 1 to 10 bar. These conditions can be characterized in terms of a Reynolds number based on channel diameter and inlet flow conditions, which is varied over the range 20 < Rej = V nd/v < 2000. The upper limit of Rej = 2000 is chosen to ensure laminar flow, hence removing the need to model turbulence. It should be noted that the validity of the boundary-layer approximations improve as the Reynolds number increases. 

In Section 17.8.3 we discussed the catalytic combustion of methane within a single one of the tubes in a honeycomb catalyst, illustrated in Fig. 17.18. The high velocity, and thus the dominance of convective over diffusive transport, makes the boundary layer approximations valid for this system. We will model the catalytic combustion performance in one of the honeycomb channels in this problem. 

Referring again to Fig. 1.5, it is seen that the wafer may rotate. Discuss the role of wafer ratation in the context of the boundary-layer simulation and oxidation uniformity. Is there an inherent conflict between boundary-layer approximations and wafer rotation Discuss the circumstances under which a boundary-layer approximation may be considered appropriate. Considering wafer rotation, what are the characteristics of atomic-oxygen profile along the lower wall that lead to oxidation uniformity on the wafer ... 

The general notion of a boundary later is found in many aspects of modeling physical systems. Recognizing boundary-layer behavior can very often lead to important simplifications in the analysis and modeling of such systems. Certainly the analysis and study of fluid mechanics is greatly facilitated by the exploitation of boundary-layer approximations. 

Scaling arguments are used to establish the circumstances where the boundary-layer behavior is valid. These arguments, which are usually made for external flows over surfaces, may be found in many texts on fluid mechanics (e.g., [350]). The essential feature of the boundary-layer approximation is that there is a principal flow direction in which the convective effects significantly dominate the diffusive behavior. As a result the flow-wise diffusion may be neglected, while the cross-flow diffusion and convection are retained. Mathematically this reduction causes the boundary-layer equations to have essentially parabolic characteristics, whereas the Navier-Stokes equations have essentially elliptic characteristics. As a result the computational simulation of the boundary-layer equations is much simpler and more efficient. 

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