Boundary Layer Analysis

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

The pressure term has been retained in the thermal-energy equation, although in low-speed flow there is very little energy content in this term. Indeed, the term is very often neglected in boundary-layer analysis. For high-speed flow (i.e., Ma > 0.3), however, the thermal energy can be substantially affected by pressure variations. 

The details of the transitions and the vortex behavior depend on the actual channel dimensions and wall-temperature distributions. In general, however, for an application like a horizontal-channel chemical-vapor-deposition reactor, the system is designed to avoid these complex flows. Thus the ideal boundary-layer analysis discussed here is applicable. Nevertheless, one must exercise caution to be sure that the underlying assumptions of one s model are valid. 

Techniques from boundary layer analysis can be used to construct a series solution to equations 29 and 30 of the form... 

The multicellular structure of laminar convection in small-scale crystal growth systems complicates the interpretation of the boundary layer analysis,... 

Further laminar boundary layer analysis is given by Crane (Z. Angew. Math. Phys., 23, 201-212 [1972]). 

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... 

Figure 3 summarizes the results obtained when the contribution of London forces In the transport rate is negligible. When the boundary-layer analysis of Levich and Lighthill is valid (Case la) the Sherwood number is given by... 

Hull and Kitchener (2) measured the rate of deposition of 0.3- an-diameter polystyrene latex particles onto a rotating disk coated with a film of polyvinyl formaldehyde. In electrolytes of high ionic strength (where the double-layer repulsion is negligible), they found close agreement between experiments and the prediction of Levich s boundary-layer analysis (Eq. 3]), indicating that a diffusion boundary layer exists and that its thickness is large compared to the domain of van der Waais and hydrodynamic interactions. These are neces-... 

Electrophoretic problems can be greatly simplified when the distortion of double layer due to the external field and particle movement is negligible. A boundary layer analysis for the ion cloud [6,7] shows that the effect of ion cloud distortion remains insignificant provided that... 

Upon comparing Eqs. (5-54) and (5-55), we note that the right sides are alike except for a difference of about 3 percent in the constant, which is the result of the approximate nature of the integral boundary-layer analysis. We recognize this approximation and write... 

While the engineer may frequently be interested in the heat-transfer characteristics of flow systems inside tubes or over flat plates, equal importance must be placed on the heat transfer which may be achieved by a cylinder in cross flow, as shown in Fig. 6-7. As would be expected, the boundary-layer development on the cylinder determines the heat-transfer characteristics. As long as the boundary layer remains laminar and well behaved, it is possible to compute the heat transfer by a method similar to the boundary-layer analysis of Chap. 5. It is necessary, however, to include the pressure gradient in the analysis because this influences the boundary-layer velocity profile to an appreciable extent. In fact, it is this pressure gradient which causes a separated-flow region to develop on the back side of the cylinder when the free-stream velocity is sufficiently large. 

For a liquid droplet in a gaseous medium, the densities differ significantly. Whereas the linearized treatment can still be extended to the liquid-phase analysis, for the gas phase the conventional boundary layer analysis should be used (82). For high Reynolds number flow over a solid sphere, approximate solutions have been obtained using both the Blasius series and the momentum integral techniques (82). 

That is, l/ie variation of pressure in the direction normal to the surface is negligible, and thus P = P(.x) and dP/dx = dPIdx. Then it follows that for a given X, die pressure in the boundary layer is equal to the pressure in the free stream, and the pressure determined by a separate analysis of fluid flow in the free stream (which is typically easier because of the ab.sence of viscous effects) can readily be used in the boundary layer analysis. 

The simple one-dimensional models of multicomponent condensation and cocurrent separation processes described in this chapter are well able to model the performance of a wetted-wall column operated by Modine (1963) and a vertical tube condenser operated by Sardesai (1979). The results obtained with the one-dimensional model are probably good enough for design purposes it is doubtful if a more sophisticated boundary layer analysis could yield any better results. 

For the turbulent part of the inner boundary layer close to solid walls we usually adopt another closure for the Reynolds stresses and neglect the molecular diffusion term. The starting point for the boundary layer analysis, to be discussed shortly, is thus the fc-equation in the following form ... 

Actually, several possibilities exist formulating the wall friction force. The natural boundary layer shear stress definition to use is the one deduced from the fundamental equilibrium boundary layer analysis. The wall shear stress is thus defined as —Om =... 

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. 

J. Boundary-Layer Analysis of Heat Transfer From a Solid Sphere... 

It is perhaps timely to stop and reflect upon the nature of the thermal boundary-layer analysis to determine whether other generalizations of the basic result (9-230) may be possible. In particular, heat transfer from solid bodies occurs frequently when the fluid motion seen by the body cannot be approximated as a uniform streaming flow, and the reader may ask whether the correlation (9-230) can be applied in these cases with a proper choice for the characteristic velocity that appears in Pe. It is especially interesting, in this regard, to compare the present analysis with the corresponding low-Peclet-number problem that appeared earlier in this chapter. 

The question at hand is whether circumstances exist for this rather simple situation in which the conditions (1) and (2) are satisfied so that boundary-layer analysis can be applied. So far as the first condition is concerned, the only flows of (9-266) that have open streamlines are those with X > 0 (which includes simple shear flow). On the other hand, there is a nonzero hydrodynamic torque on the sphere that causes it to rotate for all flows in this subgroup except X = 1. Thus, for a sphere in the general linear 2D flow, given by (9-266), there are only two cases that satisfy the conditions for applicability of boundary-layer theory ... 

These are the only cases for a sphere in a linear velocity field for which a boundary-layer analysis is expected to apply. In other cases, the sphere is surrounded by a zone of closed streamlines. 

The functional dependence of Cd on Re ]/1 is a feature common to all streamlined bodies for which the boundary-layer analysis is valid over the complete body surface (i.e., the flow does not separate). The correlation... 

The cause of large drag in the case of a body like a circular cylinder is the asymmetry in the velocity and pressure distributions at the cylinder surface that results from separation. All bodies in laminar streaming flow at large Reynolds number are subjected to viscous stresses that boundary-layer analysis shows must be... 

Experimental observations of the flow past a circular cylinder show that separation does indeed occur, with a separation point at 0S — 110 . It should be noted, however, that steady recirculating wakes can be achieved, even with artificial stabilization,24 only up to Re 200, and it is not clear that the separation angle has yet achieved an asymptotic (Re —> oo) value at this large, but finite, Reynolds number. In any case, we should not expect the separation point to be predicted too accurately because it is based on the pressure distribution for an unseparated potential flow, and this becomes increasingly inaccurate as the separation point is approached. The important fact is that the boundary-layer analysis does provide a method to predict whether separation should be expected for a body of specified shape. This is a major accomplishment, as has already been pointed out. 

The amazing feature of (10-260) is that it is obtained entirely from the inviscid flow solution - the boundary-layer analysis does play an important role in demonstrating that the volume integral of 4>, based on the inviscid velocity field, will provide a valid first approximation to the total viscous dissipation but does not enter directly. 

Moore32 generalized the boundary-layer analysis for a spherical bubble to include the deformation to an oblate ellipsoidal shape. His analysis is not reproduced here, but it is worth recording the final result for the drag coefficient, which takes the form... 

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