Boundary layer assumptions

A simplified model usiag a stagnant boundary layer assumption and the one-dimension diffusion—convection equation has been used to calculate wall concentration ia an RO module. The iategrated form of this equation, the widely appHed film theory (41), is given ia equation 8. 

For contact angle <(> = 0 and 8 << R, Eq. (2-58) should be reduced to Plesset-Zwick s equation based on the thin boundary layer assumption thus, 4>e = tt/2. 

Levich (L3) obtained an asymptotic solution to Eq. (3-39) for Pe oo, using the thin concentration boundary layer assumption discussed in Chapter 1. Curvature of the boundary layer and angular diffusion are neglected (i.e., the last term in Eq. (3-39) is deleted), so that the solution does not hold at the rear of the sphere where the boundary layer thickens and angular diffusion is significant. The asymptotic boundary layer formula, Eq. (1-59), reduces for a sphere to ... 

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 ... 

When the shape of a particle oscillates, the surface area changes with time. This situation has been modeled by neglecting the motion adjacent to the surface due to the terminal velocity of the particle, i.e., by considering the particle to be oscillating but stationary, with material transferred by transient molecular diffusion over a time equal to the period of oscillation. For Sc 1 the thin concentration boundary layer assumptions are invoked (see Chapter 1). 

Sherwood number under the thin concentration boundary layer assumption through Eq. (3-46). The results are plotted in terms of and in Fig- 9.7. For a rigid sphere in creeping flow, the relationship between these quantities and the velocity ratio K is... 

Mass transfer to the liquid phase around a slug can be treated with the thin concentration boundary layer assumption through Eq. (1-63). Van Heuven and Beek (VI) completed these calculations for a slug with viscous and surface tension forces negligible (Eod > 100, M < 10 ). The results can be represented by... 

It is apparent that as the product RePr increases, the effect of the axial diffusion is significantly diminished. The boundary-layer assumptions call for neglecting the axial diffusion altogether, yielding the following governing equation ... 

Such flow can be treated adequately using the boundary layer assumptions but the freestream velocity gradients exist purely because of the boundary layer growth and the boundary layer and inviscid core flows must be simultaneously considered. There are, nevertheless, many important practical problems in which such interactions can be ignored. 

As was the case with the full equations, these contain beside the three mean flow variables u, v, and T (the pressure is, of course, by virtue of Eq. (2.157) again determined by the external in viscid flow) additional terms arising as a result of the turbulence. Therefore, as previously discussed, in order to solve this set of equations, there must be an additional input of information, i.e., a turbulence model must be used. Many turbulence models are based on the turbulence kinetic energy equation that was previously derived. When the boundary layer assumptions are applied to this equation, it becomes ... 

When only approximate values of the overall features of the flow, such as the surface heat transfer rate and surface shearing stress, are required, it is possible to apply the boundary layer assumptions in a different way to obtain, relatively easily, approximate solutions for these quantities. The derivation of the equations required in this approximate solution procedure will be discussed in the present section (21,[7),[13]. 

It will be assumed that the Reynolds number is large enough for the boundary layer assumptions to be applicable. It will further be assumed that the flow is two-dimensional which means that the plate is assumed to be wide compared to its longitudinal dimension. As a result of these assumptions, the equations governing the problems are ... 

It was shown in the derivation of the boundary layer assumptions that for the boundary layer assumptions to apply it is necessary that ... 

Attention will mainly be given to turbulent boundary layer flows in this chapter. A brief discussion of the analysis of flows in which it is not possible to use these boundary layer assumptions will be given at the end of the chapter. 

As in the case with forced convective flows, there are many free convective flows that can be analyzed with sufficient accuracy by adopting the boundary layer assumption. Essentially this boundary layer assumption is that the flow consists of two regions ... 

Because the boundary layer assumptions are being adopted, the equations governing the flow ire, as discussed above ... 

Consider, for example, assisting turbulent mixed convection over a vertical flat plate. This situation is schematically shown in Fig. 9.20. If it is assumed that the boundary layer assumptions apply, the governing equations for the mean velocity components and temperature are ... 

While this is rather small to apply the boundary layer assumptions it will, nevertheless, be assumed that the mean Nusselt number is given by ... 

It is to be noted that the values of and obtained by an inviscid analysis are used here to calculate the impressed pressure over the shear layer. If F is considered to be small, then the impressed pressure gradient is going to be negligibly small and not cause flow separation. In the absence of separation, boundary layer assumption holds and the impressed pressure remains the same across the shear layer and is the reason that the analysis results using inviscid pressure distribution provide vital clue to the receptivity route. 

Professor L. A. Skinner, in a private communication (SI 3b), makes some interesting comments, based upon Fig. 2, on the validity of the thin thermal boundary layer assumption. Approximating the curve by a straight line in the interval 0.2 < l/j8 < oo, one obtains... 

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