Boundary-layers curvature terms

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 ... 

For the case of high Peclet numbers Pe 1), the concentration boundary layer is very thin compared to the local radius of curvature of the particle, and, thus, the curvature term as well as the tangential diffusion terms can be neglected as it has already been shown [1,8,11]. In that case, eq. (2) becomes parabolic on 0 and can be solved analytically in a manner quite similar to that of Levich, providing concentration profiles in the fluid phase of the form... 

One interesting feature is that the operator V20, which is expressed on the left-hand side of (9-222) in terms of rescaled spherical coordinate variables, takes a form in the limiting approximation (9-225) that appears to be just the normal derivative term in V20 for a Cartesian coordinate system. In fact, we shall see that boundary-layer equations always can be expressed in terms of a local Cartesian coordinate system, with one variable normal to the body surface at each point (Y in this case) and the others tangent to it. This reduction of the equations in the boundary layer to a local, Cartesian form is due to the fact that the dimension of the boundary layer is so small relative to that of the body that surface curvature effects play no role. 

Because this inner boundary-layer region is infinitesimal in thickness relative to a, all curvature terms that appear when the equations of motion are expressed in curvilinear coordinates will drop out to first order in Re thus leaving boundary-layer equations that are effectively expressed in terms of a local Cartesian coordinate system. 

Thin-Layer Approximation. Laminar analyses often make the further approximation that the boundary layer is so thin that when the simplified equations of motion are rewritten in terms of local surface coordinates, i.e., in terms of the x and y of Fig. 4.3a, several terms normally associated with curvature effects can be dropped. The Nusselt number equation, based on solutions to such laminar thin-layer equation sets, always takes the form... 

Hence, neglecting the curvature correction factors of in the radial term of the equation of continuity corresponds to a 14% underestimate of (l/r )[d(r Vr)/dr] at the outer edge of the mass transfer boundary layer on the equator of the pellet. 

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. 

The order of magnitude for the boundary layer length scale is obtained assuming that convective and diffusive terms are comparable in magnitude, that is, the scaling coefficients must be equal (the curvature term of diffusion of 0(d) is much smaller than 1 and can be ignored) ... 

For p 1 the deformations of layers are restricted to a region of volume a and do not modify the orientation at the boundaries. The nucleation is defined by the balance of the elastic and field terms. The elastic energy scales linearly with TFCD radius, hp, while the field term should scale as -- h p3, which is the volume of significant layers curvature. More... 

One of the most important points to be discussed in this section is the mutual influence of the bulk and the boundary part of the medium confined by the curved interface on their melting behavior. From Equation 9.38 it follows that the shift of the triple point temperature for the sublayers of a curved boundary phase will differ from that for the case of plane interface (as described by Equation 9.22), first fall due to the effect of curvature described by the terms in the second set of brackets (within []). This shift, in turn, affects the values of y l and nl (because of their temperature dependence), which should be substituted into the first set of brackets and the magnitude of the effective latent heat of fusion (see the discussion after Equation 9.36). As a mle, the terms in the first and in the second sets of brackets act in the same direction (while it is not necessary for the general case). This synergetic action makes the sequential phase transitions in boundary sublayers more pronounced and more separated on the temperature scale compared to the case of plane interfaces. Therefore, the phase transitions in deeper boundary sublayers become experimentally detectable. This effect is revealed in most experimental methods as an apparent increase in the thickness of a skin layer (melted layer in the case of premelting) for the curved interfaces [11, 60]. 

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