Thin films gradient theory

There are several theories concerned with mass transfer across a phase boundary. One of the most widely used is Whitman s two-film theory in which the resistance to transfer in each phase is regarded as being located in two thin films, one on each side of the interface. The concentration gradients are assumed to be linear in each of these layers and zero elsewhere while at the interface itself, equilibrium conditions exist (Fig. 5). Other important theories are Higbie s penetration theory and the theory of surface renewal due to Danckwerts. All lead to the conclusion that, in... 

We shall see that there are certain generic features of the thin-film approximation that have already been illustrated by the journal-bearing analysis, which will appear whenever the thin-film approximation is used. The most significant feature of the journal-bearing analysis, which is shared by all thin-film theories, is the assumption that velocity gradients in the direction parallel to the boundaries are asymptotically small compared with those... 

Although the solution (5 74) seems to be complete, the key fact is that the pressure gradient V.s//0) in the thin gap, and thus p(0 xs, 0, is unknown. In this sense, the solution (5-74) is fundamentally different from the unidirectional flows considered in Chap. 3, where p varied linearly with position along the flow direction and was thus known completely ifp was specified at the ends of the flow domain. The problem considered here is an example of the class of thin-film problems known as lubrication theory in which either h(xs) and us, or h(xs, 0) and uz are prescribed on the boundaries, and it is the pressure distribution in the thin-fluid layer that is the primary theoretical objective. The fact that the pressure remains unknown is, of course, not surprising as we have not yet made any use of the continuity equation (5-69) or of the boundary conditions at z = 0 and h for the normal velocity component ui° ... 

Mitlin, V.S., and M.M. Sharma. 1993. A local gradient theory for structural forces in thin fluid films. J. Colloid Interface Sci. 157 447-464. 

Two major variables which effect E are the film thickness and surface concentration. The concentration of surface-active material influences the concentration gradients. For a freshly produced foam to survive, then surface tension gradients are necessary. However, the main deficiency in the early studies on Gibbs elasticity was that it applies to thin films and the diffusion effects from bulk solution were neglected. In fact, the Gibbs theory only applies to a hypothetical equilibrium state (i.e. it is assumed that there is insufficient surfactant in the film to diffuse to the surface and lower the surface tension). 

Chapter 1 introduced the reader to the notion of a mass transfer coefficient and has shown the connechon to what is termed film theory. In essence, this approach assumes the resistance to mass transfer to be confined to a thin film in the vicinity of an interface in which the actual concentration gradient is replaced by a linear approximation. The result is that the rate of mass transport can be represented as the product of a mass transfer coefficient and a linear concentration difference, or concentration driving force. Thus,... 

Although mass transfer across the water-air interface is difficult in terms of its application in a sewer system, it is important to understand the concept theoretically. The resistance to the transport of mass is mainly expected to reside in the thin water and gas layers located at the interface, i.e., the two films where the gradients are indicated (Figure 4.3). The resistance to the mass transfer in the interface itself is assumed to be negligible. From a theoretical point of view, equilibrium conditions exist at the interface. Because of this conceptual understanding of the transport across the air-water boundary, the theory for the mass transport is often referred to as the two-film theory (Lewis and Whitman, 1924). 

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