Film Problems

A reagent in solution can enhance a mass transfer coefficient in comparison with that of purely physical absorption. The data of Tables 8.1 and 8.2 have been cited. One of the simpler cases that can be analyzed mathematically is that of a pseudo-first order reaction that goes to completion in a liquid film, problem P8.02.01. It appears that the enhancement depends on the specific rate of reaction, the diffusivity, the concentration of the reagent and physical mass transfer coefficient (MTC). These quantities occur in a group called the Hatta number,... 

If transport of A, through the film of Cg is very fast compared to r , then the rate of the process is exacfly the same as if no solid film were formed, and this would be the growing (or dissolving) film problem we considered previously. However, now we have the possibility that the rate of the reaction slows down as the film of product thickens. 

If the plane source is on the surface of a semi-infinite medium, the problem is said to be a thin-film problem. The diffusion distance stays the same, but the same mass is distributed in half of the volume. Hence, the concentration must be twice that of Equation 3-45a ... 

The principal physical error is probably geometrical. Compared to this, the above assumptions are not unduly restrictive, although extremely fast high-power excursions at low pressures are ruled out by assumptions (2) and (3). Assumption (2) is more nearly fulfilled at high pressures with low liquid heads. Assumption (3) is acceptable in the vapor-film problem even when the radiative flux from the solid surface is appreciable, provided that the liquid (and, of course, vapor) is nearly transparent. 

Generally all these considerations are also valid for the second fluid film phase, provided that reactions occur there (135). Both analytical and numerical solutions of the coupled diffusion-reaction film problem are analyzed at full length in Ref. 167 their particular applications are considered in Section 3. 

Basic issues such as surface reactions, surface film formation, passivation, ionic and electronic transport phenomena through surface films, problems in uniformity of deposition and dissolution processes, correlation between surface chemistry, morphology, and electrochemical properties are common to all active metal electrodes in nonaqueous solutions and are dealt with thoroughly in this chapter. It is believed that many conclusions related to Li, Mg, Ca, and A1 electrodes can be extended to other active metal electrodes as well. 

The second basic class of thin-film problems involves the dynamics of films in which the upper surface is an interface (usually with air). In this case, the same basic scaling ideas are valid, but the objective is usually to determine the shape of the upper boundary (i.e., the geometry of the thin film), which is usually evolving in time. 

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° ... 

On the other hand, if a = 0, and the dynamics of the film is still dominated by body forces, then it appears from (6-3) that uc = eil1cpg/ii. In other cases, however, gravitational forces may play only a secondary role in the motion of the film, which is instead dominated by capillary forces. Then the appropriate choice for uc would involve the surface tension rather than either of the choices previously listed and the body-force terms in both (6-2) and (6-3) would be asymptotically small for the limit e -> 0. This then is a fundamental difference between this class of thin-film problems and the lubrication problems of the previous chapter. Here, the characteristic velocity will depend on the dominant physics, and if we want to derive general equations that can be used for more than one problem, we need to temporarily retain all of the terms that could be responsible for the film motion and only specify uc (and thus determine which terms are actually large or small) after we have decided which particular problem we wish to analyze. 

A generic problem, which is mathematically analogous to a number of thin-film problems for the shape function h, is the evolution of the radially symmetric concentration distribution that evolves at large times from a pulselike initial source. At very large times, the form of the concentration distribution becomes insensitive to its initial shape, i.e., it is independent of the details of the initial spatial distribution of c and depends on only the dimension of the distribution. In d dimensions, the form of the diffusion equation that describes the evolution of this radially symmetric concentration distribution is... 

Stone2 has summarized a generalization of the solution for this simple linear problem, due to Pattle3 and Pert4, which is extremely useftd in the analysis of thin-film problems. This is the development of similarity solutions for the (/-dimensional symmetric diffusion equation with a diffusivity that depends on the concentration,... 

It is also convenient to express (12-125) and (12-126) in dimensional form because the characteristic scales that were used in nondimensionalizing these equations are not the most appropriate choice for the thin-film problem (for example we used lc = R, whereas the gap width is a much more appropriate choice for a characteristic length scale in the narrow gap problem). Hence, reversing the nondimensionalization (12-119) and introducing the approximation (12-142), we have... 

The problem of poor adhesion to the substrate may be mitigated by the choice of an appropriate material. Especially metals with the ability to form carbides are suitable for this purpose. However, carbide formation alone is not sufficient to obtain films rehably adhering to the respective substrate. Films deposited on molybdenum, for instance, tend toward spontaneous delamination upon stress. Hence the film should be sought to bear the least possible stress to achieve good adhesion. The so-called stress is a quantity hard to assess for being influenced by a multitude of parameters Hke the thickness of the film, problems related to the grain boundaries, lattice mismatch, and thermal strain. 

Booth and Hirst [10] examined the squeeze film problem for two rigid circular parallel plates of radius separated by an oil film of thickness fi (h<Starting with the Navier-Stokes equations in cylindrical coordinates, they obtained the relations... 

The rate of thinning (drainage) of liquid films is drastically influenced by the rheological properties of the related adsorption layer. We will restrict ourselves to just a few examples. A detailed description of various sites of thin film problems is given for example by Ivanov (1988) and Hunter (1993). The immobilisation of a cylindrical plane film is a precondition for... 

As will be shown later, heat transfer is considerably more important than mass transfer in the external film. Detailed theoretical analyses of the external film problem are available (Carberry and Kulkarni, 1973 Carberry,l975) and will not be considered here. 

How to Solve Blown Film Problems, (Lyondell Chemical Company, Cincinnati Technology Center, Cincinnati, OH, USA, 2013)... 

The inversion of the Drude equations, that is the estimation of unknown thicknesses or optical constants from ellipsometric measurements, relies upon the application of computer-intensive search and optimization methods, which are well within the capabilities of personal computers. The software for solving a wide variety of film problems is now available as part of the instrumentation package from a good number of ellipsometer manufacturers. This has resulted in the fast-widening scope of ellipsometry as reflected in the number of publications in which the technique is dominant. 

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