Linearized theory film model

The most comprehensive study of the combined effects of axial dispersion and mass transfer resistance under constant pattern conditions has been done by Rhee and Amimdson [17,18] using the shock layer theory. These authors assumed a solid film linear driving force model (Eq. 14.3) and wrote the mass balance equation as... 

As our first application of the linearized theory we consider steady-state, one-dimensional diffusion. This is the simplest possible diffusion problem and has applications in the measurement of diffusion coefficients as discussed in Section 5.4. Steady-state diffusion also is the basis of the film model of mass transfer, which we shall discuss at considerable length in Chapter 8. We will assume here that there is no net flux = 0. In the absence of any total flux, the diffusion fluxes and the molar fluxes are equal = J. ... 

A number of investigators used the wetted-wall column data of Modine to test multicomponent mass transfer models (Krishna, 1979, 1981 Furno et al., 1986 Bandrowski and Kubaczka, 1991). Krishna (1979b, 1981a) tested the Krishna-Standart (1976) multicomponent film model and also the linearized theory of Toor (1964) and Stewart and Prober (1964). Furno et al. (1986) used the same data to evaluate the turbulent eddy diffusion model of Chapter 10 (see Example 11.5.3) as well as the explicit methods of Section 8.5. Bandrowski and Kubaczka (1991) evaluated a more complicated method based on the development in Section 8.3.5. The results shown here are from Furno et al. (1986). 

Develop the film model for simultaneous mass and energy transfer including Soret and Dufour effects. Use the Toor-Stewart-Prober linearized theory in developing the model. An example of a process where thermal diffusion effects cannot be ignored is chemical vapor deposition. Use the model to perform some sample calculations for a system of practical interest. You will have to search the literature to find practical systems. To get an idea of the numerical values of the transport coefficients consult the book by Rosner (1986). 

Taylor, R., Film Models for Multicomponent Mass Transfer Computational Methods II—The Linearized Theory, Comput. Chem. Eng., 6, 69-75 (1982b). 

Contemporary understanding of liquid film rupture is based on the Linear Stability Theory and the concept of existence of fluctuational waves on liquid surfaces [81]. According to this model the film is ruptured by unstable waves, i.e. waves the amplitudes of which increase with time. The rupture occurs at the moment when the amplitude Ah or the root mean... 

Chao et al. [19] proposed a model that explains the growth of a film under steady-state conditions. It was considered that the passive film contains a high concentration of no recombining point defects. Metal/film and film/solution interfaces were assumed to be at electrochemical equilibrium. This theory successfully accounts for the linear dependencies of both the steady-state film thickness and the logarithm of the passive current on the applied voltage. 

Fig. 2 shows of the g(T) in SWCNT film measured by Bae et al. [8, Fig. 1], fitted to the theoretical Wj(E,T) computed using the equation (1). The theory describes well the experimental data. For an explanation of these results in the framework of LL model the authors of [8] involved the additional term with linear temperature dependence. The PhAT model can also explain the crossover of g(T) from semi-conducting-like to metallic-like observed in some works [5,16]. Since the PhAT theory includes an absorption/emission of phonons in the carrier tunneling, the variation of g(T) will be determined by the competition of the absorption and emission of phonons. 

The MO measurements provide information about the angular distribution of molecules in the x, y, and z film coordinates. To extract MO data from IR spectra, the general selection rule equation (1.27) is invoked, which states that the absorption of linearly polarized radiation depends upon the orientation of the TDM of the given mode relative to the local electric field vector. If the TDM vector is distributed anisotropically in the sample, the macroscopic result is selective absorption of linearly polarized radiation propagating in different directions, as described by an anisotropic permittivity tensor e. Thus, it is the anisotropic optical constants of the ultrathin film (or their ratios) that are measured and then correlated with the MO parameters. Unlike for thick samples, this problem is complicated by optical effects in the IR spectra of ultrathin films, so that optical theory (Sections 1.5-1.7) must be considered, in addition to the statistical formulas that establish the connection between the principal values of the permittivity tensor s and the MO parameters. In fact, a thorough study of the MO in ultrathin films requires judicious selection not only of the theoretical model for extracting MO data from the IR spectra (this section) but also of the optimum experimental technique and conditions [angle(s) of incidence] for these measurements (Section 3.11.5). 

Of particular interest in Eq. (2.3-65), as well as in the earlier results of film theory and uansiem diffusion into a fluid, is the predicted dependence of the flux on geometry, bydrodymuntcs (6. y, or Re), physical properties (Dam and Sc), and composition driving force. In all cases the flux varies linearly with the composition driving force but the dependence of flux on diflusivity ranges from the first to the one-half power. These simple models can be used to guide or interpret mass transfer rate observations in mote complex situations. 

The theories of the electronic and ionic currents have some features in common. One may formulate models in which the current is limited by the injection into the film from the contacts of positively or negatively charged carriers, or one may consider an equilibrium state to exist across either or both interfaces. One may postulate space-charge limited currents, trapping, and recombination processes. One of the chief differences between the ionic and the electronic currents is that the average velocity of the ions is approximately exponentially dependent on the field for fields which produce experimentally observable ionic currents, whereas the average velocity of electrons is linearly dependent on the field at low fields with different types of nonlinearity at high fields. 

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