Film theory effective

Unfortunately none of the various proposed forms of the potential theory satisfy this criterion Equation XVII-78 clearly does not Eq. XVII-79 would, except that / includes the constant A, which contains the dispersion energy Uo, which, in turn, depends on the nature of the adsorbent. Equation XVII-82 fares no better if, according to its derivation, Uo reflects the surface polarity of the adsorbent (note Eq. VI-40). It would seem that after one or at most two layers of coverage, the adsorbate film is effectively insulated from the adsorbent. 

Film Theory. Many theories have been put forth to explain and correlate experimentally measured mass transfer coefficients. The classical model has been the film theory (13,26) that proposes to approximate the real situation at the interface by hypothetical "effective" gas and Hquid films. The fluid is assumed to be essentially stagnant within these effective films making a sharp change to totally turbulent flow where the film is in contact with the bulk of the fluid. As a result, mass is transferred through the effective films only by steady-state molecular diffusion and it is possible to compute the concentration profile through the films by integrating Fick s law ... 

Equation 39 can often be simplified by adopting the concept of a mass transfer unit. As explained in the film theory discussion eadier, the purpose of selecting equation 27 as a rate equation is that is independent of concentration. This is also tme for the Gj /k aP term in equation 39. In many practical instances, this expression is fairly independent of both pressure and Gj as increases through the tower, increases also, nearly compensating for the variations in Gj. Thus this term is often effectively constant and can be removed from the integral ... 

Effects of Total Pressure on Uq and The influence of total system pressure on the rate of mass transfer from a gas to a licniid or to a solid has been shown to be the same as would be predicted from stagnant-film theory as defined in Eq. (5-285), where... 

There have been many studies of the effect of boundary films on mass and heat transfer to single pellets and in packed beds, including the work of Ranz and Marshall 27 and Dwivedi and Upadhey(28). Other theories of mass and heat transfer are discussed in Volume 1, Chapter 10, although only the steady-state film-theory is considered here. It is assumed that the difference in concentration and temperature between the bulk fluid and the external surface of a pellet is confined to a narrow laminar boundary-layer in which the possibility of accumulation of adsorbate or of heat is neglected. 

In the present study, EXAMS was used to calculate volatilization rate constants from water, wet soil, and a water-soil mixture. EXAMS uses the two-film theory to calculate volatilization rates from the 10 cm wind speed as discussed above. EXAMS requires as a minimum environment at least one littoral (water) and one benthic (sediment) compartment. A very small benthic compartment for the water system and a very small littoral compartment for the wet soil system (7.09 x 10 m3 volume and 1 x 10 8 m depth in both cases) was used, so that these compartments and their input parameters had a negligible effect on the calculated rates. For the water-soil system, the same proportions were used as in the laboratory experiment. Transfer rates between soil and water were assumed to be rapid relative to volatilization rates, and were set as recommended in the EXAMS manual (24). The input data needed by EXAMS in order to calculate volatilization rates from a water-soil system, using parathlon as an example, are shown in Table IV. 

The film model referred to in Chapters 2 and 5 provides, in fact, an oversimplified picture of what happens in the vicinity of interface. On the basis of the film model proposed by Nernst in 1904, Whitman [2] proposed in 1923 the two-film theory of gas absorption. Although this is a very useful concept, it is impossible to predict the individual (film) coefficient of mass transfer, unless the thickness of the laminar sublayer is known. According to this theory, the mass transfer rate should be proportional to the diffusivity, and inversely proportional to the thickness of the laminar film. However, as we usually do not know the thickness of the laminar film, a convenient concept of the effective film thickness has been assumed (as... 

The treatment can be modified to include effects of the temperature development and tilting of the susceptor by using the temperature dependence of the diffiision coefficient and adjusting d and (191). In this manner, the experimental data can be correlated, but the model has limited capability for predicting behavior beyond the particular set of experiments used to fit the model. In fact, because of the low values of the Reynolds number (<50) in typical horizontal CVD reactors, film theory and simple... 

The so-called Two Film Theory (Lewis and Whiteman, 1923-24) assumes the formation of laminar boundary layers on both sides of the interphase. Mass transfer through these boundary layers can only be effected by means of diffusion, while the phase transition is immeasurably fast, Fig. 86. Consequently, an equilibrium predominates in the interphase and the saturation concentration cG of the gas in the interphase ( ) obeys Henry s law ... 

In Chap. 2, the gas liquid-solid reaction process based on the film theory was analyzed. In this chapter, some of the reported models for three-phase reactors are presented. Some models consider only the effectiveness of contact between the liquid and solid, while others consider the roles of the gas-liquid and liquid-solid mass-transfer resistances in three-phase gas-liquid-solid reactors. A large number of models consider the role of the RTD on the reactor performance. Both isothermal and non-isothcrmal models are considered here. 

The rate of decompression can also have an effect on the ability of the compacts to consolidate (form bonds). Based on the liquid-surface film theory, the rate of crystallization or solidification should have an effect on the strength of the bonded surfaces. The rate of crystallization is affected by the pressure (and the rate at which the pressure is removed). High decompression rates should result in high rates of crystallization. Typically, slower crystallization rates result in stronger crystals. Therefore, if bonding occurs by these mechanisms, lower machine speeds (lower rates of... 

To describe gas absorption with simultaneous chemical reaction, the film theory will be used for purposes of illustration, because the predictions based on the three models are quite similar (except in regard to the effect of solute gas and reactant diffusivities on the rate of absorption). 

Even the film theory discussed in Chapter 8 falls neatly into the framework provided by Eq. 10.3.22. If we assume that the mass transfer process is governed by molecular transport within an effective film of thickness and that the level of turbulence is such as to wash out completely all composition gradients beyond this distance, then we see that... 

The heat transfer coefficients estimated from correlations or analogies are the low flux coefficients and, therefore, need to be corrected for the effects of finite transfer rates before use in design calculations. We recommend the film theory correction factor given by Eq. 11.4.12. 

Numerical simulations of Sardesai s experiments are discussed by Webb and Sardesai (1981) and Webb (1982) (who used the Krishna-Standart (1976), Toor-Stewart-Prober (1964) and effective diffusivity methods to calculate the condensation rates), McNaught (1983a, b) (who used the equilibrium model of Silver, 1947), and Furno et al. (1986) (who used the turbulent diffusion models of Chapter 10 in addition to methods based on film theory). It is the results of the last named that are presented here. 

Convective effects in the reactor space external to the feature are included by making use of the film theory. The effects of the bath hydrodynamics external to the wafer are included by assuming a thin concentration boundary layer adjacent to the wafer. 

The agreement between the experimental results and the calculated spectra demonstrates that Maxwell-Garnett effective medium theory effectively accounts for such dipole-dipole interactions. Similarly, the reflectance properties of the films are also affected by the interparticle spacing. This is supported by the data shown in Fig. 3, where the specular reflectance for films with different interparticle separation are compared to the spectra calculated by means of Eqs. (11), (18) and (19) [13]. Again, as the particles approach each other, the reflectance red-shifts and broadens. The effect is not as dramatic as seen for absorption, but the film changes from a green reflected hue to a metallic gold as (j) increases. 

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