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Film: diffusion penetration theory

Further retaxation of the assumptions of the film aed penetration theories was suggested by Danckwerta1 who viewed the process as one of transient one-dimensioual diffusion to packets or elements of fluid that reside at the phase interface for varying periods of time. Therefore, the model is that of the penetration theory with a distribution of conract times, The surface age distribution p(t) is defined such that 4(i)tb is the fraction of surface that has seslded at the interface for a time between I and t + di. The mass transfer flua for the entire surface is obtained by integration or the instantaneous flux over all exposure times ... [Pg.105]

Thus we have two models represented by Equations 4.81 and 4.82 for the film and penetration theories, respectively, to choose from. In making a choice, note that one is often concerned with the ratio of diffusivities of the two reactants of a two-phase system (see Chapter 14), and not the diffusivity of just one diffusing component. Because the diffusion coefficients of organic compounds in many of the solvents normally used do not greatly differ from one another, the difference... [Pg.79]

This is a regime in which the diffusion coefficients of A and B in the liquid are the controlling parameters, and chemical reaction plays practically no part. Thus it has frequently been used to compare various theories of mass transfer to and from solid surfaces. The main conclusion is that the value of the exponent p in D /Dpy is different for different theories. The value of n for the boundary layer theory is 2/3. Recalling the values for the film and penetration theories,... [Pg.481]

ODES in Eqns (90), (91) and (92). However, they have not considered the way in which the diffusion and reaction interact with the overall material balances on the gas and liquid phases, and the equations were solved on the basis of arbitrary film/bulk boundary conditions. Instead, they compared reaction factors E as calculated using the film and penetration theories to describe the diffusion and reaction. Also, they calculated film yields according to both the film and penetration theories and showed that the differences in film yields are somewhat greater than the differences in reaction factors. The authors also stressed that their computed yields were point yields, and that differences between film and penetration theories for an overall reactor yield would indeed be magnified. [Pg.273]

However, in practice, our success is limited. As we explained at the start of the chapter, we expect that the mass transfer coefficient k should vary with the square root of the diffusion coefficient D (cf. Equation 9.0-1). This is consistent with the penetration theory. We also expect that k should vary with the two-thirds power of the fluid velocity V. This is larger than that expected by both film and penetration theories. This shortcoming will be explored more in Section 9.3. [Pg.279]

The mass transfer theories developed in the previous sections of this chapter are not especially successful. To be sure, the penetration and surface-renewal theories do predict that mass transfer does vary with the square root of the diffusion coefficient, consistent with many correlations. However, neither the film theory nor the surface-renewal theory predicts how mass transfer varies with flow. The penetration theory predicts variation with the square root of flow, less than that indicated by most correlations. This failure to predict the variation of mass transfer with flow is especially disquieting the film and penetration theories should bracket all behavior because a thin film and a semi-infinite slab bracket all possible geometries. [Pg.281]

Other Models for Mass Transfer. In contrast to the film theory, other approaches assume that transfer of material does not occur by steady-state diffusion. Rather there are large fluid motions which constantiy bring fresh masses of bulk material into direct contact with the interface. According to the penetration theory (33), diffusion proceeds from the interface into the particular element of fluid in contact with the interface. This is an unsteady state, transient process where the rate decreases with time. After a while, the element is replaced by a fresh one brought to the interface by the relative movements of gas and Uquid, and the process is repeated. In order to evaluate a constant average contact time T for the individual fluid elements is assumed (33). This leads to relations such as... [Pg.23]

HARRIOTT 25 suggested that, as a result of the effects of interfaeial tension, the layers of fluid in the immediate vicinity of the interface would frequently be unaffected by the mixing process postulated in the penetration theory. There would then be a thin laminar layer unaffected by the mixing process and offering a constant resistance to mass transfer. The overall resistance may be calculated in a manner similar to that used in the previous section where the total resistance to transfer was made up of two components—a Him resistance in one phase and a penetration model resistance in the other. It is necessary in equation 10.132 to put the Henry s law constant equal to unity and the diffusivity Df in the film equal to that in the remainder of the fluid D. The driving force is then CAi — CAo in place of C Ao — JPCAo, and the mass transfer rate at time t is given for a film thickness L by ... [Pg.613]

As noted previously, for equimolecular counterdiffusion, the film transfer coefficients, and hence the corresponding HTUs, may be expressed in terms of the physical properties of the system and the assumed film thickness or exposure time, using the two-film, the penetration, or the film-penetration theories. For conditions where bulk flow is important, however, the transfer rate of constituent A is increased by the factor Cr/Cgm and the diffusion equations can be solved only on the basis of the two-film theory. In the design of equipment it is usual to work in terms of transfer coefficients or HTUs and not to endeavour to evaluate them in terms of properties of the system. [Pg.625]

Two rather similar models have been devised to remedy the problems of simple film theory. Both the penetration theory of Higbie and the surface renewal theory of Danckwerts replace the idea of steady-state diffusion across a film with transient diffusion into a semi-inhnite medium. We give here a brief account of surface renewal theory. [Pg.410]

The devolatilization of a component in an internal mixer can be described by a model based on the penetration theory [27,28]. The main characteristic of this model is the separation of the bulk of material into two parts A layer periodically wiped onto the wall of the mixing chamber, and a pool of material rotating in front of the rotor flights, as shown in Figure 29.15. This flow pattern results in a constant exposure time of the interface between the material and the vapor phase in the void space of the internal mixer. Devolatilization occurs according to two different mechanisms Molecular diffusion between the fluid elements in the surface layer of the wall film and the pool, and mass transport between the rubber phase and the vapor phase due to evaporation of the volatile component. As the diffusion rate of a liquid or a gas in a polymeric matrix is rather low, the main contribution to devolatilization is based on the mass transport between the surface layer of the polymeric material and the vapor phase. [Pg.813]

The theories vary in the assumptions and boundary conditions used to integrate Fick s law, but all predict the film mass transfer coefficient is proportional to some power of the molecular diffusion coefficient D", with n varying from 0.5 to 1. In the film theory, the concentration gradient is assumed to be at steady state and linear, (Figure 3-2) (Nernst, 1904 Lewis and Whitman, 1924). However, the time of exposure of a fluid to mass transfer may be so short that the steady state gradient of the film theory does not have time to develop. The penetration theory was proposed to account for a limited, but constant time that fluid elements are exposed to mass transfer at the surface (Higbie, 1935). The surface renewal theory brings in a modification to allow the time of exposure to vary (Danckwerts, 1951). [Pg.82]

Adoption and Use of Modeling Framework The rate of diffusion and species generation by chemical reaction can be described by film theory, penetration theory, or a combination of the two. The most popular description is in terms of a two-film theory, which is... [Pg.25]

However, it must be noted that as the film flows over the disc, the film thickness progressively decreases, provided the liquid fully wets the disc. As this occurs, the concentration profiles normal to the disc plane are compressed, thereby causing a proportionate enhancement of the solute diffusion rate beyond that predicted by penetration theory. Thus the local value of kL can be corrected to account approximately for the steepened concentration gradients by multiplying by a factor sx/s, where. v, is the film thickness at a radius i as given by Eq. (9). The corrected local value of kL is then... [Pg.100]

Ammonia is absorbed in a falling film of water in an absorption apparatus and the film is disrupted and mixed at regular intervals as it flows down the column. The mass transfer rate is calculated from the penetration theory on the assumption that all the relevant conditions apply. It is found from measurements that the mass transfer rate immediately before mixing is only 16 per cent of that calculated from the theory and the difference has been attributed to the existence of a surface film which remains intact and unaffected by the mixing process. If the liquid mixing process takes place every second, what thickness of surface film would account for the discrepancy Diffusivity of ammonia in water = 1.76 x 10 9 m2/s. [Pg.251]

The mass transfer model. In our previous work [6] the mass transfer model equations and their mathematical treatment have been described extensively. The relevant differential equations, describing the process of liquid-phase diffusion and simultaneous reactions of the species according to the penetration theory, are summarized in table 1. Recently we derived from this penetration theory description a film model version, which is incorporated in the evaluation of the experimental results. Details on the film model version are given elsewhere [5]. [Pg.379]

Mass-Transfer Coefficient Denoted by kc, kx, Kx, and so on, the mass-transfer coefficient is the ratio of the flux to a concentration (or composition) difference. These coefficients generally represent rates of transfer that are much greater than those that occur by diffusion alone, as a result of convection or turbulence at the interface where mass transfer occurs. There exist several principles that relate that coefficient to the diffusivity and other fluid properties and to the intensity of motion and geometry. Examples that are outlined later are the film theory, the surface renewal theory and the penetration theory, all of which pertain to idealized cases. For many situations of practical interest like investigating the flow inside tubes and over flat surfaces as well as measuring external flow through banks of tubes, in fixed beds of particles, and the like, correlations have been developed that follow the same forms as the above theories. Examples of these are provided in the subsequent section on mass-transfer coefficient correlations. [Pg.45]

The penetration theory predicts that k L should vary by the square root of the molecular diffusivity, as compared with film theory, which predicts a first-power dependency on D. Various investigators have reported experimental powers of D ranging from 0.5 to 0.75, and the Chilton-Colburn analogy suggests a 2/6 power. [Pg.62]

Penetration theory often is used in analyzing absorption with chemical reaction because it makes no assumption about the depths of penetration of the various reacting species, and it gives a more accurate result when the diffusion coefficients of the reacting species are not equal. When the reaction process is very complex, however, penetration theory is more difficult to use than film theory, and tne latter method normally is preferred. [Pg.62]

The stagnant-film model discussed previously assumes a steady state in which the local flux across each element of area is constant i.e., there is no accumulation of the diffusing species within the film. Higbie [Trans. Am. Inst. Chem. Eng., 31,365 (1935)] pointed out that industrial contactors often operate with repeated brief contacts between phases in which the contact times are too short for the steady state to be achieved. For example, Higbie advanced the theory that in a packed tower the liquid flows across each packing piece in laminar flow and is remixed at the points of discontinuity between the packing elements. Thus, a fresh liquid surface is formed at the top of each piece, and as it moves downward, it absorbs gas at a decreasing rate until it is mixed at the next discontinuity. This is the basis of penetration theory. [Pg.430]

The value of a varies with the system under consideration. For example, in equimolar counter diffusion, Na and Nb are of the same magnitude, but in opposite direction. As a result, a is equal to 1 and hence, Eq. (2) reduces to Eq. (1), where is equal to Convective mass transfer coefficients are used in the design of mass transfer equipment. However, in most cases, these coefficients are extracted from empirical correlations that are determined from experimental data. The theories, which are often used to describe the mechanism of convective mass transfer, are the film theory, the penetration theory, and the surface renewal theory. [Pg.1163]


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See also in sourсe #XX -- [ Pg.644 ]




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