Mass transfer across film resistance

Mass transfer across a film resistance from the bulk gas phase to the external surface of the porous catalyst. 

The concentration of gas over the active catalyst surface at location / in a pore is ai [). The pore diffusion model of Section 10.4.1 linked concentrations within the pore to the concentration at the pore mouth, a. The film resistance between the external surface of the catalyst (i.e., at the mouths of the pore) and the concentration in the bulk gas phase is frequently small. Thus, a, and the effectiveness factor depends only on diffusion within the particle. However, situations exist where the film resistance also makes a contribution to rj so that Steps 2 and 8 must be considered. This contribution can be determined using the principle of equal rates i.e., the overall reaction rate equals the rate of mass transfer across the stagnant film at the external surface of the particle. Assume A is consumed by a first-order reaction. The results of the previous section give the overall reaction rate as a function of the concentration at the external surface, a. ... 

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). 

Equation (4.20) expresses that the total resistance to mass transfer across the air-water boundary is equal to the sum of the resistances across the liquid film and the gas film. The importance of the magnitude of Henry s constant is, in this respect, evident. For high values of HA, e.g., exemplified by 02, the resistance mainly exists in the water film, and turbulence in a sewer will, therefore, enhance the water-air transfer process. The importance of turbulence in the water phase is reduced for odorous components with a relatively low HA value, and turbulence in the air phase will correspondingly increase the release rate (Table 4.1). As seen from Equations (4.20) and (4.21), these facts also depend on the k1A/k2A ratio that varies according to system characteristics. 

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... 

Research with pilot scale units has shown that the major resistances to mass transfer of reactant to catalyst are within the liquid film surrounding the wetted catalyst particles and also intraparticle diffusion. A description of these resistances is afforded by Fig. 14. Equating the rate of mass transfer across the liquid film to the reaction rate, first order in hydrogen concentration... 

Lewis and Whitman (1924) proposed that this resistance to mass transfer across an interface is the sum of the resistances in each phase. They called this concept the two-film theory. As Treybal (1968) pointed out, their two-film theory does not depend on which model is used to describe the mass transfer in each phase, therefore, the two-resistance theory would be a more appropriate name. It would also cause less confusion, since the names film theory (mass transfer in one phase) and two-film theory (mass transfer between... 

Several models have been proposed to represent the mechanics of mass transfer across the phases. In the two-film theory, the resistance to mass transfer is assumed to take place in a liquid film and a vapor film at the phase boundary interface, as shown schematically in Figure 15.3b. The coordinates X and Y in Figure 15.3a represent the mole fractions of a given component in the liquid and vapor, respectively. For simplicity, component subscripts are dropped and the discussions refer to one... 

Some implications of this resistance term can be observed immediately. First, the larger the value of k, the smaller the resistance. Second, the value of each resistance can be different. When one resistance is significantly larger than the other (or the mass transfer coefficient for one phase is significantly smaller), it is dominant and is termed the controlling resistance. Mass transfer across both films is controlled (limited) by the dominant resistance. Third, the larger the value of m, the larger the liquid-phase resistance (can you see why physically ). 

Here, we are concerned with regimes in which the reaction occurs exclusively in the bulk and is controlled by either chemical reaction (regime 1) or diffusion, that is, mass transfer across the liquid film (regime 2), as well as with the intermediate regime in which there is a mass transfer resistance in the film but the reaction still occurs exclusively in the bulk. 

In many systems, particularly liquid systems, the resistance due to diffusion in the pores is much more important than the resistance due to mass transfer across the film is small and hence the second... 

In the case of external mass transfer resistance, the rate of substrate depletion, -va, is controlled by the mass transfer across the stagnant film ... 

The two-film theory still is widely employed to explain mass transfer operations. The boundary between the gas phase and the liquid phase is presumed to consist of a gas film adjacent to a liquid film. Flow in both of these films is assumed to be laminar or stagnant. The main-body gas phase, as well as the main-body liquid phase, are assumed to be completely mixed in turbulent flow so that no concentration gradient exists in the main body of either phase. Further, the solute concentration in the gas film at the interface is assumed to be in equilibrium with the solute concentration in the liquid film at the interface, and there is no resistance to mass transfer across the interface. There is a solute concentration gradient across both the gas film and the liquid film. 

FIGURE 4.19 (a) Mass transfer across a polymer film-fluid interface. (b) Mass transfer across an interface with two resistances present the first in one side of the interface (C oo to )> the second in the other side of the interface (Ca to Caoo)- Henry s law is applicable at the interface, between Ca and C -. 

As an example, it may be supposed that in phase 1 there is a constant finite resistance to mass transfer which can in effect be represented as a resistance in a laminar film, and in phase 2 the penetration model is applicable. Immediately after surface renewal has taken place, the mass transfer resistance in phase 2 will be negligible and therefore the whole of the concentration driving force will lie across the film in phase 1. The interface compositions will therefore correspond to the bulk value in phase 2 (the penetration phase). As the time of exposure increases, the resistance to mass transfer in phase 2 will progressively increase and an increasing proportion of the total driving force will lie across this phase. Thus the interface composition, initially determined by the bulk composition in phase 2 (the penetration phase) will progressively approach the bulk composition in phase 1 as the time of exposure increases. 

Mass transfer of the products across a film resistance into the bulk gas phase. 

It may be appropriate here to introduce film theory. As mentioned in reference to the steady diffusion across a thin film, we often hypothesize a film called an unstirred layer to account for the aqueous diffusion resistance to mass transfer. Film theory is valuable not only because of its simplicity but also because of its practical utility. However, the thickness of the film is often difficult to determine. In the following, we try to answer the question, What does the thickness of the film represent ... 

Figure 9.7 shows concentration profiles schematically for A and B according to the two-film model. Initially, we ignore the presence of the gas film and consider material balances for A and B across a thin strip of width dx in the liquid film at a distance x from the gas-liquid interface. (Since the gas-film mass transfer is in series with combined diffusion and reaction in the liquid film, its effect can be added as a resistance in series.)... 

The transport process, according to the two-film theory, of a volatile component across the air-water interface is depicted in Figure 4.3. The figure illustrates a concept that concentration gradients in both phases exist and that the total resistance for mass transfer is the sum of the resistance in each phase. 

The rate of mass transfer of a snbstance across a water-gas bonndary is controlled by the diffnsion film model as well. Gas transfer from a water sonrce is faster than from a solid sonrce, and the chemical does not nndergo a chemical reaction during the transfer process. Under these conditions, the interface concentration may be interpreted in terms of the Henry constant (K ), which indicates whether the controlling resistance is in the liqnid or the gas film. When 5, a water film is the controlling factor, while a gas film controls the behavior when K >500. 

One objection to a Forster and Zuber assumption has been given by Zwick (Zl). Forster and Zuber state that the principal mechanism for heat transfer to a growing bubble is conduction across the film resistance. Zwick points out that heat can also flow by mass transport and that this convection should be included in the equations. 

The apparent rates of adsorption into adsorbent particles usually involve the resistances for mass transfer of adsorbate across the fluid film around adsorbent particles and through the pores within particles. Adsorption perse at adsorption sites occurs very rapidly, and is not the rate-controlling step in most cases. 

In most common separation processes, the main mass transfer is across an interface between a gas and a liquid or between two liquid phases. At fluid-fluid interfaces, turbulence may persist to the interface. A simple theoretical model for turbulent mass transfer to or from a fluid-phase boundary was suggested in 1904 by Nernst, who postulated that the entire resistance to mass transfer in a given turbulent phase lies in a thin, stagnant region of that phase at the interface, called a him, hence the name film theory.2 4,5 Other, more detailed, theories for describing the mass transfer through a fluid-fluid interface exist, such as the penetration theory.1,4... 

The film theory was originally proposed by Whitman,195 who obtained his idea from the Nernst117 concept of the diffusion layer. It was first applied to the analysis of gas absorption accompanied by a chemical reaction by Hatta.85,86 It is a steady-state theory and assumes that mass-transfer resistances across the interface are restricted to thin films in each phase near the interface. If more than one species is involved in a multiphase reaction process, this theory assumes that the thickness of the film near any interface (gas-liquid or liquid-solid) is the same for all reactants and products. Although the theory gives a rather simplified description of the multiphase reaction process, it gives a good answer for the global reaction rates, in many instances, particularly when the diffusivities of all reactants and products are identical. It is simple to use, particularly when the... 

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