The two-film theory

The two-film theory of WHITMAN119 was the first serious attempt to represent conditions occurring when material is transferred from one fluid stream to another. Although it does not closely reproduce the conditions in most practical equipment, the theory gives expressions which can be applied to the experimental data which are generally available, and for that reason it is still extensively used. 

The mass transfer is treated as a steady state process and therefore the theory can be applied only if the time taken for the concentration gradients to become established is very small compared with the time of transfer, or if the capacities of the films are negligible. 

From equation 10.22 the rate of transfer per unit area in terms of the two-film theory for equimolecular counterdiffusion is given for the first phase as  

For the second phase, with the same notation, the rate of transfer is  

Because material does not accumulate at the interface, the two rates of transfer must 

Our considerations so far have been limited to transport through a single phase i.e., in postulating the film theory it was assumed that a single film resistance was operative. This was the case for a pure liquid evaporating into a gas stream, or when a solid dissolved into a solvent stream. We now consider the extension of this process to encompass simultaneous transport in two adjacent phases. This leads to the formation of two film resistances, and brings us to the so-called two-film theory, which is taken up below. 

The two-film concept (a) mass transfer (b) heat transfer. 

These expressions, while valid imder the constraints of two-film theory, are nevertheless ill-suited for practical use, as neither of the interfacial concentrations x i or is generally known. We avoid this difficulty by postulating an equivalent rate law, given by 

The reciprocal terms appearing in these expressions may be regarded as resistances to the mass transfer process. Thus, in Equation 1.22a, l/fcy is the resistance due to the gas film, m/k represents the liquid film resistance, and the sum of the two yields the overall resistance 1/K. This is often referred to as the law of additivity cf resistances, which was previously encountered in Equation 1.7g. Similar arguments apply to Equation 1.22b. Note that when m is small compared to k, we obtain die approximate relation 

This implies that the gas is highly soluble and that most of the resistance is likely to reside in the gas phase. We speak of the process as being gas-film controlled. Conversely, if m is large, i.e., the gas is sparsely soluble, we obtain 

Mass Transfer and Separation Processes Principles and Applications 

Two limiting cases are of note When m is small compared to k, one obtains the approximate relahon 

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Fig. 2. Schematic representation for the two-film theory of gas transfer = partial pressure of gas Pj = partial pressure of the gas at the interface Cj = concentration of gas at time t, Cj = initial concentration of gas at the interface Cg = initial concentration of gas at t = 0 and S = gas saturation.

F = Function of the molecular volume of the solute. Correlations for this parameter are given in Figure 7 as a function of the parameter (j), which is an empirical constant that depends on the solvent characteristics. As points of reference for water, (j) = 1.0 for methanol, (j) = 0.82 and for benzene, (j) = 0.70. The two-film theory is convenient for describing gas-liquid mass transfer where the pollutant solute is considered to be continuously diffusing through the gas and liquid films. 

Because of the difficulties in determining x, the thickness of the film between the two vapor pressures, an overall transfer coefficient is introduced. Based on the two film theory, the overall transfer coefficient is used. In the case of water evaporation, the gas film is the controlling mechanism and the resulting equation is... 

If it is assumed that each element resides for the same time interval te in the surface, equation 10.115 gives the overall mean rate of transfer. It may be noted that the rate is a linear- function of the driving force expressed as a concentration difference, as in the two-film theory, but that it is proportional to the diffusivity raised to the power of 0.5 instead of unity. 

When the film theory is applicable to each phase (the two-film theory), the process is steady state throughout and the interface composition does not then vary with time. For this case the two film coefficients can readily be combined. Because material does not accumulate at the interface, the mass transfer rate on each side of the phase boundary will be the same and for two phases it follows that ... 

These relations between the various coefficients are valid provided that the transfer rate is linearly related to the driving force and that the equilibrium relationship is a straight line. They are therefore applicable for the two-film theory, and for any instant of time for the penetration and film-penetration theories. In general, application to time-averaged coefficients obtained from the penetration and film-penetration theories is not permissible because the condition at the interface will be time-dependent unless all of the resistance lies in one of the phases. 

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. 

Figure 1.29. Concentration gradients according to the Two-Film theory.

Note that the transfer rate equation is based on an overall concentration driving force, (X-X ) and overall mass transfer coefficient, Kl. The two-film theory for interfacial mass transfer shows that the overall mass transfer coefficient, Kl, based on the L-phase is related to the individual film coefficients for the L and G-phase films, kL and ko, respectively by the relationship... 

The experiments were conducted at four different temperatures for each gas. At each temperature experiments were performed at different pressures. A total of 14 and 11 experiments were performed for methane and ethane respectively. Based on crystallization theory, and the two film theory for gas-liquid mass transfer Englezos et al. (1987) formulated five differential equations to describe the kinetics of hydrate formation in the vessel and the associate mass transfer rates. The governing ODEs are given next. 

The approach taken is semi-empirical. Point efficiencies are estimated making use of the two-film theory , and the Murphree efficiency estimated allowing for the degree of mixing likely to be obtained on real plates. 

The two-film theory considering molecular diffusion through stagnant liquid and gas films is the traditional way of understanding mass transfer across the air-water boundary. As briefly described, other theories exist. However, the two-film theory gives an understanding of fundamental phenomena that may lead to simple empirical expressions for use in practice. 

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. 

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). 

According to the two-film theory, it is appropriate to consider the transport of volatile components between the water phase and the air phase in two steps from the bulk water phase to the interface and from the interface to the air, or vice versa. The driving force for the transfer of mass per unit surface area from the water phase to the interface and from the interface to the air phase is determined from the difference between the actual molar fractions, xA and yA, and the corresponding equilibrium values, xA and yA ... 

A relation between dy/dZ and (Ay)/ may be obtained on the basis of the two-film theory of mass transfer. For the vapour film, Fick s law, Volume 1, Chapter 10, gives ... 

The preceding analysis of the process of absorption is based on the two-film theory of Whitman 11. It is supposed that the two films have negligible capacity, but offer all the resistance to mass transfer. Any turbulence disappears at the interface or free surface, and the flow is thus considered to be laminar and parallel to the surface. 

All three of these proposals give the mass transfer rate N A directly proportional to the concentration difference (CAi — CAL) so that they do not directly enable a decision to be made between the theories. However, in the Higbie-Danckwerts theory N A a s/Dj whereas NA film theory. Danckwerts applied this theory to the problem of absorption coupled with chemical reaction but, although in this case the three proposals give somewhat different results, it has not been possible to distinguish between them. 

As Sherwood and Pigford(3) point out, the use of spray towers, packed towers or mechanical columns enables continuous countercurrent extraction to be obtained in a similar manner to that in gas absorption or distillation. Applying the two-film theory of mass transfer, explained in detail in Volume 1, Chapter 10, the concentration gradients for transfer to a desired solute from a raffinate to an extract phase are as shown in Figure 13.19, which is similar to Figure 12.1 for gas absorption. 

Let ns now consider the systems in a diffnsional regime. Given the simplified picture of the interface described by the two-film theory (see Fig. 5.1), we can now visnaUze that any species going from one phase to the other, crossing the liquid-Uqnid interface, will enconnter a total resistance R, which is the sum of three separate resistances. Two of these resistances are of a diffnsional nature and depend on the fact that diffnsion throngh the stagnant interfacial films can be a slow process. These resistances are indicated in Fig. 5.1 d R and R . R ... 

It is apparent from Table II that variations in the experimental rate constants (k) are essentially controlled by the Henry s law constant, in agreement with the two-film theory prediction. A plot of kys. H for the five pesticides gave an intercept of 5.4 x 10 hr, a slope of 6.9 x 10 mol/(hr atm m" ), and a correlation coefficient of 0.969. Thus, it seems that Henry s law values could be used to predict relative volatilization rates of the pesticides, and an absolute volatilization rate for one pesticide can be calculated if the volatilization rate is known for another and Henry s law constants are known for both ... 

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 two-film theory is not always applicable to heterogeneous processes and, for example, in gas—solid reactions, only the gas film is considered. 

To develop the rate equation, let us draw on the two-film theory, and let us use the following nomenclature ... 

Since reactant A must move from gas to liquid for reaction to occur, diffusional resistances enter the rate. Here we will develop everything in terms of the two-film theory. Other theories can and have been used however, they give essentially the same result, but with more impressive mathematics. 

Figure 23.1 Setting up the rate equation for straight mass transfer based on the two film theory.

Figure 23.5 Concentration of reactants as visualized by the two-film theory for an infinitely fast irreversible reactions of any order, A + bB - products. Case A high Cg (see Eq. 17).

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... 

This chapter will first provide some basics on ozone mass transfer, including theoretical background on the (two-) film theory of gas absorption and the definition of over-all mass transfer coefficients KLa (Section B 3.1) as well as an overview of the main parameters of influence (Section B 3.2). Empirical correction factors for mass transfer coefficients will also be presented in Section B 3.2. These basics will be followed by a description of the common methods for the determination of ozone mass transfer coefficients (Section B 3.3) including practical advice for the performance of the appropriate experiments. Emphasis is laid on the design of the experiments so that true mass transfer coefficients are obtained. 

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... 

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