Two-Resistance Theory

To get around this problem, Lewis and Whitman (1924) assumed that the only diffusional resistances are those residing in the fluids themselves. There is then no resistance to solute transfer across the interface separating the phases, and as a result the concentrations yAi and xAi are equilibrium values given by the system s equilibrium-distribution curve. This concept has been called the two-resistance theo -O - 

---

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... 

Figure 3-2 Two-film or two-resistance theory with linear concentration gradients (Lewis and Whitman, 1924).

The practical applications provided here all involve two phases, with molecules transferring between them. Thus, there are two resistances to transfer, plus possibly a third resistance at the interface itself. We have just discussed transfer within a phase and ending at a phase boundary, such as an interface. It is necessary to couple individual phase resistances to characterize the overall transfer process. The first attempt at this, and indeed a lasting one, was presented by Lewis and Whitman [19] as the two-film theory. More recently it has been called simply the two-resistance theory, eliminating the reqnirement that transport in each phase be handled by the film concept. 

The efficiency concept is based on two-resistance theory and is usually expressed in terms of the molar rate of diffusion ... 

Figure 3.4 Interfacial concentrations as predicted by the two-resistance theory.

This general equation reduces to the Whitman two-resistance theory for air-water exchange in which the interfacial resistance Ri is assumed to be negligible. The liquid and gas resistances are equivalent to 1/ l- l and MKqZq, then since/l is Cl/Zl and /g is Pq, the partial pressure,... 

This shows that the relationship between the individual-phase transfer coefficients and the overall coefficient takes the form of addition of resistances (hence two-resistance theory). In similar fashion, is a measure of q and can be used to define another overall coefficient K... 

For purposes of establishing the nature of the two-resistance theory and the overall coefficient concept, gas absorption was chosen as an example. The principles are applicable to any of the mass-transfer operations, however, and can be applied in terms of /c-type coefficients using any concentration units, as listed in Table 3.1 (the F-type coefficients are considered separately below). For each case, the values of m and m" must be appropriately defined and used consistently with the coefficients. Thus, if the generalized phases are termed (with concentrations expressed as /) and R (with concentrations expressed asy),... 

The two-film (or two-resistance) theory was first proposed in a landmark paper by W.K. Lewis and W.G. Whitman over 80 years ago Ind. Eng. Chem. 16,1215 [1924]). In spite of sporadic criticism, particularly of the assumption of interfacial equilibrium, it has survived remarkably well and continues to serve as one of the mainstays in the design of industrial separation processes. 

The mathematics of mass diffusion within a single phase has thus been well estab-Ushed. Diffusion between phases such as air and water was not fully understood until 1923 when Whitman proposed the two-film theory in which transfer is expressed using two mass transfer coefficients in series, one for each phase (Whitman, 1923). This concept has been rediscovered in pharmacology and probably in other areas and is now more correctly termed the two-resistance theory. ... 

Conventional filtration theory has been challenged a two-phase theory has been appHed to filtration and used to explain the deviations from paraboHc behavior in the initial stages of the filtration process (10). This new theory incorporates the medium as an integral part of the process and shows that the interaction of the cake particles with the medium controls filterabiHty. It defines a cake-septum permeabiHty which then appears in the slope of the conventional plots instead of the cake resistance. This theory, which merely represents a new way of interpreting test data rather than a new method of siting or scaling filters, is not yet accepted by the engineering community. 

It seems probable that a fruitful approach to a simplified, general description of gas-liquid-particle operation can be based upon the film (or boundary-resistance) theory of transport processes in combination with theories of backmixing or axial diffusion. Most previously described models of gas-liquid-particle operation are of this type, and practically all experimental data reported in the literature are correlated in terms of such conventional chemical engineering concepts. In view of the so far rather limited success of more advanced concepts (such as those based on turbulence theory) for even the description of single-phase and two-phase chemical engineering systems, it appears unlikely that they should, in the near future, become of great practical importance in the description of the considerably more complex three-phase systems that are the subject of the present review. 

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. 

Carbon dioxide is absorbed in water from a 25 per cent mixture in nitrogen. How will its absorption rate compare with that from a mixture containing 35 per cent carbon dioxide, 40 per cent hydrogen and 25 per cent nitrogen It may be assumed that the gas-film resistance is controlling, that the partial pressure of carbon dioxide, at the gas-liquid interface is negligible and that the two-lilm theory is applicable, with the gan film thickness the same in the two cases. 

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). 

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. 

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 ... 

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... 

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. 

The connection between the film mass transfer coefficients and the over-all mass transfer coefficients is provided by the two-film theory from Lewis and Whitman (1924) the total resistance to mass transfer is the sum of the resistances in each phase. 

The two-film theory is one of the mechanisms suggested to represent the conditions in the region of the phase boundary. This model suggested that the resistance to transfer in each phase could be regarded as lying in a thin laminar film close to the interface (Figure 6.1-2). 

The two-film theory supposes that the entire resistance to transfer is contained in two fictitious films on either side of the interface, in which transfer occurs by molecular diffusion. This model leads to the conclusion that the mass-transfer coefficient kL is proportional to the diffusivity DAB and inversely proportional to the film thickness Zy as... 

As an introduction, it is worth qualitatively analysing some common industrial situations. In an analogy to the two film theory (Section 9.3.1), we consider three contributions to the resistance against heat transport [1] ... 

The rate-based models usually use the two-film theory and comprise the material and energy balances of a differential element of the two-phase volume in the packing (148). The classical two-film model shown in Figure 13 is extended here to consider the catalyst phase (Figure 33). A pseudo-homogeneous approach is chosen for the catalyzed reaction (see also Section 2.1), and the corresponding overall reaction kinetics is determined by fixed-bed experiments (34). This macroscopic kinetics includes the influence of the liquid distribution and mass transfer resistances at the liquid-solid interface as well as dififusional transport phenomena inside the porous catalyst. 

In rate-based multistage separation models, separate balance equations are written for each distinct phase, and mass and heat transfer resistances are considered according to the two-film theory with explicit calculation of interfacial fluxes and film discretization for non-homogeneous film layer. The film model equations are combined with relevant diffusion and reaction kinetics and account for the specific features of electrolyte solution chemistry, electrolyte thermodynamics, and electroneutrality in the liquid phase. 

Point efficiency is usually discussed in terms of the two-film theory. The theory postulates resistances to mass transfer in both the vapor and liquid films near a vapor-liquid ihterface (Fig. 7.2a). The molar rate of diffusion, N (moles/s), is given by... 

---