Higbie theory

Using Davidson s theoretical expression for bubble-cloud circulation and the Higbie theory for cloud-emulsion diffusion the interchange of gas between bubble and cloud is then found to be... 

The absorption rate I>(e) (in gm moles/sec or cm /sec of component A), is measured experimentally, and Q 6)l6 is calculated for different kinetic regimes from the Higbie theory. The contact time 6, calculated from Eq. (86), can be altered by altering and the geometric parameters of the equipment. Thus, by carrying out experiments with the same chemical systems and with the same kinetic regime used to determine mass-transfer parameters in the industrial gas-liquid reactor, I>(6) can be determined as a function of 6 in the laboratory equipment and the variations of Q(6) = 6 6)/Am for different values of 6 serve to determine the parameters. 

Danckwerts [13] pointed out that the Higbie theory /ith its constant time of exposure of the eddies of fluid at the surface is a special case of what may be a more realistic picture, where the eddies are exposed for varying lengths of time. The liquid-gas interface is then a. mosaic of surface... 

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. Jn.st. 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 theoiy. 

The work of Higbie laid the basis of the penetration theory in which it is assumed that the eddies in the fluid bring an element of fluid to the interface where it is exposed to the second phase for a definite interval of time, after which the surface element is mixed with the bulk again. Thus, fluid whose initial composition corresponds with that of the bulk fluid remote from the interface is suddenly exposed to the second phase. It is assumed that equilibrium is immediately attained by the surface layers, that a process... 

Given that, from the penetration theory for mass transfer across an interface, the instantaneous rale ol mass transfer is inversely proportional to the square root of the time of exposure, obtain a relationship between exposure lime in the Higbie mode and surface renewal rate in the Danckwerts model which will give the same average mass transfer rate. The age distribution function and average mass transfer rate from the Danckwerts theory must be deri ved from first principles. 

In calculating Ihe mass transfer rate from the penetration theory, two models for the age distribution of the surface elements are commonly used — those due to Higbie and to Danckwerts, Explain the difference between the two models and give examples of situations in which each of them would be appropriate. 

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. 

Penetration theories (Higbie, 1935 Danckwerts, 1951 Dobbins, 1956) according to the penetration theory, diffusion of gases takes place into elements of water transported by turbulence to the surface. 

An alternative theory described in detail in Volume 1, Chapter 10, has been put forward by Higbie 2 , and later extended by Danckwerts and Danckwerts and Kennedy14 in which the liquid surface is considered to be composed of a large number of small elements each of which is exposed to the gas phase for an interval of time, after which they are replaced by fresh elements arising from the bulk of the liquid. 

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. 

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... 

Alternatives to the film theory are also in use. These models [Higbie (1935) Danckwerts (1950, 1955)] view that the liquid at the interface is continually washed away and replaced by fresh fluid from the main body of the liquid, and that this is the means of mass transport. These unsteady-state surface renewal theories all predict... 

The penetration theory is attributed to Higbie (1935). In this theory, the fluid in the diffusive boundary layer is periodically removed by eddies. The penetration theory also assumes that the viscous sublayer, for transport of momentum, is thick, relative to the concentration boundary layer, and that each renewal event is complete or extends right down to the interface. The diffusion process is then continually unsteady because of this periodic renewal. This process can be described by a generalization of equation (E8.5.6) ... 

Figure 8.9. Concentration boundary layer growth as visualized by Higbie s penetration theory.

Instead of determining 4, in equation (8.33), we must determine r in equation (8.34). Although the difference between Higbie s penetration theory and Danckwerts surface renewal theory is not great, the fact that a statistical renewal period would have a similar result to a fixed renewal period brought much credibility to Higbie s penetration theory. Equation (8.34) is probably the most used to date, where r is a quantity that must be determined from the analysis of experimental data. 

Compared with the Higbie penetration theory, the main difference consists in the fact that both the spatial and temporal dependence of the concentration are taken into account. In other words, the elements of liquid are characterized not only by their time of contact A with the interface, but also by their dimension x0 along the main flow direction. The average mass transfer coefficient should be, therefore, defined as... 

This provides an equation which has the same form as that obtained on the basis of the Higbie penetration theory ... 

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). 

Several workers (Kolbel et al. [40, 41], Deckwer et al. [17], Michael and Reicheit [42]) have investigated the heat transfer in BSCR versus solid concentration and particle diameters. Deckwer et al. [17] applied Kolmogoroff s theory of isotropic turbulence in combination with the surface renewal theory of Higbie [43] and suggested the following expression for the heat transfer coefficient in the Fischer-Tropsch synthesis in BSCR ... 

Penetration theory (Higbie, 1935)assumes that turbulent eddies travel from the bulk of the phase to the interface where they remain for a constant exposure time te. The solute is assumed to penetrate into a given eddy during its stay at the interface by a process of unsteady-state molecular diffusion. This model predicts that the mass-transfer coefficient is directly proportional to the square root of molecular diffusivity... 

As can be seen from Figure 8, if Fo < 0.02, the concentration changes within the film are confined largely to the surface layer and the local mass transfer coefficient is given by the Higbie penetration theory (9) as... 

Ravetkar and Kale (1981) have shown that Higbie s surface-renewal theory can be successfully employed for the calculation of the liquid-side mass-transfer coefficients for Newtonian and non-Newtonian liquids as... 

Two other theories for predicting ki,/ki" for very fast reactions lead to similar results. Higbies penetration theory (i9), which is based on unsteady state diflFusion into a laminar fluid stream, leads to the expression (11) ... 

The mass-transfer coefficient in each film is expected to depend upon molecular diffusivity, and this behavior often is represented by a power-law function k . For two-film theory, n = 1 as discussed above [(Eq. (15-62)]. Subsequent theories introduced by Higbie [Trans. AIChE, 31, p. 365 (1935)] and by Dankwerts [Ind. Eng. Chem., 43, pp. 1460-1467 (1951)] allow for surface renewal or penetration of the stagnant film. These theories indicate a 0.5 power-law relationship. Numerous models have been developed since then where 0.5 < n < 1.0 the results depend upon such things as whether the dispersed drop is treated as a rigid sphere, as a sphere with internal circulation, or as oscillating drops. These theories are discussed by Skelland [ Tnterphase Mass Transfer, Chap. 2 in Science and Practice of Liquid-Liquid Extraction, vol. 1, Thornton, ed. (Oxford, 1992)]. 

The film and boundary layer theories presuppose steady transport, and can therefore not be used in situations where material collects in a volume element, thus leading to a change in the concentration with time. In many mass transfer apparatus fluids come into contact with each other or with a solid material for such a short period of time that a steady state cannot be reached. When air bubbles, for example, rise in water, the water will only evaporate into the bubbles where it is contact with them. The contact time with water which surrounds the bubble is roughly the same as that required for the bubble to move one diameter further. Therefore at a certain position mass is transferred momentarily. The penetration theory was developed by Higbie in 1935 [1.31] for the scenario described here of momentary mass transfer. He showed that the mass transfer coefficient is inversely proportional to the square root of the contact (residence) time and is given by... 

The first is the penetration theory of Higbie (1935). If the liquid immediately adjacent to a rising bubble is assumed to rise with the bubble, i.e., the relative velocity between the bubble and the liquid is 0, the mass transfer conditions are those of unsteady-state molecular diffusion. The mathematical solution of this problem leads to... 

---