Higbie’s penetration theory

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... 

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. 

The addition of various surfactants and micelle forming agents on the biphasic hydroformylation of olefins was also considered as a tool for enhancement of the reaction rate. The relation between the extent of emulsification of the reaction mixture and the performance of hydroformylation reaction was also investigated. Mass transfer effects in biphasic hydroformylation of 1-octene in the presence of cetyltri-methylammoniumbromide (CTAB), was studied by Lekhal etal. [33], A mass-transfer model based on the Higbie s penetration theory was proposed to predict the rate of hydroformylation in a heterogeneous gas-liquid-liquid system under... 

Higbie s penetration theory was modified by Danckwerts, who assumed a residence time distribution for the active eddies at the interface. The same square root dependence of the k/ or kg value on diffusivity as in the penetration theory is obtained, but the resulting model is much more intellectually satisfying than the original penetration model. 

We note that Higbie s penetration theory (HI5), with contact time assumed as that required by the drop to traverse a distance of one diameter (W6), gives an expression identical with Eq. (18). Although potential-flow theory, unlike the penetration theory, takes interfacial acceleration into account, the two are actually physically identical, both being based on diffusion into an element of fluid sliding over the constant-temperature interface. 

The transfer efficiency in the drop-formation region varies approximately as the square root of the drop-formation time and inversely as the drop diameter. Since the drop acceleration interval is quite short, acceleration effects are normally combined with drop-formation effects. Heertjes (HI3), basing his analysis on Higbie s penetration theory, suggested equations for the drop-formation and coalescence regions. For the drop-formation region. 

Danckwerts and co-workers have interpreted the data in terms of contact or exposure time and Higbie s penetration theory as follows. For a pseudo-first-order reaction i given by Eq. 6.4.b-4, the amount absorbed during the... 

Higbie s penetration theory A conceptual approach to mass transfer at the interface between a liquid and a gas. The gas liquid interface is assumed not to be at steady state but is continually being refreshed by liquid from the bulk below. A theory assumes that an element of liquid teaches the interface for a fixed time depending on the hydrodynamic conditions and arrives with a particular concentration. The residence time is assumed to be short enough for the transferred component not to penetrate the element far enough to affect the bulk concentration. It was proposed by R. Higbie in 1935. 

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). 

In surface renewal models the liquid surface is assumed to consist of a mosaic of elements with different age at the surface. The rate of absorption at the surface is then an average of the rates of absorption in each element, weighted with respect to a distribution function (t)—see Eq. 6.2-5. Under this heading of surface renewal theory we will also occasionally mention results of Higbie s [23] so-called penetration-theory, which can be considered as a special case in which every element is exposed to the gas for the same length of time before being replaced. The main emphasis of this section is on the Danckwerts [24] approach using the distribution function for completely random replacement of surface elements ... 

Penetration theory can also be applied to turbulent conditions by assuming the turbulence spectrum to consist of large eddies, capable of surface renewal, and small eddies responsible for the presence of eddy diffusivity The small eddies are damped when an element of liquid reaches the interface so that, during its residence time there, mass transfer occurs in accordance with the assumptions of the penetration theory If all the eddies stay at the interface for the same interval of time we talk about penetration theory with regular surface renewal or the Higbie model If there is random distribution of residence times with an age-independent fractional rate of surface renewal, s, the term penetration theory with random surface renewal, or the Danckwerts nK)del, is employed In the case of the Higbie model, the mass transfer coefficient is the same as that given by eqn (18). For the Danckwerts model it takes the form... 

The work of Higbie in 1935 provided the foundation for the penetration theory, which supposes that tuibulence transports eddies from the bulk of the phase to the interface, where they remain for a short but constant time before being displaced back into the interior of the phase to be mixed with the bulk fluid. Solute is assumed to penetrate" into a given eddy during its stay at the interface by a process of unsteady-state molecular diffusion, in accordance with Pick s Second Law and appropriate boundaiy conditions ... 

Whereas Higbie assumed a constant renewal time, Dankwert s (1951) extension of the penetration theory employs a wide spectrum of eddy contact times. These eddy-like fluid packets are assumed to remain in contact with the interface for variable times from zero to infinity. He assumed a surface-age distribution function, which skews the contact times to small values it is... 

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