Mass penetration theory

Other Models for Mass Transfer. In contrast to the film theory, other approaches assume that transfer of material does not occur by steady-state diffusion. Rather there are large fluid motions which constantiy bring fresh masses of bulk material into direct contact with the interface. According to the penetration theory (33), diffusion proceeds from the interface into the particular element of fluid in contact with the interface. This is an unsteady state, transient process where the rate decreases with time. After a while, the element is replaced by a fresh one brought to the interface by the relative movements of gas and Uquid, and the process is repeated. In order to evaluate a constant average contact time T for the individual fluid elements is assumed (33). This leads to relations such as... 

Simplified Mass-Transfer Theories In certain simple situations, tne mass-transfer coefficients can be calculated from first principles. The film, penetration, and surface-renewal theories are attempts to extend tnese theoretical calculations to more complex sit-... 

The rate of mass transfer in the liquid phase in wetted-waU columns is highly dependent on surface conditions. When laminar-flow conditions prevail without the presence of wave formation, the laminar-penetration theory prevails. When, however, ripples form at the surface, and they may occur at a Reynolds number exceeding 4, a significant rate of surface regeneration develops, resulting in an increase in mass-transfer rate. 

In an experimental wetted wall column, pure carbon dioxide, is absorbed in water. The mass transfer rate is calculated using the penetration theory, application of which is limited by the fact that the concentration should not teach more than 1 per cent of the saturation value at a depth below the surface at which the velocity is 95 per cent of the surface velocity. What is the maximum length of column to which the theory can be applied if the flowrate of water is 3 cm3/s per cm of perimeter ... 

The penetration theory has been used to calculate the rate of mass transfer across an interface for conditions where the concentration CAi of solute A in the interfacial layers (y = 0) remained constant throughout the process. When there is no resistance to mass transfer in the other phase, for instance when this consists of pure solute A, there will be no concentration gradient in that phase and the composition at the interface will therefore at all Limes lie the same as the bulk composition. Since the composition of the interfacial layers of the penetration phase is determined by the phase equilibrium relationship, it, too. will remain constant anil the conditions necessary for the penetration theory to apply will hold. If, however, the other phase offers a significant resistance to transfer this condition will not, in general, be fulfilled. 

HARRIOTT 25 suggested that, as a result of the effects of interfaeial tension, the layers of fluid in the immediate vicinity of the interface would frequently be unaffected by the mixing process postulated in the penetration theory. There would then be a thin laminar layer unaffected by the mixing process and offering a constant resistance to mass transfer. The overall resistance may be calculated in a manner similar to that used in the previous section where the total resistance to transfer was made up of two components—a Him resistance in one phase and a penetration model resistance in the other. It is necessary in equation 10.132 to put the Henry s law constant equal to unity and the diffusivity Df in the film equal to that in the remainder of the fluid D. The driving force is then CAi — CAo in place of C Ao — JPCAo, and the mass transfer rate at time t is given for a film thickness L by ... 

Thus either the penetration theory or the film theory (equation 10.144 or 10.145) respectively can be used to describe the mass transfer process. The error will not exceed some 9 per cent provided that the appropriate equation is used, equation 10.144 for L2 jDt > n and equation 10.145 for L2/Dt < n. Equation 10.145 will frequently apply quite closely in a wetted-wall column or in a packed tower with large packings. Equation 10.144 will apply when one of the phases is dispersed in the form of droplets, as in a spray tower, or in a packed tower with small packing elements. 

The penetration and film-penetration theories have been developed for conditions of equimolecular counterdiffusion only the equations are too complex to solve explicitly for transfer through a stationary carrier gas. For gas absorption, therefore, they apply only when the concentration of the material under going mass transfer is low. On the other hand, in the two-fihn theory the additional contribution to the mass transfer which is caused by bulk flow is easily calculated and hp (Section 10.23) is equal to (D/L)(Cr/Cum) instead of D/L. 

According to the simple penetration theory the instantaneous mass flux, (AfOt is ... 

What are the general principles underlying the two-film, penetration and film-penetration theories for mass transfer across a phase boundary Give the basic differential equations which have to be solved for these theories with the appropriate boundary conditions. 

According ro the penetration theory, the instantaneous rate of mass transfer per unit area LV, > at some time i after the commencement of transfer is given by ... 

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. 

Ammonia is absorbed in a falling film of water in an absorption apparatus and the film is disrupted and mixed at regular intervals as it flows down the column. The mass transfer rate is calculated from the penetration theory on the assumption that all the relevant conditions apply. It is found from measurements that the muss transfer rate immediately before mixing is only 16 pet cent of that calculated from the theory anil the difference has been attributed to the existence of a surface film which remains intact and unaffected by the mixing process. If the liquid mixing process lakes place every second, what thickness of surface film would account for the discrepancy, ... 

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. 

In the Danckwerts model, it is assumed that elements of the surface have an age distribution ranging from zero to infinity. Obtain the age distribution function for this model and apply it to obtain the average, mass Iransfer coefficient at the surface, given that from the penetration theory the mass transfer coefficient for surface of age t is VlD/(7rt, where D is the diffusivity. 

Explain the basis of the penetration theory for mass transfer across a phase boundary. What arc the assumptions in the theory which lead to the result that the mass transfer rate is inversely proportional to the square root of the time for which a surface element has been expressed (Do not present a solution of the differential equal ion.) Obtain the age distribution function for the surface ... 

On the. assumptions involved in the penetration theory of mass transfer across a phase boundary, the concentration Ca of a solute A at a depth v below the interface at a time l after the formation of the interlace is given by ... 

In a liquid-liquid extraction unit, spherical drops of solvent of uniform size are continuously fed to a continuous phase of lower density which is flowing vertically upwards, and hence countercurrently with respect to the droplets. The resistance to mass transfer may be regarded as lying wholly within the drops and the penetration theory may be applied. The upward velocity of the liquid, which may be taken as uniform over the cross-section of the vessel, is one-half of the terminal falling velocity of the droplets in the still liquid. 

According to the penetration theory for mass transfer across an interface, the ratio of the concentration Ca at a depth y and time r to the surface concentration Ca. if the liquid is initially free of solute, is giver by ... 

It may be assumed that the penetration model may be used to represent the mass transfer process. The depth of penetration is small compared with the radius of the droplets and the effects of surface curvature may he neglected. From the penetration theory, the concentration C, at a depth y below the surface at time r is given by ... 

Wbat is the penetration theory for mass transfer across a phase boundary Give deiails of she underlying... 

From the penetration theory, the mass transfer rate per unit area N, is given in terms of the concentration difference AC, between the interface and the bulk fluid, the molecular diffusivity D and the age t of the. surface clement by ... 

The devolatilization of a component in an internal mixer can be described by a model based on the penetration theory [27,28]. The main characteristic of this model is the separation of the bulk of material into two parts A layer periodically wiped onto the wall of the mixing chamber, and a pool of material rotating in front of the rotor flights, as shown in Figure 29.15. This flow pattern results in a constant exposure time of the interface between the material and the vapor phase in the void space of the internal mixer. Devolatilization occurs according to two different mechanisms Molecular diffusion between the fluid elements in the surface layer of the wall film and the pool, and mass transport between the rubber phase and the vapor phase due to evaporation of the volatile component. As the diffusion rate of a liquid or a gas in a polymeric matrix is rather low, the main contribution to devolatilization is based on the mass transport between the surface layer of the polymeric material and the vapor phase. 

The experimental results imply that the main reaction (eq. 1) is an equilibrium reaction and first order in nitrogen monoxide and iron chelate. The equilibrium constants at various temperatures were determined by modeling the experimental NO absorption profile using the penetration theory for mass transfer. Parameter estimation using well established numerical methods (Newton-Raphson) allowed detrxmination of the equilibrium constant (Fig. 1) as well as the ratio of the diffusion coefficients of Fe"(EDTA) andNO[3]. 

On the basis of the simplified view of the flow patterns just described, a model for predicting mass transfer rates can be developed using penetration theory and the fact that mass is transferred simultaneously from both the nip and the wiped film. We can therefore write that the total molar mass transfer rate from an element of fluid over a length dk in the extruder is... 

A specific expression for the mass transfer rate in Eq. (11) was first developed by Latinen (1962) in a classic paper that showed how penetration theory can be applied to the analysis of devolatilization processes in single-screw extruders. The derivation presented here parallels that by Latinen but differs in some respects for reasons of clarity. 

For the mass transfer coefficient on the outside of the drop kc, Eq. (9.34), according to the penetration theory by Highbie [19], obtains the contact time t as the quotient between the rising distance between two stages and the rising velocity. 

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... 

In the penetration theory, the mass transport equation takes the form... 

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... 

In many of these experiments, interfacial turbulence was the obvious visible cause of the unusual features of the rate of mass transfer. There are, however, experimental results in which no interfacial activity was observed. Brian et al. [108] have drawn attention to the severe disagreement existing between the penetration theory and data for the absorption of carbon dioxide in monoethanolamine. They have performed experiments on the absorption of C02 with simultaneous desorption of propylene in a short, wetted wall column. The desorption of propylene without absorption of C02 agrees closely with the predictions of the penetration theory. If, however, both processes take place simultaneously, the rate of desorption is greatly increased. This enhancement must be linked to a hydrodynamic effect induced by the absorption of C02 and the only one which can occur appears to be the interfacial turbulence caused by the Marangoni effect. No interfacial activity was observed because of the small scale and small intensity of the induced turbulence. 

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