Mass transfer rate penetration theory

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 ... 

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 ... 

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. 

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 ... 

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 ... 

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. 

State the assumptions made in the penetration theory for the absorption of a pure gas into a liquid. The surface of an initially solute-free liquid is suddenly exposed to a soluble gas and the liquid is sufficiently deep for no solute to have time to reach the far boundary of the liquid. Starting with Fick s second law of diffusion, obtain an expression for (i) the concentration, and (ii) the mass transfer rate at a time t and a depth y below the surface. 

E. Laminar, inclined, plate Vtog-0.783NJ lllmtf /p/sina J3 < 2000 D [T] Constant-property liquid film with low mass-transfer rates. Use arithmetic concentration difference. Newtonian fluid. Solute does not penetrate past region of linear velocity profile. Differences between theory and experiment. tv = width of plate, 5y= film thickness, a = angle of inclination, x = distance from start soluble surface. [141] p. 130 [138] p. 209... 

These values are within 5% of the values calculated with the penetration theory correction factor matrix and support our earlier suggestion that it is sufficient to use the simpler film model correction factor matrix in multicomponent mass transfer calculations at high mass transfer rates. ... 

Thus to predict the mass transfer rate in the interface, the chemical kinetics and the local flow dynamics must be known. For determining the reaction kinetics, special laboratory reactors are occasionally necessary [536]. The problem of flow dynamics can be circumvented, if the film and penetration theories are fallen back upon. 

The instantaneous mass transfer rate is expressed as a function of time. In order to calculate an average mass transfer coefEcient we need to average the instantaneous coefEcient over the total exposure time period. To do this we need to know the age distribution function, which represents the fraction of elements having ages between t and t + dt t the surface. In the penetration theory, it is assumed that all the elements reside at the interface for a time period of the same length. As a consequence of this assumption the age distribution function is [6] ... 

Ni)ts average mass transfer rate in surface renewal and penetration theories mol / m s)... 

The various forms of the penetration theory can be classified as surface-renewal models, implying either formation of new surfaee at frequent intervals or replacement of fluid elements at the surface with fresh fluid from the bulk. The time or its reciprocal, the average rate of renewal, are functions of the fluid velocity, the fluid properties, the the geometry of the system and can be accurately predicted in only a few special cases. However, even if tj must be determined empirically, the surface-renewal models give a sound basis for correlation of mass-transfer data in many situations, particularly for transfer to drops and bubbles. The similarity between Eqs. (21.44) and (15,20) is an example of the close analogy between heat and mass transfer. It is often reasonable to assume that tj-is the same for both processes and thus to estimate rates of heat transfer from measured mass-transfer rates or vice versa. 

The addition of various surfactants and micelle-forming agents in the biphasic hydroformylation of olefins was also considered as a tool for enhancement of the reaction rates (see Section 2.3.4). Whereas the presence of a surfactant leads to a lower droplet size in the dispersed phase, thus increasing the liquid-liquid interfadal area and hence the mass-transfer rate, the formation of emulsions is considered as a maj or drawback of this system. Mass-transfer effects in biphasic hydroformylation of 1-octene in the presence of cetyltrimethylammonium bromide (CTAB) was studied by Lekhal et al. [37]. A mass-transfer model based on Higbie s penetration theory was proposed to predict the rate of hydroformylation in a gas-liquid-liquid system. 

The Sherwood number Sh = kL/D can be regarded as the ratio of the diffusion time in the absence of flow, L fD, to the actual mass transfer time, 1/ka ( I/k). In the following sections of this chapter, we discuss various fluid-fluid microreactor configurations and discuss how mass transfer rates in them may be predicted using penetration theory and film theory. In the following paragraph, we briefly discuss experimental methods to determine mass transfer rates. 

High mass transfer rates will influence not only the mass transfer coefficient but also the heat transfer coefficients and friction factor. Analysis of film theory penetration theory and boundary layer theory (21) show that the relation of the various coefficients at high (k ) and low mass transfer (kj ) can be given by 0 s ... 

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