Model mass transfer rates

Adsorption Dynamics. An outline of approaches that have been taken to model mass-transfer rates in adsorbents has been given (see Adsorption). Detailed reviews of the extensive Hterature on the interrelated topics of modeling of mass-transfer rate processes in fixed-bed adsorbers, bed concentration profiles, and breakthrough curves include references 16 and 26. The related simple design concepts of WES, WUB, and LUB for constant-pattern adsorption are discussed later. 

The predictions of correlations based on the film model often are nearly identical to predictions based on the penetration and surface-renewal models. Thus, in view of its relative simphcity, the film model normally is preferred for purposes of discussion or calculation. It should be noted that none of these theoretical models has proved adequate for maldug a priori predictions of mass-transfer rates in packed towers, and therefore empirical correlations such as those outlined later in Table 5-28. must be employed. 

Correlations of heat and mass-transfer rates are fairly well developed and can be incorporated in models of a reaction process, but the chemical rate data must be determined individually. The most useful rate data are at constant temperature, under conditions where external mass transfer resistance has been avoided, and with small particles... 

Gal-Or and Resnick (Gl) have developed a simplified theoretical model for the calculation of mass-transfer rates for a sparingly soluble gas in an agtitated gas-liquid contactor. The model is based on the average gas residencetime, and its use requires, among other things, knowledge of bubble diameter. In a related study (G2) a photographic technique for the determination of bubble flow patterns and of the relative velocity between bubbles and liquid is described. 

This approach to the problem is purely theoretical, since this model is based on a characteristic stagnant film thickness which is difficult to estimate or to measure. In addition, this model does not give any information as to the value of A, which must be determined separately by some other method. As a result, it is impossible to estimate the total mass-transfer rate in the disperser with the aid of this model only. 

Experimental work was undertaken (G8) to provide the information necessary to permit a test of this theoretical model. The system used bore complete geometrical and chemical similarity to that used by Cooper et al. (C9) so that their mass-transfer rate measurements, along with the average residence-time and power-consumption results determined in the experimental work (see Section II,D), were used to compare the experimental values with the model. 

In conclusion, therefore, the model proposed by Gal-Or and Resnick predicts mass-transfer rates that correlate reasonably well with the experimental data now available. Further experimental work as well as accurate values for the reaction-rate constants, diffusivities, and solubilities for other... 

Comparison of Eq. (184) with Eq. (183) shows the effect of size distribution for the case of fast chemical reaction with simultaneous diffusion. This serves to emphasize the error that may arise when one applies uniform-drop-size assumptions to drop populations. Quantitatively the error is small, because 1 — is small in comparison with the second term in the brackets [i.e., kL (kD)112). Consequently, Eq. (184) and Eq. (183) actually give about the same result. In general, the total average mass-transfer rate in the disperser has been evaluated in this model as a function of the following parameters ... 

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. 

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. 

Using a steady-state film model, obtain an expression for the mass transfer rate across a laminar film of thickness /. in the vapour phase for the more volatile component in a binary distillation process ... 

Main axis experimental conversions ( ) intrinsic kinetic model (solid line) kinetic model + mass transfer kinetics (dashed line). Secondary axis variation of computed k,o with flow rate. 

Experimental gas-solid mass-transfer data have been obtained for naphthalene in CO2 to develop correlations for mass-transfer coefficients [Lim, Holder, and Shah, Am. Chem. Soc. Symp. Ser, 406, 379 (1989)]. The mass-transfer coefficient increases dramatically near the critical point, goes through a maximum, and then decreases gradually. The strong natural convection at SCF conditions leads to higher mass-transfer rates than in liquid solvents. A comprehensive mass-transfer model has been developed for SCF extraction from an aqueous phase to CO2 in countercurrent columns [Seibert and Moosberg, Sep. Sci. Techrwl, 23, 2049 (1988) Brunner, op. cit.]. 

The modelling approach to multistage countercurrent equilibrium extraction cascades, based on a mass transfer rate term as shown in Sec. 1.4, can therefore usefully be applied to such types of extractor column. The magnitude of the... 

Improve the model by assuming a mass transfer rate at the solid surface, such that M2 = Kx A Cl. Assume a suitable value for Kj. 

Part I treats the fundamentals of modelling (mass balance ei ualions, involving reaction kinetics and mass-transfer rates), making them readily understandable to those new in the field. 

In Ref. 30, the transfer of tetraethylammonium (TEA ) across nonpolarizable DCE-water interface was used as a model experimental system. No attempt to measure kinetics of the rapid TEA+ transfer was made because of the lack of suitable quantitative theory for IT feedback mode. Such theory must take into account both finite quasirever-sible IT kinetics at the ITIES and a small RG value for the pipette tip. The mass transfer rate for IT experiments by SECM is similar to that for heterogeneous ET measurements, and the standard rate constants of the order of 1 cm/s should be accessible. This technique should be most useful for probing IT rates in biological systems and polymer films. 

Of considerable interest is the use of small isolated electrodes, in the form of strips or disks embedded in the wall, to measure local mass-transfer rates or rate fluctuations. Mass-transfer to spot electrodes on a rotating disk is represented by Eqs. (lOg-i) of Table VII. Analytical solutions in this case have to take account of curved streamlines. Despic et al. (Dlld) have proposed twin spot electrodes as a tool for kinetic studies, similar to the ring-disk electrode applications of disk and ring-disk electrodes for kinetic studies are discussed in several monographs (A3b, P4b). In fully developed channel or pipe flow, mass transfer to such electrodes is given by the following equation based on the Leveque model ... 

When electrically insulated strip or spot electrodes are embedded in a large electrode, and turbulent flow is fully developed, the steady mass-transfer rate gives information about the eddy diffusivity in the viscous sublayer very close to the electrode (see Section VI,C below). The fluctuating rate does not give information about velocity variations, and is markedly affected by the size of the electrode. The longitudinal, circumferential, and time scales of the mass-transfer fluctuations led Hanratty (H2) to postulate a surface renewal model with fixed time intervals based on the median energy frequency. 

The modelling approach to multistage countercurrent equilibrium extraction cascades, based on a mass transfer rate term as shown in Section 1.4, can therefore usefully be applied to such types of extractor column. The magnitude of the mass transfer capacity coefficient term, now used in the model equations, must however be a realistic value corresponding to the hydrodynamic conditions, actually existing within the column and, of course, will be substantially less than that leading to an equilibrium condition. 

Many wastewater flows in industry can not be treated by standard aerobic or anaerobic treatment methods due to the presence of relatively low concentration of toxic pollutants. Ozone can be used as a pretreatment step for the selective oxidation of these toxic pollutants. Due to the high costs of ozone it is important to minimise the loss of ozone due to reaction of ozone with non-toxic easily biodegradable compounds, ozone decay and discharge of ozone with the effluent from the ozone reactor. By means of a mathematical model, set up for a plug flow reactor and a continuos flow stirred tank reactor, it is possible to calculate more quantitatively the efficiency of the ozone use, independent of reaction kinetics, mass transfer rates of ozone and reactor type. The model predicts that the oxidation process is most efficiently realised by application of a plug flow reactor instead of a continuous flow stirred tank reactor. 

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... 

The authors applied this model to the situation of dissolving and deposited interfaces, involving chemically interacting species, and included rate kinetics to model mass transfer as a result of chemical reactions [60]. The use of a stochastic weighting function, based on solutions of differential equations for particle motion, may be a useful method to model stochastic processes at solid-liquid interfaces, especially where chemical interactions between the surface and the liquid are involved. 

The model described in the previous sections is very complicated and therefore in Part I only isothermal studies will be reported. This limiting situation can be obtained by putting //, = 0 (i = 1,. ..,n) and setting the bulk temperatures equal to To in eq. (16b). In this paper the influence of multicomponent transport phenomena on the mass transfer rate between a gas/vapour and a liquid will be studied in detail for both non-reactive and reactive conditions. It should be stressed that the validity of the model developed in... 

Sivasubramanian, M. S. and Boston, J. F., 1990, The heat and mass transfer rate-based approach for modelling multicomponent separation processes, in Computer Applications in Chemical Engineering, pp. 331-336. Elsevier, Amsterdam. 

For simplicity, this section discusses only the mass transfer of one component in a liquid-liquid system with negligible miscibility of both liquids and with one transitional component. On the other hand, calculations must consider mass transfer rates of several components and more or less strong variation in the mass flows along the column, where both complicate the equation considerably [21-23]. Chemical reactions may cause further complications. Their kinetics can enhance the mass transfer coefficients and, therefore, the reaction equations have to be part of the mathematical model of the extractor [24,25]. 

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