Mass transfer Theoretical study

Models are applied to a system, or a portion of the observable universe separated by well-defined boundaries for the purpose of investigation. A chemical model is a theoretical construct that permits the calculation of chemical properties and processes, such as the thermodynamic, kinetic, or quantum mechanical properties of a system. A geochemical model is a chemical model developed for geologic systems. Geochemical models often incorporate chemical models such as ion association and aqueous speciation together with mineralogical data and assumptions about mass transfer to study water-rock interactions. 

Experimental Mass Transfer Coefficients. Hundreds of papers have been pubHshed reporting mass transfer coefficients in packed columns. For some simple systems which have been studied quite extensively, mass transfer data may be obtained directiy from the Hterature (6). The situation with respect to the prediction of mass transfer coefficients for new systems is stiU poor. Despite the wealth of experimental and theoretical studies, no comprehensive theory has been developed, and most generalizations are based on empirical or semiempitical equations. 

R. W. Schrage, Theoretical Study of Interface Mass Transfer, Columbia University, New York, 1953. 

Research and development efforts have been directed toward improved ceU designs, theoretical electrochemical studies of magnesium ceUs, and improved cathode conditions. A stacked-type bipolar electrode ceU has been operated on a lab scale (112). Electrochemical studies of the mechanism of magnesium ion reduction have determined that it is a two-electron reversible process that is mass-transfer controUed (113). A review of magnesium production is found ia Reference 114. 

Several reported chemical systems of gas-liquid precipitation are first reviewed from the viewpoints of both experimental study and industrial application. The characteristic feature of gas-liquid mass transfer in terms of its effects on the crystallization process is then discussed theoretically together with a summary of experimental results. The secondary processes of particle agglomeration and disruption are then modelled and discussed in respect of the effect of reactor fluid dynamics. Finally, different types of gas-liquid contacting reactor and their respective design considerations are overviewed for application to controlled precipitate particle formation. 

Most theoretical studies of osmosis and reverse osmosis have been carried out using macroscopic continuum hydrodynamics [5,8-13]. The models used include those that treat the wall as either nonporous or porous. In the nonporous models the membrane surface is assumed homogeneous and nonporous. Transport occurs by the molecules dissolving in the membrane phase and then diffusing through the membrane. Mass transfer across the membrane in these models is usually described using the solution-diffusion... 

Recent publications on mass transfer from single bubbles include a theoretical study by Ruckenstein (R4) and theoretical and experimental studies by Lochiel and Calderbank (C4, L7). Bowman and Johnson (B8, J3) and Li et al. (L5) have studied mass transfer from streams of bubbles rising in water. 

Calderbank et al. (C6) studied the Fischer-Tropsch reaction in slurry reactors of 2- and 10-in. diameters, at pressures of 11 and 22 atm, and at a temperature of 265°C. It was assumed that the liquid-film diffusion of hydrogen from the gas-liquid interface is a rate-determining step, whereas the mass transfer of hydrogen from the bulk liquid to the catalyst was believed to be rapid because of the high ratio between catalyst exterior surface area and bubble surface area. The experimental data were not in complete agreement with a theoretical model based on these assumptions. 

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. 

Recent studies on heat- and mass-transfer to and from bubbles in liquid media have primarily been limited to studies of the transfer mechanism for single moving bubbles. Transfer to or from swarms of bubbles moving in an arbitrary liquid field is very complex and has been analyzed theoretically in certain simple cases only (G3, G5, G6, G8, M3, R9, Wl). 

Most theoretical studies of heat or mass transfer in dispersions have been limited to studies of a single spherical bubble moving steadily under the influence of gravity in a clean system. It is clear, however, that swarms of suspended bubbles, usually entrained by turbulent eddies, have local relative velocities with respect to the continuous phase different from that derived for the case of a steady rise of a single bubble. This is mainly due to the fact that in an ensemble of bubbles the distributions of velocities, temperatures, and concentrations in the vicinity of one bubble are influenced by its neighbors. It is therefore logical to assume that in the case of dispersions the relative velocities and transfer rates depend on quantities characterizing an ensemble of bubbles. For the case of uniformly distributed bubbles, the dispersed-phase volume fraction O, particle-size distribution, and residence-time distribution are such quantities. 

Most studies on heat- and mass-transfer to or from bubbles in continuous media have primarily been limited to the transfer mechanism for a single moving bubble. Transfer to or from swarms of bubbles moving in an arbitrary fluid field is complex and has only been analyzed theoretically for certain simple cases. To achieve a useful analysis, the assumption is commonly made that the bubbles are of uniform size. This permits calculation of the total interfacial area of the dispersion, the contact time of the bubble, and the transfer coefficient based on the average size. However, it is well known that the bubble-size distribution is not uniform, and the assumption of uniformity may lead to error. Of particular importance is the effect of the coalescence and breakup of bubbles and the effect of these phenomena on the bubble-size distribution. In addition, the interaction between adjacent bubbles in the dispersion should be taken into account in the estimation of the transfer rates... 

The treatment of the two-phase SECM problem applicable to immiscible liquid-liquid systems, requires a consideration of mass transfer in both liquid phases, unless conditions are selected so that the phase that does not contain the tip (denoted as phase 2 throughout this chapter) can be assumed to be maintained at a constant composition. Many SECM experiments on liquid-liquid interfaces have therefore employed much higher concentrations of the reactant of interest in phase 2 compared to the phase containing the tip (phase 1), so that depletion and diffusional effects in phase 2 can be eliminated [18,47,48]. This has the advantage that simpler theoretical treatments can be used, but places obvious limitations on the range of conditions under which reactions can be studied. In this section we review SECM theory appropriate to liquid-liquid interfaces at the full level where there are no restrictions on either the concentrations or diffusion coefficients of the reactants in the two phases. Specific attention is given to SECM feedback [49] and SECMIT [9], which represent the most widely used modes of operation. The extension of the models described to other techniques, such as DPSC, is relatively straightforward. 

The reader is referred to basic studies of mass transfer in freeze-drying by Pikal and coworkers for in-depth treatment of the theoretical and practical aspects of mass transfer [29,32], Briefly, the rate-limiting step in mass transfer is transfer of water vapor through the partially dried matrix of solids. Resistance of the dried layer increases in a more or less continuous fashion as the depth of the dried layer increases, and the resistance also increases with the concentration of solids in the dried layer. Other factors can also affect the resistance of the dried layer, such as the method of freezing faster freezing tends to create a higher resistance in the dried layer. 

Another survey by Ibl (13) in 1963 listed 13 mass-transfer correlations established by the limiting-current method, only four of which were derived from quantitative considerations. At the time of writing the total number of publications is more than 200. The majority of these concern flow conditions under which theoretical predictions are, at best, qualitative. More recently, an increasing number of publications deal with model hydrodynamic studies of more complex situations, for example, packed and fluidized beds. 

In a study in which styrene was stripped from polystyrene, Latinen (1962) concluded that his theory correctly described the dependence of mass transfer rates on screw speed and flow rate. This conclusion was based on the agreement obtained between the measured and predicted exit concentration of styrene over a broad range of screw speeds and flow rates (Fig. 8). But, agreement between the theoretical expression and the experimental data was obtained using a diffusion coefficient of the order of 3 X 10 m sec , at 2(X)°C a value which is unrealistically high for this system. If the system ethylbenzene-polystyrene—which has a diffusion... 

In a sense, all the present papers treat problems in interphase contacting. On the theoretical and observational sides, respectively, Davies and Kintner explore the properties of two-phase systems undergoing mass transfer. In a third study, both the descriptive and the theoretical properties of cocurrent two-phase flow systems are presented by Scott. Longitudinal dispersion (or axial mixing), which has only recently been identified and analyzed as a substantial factor in equipment performance, is reviewed by Levenspiel and Bischoff. 

A rotating disk electrode (RDE) [7] is used to study electrode reactions, because the mass transfer to and from the electrode can be treated theoretically by hydrodynamics. At the RDE, the solution flows toward the electrode surface as shown in Fig. 5.22, bringing the substances dissolved in it. The current-potential curve at the RDE is S-shaped and has a potential-independent limiting current region, as in Fig. 5.6. The limiting current (A) is expressed by Eq. (5.33), if it is controlled by mass transfer ... 

Kramers and Kreyger (K24), 1956 Experimental and theoretical study of mass transfer between a soluble wall surface and film flowing on it. Experiments carried out at low ARo on inclined plane surface. 

The effect of condensation upon transfer rates with application to flue-gas washing plants and cooling towers are discussed. Theoretical models were developed for determining the rate of heat and mass transfer under conditions where fog formation prevails. Derived relationships are functions of the vapor and liquid equilibria and local heat and mass transfer of driving forces. They were used for a numerical study of the amount of fog formation as a function of the operational variables of a flue-gas washing plant in which the inlet gas temperature is typically... 

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