Mass transfer analysis

In Temkin s analysis, mass transfer is assumed to be absent in the two-component system. He found that an attenuation coefficient, A, is function of the relaxation times td and t, and the mass fration of suspension, x . 

The square of the uranyl nitrate concentration is required in the kinetic analysis. Mass transfer with chemical reaction (16, 83) is, however, too specialized a subject to be discussed fully here. 

This chapter is a sinplified introduction to the fascinating and valuable sorption separation methods. The first three-fourths of the chapter relies on equilibrium analysis—mass transfer is introduced in Section 18.6. The development is similar, but at a more introductory level, to that in Wankat (1986 19901 Once you understand this chapter, you will be able to discuss these techniques with experts and will be prepared to begin more detailed explorations of these methods in more advanced books (e.g.. Do. 1998 ... 

The main conclusion to be drawn from these studies is that for most practical purposes the linear rate model provides an adequate approximation and the use of the more cumbersome and computationally time consuming diffusing models is generally not necessary. The Glueckauf approximation provides the required estimate of the effective mass transfer coefficient for a diffusion controlled system. More detailed analysis shows that when more than one mass transfer resistance is significant the overall rate coefficient may be estimated simply from the sum of the resistances (7) ... 

Static mixing of immiscible Hquids can provide exceUent enhancement of the interphase area for increasing mass-transfer rate. The drop size distribution is relatively narrow compared to agitated tanks. Three forces are known to influence the formation of drops in a static mixer shear stress, surface tension, and viscous stress in the dispersed phase. Dimensional analysis shows that the drop size of the dispersed phase is controUed by the Weber number. The average drop size, in a Kenics mixer is a function of Weber number We = df /a, and the ratio of dispersed to continuous-phase viscosities (Eig. 32). 

The effects of a solvent on growth rates have been attributed to two sets of factors (28) one has to do with the effects of solvent on mass transfer of the solute through adjustments in viscosity, density, and diffusivity the second is concerned with the stmcture of the interface between crystal and solvent. The analysis (28) concludes that a solute-solvent system that has a high solubiUty is likely to produce a rough interface and, concomitandy, large crystal growth rates. 

Mass Transport. Probably the most iavestigated physical phenomenon ia an electrode process is mass transfer ia the form of a limiting current. A limiting current density is that which is controlled by reactant supply to the electrode surface and not the appHed electrode potential (42). For a simple analysis usiag the limiting current characteristics of various correlations for flow conditions ia a parallel plate cell, see Reference 43. 

According to this analysis one can see that for gas-absorption problems, which often exhibit unidirectional diffusion, the most appropriate driving-force expression is of the form y — y tyBM,. ud the most appropriate mass-transfer coefficient is therefore kc- This concept is to he found in all the key equations for the design of mass-transfer equipment. 

All these processes are, in common, liquid-gas mass-transfer operations and thus require similar treatment from the aspects of phase equilibrium and kinetics of mass transfer. The fluid-dynamic analysis ofthe eqmpment utihzed for the transfer also is similar for many types of liquid-gas process systems. 

End Effects Analysis of the mass-transfer efficiency of a packed cohimn should take into account that transfer which takes place outside the bed, i.e., at the ends of the packed sections. Inlet gas may veiy weU contact exit liquid below the bottom support plate, and exit gas can contact liquid from some types of distributors (e.g., spray nozzles). The bottom of the cohimn is the more hkely place for transfer, and SU-vey and KeUer [Chem. Eng. Prog., 62(1), 68 (1966)] found that the... 

If no wave formations are present, analysis of behavior of the liquid-film mass transfer as developed by Hatta and Katori [J. Soc. Chem. Ind., 37, 280B (1934)] indicates that... 

Methods for analysis of fixed-bed transitions are shown in Table 16-2. Local equilibrium theoiy is based solely of stoichiometric concerns and system nonlinearities. A transition becomes a simple wave (a gradual transition), a shock (an abrupt transition), or a combination of the two. In other methods, mass-transfer resistances are incorporated. 

Two dimensionless variables play key roles in the analysis of single transition systems (and some multiple transition systems). These are the throughput parameter [see Eq. (16-129)] and the number of transfer units (see Table 16-13). The former is time made dimensionless so that it is equal to unity at the stoichiometric center of a breakthrough cui ve. The latter is, as in packed tower calculations, a measure of mass-transfer resistance. 

A model of a reaction process is a set of data and equations that is believed to represent the performance of a specific vessel configuration (mixed, plug flow, laminar, dispersed, and so on). The equations include the stoichiometric relations, rate equations, heat and material balances, and auxihaiy relations such as those of mass transfer, pressure variation, contac ting efficiency, residence time distribution, and so on. The data describe physical and thermodynamic properties and, in the ultimate analysis, economic factors. 

For the analysis heat and mass transfer in concrete samples at high temperatures, the numerical model has been developed. It describes concrete, as a porous multiphase system which at local level is in thermodynamic balance with body interstice, filled by liquid water and gas phase. The model allows researching the dynamic characteristics of diffusion in view of concrete matrix phase transitions, which was usually described by means of experiments. 

HETP of a TLC plate is taken as the ratio of the distance traveled by the spot to the plate efficiency. The same three processes cause spot dispersion in TLC as do cause band dispersion in GC and LC. Namely, they are multipath dispersion, longitudinal diffusion and resistance to mass transfer between the two phases. Due to the aforementioned solvent frontal analysis, however, neither the capacity ratio, the solute diffusivity or the solvent velocity are constant throughout the elution of the solute along the plate and thus the conventional dispersion equations used in GC and LC have no pertinence to the thin layer plate. 

The Sauter mean is often used in sieve analysis, and in situations in whieh the surfaee area is important, e.g. mass transfer ealeulations, ete. 

The optimum flow rate for most SEC separations using conventional PLgel column dimensions (internal diameter 7.5 mm) is 1.0 ml/min. It may be of some benefit to work with lower flow rates, particularly for the analysis of higher molecular weight polymers where the reduced flow rate improves resolution through enhanced mass transfer and further reduces the risk of shear... 

Bolles and Fair [129] present an analysis of considerable data in developing a mass-transfer model for packed tower design however, there is too much detail to present here. 

Only one publication on gas-liquid mass transfer in bubble-column slurry reactors has come to the author s attention. However, a relatively large volume of information regarding mass transfer between single bubbles or bubble swarms and pure liquid containing no suspended solids is available, and this information is probably of some relevance to the analysis of systems... 

Mass transfer across the liquid-solid interface in mechanically agitated liquids containing suspended solid particles has been the subject of much research, and the data obtained for these systems are probably to some extent applicable to systems containing, in addition, a dispersed gas phase. Liquid-solid mass transfer in such systems has apparently not been studied separately. Recently published studies include papers by Calderbank and Jones (C3), Barker and Treybal (B5), Harriott (H4), and Marangozis and Johnson (M3, M4). Satterfield and Sherwood (S2) have reviewed this subject with specific reference to applications in slurry-reactor analysis and design. 

This brief discussion of some of the many effects and interrelations involved in changing only one of the operating variables points up quite clearly the reasons why no exact analysis of the dispersion of gases in a liquid phase has been possible. However, some of the interrelationships can be estimated by using mathematical models for example, the effects of bubble-size distribution, gas holdup, and contact times on the instantaneous and average mass-transfer fluxes have recently been reported elsewhere (G5, G9). 

Gal-Or and Hoelscher (G5) have recently developed a fast and simple transient-response method for the measurement of concentration and volumetric mass-transfer coefficients in gas-liquid dispersions. The method involves the use of a transient response to a step change in the composition of the feed gas. The resulting change in the composition of the liquid phase of the dispersion is measured by means of a Clark electrode, which permits the rapid and accurate analysis of oxygen or carbon dioxide concentrations in a gas, in blood, or in any liquid mixture. 

Although the absorption of a gas in a gas-liquid disperser is governed by basic mass-transfer phenomena, our knowledge of bubble dynamics and of the fluid dynamic conditions in the vessel are insufficient to permit the calculation of mass-transfer rates from first principles. One approach that is sometimes fruitful under conditions where our knowledge is insufficient to completely define the system is that of dimensional analysis. 

Pavlushenko et al. (P4) in their dimensional analysis considered Ks, the volumetric mass transfer coefficient, to be a function of pc, pc, L, Dr, N, Vs, and g. They determined the following relationship for the dimensionless groupings ... 

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