Mass transfer coefficient correlations

Earlier studies in mass transfer between the gas-liquid phase reported the volumetric mass-transfer coefficient kLa. Since kLa is the combination of two experimental parameters, mass-transfer coefficient and mterfacial area, it is difficult to identify which parameter is responsible for the change of kLa when we change the operating condition of a fermenter. Calderbank and Moo-Young (1961) separated kta by measuring interfacial area and correlated mass-transfer coefficients in gas-liquid dispersions in mixing vessels, and sieve and sintered plate column, as follows ... 

The development of mass transfer models require knowledge of three properties the diffusion coefficient of the solute, the viscosity of the SCF, and the density of the SCF phase. These properties can be used to correlate mass transfer coefficients. At 35 C and pressures lower than the critical pressure (72.83 atm for CO2) we use the diffusivity interpolated from literature diffusivity data (2,3). However, a linear relationship between log Dv and p at constant temperature has been presented by several researchers U>5) who correlated diffusivities in supercritical fluids. For pressures higher than the critical, we determined an analytical relationship using the diffusivity data obtained for the C02 naphthalene system by lomtev and Tsekhanskaya (6), at 35 C. 

Mass Transfer Correlation. The development of a correlation for the mass transfer coefficient is based upon the mass transfer coefficients which were obtained through the use of the cell model proposed by Kramers and Alberda ( [). Buoyant effects become important under supercritical conditions because of the small kinematic viscosities which are a consequence of the high densities and low viscosities. Consequently it is necessary to consider both forced and natural convection when attempting to correlate mass transfer coefficients under supercritical conditions. 

In an extensive investigation, Billet and Schultes (1991b) measured and correlated mass-transfer coefficients for 31 different binary and ternary systems with 67... 

In the preceding discussion, a very simple model of the absorption process has been used the film model. Other models of increased complexity have been proposed for calculating and correlating mass transfer coefficients for fluid properties and flow regimes. The rationale and assumptions for the more important mass transfer models are now very briefly discussed. 

The usual Sherwood number or correlations have been used to correlate mass transfer coefficients with system properties and flow characteristics. Scores of such correlations have been reported covering a wide range of electrode-electrolyte configurations. A few of these that represent mass transfer in the more common electrochemical reactors are listed in Prentice (1991). 

Film Theory. Many theories have been put forth to explain and correlate experimentally measured mass transfer coefficients. The classical model has been the film theory (13,26) that proposes to approximate the real situation at the interface by hypothetical "effective" gas and Hquid films. The fluid is assumed to be essentially stagnant within these effective films making a sharp change to totally turbulent flow where the film is in contact with the bulk of the fluid. As a result, mass is transferred through the effective films only by steady-state molecular diffusion and it is possible to compute the concentration profile through the films by integrating Fick s law ... 

To use all of these equations, the heights of the transfer units or the mass transfer coefficients and must be known. Transfer data for packed columns are often measured and reported direcdy in terms of and and correlated in this form against and... 

Other correlations based partially on theoretical considerations but made to fit existing data also exist (71—75). A number of researchers have also attempted to separate from a by measuring the latter, sometimes in terms of the wetted area (76—78). Finally, a number of correlations for the mass transfer coefficient itself exist. These ate based on a mote fundamental theory of mass transfer in packed columns (79—82). Although certain predictions were verified by experimental evidence, these models often cannot serve as design basis because the equations contain the interfacial area as an independent variable. 

Correlations for the mass-transfer coefficient, as the Sherwood number for various membrane geometries have been reviewed (39). 

Mass-Transfer Coefficient Denoted by /c, K, and so on, the mass-transfer coefficient is the ratio of the flux to a concentration (or composition) difference. These coefficients generally represent rates of transfer that are much greater than those that occur by diffusion alone, as a result of convection or turbulence at the interface where mass transfer occurs. There exist several principles that relate that coefficient to the diffusivity and other fluid properties and to the intensity of motion and geometry. Examples that are outlined later are the film theoiy, the surface renewal theoiy, and the penetration the-oiy, all of which pertain to ideahzed cases. For many situations of practical interest like investigating the flow inside tubes and over flat surfaces as well as measuring external flowthrough banks of tubes, in fixed beds of particles, and the like, correlations have been developed that follow the same forms as the above theories. Examples of these are provided in the subsequent section on mass-transfer coefficient correlations. 

The important point to note here is that the gas-phase mass-transfer coefficient fcc depends principally upon the transport properties of the fluid (Nsc) 3nd the hydrodynamics of the particular system involved (Nrc). It also is important to recognize that specific mass-transfer correlations can be derived only in conjunction with the investigator s particular assumptions concerning the numerical values of the effective interfacial area a of the packing. 

With complicated geometries, the product of the interfacial area per volume and the mass-transfer coefficient is required. Correlations of kop or of HTU are more accurate than individual correlations of k and since the measurements are simpler to determine the produc t kop or HTU. 

To determine the mass-transfer rate, one needs the interfacial area in addition to the mass-transfer coefficient. For the simpler geometries, determining the interfacial area is straightforward. For packed beds of particles a, the interfacial area per volume can be estimated as shown in Table 5-27-A. For packed beds in distillation, absorption, and so on in Table 5-28, the interfacial area per volume is included with the mass-transfer coefficient in the correlations for HTU. For agitated liquid-liquid systems, the interfacial area can be estimated... 

It is important to recognize that the effects of temperature on the liquid-phase diffusion coefficients and viscosities can be veiy large and therefore must be carefully accounted for when using /cl or data. For liquids the mass-transfer coefficient /cl is correlated in terms of design variables by relations of the form... 

In developing correlations for the mass-transfer coefficients Icq and /cl, the various authors have assumed different but internally compatible correlations for the effective interfacial area a. It therefore would be inappropriate to mix the correlations of different authors unless it has been demonstrated that there is a valid area of overlap between them. 

Experimental values of Hqg -nd Hql for a number of distillation systems of commercial interest are also readily available. Extrapolation of the data or the correlations to conditions that differ significantly from those used for the original experiments is risky. For example, pressure has a major effect on vapor density and thus can affect the hydrodynamics significantly. Changes in flow patterns affeci both mass-transfer coefficients and interfacial area. 

The unknown intermediate concentration C, has been mathematically ehminated from the last term. In this case, r can be solved for explicitly, but that is not always possible with surface rate equations of greater complexity. The mass transfer coefficient /ci is usually obtainable from correlations. When the experimental data are of (C, r) the other constants can be found by linear plotting. 

Example 8 Calculation of Rate-Based Distillation The separation of 655 lb mol/h of a bubble-point mixture of 16 mol % toluene, 9.5 mol % methanol, 53.3 mol % styrene, and 21.2 mol % ethylbenzene is to be earned out in a 9.84-ft diameter sieve-tray column having 40 sieve trays with 2-inch high weirs and on 24-inch tray spacing. The column is equipped with a total condenser and a partial reboiler. The feed wiU enter the column on the 21st tray from the top, where the column pressure will be 93 kPa, The bottom-tray pressure is 101 kPa and the top-tray pressure is 86 kPa. The distillate rate wiU be set at 167 lb mol/h in an attempt to obtain a sharp separation between toluene-methanol, which will tend to accumulate in the distillate, and styrene and ethylbenzene. A reflux ratio of 4.8 wiU be used. Plug flow of vapor and complete mixing of liquid wiU be assumed on each tray. K values will be computed from the UNIFAC activity-coefficient method and the Chan-Fair correlation will be used to estimate mass-transfer coefficients. Predict, with a rate-based model, the separation that will be achieved and back-calciilate from the computed tray compositions, the component vapor-phase Miirphree-tray efficiencies. 

Interfacial Area This consideration in agitated vessels has been reviewed and summarized by Tatterson (op. cit.). Predictive methods for interfacial area are not presented here because correlations are given for the overall volumetric mass transfer coefficient liquid phase controlhng mass transfer. 

Overall Mass-Transfer Coejficient Tatterson (op. cit.) and Zlokarnik (op. cit.) have summarized the hterature covering overall mass-transfer coefficients. There is much scatter in the experimental data because the presence of surface-ac tive agents and electrolytes have a significant effect on the mass transfer. The correlation of Van t Riet [Ind. Eng Chem. Process Des. Dev., 18(3), 357 (1979)] is recommended ... 

The mass-transfer coefficients depend on complex functions of diffii-sivity, viscosity, density, interfacial tension, and turbulence. Similarly, the mass-transfer area of the droplets depends on complex functions of viscosity, interfacial tension, density difference, extractor geometry, agitation intensity, agitator design, flow rates, and interfacial rag deposits. Only limited success has been achieved in correlating extractor performance with these basic principles. The lumped parameter deals directly with the ultimate design criterion, which is the height of an extraction tower. 

TABLE 16-9 Recommended Correlations for External Mass Transfer Coefficients in Adsorption Beds (Re = evdp/v. Sc = v/D)... 

FIG. 16"10 Sherwood mimher correlations for external mass-transfer coefficients in packed beds for e = 0.4 (adapted from Siiziild, gen. refs.). 

Not only is the type of flow related to the impeller Reynolds number, but also such process performance characteristics as mixing time, impeller pumping rate, impeller power consumption, and heat- and mass-transfer coefficients can be correlated with this dimensionless group. 

Ga.s-Lic(uid Ma.s.s Tran.sfer Gas-liqiiid mass transfer norrnallv is correlated bv means of the mass-transfer coefficient K a ersiis powder le el at arioiis superficial gas elocities. The superficial gas clocitv is the ohirne of gas at the a erage temperature and pressure at the midpoint in the tank di ided bv the area of the essel. In order to obtain the partial-pressure dri ing force, an assumption must be made of the partial pressure in equilibrium wdth the concentration of gas in the liquid, Manv times this must be assumed, but if Fig, 18-26 is obtained in the pilot plant and the same assumption principle is used in e ahiating the mixer in the full-scale tank, the error from the assumption is limited. 

Experimental gas-solid mass-transfer data have been obtained for naphthalene in CO9 to develop correlations for mass-transfer coefficients [Lim et al., Am. Chem. Soc. Symp. Ser, 406, 379 (1989)]. The data were correlated over a wide range of conditions with the following equation for combined natural and forced convection ... 

For purely physical absorption, the mass-transfer coefficients depend on trie hydrodynamics and the physical properties of the phases. Many correlations exist for example, that of Dwivedi and Upadhyay (Ind. Eng. Chem. Proc. De.s. izDev.,... 

With a reactive solvent, the mass-transfer coefficient may be enhanced by a factor E so that, for instance. Kg is replaced by EKg. Like specific rates of ordinary chemical reactions, such enhancements must be found experimentally. There are no generalized correlations. Some calculations have been made for idealized situations, such as complete reaction in the liquid film. Tables 23-6 and 23-7 show a few spot data. On that basis, a tower for absorption of SO9 with NaOH is smaller than that with pure water by a factor of roughly 0.317/7.0 = 0.045. Table 23-8 lists the main factors that are needed for mathematical representation of KgO in a typical case of the absorption of CO9 by aqueous mouethauolamiue. Figure 23-27 shows some of the complex behaviors of equilibria and mass-transfer coefficients for the absorption of CO9 in solutions of potassium carbonate. Other than Henry s law, p = HC, which holds for some fairly dilute solutions, there is no general form of equilibrium relation. A typically complex equation is that for CO9 in contact with sodium carbonate solutions (Harte, Baker, and Purcell, Ind. Eng. Chem., 25, 528 [1933]), which is... 

Mass-transfer coefficients seem to vary as the 0.7 exponent on the power input per unit volume, with the dimensions of the vessel and impeller and the superficial gas velocity as additional factors. A survey of such correlations is made by van t Riet (Ind Eng. Chem Proc Des Dev., IS, 3.57 [1979]). Table 23-12 shows some of the results. 

TABLE 23-12 Correlations of Mass-Transfer Coefficients in Stirred Tanks... 

Two complementai y reviews of this subject are by Shah et al. AIChE Journal, 28, 353-379 [1982]) and Deckwer (in de Lasa, ed.. Chemical Reactor Design andTechnology, Martinus Nijhoff, 1985, pp. 411-461). Useful comments are made by Doraiswamy and Sharma (Heterogeneous Reactions, Wiley, 1984). Charpentier (in Gianetto and Silveston, eds.. Multiphase Chemical Reactors, Hemisphere, 1986, pp. 104—151) emphasizes parameters of trickle bed and stirred tank reactors. Recommendations based on the literature are made for several design parameters namely, bubble diameter and velocity of rise, gas holdup, interfacial area, mass-transfer coefficients k a and /cl but not /cg, axial liquid-phase dispersion coefficient, and heat-transfer coefficient to the wall. The effect of vessel diameter on these parameters is insignificant when D > 0.15 m (0.49 ft), except for the dispersion coefficient. Application of these correlations is to (1) chlorination of toluene in the presence of FeCl,3 catalyst, (2) absorption of SO9 in aqueous potassium carbonate with arsenite catalyst, and (3) reaction of butene with sulfuric acid to butanol. 

For conditions in industrial production reactors and in corresponding recycle reactors, the mass transfer coefficients of Gamson et al (1943) will be used. These are approximately correct and simple to use. There may be better correlations for specific cases and especially for larger molecules, where diffiisivity is low and Schmidt number is high. In such cases literature referring to given conditions should be consulted. 

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