Overall Mass-Transfer Coefficient In systems with relatively sparing soluble gases, where the gas-phase resistance is negligible, the mass-transfer rate can be determined by using the concept of an overall volumetric mass-transfer coefficient kLa as follows:
where Ms = solute molar mass-transfer rate, kgmol/s; kLa = overall mass-transfer coefficient, 1/s; C* = solute concentration in equilibrium with the liquid phase, kg-mol/s; and C,b = solute concentration in bulk of liquid.
Bakker et al. (op. cit.) have given a correlation for kLa for aqueous systems in the absence of significant surface active agents.
where CkLa = 0.015, 1/s; Eq. (14-219) applies for both 6BD and CD-6.
Interfacial Phenomena These can significantly affect overall mass transfer. Deckwer, Bubble Column Reactors, Wiley, Hoboken, N.J., 1992, has covered the effect of surfactants on mass transfer in bubble columns. In fermentation reactors, small quantities of surface-active agents (especially antifoaming agents) can drastically reduce overall oxygen transfer (Aiba et al., op. cit., pp. 153, 154), and in aerobic mechanically aerated waste-treatment lagoons, overall oxygen transfer has been found to be from 0.5 to 3 times that for pure water from tests with typical sewage streams (Eckenfelder et al., op. cit., p. 105).
One cannot quantitatively predict the effect of the various interfacial phenomena; thus, these phenomena will not be covered in detail here. The following literature gives a good general review of the effects of interfacial phenomena on mass transfer: Goodridge and Robb, Ind. Eng. Chem. Fund., 4, 49 (1965); Calderbank, Chem. Eng. (London), CE 205 (1967); Gal-Or et al., Ind. Eng. Chem., 61(2), 22 (1969); Kintner, Adv. Chem. Eng., 4 (1963); Resnick and Gal-Or, op. cit., p. 295; Valentin, loc. cit.; and Elenkov, loc. cit., and Ind. Eng. Chem. Ann. Rev. Mass Transfer, 60(1), 67 (1968); 60(12), 53 (1968); 62(2), 41 (1970). In the following outline, the effects of the various interfacial phenomena on the factors that influence overall mass transfer are given. Possible effects of interfacial phenomena are tabulated below:
1. Effect on continuous-phase mass-transfer coefficient a. Impurities concentrate at interface. Bubble motion produces circumferential surface-tension gradients that act to retard circulation and vibration, thereby decreasing the mass-transfer coefficient.
b. Large concentration gradients and large heat effects (very soluble gases) can cause interfacial turbulence (the Marangoni effect), which increases the mass-transfer coefficient.
2. Effect on interfacial area a. Impurities will lower static surface tension and give smaller bubbles.
b. Surfactants can electrically charge the bubble surface (produce ionic bubbles) and retard coalescence (soap stabilization of an oil-water emulsion is an excellent example of this phenomenon), thereby increasing the interfacial area.
c. Large concentration gradients and large heat effects can cause bubble breakup.
3. Effect on mean mass-transfer driving force a. Relatively insoluble impurities concentrate at the interface, giving an interfacial resistance. This phenomenon has been used in retarding evaporation from water reservoirs.
b. The axial concentration variation can be changed by changes in coalescence. The mean driving force for mass transfer is therefore changed.
Gas Holdup (e) in Bubble Columns With coalescing systems, holdup may be estimated from a correlation by Hughmark [Ind. Eng Chem. Process Des. Dev., 6, 218-220 (1967)] reproduced here as Fig. 14-104. For noncoalescing systems, with considerably smaller bubbles, £ can be as great as 0.6 at Usg = 0.05 m/s, according to Mersmann [Ger Chem. Eng., 1, 1 (1978)].
It is often helpful to use the relationship between £ and superficial gas velocity (Usg) and the rise velocity of a gas bubble relative to the liquid velocity (Ur + UL, with UL defined as positive upward):
Rise velocities of bubbles through liquids have been discussed previously.
For a better understanding of the interactions between parameters, it is often helpful to calculate the effective bubble rise velocity Ur from measured valves of £; for example, the data of Mersmann (loc. cit.) indicated £ = 0.6 for Usg = 0.05 m/s, giving Ur = 0.083 m/s, which agrees with the data reported in Fig. 14-43 for the rise velocity of bubble clouds. The rise velocity of single bubbles, for db ~ 2 mm, is about 0.3 m/s, for liquids with viscosities not too different from water. Using this value in Eq. (14-220) and comparing with Fig. 14104, one finds that at low values of Usg, the rise velocity of the bubbles is less than the rise velocity of a single bubble, especially for small-diameter tubes, but that the opposite occurs for large values of Usg.
More recent literature regarding generalized correlational efforts for gas holdup is adequately reviewed by Tsuchiya and Nakanishi [Chem. Eng Sci., 47(13/14), 3347 (1992)] and Sotelo et al. [Int. Chem. Eng., 34(1), 82-90 (1994)]. Sotelo et al. (op. cit.) have developed a dimensionless correlation for gas holdup that includes the effect of gas and liquid viscosity and density, interfacial tension, and diffuser pore diameter. For systems that deviate significantly from the waterlike liquids for which Fig. 14-104 is applicable, their correlation (the fourth numbered equation in the paper) should be used to obtain a more accurate estimate of gas holdup. Mersmann (op. cit.) and Deckwer et al. (op. cit.) should also be consulted.
Liquid-phase mass-transfer coefficients in bubble columns have been reviewed by Calderbank ("Mixing," loc. cit.), Fair (Chem. Eng., loc. cit.), Mersmann [Ger. Chem. Eng. 1, 1 (1978), Int. Chem. Eng., 32(3) 397-405 (1991)], Deckwer et al. [Can. J. Chem. Eng, 58, 190 (1980)], Hikita et al. [Chem. Eng. J., 22, 61 (1981)] and Deckwer and Schumpe [Chem. Eng. Sci., 48(5), 889-911 (1993)]. The correlation of Ozturk, Schumpe, and Deckwer [AIChE J., 33,1473-1480 (1987)] is recommended. Deckwer et al. (op. cit.) have documented the case for using the correlation:
Ozturk et al. (1987) developed a new correlation on the basis of a modification of the Akita-Yoshida correlation suggested by Nakanoh and Yoshida (1980). In addition, the bubble diameter db rather than the column diameter was used as the characteristic length as the column diameter has little influence on kLa. The value of db was assumed to be approximately constant (db = 0.003 m). The correlation was obtained by nonlinear regression is as follows:
where kLa = overall mass-transfer coefficient, dB = bubble diameter = 0.003 m, Dl = diffusivity of gas in liquid, p = density, ^ = viscosity, a = interfacial tension, g = gravitational acceleration.
As mentioned earlier, surfactants and ionic solutions significantly affect mass transfer. Normally, surface affects act to retard coalescence and thus increase the mass transfer. For example, Hikata et al. [Chem. Eng. J., 22,61-69 (1981)] have studied the effect of KCl on mass transfer in water. As KCI concentration increased, the mass transfer increased up to about 35 percent at an ionic strength of 6 gm/l. Other investigators have found similar increases for liquid mixtures.
Axial Dispersion Backmixing in bubble columns has been extensively studied. Wiemann and Mewes [Ind. Eng. Chem. Res., 44, 4959 (2005)] and Wild et al. [Int. J. Chemical Reactor Eng., 1, R7 (2003)] give a long list of references pertaining to backmixing in bubble columns. An excellent review article by Shah et al. [AIChE J., 24, 369 (1978)] has summarized the literature prior to 1978. Works by Konig et al. [Ger. Chem. Eng., 1,199 (1978)], Lucke et al. [Trans. Inst. Chem. Eng., 58, 228 (1980)], Riquarts [Ger Chem. Eng., 4, 18 (1981)], Mersmann (op. cit.), Deckwer (op. cit.), Yang et al. [Chem. Eng. Sci., 47(9-11), 2859 (1992)], and Garcia-Calvo and Leton [Chem. Eng. Sci., 49(21), 3643 (1994)] are particularly useful references.
Axial dispersion occurs in both the liquid and the gas phases. The degree of axial dispersion is affected by vessel diameter, vessel internals, gas superficial velocity, and surface-active agents that retard coalescence. For systems with coalescence-retarding surfactants the initial bubble size produced by the gas sparger is also significant. The gas and liquid physical properties have only a slight effect on the degree of axial dispersion, except that liquid viscosity becomes important as the flow regime becomes laminar. With pure liquids, in the absence of coalescence-inhibiting, surface-active agents, the nature of the sparger has little effect on the axial dispersion, and experimental results are reasonably well correlated by the dispersion model. For the liquid phase the correlation recommended by Deckwer et al. (op. cit.), after the original work by Baird and Rice [Chem. Eng. J., 9, 171(1975)] is as follows:
where EL = liquid-phase axial dispersion coefficient, UG = superficial velocity of the gas phase, D = vessel diameter, and g = gravitational acceleration.
The recommended correlation for the gas-phase axial-dispersion coefficient is given by Field and Davidson (loc. cit.):
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