Hatta theory

Figure 6.3b shows the idealized concentration profile of an absorbed component A, obtained by the Hatta theory, for the case of a relatively slow reaction that is either first-order or pseudo first-order with respect to A. As A is consumed gradually while diffusing across the film, the gradient of concentration of A that is required for its diffusion gradually decreases with increasing distance from the interface. The enhancement factor for such cases is given by the Hatta theory as r... 

Hatta [5] derived a series of theoretical equations for E, based on the film model. Experimental values of E agree with the Hatta theory, and also with theoretical values of E derived later by other investigators, based on the penetration model. 

Figure 6.3a shows the idealized sketch of concentration profiles near the interface by the Hatta model, for the case of gas absorption with a very rapid second-order reaction. The gas component A, when absorbed at the interface, diffuses to the reaction zone where it reacts with B, which is derived from the bulk of the liquid by diffusion. The reaction is so rapid that it is completed within a very thin reaction zone this can be regarded as a plane parallel to the interface. The reaction product diffuses to the liquid main body. The absorption of C02 into a strong aqueous KOH solution is close to such a case. Equation 6.21 provides the enhancement factor E for such a case, as derived by the Hatta theory ... 

In Chapter 7 we discussed the basics of the theory concerned with the influence of diffusion on gas-liquid reactions via the Hatta theory for flrst-order irreversible reactions, the case for rapid second-order reactions, and the generalization of the second-order theory by Van Krevelen and Hofitjzer. Those results were presented in terms of classical two-film theory, employing an enhancement factor to account for reaction effects on diffusion via a simple multiple of the mass-transfer coefficient in the absence of reaction. By and large this approach will be continued here however, alternative and more descriptive mass transfer theories such as the penetration model of Higbie and the surface-renewal theory of Danckwerts merit some attention as was done in Chapter 7. 

Additional experiments in a loop reactor where a significant mass transport limitation was observed allowed us to investigate the interplay between hydrodynamics and mass transport rates as a function of mixer geometry, the ratio of the volume hold-up of the phases and the flow rate of the catalyst phase. From further kinetic studies on the influence of substrate and catalyst concentrations on the overall reaction rate, the Hatta number was estimated to be 0.3-3, based on film theory. 

The expression for the enhancement factor E, eq. (35), has first been derived by van Krevelen and Hof-tijzer in 1948. These authors used Pick s law for the description of the mass transfer process and approximated the concentration profile of component B by a constant Xb, over the entire reaction zone. It seems worthwhile to investigate whether the same equation can be applied in case the Maxwell-Stefan theory is used to describe the mass transfer process. To evaluate the Hatta number, again an effective mass transfer coefficient given by eq. (34), is required. The... 

Figures 7(a)-(c) show a comparison between the numerically computed absorption flux and the absorption flux obtained from expression (31), using eqs (24), (30) and (34)-(37). From these figures it can be concluded that for both equal and different binary mass transfer coefficients absorption without reaction can be described well with eq. (24), whereas absorption with instantaneous reaction can be described well with eq. (30). If the Maxwell-Stefan theory is used to describe the mass transfer process, the enhancement factor obeys the same expression as the one obtained on the basis of Fick s law [eq. (35)]. Finally, from Figs 7(b) and 7(c) it appears that the use of an effective mass transfer coefficient m the Hatta number again produces satisfactory results.

Hatta (H3), Hatta and Katori (H4), 1934 Theoretical and experimental work on absorption of CO2 by water film on channel 1.5 cm. X various lengths, slopes l°-90°. Shows inapplicability of two film theory to most cases of gas absorption by liquid films. 

This view is been confirmed by an electrochemical product study (Hatta et al. 2001) that is discussed below. The pfCa value of the Thy radical cation has been determined at 3.2 (Geimer and Beckert 1998). When the position at N( ) is substituted by a methyl group and deprotonation of the radical cation can no longer occur at this position, deprotonation occurs at N(3) (Geimer and Beckert 1999 for spin density calculations using density functional theory (DFT) see Naumov et al. 2000). This N(3) type radical is also produced upon biphotonic photoionization of N(l)-substituted Thy anions [reaction (7)] in basic 8 molar NaC104 D20 glasses which allowed to measure their EPR spectra under such conditions (Sevilla 1976). 

As the Hatta number increases, the effective liquid-phase mass-transfer coefficient increases. Figure 14-13, which was first developed by Van Krevelen and Hoftyzer [Rec. Trav. Chim., 67, 563 (1948)] and later refined by Perry and Pigford and by Brian et al. [AlChE J., 7,226 (1961)], shows how the enhancement (defined as the ratio of the effective liquid-phase mass-transfer coefficient to its physical equivalent q = ki/kl) increases with NHa for a second-order, irreversible reaction of the kind defined by Eqs. (14-60) and (14-61). The various curves in Fig. 14-13 were developed based upon penetration theory and... 

According to the film theory, in reactive-absorption processes the resistance to mass transfer is concentrated in a small region near the gas/liquid interface. The ratio between tbe rate of chemical reaction and liquid-phase mass transfer is given by the Hatta number. For a second-order reaction (12.1), the Hatta number is defined as ... 

Approximate vs. Numerical Solution. The accuracy of the approximate reaction factor expression has been tested over wide ranges of parameter values by comparison with numerical solutions of the film-theory model. The methods of orthogonal collocation and orthogonal collocation on finite elements (7,8) were used to obtain the numerical solutions (details are given by Shaikh and Varma (j>)). Comparisons indicate that deviations in the approximate factor are within few percents (< 5%). It should be mentioned that for relatively high values of Hatta number (M >20), the asymptotic form of Equation 7 was used in those comparisons. 

The film theory was originally proposed by Whitman,195 who obtained his idea from the Nernst117 concept of the diffusion layer. It was first applied to the analysis of gas absorption accompanied by a chemical reaction by Hatta.85,86 It is a steady-state theory and assumes that mass-transfer resistances across the interface are restricted to thin films in each phase near the interface. If more than one species is involved in a multiphase reaction process, this theory assumes that the thickness of the film near any interface (gas-liquid or liquid-solid) is the same for all reactants and products. Although the theory gives a rather simplified description of the multiphase reaction process, it gives a good answer for the global reaction rates, in many instances, particularly when the diffusivities of all reactants and products are identical. It is simple to use, particularly when the... 

This expression is based on Hatta s film concept. Similar expressions result from penetration theory. Figure 3 illustrates the concentration profiles for A and B in and around the fluid film for Regime II. 

Film Theory and Gas-Liquid and Liquid-Liquid Mass Transfer. The history and literature surrounding interfacial mass transfer is enormous. In the present context, it suffices to say that the film model, which postulates the existence of a thin fluid layer in each fluid phase at the interface, is generally accepted (60). In the context of coupled mass transfer and reaction, two common treatments involve 1) the Hatta number and (2) enhancement factors. Both descriptions normally require a detailed model of the kinetics as well as the mass transfer. The Hatta number is perhaps more intuitive, since the numbers span the limiting cases of infinitely slow reaction with respect to mass transfer to infinitely fast reaction with respect to mass transfer. In the former case all reaction occurs in the bulk phase, and in the latter reaction occurs exclusively at the interface with no bulk reaction occurring. Enhancement factors are usually categorized in terms of reaction order (61). In the context of nonreactive systems, a characteristic time scale (eg, half-life) for attaining vapor-liquid equilibrium and liquid-liquid equilibrium, 6>eq, in typical laboratory settings is of the order of minutes. 

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