Hatta 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 ... 

GL 26] [R 3] [P 28] A simple reactor model was developed assuming isothermal behavior, confining mass transport to only from the gas to the liquid phase, and a sufficiently fast reaction (producing negligible reactant concentrations in the liquid phase) [10]. For this purpose, the Hatta number has to be within given limits. 

The first equation gives the rate of gas consumption as moles of gas (n) versus time. This is the only state variable that is measured. The initial number of moles, nO is known. The intrinsic rate constant, K is the only unknown model parameter and it enters the first model equation through the Hatta number y. The Hatta number is given by the following equation... 

A knowledge of the velocity profiles within falling films under various flow conditions would be of very great value, making it possible to calculate the rates of convective heat and mass transfer processes in flowing films without the need for the simplified models which must be used at present. For instance, the analyses of Hatta (H3, H4) and Vyazovov (V8, V9) indicate clearly the differences in the theoretical mass-transfer rates due to the assumption of linear or semiparabolic velocity profiles in smooth... 

As for the cardiovascular system, the cardioprotective effects of selective H3-receptor agonists, demonstrated in models of protracted myocardial ischemia (Imamura et al., 1994, 1995, 1996a Hatta et al., 1996, 1997), could be predictive of beneficial effects in coronaropatic patients. Hence, the attenuation of carrier-mediated noradrenaline release in hypoxic and/or ischemic myocardium by H3-agonists would limit the sympathetic overactivity and the associated incidence of ventricular arrhythmias and angina, as well as the increase of metabolic demand by the myocardium, thus preventing further damage and cardiac failure. 

Hatta, E, Imamura, M., Yasuda, K., Levi, R., 1996. Activation of histamine H3-receptors inhibits carrier-mediated norepinephrine release in a human model of protracted myocardial ischemia. Circulation 94,1-474. 

The enhancement factors are either obtained by fitting experimental results or are derived theoretically on the grounds of simplified model assumptions. They depend on reaction character (reversible or irreversible) and order, as well as on the assumptions of the particular mass transfer model chosen [19, 26]. For very simple cases, analytical solutions are obtained, for example, for a reaction of the first or pseudo-first order or for an instantaneous reaction of the first and second order. Frequently, the enhancement factors are expressed via Hatta-numbers [26, 28]. They can be used in combination with the HTU/NTU-method or with a more advanced mass transfer description method. However, it is generally not possible to derive the enhancement factors properly from binary experiments, and a theoretical description of reversible, parallel or consecutive reactions is based on rough simplifications. Thus, for many reactive absorption processes, this approach appears questionable. 

Biofilters are chemically enhanced absorbers, and therefore mass transfer limited (see Absorption with Chemical Reaction in Sec. 14). The magnitudes for the Hatta [= Damkohler II = (Thiele modulus)2] numbers are quite low, perhaps below 5. Nevertheless, for design simplicity, mass-transfer limitation is generally assumed to be in the liquid phase (the biofilm). For a single-component biofilter, the simplified biofilter model and design equation is... 

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. 

Fluid-fluid reactions are reactions that occur between two reactants where each of them is in a different phase. The two phases can be either gas and liquid or two immiscible liquids. In either case, one reactant is transferred to the interface between the phases and absorbed in the other phase, where the chemical reaction takes place. The reaction and the transport of the reactant are usually described by the two-film model, shown schematically in Figure 1.6. Consider reactant A is in phase I, reactant B is in phase II, and the reaction occurs in phase II. The overall rate of the reaction depends on the following factors (i) the rate at which reactant A is transferred to the interface, (ii) the solubihty of reactant A in phase II, (iii) the diffusion rate of the reactant A in phase II, (iv) the reaction rate, and (v) the diffusion rate of reactant B in phase II. Different situations may develop, depending on the relative magnitude of these factors, and on the form of the rate expression of the chemical reaction. To discern the effect of reactant transport and the reaction rate, a reaction modulus is usually used. Commonly, the transport flux of reactant A in phase II is described in two ways (i) by a diffusion equation (Pick s law) and/or (ii) a mass-transfer coefficient (transport through a film resistance) [7,9]. The dimensionless modulus is called the Hatta number (sometimes it is also referred to as the Damkohler number), and it is defined by... 

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. 

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. 

Different model assumptions reflect the relation between the mass transfer and reaction rates. The definition of the Hatta number, representing the maximum reaction rate with reference to that of the mass transfer, helps to discriminate between very fast, fast, average and slow chemical reactions [56, 57]. 

One further difficulty is the variation in thermal contract between calorimeter and pan (and/or between pan and sample) from run to run. Hatta and co-workers [21,22] produced a method to account for varying thermal resistance taking an inert sample (Cp real and positive) and a simple model for a calorimeter. The same method can be extended and combined with the above general model to account for an uncertain heat transfer coefficient between the sample and its pan, but assuming good thermal contact between the pan and the calorimeter. (If heat transfer coefficients between the pan and its contents and between the pan and its environs are both unknown—and finite—correction will be significantly harder.)... 

The following expression for this case was first derived by Hatta based on the film model ... 

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. 

In the frame of this model, the mass transfer time for A across the film is tg = 6 / and, assuming a second order reaction, the reaction time is tj = 1/(k Cg), The competition between transfer and reaction is controlled by the Hatta number such that... 

Many filler particles are in the shape of flakes, particles whose basic geometry is that of a disc or plate. These particles have a diaracteristic aspect ratio of some length parameter divided by thickness. Common flake filler particles are mica and talc. Metal flakes have also been inve gated a means of providing electrical ccmductivity to polymers. Hatta and Taya developed a model to describe the ttermal conductivity of a compete filled with flakes [27]. That model uses the same basic equation they devdoped for spherical and irregular filler partides ... 

There is little available data to analyze the predictive at ty of these modek to evaluate the through-the-plane conductivity of composites in which tte fibers are randomly oriented within a plane. In Delmonte s text there is limited data on the thermal conductivity of carbon fiber reinforced polyamide 66 composites, measured through the thickness of the plane of orientation of the fibers [26]. This data is summarized in Table 3. The fiber aspect ratios were taken from Fig. 8, based on the assumption that sufficient 3-D mixing occurred in the compounding and injection molding machines to reduce the fibers to an aspect ratio compatible with the maximum fraction associated with a random isotropic condition. The Hatta and Taya model does quite well in predicting the thermal conductivity across the plane of fiber orientation for these composites. 

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