Liquid film diffusion

Calderbank et al. (C6) studied the Fischer-Tropsch reaction in slurry reactors of 2- and 10-in. diameters, at pressures of 11 and 22 atm, and at a temperature of 265°C. It was assumed that the liquid-film diffusion of hydrogen from the gas-liquid interface is a rate-determining step, whereas the mass transfer of hydrogen from the bulk liquid to the catalyst was believed to be rapid because of the high ratio between catalyst exterior surface area and bubble surface area. The experimental data were not in complete agreement with a theoretical model based on these assumptions. 

The main mass transport resistance in liquid fluidized beds of relatively small particles lies in the liquid film. Thus, for ion exchange and adsorption on small particles, the mass transfer limitation provides a simple liquid-film diffusion-controlled mass transfer process (Hausmann el al., 2000 Menoud et al., 1998). The same holds for catalysis. 

According to their analysis, if is zero (practically much lower than 1), then the liquid-film diffusion controls the process rate, while if tfis infinite (practically much higher than 1), then the solid diffusion controls the process rate. Essentially, the so-called mechanical parameter represents the ratio of the diffusion resistances (solid and liquid film). The authors did not refer to any assumption concerning the type of isotherm for the derivation of the above-mentioned criterion it is sufficient to be favorable (not only rectangular). They noted that for >1.6, the particle diffusion is more significant, whereas if < 0.14, the external mass transfer controls the adsorption rate. 

Eq. (4.140) is for liquid-film diffusion control and eq. (4.141) for solid diffusion control. The following equation is a solution of the fixed-bed model under the constant pattern and plug-flow assumption, for fluid-film diffusion control and the favorable Freundlich... 

In Figure 4.23, the model results for solid diffusion control (eq. (4.141)) and two different values of the Langmuir constant La) are presented. In Figure 4.24, the model results for solid diffusion and liquid-film diffusion control (eq. (4.140)) for La = 0.5 are presented. 

In Figure 4.27, some examples of theoretical breakthrough curves calculated from the analytical solutions for the Freundlich isotherm (Fr = 0.5) are presented. As is clear, the curve corresponds to the case of equal and combined solid and liquid-film diffusion resistances ([ = 1) which is between the two extremes, i.e. solid diffusion control (l = 10,000) and liquid-film diffusion control ( = 0.0001). 

In the above equations, H0, //p, and H are the plate-height contributions due to the finite particle size, solid diffusion, and liquid-film diffusion, respectively. CGS units are used in these equations. Obviously, the bigger the height of the plate, the higher the resistance to the diffusion and the lower the uptake rate. 

Rate equation for liquid-film diffusion (4.131) dp, e, and particle shape... 

For liquid film diffusion control, the following analogous expressions can be used ... 

Extraction efficiency decreases on increasing the initial metal concentration. This fact shows a change in the bed if there is no saturation process the mass transfer is controlled by liquid film diffusion if there is a saturation process the mass transfer is controlled by surface diffusion. The fluidized bed improves metal extraction with regard to the fixed bed, as can be seen in Fig. 36 for the extraction of Zn(II) with XAD2-DEHPA resins. 

In Chapter 14, we formulated a number of regimes with corresponding conditions and governing rate equations. In the present chapter, we recast the rate equations in a general form that indicates the relative roles of reaction, liquid film diffusion, and gas film diffusion. Then we briefly discuss the design principles of the more common classes of fluid-fluid reactors. Detailed treatments of design may be found in the books of Astarita (1967), Danckwerts (1970), Shah (1979), Levenspiel (1972, 1993), Doraiswamy and Sharma (1984b), Bisio et al. (1985), and Kastanek et al. (1993). 

Models Considering Membrane and Liquid Film Diffusion. Models considering membrane and liquid film diffusion are quite complex as they are of second order in nature, and the solution to these models require numerical analysis or a method of moments due to their complexity (Sobotka et al., 1982). Linek et al. (1985), Ruchti et al. (1981), and Dang et al. (1977) suggested that while these models are more complex and involved, their solutions are much superior to any first-order model. However, due to their complexity, they are typically not used and the reader is referred to the literature for more information concerning these models. 

Diffusion in the liquid film Diffusion in the adsorption layer eoc Ds... 

The same numerical methods as those used to solve the homogeneous reactor models (PFR, BR, and stirred tank reactor) as well as the heterogeneous catalytic packed bed reactor models are used for gas-Uquid reactor problems. For the solution of a countercurrent column reactor, an iterative procedure must be applied in case the initial value solvers are used (Adams-Moulton, BD, explicit, or semi-implicit Runge-Kutta). A better alternative is to solve the problem as a true boundary value problem and to take advantage of a suitable method such as orthogonal collocation. If it is impossible to obtain an analytical solution for the liquid film diffusion Equation 7.52, it can be solved numerically as a boundary value problem. This increases the numerical complexity considerably. For coupled reactions, it is known that no analytical solutions exist for Equation 7.52 and, therefore, the bulk-phase mass balances and Equation 7.52 must be solved numerically. 

In regime 3, the reaction is sufficiently fast to consume the dissolved solid reactant completely in the liquid film. Diffusion and reaction occur simultaneously in a parallel fashion in the liquid film. The mass transfer coefficient... 

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