Solid-phase mass transfer coefficient

The parameters Z and W are dimensionless parameters proportional to the fluid and solid phase mass transfer coefficients, respectively, according to Equations 5.14 and 5.15 ... 

In principle, mass transfer in solid particles is distributed in both time and distance, calling for the use of a PDE (Pick s equation) for a rigorous description of the process. In an elegant study conducted in the 1950s, it was shown by Glueckauf that the results of the formal treatment can be approximated by a volumetric solid-phase mass transfer coefficient, given by... 

We start by noting that the use of a solid-phase mass transfer coefficient calls for a driving force based on the internal concentration X (kg solute/kg solid). A solute mass balance, performed about a single particle assumed to be a sphere, leads to the expression... 

The preceding method is useful if both fluid- and solid-phase mass-transfer coefficients are available. The solid-phase coefficient particularly is not usually available for most industrial developments. In such cases the following extension is a useful approximation, also limited to cases of favorable adsorption equilibrium [14]. 

The point of view based on a physical model started with the 1935 paper of Higbie [30], While the main problem treated by Higbie was that of the mass transfer from a bubble to a liquid, it appears that he had recognized the utility of his representation for both packed beds and turbulent motion. The basic idea is that an element of liquid remains in contact with the other phase for a time A and during this time, absorption takes place in that element as in the unsteady diffusion in a semiinfinite solid. The mass transfer coefficient k should therefore depend on the diffusion coefficient D and on the time A. Dimensional analysis leads in this case to the expression... 

Kl = the overall gas-phase mass transfer coefficient as defined in equation (3.59) kf = the liquid-solid mass transfer coefficient... 

Any form of convection, of course, increases the value of Ks. In slurry operation with no liquid flow, gas flow induces convection. In an agitated slurry reactor, stirring causes convection. In a pulsating slurry reactor, pulsation of the slurry induces convection and in a three-phase fluidized bed, the movements of both gas and liquid phases cause convection. Any one or more modes of convection will increase the value of the solid-liquid mass-transfer coefficient. In broad terms, the convective liquid-solid mass-transfer coefficient is correlated by-two steady state theories. Here we briefly review and compare them. 

A solid-liquid mass transfer coefficient of 0.015 cm/s was found by comparing the predictions of [S(IV)] to experimental results obtained under conditions in which the liquid phase kinetics were fast. The model was then applied to slurry oxidation under more general conditions by using liquid phase reaction rate kinetics obtained in clear solutions. The results of the model agree with experimental findings for the total rate of oxidation. 

Constant diffusivity (D ) and the solid-liquid mass-transfer coefficient are assumed for QX and QY. The model equations are nondimensionalized in terms of the Thiele parameter (f>, Biot number for mass transfer Bi, and nondi-mensional time and distance. An important conclusion from the subsequent analysis of the model simulations is the importance of the solid phase on the conversion of the organic substrate in the organic phase. Results of their simulation are shown in Figure 11. It can be noticed that at low (f>, corresponding to low diffusional limitations, the overall organic reagent conversion is lower than at higher values o <. This result is the exact opposite of what is observed in analysis of... 

Liquid-solid. Transport between the liquid and solid (catalyst) phases in trickle-bed reactors is at least a first cousin to transport in more conventional fixed beds, and our understanding of the liquid phase mass-transfer coefficient here benefits from the decades of research devoted to that topic. A good correlation was reported as far back as 1948 by Van Krevelen and Krekels [D.W. Van Krevelen and J.T.C. Krekels, Rec. Trav. Chim. Pays-Bas, 67, 512 (1948)], who proposed... 

Now, it is instructive to re-analyze the unsteady-state macroscopic mass balance on an isolated solid pellet of pure A with no chemical reaction. The rate of output due to interphase mass transfer from the solid particle to the liquid solution is expressed as the product of a liquid-phase mass transfer coefficient c, liquids a Concentration driving force (Ca, — Ca), and the surface area of one spherical pellet, 4nR. The unsteady-state mass balance on the solid yields an ordinary differential equation for the time dependence of the radius of the peUet. For example,... 

In the absence of convective mass transfer and chemical reaction, calculate the steady-state liquid-phase mass transfer coefficient that accounts for curvature in the interfacial region for cylindrical liquid-solid interfaces. An example is cylindrical pellets that dissolve and diffuse into a quiescent liquid that surrounds each solid pellet. The appropriate starting point is provided by equation (B) in Table 18.2-2 on page 559 in Bird et al. (1960). For one-dimensional diffusion radially outward, the mass transfer equation in cylindrical coordinates reduces to... 

As in any solid-liquid reaction, when the solid is sparingly soluble, reaction occurs within the solid by diffusion of the liquid-phase reactant into it across the liquid film surrounding the solid. Thus two diffusion parameters are operative, the solid-liquid mass transfer coefficient sl and the effective diffusivity D. of the reactant in the solid. A reaction in the solid can occur by any of several mechanisms. The simpler and more common of these were briefly explained in Chapter 15. For reactions following the sharp interface model, ultrasound can enhance either or both these constants. Indeed, in a typical solid-liquid reaction such as the synthesis of dibenzyl sulfide from benzyl chloride and sodium sulfide ultrasound enhances SL by a factor of 2 and by a factor of 3.3 (Hagenson and Doraiswamy, 1998). Similar enhancement in was found for a Michael addition reaction (Ratoarinoro et al., 1995) and for another mass transfer-limited reaction (Worsley and Mills, 1996). 

Koide, K., Horibe, K., Kawabata, H., and Ito, S. (1985), Gas holdup and volumetric liquid-phase mass transfer coefficient in solid-suspended bubble column with draught tube, Journal of Chemical Engineering of Japan, 18(3) 248-254. 

By comparison to solid particles, drops are not only subject to deformation but also to internal circulation and oscillation. This affects not only the values of the continuous but also of the dispersed phase mass transfer coefficients. Relevant theoretical and empirical correlations are collected in literature (24, 25). For oscillating drops the equations of Clift et al. (26) give usually a good prediction (27)... 

Three phase slurry reactors are characterized by a gas-liquid (K,a) and liquid-solid (k ) mass transfer coefficient. These coefficients were determined for the rotating disc reactor at the appropriate operation conditions ... 

For type B experiments. Criteria I-VIII should be checked. If one of these criterion are not met, then consider decreasing the mixing time 9m or increasing the mass-transfer rates between phases, that is i g-i, and i s i by some means. For example, if criteria VII and VIII are not met, the primary options are (a) use more finely divided material or (b) add baffles to the vessel, since a dramatic increase in the solid-liquid mass-transfer coefficient will result. 

It is worth pointing out that few studies took into account the pseudoplastic behavior of the polymeric dispersions [74, 75). There is little and sometimes contradictory information about the presence of the additional gas phase on the solid-liquid mass-transfer coefficients. Sanger et al. [76] observed that in a bubble column, Kp increased upon increasing the gas flow rate. Grisafi et al. [77], observed just the opposite in a stirred tank this was attributed to reduction of the effective power... 

Contribution of the solid-phase mass transfer to the kinetics of selective dissolution increases with increasing concentration and diffiisivity of dissolving metal ions in a solution as well as with reducing diffusion coefficient of this metal and its atomic fraction in an alloy. 

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