Catalytic Sherwood numbers

Traditionally, an average Sherwood number has been determined for different catalytic fixed-bed reactors assuming constant concentration or constant flux on the catalyst surface. In reality, the boundary condition on the surface has neither a constant concentration nor a constant flux. In addition, the Sh-number will vary locally around the catalyst particles and in time since mass transfer depends on both flow and concentration boundary layers. When external mass transfer becomes important at a high reaction rate, the concentration on the particle surface varies and affects both the reaction rate and selectivity, and consequently, the traditional models fail to predict this outcome. 

Open-channel monoliths are better defined. The Sherwood (and Nusselt) number varies mainly in the axial direction due to the formation ofa hydrodynamic boundary layer and a concentration (temperature) boundary layer. Owing to the chemical reactions and heat formation on the surface, the local Sherwood (and Nusselt) numbers depend on the local reaction rate and the reaction rate upstream. A complicating factor is that the traditional Sherwood numbers are usually defined for constant concentration or constant flux on the surface, while, in reahty, the catalytic reaction on the surface exhibits different behavior. 

By analogy with the Graetz problem for the Nusselt number, determine quantitatively the behavior of the Sherwood number as a function of z, the axial distance from the start of the catalytic section... 

Liquid-solid mass transfer is typically not limiting due to the small particle size resulting in large particle surface area/volume of reactor, unless the concentration of the particles is very low, and or larger particles are used. In the latter case, intraparticle mass-transfer limitations would also occur. Ramachandran and Chaudhari (Three-Phase Catalytic Reactors, Gordon and Breach, 1983) present several correlations for liquid-solid mass transfer, typically as a Sherwood number versus particle Reynolds and Schmidt numbers, e.g., the correlation of Levins and Glastonbury [Trans. Inst. Chem. Engrs. 50 132 (1972)] ... 

We further mention that at low values of the Reynolds number (that is at very low fluid velocities or for very small particles) for flow through packed beds the Sherwood number for the mass transfer can become lower than Sh = 2, found for a single particle stagnant relative to the fluid [5]. We refer to the relevant papers. For the practice of catalytic reactors this is not of interest at too low velocities the danger of particle runaway (see Section 4.3) becomes too large and this should be avoided, for very small particles suspension or fluid bed reactors have to be applied instead of packed beds. For small particles in large packed beds the pressure drop become prohibitive. Only for fluid bed reactors, like in catalytic cracking, may Sh approach a value of 2. 

Considerations along the above lines lead to analogous correlations for the Sherwood number for the description of mass transfer in a single channel. The application of the rather simple Nusselt and Sherwood number concept for monolith reactor modeling implies that the laminar flow through the channel can be approached as plug flow, but it is always limited to cases in which homogeneous gas-phase reactions are absent and catalytic reactions in the washcoat prevail. If not, a model description via distributed flow is necessary. 

Cybulski and Moulijn [27] proposed an experimental method for simultaneous determination of kinetic parameters and mass transfer coefficients in washcoated square channels. The model parameters are estimated by nonlinear regression, where the objective function is calculated by numerical solution of balance equations. However, the method is applicable only if the structure of the mathematical model has been identified (e.g., based on literature data) and the model parameters to be estimated are not too numerous. Otherwise the estimates might have a limited physical meaning. The method was tested for the catalytic oxidation of CO. The estimate of effective diffusivity falls into the range that is typical for the washcoat material (y-alumina) and reacting species. The Sherwood number estimated was in between those theoretically predicted for square and circular ducts, and this clearly indicates the influence of rounding the comers on the external mass transfer. 

The boundary condition of the governing partial differential equation that describes mass transfer in a reactor with catalytic walls, differs from the standard boundary conditions assumed in most texts on heat transfer in rod bundles (either constant wall temperature/concentration or constant heat/mass flux). However, the Sherwood number of a reactor with catalytic walls will lie between the values obtained for these two standard boundary conditions, which deviate less than 30% for relative pitches higher than 1.1. 

Figure 9.7. Dependence of the utlization factor as a function of at different Sherwood numbers (F. Kapteijn, G.B. Marin, J.A. Moulijn, Catalytic reaction engineering, in Catalysis an integrated approach, Elsevier, 1999).

If the mass transfer is accompanied by a chemical reaction at the catalyst surface on the reactor wall, the mass transfer depends on the reaction kinetics [29]. For a zero-order reaction the rate is independent of the concentration and the mass flow from the bulk to the wall is constant, whereas the reactant concentration at the catalytic wall varies along the reactor length. For this situation the asymptotic Sherwood number in circular tube reactors becomes Sh a = 4.36 [29]. The same value is obtained when reaction rates are low compared vdth the rate of mass transfer. If the reaction rate is high (very fast reactions), the concentration at the reactor wall can be approximated to zero within the whole reactor and the final value for Sh is Shoo = 3.66. As a consequence, the Sherwood number in the reaction system depends on the ratio of the reaction rate to the rate of mass transfer characterized by the second Damkohler number defined in Equation (15.22). 

Pig. 13. Influence of the Damkohler number. Da, and of the monoHth channel geometry on the as5miptotic Sherwood number. Shoo, for circular, square, and equilateral triangular channels. Da represents the ratio of the rate of the catalytic reaction to the rate of gas/solid... 

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