Kinetics Sherwood number

Considering these Biot numbers, we can observe that they are similar to the Nusselt and Sherwood numbers. The only difference between these dimensionless numbers is the transfer coefficient property characterizing the Biot numbers transfer kinetics for the external phase (a x heat transfer coefficient for the external phase, k ex- mass transfer coefficient for the external phase). We can conclude that the Biot number is an index of the transfer resistances of the contacting phases. 

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 process of formulating mesoscale models from the microscale equations is widely used in transport phenomena (Ferziger Kaper, 1972). For example, heat transfer between the disperse phase and the fluid depends on the Nusselt number, and mass transfer depends on the Sherwood number. Correlations for how the Nusselt and Sherwood numbers depend on the mesoscale variables and the moments of the NDF (e.g. mean particle temperature and mean particle concentration) are available in the literature. As microscale simulations become more and more sophisticated, modified correlations that are based on the microscale results will become more and more common (Beetstra et al, 2007 Holloway et al, 2010 Tenneti et al, 2010). Note that, because the kinetic equation requires mesoscale models that are valid locally in phase space (i.e. for a particular set of mesoscale variables) as opposed to averaged correlations found from macroscale variables, direct numerical simulation of the microscale model is perhaps the only way to obtain the data necessary in order for such models to be thoroughly validated. For example, a macroscale model will depend on the average drag, which is denoted by... 

It was shown in [ 166,351 ] that Eq. (5.1.5) provides several valid initial terms of the asymptotic expansion of the Sherwood number as Pe —> 0 for any kinetics of the surface chemical reaction. (Specifically, one obtains three valid terms for the translational Stokes flow and four valid terms for an arbitrary shear flow.)... 

For an arbitrary dependence of the kinetic function on the concentration, the mean Sherwood number for a spherical particle in a stagnant fluid can be calculated [360] by using the expression... 

For nonspherical particles in the case of a more complicated kinetic function /v(c), to calculate the Sherwood number, one can use formula (5.3.5), where the first summand on the right-hand side (equal to 1) must be replaced by Sho-... 

The catalyst is ignited on the fiill range of temperature. At low temperature (340°C), the rate constant obeys Arrhenius law with a consistent apparent activation energy. The corresponding Sherwood number is very low and it does not make sense. At high temperature (520 C), E a is still nonzero and tire corresponding Sherwood niunber is still smaller than 3. This shows that obtaining full mass transfer control is difficult, and that unusually low Sherwood numbers can be due to a partial kinetic control. 

The overall gas-liquid effectiveness factor is thus proportional to the inverse of the mass transfer Sherwood number, and we have a nice theoretical resolution for the analysis of gas-liquid reaction for the irreversible first-order kinetics considered. 

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). 

A knowledge of mass transfer is essential for the understanding of the mechanism of combustion of coal in a turbulent fluidized bed. If the kinetic rate of combustion of the fuel is known, one can estimate the burning rate using the information on the mass transfer rate. The rate of transfer of oxygen from the bulk of the bed to the particle surface, k, is often expressed as the dimensionless Sherwood number, Sh = kgdp/Dg- For diffusion to a fixed single sphere in an extensive fluid, Sherwood number may be expressed as [28, 29]... 

Figure 5.6 Kinetics of the agent consumed in food with the process controlled by diffusion, with Sherwood number 100, for various values of K and SN, with L = 0.05 cm D = 10 cmVs h = 2-10 cm/s.

Where Sh is Sherwood number [-], Re is Reynolds number [-], Sc is Schmidt number [-], is liquid-phase-mass-transfer coefficient [m-s ], d] is diameter of a stirrer [m], D is diffusion coefficient [m - s ], n is agitation speed [s ], V is kinetic viscosity of water [m - s ] and i is chemical species. Sc was varied by adding propylene glycol in the solution. Then the value of At,i for limestone dissolution was evaluated by Eq. 4, in which D[ and n were varied with experimental condition while V and c/l were constant. 

For vapors the Schmidt number can be estimated from kinetic theory f Sherwood et al.. 197S. pp. 17-24). The equation is... 

To evaluate the potential of carbon formation in a steam reformer, it is therefore essential to have a rigorous computer model, which contains kinetic models for the process side (reactor), as well as heat transfer models for the combustion side (furnace). The process and combustion models must be coupled together to accurately calculate the process composition, pressure, and temperature profiles, which result from the complex interaction between reaction kinetics and heat transfer. There may also be a temperature difference between bulk fluid, catalyst surface, and catalyst interior. Lee and Luss (7) have derived formulas for this temperature difference in terms of directly observable quantities The Weisz modulus and the effective Sherwood and Nusselt numbers based on external values (8). 

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