Reaction Carberry number

To estimate the average gradient, the concentration difference should be divided by the unknown boundary layer depth 5. While this is unknown, the Carberry number (Ca) gives a direct estimate of what concentration fraction drives the transfer rate. The concentration difference tells the concentration at which the reaction is really running. 

In unicellular organisms, the progressive doubling of cell number results in a continually increasing rate of growth in the population. A bacterial culture undergoing balanced growth mimics a first-order autocatalytic chemical reaction (Carberry, 1976 Levenspiel, 1972). Therefore, the rate of the cell population increase at any particular time is proportional to the number density (CN) of bacteria present at that time ... 

Criteria are usually derived so that deviations from the ideal situation are not larger than 5%. In order for external mass transfer limitations to be negligible, for an isothermal, n order irreversible reaction in a spherical particle, a criterion for the Carberry number can be derived, which assures that the observed rate does not deviate more than 5% from the ideal rate ... 

Whereas Figure 2.20 is quite instructive, it is not of practical use for estimating the importance of the mass transfer influence from experimental data, as the intrinsic rate constant is normally unknown. Replotting the effectiveness factor as function of the ratio between observed reaction rate to the maximum mass transfer rate called as Carberry number Ca) allows estimating the external effectiveness factor plotted in Figure 2.21. 

For a given system the temperature difference between bulk and surface depends on the reactant concentration via AT j, the ratio between Prandtl and Schmidt number, and the Carberry number. The temperature difference is maximum for reactions limited by mass transfer (Ca = >l). As for gases the Schmidt and Prandtl numbers are approximately unity Pr os Scoi 1), the temperature difference can reach the adiabatic temperature T - — T ). 

The latter is a well-known quantity in the reaction-diffusion analysis in catalytic media (see Section 8.2.3) and can be written as the ratio between the average reaction rate over the washcoat cross-sectional area at a given axial position and its value at the surface. The former compares the driving force for mass transfer toward the coating, with the total potential for concentration decay (due to mass transfer and surface reaction). For a first-order reaction, 0 reduces to the Carberry number (see Chapter 3), and >/ is a concentration ratio between the averaged value inside the washcoat and the one at the surface. 

Clearly, in the absence of a radial temperature or velocity gradient, no radial mass transfer can exist unless, of course, a reaction occurs at the bed wall. When a system is adiabatic, a radial temperature and concentration gradient cannot exist unless a severe radial velocity variation is encountered (Carberry, 1976). Radial variations in fluid velocity can be due to the nature of flow, e.g. in laminar flow, and in the case of radial variations in void fraction. In general, an average radial velocity independent of radial position can be assumed, except from pathological cases such as in very low Reynolds numbers (laminar flow), where a parabolic profile might be anticipated. 

The various volumetric mass-transfer coefficients are defined in a manner similar to that discussed for gas-liquid and fluid-solid mass transfer in previous sections. There are a large number of correlations obtained from different gas-liquid-solid systems. For more details see Shah (Gas-Liquid-Solid Reactor Design, McGraw-Hill, 1979), Ramachandran and Chaudhari (Three-Phase Catalytic Reactors, Gordon and Breach, 1983), and Shah and Sharma [Gas-Liquid-Solid Reactors in Carberry and Varma (eds.), Chemical Reaction and Reactor Engineering, Marcel Dekker, 1987],... 

Number of models have been proposed for gas-solid noncatalytic reactions in the literature. Most of the workers have limited their models by neglecting the structural changes as the reaction proceeds. Microscopic consideration of pore size change has been considered by Petersen (1), White and Carberry (2), Schechter and Gidley (3), Szekelly and Evans (4), Ramachandran and Smith (5, 6), Do gu (7), and Orbey et al. (8). 

A relatively large number of models can be written down for a packed-bed reactor, depending on what is accounted for in the model. These models, however, basically fall into two categories pseudohomogeneous models and heterogeneous models. The various models are described in standard reaction engineering texts — such as those of Carberry ( ), Froment and Bischoff ( ), and Smith ( ), to cite just a few — and in review articles (cf., and so details of their equations will not be reported here. We will, instead, only make some qualitative remarks about the models. 

The notion of turnover numhera was raised by Boudart, but Carberry has emphasised Chat generally we attempt to measure the number of catalyst sites (by chemisorption or titration) rather than the wore relevant number of sites c f catalysis and these are unlikely to be equal. Possibly the alkene titration using reactant cyclohexene would assess the summation of active sites as the active surface area under reaction conditions. An interesClng alternative approach is to use pulsed isothermal alkane titrations Co deduce the numbers of different surface Ptg sites Involved later In hydrogenation. If such a partial deconvolution of total average turnover numbers is really possible, then potentially it Is excemely useful. 

The observed reaction rates of benzene and of toluene compared with their solubilities in water at 423 K and 5.0 MPa. Carberry and Wheeler-Weisz numbers in the hydrogenation of benzene and of toluene over a ruthenium catalyst. (T =423 K P = 5.0 MPa 0.2 g ruthenium catalyst 200 cm substrate 75 cm water and 2.1 mmol ZnS04-7H20.)... 

The single CSTR has been used for many years in the laboratory for the study of kinetics of liquid phase reactions, and is now increasingly being employed for the measurement of gas/solid heterogeneous catalytic kinetics as well [early developments in the latter application are described by J.J. Car berry, Ind. Eng. Chem., 56, 39 (1964) D.J. Tajbl, J.B. Simons and J.J. Carberry, Ind. Eng. Chem. Eundls., 5, 171 (1966). A number of related designs based on internal recirculation of the reaction mixture through a small fixed bed of catalysts may also be treated conceptually as... 

Figure 7.10 Ratio of external to total AT vs. the observable quantity eDa for solid-catalyzed reactions, with the ratio of mass to heat transfer Biot numbers B as parameter (Carberry, 1975).

Figure 6.13 Effectiveness factor versus Weisz modulus for different Biot numbers. (Adapted from Carberry, J.J., Chemical and Catalytic Reaction Engineering, McGraw-Hill, New York, 1976.)...

Furthermore, during the cracking process there is an increase in the number of moles of products being formed from the gas oil. In a riser, which is normally modeled as a plug flow reactor, this results in a decrease in the massic vapour density. If the concentration of the reactants is defined at a constant volumetric flow, the apparent order also increases because the actual concentration decreases much faster than expected as the conversion increases. A pseudo-second-order reaction has been normally used to account for these non-linear effects (Weekman, 1968). However, the reaction order should be independent of the reaction system used and an additional term to account for the increased vapour velocity in a flow unit should be used to define the formal kinetics (Shaikh and Carberry, 1984). 

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