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. 

Page 3 gives a summary of the most important result in a figure illustrating in a semi-quantitative way the conditions in the specified CSTR. As can be seen on line 74, Dar is somewhat larger than the critical value but the concentration difference on line 75 is small, so this result can be accepted with some reservations. The Carberry number is also larger than the criteria, therefore these experimental results are marginal for Nox abatement... 

The unnamed number C is now called the Carberry number, and D is identical with Daiv-=Dav. 

A dimensionless number Ca, the Carberry number [8], is introduced as follows ... 

Both the calculation of the external effectiveness factor and of the Dfln number require the knowledge of the intrinsic rate expression, rViP(CA). The Carberry number [1] ... 

Typical values for the mass transfer coefficient lie around 10-3 m mf2 s 1 which is sufficiently high to allow neglect of the effects of mass transfer from the bulk of the liquid to the external pellet surface on the measured rates. In order to verify this it is sufficient to substitute the calculated mass transfer coefficient in Eqn. 7.74 for the Carberry number, bearing Eqn. 7.75 in mind. 

Substitution of the calculated value for the mass transfer coefficient, kfA, in expression 7.74 for the Carberry number again allows us to assess whether or not mass transfer effects can be neglected. 

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 ... 

The extent of external and internal mass transfer limitation can be estimated by the methods introduced by Carberry and by Wheeler-Weisz [12]. A Carberry number (Ca) smaller than 0.05 means that diffusional retardation by external mass transport may be neglected. A Wheeler-Weisz group (WW) smaller than 0.1 means that pore diffusion limitation is negligible [13,14]. 

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. 

Figure 2.21 Effectiveness factor as function of the observable variabie the Carberry number (Adapted from Ref. [15], Figure 4.8 Copyright 0 2012, Wiley-VCH GmbH Co. KGaA)...

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. 

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