Interphase effectiveness factors

By comparing this relationship with the solution for the effectiveness factor in absence of interphase concentration gradients (cq 51), it becomes obvious that the overall effectiveness factor t] can be expressed as the product of separate pore and external (film) effectiveness factors ... 

Figure 9. Effectiveness factor ij as a function of the Weisz modulus iji. Combined influence of intraparticle and interphase mass transfer on the effective reaction rate (isothermal, first order, irreversible reaction in a sphere, Biot number Bim as a parameter).

From this figure, it can be concluded that the reduction of the effectiveness factor at large values of becomes more pronounced as the Biot number is decreased. This arises from the fact that the reactant concentration at the external pellet surface drops significantly at low Biot numbers. However, a clear effect of interphase diffusion is seen only at Biot numbers below 100. In practice, Bim typically ranges from 100 to 200. Hence, the difference between the overall and pore effectiveness factor is usually small. In other words, the influence of intraparticle diffusion is normally by far more crucial than the influence of interphase diffusion. Thus, in many practical situations the overall catalyst efficiency may be replaced by the pore efficiency, as a good approximation. 

The first term in parentheses in eq 75, together with the preceding factor 3j, denotes the isothermal intraparticle effectiveness factor (see eq 51). The second term in parentheses is identical to the isothermal interphase effectiveness factor (see eq 60). The exponential factor between the two terms describes the influence of the deviating catalyst temperature. 

Any of the curves in Fig. 10, which refer to different values of the modified Prater number fi, tend to approach a certain limiting value of the Weisz modulus for which the overall effectiveness factor obviously becomes infinitely small. This limit can be easily determined, bearing in mind that the effective reaction rate can never exceed the maximum interphase mass transfer rate (the maximum rate of reactant supply) which is obtained when the surface concentration approaches zero. To show this, we formulate the following simple mass balance, analogous to eq 62 ... 

In the most general case, i.e. when intraparticlc and interphase transport processes have to be included in the analysis, the effectiveness factor depends on five dimensionless numbers, namely the Thiele modulus the Biot numbers for heat and mass transport Bih and Bim, the Prater number / , and the Arrhenius number y. Once external transport effects can be neglected, the number of parameters reduces to three, because the Biot numbers then approach infinity and can thus be discarded. 

Although multiplicities of the effectiveness factor have also been detected experimentally, these are of minor importance practically, since for industrial processes and catalysts, Prater numbers above 0.1 are less common. On the contrary, effectiveness factors above unity in real systems are frequently encountered, although the dominating part of the overall heat transfer resistance normally lies in the external boundary layer rather than inside the catalyst pellet. For mass transfer the opposite holds the dominating diffusional resistance is normally located within the pellet, whereas the interphase mass transfer most frequently plays a minor role (high space velocity). 

The Damkohler number indicates which characteristic first-order process is faster, external diffusion or reaction. For very large values of Da (ks the surface concentration of reactant approaches zero, whereas for very small values of Da ks the surface concentration approaches the bulk fluid concentration. An interphase effectiveness factor, Tj, is defined as the reaction rate based on surface conditions divided by the rate that would be observed in the absence of diffusional limitations ... 

The overall effectiveness factor is actually comprised of the individual effectiveness factors for intraphase and interphase transport ... 

For industrial reactors, the effectiveness factor (i]) is used to provide a measure of the actual reaction rate, as affected by operating conditions, in comparison to the intrinsic reaction kinetics. Assuming that the trickle bed reactor shown in Fig. 4 is operated so that interphase transport of one of the reactants, steps 2 or 8 above, is controlling. 

FIGURE 5.50 Interphase and intraphase non-isothermai effectiveness factor for LHHW kinetics for the reaction C — P, y = 40. 

Since the major thermal resistances in nonisothermal reaction systems are encountered in the boundary layer, while the major mass transfer resistances occur within the particle, we can entertain some simplification of the overall effectiveness factor problem we have been considering. This simplified model envisions interphase temperature gradients and intraphase concentration gradients only. For this case... 

The latter concentration driving force (i.e., CA,buik gas — CA,sur ce) is used to construct an expression for the flux of reactant A into the pellet via interphase mass transfer. One employs equation (20-1) for the definition of the effectiveness factor E ... 

Cassiere, G. and Carberry, J.J. (1973) Interphase catalytic effectiveness factor activity, yield and non-isothermality. Chem. Eng. Educ., 7 (1), 22—26. 

Equation 3.53 reduces to several asymptotic results presented in the literature [80, 101], as they represent a shape and kinetic normalization at low reaction rates with nonzero interphase resistances. The application of these estimates for the effectiveness factor is shown in Figure 3.3 for negligible external mass transfer resistance (Dirichlet problem) and in Figure 3.4 for finite interphase resistance. 

The same asymptote is shown in Figure 3.4 for the region mainly limited by intraphase diffusion. The regime of strong interphase mass transfer resistance is also depicted in the same representation for two values of the mass Biot number. The asymptotic behavior of the effectiveness factor in this limit 1 and low is obtained from (power-law... 

The same cannot be said under mass transfer control, where the surface distribution of concentration and reaction rates included in the Thiele modulus also depend on the mass transfer problem in the channel. This only happens for nonlinear kinetics and gives origin the distinction between an overall regime (with low wall concentration but still allowing for strong gradients to develop in the coating) and a purely interphase resistance (where channel concentration is so low that the effectiveness factor may even approach 1). 

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