External Arrhenius number

Whether or not such an effect occurs in a practical situation and if so, how pronounced it will be, depends basically on the modified Prater number fi (see eq 71), that is on the maximum amount of heat effectively produced inside the pellet, as compared to the maximum amount of heat transported across the external boundary layer. Additionally, the Arrhenius number plays an important role which, as a normalized form of the activation energy, is a measure for the increase of the reaction rate due to an increase of temperature. 

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. 

One first-order irreversible chemical reaction occurs within a porous catalyst that exhibits rectangular symmetry. The center of the catalyst corresponds to rj = 0, and the external surface is at = 1. The intrapellet Damkohler number for reactant A is 1, and the Arrhenius number is 8.6. 

Figure 2.23 Nonisothermal external effectiveness factor as function of the Arrhenius number, Y and the Carberry number pg = 0.5). (Adapted from Ref. [16], Figure 4.10 Copyright 2012, Wiley-VCH GmbH Co. KGaA.)...

Figure 4.5.15 External effectiveness factor as a function ofthe ratio ofthe (measurable) effective reaction rate to the maximum rate (complete control by external mass transfer) fora constant Arrhenius number of 20. For a Prater number jSex < 0, the reaction is endothermic, for /Sex > 0 exothermic, and for /Sex = 0 we have isothermal conditions. Arrows and dashed line indicate ignition, as explained in the text.

Figure 2. Arrhenius plot of K efflux from Chlorella under Og stress. Rate of efflux was measured by dividing the time rate of change of external K (measured as in Figure 1) by the number of cells leaking Both rates (control 0— +O5 O—O) cltc the steady-state rates of efflux. The energies of activation are 14 4 and 15 L 3 kcal/mole for the control and - -Og, respectively.

This effect will be particularly emphasized at small values of the Thiele modulus where the intrinsic rate of reaction and the effective rate of diffusion assume the same order of magnitude. At large values of , the effectiveness factor again becomes inversely proportional to the Thiele modulus, as observed under isothermal conditions (Section 6.2.3.1). Then the reaction takes place only within a thin shell close to the external pellet surface. Here, controlled by the Arrhenius and Prater numbers, the temperature may be distinctly higher than at the external pellet surface, but constant further towards the pellet center. 

Operative. For the non isothermal case, effectiveness factors greater than unity are possible. Weisz and Hicks have considered this problem in some detail and constructed a number of graphs for various heats of reaction and activation energies. When a reaction is limited by pore diffusion, the reaction rate is proportional to yjky. If the temperature effects can be expressed as a simple Arrhenius relationship = A txp —E/RT), then the measured activation energy E will be about half the true activation energy. Very low values of the activation energy, i.e, 1-2 kcal. mole are only observed when mass transfer to the external catalyst surface is limiting the rate. 

KAPRAL - There are a number of examples of such phenomena in models for chemical and biological systems, and one can consider external noise processes that induce transitions between a chaotic state and another attractor. One example that we (R. Kapral, M. Schell and S. Frazer, 3. Phys. Chem. 86, 2203, 1982) have considered is a forced non linear oscillator in the presence of external noise. The deterministic dynamics exhibits bistability between a chaotic state and a periodic state. The rates of transition between the chaotic state and the periodic state were studied as a function of the amplitude of the external noise and, in some circumstances, interesting "non-Arrhenius" behavior was observed. 

The basic approximations made in arriving at the reactor point effectiveness are (1) isothermal pellet, (2) negligible external mass transfer resistance, and (3) estimation of the pellet center concentration by a simple relationship when the reaction is not severely diffusion-limited. The first two approximations are quite adequate in view of the fact that the mass Biot number is of the order of hundreds under realistic reaction conditions. Both theoretical and experimental justifications for these approximations have been given in Chapter 4. The first approximation will be relaxed when reactions affected by pore-mouth poisoning are considered since a definite temperature gradient then exists within the pellet. An additional approximation is the representation of the difference between the Arrhenius exponentials evaluated at the pellet surface and the bulk-fluid temperatures by a linear rela-... 

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