External Prater number

The maximum relative temperature difference (external Prater number) is given by ... 

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. 

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. 

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). 

Since the Prater numbers of RD processes were found to have values p < 10, the internal heat-transport effects can be neglected. The same is true for external heat transport effects [33]. 

A steady-state one-dimensional heterogeneous plug-flow model has been used to represent the behavior of the laboratory reactor. The internal and external mass transfer resistances are taken into account. For different operating conditions, very small Prater numbers were calculated. For this reason, the thermal gradients inside the washcoat were not included in the reactor model (Froment and Bischoff, 1990). The axial solid temperature profiles were measured for each experiment, therefore only the mass balances for gas and solid phases are needed to model the reactor. The equations that represent the system are the following ... 

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.

The dimensionless quantities (sometimes called Prater number) and relate the concentration differences to the temperature differences within the pellet and external to the pellet, respectively, as will be seen later. 

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