Sherwood number Thiele modulus

The PFR is efficient for screening solid catalyst in a single fluid phase. It can also be used in later research stages to assess commercial criteria. Consider the evaluation of the ultimate commercial performance of a newly developed fixed-bed catalyst. The theory of similarity teaches that for the laboratory and the industrial reactor, the Damkohler number (NDa), the Sherwood number (Nsh), and the Thiele modulus (<)>) need to be kept constant (Figure 2). As a result, the laboratory reactor must have the same length as the envisioned commercial reactor (7). In this case, scale up is done by increasing the diameter of the reactor. This example further illustrates that laboratory reactors are not necessarily small in size. 

If the particle diameter is small, the effectiveness factor is practically independent of Thiele modulus, i.e., = 1, as we have seen earlier. In this case, the pore diffusion does not affect the rate and the resistance is due to the chemical reaction surface, which is the limiting step of the process. It is known that the prodnct kiai depends on the particle diameter and diffusion, which are usnally represented by Sherwood number. Thus,... 

This means that if the particle diameter is small, the resistance to mass transfer in liquid phase becomes negligible. On the other hand, when analyzing the same expressions for particle diameters is larger, we see that the effectiveness factor decreases and the Sherwood number increases. Knowing that the effectiveness factor is inversely proportional to the Thiele modulus and Sherwood number is directly proportional to the diameter of the particle, we obtain such combined effects. Therefore ... 

The dialytic regime is characterized by high surface reaction rate coefficients and by rate-limiting diffusion. The Sherwood number (Sh) characterizes the regimes. Sh is defined as the ratio of the driving force for diffusion in the boundary layer to the driving force for surface reaction alternatively, it is the ratio of the resistivity for diffusion to the resistivity for chemical reaction (reciprocal reaction rate coefficient). Diffusion limitation is the regime at Sh 1 and reaction limitation means Sh 1. The Sherwood number is closely related to the Biot, Nusselt, and Damkohler II numbers and the Thiele modulus. Some call it the CVD number. In the boundary-layer model it is a simple function of the thickness of the boundary layer, the diffusion coefficient, and the reaction rate coefficient. For simplicity a first-order reaction will be considered in the derivation below. 

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