Thiele-Damkohler theory

Mass transfer and reaction within catalyst particle (Thiele-Damkohler theory [16,17])... 

Granted the premises of the classic Thiele-Damkohler theory, the dependence of the overall rate on the Thiele modulus and thus on particle size can be predicted from experiments at two different particle sizes. In a plot of lnij versus ln, a ratio of particle sizes corresponds to a horizontal distance a ratio of rates, to a vertical distance. A rectangular triangle formed by these two distances can be fitted to the theoretical curve (see Figure 9.3). This allows the Thiele moduli at both particle sizes to be established and, with these, the effectiveness at other particle sizes to be predicted. The method fails if the two rates are equal, or if the ratios of the rates and reciprocal particle sizes are equal (straight-line portions of the graph). 

The ideal Thiele-Damkohler theory assumes isothermal behavior of the catalyst particle. However, if the particle is large and the reaction is highly exothermic, heat transfer may not be fast enough to remove the heat of reaction from the interior. The particle center then heats up, causing the reaction rate to be higher than under isothermal conditions. The effectiveness factor can be larger than unity This may occur if the structural material of the particle and the imbibed fluid are poor heat conductors, as might be the case, for example, in gas reactions in silica-ceous particles. 

The ideal Thiele-Damkohler theory also assumes that mass transfer in the particle occurs exclusively by diffusion. In a gas reaction, however, the volume of the reacting mixture expands if the mole number increases, and contracts if the mole number decreases. If it expands, forced convection out of the particle counteracts reactant diffusion into it and thereby slows the reaction down. If the volume contracts, forced convection sucks reactant into the particle and speeds the reaction up [16,28]. 

The classic Thiele-Damkohler theory accounts for these effects, but is restricted to isothermal behavior and intraparticle mass transfer only by diffusion. If the reaction is highly exothermic and the particle is a poor heat conductor, the temperature in the particle center may rise above that in the contacting fluid and cause the overall rate to be higher than in the absence of heat- and mass-transfer limitations. Moreover, gas-phase reactions with change in mole number cause forced inward or outward convection that assists or counteracts reactant penetration into the particle and so enhances or depresses the rate. 

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. 

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