Weisz-Prater number

Unfortunately, the precise value of the intraparticle diffusivity, D, for butene in the USHY catalyst particles is not known. One should expect D to be smaller than the bulk diffusivity (typically lO" cnfl/s). It appears reasonable to anticipate D to be in the range of 10"7 to 10 5 cm /s. For dp=91 pm, the effective rate constant (kapp) was found to be 19.3 s". The Weisz-Prater numbers can subsequently be calculated for the estimated range of diffusivities. 

If a second-order reaction is chosen as a reasonable upper limit, then with n = 2, p = 0.95, and the dimensionless Weisz criterion, Nw-p, frequently referred to as the Weisz-Prater number [21], is obtained for negligible diffu-sional limitations ... 

The final result is again a dimensionless number that compares the rate of reaction to the rate of mass transfer to the active sites in the pores. This is a conservative estimate because the concentration gradient depicted by curve 1 is likely to be steeper than the linear plot shown if the rate is very high. Furthermore, for a value of t s 0.95, the inequalities required in equation 4.88 for l -order and zero-order reactions are 0.6 and 6, respectively therefore, if a Weisz-Prater criterion (equation 4.93) of 0.3 or less is used, rates for all reactions with an order of 2 or less should have negligible mass transfer limitations. In addition, if the Weisz-Prater number is greater than 6, pore diffusion limitations definitely exist [20]. This transition region from kinetic to diffusion control occurs frequently around a TOF 1 s for many supported metal catalysts [22]. 

The Weisz-Prater criterion can be used for detecting drop in effectiveness due to internal diffusion. This is a number representing the ratio of actual reaction rate to a diffusion rate, and is given by ... 

According to eq 71 the temperature of the catalyst pellet can be calculated as a function of the Weisz modulus, for given values of the modified Prater number and the Biot number for mass transport. 

Figure 10. Effectiveness factor ij as a function of the Weisz modulus ji. Combined influence of intraparticle and interphase mass transfer and interphase heat transfer on the effective reaction rate (first order, irreversible reaction in a sphere, Biot number Bim = 100, Arrhenius number y — 20, modified Prater number ( as a parameter).

This procedure yields the curves depicted in Fig. 10 for fixed values of Bim and y. and the modified Prater number fi" as a parameter. From this figure, it is obvious that for exothermal reactions (fi > 0) and large values of the Weisz modulus, effectiveness factors well above unity may be observed. The reason for this is that the decline of the reactant concentration over the... 

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 ... 

Since the equations are nonlinear, a numerical solution method is required. Weisz and Hicks calculated the effectiveness factor for a first-order reaction in a spherical catalyst pellet as a function of the Thiele modulus for various values of the Prater number [P. B. Weisz and J. S. Hicks, Chem. Eng. Sci., 17 (1962) 265]. Figure 6.3.12 summarizes the results for an Arrhenius number equal to 30. Since the Arrhenius number is directly proportional to the activation energy, a higher value of y corresponds to a greater sensitivity to temperature. The most important conclusion to draw from Figure 6.3.12 is that effectiveness factors for exothermic reactions (positive values of j8) can exceed unity, depending on the characteristics of the pellet and the reaction. In the narrow range of the Thiele modulus between about 0.1 and 1, three different values of the effectiveness factor can be found (but only two represent stable steady states). The ultimate reaction rate that is achieved in the pellet... 

To calculate ripom, the mass and heat balances must be solved simultaneously. Analytical and numerical solutions are given by Petersen (1962), Tinkler and Pigford (1961), Carberry (1961), Tinkler and Metzner (1961), and Weisz and Hicks (1962). The behavior of a non-isothermal pellet in the regime of pore diffusion limitation is governed by the Thiele modulus (f> (related to Tsurface)> the Prater number and the Arrhenius number /int ... 

Figure 4.5.23 Effectiveness factor of a non-isothermal catalyst particle as a function of the Thiele modulus 0 (at 7 ) and the Prater number for an Arrhenius number of 20 (for solutions for other Yi x values see Weisz and Hicks, 1962 Levenspiel, 1996).

This is known as the Weisz-Prater (1954) criterion. The usefulness of the criterion lies in the fact that all quantities appearing in the right hand side of q. 4.145 are measurable. While this relationship has been derived for a first-order reaction, it is nevertheless applicable to many reactions for which the kinetics are not known. Mears (1971), for instance, suggested the use of l/ in place of 1 in Eq. 4.145 for an n -order reaction. A conservative number, say 0.1, may be used in the criterion for reactions with unknown kinetics. It should be recognized, however, that the criterion does not necessarily work in all cases. This is particularly so when the reaction is strongly inhibited by one of the products (Froment and Bischoff 1979). 

Figure 4.7 Variation of the Dimensionless Numbers P (Transverse Peclet Number) and C (Weisz-Prater Modulus) in the Temperature Range of 200-575 C for the Standard SCR Reaction on Fe-Zeolite Monolithic Catalysts. Extracted from Metkaretal. [129], with permission...

The analytical problems associated with differential reactors can be overcome by the use of the recirculation reactor. A simplified form, called a Schwab reactor, is described by Weisz and Prater . Boreskov.and other Russian workers have described a number of other modifications " . The recirculation reactor is equivalent kinetically to the well-stirred continuous reactor or backmix reactor , which is widely used for homogeneous liquid phase reactions. Fig. 28 illustrates the principle of this system. The reactor consists of a loop containing a volume of catalyst V and a circulating pump which can recycle gas at a much higher rate, G, than the constant feed and, withdrawal rates F. 

The curves shown were obtained by numerical integration by Weisz and Hicks [30]. Efficiency factors higher than one can be expected at relatively low Thiele modulus and high Prater and Arrhenius numbers. At large values of [Pg.77]

---