Weisz—Prater modulus

Equations (5.67) and (5.68), termed the Weisz-Prater modulus, contain only observable quantities. A quick test for internal diffusion can be performed by varying L and observing the effect on r if no effect is detected, pore diffusion cannot be important in this situation. Figure 5.11 elucidates the possible influence of pore diffusion on the observed reaction rate. 

On the other hand, the Weisz-Prater modulus ( P) provides an estimate of the extent of the catalytic layer diffusion limitations (Equation (4.7)). This is defined as follows is the observed reaction rate) ... 

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...

Transport Limitation For the estimation of the mass transport limitation, Equation (20) has an important drawback. In many cases neither the rate constant k nor the reaction order n is known. However, the Weisz-Prater criterion, cf. Equation (21), which is derived from the Thiele modulus [4, 8], can be calculated with experimentally easily accessible values, taking < < 1 for any reaction without mass transfer limitations. However, it is not necessary to know all variable exactly, even for the Weisz-Prater criterion n can be unknown. Reasonable assumptions can be made, for example, n - 1, 2, 3, or 4 and / is the particle diameter instead of the characteristic length. For the gas phase, De can be calculated with statistical thermodynamics or estimated common values are within the range of 10-5 to 10 7 m2/s. In the liquid phase, the estimation becomes more complicated. A common value of qc for solid catalysts is 1,300 kg/m3, but if the catalyst is diluted with an inert material, this... 

However, since no intrinsic kinetics are measured, one cannot determine whether criterion (8.130) is satisfied. Introduction of the Wheeler-Weisz modulus, Eqn. (8.78), and series expansion, leads to the Weisz-Prater criterion to assess the importance of internal diffusion limitations. For an irreversible n order reaction and for a spherical particle, this criterion is ... 

Closure After completing this chapter, the reader should be able to derive differential equations describing diffusion and reaction, discuss the meaning of the effectiveness factor and its relationship to the Thiele modulus, and identify the regions of mass transfer control and reaction rate control. The reader should be able to apply the Weisz-Prater and Mears criteria to identify gradients and diffusion limitations. These principles should be able to be applied to catalyst particles as well as biomaierial tissue engineering. The reader should be able to apply the overall effectiveness factor to a packed bed reactor to calculate the conversion at the exit of the reactor. The reader should be able to describe the reaction and transport steps in slurry reactors, trickle bed reactors, fluidized-besd reactors, and CVD boat reactors and to make calculations for each reactor. 

The Weisz-Prater criterion makes use of observable quantities like -Ra)p, the measured global rate (kmol/kg-s) dp, the particle diameter (m) pp, the particle density (kg/m ) Dg, the effective mass diffusivity (m /s) and the surface concentration of reactant (kmol/m ). The intrinsic reaction rate constant ky need not be known in order to use the Weisz-Prater criterion. If external mass transfer effects are eliminated, CAb can be used, and the effective diffusivity can be estimated using catalyst and fluid physical properties. The criterion can be extended to other reaction orders and multiple reactions by using the generalized Thiele modulus, and various functional forms are quoted in the literature [17, 26, 28]. 

The second method is based upon the Weisz-Prater criterion [1954]. It is arrived at by rewriting the Thiele modulus (3.6.1-16) ... 

Using the generalized modulus, the Weisz-Prater criterion (3.7-5) was extended by Petersen [1965a, b] and Bischoff [1967] to the case where the reaction rate is not first order and may be written as... 

Another criteria due to Satterfield [65] is one that is based on work by Petersen [56], as developed further by Froment and Bischoff [16], which has as a starting point the observation that the film mass transfer coefficient in a catalytic system cannot limit unless pore diffusion is also limiting. Hence, a mass transfer limitation on the outside of the porous catalyst pellet can only be important, for example, if "the Weisz and Prater modulus is greater than about 3 to 10 which corresponds to an effectiveness factor, n, of 0.3 to about 0.7. Thus, the criteria suggests that if the effectiveness factor of the catalyst in question in a reactor is close to unity mass transfer limitations cannot be important. 

Surface tension of water and liquid, mN m = Weisz and Prater Modulus, dimensionless... 

To assess whether a reaction is influenced by intraparticle diffusion effects, Weisz and Prater [11] developed a criterion for isothermal reactions based upon the observation that the effectiveness factor approaches unity when the generalised Thiele modulus is of the order of unity. It has been shown that the effectiveness factor for all catalyst geometries and reaction orders (except zero order) tends to unity when... 

In assessing whether a reactor is influenced by intraparticle mass transfer effects WeiSZ and Prater 24 developed a criterion for isothermal reactions based upon the observation that the effectiveness factor approaches unity when the generalised Thiele modulus is of the order of unity. It has been showneffectiveness factor for all catalyst geometries and reaction orders (except zero order) tends to unity when the generalised Thiele modulus falls below a value of one. Since tj is about unity when 0 < ll for zero-order reactions, a quite general criterion for diffusion control of simple isothermal reactions not affected by product inhibition is < 1. Since the Thiele modulus (see equation 3.19) contains the specific rate constant for chemical reaction, which is often unknown, a more useful criterion is obtained by substituting l v/CAm (for a first-order reaction) for k to give ... 

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... 

It is not always possible to determine the modulus directly, because the intrinsic rate constant ky is not known. Weisz and Prater have suggested a method for the determination of t which does not require a knowledge of ky The observed reaction rate — dn/dr can be expressed as follows... 

The coefficients k and rj in Eq. (42) have the same physical significance as in the case of macroporous catalysts, although the mechanism of diffusion is different. We have r] = tanh 9/9 for linear geometry and first order however, very similar behavior results for different geometry as pointed out by Weisz and Prater (59). The influence of the modulus

physical explanation, as pointed out by... 

Probably the most widely applied criterion is the one for internal mass transfer limitations in an isothermal catalyst particle, e.g. for pore diffusion. Due to Weisz and Prater (Advances in Catalysis 6 (1954) 143) no pore diffusion limitation occurs, if the Weisz modulus... 

Figure 7.7 gives some experimental results, obtained for the cracking of cumene on silica/alumina catalysts, in which the diffusional modulus has been changed by variation of the catalyst pellet dimension [P.B. Weisz and C.D. Prater, Advan. Catalysis, 6, 144 (1954)]. It is seen that the slope of the In r versus /T plot for the severely diffusionally limited pellets of 0.175-cm radius is approximately one-half that of the smaller 0.0056-cm pellets. 

This modulus, named for Weisz who first proposed it along with Prater in 1954, can easily be prepared from the more common e- plot. 

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]

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).

Petersen (1965) used the data of Austin and Walker (1963) for the reaction, C + CO2 — 2CO. Use the result of Problem 4.15 and the following data to check the importance of diffusion limitation, first using the Weisz and Prater criterion and then the criterion based on the generalized modulus. 

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