The Weisz Modulus

The problem of evaluating the influence of pore diffusion on an experimental result can be simplified through some transformations of the previous equations. Suppose that a reaction rate has been measured in some kind of experimental reactor, preferably an ideal CSTR or a differential PFR. From the experimental data, a rate of reaction per unit of geometrical catalyst volume, designated —Ra,y, can be calculated. The v in the subscript indicates that this is a volumetric raLtc of reaction. The measured rate (—Ra.v) is not necessarily the same as the intrinsic rate, expressed on a volumetric basis (—rA,v)-The measured rate may reflect internal transport effects, whereas the intrinsic rate does not. 

The measured rate can be expressed in terms of the effectiveness factor. 

Since —Ra, is the measured rate of reaction, the value of can be calculated directly from experimental data, as long as the other parameters in Eqn. (9-30) are known or can be estimated. It is not necessary to know the value of the intrinsic rate constant ky in order to calculate 

The use of the Weisz modulus to estimate the effectiveness factor directly from experimental data is illustrated in the following example. 

Satterfield et al. studied the liquid-phase hydrogenation of a-methyl styrene to cumene over a 1 wt.% Pd/alumina catalyst. The catalyst was spherical, with a diameter of 0.825 cm and a porosity e of 0.50. The tortuosity Tp was measured and found to be approximately 8. 

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GP 9[ [R 16]The extent of internal transport limits was analysed for the wide fixed-bed reactor, using experimental data on carbon monoxide conversion and matter and process parameter data for the reactants [78]. The analysis was based on the Weisz modulus and the Anderson criterion for judging possible differences between observed and actual reaction rates. As a result, it was found that the small particles eliminate internal transport limitations. 

Then the Weisz modulus which only includes observables is13... 

Fig. 82 Catalyst effectiveness factor T]p as a function of the Weisz modulus, A r]p DaM, and the reaction order, m.

Comparing eqs 56 and 27, and recalling the definition of the effectiveness factor according to cq 40, yields the following simple relationship between the Thiele modulus and the Weisz modulus ... 

A plot of the effectiveness factor from cq 53 against the Weisz modulus 1ppn from cq 58 gives the curve depicted in Fig. 8 for a first order reaction (flat plate). On the basis of this diagram, the effectiveness factor can be determined easily once the effective reaction rate and the effective diffusivity arc known. 

Figure 9. Effectiveness factor ij as a function of the Weisz modulus iji. Combined influence of intraparticle and interphase mass transfer on the effective reaction rate (isothermal, first order, irreversible reaction in a sphere, Biot number Bim as a parameter).

For practical purposes however, eq 60 again suffers from the disadvantage that the Thiele modulus must be specified in order to calculate the catalyst efficiency. Thus, the intrinsic rate constant must be known. In this situation, instead of directly plotting eq 60, it is more convenient to relate the effectiveness factor to the Weisz modulus, calculated from eq 58. For selected values of the Biot number Bim, such a diagram is given in Fig. 9. 

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. 

Again, eq 75 cannot be used immediately to calculate the overall effectiveness factor, since the modulus fi, which is related to the unknown catalyst temperature, can only be determined when the overall efficiency has been specified (see eqs 71 and 72). Therefore, both sides of eq 74 arc multiplied by 2, resulting in an expression which relates the Weisz modulus ift to the modulus . Then, for a given value of fi, the corresponding value of ij/ is calculated, and from ij/ the unknown catalyst temperature 0S (eq 71). This temperature is substituted into eq 72 to obtain the corresponding value of the Thiele modulus <)>. Dividing ij/ by fi finally yields the overall effectiveness factor which is then plotted against i/f. 

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 ... 

The maximum effective reaction rate is obtained for the limiting value of cs = 0. This means that the product of the effectiveness factor and the second Damkohler number can never exceed unity. A comparison of the definition of the Weisz modulus (eq 56) with the definition of Dan (eq 78) gives the equivalence... 

Figure 18. Effectiveness factor rj of a first-order reversible reaction versus the Weisz modulus ip (related to the forward rate constant k+). Influence of intraparticle diffusion on the effective reaction rate (isothermal reaction in a sphere, equal diffusivitics i,e = Die, equilibrium constant as a parameter).

Roberts and Satterfield [87, 88] analyzed this type of reaction. On the basis of numerical calculations for a flat plate, these authors presented a solution in the form of effectiveness factor diagrams, from which the effectiveness factor can be determined as a function of the Weisz modulus as well as an additional parameter Kp s which considers the influence of the different adsorption constants and effective diffusivities of the various species [91], The constant K involved in this parameter is defined as follows ... 

In Fig. 19, calculated curves of the effectiveness factor versus the Weisz modulus are shown for different values of Kpis [91]. For comparison, this diagram also contains the curves corresponding to the results which apply to simple, irreversible power rate laws of zeroth, first and second order. From this figure it is obvious that a strong adsorption of at least one of the products leads to a similar decrease of the effectiveness factor as it is observed in the case of a reversible reaction. 

Inspection of the curves of the effectiveness factor versus the Weisz modulus for different values of Kp. s and E reveals two interesting phenomena when E > 0 (Fig. 20) [87, 88, 91]. At first, for large values of Ay i.s (10-100) effectiveness factors above unity may occur even though isothermal conditions prevail. This can be explained by the fact that the reaction rate given by eq 103 has a maximum for certain combinations of p and P2. This maximum results from the assumption that the rate is proportional to the concentration of the adsorbed reactants At and A2 which compete for adsorption sites on the active (inner) surface. When, for example, Ai is adsorbed more strongly than A2, then a raised partial pressure of Ai, at constant partial pressure of A2, will lead to a displacement of A2 from the surface, and hence to a lowered reaction rate. By a quantitative analysis, it can be shown that effectiveness factors above unity will appear whenever Kp, is greater than (E + 2)/E [91]. 

The effectiveness factor versus the Weisz modulus according to Kao and Satterfield [61] is shown in Fig. 21 for C = 0.5 and different values of B. From this diagram, a similar behavior is seen as in the case of a simple, first order, reversible reaction (see Fig. 18) with decreasing value of B, the effectiveness factor is reduced. A decline of the effectiveness factor is also observed for a rise of the parameter C, which corresponds to a shift towards the chemical equilibrium, and hence to a reduction of the net reaction rate [91]. 

To compare the change in overall selectivity, eqs 135 and 152 would have to be integrated over a range of conversions. However, the resulting lengthy expressions are not given here. Instead, Fig. 23 shows a plot of the relative point selectivity obtained from eq 153 versus the Weisz modulus... 

The Thiele modulus requires knowledge of the true rate constant, which may not always be available. It is more conveniently defined based on the actual measured rate of a reaction. This is called the Weisz modulus (1954) and, for a general shape with i as the diffusion length, is defined as... 

Diffusional limitations are often analysed through the use of the Weisz modulus, 0, which compares the observed reaction rate to the difiusion rate [18]. When 0 1, the diffusion phenomenon is not significant and the observed reaction rate is equal to the intrinsic reaction rate. When 1, diffusional limitations modify the apparent kinetics, and the observed reaction rate can be very different fi om the intrinsic reaction rate. Since carbon xerogels are composed of two distinct levels, i.e. the pellet level and the microporous nodules level, both with their own pore size and length scale, 0 must be calculated at both levels. This was the object of a complete study [19]. 

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... 

Note that the Thiele modulus used is the generalized Thiele modulus . A shortcoming of the modulus defined by Equation 20.8 is that values of V, and ATm may not always be available. To overcome this practical deficiency, we turn to the Weisz modulus of Chapter 7 based on the observed reaction rate r a which... 

The relationship shown in Equation 2.207 suffers from the fact that the Thiele modulus must be specified to estimate the catalyst efficiency. This is, in general, not possible as the intrinsic kinetics is not known. It is, therefore, more convenient to relate the overall effectiveness factor to the Weisz modulus, which is based only on observable parameters. 

Figure 2.30 Overall effectiveness factor as a function of the Weisz modulus for different mass Biot numbers (isothermal, irreversible first order reaction in a porous slab). (Adapted from Ref. [16], Figure 4.17 Copyright 2012, Wiley-VCH GmbH Co. KGaA.)...

When internal mass transport limitations are negligible (jj = 1), the Weisz modulus is approximately equal to... 

As a result, high values for the Thiele modulus correspond to high values for the Weisz modulus and low values for the Thiele modulus also correspond to low values for the Weisz modulus. It can be shown that for nth-order kinetics, the following expression is obtained for the Weisz modulus (2) ... 

To evaluate the potential of carbon formation in a steam reformer, it is therefore essential to have a rigorous computer model, which contains kinetic models for the process side (reactor), as well as heat transfer models for the combustion side (furnace). The process and combustion models must be coupled together to accurately calculate the process composition, pressure, and temperature profiles, which result from the complex interaction between reaction kinetics and heat transfer. There may also be a temperature difference between bulk fluid, catalyst surface, and catalyst interior. Lee and Luss (7) have derived formulas for this temperature difference in terms of directly observable quantities The Weisz modulus and the effective Sherwood and Nusselt numbers based on external values (8). 

The Weisz modulus [11] allows for the estimation of intraparticle diffusion limitations in packed bed microchannel reactors ... 

The determination of ijpore still leads to the problem that rate constants are measured values and thus the question arises as to whether and to what extent the kinetic parameters are already influenced hy transport processes. With the equations given so far, this question cannot he answered, since the Thiele modulus is defined based on the intrinsic constant and not on km eff- This problem can he solved hy a modulus that contains only km.eff- This modulus is known as the Weisz modulus i/r and is defined for an arbitrary particle shape and order n as ... 

Lee and Luss [1969] derived formulas for the maximum temperature differences between bulk fluid, catalyst surface, and catalyst interior in terms of directly observable quantities the Weisz modulus and the effective Sherwood and Nusselt numbers based on external values. The steady-state mass and heat balances for an arbitrary reaction, using slab geometry, are... 

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