Thiele modulus calculations

The value 1.693 suggests that the internal diffusion should only slightly affect the catalyst effectiveness. From a Thiele-modulus calculation, the effectiveness should be at least 90%. On the other hand, because of rapid acid consumption on down-... 

The analysis of the effects of transport on catalysis has focused on a comparison of the availability of reacting species by diffusion to the rate of reaction on the catalytic sites. High-surface-area catalysts are usually porous. Comparison of transport to reaction rates has usually been based on Knudsen diffusion (by constricted collision with the pore walls) as the dominant mode of transport. DeBoer has noted that for small pores surface diffusion may dominate transport (192). Thiele modulus calculations may therefore not be valid if they are applied to systems where surface diffusion can be significant. This may mean that the direct participation of spillover species in catalysis becomes more important if the catalysts are more microporous. Generalized interpretations of catalyst effectiveness may need to be modified for systems where one of the reactants can spill over and diffuse across the catalyst surface. 

The Thiele modulus, calculated from reaction rate data, diffusion length, and diffusion coefficient, can be used to evaluate internal mass transfer effects. Expressed with the Michaelis constant and maximum reaction velocity, the Thiele modulus is given by... 

The values of Thiele modulus calculated for studied samples (Table 4) are much higher (values of p much lower) as compared with those determined earlier for various Fe-containing catalysts [20, 24, 25]. Thus, in [20] the same equation for p calculation was used as in this work. However, the dependence of Thiele modulus on the particle size was different due to a lower and varied porosity of the AC particles in the present work. Another relation between ( ) and r] was applied in [24] however, in that work more complex kinetic model for the less active catalyst was used. 

Catalyst Effectiveness. Even at steady-state, isothermal conditions, consideration must be given to the possible loss in catalyst activity resulting from gradients. The loss is usually calculated based on the effectiveness factor, which is the diffusion-limited reaction rate within catalyst pores divided by the reaction rate at catalyst surface conditions (50). The effectiveness factor E, in turn, is related to the Thiele modulus,

first-order rate constant, a the internal surface area, and the effective diffusivity. It is desirable for E to be as close as possible to its maximum value of unity. Various formulas have been developed for E, which are particularly usehil for analyzing reactors that are potentially subject to thermal instabilities, such as hot spots and temperature mnaways (1,48,51). 

The relationship between effectiveness factor p and Thiele modulus < >l may be calculated for several other regular shapes of particles, where again the characteristic dimension of the particle is defined as the ratio of its volume to its surface area. It is found that... 

It depends only on J sJkj A, which is a dimensionless group known as the Thiele modulus. The Thiele modulus can be measured experimentally by comparing actual rates to intrinsic rates. It can also be predicted from first principles given an estimate of the pore length =2 . Note that the pore radius does not enter the calculations (although the effective diffusivity will be affected by the pore radius when dpore is less than about 100 run). 

Using this definition of the Thiele modulus, the reaction rate measurements for finely divided catalyst particles noted below, and the additional property values cited below, determine the effectiveness factor for 0.5 in. spherical catalyst pellets fabricated from these particles. Comment on the reasons for the discrepancy between the calculated value of rj and the ratio of the observed rate for 0.5 in. pellets to that for fine particles. 

The pseudo Thiele modulus may be calculated from equation A. 

Fig. 5 Comparison of phase diagrams calculated for the melt of a proteinlike heteropolymer (b) with the phase diagram of a Markovian copolymer according to criterion II (a) and criterion I (c). Proteinlike heteropolymer consisting of / = 103 units is obtained for polymeranalogous reaction in a homopolymer globule at the value of the Thiele modulus h equal to 35... 

For comparison reasons, the results derived from the simulation were additionally calculated by means of the Thiele modulus (Equation 12.12), i.e., for a simple first-order reaction. The reaction rate used in the model is more complex (see Equation 12.14) thus, the surface-related rate constant kA in Equation 12.12 is replaced by... 

The Thiele modulus and the effectiveness factor, respectively, were calculated for the three CO conversions X = 5,40, and 80%. The H2, CO, and H20 gas phase concentrations as well as the respective H2 concentration at the gas-wax phase boundary were taken from Table 12.3. The value of the diffusion coefficient Dm, is listed in Table 12.1. 

The effectiveness factors calculated by the Thiele modulus as well as the findings obtained from the simulation are shown in Table 12.4. 

Comparison of the Effectiveness Factors Modeled with Presto Kinetics and Calculated by Means of the Thiele Modulus Lpore = 6.75-10-4 m -%... 

In Figure 12.4, we clearly see that the effective reaction rate is smaller in the simulation than that calculated by the Thiele modulus, which causes a higher effectiveness factor (see also Equations 12.11 and 12.20). 

FIGURE 12.4 Concentration profiles of H2 in a wax-filled pore (Lpore = 6.75-1CH m - dparticie.cyi = Lpore-4 = 2.7-10-3 m) at a CO conversion of 40% modeled with Presto Kinetics and calculated by the Thiele modulus. 

The data derived from modeling at different conversion degrees (X = 5, 40, and 80%) were also compared to the results obtained from the calculation of the classical Thiele modulus. The calculated (by the Thiele modulus) and modeled (by Presto Kinetics) effectiveness factors showed comparable values. Hence, the usage of simulation software is not required to get a first impression of the diffusion limitations in a Fischer-Tropsch catalyst pore. Nevertheless, modeling represents a valuable tool to better understand conditions within a catalyst pore. 

The presence (or absence) of pore-diffusion resistance in catalyst particles can be readily determined by evaluation of the Thiele modulus and subsequently the effectiveness factor, if the intrinsic kinetics of the surface reaction are known. When the intrinsic rate law is not known completely, so that the Thiele modulus cannot be calculated, there are two methods available. One method is based upon measurement of the rate for differing particle sizes and does not require any knowledge of the kinetics. The other method requires only a single measurement of rate for a particle size of interest, but requires knowledge of the order of reaction. We describe these in turn. 

This example illustrates calculation of the rate of a surface reaction from an intrinsic-rate law of the LH type in conjunction with determination of the effectiveness factor (rj) from the generalized Thiele modulus (G) and Figure 8.11 as an approximate representation of the 7]-Q relation. We first determine G, then 17, and finally (—rA)obs. 

Equation 21.3-16 may be used to convert rate constants from a mass to a particle volume basis for calculation of the Thiele modulus (eg., equation 8.5-20b). In this chapter, (—rA) without further designation means ( -rA)m. 

The Thiele modulus is also calculated and explained in the program. 

For reasons of simplicity, the Thiele modulus will be defined and calculated for a catalyst plate with pore access at both ends of the plate and not at the bottom or top. Note that for most cases in real-life applications the assumptions have to be modified using polar coordinates for the calculations. The Thiele modulus q> is therefore defined as the product of the length of the catalyst pore, /, and the square root of the quotient of the constant of the speed of the reaction, k. divided by the effective diffusion coefficient DeS ... 

A major part of the fast coke is probably desorbed from the catalyst bed and burned in the gas phase. Even if none of the fast coke was desorbed, a calculation of the Thiele modulus tj for conditions in the plume burner and the top of the first zone of the kiln shows that rj is in the range 0.92-0.99. Thus, the fast coke can be assumed to bum without significant diffusion limitation. 

Calculation of the internal effectiveness factor for spherical pellets and fust order reaction The Thiele modulus is... 

Then the classical Thiele modulus ( ) and the effectiveness factor (t/), expressing the extent of internal diffusion limitations in the catalytic washcoat layer of thickness 8, can be calculated according to (cf. Aris, 1975 Froment and Bischoff, 1979, 1990)... 

The value of the Thiele modulus is calculated from Equation 7.21. 

A great deal of attention has been devoted to this topic because of the interesting and often solvable mathematical problems that it presents. Results of such calculations for isothermal zero-, first-, and second-order reactions in uniform cylindrical pores are summarized in Figure 17.6. The abscissa is a modified Thiele modulus whose basic definition is... 

The effectiveness factor for the slab model may also be calculated for reactions other than first order. It turns out that when the Thiele modulus is large the... 

When the Thiele modulus is large Cam is effectively zero and the maximum difference in temperature between the centre and exterior of the particle is (- AH)DeCAJke. Relative to the temperature outside the particle this maximum temperature difference is therefore 0. For exothermic reactions 0 is positive while for endothermic reactions it is negative. The curve in Fig. 3.6 for 0 = 0 represents isothermal conditions within the pellet. It is interesting to note that for a reaction in which -AH- 10 kJ/kmol, ke= lW/mK, De = 10 5m2/s and CAa> = 10 1 kmol/m3, the value of Tu - Tx is 100°C. In practice much lower values than this are observed but it does serve to show that serious errors may be introduced into calculations if conditions within the pellet are arbitrarily assumed to be isothermal. 

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