Thiele modulus constant

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 result is shown in Figure 10, which is a plot of the dimensionless effectiveness factor as a function of the dimensionless Thiele modulus ( ), which is R.(k/Dwhere R is the radius of the catalyst particle and k is the reaction rate constant. The effectiveness factor is defined as the ratio of the rate of the reaction divided by the rate that would be observed in the absence of a mass transport influence. The effectiveness factor would be unity if the catalyst were nonporous. Therefore, the reaction rate is... 

Figure 10 shows that Tj is a unique function of the Thiele modulus. When the modulus ( ) is small (- SdSl), the effectiveness factor is unity, which means that there is no effect of mass transport on the rate of the catalytic reaction. When ( ) is greater than about 1, the effectiveness factor is less than unity and the reaction rate is influenced by mass transport in the pores. When the modulus is large (- 10), the effectiveness factor is inversely proportional to the modulus, and the reaction rate (eq. 19) is proportional to k ( ), which, from the definition of ( ), implies that the rate and the observed reaction rate constant are proportional to (1 /R)(f9This result shows that both the rate constant, ie, a measure of the intrinsic activity of the catalyst, and the effective diffusion coefficient, ie, a measure of the resistance to transport of the reactant offered by the pore stmcture, influence the rate. It is not appropriate to say that the reaction is diffusion controlled it depends on both the diffusion and the chemical kinetics. In contrast, as shown by equation 3, a reaction in solution can be diffusion controlled, depending on D but not on k. 

Diffusion effects can be expected in reactions that are very rapid. A great deal of effort has been made to shorten the diffusion path, which increases the efficiency of the catalysts. Pellets are made with all the active ingredients concentrated on a thin peripheral shell and monoliths are made with very thin washcoats containing the noble metals. In order to convert 90% of the CO from the inlet stream at a residence time of no more than 0.01 sec, one needs a first-order kinetic rate constant of about 230 sec-1. When the catalytic activity is distributed uniformly through a porous pellet of 0.15 cm radius with a diffusion coefficient of 0.01 cm2/sec, one obtains a Thiele modulus y> = 22.7. This would yield an effectiveness factor of 0.132 for a spherical geometry, and an apparent kinetic rate constant of 30.3 sec-1 (106). 

Currently, benzene alkylation to produce ethylbenzene and cumene is routinely carried out using zeohtes. We performed a study comparing a zeohte Y embedded in TUD-1 to a commercial zeolite Y for ethylbenzene synthesis. Two different particle diameters (0.3 and 1.3 mm) were used for each catalyst. In Figure 41.7, the first-order rate constants were plotted versus particle diameter, which is analogous to a linear plot of effectiveness factor versus Thiele modulus. In this way, the rate constants were fitted for both catalysts. 

In terms of the equilibrium constant for the reaction, the Thiele modulus becomes... 

Since the factor (K + 1 )/K is always greater than unity, Lrev will always be greater (and r less) than the corresponding Thiele modulus for the forward reaction alone, other conditions remaining constant. 

When the effectiveness factors for both reactions approach unity, the selectivity for two independent simultaneous reactions is the ratio of the two intrinsic reaction-rate constants. However, at low values of both effectiveness factors, the selectivity of a porous catalyst may be greater than or less than that for a plane-catalyst surface. For a porous spherical catalyst at large values of the Thiele modulus s, the effectiveness factor becomes inversely proportional to (j>S9 as indicated by equation 12.3.68. In this situation, equation 12.3.133 becomes... 

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... 

Keeping the concentration ratio of H20 and CO in the simulation model constant (according to the Thiele modulus see Equation 12.21) leads to equal concentration profiles of H2, as shown in Figure 12.4, and consequently to equal effectiveness factors for both methods (Thiele modulus and simulation). In fact, the concentrations of H2, CO, and H20 change inside the pore, as considered in the simulation. Therefore, the results obtained by the software used represent reality best. 

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. 

Note that if ( -rA) or a corresponding rate constant kA is specified on a mass basis, kA must be multiplied by tbe particle density to obtain tbe form of the rate constant to be used in the Thiele modulus that is, fcA = (kA)p = pp(ka)m, which follows from equation 21.3-16. 

When the various constants are known, the Thiele modulus is used in the form... 

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 ... 

Thus the rate of reaction in a porous catalyst is equal to the rate if the concentration had remained constant at Cah multiplied by rj, r" = k"CAs - We have defined the dimensionless quantify (j) which is called the Thiele modulus. This function is plotted on linear and log-log... 

Xi+ and X2+ are diffusion-disguised rate constants 4>i = Thiele modulus R = effective crystallite radius (apparent agglomerate size) (Figure 3)... 

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 ... 

Calculate the isothermal effectiveness factor rj for the porous catalyst pellet in problem 1 as a function of the Thiele modulus d> for the first reaction A —> B utilizing the fact that the rate constant of the second reaction B —> C is half the rate constant of A —> B, the pellet is isothermal, and the external mass transfer resistance is negligible. 

Once the theoretical curves have been fitted (Figs. 2.10 and 2.11), it is possible to plot the concentration profiles of all the species included in the model and to determine the optimum thickness of the enzyme layer (Fig. 2.12). Because the Thiele modulus is the controlling parameter in the diffusion-reaction equation, it is obvious from (2.22) that the optimum thickness will depend on the other constants and functions included in the Thiele modulus. For this reason, the optimum thickness will vary from one enzyme and one kinetic scheme to another. 

To have a quantitative idea of the problem of intraparticle diffusion, effectiveness factors for the two catalysts were calculated from the observed second order rate constants (based on surface area) using the "triangle method" suggested by Saterfield (4). The effectiveness factors for Monolith and Nalcomo 474 catalysts on Synthoil liquid at 371°C (700 F) were calculated to be 0.94 and 0.216, respectively. In applying the relationship between the "Thiele Modulus," 4>> and the "effectiveness factor," n> the following simplifying assumptions were made ... 

The difficulty in determining the Thiele modulus is that the intrinsic rate constant (without diffusion limitations) is required, while the Thiele modulus is used to determine if diffusion limitations are present in the first place. This problem can be avoided by using the iterative procedure described by van Donk et al39, where the observed rate constant can be used for determination of the Thiele modulus and effectiveness. 

Table 7 the effective diffusion De, Thiele modulus < ) and effectiveness T as a function of support at 75°C. For the observed rate constant k0bs the most active catalyst for each support material was taken. 

However, in many practical situations the problem exists that effective rate constants and activation energies have been derived on the basis of laboratory experiments. The question arises then as to whether or not these parameters arc influenced by transport effects. With the relations given so far, this question cannot be answered yet, since according to its definition by cq 27 the Thiele modulus is based on the intrinsic rate constant k. This problem can be solved by introducing a new modulus, which in contrast to only contains observable (effective) quantities, and thus can... 

However, the boundary condition at the external pellet surface is now defined by eq 36 instead of eq 38. As a consequence, a different expression for the integration constant C results, which is not only a function of the Thiele modulus , but also depends on the Biot number for mass transport Bim. Hence, a complete characterization of this problem already requires two parameters. 

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. 

This effect will be particularly emphasized at small values of the Thiele modulus where the intrinsic rate of reaction and the effective rate of diffusion assume the same order of magnitude. At large values of , the effectiveness factor again becomes inversely proportional to the Thiele modulus, as observed under isothermal conditions (Section 6.2.3.1). Then the reaction takes place only within a thin shell close to the external pellet surface. Here, controlled by the Arrhenius and Prater numbers, the temperature may be distinctly higher than at the external pellet surface, but constant further towards the pellet center. 

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