Thiele modulus reactors

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). 

Example 5 Application of Effectiveness For a second-order reaction in a plug flow reactor the Thiele modulus is ( ) = SVQ, and inlet concentration is C50 = 1.0. The equation will he integrated for 80 percent conversion with Simpsons rule. Values of T) are... 

Estimate the Thiele modulus and the effectiveness factor for a reactor in which the catalyst particles are ... 

Microlevel. The starting point in multiphase reactor selection is the determination of the best particle size (catalyst particles, bubbles, and droplets). The size of catalyst particles should be such that utilization of the catalyst is as high as possible. A measure of catalyst utilization is the effectiveness factor q (see Sections 3.4.1 and 5.4.3) that is inversely related to the Thiele modulus (Eqn. 5.4-78). Generally, the effectiveness factor for Thiele moduli less than 0.5 are sufficiently high, exceeding 0.9. For the reaction under consideration, the particles size should be so small that these limits are met. 

The reactor feed mixture was "prepared so as to contain less than 17% ethylene (remainder hydrogen) so that the change in total moles within the catalyst pore structure would be small. This reduced the variation in total pressure and its effect on the reaction rate, so as to permit comparison of experiment results with theoretical predictions [e.g., those of Weisz and Hicks (61)]. Since the numerical solutions to the nonisothermal catalyst problem also presumed first-order kinetics, they determined the Thiele modulus by forcing the observed rate to fit this form even though they recognized that a Hougen-Watson type rate expression would have been more appropriate. Hence their Thiele modulus was defined as... 

This approach is analytically correct for isothermal reactors and first-order rate laws, since concentration does not appear in the expression for the Thiele modulus. For other (nonlinear) rate laws, concentration changes along the reactor affect the Thiele modulus, and hence produce changes in the local effectiveness factor, even if the reaction is isothermal. Problem 21-15 uses an average effectiveness factor as an approximation. 

A second order reaction takes place in a flow reactor with catalyst particles in the shape of lamellae. At the inlet the concentration is 2 lbmol/cuft and the Thiele modulus is

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Feed rate is 100 cfh, inlet concentration is 0.5 lbmol/cuft and the Thiele modulus at the inlet of the reactor is 15. Find Vr/V for 90% conversion. 

The concentration and temperature Tg will, for example, be conditions of reactant concentration and temperature in the bulk gas at some point within a catalytic reactor. Because both c g and Tg will vary with position in a reactor in which there is significant conversion, eqns. (1) and (15) have to be coupled with equations describing the reactor environment (see Sect. 6) for the purpose of commerical reactor design. Because of the nonlinearity of the equations, the problem can only be solved in this form by numerical techniques [5, 6]. However, an approximation may be made which gives an asymptotically exact solution [7] or, alternatively, the exponential function of temperature may be expanded to give equations which can be solved analytically [8, 9]. A convenient solution to the problem may be presented in the form of families of curves for the effectiveness factor as a function of the Thiele modulus. Figure 3 shows these curves for the case of a first-order irreversible reaction occurring in spherical catalyst particles. Two additional independent dimensionless paramters are introduced into the problem and these are defined as... 

Thiele modulus, useful for predicting reactor behavior from known kinetic information, thus known k" ... 

A certain spherical porous catalyst with a pellet diameter of 1/8 in. has a Thiele modulus of 0.5 for a first-order reaction and gives 90% conversion in a packed bed reactor. It is proposed to... 

The catalyst in the previous problem had a Thiele modulus of 0.1. A new catalyst was found to have a Thiele modulus of 5 with all other properties unchanged. What will be the conversion in the above reactor with this new catalyst What will be the appropriate kinetics ... 

By a reactor model, we mean a system of equations (algebraic, ordinary, or partial differential, functional or integral) which purports to represent a chemical reactor in whole or in part. (The adequacy of such a representation is not at issue here.) It will be called linear if all its equations are linear and simple if its input and output can be characterized by single, concentration-like variables, Uo and u. The relation of input and output will also depend on a set of parameters (such as Damkohler number. Thiele modulus, etc.) which may be denoted by p. Let A(p) be the value of u when w0 = 1. Then, if the input is a continuous mixture with distribution g(x) over an index variable x on which some or all of the parameters may depend, the output is distributed as y(x) = g(x)A(p(jc)) and the lumped output is... 

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 ... 

We will now consider the design of an agitated tank slurry reactor which might be used industrially for this reaction. We will choose a particle size of 100 fim rather than the very small size particles used in the laboratory experiments, the reason being that industrially we should want to be able easily to separate the catalyst particles from the liquid products of the reaction. If we use spherical particles (radius ro, diameter dp), the Thiele modulus (see Chapter 3) is given by ... 

In this section we have presented and solved the BVPs associated with the diffusion and reaction that take place in the pores of a porous catalyst pellet. The results were expressed graphically in terms of the effectiveness factor rj versus the Thiele modulus d> for two cases One with negligible external mass and heat transfer resistances, i.e., when Sh and Nu —> oo, and another with finite Sh and Nu values. This problem is very important in the design of fixed-bed catalytic reactors. The sample results presented here have shown that for exothermal reactions multiple steady states may occur over a range of Thiele moduli d>. Efficient numerical techniques have been presented as MATLAB programs that solve singular two-point boundary value problems. 

Fig. 7.17. Extension of cycle times [t) due to microstructuring of hybrid catalyst-adsorbent pellets for the Claus and water-gas shift reactions compared to the use of comparable distinct catalyst and adsorbent pellets as a function of the Thiele modulus (< )) and Stanton number (St) [52]. In the water-gas shift reaction preloaded adsorbent is used to enhance the level of excess steam both with and without an additional steam supply in the reactor inlet.

Figure 5 shows the dependence of the effectiveness factor on the Thiele modulus for the different pellet shapes. At small values of 4> the effectiveness factor approaches unity in all cases. Here, the chemical reaction constitutes the rate determining step—the corresponding concentration profiles over the pellet cross-section arc flat (sec Fig. 4). This situation may occur at low catalyst activity (k is small), large pore size and high porosity (Dc is large), and/or small catalyst pellets (R is small, i.c. in fluidized bed reactors R is typically around 50 /im).

Figure 30. Simulated para selectivity y°ul at the reactor outlet as a function of the isomerization rate constant Ari. Dependence on the Thiele modulus of the disproportionation reaction F.

The PFR is efficient for screening solid catalyst in a single fluid phase. It can also be used in later research stages to assess commercial criteria. Consider the evaluation of the ultimate commercial performance of a newly developed fixed-bed catalyst. The theory of similarity teaches that for the laboratory and the industrial reactor, the Damkohler number (NDa), the Sherwood number (Nsh), and the Thiele modulus (<)>) need to be kept constant (Figure 2). As a result, the laboratory reactor must have the same length as the envisioned commercial reactor (7). In this case, scale up is done by increasing the diameter of the reactor. This example further illustrates that laboratory reactors are not necessarily small in size. 

A prominent trade-off in fixed-bed reactor design concerns the catalyst particle size. What is the basis for the choice of a certain particle size When the catalyst performance is to be optimized, the application of the Thiele model helps to provide an answer (Figure 7). The Thiele modulus accounts for the competition between the chemical reaction and the limitation of transport of reactants by diffusion in a porous catalyst particle. It is defined as the square root of the ratio of the characteristic diffusion time fo = L /D and the characteristic reaction time (r. For a... 

Recommendations For large particles (dp > 3 mm) the correlations of Sato et al.74 and Lemay et al.50 should be useful. For Gc < 0.01 g cm-2 s Eq. (6-62) would also be useful. For smaller particles, the relations given by Goto and Smith34 and Goto et al.35 would be useful. Under trickle-flow conditions, it, however, appears that the gas -liquid or liquid-solid mass-transfer resistances are less important than the intraparticle resistances.78 Specchia et al.91 showed that gas-liquid and liquid -solid mass-transfer resistances in a trickle-bed reactor can be neglected if the Thiele modulus for the catalyst is less than unity. 

If the catalyst is dispersed throughout the pellet, then internal diffusion of the species within the pores of the pellet, along with simultaneous reaction(s) must be accounted for if the prevailing Thiele modulus > 1. This aspect gives rise to the effectiveness factor" problem, to which a significant amount of effort, summarized by Aris ( ), has been devoted in the literature. It is important to realize that if the catalyst pellet effectiveness factor is different from unity, then the packed-bed reactor model must be a heterogeneous model it cannot be a pseudohomogeneous model. 

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