Particle diameter Thiele modulus

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. 

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 ... 

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... 

We observe that as the particle diameter becomes very small. decreases, so that the effectiveness factor approaches 1 and the reaction is surface-reaction-limited. On the other hand, when the Thiele modulus < ) is large (- 30), the internal effectiveness factor v) is small (i.e., < 1), and the reaction is diffu-... 

We also know that for small diameters of the particles, the efficiency is equal to 1. While for larger particle sizes, the efficiency is inversely proportional to the Thiele modulus ... 

If the particle diameter is small, the effectiveness factor is practically independent of Thiele modulus, i.e., = 1, as we have seen earlier. In this case, the pore diffusion does not affect the rate and the resistance is due to the chemical reaction surface, which is the limiting step of the process. It is known that the prodnct kiai depends on the particle diameter and diffusion, which are usnally represented by Sherwood number. Thus,... 

This means that if the particle diameter is small, the resistance to mass transfer in liquid phase becomes negligible. On the other hand, when analyzing the same expressions for particle diameters is larger, we see that the effectiveness factor decreases and the Sherwood number increases. Knowing that the effectiveness factor is inversely proportional to the Thiele modulus and Sherwood number is directly proportional to the diameter of the particle, we obtain such combined effects. Therefore ... 

Figure 4.36. Graphical representation of the concept of effectiveness factors (a) The effectiveness factor of reaction, rj, as a function of the Damkoehler number of the second kind, Dan, or of the Thiele modulus, (j) (cf. Equ. 4.74). (b) The effective reaction rate, r ff, as a function of the diameter of the particle, d. Part (b) can be used to obtain the value of for a given average diameter, of a population of floes with a distribution d. The range of validity of kinetic control (r js) and of diffusion control (Wfds) is indicated in part a.

These numbers reveal that the (intrinsic) alkylation reaction is exceedingly fast. Using the value of 10- cm /s for the diffusivity, it was found that the generalized Thiele modulus will be unity at 100 for a particle with a 3.1 pm diameter. This diameter is approximately the same as that of the USHY single crystals Thus it can be seen that, practically, the reaction can not be run at 100 °C in the liquid phase without diffusion limitations masking the intrinsic kinetics. 

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]. 

A low effectiveness factor — see Figure 3.23 — implies that the effectiveness factor and thus the effective rate for the steam reforming of methane is inversely proportional to the Thiele modulus [199] and hence the equivalent particle diameter assuming that the particle is isotherm. For a first-order equilibrium rate expression, a general effectiveness factor can be evaluated as shown in [199] [389]. For a large equilibrium constant, this equation can be simplified to ... 

Figure 4.5.26 shows the radial concentration distribution in a porous spherical particle with diameter 2tp according to Eq. (4.5.115) for two values of the Thiele modulus ( reversible fot the example of a gas phase free of B (cB,g = 0) and Defr/O tp) = 0.05 and K = l. Note that in the case of high values of ( reversible (S>5 in Figure 4.5.26), the external mass transfer determines the effective reaction rate, that is, the equilibrium concentrations are almost reached within the porous particle (for the example of Figure 4.5.26, K<, = 1 and Ca,equilibrium = Cb,equilibrium = 0.5c J, and the concentrations vary strongly in the boundary layer, for example,...

Figure 4.6.4 clearly shows that by the measurement of the conversion of the solid with time (the dimensionless time t/tfin) we can decide whether the rate is controlled by diffusion through the solid product layer. However, we see that it is hard to distinguish between the other two cases depicted in Figure 4.6.4, and hence we then need more experimental data and further calculations. For example, the variation of particle size is helpful, as the final time for conversion is proportional to the initial particle diameter for control by the chemical reaction, Eq. (4.6.30), whereas tfin Tp if a gaseous product is formed and the rate is controlled by external diffusion, Eq. (4.6.35). Additional calculations are also helpful, for example, to estimate the Thiele modulus or to compare the measured rate constant and the expected value, if external diffusion alone limits the rate. 

Assume that the reaction and diffusion model is applicable, as presented in section 5,4,3.1, When the Thiele modulus cp > 3, only a fraction of l/(p of the porous particle, close to the outer surface, will take part in the reaction. As the solid is converted to a solid product with a larger volume, the pore diameter will decrease. The effective diffusivity is thereby reduced, so that the thickness of the reaction zone is also reduc. The higher the reaction rate constant, the higher the Thiele modulus was at the beginning, and the sooner the pore mouth will be plugged. A rough estimate indicates that the maximum degree of conversion of the solid reactant will be smaller than the effectiveness factor, which is tank (p/(p. 

It is considered to carry out the same reaction in an upflow column, with a catalyst of the same material, but with a diameter of 4 mm. Estimate the required reaction volume, for a mean residence time of the liquid phase of 30 minutes (it is expected that the conversion will be the same as in the batch reactor). The Thiele modulus is proportional to the particle diameter, so for the larger catalyst particles is approximately 16, and the effectiveness factor is 1/16 (see eqs. (5.48) and (5.50)). This means that the required catalyst volume is 16 times larger, that is 16 x 0.1 x 10 = 16 m. When the bed has a void fraction of 0.5, die total effective reactor volume has to be 32 m. But this is only correct if the process rate is still determined by chemical kinetics. The gas/liquid mass transfer would be a possible limiting factor. One can make the following estimate The bubble hold-up in a stirred tmik and in an upflow column will both be on the order of 0.2. The bubble diameter will be on the order of 1 mm in the stirred tank, and 2 mm in the upflow column (with particles of 4 mm). 

As discussed above, the transport properties of porous catalyst particles of ca 3 to 100 pm are extremely important for the selectivity of catalytic reactions in which the desired initial products are liable to further reaction to undesired material. The ratio of the rate of catalytic reaction to that of transport within the pore system of catalyst particles is represented by Thiele s modulus [1], which is proportional to the pore length and to the square root of the diameter of the pores. Accordingly reducing the size of the catalyst particles is more elfective than increasing the diameter of the pores. 

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