Effectiveness factor for the pellet

A first-order chemical reaction occurs isothermally in a reactor packed with spherical catalyst pellets of radius R. If there is a resistance to mass transfer from the main fluid stream to the surface of the particle in addition to a resistance within the particle, show that the effectiveness factor for the pellet is given by ... 

In the present work, it has been conservatively assumed that the change in particle size from 50 um to 1 ym does not modify the overall kinetic rate expression given by eq, (3)o As a consequence, the effectiveness factor for the pellet is determined by the diffusional resistance in the macropores formed by the spacing between the particleso... 

ILLUSTRATION 12.3 DETERMINATION OF CATALYST EFFECTIVENESS FACTOR FOR THE CUMENE CRACKING REACTION USING THE EFFECTIVE DIFFUSIVITY APPROACH Use the effective diffusivity approach to evaluate the effectiveness factor for the silica-alumina catalyst pellets considered in Illustration 12.2. 

For a more detailed analysis of measured transport restrictions and reaction kinetics, a more complex reactor simulation tool developed at Haldor Topsoe was used. The model used for sulphuric acid catalyst assumes plug flow and integrates differential mass and heat balances through the reactor length [16], The bulk effectiveness factor for the catalyst pellets is determined by solution of differential equations for catalytic reaction coupled with mass and heat transport through the porous catalyst pellet and with a film model for external transport restrictions. The model was used both for optimization of particle size and development of intrinsic rate expressions. Even more complex models including radial profiles or dynamic terms may also be used when appropriate. 

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

The main objective of the pellet model is to calculate the effectiveness factors for the six reactions that take place inside the reactor. These factors are defined as the ratios of the actual rates occurring inside the pellet and the rates occurring in the bulk phase, i.e., inside the pellet when diffusional resistances are negligible. 

The concentration and temperature profiles are calculated from the above non-linear equations using the Broyden quasi-Newton method. The effectiveness factors for the catalyst pellet may be expressed as... 

Based on the studies on the KD306-type sulfur-resisting methanation catalyst, the non-isothermal one-dimensional and two-dimensional reaction-diffusion models for the key-components have been established, which were solved using an orthogonal collocation method. The simulation values of the effectiveness factors for the methanation reaction ch4 and the shift reaction fco2are in fair agreement with the experimental values, which indicates that both models are able to predict intraparticle transport and reaction processes within catalyst pellets. 

To give an example, Fig. 20 shows a diagram for E = 10 and various selected values of Kp fi. The dashed lines indicate the range over which multiple steady states of t/(ip) occur. Here, by means of numerical methods it is not possible to determine a unique solution of the effectiveness factor of the pellet for given conditions at the external pellet surface [91]. Which operating point will be observed in a real situation again depends upon the direction from which the stationary state is approached [91]. 

An experimental test to verify the absence of significant concentration gradients inside the catalyst pellet is based on the inverse proportional relation between the effectiveness factor and the pellet diameter for strong internal diffusion limitations. Hence, a measured rate which is independent of the pellet size indicates that internal diffusion limitations can be neglected. Care should be taken to avoid artifacts. External heat transfer effects also depend on pellet size and for exothermic reactions might compensate the internal diffusion limitations. If the catalyst pellet consists of a support with an non-uniformly distributed active phase, crushing and sieving to obtain smaller pellets is hazardous. 

When gum formation proceeds, the minimum temperature in the catalyst bed decreases with time. This could be explained by a shift in the reaction mechanism so more endothermic reaction steps are prevailing. The decrease in the bed temperature speeds up the deactivation by gum formation. This aspect of gum formation is also seen on the temperature profiles in Figure 9. Calculations with a heterogenous reactor model have shown that the decreasing minimum catalyst bed temperature could also be explained by a change of the effectiveness factors for the reactions. The radial poisoning profiles in the catalyst pellets influence the complex interaction between pore diffusion and reaction rates and this results in a shift in the overall balance between endothermic and exothermic reactions. 

Both catalysts were activated at the optimum conditions determined using TPR. The rates at the maximum selectivity to benzyl alcohol were compared. In the presence of particulate catalyst the rate amounted to 0.0009S mol/(gNi./ min), while for monolithic catalyst the rate was approximately 0.00175 mol/(gNi min), i.e., about two times more. The diffusion length in the nickel monolith is much shorter than in the 3.2effectiveness factor for the nickel pellets, and hence in a lower reaction rate. Selectivity of both catalysts with respect to benzyl alcohol was nearly the same, at least within the precision of analytical methods used 94.9% for pelleted catalyst and 95.1% for monoliAic catalyst. We may therefore conclude that the selectivity is not controlled by internal diffusion but by the surface properties of the catalysts. 

Problems with multiple steady states are interesting to solve numerically. Computational effort required for solving these problems can be highly demanding. Multiple steady states in a rectangular catalyst pellet were analyzed in example 3.2.2, 3.2.5 and 3.2.9. One has to provide an approximate solution or a guess value to predict the three multiple solutions. It is difficult to predict the effectiveness factor of the pellet as a function of O or y using the numerical approaches described earlier in this chapter. In the next example, this boundary value problem will be solved as an initial value problem. 

Example 11-10 From the experimental data jgmn in Example 11-9 evaluate the effectiveness factor for the catalyst pellet. Also predict 4, using the results of Weisz and Hicks. ... 

Test the suitability of the proposed rate equation and evaluate the constants and K. Also calculate effectiveness factors for the -in. pellets. The following additional data are available ... 

P12 12 Derive the eoncentration profile and effectiveness factor for cylindrical pellets 0.2 cm in diameter and 1.5 cm in length. Neglect diffusion through the ends of the pellet. 

These effects will be reflected in the effectiveness factor for the catalyst pellet. Some of the deposits can obviously affect both the intrinsic kinetics as well as the diffusivities e.g. when deposits occur on the active site (thus decreasing the intrinsic kinetics) then grow and block the pore thus affecting the diffusivities as well (Beeckman et ai, 1978 Beeckman and Froment, 1979, 1982). 

Effectiveness factor for the catalyst pellets of ammonia converters... 

The effectiveness factors for the o-Xylene consumption and phthalic anhydride production are shown in Figures 6.38 and 6.39 which show values very close to unity. The values of the effectiveness factor increase slowly along the length of the reactor which indicate that a small difference between the catalyst pellet surface and the bulk conditions do exist. From the figures, effectiveness factor increases as the oXylene feed mole fraction increases which was also noted in the single pellet parametric investigation in chapter 5. 

The integration of equation (19) alone leads to the following expression for effectiveness factor for spherical pellet ... 

Figure I Correlation of effectiveness factors for the 6.3-mm pellets at 207 bars. The solid line is calculatedfrom the data for the 0.4-mm particles by the method of Bischoff. A value of 7.2 was used for t (from Brown and Bennett [97]). 

In engineering work it is desirable to express the total reaction rate in a catalyst pellet in terms of the surface conditions alone. For this reason, the effect of the concentration and temperature profiles within the particle are incorporated in a single quantity, the effectiveness factor. For the case of a single reaction, the effectiveness factor is defined as ... 

Intraparticle effectiveness factors for either f, f" have the same functional form, because of the assumption of isotropy The effectiveness factor for spherical pellets is ... 

What we now wish to do is derive an effectiveness factor for the biofilm to help us establish conditions for maximum removal efficiency. The procedure is the same as that used for the catalyst pellet (Illustration 4.10). Equation 426d is integrated twice to obtain the concentration profile in the film Q = /(x), which is then differentiated to obtain the flux into the biofilm. The result, the derivation of which is foimd in Practice Problem 4.23, is given by... 

For the nonisothermal pellet being considered, /c, in Eq. 5.38 has been replaced by ka, i.e., the rate constant evaluated at the conditions at the boundary. With the assumption of negligible external mass transfer resistance, C, has also been replaced by the bulk fluid concentration Ct. The internal effectiveness factor for the fresh inner core is rewritten as ... 

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