Nonisothermal effectiveness factor

Generation of heat inside a pellet due to reaction and its transport through the pellet can greatly affect the reaction rate. For endothermic reactions there is a fall in temperature within the pellet. As a result, the rate falls, thus augmenting the retarding effect of mass diffusion. On the other hand, for exothermic reactions, there is a rise in temperature within the pellet. This leads to an increase in rate that can more than offset the decrease due to lowered concentration. Thus, the effectiveness factor can actually be greater than 1. 

In analyzing the effect of thermal diffusion within the pellet, the meth- The energy balance within the odological solution involves the solution of equation of continuity and 

We now consider a differential section of a pellet of any shape and write the following continuity equation  

Multicomponent diffusion The mathematical description of diffusion of more than one component is a complex problem and not germane to our subject. However, with the present trend toward increasing use of solid catalysts in chemical synthesis, many situations do arise in which multicomponent diffusion is involved. We left the prediction of the multicomponent diffusivity outside the scope of this book, which can be found elsewhere (Doraiswamy, 2001). Once the multicomponent diffusivity is determined, the effective diffusivity can then be found from Equation 6.7. 

Miscellaneous effects A number of factors can influence the effectiveness factor, some of which are particle size distribution in a mixture of particles/pellets, change in volume upon reaction, pore shape and constriction (such as ink-bottle-type pores), radial and length dispersion of pores, micro-macro pore structure, flow regime (such as bulk or Knudsen), surface diffusion, nonuniform environment around a pellet, dilution of catalyst bed or pellet, distribution of catalyst 

Even when is low, the center and surface temperatures may differ appreciably, because catalyst pellets have low thermal conductivities (Sec. 11-5). The combined effect of mass and heat transfer on can still be represented by the general definition of the effectiveness factor, according to Eq. (11-41). Hence Eq. (11-42) may be used to find r, provided rj is the nonisothermal effectiveness factor. The nonisothermal 17 may be evaluated in the same way as the isothermal 77, except that an energy balance must be combined with the mass balance. 

Consider the same irreversible first-order reaction A B used in Sec. 11 -6 to. obtain the isothermal rj. If the effect of temperature on D is neglected, the differential mass balance and boundary conditions, Eqs. (11-46) to (11-48), are still applicable. The energy balance over the spherical shell of thickness Ar (see Fig. 11-6) is 

The solution of Eqs. (11-46) to (11-48) and (11-72) to (11-74) gives the concentration and temperature profiles within the pellet. A numerical solution is necessary because Eqs. (11-46) arid (11-72) are coupled through the nonlinear dependence of k on temperature k = A Neverthe- 

CHAPTER 11 REACTION AND DIFFUSION WITHIN POROUS CATALYSTS 

If we integrate this equation once, using Eqs. (11-47) and (11-73) for the boundary conditions, and then integrate a second time, using Eqs. (11-48) and (11-74), we obtain  

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FIGURE 10.3 Nonisothermal effectiveness factors for first-order reactions in spherical pellets. (Adapted from Weisz, P. B. and Hicks, J. S., Chem. Eng. Sci., 17, 265 (1962).)... 

Do an order-of-magnitude calculation for the nonisothermal effectiveness factor. 

Figure 18.8 Nonisothermal effectiveness factor curve for temperature variation within the particle. Adapted from Bischoff (1967).

Let us return to the nonisothermal effectiveness factor. Weisz and Hicks solved Eqs. (11-46) and (11-72) numerically to obtain the concentration profile within the pellet. Then r was obtained from Eq. (11-51), which is not limited to isothermal conditions, provided is evaluated at the surface temperature. The results expressed 17 as a function of three dimensionless parameters ... 

Fig. 11-10 Nonisothermal effectiveness factors for first-order reactions in spherical catalyst pellets...

Do an order-of-magnitude calculation for the nonisothermal effectiveness factor. Hint Use the pore model to estimate an isothermal effectiveness factor and obtain from that. Assume k ff = 0.15 J m s K. 

Figure 7.14 Two examples of nonisothermal effectiveness factor results on inter- and intra-phase gradients. [After D.L. Cresswell, Chem. Eng. ScL, 25, 267, with permission of Pergamon Press, Ltd., London, England, (1970).]...

THERMAL ENERGY BALANCE IN MULTICOMPONENT MIXTURES AND NONISOTHERMAL EFFECTIVENESS FACTORS VIA COUPLED HEAT AND MASS TRANSFER IN POROUS CATALYSTS... 

What is the reason for the multiple values for the nonisothermal effectiveness factors What is the reason for the values of effectiveness factors > 1 What physical significance does this bear ... 

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