The Nonisothermal Catalyst Pellet

For the most adroit scaling we shall need the best estimates of the variables that we can get, and in any case, it is important get as much a priori information as possible.14 Some bounds are physically obvious, as when we have a system in which a substance is disappearing by an irreversible reaction and its concentration cannot exceed the value at the inlet. A nice illustration of a less trivial estimate is provided by the nonisothermal catalyst pellet, the equations of which are here transcribed from (Eqs. 112-114 and 123-125) for u(x), the concentration of the reactant, 

We shall use these equations bearing in mind that, unless specifically excepted, all the results are valid for any shape of particle. 

13 Actually, the problem was worked out for chromatography when I was working for Imperial Chemical Industries, but the company had been a bit cautious about letting me publish before and, when I came to write the work up, I made the change to avoid having to submit the paper for permission to publish. 

14 Gavalas was the first to recognize the value of a priori estimates. See his Nonlinear Equations of Chemically Reacting Systems, Heidelberg Springer-Verlag, 1968. 

We observe that the combination w = fiu + v satisfies Laplace s equation. If fi and v are both very large so that we have u = v = 1, then w = 1 + over the whole bounding surface, in this case, x = 1. Now, it is well known that a potential function takes its maximum on the boundary of its domain, and so, if it is constant there, it is the same constant value within. It follows that 

---

There are several factors that may be invoked to explain the discrepancy between predicted and measured results, but the discrepancy highlights the necessity for good pilot plant scale data to properly design these types of reactors. Obviously, the reaction does not involve simple first-order kinetics or equimolal counterdiffusion. The fact that the catalyst activity varies significantly with time on-stream and some carbon deposition is observed indicates that perhaps the coke residues within the catalyst may have effects like those to be discussed in Section 12.3.3. Consult the original article for further discussion of the nonisothermal catalyst pellet problem. 

For the nonisothermal catalyst pellet with negligible external mass and heat transfer resistances, i.e., with Sh —> 00 and Nu —> 00 and for a first-order reaction, the dimensionless concentration and temperature are governed by the following couple of boundary value differential equations... 

Assuming a steady state, for first-order reaction-diffusion system A -> B under nonisothermal catalyst pellet conditions, the mass and energy balances are... 

Effectiveness factors for a first-order reaction in a spherical, nonisothermal catalysts pellet. (Reprinted from R B. Weisz and J. S. Hicks, The Behavior of Porous Catalyst Particles in View of Internal Mass and Heat Diffusion Effects, Chem. Eng. Sci., 17 (1962) 265, copyright 1962, with permission from Elsevier Science.)... 

B under nonisothermal catalyst pellet conditions, the mass and energy... 

The exciting issue of steady-state multiplicity has attracted the attention of many researchers. First the focus was on exothermic reactions in continuous stirred tanks, and later on catalyst pellets and dispersed flow reactors as well as on multiplicity originating from complex isothermal kinetics. Nonisothermal catalyst pellets can exhibit steady-state multiplicity for exothermic reactions, as was demonstrated by P.B. Weitz and J.S. Hicks in a classical paper in the Chemical Engineering Science in 1962. The topic of multiplicity and oscillations has been put forward by many researchers such as D. Luss, V. Balakotaiah, V. Hlavacek, M. Marek, M. Kubicek, and R. Schmitz. Bifurcation theory has proved to be very useful in the search for parametric domains where multiple steady states might appear. Moreover, steady-state multiplicity has been confirmed experimentally, one of the classical papers being that of A. Vejtassa and R.A. Schmitz in the AIChE Journal in 1970, where the multiple steady states of a CSTR with an exothermic reaction were elegantly illustrated. 

To evaluate the effectiveness factor for a first-order, isobaric, nonisothermal, flat plate catalyst pellet, the material and energy balances must be solved simultaneously. As shown previously, the mole balance in a slab is given by ... 

Yeung et al. [1994] extended the studies to a general case of a bed of catalyst pellets on the feed side of a membrane reactor where the membrane is catalytically inert for an arbitrary number of reactions with arbitrary kinetics under nonisothermal conditions. Their conclusions are similar to those for the case of pellets in a fixed bed reactor [Baratti et al., 1993]. It appears that the presence of a catalytically inert membrane and a permeate su-eam do not affect the nature of the optimal catalyst distribution but may... 

P12-16c Determine the effectiveness factor for a nonisothermal spherical catalyst pellet in which a first-order isomerization is taking place. 

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. 

Close examination of the numerical solutions indicates that under conditions where t] 1, there is a large temperature gradient at the periphery of the pellet, but the gradient flattens ont considerably as one approaches the center of the pellet. For fast exothermic reactions (large < ), p > 0), the interior of the pellet will be at a relatively uniform high temperatnre, bnt there will be a fairly sharp decrease in temperature as one nears the exterior surface. In this regime t) becomes inversely proportional to as in the isothermal case, and the large majority of the reaction takes place in the nonisothermal shell of catalyst at its periphery. 

To illustrate the above points in a simple manner, let us consider a nonisothermal adiabatic fixed-bed reactor packed with nonporous catalyst pellets for a simple exothermic reaction... 

Derive the energy balance for a flat-plate catalyst pellet operating under nonisothermal conditions (first-order exothermic reaction). Give a plausible argument why the effectiveness factor can in this case exceed unity. 

Heat Effects in a Catalyst Pellet The Nonisothermal Effectiveness Factor... 

In the transition region between regimes I and JJ where the chemical reaction and diffusion present a comparable resistance to the overall progress of reaction, multiple solutions may occur and the possibility of instability arises when the reaction is exothermic [15]. The criteria for the existence of multiple steady state for chemical reactions in porous catalyst pellets have been studied extensively [17-21]. The effect of net gas generation or consumption on nonisothermal reaction in a porous solid was analyzed by Weekman [22]. 

Consider the case of a nonisothermal reaction A B occurring in the interior of a spherical catalyst pellet of radius R (Figure 6.4). We wish to compute the effect of internal heat and mass transfer resistance upon the reaction rate and the concentration and temperature profiles within the pellet. If Z)a is the effective binary diffusivity of A within the pellet, and we have first-order kinetics, the concentration profile CA(f) is governed by the mole balance... 

So far it has been assumed that both reactions are first order and the pellet can be treated as isothermal. It may be obvious to note that under nonisothermal conditions the ratio of the intrinsic activation energies and, if necessary, the ratio of the external heat transfer coefficients will also affect the apparent selectivity of the catalyst. In addition, if the kinetic orders of the two reactions are different, this will also influence selectivity. 

It is desired to conduct the transesterification of benzyl chloride with sodium acetate using polymer-supported tributylmethylammonium chloride as a triphase catalyst. The reactor is maintained at isothermal conditions, but temperature gradients can exist within the pellets. Using the following data, generate plots of reactant concentration as a function of time for both isothermal and nonisothermal pellets. 

Nonisothermal Pellet For highly endothermic or exothermic reactions, the temperature ofthe catalyst surface can be considerably different from the temperature of the surrounding fluid. 

For the design and analysis of fixed-bed catalytic reactors as well as the determination of catalyst efficiency under nonisothermal conditions, the effective thermal conductivity of the porous pellet must be known. A collection of thermal conductivity data of solids published by the Thermophysical Properties Research Centre at Purdue University [ ] shows "a disparity in data probably greater than that of any other physical property ". Some of these differences naturally can be explained, as no two samples of solids, especially porous catalysts, can be made completely identical. However, the main reason is that the assumed boundary conditions for the Fourier heat conduction equation... 

---