Nonisothermal Pellet

A large number of catalytic reactions are exothermic and are accompanied by thermal effects. For relatively fast intrinsic kinetics as compared to the mass and heat transfer phenomena, the development of internal temperature gradients can be expected. Heat and mass transfer balances have to be solved simultaneously to estimate concentration and temperature profiles under steady-state conditions. 

As the reaction rate depends exponentially on temperature, the resulting temperature and concentration profiles have to be calculated by numerical methods. 

With the surface concentration and temperatures and T, we obtain after integration a linear relationship between internal temperature and reactant concentration  

The largest possible temperature difference in the particle is attained, when the concentration in the particle center becomes Cj = 0. 

Obviously, the maximum temperature difference will depend on the reaction enthalpy and the ratio between effective diffusion and effective thermal conductivity. 

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Generally, under either isothermal or noniso-thermal conditions, intrapartiole diffusional limitations are undesirable because they reduce the selectivity below that which can be achieved in their absence. The exception to this generalization is a set of endothermic reactions that take place in nonisothermal pellets where the second reaction has an activation energy that is greater than that of the first. 

For bimolecular reactions with nonisothermal pellets, only numerical solutions are presented in literature [23]. Karanth et al. [24] defined the parameter ... 

The above suggests that the discussion of the Aris numbers for simple reactions also holds for nonisothermal pellets. For example, effectiveness factors larger than one are found if the number An, becomes negative. According to Equation 7.14 this is the case if... 

Nonconstant density, 136 Nonhomogeneous equation, 117 Nonideal mixtures, 82 Nonisothermal pellet, 393 Nonlinear differential equation, 119 nth-order... 

The treatment can be extended to reactions involving more than one adsorbed species provided that the number of additional parameters (i.e., Kb[B], Kr[R], etc.) can be reduced, preferably to one. An elegant method has been proposed by Roberts and Satterfield (1965, 1966) in which precisely this is accomplished all of the adsorption parameters are combined into a single parameter A>as- This method has been further generalized (Rajadhyaksha et al., 1976) by extending it to nonisothermal pellets, but graphical representation in this case becomes unwieldy because now two additional (thermal) parameters ag and must be accommodated. 

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. 

Mass and Heat Transfer Effects on Heterogenous Catalytic Reactions 77 2.6.1.2 Nonisothermal Pellet... 

Concentration and temperature profiles in nonisothermal pellets are shown in Figures 10a and 10b for the corresponding situations. 

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 ... 

The existence of internal resistances complicates the analysis of transport effects for trickle-beds since the pellet cannot necessarily be assumed isothermal. Reactions in which the heat effect is negligible are considered first, and the case of a nonisothermal pellet will be treated in the following section. For arbitrary kinetics kfiC), the internal, isothermal effectiveness factor (Chapter 4) is ... 

Derive a criterion of negligible transport effects for a first-order reaction taking place in trickle-bed. Note that one only needs to replace Qq in Eq. 7.21 by Cl if kCt is taken as the intrinsic rate. First obtain the criterion for an isothermal pellet and then for a nonisothermal pellet. 

The intraparticle deactivation of nonisothermal pellets has been analyzed by Ray (21) using a pore mouth poisoning, slab geometry model. Heat flux at the boundary of the active and inactive portions of the particle (Figure 2a) was computed via Bischoff s ( ) asymptotic solution for large Thiele modulus. An effective diffusional modulus for this case can be defined as ... 

Figure 2, (a) Geometry of partially deactivated pellet (b) effective activity factor vs. dtffusUmal modulus for aeactivation of nonisothermal pellet (21)... 

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).)... 

ILLUSTRATION 12.4 EFFECTIVENESS FACTOR DETERMINATION FOR A NONISOTHERMAL CATALYST PELLET-EXOTHERMIC REACTION... 

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. 

When reaction is so fast that the heat released (or absorbed) in the pellet cannot be removed rapidly enough to keep the pellet close to the temperature of the fluid, then nonisothermal effects intrude. In such a situation two different kinds of temperature effects may be encountered ... 

Nonisothermal effects within pellets may also cause variations in deactivation with location, especially when deactivation is caused by surface modifications due to high temperatures. 

I learned about chemical reactors at the knees of Rutherford Aris and Neal Amundson, when, as a surface chemist, I taught recitation sections and then lectures in the Reaction Engineering undergraduate course at Minnesota. The text was Aris Elementary Chemical Reaction Analysis, a book that was obviously elegant but at first did not seem at all elementary. It described porous pellet diffusion effects in chemical reactors and the intricacies of nonisothermal reactors in a very logical way, but to many students it seemed to be an exercise in applied mathematics with dimensionless variables rather than a description of chemical reactors. 

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,... 

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... 

Polymer processing operations, by and large, are nonisothermal. Plastics pellets are compacted and heated to the melting point by interparticle friction, solid deformation... 

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. 

So far, only isothermal operation has been considered. Under nonisothermal conditions, in Type II reactions a change of the apparent selectivity, caused by temperature variations across the pellet and/or the interphase boundary layer, may also be expected once the two reactions exhibit different activation energies. This holds irrespective of whether the kinetic order of the two reactions is equal or not. Nonisothermal Type II problems have been analyzed, for example, by 0stergaard [81]. 

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

The nondimensional parameter /) (positive for exothermic reactions) is a measure of nonisothermal effects and is called the heat generation function. It represents the ratio between the rate of heat generation due to the chemical reaction and the heat flow by thermal conduction. Nonisothermal effects may become important for increasing values of /3, while the limit (3 - 0 represents an isothermal pellet. Table 9.1 shows the values of [3 and some other parameters for exothermic catalytic reactions. For any interior points within the pore where the reactant is largely consumed, the maximum temperature difference for an exothermic reaction becomes... 

Hie Aris numbers Anx and An0 can be applied to nonisothermal catalyst pellets. Three items will be discussed ... 

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