Effectiveness, nonisothermal

Catalyst supports such as silica and alumina have low thermal conductivities so that temperature gradients within catalyst particles are likely in all but the finely ground powders used for infrinsic kinetic studies. There may also be a film resisfance fo heaf fransfer af fhe exfemal surface of the catalyst. Thus the internal temperatures in a catalyst pellet may be substantially different than the bulk gas temperature. The definition of the effectiveness factor, Equation 10.23, is unchanged, but an exothermic reaction can have reaction rates inside the pellet that are higher than would be predicted using the bulk gas temperature. In the absence of a diffusion limitation, rj 1 would be expected for an exothermic reaction. (The case 1 is also possible for some isothermal reactions with weird kinetics.) Mass transfer limitations may have a larger 

The theory of nonisothermal effectiveness is sufficiently well advanced to allow estimates for rj. The analysis requires simultaneous solutions for the concentration and temperature profiles within a pellet. The solutions are necessarily numerical. Solutions are feasible for actual pellet shapes (such as cylinders) but are significantly easier for spherical pellets since this allows a one-dimensional form for the energy equation  

Catalyst pellets often operate with internal temperatures that are substantially different from the bulk gas temperature. Large heats of reaction and the low thermal conductivities typical of catalyst supports make temperature gradients likely in all but the hnely ground powders used for intrinsic kinetic studies. There may also be a him resistance to heat transfer at the external surface of the catalyst. 

The theory of nonisothermal effectiveness is sufficiently well advanced to allow order-of-magnitude estimates for rj. The analysis requires simultaneous 

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).) 

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

The discussion of nonisothermal effects has been delayed until this section because the models that account for such effects constitute a relatively small subset of the total number of models and have already been discussed above in other... 

As mentioned, to include nonisothermal effects, an overall thermal energy balance needs to be added to the set of governing equations. The energy conservation equation can be written for phase k in the... 

This review has highlighted the important effects that should be modeled. These include two-phase flow of liquid water and gas in the fuel-cell sandwich, a robust membrane model that accounts for the different membrane transport modes, nonisothermal effects, especially in the directions perpendicular to the sandwich, and multidimensional effects such as changing gas composition along the channel, among others. For any model, a balance must be struck between the complexity required to describe the physical reality and the additional costs of such complexity. In other words, while more complex models more accurately describe the physics of the transport processes, they are more computationally costly and may have so many unknown parameters that their results are not as meaningful. Hopefully, this review has shown and broken down for the reader the vast complexities of transport within polymer-electrolyte fuel cells and the various ways they have been and can be modeled. 

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 ... 

We next ask which form of nonisothermal effect, if any, may be present. The following simple calculations tell. 

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

However, for gas-solid systems Hutchings and Carberry (1966) and McGreavy and coworkers (1969, 1970) show that if reaction is fast enough to introduce nonisothermal effects, then the temperature gradient occurs primarily across the gas film, not within the particle. Thus we may expect to find a significant film AT, before any within-particle AT becomes evident. 

Nonisothermal Effects. We may expect temperature gradients to occur either across the gas film or within the particle. However, the previous discussion indicates that for gas-solid systems the most likely effect to intrude on the rate will be the temperature gradient across the gas film. Consequently, if experiment shows that gas film resistance is absent then we may expect the particle to be at the temperature of its surrounding fluid hence, isothermal conditions may be assumed to prevail. Again see Example 18.1. 

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

The classic extrusion model gives insight into the screw extrusion mechanism and first-order estimates. For more accurate design equations, it is necessary to eliminate a long series of simplifying assumptions. These, in the order of significance are (a) the shear rate-dependent non-Newtonian viscosity (b) nonisothermal effects from both conduction and viscous dissipation and (c) geometrical factors such as curvature effects. Each of these... 

Distributed Parameter Models Both non-Newtonian and shear-thinning properties of polymeric melts in particular, as well as the nonisothermal nature of the flow, significantly affect the melt extmsion process. Moreover, the non-Newtonian and nonisothermal effects interact and reinforce each other. We analyzed the non-Newtonian effect in the simple case of unidirectional parallel plate flow in Example 3.6 where Fig.E 3.6c plots flow rate versus the pressure gradient, illustrating the effect of the shear-dependent viscosity on flow rate using a Power Law model fluid. These curves are equivalent to screw characteristic curves with the cross-channel flow neglected. The Newtonian straight lines are replaced with S-shaped curves. 

Next, we explore some nonisothermal effects on of a shear-thinning temperature-dependent fluid in parallel plate flow and screw channels. The following example explores simple temperature dependent drag flow. 

Galili and Takserman-Krozer (20) have proposed a simple criterion that signifies when nonisothermal effects must be taken into account. The criterion is based on a perturbation solution of the coupled heat transfer and pressure flow isothermal wall problem of an incompressible Newtonian fluid. 

The first milestone in modeling the process is credited to Pearson and Petrie (42—44). who laid the mathematical foundation of the thin-film, steady-state, isothermal Newtonian analysis presented below. Petrie (45) simulated the process using either a Newtonian fluid model or an elastic solid model in the Newtonian case, he inserted the temperature profile obtained experimentally by Ast (46), who was the first to deal with nonisothermal effects and solve the energy equation to account for the temperature-dependent viscosity. Petrie (47) and Pearson (48) provide reviews of these early stages of mathematical foundation for the analysis of film blowing. 

Following Gaskell s work, a great deal of effort was invested by numerous researchers in the field to improve on his model. Most of this effort, however, basically concentrated on solving the Gaskell model with more realistic, constitutive equations and attempts to account for nonisothermal effects. In the original Gaskell model, a purely viscous (nonelastic and time-independent) fluid model is assumed, with specific... 

Mechanistic studies with real catalysts near atmospheric pressure conditions are complicated by several factors the surface structure and composition will be inhomogeneous and hence also the reactivity may be spatially different. In addition, the heat released by the reaction may change the (local) temperature, and as a consequence, kinetic oscillations are frequently associated with strong nonisothermal effects. These prob-... 

The generalized form can represent slow transport processes and nonisothermal effects, and satisfies the detailed balance at thermodynamic equilibrium. The exchange current is... 

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... 

Table 9.1 shows some of the experimental and assumed values of the parameters considered for catalytic oxidation of CH3OH to CH20 with 13 = 0.0109 and hence display relatively fewer nonisothermal effects. The thermal diffusion coefficient is usually smaller by a factor of 102—103 than the ordinary diffusion coefficient for nonelectrolytes and gases. Therefore, for the present analysis the values for e and co are assumed to be 0.001. 

The value of jS is a measure of nonisothermal effects. As j8 approaches zero, the system becomes isothermal. 

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