Nonisothermal Axial Dispersion

The axial dispersion model is readily extended to nonisothermal reactors. The turbulent mixing that leads to flat concentration profiles will also give flat temperature profiles. An expression for the axial dispersion of heat can be written in direct analogy to Equation (9.14)  

The boundary conditions associated with Equation (9.24) are of the Danckwerts type  

Correlations for E are not widely available. The more accurate model given in Section 9.1 is preferred for nonisothermal reactions in packed-beds. However, as discussed previously, this model degenerates to piston flow for an adiabatic reaction. The nonisothermal axial dispersion model is a conservative design methodology available for adiabatic reactions in packed beds and for nonisothermal reactions in turbulent pipeline flows. The fact that E D provides some basis for estimating E. Recognize that the axial dispersion model is a correction to what would otherwise be treated as piston flow. Thus, even setting E=D should improve the accuracy of the predictions. 

Only numerical solutions are possible when Equation (9.24) is solved simultaneously with Equation (9.14). This is true even for first-order reactions because of the intractable nonlinearity of the Arrhenius temperature dependence. 

Correlations for E are not widely available. The more accurate model given in Section 

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The next example illustrates the use of reverse shooting in solving a problem in nonisothermal axial dispersion and shows how Runge-Kutta integration can be applied to second-order ODEs. 

Example 9.6 Compare the nonisothermal axial dispersion model with piston flow for a first-order reaction in turbulent pipeline flow with Re= 10,000. Pick the reaction parameters so that the reactor is at or near a region of thermal runaway. 

Solve the same nonisothermal axial dispersion problem when the reaction is second-order with Pen = 4 and P m = 8. 

Here let us consider the nonisothermal axial dispersion model in extension of equations (6-97) and (6-98) as... 

In particular, equation (7-146) expresses that there is no mass transfer at the wall, since the concentration derivative is zero, and that heat transfer occurs with a constant wall temperature, Tw, and a local heat-transfer coefficient, This heat-transfer coefficient is now appearing in a boundary condition and is not equivalent to the overall heat-transfer coefficient used in nonisothermal axial dispersion models. The radial dispersion coefficient, Z) is, as the name implies, the radial counterpart to the axial dispersion coefficient, and while we expect a different correlation for it there are no new conceptual boundaries set here. The effective bed thermal conductivity, A however, is another matter altogether and we will worry about it more later. 

The solution of Eq. (173) poses a rather formidable task in general. Thus the dispersed plug-flow model has not been as extensively studied as the axial-dispersed plug-flow model. Actually, if there are no initial radial gradients in C, the radial terms will be identically zero, and Eq. (173) will reduce to the simpler Eq. (167). Thus for a simple isothermal reactor, the dispersed plug flow model is not useful. Its greatest use is for either nonisothermal reactions with radial temperature gradients or tube wall catalysed reactions. Of course, if the reactants were not introduced uniformly across a plane the model could be used, but this would not be a common practice. Paneth and Herzfeld (P2) have used this model for a first order wall catalysed reaction. The boundary conditions used were the same as those discussed for tracer measurements for radial dispersion coefficients in Section II,C,3,b, except that at the wall. 

For packed bed reactors, Carberry and Wendel (1963), Hlavacek and Marek (1966), and Carberry and Butt (1975) report that axial dispersion effects are negligible if the reactor length is sufficient. These and other researchers (Young and Finlayson, 1973 Mears, 1976) have developed criteria based on the reactor length for conditions where axial dispersion can safely be neglected. The criterion shown in Table V is a classic criterion for neglecting axial mass dispersion. The works by Young and Finlayson (1973) and Mears (1976) provide more detailed criteria to predict when axial dispersion is unimportant in nonisothermal packed bed reactors. 

Develop the boundary conditions for the heat balance design equation of axial dispersion in the nonisothermal case as suggested on p. 262. 

Potential pitfalls exist in ranking catalysts based solely on correlations of laboratory tests (MAT or FFB) to riser performance when catalysts decay at significantly different rates. Weekman first pointed out the erroneous conversion ranking of decaying catalysts in fixed bed and moving bed isothermal reactors (1-3). Phenomena such as axial dispersion in the FFB reactor, the nonisothermal nature of the MAT test, and feedstock differences further complicate the catalyst characterization. In addition, differences between REY, USY and RE-USY catalyst types exist due to differences in coke deactivation rates, heats of reaction, activation energies and intrinsic activities. 

The axial dispersion of heat in large-scale reactors should be measured. This information would be useful in modeling large-scale nonisothermal reactors. 

If the flow rate is sufficiently high to create turbulent flow, then Pe is a constant and the magnitude of the right-hand side of the equation is determined by the aspect ratio, L/d. By solving Equation, (8.4.12) and comparing the results to the solutions of the PER [Equation (8.4.3)], it can be shown that for open tubes, L/d, > 20 is sufficient to produce PER behavior. Likewise, for packed beds, L/d, > 50 (isothermal) and L d, >150 (nonisothermal) are typically sufficient to provide PER characteristics. Thus, the effects of axial dispersion are minimized by turbulent flow in long reactors. 

The influence of activity changes on the dynamic behavior of nonisothermal pseudohomogeneoiis CSTR and axial dispersion tubular reactor (ADTR) with first order catalytic reaction and reversible deactivation due to adsorption and desorption of a poison or inert compound is considered. The mathematical models of these systems are described by systems of differential equations with a small time parameter. Thereforej the singular perturbation methods is used to study several features of their behavior. Its limitations are discussed and other, more general methods are developed. 

What models should be used either for scaleup or to correlate pilot plant data Section 9.1 gives the preferred models for nonisothermal reactions in packed beds. These models have a reasonable experimental basis even though they use empirical parameters D, hr, and Kr to account for the packing and the complexity of the flow field. For laminar flow in open tubes, use the methods in Chapter 8. For highly turbulent flows in open tubes (with reasonably large L/dt ratios) use the axial dispersion model in both the isothermal and nonisothermal cases. The assumption D = E will usually be safe, but do calculate how a PFR would perform. If there is a substantial difference between the PFR model and the axial dispersion model, understand the reason. For transitional flows, it is usually conservative to use the methods of Chapter 8 to calculate yields and selectivities but to assume turbulence for pressure drop calculations. 

Develop a computational procedure for the solution of the nonisothermal, onedimensional axial dispersion reactor model. 

If one were to attempt to determine any communality in the discussion of models given in this chapter, about the best would be to say that the parameters invoked are derivatives of the model, as would be inferred from the titles of the previous sections. For example, there is the overall heat-transfer coefficient, h, that appears in the nonisothermal, one-dimensional axial dispersion model, which is not to be confused with the wall heat transfer coefficient, a y, that belongs to the radial dispersion model. Similarly, would the bed thermal conductivity be the same in an axial dispersion model as in a radial dispersion model What is the difference between a mass Peclet number and a thermal Peclet number and so on. In fact, let us take a moment... 

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