Solution of Nonisothermal Plug-Flow Reactor

Example 5.3 Solution of Nonisothermal Plug-Flow Reactor. Write general MATLAB functions for integrating simultaneous nonlinear differential equations using the Euler, Euler predictor-corrector (modified Euler), Runge-Kutta, Adams, and Adams-Moulton methods. Apply these functions for the solution of differential equations that simulate a nonisotherm plug flow reactor, as described below.  

Vapor-phase cracking of acetone, described by the following endothermic reaction  

Overall heat transfer coefficient Heat transfer area  

Determine the temperature profile of the gas along the length of the reactor. Assume constant pressure throughout the reactor. 

Method of Solution In order to calculate the temperature profile in the reactor, we have to solve the material balance and energy balance equations simultaneously  

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Example 5.3 Solution of Nonisothermal Plug-Flow Reactor... 

The difficulty in this is the awkward form of equation (6-140) with respect to 7). One likes to compute in sequence through the series of cells, but here we face the implicit form of Ti as a function of r, i. This is the same basic difficulty that limits the utility of the CSTR sequence as an analytical model for nonisothermal reactors. For the case here though, where we employ a relatively large value of the index n in approximation of a plug-flow reactor, and where the solution will be via numerical methods anyway, we will strong-arm the problem with the approximation T,- r,- ] in the exponentials, so that... 

Nonisothermal reactor design requires the simultaneous solution of the appropriate energy balance and the species material balances. For the batch, semi-batch, and steady-state plug-flow reactors, these balances are sets of initial-value ODEs that must be solved numerically, in very limited situations (constant thermodynamic properties, single... 

In the previous examples, we have exploited the idea of an effectiveness factor to reduce fixed-bed reactor models to the same form as plug-flow reactor models. This approach is useful and solves several important cases, but this approach is also limited and can take us only So far. In the general case, we must contend with multiple reactions that are not first order, nonconstant thermochemical properties, and nonisothermal behavior in the pellet and the fluid. For these cases, we have no alternative but to solve numerically for the temperature and species concentrations profiles in both the pellet and the bed. As a final example, we compute the numerical solution to a problem of this type. 

Stability analysis could prove to be useful for the identification of stable and unstable steady-state solutions. Obviously, the system will gravitate toward a stable steady-state operating point if there is a choice between stable and unstable steady states. If both steady-state solutions are stable, the actual path followed by the double-pipe reactor depends on the transient response prior to the achievement of steady state. Hill (1977, p. 509) and Churchill (1979a, p. 479 1979b, p. 915 1984 1985) describe multiple steady-state behavior in nonisothermal plug-flow tubular reactors. Hence, the classic phenomenon of multiple stationary (steady) states in perfect backmix CSTRs should be extended to differential reactors (i.e., PFRs). 

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. 

Most of the models assume that neutral-species transport can be represented with either a well-mixed model or a plug flow model. The major drawback to these assumptions is that important inelastic rate processes such as molecular dissociation are usually localized in space in the reactor and are often fast compared with rates of diffusion or convection. As a result, the spatial variation of fluid flow in the reactor must be accounted for. This variation introduces a major complication in the model, because the solution of the nonisothermal Navier-Stokes equations in multidimensional geometries is expensive and difficult. 

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