A Nonisothermal Distributed System

The problem of this section stems from a typical distributed system. In a distributed system the dependent state variables such as concentration and temperature vary in 

For a distributed system the discrete mass balance design equation has the form 

If we take the limit of Al — 0 in (4.2), we arrive at the equivalent differential equation 

For a single reaction, i.e., with N = 1, the above equation becomes 

To obtain the heat balance design equation, an enthalpy balance over the Al element gives 

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Here we consider a nonisothermal, nonadiabatic tubular reactor as a distributed system. Our objective is to find its steady-state concentration and temperature profiles. 

The same very simple principles apply to the heat balance equations for a nonisothermal system and also for distributed systems as shown in the following section. It also applies to the heterogeneous system, as shown in Chapter 6. 

Further the pressure and temperature dependences of all the transport coefficients involved have to be specified. The solution of the equations of change consistent with this additional information then gives the pressure, velocity, and temperature distributions in the system. A number of solutions of idealized problems of interest to chemical engineers may be found in the work of Schlichting (SI) there viscous-flow problems, nonisothermal-flow problems, and boundary-layer problems are discussed. 

Section 6.3 treats distributed nonreacting systems and specifically packed bed absorption, while Section 6.4 studies a battery of nonisothermal CSTRs and its dynamic behavior. 

The rational design of a reaction system to produce a desired polymer is more feasible today by virtue of mathematical tools which permit one to predict product distribution as affected by reactor type and conditions. New analytical tools such as gel permeation chromatography are beginning to be used to check technical predictions and to aid in defining molecular parameters as they affect product properties. The vast majority of work concerns bulk or solution polymerization in isothermal batch or continuous stirred tank reactors. There is a clear need to develop techniques to permit fuller application of reaction engineering to realistic nonisothermal systems, emulsion systems, and systems at high conversion found industrially. A mathematical framework is also needed which will start with carefully planned experimental data and efficiently indicate a polymerization mechanism and statistical estimates of kinetic constants rather than vice-versa. 

The mathematical complexity involved with temperature variations has limited most of the studies cited in this paper to the isothermal case. Since few commercial polymerization reactor systems can or should operate isothermally, there is a clear need to develop techniques to permit fuller application of reaction engineering to nonisothermal systems. In polymerizations as in simpler reactions, changes in temperature or temperature profile can have larger effects on rate and distribution than even reactor type. 

Both Eq. (13.5) (isothermal systems) and Eq. (13.8) (nonisothermal systems) have the same form, in both equations being given in terms of a symmetric tensor a. In the next several sections various integrals over Q involving the distribution function arise, and hence we summarize them here ... 

Together, Chapters 3 and 4 provide systematic, easy-to-understand coverage of all types of homogeneous models, both lumped/distributed and isothermal/nonisothermal systems. Both chapters can also be used as the necessary materials for a thorough course on chemical reaction engineering based on a well-organized approach utilizing system theory. 

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