Theory nonisothermal

Local equilibrium theory Shows wave character—simple waves and shocks Usually indicates best possible performance Better understanding Mass and heat transfer very rapid Dispersion usually neglected If nonisothermal, then adiabatic... 

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

It was felt that a nonisothermal policy might have considerable advantages in minimizing the reaction time compared to die optimal isothermal policy. Modem optimal control theory (Sage and White (1977)), was employed to minimize the reaction time. The mathematical development is presented below. 

Cleland and Wilhelm (C18) used a finite-difference technique which could be used for nonlinear reactions, but they limited their study to a first-order reaction. Experiments were also performed to test the results of the theory. In a small reaction tube, the two checked quite well. In a large tube there were differences which were explained by consideration of natural convection effects which were due to the fact that completely isothermal conditions were not maintained. This seems to be the only experimental data in the literature to date, and shows another area in which more work is needed. The preceding discussion considered only isothermal conditions except for Chambre (C12) who presented a general method for nonisothermal reactors. 

The special form of second-order equation in which the right-hand side is a function only of the dependent variable also turns up in the theory of diffusion and reaction in a slablike particle. Corresponding to equations (123-125) for the sphere, we would have, thanks to the reduction described in Chapter 2 and the example of a first-order nonisothermal reaction given by Eq. (129),... 

A theory has been developed which translates observed coke-conversion selectivity, or dynamic activity, from widely-used MAT or fixed fluidized bed laboratory catalyst characterization tests to steady state risers. The analysis accounts for nonsteady state reactor operation and poor gas-phase hydrodynamics typical of small fluid bed reactors as well as the nonisothermal nature of the MAT test. Variations in catalyst type (e.g. REY versus USY) are accounted for by postulating different coke deactivation rates, activation energies and heats of reaction. For accurate translation, these parameters must be determined from independent experiments. 

To formalize the main hypothesis, the nonisothermal functional given by Eq. (16.1) is modified by a new isothermic functional with a modified time scale to account for the temperature history. Now, according to the basic hypothesis of linear theory, the specific form of the stress-strain relationship can be written as... 

Instability typically arises from the interaction of two phenomena with different dependences on a reaction parameter In a nonisothermal reaction, the dependence on temperature is exponential for heat generation by the reaction, but linear for heat loss to the cooling coil or environment in a reaction with chain branching, the dependence on radical population is exponential for acceleration by branching, but quadratic for chain termination. A reaction is unstable if acceleration outruns retardation. This can cause an explosion or, in a CSTR, lead to multiple steady states. Feinberg s network theory can help to assess whether an isothermal reaction admits multiple steady states in a CSTR. 

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 ... 

The basic theory for nonisothermal prediction of degradation rate was established in the 1950s.320-321 Its application to the stability prediction of pharmaceuticals was reported by Rogers322 and extended by Eriksen and Stelmach.323 Initial temperature programs or algorithms used relationships that could easily be integrated when inserted into Eq. (2.84). 

The theory of Avrami is limited to isothermal processes. Since polymer processing is mostly performed under nonisothermal conditions, the theory has been extended [Ziabicki, 1967 Ozawa, 1971 Ziabicki, 1976]. 

There are methods that exist that address these constraints however, these require additional theory to understand. In the absence of an explicit temperature expression, it is often easier to tackle the problem numerically, with the aid of an automated AR construction scheme. In Chapter 8, a number of AR construction methods are discussed that may be used for nonisothermal systems. Although these methods often do not suggest an optimal reactor structure, knowledge of the limits of achievability for a nonisothermal is often sufficient for setting design targets. 

In this chapter, a number of simplified examples have been outlined to demonstrate how AR theory may be used to answer common reactor synthesis problems related to adiabatic systems and minimum residence time. A number of natural extensions to these discussions may be carried out that enhance the use of AR theory to nonisothermal systems. For the interested reader, two notable papers are available that extend on the ideas discussed here. Nicol et al. (1997) show how the AR for an exothermic reaction may be generated that incorporates external heating or cooling, whereas Glasser et al. (1992) extend the two-dimensional preheating examples, shown in this chapter, to involving x-T-r space. 

We will begin by discussing a number of important formulae for converting common process variables involving moles to equivalent quantities involving mass fraction. These concepts are not difficult to understand, however, they are fundamental to how the computation of ARs in mass fraction space must be organized. Discussion of how the stoichiometric subspace may be computed and how residence time may be incorporated in mass fraction space is also provided. From this, a number of examples are provided that demonstrate the theory. In particular, isothermal and nonisothermal unbounded gas phase systems shall be investigated. 

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