Isothermal/nonisothermal

The models discussed thus far were essentially isothermal. Nonisothermal models as described below contain an additional degree of complexity that is always related to the characteristics of a particular reactor and the form of a particular catalyst. On the other hand, it is usually easier to construct a nonisothermal model that oscillates, because the exponential term in the Arrhenius law provides a very strong nonlinearity. Nonisothermal models can be classified into two categories surface blocking/reactivation models and models with additional bulk effects. 

For batch, plug flow, and CSTR. Includes gas-phase isothermal, nonisothermal, and nonisobaric reactions, heterogeneous catalysis, and thermochemical database for calculation of equilibrium constants. Many subprograms for special situations (shock waves, flames, partially stirred reactors, etc.) are available. 

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. 

They differ by the kinematical assumptions (ID uniform section reduction 2D dog-bone section with a membrane approximation, fully 3D), the rheological behaviour (Newtonian, viscoelastic), and the thermal hypotheses (isothermal, nonisothermal). 

Nonvolatile Solvents. In practice, some gases tend to Hberate such large amounts of heat when they are absorbed into a solvent that the operation caimot be assumed to be isothermal, as has been done thus far. The resulting temperature variations over the tower will displace the equiUbrium line on 2tj—x diagram considerably because the solubiUty usually depends strongly on temperature. Thus nonisothermal operation affects column performance drastically. 

For an isothermal system the simultaneous solution of equations 30 and 31, subject to the boundary conditions imposed on the column, provides the expressions for the concentration profiles in both phases. If the system is nonisotherm a1, an energy balance is also required and since, in... 

Phenomena of multiple steady states and instabilities occur particularly with nonisothermal CSTRs. Some isothermal processes with hyperbohc rate equations and processes with porous catalysts also can have such behavior. 

Minimum reactor volumes of isothermal and nonisothermal cascades by dynamic programming... 

The downflow condenser is used mainly for nonisothermal condensation. Vapors enter through a header at the top and flow downward. The refliix condenser is used for isothermal and small-temperature-change conditions. Vapors enter at the bottom of the tubes. 

Isothermal jets are not influenced much by small changes in flow rate the size of the influence can be seen from the different equations. Nonisothermal jets could be changed substantially by small differences in both outlet velocity (flow rate) and/or temperature. 

In order to get a quantitative idea of the magnitude of the effects of these temperature variations on molecular structure and morphology an experimental study was undertaken. Two types of polymerizations were conducted. One type was isothermal polymerization at fixed reaction time at a series of temperatures. The other type was a nonisothermal polymerization in the geometry of a RIM mold. Intrinsic viscosities, size exclusion chromotograms (gpc) and differential scanning calorimetry traces (dsc) were obtained for the various isothermal products and from spatially different sections of the nonisothermal products. Complete experimental details are given below. 

The sum of squares as defined by Equation 7.8 is the general form for the objective function in nonlinear regression. Measurements are made. Models are postulated. Optimization techniques are used to adjust the model parameters so that the sum-of-squares is minimized. There is no requirement that the model represent a simple reactor such as a CSTR or isothermal PER. If necessary, the model could represent a nonisothermal PFR with variable physical properties. It could be one of the distributed parameter models in Chapters 8 or 9. The model... 

Most kinetic experiments are run in batch reactors for the simple reason that they are the easiest reactor to operate on a small, laboratory scale. Piston flow reactors are essentially equivalent and are implicitly included in the present treatment. This treatment is confined to constant-density, isothermal reactions, with nonisothermal and other more complicated cases being treated in Section 7.1.4. The batch equation for component A is... 

This technique should give reasonable results for isothermal, first-order reactions. It and other modeling approaches are largely untested for complex and nonisothermal reactions. 

Previous chapters have discussed how isothermal or adiabatic reactors can be scaled up. Nonisothermal reactors are more difficult. They can be scaled by maintaining the same tube diameter or by the modeling approach. The challenge is to increase tube diameter upon scaleup. This is rarely possible and when it is possible, scaleup must be based on the modeling approach. If the predictions are satisfactory, and if you have confidence in the model, proceed with scaleup. 

The steady-state design equations (i.e., Equations (14.1)-(14.3) with the accumulation terms zero) can be solved to find one or more steady states. However, the solution provides no direct information about stability. On the other hand, if a transient solution reaches a steady state, then that steady state is stable and physically achievable from the initial composition used in the calculations. If the same steady state is found for all possible initial compositions, then that steady state is unique and globally stable. This is the usual case for isothermal reactions in a CSTR. Example 14.2 and Problem 14.6 show that isothermal systems can have multiple steady states or may never achieve a steady state, but the chemistry of these examples is contrived. Multiple steady states are more common in nonisothermal reactors, although at least one steady state is usually stable. Systems with stable steady states may oscillate or be chaotic for some initial conditions. Example 14.9 gives an experimentally verified example. 

In this paper we present a meaningful analysis of the operation of a batch polymerization reactor in its final stages (i.e. high conversion levels) where MWD broadening is relatively unimportant. The ultimate objective is to minimize the residual monomer concentration as fast as possible, using the time-optimal problem formulation. Isothermal as well as nonisothermal policies are derived based on a mathematical model that also takes depropagation into account. The effect of initiator concentration, initiator half-life and activation energy on optimum temperature and time is studied. 

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. 

Nonisothermal policies were computed using a gradient method as outlined in Appendix A. The optimal isothermal proflle was used as an initial guess. 

Figure 6 shows the temperature proflle that should be used with the initiator monomer system described in the caption to reduce the monomer concentration from 0.47 mol/L to 0.047 mol/L. The optimal nonisothermal policy consists of decreasing temperature from a temperature above the optimal isothermal temperature to one below it. The rate of polymerization could be increased, as expected, by an initially higher temperature, but the temperature must be decreased to avoid depletion of initiator and depolymerization. However, the amount of time saved by this policy does not seem to be significant in comparison to the isothermal policy for this case. 

Figure 7 shows results from a nonisothermal policy obtained if a monomer with high (-AH) values were used. The policy was similar to the one shown in Figure 6. However this policy resulted in a time saving of 15 percent compared to the isothermal policy.

A closer look at the nonisothermal and isothermal policy results reveals some additional interesting features with regard to optimization. As mentioned earlier, isothermal policies were determined by two factors. One was the M, value and the other was the dead end polymerization caused by depletion of initiator. It was also observed that the minimum time from a nonisothermal policy was considerably less than the minimum time due to the isothermal policy whenever H>, was the controlling factor in the isothermal policy when the isothermal policy was controlled by initiator depletion, a nonisothermal policy did not show significant improvement in minimum time relative to the isothermal one. 

In this paper we formulated and solved the time optimal problem for a batch reactor in its final stage for isothermal and nonisothermal policies. The effect of initiator concentration, initiator half-life and activation energy on optimum temperature and optimum time was studied. It was shown that the optimum isothermal policy was influenced by two factors the equilibrium monomer concentration, and the dead end polymerization caused by the depletion of the initiator. When values determine optimum temperature, a faster initiator or higher initiator concentration should be used to reduce reaction time. 

Comparison of isothermal and nonisothermal policies revealed some interesting features of the polymer system. When M , values determine the isothermal policy, a nonisothermal operation reduces the minimum time compared to isothermal operation (by about 15%). However, when dead-end polymerization influences isothermal operation, a nonisothermal operation does not offer significant improvement. 

Obtaining Kinetic Samples for Reactive Extrusion. To develop and test kinetic models, homogeneous samples with a well defined temperature-time history are required. Temperature history does not necessarily need to be isothermal. In fact, well defined nonisothermal histories can provide very good test data for models. However, isothermal data is very desirable at the initial stages of model building to simplify both model selection and parameter estimation problems. 

---