Nonisothermal policy

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. 

Butala et al. [43] applied optimization techniques to styrene polymerization initiated by BPO (dibenzoyl peroxide, 1 h half-life time, 91°C) and TBPB (tert-butyl perbenzoate, 1 h half-life time, 124°C). As mentioned before, the batch time can be minimized by using nonisothermal temperature profiles. Three independent runs with different optimization policies were performed. The detailed control policies are listed in Table 5.1. 

It is seen that the optimal policy would be that of maintaining the conversion constant if A xa- is zero or a function solely of x since then the left hand side of Eq. 13.37 should be constant, which implies constant x. It can be shown (Chou et al. 1967) that Eq. 13.37 leads to the optimal policy of maintaining a constant conversion for an irreversible reaction in which only one rate constant and temperature dependence is involved, and that the policy is also applicable to nonisothermal reactors. The limitations under which the policy of a constant conversion is valid are embodied in Eq. 13.37 x should be a function of A and A and not dependent explicitly on t, and A xa should be a function solely of x. The policy gives the maximum of P when an optimal temperature is chosen and hence requires a search for the optimal temperature. 

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