A computer model was developed for tubular high-pressure polyethylene reactors. Plug flow and absence of axial mixing were assumed. Emphasis was placed on realistic modeling of the reaction kinetics and the variation of physical properties along the reaction coordinate. A good simulation of axial temperature profiles, conversion, molecular weights, molecular weight distribution, and transport properties along the reaction coordinate is believed to have been achieved. The model can be extended readily to cases where radial diffusion is significant. SCOPEA computer model was developed for tubular highpressure polyethylene reactors. Simplifications (plug flow, absence of axial mixing) were made which are shown to be justifiable in high throughput reactors. However, the steady state free radical approximation was not made, and variations in the density and viscosity of the fluid along the reaction coordinate were taken into account explicitly. Emphasis was placed on realistic kinetics including long chain branching, use of the best rate constants available, and realistic simulation of all physical properties along the reaction coordinate. Temperature profiles, monomer and initiator conversion, the number average molecular weight (sn), and the molecular weight distribution (MWD) were calculated for a typical set of operating conditions as they might exist in a simple, high throughput reactor. The response of the model to changes in operating variables and kinetic constants was then examined. The model can be extended readily to simulate reactors with multiple initiator or monomer injection and to reactors in which laminar conditions prevail in the high conversion zone. CONCLUSIONS AND SIGNIFICANCEThis model permits a calculation, for any point along the reaction coordinate, of transport properties, the first several moments of the radical and polymer size distributions, and of long chain branching. Considerable insight is therefore gained into the kinetics of this most important vinyl polymerization, as carried out under commercial conditions, and into the physical properties of the product. The reaction is found to be rapid, and the concentration of the active intermediates (radicals) is larger by about two orders of magnitude than that encountered in the much more familiar isothermal case, where the steady state approximation for the free radicals is normally made.Because the reaction is exothermic and the rate of initiation has a large positive temperature coefficient, the polymerization reaction accelerates rapidly after a certain rate of initiation has been achieved. The temperature profile is therefore S-shaped between the reactor inlet and the point at which the maximum temperature is reached, with the appearance of a point of inflection at which the reaction may be said to take off. Beyond that point, little heat is exchanged with the environment, and only limited control can be exerted over the course of the reaction. To a first approximation, therefore, conversion and polymer properties are determined by...

No abstract

Q Q,N while the LGL method is restricted to systems linear in the control.One definite advantage held by the instantaneous minimization method is the fact that the actual performance functional kernel is used, whereas in a Lyapunov scheme the Lyapunov functional may bear little relation to the actual performance functional, thus degrading the quality of the resulting suboptimal control. Part II. Discussion of Test Systems and Numerical ResultsThe suboptimal control algorithms developed in Part I are tested on four example problems to obtain some indication of their performance. Results are presented and compared with results obtained from the application J. GREGORY VERMEYCHUKof classical open-loop optimal control synthesis algorithms to the same test and LEON LAPIDUS problems. Department of Chemical EngineeringThe suboptimal feedback control algorithms are found to give acceptprinceton university, princeton, J, 08540 able performance on both linear and nonlinear systems at the expense of very little computational effort. SCOPEThe performance of the suboptimal control algorithms developed in Part I of this work is to be evaluated by direct comparison of the effect of suboptimal control to the effect . _ of classically optimal control upon four specific These four classes encompass a great number of the distributed systems encountered in engineering practice and are thus sufficient to provide the indications of performance we desire. test problems.The basic test systems are a linear parabolic system, a linear first-order hyperbolic system, a linear parabolic It should be noted that a rigorously optimal control for a linear system with a quadratic performance functional system with variable coefficients, and a nonlinear para-may be obtained in feedback form but O~Y at the cost of d l i c system. extensive computation. No method for the computation ofConsider the system depicted in Figure 1. The sheet, entering the furnace at velocity a with temperature ug, is to be heated to an exit temperature T. The temperature inside the furnace is u (t), and the heat trans-

A new method for suboptimal feedback control of certain classes of distributed systems based on successive instantaneous minimization of the performance functional kernel is presented. This method is applicable to both linear and nonlinear systems.The Lyapunov functional approach is extended to distributed systems, yielding two new suboptimal feedback control techniques. Also, an application of multilevel bang-bang control to the first-order hyperbolic system is presented and generalized.Advantages and disadvantages of the suboptimal control techniques vis d vis open loop optimal control methods are listed, as well as comparative features of the suboptimal control methods. SCOPEMany processes encountered in engineering practice possess significantly nonuniform spatial distributions in the values of their state variables. In modeling the unsteady state behavior of such systems, the use of partial differential equations is very desirable. Although ensuring an accurate representation of system behavior, the use J. G. Vermeychuk is with the of a distributed model makes the synthesis of an optimal control policy for the system extremely laborious. Furthermore, except in one special case, the optimal control must be applied to the system in a feedfonvard manner. Feedfonvard control is plagued by many shortcomings, rendering it unsatisfactory for many practical applications.The objective of this work is to develop feedback control algorithms for distributed systems which provide nearoptimal control, yet which are conceptually simple and flexible enough to have potential for practical use.