Optimal steady state necessary conditions

Let the pair (y, u) denote the optimal steady state solution with the corresponding multipliers A, fi, and P. The necessary conditions for this solution are provided in Appendix 8.A (p. 260). Since this solution minimizes J, satisfying the state equation and the constraints,... 

A real-time optimization (RTO) system determines set-point changes and implements them via the computer control system without intervention from unit operators. The RTO system completes all data transfer, optimization calculations, and set-point implementation before unit conditions change that may invalidate the computed optimum. In addition, the RTO system should perform all tasks without upsetting plant operations. Several steps are necessary for implementation of RTO, including determination of the plant steady state, data gathering and validation, updating of model parameters (if necessary) to match current operations, calculation of the new (optimized) set points, and the implementation of these set points. 

Tubular reactors do not necessarily operate under isothermal conditions in industry, be it for reasons of chemical equilibrium or of selectivity, of profit optimization, or simply because it is not economically or technically feasible. It then becomes necessary to consider also the energy equation, that is, a heat balance on a differential volume element of the reactor. For reasons of analogy with the derivation of Eq. 9.1-1 assume that convection is the only mechanism of heat transfer. Moreover, this convection is considered to occur by plug flow and the temperature is completely uniform in a cross section. If heat is exchanged through the wall the entire temperature difference with the wall is located in a very thin film close to the wall. The energy equation then becomes, in the steady state ... 

Let u be the optimal control under steady state with the corresponding state y, and multipliers A, p, and i> all of which are time invariant. According to the John Multiplier Theorem (Section 4.5.1, p. 113), the necessary conditions for the minimum of I subject to the equality and inequality constraints are as follows ... 

Chapter 2 treats the topic of steady-state optimization. Necessary conditions for extrema of functions are derived using variational principles. These steady-state optimization techniques are used for the determination of optimal setpoints for regulators used in supervisory computer control. 

The biggest advantage of a stirred tank reactor is that it operates in a continuous manner under stationary conditions after the steady state has been attained. The product quality is therefore very even. Good heat exchange can be achieved, as reactants are continuously being fed into the reactor. The reactant in-flow can be either cooled or preheated to a suitable temperature as necessary. For specific reactions, such as reversible exothermic reactions, the optimal reaction temperature can be calculated a priori, and the reaction conditions can thereafter be chosen in such a way that the reactor actually operates at this temperature. 

Although the economic benefits from feedback control are not always readily quantifiable, RTO offers a direct method of maximizing the steady-state profitability of a process or group of processes. The optimization of the set points is performed as frequently as necessary, depending on changes in operating conditions or constraints. It is important to formulate the optimization problem carefully a methodology for formulation and solution of optimization problems is presented in this... 

We have emphasized that the goal of this steady-state optimization is to determine ysp and Usp, the set points for the control calculations in Step 6 of Fig. 20.9. But why not use yref and iiref for this purpose The reason is that yref and iiref are ideal values that may not be attainable for the current plant conditions and constraints, which could have changed since yref and iiref were calculated. Thus, steady-state optimization (Step 5) is necessary to calculate ysp and Usp, target values that more accurately reflect current conditions. In Eq. 20-68, ysp and Usp are shown as the independent values for the optimization. However, ysp can be eliminated by substituting the steady-state model, ysp = Kusp. 

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