Balance equations optimal control

It is becoming common practice, in today s chemical plants, to incorporate some kind of technique to rectify or reconcile the plant data. These techniques allow adjustment of the measurement values so that the corrected measurements are consistent with the corresponding balance equations. In this way, the simulation, optimization, and control tasks are based on reliable information. Figure 3 shows schematically a typical interconnection between the previous mentioned activities (Simulation Sciences Inc., 1989). 

We have used sensitivity equation methods (Leis and Kramer, 1985) for gradient evaluation as these are simple and efficient for problems with few parameters and constraints. In general, the balance in efficiency between sensitivity and adjoint methods depends on the type of problem being addressed. Adjoint methods are particularly advantageous for optimal control problems in which the inputs are represented as a large number of piecewise constant input values and few interior point constraints exist. Sensitivity methods are preferable for problems with few parameters and many constraints. 

A multiobjective optimization problem is formulated for the MTBE RD column with respect to economic performance and exergy efficiency. The formulation includes the balance equations 8.1-8.5 and 8.29, the criteria definitions 8.42 and the optimization formulation 8.62. Constraints imposed by operating conditions and product specifications are included. For the sake of controlled built-up of optimization complexity the first approximation of this approach omits the response time constant as objective function. 

We clarify this for stationary plug flow along a tubular reactor. In optimal control theory one rewrites the balance equations in the form... 

As indicated in the previous chapter, models are used for a variety of applications, such as study of the dynamic behavior, process design, model-based control, optimization, controllability study, operator training and prediction. These models are usually based on physical fundamentals, conservation balances and additional equations. In this chapter the conservation balances for mass, momentum, energy and components are introduced. 

Physical models, based on balance equations empirical relations of the crystallisation kinetics are used to design feedback controllers that steer the unseeded batch crystalliser from an initial state to a final optimal state. 

Process simulators contain the model of the process and thus contain the bulk of the constraints in an optimization problem. The equality constraints ( hard constraints ) include all the mathematical relations that constitute the material and energy balances, the rate equations, the phase relations, the controls, connecting variables, and methods of computing the physical properties used in any of the relations in the model. The inequality constraints ( soft constraints ) include material flow limits maximum heat exchanger areas pressure, temperature, and concentration upper and lower bounds environmental stipulations vessel hold-ups safety constraints and so on. A module is a model of an individual element in a flowsheet (e.g., a reactor) that can be coded, analyzed, debugged, and interpreted by itself. Examine Figure 15.3a and b. 

In order for a process to be controllable by machine, it must represented by a mathematical model. Ideally, each element of a dynamic process, for example, a reflux drum or an individual tray of a fractionator, is represented by differential equations based on material and energy balances, transfer rates, stage efficiencies, phase equilibrium relations, etc., as well as the parameters of sensing devices, control valves, and control instruments. The process as a whole then is equivalent to a system of ordinary and partial differential equations involving certain independent and dependent variables. When the values of the independent variables are specified or measured, corresponding values of the others are found by computation, and the information is transmitted to the control instruments. For example, if the temperature, composition, and flow rate of the feed to a fractionator are perturbed, the computer will determine the other flows and the heat balance required to maintain constant overhead purity. Economic factors also can be incorporated in process models then the computer can be made to optimize the operation continually. 

A linear model predictive control law is retained in both cases because of its attracting characteristics such as its multivariable aspects and the possibility of taking into account hard constraints on inputs and inputs variations as well as soft constraints on outputs (constraint violation is authorized during a short period of time). To practise model predictive control, first a linear model of the process must be obtained off-line before applying the optimization strategy to calculate on-line the manipulated inputs. The model of the SMB is described in [8] with its parameters. It is based on the partial differential equation for the mass balance and a mass transfer equation between the liquid and the solid phase, plus an equilibrium law. The PDE equation is discretized as an equivalent system of mixers in series. A typical SMB is divided in four zones, each zone includes two columns and each column is composed of twenty mixers. A nonlinear Langmuir isotherm describes the binary equilibrium for each component between the adsorbent and the liquid phase. 

The second factor arises from the need to balance maximization of directed separation versus minimization of random ion mixing due to diffusion, space charge effects, and flow and field imperfections, as discussed above. Unlike the last two, the diffusion depends on E/N (2.2.4) and thus minimizing it is a part of the F(t) optimization. Our concern here is with the diffusion along E controlled by the longitudinal diffusion coefficient, Du (2.2.4). Using Equations 2.22 and 2.27, the mean Du over the F(t) by Equation 3.9 is... 

The examples presented have shown that the construction of simple unstructured models based on some notion of relevant kinetic equations in combination with the very important concept of elemental composition and enthalpy balance is of great help in understanding factors relevant in the optimization and control of even such complex fermentation processes as the production of antibiotics. 

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