(607c) Reactor Network Design With Guaranteed Robust Dynamic Properties

This work presents a process synthesis method for designing reactor networks with guaranteed robust dynamic properties. Plug flow reactors (PFR), continuous stirred tank reactors (CSTR) and single-loop PI controllers are interconnected in a superstructure comprising all possible reactor networks and its decentralized control system. An economical cost function complemented by a regularization term to penalized excessive control action is minimized. The objective function is subject to mass and energy balances for the reactors, the control law for the controllers and general robust eigenvalue constraints. Parametric uncertainty may either result from model uncertainties, e.g., reaction kinetics or heat and mass transfer coefficients, or from process uncertainties, e.g., slow disturbances in loads or quality of the raw materials. The objective of this paper is to find an optimal reaction process with desired dynamic properties, including the structure of the reactor network and its control system, optimal equipment parameters and controller settings as well as an optimal operating point. The desired dynamic properties are reflected by eigenvalue locations, which are required to be located in a certain region of the complex plane despite parametric uncertainty. PI controllers are used to stabilize the process and to reject disturbances. The work extends the methodology of structured dynamic system (SDS) [ZM11, ZM13] to treat structural design problem with subsystems, which are modeled by partial differential equations (PDE). A case study of allyl-chloride manufacturing is used for demonstrating this new approach [PKN01].

For reactor network design problems, superstructure approaches [Flo95, YG99, Gro05] have been developed for more than 20 years. Although the superstructure approach has been shown to be promising in many applications, few designers take the dynamic properties of the process explicitly into account during the design stage. Hence, the designed reaction process may be difficult to control or to operate. The problem can be overcome if dynamic properties and feedback control are already considered during process design. This approach has been called integration of design and control previously (cf. [BPP+00, KP01, NP94, MPP96, MPP97, BB03, RS08] for a few selected publications). 

The dynamic properties of a process can be characterized locally by its eigenvalues. For example, local stability of an operating point requires all eigenvalues to be located in the left hand side of the complex plane. It is also desirable that, all eigenvalues are not too close to the imaginary axis, such that the system can properly respond to disturbances. In this paper, a favorable dynamic response is supposed to be guaranteed despite the presence of parametric uncertainty by requiring that all the eigenvalues appear only in certain desired region of the complex plane [MM02, MM05]. 

The integration of optimally designing reactor networks and their control system with guaranteed favorable dynamic properties is difficult, since a large number of structural alternatives exist and a direct treatment of eigenvalues is theoretically and computationally involved [LO96, KQTY11]. Since constraints which are involving the eigenvalues of a state- or parameter-dependent matrix do not satisfy the smoothness requirements of nonlinear programming, established numerical techniques cannot be applied straightforwardly. 

The optimal design of reactor networks with guaranteed dynamic properties in the presence of parametric uncertainty can be accomplished by applying the structured dynamic systems (SDS) modeling concept [ZM11, ZM13]. Structured dynamic systems (SDS) refers to a modeling concept, in which a dynamic system consists of several ”subsystems”, and those subsystems are connected with each other through their inputs and outputs [ZM13]. Each connection may either exist or not. For reactor network design problems, each reactor can be seen as a dynamic subsystem, possibly connected by material and energy streams. Once we know the status of these connections, the structure of the flowsheet is fixed. 

The design of closed-loop reactor networks involving PFR and CSTRs is an extend application of SDS [ZM13], where PFR are modeled by PDE, CSTR and controllers are modeled by ODE systems. In order to treat the subsystems modeled by PDE as standard subsystems in SDS, some semi-discretization methods such as a variant of the method of lines has to be applied. After discretization, we obtain a SDS with only subsystems modeled by ODE, and all results proved in [ZM13] can be applied straightforwardly. The problem is solved by applying the normal vector approach [MM02, MM05]. The solution results in an optimal structure of the reactor network and its control system, an optimal operating point and the associated equipment and controller parameters. All eigenvalues of the process are located in the desired region despite parametric uncertainty. 

The methodology is applied to a simple case study of allylchloride manufacturing. Allylchloride is manufactured by means of non-catalytic chlorination of propylene in the vapor phase [PKN01]. A reactor network superstructure comprising a PFR and a CSTR is set up accounting full connectivity. The objective is to maximize an objective function, which includes the revenue of the product and the cost of raw materials. In order to compare the results with or without eigenvalue constraints, we solved the problem first without and afterwards with eigenvalue constraints. The proposed methodology guarantees an economical optimal solution, and makes sure that all eigenvalues of the closed-loop system are located in a required region of the complex plane despite parametric uncertainty. Further work will consider a distillation section and will rigorously cover the control system structure problem in the process design.

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