Nonlinear Control Of A Shell And Tube Heat Exchanger

Abstract

The paper deals with design and simulation of nonlinear adaptive control of a shell and tube heat exchanger. The method is based on factorization of the controller on a nonlinear static part and an adaptive linear dynamic part. The nonlinear static part is derived using inversion and subsequent exponential approximation of simulated or measured steady-state characteristics of the exchanger. The linear dynamic part is then obtained from an external linear model of nonlinear elements of the closed-loop. The parameters of the external linear model are recursively estimated via a corresponding delta model. The control law in the 1DOF and 2DOF control system structures is derived using the polynomial approach. INTRODUCTION Heat exchangers are an essential part of many technologies in energy and chemical industry, polymer manufacturing, petroleum refineries, and many others. By construction, heat exchangers can be classified into exchangers with direct contact, various types of plate exchangers, and, shell and tube heat exchangers (STHEs), see, e.g. (Smith 2005; Hewitt et al. 1994; Incropera et al. 2011). As known, STHEs are most common types of heat exchangers. From the system theory, they belong to a class of nonlinear distributed parameter systems with mathematical models in the form of nonlinear partial differential equations. Modelling and simulation of such processes are described in many publications, e.g. in (Luyben 1989; Corriou 2004; Babu 2004; Ogunnaike and Rao 1994). As known, these processes can be hardly controllable by conventional methods that can lead to control of a poor quality. In this case, some advanced control methods should be used such as adaptive, predictive, optimal or nonlinear control, and some others. Obviously, control design always requires a preliminary steady-state and dynamic analysis of the process by simulation tools. Some methods of numerical mathematics used to build simulation models can be found e.g. in (Nevriva et al. 2009; Cook 2002). The aim of the paper is an application of nonlinear control and subsequent control simulation of a simple type of the shell and tube heat exchanger. The control strategy is based on the idea of factorization of the controller on a nonlinear static part (NSP) and an adaptive linear dynamic part (LDP). Similar approaches can be found e.g. in (Chen et al. 2006; Dostal et al. 2011b). The nonlinear static part is obtained from simulated or measured steady-state characteristic of the STHE, its inversion, exponential approximation, and, subsequently, its differentiation. On behalf of development of the linear part, the NSP including the nonlinear model of the STHE is approximated by a continuous-time external linear model (CT ELM). For the CT ELM parameter estimation, an external delta model with the same structure as the CT model is used. The basics of delta models have been described e.g. in (Mukhopadhyay et al. 1992; Garnier and Wang 2008). Although delta models belong into discrete models, they do not have such disadvantageous properties connected with shortening of a sampling period as discrete zmodels. In addition, parameters of delta models can directly be estimated from sampled signals. Moreover, it can be easily proved that these parameters converge to parameters of CT models for a sufficiently small sampling period (compared to the dynamics of the controlled process), as shown in (Stericker and Sinha 1993; Dostal et al. 2004). The 1DOF and the 2DOF control system structures are considered. In the first case, the control system includes only a feedback controller, in the second case, the controller consist of a feedback and a feedforward part. Such structures were described and applied e.g. in (Dostal et al. 2011a; Grimble 1993). Then, resulting CT controllers are derived using the polynomial approach and the pole placement method (Kucera 1993; Mikles and Fikar 2004). The simulations are performed on a nonlinear model of the STHE. MODEL OF THE STHE Consider an ideal plug-flow shell and tube heat exchanger in the fluid phase and with the counterflow cooling. The fluid flowing in tubes is cooled by a fluid flowing in the shell as shown in Fig. 1. Heat losses and heat conduction along the metal walls of tubes are Proceedings 28th European Conference on Modelling and Simulation ©ECMS Flaminio Squazzoni, Fabio Baronio, Claudia Archetti, Marco Castellani (Editors) ISBN: 978-0-9564944-8-1 / ISBN: 978-0-9564944-9-8 (CD) assumed to be negligible, but dynamics of the metal walls of tubes are significant. All densities, heat capacities, and heat transfer coefficients are assumed to be constant. Under above assumptions, the STHE model can be described by three partial differential equations (PDEs) in the form