On POD reduced models of tubular reactor with periodic regimes

Abstract

The distributed model of a tubular reactor with recycle is reduced by approximating it first with a Continuous Stirred Tank Reactor (CSTR) cascade. Proper orthogonal decomposition (POD) with Galerkin projection is introduced to study oscillatory regimes. The dynamics of the resulting models is studied via numerical simulation, and solutions in the form of time series and phase plot diagrams are compared to those obtained for the original CSTR cascade model. Different methods to choose the minimum number of basis functions are compared and discussed. Solution diagrams built with POD models are compared with those from CSTR cascade, as a function of the Damköhler number. Features and limitations of POD models are discussed for different snapshot sampling policies and for different values of the Péclet number. Qualitative performance increases and quantitative performance decreases as samples from steady states are considered along with periodic solution samples. Performance becomes worse as the Péclet number increases.

Introduction

Many chemically reactive systems are characterized by parametric sensitivity that results in the occurrence of a wide variety of static and dynamic phenomena (Aris, Aronson, & Swinney, 1991). Depending on the operating conditions or on the values assumed by different parameters of the model which describes the physical and chemical properties of the system, static equilibrium as well as more complex time-asymptotic regimes such as periodic, quasi-periodic or chaotic oscillations can be observed. To describe all possible states and determine values of parameters for which dramatic qualitative or quantitative changes of the solution occur, it is necessary to conduct an accurate bifurcation analysis in the parameter space. In order to accomplish this, the original PDE evolutionary equations, which in most cases constitute the model, are to be reduced to a finite-dimensional dynamical system. This is typically done by numerical discretization of the spatial differential operators with finite difference schemes or various spectral methods (see, for example, Quarteroni & Valli, 1994, chap. 5). The resulting reduced model can then be analyzed in its dynamical behaviour via bifurcation analysis (Kuznetsov, 1998). In spite of the availability of various packages for performing bifurcation analysis (Doedel et al., 2001), often the problem becomes very difficult because the order of the dynamical model required to properly describe the system is very high. This, coupled with obvious limitations of software and hardware, leads to the need of expressing the original model as a set of ordinary differential equations of the lowest possible order. Among classical numerical approaches, finite difference methods are simple but require a relatively large number of ODE as compared, for instance, to orthogonal collocation (Villadsen & Michelsen, 1978). Other spectral methods have been used for reactive systems such as Galerkin projection (see, for example, Oran & Boris, 1987). Projection methods provide an interesting framework in that, by proper choice of the functional basis, one can reduce the number of ODEs necessary to accurately describe the dynamics of the original PDE model. Many applications adopt as functional basis a set of eigenfunctions of the main differential operator in the model equations (Adrover, Cerbelli, & Giona, 2002) which by the way in our case (Laplacian expressing molecular diffusion) is a self-adjoint operator and thus it has a set of orthonormal eigenfunctions.

The number of basis functions chosen for the approximation is rather critical: Graham and Kevrekidis (1996) showed that an insufficient number of basis functions – in their case Chebyshev polynomials – can cause that significant phenomena such as period doubling are not detected. Hence, they proposed to use basis determined by proper orthogonal decomposition, with some extension of the original procedure that yield optimal basis for parameter varying applications. Proper orthogonal decomposition (POD) is a procedure that delivers an optimal set of empirical basis functions from an ensemble of observations obtained either experimentally or from numerical simulation, which characterize the spatio-temporal complexity of the system (see, for example, Holmes & Lumley, 1996). Obtained orthogonal functions can be afterwards used in a Galerkin projection of the original system and as a result a low-dimensional model can be developed (Lumley, 1967). POD has been widely used in disciplines such as control of chemical reactors (Padhi & Balakrishnan, 2003; Shvartsman et al., 2000), modelling of non-reactive fluidized beds (Cizmas, Palacios, O’Brien, & Syamlal, 2003; Yuan, Cizmas, & O’Brien, 2005), as well as in reducing complex chemical kinetic models (Danby & Echekki, 2005). Besides, it has been applied in many engineering fields, among other things to damage detection (Joyner, 2004), low-order modelling of flows (Sirsup, Karniadakis, Xiu, & Kevrekidis, 2005) or microelectromechanical systems (Liang, 2002) and, depending on the field of application, it is also known as Principal Components Analysis (PCA) (Jolliffe, 2002), Singular Value Decomposition (SVD) (Grossman, Kamath, & Kegelmeyer, 2001, chap. 22), Karhunen–Loève (KL) decomposition (Kirby & Sirovich, 1990) or Hotelling transformation (Eriksson, Jiménez, Bühler, & Murtagh, 2002). Some extensions of the procedure, such as application of POD to moving boundary problems of fluid flow was studied by Utturkar, Zhang, and Shyy (2004), whereas Gokulakrishnan, Lawrence, McLellan, and Grandmaison (2006), introduced a functional approach (functional-PCA) with application to oxidation processes.

In this paper, we consider a model of pseudohomogeneous tubular reactor with mass recycle. This system exhibits the so-called “relaxation oscillations” (Phillipson & Schuster, 2001; Smuła, 2006), an oscillatory regime in which a fast variation of the state is followed by a slow “relaxation” phase. Such type of regime is difficult to handle from a numerical point of view. Moreover, periodic regimes retain both fast and slow modes. These two features identify this system as an appropriate benchmark for testing reduced models. We apply POD/Galerkin and analyze the performance of the method by comparing solutions from the reduced model with a “reference” numerical solution.