Chemical Engineering Science 57 (2002) 4145–4159 www.elsevier.com/locate/ces Perturbation techniques for accelerated convergence of cyclic steady state (CSS) in oxygen VSA simulations Simon J. Wilson, Paul A. Webley * Department of Chemical Engineering, Monash University, Clayton, Vic. 3168, Australia Received 11 October 2001; accepted 6 April 2002 Abstract Non-isothermal PSA bulk gas separations are notoriously slow to converge to cyclic steady state (CSS). Most numerical simulators that model the slow transient behaviour of these processes share this slow pace of convergence. This diculty with numerical simulators has long been recognised, and has hampered the optimisation of oxygen VSA. This paper outlines the application of perturbation techniques to enable more rapid determination of CSS temperature proles. A number of dierent techniques are proposed. The simplest approach based on experimental observations involves a two-time-scale decomposition of the adsorbent temperature prole. This decomposition of the temperature term enables the rapid numerical determination of the fast (or adsorptive) component of the temperature, followed by the direct determination of an estimated cyclic steady-state slow time-scale (or convective) temperature. It is demonstrated that this approach is the same as a zeroth-order multiple scale analysis (MSA) approach and a simple application of the Krylov–Bogoliubov (K–B) method of averaging. This study compares the results of this acceleration technique for determining CSS with a full numerical model. The results indicate that the proposed acceleration technique based on perturbation methods provides fast and reasonable estimates of the CSS temperature prole with some limitations. Recognising these limitations of the zeroth-order approach, a rst-order MSA is developed. This compares better with full numerical simulations, and could be used to augment and complement existing acceleration techniques. It is shown that a rst-order MSA approach can also be used to capture the dynamic temperature response of the oxygen VSA process, in addition to the CSS axial temperature prole. The perturbation techniques presented here are limited to a series expansion of the temperature term in the energy balance, and mass balance is ignored in this analysis. A procedure is outlined where a K–B approach can be used to incorporate a perturbation series expansion in the mass balance. We show that these perturbation techniques not only oer potential for simulation acceleration, but also oer useful insights into the thermal convergence to CSS in oxygen VSA. ? 2002 Elsevier Science Ltd. All rights reserved. Keywords: Oxygen VSA; Cyclic steady state; Temperature proles; Accelerated convergence; Numerical simulations; Multiple scale analysis; Perturbation techniques 1. Introduction Oxygen VSA represents an example of a non-isothermal, cyclic adsorption, bulk gas separation which is notoriously slow to converge to cyclic steady state (CSS). Typically, the process requires in excess of 1000 cycles (12–15 h) to converge to CSS within a reasonably loose tolerance in proleerror.Thisslowpaceofconvergenceisalsoevidentin most numerical models that rely on successive substitution. The inability to accelerate numerical solutions has hampered ∗ Corresponding author. Tel.: +61-3-9905-1874; fax: +61-3-9905-5686. E-mail address: paul.webley@eng.monash.edu.au (P.A. Webley). the optimisation of oxygen VSA design and operation. With many existing numerical simulators, it is a very time con- suming and arduous process to simulate dierent oxygen VSA designs and process conditions to enable a direct cost comparison of dierent options. This problem has long been recognised and this study seeks to contribute to the eld of acceleration techniques for oxygen VSA and other PSA separations based on physi- cal observations and perturbation techniques rather than nu- merical observations. The benet of the proposed scheme over previous acceleration techniques is that it oers ad- ditional physical insight into mechanisms underlying the convergence rate of cyclic adsorption systems. By way of introduction, we briey review some existing acceleration techniques. 0009-2509/02/$ - see front matter ? 2002 Elsevier Science Ltd. All rights reserved. PII:S0009-2509(02)00365-2