Fast solution-adaptive finite volume method for PSA/VSA cycle simulation; 1 single step simulation

Abstract

Accurate numerical solution of cyclic adsorption problems can be challenging and time consuming due to the inherent non-linearity of the system conservation equations and the presence of strong temporal and spatial gradients. In addition, the wide variety of possible boundary conditions makes solution of a general cycle problematic. In this study we present a quadratic upwind finite volume technique that is at least second-order accurate in the spatial dimension and can easily handle any type of boundary condition (both one and two point) at either end of the adsorption column. The effect of bed pressure drop is readily accommodated. The oscillations at sharp gradients common in high-order upwind methods is significantly reduced by a solution-adaptive strategy which does not introduce significant diffusion unlike first order upwind methods. To ensure accurate simulation of sharp fronts, we find that the finite volume node width should be approximately one tenth of the width of the mass transfer zone. A comparison is made for linear isotherm systems with the analytic solution and excellent agreement is produced for very moderate grid sizing. The method is fast and robust and is ideally suited to cycle simulation.

Introduction

Pressure swing adsorption (PSA) and, more recently, vacuum swing adsorption (VSA) has become a popular gas separation technology for small-scale production of oxygen and nitrogen as well as purification of hydrogen and natural gas. Currently, design of PSA systems relies heavily on experimental data for the system of interest. Experiments are conducted over a wide variety of process conditions (temperature, pressure, purge/feed rates, steps times, etc.) for a specified adsorbent(s) and cycle. Performance data are collected and the procedure is repeated for additional cycles and/or adsorbents. Changing the column configuration experimentally and repeating the above experiments allows a good understanding to be achieved of the relative importance of all the design variables. However, the large number of experimental variables (even with a well-designed experimental test) and the long times required for achieving cyclic steady state implies a long and expensive test program. This is especially true if multicomponent systems are to be studied and the effect of feed composition on system performance evaluated. The use of a computational model rapidly becomes a necessity. However, the computational model must be accurate enough to match the experimental system closely and must be faster that the real time taken to perform experiments.

Many workers have developed detailed numerical models to simulate simplified PSA processes. A variety of numerical techniques have been employed to solve the coupled non-linear conservation PDE's such as moving and fixed finite element methods (Yu & Wang, 1989), moving grid finite difference schemes (Sun & Meunier, 1991), combined finite difference/method of characteristic methods (Wang, Liou & Chang, 1980), orthogonal collocation (Sristava & Joseph, 1984), fast Fourier transform (Mees, Gerritsen & Verheijen, 1989Sun & Costa, 1992), characteristics (MOC) (Loureiro & Rodrigues, 1991) and the method of lines (MOL) (Kumar et al., 1994). In general, sophisticated models exist for adsorption simulation but are computationally expensive frequently taking longer than real time to converge to a cyclic steady state. There is a need for a flexible and validated PSA model, which is sufficiently complex to capture real effects present in industrial systems, yet simple enough to permit efficient computation. Most importantly, the model must be capable of simulating the actual boundary conditions used in operating a PSA process. Since pressure history (P vs. t) is not usually specified when operating a cycle (other than end of step pressures), the simulation should produce this history as a consequence of system boundary conditions, rather than requiring P(t) as input.

In this paper, a finite volume method is described for solution of the PSA/VSA equations. Quadratic upstream interpolation is used to approximate wall values of composition but it is shown that for simulation of systems close to equilibrium, an additional solution-adaptive strategy is required to yield an approximately monotonic solution to the simple step input. While solution schemes which are higher than first order (in spatial discretization) cannot be strictly monotonic, the scheme implemented here gives satisfactory solution to the step response without introducing excessive dispersion so characteristic of simple first order upwind interpolation. We also show that our finite volume formulation with appropriate extrapolations is flexible enough to accommodate boundary conditions for any type of cycle step relevant to operation of a PSA/VSA system. We demonstrate the effectiveness of this technique by comparison with exact analytic solutions for linear isotherms. The technique described here is extremely fast and is suitable for implementation in a more general cycle simulator that we describe in a future work.

It is not the intent of this paper to compare the finite volume technique to other numerical techniques (e.g. orthogonal collocation on finite elements, method of characteristics, etc.). It is recognized that other techniques may offer advantages in certain respects for certain cases — for our case, an accurate and fast solution technique was required since the ultimate purpose of the model is cycle simulation, evaluation, and optimization. These requirements are adequately met by the finite volume method with suitable higher-order interpolation schemes.