Efficient Computation of Cyclic Steady States in Periodic Adsorption Processes Using the JFNK Method
Richard Pattisona, Pieter Schmalb, and Costas Pantelidesb
aMcKetta Department of Chemical Engineering, The University of Texas at Austin
bProcess Systems Enterprise Ltd.
Periodic adsorption processes (PAPs) are attractive alternatives to traditional distillation and absorption separation systems due to their relatively low capital costs and energy requirements . PAPs have found many industrial applications , and recently have gained attention as an option for hydrogen purification for electricity generation and other purposes [3-4]. Optimizing the design of these processes is critical to realizing the potential cost and energy savings.
The modeling, simulation and optimization of PAPs is challenging for several reasons. The mathematical models of these systems are described by nonlinear partial differential algebraic equations (PDAEs), with variation in both temporal and spatial domains, and the boundary conditions change several times during each cycle simulation. A particularly challenging aspect is the need to compute the cyclic steady state (CSS) behavior of a given PAP, i.e., the state in which the variable trajectories are identical in successive cycles. This is of the most practical interest for industrial purposes.
The conventional method for calculating the CSS is to simply integrate the process model until there is no change in the state trajectories between iterations. While this is an effective and proven method, it can often require the integration of many cycles (e.g., up to 4000 have been reported ) to reach the CSS. Other methods have proposed formulating the problem as a system of nonlinear equations which require the values of the state variables at the beginning and end of each cycle to be identical. However, the relation between these two sets of variable values is determined by the temporal integration of a PDAE system over a cycle; calculating the Jacobian that is required for each Newton iteration is computationally expensive, and for large systems it may be practically intractable . An alternative approach is to simultaneously discretize both the temporal and spatial variations within the underlying PDAEs, thereby obtaining a very large set of nonlinear algebraic relations which can then be solved using a sparse solver . However, this imposes severe demands on both storage and computation; moreover, using a fixed discretization scheme may fail to result in the required accuracy of solution.
In this work, we propose an algorithm for computing the CSS in PAPs based on the Jacobian-Free Newton-Krylov (JFNK)  method. The latter provides the benefit of super-linear convergence without the need to form or store the actual Jacobian. The linear system required for the Newton update is solved by an iterative linear solver which uses accurate approximations of Jacobian-vector products to minimize the linear residual. The Jacobian-vector product required at each iteration is obtained via the simulation of a single cycle of the PAP.
Whilst JFNK methods are conceptually simple, their efficient operation is predicated on obtaining sufficiently fast convergence of the iterative linear solver. This is typically achieved via the choice of an appropriate preconditioner. We propose a novel preconditioner that can result in significant improvements in performance both for single “one-off” CSS determination and in the context of parametric studies where a sequence of CSSs need to be calculated for different values of process design and operation parameters.
We illustrate the benefits of the proposed method in the simulation of PAPs using a rapid pressure swing adsorption (RPSA) process and a multi-bed PSA process as case studies.
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