Vector Acceleration Methods for Faster Convergence of Cyclic Steady State in Adsorption Process Simulations

Simulations of fixed-bed processes with in situ adsorbent regeneration, which are particularly important for adsorption-based separations, require the repeated solution of nonlinear partial differential equations over multiple cycles, starting from predefined initial conditions and continuing until a cyclic steady state (CSS) is reached. This iterative approach, typically based on successive substitution, poses a significant computational challenge, making simulation and optimization time-consuming. The present study systematically investigated several variants of Steffensen's vector acceleration methods for their ability to expedite CSS convergence in adsorption process simulations. These methods are straightforward to implement, requiring no prior knowledge of the first derivatives (or Jacobian). Three different adsorption processes, each with unique cycle dynamics, are considered to be test cases: a four-step vacuum swing adsorption (VSA) cycle, a six-step temperature swing adsorption (TSA) cycle, both designed for postcombustion CO₂ capture, and a three-step vacuum temperature swing adsorption (VTSA) cycle designed for CH₄ upgrading from atmospheric sources. The results show that these methods, particularly the Graves-Morris extrapolator, can consistently enhance the rate of CSS convergence across the different applications. More specifically, in VSA simulations, the computational times are reduced to a quarter of that needed by successive substitution. In TSA and VTSA simulations, the methods reduced times by up to 70 and 35%, respectively. The analysis demonstrated the potential for accelerating CSS convergence through simple, code-agnostic modifications.