An Inexact Trust-Region Algorithm for the Optimization of Periodic Adsorption Processes

Periodic adsorption processes have gained increasing commercial importance as an energy-efficient separation technique over the past two decades. Based on fluid solid interactions, these systems never reach steady state. Instead they operate at cyclic steady state, where the bed conditions at the beginning of the cycle match with those at the end of the cycle. Nevertheless, optimization of these processes remains particularly challenging, because cyclic operation leads to dense Jacobians, whose computation dominates the overall cost of the optimization strategy. To efficiently handle these Jacobians during optimization and reduce the computation time, this work presents a new composite step trust-region algorithm for the solution of minimization problems with both nonlinear equality and inequality constraints, and combines two approaches developed in Walther(1) and Arora and Begler.(2) Instead of forming and factoring the dense constraint Jacobian, this algorithm approximates the Jacobian of equality constraints with a specialized quasi-Newton method. Hence it is well suited to solve optimization problems related to periodic adsorption processes. In addition to allowing inexactness of the Jacobian and its null-space representation, the algorithm also provides exact second-order information in the form of Hessian-vector products to improve the convergence rate. The resulting approach(3) also combines automatic differentiation and more sophisticated integration algorithms to evaluate the direct sensitivity and adjoint sensitivity equations. A 5-fold reduction in computation is demonstrated with this approach for two periodic adsorption optimization problems: a simulated moving bed system and a nonisothermal vacuum swing adsorption O-2 bulk gas separation.