Quick robust design for simulated moving bed chromatography under comprehensive uncertainty via robust triangle theory

In this study, we propose a robust triangle theory, which integrates the widely used triangle theory for simulated moving bed (SMB) chromatography with a statistical treatment of uncertainty, enabling the determination of robust operating conditions under comprehensive uncertainty at low computational cost. Process robustness—defined as the probability that the purities of the two product streams, raffinate and extract, satisfy their respective target purities—is a key performance measure in SMB design and operation. The proposed method identifies operating conditions that maintain high process robustness under statistically quantified uncertainties in model parameters, feed concentrations, and pump flow-rate errors. The resulting robust operating conditions, expressed as four dimensionless flow-rate ratios, are obtained from closed-form algebraic expressions, allowing robust design without resorting to computationally intensive dynamic simulations, which makes the method particularly suitable for environments with limited computational resources. The effectiveness of the robust triangle theory is demonstrated through a case study using dynamic SMB simulations with a competitive Langmuir isotherm. Process robustness was estimated using the Bayesian posterior predictive distribution. As a representative result, when the target purities of both products were set to 99.9%, the proposed method identified an operating condition that achieves a joint probability of 0.909 of satisfying both purity specifications. These findings highlight the potential of the robust triangle theory as a fast and scalable design tool that enables uncertainty-aware, high-purity SMB operation without requiring extensive computational resources. • A robust triangle theory for determining SMB operating conditions is proposed. • Uncertainties in model parameters, feed concentration, and flow rate are considered. • Analytically formulated operating conditions require no SMB dynamic simulations. • Process robustness is quantified via joint probability in a Bayesian framework. • Posterior predictive analysis shows a 0.909 probability of achieving ≥99.9% purity.