(134a) Nonsmooth Models and Methods in Chemical Engineering

Nonsmooth process models occupy a useful niche between discrete models with integer variables and traditional smooth models. Nonsmoothness in a model can be used to encode certain discrete features such as transitions in thermodynamic phase or flow regime, process units that switch between discrete operating modes, and pinch analyses for heat integration.

Compared to models with integer variables, nonsmooth model enforce continuity of design variables, which provides structural information that can be exploited by numerical methods. Moreover, allowing nonsmoothness in a model may provide enough versatility to represent discrete behavior that cannot be expressed in a smooth model, and to permit the use of a wider class of methods such as aggressive penalty approaches for constrained optimization. Numerical methods for nonsmooth optimization and equation-solving have convergence properties between those of methods for smooth problems and methods for problems with integer variables.

This presentation covers recent advances in nonsmooth methods and model formulation, based on pioneering work on generalized derivatives by Clarke and Nesterov. These generalized derivatives are designed to serve a similar function to gradients of smooth systems. Recent advances include an accurate, automatable, computationally tractable method for evaluating generalized derivatives for nonsmooth functions that are compositions of simple, known functions and operations. This method has been extended to nonsmooth dynamic systems, and to implicit quantities such as the unknown time at which a relief valve opens. In turn, these methods yield new, effective approaches for designing and simulating multistream heat exchangers with pinch analyses embedded, for example, and for simulating dynamic bioreactors with pseudo-steady metabolic models embedded. Implementations, future directions, and open challenges will also be discussed.