(47e) Optimization of an Air Separation Column Using a Complementarity-Based Vapor-Liquid Equilibrium Formulation

Vapor-liquid equilibrium (VLE) calculations are present in a wide range of applications. They are used in large-scale processes like heat exchanger networks, distillation columns, emulsification and natural gas network pipelines in dynamic simulations and equation-oriented (EO) flowsheet optimization. Given their wide-ranging applications, the importance of performing VLE (or flash) calculations reliably and efficiently cannot be overstated. Flash calculations based on non-ideal thermodynamic relationships are well-known to be challenging to incorporate and solve in EO flowsheet optimization. Traditional algorithms based on iterative approaches are susceptible to failure, especially in operating conditions corresponding to the existence of a single phase, since VLE relationships are only valid in the two-phase region between the bubble point and the dew point. Moreover, these algorithms are especially not suitable when the system is defined by non-ideal thermodynamic models such as cubic equations of state.

An equation of state represents the thermodynamic relationship between two or more state functions (temperature, pressure, volume etc.) to describe the properties of fluids and their mixtures. Cubic equations of state are desirable models for VLE because they represent physical phenomena with reasonable accuracy and require only moderate computational complexity for a variety of compounds (with appropriate mixing rules and interaction parameters.) Kamath et al. (2010) introduced an optimization-based equation-oriented VLE formulation that leverages the properties of the derivatives of the cubic equation of state to identify and isolate roots corresponding to the appropriate phases, as governed by the operating conditions. The properties are incorporated as constraints along with the governing equations for VLE and an objective function that describes the disappearance of a phase in case of single-phase solutions is added. Burgard et al. (2018) introduced a smooth, square EO formulation that takes into account the calculation of bubble and dew points. This formulation is based on smooth approximations of complementarity relations and is valid for both subcritical and supercritical conditions.

In this work, we present an improved EO formulation for single-stage flash. This is a smooth, square formulation that obviates the calculation of bubble and dew points to define the two-phase region. Instead, an equilibrium temperature is defined using complementarities with the vapor and liquid flows. Moreover, the formulation identifies the appropriate phase as dictated by the operating conditions using derivatives of the cubic equation of state. The formulation shows improved tractability compared to previous works and is successfully tested on various systems for both subcritical and supercritical conditions. A performance comparison with the work of Burgard and co-workers reveals significant improvement in convergence, especially for conditions close to the critical point, as well as shorter initialization and solution times using the large-scale nonlinear equation solver IPOPT. Moreover, we embed this single-stage formulation into a tray distillation column for an air separation mixture and demonstrate its optimization to minimize the reboiler duty while satisfying purity constraints for the top and bottom products.

The tray distillation column model and the single-stage flash formulation are part of the unit model and property model library, respectively, in the IDAES Process Systems Engineering framework, an open-source platform that modularizes unit operations and flowsheet construction, and is based on Pyomo, an open-source framework for modeling and optimization. Using AMPL Solver Library’s (ASL) external functions, the calculation and isolation of the cubic roots and derivatives is kept outside the modeling framework. This ensures that no additional variables or constraints are added to the flash formulation. It also ensures that only the correct root impacts the solution, and that the model stays smooth and square. We incorporate the VLE formulation into IDAES’ tray distillation column and demonstrate the steady-state optimization of the column for an air separation mixture. Specifically, the reboiler duty, a major operating cost, is minimized while ensuring that purity constraints for the top and bottom products are satisfied. The stages of the distillation column are represented by the proposed EO formulation for flash calculations. To demonstrate the handling of missing phases, we present several case studies with differing minimum purity specifications for the top and bottom products.

References:

Kamath, R. S., Biegler, L. T., & Grossmann, I. E. (2010). An equation-oriented approach for handling thermodynamics based on cubic equation of state in process optimization. Computers & Chemical Engineering, 34(12), 2085-2096

Burgard, A. P., Eason, J. P., Eslick, J. C., Ghouse, J. H., Lee, A., Biegler, L. T., & Miller, D. C. (2018). A smooth, square flash formulation for equation-oriented flowsheet optimization. In Computer aided chemical engineering (Vol. 44, pp. 871-876). Elsevier