Over the last two decades, a succession of thermodynamic modules (EQUIL, MARS, SEQUILIB, SCAPE, SCAPE2, EQUISOLV II and ISORROPIA, has been developed to predict the phase transition and multistage growth phenomena of inorganic aerosols. These modules calculate the composition of atmospheric aerosols by solving a set of nonlinear algebraic equations derived from chemical equilibrium relations. One of the most challenging parts for them is the prediction of the partitioning of the inorganic aerosol components between aqueous and solid phases. A major weakness in these modules lies in the way they treat the transition between aqueous and solid aerosol phases. By relying on a priori and often incomplete knowledge of the presence of solid phases at a certain relative humidity and overall composition, these modules often fail to accurately predict the phase state and composition and the multistage growth phenomena of inorganic aerosols.
On the other hand, thermodynamic models that are based on the minimization of the Gibbs free energy (GFEMN or AIM) implicitly predict phase transition and multistage aerosol growth without any a priori knowledge of the behavior of inorganic aerosols. However, such direct minimization of the Gibbs free energy is computationally intensive making its use in 3D air quality models infeasible.
The goal of this project is to develop an algorithm for the accurate and computationally efficient prediction of partitioning. A primal-dual active set algorithm for the efficient and accurate prediction of the phase transition and multistage growth phenomena of inorganic aerosols is developed.
The mathematical framework for modeling solid-liquid equilibrium reactions is based on the canonical stoichiometry of inorganic aerosols.
The canonical form is elucidated from the analysis of the algebraic structure of the aqueous electrolyte solution system and the Karush-Kuhn-Tucker (KKT) conditions for the constrained minimization of the Gibbs free energy. In the canonical stoichiometry, the concentrations of solid species in solid-liquid equilibrium are interpreted as the Lagrange multipliers of dual linear inequality constraints. This primal-dual relation is the key for the development of our primal-dual active set algorithm, whose principle features can be summarized as follows:
- The algorithm applies Newton's method to the reduced KKT system of equations projected on an active set of solid phases to find the next primal-dual approximation of the solution.
- The active set method permits us to add a solid salt when it reaches saturation, while delete a solid phase from the active set when its concentration does not satisfy the non-negativeness constraint.
- The active set method permits us to add/delete salts to/from a working set of saturated salts until the equilibrium set of solid phases is obtained.
- The linear inequality constraints are enforced on the dual variables such that salts remain sub-saturated with respect to the aqueous electrolyte solution, until the saturation is reached at an iteration, where the corresponding inequality constraint becomes active and the saturated salt is added into the active set.
- The concentrations of the saturated salts in the active set are the Lagrange multipliers of the dual active constraints so that their non-negativeness is enforced by deleting a saturated salt from the active set when its concentration becoming negative.
- A second order stability criterion is implemented by keeping the reduced Hessian of the Gibbs free energy positive definite so that the algorithm converges to a stable equilibrium rather than any other first order optimality point such as a maximum or a saddle point.
The more precise description of the method can found in the article "A Primal-Dual Active Set Algorithm for Chemical Equilibrium Problems Related to the Modeling of Atmospheric Inorganic Aerosols".
The UHAERO module 1 software for thermodynamical equilibrium of inorganic aerosols can be found here.
