## A Primal-Dual Active-Set Method and Algorithm for Chemical Equilibrium Problem Related To Modeling of Atmospheric Inorganic Aerosols

##### Abstract

In this dissertation, we studied a general equilibrium model for multiphase multicomponent inorganic atmospheric aerosols. We developed the thermodynamic model to predict the phase transition and multistage growth phenomena of inorganic aerosols. The thermodynamic equilibrium is given by the minimum of the Gibbs Free Energy for a system involving an aqueous phase, a gas phase and solid salts. A primal-dual algorithm for solving the {\em Karush-Kuhn-Tucker} conditions is one of the main focus of the model. We applied an active set and the Newton method to compute the minimum of energy and determined if solid salts exist or not at the equilibrium.
We presented that the model were set up based on the mass balance equations and the minimization of the Gibbs Free Energy. We developed a mathematical framework for modeling solid-liquid equilibrium reactions that was based on the canonical stoichiometry of the inorganic aerosols. We showed detailed work on how to model a typical system of inorganic aerosols at equilibrium. We demonstrated how the active set method was applied in two modeling problems. One was for general chemical equilibrium problem. Another one was to extend the current modeling problem to investigate the system at fixed relative humidity.
Numerical results of the model are included to show the efficiency of the algorithm for the prediction of multiphase multireaction chemical equilibria. We used typical inorganic aerosol systems for this purpose. One system was the sulfate aerosols which included ammonium sulfate $\displaystyle{(\mathrm{NH_4})_2\mathrm{SO}_4}$, sulfuric acid $\mathrm{H}_2\mathrm{SO}_4$
and water $\mathrm{H}_2\mathrm{O}$. Another system had two type of aerosols: urban and remote continental. This system consisted of water $\mathrm{H}_2\mathrm{O}$, sulfuric acid $\mathrm{H}_2\mathrm{SO}_4$, nitric acid $\mathrm{H}\mathrm{NO_3}$, and ammonia $\mathrm{NH_3}$. From the results, we demonstrated that the model was capable of computing phase behavior of inorganic aerosols efficiently and rigorously. It also computed the deliquescent behavior of the system.