Sequential optimal experimental design for vapor-liquid equilibrium modeling

Abstract

We propose a general methodology of sequential locally optimal design of experiments for explicit or implicit nonlinear models, as they abound in chemical engineering and, in particular, in vapor-liquid equilibrium modeling. As a sequential design method, our method iteratively alternates between performing experiments, updating parameter estimates, and computing new experiments. Specifically, our sequential design method computes a whole batch of new experiments in each iteration and this batch of new experiments is designed in a two-stage locally optimal manner. In essence, this means that in every iteration the combined information content of the newly proposed experiments and of the already performed experiments is maximized. In order to solve these two-stage locally optimal design problems, a recent and efficient adaptive discretization algorithm is used. We demonstrate the benefits of the proposed methodology on the example of of the parameter estimation for the non-random two-liquid model for narrow azeotropic vapor-liquid equilibria. As it turns out, our sequential optimal design method requires substantially fewer experiments than traditional factorial design to achieve the same model precision and prediction quality. Consequently, our method can contribute to a substantially reduced experimental effort in vapor-liquid equilibrium modeling and beyond.

Figures (8)

• Figure 1: Schematic of the proposed sequential locally optimal experimental design method.
• Figure 2: Illustration of the experimental points from the initial design $\texttt{init}$.
• Figure 3: Illustration of the experimental points from the factorial designs $\texttt{fed}^1$ (blue triangles), $\texttt{fed}^2 = \texttt{fed}^1 \,\&\, \texttt{fed}^{1,+}$, and $\texttt{fed}^3 = \texttt{fed}^2 \,\&\, \texttt{fed}^{2,+}$. The experimental points of the designs $\texttt{fed}^{1,+}$ and $\texttt{fed}^{2,+}$ are the red pentagons and the green stars, respectively.
• Figure 4: Illustration of the experimental points from the optimal designs $\texttt{oed}^0 = \texttt{init}$ (gray circles), and $\texttt{oed}^1 = \texttt{oed}^0 \,\&\, \texttt{oed}^{0,+}$, $\texttt{oed}^2 = \texttt{oed}^1 \,\&\, \texttt{oed}^{1,+}$, and $\texttt{oed}^3 = \texttt{oed}^2 \,\&\, \texttt{oed}^{2,+}$. The experimental points of the designs $\texttt{oed}^{0,+}$, $\texttt{oed}^{1,+}$ and $\texttt{oed}^{2,+}$ are the blue diamonds, the green pluses and the red triangles, respectively.
• Figure 5: The predicted and measured bubble- and dew-point temperatures for propanol and propyl acetate at the pressures $1\, \mathrm{bar}$ (bottom-most curves), $2\, \mathrm{bar}$ (middle curves), and $3\, \mathrm{bar}$ (top-most curves). The predictions are based on the final least-squares estimate $\widehat{\theta}^\texttt{tot}$ from Table \ref{['tab:lse-tot']}.