(413b) Parameter Estimation of Complicated Thermodynamic Models for Accurate Brine Separation

Authors: Pengfei Xu, Robert Gottlieb, and Matthew D. Stuber

AIChE 2023 Annual Meeting Abstract

Session: Industrial Applied Mathematics

Motivated by the need for sustainable water generation, purification, and reuse, the development of more efficient water separations, desalination, and brine concentrator systems are important for many industrial processes. To model these systems, accurate and open-source thermodynamic models are critical. The refined electrolyte non-random two-liquid model (r-eNRTL [1]) model has been shown to have exceptional prediction performance for properties of concentrated brine solutions as compared to the original eNRTL model [2]. However, due to the complexity of the r-eNRTL model, solving parameter estimation problems for this model (i.e., fitting it to data) to global optimality is currently out of reach. One challenge with this model is in the coding implementation itself. The r-eNRTL Gibb’s free energy expression and its derivatives, corresponding to the activity coefficient (and other thermodynamic properties), are complicated and prone to errors in their derivation and implementation in code. Another significant challenge is due to numerically challenging terms present in the objective function (activity coefficient), which lead to extreme overestimation of interval bounding methods as well as convex/concave relaxations that are needed for deterministic global optimization.

To overcome the first challenge, we propose an approach using symbolic algebra and source-code generation/transformation packages Symbolics.jl [3] and SymbolicUtils.jl [4] in the Julia programming language [5], to generate symbolic expressions of the r-eNRTL Gibbs free energy expression. We then apply automatic differentiation (AD) [6] to automatically generate the activity coefficient expression needed for parameter estimation. With this approach, the modeler never has to code such complicated expressions explicitly, which saves labor effort and enables more flexibility in model development.

To overcome the second challenge, we develop a novel, tight interval extension rule for the problematic multivariate quotient term where a summation of exponentials appears in both the numerator and denominator. Due to the dependency problem of interval arithmetic (i.e., every appearance of an interval X in an interval-valued function is treated independently), current interval extensions of this term are wildly expansive. For example, given an interval X= [-5,5], the natural interval extension of the function f(x)=exp(x)/exp(x) is given by F(X)=[-4.540e-5,2.203e4], whereas the range is just the value 1. With the new interval bounds, tighter McCormick-based relaxations are also calculable for use within deterministic global optimization.

Finally, we demonstrate how these new rules, source code generation, and AD are to be used with complicated models embedded in optimization formulations (e.g., parameter estimation problems). We detail how such problems are naturally accessible for the EAGO.jl solver [7-9] and how complexity and scalability may be addressed using our new SourceCodeMcCormick.jl package [10]. With these developments and proposed approaches, we aim to solve the r-eNRTL parameter estimation problem to guarantee global optimality for nontrivial multi-electrolyte solutions. Together with an open-source software implementation, we aim to provide industry with highly accurate and flexible modeling, simulation, and optimization tools to enable the design of sustainable brine treatment systems.

References

[1] Bollas, G. M., Chau-Chyun Chen, and P. I. Barton. "Refined electrolyte‐NRTL model: Activity coefficient expressions for application to multi‐electrolyte systems." AIChE Journal 54, no. 6 (2008): 1608-1624.

[2] Song, Yuhua, and Chau-Chyun Chen. "Symmetric electrolyte nonrandom two-liquid activity coefficient model." Industrial & Engineering Chemistry Research 48, no. 16 (2009): 7788-7797.

[3] Gowda, Shashi, Yingbo Ma, Alessandro Cheli, Maja Gwóźzdź, Viral B. Shah, Alan Edelman, and Christopher Rackauckas. "High-performance symbolic-numerics via multiple dispatch." ACM Communications in Computer Algebra 55, no. 3 (2022): 92-96.

[4] Shashi Gowda, Yingbo Ma, Mason Protter, Julia Computing, JuliaSymbolics/SymbolicUtils.jl, 2020. URL: https://github.com/JuliaSymbolics/SymbolicUtils.jl

[5] Bezanson, Jeff, Alan Edelman, Stefan Karpinski, and Viral B. Shah. "Julia: A fresh approach to numerical computing." SIAM review 59, no. 1 (2017): 65-98.

[6] Corliss, George, Christele Faure, Andreas Griewank, Laurent Hascoet, and Uwe Naumann, eds. Automatic differentiation of algorithms: from simulation to optimization. Springer Science & Business Media, 2002.

[7] Wilhelm, Matthew E., and Matthew D. Stuber. "EAGO. jl: easy advanced global optimization in Julia." Optimization Methods and Software 37, no. 2 (2022): 425-450.

[8] Matthew E. Wilhelm, Robert X. Gottlieb, and Matthew D. Stuber. PSORLab/McCormick.jl, 2020. URL: https://github.com/PSORLab/McCormick.jl.
[9] Matthew E. Wilhelm and Matthew D. Stuber. Easy advanced global optimization (EAGO): An open-source platform for robust and global optimization in Julia. In AIChE Annual Meeting. AIChE, 2017.

[10] Gottlieb, Robert X., Pengfei Xu, and Matthew D. Stuber. "Automatic Source Code Generation of Complicated Models for Deterministic Global Optimization with Parallel Architectures."