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Keywords

Mathematical Modelling, Chemical Modelling, Optimization, Ordinary Differential Equations

Abstract

Maintaining optimal pH is crucial in chemical production, as it directly influences formation, quality, and quantity of the product. While pH can be readily measured in physical systems, digital twin systems or in-silico experimentation require an estimation of pH (Gustafsson et al 1995).

The model  introduces a pH predictor specifically designed for known chemical systems, utilizing equality constrained problems, estimated Gibbs free energy (Dick 2019) and the ideal gas law for accurate predictions of pH over time. The model utilizes stoichiometric matrices, making it adaptable for  both small and large chemical systems.

The problems are defined in a well-stirred tank with a constant volume, where in and outflow can be defined. This allows the system to be described by ordinary differential equations, allowing for tracking of the various concentrations of chemical substances as well as the pH over time.

The model includes examples of methanol synthesis in water and phosphoric acid in water with ammonia, carbonic acid, nitric acid, and sodium hydroxide as the control. These examples demonstrate the model’s ability to accurately estimate the pKa-values of PO43- to H3PO4, CO3-2 to H2CO3 and  NH3 to NH4+, in a single system with a total of 18 species.

The examples are written as optimization problems in Julia with JuMP (Lubin et al. 2023) as well as our own implementation of optimization algorithms. The model is written in Julia to prioritize speed and open-source accessibility, enabling efficient problem-solving.

The model offers a fast and reliable solution for optimizing pH levels in chemical production. Its adaptability and computational efficiency make it an asset for improving production efficiency and product quality in closed loop well-stirred chemical tanks.

Acknowledgement

Peter Emil Carstensen is funded by the The Novo Nordisk Foundation Center for Biosustainability, Technical University of Denmark, NNF20CC0035580.

References

  • Gustafsson, T. K., Skrifvars, B. O., Sandstrom, K. V., & Waller, K. V. (1995). Modeling of pH for control. Industrial and Engineering Chemistry Research, 34(3), 820–827. doi.org/10.1021/ie00042a014
  • Lubin, M., Dowson, O., Garcia, J. D., Huchette, J., Legat, B., & Vielma, J. P. (2023). JuMP 1.0: recent improvements to a modeling language for mathematical optimization. Mathematical Programming Computation, 15(3), 581–589. doi.org/10.1007/s12532-023-00239-3
  • Dick, J. M. (2019). CHNOSZ: Thermodynamic calculations and diagrams for geochemistry. Frontiers in Earth Science, 7, 180. doi.org/10.3389/feart.2019.00180