Mathematical Modeling of a pH Swing Precipitation Process and its Optimal Design

TL;DR

This paper develops a comprehensive mathematical model for the pH swing precipitation process of calcium carbonate, incorporating stochastic PDEs and SDEs, and validates it with experimental data using advanced fitting methods including deep neural networks.

Contribution

It introduces a novel coupled stochastic PDE and SDE model for precipitation dynamics and compares three data-fitting methods, highlighting the effectiveness of DNN-based approach.

Findings

The DNN method achieves the lowest fitting error.

The model accurately captures the long-term behavior of the process.

The coupled system is mathematically well-posed and validated with lab data.

Abstract

In this work we consider the semi-batch process of precipitation of calcium carbonate solids from a solution containing calcium ions by adjusting the pH of the solution. The change in pH is induced either by the addition of alkaline solution such as sodium hydroxide (NaOH) or by the addition of a carbon dioxide gas (CO$_2$) to the given ionic solution. Under this setup we propose a system of degenerate stochastic partial differential equations that is able to explain the dynamical behavior of the key components of precipitation process. In particular, we propose a semi-linear advection equation for the dynamics of particle size distribution (PSD) of the precipitated particles. This is in turn coupled with a system of stochastic differential equations (SDEs) that is able to explain the chemical kinetics between calcium ions (Ca$^{+2}$), calcium carbonate (CaCO$_3$) in aqueous state and pH of the solution. The resulting coupled system is first mathematically studied, in particular conditions for the existence of a mild-solution is established and also the long time behavior of the system is established. Following this we consider the validation of the model for which we consider experimentally obtained lab-scale data. Using this data we propose three methods to fit the model with the data which then also validates the suitability of the proposed model. The three methods include manual intuitive tuning, classical forward backward SDE (FBSDE) method and finally a DNN based method. The FBSDE method is based on the stochastic optimal control formulation for which we provide the necessary and sufficient condition for the existence of an optimal solution. Lastly, we compare the three methods and show that DNN method is the best in terms of lowest error and as the most economical in terms of the compute resources necessary during online use.