Closed-Form Solutions for Control of Calcium Carbonate Precipitation

We develop a control-oriented model for manipulating calcium carbonate (CaCO3) precipitation through pH and/or dissolved CO2(aq) adjustments in a semi-batch process. We present two open-loop control strategies: first, a closed-form optimal control solution derived via Pontryagin’s Minimum Principle; and second, an orbital-flatnessbased method for boundary condition control of the particle size distribution. Despite its theoretical emphasis, the proposed methodology is readily applicable either as a direct open-loop strategy for process control or as a trajectory planning in combination with feedback in practical implementations.