En route towards a comprehensive dimensionless representation of precipitation processes

Precipitation of (nano-) particles is a unit operation for the preparation of suspensions. Despite its widespread use, no general understanding is established of how different operating conditions, such as mixing rates or reactant concentrations, affect the precipitation outcome (e.g. the particle size distribution). We obtain overarching relations between the operating conditions and the precipitation outcome by deriving dimensionless numbers governing the precipitation process. In particular, we consider the interdiffusion of two chemical compounds reacting to a sparingly soluble salt by coupling the reaction-diffusion equations for the solute concentrations, to a population balance equation for the evolution of the dispersed phase including nucleation and growth. We vary the relevant process parameters for different chemical systems and uncover three dimensionless numbers, which fully determine the precipitation outcome under the assumption of fast chemical reactions. Two of these dimensionless numbers are Damkohler numbers, which put the relevant mixing kinetics in relation to the relevant time scales for the nucleation and the growth kinetics, respectively. The third dimensionless number corresponds to a dimensionless solid concentration. Simple functional expressions for the Damkohler numbers valid for any chemical system allow estimating whether the precipitation product is controlled by mixing, or solely by the kinetics of solid formation. In the mixing-controlled regime, scaling laws relate the mean particle sizes and the number of particles to the Damkohler number.