Numerical approach for solving the equations of the ISD3 model

The ISD3 model, which is based on the population balance equation (Eq. 20), looks similar to the ISDD model [14], except for the third term (on its right-hand side) that accounts for the change in particle size due to dissolution. Instead of solving for the particle mass concentration (as in ISDD), the ISD3 model solves for particle number density,N(D_p; x, t), to simultaneously track changes in both the size and spatial distribution, and the number, of the particles in the liquid medium due to diffusion, sedimentation, and dissolution.

The final rate equations for ions (Eqs. 12 and 13) and particle number density (Eq. 20) are numerically solved along with the boundary (Eqs. 21a and 21b) and initial (Eqs. 24, 25, 27, and 28) conditions to obtain solutions to the particle number density and the ion concentrations. The numerical algorithm used for solving the equations was based on the finite difference method and was implemented in MATLAB – the details are given in the Supporting Information. The numerical solution for the particle number density was then used to calculate the mass, number, size distribution and surface area of particles, both in the liquid media and in the cells; the formulas for these calculations are given below.