Relationship between number and sizes of crystals growing in batch crystallization: Nuclei number density, nucleation kinetics and crystal polydispersity

•Clues for controlling the number of crystals growing in batch crystallization are provided.•The crystal number resulting from both nucleation time and supersaturation is calculated.•Crystal polydispersity and crystal size distribution (CSD) arise during the nucleation stage.•Convection driven plumes and crystal sedimentation impact the CSD of the grown crystals.•Practical recommendations for growing crystals of the desired number (and size) are given. Calculation of the crystal nuclei number density as a function of both nucleation time and supersaturation is enabled by newly devised formula. A theoretical treatment of crystal polydispersity and size distribution is presented. It is supposed that the prime cause for the inherent propensity to polydispersity of assembles of growing crystals is the prolonged nucleation period. During the subsequent crystal growth in unstirred solutions, the polydispersity arising out of the nucleation stage is increased due to two gravity instigated effects: (1) solution replenishment resulting from the convection driven plumes that form above sufficiently large crystals, and (2) sedimentation of such crystals, which brings them in non-depleted solution areas. It is shown that, although increasing the polydispersity, these factors hardly alter the bell-shaped distribution of crystal number density on crystal sizes inherited from the nucleation stage. This conclusion is in accord with experimentally determined bell-shaped normalized numbers of lactose crystals vs. their diameters. Also, the mean size of the growing crystals is calculated, and the effect of the convection plumes and crystal sedimentation on the crystal size distribution (CSD) and on the quality of the grown crystals is considered.