Computationally Efficient Algorithm for Solving Population Balances with Size-Dependent Growth, Nucleation, and Growth-Dissolution Cycles

Abstract

A computationally efficient algorithm for solving population balance equations (PBEs) for describing the evolution of crystal size distribution (CSD) in seeded batch crystallization processes is developed. The algorithm is particularly suitable for solving problems with size-dependent growth and dissolution kinetics where temperature cycling is applied to modify the CSD. It is much faster than the quadrature method of moments (QMOM) which has been widely applied to solve this type of problem and has the added benefit that it provides the complete crystal size distribution rather than only the moments. These features make it especially well suited for solving problems in optimization and determination of feasible regions, where many batch simulations are required. The algorithm is applied to determine attainable regions for seed-grown product crystals (characterized by the mean and variance of the seed-grown product CSD) and analyze the trade-off between the objectives of minimizing batch time and nucleated crystal volume in a batch crystallization process with temperature cycling. Two main conclusions are drawn from the results. First, the attainable region expands with an increasing number of growth-dissolution cycles as expected, but the extent of the increase or decrease in the attainable variance depends on the sensitivity of the growth and dissolution rates to the crystal size. Second, the trade-off between the abovementioned objectives subject to constraints on the seed-grown crystal mean size and variance is significant and an outcome representing a good compromise can be achieved by specifying proper supersaturation and undersaturation set points.