(646f) Stability Condition and Discretization Scheme for the Population Balance

The main contribution of this work is two-fold. First, a global stability analysis is developed to derive the stability condition for population balance in the infinite dimensional space. The general form of population balance is a partial differential equation in the particle phase space which consists of external and internal coordinates. The external coordinates show the spatial location of each particle whereas the internal coordinates provide the unique information of particle such as size, shape, change of distribution. An entropy based energy function is developed to analyze the stability property of population balance. In order to formulate the entropy function, extensive variables and corresponding intensive variables are defined based on the characteristics of population balance. It is proven that the resulting energy function is a valid energy function and stability is demonstrated using classical  energy methods. The resulting stability conditions provide guidelines for inventory control system design.

Second a novel discretization scheme is presented. The discretization scheme approximates population balance by establishing mass balance and number balance in each discrete interval. The continuous infinite dimension of particle phase space is reduced to finite dimension such that the partial differential equation is replaced with a set of ordinary different equations. Matrix analysis is employed to derive the stability property of the discretized population balance and the result is compared with that obtained for the entropy based energy analysis of the continuous population balance. Numerical simulations show that the matrix condition converges to analytical condition for continuous population balance if the number of discrete size intervals is large than a threshold value which can be obtained via simulations. Simulations results support the analytical stability condition derived from both approaches. The results of the paper are applied to crystallization and compared with Doherty’s results.