Crystallization and precipitation engineering-VI. Solving population balance in the case of the precipitation of silver bromide crystals with high primary nucleation rates by using the first order upwind differentiation

High resolution algorithms for multidimensional population balance equations

“…In the method of finite differences/discretized population balances, the population balance equation is approximated by a finite difference scheme (Kumar and Ramkrishna, 1996). Numerous discretizations of the PBE with different orders of accuracy have been investigated and applied to various particulate systems (Gelbard et al, 1980;Hounslow et al, 1988;Marchal et al, 1988;Hounslow, 1990;Muhr et al, 1996;Kumar and Ramkrishna, 1997). …”

A Comprehensive Experimental and Theoretical Investigation of the Styrene/2‐Ethylhexyl Acrylate Emulsion Copolymerization

“…These include finite-difference methods, the method of moments, [23,45] orthogonal collocation on finite elements, [20,37,46] geometric discretization methods, [47][48][49][50] etc. In the present investigation, the discretized population balance method of Hounslow et al [50] was employed.…”

“…In the present investigation, the discretized population balance method of Hounslow et al [50] was employed. Following the developments of Hounslow et al [48] , the time evolution of the full particle size distribution can be determined using a discretized population balance equation, assuming that the number radius density function f i (r, t) in the interval r to (r þ dr) remains constant. Using a fractional geometric discretization grid [50] (e.g., V i þ 1 ¼ 2 1/q V i ), the population balance Equation (44) can be approximated by the following system of ordinary differential equations, which ensures the accurate estimation of the zero and third moments (particle number and volume) of the PSD: [51] where…”

Development of Novel Crystallizer Design and Optimization Tools for Solution Crystallization

“…Depending on the philosophy of this discretization, different methods have been developed for ag-Ž gregation and breakage Kumar and Ramkrishna, 1996a,b; . Ž Vanni, 1999 andaggregation andcrystal growth Marchal et al, 1988;Muhr et al, 1996;Litster et al, 1995; In general, the DPB approach describes the population balance accurately, but can involve an extremely high number of scalars. If the final application is the implementation of the population balance in a CFD code, then the solution of the scalars has to be done in every cell of the computational domain, resulting in a very high calculation time and memory problems.…”