A new solution for closure problem in crystallization modeling using moments method

Abstract

Modeling of crystallization processes taking simultaneously into account nucleation, crystal growth, agglomeration and breakage gives complex equations very difficult to solve numerically. Furthermore, in most cases, the prediction of mean particle size and coefficient of variation are enough for practical use. For these two reasons, the resolution of crystallization equations is not directly realized from the population density but rather from the moments of the first one. For example, crystallization in non-stationary regime taking into account nucleation, crystal growth and agglomeration is modeled using the following equations where k = 0, 1, 2, …: d m k d t = r N · L 0 k + k ⁢ G ⁢ m k − 1 + ∫ 0 ∞ ∫ 0 ∞ [ 1 2 ⁢ ( L 3 + λ 3 ) k / 3 − L k ] · β ⁡ ( L , λ ) · n ⁡ ( L , t ) · n ⁡ ( λ , t ) d L d λ For certain agglomeration kernels β( L, λ), differential equations system can be obtained explicitly when k / 3 is a whole number. However, intermediate moments take part in crystallization models and must be also determined. For example, for moments of order 0, 3 and 6, we can obtain three differential equations where moments of order 2 and 5 also appear. Consequently, this system is not closed, because it contains 5 unknowns. To close it, it is necessary to express moments 2 and 5 as a function of moments 0, 3 and 6. An original idea for closuring the system consists in determining these moments by interpolation from the moments of order 0, 3, 6, …. Because of large variation values between moments 0, 3, 6, it is necessary to use very stiff interpolation functions. These interpolation functions can be used to express moments of any order (even rational moments in the case of crystal breakage) according to moments m 0, m 3 and m 6. Simple relations are given to illustrate this method which allows to simplify and solve complex prediction models for mean size and coefficient of variation of particle size distributions obtained in crystallization processes.