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

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 m0, m3 and m6. 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.