Comments on Population Balance and Modeling

While the population balance [9] forms the basis for the design and control of crystal sizes and crystal size distributions in continuous crystallizers, there is hardly any direct application in practice. The main reason for this is the poor transferability of the measured (apparatus-specific) proportionality constant (from = k MG ) in the following equation, presented here for the simple MSMPR (10], to the large-scale plant that is to be planned  

In the equations, according to Larson and Garside [10], j denotes the exponent for the suspension density M, i is the quotient of the exponents m (for the supersaturation s at the nucleation rate and n (for the supersaturation s at the crystal growth rate G), Ip is the dominant particle size of the population balance, fey is the volume shape factor for the crystallized mass produced, and is the quotient of the proportionality constants of the secondary nucleation rate B and k of the crystal growth rate G. 

All three characteristic variables i,j, and k depend on the specific solubility system with its various and changing impurities (this refers to additives at the lowest concentrations). 

While in the ideal world the exponents i andj are not functions of the temperature or the stirrrng/recrrculation (i.e., energy input), the speed constant kr is dependent on them. In k there are therefore apparatus-specific properties such as size, energy input, levels of efficiency, and so on, which change significantly within the framework of the scale-up and cannot be transferred on a one-to-one basis. 

Both the merely specific validity of the measurement results from pilot trials and the deviations from reality of an MSMPR from the FC concept lead to an unsatisfactory correlation of the results between trials and large-scale plants that are not better than the current fact that at least the mean crystal size produced in the laboratory in a small glass reactor from original solution can reliably be obtained in the large-scale FC crystallizers without the need for further measures. 

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