Stability and dynamics of isothermal CMSMPR crystallizers

Abstract

Detailed analysis of stability and bifurcation properties and of the dynamic behaviour of isothermal continuous mixed suspension, mixed product removal (CMSMPR) crystallizers is presented using the moment equation model. It is shown that isothermal CMSMPR crystallizers may exhibit not only limit cycle oscillations but, in the case of magma-dependent nucleation, also saddle point instabilities. Applying the Mikhailov stability criterion, two equivalent sets of necessary and sufficient conditions are derived for checking the stability of steady states. Stability maps are presented in the planes of different pairs of system parameters. It is shown that saddle node and boundary bifurcation, as well as bifurcation at infinity may arise in the case of magma-dependent secondary nucleation. Oscillations of the population density and distribution functions are also presented and analysed, approximating the size distribution by gamma distribution. In dynamic states, the crystal size distribution may differ significantly from the exponential one, while the mean values of oscillations of supersaturation are lower than the corresponding steady state values. A causal loop diagram between the variables reveals that instabilities are created by the interactions of the autoinhibition generated negative feedback and a positive feedback between the four leading moments of crystal size.