Dynamic instability of two-step continuous crystallization with applications to beta-lactoglobulin, lysozyme and insulin crystal growth

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Abstract This study is concerned with a continuous operation mode of bulk crystallization from a supersaturated solution with allowance for the two-step mechanism of nucleation and growth of crystals. The integro-differential system of governing equations for the crystal-size distribution function and liquid supersaturation is formulated accounting for mass input into the saturated solution and withdrawal rate of product crystals from the crystallizer (mass exchange with the environment). This system contains the particles growth rate for the two-step nucleation mechanism, which is only a function of crystal radius when considering $\beta$-lactoglobulin, lysozyme and insulin crystal growth. The steady-state analytical solutions of this system are found for the Weber-Volmer-Frenkel-Zel'dovich and Meirs nucleation kinetics. The crystal-size distribution function for unsteady crystallization mode is found by means of the integral Laplace transform method. This function enables us to derive a single unsteady integro-differential equation for the liquid supersaturation. A linear instability analysis based on this equation is carried out. The amplification rate and frequency of small perturbations following from this analysis enable us to find the neutral stability curve and the domains of stable and unstable crystallization. We show that the frequency of perturbations stabilizes near the neutral stability curve when dealing with the stable mode of continuous crystallization.