On the Theory of Unsteady-State Operation of Bulk Continuous Crystallization

Abstract

Motivated by an important application in the chemical and pharmaceutical industries, we consider the non-stationary growth of a polydisperse ensemble of crystals in a continuous crystallizer. The mathematical model includes the effects of crystal nucleation and growth, fines dissolution, mass influx and withdrawal of product crystals. The steady- and unsteady-state solutions of kinetic and balance equations are analytically derived. The steady-state solution is found in an explicit form and describes the stationary operation mode maintained by the aforementioned effects. An approximate unsteady-state solution is found in a parametric form and describes a time-dependent crystallization scenario, which tends toward the steady-state mode when time increases. It is shown that the particle-size distribution contains kinks at the points of fines dissolution and product crystal withdrawal. Additionally, our calculations demonstrate that the unsteady-state crystal-size distribution has a bell-shaped profile that blurs with time due to the crystal growth and removal mechanisms. The analytical solutions found are the basis for investigating the dynamic stability of a continuous crystallizer.

Desupersaturation dynamics in solutions with applications to bovine and porcine insulin crystallization

“…The analytical theory developed in this paper can be extended to the processes of heat and mass transfer between the solution/melt and the environment (e.g. when an impurity is continuously fed into the liquid, small crystals dissolve and the product crystals withdraw [42][43][44][45]).…”

Analysis of an integro‐differential model for bulk continuous crystallization with account of impurity feeding, dissolution/growth of nuclei and removal of product crystals

“…These include, for example, (i) crystal growth/dissolution rate [12][13][14][15][16], (ii) crystal geometry and structure [17][18][19][20][21][22], (iii) mass and heat exchange with the environment [23][24][25][26], (iv) crystallizer design Eugenya V. Makoveeva and Alexander A. Ivanov are equally contributing authors. [27,28], (v) mixing process of supersaturated/supercooled liquid [29][30][31], (vi) withdrawal of product crystals [32,33], and (vii) interaction between crystals (their coalescence, agglomeration, and fragmentation) [34][35][36][37][38]. A mathematical model that takes into account all of the above processes is extremely difficult both to formulate and to analyze.…”

“…Let us especially emphasize that there are many parameters and external processes having a considerable impact on the bulk crystallization phenomenon. These include, for example, (i) crystal growth/dissolution rate [12–16], (ii) crystal geometry and structure [17–22], (iii) mass and heat exchange with the environment [23–26], (iv) crystallizer design [27, 28], (v) mixing process of supersaturated/supercooled liquid [29–31], (vi) withdrawal of product crystals [32, 33], and (vii) interaction between crystals (their coalescence, agglomeration, and fragmentation) [34–38]. A mathematical model that takes into account all of the above processes is extremely difficult both to formulate and to analyze.…”

The boundary integral equation describing a curvilinear solid/liquid interface with nonlinear phase transition temperature

“…The evolution of the curvilinear liquid-solid interface in the crystallization processes of undercooled melts and supersaturated solutions is responsible for the properties of solidifying materials [1][2][3][4][5][6][7][8][9]. Thus, for example, the crystallization front displaces a part of the dissolved impurity into the liquid phase and another part of the impurity is absorbed into E. A. Titova and L. V. Toropova contributed equally to this work.…”