Steady-state crystallization with a mushy layer: a test of theory with experiments

Motivated by important applications in materials science and geophysics, I develop a non-linear heat and mass transfer model describing a steady-state crystallization scenario with a mushy layer. An exact analytical solution of mushy layer equations is derived in a parametric form where the temperature being the decision variable. The volume fraction of ice, the mass fraction of solute in the interstitial liquid, the spatial coordinate are found as functions of temperature in the mushy layer. Its thickness and temperature distribution in the liquid phase are defined analytically as well. The analytical theory developed well describes experiments on NaCl–H$$_2$$O solutions

The bulk crystal growth in binary supercooled melts with allowance for heat removal

A mathematical model describing crystal growth in a supercooled binary melt in the presence of heat removal and withdrawal of product crystals from the working liquid of the crystallizer is formulated. An approximate analytical solution of the integro-differential system of the kinetic and balance equations is constructed using the separation of variables method and the saddle point technique to calculate the Laplace-type integral. The particle size distribution function and the dynamics of desupercooling in the metastable system taking into account the Meirs nucleation kinetics are found. It is shown that the distribution function increases with decreasing the impurity concentration

On the theory of phase transformation process in a binary supercooled melt

The processes of bulk crystallization of a supercooled binary melt where the nucleation and growth of solid particles occur are considered with allowance for different mechanisms of nucleation kinetics. A nonlinear set of integro-differential equations is analytically solved by means of the saddle-point technique. The crystal-radius distribution function, temperature, solute concentration, and supercooling of a binary liquid are found taking into account the nonlinear growth rates of spherical crystals. The supercooling represents a decreasing function of time while the distribution function increases up to a maximum size of crystals in a metastable melt

How the shift in the phase transition temperature influences the evolution of crystals during the intermediate stage of phase transformations

The influence of the phase transition temperature shift on the growth dynamics of a polydisperse ensemble of spherical crystals in metastable melts and solutions is studied. This shift is connected with the Gibbs–Thomson effect and the attachment kinetics of atoms at the phase transition interfaces of evolving crystals. The nonlinear model of kinetic and balance equations with allowance for the particle “diffusion” term is solved analytically. The obtained solution is compared with the case when this temperature shift is not taken into account. It is shown that the Gibbs–Thomson and attachment kinetics effects slightly accelerate the system desupercooling for a single-component titanium melt. This shifts the particle-size distribution function and changes the shape of its tail, which is responsible for the concluding stage of Ostwald ripening

Effects of external heat/mass sources and withdrawal rates of crystals from a metastable liquid on the evolution of particulate assemblages

A complete analytical solution of an integro-differential model describing the transient nucleation of solid particles and their subsequent growth at the intermediate stage of phase transitions in metastable systems is constructed. A functional Fokker–Plank type equation for the density distribution function is solved by the separation of variables method for the Weber–Volmer–Frenkel–Zeldovich nucleation kinetics. A non-linear integral equation with memory kernel connecting the density distribution function and the system supercooling/supersaturation is analytically solved on the basis of the saddle point method for the Laplace integral. The analytical solution obtained shows that the transient phase transition process attains its steady-state solution at large times. An exact analytical solution for the steady-state problem is found too. It is demonstrated that the crystal-size distribution function increases with increasing the intensity of external sources. In addition, the number of larger (smaller) particles decreases (increases) with increasing the withdrawal rate of crystals from a metastable liquid

Morphological stability analysis of a planar crystallization front with convection

A linear theory of morphological stability of flat crystallization front is constructed with allowance for convective motions in liquid. The cases of slow and intense convection described by conductive and convective heat and mass transfer boundary conditions are considered. The dispersion relations defining the perturbation frequency as a function of wavenumber (wavelength) and other process parameters are derived. The neutral stability curve found in the case of slow convection substantially depends on extension rate at the phase interface. This curve divides the domains of morphological instability (MI) and morphological stability (MS). In both of these domains, the constitutional supercooling (CS) condition takes place. Therefore, we arrive at two various crystallization regimes (1) CS and MI, and (2) CS and MS. These cases respectively correspond to the mushy and slurry layers developing ahead of the crystallization front. In addition, when the fluid flows from the front, it is morphologically unstable for various perturbation wavelengths. When the fluid flows to the front, it is stable for large extension rates and unstable for smaller extension rates. The dispersion relation found in the case of intense convection shows that the perturbation frequency is always negative and small morphological perturbations decay with time. It means that the crystallization process with intense convection in liquid is absolutely stable

Unsteady, steady, and self-oscillatory modes of the bulk continuous crystallization with mass influx and withdrawal of product crystals

This paper is devoted to the study of operating modes of the crystallizer with allowance for the processes of nucleation and evolution of particles, removal of product crystals, and fines dissolution. The stationary and non-stationary analytical solutions describing various crystallization scenarios are derived. We show that the unsteady-state distributions of crystals approach their stationary distribution over time. In addition, the linear analysis of dynamic instability shows that the regions of oscillatory instability and absolute stability exist. These regions are divided by the neutral stability curve dependent of the physical parameters of crystallization process. We show that the oscillatory instability domain becomes wider when increasing the nucleation and growth rate constants. As a result, the process of volumetric crystallization under consideration can operate in a mode of self-oscillations sustained by a feedback between the nucleation rate and the process driving force (supersaturation)

Wavy Ice Patterns as a Result of Morphological Instability of an Ice–Water Interface with Allowance for the Convective–Conductive Heat Transfer Mechanism

In this research, the wavy ice patterns that form due to the evolution of morphological perturbations on the water–ice phase transition interface in the presence of a fluid flow are studied. The mathematical model of heat transport from a relatively warm fluid to a cold wall includes the mechanism of convective–conductive heat transfer in liquid and small sinusoidal perturbations of the water–ice interface. The analytical solutions describing the main state with a flat phase interface as well as its small morphological perturbations are derived. Namely, the migration velocity of perturbations and the dispersion relation are found. We show that the amplification rate of morphological perturbations changes its sign with variation of the wavenumber. This confirms the existence of two different crystallization regimes with (i) a stable (flat) interfacial boundary and (ii) a wavy interfacial boundary. The maximum of the amplification rate representing the most dangerous (quickly growing) perturbations is found. The theory is in agreement with experimental data

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

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

Directional crystallization with a mushy region. Part 2: nonlinear analysis of dynamic stability

In this paper, we develop a nonlinear theory of self-oscillatory solidification mode during directional crystallization in the presence of a quasi-equilibrium two-phase region of constitutional supercooling. This study is based on the linear stability theory (Part 1), where we demonstrated that the indicated regime can be formed due to the oscillatory instability at certain values of physical and operating parameters of the system. The development of oscillatory instability is based on a new frontal model of crystallization with a two-phase region, the main feature of which is the replacement of real two-phase region by a discontinuity surface between purely solid and liquid phases. We derive a nonlinear system of equations for determining frequencies and amplitudes of perturbations responsible for the development of oscillatory instability. The solution of this system allows one to analytically determine the fundamental and secondary harmonics of perturbations and calculate the resulting self-oscillations of the crystallization velocity and impurity distribution. The impurity concentration and period of its layered distribution in the solid phase are in good agreement with experimental data