A complete analytical solution of unsteady coagulation equations and transition between the intermediate and concluding stages of a phase transformation

In this paper, a complete analytical solution to the unsteady kinetic equation of particle coagulation in a metastable liquid is constructed with allowance for an arbitrary initial particle-volume distribution function. Based on the analytical solution found, a theory of the transition of the phase transformation process from the intermediate stage to the coagulation stage is constructed. For this purpose, an exact analytical solution describing the intermediate stage of a phase transformation is used. It is shown that the maximum of the particle-volume distribution function decreases, and its right branch (“tail”) grows with increasing time during the coagulation stage

Towards the theory of phase transformations in metastable liquids. Analytical solutions and stability analysis

The process of ongoing nucleation and evolution of particles in a metastable liquid caused by external mass sources and crystal withdrawal mechanism is considered. The steady-state and unsteady-state analytical solutions are found analytically. The crystal-size distribution function, as well as the standard Cauchy problem for the liquid supersaturation, are derived. The linear dynamic stability theory of steady-state solutions is developed. The neutral stability surface as a function of system parameters is determined

Approximate analytical solutions of the kinetic and balance equations for intense boiling

The process of intense boiling is theoretically studied on the basis of kinetic and balance equations for the bubble-size distribution function and system temperature. The kinetic equation for the bubble-size distribution function represents the first-order partial differential equation with a source term. The heat balance is spatially homogeneous and takes into account the heat exchange of the system with the external environment. This non-linear system is supplemented by the initial and boundary conditions. Namely, the initial bubble-size distribution and temperature are regarded as known, and the flux of bubbles overcoming the critical size is defined by the rate of nuclei formation. A non-linear integro-differential system of model equations is solved analytically by the integral Laplace transform and saddle-point methods. It is shown that the solution has a different form at $$x\ge \tau$$ and $$x<\tau$$ (here, x and $$\tau$$ are the dimensionless spatial and temporal variables). Also, we show that the initial bell-shaped distribution function decreases, and the liquid temperature increases with increasing time

Unsteady-state particle-size distributions at the coagulation stage of phase transformations

In this paper, a new analytical solution of nonstationary integro-differential coagulation equation, which describes the concluding stage of a phase transformation process in metastable liquids, is deduced. This solution takes into account the initial distribution function, which is found on the basis of previously developed theory for the intermediate stage. It is demonstrated that the derived particle-size distribution function representing a bell-shaped curve decreases with time and describes the long-term experimental data. The developed theoretical approach describes the transition of a phase transformation process from its intermediate stage to the stage of particle coagulation

Exact analytical solutions of a steady-state mushy layer model containing heat exchange with the environment

A set of nonlinear equations describing the crystallization of a binary melt in the presence of a quasi-equilibrium mushy layer is analytically solved in the case of heat exchange with the environment. Solute concentration, temperature distributions, and a bulk fraction of the solid phase in a mushy region are found. In addition, the average interdendritic distance in the two-phase region, which characterizes the structural-phase transition and porosity of the material, was analytically determined. The mushy layer’s thickness was also found as a function of given the physical and operating parameters of the solidification process. The analytical solutions obtained are compared with experimental data

Directional crystallization with a mushy region. Part 1: linear analysis of dynamic stability

In this paper, a linear analysis of dynamic stability of the directional solidification process with a two-phase region is carried out. We show the possibility of oscillatory mode of instability development in relation to the steady-state crystallization process with a constant velocity. We determine the steady-state solutions and derive evolutionary equations for the perturbations, derive an equation for the neutral stability curve and obtain the parametric regions of stable/unstable crystallization. It is shown that the regions of monotonous/oscillatory instability and stability can exist. The boundaries separating these regions are defined. We demonstrate that a transition between oscillatory and monotonous instabilities occurs abruptly. In addition, we show that the crystallization process with a two-phase region stabilizes the dynamic perturbations with respect to the crystallization process with a flat front

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

On the theory of directional crystallization with a two-phase region with vigorous convection

A time-dependent process of directional crystallization with a two-phase (mushy) region with allowance for vigorous convection in the liquid phase is analytically described. An exact solution of non-linear mushy layer equations is found in a parametric form. The temperature and solute concentration distributions, as well as the solid phase fraction profile, are determined in the two-phase and liquid regions. The moving boundary of phase transition, which is placed between the two-phase zone and liquid is defined as a function of time. The present analytical solutions are in good agreement with laboratory experiments on ice grown from aqueous solutions of isopropanol

The evolution of a polydisperse ensemble of ellipsoidal particles in the form of prolate and oblate ellipsoids of revolution

The evolution of a polydisperse ensemble of prolate and oblate ellipsoidal crystals in a supercooled one-component melt is theoretically studied. The volume growth rates for prolate and oblate ellipsoids are analytically found and compared at the same melt supercooling. We show that prolate crystals evolve faster than the oblate ones and the difference between their growth rates increases with increasing the melt supercooling. Then taking these volume growth rates into account, we formulate the model describing the evolution of an ensemble of prolate/oblate ellipsoidal particles. The analytical solution to this integrodifferential model is found for two nucleation mechanisms in cases of prolate and oblate ellipsoids using the saddle-point method. Our solution demonstrates that an ensemble of prolate particles grows and removes the melt supercooling faster than an ensemble of oblate particles. As a result, the particle-volume distribution function for prolate crystals is shifted to larger crystal volumes than the same distribution for oblate crystals. Keeping this behavior in mind, we conclude that the shape of crystals plays a decisive role in the melt supercooling dynamics and their volume distribution