Evolution of a Polydisperse Ensemble of Spherical Particles in a Metastable Medium with Allowance for Heat and Mass Exchange with the Environment

Abstract

Motivated by a wide range of applications in various fields of physics and materials science, we consider a generalized approach to the evolution of a polydisperse ensemble of spherical particles in metastable media. An integrodifferential system of governing equations, consisting of a kinetic equation for the particle-size distribution function (Fokker–Planck type equation) and a balance equation for the temperature (concentration) of a metastable medium, is formulated. The kinetic equation takes into account fluctuations in the growth/reduction rates of individual particles, the velocity of particles in a spatial direction, the withdrawal of particles of a given size from the metastable medium, and their source/sink term. The heat (mass) balance equation takes into account the growth/reduction of particles in a metastable system as well as heat (mass) exchange with the environment. A generalized system of equations describes various physical and chemical processes of phase transformations, such as the growth and dissolution of crystals, the evaporation of droplets, the boiling of liquids and the combustion of a polydisperse fuel. The ways of analytical solution of the formulated integrodifferential system of equations based on the saddle-point technique and the separation of variables method are considered. The theory can be applied when describing the evolution of an ensemble of particles at the initial and intermediate stages of phase transformation when the distances between the particles are large enough, and interactions between them can be neglected.

Mathematical Modeling of the Solid–Liquid Interface Propagation by the Boundary Integral Method with Nonlinear Liquidus Equation and Atomic Kinetics

“…An important task of crystallization theory is to derive the boundary integral while taking into account the nucleation and growth of particles in an undercooled liquid ahead of the phase transformation boundary. This problem can be solved by combining the present analysis and the theory of crystal growth at the intermediate stage of phase transformation [31][32][33][34][35][36][37][38]. Another important task is to determine the boundary integral equation for the directional crystallization process with a two-phase region.…”

Analysis of Smoluchowski’s Coagulation Equation with Injection

“…The agglomerate removal is described by the linear term γn(x, t). For example, this term describes the removal of product crystals when considering the bulk crystal growth in supercooled melts and supersaturated solutions [20,21]. In living cells [10], early endosomes carrying cargo disappear from the system by undergoing conversion to late endosomes at the rate γ.…”

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

“…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][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].…”

“…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.…”