scholarly journals Approximate analytical solution of the integro-differential model of bulk crystallization in a metastable liquid with mass supply (heat dissipation) and crystal withdrawal mechanism

2021 ◽  
Author(s):  
Irina Nizovtseva ◽  
Alexandr Ivanov ◽  
Irina Alexandrova
Keyword(s):  
Distribution Function ◽  
Analytical Solution ◽  
Size Distribution ◽  
Mass Exchange ◽  
Heat Dissipation ◽  
Approximate Analytical Solution ◽  
Size Distribution Function ◽  
Saddle Point Method ◽  
Heat And Mass Exchange ◽  
Differential Model

This paper is devoted to an approximate analytical solution of an integro-differential model describing the process of nucleation and growth of particles in crystallizers, taking into account the thermal-mass exchange with the environment and the removal of product crystals from the metastable medium. The method developed in this work for solving model equations (kinetic equation for the particle size distribution function and balance equations for temperature/impurity concentration) is based on using the saddle point method for calculating the Laplace-type integral. It is shown that the degree of metastability of the liquid decreases with time at a fixed value of the mass inflow from the outside (heat flow to the outside). The crystal size distribution function has the form of an irregular bell-shaped curve, which increases with the intensification of heat and mass exchange with the environment.

scholarly journals Analytical solution of integro-differential equations describing the process of intense boiling of a superheated liquid

2021 ◽  
Author(s):  
Irina Alexandrova ◽  
Alexander Ivanov ◽  
Dmitri Alexandrov
Keyword(s):  
Distribution Function ◽  
Analytical Solution ◽  
Approximate Analytical Solution ◽  
Initial Distribution ◽  
Size Distribution Function ◽  
Superheated Liquid ◽  
Balance Equations ◽  
The Laplace Transform ◽  
And Shifts ◽  
Initial Distribution Function

In this article, an approximate analytical solution of an integro-differential system of equations is constructed, which describes the process of intense boiling of a superheated liquid. The kinetic and balance equations for the bubble-size distribution function and liquid temperature are solved analytically using the Laplace transform and saddle-point methods with allowance for an arbitrary dependence of the bubble growth rate on temperature. The rate of bubble appearance therewith is considered in accordance with the Dering-Volmer and Frenkel-Zeldovich-Kagan nucleation theories. It is shown that the initial distribution function decreases with increasing the dimensionless size of bubbles and shifts to their greater values with time.

scholarly journals Effects of nonlinear growth rates of spherical crystals and their withdrawal rate from a crystallizer on the particle-size distribution function

2019 ◽  
Vol 377 (2143) ◽  
pp. 20180210 ◽  
Author(s):  
Eugenya V. Makoveeva ◽  
Dmitri V. Alexandrov
Keyword(s):  
Distribution Function ◽  
Size Distribution ◽  
Growth Rates ◽  
Separation Of Variables ◽  
Size Distribution Function ◽  
Solid Particles ◽  
Supersaturated Solution ◽  
Metastable Liquid ◽  
Differential Model ◽  
Nonlinear Growth

In this paper, we show that the nonlinear growth rate of particles in a supersaturated solution or supercooled melt, as well as the rate of removal of crystals from the metastable liquid of a crystallizer, significantly change the size-distribution function of crystals. Taking these rates into account, we present a complete analytical solution of the integro-differential model describing the transient nucleation of solid particles and their evolution in a metastable liquid. The distribution function and metastability degree (supersaturation or supercooling) are found by means of the separation of variables and saddle-point methods. The nonlinear growth rates of crystals in supersaturated solutions and supercooled melts (single-component and binary) are summarized and compared with experimental data. This article is part of the theme issue ‘Heterogeneous materials: metastable and non-ergodic internal structures’.

Informational Measures of the Shape of the Amoroso Distributions for the Research of Dispersed Materials

2021 ◽  
Vol 1031 ◽  
pp. 58-66
Author(s):  
Vitaly Polosin
Keyword(s):  
Particle Size ◽  
Distribution Function ◽  
Particle Size Distribution ◽  
Size Distribution ◽  
Applied Research ◽  
Particle Distribution ◽  
Size Distribution Function ◽  
Distribution Model ◽  
Information Uncertainty ◽  
Particle Size Distribution Function

For the particle size distribution function various forms of exponential models are used to construct models of the properties of dispersed substance. The most difficult stage of applied research is to determine the shape of the particle distribution model. For the particle size distribution function various forms of exponential models are used to construct models of the properties of dispersed substance. The most difficult stage of applied research is to determine the shape of the particle distribution model. The article proposes a uniform model for setting the interval of information uncertainty of non-symmetric particle size distributions. Based on the analysis of statistical and information uncertainty intervals, new shape coefficients of distribution models are constructed, these are the entropy coefficients for shifted and non shifted distributions of the Amoroso family. Graphics of dependence of entropy coefficients of non-symmetrical distributions show that distributions well-known are distinguish at small of the shapes parameters. Also it is illustrated for parameters of the form more than 2 that it is preferable to use the entropy coefficients for the unshifted distributions.The material contains also information measures for the well-known logarithmic normal distribution which is a limiting case of distribution Amorozo.

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