Mathematical modeling of phase transition  and separation in fluids: A unified approach

The article deals with constructing and implementing mathematical models of non-isothermal moisture transfer during drying of anisotropic capillary-porous materials, in particular wood, taking into account the movement of the evaporation zone for non-steady drying schedules, as well as to the development of effective analytical and numerical methods for their implementation. An analytical-numerical method for the determination of non-isothermal moisture transfer under non-steady schedules of the drying process has been developed, taking into account the dynamics of the phase transition boundary change. Calculation relationships are established for determining the phase transition temperature taking into account transport gradients and time for which the relative saturation reaches the boundaries of the phase transition.

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Mathematical modeling of bulk and directional crystallization with the moving phase transition layer

10.22541/au.163250947.73804831/v1 ◽

2021 ◽

Author(s):

Liubov Toropova ◽

Danil Aseev ◽

Sergei Osipov ◽

Alexander Ivanov

Keyword(s):

Mathematical Modeling ◽

Phase Transition ◽

Transition Layer ◽

Solid Phase ◽

Phase Region ◽

Mushy Region ◽

Solidification Velocity ◽

Mushy Layer ◽

Two Phase ◽

Phase Layer

This paper is devoted to the mathematical modeling of a combined effect of directional and bulk crystallization in a phase transition layer with allowance for nucleation and evolution of newly born particles. We consider two models with and without fluctuations in crystal growth velocities, which are analytically solved using the saddle-point technique. The particle-size distribution function, solid-phase fraction in a supercooled two-phase layer, its thickness and permeability, solidification velocity, and desupercooling kinetics are defined. This solution enables us to characterize the mushy layer composition. We show that the region adjacent to the liquid phase is almost free of crystals and has a constant temperature gradient. Crystals undergo intense growth leading to fast mushy layer desupercooling in the middle of a two-phase region. The mushy region adjacent to the solid material is filled with the growing solid phase structures and is almost desupercooled.

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SELECTION OF STATES AND FLUCTUATION UNDER THE FIRST ORDER PHASE TRANSITIONS

International Journal of Modern Physics B ◽

10.1142/s021797920302274x ◽

2003 ◽

Vol 17 (27) ◽

pp. 4913-4933 ◽

Cited By ~ 3

Author(s):

BOHDAN LEV ◽

HIROSHI YOKOYAMA

Keyword(s):

Phase Transition ◽

Phase Transitions ◽

Stable State ◽

Unified Approach ◽

Closed Expression ◽

First Order ◽

First Order Phase Transition ◽

First Order Phase ◽

New Phase ◽

Selection Of

A unified approach is proposed which provides both microscopic and phenomenological description of the first-order phase transition accompanied by the formation of thermodynamically stable bubbles of the new phase. A procedure is proposed for selecting the probable states whose contribution in the thermodynamic behavior of the system is dominant. This technique makes it possible to take into account fluctuation of any scales and to obtain a closed expression for the parameters of the new phase formation, in particular its dimensions, lifetime and time of relaxation towards a thermodynamically stable state. The feasibility of experimental observation the parameters dependence of the new phase bubble on the governing quantities is discussed.

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A Unified Approach to Analyze Vibration of Axisymmetric Rotating Structures with Flexible Stationary Parts

Journal of Vibration and Acoustics ◽

10.1115/1.1857917 ◽

2005 ◽

Vol 127 (2) ◽

pp. 125-138 ◽

Cited By ~ 6

Author(s):

Chaw-Wu Tseng ◽

Jr-Yi Shen ◽

Hyunchul Kim ◽

I. Y. Shen

Keyword(s):

Mathematical Modeling ◽

Boundary Conditions ◽

Experimental Verification ◽

Complex Geometry ◽

Lagrange Equation ◽

Housing System ◽

Mode Shapes ◽

Vibration Modes ◽

Unified Approach ◽

Arbitrary Geometry

This paper demonstrates a unified approach to analyze linear vibration of rotating machines with arbitrary geometry and complexity. In this formulation, the rotating machine consists of three components: a rotating part (rotor), a stationary part (stator or housing), and multiple bearings. The rotor is assumed axisymmetric and spinning at constant speed. Moreover, the rotor and the housing have arbitrary geometry and complexity. The bearings connecting the rotor and housing could be rolling-element bearings or hydrodynamic bearings. The paper consists of three major sections: mathematical modeling, integration with finite element analysis (FEA), and experimental verification. For the mathematical modeling, a stationary rotor with free boundary conditions is first discretized to obtain its normal vibration modes and modal parameters. In the meantime, the housing with its actual boundary conditions (but no bearings) is also discretized. The discretization can be achieved, for example, through FEA to accommodate arbitrary and complex geometry of the rotor and the housing. Because these vibration modes are complete, modal response of each mode can serve as a generalized coordinate to describe vibration of the actual spinning rotor and housing system. With these generalized coordinates, gyroscopic effects of the spinning rotor can be derived through material derivatives for a ground-based observer. As a result, application of Lagrange equation leads to a set of gyroscopic equations of motion with constant coefficients. These coefficients, however, contain complicated volume integrals of the mode shapes and their spatial derivatives. Therefore, algorithms are developed to calculate these coefficients explicitly from FEA. For the experimental verification, a ball-bearing spindle carrying a cylinder closed at one end is used to validate the mathematical model. Frequency response functions of the spindle/cylinder system are measured for spin speed ranging from 0 to 6000 rpm. Natural frequencies measured from the experiments agree very well with the theoretical predictions from the unified approach up to 2 kHz.

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