Bubble transport in three-dimensional laminar gravity-driven flow - mathematical formulation

This paper presents a complete set of coupled equations that govern the bubble transport in three-dimensional gravity-driven flow. The model accounts for bubble growth or shrinkage due to pressure and temperature changes as well as for multiple gas diffusion in and out of the bubbles but neglects bubble coalescence, break-up, and nucleation. The model applies to glass melting furnaces but it could be extended to other two-phase flow applications such as metal and polymer processing, passive cooling systems, and two-phase flow around naval surface ships. Governing equations are given for the key variables which are, in the present case, (1) the refining agent concentration, (2) the gas species dissolved in the liquid phase, and (3) the bubble radius, gas molar fraction, and density function. The method of solution based on the backward method of characteristics is briefly discussed.     

Explicit solutions for boundary value problems in diffusion of heat and mass

The Fourier's infinite series solutions of transient diffusion in finite solids, with convection boundary conditions, contain eigenvalues, which are defined by implicit transcendental equations. The eigenvalue equations have to be evaluated numerically, making the use of the Fourier's solution difficult. Using the successive substitution method, explicit expressions were derived for the first eigenvalue λ₁, in plates, cylinders and spheres, as a function of the Biot number. The derived expressions for λ₁(Bi), together with the one-term Fourier series solution, yield simple explicit solutions for temperature or concentration profiles in plates, cylinders and spheres. The explicit solutions are valid for Fo>0.2.     

Behavior of a nematic liquid crystal in oscillatory flow at weak surface anchoring

The effect of the surface viscosity on the orientational dynamics of a nematic liquid crystal in an oscillatory flow has been studied. For the shear and the Poiseuille flows, the approximate analytical solutions of the nematodynamic equations are obtained at small flow amplitudes. The frequency range of the oscillatory flow is determined, in which the surface viscosity strongly affects the optical response of a nematic layer. The results obtained are compared with the data of direct numerical simulation of the nematodynamic equations and the method for the experimental determination of surface viscosity is proposed.     

Three‐dimensional numerical investigation of the effects of an open window on the convective heat transfer of an oxide Czochralski system

A three‐dimensional numerical analysis was carried out for a real Czochralski crystal growth furnace containing only gas and without any melt and crystal in order to investigate the effects of a small observation window on the temperature and flow field of the system. For this approach, the induction heating equations, the Navier‐Stokes equation with Boussinesq approximation, the continuity and energy equations have been solved in cylindrical coordinates using the finite element method. It has been found that the flow and thermal fields in the system are obviously three‐dimensional and non‐axisymmetric. The gas enters the system through the window is directed towards the opposite side wall where it is divided into two parts of vertical direction as well as expands in horizontal direction. Consequently, there is a spiral gas flow in the crucible and afterheater which rotates upwards in azimuthal direction along the walls. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Generalization of the nonstandard approach in the dynamic theory of diffraction for deformed crystals

The nonstandard theory of X-ray scattering in a deformed crystal has been generalized. The vector of atomic-plane displacement is introduced into the crystal polarizability model like in the generalized Takagi dynamic theory. The solution to the wave equation is sought for using the procedure of expanding the field amplitude and vector operators in the Fourier components of polarizability χ _H in a series according to the multiscale method. It is shown that considering lattice strain generally calls for introducing various characteristic spatial regions for the diffraction equation, which is in complete agreement with the main concept of the multiscale method. A particular case of a strain field depending on one scale is considered. If a relative change in strain occurs at a length on the order of the extinction length, one can obtain equations generalizing the Takagi equations to the case of arbitrary diffraction geometries.     

Optimization of surface temperature distribution for control of point defects in the silicon single crystal

Optimization of crystal surface temperature distribution is performed for the control of point defects in a silicon single crystal grown by the Czochralski process. In the optimization problem, we seek an optimal solution that minimizes the level of point defects while the radial uniformity of temperature is maximized. In order to solve the optimization problem with the equality constraints described by partial differential equations, the variational principle is used. Based on the calculus of variations and the method of Lagrange multiplier, the Euler–Lagrange equations are derived and solved by the finite difference method. In order to handle inequality constraints, the penalty function method is also applied. The optimal distribution of the crystal surface temperature is expected to provide an insight for the design of thermal surroundings such as the thermal shield configuration and the heater/cooler position.     

Acid-base reactions in molten silicates

The principles of the acid-base theory are transposed to molten silicates and hence to vitreous silicates; the classical concepts of equilibrium constants are applied to the various species involved in the neutralization of silica, regarded as a polyacid.The general system appears as a group of ten reactions to which correspond ten equilibrium constants which are reducible to four independent values. It results in a group of six non-linear equations for six unknown quantities. The problem is solved by a numerical method.It is thus possible to determine the distribution of species as a function of the neutralization rate and plot the neutralization curves pO = f(x) according to the pK values.The previous theories of Toop and Samis and also Masson appear to be particular cases of the general theory and correspond to various arbitrary simplifications of the group of constants. The true values can be obtained when the experimental data of free energy of mixing are taken into account.As the method is valid for a wide range of compositions, some difficulties inherent in earlier theories are overcome. It is possible in particular to deal with the case of melts close to metasilicate, and to extend the method to multicomponent systems.     

Comparative analysis of two methods of calculation of the orientation of domain walls in ferroelastics

Two types of domain-wall equations are analyzed: the equations derived by the Sapriel method and the equations obtained by interface matching of the thermal-expansion tensor. It is shown that, for W-type domain walls, these methods yield the same equations. For W′-type domain walls, the equations obtained by different methods coincide for proper ferroelastics and differ for improper ferroelastics.     

Determination of refractive indexes of dielectric films by the use of immersion liquids

Examples of using immersion liquids in ellilipsometry and multiple-beam interferometry are given. The refractive index of very thin (∼ 5.0 nm) silica films on silicon and on films of silicon nitride are determined by the ellipsometry method. The method of precise determination of refractive index of monolayer dielectric films is described when a precision equal to ± 3 · 10⁻⁴ was reached. A method of determination film thicknesses near the substrate surface in a twolayer system is suggested. The construction of different prisms is given by means of which the immersion liquids are inserted. The selection of the equations for the determination of refractive index in the multiple-beam interferometry is discussed. Two versions of the use of immersion liquids for the determination of refractive indexes of dielectric films by interferometry method are given.     

石英晶体板非线性高频振动的拓展伽辽金法分析

针对考虑几何和材料非线性的石英晶体板厚度剪切振动和弯曲振动的方程组,利用扩展伽辽金法对该方程组进行转化和求解,分别获得了强烈耦合的厚度剪切振动模态和弯曲振动模态的频率响应关系,绘制了不同振幅比和不同驱动电压影响下的频率响应曲线图。数值计算结果表明可以选取石英晶片的最佳长厚比尺寸来避免两种模态的强烈耦合。驱动电压的变化将引起石英晶体谐振器厚度剪切振动频率的明显改变,必须将振动频率的漂移值控制在常用压电声波器件的允许值之内。扩展伽辽金法对石英晶体板非线性振动方程组的求解为非线性有限元分析和偏场效应分析奠定了基础。     

Design of a continuous crystallizer by evaporation as unit operation

We have examined the continuous crystallization by evaporation as a unit operation, a process which is normally empirically designed and dimensioned. Through the mass balances and process equations (concerning the crystal growth, the secondary nucleation, the crystals dimensional and numeric balances) six equations were obtained upon which the calculation of the operating variables depends. The resulting mathematical system offers two degrees of freedom, and by defining the two independent variables it was possible to calculate — as shown in the graphics — the trends of the other six variables. One of the chosen independent variables was the diameter of the formed nuclei (as this is basically an experimental date), whereas the other was the variable τ which defines the residence time.     

不同磁场对大直径单晶硅生长中的动量与热量输运影响的数值分析

本文采用紊流模型对大直径单晶硅在垂直磁场及勾形磁场作用时熔体内动量及热量输运作了数值模拟.采用有限体积法离散控制方程,采用SIMPLE((Semi-implicit Method for Pressure Linked Equations)算法耦合压力和速度场.对无磁场、垂直磁场及勾形磁场作用下熔体内的传输特性进行了比较.数值计算结果表明,垂直磁场对动量及热量的分布具有双重效应.垂直磁场强度过大,不利于晶体生长.随着勾形磁场强度的增加,熔体内子午面上的流动减弱,并且紊流强度也相应降低.     

采用延时修正法实施QUICK格式模拟提拉单晶体的生长

为了数值模拟提拉(又名Czochralski)法获得单晶体的生长过程,本文采用有限容积法离散控制方程,采用非均匀的交错网格避免不合理的振荡压力场,采用三阶精度QUICK (Quadratic Upwind Interpolation of Convective Kinematics)格式离散对流项,采用延时修正来实施QUICK格式获得满足主对角占优的代数方程组,采用SIMPLE(Semi-implicit Method for Pressure Linked Equations)算法耦合压力和速度场,给出了基于上述方法的方程、算法,并发展了程序,计算了Wheeler标准问题,计算结果与文献相当一致,同时本算法能模拟计算高葛拉晓夫数时的流动,显示出非均匀网格QUICK格式模拟晶体生长的优越性;另外本文将这一算法运用到单晶硅的数值模拟中,计算结果令人满意.     

Eine neue absolute sin ϕi/sin ϕj-Methode für die Bestimmung von Gitterparametern

The reciprocal squares of lattice parameters are determined for one-, two- and three parameter lattices by solving of linear equations. The systematic deviations are suppressed totally at this work method. The method is applicable universally with many diffraction procedures.     

Prediction of immiscibility boundaries of multi-component silicate glasses

The principle and method which have previously been used to estimate the immiscibility boundary of a ternary silicate system are also applicable to the estimation of the immiscibility boundaries of quaternary and higher order multi-component silicate glasses. The method is based upon the regular mixing equations derived for quaternary and multi-component silicate systems and the superposition principle of the immiscibility boundary. As an example, the Li₂ONa₂OK₂OSiO₂ system is used to demonstrate the success of the estimation method. Expressions for higher order multi-component silicate glasses are also shown. A polynomial expression is obtained for the master curve of the normalized immiscibility boundary to facilitate the immiscibility temperature estimation.     

Systematic Solution of the Mean Field Equation for Liquid Crystals

Starting with a pairwise, spatially and orientationally dependent intermolecular potential of the Kobayashi-McMillan form, we carry out a systematic solution of the mean field equation for liquid crystals. The mean field equation, presented as first of a hierarchy of BBGKY equations, is first reduced to a set of coupled integrodifferential equations by means of expanding the distribution function f (r, δ) and In f (r, δ) in Legendre polynomials and the reciprocal lattice space. In the first level of approximation, the expansion retains only the lowest-order coefficients, permitting a complete decoupling of the equations. In the second level of approximation, the leading coefficient which couples spatial order to orientational order is included. In the third level of approximation two more higher order coefficients are included. At each level, the free energy functional is evaluated to determine the equilibrium phase at given temperatures and chainlengths of a homologous series. It is shown that the expansion converges very rapidly, the second level of approximation being entirely sufficient. This lends support to our earlier variational calculation which contained only three variational parameters.     

相场方法及其在晶体生长中的应用

相场方法已被发掘出用于直接求解时的自由边界问题－著名的斯特藩方程。该方法作为晶体生长过程中模拟复杂图形成图的计算工具，已呈现出强有力的生命力。目前的研究在于努力发展精巧的计算技术，以便于晶体生长和金属凝固过程进行了理论模拟，而这些技术将有可能方法地应用于工业流程。相场方法之所以具有吸引力，基于如下事实：在计算机模拟过程中，即可避免对于边界实时追踪，又不需要反复判别是否满足显式边界条件。在过去的10年中，它已逐步被用于研究晶体生长的基础课题。诸如；热质输运，晶体生长动力学，二维和三维枝晶生长，图形选择，生长形态和显微结构等。本文对相场方法进行评述，同时给出其最新应用结果。     

A Kinetic Theory of Incoherent Exciton Annihilation

The properties of an exciton system with the high concentration are investigated on the basis of microscopical equations for the density matrix of the type of BBGKY hierarchy. A new recipe for decoupling a chain of equations is proposed, which takes into account the effect of three-particle collisions on the definition of a two-particle distribution function of an exciton sub-system. Using this decoupling procedure and making no assumption about the smallness of the quadratic concentration effects the set of equations for density matrix reduced to the known phenomenological equation describing exciton annihilation under steady-state excitation.

The kinetic equations applicable to an essentially larger concentration region are also derived taking into account high concentration effects. There use for the case of excitons with a great diffusion length permits us to predict a new effect of additional luminescence quenching at high excitation levels. This effect turns to be very appreciable in two-, and particularly, in one-dimensional systems and may manifest itself in the kinetics of nonlinear luminescence quenching. A qualitatively new-behaviour of the quantum yield dependence on excitation intensity is predicted, which gives reasonable explanation of recent experimental results.

The limits on concentration and intensity values where the present theory is good are pointed out.

The results of the paper may be applied to other quantum many-particle systems where the coordinates of a quasi-particle are good quantum numbers, and also to chemical reaction theory, to coagulation and the related problems.     

Unsteady three-dimensional buoyancy-driven convection in a circular cylindrical cavity and its damping by magnetic field

The oscillatory convection in an electrically conducting liquid-metal confined in a cylindrical cavity of moderate length is investigated using a spectral numerical method involving direct resolution of the Navier-Stokes equations for three-dimensional flows. Time signals, symmetry properties and energy budgets are presented in order to characterize the flow oscillations. The damping of the oscillatory flows by a vertical magnetic field is then studied up to the stabilization to a steady flow.