An ultrasonic technique for the determination of casting-chill contact during solidification

Heat transfer between a solidifying aluminium alloy casting and a mould is dominated by the thermal resistance created by the interface. Interfacial heat transfer occurs by conduction through the atmosphere between the two surfaces and by conduction through the points of actual contact. (Heat transfer by radiation is probably significant only for ferrous castings.) The extent of real physical contact between two surfaces is difficult to quantify. This paper explains a method, using ultrasonic flaw detection techniques, whereby an estimate of the propagation of an ultrasonic signal through a casting-chill interface is used to infer the degree of actual contact occurring between them.In experiments involving casting and solidification of an aluminium alloy onto a copper chill the technique was found to give information for the first two seconds of the casting process only. In this time a peak in ultrasound transmission was observed, correlating to a maximum in the area of casting-chill contact, followed by a decrease in the ultrasound transmission that corresponded to actual contact areas between the casting and the chill in the region of 5 to 10%.     

An ultrasonic technique for the determination of casting-chill contact during solidification

Heat transfer between a solidifying aluminium alloy casting and a mould is dominated by the thermal resistance created by the interface. Interfacial heat transfer occurs by conduction through the atmosphere between the two surfaces and by conduction through the points of actual contact. (Heat transfer by radiation is probably significant only for ferrous castings.) The extent of real physical contact between two surfaces is difficult to quantify. This paper explains a method, using ultrasonic flaw detection techniques, whereby an estimate of the propagation of an ultrasonic signal through a casting-chill interface is used to infer the degree of actual contact occurring between them.In experiments involving casting and solidification of an aluminium alloy onto a copper chill the technique was found to give information for the first two seconds of the casting process only. In this time a peak in ultrasound transmission was observed, correlating to a maximum in the area of casting-chill contact, followed by a decrease in the ultrasound transmission that corresponded to actual contact areas between the casting and the chill in the region of 5 to 10%.     

Inward freezing of water in cylinders

The problem of one-dimensional inward solidification of a saturated liquid in a circular cylinder has been investigated both experimentally and theoretically with the boundary surface maintained at a constant temperature. A mathematical model has been developed using the integral heat balance method used by Bell. Experimental measurements of the location of the interface boundary and the temperature distribution in the solid region as a function of time have been compared with the new model and with those of other investigators.     

On the freezing time of cylinders and spheres

The Stefan problem is considered for freezing of spherical and cylindrical bodies submerged in a cooling agent. A method is suggested for analytical solution of this problem. Formulas are obtained for the freezing time and volume-average temperatures of a cylinder and sphere after completion of the freezing process. Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 70, No. 2, pp. 308–313, March–April, 1997.     

Function space theory of dynamic nucleation during solidification at high cooling rates

An analytical theory is proposed to study the dynamic nucleation of crystals from melt at very high cooling rates (10⁻⁶ to 10⁻¹² °K/sec). The mathematical framework is found to be isomorphic with the function space theory, wave and matrix mechanics, which enables application of various approximate methods of the latter disciplines. In principle, the mathematical apparatus and concepts of function space and wave mechanics can be utilized to study the time varying nucleation process. The Arrhenius law has been used to extrapolate the self-diffusion coefficient as a function of temperature above the melting point than those below. Since, applicability of Arrhenius equation at very high degrees of supercooling is not known and has to be substituted with appropriate constitutive relationship based on free volume theory of transport, the conclusion derived from the present analysis will not be unique with respect to the certainty of crystallization during the solidification process.     

A solution via generalised intergral transform technique for the simultaneous transport processes during combustion of wood cylinders

A non‐linear moving boundary diffusion problem is proposed as a simple model for the heat transfer during combustion of wood cylinders. Such a problem is solved here by applying the generalized integral transform technique. A new filtering strategy, denoted as local‐instantaneous filter, is used in order to accelerate the convergence of the series‐solution obtained with the present hybrid numerical–analytical technique. We show that the use of such filtering approach reduces the stiffness of the system of ordinary differential equations, resultant from the integral transformation of the original problem. Hence, subroutines based on simpler and faster methods can be used for the solution of such systems. Results are presented in the paper for the combustion of cylinders of different sizes and involving different initial moisture contents and densities. The effects on the solution of different models available in the literature for the evaluation of thermal conductivity and specific heat are also addressed on the paper. Copyright © 2000 John Wiley & Sons, Ltd.     

Segregation in Pb1−xSnx Te during solidification by the Bridgman technique

The influence of the growth parameters on the concentration gradient of Pb_1–xSn_xTe monocrystals grown by the vertical Bridgman technique is described. The concentration profiles of all three constituents (Pb, Sn, Te) have been analysed in ingots having different Pb-Sn ratios and grown under different conditions. The presumed role played by free convection in the Pb-Sn ratio is shown. Te was found to be constant in all ingots and not dependent upon the growth conditions.     

Analytical method for predicting freezing times of rectangular blocks of foodstuffs

For irregular geometries, the distance from the surface to the thermal centre is not uniform. This makes the calculation of freezing time impossible. The difficulty can be overcome by defining a ‘mean conducting path’ D_m/2, which can then be used to calculate Biot numbers. D_m/2 is considered to be the mean length of heat conducting paths from the surface of the body to the thermal centre. For rectangular blocks, experimental data suggest that D_m is proportional to the geometric mean of the two shorter sides. An equation for calculating the effective Biot number is presented, which incorporates one empirical coefficient. For squat-shaped bodies a good estimate of D_m can be obtained by averaging the shortest and longest dimensions. Pham's freezing time prediction method applied to brick-shaped blocks yields about the same accuracy as for the basic shapes (infinite slabs, infinite cylinders and spheres), and is more accurate than previous methods, including finite-different methods.     

Analytical phase-measuring technique for retardation measurements

An analytical method for phase detection in retardation measurements is proposed. The experimental setup is based on a simple linear polariscope with a λ plate. The intensity modulation at the output of the polariscope is measured when the wavelength is changed and a grid of phase-shifted intensity values is recorded. The phase difference between the components of the light propagating along the principal axes of the birefringent sample is determined with Greivenkamp's algorithm employed in phase-stepping interferometry. Error analysis for the new method is performed. Simplified algorithms for faster data analysis are proposed. The accuracy attained is comparable with the accuracy of known phase-detection methods used in retardation measurements.     

Analytical elastic solutions for pressurized hollow cylinders with internal functionally graded coatings

The main objective of this work is to obtain analytical solutions for thick-walled cylinders subjected to internal and external pressure in which the entire wall is made of functionally graded material or of only a thin functionally graded coating present on the internal homogeneous wall. We assume that the materials are isotropic with constant Poisson’s ratio; as far as the Young modulus is concerned, we consider a power and an exponential. The proposed analytical solutions show the effects of the different profiles describing the graded properties of the materials on the stress and displacement fields; in addition, comparisons between graded coating and conventional homogeneous coating highlight the advantage of the graded material on the interface stress reduction. Furthermore, we show how even a thin graded coating can be useful to satisfy the requirements of a specific application without having to make an entire wall with graded properties. This investigation permits us to optimize the elastic response of cylinders under pressure by tailoring the thickness variation of the elastic properties and to reduce manufacturing costs given by the technological limitations that occur to produce entire functionally graded walls.     

Analytical solutions for functionally graded incompressible eccentric and non-axisymmetrically loaded circular cylinders

We study analytically plane strain static deformations of functionally graded eccentric and non-axisymmetrically loaded circular cylinders comprised of isotropic and incompressible linear elastic materials. Normal and tangential surface tractions on the inner and the outer surfaces of a cylinder may vary in the circumferential direction. The shear modulus is taken to vary either as an exponential function or as a power law function of the radius only. The radial and the circumferential displacements, and the hydrostatic pressure are expanded in Fourier series in the angular coordinate, and expressions for their coefficients are derived from equations expressing the balance of mass (or the continuity equation) and the balance of linear momentum. Boundary conditions are satisfied in the sense of Fourier series. For the exponential variation of the shear modulus, the method of Frobenius series is used to solve 4th-order ordinary differential equations for coefficients of the Fourier series. It is shown that the series solutions for displacements and the hydrostatic pressure converge rapidly. Results for eccentric cylinders and non-axisymmetrically loaded circular cylinders are computed and exhibited graphically. Effects on stress distributions of the eccentricity in the cylinders and of the gradation in the shear modulus are illuminated. It is found that in a thin cylinder subjected to cosinusoidally varying pressure on the inner surface, segments of the cylinder between two adjacent cusps in the pressure deform due to bending rather than stretching.     

Control of the freezing interface motion in two-dimensional solidification processes using the adjoint method

The aim of this work is to calculate the optimum history of boundary cooling conditions that, in two-dimensional conduction driven solidification processes, results in a desired history of the freezing interface location/motion. The freezing front velocity and heat flux on the solid side of the front, define the obtained solidification microstructure that can be selected such that desired macroscopic mechanical properties and soundness of the final cast product are achieved. The so-called two-dimensional inverse Stefan design problem is formulated as an infinite-dimensional minimization problem. The adjoint method is developed in conjunction with the conjugate gradient method for the solution of this minimization problem. The sensitivity and adjoint equations are derived in a moving domain. The gradient of the cost functional is obtained by solving the adjoint equations backward in time. The sensitivity equations are solved forward in time to compute the optimal step size for the gradient method. Two-dimensional numerical examples are analysed to demonstrate the performance of the present method.     

Two-dimensional numerical model for the analysis of macrosegregation during solidification

This paper presents a 2D extension of a one-dimensional model proposed initially for the analysis of macrosegregation of multicomponent alloys. Some simulations were carried out to compare the performance of an explicit/implicit time integration scheme for coupling thermal-solutal fields with a fully implicit one. Simulations carried out for the analysis of a two-dimensional solidification problem has fitted the unidirectional and two-dimensional experimental data very well.     

降温速率对食品材料冻干特性的影响

用鲜胡萝卜作试验材料,研究了四种降温速率对水分升华率的影响。结果表明,当表层材料通过最大冰晶生长带的时间小于约25min时,材料表层对水蒸气的渗透阻力明显增大,在冻干中表层以下的材料温升较快并出现融化。若在冻干之前刮除表层或降低降温速率,冻干可顺利完成。     

Analytical solutions for multilayered composite cylinders with harmonic quadratic eigenstrain in arbitrary layers

The eigenstrain problem of multilayered hollow and solid composite cylinders working in a constant magnetic field is investigated analytically in this paper. Each layer of the composite cylinder can undergo a harmonic and spatially varying eigenstrain. The eigenstrain is assumed to be a quadratic polynomial function of the radial coordinate. The closed-form elastic solutions are obtained by solving the inhomogeneous governing Bessel differential equations. Then, the effects of eigenstrain distribution, angular frequency and the intensity of the magnetic field on the radial displacement, radial stress, hoop stress, and axial stress are presented graphically.     

Control of the freezing interface morphology in solidification processes in the presence of natural convection

This paper presents a methodology for the solution of an inverse solidification design problem in the presence of natural convection. In particular, the boundary heat flux q₀ in the fixed mold wall, δΩ₀, is calculated such that a desired freezing front velocity and shape are obtained. As the front velocity together with the flux history q_ms on the solid side of the freezing front play a determinant role in the obtained cast structure, the potential applications of the proposed methods to the control of casting processes are enormous. The proposed technique consists of first solving a direct natural convection problem of the liquid phase in an a priori known shrinking cavity, Ω_L(t), before solving an ill-posed inverse design conduction problem in the solid phase in an a priori known growing region, Ω_S(t). The direct convection problem is used to evaluate the flux q_ml in the liquid side of the freezing front. A front tracking deforming finite element technique is employed. The flux q_ml can be used together with the Stefan condition to provide the freezing interface flux q_ms in the solid side of the front. As such, two boundary conditions (flux q_ms and freezing temperature θ_m) are especified along the (known) freezing interface δΩ_I(t). The developed design technique uses the adjoint method to calculate in L₂ the derivative of the cost functional, ∥θ_m – θ( x , t; q₀)∥, that expresses the square error between the calculated temperature θ( x , t; q₀) in the solid phase along δΩ_I(t) and the given melting temperature. The minimization of this cost functional is performed by the conjugate gradient method via the solutions of the direct, sensitivity and adjoint problems. A front tracking finite element technique is employed in this inverse analysis. Finally, an example is presented for the solidification of a superheated incompressible liquid aluminium, where the effects of natural convection in the moving interface shape are controlled with a proper adjustment of the cooling boundary conditions.