Qualitative Behavior of Solutions for Thermodynamically Consistent Stefan Problems with Surface Tension

We study the qualitative behavior of a thermodynamically consistent two-phase Stefan problem with surface tension and with or without kinetic undercooling. It is shown that these problems generate local semiflows in well-defined state manifolds. If a solution does not exhibit singularities in a sense made precise herein, it is proved that it exists globally in time and its orbit is relatively compact. In addition, stability and instability of equilibria are studied. In particular, it is shown that multiple spheres of the same radius are unstable, reminiscent of the onset of Ostwald ripening.     

Intense Condensation of a Molecular Gas

The kinetic problem of intense subsonic condensation of a polyatomic gas on a plane surface is solved by the method of semispatial moments. The contribution of the internal degrees of freedom to the total heat capacity is taken into account. The domains of existence of the Boltzmann equation solution are determined for the given method. Analytic expressions for calculating the gas concentration behind the Knudsen layer depending on the Mach number and temperature are derived for the cases of mon-, di-, and triatomic gases.     

Application of Fractional Derivatives in Thermal Analysis of Disk Brakes

This paper presents a Fractional Derivative Approach for thermal analysis of disk brakes. In this research, the problem is idealized as one-dimensional. The formulation developed contains fractional semi integral and derivative expressions, which provide an easy approach to compute friction surface temperature and heat flux as functions of time. Given the heat flux, the formulation provides a means to compute the surface temperature, and given the surface temperature, it provides a means to compute surface heat flux. A least square method is presented to smooth out the temperature curve and eliminate/reduce the effect of statistical variations in temperature due to measurement errors. It is shown that the integer power series approach to consider simple polynomials for least square purposes can lead to significant error. In contrast, the polynomials considered here contain fractional power terms. The formulation is extended to account for convective heat loss from the side surfaces. Using a simulated experiment, it is also shown that the present formulation predicts accurate values for the surface heat flux. Results of this study compare well with analytical and experimental results.     

Application of Fractional Derivatives in Thermal Analysis of Disk Brakes

This paper presents a Fractional Derivative Approach for thermal analysis of disk brakes. In this research, the problem is idealized as one-dimensional. The formulation developed contains fractional semi integral and derivative expressions, which provide an easy approach to compute friction surface temperature and heat flux as functions of time. Given the heat flux, the formulation provides a means to compute the surface temperature, and given the surface temperature, it provides a means to compute surface heat flux. A least square method is presented to smooth out the temperature curve and eliminate/reduce the effect of statistical variations in temperature due to measurement errors. It is shown that the integer power series approach to consider simple polynomials for least square purposes can lead to significant error. In contrast, the polynomials considered here contain fractional power terms. The formulation is extended to account for convective heat loss from the side surfaces. Using a simulated experiment, it is also shown that the present formulation predicts accurate values for the surface heat flux. Results of this study compare well with analytical and experimental results.     

Study of the analytical solution to the heat transfer problem and surface temperature in a semi-infinite body with a constant heat flux at the surface and an initial temperature distribution

In many practical cases, one heats a semi-infinite solid with a constant heat flux source. For such an unsteady heat transfer problem, if the body has a uniform initial temperature, the analytical solution has been given by Carslaw and Jaeger. The surface temperature of the semi-infinite body follows the $\sqrt t $ -rule, that is, the surface temperature changes in proportion to square root of heating time. But if, instead of the uniform initial temperature, the body has a temperature distribution at the beginning of heating, the analytical solution has not yet been developed. Analytical solutions to the same problem with an exponential or a linear initial temperature distribution are obtained in this paper. It is shown, that in the case of a linear initial temperature distribution the surface temperature also changes according to $\sqrt t $ -rule Approximating the initial temperature distribution near the surface by its tangent at the surface, it is found that the surface temperature within a short time after the start of heating should also satisfy the $\sqrt t $ -rule, in spite of an arbitrary initial temperature distribution. The experimental data support this argument. Furthermore, the constant heat flux can be calculated after relationship between the surface temperature and heating time according to the equation derived in this paper, if the initial temperature distribution or its first-order derivative at the surface is known.     

Flow and heat transfer in a second grade fluid over a stretching sheet

An analysis is carried out to study the flow and heat transfer characteristics in a second grade fluid over a stretching sheet with prescribed surface temperature including the effects of frictional heating, internal heat generation or absorption, and work due to deformation. In order to solve the fourth-order non-linear differential equation, associated with the flow problem, a fourth boundary condition is augmented and a proper sign for the normal stress modulus is used. It is observed that for a physical flow problem the solution is unique. The solutions for the temperature and the heat transfer characteristics are obtained numerically and presented by a table and graphs. Furthermore, it is shown that the heat flow is always from the stretching sheet to the fluid.     

An application of Discontinuous Galerkin space and velocity discretisations to the solution of a model kinetic equation

An approach based on a Discontinuous Galerkin discretisation is proposed for the Bhatnagar–Gross–Krook model kinetic equation. This approach allows for a high-order polynomial approximation of molecular velocity distribution function both in spatial and velocity variables. It is applied to model one-dimensional normal shock wave and heat transfer problems. Convergence of solutions with respect to the number of spatial cells and velocity bins is studied, with the degree of polynomial approximation ranging from zero to four in the physical space variable and from zero to eight in the velocity variable. This approach is found to conserve mass, momentum and energy when high-degree polynomial approximations are used in the velocity space. For the shock wave problem, the solution is shown to exhibit accelerated convergence with respect to the velocity variable. Convergence with respect to the spatial variable is in agreement with the order of the polynomial approximation used. For the heat transfer problem, it was observed that convergence of solutions obtained by high-degree polynomial approximations is only second order with respect to the resolution in the spatial variable. This is attributed to the temperature jump at the wall in the solutions. The shock wave and heat transfer solutions are in excellent agreement with the solutions obtained by a conservative finite volume scheme.     

Boundary conditions for gases on a body surface

The influence of boundary conditions on the distribution of macroscopic parameters of various types is studied using the problem of heat transfer for monatomic and diatomic gases as an example. The calculations are made on the basis of model kinetic equations [1, 2]. A boundary condition that takes into account mutual transitions of rotational and translational energies of particles when they collide with a body surface is proposed for diatomic gas. Comparison with the experimental results made it possible to determine the coefficients appearing in the boundary conditions.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 141–148, September–October, 1986.     

Stability of a vertical liquid film with consideration of the marangoni effect and heat exchange with the environment

The stability of a free vertical liquid film under the combined action of gravity and thermocapillary forces has been studied. An exact solution of the Navier-Stokes and thermal conductivity equations is obtained for the case of plane steady flow with constant film thickness. It is shown that if the free surfaces of the film are perfectly heat insulated, the liquid flow rate through the cross section of the layer is zero. It is found that to close the model with consideration of the heat exchange with the environment, it is necessary to specify the liquid flow rate and the derivative of the temperature with respect to the longitudinal coordinate or the flow rate and the film thickness. The stability of the solution with constant film thickness at small wave numbers is studied. A solution of the spectral problem for perturbations in the form of damped oscillations is obtained.     

Pulsed laser heating – Fourier and electron kinetic theory approaches

Laser surface pulse heating of engineering metals is in demand in the metal industry and investigation into laser pulse heating becomes fruitful in this regard. Application of Fourier theory to heat conduction due to high power laser irradiation may give closed form solution to the problem. On the other hand, the heat flux through a given plane depends on the electron energy distribution through the material and at the scale of distance required to examine the problem, the material can no longer be considered as being homogeneous continuum, therefore, errors may occur when considering the Fourier theory in laser heating process. The problem requires to be examined in the quantum field. The present study examines the pulse laser heating process when considering both Fourier conduction and electron-kinetic theory approaches. Analytical solution to Fourier conduction equation is obtained for intensity exponential pulses while numerical scheme is introduced to solve the heat transfer equation resulted from kinetic theory approach. It is found that both Fourier and electron kinetic theory approaches result in similar temperature profiles for the pulses having the same energy content. In the case of electron kinetic theory approach the rise time for surface temperature to reach the melting point is shorter than that obtained from the analytical solution. Received on 23 February 1998     

Anisotropic thermopiezoelectric solids with an elliptic inclusion or a hole under uniform heat flow

The two-dimensional problem of a thermopiezoelectric material containing an elliptic inclusion or a hole subjected to a remote uniform heat flow is studied. Based on the extended Lekhnitskii formulation for thermopiezoelectricity, conformal mapping and Laurent series expansion, the explicit and closed-form solutions are obtained both inside and outside the inclusion (or hole). For a hole problem, the exact electric boundary conditions on the hole surface are used. The results show that the electroelastic fields inside the inclusion or the electric field inside the hole are linear functions of the coordinates. When the elliptic hole degenerates into a slit crack, the electroelastic fields and the intensity factors are obtained. The effect of the heat flow direction and the dielectric constant of air inside the crack on the thermal electroelastic fields are discussed. Comparison is made with two special cases of which the closed solutions exist and it is shown that our results are valid.     

Impinging sweeping jet and convective heat transfer on curved surfaces

The application of an impinging sweeping jet, which oscillates periodically with a large angle, to convective heat transfer has received attention owing to its capability to provide a more spatially uniform and enhanced heat removal rate when compared to a steady jet. Herein, we study how the surface curvature affects the heat transfer performance of a sweeping jet and couple it with the representative flow characteristics. Heat transfer measurement and quantitative flow visualization are conducted experimentally for concave and convex surfaces as well as a flat surface. Whereas concave surfaces have a better heat transfer rate than a flat surface, the enhancement of the heat transfer is relatively small for a convex surface. For both concave and convex surfaces, the Nusselt number does not increase monotonically with the curvature magnitude but has a peak for a moderate curvature. The variation in heat transfer performance with the surface curvature is correlated with the phase-averaged velocity profile of the wall jet deflected after an impingement and the turbulence kinetic energy inside the jet. For both concave and convex surfaces, the wall jet becomes thinner than a flat surface in general, which contributes to improved heat transfer. However, whereas the turbulence kinetic energy is significantly larger for a concave surface of a moderate curvature than that of a flat surface, the turbulence kinetic energy for a convex surface is reduced from that of a flat surface, resulting in degradation of the heat transfer performance.     

Effects of mesh resolution on hypersonic heating prediction

Aeroheating prediction is a challenging and critical problem for the design and optimization of hypersonic vehicles. One challenge is that the solution of the Navier-Stokes equations strongly depends on the computational mesh. In this letter, the effect of mesh resolution on heat flux prediction is studied. It is found that mesh-independent solutions can be obtained using fine mesh, whose accuracy is confirmed by results from kinetic particle simulation. It is analyzed that mesh-induced numerical error comes mainly from the flux calculation in the boundary layer whereas the temperature gradient on the surface can be evaluated using a wall function. Numerical schemes having strong capability of boundary layer capture are therefore recommended for hypersonic heating prediction.     

3-D analysis of temperature distribution in the material during pulsed laser and material interaction

Recently, lasers are being increasingly used in the industry owing to their precision and low cost. Material is heated and evaporated during laser and material interaction due to the absorption of laser beams by the material. In this study, a 3-D Laser heating model including evaporation has been solved using the electron- kinetic theory approach. The basis in examining the problem using the kinetic theory approach is to describe the heat conduction through electron-phonon and molecule-phonon collisions. The problem is solved by using the electron-kinetic theory approach in such a way that heat conduction is taken into account until the material is heated to its melting temperature and non-conduction limited heat transfer is considered after the melting temperature is reached. Non-conduction limited heat transfer through the phase change process is resulted from vacancy-molecule collisions. A numerical scheme is introduced to solve the governing equation, owing to the fact that the energy equation resulted is in the form of integro-differential equation. Four different materials, namely iron, nickel, tantalum and titanium are chosen in this study determine the material response to laser pulse heating. For each material, time dependent temperature distribution through the depth of the material and on the surface of the material is computed and analyzed for four different materials.     

An inverse problem to estimate temperature dependent heat capacity under convection processes

Solutions of the heat capacity versus temperature in a one-dimensional slab have been studied for different types of dependency (lineal, sinusoidal, piece-wise and rectangular) under boundary conditions of natural and forced convection on both sides of the slab. The input data of this inverse problem are the temperature history ("measurements") at a particular location within the slab, obtained by adding a specified random error to the set of temperatures which are the solution of the direct problem. No prior information is used as regards the temperature-dependent functional forms of the unknown heat capacity. In all cases, a piece-wise function is used to approach the solution. Using a programming routine that minimises a classical predefined functional, successive stretches of this piece-wise function are obtained step by step by (i) fixing its length and (ii) increasing or decreasing its slope. The Network Simulation Method is used to solve both the direct and inverse problems. No mathematical manipulations of the finite-difference differential equations are required by the programmer, since they are contained in the computer code used in the method. The basic network for the inverse problem, which is basically the same as for the direct problem, is easy to design and has very few devices. Several examples are shown to prove the accuracy and effectiveness of the proposed method.     

Melting or sublimation of solids with volume heat sources

Summary The problem of the phase change produced in semi-infinite media by volume heat sources is considered for quasi steady state conditions and where the molten or sublimated material is removed as soon as it is formed. A general formal solution for the temperature distribution and the velocity of the phase change surface is presented, and it is shown that for source functions of physical interest the velocity of the surface is the ratio of the total heat flux to the energy required to produce the phase change of a unit volume of material. Several examples are considered.     

Heat Transfer and Evaporation from a Plane Surface into a Half-Space upon a Sudden Increase in Body Temperature

The one-dimensional time-dependent problem of evaporation from a plane body surface into a half-space filled by a gas (condensed phase vapor) upon a sudden increase in the body surface temperature is studied. The evaporation coefficient is the problem parameter and may take arbitrary values within the limits from zero to unity. The problem is formulated for the kinetic equation and solved by the finite-difference method. It is shown that a deviation of the evaporation coefficient from unity considerably modifies the gas phase flow pattern. However, the evaporation rate divided by the rate of evaporation into a vacuum at the given surface temperature is only weakly dependent on the evaporation coefficient.     

Phase Behavior of Methane-Pentane Mixture in Bulk and in Porous Media

The adiabatic calorimetry method is used to measure the isochoric heat capacity and the dew-bubble curves for the methane-normal pentane mixtures in bulk for three different mixture compositions of n-pentane. The near-critical behavior of the mixture heat capacity indicates that at low n-pentane concentrations, this system is close to the special point of the critical locus. The effect of porous medium has been investigated for one of the mixtures studied in bulk. It has been shown that in porous medium the essential shift of the dew-bubble curve takes place. In our opinion this shift is caused by the formation of wetting film on the surface of porous medium. The estimation of film thickness resulting from the data obtained yields the value 7–8 nm.