Approximate solutions to the Stefan problem with internal heat generation

Using a quasi-static approach valid for Stefan numbers less than one, we derive approximate equations governing the movement of a phase change front for materials which generate internal heat. These models are applied for both constant surface temperature and constant surface heat flux boundary conditions, in cylindrical, spherical, plane wall and semi-infinite geometries. Exact solutions with the constant surface temperature condition are obtained for the steady-state solidification thickness using the cylinder, sphere, and plane wall geometries which show that the thickness depends on the inverse square root of the internal heat generation. Under constant surface heat flux conditions, closed form equations can be obtained for the three geometries. In the case of the semi-infinite wall, we show that for constant temperature and constant heat flux out of the wall conditions, the solidification layer grows then remelts.     

Composite heat transfer in case of steady laminar flow of a gray fluid with large optical density past a horizontal plate embedded in a saturated porous medium

The present paper discusses the problem of composite heat transfer and viscous friction of a moving gray medium with large optical density. Expressions for temperature and velocity distributions and the ratio of the radiative component to convective component of heat flux are obtained. It is observed that for a given value ofB the ratio of radiative heat flux to convective heat flux is maximum at the edge of the boundary layer and tends to an asymptotic value as the boundary is reached. However, for a given value ofK δ, the ratio of heat fluxes increases with increase inB (the porous parameter). The results also show that as the wall temperature approaches the value of free stream temperature, the ratio of heat fluxes decreases.     

Stagnation point heat transfer from laminar,high temperature methane flames

Combustion of methane-rich fuels frequently provides forced convective heating in industry, and the ability to predict the rate of heat transfer from such flames to solid surfaces is often desirable. Mathematical modelling of stagnation point heat flux has been achieved by numerical solution of the boundary layer equations, and by an analytical equation modified to include the effects of chemical reaction in the free stream flow and to allow for the enhancement in heat flux caused by the diffusion and exothermic recombination of reactive species in the boundary layer surrounding the heat receiving body. Predictions from these models have been compared with experimental data obtained in high temperature methane flames of various equivalence ratios. Within the equilibrium region of these flames, predictions from the modified analytical equation based on total Lewis numbers equal to and greater than one form a tight envelope around the experimental results, and hence provide a relatively simple method of predicting heat flux. Numerical solutions tend to slightly underestimate predictions from the analytical equation and experimental data, although agreement with the alternative prediction method increases with the surface temperature of the heat receiving body     

Effect of Low Rotation Rate on Steady Convection During the Solidification of a Ternary Alloy

We consider the problem of steady convective flow during the directional solidification of a horizontal ternary alloy system rotating at a constant and low rate about a vertical axis. Under the limit of large far-field temperature, the flow region is modeled to be composed of two horizontal mushy layers, which are referred to here as a primary layer over a secondary layer. We first determine the basic state solution and then carry out linear stability analysis to calculate the neutral stability boundary and the critical conditions at the onset of motion. We find, in particular, that there are two flow solutions and each solution exhibits two neutral stability boundaries, and the flow can be multi-modal in the low rotating rate case with local minima on each neutral boundary. The critical Rayleigh number and the wave number as well as the vertical volume flux increase with the rotation rate, but the flow is found to be less stabilizing as compared to the binary alloy counterpart flow. The effects of low rotation rate increase the solid fraction and the liquid fraction at certain vertically oriented fluid lines, and the highest value of such increase is at a horizontal level close to the interface between the two mushy layers.     

Convective heat transfer in a locally heated plane incompressible fluid layer

The problem of convection in a plane horizontal layer of incompressible fluid with rigid boundaries when the temperature is constant on the lower boundary and has a parabolic profile on the upper boundary can be reduced to solution of a system of time-dependent one-dimensional equations. An analytic solution of the problem is obtained directly at the extremum point. Together with the wellknown solutions which describe heat transfer for the linear temperature distribution on the boundaries, the results obtained make it possible to calculate the heat flux through a thin slit for an arbitrary given heating of a thin fluid layer between heat-conducting bodies.     

Laminar forced convection conjugate heat transfer over a flat plate

Coupled heat transfer between laminar forced convection along and conduction inside a flat plate wall is theoretically studied. The laminar convective boundary layer is analyzed by employing the integral technique. The energy equations for the fluid and the plate wall are combined under the condition of the continuity in the temperature and heat flux at the fluid-solid interface. The analysis results in a simple formal solution. Expressions have been obtained for calculating local Nusselt number, wall heat flux and temperature along the plate, all are functions of the local Brun number, Br _x , which is a measure of the ratio of the thermal resistance of the plate to that of the convective boundary layer. The results indicate that for Br _x ≥0.15, neglecting the plate resistance will results in an error of more than 5% in Nusselt number. Comparison of the present solution with other previous studies has been made. The solution may be of a considerable theoretical and practical interest. Received on 19 August 1998     

Periodic heat conduction with relaxation time in cylindrical geometry

Steady-periodic heat conduction with relaxation time in an infinitely long hollow cylinder is considered. Four boundary value problems, with boundary conditions of the first and of the second kind, are solved analytically. The solution for a solid cylinder with a sinusoidally varying surface temperature is obtained as a special case of a solution found for the hollow cylinder. The effects of the relaxation time on the steady-periodic temperature field are analysed, in details, for a solid cylinder with a sinusoidally varying surface temperature and for a hollow cylinder with a sinusoidally varying heat flux at the inner surface and with a constant temperature at the outer surface. The results show that thermal resonances may occur and suggest that accurate measurements of the relaxation time could be obtained by means of experiments on steady-periodic heat conduction in cylindrical geometry. Received on 15 April 1997     

Effects of temperature dependent fluid properties and variable Prandtl number on the transient convective flow due to a porous rotating disk

In this paper we have studied the effects of temperature dependent fluid properties such as density, viscosity and thermal conductivity and variable Prandtl number on unsteady convective heat transfer flow over a porous rotating disk. Using similarity transformations we reduce the governing nonlinear partial differential equations for flow and heat transfer into a system of ordinary differential equations which are then solved numerically by applying Nachtsheim–Swigert shooting iteration technique along with sixth-order Runge–Kutta integration scheme. Comparison with previously published work for steady case of the problem were performed and found to be in very good agreement. The obtained numerical results show that the rate of heat transfer in a fluid of constant properties is higher than in a fluid of variable properties. The results further show that consideration of Prandtl number as constant within the boundary layer for variable fluid properties lead unrealistic results. Therefore, modeling thermal boundary layers with temperature dependent fluid properties Prandtl number must treated as variable inside the boundary layer.     

Retrieving the heat transfer coefficient for jet impingement from transient temperature measurements

Algorithm of retrieving the heat transfer coefficient (HTC) from transient temperature measurements is presented. The unknown distributions of two types of boundary conditions: the temperature and heat flux are parameterized using a small number of user defined functions. The solutions of the direct heat conduction problems with known boundary temperature and flux are expressed as a superposition of auxiliary temperature fields multiplied by unknown parameters. Inverse problem is formulated as a least squares fit of calculated and measured temperatures and is cast in a form of a sum of two objective functions. The first results originates from an inverse problem for retrieving the boundary temperature the second comes from the inverse problem for reproducing the boundary heat flux. The final form of the objective function is obtained by enforcing constant in time value of the heat transfer coefficient. This approach leads to substantial regularization of the results, when compared with the standard technique, where HTC is calculated from separately reconstructed temperature and heat flux on the boundary. The validation of the numerical procedure is carried out by reconstructing a known distribution of the HTC using simulated measurements laden by stochastic error. The proposed approach is also used to reconstruct the distribution of the HTC in a physical experiment of heating a cylindrical sample using an impinging jet.     

Freezing over a cylinder with finite heat transfer at the wall

A simple and straightforward implicit numerical technique for phase change problems in cylindrical geometry is proposed. When implicit schemes are used the moving interface along with the convective or radiative boundary condition pose a problem because of the requirement to calculate the interface location and boundary temperature implicitly. Due to this difficulty some of the methods available in literature used approximations near the wall and the rest used iterative methods. The present technique isolates the nonlinearity associated with the moving interface as well as that of any nonlinear wall boundary condition and permits the simultaneous evaluation of the unknown interface location and the wall temperature at the new time level. Thereafter the temperatures at the other nodes are obtained without nodal iterations. Numerical results are obtained for outward solidification with both finite and infinite heat transfer rates at the wall. Good agreement is obtained between the present numerical results and the results available in literature for the limiting cases.     

Locally similar solutions for hydromagnetic and thermal slip flow boundary layers over a flat plate with variable fluid properties and convective surface boundary condition

This paper presents heat transfer process in a two-dimensional steady hydromagnetic convective flow of an electrically conducting fluid over a flat plate with partial slip at the surface of the boundary subjected to the convective surface heat flux at the boundary. The analysis accounts for both temperature-dependent viscosity and temperature dependent thermal conductivity. The local similarity equations are derived and solved numerically using the Nachtsheim-Swigert iteration procedure. Results for the dimensionless velocity, temperature and ambient Prandtl number within the boundary layer are displayed graphically delineating the effect of various parameters characterizing the flow. The results show that momentum boundary layer thickness significantly depends on the surface convection parameter, Hartmann number and on the sign of the variable viscosity parameter. The results also show that plate surface temperature is higher when there is no slip at the plate compared to its presence. For both slip and no-slip cases surface temperature of the plate can be controlled by controlling the strength of the applied magnetic field. In modelling the thermal boundary layer flow with variable viscosity and variable thermal conductivity, the Prandtl number must be treated as a variable irrespective of flow conditions whether there is slip or no-slip at the boundary to obtain realistic results.     

Analysis of laminar flow forced convection heat transfer in the entrance region of a circular pipe

A numerical solution, for incompressible, steady-state, laminar flow heat transfer in the combined entrance region of a circular tube is presented for the case of constant wall heat flux and constant wall temperature. The development of velocity profile is obtained from Sparrow's entrance region solution. This velocity distribution is used in solving the energy equation numerically to obtain temperature profiles. Variation of the heat transfer coefficient for these two different boundary conditions for the early stages of boundary layer formation on the pipe wall is obtained. Local Nusselt numbers are calculated and the results are compared with those given byUlrichson andSchmitz. The effect of the thermal boundary conditions is studied by comparing the uniform wall heat flux results with uniform wall temperature.     

Convective heat transfer for viscoelastic fluid in a curved pipe

In this paper, fully developed convective heat transfer of viscoelastic flow in a curved pipe under the constant heat flux at the wall is investigated analytically using a perturbation method. Here, the curvature ratio is used as the perturbation parameter and the Oldroyd-B model is applied as the constitutive equation. In the previous studies, the Dirichlet boundary condition for the temperature at the wall has been used to simplify the solution, but here exactly the non-homogenous Neumann boundary condition is considered to solve the problem. Based on this solution, the non-axisymmetric temperature distribution of Dean flow is obtained analytically and the effect of flow parameters on the flow field is investigated in detail. The current analytical results indicate that increasing the Weissenberg number, viscosity ratio, curvature ratio, and Prandtl number lead to the increase of the heat transfer in the Oldroyd-B fluid flow.     

An Enthalpy Control Volume Method for Transient Mass and Heat Transport with Solidification

A single domain enthalpy control volume method is developed for solving the coupled fluid flow and heat transfer with solidification problem arising from the continuous casting process. The governing equations consist of the continuity equation, the Navier–Stokes equations and the convection–diffusion equation. The formulation of the method is cast into the framework of the Petrov–Galerkin finite element method with a step test function across the control volume and locally constant approximation to the fluxes of heat and fluid. The use of the step test function and the constant flux approximation leads to the derivation of the exponential interpolating functions for the velocity and temperature fields within each control volume. The exponential fitting makes it possible to capture the sharp boundary layers around the solidification front. The method is then applied to investigate the effect of various casting parameters on the solidification profile and flow pattern of fluids in the casting process.     

Mixed Convection in a Darcy–Brinkman Porous Medium with a Constant Convective Thermal Boundary Condition

The characteristics of the boundary layer flow past a plane surface adjacent to a saturated Darcy–Brinkman porous medium are investigated in this paper. The flow is driven by an external free stream moving with constant velocity. The surface is heated with a convective boundary condition with constant heat transfer coefficient. The problem is non-similar and is investigated numerically by a finite difference method. The problem is governed by four non-dimensional parameters, that is, the convective Darcy number, the convective Grashof number, the Prandtl number, and the axial distance along the plate. The influence of these parameters on the results is investigated, and the results are presented in tables and figures. The Darcy term and the Grashof term in the momentum equation contradict each other and this contradiction makes the problem complicated. However, the wall shear stress and the wall temperature increase continuously along the plate and the wall temperature always tends to 1.     

An analysis of the effect of the solid layer for the modified chemical vapor deposition process

The heat transfer problem relative to the modified chemical vapor deposition process has been analyzed and the effects of solid layer thickness, torch speed and tube rotation are studied. The quasi-steady three-dimensional energy equations have been solved for the temperature fields in the gas and the solid layer with a Gaussian heat flux boundary condition on the outer surface. Of particular interest is the effect of the solid layer thickness and the torch speed on inner surface temperature, gas temperature and thermophoretic velocity. The large change of the axial temperature distribution of the surface occurs for different solid layer thicknesses or torch speeds. The presence of the solid layer and tube rotation reduce the effects of nonuniform torch heating in the circumferential direction and the resulting surface temperatures are very uniform in this direction.     

Heat wave phenomena in solids subjected to time dependent surface heat flux

Hyperbolic heat conduction in a plane slab, infinitely long solid cylinder and solid sphere with a time dependent boundary heat flux is analytically studied. The solution is based on the separation of variables method and Duhamel’s principle. The temperature distribution, the propagation and reflection of the temperature wave and the effect of geometry on the shape of the wave front are studied for the case of a rectangular pulsed boundary heat flux. Comparisons with the solution obtained for Fourier heat conduction are performed by considering the limit of a vanishing thermal relaxation time.     

Effect of the nonmonotonic temperature dependence of water density on convection under uniform heating

This paper presents the results of an experimental study of free convection in a closed rectangular tank with fresh water whose initial temperature is lower than the constant temperature of air outside the tank. Convective instability is shown to play a significant role in the process of heat transfer from the boundary layers on the walls to the bulk of water. In the case where the initial water temperature is higher than the temperature at which the density is maximal (about 4°C), convective instability occurs only in the boundary layer at the bottom of the tank. At a water temperature below 4°C convective instability also exists for some time in the boundary layer on the cover. Quantitative information is given on the variations in temperature and water density in time and on the vertical coordinate.     

Network simulation method for solving phase-change heat transfer problems with variable thermal properties

 The transient heat conduction equation in a finite slab undergoing phase change (two-phase problem of melting and solidification), with isothermal, adiabatic or convective boundary conduction is studied by the network simulation method; solid phase conductivity and specific heat are assumed to be dependent on temperature. Ablation, as a particular case, is also analysed. A network model is established for a cell and boundary conditions are added to complete the whole network model. No restrictions exist, as to the kinds of linear and non-linear boundary conditions, Stefan number values or the initial conditions (when hypotheses concern of the Stefan problem, numerical and exact solutions are compared for a large interval of Stefan numbers; simulation values show good agreement). Movement of the solid–liquid boundary and thermal fields are determined in all cases. Received on 10 May 2000 / Published online: 29 November 2001     

Investigation of double-diffusive convection during the solidification of a binary mixture (NH4Cl–H2O) in a trapezoidal cavity

The solidification of binary mixture (NH₄Cl–H₂O) inside a trapezoidal cavity is investigated experimentally in this study. The effect of the initial concentration of ammonium chloride (0–19.8%) and boundary temperatures (−30 to 0°C) on the solidification process was investigated. Particle image velocimetry (PIV) technique was used for the visualization of the dynamic field in the melt. Thirty-two thermocouples were used to monitor the temperature distribution inside the cavity and on the cooling walls. The convective flow field, the temperature distribution, the frozen layer thickness and the moving solid/liquid interface were obtained for different initial concentrations of ammonium chloride and various boundary temperatures. The results obtained in the course of this study reveal that: (1) the process of solidification is slower with an increase in initial concentration levels of the binary solution: as the concentration increases, the time needed to get the same thickness of frozen layer increases; (2) an increase in the initial concentration of ammonium chloride solution reduce significantly the temperature in the melt; and (3) the initial concentration play a significant role in the evolution of convection flow patterns.