Analysis of non-Fourier heat conduction in a solid sphere under arbitrary surface temperature change

In this paper, the non-Fourier heat conduction in a solid sphere under arbitrary surface thermal disturbances is solved analytically. Four cases including sudden, simple harmonic periodic, triangular and pulse surface temperature changes are investigated step-by-step. The analytical solutions are obtained using the separation of variables method and Duhamel’s principle along with the Fourier series representation of an arbitrary periodic function and the Fourier integral representation of an arbitrary non-periodic function. Using these analytical solutions, the temperature profiles of the solid sphere are analyzed, and the differences in the temperature response between the “hyperbolic” and “parabolic” are discussed. These solutions can be applicable to all kinds of non-Fourier heat conduction analyses for arbitrary boundary conditions occurred in technology. And as application examples, particular attention is devoted to the cases of triangular surface temperature change and pulse surface temperature change. The examples presented in this paper can be used as benchmark problems for future numerical method validations.     

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.     

The temperature distribution in a semi-infinite insulated cylinder due to an arbitrary heat flux

The temperature distribution in a semi-infinite insulated cylinder with linear temperature dependent heat conductivity and with arbitrary initial temperature subjects at its base to an azimuthal symmetric arbitrary heat flux is found.This is done in two stages: first the problem is solved by assuming a constant initial temperature and constant thermal properties and then the solution is extended to the case in which the heat conductivity varies linearly with temperature provided that the diffusivity is constant and the initial temperature is an arbitrary function. Several particular cases are then checked and found to be in agreement with known solutions.Because of the complexity of the above mentioned solution a more simple solution is developed which corresponds to the case in which the cylinder base can be considered as semi-infinite. Then, the case in which the heat flux has the form (A exp [–r ²/r ₀ ² ]+G) sin t is considered. Two particular cases are considered correspond to narrow and wide beams of heat flux density. In each case the time of maximum temperature and the maximum temperature at the base centre is found.     

Hyperbolic heat conduction in the semi-infinite body with the heat source which capacity linearly depends on temperature

It is commonly taken for granted that long-time solutions of Cattaneo's hyperbolic heat conduction equation tend to overlap the related parabolic solutions. Usually, for small times considerable qualitative as well as quantitative differences between the hyperbolic and parabolic solutions are observed which vanish completely for large times. However, in the case of a heat conducting body with the temperature dependent inner heat generation, the quantitative differences may grow with time. This arises from the feedback between the temperature and the source capacity. To illustrate this effect, Cattaneo's hyperbolic equation for the semi-infinite body, with the heat source which capacity is linearly dependent on temperature, is solved analytically by the Laplace transforms method. Received on 1 July 1997     

Flow and heat transfer of an electrically conducting third grade fluid past an infinite plate with partial slip

The effects of partial slip on the steady flow and heat transfer of an electrically conducting, incompressible, third grade fluid past a horizontal plate subject to uniform suction and blowing is investigated. Two distinct heat transfer problems are studied. In the first case, the plate is assumed to be at a higher temperature than the fluid; and in the second case, the plate is assumed to be insulated. The momentum equation is characterized by a highly nonlinear boundary value problem in which the order of the differential equation exceeds the number of available boundary conditions. Numerical solutions for the governing nonlinear equations are obtained over the entire range of physical parameters. The effects of slip, magnetic parameter, non-Newtonian fluid characteristics on the velocity and temperature fields are discussed in detail and shown graphically. It is interesting to find that the velocity and the thermal boundary layers decrease with an increase in the slip, and as the slip increases to infinity, the flow behaves as though it were inviscid.     

Momentum and heat transfer of a special case of the unsteady stagnation-point flow

This paper investigates the unsteady stagnation-point flow and heat transfer over a moving plate with mass transfer,which is also an exact solution to the unsteady Navier-Stokes(NS)equations.The boundary layer energy equation is solved with the closed form solutions for prescribed wall temperature and prescribed wall heat flux conditions.The wall temperature and heat flux have power dependence on both time and spatial distance.The solution domain,the velocity distribution,the flow field,and the temperature distribution in the fluids are studied for different controlling parameters.These parameters include the Prandtl number,the mass transfer parameter at the wall,the wall moving parameter,the time power index,and the spatial power index.It is found that two solution branches exist for certain combinations of the controlling parameters for the flow and heat transfer problems.The heat transfer solutions are given by the confluent hypergeometric function of the first kind,which can be simplified into the incomplete gamma functions for special conditions.The wall heat flux and temperature profiles show very complicated variation behaviors.The wall heat flux can have multiple poles under certain given controlling parameters,and the temperature can have significant oscillations with overshoot and negative values in the boundary layers.The relationship between the number of poles in the wall heat flux and the number of zero-crossing points is identified.The difference in the results of the prescribed wall temperature case and the prescribed wall heat flux case is analyzed.Results given in this paper provide a rare closed form analytical solution to the entire unsteady NS equations,which can be used as a benchmark problem for numerical code validation.     

Numerical investigation of premixed combustion in a porous burner with integrated heat exchanger

In this paper, we perform a numerical analysis of a two-dimensional axisymmetric problem arising in premixed combustion in a porous burner with integrated heat exchanger. The physical domain consists of two zones, porous and heat exchanger zones. Two dimensional Navier–Stokes equations, gas and solid energy equations, and chemical species transport equations are solved and heat release is described by a multistep kinetics mechanism. The solid matrix is modeled as a gray medium, and the finite volume method is used to solve the radiative transfer equation to calculate the local radiation source/sink in the solid phase energy equation. Special attention is given to model heat transfer between the hot gas and the heat exchanger tube. Thus, the corresponding terms are added to the energy equations of the flow and the solid matrix. Gas and solid temperature profiles and species mole fractions on the burner centerline, predicted 2D temperature fields, species concentrations and streamlines are presented. Calculated results for temperature profiles are compared to experimental data. It is shown that there is good agreement between the numerical solutions and the experimental data and it is concluded that the developed numerical program is an excellent tool to investigate combustion in porous burner.     

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.     

Buoyant plume above a line source of heat in an anisotropic porous medium

Buoyancy-induced convection arising from a horizontal line heat source embedded in an anisotropic porous medium is investigated analytically. The porous medium is anisotropic is permeability with its principal axes oriented in a direction that is oblique to the gravity vector. Assuming the boundary layer approximation, closed-form exact similarity solutions for both flow and temperature fields are presented and compared with those of isotropic case. Scale analysis is applied to predict the order of magnitudes involved in the boundary layer regime for which the conditions of validity are obtained. Effects of both anisotropic parameters (K* and %) and Rayleigh number Ra_L are observed to be strongly significant. It is demonstrated that a minimum (maximum) intensity of the thermal convective plume above the line source of heat can be obtained if the porous matrix is oriented with its principal axis with higher permeability parallel (perpendicular) to the vertical direction.     

Note on the Heat Transfer of Flows over a Stretching Wall in Porous Media: Exact Solutions

In this note, heat transfer over a stretching sheet with mass transfer in a porous medium is revisited. Analytical solutions are presented for two cases including a prescribed power-law wall temperature case and a prescribed power-law wall heat flux case. The solutions are expressed by the Kummer’s function. Closed-form solutions are found and presented for some special parameters. The solutions might offer more insights of the heat transfer characteristics compared with the numerical solutions.     

Flow and heat transfer of an electrically conducting fluid of second grade in a porous medium over a stretching sheet subject to a transverse magnetic field

An analysis is performed for flow and heat transfer of a steady laminar boundary layer flow of an electrically conducting fluid of second grade in a porous medium subject to a transverse uniform magnetic field past a semi-infinite stretching sheet with power-law surface temperature or power-law surface heat flux. The effects of viscous dissipation, internal heat generation of absorption and work done due to deformation are considered in the energy equation. The variations of surface temperature gradient for the prescribed surface temperature case (PST) and surface temperature for the prescribed heat flux case (PHF) with various parameters are tabulated. The asymptotic expansions of the solutions for large Prandtl number are also given for the two heating conditions. It is shown that, when the Eckert number is large enough, the heat flow may transfer from the fluid to the wall rather than from the wall to the fluid when Eckert number is small. A physical explanation is given for this phenomenon.     

Non-Fourier heat conduction by axisymmetric thermal waves in an infinite solid medium

The non-stationary heat conduction in an infinite solid medium internally bounded by an infinitely long cylindrical surface is considered. A uniform and time- dependent temperature is prescribed on the boundary surface. An analytical solution of the hyperbolic heat conduction equation is obtained. The solution describes the wave nature of the temperature field in the geometry under consideration. A detailed analysis of the cases in which the temperature imposed on the boundary surface behaves as a square pulse or as an exponentially decaying pulse is provided. The evolution of the temperature field in the case of hyperbolic heat conduction is compared with that obtained by solving Fourier's equation. Received on 28 January 1998     

Unsteady flow and heat transfer past an axisymmetric permeable shrinking sheet with radiation effect

In this paper, the problem of unsteady axisymmetric boundary layer flow and heat transfer induced by a permeable shrinking sheet in the presence of radiation effect is studied. The transformed boundary layer equations are solved numerically by an implicit finite‐difference scheme known as the Keller‐box method. The influence of radiation, unsteadiness and mass suction parameters on the reduced skin friction coefficient f′′(0) and the heat transfer coefficient ?θ′(0), as well as the velocity and temperature profiles are presented and discussed in detail. It is found that dual solutions exist and suction parameter delays the separation of boundary layer. Copyright © 2010 John Wiley & Sons, Ltd.     

Thermal waves propagation due to localised heat inputs – the Laplace transforms method analysis

 Propagation of thermal waves due to localised heat inputs is analysed by the Laplace transforms method for the case of constant thermophysical parameters of the heat-conducting media and a temperature dependent internal heat source. Both the hyperbolic and parabolic models of heat conduction are used and compared. For the hyperbolic model, an energy pulse travelling along the conductor is observed. Long time hyperbolic solutions do not differ qualitatively from the related parabolic solutions, but the quantitative differences between the two solutions do not vanish with time because of the temperature dependent heat source. The cases considered in the paper describe the evolution of normal zones in technical superconductors. Received on 10 May 2000 / Published online: 29 November 2001     

Flow of a viscous incompressible fluid of temperature dependent viscosity past a permeable wedge with uniform surface heat flux

 The effect of uniform suction on the steady two-dimensional laminar forced flow of a viscous incompressible fluid of temperature dependent viscosity past a wedge with uniform surface heat flux is considered. The governing equations for the flow are obtained by using suitable transformations and are solved by using an implicit finite difference method. Perturbation solutions are also obtained near the leading edge and in the downstream regime. The results are obtained in terms of the local skin friction coefficient and the rate of heat transfer for various values of the pertinent parameters, such as the Prandtl number, Pr, the velocity gradient parameter, m, the local suction parameter, ξ, and the viscosity variation parameter, ɛ. Perturbation solutions are compared with the finite difference solutions and are found to be in excellent agreement. The effect of ξ, m and ɛ on the dimensionless velocity profiles and viscosity distribution are also presented graphically for Pr = 0.7 and 7.0, which are the appropriate values for gases and water respectively. Received on 22 July 1999     

Thermal stresses in an isotropic trimaterial interacted with a pair of point heat source and heat sink

A general analytical solution for an isotropic trimaterial interacted with a point heat source is provided in this paper. Based on the method of analytical continuation in conjunction with the alternating technique, the solutions to heat conduction and thermoelasticity problems for three dissimilar media are first derived. A rapidly convergent series solution for both the temperature and stress functions, which is expressed in terms of an explicit general term of the complex potential of the corresponding homogeneous problem, is obtained in an elegant form. As a numerical illustration, the distributions of thermal stresses along the interface are presented for various material combinations and for different positions of the applied heat source and heat sink.     

疏松金属材料冲击温度理论分析

以热力学原理和固态物质的三项式物态方程为基础,由密实物质的冲击绝热线和热力学状态,通 过等容线法推导出了疏松金属材料的冲击温度理论计算方法。以铁为例,分析了几种物理参数对该模型计算 结果的影响。计算和分析结果显示,利用新模型得到的计算结果与已有实验结果吻合较好,误差均在5%以 内。疏松金属材料的冲击温度受Grneisen系数、电子Grneisen系数影响不大,而密实度、冲击压力和电子 比热系数则会对疏松金属材料冲击温度产生较大影响。     

Natural convection cooling of a heat source embedded on the bottom of an enclosure filled with Cu-water nanofluid: effects of various thermal boundary conditions

Steady natural convection cooling of a localized heat source at the bottom wall of an enclosure filled with Cu-water nanofluid for a variety of thermal boundary conditions at the sidewalls has been studied in the present paper. Finite difference method was employed to solve the dimensionless governing equations of the problem. The effects of governing parameters, namely, solid volume fraction, the different values of the heat source length and the different locations of the heat source on the streamlines and isotherms contours as well as maximum temperature, Nusselt number and average Nusselt number along the heat source were considered. The present results are validated by favorable comparisons with previously published results. The results of the problem are presented in graphical and tabular forms and discussed.     

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