Solution Of Heat Conduction Research Articles

Analytical Solution of Nonequilibrium Heat Conduction in Porous Medium Incorporating a Variable Porosity Model With Heat Generation

This paper is concerned with the conduction heat transfer between two parallel plates filled with a porous medium with uniform heat generation under a nonequilibrium condition. Analytical solution is obtained for both fluid and solid temperature fields at constant porosity incorporating the effects of thermal conductivity ratio, porosity, and a nondimensional heat transfer coefficient at pore level. The two coupled energy equations for the case of variable porosity condition are transformed into a third order ordinary equation for each phase, which is solved numerically. This transformation is a valuable solution for heat conduction regime for any distribution of porosity in the channel. The effects of the variable porosity on temperature distribution are shown and compared with the constant porosity model. For the case of the exponential decaying porosity distribution, the numerical results lead to a correlation incorporating conductivity ratio and interstitial heat transfer coefficient.

Analytical Solution to Transient Asymmetric Heat Conduction in a Multilayer Annulus

In this paper, we present an analytical double-series solution for the time-dependent asymmetric heat conduction in a multilayer annulus. In general, analytical solutions in multidimensional Cartesian or cylindrical (r,z) coordinates suffer from existence of imaginary eigenvalues and thus may lead to numerical difficulties in computing analytical solution. In contrast, the proposed analytical solution in polar coordinates (2D cylindrical) is “free” from such imaginary eigenvalues. Real eigenvalues are obtained by virtue of precluded explicit dependence of transverse (radial) eigenvalues on those in the other direction.

Multidimensional Combustion Simulation and Analytical Solution of Wall Heat Conduction in DI Diesel Engine

Multidimensional Combustion Simulation and Analytical Solution of Wall Heat Conduction in DI Diesel Engine

Solutions for Transient Heat Conduction With Solid Body Motion and Convective Boundary Conditions

The analytical solution for the problem of transient thermal conduction with solid body movement is developed for a parallelepiped with convective boundary conditions. An effective transformation scheme is used to eliminate the flow terms. The solution uses Green’s functions containing convolution-type integrals, which involve integration over a dummy time, referred to as “cotime.” Two types of Green’s functions are used: one for short cotimes comes from the Laplace transform and the other for long cotimes from the method of separation of variables. A primary advantage of this method is that it incorporates internal verification of the numerical results by varying the partition time between the short and long components. In some cases, the long-time solution requires a zeroth term in the summation, which does not occur when solid body motion is not present. The existence of this zeroth term depends on the magnitude of the heat transfer coefficient associated with the convective boundary condition. An example is given for a two-dimensional case involving both prescribed temperature and convective boundary conditions. Comprehensive tables are also provided for the nine possible combinations of boundary conditions in each dimension.

Thermal conductivity measurement of anisotropic material using photothermal deflection method

A complete theoretical treatment of photothermal deflection spectroscopy has been performed for the measurement of thermal conductivities in an anisotropic medium. An analytical solution of three-dimensional heat conduction was obtained by using 2D Fourier Transforms for an anisotropic material irradiated by a laser beam. Thermal conductivity was determined by using the phase angle of deflection at relative positions between the heating and probe beams. Excellent agreement between theoretical and experimental photothermal deflections was obtained. Also, the thermal conductivity in an arbitrary measurement direction for anisotropic materials (Pyrolytic graphite) was measured.

A Two-Dimensional Analytical Solution for the Transient Short-Hot-Wire Method

Unlike the conventional transient hot-wire method for measuring thermal conductivity, the transient short-hot-wire method uses only one short thermal-conductivity cell. Until now, this method has depended on numerical solutions of the two-dimensional unsteady heat conduction equation to account for end effects. In order to provide an alternative and to confirm the validity of the numerical solutions, a two-dimensional analytical solution for unsteady-state heat conduction is derived using Laplace and finite Fourier transforms. An isothermal boundary condition is assumed for the end of the cell, where the hot wire connects to the supporting leads. The radial temperature gradient in the wire is neglected. A high-resolution finite-volume numerical solution is found to be in excellent agreement with the present analytical solution.

Analytical and numerical solutions of transient heat conduction in monolayer-coated tools

The heat generation during metal cutting processes affects accuracy of the machined surface and strongly influences tool wear and tool life. Knowledge of the ways in which the tool material affects the temperature distribution is therefore essential for the study of thermal effects on tool life and workpiece quality. Many studies have been done on simulation temperature distribution in coated cutting tools by means of the finite element method or the finite difference method. In this study, a thermal analytical model is firstly developed to determine temperature distribution in monolayer-coated cutting tools during orthogonal metal cutting. In the analytical model one equivalent heat source applied on the coating layer boundary substitutes for the heat generation introduced from the primary deformation zone, the secondary deformation and the frictional zone along the tool–chip interface as well as the tertiary or the sliding frictional zone at tool–workpiece interface. A mathematical model of the transient heat conduction in monolayer-coated tools is then proposed. The temperature distribution formulations in monolayer-coated tools are obtained using Laplace transform. The influence of different parameters including thermophysical properties of tool coating and tool substrate and thickness of the coating layer on temperature distribution in monolayer-coated tools is lastly discussed and illustrated.

Conduction in rectangular plates with boundary temperatures specified

Steady-state components of heat conduction solutions may have very slowly convergent series for temperatures and non-convergent heat fluxes for temperature boundary conditions. Previous papers have proposed methods to remove these convergence problems. However, even more effective procedures based on insights of Morse and Feshbach are given herein. In some cases it is possible to replace poorly-convergent or non-convergent series by closed-form algebraic solutions. Examples are given.

Semi-analytical solutions for the dual phase lag heat conduction in multilayered media

A semi-analytical solution procedure for transient heat transfer in composite mediums consisting of multi-layers within the framework of the dual phase lag model is presented. The procedure is then used to derive solutions for the temperature-, temperature gradient-, and heat flux distributions in a two-layer composite planar slab, a bi-layered solid-cylinder and sphere. The solutions obtained are applicable to the classical Fourier heat diffusion, hyperbolic heat conduction, phonon–electron interaction, and phonon scattering models with perfect or imperfect contact and with layers of different materials. The interfacial contact resistance, the heat flux and temperature gradient phase lags, thermal diffusivities and conductivities, initial temperatures of the composite medium and a general time-dependent boundary heat flux enter the solutions as parameters, allowing the solutions obtained to be applicable to a wide range of arrangements including perfect and imperfect contact. Analysis of thermal wave propagation, transmission and reflection in planar, cylindrical and spherical geometries with imperfect interfaces are presented, and geometrical—as well as the temperature gradient phase lag—effects on the thermal lagging behavior in different layered media are discussed.

A method of fundamental solutions for transient heat conduction

In this paper we investigate an application of the method of fundamental solutions (MFS) to transient heat conduction. In almost all of the previously proposed MFS for time-dependent heat conduction the fictitious sources are located outside the time-interval of interest. In our case, however, these sources are instead placed outside the space domain of interest in the same manner as is done for stationary heat conduction. A denseness result for this method is discussed and the method is numerically tested showing that accurate numerical results can be obtained. Furthermore, a test example with boundary singularities shows that it is advisable to remove such singularities before applying the MFS.

Analytical Solution for Heat Conduction Near an Encapsulated Sphere With Heat Generation

Analytical Solution for Heat Conduction Near an Encapsulated Sphere With Heat Generation

DETERMINATION OF AN EFFECTIVE HEAT TRANSFER COEFFICIENT FOR IMMERSION WATER COOLING OF CANNED FOODS

ABSTRACT Transient temperature measurement inside cans packed with tomato purees were used to estimate can surface heat flux as a function of time during the immersion cooling cycle of the thermal processing of canned foods. The estimation was carried out using a developed computer program based on a sequential function specification algorithm. The estimated heat fluxes were used to solve the direct heat conduction problem for the cans' internal temperature distributions at any position and time including internal surface temperature. The accuracy of the calculated heat flux was verified using the root mean square error for the temperatures at the sensor position. An effective heat transfer coefficient as a function of time was calculated from the knowledge of surface temperatures and heat flux history. A time‐averaged effective heat transfer coefficient of 137 ± 7 (W/m2·C) was obtained in this study, which was comparable to a published value obtained using pilot scale data.PRACTICAL APPLICATIONSThis research work provides a systematic approach for estimating heat transfer coefficients during food processing operations. The method was based on both the inverse and the direct solution of heat conduction in cylindrical geometry. The experimental setup showed that with a single transient temperature measurement, an accurate estimate of an effective heat transfer coefficient can be achieved. This research also showed how heat transfer coefficient changes with time during immersion cooling. The effective heat transfer coefficient accounts for the internal lacquer coating and external polymeric printing coating thermal resistances. The heat transfer coefficient obtained in this research is an addition to scarce data on such an important parameter, which will help in the optimization of the thermal processing of canned foods by accounting for the cooling phase lethality.

Thermal Transport Regimes and Effects of Prandtl Number in Molten Pool Transport in Laser Surface Melting Processes

The effects of Prandtl number of the molten metal in weld pool convection are studied based on a systematic scaling analysis. It is demonstrated that for a very small Prandtl number the weld pool morphology is predominantly determined by heat conduction whereas convection has a marginal effect on the molten pool morphology. By contrast, for relatively greater values of Prandtl number the convection pattern in the molten pool significantly affects the pool shape and size. The criteria, which demarcate these two regimes, are estimated by a detailed scaling argument. It is also shown from scaling analysis that turbulence in the molten pool affects momentum and thermal transport differently. Turbulence may also significantly affect the momentum transport of the pool whereas the thermal transport is marginally affected. A regime diagram is constructed based on scaling arguments where the above flow regimes are clearly indicated. In order to validate the scaling arguments on which the regime diagram is based, two three dimensional simulations are carried out for laser welding of iron (for Fe, Prandtl number in order of 0.1) and copper (for Cu, Prandtl number in the order of 0.01). It is found that in the case of Cu, the weld pool obtained from conduction solution is almost the same as the pool morphology obtained using laminar and turbulent transport models. By contrast, in the case of Fe, the weld pools obtained from laminar and turbulent transport models are significantly different from each other, and are also different from the pool obtained from the heat conduction solution.

The flux-correct Green element formulation for linear, nonlinear heat transport in heterogeneous media

The singular integral equation, obtained from applying Green's identity to the stationary part of the linear diffusion equation operator (the Laplace operator), is implemented in an element-by-element fashion for the solution of nonlinear heat conduction in heterogeneous media so that both the temperature and its normal directional distribution (the flux) are calculated directly at every nodal point. This formulation, referred to as the flux Green element method (flux GEM), is used to solve linear and nonlinear transient heat transfer problems in two-dimensional heterogeneous domains. The flux-based Green element formulation takes advantage of the ability of the boundary element theory to correctly calculate the fluxes so that the solution in each element is complete in the sense that any solution (temperature or its directional derivative) required at any other point other than the grid points is achieved by implementing the integrations only in the element in which the point is located. One apparent drawback of the formulation is increased number of degrees of freedom (unknowns) that are generated, but this is compensated by the fact that high accuracy is achieved with coarse discretization. Furthermore, this paper presents a novel way of addressing the closure problem associated with having more unknowns than discretized equations at internal nodes. Here, an additional compatibility equation is presented for the internal normal directional derivatives of the temperature. This equation has a universal appeal in the sense that it applies not only to this heat equation but to any situation where such directional derivatives exist at a point. For the nonlinear transient heat conduction, the Picard and Newton–Raphson algorithms are applied in linearizing the discrete equations. In all examples examined in this paper, the Newton–Raphson algorithm shows superior convergence characteristics. The flux GEM is applied to four examples, and in all cases, the flux GEM achieves comparable accuracy with between 20% and 33% of the grid of the earlier Green element formulation which differences the internal fluxes.

Non-Fourier heat conduction in a finite medium subjected to arbitrary periodic surface disturbance

The non-Fourier transient heat conduction in a finite medium under arbitrary periodic surface thermal disturbance is investigated analytically. In order to obtain the desired temperature field from the known solution for non-Fourier heat conduction under a harmonic disturbance, the principle of superposition along with the Fourier series representation of an arbitrary periodic function is employed. The developed method can be applied for more realistic periodic boundary conditions occurred in nature and technology.

Temperature Prediction of Concrete-Filled Rectangular Hollow Sections in Fire Using Green’s Function Method

An efficient numerical approach using the Green's function solutions of transient heat conduction for predictions of thermal response inside a concrete-filled rectangular hollow section subjected to fire is proposed in this paper. Thermal properties of construction materials are assumed to be isotropic and homogeneous. The Green's function approach adopts different series expansions for small and large time solutions, therefore the desirable convergence properties can be achieved at any range of time by using the time partitioning strategy. A useful analytical relation in terms of step Green's functions is derived in this paper to incorporate the multidimensional effect, in particular, for Neumann prescribed boundary conditions. A modified lumped capacitance method, together with an orthogonal flux concept, are employed to deal with spatially varying heat at the steel-concrete interface, where Duhamel's theorem is applied in piecewise manner along the interface to incorporate the fire boundary conditions. No spatial discretization is required in the numerical algorithms based on the Green's function approach.

Numerical solution of hyperbolic heat conduction in thin surface layers

The purpose of the present paper is to propose a new hybrid method investigating the effect of the surface curvature of a solid body on hyperbolic heat conduction. The difficulty encountered in the numerical solutions of hyperbolic heat conduction problems is the numerical oscillation in vicinity of sharp discontinuities. In the present study, we have developed a new hybrid method combined the Laplace transform, the weighting function scheme [Shong-leih Lee, Weighting function scheme and its application on multidimensional conservation equations, Int. J. Heat Mass Transfer 32 (1989) 2065–2073], and the hyperbolic shape function for solving time dependent hyperbolic heat conduction equation with a conservation term. Four different examples have been analyzed by the present method. It is found from these examples that the present method is in good agreement in the analytical solutions [Tsai-tse Kao, Non-Fourier heat conduction in thin surface layers, J. Heat Transfer 99 (May) (1977) 343–345] and does not exhibit numerical oscillations at the wave front and the surface temperature is modified by the surface curvature during the short period when the non-Fourier effect is significant. The curvature will increase or decrease the temperature of the wave front, depending on whether the surface is concave or convex.

Temperature prediction for multi‐dimensional domains in standard fire

AbstractThe existing 1D analytical solution for heat conduction inside structural members plays an important role in the estimation of temperature field and prediction of member strength in fire. In this paper, the novelty lies in a set of proposed analytical formulations for the conduction heat transfer within multi‐dimensional (2D and 3D) domains subjected to a particular time‐varying boundary condition, i.e. the standard fire conditions. Solutions of multi‐dimensional conduction are based on the eigenfunction approach using the technique of separation of variables. The multi‐dimensional effect is incorporated by the product of respective solutions of 1D transient heat conduction (1D step functions) of the plate and the infinite cylinder that serve as a set of basis for derivation of analytical solutions. Fire boundary conditions are incorporated using the Duhamel's principle, where the time‐varying fire temperature curve is imposed on the step functions. In this paper, the boundary conditions of the first and the third kind are considered, respectively. The analytical solutions are compared with results obtained from finite element analysis for verification. Copyright © 2006 John Wiley & Sons, Ltd.

Uncertainty in Thermal Basin Modeling: An Interval Finite Element Approach

Uncertainty assessment in basin modeling and reservoir characterization is traditionally treated by geostatistical methods which are normally based on stochastic probabilistic approaches. In this talk, an alternative interval-based approach will be present. A solution for the transient heat conduction in sedimentary basins will be introduced using an interval finite element approach. For this purpose, a novel formulation is developed to deal with both the special interval arithmetic properties and the transient term in the differential equation governing heat transfer. In this formulation, the "stiffness" matrix resulting from the discretization of the heat conduction equation is assembled using an element-by-element technique in which the finite elements are globally independent and lagrange multipliers are used to enforce continuity. This formulation is suggested as an alternative to traditional Monte Carlo method, where repetitive simulations are required to handle uncertainty and worst case system response is underestimated. The newly developed technique is applied to a one-dimensional thermal basin simulation to assess its potential and limitations. Numerical results will be introduced and their quality assessed.

Determining heat flows and radiogenic heat generation in the crust and lithosphere based on seismic data and surface heat flows

The balance of the Earth’s energy involves the generation of radiogenic, gravitational, chemical, and other types of energy and the heat loss from the Earth’s surface. In the quasi-stationary formulation, heat losses through the surface are equal to the sum of energy generated by all sources in the Earth’s interiors. The flow from the continental surface is estimated at 47‐ 49 mW/m 2 [1]. Estimates (accomplished by various researchers) of the mantle component of the heat flows range from 3 to 25 mW/m 2 [1]. Mantle flows are usually assayed on the basis of extensive geothermal measurements at numerous sites around the world. The heat flow through the Earth’s surface Q 0 can be expressed as the sum of the mantle flow Q M and the contribution of radiogenic energy released within the crust Q Cr . The value of Q Cr is determined based on geochemical models currently adopted for the distribution of radiogenic elements and the solution of stationary heat conduction [1, 2]. At the same time, the mantle flow Q M is, in fact, calculated as the difference between Q 0 and Q Cr . The distribution of K, U, and Th are still known relatively poorly, so that these estimates for both the crust and the mantle broadly differ, and this problem is actively debated in the literature [3, 4]. Thermal models for the mantle are the least definite, mostly because of the uncertainties in the crustal and lithospheric contributions to the overall heat generation [1‐4]. The solution to the problem of determining the temperature and/or the chemical composition of the mantle from the velocities of seismic waves was debated in [5‐9]. It is quite difficult to derive mantle geotherms from the results obtained by measuring the surface heat flow because of the lack of information on the mantle contribution. This publication proposes a method for assaying the conductive heat transfer of the lithosphere of both mantle and crustal heat flows on the basis of the data of seismic models and the heat flow through the Earth’s surface Q 0 . As an example of such a calculation, we used the profiles of velocities from the IASP91 [10] global reference model (for the “averaged” continental crust) and the BP11A regional model [11], which was developed for the mantle beneath the Kaapvaal craton in South Africa. The solution routine was provisionally subdivided into two stages. During the first of them, the temperature of the mantle ( T P and T S ) is determined from the velocities of seismic P and S waves. The procedure of the conversion of seismic profiles into thermal ones is conducted based on the equations of state for the material with regard for anharmonic and anelastic effects. The temperature profile T P , S was then made consistent with the conductive heat transfer model for the crust and mantle, and this enabled us to derive an analytical expression for temperature variations with depth, the values of crustal heat sources, and the components of heat flows in the crust and mantle. Determining temperature in the mantle from the velocities of seismic waves. Geochemical, seismic, and thermal models for the Earth’s upper mantle were made mutually consistent using the THERMOSEISM computer program complex and database, by methods of physicochemical simulations, which make it possible to convert compositional models into physical characteristics and velocity profiles into models for the temperature distribution [5, 9, 12, 13]. The temperature T P in the mantle is determined by solving the inverse problem for the velocities of primary (and/or secondary) seismic waves and the bulk composition of the rock at any point in a depth profile. The equilibrium mineral composition of the rock is determined from the bulk composition by minimizing the Gibbs free energy for the system Na 2 O–TiO 2 –eaO–FeO–MgO–Al 2 O 3 –SiO 2 with solid solutions of the following minerals: binary solutions of Fe‐Mg olivine, spinel, and ilmenite; garnet ( Gar of the pyrope‐almandine‐grossular series); orthopyroxene ( Opx — MgSiO 3 , FeSiO 3 , Ca 0.5 Mg 0.5 SiO 3 , Ca 0.5 Fe 0.5 SiO 3 , Al 2 O 3 ); and clinopyroxene ( Cpx —same components plus the jadeite end member) [12, 13]. The