Fast-converging steady-state heat conduction in a rectangular parallelepiped

A Green's function approach for precisely computing the temperature and the three components of the heat flux in a rectangular parallelepiped is presented. Each face of the parallelepiped may have a different, but spatially uniform, boundary condition. Uniform volume energy generation is also treated. Three types of boundary conditions are included: type 1, a specified temperature; type 2, a specified flux; or type 3, a specified convection boundary condition. A general form of the Green's function covering all three types of boundary conditions is given. An algorithm is presented to obtain the temperature and flux at high accuracy with a minimal number of calculations for points in the interior as well as on any of the faces. Heat flux on type 1 boundaries, impossible to evaluate with traditional Fourier series, is found by factoring out lower-dimensional solutions. A numerical example is given. This research and resulting computer program was part of a code verification project for Sandia National Laboratories.     

Heat exchange between two coupled fixed beds by fluid flow

Heat exchange by forced fluid flow between two coupled fixed beds containing solids can be used to recover heat from hot products to cool input in many industrial processes. The heat exchange between the fixed beds is studied. Analytical solutions for the transient fluid and solid temperature distributions and heat recovery effectiveness are derived. The intra-particle transient temperature distributions are accounted for in the more accurate analysis and the results are compared to lumped analysis. It is shown that the heat recovery effectiveness reaches maximum at certain optimum time instant at which the fluid circulation should be stopped. Damping of cyclic oscillations in fluid temperature or concentration is considered and analytical solution for the damper is presented.     

Solution of an initial-boundary value problem for heat conduction in a parallelepiped by time partitioning

An initial-boundary value problem for transient heat conduction in a rectangular parallelepiped is studied. Solutions for the temperature and heat flux are represented as integrals involving the Green's function (GF), the initial and boundary data, and volumetric energy generation. Use of the usual GF obtained by separation of variables leads to slowly convergent series. To circumvent this difficulty, the dummy time interval of integration is partitioned into a short time and a long time subintervals where the GFs are approximated by their small and large time representations. This paper deals with the analysis and implementation of this time partitioning method.     

Validity ranges of three analytical solutions to heat transfer in the vicinity of single boreholes

In ground-coupled heat pump systems, accurate prediction of transient ground heat transfer is important to establish the required borehole length and to determine precisely the resulting fluid temperature. Three analytical solutions to transient heat transfer in the vicinity of geothermal boreholes are presented. These solutions are referred to as the infinite line source (ILS), the infinite cylindrical source (ICS) and the finite line source (FLS) models, which vary in complexity and are based on simplifications of the borehole geometry. The results of these models are compared and their validity domains are determined.     

THERMOELASTIC TOPOLOGY OPTIMIZATION FOR PROBLEMS WITH VARYING TEMPERATURE FIELDS

Elastic structures that exist in a thermal environment usually experience complex steady-state or transient heat conduction, whereby operational temperatures and stresses may change with time, heat sources, and thermal or kinematic boundary conditions. This article proposes an evolutionary optimization procedure for topology design involving thermoelasticity in which finite element heat analysis, finite element thermoelastic analysis, and subsequently design modification are iteratively carried out. To achieve as efficacious a material usage as possible, the relative efficiency of an element is defined in terms of its thermal stress level. In this article, design cases with uniform temperature fields, nonuniform temperature fields subjected to single or multiple heat load cases, and transient temperature fields are studied. The examples presented show the capabilities of the proposed procedure to solve various thermoelastic problems under varying temperature fields.     

Comparison of the periodic solution method with TRNSYS and SUNCODE for thermal building simulation

Based on periodic solutions of the governing heat conduction equations in a single zone building, computer software ADMIT has been developed for thermal simulation of buildings. Standard computer software, namely TRNSYS and SUNCODE, have also been used to simulate the same building under similar conditions. Simulations have been performed for three different climatic zones in India for light and heavy constructions under conditions of glazed/unglazed areas and ventilation rates. The results are presented in terms of the hourly variation of the room temperature. For insulated heavy construction, the results of different models are significantly different. This difference is due to the use of different approaches to solve the heat conduction equations. SUNCODE depends on the RC network approach and underestimates the heat losses. TRNSYS uses the transfer function approach, which is sensitive to the initially assumed value of the room temperature. ADMIT represents a quasi-steady-state periodic variation and is not suitable for transient variations. For insulated light buildings, the heat transfer mechanisms used in the mathematical models are not the governing factors. The models also differ in treating the penetration of solar radiation through a glazed window and the subsequent heat-transfer mechanism. For a south window and air changes in an insulated building, the results obtained by SUNCODE and ADMIT are in good agreement, but the results obtained by TRNSYS are considerably different. The reason for this needs detailed analysis.     

Thermal impedance and transient temperature due to a spot of heat on a half-space

Analytical solutions are proposed in this paper to calculate the thermal impedance and transient temperature in a semi-infinite body subjected to a heat spot. Two cases of uniform and non-uniform heat flux are considered. These solutions are developed using integral transforms and are given in exact expression forms, without any restrictive hypothesis. They include special functions such as Bessel, Struve and hypergeometric functions. Many softwares to treat these functions are available (eg. Maple, Mathematica and others). The solutions are validated through comparisons with available models treating particular cases. The transient temperatures and impedances are presented for different spatial distributions of the heat flux dissipated by the spot.     

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.     

Analysis of flux-base fins for estimation of heat transfer coefficient

Exact solutions are given for the transient temperature in flux-base fins with the method of Green’s functions (GF) in the form of infinite series for three different tip conditions. The speed of convergence is improved by replacing the steady part by a closed-form steady solution. For the insulated-tip case, a quasi-steady solution is presented. Numerical values are presented and the conditions under which the quasi-steady solution is accurate are determined. An experimental example is given for estimation of the heat transfer coefficient (HTC) on a non-rotating roller bearing, in which the outer bearing race is treated as a transient fin.     

Unsteady heat conduction in two-dimensional two slab-shaped regions. Exact closed-form solution and results

The unsteady heat conduction analysis for multi-directional piecewise-homogeneous bodies is generally held to be complex and demanding, possibly explaining why practical guidelines for thermal field calculation are few and far between. The proposed solution method represents an extension of the new, ‘natural’ analytic approach derived in companion papers for solving one-dimensional multi-layer problems of time-dependent heat conduction. As the approach is new, it is presented in full, together with the complete temperature double-series solution prepared for computer implementation. By setting thermal diffusivity ratio unitary and assuming a uniform distribution of initial temperature, it emerges that, all other things being equal, the transient thermal response can be expressed as the product of two, separated, one-directional solutions, one across the layers and the other along the composite slab. The formulation deals properly with thermal conductivity ratios of all magnitudes. An efficient and accurate procedure of computing eigenvalues is given. Graphical and numerical output is presented and discussed.     

One‐dimensional transient heat conduction in a bilayer finite slab: A case study in the printing process

In this work, a typical case of heat distribution is examined during a paper printing process, based on one‐dimensional transient heat conduction in two‐layer finite slabs with an insulated free surface, and a constant temperature free surface. Analytical solutions were obtained in non‐dimensional form. Various examples of applying these solutions are presented. The accuracy of the solutions, with respect to time, is analyzed considering the eigenvalues of their infinite solutions. It is observed that the larger the number of eigenvalues in consideration, the better the accuracy of the solutions. The model related to a two‐layer slab describes the simplified case in which all heat transfer occurs only by conduction. The solutions obtained are finally compared with the solutions for heat conduction in two semi‐infinite solids. The comparison between the two solutions shows that results are in good agreement only during short time scales. The heat distribution study is expected to be helpful in knowing the effectiveness of various mediums to be used as the reciever during the printing process; however, there is scope for development of more robust models.     

Investigation of thermal performance of a ground coupled heat pump system with a cylindrical energy storage tank

An analytical and computational model for a solar assisted heat pump heating system with an underground seasonal cylindrical storage tank is developed. The heating system consists of flat plate solar collectors, an underground cylindrical storage tank, a heat pump and a house to be heated during winter season. Analytical solution of transient field problem outside the storage tank is obtained by the application of complex finite Fourier transform and finite integral transform techniques. Three expressions for the heat pump, space heat requirement during the winter season and available solar energy are coupled with the solution of the transient temperature field problem. The analytical solution presented can be utilized to determine the annual variation of water temperature in the cylindrical store, transient earth temperature field surrounding the store and annual periodic performance of the heating system. A computer simulation program is developed to evaluate the annual periodic water and earth temperatures and system performance parameters based on the analytical solution. Copyright © 2003 John Wiley & Sons, Ltd.     

Transverse temperature distribution and heat generation rate in composite superconductors subjected to constant thermal disturbance

Analytical solution of one-dimensional, transient heat conduction with a distributed heat source is obtained to predict the transverse temperature distribution and heat generation rate per unit volume of the composite superconductor. The solution indicates that temperature distribution and heat generation rate depend on three dimensionless parameters: the dimensionless external disturbance w₀, the dimensionless interface temperature θ¹, and the dimensionless parameter φ that is dependent on the thickness and the thermal conductivity of the superconductor. Results of transient and steady-state solutions are presented. It is shown that the heat generation rate per unit volume of the composite, Q/Qc, is directly proportional to the current in the stabilizer.     

Heat conduction in a semi-infinite solid subject to steady and non-steady periodic-type surface heat fluxes

An analytical solution for the temperature and heat flux distribution in the case of a semi-infinite solid of constant properties is investigated. The solutions are presented for time-dependent, surface heat fluxes of the forms: (i) Q₁(t) = Q₀(1+a cos ωt); and (ii) Q₂(t) = Q_o(1+bt cos ωt), where a and b are controlling factors of the periodic oscillations about the constant surface heat flux Q₀. The dimensionless (or reduced) temperature and heat flux solutions are presented in terms of decompositions C_r and S_r of the generalized representation of the incomplete Gamma function. It is demonstrated that the present analysis covers the limiting case for large times which is discussed in several textbooks, for the case of steady periodic-type surface heat fluxes. In addition, an illustrative example problem on heating of malignant tissues, making use of transient and long-time solutions, is also presented.     

Analytical investigations for heat conduction problems in anisotropic thin-layer media with embedded heat sources

This article develops the analytical rigorous solution of a fundamental problem of heat conduction in anisotropic media. The steady-state temperature and heat flux fields in a thin-layer medium with anisotropic properties subjected to concentrated embedded heat sources or prescribed temperature on the surface are analyzed. A linear coordinate transformation is used to transform anisotropic thin-layer problems into equivalent isotropic problems without complicating the geometry and boundary conditions of the problem. By using the Fourier transform and the series expansion technique, exact closed-form solutions of the specific problems are presented in series forms. The complete solutions of heat conduction problems for the thin-layer medium consist only of the simplest solutions for an infinite homogeneous medium with concentrated heat sources. The numerical results of the temperature and heat flux distributions are provided in full-field configurations.     

Transient heat transfer analysis of anisotropic material by using Element-Free Galerkin method

A transient heat transfer model for anisotropic material is established based on Element-Free Galerkin (EFG) method. The model can simplify the pretreatment for anisotropic material, which make the thermal conductivity executed by orthotropic factor and off-angle. The proposed numerical model and MATLAB programs are verified by complex engineering heat transfer problems. The effects of orthotropic factors and off-angles on solutions are investigated. The appropriate orthotropic factors along the large size of geometries and appropriate off-angle make the temperature distribution more uniform. The solutions obtained by EFG show a higher calculation precision than the FEM solutions for the transient heat transfer problems in practical problems.     

Heat transfer analysis of pile geothermal heat exchangers with spiral coils

This paper investigates the transient heat conduction around the buried spiral coils which could be applied in the ground-coupled heat pump systems with the pile foundation as a geothermal heat exchanger. A transient ring-coil heat source model is developed, and the explicit analytical solutions for the temperature response are derived by means of the Green’s function theory and the image method. The influences of the coil pitch and locations are evaluated and discussed according to the solutions. In addition, comparisons between the ring-coil and cylindrical source models give that the improved finite ring-coil source model can accurately describe the heat transfer process of the pile geothermal heat exchanger (PGHE). The analytical solutions may provide a desirable and better tool for the PGHE simulation/design.     

A general analytical solution to the equation of transient forced convection with fully developed flow

A general solution to the energy equation under zero wall temperature or zero heat flux boundary condition for the decay of an inlet and initial temperature distribution of an incompressible transient turbulent flow heat transfer between two parallel plates is given. It is shown that these solutions may then be used to obtain solutions due to unit steps in wall temperature or wall heat flux which is sufficient to sort out prescribed wall temperature and prescribed wall heat flux boundary condition. The results are confirmed experimentally by the frequency method. An experimental apparatus has been designed, built and used for this purpose.     

TRANSIENT THERMOELASTIC ANALYSIS OF AN ANNULAR FIN WITH COUPLING EFFECT AND VARIABLE HEAT TRANSFER COEFFICIENT

A hybrid numerical method of the Laplace transformation and the finite difference method is applied to solve the transient thermoelastic problem of an annular fin, in which the thermomechanical coupling effect is taken into account in the governing equation of heat conduction and the heat transfer coefficient is a function of the radius of the fin. The general solutions of the governing equations are first solved in the transform domain. Then the inversion to the real domain is completed via the method of matrix similarity transformation and Fourier series technique. The transient distributions of temperature increment and thermal stresses of the fin in the real domain are calculated numerically. The presented method is more efficient in computing time and is applicable to other types of boundary conditions.     

Transient Piezothermoelasticity of a Two-Layered Composite Hollow Cylinder Constructed of Isotropic Elastic and Piezoelectric Layers Due to Asymmetrical Heating

This article is concerned with the theoretical treatment of transient piezothermoelastic problem involving a two-layered hollow cylinder constructed of isotropic elastic and piezoelectric layers due to asymmetrical heat supply. The transient two-dimensional temperature is analyzed by the method of Laplace transformation. By using the exact solutions for piezoelectric hollow cylinder and isotropic hollow cylinder, the theoretical analysis of transient piezothermoelasticity is developed for a two-layered composite hollow cylinder under the state of plane strain. As an example, numerical calculations are carried out for an isotropic elastic hollow cylinder made of steel, bonded to a piezoelectric layer of cadmium selenide. Some numerical results for the temperature change, the stress and the electric potential distributions in a transient state are shown in figures. Furthermore, the influence of thickness of the piezoelectric layer or the isotropic elastic layer upon the temperature change, stresses and electric potential is investigated.