Solution Of Transient Heat Conduction Research Articles

Exact analytical solution of unsteady axi-symmetric conductive heat transfer in cylindrical orthotropic composite laminates

This study presents an exact analytical solution of transient heat conduction in cylindrical multilayer composite laminates. This solution is valid for the most generalized linear boundary conditions consisting of the conduction, convection and radiation heat transfer. Here, it is supposed that the fibers are winded around the cylinder and their direction can be changed in each lamina. Laplace transformation is applied to change the domain of the solutions from time into the frequency. An appropriate Fourier transformation has been derived using the Sturm–Liouville theorem. Here, a set of equations for Fourier coefficients are obtained based on the boundary conditions both inside and outside the cylinder, and the continuity of temperature and heat flux at boundaries between adjacent layers. The exact solution of this set of equations is obtained using Thomas algorithm and Fourier coefficients are expressed by recessive relations. Due to the difficulty of applying the inverse Laplace transformation, the Meromorphic function method is utilized to find the transient temperature distribution in laminate. Some industrial examples are presented to investigate the ability of current solution for solving the wide range of applied steady and unsteady problems.

3D transient heat conduction in multilayer systems – Experimental validation of semi-analytical solution

This article presents an experimental validation of a semi-analytical solution for transient heat conduction in multilayer systems. The semi-analytical solution is obtained using the heat conduction equation Green’s functions in the frequency domain. These solutions are obtained after time and spatial Fourier transforms are then applied in the two horizontal directions when the space domain has no kind of discretization. The problem is recast in the time domain by means of inverse Fourier transforms, using complex frequencies to avoid aliasing.The multilayer system used in the experimental validation is built by superimposing layers of different materials whose thermal properties were previously estimated experimentally. The external surfaces of the full system are then subjected to heat variation. The generated variation of the temperature field across the multilayer system was recorded using thermocouples. The full system was also simulated using the semi-analytical solution. Comparison of the results showed that the semi-analytical solution agrees with the experimental one.

A method of fundamental solutions for two-dimensional heat conduction

We investigate an application of the method of fundamental solutions (MFS) to heat conduction in two-dimensional bodies, where the thermal diffusivity is piecewise constant. We extend the MFS proposed in Johansson and Lesnic [A method of fundamental solutions for transient heat conduction, Eng. Anal. Bound. Elem. 32 (2008), pp. 697–703] for one-dimensional heat conduction with the sources placed outside the space domain of interest, to the two-dimensional setting. Theoretical properties of the method, as well as numerical investigations, are included, showing that accurate results can be obtained efficiently with small computational cost.

Thermal wave propagation in a finite medium irradiated by a heat source with Gaussian distribution in both the temporal and spatial domain

This paper presents an analytical solution of transient heat conduction in a thin film irradiated by symmetrical heat source with Gaussian distribution in both the temporal and spatial domain employing the dual-phase-lag model. Heating is modeled as an internal heat source. Calculations are performed to exhibit various lagging thermal behavior.

On the solution of transient heat conduction of multi‐layer slab and filtering temperature measurements

PurposeThe purpose of this paper is to show how a system of differential equations of one‐dimensional transient cooling heat conduction of different multi‐layer slabs has been solved numerically. A simple deterministic filtering matrix has been developed to remove errors involved in the experimental temperature measurements.Design/methodology/approachThe system of differential equations is solved through Crank‐Nicolson method using a developed computer code. The developed matrix is based on the available information about the system and is strong enough to detect and remove errors from the measured temperatures.FindingsThe filtering algorithm is very straightforward and easy to implement and needs to be developed once for a given system.Originality/valueThis paper shows how the governing equations of transient heat conduction of multi‐layer slabs have been solved through Crank‐Nicolson method using a developed computer code. The code is user friendly and solves a large system of simultaneous differential equations for any given composite slab configurations. Furthermore, a matrix filter has been suggested to remove experimental errors. The filter is based on the available information about the considered system. The developed matrix filter is shown to be a powerful technique to detect the noisy data and correct them while intelligent enough not to harm good data. The filtering algorithm is very straightforward and easy to implement and needs to be developed once for a given system.

Analytical solution of non-Fourier temperature response in a finite medium symmetrically heated on both sides

This paper presents an analytical solution of transient heat conduction in a thin film subjected to symmetrical laser heating, employing the dual-phase lag model. Laser heating is modeled as an internal heat source. The analytical solution is derived using the finite integral transforms and the method of variation of parameters. The solutions for different time characteristics of heat sources capacity (instantaneous, rectangular, light, and Gaussian) are discussed. The effect of the control parameter B =τ T /2r q on the temperature profiles is examined in details.

A method of fundamental solutions for transient heat conduction in layered materials

In this paper we investigate an application of the method of fundamental solutions (MFS) to transient heat conduction in layered materials, where the thermal diffusivity is piecewise constant. Recently, in Johansson and Lesnic [A method of fundamental solutions for transient heat conduction. Eng Anal Boundary Elem 2008;32:697–703], a MFS was proposed with the sources placed outside the space domain of interest, and we extend that technique to numerically approximate the heat flow in layered materials. Theoretical properties of the method, as well as numerical investigations are included.

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.

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.

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.

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.

Finite element solution of transient heat conduction using iterative solvers

PurposeTo provide an analysis of transient heat conduction, which is solved using different iterative solvers for graduate and postgraduate students (researchers) which can help them develop their own research.Design/methodology/approachThree‐dimensional transient heat conduction in homogeneous materials using different time‐stepping methods such as finite difference (Θ explicit, implicit and Crank‐Nicolson) and finite element (weighted residual and least squared) methods. Iterative solvers used in the paper are conjugate gradient (CG), preconditioned gradient, least square CG, conjugate gradient squared (CGS), preconditioned CGS, bi‐conjugate gradient (BCG), preconditioned BCG, bi‐conjugate gradient stabilized (BCGSTAB), reconditioned BCGSTAB and Gaussian elimination with incomplete Cholesky factorization.FindingsProvides information on which time‐stepping method is the most accurate, which solver is the fastest to solve a symmetric and positive system of linear matrix equations of all those considered.Practical implicationsFortran 90 code given as an abstract can be very useful for graduate and postgraduate students to develop their own code.Originality/valueThis paper offers practical help to an individual starting his/her research in the finite element technique and numerical methods.

Green’s function solution for transient heat conduction in concrete-filled CHS subjected to fire

The heat transfer analysis of concrete-filled steel circular hollow sections (CHS) subjected to fire is essentially classified as the process of deriving a solution for transient heat conduction in a composite medium in a cylindrical coordinate system. In this paper, an efficient numerical approach based on an analytical Green’s function (GF) solution is developed, where the spatial variable is analytically tractable without recourse to spatial discretization. The thin steel layer is conveniently treated as a lumped heat capacitance with uniform temperature distribution to avoid tedious analytical solutions for hollow cylinders. A computational model is developed based on Duhamel’s principle, incorporating time-varying fire conditions. The choice of sampling time is not constrained by numerical stability and can be as large as 30 min in one time step. The numerical approach can be used to predict the temperature field inside the entire composite domain and the heat flux at the fire and the steel-concrete interfaces, as well as to verify more sophisticated numerical codes, e.g. finite element codes, of heat transfer.

Study of transient heat conduction in 2.5D domains using the boundary element method

This paper presents the solution for transient heat conduction around a cylindrical irregular inclusion of infinite length, inserted in a homogeneous elastic medium and subjected to heat point sources placed at some point in the host medium. The solution is computed in the frequency domain for a wide range of frequencies and axial wavenumbers, and time series are then obtained by means of (fast) inverse Fourier transforms into space–time. The method and the expressions presented are implemented and validated by applying them to a cylindrical circular inclusion placed in an infinite homogeneous medium and subjected to a point heat source, for which the solution is calculated in closed form. The boundary elements method is then used to evaluate the temperature field generated by a point source in the presence of a cylindrical inclusion, with a non-circular cross-section, inserted in an unbounded homogeneous medium. Simulation analyses using this model are then performed to study the transient heat conduction in the vicinity of these inclusions.

Analytical solution of transient heat conduction in a two-layer anisotropic cylindrical slab excited superficially by a short laser pulse

This paper presents an exact analytical solution of 2D transient temperature, caused by a non-periodical pulse heating of a cylindrical two-layer slab. The corresponding partial differential equations are solved step-by-step, using the separation of variables technique. Analytical/computational procedure for eigenvalue derivation is considered separately. General initial assumptions, such as existence of thermal contact resistance, thermal conduction and diffusion anisotropy, radiative heat losses, non-uniform heating, and finite absorption depth are accounted in the solution. Numerical simulations of temperature distribution throughout the sample are given at the end of the paper.

Non-similarity solution and correlation of transient heat transfer in laminar boundary layer flow over a wedge

With a comprehensive and rigorous method, this paper has successfully examined the transient heat transfer in a steady and two-dimensional (2D) laminar boundary layer flow on a wedge with sudden change of thermal boundary conditions of uniform wall temperature (UWT) and heat flux (UHF). Additionally, a correlation of unsteady forced convection was also formulated through an exact solution of transient heat conduction ( ξ=0) and the similarity solutions of a steady forced convection on a wedge ( ξ=1) in this study. Particularly, for the wedge with −0.198838⩽ ξ⩽1, the deviation of the wall temperatures estimated by correlation is less than 7.5% within full time of 0⩽ ξ⩽1 comparing with numerical results in the case of UHF ranging from Pr=10 −4 to 10 4.

Verification of a formula for estimating average quality factor degradation in thermally processed foods in cylindrical containers

Summary A formula suggested by Jen et al. (1971) may be suitable for estimating the average retention of quality factors in conduction heated foods. This hypothesis was tested by comparing the results obtained for thiamine retention with calculations based on analytical and finite difference solutions of transient heat conduction in cylindrical containers. For a high Biot number, agreement was within 10.8% over a wide range of processing conditions (can size, retort temperature, initial temperature, etc).

Bulk Average Heat Transfer Coefficient of Multiple Particles Flowing in a Holding Tube

Amethodology was developed to measure the bulk average heat transfer coefficient of a large number of particles flowing in a holding tube. This technique is based on energy balance on the fluid and the analytical solution of transient heat conduction within a sphere. Particles at concentrations up to 18% v/v and fluid, at different initial temperatures, were mixed while flowing in the holding tube. The range of measured heat transfer coefficient ( h fp ) was from 180 to 800Wm− 2 2 K, based on particle concentration and flow rate. The value of h fp was found to decrease significantly as the particle concentration increased. This could be attributed to the restriction of particle movement in the holding tube as the particle concentration increased. Although bulk flow rate, as expected, was found to increase h fp , the increase was not significant at all levels.

TRANSIENT HEAT CONDUCTION DURING QUENCHING OF DOWNWARD FACING COPPER AND STAINLESS STEEL CONVEX SURFACES

Abstract Quenching experiments were performed to determine the effect of watt material properties on pool boiling heat transfer for downward facing convex surfaces( radius of curvature 218.5 mm) in saturated water. Experiments employed 303e-type stainless steel and copper test sections having identical dimensions ( 75 mm in diameter and 20 mm thick). Pool boiling curves were constructed on the basis of two-dimensional numerical solution of transient heat conduction in spherical coordinates( r, 9) in test sections during quenching. The measured temperature histories at nine interior locations near the boiling surface ( ∼0.5 mm) provided a time-dependent boundary condition for the numerical solution. To ensure stability and reduce both computer storage and execution time, the numerical solution used the alternating direction implicit ( ADD method with control volume representations. A sensitivity analysis was conducted to assess the effect of grid size on computation time as well as the accuracy of calculat...

Analytical solution for transient heat conduction in hollow cylinders containing well-stirred fluid with uniform heat sink

Analytical solution for transient heat conduction in hollow cylinders containing well-stirred fluid with uniform heat sink