Temperature Distribution in Flux Channels with Discrete Contact Boundary Conditions

An analytical approach for the thermal behavior of two-dimensional rectangular flux channels with arbitrary boundary conditions on the source plane is presented. The boundary condition along the source plane can be a combination of the first kind boundary condition (Dirichlet or prescribed temperature) and the second kind boundary condition (Neumann or prescribed heat flux). To model the boundary conditions along the source plane, the method of least squares is used. The proposed solution is in the form of Fourier series expansion and can be applied to both symmetrical and non-symmetrical channels. This method is more general than other approaches and there is no need to use equivalent heat flux distributions to model isothermal heat sources. The general approach for obtaining the multidimensional temperature profile in flux channels and the advantages of the least-square method is discussed. The proposed solution can be used to calculate the temperature at any specified point in the flux channel. Two case studies are presented. The first case study is a flux channel with five discretely specified contact temperatures along the source plane. The second case study has both of the first kind and second kind boundary conditions on the source plane. The analytical results for both systems are compared with finite element method using a commercial software package. It is shown that the proposed approach can precisely model the temperature profile over the flux channel.     

An analytical solution for two-dimensional inverse heat conduction problems using Laplace transform

An analytical method has been developed for two-dimensional inverse heat conduction problems by using the Laplace transform technique. The inverse solutions are obtained under two simple boundary conditions in a finite rectangular body, with one and two unknowns, respectively. The method first approximates the temperature changes measured in the body with a half polynomial power series of time and Fourier series of eigenfunction. The expressions for the surface temperature and heat flux are explicitly obtained in a form of power series of time and Fourier series. The verifications for two representative testing cases have shown that the predicted surface temperature distribution is in good agreement with the prescribed surface condition, as well as the surface heat flux.     

Improvement of inverse heat conduction solution using Laplace transformation: Method of partial division of time

Any solution for an inverse heat conduction problem makes the estimation of surface temperature and surface heat flux worsen in the case where these values behave like a triangular shape change with time. In order to compensate for this defect, Monde and colleagues, who succeeded in obtaining analytical inverse solutions using the Laplace transform technique, introduce a new idea where these changes over the entire measurement time can be split into several parts depending on the behavior. Therefore, an approximate equation to trace the measured temperature change can be derived, resulting in good estimation of surface temperature and surface heat flux even in the case of the triangular shape change and sharp change. © 2003 Wiley Periodicals, Inc. Heat Trans Asian Res, 32(7): 630–638, 2003; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/htj.10117     

Effects of an unsteady thermal boundary condition on forced convection heat transfer

A numerical analysis based on adjoint formulation of unsteady forced convection heat transfer is proposed to generally evaluate effects of the thermal boundary condition on the heat transfer characteristics. A numerical solution of the adjoint problem enables us to predict the heat transfer characteristics, such as the total heat transfer rate or the temperature at a specific location, when the thermal boundary conditions change arbitrarily with time. Moreover, using the numerical solution of the adjoint problem, we can obtain the optimal thermal boundary conditions in both time and space to maximize the heat transfer at any arbitrary time. Numerical solutions of the adjoint problem in a lid‐driven cavity are presented to illustrate the capability of the present method. © 2002 Wiley Periodicals, Inc. Heat Trans Asian Res, 31(3): 237–247, 2002; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/htj.10032     

Use of inverse modelling techniques for the estimation of heat transfer coefficients to fluids in cylindrical conduits

We propose a method for the estimation of the overall heat transfer coefficient to/from a fluid in a cylindrical pipe at high Peclet numbers from/to a medium with unknown temperature profile in the axial direction. The method uses an analytical solution of the heat equation in cylindrical coordinates subject to a Robin boundary condition and a parametrised, piecewise linear approximation of the external temperature profile. The velocity profiles of both viscous and turbulent flows are considered and compact solutions of the temperature profile of the fluid are evaluated in both cases as functions of the heat transfer coefficient and of the unknown external temperature profile.It is shown that these solutions provide an efficient method for the estimation of the heat transfer coefficient from fluid temperature data based on separable least squares. The overall procedure is illustrated by a numerical example using simulated data.     

MHD boundary‐layer flow over a stretching surface with internal heat generation or absorption

This article is concerned with the steady laminar magnetohydrodynamic boundary‐layer flow past a stretching surface with uniform free stream and internal heat generation or absorption in an electrically conducting fluid. A constant magnetic field is applied in the transverse direction. A uniform free stream of constant velocity and temperature is passed over the sheet. The effects of free convection and internal heat generation or absorption are also considered. The governing boundary layer and temperature equations for this problem are first transformed into a system of ordinary differential equations using similarity variables, and then solved by a new analytical method and numerical method, by using a fourth‐order Runge–Kutta and shooting method. Velocity and temperature profiles are shown graphically. It is shown that the differential transform method solutions are only valid for small values of independent variables but the results obtained by the DTM‐Padé are valid for the entire solution domain with high accuracy. © 2013 Wiley Periodicals, Inc. Heat Trans Asian Res; Published online in Wiley Online Library (wileyonlinelibrary.com/journal/htj). DOI 10.1002/htj.21054     

A view of the thermal wave behaviors in thin plates

The objective of this research is to study the temperature variation in thin dielectric materials. The thermal wave model is being used as the classical Fourier law of heat conduction breaks down when a dielectric material of sub-micron geometry is heated rapidly. The first part of the work reviews the temperature distribution due to thermal wave in semi-infinite bodies. The main emphasis of this paper is to accurately determine the temperature profile in a finite plate made up from a dielectric material. The boundary conditions of the first and the second kind are selected for this study. Using the classical Laplace transform technique, analytical solutions are obtained for finite bodies with different boundary conditions the first kind and the second kind. Due to complexity of the problem, a symbolic algebra provided the solution for examining the thermal behavior of dielectric materials during rapid heating.     

Two-Dimensional Heat Conduction Problem in a Functionally Graded Plate with a Slanting Boundary to the Functional Gradation

This article presents analytical results for temperature in a functionally graded material plate (FGMP) with a slanting boundary to the functional gradation subjected to a partial heating. The heat conductivity is expressed in terms of a exponential function of the position. The general solution of the heat conduction equation for FGMP with a slanting boundary to the functional gradation is derived by use of the variable separation method and the analytical solution, which satisfies the boundary condition is obtained. Numerical calculations are carried out for ZrO₂/Ti-6Al-4V and ZrO₂/stainless (SUS304) functionally graded plates, when the ceramic surface is partially heated. Temperature and heat flux are graphically displayed for these two cases.     

Computational modeling and analysis of thermal characteristics of a rotating spacecraft subjected to solar radiation

This paper presents a computational modeling for the temperature distribution of a rotating thick‐walled cylindrical spacecraft subjected to solar radiation. The inner surface of the spacecraft is thermally insulated while the outer surface is subjected to concurrent events of solar incidence and radiative heat dissipation to space. The governing equation for the normalized temperature is discretized using a finite difference scheme and Successive Line Over‐Relaxation (SLOR) is used to solve the resulting system of algebraic equations. Numerical simulations of temperature distribution on the spacecraft for different spinning speeds, angular positions, and different radii are discussed and evaluated for both linearized and nonlinear boundary conditions. Comparative analysis between the computational modeling and the exact analytical solution for the linearized boundary condition is presented. The results indicate that the outer surface temperature distribution of the spacecraft is nearly independent of the angular position; at sub‐cylindrical surface, this independence is achieved at low angular velocity. Moreover, numerical simulations show that the use of the linearized boundary condition at the outer surface presents a good approximation for the case of high‐speed spinning spacecrafts, while it results in significant errors in the temperature field in the case of stationary and low‐speed spinning spacecrafts. © 2011 Wiley Periodicals, Inc. Heat Trans Asian Res; Published online in Wiley Online Library ( wileyonlinelibrary.com/journal/htj ). DOI 10.1002/htj.20368     

Transient heat conduction in an infinite medium subjected to multiple cylindrical heat sources: An application to shallow geothermal systems

In this paper, we introduce analytical solutions for transient heat conduction in an infinite solid mass subjected to a varying single or multiple cylindrical heat sources. The solutions are formulated for two types of boundary conditions: a time-dependent Neumann boundary condition, and a time-dependent Dirichlet boundary condition. We solve the initial and boundary value problem for a single heat source using the modified Bessel function, for the spatial domain, and the fast Fourier transform, for the temporal domain. For multiple heat sources, we apply directly the superposition principle for the Neumann boundary condition, but for the Dirichlet boundary condition, we conduct an analytical coupling, which allows for the exact thermal interaction between all involved heat sources. The heat sources can exhibit different time-dependent signals, and can have any distribution in space. The solutions are verified against the analytical solution given by Carslaw and Jaeger for a constant Neumann boundary condition, and the finite element solution for both types of boundary conditions. Compared to these two solutions, the proposed solutions are exact at all radial distances, highly elegant, robust and easy to implement.     

Influence of thermal and flow boundary perturbations on convection heat transfer characteristics: Numerical analysis based on adjoint formulation

A numerical approach based on adjoint formulation of convection heat transfer is proposed to predict the change of heat transfer characteristics for arbitrary thermal and flow boundary perturbations. In order to obtain the adjoint system of the convection heat transfer problem, we formally linearize the governing equations by the perturbation method and then derive the adjoint system for the perturbation system. As a result, it is shown that the numerical solutions of the base and the adjoint problems enable us to predict the changes of heat transfer characteristics, such as the change of total heat transfer rate or the change of temperature at a specific location, when the thermal and flow boundary conditions are perturbed. An application example is presented to demonstrate the proposed method. © 2002 Wiley Periodicals, Inc. Heat Trans Asian Res, 32(1): 1–12, 2003; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/htj.10065     

Models for transient conduction in a flat plate subjected to a variable heat flux

This work presents analytical models allowing to identify the transient temperature distribution in a flat plate. The plate is exposed to a convective heat transfer on a face and to a heat flux on the other one. The heating flux is Heaviside (crenel type) and is maintained during a t₁ time. The heating phase is followed by a relaxation one. The theoretical method is original because it uses Green’s functions method to determine the analytical solutions of the heat propagation equation in the plate during the heating and relaxation phases. These analytical solutions allow to identify the temperature distribution as well as wall heat flux versus time. The results of our work can be useful at different levels: during the identification of parameters (such as the thermal conductivity or the thermal diffusivity of a plate), during the identification of the boundary conditions (like the heating flux or the convection coefficient) in industrial processes using this kind of systems, or even with educational intents for teaching transient conduction.     

An investigation about the relations between the results of heat conduction problems with and without phase change

There are physical relations between the velocities of the isotherms for two heat conduction problems having the same initial and boundary conditions, but one of them has phase change while the other one has not. The main idea of this study is to find these relations mathematically, which stand on a physical base, depending on the physical properties of the phases and the common parameters of these two different problems. If such kind of relations are known, then it will be possible to find the position of solid-liquid interface by using the analytical solutions and the relation itself, not solving the phase change problem. This idea is applied to heat conduction problem in semi infinite media and one dimensional phase change problem is solved for three different kind of boundary conditions via the Enthalpy Method. The results are correlated with the analytical solutions of the problem having same geometry and conditions but without phase change and obtained some relations having correlation coefficients changing between 0.91 and 0.98. This result shows that the regression analysis made is statistically meaningful.     

Traveling wave solutions to incompressible unsteady 2-D laminar flows with heat transfer boundary

Analytical solutions play important roles in the understanding of fluid dynamics and heat transfer related problems. Some analytical solutions for incompressible steady/unsteady 2-D problems have been obtained in literature, but only a few of those are found under heat transfer conditions (which brings more complexities into the problem). This paper is focused on the analytical solutions to the basic problem of incompressible unsteady 2-D laminar flows with heat transfer. By using the traveling wave method, fluid dynamic governing equations are developed based on classical Navier–Stokes equations and can be reduced to ordinary differential equations, which provide reliable explanations to the 2-D fluid flows. In this study, a set of analytical solutions to incompressible unsteady 2-D laminar flows with heat transfer are obtained. The results show that both the velocity field and the temperature field take an exponential function form, or a polynomial function form, when traveling wave kind solution is assumed and compared in such fluid flow systems. In addition to heat transfer problem, the effects of boundary input parameters and their categorization and generalization of field forming or field evolutions are also obtained in this study. The current results are also compared with the results of Cai et al. (R. X. Cai, N. Zhang. International Journal of Heat and Mass Transfer, 2002, 45: 2623-2627) and others using different methods. It is found that the current method can cover the results and will also extend the fluid dynamic model into a much wider parameter ranges (and flow situations).     

Explicit full field analytic solutions for two-dimensional heat conduction problems with finite dimensions

This study presents explicit analytical solutions of heat conduction problems for isotropic media with finite dimensions. The geometry configurations considered in this study include composite layer, wedge and circular media. The boundary conditions are assumed to be either thermal isolation or isothermal. The full field analytical solutions of temperature and heat fluxes for the composite layered media subjected to an embedded heat source are derived first by Fourier transform technique in conjunction with the image method. The corresponding problems of composite wedge and circular media are constructed by conformal mapping and the solutions of composite layer media. The explicit full field solutions are expressed in simple closed-forms which can be easily used in engineering applications. The numerical calculations of the temperature and heat fluxes distributions are provided in full field configuration base on the available analytic solutions.     

Analytical modelling of transient phase‐change problems

In this paper, an analytical solution for the temporal location of moving solid–liquid interface of a phase‐change process, occurring in parallel plate channels, is presented. The motion of the solid–liquid interface is governed by the convection from the surface of one of the plates, while constant heat supply is assumed to occur on the surface of the other plate. The steady location of the solid–liquid interface is also determined. The variation of the Biot number versus the Fourier number is investigated. The results of this study indicate that simple analytical solutions for transient phase‐change problems with heat flux and convective boundary conditions that are of practical importance to the people working in the field can be obtained. Copyright © 2000 John Wiley & Sons, Ltd.     

Transient thermoelastic analysis of infinite plates with time-periodic heat transfer coefficient: Frequency response of thermal stress

An analytical method is presented for one-dimensional transient heat conduction and associated thermal stress problems of a layered infinite plate whose heat transfer coe?cient (HTC) on one external boundary is an arbitrary function of time. A shifting function method and an orthogonal expansion technique are used to obtain analytical solutions to the problems. Using the solutions, the structural response to sinusoidal HTC oscillation is investigated in single and bilayered plates, the other boundary of which is kept at constant temperature. Numerical results demonstrate the effects of average HTC (measured by the Biot number) and mechanical boundary conditions on the frequency responses of temperature and thermal stress. It is found that the range of stress oscillation decreases with an increase in the frequency of HTC oscillation except for single-layered plates without surface tractions.     

Boundary layer flow of an Oldroyd‐B fluid with convective boundary conditions

This study is presented for the flow of an Oldroyd‐B fluid subject to convective boundary conditions. The two‐dimensional equations are simplified by using boundary layer approximations. The analytic solutions in the whole spatial domain (0 ≤ η < ∞) are derived by a homotopy analysis method (HAM). Interpretation of various emerging parameters is assigned through graphs for velocity and temperature distributions and tables for surface heat transfer. The present results are compared with the previous studies in limiting cases and results are found in very good agreement. © 2011 Wiley Periodicals, Inc. Heat Trans Asian Res; Published online in Wiley Online Library ( wileyonlinelibrary.com/journal/htj ). DOI 10.1002/htj.20381     

Determination of skin temperature distribution and heat flux during simulated fires using Green's functions over finite-length scales

One of the current practices for measuring heat flux during flash fire testing, forest fires, and other industrial cases focuses on the use of semi-infinite models to predict the heat flux during exposure through surface temperature measurements on simulated skin sensors. For short time frames, these models can be shown to have acceptable accuracy. However, when considering longer time exposures at reduced heat fluxes, such as with firefighters in a forest fire, the accuracy of these models could be brought into question. A one-dimensional, finite length scale, transient heat conduction model was developed using a Green's function approach on a rectangular sensor. The model was developed using transient temperature boundary conditions to avoid the use of complicated radiation and convection conditions at each boundary. For comparison, a semi-infinite model utilizing the same boundary condition on the exposed face was solved using both the Laplace transform method and Green's function method. Experimental data was obtained during exposure to a cone calorimeter. All measurements were taken for a minimum duration of 2 min. This temperature data was used to develop appropriate curves for the boundary conditions and validate the analytical models. It was found that the temperature obtained from the one-dimensional transient heat conduction model based on Green's functions agreed well with the experimental results over longer exposure times, and with reduced error when compared with the semi-infinite model. This suggests that modeling the problem on a finite-length scale will produce more accurate or more conservative temperature and heat flux results over extended periods of exposure in high heat load applications.