Solution Of Heat Conduction Research Articles

Measurement of anisotropic thermal conductivity of a dense forest of nanowires using the 3ω method.

The 3ω method is a dynamic measurement technique developed for determining the thermal conductivity of thin films or semi-infinite bulk materials. A simplified model is often applied to deduce the thermal conductivity from the slope of the real part of the ac temperature amplitude as a function of the logarithm of frequency, which in-turn brings a limitation on the kind of samples under observation. In this work, we have measured the thermal conductivity of a forest of nanowires embedded in nanoporous alumina membranes using the 3ω method. An analytical solution of 2D heat conduction is then used to model the multilayer system, considering the anisotropic thermal properties of the different layers, substrate thermal conductivity, and their thicknesses. Data treatment is performed by fitting the experimental results with the 2D model on two different sets of nanowires (silicon and BiSbTe) embedded in the matrix of nanoporous alumina templates, having thermal conductivities that differ by at least one order of magnitude. These experimental results show that this method extends the applicability of the 3ω technique to more complex systems having anisotropic thermal properties.

Monitoring of thermal stresses in pressure components based on the wall temperature measurement

Thick-wall boiler components limit maximum heating and cooling rates during start-up or shut-down of the boiler. First, the allowable heating rates of the critical pressure components of the boiler were determined, and the temperature of the fluid was determined. The rate of change of the wall temperature of the pressure element and the thermal stress on the inner surface are controlled online and compared with the permissible values. Boiler manufacturers designate thermal stresses on the inner surface of the pressure component on the edge of the hole based on the measurement of the wall temperature at two points located inside it. Due to the low accuracy of the method used by boiler manufacturers, a new stress-determination method has been proposed in this paper in which only the internal temperature measurement point is used to determine the stresses on the inner surface of the component. In the method proposed in the paper, first, the internal surface temperature is determined from the inverse heat conduction solution, and then the stresses are calculated. Numerous computational tests were performed for cylindrical and spherical elements. Thermal stresses on the inner surface were also determined based on actual measurement data. Thermal stresses can be monitored at small time intervals. The advantage of the method is its high accuracy even at rapid changes in the fluid temperature.

Solving Non-Fourier Heat Conduction-Radiation Problem, a Semi-Analytical Approach

Objectives: In this study, the thermal analysis of a thin film exposed to laser heating, was investigated using Cattaneo Vernotte non-Fourier heat conduction model. Methods/Statistical Analysis: Nonfourier equations were derived based on radiative heat loss and also variable thermal conductivity. Then Variational Iteration Method (VIM) and Reduced Differential Transformation Method (RDTM) were used to solve these equations. Findings: VIM is an efficient choice to find an approximate solution of nonlinear non-Fourier heat conduction and its accuracy is higher than RDTM. Considering the thermal conductivity vary with temperature creates a significant difference in the temperature profiles, which shows the importance of nonlinear analysis of the problem. In certain values of relaxation time, temperature variations take place and out of that certain domain, variations of relaxation time doesn’t effect on temperature at all. Application/Improvements: There is a remarkable difference between the Fourier and non-Fourier solutions of heat conduction equations that demonstrate the importance of non-Fourier analysis of this type of equations. Keywords: Conduction-Radiation, Laser Heating, Non-Fourier, Semi-Analytical, RDTM, VIM

Sensitivity analysis of temperature changes for determining thermal properties of partially frozen soil with a dual probe heat pulse sensor

Determining thermal conductivity (λ) and volumetric heat capacity (C) of partially frozen soils with a dual probe heat pulse (DPHP) sensor is challenging because an applied heat pulse melts ice surrounding the heater probe. Examining DPHP temperature changes with a commonly-used analytical solution that only accounts for heat conduction leads to inaccurate λ and C estimates for partially frozen soils at temperatures between −5 °C and 0 °C. In order to determine λ and C accurately and simultaneously, it is necessary to understand how various properties of partially frozen soil influence the temperature changes produced by DPHP sensors. The objective of this study is to determine the sensitivity of DPHP temperature changes to soil conditions and soil thermal properties. A numerical solution for radial heat conduction with soil freezing and thawing is developed. A series of simulations are performed, in which various errors are imposed onto a selected model parameter while other model parameters are held constant, and sensitivity coefficient values (φ) of the time of maximum probe temperature (tm) and of the maximum probe temperature rise (Tm) for each parameter are calculated. Temperature changes at the measurement probe are quite sensitive to initial soil temperature (φ values for tm and for Tm are −0.99 and 0.99, respectively), λ (φ value for tm is −0.93), and parameters determining the shape of the soil freezing characteristic (FC) curve, i.e., saturated water content θs (φ values for tm and for Tm are 0.59 and −0.73, respectively) and n (φ values for tm and for Tm are −2.7 and 2.4, respectively). Temperature changes are not very sensitive to C (φ values for tm and for Tm are 0.034 and −0.15, respectively). Although previous investigations tried to determine C by inverse analysis, this sensitivity analysis shows that the influence of C on temperature response to a heat pulse is masked by that of the FC. Thus, λ and FC parameters are the best candidate parameters to be determined by inverse analysis of DPHP data. This new result will guide further testing of DPHP sensors in partially frozen soils.

A Useful Equation for Gas Turbine Design

Abstract This article presents three different gas turbine phenomena and design cases. The sketch in the article shows a schematic of a combined cycle powerplant consisting of a Brayton cycle (gas turbine) whose exhaust provides energy to a Rankine cycle (steam turbine). Frequently, one can use simple but exact one-dimensional (1D) heat conduction solutions to estimate the heat loss or gain of gas turbine components under transient conditions. These easy-to-use solutions are found in most undergraduate heat transfer texts. The article suggests that those three widely different gas turbine phenomena and design cases all have the simple, nonlinear superposition form.

Applying Petroleum the Pressure Buildup Well Test Procedure on Thermal Response Test—A Novel Method for Analyzing Temperature Recovery Period

The theory of Thermal Response Testing (TRT) is a well-known part of the sizing process of the geothermal exchange system. Multiple parameters influence the accuracy of effective ground thermal conductivity measurement; like testing time, variable power, climate interferences, groundwater effect, etc. To improve the accuracy of the TRT, we introduced a procedure to additionally analyze falloff temperature decline after the power test. The method is based on a premise of analogy between TRT and petroleum well testing, since the origin of both procedures lies in the diffusivity equation with solutions for heat conduction or pressure analysis during radial flow. Applying pressure build-up test interpretation techniques to borehole heat exchanger testing, greater accuracy could be achieved since ground conductivity could be obtained from this period. Analysis was conducted on a coaxial exchanger with five different power steps, and with both direct and reverse flow regimes. Each test was set with 96 h of classical TRT, followed by 96 h of temperature decline, making for almost 2000 h of cumulative borehole testing. Results showed that the ground conductivity value could vary by as much as 25%, depending on test time, seasonal period and power fluctuations, while the thermal conductivity obtained from the falloff period provided more stable values, with only a 10% value variation.

Optimal Thermal Placement and Loss Estimation for Power Electronic Modules

In this paper, a study on loss estimation and optimal thermal placement of power electronic modules is presented. The module is a high-voltage inverter with three separated legs, mostly applicable in electric and hybrid vehicles. The loss is estimated using behavioral model and output waveforms obtained from the simulation in MATLAB. This model has used manufacturer’s datasheets in order to facilitate the calculation of heat generation power of both the insulated-gate bipolar transistor and diode. The finite-element (FE) method is used to solve the conductive heat transfer in complex geometry of a three-phase inverter. The heat conduction solution is combined with genetic algorithm (GA) to obtain an effective optimization process of thermal chip placement. In addition, an innovative technique is employed to change the design variables from continuous to discontinuous state and to effectively improve the GA convergence as well as reduce the number of FE simulations. The effectiveness of the proposed method is compared with that of the conventional method, and it shows that both methods resulted in the same optimal placement, whereas FE analysis of the proposed method is about 11.15% of those of conventional methods.

On axisymmetric heat conduction problem for multilayer graded coated half-space

The technique of integral Hankel transform to find the solution of heat conduction in half- -space coated by a multilayered package of homogenous laminae is applied. The half-space is heated by the given heat flux on the boundary surface. The temperature and heat flux distribution in the radial direction is analyzed for two types of coatings: 1) when the heat conductivity coefficient is described by a power or exponential function of the distance to the boundary surface; 2) multilayered coating has a periodic structure.

An Analytical Solution for Transient Heat Conduction in a Composite Slab with Time-Dependent Heat Transfer Coefficient

An analytical solution is derived for one-dimensional transient heat conduction in a composite slab consisting of n layers, whose heat transfer coefficient on an external boundary is an arbitrary function of time. The composite slab, which has thermal contact resistance at n-1 interfaces, as well as an arbitrary initial temperature distribution and internal heat generation, convectively exchanges heat at the external boundaries with two different time-varying surroundings. To obtain the analytical solution, the shifting function method is first used, which yields new partial differential equations under conventional types of external boundary conditions. The solution for the derived differential equations is then obtained by means of an orthogonal expansion technique. Numerical calculations are performed for two composite slabs, whose heat transfer coefficient on the heated surface is either an exponential or a trigonometric function of time. The numerical results demonstrate the effects of temporal variations in the heat transfer coefficient on the transient temperature field of composite slabs.

Rectification of depth measurement using pulsed thermography with logarithmic peak second derivative method

Logarithmic peak second derivative (LPSD) method is the most popular method for depth prediction in pulsed thermography. It is widely accepted that this method is independent of defect size. The theoretical model for LPSD method is based on the one-dimensional solution of heat conduction without considering the effect of defect size. When a decay term considering defect aspect ratio is introduced into the solution to correct the three-dimensional thermal diffusion effect, we found that LPSD method is affected by defect size by analytical model. Furthermore, we constructed the relation between the characteristic time of LPSD method and defect aspect ratio, which was verified with the experimental results of stainless steel and glass fiber reinforced plate (GFRP) samples. We also proposed an improved LPSD method for depth prediction when the effect of defect size was considered, and the rectification results of stainless steel and GFRP samples were presented and discussed.

Similarity type of general solution for one-dimensional heat conduction in the cylindrical coordinate

One-dimensional heat conduction process in the cylindrical coordinate is investigated, and a similarity type of general solution is developed using the Kummer functions. The limiting behaviors of the general solution are studied using the properties of the Kummer functions, and some useful identities are deduced. As applications of the general solution, an infinite line source problem under power-type initial and boundary conditions and a one-phase Stefan problem with space-dependent latent heat in the cylindrical coordinate are studied. The analytical solutions for these two problems are established using the general solution directly. Computational examples for the analytical solutions are presented. For the infinite line source problem, the computational results are compared with those of the solid cylindrical surface model, and the computational error caused by neglecting the radial dimension of the heat source under time-varying source intensity is investigated. For the one-phase Stefan problem, both the coefficient in the solution and the development of the temperature field are presented; the computational results can be used to verify the accuracy of numerical solutions for Stefan problems.

A mesh-free Monte-Carlo method for simulation of three-dimensional transient heat conduction in a composite layered material with temperature dependent thermal properties

A new solution for the three-dimensional transient heat conduction from a homogeneous medium to a non-homogeneous multi-layered composite material with temperature dependent thermal properties using a mesh-free Monte-Carlo method is proposed. The novel contributions include a new algorithm to account for the impact of thermal diffusivities from source to sink in the calculation of the particles’ step length (particles are represented as bundles of energy emitted from each source), and a derivation of the three-dimensional peripheral integration to account for the influence of material properties around the sink on its temperature. Simulations developed using the proposed method are compared against both experimental measurements and results from a finite element simulation.

A mechanistic model for swelling kinetics of waxy maize starch suspension

The evolution of 8% suspension of waxy maize starch granule size distribution when subjected to heating to 65, 70, 75, 80, 85 and 90 °C at a heating rate of 15 °C/min were measured. Granule swelling was more pronounced at higher temperatures with an increase in the average size from 13 μm to 25–28 μm, eventually approaching equilibrium. Cryo SEM images revealed porous structure of swollen granule. The proposed model for starch swelling accounts for entropy of mixing, enthalpy of water-starch interaction and elastic restoring force. The second virial coefficient of waxy maize in aqueous medium was characterized by static light scattering. Chemical potential profile as well as the temperature profile within the granule were predicted from the solution of unsteady state diffusion and heat conduction respectively. The granule size was then inferred from the chemical potential profile. The proposed model predicts the evolution of average granule size and granule size distribution accurately.

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

AbstractIn 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.

Analytical solution of non-Fourier heat conduction in a square plate subjected to a moving laser pulse

In this paper, the C-V wave model was adopted to study the thermal response of a square plate which is irradiated by a moving temporally non-Gaussian laser pulse. The governing equation was solved by using the method of mode superposition. The propagation and rebounding of thermal wave was confirmed. The dependence of the wave velocity on relaxation time was discussed. The influences of scanning velocity and wave velocity on the temperature distribution were also indicated.

SEMI-ANALYTICAL SOLUTION OF THE HEAT CONDUCTION IN A PLATE WITH HEAT GENERATION

In the present work, a formulation for the solution of the two-dimensional steady state heat conduction with heat generation is presented. The classical integral transform technique (CITT) is used to solve the problem in a semi- analytical manner. CITT deals with expansions of the sought solution in terms of infinite orthogonal basis of eigenfunctions, keeping the solution process always within a continuous domain. For the particular problem, the resulting system is composed of a set of uncoupled differential equations which can be solved analytically. However, a truncation error is involved since the infinite series must be truncated to obtain numerical results. For comparison and validation purposes, the second order central finite difference method (FDM) is also implemented. The convergence analysis showed that CITT has a greater performance having no difficulties to obtain accurate results with very few terms in the solution summation. The FDM had convergence troubles specially for the positions near the center and for high concentration of heat generation in the center of the plate.

An exact analysis for transient anisotropic heat conduction in truncated composite conical shells

This paper presents an exact analytical solution for transient anisotropic heat conduction in a truncated composite conical shell for the first time. The fibers of composite conical shell are winded around the body in any arbitrary direction. Heat convection between the ambient fluid flow and composite conical shell is modeled and the exact solution of heat conduction equation is obtained by combining two popular analytical methods. First, by means of an integral transform in angular direction, the heat conduction equation is transformed and after that the transferred heat equation is solved using separation of variables method. Eventually, an inverse transformation is used in order to achieve the final exact solution of heat conduction equation. Validation of the present analytical solution is performed by comparing the analytical results with the solution of second order finite different method. Besides, the capability of present solution in solving the industrial cases is tested. The present solution is applicable in some industrial cases such as cooling pin fins and aeronautical instruments.

Two-dimensional laser-induced thermal ablation modeling with integrated melt flow and vapor dynamics

A new detailed, physics-based computational framework for the direct prediction of material ablation under nanosecond pulsed laser irradiation is presented. The framework, named directed energy illumination and visualization (DEIVI), integrates two-dimensional (axisymmetric) models for the following physical processes within a single framework for the first time: (i) coupling of the laser irradiation with the solid, (ii) transient heat conduction in the condensed (solid and melt) phases including melt front tracking, (iii) flow of the melt layer under the action of recoil pressure, (iv) vaporization of the melt and development of recoil pressure, and (v) dynamics of the vapor/gas. The solution for transient heat conduction and melt interface tracking using the enthalpy method is similar to the development by Ho et al. [J. Appl. Phys. 78, 4696–4709 (1995)]; however, they did not model the flow of the melt layer and modeled one-dimensional heat flow in the solid. To model the melt flow, DEIVI solves the incompressible, viscous Navier–Stokes equations that include a pressure correction solver, treatment of spatial derivatives using upwind or hybrid schemes, and a staggered grid. The compressive interface capturing scheme for arbitrary mesh volume of fluid method [O. Ubbink, Ph.D. thesis, University of London (1997)] is implemented for tracking the melt/vapor interfaces. The recoil pressure is applied by setting a fixed-pressure boundary condition in grid cells located at the melt-vapor interface. Surface tension is also included in order to ensure realistic surface morphology behavior at the melt-vapor interface. The vaporization process is treated using a quasiequilibrium convection-dominated approach combined with Knudsen-type jump conditions [S. I. Anisimov, Sov. Phys. JETP 27, 182–183 (1968) and C. J. Knight, AIAA J. 17, 519–523 (1979)] to obtain the input conditions for solving the two-dimensional (axisymmetric) gasdynamics equations for the vapor. An operator splitting procedure is used to solve the Euler equations using a Lax–Wendroff or Colella and Woodward [J. Comput. Phys. 54, 174–201 (1984).] piecewise parabolic scheme. This paper presents key aspects of the integrated framework as well as verification analyses performed to ensure the accuracy of individual modules.

Analytical solution of transient heat conduction in a bi-layered circular plate irradiated by laser pulse

In this study, the heat conduction behavior of a bi-layered circular plate during pulsed laser heating is investigated. A laser pulse with non-Gaussian temporal form is applied to the center of the top surface of the plate at the beginning of time. The spatial distribution of the heat energy of the laser pulse is also considered. The Green’s function for heat conduction equation of two-dimensional laminated structure is determined and is used to derive the analytical expression of temperature increment of the problem. The validation of this theoretical model is obtained through comparison with numerical results obtained by the ANSYS software package. The spatial and temporal variations of the temperature increment are illustrated. The influence of the duration time of the laser pulse on the temperature increment is also analyzed.

Closed-form analytical solutions of transient heat conduction in hollow composite cylinders with any number of layers

Conventional analytical methods for transient heat conduction solutions, due to complicated derivations and formidable calculations, have been limited to composite bodies with two or three layers. Developed in this work is a new analytical solution method for transient heat conduction in hollow composite cylinders with an arbitrary number of layers and subject to general boundary conditions. In this method, a distributed transfer function formulation gives an s-domain solution of a composite cylinder and inverse Laplace transform of the s-domain solution via a newly derived residue formula yields the transient solution of the cylinder in an explicit and closed form. Unlike conventional analytical methods, the proposed method does not require different derivations for different cylinder configurations (such as number of layers, boundary conditions and thermal resistance at layer interfaces). Because it only involves two-by-two matrices in the solution process, the proposed method is highly efficient in computation, as demonstrated in numerical examples.