Two-dimensional Thermal Conduction Research Articles

Thermal spreading in a multilayer geometry with convective cooling along the side wall of each layer

Two-dimensional thermal conduction between a source/sink and a body of larger size results in thermal spreading and constriction, which are of much technological importance in problems such as semiconductor thermal management. While the traditional analysis of thermal spreading and constriction assumed adiabatic side walls, there is growing interest in convective heat removal from the side walls, such as in multilayer three-dimensional integrated circuits (3D ICs) and stacked Li-ion cells in a battery pack. This work presents theoretical analysis of thermal spreading in a multilayer body with distinct convective heat transfer coefficients along the side wall for each layer, in addition to anisotropic thermal conductivity in each layer and thermal contact resistance at interfaces between layers. A series solution for the temperature distributions in each layer is derived, with a unique set of eigenvalues for each layer. A set of linear algebraic equations that govern the coefficients of the series solutions is derived. Results are shown to agree well with past work for special cases, as well as with numerical simulations for general problems. The impact of source size, thermal anisotropy and Biot numbers of various layers is analyzed in detail. In particular, the interplay between Biot numbers along the sidewall and at the sink plane in determining total thermal resistance is investigated in detail. This work contributes towards a fundamental understanding of theoretical thermal conduction in problems of practical relevance, such as advanced microelectronics and Li-ion cells. Results may find application in the design and optimization of these and other engineering systems where thermal spreading or constriction play an important role.

Thermal spreading resistance from an isothermal source into a finite-thickness body

Thermal spreading resistance is encountered during two-dimensional thermal conduction from a hot surface into a body. Thermal spreading and the opposite problem of thermal constriction resistance are of much technological relevance in the design of heat sinks and thermal spreaders for thermal management of microelectronics and other heat-generating devices. Much of the past work in the theoretical analysis of thermal spreading is based on a source with given heat flux. In contrast, the isothermal source problem presents difficulties due to the resulting mixed nature of the boundary condition, due to which, only approximate solutions are available. This work derives the steady-state thermal spreading resistance from an isothermal source into a finite-thickness slab or cylinder. The mixed boundary condition is handled by posing it in the form of a spatially varying convective boundary condition, with a sufficiently large Biot number over the source to represent its isothermal nature. A series solution for the problem is derived, along a sufficient set of linear algebraic equations to determine the series coefficients. Results are shown to agree well with finite-element simulations. Results are compared with previously-reported approximate solutions within the parametric range of validity of the approximate solutions. The impact of key non-dimensional parameters on thermal spreading resistance is quantified. It is shown that thermal spreading resistance increases with decreasing size of the isothermal heat source, as expected. A third-order polynomial correlation with very good accuracy is proposed. This work advances the theoretical understanding of a problem for which only approximate solutions have been reported in the past. Results presented here offer a practical tool for thermal design and optimization of a variety of practical thermal management problems involving spreading or constriction.

Analysis of heat transfer in an aquifer thermal energy storage system: On the role of two-dimensional thermal conduction

Aquifer thermal energy storage (ATES) system plays an important role in application of renewable energy, efficient energy utilization, and reduction of CO2 emission. To realize the successful operation of ATES systems, it is essential to understand the thermal behaviors thoroughly during heat injection. Serving as a powerful tool, analytical model is widely adopted while the existing analytical models usually simplify ATES systems by ignoring aquifer transverse thermal conduction and rock longitudinal thermal conduction. This study develops a novel analytical model for an ATES system, which accounts for thermal convection, dispersion, longitudinal and transverse thermal conduction in the aquifer, and longitudinal and transverse thermal conduction in the surrounding rocks. The Green's function method is applied to derive the semi-analytical solutions of temperature. A global sensitivity analysis is conducted to analyze the relative importance of each parameter. The results indicate that ignoring the aquifer transverse thermal conductivity will overestimate the temperature in the aquifer and ignoring the caprock longitudinal thermal conductivity may underestimate the heat loss in ATES systems. Similar conclusion can be made about the bedrock longitudinal thermal conductivity. The global sensitivity analysis indicates that temperature is sensitive to the changes in the flow velocity, thermal dispersivity and aquifer thickness.

One-dimensional thermal equation modeled on a two-dimensional heat exchanger with variable solid-fluid interface

In this study, we are to present that a one-dimensional equation for vertically averaged temperature, modeled on a vertically thin, two-dimensional heat exchanger with variable top solid-fluid interface, recovers the two-dimensional thermal information, i.e. steady temperature and flux distribution on the top and temperature-fixed bottom faces. The relative error of these quantities is less than 5% with the maximum gradient of the height kept approximately below 0.5, while the computational time is reduced to 0.1–5%, when compared with direct two-dimensional computations, depending on the shape of the top face. The model equation, derived by the vertical averaging of the two-dimensional thermal conduction equation, is closed by an approximation that the heat exchanger is sufficiently thin in the sense that the second derivative of temperature with respect to the horizontal coordinate depends only on the coordinate. In this model equation, the fluid equation above the exchanger is decoupled by a conventional equation for the normal heat flux on the top surface. In principle, however, the coupling of the model and the fluid equation is possible through the temperature and heat flux on the top interface, recovered by the model equation. The type of mathematical modeling can be applicable to a wide variety of bodies with extremely small dimensions in some (coordinate-transformed) directions.

Evaluation of thermoelastic damping in micromechanical resonators with axial pretension: An analytical model accounting for two-dimensional thermal conduction

It has been reported that application of tensile axial stress can simultaneously increase quality factor and resonant frequency for micromechanical resonators. In this study, we formulate an analytical model for evaluating thermoelastic damping in micromechanical resonators based on the thermal energy method, in which thermal conductions in both thickness direction and axial direction are considered. An explicit expression for thermoelastic damping in the form of infinite series has been obtained. The proposed analytical model is further validated by finite element analysis. Results of the present study demonstrate that the 2D model needs to be adopted in order to accurately evaluate thermoelastic damping of micromechanical resonators with axial pretension. In addition, the 2D model proposed in the present study eliminates the inherent inconsistency entailed in the 1D model.

Thermoelastic damping in micromechanical resonators operating as mass sensors

This paper presents an analytical study on evaluation of thermoelastic damping (TED) in micromechanical resonators operating as mass sensors by means of the thermal energy approach. Both quasi-one-dimensional model and two-dimensional model are proposed and analytical expressions of TED in infinite series form for micro-beam resonator-based mass sensors are derived. We apply the proposed models to examine effects of various factors on TED, such as the position of attached mass particle, boundary conditions, geometry of resonators and mode shape of vibration. Numerical results show that thermoelastic damping of micro-beam resonator-based mass sensors increases obviously with the increase of the mass ratio of the attached mass to mass of the resonator. Compared with results of TED obtained by 2-D model accounting for the two-dimensional thermal conduction in micro-beam resonators, the quasi-1-D model overestimates TED in the high frequency range, and underestimates TED in the low frequency range. Noticeable differences are observed in the results of TED between quasi-1-D model and 2-D model. We also obtain the expressions of mass sensitivity and minimum detectable mass imposed by thermomechanical noise process for resonant mass sensors of bridge and cantilever configurations.

Enhancement of condensation heat transfer on grooved surfaces: Numerical analysis and experimental study

The present paper reported numerical analyses and experimental studies on fluid flow and heat transfer for laminar film condensation on various grooved surfaces. Different from the previous literatures, the coupled non-linear governing equations for the fluid flow, mass transfer and two-dimensional thermal conduction were developed base on some reasonable assumptions. Through analyzing the influences of groove shape parameters on film thickness and heat flux, we found that both of them were significantly influenced by the pitch, height and profile radius of the groove. The velocity distributions on the condensate-vapor interface and distributions of wall temperature on various grooves were also studied systematically. The calculations indicated that the surface tension gradient of the liquid film drove the liquid to flow from the crest into the trough. The horizontal velocity decreased first and then increased gradually on the crest and reached a maximum value on profile, and then decreased to zero in the trough region. On the contrary, the condensate flowed at very low vertical velocities on the crest and profile of groove, but in trough region these velocities increased rapidly to a relatively high level. The results of comparison showed that both the distributions of wall temperature and heat flux on involute groove were more favorable to reduce thermal resistance and enhance heat transfer than other grooves. In order to validate the feasibility and reliability of the present analyses, experiments were carried out with some involute grooved surfaces for various physical dimensions, a trapezoid grooved surface and a smooth surface respectively. Both the numerical results and experimental data indicated that involute groove can provide a better heat transfer performance, and the heat flux of involute grooved surface was at least 20% and 50% higher than that of trapezoid grooved and rectangular grooved surfaces. The present analyses were feasible, and could be used in the parameter design and heat transfer calculation of involute grooved surfaces as well as trapezoid grooved surfaces.

Numerical study on condensation heat transfer of trapezoid grooved surfaces

This article presents a numerical analysis and experimental study on condensation heat transfer and fluid flow for filmwise condensation on trapezoid grooved surfaces. First, a physical model was properly simplified based on some reasonable assumptions. Then, the coupled non-linear governing equations for the mass transfer, fluid flow, and two-dimensional thermal conduction were developed. The relationship between z-coordinate and heat transfer was obtained by solving the equations numerically. The influences of groove length and basic angle were discussed. The calculation results showed that the heat flux decreased with increase in groove length, and the decline range also decreased gradually. The calculation results also suggested that the heat flux through groove with α = 60° was lower than the groove with α = 75° at the top of the groove, while the opposite conclusion was obtained at the low parts. The distributions of wall temperature and heat flux on trapezoid groove were also studied systematically. The distribution of surface temperature and heat flux presents obvious lateral inhomogeneity, and the maximum wall temperature and heat flux were both obtained in region II. The thermal resistance of groove with α = 60° was lower but the liquid-discharged ability was better than that of groove with α = 75°. In order to validate the feasibility and reliability of the present analyses and to further investigate the heat transfer performance of trapezoid grooved surfaces, experiments were carried out with three condensing plates including two trapezoid grooved surfaces in different physical dimensions and one smooth surface. The experimental data obtained under various schooling were compared with the calculations, and the experimental results for different condensing plates are all in good agreement with the numerical model, with a maximum deviation less than 15%. Moreover, the trapezoid grooves can enhance the condensation heat transfer by 1.5–2.5 times higher than the smooth surface. The present analyses are feasible and can be used in the parameter design and heat transfer calculation of trapezoid grooved surfaces.

Simultaneous estimation of thermal parameters for large-format laminated lithium-ion batteries

In-situ determination of the battery thermal parameters is important to provide accurate inputs for battery thermal models. This paper develops a method to estimate the multiple thermal parameters of large-format laminated lithium-ion batteries both simultaneously and in-situ. The central area of one battery surface is heated with a circular planar heater, while the temperature responses on the opposite surface at multiple strategically-chosen locations are recorded with the attached thermocouples. This thermal system is modeled in COMSOL v4.2 using a two-dimensional axially-symmetric thermal conduction equation containing thermal parameters such as the thermal capacity, anisotropic thermal conductivities, and thermal interfacial conductance between the Al-plastic film package and the electrode core. Using optimization techniques, these thermal parameters are adjusted step by step till the difference between the simulated and the experimental temperature responses at the corresponding locations reaches a minimum. As one validation of the developed method, the estimated specific heat capacity agreed with the value measured with an accelerating rate calorimeter within 10%. The proposed method can be applied to simultaneously determine the thermal parameters of generic objects consisting of anisotropic internal materials and an outer packaging made of different material.

Mathematical modeling of heat and mass transfer processes at the ignition of a liquid condensed substance by an immersed hot particle

A numerical investigation of heat and mass transfer processes at the heating of combustible liquid was carried out at the interaction of small-size steel particle with gasoline. Developed mathematical model considers at two-dimensional statement thermal conduction, thermal convection, transfer of energy by phase change (evaporation of liquid fuel and crystallization of particle material), partial immersion of particle in liquid fuel, forming of vapor gap between particle and liquid fuel, diffusion of fuel vapors in oxidizer, dependence of thermophysical characteristics of interactive substances on temperature. It was established that the highest rates of heat and mass transfer processes in a system hot particle - gasoline - air are possible at temperature of particle higher than melting temperature of it material due to the additional heat released at the crystallization of material.

Research on Thermo Effects of Silver Modified by High-Intensity Pulsed Ion Beam

We report a modification method for Silver target by high-intensity pulsed ion beam (HIPIB) irradiation. Based on the temporal and spatial distribution models of the ion beam density detected by Faraday cup in the chamber and the ions accelerating voltage, the energy deposition of the beam ions in Ag is calculated by Monte Carlo method. Taking this time-dependent nonlinear deposited energy as the source term of two-dimensional thermal conduction equation, we obtain the temporal and spatial ablation process of metal Ag during a pulse time. The top-layer silver material in thickness of about 0.33μm is ablated by vaporization and the layer in thickness of 1.6μm is melted after one shot at the ion beam density of 300 A/cm2.

The Irradiation Effects of Metal Gold Surface by Intense Pulsed Ion Beam

We report a modification method for Gold target by intense pulsed ion beam (IPIB) irradiation. Based on the temporal and spatial distribution models of the ion beam density detected by Faraday cup in the chamber and the ions accelerating voltage, the energy deposition of the beam ions in Au is calculated by Monte Carlo method. Taking this time-dependent nonlinear deposited energy as the source term of two-dimensional thermal conduction equation, we obtain the temporal and spatial ablation process of metal Au during a pulse time. The top-layer Gold material in thickness of about 0.25μm is ablated by vaporization and the layer in thickness of 1.40μm is melted after one shot at the ion beam density of 300 A/cm2.

强流脉冲离子束辐照铜靶热效应数值研究

为了研究强脉冲离子束辐照改性铜靶的行为，建立了基于靶室中法拉第杯检测的离子束流密度和离子加速电压时空分布模型；采用蒙特卡罗方法计算了束流在铜靶中的能量沉积，并将这随时间变化的非线性沉积能量作为热传导方程的热源项，计算得到了脉冲期间靶材内部温度场的演化规律。一次脉冲后与液化烧蚀相比，靶材表面较小区域的物质气化烧蚀掉，辐照中心处熔化的厚度最大；气、液化烧蚀厚度随偏离中心位置而减小。当束流密度为300 A/cm2时，中心区域气化烧蚀和液化烧蚀的厚度分别达到0.2微米和1.3微米；脉冲期间热量延纵深方向的传播距离小于2微米。 A modification method for Copper target by intense pulsed ion beam (IPIB) irradiation has been reported. Based on the temporal and spatial distribution models of the ion beam density detected by Faraday cup in the chamber and the ions accelerating voltage, the energy deposition of the beam ions in Cu is calculated by Monte Carlo method. Taking this time-dependent nonlinear deposited energy as the source term of two-dimensional thermal conduction equa- tion, the temporal and spatial ablation process of metal Cu during a pulse time was obtained. Only a little area of mate- rial is ablated off due to the vaporization compared with the melting ablation area on the surface after a shot. The most deep melting depth is in the irradiation centre. The melting and vaporizing thicknesses decrease with the increasing dis- tance from the irradiation centre. The top-layer Copper material in thickness of about 0.2 μm is ablated by vaporization and the layer in thickness of 1.3 μm is melted after one shot under the ion beam density of 300 A/cm2. The thermal transportation along the longitudinal direction in the specimens is less than 2.0 μm during a pulse.

Online Calculating Models of Temperature and Thermal Stresses for Turbine Rotors

This paper presents the faster calculating models of temperature and thermal stress in a rotor based on two-dimensional axis-symmetry thermal conduction and thermo-elastic theory. The models can deal with nonlinear factors caused by temperature-dependant properties using modified Newton-Raphson iterative method. An example has been given to compare the results calculated by the models with those calculated by finite element method. It is shown that the analytical models gained have high accuracy. The iterative models can be used for online monitoring and control of turbine.

Numerical Study on the Ablation Effects of Tungsten Irradiated by High-intensity Pulsed Ion Beam

Based on the ion current density model of TEMP type ion beam accelerator, and utilizing the deposited energy of the high-intensity pulsed ion beam (HIPIB) in tungsten by Monte Carlo method as source term, a two-dimensional thermal conduction model controlling the irradiation process was built. The evolution of tungsten temperature during the irradiation period was obtained and analyzed. The results show that the melted thickness of tungsten surface is about 0.3μm at shooting center of target irradiated by HIPIB at the end of a pulse under the ion peak current density of 100 A/cm2, the materials of tungsten surface in thickness of about 0.2μm reach boiling point; and among the ion peak current density of 50-200 A/cm2, the melting ablation depth is less than 1μm according to the calculation. While the ion peak current density increased to about 330 A/cm2, the vaporizing ablation phenomenon appeared on the shooting center of tungsten surface.

Numerical Study on the Two-Dimensional Temperature Fields of Titanium/Aluminum Double-Layer Target Irradiated by High-Intensity Pulsed Ion Beam

Interaction between high-intensity pulsed ion beam (HIPIB) and a double-layer target with titanium film on top of aluminum substrate was simulated. The two-dimensional nonlinear thermal conduction equations, with the deposited energy in the target taken as source term, were derived and solved by finite differential method. As a result, the two-dimensional spatial and temporal evolution profiles of temperature were obtained for a titanium/aluminum double-layer target irradiated by a pulse of HIPIB. The effects of ion beam current density on the phase state of the target materials near the film and substrate interface were analyzed. Both titanium and aluminum were melted near the interface after a shot when the ion beam current density fell in the range of 100 A/cm2 to 200 A/cm2.

Solutions to Eshelby’s problems of non-elliptical thermal inclusions and cylindrical elastic inclusions of non-elliptical cross section

Eshelby’s inclusion problem is solved for non-elliptical inclusions in the context of two-dimensional thermal conduction and for cylindrical inclusions of non-elliptical cross section within the framework of generalized plane elasticity. First, we consider a two-dimensional infinite isotropic or anisotropic homogeneous medium with a non-elliptical inclusion subjected to a prescribed uniform heat flux-free temperature gradient. Eshelby’s conduction tensor field and its area average are first expressed compactly in terms of two boundary integrals avoiding the usual singularity and then specified analytically for arbitrary polygonal inclusions and for inclusions characterized by the finite Laurent series. Next, we are interested in a three-dimensional infinite isotropic or transversely isotropic homogeneous medium with a cylindrical inclusion of a non-elliptical cross section that undergoes uniform generalized plane eigenstrains. The solution to this problem is obtained by decomposing a generalized plane eigenstrain tensor into a plane strain part and an anti-plane strain part, exploiting the mathematical similarity between two-dimensional thermal conduction and anti-plane elasticity, and combining the relevant results of Zou et al. (Zou et al. 2010 J. Mech. Phys. Solids 58 , 346–372. ( doi:10.1016/j.jmps.2009.11.008 )) with those derived in the present work for Eshelby’s conduction tensor field and its area average.

Design sensitivity analysis and optimization of interface shape for zoned-inhomogeneous thermal conduction problems using boundary integral formulation

A generalized formulation of the shape design sensitivity analysis for two-dimensional steady-state thermal conduction problem as applied to zoned-inhomogeneous solids is presented using the boundary integral and the adjoint variable method. Shape variation of the external and zone-interface boundary is considered. Through an analytical example, it is proved that the derived sensitivity formula coincides with the analytic solution. In numerical implementation, the primal and adjoint problems are solved by the boundary element method. Shape sensitivity is numerically analyzed for a compound cylinder, a thermal diffuser and a cooling fin problem, and its accuracy is compared with that by numerical differentiation. The sensitivity formula derived is incorporated to a nonlinear programming algorithm and optimum shapes are found for the thermal diffuser and the cooling fin problem.

Numerical heat transfer analysis of laminar film condensation on a vertical fluted tube

The purpose of this study is to find a convenient and practical procedure for calculating heat transfer of laminar film condensation on a vertical fluted tube. The condensate film on the tube surface along the axial direction was divided into two portions: the initial portion and the developing portion. The developing portion was analyzed in details. The film thickness equation of condensate film over the crest and the momentum equation of condensate film in the trough were established respectively after some simplifications and coupled with two-dimensional thermal conduction equation. The relationship between the heat transfer rate and the length of the flute was obtained through solving the equations numerically. The present procedure was tested on a sinusoidal fluted tube. The amount of heat transfer rates Qt of the tube were calculated at different temperature differences by using this procedure. The calculation was compared with the experimental data quoted. The results were in good agreement with a maximum deviation of 18%. So the present procedure is reliable and can be used in the parameter design of sinusoidal fluted tubes.

Temperature imaging of sub-millimeter-thick water using a near-infrared camera

This paper presents a method of imaging temperature distributions of sub-millimeter-thick water using a near-infrared camera and optical narrow-bandpass filter. The principle is based on the temperature dependence of the ν1+ν3 absorption band of water. Temperature images are constructed by measuring the absorbance of water at the wavelength of 1412nm through the filter for all pixels of the camera. From calibration measurements on 0.5-mm thick water at temperatures from 26.0°C to 40.0°C, the temperature coefficient was 6.3×10−4K−1 and the standard deviation of absorbance was 1.9×10−4. Thermal diffusion in 0.5-mm thick water caused by a thin heating wire was visualized with this method. The obtained images were verified against temperature distributions calculated by solving a two-dimensional thermal conduction model. This method would be useful for temperature measurement applications and control of aqueous solutions in microchips.