Three-dimensional Heat Conduction Equation Research Articles

Thermal management of a battery pack using fin-embedded phase change material

Abstract The present study focuses on the numerical investigation of heat transfer from a battery cell surrounded by phase change material (PCM) with longitudinal fins embedded in it. Three-dimensional transient heat conduction equation is solved numerically in the battery domain, whereas the solidification-melting model adopting Enthalpy-Porosity approach is solved in PCM to obtain liquid fraction and temperature distribution. PCM with 16 fins was found to be quite effective in reducing the battery surface temperature compared to 12, 8, and 4 fins. However, the effectiveness of 16 fins is observed up to 1500 sec till the PCM completely melts into liquid. A maximum reduction of 25 K has been achieved at 1500 sec by adding 16 fins to PCM. Beyond 1500 sec, the temperature rises sharply, exceeding all other cases at 2200 sec. Fins play an essential role in augmenting heat transfer, which benefits in achieving the phase change process in less time to restrict the temperature rise; however, at the same time, when the phase change process completes, fins become detrimental to the system since the temperature of the battery starts rising sharply.

Heat transport during drop impact onto a heated wall covered with an electrospun nanofiber mat: The influence of wall superheat, impact velocity, and mat thickness

Nanofiber surface coating is a promising method for the enhancement of heat transfer during spray cooling. In the present work, the drop dynamics as well as local and overall heat transfer during single drop impact onto a heated wall covered with a polyacrylonitrile (PAN) nanofiber mat are investigated to obtain insight into the mechanisms governing the heat transport enhancement. The influence of wall superheat, drop impact velocity, and mat thickness on the hydrodynamics and heat transfer from the heated wall to the fluid is studied. The experiments were conducted inside a temperature-controlled test cell with a pure vapor atmosphere maintained with refrigerant FC−72 (perfluorohexane). The temperature field at the solid–fluid interface was observed with a high-speed infrared camera, and the heat flux field was derived by solving a three-dimensional transient heat conduction equation within the substrate. The presence of the nanofiber mat on the heater surface suppresses the drop receding phase due to the pinning of the contact line at the end of the spreading phase. Two different heat transfer scenarios are observed depending on the wall superheat and drop impact velocity: scenario (I), in which the liquid drop completely penetrated through the porous nanofiber mat and made contact with the heater surface; and scenario (II) in which the vapor produced inside the pores of the nanofiber mat prevented the liquid drop from touching the heater surface. In scenario (I), an up 450% heat flow enhancement during the sessile drop evaporation stage has been observed. In scenario (II), a 80% heat flow enhancement at this stage has been registered.

Thermally-Induced Stresses in a Pre-Buckling State of a Circular Plate within the Fractional-Order Framework

This paper considers a transient thermoelastic problem in an isotropic homogeneous elastic thin circular plate with clamped edges subjected to thermal load within the fractional-order theory framework. The prescribed ramp-type surface temperature is on the plate's top face, while the bottom face is kept at zero. The three-dimensional heat conduction equation is solved using a Laplace transformation and the classical solution method. The Gaver–Stehfest approach was used to invert Laplace domain outcomes. The thermal moment is derived based on temperature change, and its bending stresses are obtained using the resultant moment and resultant forces per unit length. The results are illustrated by numerical calculations considering the material to be an Aluminum-like medium, and corresponding graphs are plotted.

SIMULATION OF TEMPERATURE EFFECTS IN MASSIVE BODIES USING THE MODIFIED METHOD OF LINES

This article presents the main ideas and possibilities of the modified straight line method for modeling temperature effects in massive bodies. The calculation model of a rectangular beam with a defined thermal state is adopted. The procedure for reducing the dimensionality of the original equations using the modified method of straight lines and the Bubnov-Galyorkin-Petrov method for boundary conditions is applied. The reduced equation of thermal conductivity and the reduced equation of heat balance are given. Conclusions are made regarding the addition of reduced second-order equations with components from the boundary conditions on the lateral surfaces, and finally reduced second-order equations in spatial variables are given.
 We are considering a beam of rectangular cross-section, the three dimensions of which are of the same order. The thermal state of such an object is three-dimensional and is therefore described systematically using three-dimensional heat conduction equations. When applying the modified method of straight lines to reduce the dimensionality of the original equations, it is convenient to use them in the form of a system of partial differential equations of the first order in spatial coordinates.
 According to the modified method of straight lines, two systems of straight lines parallel to the coordinate axes Oy and Oz are applied to the beam (a decrease in dimensionality in terms of spatial variables y and z is assumed). The choice of such straight lines is due to the fact that the reduced equations will depend on the variables x and t, that is, they will resemble the one-dimensional equations of the one-dimensional non-stationary problem of thermal conductivity, for which efficient and convenient numerical methods have been developed in the literature of mathematical physics at the moment, to which it is easy to adapt the reduced equation.

Use Different Mathematical Methods to Solve Three Dimensional Conduction Heat Equation in Cartesian Coordinate

In this paper three-dimensional heat conduction equation in cartesian coordinate has been solved in two different methods one of which depends on the separation of variables and the other depends on the integral transform .The results are got and plotted by using Matlab. And the results obtained showed the difference between the two methods that were used in the solution . That difference is evident in the illustrations . According to the results it was concluded that the integral transform method is the best because it has fewer steps to reached to the solution

Numerical Solution of Two and Three-Dimensional Fractional Heat Conduction Equations via Bernstein Polynomials

Numerical Solution of Two and Three-Dimensional Fractional Heat Conduction Equations via Bernstein Polynomials

A fast interpolating meshless method for 3D heat conduction equations

A fast interpolating meshless (FIM) method for three-dimensional (3D) heat conduction equations is presented. Transforming a 3D problem into the relevant two-dimensional (2D) problems using the dimension splitting method (DSM) is the main idea of FIM method. The improved interpolating moving least-squares (IIMLS) method is applied in 2D problems to obtain required approximation function with interpolation property. Finite difference method (FDM) is utilized in time domain and the direction of splitting. Take the improved element-free Galerkin (IEFG) method as a comparison, difficulties created by the singularity of weight functions, such as truncation error and calculation inconvenience, are overcome by the FIM method. And it can directly implement the Dirichlet boundary conditions. To prove the advantages of the new method, three examples are selected and solved by the FIM method. Comparing and analyzing the calculation results, it can be shown that the FIM method effectively improves computation speed and precision.

Photo-thermal modeling of laser–phosphor interaction in laser lighting

In laser lighting, the heat generated in a phosphor has a great impact on the optical and thermal properties of the phosphor-converted white laser diode (PC-WLD), which is hard to be calculated and measured precisely. We, therefore, propose a photo-thermal modeling of laser–phosphor interaction in laser lighting (PTML) to simulate the illumination energy and temperature distribution of a phosphor. PTML consists of two parts: the modified Monte Carlo method (MCM) and the numerical calculation method based on the three-dimensional heat conduction equation. With full consideration of light scattering, fluorescence reabsorption, and thermal quenching effect, PTML is able to simulate the absorbed energy and temperature distribution inside a phosphor. Experiments with transparent and scattering Ce:YAG phosphor samples are carried out to validate this method. The results show that the difference between the PTML simulated and experimental light transmission energy (laser and fluorescence) is less than 10%. The temperature difference between the PTML simulation and experiment is less than 1.5°C when the laser irradiates on the Ce:YAG single crystal at pump power 1420 mW. Moreover, this method also has potential application in the photo-thermal interaction between bio-tissue and lasers.

Numerical Solution of Three-Dimensional Transient Heat Conduction Equation in Cylindrical Coordinates

Heat equation is a partial differential equation used to describe the temperature distribution in a heat-conducting body. The implementation of a numerical solution method for heat equation can vary with the geometry of the body. In this study, a three-dimensional transient heat conduction equation was solved by approximating second-order spatial derivatives by five-point central differences in cylindrical coordinates. The stability condition of the numerical method was discussed. A MATLAB code was developed to implement the numerical method. An example was provided in order to demonstrate the method. The numerical solution by the method was in a good agreement with the exact solution for the example considered. The accuracy of the five-point central difference method was compared with that of the three-point central difference method in solving the heat equation in cylindrical coordinates. The solutions obtained by the numerical method in cylindrical coordinates were displayed in the Cartesian coordinate system graphically. The method requires relatively very small time steps for a given mesh spacing to avoid computational instability. The result of this study can provide insights to use appropriate coordinates and more accurate computational methods in solving physical problems described by partial differential equations.

グラフニューラルネットを用いた粒子法の代替モデリング

For the purpose of accelerating numerical analysis, we propose a surrogate model for a particle method based on IsoGCNs, which are variants of graph neural networks (GNNs). In mechanical phenomena, when transformations such as rotations, translations, and reflections are applied to the input, the output takes on the same transformations. This property is called equivariance. IsoGCNs improve learning efficiency by introducing the equivariance into GNNs, and have been shown to be useful for finite element methods. This study shows an IsoGCN formulation for the particle method by considering relationships between particles as a graph. The proposed method is verified by comparing the errors and computation time of numerical and surrogate analyses of the three-dimensional heat conduction equation. In this paper, the problem of estimating the temperature distribution after one second from the initial temperature distribution is considered. As a result, the surrogate analysis requires more computation time than the numerical analysis with a time step size of one second. However, the accuracy is comparable, and it confirms that it can learn three-dimensional heat conduction equations.

Experimental investigation of hydrodynamics and heat transport during horizontal coalescence of two drops impinging a hot wall

This paper presents an experimental study on hydrodynamics and heat transport during the horizontal coalescence of two drops impinging a hot wall. The study addresses the influence of distance between impact locations, the time interval between drop impact, and wall superheat on the transport processes. The experiments were conducted under a pure vapor atmosphere with the refrigerant FC-72 at a saturation temperature of 54°C, corresponding to a system pressure of 0.94 bar. The drops were generated with a constant diameter and a constant impact velocity. The temperature field at the surface of the heater was measured by an infrared camera with a high spatial and temporal resolution. The local heat flux distribution was derived from the temperature field by solving the transient three-dimensional heat conduction equation within the substrate. The total heat flow was evaluated by integrating the local heat fluxes over the footprint of the drop. The impact parameters (drop size and impact velocity), the time interval between drop impact, and the distance between impact locations were evaluated through post-processing of the black-and-white images captured employing a high-speed camera. The results for simultaneous drop impingement reveal that the heat transported from the wall to the fluid is not affected by the presence of neighboring drop as long as the drops do not contact each other. In the case of horizontal coalescence of drops, the heat flow is reduced both during the spreading and receding phases and during the sessile drop evaporation phase. This can be explained by the reduction of solid–liquid interface and of three-phase contact line length after the coalescence. An increase in wall superheat leads to reducing the footprint of the drop and increasing of heat flow. Asymmetric behavior of the drop hydrodynamics and heat flux is observed during the non-simultaneous drop impingement.

Effect of Thermal Stresses and Reduced Differential Transform on Bending of The Rectangular Plate

The bending of the rectangular plate exists for a large length comparing with its width as we have considered the simply supported plate. Also, it gets the accuracy, if the plate is attached between enough distance from the ends. This paper concern with the study of bending moment in the rectangular plate by using the thermal stresses. In this paper, we have considered the mathematical model of the unsteady state three-dimensional heat conduction equation for a thick rectangular plate with non-zero temperature initially. We have determined the thermal stresses and the bending moment in the rectangular plate by using the reduced differential transform method. The results are discussed by considering the special cases for copper material. Also, results are shown graphically. The graphs are drawn by using mathematical software Scilab.

Research on Measuring Thermal Conductivity of Quartz and Sapphire Glass Using Rear-Side Photothermal Deflection Method

As the display industry continues to advance, various new materials are being developed for utilizing microtechnology and nanotechnology in display panels. Among these, transparent materials have been widely applied to the internal wiring of displays and flexible substrates, owing to their high optical transmittance, isotropy, and anisotropy. Thus, measurement of the thermophysical properties of various transparent materials is important. This study measured thermal conductivity by selecting quartz, a transparent isotropic material, and sapphire glass, a transparent anisotropic material, as measurement target materials using a rear-side photothermal deflection method. Measurements were made via a three-dimensional unsteady heat conduction equation, to which complex transformation was applied and numerically analyzed using COMSOL Multiphysics. Phase delays for a pump beam and a probe beam for a relative position were derived through a deflection analysis. From the derived phase delays between the numerical analysis and experimental result with optical alignment, the absolute and relative errors of quartz were appropriately confirmed to be 0.069 W/m-K and 5%, respectively, while those of the sapphire glass were likewise confirmed to be 0.55 W/m-K and 1.5%, respectively.

Two Nonlinear Positivity-Preserving Finite Volume Schemes for Three-Dimensional Heat Conduction Equations on General Polyhedral Meshes

Two Nonlinear Positivity-Preserving Finite Volume Schemes for Three-Dimensional Heat Conduction Equations on General Polyhedral Meshes

Lucid characterization of unsteady heat conduction in large, square crossbars and cubes subject to uniform surface heat flux by way of “quasi–steady” Helmholtz equations

The technical paper addresses unsteady heat conduction in large, square crossbars and cubes heated continually with uniformly surface heat flux. The uniform surface heat flux is customarily provided by electrical, radiative and pool fire heating in engineering applications. This type of heating fall under the category of non-homogeneous Neumann boundary conditions. The existing traditional method for solving the unsteady, heat conduction equation in two and three dimensions with constant thermo-physical properties and subject to non-homogeneous uniform heat flux boundary conditions is the principle of superposition of partial solutions of simpler unsteady, heat conduction equations. For the latter, the exact, analytical temperature distributions are customarily expressed by infinite series regardless of the solution method. For the numerical evaluation of spatio-temporal temperatures, the main inconvenience of the infinite series lies in the poor convergence when the time is small. To prevent the impeding obstacle, the Transversal Method Of Lines (TMOL) is implemented in this work to transform the unsteady, two- and three-dimensional heat conduction equations into degraded “quasi steady”, two- and three-dimensional heat conduction equations, namely the homogeneous Helmhotz equations. Thereby, the exact, analytical solutions of the converted partial differential equations of elliptic type and homogeneous were determined by the standard method of separation of variables. The resulting approximate, semi-analytical two- and three-dimensional temperature distributions for large, square crossbars and cubes do not contain elaborate infinite series. Instead, the temperature disdtributions contain hyperbolic functions elucidating worthy compactness and superb quality, regardless of the time, either “small time”, “moderate time” or “large time”.

Thermoelastic large deflection bending analysis of elliptical plate resting on elastic foundations

In this paper, a three-dimensional transient heat conduction equation with an internal source as a thermal flux outflow, decreasing linearly with time from the surface of an elliptic plate is analyzed. Applying theory of integral transformations, the heat conduction model is explained, and its expression is obtained in the form of Mathieu functions. The large deflection of a plate when placed on the elastic foundation is formulated from the potential energy equation neglecting the second strain invariant. A new elegance of modified total strain energy is obtained in these formulations by incorporating the resulting moment and force within the energy term, thus reducing the computation step. The numerical calculations of distribution of the transient temperature, thermal deflection and the maximum normal bending stress are carried out on the outer elliptic boundary and illustrated graphically. Lastly, the corresponding results for circular plates have been presumed when the ellipse degenerates to a circle. The results reveal that the highest tensile stress exists on the major axis of the circular core relative to the elliptical core, which suggests the propagation of low heating due to inadequate heat penetration into the elliptical surface.

Experimental investigation of hydrodynamics and heat transport during vertical coalescence of multiple successive drops impacting a hot wall under saturated vapor atmosphere

In this study, hydrodynamics and heat transport during the vertical coalescence of multiple successive drops impacting a hot wall are analyzed experimentally. This study addresses the influence of wall superheat and the frequency of drop generation on the hydrodynamics and heat transport. The experiments are conducted under a pure vapor atmosphere with the refrigerant FC-72 at a saturation temperature of 54.5°C, corresponding to a system pressure of 0.94 bar. The drops are generated with a constant diameter of 1.14 mm and a constant impact velocity of 0.54 m s−1. An infrared camera with a high spatial and temporal resolution is used to capture the temperature field on the surface of the heater. The local heat flux distribution is derived from the temperature field by solving the transient three-dimensional heat conduction equation within the substrate. The total heat flow is evaluated by integrating the local heat flux over the footprint of the drop. The impact parameters (drop size and impact velocity) are evaluated through post-processing of the black/white images captured using a high-speed camera. The maximum spreading radius and maximum heat flow observed after the impact of each successive drop are higher than those observed after the impact of the previous drop. For instance, in the case of a wall superheat of 12.4 K and an impact frequency of 10 Hz, the maximum spreading radius and maximum heat flow observed after the impact of the fourth drop increased by approximately 35% compared with those observed after the impact of the first drop on a dry wall. The distribution of wall heat flux during the spreading phase of the second impacting drop is characterized by the appearance of two thin ring-shaped zones of high heat flux. The time evolutions of the drop shape and heat flow depend on the wall superheat and are independent of the frequency of drop generation within the studied range of parameters.

Three-Dimensional Transient Heat Conduction Equation Solution for Accurate Determination of Heat Transfer Coefficient

Abstract Accurate quantification of local heat transfer coefficient (HTC) is imperative for design and development of heat exchangers for high heat flux dissipation applications. Liquid crystal and infrared thermography (IRT) are typically employed to measure detailed surface temperatures, where local HTC values are calculated by employing suitable conduction models, e.g., one-dimensional (1D) semi-infinite conduction model on a material with the low thermal conductivity and low thermal diffusivity. Often times, this assumption of 1D heat diffusion and ignoring its associated lateral conduction effects leads to significant errors in HTC determination. Prior studies have identified this problem and quantified the associated errors in HTC determination for some representative cooling concepts, by accounting for lateral heat diffusion. In this paper, we have presented a procedure for solution of three-dimensional (3D) transient conduction equation using alternating direction implicit (ADI) method and an error minimization routine to find accurate HTCs at relatively lower computational cost. Representative cases of a single jet and an array jet impingement under maximum crossflow condition have been considered here, for IRT and liquid crystal thermography, respectively. Results indicate that the globally averaged HTC obtained using the 3D model was consistently higher than the conventional 1D model by 7–14%, with deviation levels reaching as high as 20% near the stagnation region. Proposed methodology was computationally efficient and is recommended for studies aimed toward local HTC determination.

Heat accumulated at end faces of Nd:YAG laser crystal transferred by air and its effect on temperature field of the crystal

This study presents the actual temperature field of a Nd:YAG diode-end-pumped laser crystal. In contrast to that under an adiabatic condition, a thermal transfer occurs between air and laser crystal. Using an analytical method, the temperature distribution is obtained by solving the three-dimensional heat-conduction equation. The maximum temperature at the center of the incident end face is 382.326 K at pump power of 40 W. The difference between this value and that obtained under an adiabatic condition is 1.042 K. The actual temperature rise is linearly approximately to the pump power. A super-Gaussian order k=2 results in a peak temperature value, which indicates better concentration than those of the other super-Gaussian orders. The focal length of the thermal lens is obviously influenced by the difference in the optical path due to the pump power. The maximum OPD is 23.5 μm and f here is 272.98 mm. This research offers a theoretical foundation to accurately study the thermal problems of solid-state lasers.

Source strength identification problem for the three-dimensional inverse heat conduction equations

ABSTRACTIn this paper, we consider the source strength identification problem for the three-dimensional inverse heat conduction equations. The problem is to determine an unknown heat source strength from the measurement data for a specified location. In this process, the direct problem is solved by applying the Green’s function method. Then, this problem can be converted into a Volterra integral equation of the first kind. Further, the Tikhonov and truncated singular value decomposition regularization methods are developed to identify the unknown source strength based on the discrepancy principle for choosing the regularization parameter. Finally, numerical examples are presented to show the feasibility and efficiency of the proposed method.