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While numerous applied studies have successfully demonstrated the feasibility of unsteady cooling solutions, a consensus has yet to be reached on the local instantaneous conditions that result in heat transfer enhancement. This work aims to experimentally validate a recent analytical solution (on a local time-dependent basis) for the common flow condition of a fully developed incompressible pulsating flow in a uniformly heated vessel. The experimental setup is found to approximate the ideal constant heat flux boundary condition well, especially for the decoupled unsteady scenario where the amplitude of the most significant secondary contributions (capacitance and lateral conduction) amounts to 1.2% and 0.2% of the generated heat flux, respectively. Overall, the experimental measurements for temperature and heat flux oscillations are found to coincide well with a recent analytical solution to the energy equation by the authors. Furthermore, local time-dependent heat flux enhancements and degradations are observed to be qualitatively similar to those of wall shear stress from a previous study, suggesting that the thermal performance is indeed influenced by hydrodynamic behavior.

The Fourier and the hyperbolic heat conduction equations were solved numerically to simulate a frequency-domain thermoreflectance (FDTR) experiment. Numerical solutions enable isolation of pump and probe laser spot size effects and use of realistic boundary conditions. The equations were solved in time domain and the phase lag between the temperature of the transducer (averaged over the probe laser spot) and the modulated pump laser signal was computed for a modulation frequency range of 200 kHz–200 MHz. Numerical calculations showed that extracted values of the thermal conductivity are sensitive to both the pump and probe laser spot sizes, while analytical solutions (based on Hankel transform) cannot isolate the two effects. However, for the same effective (combined) spot size, the two solutions are found to be in excellent agreement. If the substrate (computational domain) is sufficiently large, the far-field boundary conditions were found to have no effect on the computed phase lag. The interface conductance between the transducer and the substrate was found to have some effect on the extracted thermal conductivity. The hyperbolic heat conduction equation yielded almost the same results as the Fourier heat conduction equation for the particular case studied. The numerically extracted thermal conductivity value (best fit) for the silicon substrate considered in this study was found to be about 82–108 W/m/K, depending on the pump and probe laser spot sizes used.

This article proposes the closed-form solution of the generalized non-Fourier model-based bioheat transfer equation (BHTE) in Cylindrical coordinates to understand the thermal behavior of living tissue heated by a pulsed laser. The axisymmetric living tissue exposed to the non-Gaussian temporal profile of laser heating has been considered to investigate the non-Fourier bioheat transfer phenomena. The closed-form solution of the generalized non-Fourier model-based BHTE with time-dependent thermal energy generation has been obtained through the finite integral transform (FIT) technique. The analytical solution was juxtaposed to the corresponding numerical solution in order to determine its reliability. The numerical solution of the aforementioned governing equation has been obtained by the finite volume method (FVM). The results of both analytical and numerical solutions have been verified using results given in published literature. Subsequently, the dual-phase-lag (DPL) model's findings were juxtaposed to those obtained using the hyperbolic and traditional Fourier models. The effect of different parameters like relaxation times corresponding to the temperature gradient and heat flux, metabolic energy generation, and blood perfusion on the resultant temperature distribution inside the axisymmetric living tissue exposed to pulsed laser heating has been discussed. The importance of this study might be found in various applications such as laser-based-photothermal therapy, melting of the surface of metal and alloys by laser heating.

Predictions of coupled conduction-radiation heat transfer processes in periodic composite materials are important for applications of the materials in high-temperature environments. The homogenization method is widely used for the heat conduction equation, but the coupled radiative transfer equation is seldom studied. In this work, the homogenization method is extended to the coupled conduction-radiation heat transfer in composite materials with periodic microscopic structures, in which both the heat conduction equation and the radiative transfer equation are analyzed. Homogenized equations are obtained for the macroscopic heat transfer. Unit cell problems are also derived, which provide the effective coefficients for the homogenized equations and the local temperature and radiation corrections. A second-order asymptotic expansion of the temperature field and a first-order asymptotic expansion of the radiative intensity field are established. A multiscale numerical algorithm is proposed to simulate the coupled conduction-radiation heat transfer in composite materials. According to the numerical examples in this work, compared with the fully resolved simulations, the relative errors of the multiscale model are less than 13% for the temperature and less than 8% for the radiation. The computational time can be reduced from more than 300 h to less than 30 min. Therefore, the proposed multiscale method maintains the accuracy of the simulation and significantly improves the computational efficiency. It can provide both the average temperature and radiation fields for engineering applications and the local information in microstructures of composite materials.

Lots of generalized heat conduction models have been developed in recent decades, such as local thermal nonequilibrium model, phase lagging model, and nonlocal heat conduction model. But no attempt was made to prove which model is better (or worse) than others, or whether there is a certain relationship between these different models. With this inspiration, we establish the nonlocal bioheat transfer equations with lagging time, and the two and three-temperature bioheat transfer equations with considering all the carrier's heat conduction effect are also constructed. Comparing the two (or three)-temperature equation model with the nonlocal bioheat transfer models with lagging time, one may obtain: the lagging time of temperature gradient τtand the nonlocal characteristic length λq in the space derivative items of heat flux have the same effect on heat transfer; when the heat transport occur among N energy carriers with considering the conduction effects of all carries, the heat transfer processes are dependent upon the high-order effect of τqN-1, τtN-1 and λt(2N-1) in nonlocal dual phase lag bioheat transfer model. This phenomenon is very important for biological and medical systems where numerous carriers may exist on the cellular level.

Purpose: The physiological conditions and environment have vital roles in the heat transfer in the human tissues, such as the multilayered human-eye. In this paper, a mathematical model of the human eye subjected to an exponential laser beam concerning the change in blood perfusion, porosity, evaporation rate, and ambient temperatures has been constructed based on non-Fourier heat conduction law. Methods: The human eye has been divided into six layers. Appropriate boundary and interface conditions have been considered. A separable function has been assumed, and the twelve equations have been formulated in matrix form. The solutions have been calculated by using maple 17 software. Results: The results have been shown in figures with different cases. The absolute temperature distribution based on various values of the power density of laser irradiation and relaxation times parameters have been discussed first. The effect of the blood perfusion, porosity, evaporation rate, time, and ambient temperatures have also been discussed. Conclusions: The power density of laser irradiation, blood perfusion, porosity, evaporation rate, time, and ambient temperatures significantly affects the value of the temperature passing through the human eye layers.

Conjugate heat transfer problems generally require a coupled solution of the temperature fields in the fluid and solid domains. Implementing the boundary condition at the surface of the solid using a discrete Green's function (DGF) decouples the solutions. A DGF is determined first considering only the fluid domain with prescribed thermal boundary conditions, then the temperature distribution in the solid is calculated using standard numerical methods. The only compatibility requirement is that the DGF must be specified with the same discretization as the surface of the solid. The method is demonstrated for both steady-state and transient heating of a thin plate with laminar boundary layers flowing over both sides. The resulting set of linear algebraic equations for the steady-state problem or linear ordinary differential equations for the transient problem are easily solved using conventional scientific programming packages. The method converges with nearly second-order accuracy as the discretization resolution is increased.

Analytical modeling of thermal conduction in a multilayer body is of practical importance in several engineering applications such as microelectronics cooling, building insulation, and micro-electromechanical systems. A number of analytical methods have been used in past work to determine multilayer temperature distribution for various boundary conditions. However, there is a lack of work on solving the multilayer thermal conduction problem in the presence of spatially varying convective heat transfer boundary condition. This paper derives the steady-state temperature distribution in a multilayer body with spatially varying convective heat transfer coefficients on both ends of the body. Internal heat generation within each layer and thermal contact resistance between layers are both accounted for. The solution is presented in the form of an eigenfunction series, the coefficients of which are shown to be governed by a set of linear, algebraic equations that can be easily solved. Results are shown to be in good agreement with numerical simulation and with a standard solution for a special case. The model is used to analyze heat transfer for two specific problems of interest involving spatially varying convective heat transfer representative of jet impingement and laminar flow past a flat plate. In addition to enhancing the theoretical understanding of multilayer heat transfer, this work also contributes toward design and optimization of practical engineering systems comprising multilayer bodies.

This paper proposes an efficient numerical method for transient heat conduction in a quasi-periodic structure with nonlinear defects. According to the physical features of transient heat conduction, a quasi-superposition principle for transient heat conduction in a quasi-periodic structure with nonlinear defects is presented, and then a new method is developed to separate the above nonlinear problem to be solved into a linear problem of a perfect periodic structure and nonlinear problems of some small-scale structures with defects. As the scale of nonlinear problem to be solved is significantly reduced and low computational resource is required, outstanding efficiency is achieved. Finally, a numerical example shows that the proposed method is effective and accurate.

The gray medium approximation treating all phonons with an averaged and representative mean-free-path (MFP) is an often used method in analyzing ballistic-diffusive heat conduction at nanoscale. However, whether there exists a reasonable value of the average MFP which effectively represents the entire spectrum of modal MFPs remains unclear. In this paper, phonon Monte Carlo (MC) method is employed to study the effects of the gray medium approximation on ballistic-diffusive heat conduction in silicon films by comparing with dispersion MC simulations. Four typical ways for calculating the average MFP with gray medium approximation are investigated. Three of them are based on the weighted average of the modal MFPs, and the remaining one is based on the weighted average of the reciprocals of the modal MFPs. The first three methods are found to be good at predicting effective thermal conductivity and heat flux distribution, but have difficulties in temperature profile, while the last one performs better for temperature profile than effective thermal conductivity and heat flux distribution. Therefore, none of the average MFPs can accurately characterize all the features of ballistic-diffusive heat conduction for the gray medium approximation. Phonon dispersion has to be considered for the accurate thermal analyses and modeling of ballistic-diffusive heat transport. Our work could be helpful for further understanding of phonon dispersion and more careful use of the gray medium approximation.

A bionic hierarchy generative design algorithm inspired by the leaf vein growth process is presented for the layout design of heat conduction channels. The design domain is discretized based on the element-free Galerkin (EFG) method. The generations of main channels and lateral channels are separated. The effectiveness of the developed bionic hierarchy generative design approach is investigated based on the general “volume-to-point” heat conduction problem.

Applications of impinging jets are wide-ranging from cooling to heating in industrial as well as domestic field. Most of the reported heat transfer distribution data to and from impinging jets have been found from steady-state measurements. This study utilizes the solution to three-dimensional (3D) inverse heat conduction problem to estimate transient temperatures on the impingement side. Then, the temperature gradient is determined near the impingement wall (∼0.01 mm inside) with which transient heat flux is estimated on the impingement side. Instead of steady-state values, transient heat flux and corresponding wall temperatures are utilized in a thin foil technique to find out heat transfer coefficient and reference temperature simultaneously. The scope of the present technique is examined through its application to impinging jets with various configurations such as laminar jet, turbulent jet, hot jet, cold jet, and multiple jets. In all cases, estimations are reasonably close. The application of this inverse technique can be extended to any configuration of jet impingement irrespective of geometry of nozzle (circular/rectangular), the orientation of nozzle (orthogonal/inclined), the temperature of a jet (hot/cold), Reynolds numbers (laminar/turbulent), the nozzle-to-plate spacing (any Z/d), and roughness of the plate surface. The effect of plate thickness on the accuracy of the present technique is also studied. Up to 5 mm thick plates can be used in impinging jet applications without worrying much on accuracy. The use of the present technique significantly reduces the experimental cost and time since it works on transient data of just a few seconds.

Peridynamics for transient heat conduction problems in general anisotropic materials is developed. In order to develop a new peridynamic governing equation for heat conduction problems, the microconductivity (or microdiffusivity), which contains equivalent information as the constitutive equation for classical heat conduction, is determined by directly requiring the resulting peridynamic equation to converge to a classical heat conduction equation for anisotropic materials as the generalized material horizon approaches zero. Therefore, the convergence proof is built into the theory from the perspective of the governing equation. For the application of the newly obtained peridynamic governing equation, a time-dependent three-dimensional (3D) peridynamic heat equation is analytically solved with two types of heat sources, and the results are discussed. These are believed to be the first exact analytical solutions for peridynamic heat conduction.

We theoretically study forced convection heat transfer from a single particle in uniform laminar flows. Asymptotic limits of small and large Peclet numbers Pe are considered. For Pe≪1 (diffusion-dominated regime) and a constant heat flux boundary condition on the surface of the particle, we derive a closed-form expression for the heat transfer coefficient that is valid for arbitrary particle shapes and Reynolds numbers, as long as the flow is incompressible. Remarkably, our formula for the average Nusselt number Nu has an identical form to the one obtained by Brenner for a uniform temperature boundary condition (Chem. Eng. Sci., vol. 18, 1963, pp. 109–122). We also present a framework for calculating the average Nu of axisymmetric and two-dimensional (2D) objects with a constant heat flux surface condition in the limits of Pe≫1 and small or moderate Reynolds numbers. Specific results are presented for the heat transfer from spheroidal particles in Stokes flow.

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.

This paper focuses on using an analytical method to obtain an exact solution for a long cylindrical vessel made of functionally graded materials (FGMs). Heat conduction equations are assumed to be in both radial and circumferential directions. The conduction coefficients are considered as different power-law functions of the radius. The general linear boundary conditions are adopted to make the solution applicable to the full range of problems. The obtained solution is successfully validated. Through solving illustrative test examples, the effects of material constants and boundary conditions on temperature distribution are studied. The obtained formulation can be utilized for tailoring of FGM based on the actual sophisticated thermal boundary conditions in the production process. The current analytical findings can help to manage the temperature distribution in FGMs which is an essential parameter in controlling the thermal stresses.

To estimate the thermal properties from transient data, a model is needed to produce numerical values with sufficient precision. Iterative regression or other estimation procedures must be applied to evaluate the model again and again. From this perspective, infinite or semi-infinite heat conduction problems are a challenge. Since the analytical solution usually contains improper integrals that need to be computed numerically, computer-evaluation speed is a serious issue. To improve the computation speed with precision maintained, an analytical method has been applied to three-dimensional (3D) cylindrical geometries. In this method, the numerical evaluation time is improved by replacing the integral-containing solution by a suitable finite body series solution. The precision of the series solution may be controlled to a high level and the required computer time may be minimized by a suitable choice of the extent of the finite body. The practical applications for 3D geometries include the line-source method for obtaining thermal properties, the estimation of thermal properties by the laser-flash method, and the estimation of aquifer properties or petroleum-field properties from well-test measurements. This paper is an extension of earlier works on one-dimensional (1D) and two-dimensional (2D) cylindrical geometries. In this paper, the computer-evaluation time for the finite geometry 3D solutions is shown to be hundreds of times faster than the infinite or semi-infinite solution with the precision maintained.

Monohydric alcohols have been used as promising phase change materials (PCMs) for low-temperature latent heat storage. However, the heat storage/retrieval rates are limited due to the low thermal conductivity of such alcohols. In this work, nonequilibrium molecular dynamics (NEMD) simulations were performed to study the microscopic heat conduction in example monohydric alcohols, i.e., 1-dodecanol (C12H26O), 1-tetradecanol (C14H30O), and 1-hexadecanol (C16H34O). A simplified ideal crystal model was proposed to exploit the potential for improving the thermal conductivity of monohydric alcohols. The effect of ideal crystalline structures, especially the contribution of the hydroxyl group, on the microscopic heat conduction process was analyzed. The thermal conductivity of the ideal crystals of the various monohydric alcohols was predicted to be more than twice as compared to that of their respective solids. The major thermal resistance in the ideal crystals was found around the molecular interfaces, as a result of the excellent heat conduction performance along the linear molecular chains. The calculated vibrational density of states (VDOS) and interfacial heat transfer were then investigated. When the interfaces are surrounded by hydroxyl groups as walls, strong hydrogen bond (HB) interactions were observed. The interfacial heat transfer coefficient of the ideal crystalline structures of 1-tetradecanol was found to reach up to ∼735.6 MW/m2 W. It was elucidated that the high interfacial heat transfer rate is clearly related to the stronger intermolecular interactions.

The core of high-temperature gas-cooled reactor is a dense pebble bed of random packing filled with monosized fuel spheres. Subcell radiation model (SCM) is a generic analytical approach to calculate effective thermal conductivity (ETC) of thermal radiation. For the packed bed of monosized spheres operated in various conditions, it is proven that the SCM is still applicable in the particle size ranges of 1.2–60 mm and temperature ranges of 0–1200 °C. Based on the SCM, radiation-to-conduction ratio ξ is presented and radiation becomes an essential part at ξ>0.1 for the accurate evaluation. For the beds of nonoverlapping clumped-sphere particles, the model combining with discrete element method (DEM) and SCM is presented to study the heat transfer behaviors, including effects of particle shape, emissivity distribution and pebble flow with transient heat transfer. For the experimental nuclear pebble beds, the results of SCM are in good agreement with the empirical correlation and accord well with the experimental data under high temperature range.

Accounting for the fact that thermal conductivity of fluid is much less than the thermal conductivity of solid in most of the porous medium-related applications, this study applies perturbation approach in analyzing forced convection through a parallel plate channel under local thermal nonequilibirum (LTNE) condition by denoting the thermal conductivity ratio of fluid to solid as the small parameter, suggesting leading order solutions to solve the two-equation energy model, by incorporating Darcy model and Brinkman model for large porous medium shape factor, respectively, in the presence of heat generation in both fluid and solid. This study provides important fluid temperatures, solid temperatures, and heat transfer coefficient approximations, which enables further analysis on the fluid and solid temperature gradient at the boundary and hence delineate the roles of thermal conductivities and interfacial heat transfer in LNTE mode. The results signify competition between the heat conduction from the wall through fluid conduction and interfacial heat transfer from solid to fluid in the thermal boundary layer. The effect of thermal boundary layer is intensified with the attendant increase in porous medium shape factor and heat generation in solid. The results for Brinkman model also establish conditions for temperature bifurcations to take place whereby in such cases, an increase in viscous dissipation in fluid attributes to the detachment of thermal boundary layer as the porous medium shape factor, S decreases. The phenomenon caused by insufficient convection rate to overcome viscous dissipation bears much resemblance to the separation point in the momentum boundary layer.

Precise modeling of thermoelastic cracks remains challenging due to the fact that both heat flux and stress fields have singularity issue. In the previous studies, the first author proposed different types of symplectic analytical singular element (SASE) for thermal conduction and stress analysis of cracks. It has been demonstrated that these crack-tip elements of which the interior fields are defined by analytical solutions are highly accurate and efficient. However, the thermal mechanical coupling problem of crack cannot be treated with the existing SASEs. The main difficulty is that the analytical solution of the crack problem considering arbitrary temperature distribution is not available. Approximate solution may lead to significant numerical instabilities. Moreover, the construction of a crack-tip singular element for both thermal conduction and stress analysis is complicated and requires more efforts. In this study, the governing symplectic dual equation of thermoelastic crack is restudied. The analytical solution considering arbitrary temperature distribution is obtained in close form which, to the best of the authors' knowledge, has not been found before. Then, the finite element formulation of a new SASE for thermal-mechanical fracture analysis is derived analytically through a variational approach. A two-step analysis procedure is proposed to calculate the mixed mode thermal stress intensity factors (TSIFs)) and the analysis can be done on a fixed finite element mesh. Mesh refinement around the crack tip is unnecessary, and the mixed-mode TSIFs can be solved accurately without any postprocessing.

Solidification dynamics are crucial for determining microstructure development in additively manufactured parts. Multiphysics models based on finite element or finite volume methods may help gain insight for complicated phenomena such as fluid flow, keyholing, and porosity but are too computationally expensive to use for simulating actual builds. Recent analytic and semi-analytic solutions for moving heat sources in a semi-infinite three-dimensional space provide a way to accurately estimate the solidification conditions for entire builds. The downside to these methods is that, unlike finite element or finite volume methods, they cannot use the temperature distribution of the previous timesteps to march the solution forward in time. This paper provides the mathematical formulation and implementation of a forward time stepping (FTS) approach to an existing semi-analytic solution. The speed and accuracy of the two methods are then compared for various scan patterns. The result is that, for spot melts, the forward time-stepping model provides improvements in both speed and accuracy. This is especially true for longer simulations, where the simulation can be orders of magnitude faster. The longest simulation analyzed in this paper was roughly 30× faster when using the forward time-stepping model versus the straightforward implementation of the semi-analytic solution.

A numerical study of magneto-hydrodynamic mixed convection in a cavity has been conducted to investigate the influence of magnetic field on integrated flux of thermal energy, linear momentum, and kinetic energy. Shear force through lid motion establishes the forced convection effect and buoyancy force due to differential heating of the moving lid and the stationary interface ensures the natural convection phenomenon. Additionally, conduction through the solid slab with prescribed temperature at the outer surface attached to the cavity induces conjugate heat transfer. Further, the top and bottom walls throughout the domain are kept insulated and a uniform horizontal magnetic field is applied on the interface toward left. Fluid flow and heat transfer characteristics are examined for a range of Hartmann number (Ha): 0, 10, 50, and 120 at fixed values of Reynolds number, Grashof number, and Prandtl number of 300, 9 × 104 and 0.71, respectively. The result shows that the transport of heat in the near wall regions of the fluid domain is primarily governed by diffusion, whereas advection appears stronger in the central region of the cavity. Increase in magnetic field strength from Ha = 0 to 120 gradually suppresses the recirculating structure of the flow signifying a reduction in advective strength as depicted by the decrease in the value of total integrated heat flux from 25.15×10-3 to 6.0×10-3. The reduction in heat flux, momentum fluxes, and kinetic energy fluxes with increase in magnetic field has been well correlated in the range of 0≤Ha≤120.

In this study, an analytical solution is proposed for the problem of transient anisotropic conductive heat transfer in composite cylindrical shells. The composite shells are considered to have directional heat transfer properties, which is due to the existence of fibers which can be winded in any direction. The composite shells usually show high conductivity in the direction parallel to fiber direction and low conductivity in other two orthogonal directions. To solve the heat transfer partial differential equation, finite Fourier transform and separation of variables method are used. The present solution is used to find the temperature distribution in a composite cylindrical vessel for which the composite material is graphite/epoxy and the vessel is prone to an external heat flux and also ambient flow. The analytical solution is verified perfectly by the data obtained from a second-order finite difference solution. The solution is used to investigate the effects of values of fiber angle and material conductivity coefficients on temperature distribution of the composite cylindrical vessel. The results show the important role of fiber angle values on the temperature distribution of vessel.

We consider conjugate forced-convection heat transfer in a rectangular duct. Heat is exchanged through the isothermal base of the duct, i.e., the area comprised of the wetted portion of its base and the roots of its two side walls, which are extended surfaces within which conduction is three-dimensional. The opposite side of the duct is covered by an adiabatic shroud, and the external faces of the side walls are adiabatic. The flow is steady, laminar, and simultaneously developing, and the fluid and extended surfaces have constant thermophysical properties. Prescribed are the width of the wetted portion of the base, the length of the duct, and the thickness of the extended surfaces, all three of them nondimensionalized by the hydraulic diameter of the duct, and, additionally, the Reynolds number of the flow, the Prandtl number of the fluid, and the fluid-to-extended surface thermal conductivity ratio. Our conjugate Nusselt number results provide the local one along the extended surfaces, the local transversely averaged one over the isothermal base of the duct, the average of the latter in the streamwise direction as a function of distance from the inlet of the domain, and the average one over the whole area of the isothermal base. The results show that for prescribed thermal conductivity ratio and Reynolds and Prandtl numbers, there exists an optimal combination of the dimensionless width of the wetted portion of the base, duct length, and extended surface thickness that maximize the heat transfer per unit area from the isothermal base.

Radiative and conductive heat transfer is fairly important in the nuclear pebble bed. A continuum model is proposed here to derive the effective thermal conductivity (ETC) of pebble bed. It is a physics-based equation determined by the temperature, number density, heat transfer coefficient, and the radial distribution function (RDF). Based on a concept of continuum, this model considers the conduction and thermal radiation in nuclear pebble bed through a uniform framework and the results are in good agreement with the existing model and correlations. It indicates that the local temperature in the radiation case without internal heat sources is determined by all possible surrounding pebbles weighted by a radiative kernel function. The discrete element method (DEM) packing results are in good agreement with the solution of the continuum model. Both the conductive and radiative continuum models converge to the heat conduction in continuum mechanics at size factor μ ≪ 1.

In this research paper, the cooling process of an impingement cooled spur gear is examined by means of an analytical model. The process is modeled as a coolant film, which is flung off a rotating gear tooth flank by centrifugal forces. During the process, heat is transferred from the isothermal gear tooth flank to the coolant film. With a numerical solution to the analytical model, a formulation for the transient local Nusselt number is derived. The evaluation of the numerical solution revealed that the heat transfer is dominated by heat conduction in the coolant film. The heat transfer process ends when the thermal capacity of the coolant film is reached. The transient Nusselt number is used to derive a time-averaged and a global heat transfer coefficient. Furthermore, the influence of the initial coolant film height is examined by using a modified version of the analytical model. The global heat transfer coefficient decreases toward smaller initial cooling film heights. The analytical model is then extended to include the temperature dependency of the viscosity of the coolant. A viscosity that decreases with increasing temperature yields a moderate decrease in heat transfer. A discussion is presented regarding the applicability of the analytical model toward impingement cooled spur gears. The effect of the simplifications made in the derivation of the analytical model is outlined and assessed with regard to the heat transfer mechanism.

A generalized solution for a two-dimensional (2D) transient heat conduction problem with a partial-heating boundary condition in rectangular coordinates is developed. The solution accommodates three kinds of boundary conditions: prescribed temperature, prescribed heat flux and convective. Also, the possibility of combining prescribed heat flux and convective heating/cooling on the same boundary is addressed. The means of dealing with these conditions involves adjusting the convection coefficient. Large convective coefficients such as 1010 effectively produce a prescribed-temperature boundary condition and small ones such as 10−10 produce an insulated boundary condition. This paper also presents three different methods to develop the computationally difficult steady-state component of the solution, as separation of variables (SOV) can be less efficient at the heated surface and another method (non-SOV) is more efficient there. Then, the use of the complementary transient part of the solution at early times is presented as a unique way to compute the steady-state solution. The solution method builds upon previous work done in generating analytical solutions in 2D problems with partial heating. But the generalized solution proposed here contains the possibility of hundreds or even thousands of individual solutions. An indexed numbering system is used in order to highlight these individual solutions. Heating along a variable length on the nonhomogeneous boundary is featured as part of the geometry and examples of the solution output are included in the results.

We develop a vapor chamber integrated with a microelectronic packaging substrate and characterize its heat transfer performance. A prototype of vapor chamber integrated printed circuit board (PCB) is fabricated through successful completion of the following tasks: patterning copper micropillar wick structures on PCB, mechanical design and fabrication of condenser, device sealing, and device vacuuming and charging with working fluid. Two prototype vapor chambers with distinct micropillar array designs are fabricated, and their thermal performance tested under various heat inputs supplied from a 2 mm × 2 mm heat source. Thermal performance of the device improves with heat inputs, with the maximum performance of ∼20% over copper plated PCB with the same thickness. A three-dimensional computational fluid dynamics/heat transfer (CFD/HT) numerical model of the vapor chamber, coupled with the conduction model of the packaging substrate is developed, and the results are compared with test data.

The Laplace transform (LT) is a widely used methodology for analytical solutions of dual phase lag (DPL) heat conduction problems with consistent DPL boundary conditions (BCs). However, the inversion of LT requires a series summation with large number of terms for reasonably converged solution, thereby, increasing computational cost. In this work, an alternative approach is proposed for this inversion which is valid only for time-periodic BCs. In this approach, an approximate convolution integral is used to get an analytical closed-form solution for sinusoidal BCs (which is obviously free of numerical inversion or series summation). The ease of implementation and simplicity of the proposed alternative LT approach is demonstrated through illustrative examples for different kind of sinusoidal BCs. It is noted that the solution has very small error only during the very short initial transient and is (almost) exact for longer time. Moreover, it is seen from the illustrative examples that for high frequency periodic BCs the Fourier and DPL model give quite different results; however, for low frequency BCs the results are almost identical. For nonsinusoidal periodic function as BCs, Fourier series expansion of the function in time can be obtained and then present approach can be used for each term of the series. An illustrative example with a triangular periodic wave as one of the BC is solved and the error with different number of terms in the expansion is shown. It is observed that quite accurate solutions can be obtained with a fewer number of terms.

Stagnation-point flow toward a stretching sheet with slip effects has been investigated. Unlike most classical works, Cattaneo–Christov heat flux model is utilized for the formulation of the energy equation instead of Fourier's law of heat conduction. A similarity transformation technique is adopted to reduce partial differential equations into a system of nonlinear ordinary differential equations. Numerical solutions are obtained by using shooting method to explore the features of various parameters for the velocity and temperature distributions. The obtained results are graphically presented and analyzed. It is found that fluid temperature has a converse relationship with the thermal relaxation time. A comparison of Cattaneo–Christov heat flux model and Fourier's law is also presented.

The rate of heat conduction (or mass transfer by diffusion) from a cylindrical or a spherical particle confined between two walls is determined as a function of the position and the radius of the particle. It is shown that the appropriate Green's function can be determined using the method of images even when the resulting series is divergent with the help of Shanks transformation. Asymptotic expansions for small particle radius compared to the distance between the walls are combined with the expressions for the case in which the gap between the particle and one of the walls is small compared to the particle radius to provide formulas that are surprisingly accurate for estimating the rate of heat transfer for the entire range of parameters that include the radius and the position of the particle. Results are also presented for the thermal dipole induced by a spherical or a cylindrical particle placed between two walls with unequal temperatures and these are used to predict the effective thermal conductivity of thin composite films containing spherical or cylindrical particles.

Heat transfer properties of two expanded polystyrene (EPS) samples of similar density, one without (white) and one with graphite opacifier particles (gray), are compared. Tomographic scans are used to obtain cell sizes of the foams. Using established models for closed-cell polymer foams, the extinction coefficient and the effective thermal conductivity are obtained. The effect of opacifiers is modeled using (1) an effective refractive index for the polystyrene walls within a cell model for the EPS and (2) a superposition of extinction due to a particle cloud upon extinction predicted by the cell model, where particles are modeled as oblate spheroids, or equivalent volume, surface, or hydraulic diameter spheres. Modeled effective conductivities are compared with measurements done on a guarded hot-plate apparatus at sample mean temperatures in the range from 0 °C to 40 °C. Typically, cells of the gray EPS are about 40% larger than those of the white EPS and the cell walls in the gray EPS are thicker. The refractive index mixing model and the model with graphite opacifier particles as oblate spheroids overpredict extinction, however, the mean error in the effective conductivity predicted by the oblate spheroids model is only 2.7%. Equivalent volume/surface sphere models underpredict extinction, but still yield a low mean error in effective conductivity of around 4%. While the oblate spheroids model has a lower mean error, the computationally less expensive equivalent volume or equivalent surface models can also be recommended to model the inclusions.

Sensitivity information is often of interest in engineering applications (e.g., gradient-based optimization). Heat transfer problems frequently involve complicated geometries for which exact solutions cannot be easily derived. As such, it is common to resort to numerical solution methods such as the finite element method. The semi-analytic complex variable method (SACVM) is an accurate and efficient approach to computing sensitivities within a finite element framework. The method is introduced and a derivation is provided along with a detailed description of the algorithm which requires very minor changes to the analysis code. Three benchmark problems in steady-state heat transfer are studied including a nonlinear problem, an inverse shape determination problem, and a reliability analysis problem. It is shown that the SACVM is superior to the other methods considered in terms of computation time and sensitivity to perturbation size.

There is a substantial and growing body of literature which solves Laplace's equation governing the velocity field for a linear-shear flow of liquid in the unwetted (Cassie) state over a superhydrophobic surface. Usually, no-slip and shear-free boundary conditions are applied at liquid–solid interfaces and liquid–gas ones (menisci), respectively. When the menisci are curved, the liquid is said to flow over a “bubble mattress.” We show that the dimensionless apparent hydrodynamic slip length available from studies of such surfaces is equivalent to (i) the dimensionless spreading resistance for a flat, isothermal heat source flanked by arc-shaped adiabatic boundaries and (ii) the dimensionless thermal contact resistance between symmetric mating surfaces with flat contacts flanked by arc-shaped adiabatic boundaries. This is important because real surfaces are rough rather than smooth. Furthermore, we demonstrate that this observation provides a significant source of new and explicit results on spreading and contact resistances. Significantly, the results presented accommodate arbitrary solid-to-solid contact fraction and arc geometry in the contact resistance problem for the first time. We also provide formulae for the case when each period window includes a finite number of no-slip (or isothermal) and shear free (or adiabatic) regions and extend them to the case when the latter are weakly curved. Finally, we discuss other areas of mathematical physics to which our results are directly relevant.

Invisibility has recently been achieved in optics, electromagnetics, acoustics, thermotics, fluid mechanics, and quantum mechanics; it was realized through a properly designed cloak structure with unconventional (anisotropic, inhomogeneous, and singular) material parameters, which limit practical applications. Here, we show, directly from the solution of Laplace's equation, that two or more conventional (isotropic, homogeneous, and nonsingular) materials can be made thermally invisible by tailoring the many-particle local-field effects. Our many-particle thermal invisibility essentially serves as a new class of invisibility with a mechanism fundamentally differing from that of the prevailing cloaking-type invisibility. We confirm it in simulation and experiment. As an application, the concept of many-particle thermal invisibility helps us propose a class of many-particle thermal diodes: the diodes allow heat conduction from one direction with invisibility, but prohibit the heat conduction from the inverse direction with visibility. This work reveals a different mechanism for thermal camouflage and thermal rectification by using composites, and it also suggests that besides thermotics, many-particle local-field effects can be a convenient and effective mechanism for achieving similar controls in other fields, e.g., optics, electromagnetics, acoustics, and fluid mechanics.

We recently showed theoretically that the infinite singularity of the thermal cloak designed by transformation thermodynamics could be eliminated by a new method—the path design of the heat flux without any approximation. In this paper, we present the experimental evidence of such a new strategy of thermal cloak, that is, a truly singularity-free thermal cloak. We fabricate such a transient thermal cloak device without using extreme material parameters. The experimental results show fully controlled, transient cloaking behavior, which are perfectly consistent with the theoretical derivations and simulated results. Since one can flexibly design the path of heat flux in the cloak, it has the large degree-of-freedom to construct thermal cloaks with the specific distributions of material parameters. The new method provides a new blue print for the transient thermal protection of a specific target.

Time-dependent participating volume-to-surface ratio, V(t)/A, is used to adjust the semi-infinite (SI) solid solutions to the radial systems. In cylinders and spheres, the present “radial” SI sold model extends the domain of the planar model from δ ≪ R to δ ≈ R (δ is transient penetration depth and R is radius). The corresponding increase in the time span is from 0

An inverse radiation-conduction analysis is performed for simultaneous estimation of the thermal properties in an absorbing, emitting, and linear-anisotropically scattering medium with spatially variable refractive index. The discrete ordinates method in conjugation with finite volume method is adopted to solve the direct problem. The conjugate gradient method (CGM) is employed to simultaneously estimate the conduction-radiation parameter, optical thickness, single scattering albedo, scattering phase function, and the wall emissivities from the knowledge of the exit radiation intensities over the boundaries. The effects of these parameters and the measurement errors on the precision of the inverse analysis are investigated. Results show that the proposed inverse approach can successfully retrieve the unknown parameters for different refractive index profiles.

Modeling of steady-state thermal conduction for crack and v-notch in anisotropic material remains challenging. Conventional numerical methods could bring significant error and the analytical solution should be used to improve the accuracy. In this study, crack and v-notch in anisotropic material are studied. The analytical symplectic eigen solutions are obtained for the first time and used to construct a new symplectic analytical singular element (SASE). The shape functions of the SASE are defined by the obtained eigen solutions (including higher order terms), hence the temperature as well as heat flux fields around the crack/notch tip can be described accurately. The formulation of the stiffness matrix of the SASE is then derived based on a variational principle with two kinds of variables. The nodal variable is transformed into temperature such that the proposed SASE can be connected with conventional finite elements (FE) directly without transition element. Structures of complex geometries and complicated boundary conditions can be analyzed numerically. The generalized flux intensity factors (GFIFs) can be calculated directly without any postprocessing. A few numerical examples are worked out and it is proven that the proposed method is effective for the discussed problem, and the structure can be analyzed accurately and efficiently.

This study introduces an analysis of high-order dual-phase-lag (DPL) heat transfer equation and its thermodynamic consistency. The frameworks of extended irreversible thermodynamics (EIT) and traditional second law are employed to investigate the compatibility of DPL model by evaluating the entropy production rates (EPR). Applying an analytical approach showed that both the first- and second-order approximations of the DPL model are compatible with the traditional second law of thermodynamics under certain circumstances. If the heat flux is the cause of temperature gradient in the medium (over diffused or flux precedence (FP) heat flow), the DPL model is compatible with the traditional second law without any constraints. Otherwise, when the temperature gradient is the cause of heat flux (gradient precedence (GP) heat flow), the conditions of stable solution of the DPL heat transfer equation should be considered to obtain compatible solution with the local equilibrium thermodynamics. Finally, an insight inspection has been presented to declare precisely the influence of high-order terms on the EPRs.

This paper describes a thermal homogenization approach to the application of a multiscale formulation for heat conduction with radiation problems in a porous material. The suggested formulation enables to evaluate the effective macroscopic thermal conductivity, based on the microscopic properties such as porosity, and can also provide the microscopic radiosity heat flux, based on the macroscopic temperature gradient field. This is done by scaling up and down between the microscopic and macroscopic models according to the suggested methodology. The proposed methodology involves a new iterative upscaling procedure, which uses heat conduction at macroscopic problem and heat transfer by conduction and radiation at microscopic problem. This reduces the required computational time, while maintaining the required level of accuracy. The suggested multiscale formulation has been verified by comparing its results with reference finite element (FE) solutions of porous (filled with air) materials examples; the results shows excellent agreement (up to 5% discrepancy) with reference solutions. The efficiency of the suggested formulation was demonstrated by solving a full-scale engineering transient problem.

In the microelectronics industry, the multilayered structures are found extensively where the microelectronic device/system is manufactured as a compound system of different materials. Recently, a variety of new materials have emerged in the microelectronics industry with properties superior to Silicon, enabling new devices with extreme performance. Such materials include β-Gallium-oxide (β-Ga2O3), and black phosphorus (BP), which are acknowledged to have anisotropic thermal conductivity tensors. In many of these devices, thermal issues due to self-heating are a problem that affects the performance, efficiency, and reliability of the devices. Analytical solutions to the heat conduction equation in such devices with anisotropic thermal conductivity tensor offer significant computational savings over numerical methods. In this paper, general analytical solutions for the temperature distribution and the thermal resistance of a multilayered orthotropic system are obtained. The system is considered as a multilayered three-dimensional (3D) flux channel consisting of N-layers with different thermal conductivities in the three spatial directions in each layer. A single eccentric heat source is considered in the source plane while a uniform heat transfer coefficient is considered along the sink plane. The solutions account for the effect of interfacial conductance between the layers and for considering multiple eccentric heat sources in the source plane. For validation purposes, the analytical results are compared with numerical solution results obtained by solving the problem with the finite element method (FEM) using the ANSYS commercial software package.

A desirable feature of any parameter estimation method is to obtain as much information as possible with one experiment. However, achieving multiple objectives with one experiment is often not possible. In the field of thermal parameter estimation, a determination of thermal conductivity, volumetric heat capacity, heat addition rate, surface emissivity, and convection coefficient may be desired from a set of temperature measurements in an experiment where a radiant heat source is used. It would not be possible to determine all of these parameters from such an experiment; more information would be needed. The work presented in the present research shows how thermal parameters can be determined from temperature measurements using complementary experiments where the same material is tested more than once using a different geometry or heating configuration in each experiment. The method of ordinary least squares is used in order to fit a mathematical model to a temperature history in each case. Several examples are provided using one-dimensional conduction experiments, with some having a planar geometry and some having a cylindrical geometry. The parameters of interest in these examples are thermal conductivity and volumetric heat capacity. Sometimes, both of these parameters cannot be determined simultaneously from one experiment but utilizing two complementary experiments may allow each of the parameters to be determined. An examination of confidence regions is an important topic in parameter estimation and this aspect of the procedure is addressed in the present work. A method is presented as part of the current research by which confidence regions can be found for results from a single analysis of multiple experiments.

This paper presents an analytical solution for the Joule heating problem of a segmented wire made of two materials with different properties and suspended as a bridge across two fixed ends. The paper first establishes the one-dimensional (1D) governing equations of the steady-state temperature distribution along the wire with the consideration of heat conduction and free-heat convection phenomena. The temperature coefficient of resistance of the constructing materials and the dimension of the each segmented wires were also taken into account to obtain analytical solution of the temperature. COMSOL numerical solutions were also obtained for initial validation. Experimental studies were carried out using copper and nichrome wires, where the temperature distribution was monitored using an IR thermal camera. The data showed a good agreement between experimental data and the analytical data, validating our model for the design and development of thermal sensors based on multisegmented structures.

If a steady thermal boundary layer is sufficiently thick, wall heat fluxes and associated convective heat transfer coefficients can be directly calculated from measured temperature distributions taken within it using a traversing thermocouple probe. The boundary layer can be laminar, turbulent, or transitional and on a surface of arbitrary surface temperature distribution and geometry. Herein, this technique is presented and validated in a steady, turbulent, two-dimensional boundary layer on a flat, uniform-heat-flux wall. Care is taken to properly account for radiation from the wall and conduction within the thermocouple wire. In the same setting, heat flux measurements are made for verification purposes using an energy balance on a segment of the test wall carefully designed to minimize and include radiation and conduction effects. Heat flux values measured by the boundary layer measurement technique and by the energy balance measurement agree to within 4.4% and the difference between the two lie completely within their respective measurement uncertainties of 5.74% and 0.6%.

The effect of staggered-array water impinging jets on boiling heat transfer was investigated by a simultaneous measurement between boiling visualization and heat transfer characteristics. The boiling phenomena of staggered-array impinging jets on hot steel plate were visualized by 4K UHD video camera. The surface temperature and heat flux on hot steel plate was determined by solving 2-D inverse heat conduction problem, which was measured by the flat-plate heat flux gauge. The experiment was made at jet Reynolds number of Re = 5,000 and the jet-to-jet distance of staggered-array jets of S/Dn = 10. Complex flow interaction of staggered-array impinging jets exhibited hexagonal flow pattern like as honey-comb. The calculated surface heat transfer profiles show a good agreement with the corresponding boiling visualization. The peak of heat flux accords with the location which nucleate boiling is occurred at. In early stage, the positions of maximum heat flux locate at the stagnation point of each jet as the relatively low surface temperature is shown at their positions. At the elapsed time of 10 s, the flat shape of heat flux profile is formed in the hexagonal area where the interacting flow uniformly cools down the wetted surface. After that, the wetted area continuously enlarges with time and the maximum heat flux is observed at its peripheral. These results point out that the flow interaction of staggered-array jets effectively cools down the closer area around jets and also show an expansion of nucleate boiling and suppression of film boiling during water jet cooling on hot steel plate. [This work was supported by the KETEP grant funded by the Ministry of Trade, Industry & Energy, Korea (Grant No. 20142010102910).]

Heat exchangers are important components in many engineering applications. This paper proposes a numerical two-phase heat exchanger model with simultaneous heat transfer and pressure drop calculations. The presented model provides a modeling framework compatible with numerous different correlations for both single- and two-phase flow of pure fluids and fluid mixtures. Furthermore, it considers nonconstant fluid properties as well as longitudinal heat conduction and parasitic heat loads, which is particularly relevant in mixed refrigerant cycles for cooling of low-temperature applications. The governing equations are derived and the solution strategy is presented, followed by the model validation against analytical solutions in the corresponding limits. Finally, an exemplary heat exchanger is analyzed using both homogeneous and separated flow models, and the results are compared with experimental data from literature.