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### Research Papers: Conduction

Marepalli Prabhakar, Murthy Jayathi Y., Qiu Bo, et al. Quantifying Uncertainty in Multiscale Heat Conduction Calculations J. Heat Transfer 136, 111301 (2014) (10 pages);   Paper No: HT-13-1190;   doi:10.1115/1.4027348

In recent years, there has been interest in employing atomistic computations to inform macroscale thermal transport analyses. In heat conduction simulations in semiconductors and dielectrics, for example, classical molecular dynamics (MD) is used to compute phonon relaxation times, from which material thermal conductivity may be inferred and used at the macroscale. A drawback of this method is the noise associated with MD simulation (here after referred to as MD noise), which is generated due to the possibility of multiple initial configurations corresponding to the same system temperature. When MD is used to compute phonon relaxation times, the spread may be as high as 20%. In this work, we propose a method to quantify the uncertainty in thermal conductivity computations due to MD noise, and its effect on the computation of the temperature distribution in heat conduction simulations. Bayesian inference is used to construct a probabilistic surrogate model for thermal conductivity as a function of temperature, accounting for the statistical spread in MD relaxation times. The surrogate model is used in probabilistic computations of the temperature field in macroscale Fourier conduction simulations. These simulations yield probability density functions (PDFs) of the spatial temperature distribution resulting from the PDFs of thermal conductivity. To allay the cost of probabilistic computations, a stochastic collocation technique based on generalized polynomial chaos (gPC) is used to construct a response surface for the variation of temperature (at each physical location in the domain) as a function of the random variables in the thermal conductivity model. Results are presented for the spatial variation of the probability density function of temperature as a function of spatial location in a typical heat conduction problem to establish the viability of the method.

Campo Antonio, Garza José. Transversal Method of Lines for Unsteady Heat Conduction With Uniform Surface Heat Flux J. Heat Transfer 136, 111302 (2014) (7 pages);   Paper No: HT-13-1580;   doi:10.1115/1.4028082

The transversal method of lines (TMOL) is a general hybrid technique for determining approximate, semi-analytic solutions of parabolic partial differential equations. When applied to a one-dimensional (1D) parabolic partial differential equation, TMOL engenders a sequence of adjoint second-order ordinary differential equations, where in the space coordinate is the independent variable and the time appears as an embedded parameter. Essentially, the adjoint second-order ordinary differential equations that result are of quasi-stationary nature, and depending on the coordinate system may have constant or variable coefficients. In this work, TMOL is applied to the unsteady 1D heat equation in simple bodies (large plate, long cylinder, and sphere) with temperature-invariant thermophysical properties, constant initial temperature and uniform heat flux at the surface. In engineering applications, the surface heat flux is customarily provided by electrical heating or radiative heating. Using the first adjoint quasi-stationary heat equation for each simple body with one time jump, it is demonstrated that approximate, semi-analytic TMOL temperature solutions with good quality are easily obtainable, regardless of time. As a consequence, usage of the more involved second adjoint quasi-stationary heat equation accounting for two consecutive time jumps come to be unnecessary.

Simavilla David Nieto, Venerus David C. Investigation of Anisotropic Thermal Conductivity in Polymers Using Infrared Thermography J. Heat Transfer 136, 111303 (2014) (8 pages);   Paper No: HT-14-1041;   doi:10.1115/1.4028324

A new experimental method based on infrared thermography (IRT) is developed to study deformation-induced anisotropic thermal conductivity in polymers. An analytic solution for the temperature field of samples heated by a point source is utilized with a robust fitting procedure allowing for quantitative measurement of two components of the normalized thermal conductivity tensor of uniaxially stretched samples. In order to validate the method, we compare measurements on a cross-linked polybutadiene network with those obtained from a previously developed technique based on forced Rayleigh scattering (FRS). We find excellent agreement between the two techniques. Uncertainty in the measurements using IRT method is estimated to be about 2–5%. The accuracy of the method and its potential application to nontransparent materials make it a good alternative to extend current research on anisotropic thermal transport in polymeric materials.

### Research Papers: Forced Convection

Lin Jian-Zhong, Xia Yi, Ku Xiao-Ke. Pressure Drop and Heat Transfer of Nanofluid in Turbulent Pipe Flow Considering Particle Coagulation and Breakage J. Heat Transfer 136, 111701 (2014) (9 pages);   Paper No: HT-14-1273;   doi:10.1115/1.4028325

Numerical simulations of Al2O3/water nanofluid in turbulent pipe flow are performed with considering the particle convection, diffusion, coagulation, and breakage. The distributions of particle volume concentration, the friction factor, and heat transfer characteristics are obtained. The results show that the initial uniform distributions of particle volume concentration become nonuniform, and increase from the pipe wall to the center. The nonuniformity becomes significant along the flow direction from the entrance and attains a steady state gradually. Friction factors increase with the increase of particle volume concentrations and particle diameter, and with the decrease of Reynolds number. The friction factors increase remarkably at lower volume concentration, while slightly at higher volume concentration. The presence of nanoparticles provides higher heat transfer than pure water. The Nusselt number of nanofluids increases with increasing Reynolds number, particle volume concentration, and particle diameter. The rate increase in Nusselt number at lower particle volume concentration is more than that at higher concentration. For a fixed particle volume concentration, the friction factor is smaller while the Nusselt number is larger for the case with uniform distribution of particle volume concentration than that with nonuniform distribution. In order to effectively enhance the heat transfer using nanofluid and simultaneously save energy, it is necessary to make the particle distribution more uniform. Finally, the expressions of friction factor and Nusselt number as a function of particle volume concentration, particle diameter and Reynolds number are derived based on the numerical data.

### Research Papers: Heat Exchangers

Naterer G. F., Dincer I. Transient Temperature Response of Variable Flow Heat Exchangers in a Marnoch Heat Engine J. Heat Transfer 136, 111801 (2014) (8 pages);   Paper No: HT-13-1028;   doi:10.1115/1.4028176

Within a Marnoch heat engine (MHE), a water/glycol mixture transfers heat from the heat source into a set of variable flow heat exchangers and removes heat from adjoining cold heat exchangers. The compressed dry air is used as the working medium in this heat engine. The MHE has four shell and tube heat exchangers, which operate transient and variable flow conditions. A new transient heat transfer model is developed to predict this transient behavior of the heat exchangers for different flow regimes and temperatures. The results from the model are validated against experimental results from an MHE prototype. The heat transfer model shows 85% agreement with measured data from the MHE prototype for the individual heat exchangers. This model can be used for similar shell and tube heat exchangers with straight or U-shaped tubes. The heat transfer model predicts the gas temperature on the shell side, when a step change is imposed on the liquid entering the tubes.

### Research Papers: Jets, Wakes, and Impingment Cooling

Zhang Chuanjie, Xu Guoqiang, Li Haiwang, et al. The Effect of Weak Crossflow on the Heat Transfer Characteristics of Short-Distance Impinging Cooling J. Heat Transfer 136, 112201 (2014) (11 pages);   Paper No: HT-13-1542;   doi:10.1115/1.4028081

This paper numerically and experimentally investigated the effect of weak crossflow on the heat transfer characteristics of a short-distance impinging jet. The Reynolds number of the impinging jet ranged from 6000 to 15,000, and the mass velocity ratio (M) between the crossflow and the jet varied from 0 to 0.15. The separation distance (H) between the exit of the jet nozzle and the impingement surface equals to the exit diameter (D) of the impinging jet. In the experiments, the temperature distribution on the impingement target surface was measured using a transient liquid crystal method. In the numerical simulation, a multiblock hexahedral mesh was applied to discrete the computational domain, and a commercial CFD package (Ansys cfx-12.0) with a standard $k-ɛ$ turbulence model was used for computation. It was found that compared to the impinging cooling without crossflow, the heat transfer characteristics near the impinging stagnation point remained almost constant. At the same time, the presence of crossflow decreased the heat transfer rate in the upstream region of the impinging stagnation point, while increased that in the downstream of the impinging stagnation point. Taken together, crossflow has a complex influence on the impinging cooling, which is highly dependent on the mass velocity ratio between the crossflow and the jet.

Nasif G., Barron R. M., Balachandar R. Heat Transfer due to an Impinging Jet in a Confined Space J. Heat Transfer 136, 112202 (2014) (10 pages);   Paper No: HT-13-1337;   doi:10.1115/1.4028242

A numerical investigation using unsteady three-dimensional Reynolds-averaged Navier–Stokes (RANS) equations with the k-ω SST (shear stress transport) turbulent model was conducted to determine the flow and thermal characteristics of an unsubmerged axisymmetric oil jet in air, impinging normally on to a heated flat disk with finite radius, bounded by cylindrical walls kept at constant temperature. A 10 mm thick disk subjected to a high uniform heat flux was located at impingement distances ranging from 40 to 80 mm from the nozzle exit, for nozzle exit diameters of d = 1.0, 2.0, and 4.0 mm. The volume of fluid (VOF) method with a high-resolution interface-capturing (HRIC) scheme was implemented in STAR-CCM+. A new methodology was developed to predict the stagnation zone and local heat transfer coefficients. Contrary to previous research, it is shown that the radial extent of the stagnation zone is not fixed but depends on the gradient of radial velocity along the disk. The normalized local Nusselt number profile along the disk radius is found to be weakly dependent on Reynolds number for a given nozzle size. It is also shown that the local Nusselt number is not uniform in the stagnation region as reported by experimental studies but depends on the distribution of the near-wall radial velocity gradient. Using the computational results, new correlations to predict the dimensionless radial velocity gradient and Nusselt number have been developed. The present correlations are dimensionally balanced, eliminating a deficiency in earlier correlations noted in the literature.

Halouane Yacine, Mataoui Amina, Iachachene Farida. Heat Transfer Prediction of a Jet Impinging a Cylindrical Deadlock Area J. Heat Transfer 136, 112203 (2014) (9 pages);   Paper No: HT-13-1238;   doi:10.1115/1.4028323

The turbulent heat transfer by a confined jet flowing inside a hot cylindrical cavity is investigated numerically in this paper. This configuration is found in several engineering applications such as air conditioning and the ventilation of mines, deadlock, or corridors. The parameters investigated in this work are the Reynolds number (Re, 20,000 ≤ Re ≤ 50,000) and the normalized distance Lf between jet exit and the cavity bottom (Lf, 2 ≤ Lf  ≤ 12). The numerical predictions are performed by finite volume method using the second order one-point closure turbulence model (RSM). The Nusselt number increases and attains maximum values at stagnation points, after it decreases. For an experimental test case available in the literature Lf = 8, the numerical predictions are in good agreement. Processes of heat transfer are analyzed from the flow behavior and the underlying mechanisms. The maximum local heat transfer between the cavity walls and the flow occurs at Lf = 6 corresponding to the length of the potential core. Nusselt number at the stagnation point is correlated versus Reynolds number Re and impinging distance Lf; [$Nu0=f(Re,Lf)$].

### Research Papers: Natural and Mixed Convection

Ahmadi Mehran, Mostafavi Golnoosh, Bahrami Majid. Natural Convection From Interrupted Vertical Walls J. Heat Transfer 136, 112501 (2014) (8 pages);   Paper No: HT-13-1539;   doi:10.1115/1.4028369

Steady-state external natural convection heat transfer from interrupted rectangular vertical walls is investigated. A systematic numerical, experimental, and analytical study is conducted on the effect of adding interruptions to a vertical plate. Comsol multiphysics is used to develop a two-dimensional numerical model for investigation of fin interruption effects on natural convection. A custom-designed testbed is built and six interrupted wall samples are machined from aluminum. An effective length is introduced for calculating the natural convection heat transfer from interrupted vertical walls. Performing an asymptotic analysis and using a blending technique, a new compact relationship is proposed for the Nusselt number. Our results show that adding interruptions to a vertical wall can enhance heat transfer rate up to 16% and reduce the weight of the fins, which in turn, lead to lower manufacturing and material costs.

### Research Papers: Porous Media

Iasiello Marcello, Cunsolo Salvatore, Oliviero Maria, et al. Numerical Analysis of Heat Transfer and Pressure Drop in Metal Foams for Different Morphological Models J. Heat Transfer 136, 112601 (2014) (10 pages);   Paper No: HT-13-1655;   doi:10.1115/1.4028113

Because of their light weight, open porosity, high surface area per unit volume, and thermal characteristics, metal foams are a promising material for many industrial applications involving fluid flow and heat transfer. The pressure drop and heat transfer in porous media have inspired a number of experimental and numerical studies, and many models have been proposed in the literature that correlate the pressure gradient and the heat transfer coefficient with the mean cell size and porosity. However, large differences exist among results predicted by different models, and most studies are based on idealized periodic cell structures. In this study, the true three-dimensional microstructure of the metal foam is obtained by employing x-ray computed microtomography (XCT). This is the “real” structure. For comparison, ideal Kelvin foam structures are developed in the free-to-use software “surface evolver” surface energy minimization program. These are “ideal” structures. Pressure drop and heat transfer are then investigated in each structure using the CFD module of COMSOL® Multiphysics code. A comparison between the numerical predictions from the real and ideal geometries is carried out. The predictions showed that heat transfer characteristics are very close for low values of Reynolds number, but larger Reynolds numbers create larger differences between the results of the ideal and real structures. Conversely, the differences in pressure drop at any Reynolds number are nearly 100%. Results from the models are then validated by comparing them with experimental results taken from the literature. The validation suggests that the ideal structure poorly predicts the heat transfer and pressure drops.

Negin Mortazavi S., Hassanipour Fatemeh. Effect of Apex Angle, Porosity, and Permeability on Flow and Heat Transfer in Triangular Porous Ducts J. Heat Transfer 136, 112602 (2014) (8 pages);   Paper No: HT-13-1532;   doi:10.1115/1.4028177

This paper presents an analysis of forced convection flow and heat transfer in triangular ducts containing a porous medium. The porous medium is isotropic and the flow is laminar, fully developed with constant properties. Numerical results for velocity and temperature distribution (in dimensionless format) in the channel are presented for a wide range of porosity, permeability, and apex angles. The effects of apex angle and porous media properties (porosity and permeability) are demonstrated on the velocity and temperature distribution, as well as the friction factor (fRe) and Nusselt numbers in the channel for both Isoflux (NuH) and Isothermal (NuT) boundary conditions. The consistency of our findings has been verified with earlier results in the literature on empty triangular ducts, when the porosity in our models is made to approach one.

Moeini Sedeh Mahmoud, Khodadadi J. M. Solidification of Phase Change Materials Infiltrated in Porous Media in Presence of Voids J. Heat Transfer 136, 112603 (2014) (9 pages);   Paper No: HT-13-1664;   doi:10.1115/1.4028354

Infiltration of phase change materials (PCM) into highly conductive porous structures effectively enhances the thermal conductivity and phase change (solidification and melting) characteristics of the resulting thermal energy storage (TES) composites. However, the infiltration process contributes to formation of voids as micron-size air bubbles within the pores of the porous structure. The presence of voids negatively affects the thermal and phase change performance of TES composites due to the thermophysical properties of air in comparison with PCM and porous structure. This paper investigates the effect of voids on solidification of PCM, infiltrated into the pores of graphite foam as a highly conductive porous medium with interconnected pores. A combination of the volume-of-fluid (VOF) and enthalpy-porosity methods was employed for numerical investigation of solidification. The proposed method takes into account the variation of density with temperature during phase change and is able to predict the volume shrinkage (volume contraction) during the solidification of liquids. Furthermore, the presence of void and the temperature gradient along the liquid–gas interface (the interface between void and PCM) can trigger thermocapillary effects. Thus, Marangoni convection was included during the solidification process and its importance was elucidated by comparing the results among cases with and without thermocapillary effects. The results indicated that the presence of voids within the pores causes a noticeable increase in solidification time, with a sharper increase for cases without thermocapillary convection. For verification purposes, the amount of volume shrinkage during the solidification obtained from numerical simulations was compared against the theoretical volume change due to the variation of density for several liquids with contraction and expansion during the freezing process. The two sets of results exhibited good agreement.

### Research Papers: Experimental Techniques

Sinha A., Misra J. C. Effect of Induced Magnetic Field on Magnetohydrodynamic Stagnation Point Flow and Heat Transfer on a Stretching Sheet J. Heat Transfer 136, 112701 (2014) (11 pages);   Paper No: HT-11-1549;   doi:10.1115/1.4024666

In this paper, the steady magnetohydrodynamic (MHD) stagnation point flow of an incompressible viscous electrically conducting fluid over a stretching sheet has been investigated. Velocity and thermal slip conditions have been incorporated in the study. The effects of induced magnetic field and thermal radiation have also been duly taken into account. The nonlinear partial differential equations arising out of the mathematical analysis of the problem are transformed into a system of nonlinear ordinary differential equations by using similarity transformation and boundary layer approximation. These equations are solved by developing an appropriate numerical method. Considering an illustrative example, numerical results are obtained for velocity, temperature, skin friction, and Nusselt number by considering a chosen set of values of various parameters involved in the study. The results are presented graphically/in tabular form.

### Research Papers: Radiative Heat Transfer

Cai Jian, Marquez Ricardo, Modest Michael F. Comparisons of Radiative Heat Transfer Calculations in a Jet Diffusion Flame Using Spherical Harmonics and k-Distributions J. Heat Transfer 136, 112702 (2014) (9 pages);   Paper No: HT-13-1418;   doi:10.1115/1.4026169

A new nongray radiation modeling library for combustion gases has been implemented in OpenFOAM. The spectral models for single species include gray, correlation tables and full spectrum k-distributions (FSK) assembled from a narrow-band database. Mixing models for k-distributions include the multiplication and uncorrelated mixture models. Radiative transfer equation solvers for the library include spherical harmonics such as P1, P3, SP3 and SP5 as well as the optically thin approximation. The performance of the different solution methods is compared for accuracy and speed as a tool for future model strategy selection.

### Research Papers: Two-Phase Flow and Heat Transfer

Ababneh Mohammed T., Chauhan Shakti, Chamarthy Pramod, et al. Thermal Modeling and Experimental Validation for High Thermal Conductivity Heat Pipe Thermal Ground Planes J. Heat Transfer 136, 112901 (2014) (8 pages);   Paper No: HT-14-1004;   doi:10.1115/1.4028086

Thermal ground planes (TGPs) are flat, thin (external thickness of 2 mm) heat pipes which utilize two-phase cooling. The goal is to utilize TGPs as thermal spreaders in a variety of microelectronic cooling applications. In addition to TGPs and flat heat pipes, some investigators refer to similar devices as vapor chambers. TGPs are novel high-performance, integrated systems able to operate at a high power density with a reduced weight and temperature gradient. In addition to being able to dissipate large amounts of heat, they have very high effective axial thermal conductivities and (because of nanoporous wicks) can operate in high adverse gravitational fields. A three-dimensional (3D) finite element model is used to predict the thermal performance of the TGP. The 3D thermal model predicts the temperature field in the TGP, the effective axial thermal conductivity, and the evaporation and the condensation rates. A key feature of this model is that it relies on empirical interfacial heat transfer coefficient data to very accurately model the interfacial energy balance at the vapor–liquid saturated wick interface. Wick samples for a TGP are tested in an experimental setup to measure the interfacial heat transfer coefficient. Then the experimental heat transfer coefficient data are used for the interfacial energy balance. Another key feature of this model is that it demonstrates that for the Jakob numbers of interest, the thermal and flow fields can be decoupled except at the vapor–liquid saturated wick interface. This model can be used to predict the performance of a TGP for different geometries and implementation structures. This paper will describe the model and how it incorporates empirical interfacial heat transfer coefficient data. It will then show theoretical predictions for the thermal performance of TGP's, and compare with experimental results.

### Technical Brief

Brewster M. Q. Mixed Boundary Layer Skin Friction and Heat Transfer With Abrupt Transition J. Heat Transfer 136, 114501 (2014) (5 pages);   Paper No: HT-14-1407;   doi:10.1115/1.4028442

In the form that is commonly published in introductory textbooks, the classical problem of skin friction and heat transfer for a mixed laminar–turbulent boundary-layer flow on a flat-plate with an abrupt transition is nonconservative in mass, momentum, and energy. By forcing continuity in momentum and enthalpy thicknesses, the textbook problem takes on the appearance of conserving momentum and energy. But, by doing so while retaining a turbulent virtual origin at the plate's leading edge, the textbook example omits necessary jumps in these quantities and violates conservation of mass, momentum, and energy in the flow. Here we modify this classical problem to satisfy conservation principles through the introduction of either concentrated mass/momentum/energy fluxes, at the top of the boundary layer and/or concentrated surface shear stress/heat fluxes at the bottom. Out of this simple analysis comes the intriguing idea of an entrainment flux or inflow at the top of the boundary layer over the transition region.

### Discussion

Han Kim Kyu, Lee Seung-Hyun, Jin Kim Hyun, et al. Discussion: “Analysis of Asymmetric Disk-Shaped and Flat-Plate Heat Pipes” (Vafai, K., Zhu, N., and Wang, W., 1995, ASME J. Heat Transfer, 117(1), pp. 209–218) J. Heat Transfer 136, 115501 (2014) (2 pages);   Paper No: HT-14-1056;   doi:10.1115/1.4028429

Using the in-depth integral analysis, Vafai et al. [1] presented the asymmetric vapor velocity, the vapor and liquid pressure distributions, and the vapor temperature distribution of a disk-shaped asymmetric heat pipe. Especially, nondimensionalized vapor pressure (pv+) drop equation, Eq. (16) and nondimensionalized liquid pressure (pl+) drop equation, Eq. (21) from Darcy's law in their paper were analytically obtained. But the solutions of nondimensionalized pressure drop equations appear to be in error. So, we can show that a recalculation of the nondimensionalized vapor and liquid pressure drop equations leads to different results as given by Eqs. (1) and (2).

### Closure

Vafai K., Zhu N. Closure to “Discussion of ‘On the Analysis of Asymmetric Disk-Shaped and Flat-Plate Heat Pipes’” (2014, ASME J. Heat Transfer, 136(11), p. 115501) J. Heat Transfer 136, 116001 (2014) (2 pages);   Paper No: HT-14-1500;   doi:10.1115/1.4028430

We would like to thank Kim et al. for appreciating the importance of our work. It should be noted that the introduction of a disk shaped heat pipe by us in the cited paper was done for the very first time in the literature and our comprehensive analytical results for the disk-shaped and rectangular-shaped heat pipes constitute the very first such analysis in the literature. In summary, we are going to show that all of our derivations and results are exactly correct as presented in the paper. We only have a couple of minor misprints in the paper that need to be cited. However, all of our presented analytical results and figures are correct as they are. We will rigorously show that the points raised by Kim et al. are partly due to their misunderstanding of our analysis and partly due to the errors in their work. In fact, we show that our work is indeed very much correct. In what follows, we will show that each and every one of the points brought up by Kim et al. is either a nonissue or a mistake. In summary, Kim et al. claim that