Guest Editorial

J. Heat Transfer. 2018;140(3):030301-030301-1. doi:10.1115/1.4038762.

The K-22 Heat Transfer Visualization Committee sponsored the 24th Heat Transfer Photogallery during the 2017 Summer Heat Transfer Conference at the Hyatt Regency Bellevue, Bellevue, WA, on July 9–12, 2017. Nine entries were submitted to the Photogallery sessions. A peer-reviewed evaluation was conducted by both the participants and selected HTD K-22 Technical Committee members, which identified four final entries for publication in the 2018 ASME Journal of Heat Transfer February issue. Among the selected entries, two visualize droplet phenomena, and the remainder visualizes phase change at extreme temperatures.

Topics: Heat transfer
Commentary by Dr. Valentin Fuster


J. Heat Transfer. 2018;140(3):030901-030901-1. doi:10.1115/1.4039166.

Droplet impact on vibrating solids is ubiquitous in nature and industrial applications, including impact on turbine blades, insect wings, or during spray cooling of electronic systems and steel manufacturing processes. Using high speed imaging, we demonstrate that through substrate vibration (60 – 320 Hz), droplet contact times tc, which are independent of impact speed on rigid stationary substrates, can be actively manipulated and controlled. We show that droplet dynamics and contact times are most sensitive to impact phase, followed by vibration frequency, with vibration amplitude having negligible effects (Figure 1, Figure 2b). We determine a critical impact phase φc at which contact times transition rapidly from a minimum (tc ≈ 0.5tc,th) to a maximum (tc ≈ 1.6tc,th), where tc,th is the theoretical contact time on a stationary rigid substrate (insert Figure 2a). Averaging contact times over all impact phases, we show that for low frequencies (< 80 Hz) average contact times increase relative to impact on stationary substrates, while contact times decrease for impact at higher vibration frequencies (> 100 Hz) (Figure 2a). The present findings provide guidelines for the rational design of applications where the contact time influences heat transfer. During spray cooling, for example, the per droplet heat transfer rates increase (decrease) for longer (shorter) contact times. Thus, by tailoring the vibration frequency of the substrate, the average contact time, and consequently the average heat transfer, can be actively controlled.

Commentary by Dr. Valentin Fuster
J. Heat Transfer. 2018;140(3):030902-030902-1. doi:10.1115/1.4039167.

Lubricant infused surfaces (LIS or SLIPS) have potential to enhance a variety of applications due to their non-wetting characteristics. Recently, LIS and SLIPS have been shown to greatly improve dropwise condensation heat transfer performance by enabling ultra-low contact angle hysteresis (< 5°) and droplet-surface adhesion. The low adhesion characteristics arise from the ultra-smooth liquid-liquid interface that depends on the lubricating fluid remaining infused into the surface by capillary forces. A key limiting phenomenon of LIS and SLIPS is cloaking, whereby the infused lubricant spreads and forms a very thin layer (∼100 nm) over liquid droplet residing on the substrate. This can lead to degradation due to drainage of the lubricant layer. Cloaking involves several fluid interfaces between the lubricant, droplet and air, and the resolution of the lubricant-droplet interface is essential to cloaking imaging and characterization. Our work involves goniometric measurements of advancing and receding contact angles, imaging of cloaked water droplets on LIS and SLIPS, and the characterization of cloaking during shear induced flow. The LIS and SLIPS substrates were created by spin coating lubricants of various viscosities (5 – 2712 cSt) onto superhydrophobic nanostructured boehmite (AlO(OH)). Cloaking effects were observed by inhibition of water droplet evaporation (Fig. a). Furthermore, the lubricantdroplet interface for non-cloaking droplets was resolved using lubricant-miscible dyes (Fig. a). Cloaking was also characterized by studying shear induced flow of both condensate and artificially injected droplets, resulting in lubricant drainage as shown in Figures (b) and (c). The results of this work not only provide a basis for understanding the lubricant-droplet interactions on LIS and SLIPS, but also material design guidelines for future LIS and SLIPS coatings.

Commentary by Dr. Valentin Fuster
J. Heat Transfer. 2018;140(3):030903-030903-1. doi:10.1115/1.4039168.

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).]

Commentary by Dr. Valentin Fuster
J. Heat Transfer. 2018;140(3):030904-030904-1. doi:10.1115/1.4039169.

Neutron imaging enables direct visualization of evaporation and condensation of cryogenic propellants in metal containers such as aluminum and stainless steel. CFD models of propellant behaviors inside the large tanks have shown that a thin liquid film is formed along the interior surface, but this had not been verified experimentally. In the present study, neutron imaging is used to study evaporation and condensation rates of liquid methane inside a cylindrical 10 mm, Al 6061 cell. The liquid meniscus is clearly shown, but the spatial resolution is insufficient to directly image thin liquid films that may be on the interior surface. Optical density (neutron attenuation) analysis enables quantitative measurements of these liquid films. An optical density image is formed by removing the background noise and normalizing the liquid image with that of the empty cell. Optical densities are then transformed into a liquid transmission thickness using the Beer-Lambert law. This technique enables measurement of film thicknesses smaller than the spatial resolution of the imaging system. The above graphic shows an optical density image during condensation of methane and the corresponding horizontal scan which suggests that a 11 μm film exists on the wall. The images indicate that methane undergoes film-wise condensation and is perfectly wetting to aluminum.

These experiments were conducted at the NIST Center for Neutron Research in the Neutron Imaging Facility and the relevant work is supported by an Early stage Innovations Grant from NASA's Space Technology Research Grants Program (Grant # NNX14AB05G).

Commentary by Dr. Valentin Fuster

Research Papers: Conduction

J. Heat Transfer. 2017;140(3):031301-031301-22. doi:10.1115/1.4037775.

Finite integral transform techniques are applied to solve the one-dimensional (1D) dual-phase heat conduction problem, and a comprehensive analysis is provided for general time-dependent heat generation and arbitrary combinations of various boundary conditions (Dirichlet, Neumann, and Robin). Through the dependence on the relative differences in heat flux and temperature relaxation times, this analytical solution effectively models both parabolic and hyperbolic heat conduction. In order to demonstrate several exemplary physical phenomena, four distinct cases that illustrate the wavelike heat conduction behavior are presented. In the first model, following an initial temperature spike in a slab, the thermal evolution portrays immediate dissipation in parabolic systems, whereas the dual-phase solution depicts wavelike temperature propagation—the intensity of which depends on the relaxation times. Next, the analysis of periodic surface heat flux at the slab boundaries provides evidence of interference patterns formed by temperature waves. In following, the study of Joule heating driven periodic generation inside the slab demonstrates that the steady-periodic parabolic temperature response depends on the ratio of pulsatile electrical excitation and the electrical resistivity of the slab. As for the dual-phase model, thermal resonance conditions are observed at distinct excitation frequencies. Building on findings of the other models, the case of moving constant-amplitude heat generation is considered, and the occurrences of thermal shock and thermal expansion waves are demonstrated at particular conditions.

Commentary by Dr. Valentin Fuster
J. Heat Transfer. 2017;140(3):031302-031302-6. doi:10.1115/1.4038003.

In this paper, we have studied the effect of short branches (side chains) on the thermal conductivity (TC) of a polyethylene (PE) chain. With a reverse nonequilibrium molecular dynamics (RNEMD) method, TCs of the pristine PE chain and the PE-ethyl chain are simulated and compared. It shows that the branch has a positive effect to decrease the TC of a PE chain. The TC of the PE-ethyl chain decreases with the number density increase of branches, until the density becomes larger than about eight ethyl per 200 segments, where the TC saturates to be only about 40% that of a pristine PE chain. Because of different weights, different branches will cause a different decrease of TCs, and a heavy branch will lead to a lower TC than a light one. This study is expected to provide some fundamental guidance to obtain a polymer with a low TC.

Commentary by Dr. Valentin Fuster

Research Papers: Evaporation, Boiling, and Condensation

J. Heat Transfer. 2017;140(3):031501-031501-10. doi:10.1115/1.4038194.

In this paper, the flow patterns during water flow boiling instability in pin-fin microchannels were experimentally studied. Three types of pin-fin arrays (in-line/circular pin-fins, staggered/circular pin-fins, and staggered/square pin-fins) were used in the study. The flow instability started to occur as the outlet water reached the saturation temperature. Before the unstable boiling, a wider range of stable boiling existed in the pin-fin microchannels compared to that in the plain microchannels. Two flow instability modes for the temperature and pressure oscillations, which were long-period/large-amplitude mode and short-period/small-amplitude mode, were identified. The temperature variation during the oscillation period of the long-period/large-amplitude mode can be divided into two stages: increasing stage and decreasing stage. In the increasing stage, bubbly flow, vapor-slug flow, stratified flow, and wispy flow occurred sequentially with time for the in-line pin-fin microchannels; liquid single-phase flow, aforementioned four kinds of two-phase flow patterns, and vapor single-phase flow occurred sequentially with time for the staggered pin-fin microchannel. The flow pattern transitions in the decreasing stage were the inverse of those in the increasing stage for both in-line and staggered pin-fin microchannels. For the short-period/small-amplitude oscillation mode, only the wispy flow occurred. With the increase of heat flux, the wispy flow and the vapor single-phase flow occupied more and more time ratio during an oscillation period in the in-line and staggered pin-fin microchannels.

Commentary by Dr. Valentin Fuster

Research Papers: Experimental Techniques

J. Heat Transfer. 2017;140(3):031601-031601-15. doi:10.1115/1.4038135.

Techniques based on temperature-sensitive paints (TSP) to measure time-resolved temperature and heat transfer distributions at the interface between a wall and fluid during pool and flow boiling are described. The paints are excited using ultraviolet (UV) light emitting diodes (LEDs), and changes in fluorescence intensity are used to infer local temperature differences across a thin insulator from which heat flux distribution is obtained. Advantages over infrared (IR) thermometry include the ability to use substrates that are opaque to IR (e.g., glass, plexiglass and plastic films), use of low-cost optical cameras, no self-emission from substrates to complicate data interpretation, high speed, and high spatial resolution. TSP-based methods to measure wall heat transfer distributions are validated and then demonstrated for pool and flow boiling.

Commentary by Dr. Valentin Fuster

Research Papers: Forced Convection

J. Heat Transfer. 2017;140(3):031701-031701-9. doi:10.1115/1.4037208.

Both experimental and numerical studies of a turbulent flow in a bifurcating channel are performed to characterize the dynamical behavior of the flow and its impact on the convective heat transfer on the sides of the branch. This configuration corresponds to the radial vents placed in the stator vertically to the rotor–stator air gap in the electrical machines. Indeed, our analysis focuses on the local convective heat transfer on the vents internal surface under a turbulent mass flow rate. The flow field measurements were carried out with two components particle image velocimetry (PIV) system, and the local heat transfer on the sides of the bifurcation branch was measured using an infrared thermography device. The convective heat transfer and the flow dynamics through the geometry are investigated numerically considering a three-dimensional (3D) flow. The closure system of the Navier–Stokes equations for steady and incompressible flow is based on the low-Reynolds numbers Reynolds stress model (RSM) (RSM-stress-ω). The comparison of the 3D computed results with the measurements in the xy symmetry plane is satisfactory in the vertical and horizontal channels. The numerical prediction of the secondary flow in the vertical branch was analyzed and complements the experimental results. It was particularly noticed that the accelerated flow observed at the right side of the branch's inlet allows more pronounced heat transfer comparatively to the left side. Beyond approximately 7 hydraulic diameters from the entrance of the branch, the Nusselt number curves on the two sides of the branch tend to be the same developed Nusselt number, Nud.

Commentary by Dr. Valentin Fuster

Research Papers: Heat Exchangers

J. Heat Transfer. 2017;140(3):031801-031801-10. doi:10.1115/1.4038189.

This study proposes an empirical correlation for laminar natural convection applicable to external circular finned-tube heat exchangers with wide range of configuration parameters. The transient temperature response of the heat exchangers was used to obtain the heat transfer coefficient, and the experimental data with their characteristic lengths are discussed. The data lie in the range from 1 to 1000 for Rayleigh numbers based on the fin spacing: the ratio of fin height to tube diameter ranges from 0.1 to 0.9, and the ratio of fin pitch to height ranges from 0.13 to 2.6. Sixteen sets of finned-tube electroplated with nickel–chrome were tested. The convective heat transfer coefficients on the heat exchangers were measured by elimination of the thermal radiation effect from the heat exchanger surfaces. The Nusselt number was correlated with a newly suggested composite curve formula, which converges to the quarter power of the Rayleigh number for a single cylinder case. The proposed characteristic length for the Rayleigh number is the fin pitch while that for the Nusselt number is mean flow length, defined as half the perimeter of the mean radial position inside the flow region bounded by the tube surface and two adjacent fins. The flow is regarded as laminar, which covers heat exchangers from a single horizontal cylinder to infinite parallel disks. Consequently, the result of curve fitting for the experimental data shows the reasonable physical interpretation as well as the good quantitative agreement with the correction factors.

Commentary by Dr. Valentin Fuster

Research Papers: Heat and Mass Transfer

J. Heat Transfer. 2017;140(3):032001-032001-14. doi:10.1115/1.4038133.

Detailed Nusselt number (Nu) distributions over the leading (LE) and trailing (TE) endwalls and the pressure drop coefficients (f) of a rotating transverse-ribbed two-pass parallelogram channel were measured. The impacts of Reynolds (Re), rotation (Ro), and buoyancy (Bu) numbers upon local and regionally averaged Nu over the endwall of two ribbed legs and the turn are explored for Re = 5000–20,000, Ro = 0–0.3, and Bu = 0.0015–0.122. The present work aims to study the combined buoyancy and Coriolis effects on thermal performances as the first attempt. A set of selected experimental data illustrates the isolated and interdependent Ro and Bu influences upon Nu with the impacts of Re and Ro on f disclosed. Moreover, thermal performance factors (TPF) for the tested channel are evaluated and compared with those collected from the channels with different cross-sectional shapes and endwall configurations to enlighten the relative heat transfer efficiency under rotating condition. Empirical Nu and f correlations are acquired to govern the entire Nu and f data generated. These correlations allow one to evaluate both isolated and combined Re, Ro and/or Bu impacts upon the thermal performances of the present rotating channel for internal cooling of gas turbine blades.

Commentary by Dr. Valentin Fuster

Research Papers: Jets, Wakes, and Impingment Cooling

J. Heat Transfer. 2017;140(3):032201-032201-12. doi:10.1115/1.4037786.

Liquid microjets are emerging as candidate primary or secondary heat exchangers for the thermal management of next generation photonic integrated circuits (PICs). However, the thermal and hydrodynamic behavior of confined, low Reynolds number liquid slot jets is not yet comprehensively understood. This investigation experimentally examined jet outlet modifications—in the form of tabs and chevrons—as techniques for passive control and enhancement of single-phase convective heat transfer. The investigation was carried out for slot jets in the laminar flow regime, with a Reynolds number range, based on the slot jet hydraulic diameter, of 100–500. A slot jet with an aspect ratio of 4 and a fixed confinement height to hydraulic diameter ratio (H/Dh) of 1 was considered. The local surface heat transfer and velocity field characteristics were measured using infrared (IR) thermography and particle image velocimetry (PIV) techniques. It was found that increases in area-averaged Nusselt number of up to 29% compared to the baseline case could be achieved without incurring additional hydrodynamic losses. It was also determined that the location and magnitude of Nusselt number and velocity peaks within the slot jet stagnation region could be passively controlled and enhanced through the application of outlet tabs of varying geometries and locations.

Commentary by Dr. Valentin Fuster

Research Papers: Micro/Nanoscale Heat Transfer

J. Heat Transfer. 2017;140(3):032401-032401-10. doi:10.1115/1.4038191.

The heat and mass transfer close to the apparent three-phase contact line is of tremendous importance in many evaporation processes. Despite the extremely small dimensions of this region referred to as the microregion compared to the macroscopic length scale of a boiling process, a considerable fraction of heat can be transferred in this region. Due to its small characteristic length scale, physical phenomena are relevant in the microregion, which are completely negligible on the macroscopic scale, including the action of adhesion forces and the interfacial heat resistance. In the past, models have been developed taking these effects into account. However, so far these models are based on the assumption of one-dimensional (1D) heat conduction, and the flow within the thin liquid film forming the microregion near the apparent three-phase contact line is modeled utilizing the lubrication approximation. Hence, the application of existing models is restricted to small apparent contact angles. Moreover, the effects of surface structures or roughness are not included in these lubrication models. To overcome these limitations, a direct numerical simulation (DNS) of the liquid flow and heat transfer within the microregion is presented in this paper. The DNS is employed for validation of the existing lubrication model and for investigation of the influence of surface nanostructures on the apparent contact angle and in particular on the heat transfer within the microregion.

Commentary by Dr. Valentin Fuster

Research Papers: Natural and Mixed Convection

J. Heat Transfer. 2017;140(3):032501-032501-14. doi:10.1115/1.4037875.

The continuity, momentum, and the energy conservation equation for air around a hollow cylinder with inline or staggered holes have been solved in three dimensions to assess the buoyancy driven flow and temperature field around the cylinder. From the thermal field, the average surface Nu could be computed for hollow cylinders with inline or staggered holes and the heat loss from the cylinder could be compared with that of a hollow cylinder without holes. Interesting flow and thermal plume around the hollow cylinder with holes could be seen, which could help to explain why there is more heat loss from a cylinder with staggered holes compared to a cylinder with inline holes at lower Ra of 105, whereas for higher Ra of 106 or more, there exists an optimum d/D where the heat loss from the perforated cylinder has a maximum value and thereafter it reduces. There are interesting comparisons on Nu for the hollow cylinder with inline or staggered holes and new correlations for Nu versus many different pertinent input parameters have been developed for many cases, which can be used practically in industry for designing perforated cylinder with heat loss.

Commentary by Dr. Valentin Fuster

Research Papers: Porous Media

J. Heat Transfer. 2017;140(3):032601-032601-11. doi:10.1115/1.4038006.

The interstitial heat transfer coefficient (IHTC) is a key parameter in the two-energy equation model usually employed to investigate the thermal performance of high porosity open-cell metal foam/paraffin composite phase change material. Due to the existence of weak convection of liquid paraffin through metal foam during phase change process, the IHTC should be carefully determined for a low Reynolds number range (Re = 0–1), which however has been rarely addressed in the literature. In this paper, a direct simulation at foam pore scale is carried out to determine the IHTC between paraffin and metal foam at Re = 0–1. For this purpose, the cell structures reflecting realistic metal foams are first constructed based on the three-dimensional (3D) Weaire–Phelan foam cell to serve as the representative elementary volume (REV) of metal foam for direct simulation. Then, by solving the Navier–Stokes equations and energy equation for the REV, the influences of Reynolds number (Re), Prandtl number (Pr), foam porosity (ε), and pore density (PPI) on the dimensionless IHTC, i.e., the Nusselt number Nuv, are investigated. According to the numerical results, a correlation of Nuv at Re = 0–1 is proposed for metal foam/paraffin composite material, which covers both diffusion-dominated interstitial heat transfer region (Re ≤ 0.1) and convection-dominated interstitial heat transfer region (0.1 < Re ≤ 1). Finally, the applicability of this correlation in the two-energy equation model for solid–liquid phase change of paraffin in metal foam is validated by comparing the model predicted melting front with that of experimental observations made in this study. It is found that the IHTC correlation proposed in this study can be used in the two-energy equation model for well predicting the phase change process of paraffin in metal foam.

Commentary by Dr. Valentin Fuster

Research Papers: Radiative Heat Transfer

J. Heat Transfer. 2017;140(3):032701-032701-11. doi:10.1115/1.4037823.

In this study, a new model for intumescent coatings is developed including the radiation transfer equation. So, one of the important features of this model is to give the insight of the radiative heat transfer in intumescent coating during expansion. In addition, the model equations are derived into a new coordinate system by introducing the expansion effect into the corresponding parameters. Consequently, the numerical results can be carried out by using a fixed grid system. The numerical results show that the radiative heat transfer near the exposed coating surface cannot be well simulated by the model of thermal radiation conductivity, which is widely used in the previous studies. So, it is suggested that the radiative heat transfer in the expanded char region should be formulated by a more considerate model. In addition, several parameters of coating thermal properties (thermal conductivity, extinction coefficient, and albedo) are tested and investigated under a radiant heat source. In addition to the transient response, the effects of these coating properties on the quasi steady results are also discussed. It is found that the thermal conductivity and the extinction coefficient in the expanded char region both dominate the coating performance. For the thermal properties of virgin coating, the thermal conductivity may have significant effect when the coating has large incomplete pyrolysis (expansion) region, while the extinction coefficient has little influence. Besides, the thermal conductivity and the albedo of virgin coating both alter the heating time to initial expansion but in different mechanisms.

Commentary by Dr. Valentin Fuster
J. Heat Transfer. 2017;140(3):032702-032702-12. doi:10.1115/1.4038004.

In the present work, strategies for the grouping of the spectral absorption coefficients used in multiscale (MS) multigroup (MG) full-spectrum k-distribution models were improved by considering the effects of variations in both temperature and species molar ratio on the correlated-k characteristics of the spectra of H2O–CO2-soot mixtures. The improvements in the accuracy of predictions of radiation heat transfer characteristics resulting from these new grouping strategies were evaluated using a series of semi-one-dimensional (1D) cases with significant temperature, participating species molar ratio, and pressure inhomogeneities. Finally, evaluations of grouping strategies were presented on calculation of the full-spectrum thermal images of an actual aeroengine combustor.

Commentary by Dr. Valentin Fuster

Technical Brief

J. Heat Transfer. 2017;140(3):034501-034501-6. doi:10.1115/1.4037874.

The separation of variables (SOV) can be used for all Fourier, single-phase lag (SPL), and dual-phase lag (DPL) heat conduction problems with time-independent source and/or boundary conditions (BCs). The Laplace transform (LT) can be used for problems with time-dependent BCs and sources but requires large computational time for inverse LT. In this work, the orthogonal eigenfunction expansion (OEEM) has been proposed as an alternate method for non-Fourier (SPL and DPL) heat conduction problem. However, the OEEM is applicable only for cases where BCs are homogeneous. Therefore, BCs of the original problem are homogenized by subtracting an auxiliary function from the temperature to get a modified problem in terms of a modified temperature. It is shown that the auxiliary function has to satisfy a set of conditions. However, these conditions do not lead to a unique auxiliary function. Therefore, an additional condition, which simplifies the modified problem, is proposed to evaluate the auxiliary function. The methodology is verified with SOV for time-independent BCs. The implementation of the methodology is demonstrated with illustrative example, which shows that this approach leads to an accurate solution with reasonable number of terms in the expansion.

Commentary by Dr. Valentin Fuster
J. Heat Transfer. 2017;140(3):034502-034502-4. doi:10.1115/1.4038001.

The constant flux forced convection in a rectangular duct with two highly conductive (H1) walls and two poorly conductive (H2) walls is studied for the first time. This mixed problem is solved analytically using a modified single series method. The Nusselt number is determined for various duct aspect ratios. Depending on the aspect ratio, hot spots and cold spots may occur either on the H1 walls or on the H2 walls.

Commentary by Dr. Valentin Fuster

Technical Briefs

J. Heat Transfer. 2017;140(3):034503-034503-7. doi:10.1115/1.4038134.

The flow structure, heat transfer, and entropy generation characteristics in the entrance region of mixed convection under the effect of transverse buoyancy force are investigated numerically. Results are obtained for laminar flow of uniform inlet velocity and temperature through a square duct with uniform wall temperature. The buoyancy induced-secondary flow is observed in the entrance region where flow structure and heat transfer are significantly affected. The flow entrance region is extended by buoyancy, while the thermal entrance region is shortened. The developments of Nusselt number and local entropy generation are discussed in detail for Richardson numbers of 0 ≤ Ri ≤ 10, Reynolds number Re = 100 and Prandtl number Pr = 0.7. The total heat transfer rate and global entropy generation by friction increase with buoyancy, while global entropy generation by heat convection changes a little. The effect of Reynolds number on entropy generation is also discussed.

Commentary by Dr. Valentin Fuster

Discussion: Discussions

Commentary by Dr. Valentin Fuster

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