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### Research Papers: Conduction

J. Heat Transfer. 2019;141(6):061301-061301-10. doi:10.1115/1.4042912.
OPEN ACCESS

Shape factors for steady heat conduction enable quick and highly simplified calculations of heat transfer rates within bodies having a combination of isothermal and adiabatic boundary conditions. Many shape factors have been tabulated, and most undergraduate heat transfer books cover their derivation and use. However, the analytical determination of shape factors for any but the simplest configurations can quickly come to involve complicated mathematics, and, for that reason, it is desirable to extend the available results as far as possible. In this paper, we show that known shape factors for the interior of two-dimensional objects are identical to the corresponding shape factors for the exterior of those objects. The canonical case of the interior and exterior of a disk is examined first. Then, conformal mapping is used to relate known configurations for squares and rectangles to the solutions for the disk. Both a geometrical and a mathematical argument are introduced to show that shape factors are invariant under conformal mapping. Finally, the general case is demonstrated using Green's functions. In addition, the “Yin-Yang” phenomenon for conduction shape factors is explained as a rotation of the unit disk prior to conformal mapping.

Topics: Disks , Shapes , Temperature
Commentary by Dr. Valentin Fuster

### Research Papers: Evaporation, Boiling, and Condensation

J. Heat Transfer. 2019;141(6):061501-061501-7. doi:10.1115/1.4043160.

The analytical solutions for condensation (and evaporation) rates of laminar humid air flow in the channels of a battery pack at both entrance and downstream regions are obtained/modeled. The effects of the entrance fluid velocity profile and the aspect ratio of the flow channel are taken into consideration. Initially, an analytical solution for laminar humid air flows' condensation at the flow channel with infinite aspect ratio and fully developed flow profile at the entrance is obtained. The solution is in good agreement with the simulation result obtained from a correlated fluent condensation model. After performing simulations of water vapor condensation and evaporation in the flow channel for different aspect ratios and entrance velocity profiles, the analytical solutions of condensation and evaporation rate of water vapor are presented in analytic forms by adding appropriate coefficients/correction factors to the solution for the flow in the channel with infinite aspect ratio. The resulting model accurately captures the effects of the entrance velocity profile and aspect ratio.

Commentary by Dr. Valentin Fuster

### Research Papers: Experimental Techniques

J. Heat Transfer. 2019;141(6):061601-061601-5. doi:10.1115/1.4043167.

The presence of unknown thermal contact thermal resistance has limited prior two-probe thermal transport measurements of suspended graphene samples. Here, we report four-probe thermal transport measurements of suspended seven-layer graphene. By isolating the thermal contact resistance, we are able to attribute the observed reduced thermal conductivity primarily to polymeric residue on the sample instead of the contact thermal resistance, which resulted in ambiguity in the prior experimental studies of the effect of polymer reside. The extrinsic scattering rate due to the polymer residue is extracted from the measurement results based on a solution of the Peierls-Boltzmann phonon transport equation.

Commentary by Dr. Valentin Fuster

### Research Papers: Forced Convection

J. Heat Transfer. 2019;141(6):061701-061701-7. doi:10.1115/1.4042774.

This paper investigates the flow and heat transfer of power-law fluids over a stretching sheet where the coupling dynamics influence of viscous sheet and ambient fluid is taken into account via the stress balance. A modified Fourier's law is introduced in which the effects of viscous dissipation are taken into account by assuming that the thermal conductivity is to be shear-dependent on the velocity gradient. The conditions for both velocity and thermal boundary layers admitting similarity solutions are found, and numerical solutions are computed by a Bvp4c program. The results show that the viscous sheet and rheological properties of ambient fluids have significantly influences on both velocity and temperature fields characteristics. The formation of sheet varies with the viscosity of fluid and draw ratio, which then strongly affects the relations of the local skin friction coefficient, the local Nusselt number, and the generalized Reynolds number. Moreover, for specified parameters, the flow and heat transfer behaviors are discussed in detail.

Commentary by Dr. Valentin Fuster
J. Heat Transfer. 2019;141(6):061702-061702-9. doi:10.1115/1.4043303.

The historical approach to averaging the convection coefficient in tubes of constant wall heat flux leads to quantitative errors in short tubes as high as 12.5% for convection into fully developed flows and 33.3% for convection into hydrodynamically developing flows. This mistake can be found in teaching texts and monographs on heat transfer, as well as in major handbooks. Using the correctly defined relationship between local and average convection coefficients, eight new correlations are presented for fully developed and developing flows in round tubes and between parallel plates for the constant wall heat flux condition. These new correlations are within 2% of exact solutions for fully developed flows and within 6% of first principle calculations for hydrodynamically developing flows.

Commentary by Dr. Valentin Fuster

### Research Papers: Heat and Mass Transfer

J. Heat Transfer. 2019;141(6):062001-062001-9. doi:10.1115/1.4043172.

In this present work, the influence of corrugated booster reflectors (CBR) in a centrally finned twist (CFT) inserted solar thermal collector (SC) on heat transfer and thermal performance characteristics has been approached experimentally. The experimental trials have been made with two different twist ratios (Y = 3 and 6) for typical twist (TT) and CFT under same working conditions. The results were compared with the plain tube SC with CBR plain and also with the plain tube SC with flat booster reflectors (FBR plain). The experimental result of the CBR plain has been verified with the standard equations and found the deviations within ±10.05% for Nusselt number and ±9.42% for friction factor. The CBR has 1.6% higher effective reflection area than the FBR. Hence, the CBR augmented the Nusselt number around 8.25% over the FBR. When compared to the CBR plain, the CFT of minimum twist ratio (Y = 3) offered 10.09% higher thermal efficiency. In addition, empirical correlations have been derived for predicting the Nusselt number and friction factor. The deviations of the predicted value from the experiment value fall within ±10.62% for Nusselt number and ±11.28% for friction factor.

Commentary by Dr. Valentin Fuster
J. Heat Transfer. 2019;141(6):062002-062002-27. doi:10.1115/1.4043299.

A mechanistic model of film movements is developed based on the treatments on the annular flow field. The initial conditions at the inlet are determined by adopting a validated film thickness correlation of fully developed upward annular flow in vertical pipes. The overall pressure gradient is assumed to be uniform all along the axial distance within the elbow and the static pressure is also uniform on every cross section. The axial velocities of the liquid film and the core region are both uniform on the cross-sectional plane. The droplets are assumed to travel in straight lines normal to the inlet plane until colliding on and absorbed by the liquid film surface. The liquid film motion is divided into the axial and radial directions. Energy conservation law and Newton's second law are, respectively, used in the two directions. The film motion calculation is executed by using a discrete method with an explicit solution. The average film thickness and the circumferential thickness distribution on an arbitrary cross section can be obtained for the given flow conditions. The mechanistic model is verified by comparing the predicted circumferential distribution of film thickness with three series of experimental data from the literature. Parametric studies are also conducted to investigate the parameter effects and the range of application. The present work proves that the variation and distribution of film thickness within the elbows can be efficiently described by the mechanistic model.

Commentary by Dr. Valentin Fuster

### Research Papers: Heat Transfer in Manufacturing

J. Heat Transfer. 2019;141(6):062101-062101-10. doi:10.1115/1.4042904.

In hot-forming die-quenching (HFDQ) boron manganese steel blanks are heated within a roller hearth furnace, and then simultaneously quenched and formed into fully martensitic body-in-white components. Industry needs models that can predict the instantaneous temperature and austenite phase fraction within the roller furnace to diagnose problems (e.g., incomplete austenitization), forecast costs, and optimize process settings. This paper introduces a thermometallurgical model for Al–Si coated 22MnB5, consisting of a coupled heat transfer and austenitization submodels. Two candidate austenitization submodels are considered: an empirical first-order model and a model based on the detailed austenitization kinetics. In the case of the first-order model, a detailed Monte Carlo procedure is used to construct 95% credibility intervals for the blank temperature and austenite phase fraction that accounts for uncertainties in the furnace temperature and model parameters. The models are first assessed using temperature and austenite phase fractions from Al–Si coated 22MnB5 coupons heated in a laboratory-scale muffle furnace, and then used to simulate austenitization of patched blanks within an industrial roller hearth furnace. The results show that the empirical first-order model provides a more robust estimate of austenite phase fraction compared to the detailed model.

Commentary by Dr. Valentin Fuster

### Research Papers: Micro/Nanoscale Heat Transfer

J. Heat Transfer. 2019;141(6):062401-062401-8. doi:10.1115/1.4042810.

An experimental investigation of fluid flow friction and heat transfer coefficient in simultaneously developing flow through a multiport microchannel flat tube (MMFT) was presented. The cross-sectional geometries of five tubes were rectangular with hydraulic diameters of 0.8–1.33 mm and aspect ratio of 0.44–0.94. The working fluid was water, and the Reynolds number was in the range 150–4500. The experiment result showed that friction factor was successfully predicted by classical correlation in laminar regime, whereas the laminar–turbulent transition in the developing flow was not as obvious as in the completely developed flow. The greater aspect ratio produced stronger heat transfer capacity in the developing flow, although the effect of the aspect ratio decreased at increased Reynolds numbers for heat transfer characteristics. Moreover, the scale effect improved the heat transfer performance of MMFTs, especially at high Reynolds numbers.

Commentary by Dr. Valentin Fuster
J. Heat Transfer. 2019;141(6):062402-062402-14. doi:10.1115/1.4042840.

Heat transfer from a cylinder of square cross section (either dissipating constant heat flux (qW) or maintaining at a constant temperature (TW)) placed near a plane wall under the incidence of nonuniform linear/nonlinear velocity profile is studied numerically (finite volume method (FVM), quadratic upstream interpolation for convective kinematics (QUICK), and SIMPLE). The conventional fluids are chosen as water, and ethylene glycol–water mixture. The nanoparticles are selected as Al2O3 and CuO. Roles of pressure gradient P (at the inlet), temperature of base fluids, thermal conditions (TW or qW), and nanofluids' parameters (nanoparticle concentrations (ϕ), diameter, materials, and base fluids) on the heat transfer (Nusselt number ($Nu¯M$)) of the cylinder are investigated here. $Nu¯M$ enhancement from the cylinder together with its drag coefficient reduction/increment due to addition of nanomaterials in both fluids at two different temperatures is assessed under the Couette flow. Classical fluid dynamics relationship among $Nu¯M$, Reynolds number (Re), and Prandtl number is discussed through Colburn j–factor, and hence the utility of proposed correlation between j–factor and Re toward engineering problems is also explored. The graphical observations of dependency of $Nu¯M$ on the aforesaid parameters are reconfirmed by proposed functional forms of $Nu¯M=Nu¯M(P), Nu¯M=Nu¯M(ϕ)$ and hence $Nu¯M=Nu¯M(P,ϕ)$. An effort is made to examine the effectiveness of the aforementioned parameters on the heat transfer enhancement rate.

Commentary by Dr. Valentin Fuster
J. Heat Transfer. 2019;141(6):062403-062403-10. doi:10.1115/1.4043163.

In this paper, effect of Joule heating (JH), viscous dissipations (VD), and super hydrophobic surfaces on heat transfer of water–Al2O3 and water–CuO nanofluids in a microchannel has been investigated using lattice Boltzmann method (LBM). The microchannel is under a uniform and transverse magnetic field. The lower wall of the microchannel is insulated and a uniform heat flux has been applied to the upper wall. Results are generated at constant Reynolds number of 150, volume fraction of 2%, and a diameter of 25 nm with variable Hartmann numbers ranging from 0 to 20 and nondimensional slip coefficients from 0 to 0.05. The results of the developed code are in good agreement with other analytical, numerical, and experimental reports. Moreover, the results show that in such case, ignoring the JH and VD leads to a significant error in the prediction of Nusselt number up to 62% and 56%, respectively, for water–Al2O3 and water–CuO nanofluids. It has also been shown that using a super hydrophobic surface with a slip coefficient of 0.05 leads to a significant reduction in VD; however, it increases the effect of JH. On the other hand, it is found that, despite JH and viscous dissipation effects, using super hydrophobic surfaces (up to a slip coefficient of 0.05) leads to an increase in Nusselt number and decrease in shear stress for all the studied Hartmann numbers. Finally, it has been concluded that super hydrophobic surfaces can be used as a passive tool to enhance the heat transfer rate and simultaneously decrease the pumping power demand.

Commentary by Dr. Valentin Fuster
J. Heat Transfer. 2019;141(6):062404-062404-8. doi:10.1115/1.4043301.

A proposal is made to demonstrate features of thermodynamic evaporation at the nanoscale using only an external electric field. The consequences of exposure to both uniform and nonuniform electric field on the water nanofilms are analyzed through molecular dynamics simulations. The temporal evolution of temperature and molecular nucleation under uniform electric field resembles evaporation at high heat. The temperature fluctuations of the system are analyzed from the density variation of the system, which has received no heat input from outside. Evaporation like process and nucleation from the water surface is described as a systematic polarization of the water molecules in the presence of electric field. The nucleation of the vapor bubble with a nonuniform electric field also shows similarity with heat-induced pool boiling. The reason behind isolated nucleation is analyzed from the temperature map of the system at different time instants. Possible surface instabilities due to the exposure of electric field on water nanolayer are also elaborated for both uniform and nonuniform cases.

Commentary by Dr. Valentin Fuster
J. Heat Transfer. 2019;141(6):062405-062405-11. doi:10.1115/1.4043165.

Unicellular Rayleigh–Bénard convection of water–copper nanoliquid confined in a high-porosity enclosure is studied analytically. The modified-Buongiorno–Brinkman two-phase model is used for nanoliquid description to include the effects of Brownian motion, thermophoresis, porous medium friction, and thermophysical properties. Free–free and rigid–rigid boundaries are considered for investigation of onset of convection and heat transport. Boundary effects on onset of convection are shown to be classical in nature. Stability boundaries in the $R1*$–R2 plane are drawn to specify the regions in which various instabilities appear. Specifically, subcritical instabilities' region of appearance is highlighted. Square, shallow, and tall porous enclosures are considered for study, and it is found that the maximum heat transport occurs in the case of a tall enclosure and minimum in the case of a shallow enclosure. The analysis also reveals that the addition of a dilute concentration of nanoparticles in a liquid-saturated porous enclosure advances onset and thereby enhances the heat transport irrespective of the type of boundaries. The presence of porous medium serves the purpose of heat storage in the system because of its low thermal conductivity.

Commentary by Dr. Valentin Fuster
J. Heat Transfer. 2019;141(6):062406-062406-10. doi:10.1115/1.4043175.

We present droplet growth dynamics on homogeneous and patterned surfaces (surface with hydrophilic and hydrophobic region) using two-dimensional thermal lattice Boltzmann method (LBM). In the first part, we performed 2D simulations on homogeneous hydrophobic surfaces. The result shows that the droplet grows at higher rate on a surface with higher wettability which is attributed to low conduction resistance and high solid–liquid contact area. In the later part, we performed simulations on patterned surface and observed that droplet preferentially nucleates on the hydrophilic region due to lower energy barrier and grows in constant contact line (CCL) mode because of contact line pinning at the interface of hydrophilic–hydrophobic region. As the contact angle reaches the maximum value of hydrophobic surface, contact line depins and droplet shows constant contact angle (CCA) growth mode. We also discuss the effect of characteristic width of hydrophilic region on growth of droplet. We show that contact angle of the droplet increases rapidly and reaches the contact angle of hydrophobic region on a surface with a lower width of the hydrophilic surface.

Commentary by Dr. Valentin Fuster

### Research Papers: Natural and Mixed Convection

J. Heat Transfer. 2019;141(6):062501-062501-9. doi:10.1115/1.4042905.

Turbulent natural convection in a large-scale cavity has taken a great attention thanks to its importance in many engineering applications such as building. In this work, the lattice Boltzmann method (LBM) is used to simulate turbulent natural convection heat transfer in a small room of housing heated from below by means of a heated floor. The ceiling and the four vertical walls of the room are adiabatic except for a portion of one vertical wall. This portion simulates a glass door with a cold temperature θc = 0. The cavity is filled by air (Pr = 0.71) and heated from below with uniformly imposed temperature θh = 1. The effects of the heat source length (Lr) and Rayleigh number (Ra) on the flow structure and heat transfer are studied for ranges of 0.2 ≤ Lr ≤ 0.8 and 5 × 106 ≤Ra ≤ 108. The heat transfer is examined in terms of local and mean Nusselt numbers. The results show that an increase in Rayleigh number or in heat source length increases the temperature in the core of the cavity. The flow structure shows that turbulent natural convection regime is fully developed for Ra = 108. Correlations for mean Nusselt number as a function with Ra for different values of Lr are expressly derived.

Commentary by Dr. Valentin Fuster
J. Heat Transfer. 2019;141(6):062502-062502-12. doi:10.1115/1.4042811.

The coupled phenomena of radiative–magnetohyrodynamic (MHD) natural convection in a horizontal cylindrical annulus are numerically investigated. The buoyant flow is driven by the temperature difference between the inner and outer cylinder walls, while a circumferential magnetic field induced by a constant electric current is imposed. The hybrid approach of finite volume and discrete ordinates methods (FV-DOM) is developed to solve the nonlinear integro-differential governing equations in polar coordinate system, and accordingly, the influences of Hartmann number, radiation–convection parameter, and optical properties of fluid and wall on thermal and hydrodynamic behaviors of the “downward flow,” originally occurring without consideration of radiation and magnetic field, are mainly discussed. The results indicate that both the circulating flow and heat transfer are weakened by the magnetic field, but its suppression effect on the latter is rather small. Under the influence of magnetic field, the “downward flow” pattern has not been obtained from zero initial condition even for the case of weak radiation of NR = 0.1. Besides, the variation of radiative heat transfer rate with angular positions diminishes for the fluid with strong scattering or weak absorption.

Commentary by Dr. Valentin Fuster

### Research Papers: Porous Media

J. Heat Transfer. 2019;141(6):062601-062601-11. doi:10.1115/1.4043213.

The research on water-heat transport of soil porous media has important theoretical and practical significance for the problem of agricultural production and environmental governance. In this work, the water-heat transport characteristics of sandy soil porous media are analyzed. The two-dimensional continuum physical model is constructed by continuum method, and the two-dimensional pore network physical model is constructed directly at pore scale by taking into account the complicated pore and skeleton structures of soil. Mathematical models of water-heat transport process of sandy soil are constructed based on heat-mass transfer mechanism. Mathematical models of the continuum method and pore network method are solved by ANSYS and self-designed solving algorithm, respectively. The numerical simulation results of soil temperature distributions and moisture distributions are in good agreement with the experimental results. The pore network simulation results are in good agreement with the measured data and are superior to the existing continuous scale method. The pore network simulation results can directly present the characteristics of the preferential flow and wetting front during the water-heat transport process of soil.

Commentary by Dr. Valentin Fuster

### Research Papers: Radiative Heat Transfer

J. Heat Transfer. 2019;141(6):062701-062701-7. doi:10.1115/1.4043306.

In a recent contribution, the authors show that the uncertainty in heat transfer results obtained using the Monte Carlo ray-trace (MCRT) method is related to the median of the radiation distribution factor probability density function (PDF). The value of this discovery would be significantly enhanced if the median could be known a priori without first computing the distribution factors. This would allow the user to determine the number of rays required to achieve the desired accuracy of a subsequent heat transfer analysis. The current contribution presents a correlation for the median of the distribution factor PDF as a function of emissivity and the number of surface elements defining an enclosure. The correlation involves a single parameter whose value is unique for a given enclosure geometry. We find that the radiation behavior of a given enclosure can be classified on a scale ranging from reflection-dominated to geometry-dominated. The correlation is shown to work well for reflection-dominated enclosures but less well for geometry-dominated enclosures.

Commentary by Dr. Valentin Fuster

### Technical Brief

J. Heat Transfer. 2019;141(6):064501-064501-3. doi:10.1115/1.4042488.

The H2-forced convection in a rectangular duct of large aspect ratio (>10) is studied. It is found that the short ends have non-negligible effects on the Nusselt number and the temperature distribution. Even at infinite aspect ratios, the Nusselt number depends on the net heat addition from the ends, but not how they are distributed.

Topics: Convection , Ducts , Heat , Heat flux
Commentary by Dr. Valentin Fuster
J. Heat Transfer. 2019;141(6):064502-064502-3. doi:10.1115/1.4042489.

The forced convection problem for a developing thermal boundary layer in a parallel shear flow is studied. If the shear flow has a power-law velocity profile, exact similarity thermal boundary layer solutions in terms of Gamma functions can be found. Specifically, three types of thermal boundary conditions are considered: a step temperature change, a step flux change, and a concentrated heat source. The latter is also analogous to mass diffusion form an isolated source. The mixing index for mass diffusion is found exactly.

Commentary by Dr. Valentin Fuster
J. Heat Transfer. 2019;141(6):064503-064503-9. doi:10.1115/1.4043304.

The numerical analysis for the round jet impingement over a circular cylinder has been carried out. The v2f turbulence model is used for the numerical analysis and compared with the two equation turbulence models from the fluid flow and the heat transfer point of view. Further, the numerical results for the heat transfer with original and modified v2f turbulence model are compared with the experimental results. The nozzle is placed orthogonally to the target surface (heated cylindrical surface). The flow is assumed as the steady, incompressible, three-dimensional and turbulent. The spacing between the nozzle exit and the target surface ranges from 4 to 15 times the nozzle diameter. The Reynolds number based on the nozzle diameter ranges from 23,000 to 38,800. From the heat transfer results, the modified v2f turbulence model is better as compared to the other turbulence models. The modified v2f turbulence model has the least error for the numerical Nusselt number at the stagnation point and wall jet region.

Commentary by Dr. Valentin Fuster