Surface-averaged Nusselt Number Research Articles

Heat transfer rate and uniformity of mist flow jet impingement for glass tempering

The thinning and miniaturization of components, such as liquid crystal display and solar cells, has increased the market demand for tempered ultra-thin glass. The physical tempering in glass, which is less than 2mm thick, is difficult to achieve in air jet impingement. In this study, we propose the use of mist flow jet impingement cooling technology on ultra-thin glass. The effects of the mist flow droplets diameters, nozzle inlet temperature, and mass fraction on the heat transfer rate and uniformity are numerically studied. Our performance in terms of the heat transfer rate and the uniformity is accounted for through an evaluation of the surface-averaged temperature, averaged Nusselt number, surface-averaged Nusselt number, and surface standard deviation percentage of the Nusselt number. The empirical correlation for the surface averaged Nusselt number asa function of mist flow droplet diameters is provided. The droplet diameter is the major factor affecting the heat transfer rate, while the jet inlet temperature is a minor factor. We can obtain a better heat transfer uniformity by controlling the droplet diameter of the mist flow jet impingement. Using mist flow jet impingement, we would not only greatly reduce the amount of air and save energy, but also meet the needs of sudden cooling for tempered glass with a thickness of less than 2mm.

Hydrodynamics and heat transfer of yawed circular cylinder

Hydrodynamics and heat transfer from single circular cylinder with yaw angle were examined in the current study. During the experiment the distribution of the Nusselt number on the surface and the 3D velocity field (using stereo-PIV) in the cylinder wake were acquired simultaneously. Experiments were conducted with constant wall temperature conditions, for yaw angles β/°: 90 (cross-flow), 85, 75, 70, 65, 50 and 45; and Reynolds numbers Re:104,1.3×104,2.6×104,4.7×104 and 7.9×104. The results showed the minimum in surface-averaged Nusselt number Nu‾ for yaw angles β/° between 70 and 65. For β/°⩽65 surface-averaged Nusselt number Nu‾ increases up to 32% (at β/°=45, Re=7.9×104) compared with the cross-flow case. Both decrease and increase of Nu‾ is induced by changes of a flow pattern in the wake of the cylinder and, as a result, changes in a velocity field in the vicinity of a rear cylinder surface.

The effect of instability flow for two-dimensional natural convection in a square enclosure with different arrays of two inner cylinders

This study numerically investigates the two-dimensional natural convection in a square enclosure with different arrays of two inner cylinders at Rayleigh numbers of 103≤Ra≤106. A simulation was carried out based on the immersed boundary method to obtain an accurate solution. The results were compared with those of two inner circular cylinders with different vertical locations from a previous study. Detailed analysis results are presented for the distribution of the flow and thermal fields and the time- and surface-averaged Nusselt numbers. The flow and thermal fields eventually reach steady or unsteady states, depending on the variation in the distance between the cylinders.

Drag coefficient and Nusselt number for porous particles under laminar flow conditions

This work studies three-dimensional numerical simulations of heat and fluid flow past and through porous particles. Its main objective is to investigate the influence of particle porosity on particle-averaged drag coefficient and surface-averaged Nusselt number numerically. Reynolds numbers considered are in the range of 10<Re<250 and up to 450 for specific porosities. Fluid flow features inside the particles are explored using an immersed boundary method. For this purpose, the particle porosity is modelled using agglomerates of small particles of spherical or cubical shape, respectively. This provides variation of the specific surface area while porosity is kept constant. The simulation results show a significant influence of porosity and Reynolds number on drag coefficient and Nusselt number. For these relationships new formulae are derived and compared to data from the literature.

Three-Dimensional Study of Multiple-Jet Cross Flow Cooling System with Single Array of Heat Sources

ABSTRACTThe objective of this study is to predict fluid flow and thermal characteristics associated with forced convection cooling of an array of heat sources mounted in a three-dimensional duct by impinging jets in the presence of cross flow. The equations governing fluid flow and heat transfer are solved using Streamline Upwind / Petrov - Galerkin based finite element method. The role of three dimensional flow structures, horseshoe vortex and counter rotating vortex pair in overall heat transfer distribution is examined. A parametric study is performed to analyze the performance of jets in cross flow for various Reynolds numbers (Re = 100, 250, 500) and velocity ratios (VR = 1.0, 5.0, 10.0). It is observed that with the increase in velocity ratio, the enhancement of heat transfer from heat sources occurs due to increase in the dominance of jet flow accompanied by an unfavorable reduction in heat transfer owing to the increase in size of horseshoe vortex. From numerical studies, it can be concluded that VR = 5.0 exhibits higher overall surface averaged Nusselt number (51.7% over VR = 1.0 and 44.1% over VR = 10.0 for Re = 500) compared to other velocity ratios considered in this study.

Influence of hydrodynamic slip on convective transport in flow past a circular cylinder

The presence of a finite tangential velocity on a hydrodynamically slipping surface is known to reduce vorticity production in bluff body flows substantially while at the same time enhancing its convection downstream and into the wake. Here, we investigate the effect of hydrodynamic slippage on the convective heat transfer (scalar transport) from a heated isothermal circular cylinder placed in a uniform cross-flow of an incompressible fluid through analytical and simulation techniques. At low Reynolds (Re > 1) numbers, our theoretical analysis based on Oseen and thermal boundary layer equations allows for an explicit determination of the dependence of the thermal transport on the non-dimensional slip length l(s). In this case, the surface-averaged Nusselt number, Nu transitions gradually between the asymptotic limits of Nu similar to Pe(1/3) and Nu similar to Pe(1/2) for no-slip (l(s) -> 0) and shear-free (l(s) -> infinity) boundaries, respectively. Boundary layer analysis also shows that the scaling Nu similar to Pe(1/2) holds for a shear-free cylinder surface in the asymptotic limit of Re >> 1 so that the corresponding heat transfer rate becomes independent of the fluid viscosity. At finite Re, results from our two-dimensional simulations confirm the scaling Nu similar to Pe(1/2) for a shear-free boundary over the range 0.1 <= Re <= 10(3) and 0.1 <= Pr <= 10. A gradual transition from the lower asymptotic limit corresponding to a no-slip surface, to the upper limit for a shear-free boundary, with l(s), is observed in both the maximum slip velocity and the Nu. The local time-averaged Nusselt number Nu(theta) for a shear-free surface exceeds the one for a no-slip surface all along the cylinder boundary except over the downstream portion where unsteady separation and flow reversal lead to an appreciable rise in the local heat transfer rates, especially at high Re and Pr. At a Reynolds number of 10(3), the formation of secondary recirculating eddy pairs results in appearance of additional local maxima in Nu(theta) at locations that are in close proximity to the mean secondary stagnation points. As a consequence, Nu exhibits a non-monotonic variationwith l(s) increasing initially from its lowermost value for a no-slip surface and then decreasing before rising gradually toward the upper asymptotic limit for a shear-free cylinder. A non-monotonic dependence of the spanwise-averaged Nu on l(s) is observed in three dimensions as well with the three-dimensional wake instabilities that appear at sufficiently low l(s), strongly influencing the convective thermal transport from the cylinder. The analogy between heat transfer and single-component mass transfer implies that our results can directly be applied to determine the dependency of convective mass transfer of a single solute on hydrodynamic slip length in similar configurations through straightforward replacement of Nu and Pr with Sherwood and Schmidt numbers, respectively.

Direct numerical simulations of flow and heat transfer over a circular cylinder at Re = 2000

Unsteady direct numerical simulations of the flow around a circular cylinder have been performed at Re = 2000. Both two-dimensional and three-dimensional simulations were validated with laminar cold flow simulations and experiments. Heat transfer simulations were carried out and the time-averaged local Nusselt number at the cylinder surface was obtained for various Reynolds numbers. Finally, the heat transfer of 2D and 3D simulations are compared. The average Nusselt numbers were found to be in accordance with empirical correlations. The 3D simulation gives a higher heat transfer due to the captured effects of motions in the spanwise direction compared to the 2D simulation. The irregular fluctuation of surface-averaged Nusselt number can be captured by the 3D simulation, while 2D simulation results show a regular fluctuation corresponding to the shedding from the cylinder, similar to that of a laminar flow.

A volume-shrinkage-based method for quantifying the inward solidification heat transfer of a phase change material filled in spherical capsules

In order to study quantitatively the inward solidification heat transfer of a phase change material (PCM) enclosed in spherical capsules, an experimental method was proposed in this Technical Note, which is based upon measurement on the instantaneous volume shrinkage of the PCM during solidification. The shrunk volume was compensated continuously by an extra amount of liquid PCM preserved in a tube, and the descending height in the tube was recorded to calculate the volume shrinkage. Other quantities of interest, solidification fraction and surface-averaged heat flux for example, were readily reduced as single-valued functions of the volume shrinkage. Using 1-tetradecanol as the PCM, the proposed method was realized in laboratory with a stainless steel spherical capsule having an inner diameter of approximately 60mm. Parallel tests were carried out for a set of three boundary cooling temperatures. It was shown that the measured descending heights are reproducible and are in agreement with the theoretical maximum values approaching the end of solidification. The experimental results were then discussed with an emphasis on the transient variations of solidification and heat transfer. Correlations were obtained for both solidification fraction and surface-averaged Nusselt number with respect to the product of Fourier and Stefan numbers.

CFD Analysis of Heat Transfer Due to Jet Impingement Onto a Heated Disc Bounded by a Cylindrical Wall

This computational study aims to evaluate the cooling process due to an impinging oil jet in a cylindrically confined space on both stationary and high-speed reciprocating smooth discs with uniform heat flux and without phase change. The three-dimensional Navier–Stokes and energy equations are numerically solved using a finite-volume discretization. The volume of fluid model is employed in this study to capture the interface in multiphase flow (oil–air). A conjugate heat transfer formulation is used to couple the heat transfer at the interface between the solid and fluid. For jet impingement onto a stationary disc, the stagnation zone Nusselt number is higher and the overall surface average Nusselt number is smaller at a shorter impingement distance. For jet impingement onto a high-speed reciprocating disc, the higher heat transfer coefficient is associated with a higher rotational speed of the driven machine. The higher rotational speed also produces more effective cooling on the disc surface and a lower temperature within the disc.

Effect of buoyancy-assisted flow on convection from an isothermal spheroid in power-law fluids

In this work, the coupled momentum and energy equations have been solved to elucidate the effect of aiding-buoyancy on the laminar mixed-convection from a spheroidal particle in power-law media over wide ranges of the pertinent parameters: Richardson number, 0≤Ri≤5; Reynolds number, 1≤Re≤100; Prandtl number, 1≤Pr≤100; power-law index, 0.3≤n≤1.8, and aspect ratio, 0.2≤e≤5 for the case of constant thermo-physical properties. New results for the velocity and temperature fields are discussed in terms of the streamline and isotherm contours, surface pressure and vorticity contours, drag coefficient, local and surface averaged Nusselt number. The effect of particle shape on the flow is seen to be more pronounced in the case of oblates (e 1). The propensity for wake formation reduces with the rising values of power-law index, Richardson number and slenderness of the body shape (e > 1). Also, the drag coefficient is seen to increase with the Richardson number and power-law index. All else being equal, the Nusselt number shows a positive dependence on the Richardson number and Reynolds number and an inverse dependence on the power-law index and aspect ratio of the spheroid. Limited results were also obtained by considering the exponential temperature dependence of the power-law consistency index. This factor can increase the values of the average Nusselt number by up to ~10-12% with reference to the corresponding values for the case of the constant thermo-physical properties under otherwise identical conditions. Finally, the present values of the Nusselt number have been consolidated in the form of Colburn j-factor as a function of the modified Reynolds and Prandtl numbers for each value of the aspect ratio (e). The effect of the temperature dependent viscosity is included in this correlation in terms of a multiplication factor.

Assessment of turbulence models in natural convection from two- and three-dimensional rectangular enclosures

In this study, unsteady turbulent natural convection heat transfer in two- and three-dimensional rectangular enclosures is investigated numerically. The enclosures are heated and cooled from the vertical opposing isothermal walls. All other side walls are assumed to be smooth and adiabatic. The working fluid is air (Pr = 0.71), and the flow is turbulent. Two- and three-dimensional unsteady-state continuity, Reynolds-Averaged Navier-Stokes (RANS), along with the averaged-energy equation, are solved using the commercial software – FLUENT 6.3.26®. Standard k−ε (SKE), Re-Normalization Group k−ε (RNGKE), Realizable k−ε (RKE), Reynolds Stress Model (RSM), Standard k-ω (SKW) and Shear Stress Transport k−ω (SSTKW) RANS models are used in conjunction with the two-layer (or Enhanced Wall Treatment, EWT) wall model for Rayleigh numbers ranging from 108 to 1013. The performance of the turbulence models on the heat transfer rate predictions is comparatively investigated for 2D square enclosure, as well as 3D rectangular enclosures having the slenderness ratio of H/W = 1 and H/W = 10. The heat transfer rate predictions are assessed for each case via surface-averaged mean Nusselt numbers over the hot wall, and empirical power-law correlations are derived for 2D and 3D enclosures for each turbulence model. The study reveals that 3D laminar and RANS models yield almost identical mean Nusselt number predictions up to Ra = 1010, and these predictions are compatible with those of obtained from 2D simulations. For larger Rayleigh numbers, the mean Nusselt numbers the flow becomes three-dimensional and 2D RANS models do not yield accurate predictions. The study reveals that 3D RANS models yield more accurate mean Nusselt numbers, and a mean Nusselt number correlation is proposed.

Heat transfer characteristics of aluminum metal foam subjected to a pulsating/steady water flow: Experimental and numerical approach

The microminiaturization of electronic components has led to an increase in the heat dissipation rate (per unit area) of electronic chips. This paper presents the results of an experimental and numerical study of the use of aluminum foam as a heat sink for the Intel core i7 processor. The aluminum foam was subjected to a pulsating water flow with a frequency range between 0.04 and 0.1Hz, an amplitude range between 297 and 1353, and a heat flux range between 8.5 and 13.8W/cm2. The distributions of the cycle average local surface temperature over the heater representing the electronic surface and cycle average local Nusselt number were measured and compared with the numerical data obtained using the finite element method. The numerical results were in good agreement with the experimental data for different flow amplitudes and heat flux within a maximum relative error of 0.75% for the local temperature and 1.8% for the local Nusselt number. The pressure drop across the aluminum foam heat sink was measured and the effects of the dimensionless flow frequency and the flow amplitude on the heat transfer characteristics of the pulsating flow through the aluminum foam were analyzed. An empirical correlation of the average Nusselt number as a function of the dimensionless flow frequency and flow amplitude was developed. A comparison between the heat transfer characteristics of the steady and pulsating water flows was also conducted. The experimental results revealed that the cycle average local temperature decreases along with decreasing heat flux, increasing the pulsating flow frequency and amplitude. The local temperature distribution shows a convex profile due to the reversing flow and development of a boundary layer. Results also revealed a 14% increase in the average Nusselt number for pulsating flow when compared to the steady flow. Lastly, the thermal performance of the aluminum foam heat sink was evaluated. The experimental result revealed a 73% decrease in the uniformity index for pulsating flow when compared to the steady flow, indicating that the pulsating water flow leads to a more uniform temperature distribution. This finding is very important to the field of electronic cooling since the reliability of transistors and the operating speed are dependent upon temperature uniformity along the surface.

Mixed convection from a spheroid in Bingham plastic fluids: Effect of buoyancy-assisted flow

ABSTRACTIn this work, laminar mixed convection from an isothermal spheroidal particle immersed in a Bingham plastic fluid is studied numerically in the buoyancy-assisted regime. The results reported herein encompass the following ranges of conditions: Reynolds number, 0.1 ≤ Re ≤ 100; Prandtl number, 10 ≤ Pr ≤ 100; Bingham number, 0 ≤ Bn ≤ 100; Richardson number, 0 ≤ Ri ≤ 8; and aspect ratio of the spheroid, 0.2 ≤ e ≤ 5. In particular, consideration is given to the effect of shape and orientation of the particle on the detailed flow and temperature fields (in terms of streamlines, iso-vorticity, and isotherm contours), morphology of the yielded–unyielded regions, and the local and surface-averaged Nusselt number. All else being equal, the propensity for flow separation is seen to be greater for oblates (e < 1) than that for prolates (e > 1). In both cases, this reduces with the increasing Bingham number and/or the Richardson number. Both drag coefficient and the Nusselt number show a positive dependence on the Bingham number as well as on the Richardson number. Overall, the drag coefficient increases as the particle shape changes from an oblate to prolate, whereas the reverse trend is obtained for the average Nusselt number, which is in line with the general inference that more drag corresponds to more heat transfer. Finally, the average Nusselt number is correlated with the pertinent dimensionless parameters (Re, Pr, Bn, Ri, e) via a simple correlation, thereby enabling its prediction for intermediate values of the parameters and/or in a new application.

Thermodynamic investigation and multi-objective optimization for jet impingement cooling system with Al2O3/water nanofluid

In the present study, a multi-objective optimization procedure combining finite element modeling of impingement cooling system, Response Surface Approximation (RSA) of objective functions and optimization based on Multi-Objective Genetic Algorithm (MOGA), to achieve maximum heat transfer and minimum entropy generation is demonstrated. For the purpose, numerical simulations are performed for impingement cooling system with Al2O3/water nanofluid, to investigate the influence of Reynolds number (Re), non-dimensional channel height (H/L) and nanoparticle volume fraction (ϕ) on fluid flow, heat transfer and entropy generation. The simulated results illustrate that, a secondary recirculation bubble observed on upper surfaces of fourth (at Re⩾500) and fifth (for Re⩾800) heat sources, lead to an accumulation of heat. The magnitude of local Nusselt number (Nu) is found to be maximum along stagnation region whereas in the regions of secondary recirculation a minimum value is observed. Further, an increase in overall surface averaged Nusselt number (Nuov‾) and global total entropy generation (Stot,Ω) is observed with increasing Re,ϕ and decreasing H/L. Subsequently, Nuov‾ and Stot,Ω are selected as objective functions and are modeled using RSA. Furthermore, MOGA has been implemented to obtain optimum configurations of impingement cooling system encapsulating in the functional space lying on the Pareto-optimal frontier where a trade-off between two performance parameters, Nuov‾ and Stot,Ω are obtained.

Effect of thermal boundary condition of an inner cube on three-dimensional natural convection in a cubical

We numerically investigated three-dimensional (3-D) natural convection in a vertical cubic enclosure with an inner cube for Rayleigh numbers (Ra) in the range of 103 ≤ Ra ≤ 106. For the inner cube at the center, four different thermal boundary conditions (adiabatic, neutral, and hot and cold isothermal conditions) were considered in order to investigate their effect on flow and thermal fields. For Ra = 103 and Ra = 104, single circulation appears regardless of the thermal boundary condition of the inner cube. When Ra = 105 and Ra = 106, the combined effects of the inner cube as a bluff body and the thermal condition imposed on the inner cube on the fluid flow and thermal fields are significant, and intensify the 3-D effect. Generally, for Ra = 105 and Ra = 106, the convective flow is characterized by the formation of two inner vortices embedded in the primary circulation, and by secondary vortices due to flow separation at the edge of the inner body. As Ra increases, the local Nusselt number varies rapidly in the vertical direction, which is supported by the temperature isosurfaces that form an S-shape. The total surface-averaged Nusselt numbers for the different cases have approximately the same profile with respect to the Rayleigh number as the power function.

Effect of Prandtl Number on Free Convection from Two Cylinders in a Square Enclosure

This study explores the effect of Prandtl number on the laminar natural convection heat transfer to Newtonian fluids in a square enclosure consisting of one hot circular cylinder and one cold circular cylinder. The walls of the square enclosure are maintained isothermal and at the same temperature as the cold cylinder and the fluid medium. The governing partial differential equations have been solved numerically over the following ranges of conditions: Grashof number, 10 to 105; Prandtl number, 0.7 to 100 (or the range of Rayleigh numbers as 7 to 107); and relative positioning of the cylinders, −0.25 to 0.25. However, the ratio of the radius of the cylinder to the side of the enclosure is held fixed at 0.2. Extensive results on the streamline and isotherm contours, the local Nusselt number distribution, and the average Nusselt number are discussed to delineate the influence of Grashof and Prandtl numbers on them for a given location with respect to the horizontal center line. The surface-averaged Nusselt number shows a positive dependence on Grashof and Prandtl numbers for a fixed location of the two cylinders. The heat transfer results have been correlated as a function of the Rayleigh number and geometric parameters, thereby enabling its prediction in a new application.

Laminar Free Convection in Bingham Plastic Fluids from an Isothermal Semicircular Cylinder

Laminar free convection heat transfer in Bingham plastic fluids from an isothermal semicircular cylinder has been studied numerically over the following ranges of conditions: Rayleigh number ; Prandtl number and Bingham number . New extensive results on the streamline and isotherm contours adjacent to the semicircular cylinder, yield surfaces separating the fluidlike and solidlike regions, drag coefficient, and local and surface average Nusselt number are discussed in detail, especially with reference to the corresponding behavior observed in ordinary Newtonian fluids. The extent of the unyielded regions progressively increases with Bingham number and/or with the decreasing value of the Rayleigh number. Naturally, conduction dominates heat transfer in the unyielded regions, and convection is relevant only to the yielded regions. In summary, for a given Bingham number, the Nusselt number bears a positive dependence on the Rayleigh number. On the other hand, as the value of the Bingham number is gradually increased, the average Nusselt number decreases from its maximum value in Newtonian media () to its conduction limit () when practically most of the fluid is frozen. Finally, the present numerical values of the average Nusselt number have been consolidated in terms of the modified Rayleigh number and Prandtl number , thereby enabling its prediction in a new application and/or for the intermediate values of the governing parameters.

Modelling of the mixed convection in a lid-driven cavity with a constant heat flux boundary condition

In this paper steady state two-dimensional mixed convection heat transfer problem in a lid-driven cavity heated via an uniformly distributed heat flux on the bottom wall is investigated numerically. The lid moves with constant velocity and is kept at low constant temperature, between two ideally thermally insulated vertical walls. A wide range of Prandtl Pr and Richardson Ri number is examined to study their effects on heat transfer rate and fluid flow. Governing parameters are 0.001 ≤ Ri ≤ 1.0 and 0.71 ≤ Pr ≤ 56.00. Grashof number Gr is fixed at 104. The results are presented in the form of isotherms and streamlines plots. Also, local and mean Nusselt number are depicted on charts. Numerical values of the surface averaged Nusselt number are also presented. Results show that increase of Prandtl number strongly influences enhancement of heat transfer rate and that decreasing of Richardson number increases surface averaged Nusselt number. Mechanisms responsible for intensification of heat transfer are identified and physical explanation of this phenomenon are also given.

Analysis of Mixed Convective Heat Transfer in a Ribbed Channel Using the Lattice Boltzmann Method

This article presents a two-dimensional (2D) numerical study on hydrodynamics and thermal transport in the presence of superimposed thermal buoyancy in a rectangular channel with staggered ribs mounted on both walls. This study is carried out for the Newtonian fluid at moderate Reynolds number. The Reynolds number is considered within the range (50 ≤ Re ≤ 120) with constant Prandtl number Pr = 0.7, and mixed convective heat transfer is studied for Richardson number range 0 ≤Ri≤2. The unsteady 2D governing equations are solved using the lattice Boltzmann method. The computational strategy is applied in various test cases and validated with the results reported in the literatures. The flow and heat transfer characteristics are analyzed with the streamlines and isotherm patterns at various Reynolds and Richardson numbers. The effects of superimposed thermal buoyancy on flow and isotherm patterns, drag coefficient, local Nusselt number distribution, and time- and surface-average Nusselt number are presented and discussed.

Natural convection in a square enclosure with four circular cylinders positioned at different rectangular locations

A numerical study was carried out to investigate natural convection heat transfer induced by a temperature difference between a cold outer square enclosure and four hot inner circular cylinders placed in a rectangular array. A two-dimensional solution for natural convection in an enclosure with four cylinders was obtained using an accurate and efficient immersed boundary method. The immersed boundary method based on the finite volume method was used to handle four inner cylinders placed equal distance away from each other within an enclosure moving along the diagonal centerlines for different Rayleigh numbers in the range 103⩽Ra⩽106. This study investigated the effects of the location of the cylinders in the enclosure on the heat transfer and fluid flow, when they moved along the diagonal centerlines of the enclosure. For all the cases considered in the present study, the flow and thermal fields eventually reached steady state in the range of 103⩽Ra⩽104. However, for 105⩽Ra⩽106, the flow and thermal fields became unsteady depending on the cylinders position. Also, the symmetry about the vertical centerline was affected by the cylinders position as well. The distribution of the instantaneous, time-averaged local and surface-averaged Nusselt number as a function of the Rayleigh number and the distance between inner cylinders was also presented.