A numerical study of laminar free convection heat transfer between inner sphere and outer vertical cylinder

The effects of height and radius ratio with a Newtonian fluid have been investigated numerically to determine heat transfer by natural convection between the sphere and vertical cylinder with isothermal boundary conditions. The inner sphere and outer vertical cylinder were heated and cooled in a steady change of temperature. Calculations were carried out systematically for a range of the Rayleigh numbers to determine the average Nusslet numbers which are affected by the geometric ratio parameters (HR and RR) on the flow and temperature fields. The governing equations, in terms of vorticity, stream function and temperature are expressed in a spherical polar coordinate system. Results of the parametric study conducted further reveal that the heat and flow fields are primarily dependent on the Rayleigh number and height and radius ratio, for a Prandtl number of 0.7, with the Rayleigh number ranging from 10³ to 10⁶, and the height and radius ratio varying from 1.2 to 5.0. Above all, the specification of different convective configurations has a significant effect on the average heat transfer rate across the composite annulus gap.     

Assessment of thermal characteristics of square wavy plates

In this paper, the thermal characteristics and air flow behavior over heated square plate of wavy configurations are assessed. Two cases of heated wavy configurations were considered, one having an elliptical and other with the sinusoidal cross‐sectional shape to augment heat transfer rate over the plate. To explore the impact of wavy configurations of plate on convective heat transfer, the Rayleigh numbers in the range of 10⁴ to 10¹³ were considered. The steady‐state two‐dimensional momentum and energy equations were solved using a validated numerical model. The numerical results in terms of Nusselt numbers, Rayleigh numbers, aspect ratio (AR), and flow/temperature distribution over the plate are presented. The variations in the heat transfer coefficient and temperature contours with Rayleigh number and AR of waves were concentrated for both the elliptical and sinusoidal configurations of the plate. The results obtained indicate that the enhancement in heat transfer rate from the heated plate depends on the Rayleigh number, the AR of waves, plate configuration, and the Prandtl number.     

Three-dimensional natural convection in an enclosure with a sphere at different vertical locations

Numerical calculations are carried out for the three-dimensional natural convection induced by a temperature difference between a cold outer cubic enclosure and a hot inner sphere. The immersed-boundary method (IBM) to model a sphere based on the finite volume method is used to study a three-dimensional natural convection for different Rayleigh numbers varying in the range of 10³–10⁶. This study investigates the effect of the inner sphere location on the heat transfer and fluid flow. The flow and thermal fields eventually reach the steady state for all Rayleigh numbers regardless of the sphere location. For Rayleigh numbers of 10⁵ and 10⁶, the variation of local Nusselt number of the sphere along the circumferential direction is large, showing the strong three dimensionality of the natural convection in the enclosure unlike to the cases of lower Rayleigh numbers of 10³ and 10⁴. For the highest Rayleigh number, the local peaks of the Nusselt number on the top wall of the enclosure shows the sinusoidal distribution along the circumferential direction. The flow and thermal fields, and the local and surface-averaged Nusselt numbers on the sphere and the enclosure are highlighted in detail.     

Turbulent natural convection of liquid deuterium,hydrogen and nitrogen within enclosed vessels

Quasi-steady natural convection of liquid deuterium, hydrogen, and nitrogen within a sphere, hemisphere, horizontal cylinder, and vertical cylinder has been studied experimentally for the case of a nearly uniform wall temperature. A single expression relating the Nusselt and Rayleigh numbers, Nu = 0·104Ra^0·352, fits the deuterium and nitrogen data over the range 7 × 10⁸ < Ra < 6 × 10¹¹, while the hydrogen Nusselt numbers are 8 per cent lower. The temperature field within the vessels is virtually free of horizontal temperature gradients. A single dimensionless temperature profile characterizes the vertical temperature distribution for each vessel shape, with the profiles for the sphere, hemisphere, and horizontal cylinder being nearly identical.     

Numerical investigation of natural convection phenomena in a uniformly heated circular cylinder immersed in square enclosure filled with air at different vertical locations

In this model, a numerical study of two dimensional steady natural convection is performed for a uniform heat source applied on the inner circular cylinder in a square air (Pr = 0.7) filled enclosure in which all boundaries are assumed to be isothermal (at a constant low temperature). The developed mathematical model is governed by the coupled equations of continuity, momentum and energy and is solved by finite volume method. The effects of vertical cylinder locations and Rayleigh numbers on fluid flow and heat transfer performance are investigated. Rayleigh number is varied from 10³ to 10⁶ and the location of the inner cylinder is changed vertically along the centerline of the enclosure from − 0.25 L to 0.25 L upward and downward, respectively. It is found that at small Rayleigh numbers does not have much influence on the flow field while at high Rayleigh numbers have considerable effect on the flow pattern. In addition, the numerical solutions yield a two cellular flow field between the inner cylinder and the enclosure. Also, the total average Nusselt number behaves nonlinearly as a function of locations. Results are presented in terms of the streamlines, isotherms, local and average Nusselt numbers. Detailed results of the numerical has been compared with literature ones, and it gives a reliable agreement.     

Natural convection from a confined horizontal cylinder: the optimum distance between the confining walls

The laminar natural convection from an isothermal horizontal cylinder confined between vertical walls, at low Rayleigh numbers, is investigated by theoretical, experimental and numerical methods. The height of the walls is kept constant, however, their distance is changed to study its effect on the rate of the heat transfer. Results are incorporated into a single equation which gives the Nusselt number as a function of the ratio of the wall distance to cylinder diameter, t/D, and the Rayleigh number. There is an optimum distance between the walls for which heat transfer is maximum.     

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.     

Natural convective flow in an inclined lid-driven enclosure with a heated thin plate in the middle

Natural convection heat transfer from a heated thin plate located in the middle of a lid-driven inclined square enclosure has been analyzed numerically. Left and right of the cavity are adiabatic, the two horizontal walls have constant temperature lower than the plate’s temperature. The study is formulated in terms of the vorticity-stream function procedure and numerical solution was performed using a fully higher-order compact (FHOC) finite difference scheme on the 9-point 2D stencil. Air was chosen as a working fluid (Pr = 0.71). Two cases are considered depending on the position of heated thin plate (Case I, horizontal position; Case II, vertical position). Governing parameters, which are effective on flow field and temperature distribution, are Rayleigh number values (Ra) ranging from 10³ to 10⁵ and inclination angles γ (0° ? γ < 360°). The fluid flow, heat transfer and heat transport characteristics were illustrated by streamlines, isotherms and Nusselt number (Nu). It is found that fluid flow and temperature fields strongly depend on Rayleigh numbers and inclination angles. Further, for the vertical located position of thin plate heat transfer becomes more enhanced with lower γ at various Rayleigh numbers.     

Thermal description of pseudosteady-state natural convection inside a vertical cylinder

The heat transfer correlation, minimum temperature, and mean temperature are presented for pseudosteady-state natural convection heat transfer to a fluid inside a vertical cylinder. The SIMPLER numerical method was used for calculation in the range 0.25 < H/D <2, Ra < 10⁷, and Pr = 7. This range includes conduction to weak turbulence. The overall heat transfer for the convection-dominated range was found to be correlated by Nu = 0.519Ra^0.255 where the temperature difference for both the Nusselt and Rayleigh numbers was the center temperature minus the inside wall temperature. Correlations using other temperature differences are also presented and provide a method for prediction of the mean temperature, minimum temperature, or center temperature by knowing any one of them.     

Natural convection on inclined QFN32 electronic package generating constant volumetric heat flux

Qualification and quantification of the natural convective phenomena are examined in the case of a Quad Flat Non-lead (QFN32). This active electronic package is inclined with respect to the horizontal plane by an angle varying between 0° and 90° corresponding to the horizontal and vertical position respectively. It generates during its operation a constant volumetric heat flux leading to Rayleigh numbers varying in the range 1.31x10⁷ ‐ 1.01x10⁸. The walls of the large air-filled cubic cavity containing this device are maintained isothermal. The temperature and velocity fields are presented for different combinations of the Rayleigh number and inclination angle. The convective heat transfer concerning the whole component exchange surface is determined for all the treated configurations. Correlations of Nusselt–Rayleigh type are proposed. They allow optimizing the thermal design of electronic assemblies used in various engineering domains.     

Experimental Study of Free Convection Heat Transfer from Horizontal Isothermal Cylinders Arranged in Vertical and Inclined Arrays

Laminar free convection heat transfer from vertical and inclined arrays of horizontal isothermal cylinders in air were investigated experimentally. Experiments were carried out using a Mach-Zehnder interferometer. For the vertical array, the cylinder spacing (center to center) varied from 2 to 5 cylinder diameter. The same range of vertical spacing also was used for the inclined array. The horizontal spacing varied from 0 to 2 cylinder diameter in the inclined array. The Rayleigh number based on the cylinder diameter varied between 10³ and 3× 10³. The effect of vertical and horizontal cylinder spacing and Rayleigh number on the heat transfer from each individual cylinder and the whole array were investigated. It is found that the free convection heat transfer from any individual cylinder in the array depends on its position relative to the others. Heat transfer correlations have been developed for any individual cylinder in the vertical and inclined arrays and for the arrays. Also the experiment was carried out on a single cylinder for a comparison with the results from other research.     

Lattice Boltzmann simulation of natural convection around a horizontal elliptic cylinder inside a square enclosure

Natural convection between a square outer cylinder and a heated elliptic inner cylinder has been studied numerically. The inner and outer walls are maintained at temperatures T_h and T_c, respectively, with T_h > T_c. Lattice Boltzmann method (LBM) has been used to investigate the hydrodynamic and thermal behaviors of the fluid at various vertical positions of the inner cylinder for different Rayleigh numbers ranging from 10³ to 10⁶. The results show that streamlines, isotherms, and the number, size and formation of the cells strongly depend on the Rayleigh number and the position of inner cylinder. The changes in heat transfer quantities have also been presented.     

Multi-Prandtl correlating equations for free convection heat transfer from a horizontal tube of elliptic cross-section

Steady laminar free convection from a horizontal elliptic cylinder set in unbounded space is studied numerically under the assumption of uniform surface temperature. A specifically developed computer-code based on the SIMPLE-C algorithm is used for the solution of the mass, momentum and energy transfer governing equations. Simulations are performed for ratios between the minor and major axes of the elliptic cross-section of the cylinder in the range between 0.05 and 0.98, inclination angles of the major axis of the elliptic cross-section with respect to gravity in the range between 0° and 90°, Rayleigh numbers based on the major axis of the elliptic cross-section in the range between 10 and 10⁷, and Prandtl numbers in the range between 0.7 and 700. It is found that the heat transfer rate increases with increasing the Rayleigh and Prandtl numbers, while decreases with increasing the orientation angle of the cross-section of the cylinder, i.e., passing from the slender to the blunt configuration. In addition, a noteworthy fact is that in most cases the amount of heat exchanged at the cylinder surface has a peak at an optimum axis ratio which is practically independent of the Prandtl number, while may either increase or decrease with increasing the Rayleigh number depending on whether the orientation angle of the tube is above or below a critical value of approximately 67.5°. Dimensionless correlating equations are proposed both for the optimum axis ratio for maximum heat transfer and for the heat transfer rate from the cylinder surface to the undisturbed surrounding fluid reservoir.     

Influences of Boundary Condition on Laminar Natural Convection of Bingham Fluids in Square Cross-Sectioned Cylindrical Annular Enclosures with Differentially Heated Vertical Walls

Numerical simulations have been carried out to analyze steady-state laminar natural convection of yield stress fluids obeying Bingham model in square cross-sectioned cylindrical annular enclosures with differentially heated vertical walls for both constant wall temperature and constant wall heat flux boundary conditions for active walls. The simulations have been performed under the assumption of axisymmetry for a nominal Rayleigh number range of 10³ to 10⁶ and nominal Prandtl number range of 10 to 10³ for different ratio of internal cylinder radius to cylinder height range of 0.125 to 16. The mean Nusselt number on the inner periphery for the constant wall heat flux configuration has been found to be smaller than that in the case of constant wall temperature configuration for a given set of values of nominal Rayleigh and Prandtl numbers for both Newtonian and Bingham fluid cases. The mean Nusselt number normalized by the corresponding value obtained for pure conductive transport increases with increasing internal radius before approaching the corresponding mean Nusselt number for square enclosures regardless of the boundary conditions. Detailed physical explanations have been provided for the effects of the aforementioned parameters on the mean Nusselt number on the inner periphery. Finally, the new Nusselt number correlations have been proposed for laminar natural convection of both Newtonian and Bingham fluids in square cross-sectioned cylindrical annular enclosures for both constant wall temperature and constant wall heat flux boundary conditions.     

Lattice Boltzmann simulation of natural convection heat transfer in an elliptical-triangular annulus

A numerical study for steady-state, laminar natural convection in a horizontal annulus between a heated triangular inner cylinder and cold elliptical outer cylinder was investigated using lattice Boltzmann method. Both inner and outer surfaces are maintained at the constant temperature and air is the working fluid. Study is carried out for Rayleigh numbers ranging from 1.0 × 10³ to 5.0 × 10⁵. The effects of different aspect ratios and elliptical cylinder orientation were studied at different Rayleigh numbers. The local and average Nusselt numbers and percentage of increment heat transfer rate were presented. The average Nusselt number was correlated. The results show that by decreasing the value of aspect ratio and/or increasing the Rayleigh number, the Nusselt number increases. Also the heat transfer rate increases when the ellipse positioned vertically.     

Optimal geometry of L and C-shaped channels for maximum heat transfer rate in natural convection

The present paper documents the geometric optimization of L and C-shaped channels in laminar natural convection subject to global constraints. The objective is to maximize the heat transfer rate from the hot wall to the coolant fluid. Three different configurations were considered: (i) an L-shaped asymmetric vertical heated channel with an adiabatic horizontal inlet, (ii) an asymmetric vertical heated channel with an adiabatic vertical outlet, and finally, (iii) a C-shaped vertical channel with horizontal inlet and outlet. The two first configurations are free to morph according to two degrees of freedom: the wall-to-wall spacing and inlet (or outlet) height. The third configuration is optimized with respect to the wall-to-wall spacing, and the heights of the inlet and outlet ports. The effect of the inlet or outlet horizontal adiabatic duct lengths is also investigated. The optimization is performed numerically by using the finite element technique, in the range 10⁵ < Ra < 10⁷ for Pr = 0.7, where Ra is the Rayleigh number based on a fixed total height H of the channel. The numerical results show that optimization is relevant, since the three degrees of freedom considered have a strong effect on the heat transfer delivered from the hot wall to the fluid. The optimal geometric characteristics obtained numerically (i.e., optimal spacing, optimal height and lengths) are reported and correlated within a 7.5% maximal disagreement range.     

NUMERICAL STUDY OF DEVELOPING NATURAL LAMINAR CONVECTION IN A VERTICAL HYPERBOLIC DUCT OF A FIXED LENGTH AND WITH A CONSTANT WALL TEMPERATURE

The natural laminar convection in a vertical hyperbolic duct of a fixed length and with a constant wall temperature is numerically investigated. The governing equations are solved by a finite difference method. The results are obtained for the velocity, temperature, and pressure fields, and for the mean heat transfer coefficient. The numerical calculations are fulfilled for Rayleigh numbers ( Ra ) ranging from 5 to 3 · 104 and for the numerical eccentricity ranging from 5 to 100. The effects of the numerical eccentricity and Ra are examined and the results are compared with those of a cylindrical vertical duct. It was found that the flow fields and Nusselt number ( Nu ) are affected significantly at small values of the numerical eccentricity and Ra .     

A numerical study of natural convection in a square enclosure with a circular cylinder at different vertical locations

Numerical calculations are carried out for natural convection induced by a temperature difference between a cold outer square enclosure and a hot inner circular cylinder. A two-dimensional solution for unsteady natural convection is obtained, using the immersed boundary method (IBM) to model an inner circular cylinder based on the finite volume method for different Rayleigh numbers varying over the range of 10³–10⁶. The study goes further to investigate the effect of the inner cylinder location on the heat transfer and fluid flow. The location of the inner circular cylinder is changed vertically along the center-line of square enclosure. The number, size and formation of the cell strongly depend on the Rayleigh number and the position of the inner circular cylinder. The changes in heat transfer quantities have also been presented.     

Enhancement of Free Convection Heat Transfer from a Vertical Array of Isothermal Cylinders by Flow Diverters

Free convection heat transfer from a vertical array of five horizontal isothermal cylinders separated by flow diverters is investigated experimentally through a Mach-Zehnder interferometer. Flow diverters with widths of 1, 2, and 3 cylinder-diameter are placed midway between the cylinders with a 45-degree inclination angle. The cylinders vertical center to center spacing is kept constant to three-cylinder diameter. The experiments are carried out for various Rayleigh numbers based on the cylinder diameter in the range of 10³ to 2.5× 10³ and for different flow diverter width. It is observed that the flow diverters enhance the heat transfer of the array from 10% to 27%, depending on their width. By increasing the width of the diverters and the Rayleigh number, enhancement of the heat transfer from the array increases.     

Experimental study of natural convection heat transfer of air layers in vertical annuli under high rayleigh number conditions

In this paper experimental investigations of natural convection heat transfer of air layers in vertical annuli are presented. In these experiments, the surface of the inner cylinder is maintained at a constant heat flux condition and the outer cylinder is cooled in the atmosphere. In order to obtain the convective contribution, the overall heat transfer data are corrected for thermal radiation and axial conduction losses from the end plates in the annuli. Special emphasis, in these investigations, was placed on the high Rayleigh number region where no experimental data are available in the literature. Data were obtained for Rayleigh numbers greater than 10⁹. The radius ratios studied were 2.03 and 3.92, and the aspect ratios studied were 23.94 and 66.67. Heat transfer correlations for average Nusselt numbers were developed for different Rayleigh number regions. For the low Rayleigh number region the results of this paper agree with the correlations reported in the literature. Much needed data and correlations for the high Rayleigh number region are obtained for the first time. These results improve the predictive ability for the heat transfer characteristics in the high Rayleigh number region. ©1999 Scripta Technica, Heat Trans Asian Res, 28(1): 50–57, 1999