Natural convection and conduction heat transfer in open shallow cavities with bounding walls

A numerical study has been carried out on inclined open shallow cavities, which are formed by a wall and horizontal fins. Constant heat flux is applied on the surface of the wall inside the cavity while its other surface was kept isothermal. The wall and the fins are conductive. Conjugate heat transfer by natural convection and conduction is studied by numerically solving equations of mass, momentum and energy. Streamlines and isotherms are produced, heat and mass transfer is calculated. A parametric study is carried out using following parameters: Rayleigh number from 10⁶ to 10¹², conductivity ratio from 1 to 60, open cavity aspect ratio from 1 to 0.125, dimensionless end wall thickness from 0.05 to 0.20, horizontal walls from 0.01 to 0.15 and inclination of the end wall from 90° to 45°. It is found that the volume flow rate and Nusselt number are a decreasing function of the cavity aspect ratio, horizontal fin thickness and conductivity ratio. They are an increasing function of end wall thickness and inclination angle, except in the latter case optima exist at high Rayleigh numbers.     

Natural convection in inclined rectangular enclosures with perfectly conducting fins attached on the heated wall

The natural convection heat transfer in inclined rectangular enclosures with perfectly conducting fins attached to the heated wall is numerically studied. The parameters governing this problem are the Rayleigh number (10²≤Ra≤2×10⁵), the aspect ratio of the enclosures (2.5≤A=H′/L′≤∞), the dimensionless lengths of the partitions (0≤B=?′/L′≤1), the aspect ratio of micro-cavities (A≤C=h′/L′≤0.33), the inclination angle (0≤φ≤60^°) and the Prandtl number (Pr=0.72). The results indicate that the heat transfer through the cover is considerably affected by the presence of the fins. At low Rayleigh numbers, the heat transfer regime is dominated by conduction. When B≈0.75 and C≈0.33, the heat transfer through the cold wall decreases considerably. This trend is enhanced when the enclosure is inclined. Useful engineering correlations are derived for practical applications.     

Natural convection in an inclined enclosure containing internal energy sources and cooled from below

Natural convection in an inclined enclosure from below and containing internally heated fluid has been investigated using a finite difference calculation procedure. Results have been obtained for Rayleigh number values up to 10⁶ and for inclination angles of 30 and 60°. For internal Rayleigh numbers that are much larger than the external Rayleigh number, the flow rises in the interior and moves down both the hot and cold walls. On the other hand, if the external Rayleigh number has a larger magnitude, the flow moves upwards along the hot surface and downwards along the cold surface. For the latter situation, the inner core is multicellular in nature at large external Rayleigh numbers. The average heat flux ratio along the cold surface (convective heat flux/corresponding conduction heat flux) increases with increasing external Rayleigh number and decreasing internal ratio is non-monotonic in nature. The heat flux ratio along both surfaces is observed to be strongly dependent on the inclination angle at high external Rayleigh numbers. A maximum in the local heat flux along the cold surface is obtained in the vicinity of x/L = 1 where hot fluid, either from the interior or directly from the opposite hot wall, meets the surface. Along the hot wall, a maximum in the heat flux ra flo     

Numerical and experimental studies of natural convective heat transfer from vertical and inclined narrow isothermal flat plates

Natural convective heat transfer from an isothermal narrow flat plate embedded in a plane adiabatic surface and inclined at moderate positive and negative angles to the vertical has been numerically and experimentally studied. The solution has the Rayleigh number, the dimensionless plate width, the angle of inclination, and the Prandtl number as parameters. Attention was restricted to a Prandtl number of 0.7. The numerical results have been obtained for Rayleigh numbers between 10³ and 10⁷ for dimensionless plate widths of between 0.3 and 1.2 and for angles of inclination between +45° and −45°. In the experimental study, results have been obtained for Rayleigh numbers between 4 × 10² and 10⁵ for dimensionless plate widths of 0.4 and 2.5 and for angles of inclination between +45° and −45° to the vertical. Empirical equations for the heat transfer rate have been derived.     

Lattice Boltzmann simulation of natural convection in nanofluid-filled 2D long enclosures at presence of magnetic field

In this paper, the effects of a magnetic field on natural convection flow in filled long enclosures with Cu/water nanofluid have been analyzed by lattice Boltzmann method. This study has been carried out for the pertinent parameters in the following ranges: the Rayleigh number of base fluid, Ra = 10³–10⁵, the volumetric fraction of nanoparticles between 0 and 6 %, the aspect ratio of the enclosure between A = 0.5 and 2. The Hartmann number has been varied from Ha = 0 to 90 with interval 30 while the magnetic field is considered at inclination angles of θ = 0°, 30°, 60° and 90°. Results show that the heat transfer decreases by the increment of Hartmann number for various Rayleigh numbers and the aspect ratios. Heat transfer decreases with the growth of the aspect ratio but this growth causes the effect of the nanoparticles to increase. The magnetic field augments the effect of the nanoparticles at high Rayleigh numbers (Ra = 10⁵). The effect of the nanoparticles rises for high Hartmann numbers when the aspect ratio increases. The rise in the magnetic field inclination improves heat transfer at aspect ratio of A = 0.5.     

Analytical and numerical study of natural convection heat transfer in an inclined composite enclosure

Laminar natural convection heat transfer in inclined fluid layers divided by a partition with finite thickness and conductivity is studied analytically and numerically. The governing equations for the fluid layers are solved analytically in the limit of a thin layered system with constant flux boundary conditions. The study covers of the range of Ra from 10³ to 10⁷, from 0° to 180° and the thermal conductivity ratio of partition to fluid ratioK from 10^–2 to 10⁶. The Prandtl number was 0.72 (for air). Results are obtained in terms of an overall Nusselt number as a function of Rayleigh number, angle of inclination of the system, mid layer thickness, and mid layer thermal conductivity. The critical Rayleigh number for the onset of convection in a bottom-heated horizontal system is predicted. The results are compared with the numerical results obtained by solving the complete system of governing equations, using SIMPLER method, as well as with the limiting cases in the literature.     

Effect of Conduction in Bottom Wall on Darcy–Bénard Convection in a Porous Enclosure

Conjugate natural convection-conduction heat transfer in a square porous enclosure with a finite-wall thickness is studied numerically in this article. The bottom wall is heated and the upper wall is cooled while the verticals walls are kept adiabatic. The Darcy model is used in the mathematical formulation for the porous layer and the COMSOL Multiphysics software is applied to solve the dimensionless governing equations. The governing parameters considered are the Rayleigh number (100 ≤ Ra ≤ 1000), the wall to porous thermal conductivity ratio (0.44 ≤ K _r ≤ 9.90) and the ratio of wall thickness to its height (0.02 ≤ D ≤ 0.4). The results are presented to show the effect of these parameters on the heat transfer and fluid flow characteristics. It is found that the number of contrarotative cells and the strength circulation of each cell can be controlled by the thickness of the bottom wall, the thermal conductivity ratio and the Rayleigh number. It is also observed that increasing either the Rayleigh number or the thermal conductivity ratio or both, and decreasing the thickness of the bounded wall can increase the average Nusselt number for the porous enclosure.     

Double-diffusive natural convection in inclined finned triangular porous enclosures in the presence of heat generation/absorption effects

The problem of double-diffusive convection in inclined finned triangular porous enclosures for various thermal and concentration boundary conditions and in the presence of heat source or sink was studied. The finite difference method was employed to solve the dimensionless governing equations of the problem. The effects of the governing parameters, namely the dimensionless time parameter, the inclination angle, Darcy number, heat generation/absorption parameter, the buoyancy parameter and the Rayleigh number on the streamlines, temperature and concentration contours as well as selected velocity component in the x-direction, local and average Nusselt numbers and local and average Sherwood number at the heated and concentrated wall for various values of the aspect ratio and the position of the fin were considered. The present results are validated by favorable comparisons with previously published results. All the results of the problem were presented in graphical and tabular forms and discussed.     

Natural convection with surface radiation from a planar heat generating element mounted freely in a vertical channel

Experiments and numerical simulations have been conducted to study the conjugate heat transfer by natural convection and surface radiation from a planar heat generating element placed centrally between two adiabatic vertical plates. The relevant problem dependent parameters considered in this study are modified Rayleigh number, channel aspect ratio, stream-wise location of the heat generating element, and surface emissivities of the heat generating element and the adiabatic side plates. Experiments are conducted for different values of modified Rayleigh number ranging from 3.2 × 10⁵ to 1.6 × 10⁷ and surface emissivities 0.05, 0.55, 0.75 and 0.85. The interdependence between the heat transfer mechanism and the flow field under the influence of surface radiation on natural convection is explored and discussed. Experimental correlations for total and convective Nusselt number, and dimensionless temperature in terms of relevant parameters have been developed. The mathematical model governing the problem has been numerically solved using a commercial computational fluid dynamics package FLUENT 6.3 and the numerical predictions substantiate the experimental observations.     

MHD natural convection with Joule and viscous heating effects in iso-flux porous medium-filled enclosures

In this study, the magnetohydrodynamics （MHD） natural convection heat transfer with Joule and viscous heating effects inside an iso-flux porous medium-filled inclined rectangular enclosure is studied numerically. An iso-heat flux is applied for heating and cooling the two opposing walls of the enclosure while the other walls are adiabatic. The Forchheimer extension of Darcy-Oberbeck-Boussinesq and energy equations is transformed into a dimensionless form using a set of suitable variables instead of a finite difference scheme. The governing parameters are the magnetic influence number, the modified Rayleigh number, the inclination angle, and the aspect ratio of the enclosure. The results show that viscous and Joule heating effects decrease heat transfer rates.     

Conjugate Natural Convection in a Porous Enclosure with Non-Uniform Heat Generation

Effects of a conductive wall on natural convection in a square porous enclosure having internal heating at a rate proportional to a power of temperature difference is studied numerically in this article. The horizontal heating is considered, where the vertical walls heated isothermally at different temperatures while the horizontal walls are kept adiabatic. The Darcy model is used in the mathematical formulation for the porous layer and finite difference method is applied to solve the dimensionless governing equations. The governing parameters considered are the Rayleigh number (0 ???Ra ???1000), the internal heating and the local exponent parameters (0 ????? ???5), (1 ????? ???3), the wall to porous thermal conductivity ratio (0.44 ???Kr ???9.9) and the ratio of wall thickness to its width (0.02 ???D ???0.5). The results are presented to show the effect of these parameters on the fluid flow and heat transfer characteristics. It is found a strong internal heating can generate significant maximum fluid temperature more than the conductive solid wall. Increasing value thermal conductivity ratio and/or decreasing the thickness of solid wall can increase the maximum fluid temperature. It is also found that at very low Rayleigh number, the heat transfer across the porous enclosure remain stable for any values of the thermal conductivity ratio.     

Numerical computation of natural convection in an isosceles triangular cavity with a partially active base and filled with a Cu–water nanofluid

This paper discusses the results of a study related to natural convection cooling of a heat source located on the bottom wall of an inclined isosceles triangular enclosure filled with a Cu water-nanofluid. The right and left walls of the enclosure are both maintained cold at constant equal temperatures, while the remaining parts of the bottom wall are insulated. The study has been carried out for a Rayleigh number in the range 10⁴ ≤ Ra ≤ 10⁶, for a heat source length in the range 0.2 ≤ ε ≤0.8, for a solid volume fraction in the range 0 ≤ ?≤0.06 and for an inclination angle in the range 0° ≤ δ≤45°. Results are presented in the form of streamline contours, isotherms, maximum temperature at the heat source surface and average Nusselt number. It is noticed that the addition of Cu nanoparticles enhances the heat transfer rate and therefore cooling effectiveness for all values of Rayleigh number, especially at low values of Ra. The effect of the inclination angle becomes more noticeable as one increases the value of Ra. For high Rayleigh numbers, a critical value for the inclination angle of δ = 15° is found for which the heat source maximum temperature is highest.     

Natural convection in an inclined rectangular porous cavity with uniform heat flux from the side

The problem of natural convection in an inclined rectangular porous layer enclosure is studied numerically. The enclosure is heated from one side and cooled from the other by a constant heat flux while the two other walls are insulated. The effect of aspect ratio, inclination angle and Rayleigh number on heat transfer is studied. It is found that the enclosure orientation has a considerable effect on the heat transfer. The negative orientation sharply inhibits the convection and consequently the heat transfer and a positive orientation maximizes the energy transfer. The maximum temperature within the porous medium can be considerably higher than that induced by pure conduction when the cavity is negatively oriented. The peak of the average Nusselt number depends on the Rayleigh number and the aspect ratio. The heat transfer between the two thermally active boundaries is sensitive to the effect of aspect ratio. For an enclosure at high or low aspect ratio, the convection is considerably decreased and the heat transfer depends mainly on conduction.     

Natural convection in inclined two dimensional rectangular cavities

Steady two-dimensional natural convection in fluid filled cavities is numerically investigated. The channel is heated from below and cooled from the top with insulated side walls and the inclination angle is varied. The field equations for a Newtonian Boussinesq fluid are solved numerically for three cavity height based Rayleigh numbers, Ra = 10⁴, 10⁵ and 10⁶, and several aspect ratios. The calculations are in excellent agreement with previously published benchmark results. The effect of the inclination of the cavity to the horizontal with the angle varying from 0° to 180° and the effect of the startup conditions on the flow pattern, temperature distribution and the heat transfer rates have been investigated. Flow admits different configurations at different angles as the angle of inclination is increased depending on the initial conditions. Regardless of the initial conditions Nusselt number Nu exhibits discontinuities triggered by gradual transition from multiple cell to a single cell configuration. The critical angle of inclination at which the discontinuity occurs is strongly influenced by the assumed startup field. The hysteresis effect previously reported is not always present when the calculations are reversed from 90° to 0°. A comprehensive study of the flow structure, the Nu variation with varying angle of inclination, the effect of the initial conditions and the hysteresis effect are presented.     

Effect of Discrete Heating on Natural Convection in a Rectangular Porous Enclosure

The main objective of this article is to study the effect of discrete heating on free convection heat transfer in a rectangular porous enclosure containing a heat-generating substance. The left wall of the enclosure has two discrete heat sources and the right wall is isothermally cooled at a lower temperature. The top and bottom walls, and the unheated portions of the left wall are adiabatic. The vorticity–stream function formulation of the governing equations is numerically solved using an implicit finite difference method. The effects of aspect ratio, Darcy number, heat source length, and modified Rayleigh number on the flow and heat transfer are analyzed. The numerical results reveal that the rate of heat transfer increases as the modified Rayleigh number and the Darcy number increases, but decreases on increasing the aspect ratio. The average heat transfer rate is found to be higher at the bottom heater than at the top heater in almost all considered parameter cases except for ε = 0.5. Also, the maximum temperature takes place generally at the top heater except for the case ε = 0.5, where the maximum temperature is found at the bottom heater. Further, the numerical results reveal that the maximum temperature decreases with the modified Rayleigh number and increases with the aspect ratio.     

MHD natural convection in porous media-filled enclosures

In this work, the magnetohydrodynamics （MHD） natural convection heat transfer problem inside a porous medium filled with inclined rectangular enclosures is investigated numerically. The boundary conditions selected on the enclosure are two adiabatic and two isothermal walls. The governing equations, continuity, and Forchheimer extension of the Darcy law and energy are transformed into dimensionless forms by using a set of suitable variables, and then solved by using a finite difference scheme. The governing parameters are the magnetic influence number, the Darcy Rayleigh number, the inclination angle, and the aspect ratio of the enclosure. It is found that the magnetic influence number and the inclination angle have pronounced effects on the fluid flow and heat transfer in porous media-filled enclosures.     

Natural convection flow of water near its density maximum in a rectangular enclosure having isothermal walls with heat generation

We consider unsteady laminar natural convection flow of water subject to density inversion in a rectangular cavity formed by isothermal vertical walls with internal heat generation. The top and bottom horizontal walls are considered to be adiabatic, whereas the temperature of the left vertical wall is assumed to be greater than that of the right vertical wall. The equations are non-dimensionalized and are solved numerically by an upwind finite difference method together with a successive over-relaxation (SOR) technique. The effects of both heat generation and variations in the aspect ratio on the streamlines, isotherms and the rate of heat transfer from the walls of the enclosure are presented. Investigations are performed for water taking Prandtl number to be Pr=11.58 and the Rayleigh number to be Ra=10⁵.     

Natural convection in partitioned enclosed spaces of solar collector

The two-dimensional, steady, laminar natural convection phenomena in partitioned enclosure of solar collector has been studied numerically. Heat conduction along the partition is considered. An iterative finite-difference scheme is employed to solve the governing equations in the flow field. The effects of Rayleigh number, thermal conductivity ratio, partition angle, tilt angle, and aspect ratio on both the local and average heat transfer coefficients of the solar collector have been discussed. The range of Rayleigh number tested was up to 5 × 10⁴, the thermal conductivity ratio of 4.5 and 30, partition angle from 10 deg to 170 deg, tilt angle from 10 deg to 90 deg, and aspect ratio varied between 0.2 and 10. The results indicate that the convective heat transfer is strongly affected with the aspect ratio of the enclosures.

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Freie Konvektion in unterteilten Kammern von Solarkollektoren

Zusammenfassung Die zweidimensionale, stetige, laminare freie Konvektion in unterteilten Kammern von Solarkollektoren wurde numerisch untersucht. Die Wärmeübertragung entlang dieser Kammern wurde betrachtet. Ein iteratives Finite-Differenzen-Schema wurde angewandt um die Gleichungen, welche das Strömungsfeld beschreiben, zu lösen. Der Einfluß der Rayleigh-Zahl, der thermische Leitfähigkeit, des Kammerwinkels, des Neigungswinkels und der Längenverhältnisse auf die örtlichen und durchschnittlichen Wärmeübertragungskoeffizienten des Solarkollektors wurde diskutiert. Der Bereich der Rayleigh-Zahl erstreckte sich bis zu 5 × 10⁴, das Verhältnis der thermischen Leitfähigkeit betrug 4.5 und 30, der Kammerwinkel lag zwischen 10° und 170°, der Neigungswinkel zwischen 10° und 90° und das Längenverhältnis variierte zwischen 0.2 und 10. Die Ergebnisse beinhalten, daß die konvektive Wärmeübertragung sehr stark durch das Längenverhältnis der Kammern beeinflußt wird.

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Nomenclature a slope of the partition with respect to the horizontal - A H/L=cell aspect ratio - A _w t/L=wall aspect ratio - g acceleration due to gravity - h local heat transfer coefficient - average heat transfer coefficient - H cell length - k thermal conductivity of fluid within the cell - k _w thermal conductivity of the cell wall - L plate spacing - Nu _f h L/k=local cell Nusselt number - L/k=average cell Nusselt number - overall average Nusselt number - qL/k _w t(T _h–T _c)=average wall Nusselt number - Pr /=Prandtl number - q heat transfer in the cell wall from the hot to cold plate per unit depth - Ra g L ³ T/=Rayleigh number - R _k k _w/k=ratio of wall thermal conductivity to that of the fluid - t thickness of cell wall - T _c cold plate temperature - T _f temperature in cell - T _h hot plate temperature - T _w temperature in cell wall - u, U dimension and dimensionless velocities inx-direction - v, V dimension and dimensionless velocities iny-direction - x distance measured from the bottom of the cell (Fig. 1) - X x/L=normalized distance ofx - y distance measured from hot plate (Fig. 1) - Y y/L=normalized distance ofy - x ₁ distance measured in wall (Fig. 1) - X ₁ x/L=normalized distance ofx ₁ Greek symbols thermal diffusivity of fluid - coefficient of volumetric expansion of fluid - partition angle with respect to the hot plate - _f T _f–T _c/T _h–T _c=dimensionless temperature in cell - _w T _w–T _c/T _h–T _c=dimensionless temperature in cell wall - kinematic viscosity of fluid - enclosure tilt angle from horizontal - dimensional vorticity - L ²/=dimensionless vorticity - dimensionless streamline     

Influence of the variable thermophysical properties on the turbulent buoyancy-driven airflow inside open square cavities

The effects of the air variable properties (density, viscosity and thermal conductivity) on the buoyancy-driven flows established in open square cavities are investigated, as well as the influence of the stated boundary conditions at open edges and the employed differencing scheme. Two-dimensional, laminar, transitional and turbulent simulations are obtained, considering both uniform wall temperature and uniform heat flux heating conditions. In transitional and turbulent cases, the low-Reynolds k − ω turbulence model is employed. The average Nusselt number and the dimensionless mass-flow rate have been obtained for a wide and not yet covered range of the Rayleigh number varying from 10³ to 10¹⁶. The results obtained taking into account variable properties effects are compared with those calculated assuming constant properties and the Boussinesq approximation. For uniform heat flux heating, a correlation for the critical heating parameter above which the burnout phenomenon can be obtained is presented, not reported in previous works. The effects of variable properties on the flow patterns are analyzed.     

Thermoconcentration convection under uniform lateral heating

The fine structure of thermoconvective flow near an inclined- plane heat source within a fluid with a stable density stratification due to salinity gradient is investigated using optical and probe techniques. The time dependence of the parameters of optical refraction coefficient gradient, specific electric conductivity, velocity and temperature fields is determined for different values of the heat source inclination angle. The difference in the spatial scales of variation of the different variables is demonstrated. The Nusselt numbers range from 1.8 to 5.4 for the structures under the plate and from 1.8 to 10.6 for the structures above the plate for Rayleigh numbers on the range from 2·10⁵ to 2·10⁶.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 5, pp. 112–124, September–October, 1995.