2D transient natural convection in diode cavities containing an electronic equipment with discrete active bands under constant heat flux

The thermal behavior of airborne electronic equipment submitted to natural convection in closed parallelogrammic air-filled cavities is examined in this study. The cold active wall of the enclosure is maintained isothermal. The hot wall, representing the electronic device, is composed of three parallel discrete bands generating a constant heat flux, separated by two adiabatic bands of equal dimensions. Both walls remain always vertical. The channel is considered adiabatic and the aspect ratio of the cavity is equal to unity. Many configurations are examined while varying the inclination angle of the top and bottom walls of the channel. When the angle is positive the convective heat transfer is favored in comparison with the case of the right cavity, but, on the contrary, it is reduced for negative angles. The resultant enclosures are so called diode cavities in the convective heat transfer sense of the word. The experimental part of the study is achieved with a setup based on electrical data and temperature measurements on the walls. The numerical approach using the finite volume method allows to complete the experimental results with the thermal and dynamical characteristics of the 2D flow. The temperature fields show the thermal behavior of the device during the transient phase after switching it on. The convective results concerning the imposed heat flux treated in this study differ from those corresponding to impose the temperature on the hot bands. The distribution and evolution of the Nusselt number allow to characterize the natural convection occurring in the cavity. The results of this work are consistent with previous studies and allow to predict the thermal behavior of the electronic equipment during the transient phase.     

NATURAL CONVECTION IN OPEN CAVITIES AND SLOTS

The present work is concerned with natural convection from open cavities or heated plates attached with parallel vertical strips. The bottom of the cavity is heated, and the vertical walls are assumed adiabatic. Numerical results are presented for steady, laminar natural convection for the geometry described. Effects of Rayleigh numbers from 1 × 10 ³ to 1 × 10 ⁷ , inclination angles from 10^° to 90^°, and aspect ratios of 0.5, 1.0, and 2.0 are investigated for a fixed Prandtl number (0.7). It is found that the average Nusselt number is not very sensitive to the inclination angle. Flow becomes unstable at high Rayleigh numbers and at low inclination angles. Flow pattern and heat transfer results are presented and discussed.     

Laminar mixed convection in shallow inclined driven cavities with hot moving lid on top and cooled from bottom

Laminar mixed convective heat transfer in two-dimensional shallow rectangular driven cavities of aspect ratio 10 is studied numerically. The top moving lid of the cavity is at a higher temperature than the bottom wall. Computations are performed for Rayleigh numbers ranging from 10⁵ to 10⁷ keeping the Reynolds number fixed at 408.21, thus encompassing the dominating forced convection, mixed convection, and dominating natural convection flow regimes. The fluid Prandtl number is taken as 6 representing water. The effects of inclination of the cavity on the flow and thermal fields are investigated for inclination angles ranging from 0° to 30°. Interesting behaviours of the flow and thermal fields with increasing inclination are observed. The streamline and isotherm plots and the variation of the local and average Nusselt numbers at the hot and cold walls are presented. The average Nusselt number is found to increase with cavity inclination. The rate of increase of the average Nusselt number with cavity inclination is mild for dominating forced convection case while it is much steeper in dominating natural convection case.     

Effect of surface radiation on buoyant convection in vertical triangular cavities with variable aperture angles

The present investigation deals with the numerical computation of laminar natural convection with and without surface-to-surface radiation in a class of right-angled triangular cavities filled with air. The vertical walls are uniformly heated and the inclined walls are uniformly cooled while the upper connecting walls are adiabatic. The aperture angle φ located at the lower vertex of the triangular cavities between the vertical and the inclined walls identifies the shape of each cavity. This kind of cavity finds application in the miniaturization of cabinets housing electronic components constrained by space and/or weight severely. With a view at enhancing the heat transfer rates and/or reducing the size of cabinets, the influence that surface radiation exerts upon natural convection should be scrutinized. To this end, the finite volume method is implemented to perform the computational analysis of the above-described problem(s). Numerical results are reported for the local quantities, the velocity and temperature fields encompassing aperture angles φ that extend from 15° to 45° at two extreme Rayleigh numbers, Ra = 10³ and 10⁶. Additionally, the two global quantities, the mean convective Nusselt number and the mean radiative Nusselt number are reported in tabulated and graphical forms for the same controlling parameters. Overall, it was found that the competition between surface radiation and natural convection in right-angled triangular cavities filled with air plays a preponderant role. Finally, the analysis culminates with the construction of a comprehensive correlation equation for the total Nusselt number in terms of the controlling parameters which should be useful for engineering analysis and design. This correlation equation will undoubtedly provide a fast evaluation avenue to judge the cavity thermal performance.     

Nusselt number for the natural convection and surface thermal radiation in a square tilted open cavity

In this communication, the numeric results of the heat transfer by natural convection and surface thermal radiation in a tilted 2D open cavity are presented. This study has importance in the thermal design of receivers for solar concentrators. The opposite wall to the aperture in the cavity holds a constant temperature of 500 K, while the temperature of the surrounding fluid interacting with the aperture is 300 K. The other walls are kept insulated. The results in the steady state are obtained for a Rayleigh range from 10⁴ to 10⁷ and for an inclination angles range of the cavity from 0° to 180°. The results show that the Nusselt numbers increase with the Rayleigh number except the convective Nusselt number for 180°, where it stays almost constant. The convective Nusselt number changes substantially with the inclination angle of the cavity, while the radiative Nusselt number is insensitive to the orientation change of the cavity.     

On thermal control of devices contained in inclined hemispherical cavities with dome oriented downwards and subjected to transient natural convection

In this study, relationships of Nusselt–Rayleigh–Fourier type are proposed for the case of air-filled hemispherical cavity whose dome is oriented downwards and maintained isothermal. Its disk is subjected to a constant heat flux and inclined at an angle varying between 90° (vertical position) and 180° (disk horizontal with dome oriented downwards). The numerical approach is performed in transient regime by means of the finite volume method for Rayleigh numbers in the range of 10⁴ − 5 × 10⁸. These results are confirmed at steady state by measurements done for some configurations in a previous study for the same Rayleigh and inclination ranges. Otherwise, they complete other surveys considering inclination angles varying between 0° (horizontal cavity with dome oriented upwards) and 90° (vertical cavity) for a wider range of Rayleigh numbers. The correlations allow thermal control of devices submitted to natural convection in hemispherical cavities during the time preceding the steady state after their switch on.     

Role of finite element based grids and simulations on evaluation of Nusselt numbers for heatfunctions within square and triangular cavities involving multiple discrete heaters

Nusselt number is an important non-dimensional parameter which quantifies the heat transfer rate. Local Nusselt number is useful in predicting the heat transfer rate along the various hot and cold sections of the side walls in a discretely heated enclosed cavity. In addition, the overall heat balance in an enclosed cavity (total heat delivered by the hot isothermal walls should be equal to the total heat gained by the cold isothermal walls) can be validated via the average Nusselt numbers. Current finite element based simulations and post-processing have been carried out in order to analyze the influence of the multiple heaters on the Nusselt number along various sections (hot and cold) of the side walls in discretely heated square and triangular (design 1 and design 2) cavities. The working fluid is considered to be air (Pr = 0.7) and the numerical studies have been carried out for a large range of Rayleigh number (Ra = 10³–10⁵) for four different biquadratic elements (24 × 24, 28 × 28, 32 × 32 and 34 × 34). The current work also estimates the fractional error in the heat balance (ϵ) and it is clearly inferred that ϵ is comparatively lower for 34 × 34 biquadratic elements. Current work also reveals that the fractional error (ϵ) is mainly induced due to the sharp variations in the Nusselt number at the cold-hot junctions along the side walls. The present study also involves the detailed evaluation of the heatfunction (Π) expressions along the cold-hot junctions of the side walls. The computations of the heatfunctions are intrinsically related to the Nusselt numbers of the hot-cold junctions.     

Quantification of natural convective heat transfer within air-filled hemispherical cavities. Isothermal tilted disk with dome oriented upwards and wide Ra range

This work quantifies the natural convective heat transfer occurring in hemispherical air-filled cavities whose disk is inclined at an angle varying between 0° and 90°. This active hot disk as well as the dome are maintained isothermal at different temperatures. The numerical approach by means of the control volume method allows the examination of the dynamical phenomena that occurs in many configurations obtained by varying the temperature difference between the two active walls and the radius of the hemisphere. Convective heat transfer at the hot wall is represented by the average Nusselt number associated to Rayleigh numbers varying between 10⁴ and 2.55 × 10¹². By taking into account all the studied configurations, correlations between these two dimensionless numbers are established for the set of considered inclination angles. Comparisons with results from other studies for the case of horizontal cavity show a good agreement. The relationships presented here cover the laminar, transitional and turbulent heat transfer regimes. They complement previous studies with the condition of heat flux imposed on the disk. The wide range of Rayleigh numbers considered in this survey and its association with the large inclination angle range allow the application of the correlations to various engineering fields such as nuclear technology, solar energy, building, embarked electronics, architecture, safety or domotics.     

THREE-DIMENSIONAL NATURAL CONVECTION OF AIR IN AN INCLINED CUBIC CAVITY

Three-dimensional steady and transient natural connective flow in a differentially heated inclined cubic cavity of air was numerically simulated by the projection method combined with the power law scheme, Various three-dimensional flow structures were induced, depending on the Rayleigh number and inclined angle, although the main flow is nearly two-dimensional. It is of interest to note that the flow is stronger when a vertical cavity is inclined, implying significant flow acceleration by the buoyancy component normal to the hot and cold walls. Change in the flow structure during the transient causes wavy variation of the space-averaged Nusselt number with time.     

Analysis of mixed convection of nanofluid in a 3D lid-driven trapezoidal cavity with flexible side surfaces and inner cylinder

Numerical study of mixed convection in a lid-driven 3D flexible walled trapezoidal cavity with nanofluids was performed by using Galerkin weighted residual finite element method. Effects of various pertinent parameters such as Richardson number (between 0.05 and 50), elastic modulus of the side surfaces (between 1000 and 10⁵), side wall inclination angle (between 0° and 20°) and solid particle volume fraction (between 0 and 0.04) on the fluid flow and heat transfer characteristics in a 3D lid-driven-trapezoidal cavity were numerically examined. It was observed that these characteristics are influenced when the pertinent parameters change. Flexible side surface can be used as control element for heat transfer rate. Increment and reduction in the space which are provided by the flexible side walls result in heat transfer enhancement and deterioration for side wall inclination angle of 0° and 10°. Average Nusselt number enhances by about 9.80% when the value of the elastic modulus is increased from 1000 to 10⁵ for side wall inclination angles of θ = 0°. Adding nanoparticles to the base fluid results in linear increment of heat transfer and at the highest volume fraction, 25.30% of heat transfer enhancement is obtained. A polynomial type correlation for the average Nusselt number along the hot wall was proposed and it has a fourth order polynomial dependence upon the Richardson number and first order dependence upon the solid particle volume fraction.     

Empirical Nu–Ra–Fo relationships for natural convection in air-filled hemispherical enclosures. Isothermal and inclined disk with dome oriented downwards

Correlations of Nusselt–Rayleigh–Fourier type proposed in this work allow quantifying of the transient convective heat transfer occurring in air-filled hemispherical cavities. The disk, initially at ambient temperature, is suddenly heated and kept isothermal. Throughout the heating process, the dome is maintained at ambient temperature. The radius of the cavity, associated with the temperature difference imposed between the disk and the dome, involves a large Rayleigh number range, varying between 10⁴ and 5 × 10⁸. The disk can be inclined with respect to the horizontal plane at an angle varying between 90° (vertical disk) and 180° (horizontal disk with dome downwards) by steps of 10°. The numerical approach is based on the finite volume method. The proposed empirical relationships in transient regime are linked to the steady state Nusselt–Rayleigh ones. They are obtained via the analysis of results corresponding to many combinations of the Rayleigh number and the disk inclination angle. The relationships are new since the considered geometry associated with the inclinations of the isothermal disk has not been treated previously. They constitute an important tool for the thermal design of engineering systems involved as they allow determining the convective heat transfer during the transient regime. They can be applied in several fields such as nuclear technology, solar energy, security and safety electronics, building, domotics or aeronautics.     

Computation of turbulent free convection in left and right tilted porous enclosures using a macroscopic k–ε model

Comparisons of computations for turbulent natural convection within clockwise and counter-clockwise inclined cavities, filled with a fluid saturated porous medium, are presented. The finite volume method in a generalized coordinate system is applied. Oblique walls are maintained at constant but different temperatures, whereas horizontal surfaces are kept insulated. Flow and heat transfer characteristics are investigated for Rayleigh number up to 10⁴ and inclination angles up to 45°, in both directions of rotation. Turbulent is handled using a macroscopic two-equation model with a wall function. In this work, the turbulence model is first switched off and the laminar branch of the solution is obtained. Subsequently, the turbulence model is included and the solution merges to the laminar branch for a reducing value of Ra_m. Present computations are compared with published results and the influence of the inclination angle on Ra_cr is analyzed, for both the left and right rotating directions. For Ra_m greater than around 10⁴, both laminar and turbulent flow solutions deviate, possibly indicating that a critical value for Ra_m was reached. Both left and right rotation of the hot wall reduce Nu, but rotating the hot wall on the counter-clockwise direction decreases Nu at a faster rate than when bending the cavity to the right.     

TRANSIENT BUOYANCY-INDUCED FLOW THROUGH A HEATED,VERTICAL CHANNEL OF FINITE HEIGHT

A detailed numerical analysis is carried out to study the transient natural convection in a symmetrically heated vertical plate channel of finite height due to a step change in plate temperature. Both the inlet and exit effects are included in the analysis. In particular, a numerical scheme derived from the SIMPLER algorithm is employed to solve the governing differential equations. Emphasis is placed on the temporal evolution of the flow and thermal characteristics. Results are obtained for air with Ra varying from 10³ to 10⁶ and the aspect ratio from 5 to 10. The existence of multiple cells in the channel and in the region surrounding the top end during the transient stage is predicted. Over a certain period, temperature overshoot is noted, causing a dip in the time variation of the Nusselt number. Correlation equations are proposed for the average Nusselt number and the amount of air drawn into the channel at the steady state.     

Natural convection heat transfer in partially open inclined square cavities

A numerical study has been carried out on inclined partially open square cavities, which are formed by adiabatic walls and a partial opening. The surface of the wall inside the cavity facing the partial opening is isothermal. Steady-state heat transfer by laminar natural convection in a two dimensional partially open cavity 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 10³ to 10⁶, dimensionless aperture size from 0.25 to 0.75, aperture position at high, center and low, and inclination of the opening from 0° (facing upward) to 120° (facing 30° downward). It is found that the volume flow rate and Nusselt number are an increasing function of Rayleigh number, aperture size and generally aperture position. Other parameters being constant, Nusselt number is a non-linear function of the inclination angle. Depending on the application, heat transfer can be maximized or minimized by selecting appropriate parameters, namely aperture size, aperture position and inclination angle at a given operation Rayleigh number.     

Heat transfer and pressure drop characteristics of laminar air flows moving in a parallel-plate channel with transverse hemi-cylindrical cavities

Laminar flows in parallel-plate channels are usually caused by a combination of small channel dimensions and low fluid velocities. As a consequence, the heat transfer coefficients in these channels are extremely low. The present study avoids inward protruding fins attached to the channel walls and instead focuses on the opposite arrangement. That is, molding the walls of parallel-plate channels with arrays of transverse cavities pointing outward. Two configurations were studied, one with symmetrically opposing cavities onto the bottom and upper walls and another with non-symmetric or staggered cavities onto the two parallel walls. A 120-cm-long channel contains two series of 3, 6 and 12 transverse cavities having ratios of cavity depth to cavity print diameter δ/D of 0.125, 0.25, 0.375, and 0.5. Computations are performed for Reynolds numbers based on the hydraulic diameter ranging from 1000 to 2500 for air (Pr = 0.7). The finite-volume method is used to perform the computational analysis with embedded second-order-accurate QUICK and SIMPLE schemes. It is found that the cavity/channel assemblies can achieve heat transfer enhancements of about 30% relative to the smooth channel, with pressure loss increases of 19%. In all cases examined, the outcome of the numerical simulation reveals that the heat transfer enhancement overcomes the pressure drop accretion.     

Investigation of natural circulation in cavities with uniform heat generation for different Prandtl number fluids

Natural convection in enclosures with uniform heat generation and isothermal side walls is studied here. For the rectangular enclosure, two-dimensional conservation equations are solved using SIMPLE algorithm. Parametric studies are conducted to examine the effects of orientation of the cavity, fluid properties (Pr number), and aspect ratio for Rayleigh numbers up to 10⁶. For a horizontal square cavity, the flow becomes periodically oscillating at Ra = 5 × 10⁴ and chaotic at Ra = 8 × 10⁵. With a slight increase in the inclination angle, the oscillations die and for inclination angles greater than 15⁰, the flow attain a steady state over a range of Ra. It is found that for tall cavities (aspect ratio > 1), the steady-state solution is obtained for all values of Ra considered here. However, for wide cavities (aspect ratio < 1), an oscillatory flow regime is observed. The maximum temperature within the cavity is calculated for the range of Ra, aspect ratio and Pr number. Correlations for the maximum cavity temperature is presented here. The values of critical Rayleigh number at which the convection sets in the rectangular cavity are also studied and two distinct criteria are determined to evaluate the critical Rayleigh number. Further, a three-dimensional simulation is performed for a cubic cavity. It is found that the steady state solutions are obtained for all Rayleigh number, except at Ra = 10⁶. This is in contrast to the predictions for a two-dimensional square cavity, which has an oscillatory zone from Ra = 5 × 10⁴ onwards.     

Investigation of buoyancy-driven flow and heat transfer in a trapezoidal cavity filled with water–Cu nanofluid

A numerical study is conducted to investigate the transport mechanism of free convection in a trapezoidal enclosure filled with water–Cu nanofluid. The horizontal walls of the enclosure are insulated while the inclined walls are kept at constant but different temperatures. The numerical approach is based on the finite element technique with Galerkin's weighted residual simulation. Solutions are obtained for a wide range of the aspect ratio (AR) and Prandtl number (Pr) with Rayleigh number (Ra = 10⁵) and solid volume fraction (? = 0.05). The streamlines, isotherm plots and the variation of the average Nusselt number at the left hot wall are presented and discussed. It is found that both AR and Pr affect the fluid flow and heat transfer in the enclosure. A correlation is also developed graphically for the average Nusselt number as a function of the Prandtl number as well as the cavity aspect ratio.     

Investigation of heat transfer in square porous-annulus

The current study is focused to analyze the heat transfer characteristics in a porous duct. The mathematical model of heat transfer in a porous duct was solved by converting the governing partial differential equations into a set of algebraic equations with the help of finite element method. A simple three noded triangular element is used to mesh the duct domain. The current problem consists of a square duct with outer walls being exposed to hot temperature T_h, and inner walls subjected to cool temperature T_c. Emphasis is given to investigate the effect of width ratio of cavity on heat and fluid flow characteristics inside the porous medium. The results are reported for various duct width ratios, Rayleigh number etc. It is found that the Nusselt number increases with increase in height of cavity along the vertical walls of duct; however the Nusselt number for certain values of duct ratio oscillates along the width of the porous medium at bottom wall of the cavity.     

Numerical investigation of several physical and geometric parameters in the natural convection into trapezoidal cavities

Natural convection in trapezoidal cavities, especially those with two internal baffles in conjunction with an insulated floor, inclined top surface, and isothermal left-heated and isothermal right-cooled vertical walls, has been investigated numerically using the Element based Finite Volume Method (EbFVM). In numerical simulations, the effect of three inclination angles of the upper surface as well as the effect of the Rayleigh number (Ra), the Prandtl number (Pr), and the baffle’s height (H_b) on the stream functions, temperature profiles, and local and average Nusselt numbers has been investigated. A parametric study was performed for a wide range of Ra numbers (10³ ? Ra ? 10⁶) H_b heights (H_b = H¹/3, 2H¹/3, and H¹), Pr numbers (Pr = 0.7, 10 and 130), and top angle (θ) ranges from 10 to 20. A correlation for the average Nusselt number in terms of Pr and Ra numbers, and the inclination of the upper surface of the cavity is proposed for each baffle height investigated.     

Numerical Study of Transient Natural Convection in Air in Vertical Divergent Channels

This article carries out a numerical, transient, two-dimensional analysis of natural convection in air in a divergent channel, characterized by two inclined flat plates heated at a uniform heat flux. The flow is assumed to be laminar and incompressible. Simulations allow to detect the complex structures of the flow inside and outside the channel. Results, in terms of temperature distributions, average Nusselt and Reynolds profiles, depending on time as a function of the divergence angle and channel spacing, are presented. Flow visualizations and stream function contours confirmed the disturbances inside the channel for the highest divergence angles (>5°). Correlations in terms of Nusselt numbers as a function of Rayleigh and Rayleigh modified numbers, ranging from 7.6 × 10² to 1.3 × 10⁹ and from 30 to 8.2 × 10⁸, respectively, were proposed. They were in very good agreement with the experimental relations.