Numerical heat transfer in a cavity with a solar control coating deposited to a vertical semitransparent wall

A transient two‐dimensional computational model of combined natural convection, conduction, and radiation in a cavity with an aspect ratio of one, containing air as a laminar and non‐participating fluid, is presented. The cavity has two opaque adiabatic horizontal walls, one opaque isothermal vertical wall, and an opposite semitransparent wall, which consists of a 6‐mm glass sheet with a solar control coating of SnS–Cu_xS facing the cavity. The semitransparent wall also exchanges heat by convection and radiation from its external surface to the surroundings and allows solar radiation pass through into the interior of the cavity. The momentum and energy equations in the transient state were solved by finite differences using the alternating direction implicit (ADI) technique. The transient conduction equation and the radiative energy flux boundary conditions are coupled to these equations. The results in this paper are limited to the following conditions: 10⁴≤Gr≤10⁶, an isothermal vertical cold wall of 21°C, outside air temperatures in the range 30°C≤T₀≤40°C and incident solar radiation of AM2 (750 W m⁻²) normal to the semitransparent wall. The model allows calculation of the redistribution of the absorbed component of solar radiation to the inside and outside of the cavity. The influences of the time step and mesh size were considered. Using arguments of energy balance in the cavity, it was found that the percentage difference was less than 4 per cent, showing a possible total numerical error less than this number. For Gr=10⁶ a wave appeared in the upper side of the cavity, suggesting the influence of the boundary walls over the air flow inside the cavity. A Nusselt number correlation as a function of the Rayleigh number is presented. Copyright © 2000 John Wiley & Sons, Ltd.     

Conjugate turbulent heat transfer in a square cavity with a solar control coating deposited to a vertical semitransparent wall

This paper presents a numerical study of the conjugate heat transfer (natural convection, surface thermal radiation and conduction) in a square cavity with turbulent flow. The cavity has one vertical isothermal wall, two horizontal adiabatic walls and one vertical semitransparent wall with a selective coating applied to the inner side to control the solar radiation transmission. Later on the semitransparent wall is replaced with another one without the selective coating. The mathematical model for the turbulent flow in the cavity was solved using the finite volume method. The system had the following conditions: the uniform temperature in the isothermal wall was 21 °C, the external ambient temperature was fixed at 35 °C and on the semitransparent wall the direct normal solar irradiation of 750 W/m² was considered constant. The Rayleigh number was varied in the range of 10⁹ ? Ra ? 10¹² by changing the lengths of the cavity from 0.70 m to 6.98 m, respectively. The results show that, even though the air temperature of the cavity with the solar control film coating semitransparent wall (case A) is higher compared with the one without solar film coating (case B), the total amount of heat going through the cavity is lower compared to the one going through the cavity without solar control film. The total amount of energy transferred to the air in cavity for the case A was 41.98% less than for the case B. A set of correlations for the Nusselt number was obtained for both cases considering the conjugate heat transfer.     

Effects of walls temperature variation on double diffusive natural convection of Al2O3–water nanofluid in an enclosure

The effect of wall temperature variations on double diffusive natural convection of Al₂O₃–water nanofluid in a differentially heated square enclosure with constant temperature hot and cold vertical walls is studied numerically. Transport mechanisms of nanoparticles including Brownian diffusion and thermophoresis that cause heterogeneity are considered in non-homogeneous model. The hot and cold wall temperatures are varied, but the temperature difference between them is always maintained 5 °C. The thermophysical properties such as thermal conductivity, viscosity and density and thermophoresis diffusion and Brownian motion coefficients are considered variable with temperature and volume fraction of nanoparticles. The governing equations are discretized using the control volume method. The results show that nanoparticle transport mechanisms affect buoyancy force and cause formation of small vortexes near the top and bottom walls of the cavity and reduce the heat transfer. By increasing the temperature of the walls the effect of transport mechanisms decreases and due to enhanced convection the heat transfer rate increases.     

Natural convection in partially cooled tilted cavities

Two-dimensional numerical simulations of laminar natural convection in a partially cooled, differentially heated inclined cavities are performed. One of the cavity walls is entirely heated to a uniformly high temperature (heat source) while the opposite wall is partially cooled to a lower temperature (heat sink). The remaining walls are adiabatic. The tilt angle of the cavity is varied from 0° (heated from left) to −90° (heated from top). The fast false implicit transient scheme (FITS) algorithm, developed earlier by the same authors, is modified to solve the derived variables vorticity-streamfunction formulation. The effects of aspect ratio (AR), sink–source ratio and tilt angle on the average Nusselt number are examined through a parametric study; solutions are obtained for two Grashof numbers, 10⁵ and 10⁷. Flow patterns and isotherms are used to investigate the heat transfer and fluid flow mechanisms inside the cavity. © 1998 John Wiley & Sons, Ltd.     

Turbulent natural convection in a square enclosure bounded by a solid wall with localized heating

Turbulent natural convection and conduction in a square enclosure bounded by a massive wall with a localized heating is numerically studied. The bounding solid wall has a relative thermal conductivity of 10 and a relative thickness of 0.1. Losses to the surroundings are specified using a Biot number of 500. Two-dimensional equations of conservation of mass, momentum and energy, with the Boussinesq approximation and using the κ-ε model for turbulence are solved using finite difference method. Grids are generated in a nonuniform manner so that steep gradients near the wall regions are accounted for as required. Numerical solution is obtained for Ra numbers ranging from 10⁶ to 10¹³. The position of the source is also investigated. It is found that the heat transfer by convection is the highest when the heat source is located at the upper part of the cavity. The turbulent properties show also the same conclusion. Received on 4 November 1998     

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.     

Heat transfer by conduction,natural convection and radiation across a rectangular cellular structure

This paper describes results on the effects of wall conduction and radiation heat exchange among surfaces on laminar natural convection heat transfer in a two-dimensional rectangular cavity modelling a cellular structure. Parametric heat transfer calculations have been performed, and numerical results are presented in graphical and tabular form. Local and average Nusselt numbers along the cavity walls are reported for a range of parameters of physical interest. The findings suggest that the local or the average Nusselt number is one of many parameters that control conjugate heat transfer problems. The results indicate that natural convection heat transfer in the cavity is reduced by heat conduction in the walls and radiation exchange among surfaces. The results obtaibed for the total heat transfer rate through the system using the two-dimensional model are compared with those based on a one-dimensional model.     

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 flow in a square cavity revisited: Laminar and turbulent models with wall functions

Numerical simulations have been undertaken for the benchmark problem of natural convection flow in a square cavity. The control volume method is used to solve the conservation equations for laminar and turbulent flows for a series of Rayleigh numbers (Ra) reaching values up to 10¹⁰. The k-? model has been used for turbulence modelling with and without logarithmic wall functions. Uniform and non-uniform (stretched) grids have been employed with increasing density to guarantee accurate solutions, especially near the walls for high Ra-values. ADI and SIP solvers are implemented to accelerate convergence. Excellent agreement is obtained with previous numerical solutions, while some discrepancies with others for high Ra-values may be due to a possibly different implementation of the wall functions. Comparisons with experimental data for heat transfer (Nusselt number) clearly demonstrates the limitations of the standard k-? model with logarithmic wall functions, which gives significant overpredictions.     

Interaction of surface radiation with conjugate mixed convection from a vertical channel with multiple discrete heat sources

Important results of a numerical study performed on combined conduction–mixed convection–surface radiation from a vertical channel equipped with three identical flush-mounted discrete heat sources in its left wall are provided here. The channel has walls of identical height with the spacing varied by varying its aspect ratio (AR). The cooling medium is air that is considered to be radiatively transparent. The heat generated in the channel gets conducted along its walls before getting dissipated by mixed convection and radiation. The governing equations for fluid flow and heat transfer are considered without boundary layer approximations and are transformed into vorticity–stream function form and are later normalized. The resulting equations are solved, along with relevant boundary conditions, making use of the finite volume method. The computer code written for the purpose is validated both for fluid flow and heat transfer results with those available in the literature. Detailed parametric studies have been performed and the effects of modified Richardson number, surface emissivity, thermal conductivity and AR on various pertinent results have been looked into. The significance of radiation in various regimes of mixed convection has been elucidated. The relative contributions of mixed convection and radiation in carrying the mandated cooling load have been thoroughly explored.     

Experimental and numerical study on laminar natural convection in a cavity heated from bottom due to an inclined fin

Natural convection heat transfer in an inclined fin attached square enclosure is studied both experimentally and numerically. Bottom wall of enclosure has higher temperature than that of top wall while vertical walls are adiabatic. Inclined fin has also adiabatic boundary conditions. Numerical solutions have been done by writing a computer code in Fortran platform and results are compared with Fluent commercial code and experimental method. Governing parameters are Rayleigh numbers (8.10⁵ ≤ Ra ≤ 4 × 10⁶) and inclination angle (30° ≤ and ≤ 120°). The temperature measurements are done by using thermocouples distributed uniformly at the wall of the enclosure. Remarkably good agreement is obtained between the predicted results and experimental data. A correlation is also developed including all effective parameters on heat transfer and fluid flow. It was observed that heat transfer can be controlled by attaching an inclined fin onto wall.     

Natural convection due to solar radiation over a non-absorbing plate with and without heat losses

Natural convection over a non-reflecting, non-absorbing, ideally transparent semi-infinite vertical flat plate due to absorption of incident radiation (solar radiation) is considered. The absorbed radiation acts as a distributed source which initiates buoyancy-driven flow and convection in the absorbing layer. The plate when heated by the absorbing fluid loses heat to the surroundings from its external side. Solution of the governing equations of the flow under these circumstances is non-similar because of both the heat source term in the energy equation and the temperature boundary condition at the plate. A local non-similar technique is used to obtain solutions for a wide range of the dimensionless distance along the plate and of the dimensionless loss coefficient to the surroundings. The results show that the temperature distribution has a maximum temperature in the depth of the fluid rather than on the plate. A new definition for a local heat transfer coefficient between the plate and the absorbing fluid is introduced which is based on the local maximum temperature rise in the fluid. A formula to calculate this heat transfer coefficient is given for the anticipated range of the loss coefficient.     

Local Nusselt number and temperature field in turbulent flow through a heated square-sectioned U-bend

Local heat transfer coefficients and temperature distributions within the fluid for air flow around a 180° square-sectioned bend have been measured. The ratio of bend radius to hydraulic diameter of the duct is 3.35:1 and the flow entering the bend is sensibly fully developed. Measurements of air and wall temperatures span a range of Reynolds numbers from 9.9 × 10³ to 9.2 × 10⁴ with the principal emphasis given to the case of Re ? 5.6 × 10⁴. This Reynolds number and geometric configuration coincide with that of a companion LDA study carried out by Chang et al¹ which provides detailed maps of the mean and turbulent velocity fields. The data show that by 45° into the bend the heat transfer coefficients on the inner convex wall of the bend drop markedly while those on the other walls increase. By 90° the ratio of the heat transfer coefficients at the mid positions of the concave and convex walls is more than 2:1. Nevertheless this ratio is less than would be anticipated from considering two-dimensional flow on weakly curved surfaces. There is a general consistency between the velocity and the temperatyre field data in the heated fluid     

Numerical and experimental study of heat transfer in a tall vertical closed cavity

In this work the numerical and experimental results of heat transfer in a vertical tall closed cavity are presented. The cavity has an aspect ratio of 20, one of the vertical walls receive a constant and uniform heat flux, while the opposite wall is kept at a constant temperature. The remaining walls are assumed adiabatic. The cavity is full of air. The computational fluid dynamics software Fluent 6.3 was used for the simulation and an experimental prototype was built to obtain the heat transfer coefficients. The air temperature and the fluid velocity values are higher when emissivity (ε) is 0.03 (almost pure natural convection). The experimental total heat transfer coefficient increases between 119.9 and 159.9 % when the emissivity of the walls changes from 0.03 to 0.95.     

Experiments on melting of a vertical ice layer immersed in immiscible liquid

Experiments on the melting of a vertical ice layer immersed in immiscible liquid yielded quantitative results both for the timewise evolution of the melting front and the heat transfer. Vegetable oil, which was contained in a rectangular vessel, was adopted as a testing liquid. A bubble-free ice block stuck on a cooled wall was installed vertically in the vessel. The experiments were carried out for the immiscible liquid temperatures from 7.6 to 30.0 ^°C, while for the cooled wall temperatures from 0 to ?11.5 ^°C. The flow structure of the liquid and the melting front were extensively observed and recorded photographically. It was found that the heat transfer and the rate of melting are significantly affected by a couple of fluid motions of both the water melt induced by melting of ice and the immiscible liquid based on free convection.     

Heat transfer characteristics of internally heated liquid pools at high Rayleigh numbers

Detailed numerical analysis is presented for buoyancy driven flow of a Newtonian fluid contained in a square enclosure for high Rayleigh (Ra) numbers. Natural convection is due to internal heating sources, which are assumed to be uniformly distributed within the enclosure. All walls of the cavity are maintained at constant temperature. Flow and heat transfer characteristics are investigated for a Ra number range of 10⁷ to 10¹² while Prandtl (Pr) number is taken to be 7.0. Governing equations (in primitive variables) are discretised using control volume technique based on staggered grid formulation. These equations are solved using SIMPLER algorithm of Patankar. Flow and heat transfer characteristics, streamlines, isotherms and average wall Nusselt (Nu) number, are presented for whole range of Ra number considered. Finally, present results for average wall Nu numbers are compared with experimental observations obtained from open literature. It is concluded that both results are in very good agreement, which confirmed the accuracy of the scaling used for present investigation. Received on 15 November 1999     

Entropy generation and natural convection in square cavities with wavy walls

The aim of the present work is to study the entropy generation in the natural convection process in square cavities with hot wavy walls through numerical simulations for different undulations and Rayleigh numbers, while keeping the Prandtl number constant. The results show that the hot wall geometry affects notably the heat transfer rate in the cavity. It has been found in the present numerical study that the mean Nusselt number in the case of heat transfer in a cavity with wavy walls is lower, as compared to heat transfer in a cavity without undulations. Based on the obtained dimensionless velocity and temperature values, the distributions of the local entropy generation due to heat transfer and fluid friction, the local Bejan number, and the local entropy generation are determined and plotted for different undulations and Rayleigh numbers. The study is performed for Rayleigh numbers 10³ < Ra < 10⁵, irreversibility coefficients 10^?4 < φ < 10^?2, and Prandtl numbers Pr = 0.71. The total entropy generation is found to increase with increasing undulation number.     

Natural convection and radiation heat transfer of an externally-finned tube vertically placed in a chamber

A three-dimensional numerical study was made to investigate effects of fin angle, fin surface emissivity, and tube wall temperature on heat transfer enhancement for a longitudinal externally-finned tube placed vertically in a small chamber. The numerical model was first validated through comparison with experimental measurements and the appropriateness of general boundary conditions was examined. The numerical results show that the mean Nusselt number increases with Rayleigh number for all the fin angles investigated. The maximum heat transfer rate per mass occurs when the fin angle is about 60° for fin surface emissivity between 0.7 and 0.8 and 55° when the surface emissivity increases to 0.9. With increasing tube wall temperature, both the natural convection and radiation heat transfer are enhanced, but the fraction of radiation heat transfer decreases in the temperature range studied. Radiation fraction increases with increasing fin surface emissivity. Both convection and radiation heat transfer modes are important.     

Effects of surface radiation on natural convection in a rayleigh-benard square enclosure: steady and unsteady conditions

Coupled laminar natural convection with radiation in air-filled square enclosure heated from below and cooled from above is studied numerically for a wide variety of radiative boundary conditions at the sidewalls. A numerical model based on the finite difference method was used for the solution of mass, momentum and energy equations. The surface-to-surface method was used to calculate the radiative heat transfer. Simulations were performed for two values of the emissivities of the active and insulated walls (ɛ₁=0.05 or 0.85, ɛ₂=0.05 or 0.85) and Rayleigh numbers ranging from 10³ to 2.3×10⁶ . The influence of those parameters on the flow and temperature patterns and heat transfer rates are analyzed and discussed for different steady-state solutions. The existing ranges of these solutions are reported for the four different cases considered. It is founded that, for a fixed Ra, the global heat transfer across the enclosure depends only on the magnitude of the emissivity of the active walls. The oscillatory behavior, characterizing the unsteady-state solutions during the transitions from bicellular flows to the unicellular flow are observed and discussed.     

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.