A control volume based numerical study has been carried out on trapezoidal cavity which has formed by inclined left heated wall and insulated bottom horizontal walls. The calculations are performed for Rayleigh numbers varying from 103 to 107 and orientation of heated wall is and . The results are compared for cases without opening and with different openings of the vertical wall as functions of isotherms, streamfunctions, local and average heat transfer rates. The overall Nusselt number increases with increase of Rayleigh number, but decreases with increase in angle of orientation.
1. Introduction
The natural convection in cavities is significant in many fields such as heat transfer in buildings and rooms of building, thermal management of nuclear reactors, growth of crystals etc. However, in the recent decade, the use of partially opened cavities is of great importance and more practical than closed one. The examples of such cases are open cavity solar thermal receivers with insulated strips, room heating, heat convection in rooms, extended surface in heat exchanger, thermal design of buildings, cooling of electronic components.
The free convection is most preferred method to maintain optimal working temperature of electronics equipment due simplicity in construction and low cost besides being noise free and effecting savings in energy consumption as active cooling devices such as fans need not be deployed. A limited number of studies have been presented for irregular shapes like trapezoidal cavity. As usual, the shape of the solar reflectors and enclosures of LED lighting system are trapezoidal in shape.
The more extensive study on discreet heating along the vertical wall with optimum position has been discussed. Analytical work on sources of discrete heating placed on vertical walls has been discussed by Silva et al. This study reveals that the heat sources have been placed for both entry of cold fluid and exit of heated cold fluid.
The experimental work on open cavities has been carried out by various research studies. The natural convection in a horizontal fully opened cavity is reported in. Hess and Henze have conducted experimental work on centrally placed vent of size 0.5H for a square cavity. In this work, the visualisation fluid flow characteristics is studied using laser Doppler velocimetry and local heat transfer for Rayleigh number varied between 107 to 1011.
The experimental work to measure the local heat transfer coefficient for both partially or fully opened tilted cavities with openings is noticed by Chakroun et al. The tests have been carried out for constant heat flux and isothermally heated wall for Grashoff number 5.5 x 108. The size of centrally located aperture is varied from 0.25 to 1. It is found that the heat transfer coefficient rises sharply as the angle of orientation is reducing until it reaches zero value.
The numerical work on free convection heat transfer in fully opened cavities is reported in open literature. The stream function-vorticity technique is used to study the free convection flows in a uniformly heated open square cavity by Penot. The results are presented for effect of Grashoff number and orientation of the cavity. It is noticed that unsteady solutions are observed for Gr > 105 and cavity aperture facing upward. Chan and Tien have carried out numerical simulations on square fully opened cavities using an extended domain.
A comparison of numerical work between shallow fully opened cavity and square cavity has been presented by Chan and Tien. It is found that a satisfactory heat transfer could be achieved in a uniformly heated isothermal vertical wall of a square cavity while two horizontal walls are kept adiabatic. Later, Mohamad made numerical simulation for the same problem that of Chan and Tien with different orientations of the cavity. It is found that at higher Rayleigh number and small inclination angles, the flow becomes unstable. The average heat transfer is not sensitive to the angle orientation.
A numerical work on free convection heat transfer in a partially opened square cavity is reported by Bilgen and Oztop. A parametric study is carried out for Ra ranging between 103 to 106, aperture size varied between 0.25H and 0.75H and position of aperture from the centre of top of the vertical wall. It is noticed that the rate heat transfer can be controlled by choosing the angle of orientation of cavity, size and position of the aperture.
The numerical studies on fully opened tilted shallow cavities are reported by Polat and Bilgen. The side opposite to the opening is warmed by constant heat flux, horizontal walls are insulated. The finite volume based numerical procedure is used to study the free convection in a rectangular geometry by Sharif et al. The bottom wall is heated at uniform heat flux ranging from 20% to 80% from the vertical centre of the bottom wall and remaining portion is adiabatic. The side walls are cooled uniformly by constant temperature while top wall is insulated. It is noticed that the study is restricted to rectangular cavity only.
The study of Nusselt number and rate of fluid flow in a partially and fully open square cavity with heaters placed in a vertical wall have been published. They reported that the overall conductance is increasing with respect to Ra, the size and number of heaters. The maximum heat transfer is achieved by placing the heaters close to each other and at bottom entry of the cold fluid.
In open literature, it is observed that most of the articles published on natural convection in trapezoidal enclosures are limited to temperature boundary conditions only. Few such articles are discussed below. An analytical solution to the free convection within a trapezoidal cavity is dealt by Iyican and Bayazitoglu. The critical Rayleigh number is reported for different orientations of the cavity. The grid independence study of free convection within the trapezoidal cavities with grid refinement starting with 10 × 10 cells to 160 × 160 is carried out by Peric.
Authors have studied the solution method and error analysis of grid test for temperature boundary conditions. It is found that by reducing the mesh spacing by an order of magnitude, the error reduces two orders of magnitude. Also it is noticed that absolute level of errors on the finest grid is about 0.1% by using second-order scheme. Natarajan et al. have reported the free convection for trapezoidal enclosures, when bottom wall is subjected to uniform and sinusoidal variation of temperature with insulated top wall. Again, the free convection in a trapezoidal cavity with uniform and linearly varied temperature at bottom and inclined side walls is studied numerically by using Galerkin finite element method. It is observed that the local Nusselt number is more at the centre of the uniformly heated wall. The secondary circulations are formed nearer to hot and cold walls. The experimental and numerical study on free convection in a solar concentrator of trapezoidal shape with Fresnel reflectors is carried out by Manikumar et al. They proved that a cavity with plane surface absorber is more useful as that of tube surface. Recently, Salari et al. have carried out numerical work related to the influence of using turbulent models on free convection in trapezoidal enclosure. It is found that the transition/turbulence model has better prediction for the flow and heat fields than fully turbulent models. The free convection within a trapezoidal enclosure with different angles of orientations of side heating wall is performed by Lasfer et al. The authors have noticed that heat transfer is dependent on flow field, inclination angle, cavity aspect ratio and Ra.
Several researchers have experimentally proved that the nanofluids are remarkable in enhancement of heat transfer in car radiators and high heat generating computer processors. Recently published articles are discussed. Jajja et al. have carriedout an experimental to investigate the effect of fin spacing of water cooled minichannel heat sink. A heater of 325 W is used to simulate a microprocessor heat. It is noticed that the overall heat transfer coefficient increases with decrease of fin spacing while decreasing in thermal resistance. The lowest base temperature of 40.5 °C is achieved for fin spacing of 0.2 mm. Thermal management of high heat generating computer microprocessor is studied experimentally using multiwalled carbon Nanotube nanofluids as a coolant by Jajja et al. It is found that the thermal resistance dropped with decrease of fin spacing which results in increase of overall heat transfer rate by about 15%.
Ali et al. have carriedout an experimental work to enhance heat transfer performance of a car radiator using ZnO and MgO-water nanofluids separately. The volumetric concentration of ZnO in water is ranging from 0.01% to 0.3%. The heat transfer is enhanced up to 46% with increase of volume concentration to 0.2% and it is decreasing with increase of volumetric concentration. Further, heat transfer rate is enhanced up to 4% as the fluid inlet temperature is increased from 45 °C to 55 °C. The water based MgO nanofluids enhanced 31% of heat transfer rate at 0.12% volumetric concentration. Further, the heat transfer rate enhanced by 6% with increase of inlet temperature to 8 °C.
The experimental investigations on the performance of two different Al2O3 and Cu water based nanofluids are conducted by Siddiqui et al. The experimentations were performed with two heaters of 130 W power for heat generation and the fluid flow rate ranging from 0.45 to 0.85 L per minute. It is showed that the Al2O3-water based nanofluid is superior in overall performance of heat transfer rate and lowest base temperature of 79.45 °C is achieved at Reynolds number 751. However, Cu-water nanofluids exhibited much better thermal enhancement than Al2O3-water nanofluids especially at higher ranges of Reynolds number.
In order to obtain the heat transfer performance of staggered and inline square pin fin minichannel heat sink is experimented by Ali et al. The water based rutile and anatase TiO2 nanofluids of 3.99% and 4.31% of volume concentration have tested and there results are compared with distilled water. A power of 192 W is applied at the side of heat sink. The better thermal performance is achieved by rutile nanofluids for both tested geometries anatase nanofluids. Also, the lowest base temperature of 29.4 °C is obtained with staggered pin fin heat sink for rutile nanofluid. The effect of pin fin heat sink channel is examined experimentally using water based graphene nano platelets nanofluids (GNPs) at a flow rate of 0.25–0.75 LPM. The tests are conducted for three different angles (22.5°, 45° and 90°) with respect to positive x-axis. The better thermal performance is achieved at 22.5° channel angle heat sink with lowest convective thermal resistance to that of other tested heat sinks.
Arshad and Ali have carried out experimentation on heat transfer and pressure drop characteristics associated with minichannel heat sink using GNPs and TiO2 nanofluids. For GNPs nanofluids, the lowest base temperature of 36.81 °C and maximum convective heat transfer enhancement about 23.91% is achieved at Reynolds number 972 for heat flux of 47.96 kW/m2. Also it is found that maximum pumping power of 0.04 W for GNPs nanofluids at heat flux of 47.96 kW/m2. The 15% weight concentration of TiO2-water based nanofluids is used as coolants at heating elements of 100 W, 125 W and 150 W power and its performance is compared with distilled water. The lowest wall temperature of 37.05 °C is recorded using TiO2 nanofluids at Reynolds number of 922 corresponding to 100 W heating power.
Several attempts to investigate the efficacy and entropy generation of hybrid nanofluids have been numerically studied and published in the open literature. The energy efficiency and flow characteristics of TiO2 and graphene-platinum nanoparticles in a chaotic geometry are investigated numerically by Bahiraei et al. They have demonstrated chaotic channel as higher heat transfer and pressure drop than the simple one and presents better cooling in comparison with base fluid. The convective heat transfer coefficient in a chaotic channel has a periodic trend. Due to excellent attributes of Graphene, there is a significant enhancement in heat transfer without noticeable increase in pressure drop. The value for figure of merit becomes >1.5 at low Reynolds number.
Irreversibility caused by heat transfer and friction in a minichannel having chaotic perturbations is examined by Bahiraei et al. It is noticed that frictional entropy generation in the chaotic channel is greater than in the straight channel and it is smaller at lower level of irreversibility. Bahiraei et al. investigated the flow and convective heat transfer of non-Newtonian Cu-water based nanofluids in annuli. Artificial Neural Network (ANN) model is used to predict the convective heat transfer coefficient. Genetic Algorithm (GA) technique is employed to optimise minimum pressure drop with maximum heat transfer considering designers view point. The thermal performance of novel liquid block working with a nanofluid containing Graphene Nanoplatelets decorated with silver nanoparticles for CPU cooling is reported by. It is found that novel liquid block has a super efficacy and lower irreversibility than conventional one. The merit of using the nanofluids in the liquid block is greater than pure water.
The majority of experimental and numerical studies are limited to free convection in square cavities for different opening ratios of vertical wall. In a real situation such as, the walls of the building and roofs of the industrial shops etc., it is very much essential to keep one of the sidewall inclined and other walls remain vertical. After thorough literature survey, the free convection in a side heated trapezoidal cavity, uniformly cooled and different openings of the sidewall opposite heated one has not been studied extensively. Moreover it is very essential to understand the fluid flow and heat transfer mechanism to enhance the rate of reheat transfer in a room with a large inclined surface at the summer seasons. Since, the aim of the present work is to know the effect of thermal strength, heat transfer rate and angle of inclination of heated walls with cold vertical wall/one opening at the top/inlet vent at the bottom and different positions of the outlet vent from the top adiabatic wall while remaining position of the cold wall is assumed to be adiabatic. Finally, power law equations are presented for average heat transfer rate and Grashoff number. This is to provide the effective thermal management of rooms of buildings and Fresnel solar reflectors.
Stanley R. Card 
