Natural convection in differentially heated rectangular cavities with time periodic boundary condition on one side

International Journal of Heat and Mass Transfer 129 (2019) 224–237

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Natural convection in differentially heated rectangular cavities with time periodic boundary condition on one side Hakan Karatas ⇑, Taner Derbentli Istanbul Technical University, Faculty of Mechanical Engineering, Gumussuyu, 34437 Istanbul, Turkey

a r t i c l e

i n f o

Article history: Received 16 April 2018 Received in revised form 9 September 2018 Accepted 17 September 2018

Keywords: Cavity Rectangular Square Local Natural convection Time periodic

a b s t r a c t Natural convection in differentially heated rectangular cavities has been investigated. One of the vertical walls of the cavity is cooled, and the opposing vertical wall is heated. The cold and hot vertical walls have constant and time periodic temperatures, respectively. The other four walls are adiabatic. Heat transfer is by convection only because surface emissivities of the walls have been reduced. Experiments are performed on rectangular cavities with aspect ratios of 1, 2.09, 3, 4, 5 and 6. All six cavities have a height of 340 mm and a depth of 210 mm. The cavity length is changed to obtain different rectangular cavities. The cavity is closed and filled with air. Thermocouples are used to measure the temperature. For each cavity, the temperature distribution between the cold and the hot vertical walls is obtained at 35 positions along the length, three positions along the height and one position along the depth directions. The contour maps of the dimensionless temperature are presented at the mid-depth of the cavity. The local temperature difference is highest, and the local Nusselt number is lowest at the mid-height. Heat transfer correlations are presented for the Rayleigh number range of 4.51
105–1.13
108, and for the aspect ratio range of 1–6. The Nusselt number largely decreases when the temperature on the heated wall is changed from constant to sinusoidally varying. Ó 2018 Elsevier Ltd. All rights reserved.

1. Introduction Natural convection in a differentially heated cavity has been extensively studied in the past. In most cases, the heated and cooled walls are maintained at constant temperatures of Th and Tc, respectively. However, in some engineering applications, the temperature of the heated wall periodically changes with time. The heating element is turned on when the temperature is below a predetermined value and turned off when the temperature is above a second predetermined value. The periodic operation can be in sinusoidal or square wave form. Some researchers investigated natural convection in a differentially heated cavity with a constant temperature on the cold vertical wall (Tc) and a time periodic temperature on the hot vertical wall (Th). The side walls are cooled and heated from whole surfaces. Kazmierczak and Chinoda [1] numerically studied natural convection of a fluid with the Prandtl number of 7 in a square cavity at the Rayleigh number of 1.4
105. The temperature of the heated wall was varied sinusoidally with time about a mean value. The temperature variation in time had amplitude of A and

⇑ Corresponding author. E-mail address: [email protected] (H. Karatas). https://doi.org/10.1016/j.ijheatmasstransfer.2018.09.087 0017-9310/Ó 2018 Elsevier Ltd. All rights reserved.

frequency of f. The dimensionless amplitude (a) and the dimensionless time period (p) of the hot wall temperature were changed for 0.2, 0.4 and 0.8, and 0.005, 0.01 and 0.02, respectively. Lage and Bejan [2] conducted a theoretical and numerical study on natural convection of three fluids in a square cavity. For the fluid with the Prandtl number of 0.01, the Rayleigh number was varied as 105, 106 and 107. The Rayleigh number was changed as 106, 107 and 108 for the Prandtl number of 0.7. For the Prandtl number of 7, the Rayleigh number was changed as 107, 108 and 109. The heat flux at the hot wall was fluctuated in square wave form around the mean value. Antohe and Lage [3,4] theoretically and numerically investigated natural convection in a square cavity for different cases. For a fluid with the Prandtl number of 7, the Rayleigh number was changed as 107, 108 and 109. For the porous medium with the Darcy number of 102, the Rayleigh number was varied as 106, 107 and 108. For the porous medium with the Darcy number of 104, the Rayleigh number was changed as 1010, 1011 and 1012. The Prandtl number was also changed as 0.01, 0.02 and 0.7. The heat flux at the hot wall was varied in square wave form. Kwak et al. [5] carried out a numerical study on natural convection of a fluid with the Prandtl number of 0.7 in a square cavity at the Rayleigh number of 107. The temperature of the hot wall was changed sinusoidally. The dimensionless amplitude of the hot wall temperature oscillation was varied as 0.0, 0.1, 0.5 and 1.0. The

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225

Nomenclature a A D Da f g Gr h hy H k K L NuL NuyL p P Pr Q R RaL RayL T

dimensionless amplitude aspect ratio, A = H/L; amplitude cavity depth, m Darcy number, Da ¼ K=H2 frequency, 1/s, Hz gravitational acceleration, m/s2 Grashof number average heat transfer coefficient, W/m2 K local heat transfer coefficient, W/m2 K cavity height, m thermal conductivity of fluid, W/mK permeability, m2 cavity length, m average Nusselt number based on the cavity length local Nusselt number based on the cavity length dimensionless time period time period Prandtl number, Pr ¼ m=a heat transfer rate, W total thermal resistance, K/W average Rayleigh number based on the cavity length local Rayleigh number based on the cavity length temperature, °C

dimensionless time period was varied in the range of 0.67 to 10. Shu et al. [6] numerically and experimentally studied natural convection of distilled water in a rectangular prism cavity with an aspect ratio of 1.11. Aspect ratio (A) is defined as the ratio of the cavity height (H) to cavity length (L). The cavity was 20 mm high, 18 mm long and 150 mm deep. The angle of inclination of cavity was changed for 0° and 45°. The temperature of the hot wall was changed sinusoidally. The frequency was varied from 0.02 to 0.05 Hz. In the experimental study, the surface temperatures of the hot and cold walls and the velocity field in the cavity were presented. Wang et al. [7] carried out a numerical study on natural convection in a cubical cavity for the Rayleigh number range of 106 to 107. The fluid in the cavity was a porous medium with the Prandtl number of 1. The cavity was inclined around two axes alternately. One of the angles was fixed at 0° and the other was varied in the range of 0–90°. The temperature variation of the hot wall was sinusoidal. The dimensionless oscillating frequency was changed from 5p to 90p for the optimal inclinations of 50° and 45°, and for the Darcy number of 103. Huang et al. [8] numerically studied natural convection of air in a cubical cavity. The Rayleigh number was varied from 103 to 106. The hot wall had a sinusoidally varying temperature. The pulsating amplitude and period were changed for 0.0, 0.5, 1.0 and 1.5, and in the range of 1–104, respectively. Wang et al. [9] numerically and experimentally investigated natural convection of water in a rectangular prism cavity with an aspect ratio of 0.94 at the Grashof number of 6.8
107. The cavity was 30 mm high and 32 mm long. The temperature of the heated wall was varied in square wave form. The angle of inclination of cavity was maintained at 0°, 30°, 60° and 90°. Zhang et al. [10] numerically studied conjugate conduction and natural convection in a square cavity at the Rayleigh number of 105. The fluid was air. The angle of inclination of cavity was changed from 0° to 90°. The temperature of the hot wall was varied sinusoidally. For the angle of inclination of 0°, the thermal conductivity and the thermal diffusivity ratios were maintained as 10 and 0.01, respectively, and the amplitude and the pulsating period were changed as 0.2, 0.8 and 1.5, and 1, 10, 200 and 104, respectively.

DT DTy x, y, z X, Y, Z

average temperature difference, °C local temperature difference, °C coordinates, m dimensionless coordinates, X = x/L, Y = y/H, Z = z/D

Greek symbols a thermal diffusivity of fluid, m2/s b thermal expansion coefficient of fluid, 1/K m kinematic viscosity of fluid, m2/s T  Tc h dimensionless temperature, h ¼ Th  Tc Subscripts a ambient c cold, cooled f at the mean temperature Tm fy at the local mean temperature Tmy h hot, heated L based on L m mean my local mean y at the measurement position y, local

A few researchers studied natural convection in a differentially heated cavity having partially active side walls. The cavity had a constant temperature on the cold wall and a time periodic temperature on the hot wall. Ghasemi and Aminossadati [11] numerically investigated natural convection in a square cavity for the Rayleigh number range of 103–106. The heated and cooled vertical walls were partially and fully active, respectively. The active part was half height of the cavity wall and the inactive part was adiabatic. Two fluids were studied: water and a water based nanofluid with the Prandtl number of 6.2. The variation of the heat flux at the heated wall was cosinusoidal. The dimensionless oscillation period was investigated for 0.01, 0.1, 1, 10 and 100. Nithyadevi et al. [12] numerically studied natural convection in a cavity with partially active vertical walls. The active part was half height of the cavity wall and the inactive part was adiabatic. The active part was located at the top, middle and bottom for each vertical wall and thus, nine cases were studied. The temperature of the hot wall was varied sinusoidally. The effect of the magnetic field was investigated. In the case without the magnetic field, the active parts of the hot and cold walls were fixed at the bottom and top positions, respectively, the Prandtl number, the dimensionless amplitude and the dimensionless time period were maintained at 0.71, 0.4 and 3, respectively, and the Grashof number was changed for 104, 105 and 106. In summary, many researchers investigated natural convection in a differentially heated cavity having constant temperatures on the cooled and the heated vertical walls, and presented heat transfer correlations for different ranges of aspect ratio, Rayleigh number, and Prandtl number. This is due to the heat transfer differs widely in boundary conditions, geometries, fluids and flow regimes. Extensive studies have also been conducted on time periodic boundary condition. Many other researchers studied natural convection in a differentially heated cavity with a constant temperature on the cooled vertical wall and a time periodic temperature on the heated vertical wall. However, in most of the studies, the focus was a single cavity with the aspect ratio of 1. In the studies of [1–5,10–12], the cavity was a square, and in the studies of [7,8], the cavity was a cubical. Furthermore, in the

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studies of [6,9], the aspect ratio was around 1. There are no results for a wide range of aspect ratio. On the other hand, in most cases, the study was numerical [1–5,7,8,10–12]. There are not many experimental results. In addition, in most of the works, the geometry was two-dimensional. The present study investigates natural convection of air inside a differentially heated cavity with a constant temperature on the cooled vertical wall and a time periodic temperature on the heated vertical wall. The objectives of this study are to investigate the effect of aspect ratio on heat transfer in the cavity, to study the natural convection in three-dimensional rectangular prism cavities, to experimentally study the heat transfer, to provide data for numerical studies and to obtain heat transfer correlations for different ranges of aspect ratio and Rayleigh number. In the present study, attention has also been given to the effect of the temperature boundary condition on the heated wall. The results of this study for sinusoidally varying temperature are compared with the results of the previous studies for constant temperature in the aspect ratio range of 1–5. Thus, the relationship between the time periodic and the constant temperature boundary conditions is obtained for differentially heated rectangular cavities.

2. Experimental facility In the present study, natural convection of air in closed rectangular prism cavities with aspect ratios (A) of 1, 2.09, 3, 4, 5 and 6 was investigated. All six cavities had a height (H) of 340 mm and a depth (D) of 210 mm. The length (L) was changed to obtain cavities with different aspect ratios. Therefore, each rectangular cavity had a different length. The test room was air conditioned. Fig. 1 shows the description of the cavity. The schematic, front and top sectional views of the cavity with the aspect ratio of 2.09 are presented in Fig. 1a–c, respectively. The cavity is drawn to scale. The gravitational acceleration (g) acts in the negative y direction. The test cavity was cooled from one side, and heated from the opposite side. The top, bottom, front and back walls were adiabatic. The left and right walls had thinner insulations than the adiabatic walls. The cavity was locally cooled from the left vertical wall. The local heat sink was square with a side length of 42 mm. In the y direction, the distance between the bottom edge of local sink and the bottom of cavity was 186.5 mm. In the z direction, the distance between the center of local sink and the front of cavity was 105 mm. The local sink was the cooling part of a thermoelectric device. Because the contact between the local sink and the left wall was maintained by means of thermal paste and screws, and the wall was made of aluminium, the left wall was cooled from the whole surface. The cavity was locally heated from the right vertical wall. The local heat source was square with a side length of 42 mm. The heat sink and the heat source had equal dimensions. The distance between the top edge of local source and the top of cavity was 186.5 mm in the y direction. The distance between the center of local source and the front of cavity was 105 mm in the z direction. The local source was the heating part of a thermoelectric device. The contact between the local source and the right wall was maintained by means of thermal paste and screws, and the wall was made of aluminium. Therefore, the right wall was heated from the whole surface. The details of the local heat sink are shown in Fig. 1d and e. The local heat sink was composed of three parts: hot plate, Peltier plate and cold block. The Peltier plate was mounted between the hot plate and the cold block. The Peltier plate heated the hot plate and cooled the cold block due to its hot left and cold right surfaces. The hot plate was at the outside of the cavity, and cooled with air by a fan. The hot plate had cooling fins on it. The Peltier plate and the cold block were embedded in the insulation of the cooled left

wall of the cavity. Therefore, the top, bottom, front and back walls of the Peltier plate and the cold block were insulated. The cold block was mounted to the left wall of the cavity. The cavity was cooled by the right surface of the cold block. The contacts between the hot plate, the Peltier plate and the cold block were maintained by means of thermal paste and screws. The hot plate and the cold block were made of aluminium. The left and right surfaces of the Peltier plate were ceramic. A temperature sensor was placed on the front surface of the cold block to control the temperature of the left wall of the cavity. Although much research has been done on natural convection in differentially heated cavities, less attention has been paid to the temperature distribution on the wall. The present study aims to fill this void. Most of the previous studies have concentrated on uniform temperature distributions on the cooled and the heated walls. This study aims to investigate natural convection in differentially heated cavity having non uniform temperature distributions on the cooled and the heated walls. For this purpose, a local heat sink and a local heat source were used. The cold vertical wall was cooled with a single heat sink to obtain a non uniform temperature distribution on the wall surface. The cooling part of a thermoelectric device was used to cool the sink. The surface area of the heat sink resulted from the surface area of the device. The hot vertical wall was heated with a single heat source to obtain a non uniform temperature distribution on the wall surface. The heating part of another thermoelectric device was used to heat the source. The surface area of the heat source resulted from the surface area of the device. The cooled left wall was counterbalanced with the heated right wall, and also, the local heat sink on the left wall was counterbalanced with the local heat source on the right wall. On the other hand, the cold wall was cooled from its whole surface and the hot wall was heated from its whole surface. The surface temperature of the cold wall changed along the cavity height due to the local heat sink, and the surface temperature of the hot wall changed along the cavity height due to the local heat source. The temperature difference between the cold and hot vertical walls, the temperature field inside the cavity, and the heat transfer changed along the height. The other aims of this study are to investigate the effects of the sink and the source on the temperature distributions on the cold and hot walls, respectively, to study the effects of the local sources on the temperature field, the local temperature difference between the opposing walls, and the local heat transfer along the height. The cooled and heated walls were made of aluminium and the other four walls were made of plastic. The aluminium sheet had a thickness of 1 mm and a thermal conductivity of 237 W/m K [13]. The plastic, polypropylene, sheet had a thickness of 1 mm and a thermal conductivity of 0.22 W/m K [14]. All six walls of the cavity were covered with an adhesive aluminium foil tape to prevent heat transfer by radiation. The aluminium foil had a thickness of 0.12 mm and a thermal conductivity of 235 W/m K. The polished aluminium foil had a surface emissivity of 0.05 as reported both in [13] and in the data sheet of the manufacturer. Hence, it is assumed that the heat transfer mechanism in the cavity is by natural convection only. All six walls of the cavity were insulated with various insulation materials of different thicknesses. The cooled left wall and the heated right wall were insulated with expanded polystyrene boards. The insulation thickness was 23 mm for the left and right walls. The other four walls were insulated with extruded polystyrene boards. The insulation thickness of the bottom wall was 100 mm. The insulation thicknesses of the top, front and back walls were 80 mm. The thermal conductivities were 0.039 and 0.035 W/m K for the expanded and extruded polystyrene boards, respectively. Plastic sheet with a thickness of 2 mm and a thermal conductivity of 0.22 W/m K was used on the outer surfaces of the left and right walls.

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227

Fig. 1. Description of the cavity: (a) Schematic view, (b) Front sectional view, (c) Top sectional view, and (d and e) Details of the local heat sink.

Temperature measurements were performed to investigate the average and local heat transfer inside the rectangular cavity. Type T, copper-constantan, thermocouples were used to measure surface and air temperatures in the cavity. The diameter of the thermocouple wires was 0.25 mm, American Wire Gauge (AWG) 30. In the experimental set-up, 44 thermocouples were used to measure surface and air temperatures. The ambient air temperature outside

the cavity was measured by two thermocouples, placed at the outside and the geometric centers of the front and back walls of the cavity. The set-up was designed to measure both the average and the local temperatures in the same experiment, and furthermore, to simultaneously measure the local temperatures along the x axis. Fig. 2 shows the instrumentation and the cavity. The thermocouples are not drawn to scale. The surface thermocouples are

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represented by black filled circles, and the air thermocouples are represented by white circles. The surface temperatures of the cavity walls were measured by six thermocouples. One thermocouple was installed on each of the six walls. The air temperature in the cavity was measured by one thermocouple, installed at the geometric center of the cavity. The positions of these seven thermocouples were fixed in all experiments, and are shown in the schematic view in Fig. 2a. The temperature distribution between the cooled left wall and the heated right wall, along the x axis, in a specific position in the cavity was measured by 35 thermocouples. The measurement system is shown in the front sectional view in Fig. 2b. The surface temperatures of the left and right walls were measured by two thermocouples, one placed at the left and the other placed at the right. The measurement of air temperature was performed by 33 thermocouples. Firstly, the thermocouples were installed on a measurement rod at the outside of the cavity. Then, the measurement rod was placed into the cavity. The surface thermocouples and the measurement rod were aligned. The length of the measurement rod was smaller than that of the cavity in order to prevent heat transfer by conduction between the left and right walls. In all cavities, the air thermocouples were placed 1.8 mm apart from each other for the first 18 mm from the left wall in the first region. The air thermocouples were placed 25.0, 10.0, 6.5, 4.0, 2.6 and 1.8 mm apart from each other for the cavities with aspect ratios of 1, 2.09, 3, 4, 5 and 6, respectively, and up to the cavity center in the second region. The locations of the thermocouples from the right wall were symmetric. The position of the surface thermocouples and the rod was changed in different experiments. The thermocouple wires were bundled together and fixed in the cavity. The top, bottom, front, back and cooled walls of the cavity are shown in Fig. 2c, when the heated wall is opened. The cavity walls are covered with aluminium foil. The thermocouples were connected to data loggers to complete the temperature measurement system. The data loggers were controlled by computers. The temperature measurement system was calibrated by using a constant temperature bath, a standard platinum resistance thermometer and a readout instrument. The stability of the bath at 95% confidence was ±0.005 °C at 0 °C. The accuracy of the thermometer was 0.00025 °C at 0 °C. The

temperature measurement system was calibrated in the temperature range of 1 to 40 °C, at nine points with an interval of 5 °C. The calibration coefficients for each of the 44 thermocouples were obtained by regression analysis. The dimensionless cavity length (X) is the ratio of the distance on the x axis to the cavity length. The temperature distribution between the left (X = 0) and right (X = 1) walls of the cavity was investigated for three measuring positions of the cavity height and one measuring position of the cavity depth. The dimensionless cavity height (Y) is the ratio of the distance on the y axis to the cavity height. The dimensionless cavity depth (Z) is the ratio of the distance on the z axis to the cavity depth. In all cavities, the number of thermocouples is kept constant in each of the three regions. In addition, the distance between thermocouples, in other words, the distance on the x axis is fixed in the first and the third regions. However, the dimensionless cavity length (X) changes with varying aspect ratio due to the cavity length decreases with increasing aspect ratio from A = 1 to 6. On the other hand, the distance between thermocouples decreases with decreasing cavity length in the second region. Chu et al. [15] carried out a numerical and experimental study on natural convection of air in rectangular cavities. One vertical wall was heated by one heat source and opposing vertical was isothermally cooled. Inactive parts of the wall were adiabatic. They investigated the optimum dimensionless location of the heat source for a square cavity. The optimum condition was for maximum heat transfer. The dimensionless height of the heat source was 0.2. Horizontal walls were adiabatic or isothermally cooled. In the case of adiabatic horizontal walls, the optimum dimensionless location of the heat source was found as 0.5, 0.5, 0.5 and 0.6 for the Rayleigh numbers of 6.25
103, 2.5
104, 6.25
104 and 105, respectively. The location of the heat source was the distance of the center of the source from the top wall of the cavity. The location was investigated from 0.1 to 0.9 with an interval of 0.1. Considering the distance of the bottom edge of the source from the bottom wall of the cavity, the optimum dimensionless location of the heat source was 0.4, 0.4, 0.4 and 0.3 for the Rayleigh numbers of 6.25
103, 2.5
104, 6.25
104 and 105, respectively. Dias and Milanez [16] numerically investigated natural convection of air

Fig. 2. Instrumentation and cavity: (a) Thermocouple positions in schematic view, (b) Thermocouple positions in front sectional view, and (c) Cavity walls.

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in a square cavity in the Rayleigh number range of 102 to 106. Horizontal walls were adiabatic. One vertical wall was heated by discrete heat sources and opposing vertical wall was isothermally cooled. Sources were flush mounted. Inactive parts of the wall were adiabatic. They studied one or two heat sources and varied the dissipation rates of the sources. In the case of one heat source, the optimum dimensionless location of the heat source was found 0.45 and 0.334 for the Rayleigh numbers of 103 and 106, respectively. The location of the heat source was the distance of the bottom edge of the source from the bottom wall of the cavity. The optimum condition was for maximum heat transfer. Fig. 2b also shows the positions of the local heat source and the local heat sink. In the studies of [15] and [16], the optimum dimensionless location of the heat source was found 0.3 and 0.334 for the Rayleigh numbers of 105 and 106, respectively. In the present study, the distance of the bottom edge of the local heat source from the bottom wall of the cavity is 0.328. The dimensionless location in the present study is in agreement with those in [15,16]. The local heat source is between Y = 0.328 and 0.451 at the heated right wall. The fluid motion along the cooled left wall is in the negative y direction while that along the heated right wall is in the positive y direction. Therefore, the position of the local heat sink is maintained at the symmetry of the position of the local heat source with respect to Y = 0.500. The distance of the top edge of the local heat sink from the top wall of the cavity is 0.328. The local heat sink is between Y = 0.549 and 0.672 at the cooled left wall. The centers of the local heat source and sink are positioned at the half depth of the cavity (Z = 0.500) in the z direction.

ture of the air conditioned room was maintained in the range of 21–22 °C. The investigation was carried out for the dimensionless cavity heights (Y) of 0.125, 0.500 and 0.875 at the dimensionless cavity depth (Z) of 0.500. Because each height was obtained by a separate experiment, three experiments were performed for each cavity. Therefore, a total of 18 experiments were conducted in this study. The heat transfer characteristics were defined between the cooled and heated walls. The heat transfer in the cavity was investigated in two ways: average heat transfer and local heat transfer. The average heat transfer was based on the thermocouple positions shown in the schematic view in Fig. 2a. The surface temperature of the cooled wall (Tc) was measured at x = 0, y = H/2 and z = 7D/8. The surface temperature of the heated wall (Th) was measured at x = L, y = H/2 and z = 7D/8. Because three experiments were conducted, three average heat transfer data were obtained for each cavity. The average temperature difference was the difference between the surface temperatures of the heated and cooled walls:

DT ¼ T h  T c

In the present study, natural convection of air in six rectangular cavities was investigated. The left wall of the cavity was cooled by a thermoelectric device. The right wall of the cavity was heated by another thermoelectric device. Each device had a temperature sensor and an electronic control unit to control the wall temperature. The constant and the time periodic temperatures were obtained on the cooled left and the heated right walls, respectively. For the time periodic temperature boundary condition, the effects of the amplitude and the time period were investigated in the previous studies. The amplitude, the period and the frequency were varied in the studies of [1,5–8,10,11] for a fixed aspect ratio. In this study, the dimensionless amplitude and the dimensionless time period of the local heat source on the heated wall were maintained constant, and the aspect ratio was varied. The thermoelectric device is powered by alternating current, and controlled by a temperature sensor. The device is turned on and off by means of the sensor. The temperature of the heated right wall varies with time due to the device. A sinusoidally varying temperature is obtained on the hot wall. Therefore, this study focuses on the periodic variation of the temperature. In the experiments, the positions of the thermocouples for the ambient air, the cavity temperature and the surface temperatures were kept constant while the positions of the thermocouples to obtain the temperature distribution between the left and right walls were changed. The temperature distribution across the two opposite walls was obtained for different cavity heights at a fixed cavity depth. Prior to the experiment, the measurement rod was installed at a specific height. Then, the heated wall was closed and sealed. Thus, a closed cavity was formed. After, the cooler and the heater, the two thermoelectric devices, were started. The experiment took 4–5 h. The scanning interval was one minute. The last 45 min of the recorded data was used to obtain the arithmetic average for each thermocouple measurement. The tempera-

ð1Þ

The average heat transfer coefficient was found as follows:

h¼

Qc DT HD

ð2Þ

where Qc is the heat transfer rate from the cooled wall. The heat transfer rate from the cooled wall (Qc) was calculated by the following equation:

Qc ¼ 3. Experimental method and data reduction

229

ðT a  T c Þ Rc

ð3Þ

Eq. (3) gives the heat loss through the cooled wall. The Rc is the total thermal resistance between the ambient air temperature (Ta) and the surface temperature of the cooled wall (Tc). The Rc consists of the following components: the conductive thermal resistances of the inner wall, the insulation and the outer wall, the convective and radiative thermal resistances of the ambient air outside the cavity, which are calculated by using the equations and correlations given in [13]. The cooled left wall is composed of three walls: outer wall, insulation and inner wall. The Rc is composed of the thermal resistances of the ambient air outside the cavity, the outer wall, the insulation and the inner wall. The thermal resistances of the ambient air outside the cavity, the outer wall, the insulation and the inner wall were calculated as 0.75021, 0.04682, 4.67516 and 0.00006 K/W, respectively. The total thermal resistance (Rc) for the cooled left wall of the cavity was found 5.47225 K/W. In this study, the surface temperature of the cooled wall (Tc), the ambient air temperature (Ta) and the heat transfer rate from the cooled wall (Qc) vary for each of the experiments while the Rc is constant. The average Nusselt and Rayleigh numbers based on the cavity length were calculated as follows:

NuL ¼

hL kf

RaL ¼

g b f DT L 3

mf af

ð4Þ

The thermal conductivity (kf), thermal expansion coefficient (bf), kinematic viscosity (mf) and thermal diffusivity (af) of the fluid, in Eq. (4), were calculated at the mean temperature, Tm:

Tm ¼

Th þ Tc 2

ð5Þ

The local heat transfer was based on the thermocouple positions shown in the front sectional view in Fig. 2b. For simplicity, the analysis is defined as local. The local surface temperature of the cooled wall (Tcy) was measured at x = 0 and z = D/2. The local surface temperature of the heated wall (Thy) was measured at

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x = L and z = D/2. The position of the measurement rod was changed for the dimensionless cavity heights (Y) of 0.125, 0.500 and 0.875 in the y direction. Therefore, three local data were obtained for each cavity. The local data were used to investigate the variation of heat transfer in the y direction. This is due to the varying local surface temperatures of the cooled and heated walls, Tcy and Thy, respectively, along the height. The local temperature difference was the difference between the local surface temperatures of the heated and cooled walls at the measurement position y:

DT y ¼ T hy  T cy

ð6Þ

The local heat transfer coefficient was found as follows:

hy ¼

Qc DT y HD

ð7Þ

The local Nusselt and Rayleigh numbers based on the cavity length were calculated by the following expressions:

NuyL ¼

hy L kfy

RayL ¼

g bfy DT y L3

ð8Þ

mfy afy

The thermal conductivity (kfy), thermal expansion coefficient (bfy), kinematic viscosity (mfy) and thermal diffusivity (afy) of the fluid, in Eq. (8), were calculated at the local mean temperature, Tmy:

T my ¼

T hy þ T cy 2

ð9Þ

The dimensionless temperature was defined by:

h¼

T  Tc Th  Tc

ð10Þ

In Eq. (10), T is the temperature at the measurement position (x, y). Th is the highest heated wall temperature and Tc is the lowest cooled wall temperature among the measurements for three different Y positions in each cavity. 4. Results and discussion The variations of the characteristics with time are presented in Fig. 3 for one of the 18 experiments. The plots are for the case of A = 3, Y = 0.500. The time period is the last 45 min of an experiment lasting 295 min. The scanning interval is one minute. Fig. 3a shows the time variation of the surface temperature of the cooled wall (Tc). The time variation of the surface temperature of the heated wall (Th) is presented in Fig. 3b. The temperature of the heated wall 

varies periodically with time about a mean value (T h ). The temperature variation in time has an amplitude of A. The dimensionless amplitude (a) was found as follows:

a¼

A 

ðT h  T c Þ

ð11Þ

The temperature variation in time has a time period of P. The dimensionless time period (p) was calculated by the following equation:

p¼

P ðH2 =af Þ

ð12Þ

Fig. 3c shows the time variation of the average Nusselt number based on the cavity length (NuL). The standard deviation was calculated for all thermocouple measurements and experiments. For 18 experiments and for the surface temperature of the cooled wall, the minimum, maximum and mean of the standard deviations were 0.023, 0.055 and

0.037 °C, respectively. For the 18 experiments and for the surface temperature of the heated wall, the minimum, maximum and mean of the standard deviations were 0.818, 1.352 and 0.998 °C, respectively. In the calculations, the arithmetic averaging was used to represent the measurement data which vary with time. Fig. 4 shows the temperature distribution between the cooled left wall and the heated right wall for the cavities with the aspect ratios of A = 1, 2.09, 3, 4, 5 and 6. In each aspect ratio, the temperature distribution was plotted for the dimensionless cavity heights (Y) of 0.125, 0.500 and 0.875. For each height, the temperature measurements at 35 positions along the x axis were presented. The dimensionless cavity length (X) was between 0 and 1. The graphs show that the temperature increases from the left wall to the right wall, and three regions are observed along the X axis for all cavities. The second region is from X = 0.2 to X = 0.8. The boundary between the first and the second regions becomes less pronounced as the aspect ratio increases. The same is true for the boundary between the second and the third regions. The boundaries are unclear in the cases for A = 5 and 6. In the second region, the temperature gradient is low, and the temperature profile shows a linear variation. The temperature can be considered constant because it hardly changes with the distance. The temperature profile is nearly horizontal for A = 1. The slope of the line slightly increases with increasing aspect ratio from A = 2.09 to 6. The temperature gradient is the ratio of the change in temperature to the change in distance in the direction normal to the wall. In the first and third regions, the temperature gradient is high and the temperature sharply changes. In the first region, the temperature increases from the left wall to the center. In the cases for A = 1, 2.09 and 3, the curves are increasing and concave downward for the three heights. In the cases for A = 4, 5 and 6, the curves are increasing and concave downward for Y = 0.500 and 0.875. In the third region, the temperature decreases from the right wall to the center. In the cases for A = 1, 2.09 and 3, the curves are increasing and concave upward for the three heights. In the cases for A = 4, 5 and 6, the curves are increasing and concave upward for Y = 0.500. The curves are linear for Y = 0.125 and 0.875. For each of the six cavities, the temperature gradient is nearly the same for the left and the right walls. The temperature gradient and the curvature of the profile in the first and third regions decrease with increasing aspect ratio from A = 1 to 6. In the cases of A = 4, 5 and 6, the profile is linear and the slope of the line is small between X = 0 and 1 for Y = 0.125. For all aspect ratios, the temperature decreases as Y decreases from 0.875 to 0.125. The present study differs from the existing literature in the following ways: First, the average and the local temperatures were measured in the same experiment. Second, the local temperatures between the left and right walls were measured simultaneously. In this study, one thermocouple for each position, and thus, a total of 35 thermocouples were used to instantaneously measure the temperature distribution along the x axis, between X = 0 and 1. In the previous studies, different methods have been used to measure the temperature distribution. In the works of Eckert and Carlson [17] and Nicolette et al. [18], a Mach-Zehnder interferometer was used. In the work of Corvaro et al. [19], a holographic interferometer was used. In the studies of MacGregor and Emery [20], Yin et al. [21] and Betts and Bokhari [22], a single thermocouple was moved by a traverse system to measure the temperature distribution along the x axis. Therefore, the temperatures between X = 0 and 1 were measured sequentially. The results of the present study were compared with the literature in terms of the form of the temperature profile. The concave downward curve on the left wall, the concave upward curve on the right wall, the gradients on the walls, the horizontal line in the central region, and the overall form of the profile are in agreement with those obtained in the previous studies [17–22]. It may be concluded that the thermocouples together

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231

Fig. 3. For the experiment A = 3, Y = 0.500, variations of characteristics with time: (a) Tc, (b) Th, and (c) NuL.

with the measurement rod are not very intrusive to the flow inside the cavity. Lankhorst et al. [23] experimentally studied natural convection of air in a differentially heated square cavity at the Rayleigh number of 109. They measured the mean velocity profile and also the velocity fluctuations by using the laser Doppler velocimeter. They presented the vertical velocity profiles on the cold wall for the dimensionless cavity heights (Y) of 0.70, 0.85 and 0.95. The velocity on the cold wall was in the downward direction. They also provided the vertical velocity profiles on the hot wall for Y = 0.05, 0.15 and 0.50. The velocity on the hot wall was in the upward direction. They found that the peak velocity shifted away from the wall and the thickness of the boundary layer increased with increasing dimensionless height (Y) on the hot wall. They also presented the velocity profile in the core region, and found the stratification in this region, between the cold and hot walls along the cavity length. Betts and Bokhari [22] experimentally studied natural convection of air in a rectangular cavity with an aspect ratio of 28.7 for the Rayleigh number range of 0.86
106–1.43
106. They presented the vertical velocity profile for the cold and hot walls and for the dimensionless heights (Y) from 0.05 to 0.95 by means of the laser Doppler anemometry. They observed the downward and the upward flows along the cold and the hot walls, respectively. They reported that the maximum velocity for the cold wall was at Y = 0.9 and that for the hot wall was at Y = 0.1. In the present

study, the temperature profile is used to describe the flow in the cavity. It can be expressed that the temperature profiles of this study are in agreement with the velocity profiles of Lankhorst et al. [23] and Betts and Bokhari [22]. Fig. 5 shows the isotherms for the cavities with the aspect ratios of A = 1, 2.09, 3, 4, 5 and 6. The isotherms are the contour maps of the dimensionless temperature (h). For each cavity, the temperature measurements at 35 positions along the x axis and three positions along the y axis were presented. The dimensionless cavity length (X) was between 0 and 1. The measurement positions for the dimensionless cavity height (Y) were 0.125, 0.500 and 0.875. Therefore, the Y axis was limited between 0.125 and 0.875. The local heat sink is between Y = 0.549 and 0.672 at the cooled left wall and the local heat source is between Y = 0.328 and 0.451 at the heated right wall. The minimum and maximum dimensionless temperatures occur at X = 0 and X = 1, respectively. The dimensionless temperature is lowest at the mid-height (Y = 0.500) of the cooled wall, and symmetrically increases in both upward and downward directions from Y = 0.500 to 0.875 and from Y = 0.500 to 0.125, respectively, at X = 0. In the cases for A = 1, 2.09 and 3, the isotherms are concentrated near the cooled wall, between X = 0 and X = 0.05. In the cases for A = 4, 5 and 6, the isotherms are concentrated near the cooled wall, between X = 0 and X = 0.1. The dimensionless temperature is highest near the mid-height (Y = 0.500) of the heated wall, and

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Fig. 4. Temperature distribution between the left and right walls: (a) A = 1, (b) A = 2.09, (c) A = 3, (d) A = 4, (e) A = 5, and (f) A = 6.

decreases in both upward (from Y = 0.500 to 0.875) and downward (from Y = 0.500 to 0.125) directions at X = 1. The decrease in temperature and the distribution of the isotherms in upward and downward directions are more symmetric in the cases for A = 1, 2.09 and 3 than those in the cases for A = 4, 5 and 6. The observations for the isotherms are consistent with the temperature profiles in Fig. 4. For A = 4, 5 and 6, the temperature profile for Y = 0.500 is closer to Y = 0.875 than Y = 0.125 between X = 0.7 and X = 1. This can be due to the heat transfer coefficient decreases

and the effect of geometry on heat transfer becomes larger with decreasing cavity length. The cavity length decreases with increasing aspect ratio. The isotherms are concentrated between X = 0.95 and X = 1 for A = 1, 2.09 and 3, and from X = 0.9 to X = 1 for A = 4, 5 and 6, respectively. Among the three Y positions, the isotherms are centered at Y = 0.500 for both cooled and heated walls, and for all aspect ratios. The density of the isotherms is the same for the cooled left wall and the heated right wall. In near wall regions, the contours are

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233

Fig. 5. Isotherms in the cavity: (a) A = 1, (b) A = 2.09, (c) A = 3, (d) A = 4, (e) A = 5, and (f) A = 6.

straight, vertical and parallel for A = 1, 2.09 and 3, and curved for A = 4, 5 and 6. The density and the curvature of the contours increase with increasing aspect ratio from 1 to 6 between X = 0.1 and X = 0.9. The contours are nearly horizontal for A = 1. The rate of change decreases with increasing aspect ratio. The density of the contours slightly increases with increasing aspect ratio from A = 4 to 6. The density of the isotherms in near wall regions is larger than that in the central region. Considering the diagonal line from the top left corner to the bottom right corner, the contours

are nearly symmetrical. The fluid moves from the bottom left corner to the top right corner, in a counter-clockwise direction. The contours are curved to the right and downward at the cooled wall between Y = 0.125 and 0.500. The isotherms are curved to the left and upward at the heated wall between Y = 0.500 and 0.875. In Fig. 6, three average heat transfer data for each cavity, and a total of 18 data are presented. The average Nusselt and Rayleigh numbers are based on the cavity length. The uncertainty is calculated by using the method described by Kline and McClintock

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Fig. 7. Local temperature difference along the cavity height. Fig. 6. Relationship between NuL and RaLA.

[24] and Moffat [25] at 95% confidence. For 18 data in Fig. 6, the minimum, maximum and average uncertainties for the Nusselt number (NuL) are 0.98, 1.63 and 1.15%, respectively. Each of the three data represents the result of a different experiment. Each of the experiments is performed for a different height. The dimensionless cavity height (Y) is investigated for 0.125, 0.500 and 0.875. In each experiment, the position of the measurement rod and the thermocouples for the local surface temperatures, shown in Fig. 2b, is changed while the position of the average surface temperatures, shown in Fig. 2a, is fixed. The former gives both the temperature distribution and the local heat transfer while the latter provides the average heat transfer. By using the three data for each of the six cavities, the standard deviations for the NuL are calculated as 0.94, 0.08, 0.08, 0.10, 0.08 and 0.01 for the aspect ratios of 1, 2.09, 3, 4, 5 and 6, respectively. The results show that the three data for each of the six cavities are repeatable. The results also show that the change in the position of the measurement rod does not affect the average heat transfer in the cavity for all six cases. It may be concluded that the measurement rod is not very intrusive to the flow. In this study, the Rayleigh number and the aspect ratio are changed at the same time. The change in the Rayleigh number is mainly due to the change in the aspect ratio. The average Rayleigh number was multiplied by the aspect ratio to obtain RaLA for each cavity. First, the variation of the NuL with the RaLA was found. Then, on the basis of the three heat transfer data for each cavity (thus a total of 18 data), a heat transfer correlation was obtained by regression analysis as follows:

NuL ¼ 0:0018 ðRaL AÞ0:4918

ð13Þ

The average Rayleigh number based on the cavity length is in the range of 4.51
105 to 1.13
108. The RaLA is between 2.70
106 and 1.13
108. The aspect ratio is in the range of 1–6. The regression coefficient is 0.99525860. The plot of the correlation is presented in Fig. 6. Another heat transfer correlation was obtained by using the same results:

NuL ¼ 0:1583 Ra0:25 A0:25 L

ð14Þ

The average Rayleigh number based on the cavity length is in the range of 4.51
105 to 1.13
108. The aspect ratio is in the range of 1 to 6. The regression coefficient is 0.99364927. Fig. 7 shows the local temperature difference between the cooled left wall and heated right wall along the cavity height.

Fig. 8. Local Nusselt number along the cavity height.

The local temperature difference increases with increasing Y from 0.125 to 0.500, and decreases with increasing Y from 0.500 to 0.875. The local temperature difference is highest at Y = 0.500 because the local heat sink is between Y = 0.549 and 0.672, and the local heat source is between Y = 0.328 and 0.451. The local temperature difference is nearly symmetric with respect to the midheight. On the other hand, the local temperature difference at Y = 0.875 is slightly larger than that at Y = 0.125. Fig. 8 presents the local Nusselt number along the cavity height. The Nusselt number is based on the cavity length. The change in the local Nusselt number results from the change in the local heat transfer coefficient. The reason for the change in the local heat transfer coefficient is the change in the local temperature difference. The relationship between Figs. 7 and 8 is opposite. The local Nusselt number decreases with increasing Y from 0.125 to 0.500, and increases with increasing Y from 0.500 to 0.875. The local Nusselt number is lowest at Y = 0.500. The local Nusselt number is nearly symmetric with respect to the mid-height. In addition, the local Nusselt number at Y = 0.875 is slightly smaller than that at Y = 0.125. The variation of the local Nusselt number along the cavity height decreases with increasing A from 1 to 6. The rate of variation is the largest for A = 1. This is because the square cavity

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Fig. 9. Relationship between NuymL and RaymLA.

has the greatest cavity volume. The volume of the cavity increases with decreasing aspect ratio while the powers of the local cooler and heater are maintained constant. The variation for A = 2.09 is greater than that for A = 3. For A = 4, 5 and 6, the variation of the Nusselt number along the height is nearly the same. For a fixed Y, the local Nusselt number decreases as the aspect ratio increases. The rate of decrease from A = 1 to 3 is larger than that from A = 4 to 6. The mean of the local Nusselt numbers (NuymL) is obtained by taking the weighted average of the results in Fig. 8. The weighted average is calculated for the Y range of 0 to 1, and for each of the six cavities. The weight factors are 0.25, 0.50 and 0.25 for the dimensionless cavity heights (Y) of 0.125, 0.500 and 0.875, respectively. The mean of the local Rayleigh numbers (RaymL) is found in the same way. The mean of the local Rayleigh numbers was multiplied by the aspect ratio to obtain RaymLA. Fig. 9 shows the relationship between NuymL and RaymLA. A regression analysis was done to obtain the following heat transfer correlation:

NuymL ¼ 0:0017 ðRaymL AÞ0:4908

ð15Þ 5

8

The RaymL is in the range of 5.10
10 to 1.20
10 . The RaymLA is between 3.06
106 and 1.20
108. The aspect ratio is in the range of 1–6. The regression coefficient is 0.99837592. Fig. 9 also shows the plot of the correlation. The NuymL correlation is within the 6% and 6% error limits of the NuL correlation in Eq. (13). Table 1 summarizes the comparison of the present study with the literature for square cavity, and for constant and sinusoidally varying temperatures on the cooled and heated walls, respectively. For the present study, the correlation in Eq. (13) is considered. Among twelve studies [1–12], two works reported a time averaged Nusselt number. In the work of Kazmierczak and Chinoda [1], the Prandtl number, the dimensionless amplitude and the dimensionless time period are maintained at 7, 0.4 and 0.01, respectively. In the present study and in the work of Kazmierczak and Chinoda [1], two opposing vertical walls are cooled and heated from whole

surfaces. In the study of Nithyadevi et al. [12], the cavity has partially active vertical walls, the active part is half height of the cavity wall and the inactive part is adiabatic. The active parts of the hot and cold walls are fixed at the bottom and top positions, respectively. The magnetic field is not applied. The Prandtl number, the dimensionless amplitude and the dimensionless time period are maintained at 0.71, 0.4 and 3, respectively. The discrepancy between the present study and [1,12] can be explained by the differences in the boundary conditions, the Rayleigh number, the dimensionless amplitude and the dimensionless time period. This can be supported by a comparison between two works in the literature. The Nusselt number increases from 5.41 to 15.38 while the Rayleigh number is nearly the same, when focused on [1,12]. The effects of the dimensionless amplitude and the dimensionless period were investigated in the previous studies. Kazmierczak and Chinoda [1] reported that the time averaged Nusselt number increased from 5.35 to 5.58 with increasing amplitude from 0.2 to 0.8 for the period of 0.01, and changed from 5.36 to 5.43 with increasing period from 0.005 to 0.02 for the amplitude of 0.4. Kwak et al. [5] reported that the Nusselt number did not change with varying period for the amplitude of 0.1, and the values of the gain of the time averaged Nusselt number were 0.2, 4.2 and 13.1% for the amplitudes of 0.1, 0.5 and 1.0, respectively, for the period of around 0.67. They also reported that the isotherms significantly changed when the amplitude increased to 1.0. On the basis of these results, it can be concluded that the increase in the time averaged Nusselt number would be expected in the range of 2–15 %, and also a change in the temperature field in the cavity would be expected with varying amplitude and period. Table 2 presents the comparison of this study with the literature having constant temperatures on the cooled and the heated walls. The comparison is for A = 1. For the present study, the temperatures on the cooled and heated walls are constant and sinusoidally varying, respectively, and the correlation in Eq. (13) is considered. For the studies in the literature, the temperatures on the cooled and heated walls are constant. De Vahl Davis and Jones [26] numerically studied natural convection of air in a differentially heated square cavity, and presented the Nusselt number for the Rayleigh number range of 103–106. Salat et al. [27] studied natural convection of air in a differentially heated rectangular prism cavity with an aspect ratio of 1, and presented the Nusselt number. In the numerical study, the Rayleigh number was 107. In the experimental study, hot and cold walls and top and bottom walls were polished aluminium and aluminium foil, respectively, and the Rayleigh number was 1.5
109. Markatos and Pericleous [28] numerically investigated natural convection of air in a square cavity in

Table 2 Comparison of the present study with the literature having constant temperatures on the cooled and the heated walls: A = 1. Researcher

A

RaL

NuL

The present study De Vahl Davis and Jones [26] De Vahl Davis and Jones [26] Salat et al. [27] Salat et al. [27] Markatos and Pericleous [28]

1 1 1 1 1 1

1.13
108 105 106 107 1.5
109 108

16.44 4.519 8.800 16.5 54 35.14

Table 1 Comparison of the present study with the literature for square cavity, and for constant and sinusoidally varying temperatures on the cooled and heated walls, respectively. Researcher

a

p

RaL

NuL

The present study Kazmierczak and Chinoda [1] Nithyadevi et al. [12]

0.06 0.4 0.4

0.03 0.01 3

1.13
108 1.4
105 7.1
105

16.44 5.41 15.38

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the Rayleigh number range of 103–1016. They obtained heat transfer correlations for different Rayleigh number ranges. They proposed the following heat transfer correlation for the Rayleigh number range of 106–1012:

NuL ¼ 0:082 Ra0:329 L

ð16Þ

For the work of Markatos and Pericleous [28], the correlation in Eq. (16) is considered. A comparison of [26] with [27] shows that the Nusselt number increases from 8.8 to 16.5 with increasing Rayleigh number from 106 to 107. A comparison of [27] with [28] shows that the Nusselt number increases from 16.5 to 35.14 with increasing Rayleigh number from 107 to 108. The comparison between [28] and [27] presents that the Nusselt number increases from 35.14 to 54 as the Rayleigh number increases from 108 to 1.5
109. Table 2 shows that the results in the literature are in agreement with one another, and the increasing trend of the Nusselt number is observed in the different works, in the Rayleigh number range of 105 to 1.5
109. A comparison between [28] and the present study shows that the Nusselt number largely decreases when the temperature on the heated wall is changed from constant to sinusoidally varying at the Rayleigh number of 108 for A = 1. The Nusselt number for the sinusoidally varying temperature is 46.8% of the Nusselt number for the constant temperature. Table 3 presents the comparison of this study with the literature having constant temperatures on the cooled and the heated walls. The comparison is for A = 5. For the present study, the temperatures on the cooled and heated walls are constant and sinusoidally varying, respectively, and the correlation in Eq. (13) is considered. For the study in the literature, the temperatures on the cooled and the heated walls are constant. Inaba [29] conducted an experimental study on natural convection of air in rectangular cavities with aspect ratios of 5, 10, 29, 58 and 83. The Rayleigh number was varied from 1.2
103 to 2.0
106. Hot and cold walls were polished copper with a surface emissivity of 0.06. He subtracted radiation from total heat transfer and used natural convection in results. He proposed the following heat transfer correlation for the aspect ratio range of 5–83 and the Rayleigh number range of 5
103–1.2
106:

NuL ¼ 0:271 Ra0:25 A0:21 L

ð17Þ

For the work of Inaba [29], the correlation in Eq. (17) is considered. A comparison between [29] and the present study shows that the Nusselt number highly reduces when the temperature on the heated wall is changed from constant to sinusoidally varying at the Rayleigh number of 8.05
105 for A = 5. The Nusselt number for the sinusoidally varying temperature is 55.1% of the Nusselt number for the constant temperature. The comparisons show that the effect of the changing boundary condition from constant to sinusoidally varying slightly decreases when A is increased from 1 to 5. The cavity with A = 5 is less affected than the cavity with A = 1 when the temperature on the heated wall is changed from constant to time periodic. Tables 2 and 3 show that the Nusselt number decreases by around 50% when the temperature on the heated wall is changed from constant to sinusoidally varying. The side walls are cooled and heated from whole surfaces. In the case for sinusoidally varying, the dimension-

Table 3 Comparison of the present study with the literature having constant temperatures on the cooled and the heated walls: A = 5. Researcher

A

RaL

NuL

The present study Inaba [29]

5 5

8.05
105 8.05
105

3.19 5.79

less amplitude (a) is 0.06, the dimensionless time period (p) is 0.03, and the cavity is locally cooled from one side and locally heated from the opposite side. In this study, the natural convection problem has been investigated experimentally. Some researchers investigated classic problems by using new numerical methods. Podvin and Le Quere [30] applied the proper orthogonal decomposition to the natural convection of air in a differentially heated cavity with an aspect ratio of 4. Han et al. [31] used the proper orthogonal decomposition reduced-order model on natural convection in the parallelogram cavity and the eccentric semi-annulus. Ma et al. [32,33] studied the use of neural networks to fit direct numerical simulations data to obtain closure relations for the two-fluid multiphase flow equations for a simple bubble system [32], and also for the bubbly upflow in a vertical channel [33]. In the study of Tryggvason et al. [34], direct numerical simulation was used to model the transient motion of bubbly flows in vertical channels. Varde et al. [35], proposed a computational estimation technique to obtain heat transfer coefficients as a function of temperature. 5. Conclusions Natural convection of air in rectangular cavities is investigated experimentally. The cooled and heated vertical walls have constant and time periodic temperatures, respectively. The aspect ratio is changed between 1 and 6. Heat transfer is by convection only. The investigation is conducted at the mid-depth of the cavity. The temperature profile across the cavity length is obtained for three heights. The temperature gradient in the horizontal direction is nearly the same for the cold and hot walls. The isotherms are presented for each of the six cavities. The average and the local heat transfer data are presented. The local temperature difference is nearly symmetric with respect to the mid-height. The effects of the local sink and the local source on the temperature profile and the heat transfer along the cavity height are obtained. Heat transfer correlations are presented on the basis of both average and local data. The Nusselt number increases from 2.64 to 16.44 with decreasing aspect ratio from 6 to 1. The effect of the temperature boundary condition on the heated wall on the natural convection inside the cavity is presented. The Nusselt number for the sinusoidally varying temperature is half of that for the constant temperature. The experimental results of the temperature distribution and the natural convection heat transfer can be used for comparison in numerical studies. Conflict of interest There is no conflict of interest. References [1] M. Kazmierczak, Z. Chinoda, Buoyancy-driven flow in an enclosure with time periodic boundary conditions, Int. J. Heat Mass Transfer 35 (6) (1992) 1507– 1518. [2] J.L. Lage, A. Bejan, The resonance of natural convection in an enclosure heated periodically from the side, Int. J. Heat Mass Transfer 36 (8) (1993) 2027–2038. [3] B.V. Antohe, J.L. Lage, Amplitude effect on convection induced by time-periodic horizontal heating, Int. J. Heat Mass Transfer 39 (6) (1996) 1121–1133. [4] B.V. Antohe, J.L. Lage, The Prandtl number effect on the optimum heating frequency of an enclosure filled with fluid or with a saturated porous medium, Int. J. Heat Mass Transfer 40 (6) (1997) 1313–1323. [5] H.S. Kwak, K. Kuwahara, J.M. Hyun, Resonant enhancement of natural convection heat transfer in a square enclosure, Int. J. Heat Mass Transfer 41 (1998) 2837–2846. [6] Y. Shu, B.Q. Li, B.R. Ramaprian, Convection in modulated thermal gradients and gravity: experimental measurements and numerical simulations, Int. J. Heat Mass Transfer 48 (2005) 145–160. [7] Q.W. Wang, J. Yang, M. Zeng, G. Wang, Three-dimensional numerical study of natural convection in an inclined porous cavity with time sinusoidal oscillating boundary conditions, Int. J. Heat Fluid Flow 31 (2010) 70–82.

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