natural convection heat transfer from two parallel horizontal cylinders embedded in a porous medium...
TRANSCRIPT
Vol 20, No. 3;Mar 2013
Natural Convection Heat Transfer from Two Parallel Horizontal Cylinders Embedded in a Porous Medium
inside a Horizontal Cylindrical Enclosure
Ahmed Tawfeeq Ahmed Al-Sammarraie Mechanical Engineering Department, College of Engineering, University of Tikrit
Tikrit, Iraq Tel: +964 770 3769881 E-mail: [email protected]
Abstract Natural convection heat transfer from two parallel horizontal cylinders embedded in a porous medium inside a horizontal cylindrical enclosure has been experimentally studied, under constant surface temperature boundary condition. The study has investigated the effects of medium Rayleigh number, rotation angle and spacing between two cylinders on their heat loss ability. The experimental rig consists of water container, test section and two copper cylinders with 19 mm diameter. The two cylinders embedded in a porous medium (alumina granules), with a particle diameter rate (3.818 mm). The experiments were done at the range of medium Rayleigh numbers between ( 1.186.1 Ra ), cylinders rotation angles ( ooo 90,45,0 ) and spacing ratios (S/D= 2, 2.5, 3).
The study has clearly shown that the heat loss ability is a function of medium Rayleigh number, cylinders rotation angle and spacing between them. It is noticed that this ability increased with increase of the medium Rayleigh number and reaches the maximum value at the first and second cylinder ( , ) at spacing ratio (S/D=3) and minimum value at spacing ratio (S/D=2) at rotation angle ( oo 90,45 ) for the first and ( oo 90,0 ) for second cylinder, respectively. The experimental results are related by two correlating equations each one explains the dimensionless relationship of natural convection heat transfer from each cylinder that represented by Nusselt number against medium Rayleigh number, rotation angle and spacing ratio. Keywords: Natural Convection Heat Transfer, Parallel Cylinders, Cylindrical Enclosure, Porous
Medium. 1. Introduction The process of heat transfer from or to a horizontal cylinder has a considerable importance in most engineering applications, such as in the design of heating and cooling devices, steam pipelines, heat exchangers and etc. So it had received an ample effort of theoretical and experimental research to detect their ability to absorb or dissipate heat for all flow types and in different arrangements (Al-Thaher and Yacoub, 1992). Also, the field has headed for, in the last decades, to search for catalysts can be used in a practical way that works to increase the amount of heat transmitted from the heat exchange systems such as porous media in its two types: packed beds and fluidized beds. The utilization of the porous medium, as a catalyst for heat transfer in these systems, may increases the heat transfer
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and gives the opportunity to reducing the surface area that required for the heat exchange. This leads to reducing the size of the system and its cost, as well as, to minimize the differences in temperatures, thus increase the efficiency of those systems for the same size or reducing the size for the same efficiency. On the other hand, knowledge of the distribution of heat and flow fields within a closed enclosure in its different geometrical shapes has gained a significant importance because it has extensive uses in the industry and its wide applications in multiple areas such as receptor solar collector isolation, cooling systems in nuclear reactors and the pipelines buried underground isolation. It was observed that the cables buried underground, which uses to transmit high tension electric power and cryogenic by compressed gas, were dangling as a result of thermal expansion which leads to change the concentration of the cables in the center of the insulator pipe. So that the heat transfer rate changes from these cables accordingly (Kuehn and Goldstin, 1976; Faruak and Guceri, 1982). A significant effort was enough to understand the nature of the flow and natural convection heat transfer through the gap between a heated inner concentric or eccentric cylinder and a cylindrical enclosure through several adequate and comprehensive studies. The most prominent of these studies were done by the two researchers (Kuehn and Goldstin, 1976) through carry out an experimental and theoretical study of natural convection in the annulus between horizontal concentric cylinders. Besides the numerical analysis that was carried out by (Faruak and Guceri, 1982) for the laminar and turbulent natural convection in the annulus between horizontal concentric cylinders. As well, the natural convection heat transfer between two eccentric cylinders had been investigated through an experimental study by (Kuehn and Goldstin, 1978), an experimental and numerical study by (Naylor et al, 1989) and a numerical study by (Guj and Stella, 1995). Regarding a complex geometrical shape, such as two concentric horizontal cylinders surrounded by a cylindrical enclosure, (Al-Naddawi, 1994) performed a numerical study. It was found from this study the distribution of shear stresses arising from the movement of the fluid confined, as a result of the natural convection effect, between the surfaces of the two inner cylinders and the outer cylinder. The study was under constant heat flux for the two inner cylinders and constant surface temperature for the outer one, at different values of the angle limited between the horizontal plane and the line passing the center of the cylinders ( ). It was shown that the values of the local shear stresses on the two inner cylinders surfaces increased with increase of the distance from the surface, and the flow distribution within the enclosure changed with change of the rotation angle ( ). (Ali, 2008) conducted a numerical analysis for the natural convection from two parallel horizontal cylinders enclosed by circular cylinder, under constant surfaces temperatures condition. In this study, boundary fitted coordinate system was adopted to solve the governing equations. Twenty different cases were studied for the impact of the change in the two inner cylinders position, horizontally and vertically, within the outer cylinder upon the heat transfer and the buoyant force which causes the flow, for Rayleigh numbers ( 250001000 Ra ). The results showed that the position of the two inner cylinders highly affects the heat transfer and fluid movements within the enclosure. It was observed that the average Nusselt number increased with increase of the horizontal distance between the two inner cylinders at low Rayleigh numbers, vice versa at high Rayleigh numbers, while moving the two inner cylinders together toward the bottom of the outer cylinder led to increase the average Nusselt number at all Rayleigh numbers. The above study had been reinforced by the study that carried out by (Al-Sammarraie, 2010), where an experimental study on natural convection heat transfer from two parallel horizontal cylinders in horizontal cylindrical enclosure under constant surfaces temperatures condition was conducted. The study included the impact of Rayleigh number, rotation angle ( ), and the spaces between two cylinders on their heat loss ability. An experimental rig was used for this purpose, and the experiments were done at the range of Rayleigh numbers between ( 360003000 Ra ) cylinders rotation angles ( ooo 90,45,0 ) and spacing ratios (S/D= 2, 2.5, 3). This study showed that the ability of heat loss from two
Vol 20, No. 3;Mar 2013
cylinders is a function of Rayleigh number, cylinders rotation angle and the space between them. This ability increased with increase of Rayleigh number, and it was noticed that this ability reached the maximum value at the first cylinder ( ) and the minimum value at the second cylinder ( ) at spacing ratio (S/D=3) and rotation angle ( oo 90,45 ) for the first and ( oo 90,45 ) for second cylinder, respectively.
Based on the above mentioned and through a comprehensive review for the available published researches in the field of heat transfer, it has been found that, there are no experimental or theoretical studies had dealt with the impact of rotation angle for two cylinders embedded in a porous medium inside a horizontal cylindrical enclosure and the spaces between them on the natural convection heat transfer, despite the importance of this engineering arrangement and its different practical applications. So it is planned in the present study for lighting even a small aspect of this vital subject experimentally. This has been done by conducting an experimental study on natural convection heat transfer from two parallel horizontal cylinders embedded in a porous medium inside a horizontal cylindrical enclosure and detecting the effect of rotation angle of these two cylinders and the spaces between them on the heat transfer, as shown in Figure (1) which shows the physical problem of the current research. The test section is a closed cylindrical enclosure immersed in an iced water bath, to ensure getting constant surface temperature for the inner surface of the enclosure, so that the inside fluid (air) temperature equals ( aT ), and the horizontal axis of this enclosure is parallel to the two cylinders axes. For this purpose, two cylinders made of copper were used as the first and second cylinders, which are rotated about the horizontal axis of the cylindrical enclosure with a rotation angle ( ) and ( ), respectively. The two cylinders are fixed inside the enclosure, which filled with a porous bed composite of alumina granules (aluminum oxide), with certain spacing ratios between them (S/D= 2, 2.5, 3) and a rotation angle ( o0 ). After that, they are heated and extracted their ability to dissipate heat, then changed the rotation angle to become ( o45 ) and then ( o90 ) and tested their ability to dissipate heat, and so on. 2. Experimental Rig and Calculations The experimental rig is shown schematically in Figure (2). It consists mainly of a water container, test section and two heating elements, as well as, the measurement system that consisting of the power -provided to the heaters- measuring devices and thermocouples to measure the temperatures of the two test cylinders surfaces and the air inside the test section. The used water container had been made of galvanized iron sheets with thickness of (0.8 mm) and dimensions (400x300x400 mm3) in order to put the iced water inside it even the fixed test section within the container is immersed. This leads to providing the water bath needed to cooling the test section and ensuring access to constant inner surface temperature for this section and consequently fixing the air temperature inside it. The container was isolated from the outside with layers of cork with the same dimensions of the container and thickness of (15 mm) to ensure thermal insulation between the container and the external environment. The test section was fixed inside the container centrally and horizontally. It was a cylinder made of plastic with an inner diameter of (99 mm), wall thickness (3.5 mm) and length (400 mm). The two heating elements were hollow cylinders made of copper with an outside diameter of (19 mm), wall thickness (0.7 mm) and length (400 mm) as shown in Figure (3). Each cylinder was heated by a heater consists mainly of electrical resistance (nickel-chrome) with a resistance of (100 m/ ). These two cylinders were fixed horizontally inside the test section using two Teflon plugs their diameters are the same as the inner diameter of this section, where three pairs of these plugs were made to use them in control of the spacing ratios between the two cylinders and rotation angles within the test section. The thickness of the copper cylinder gave a thermally homogeneous distribution on its surface and consequently constant surface temperature around the cylinder perimeter, where the
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temperature around its circumference is almost constant and with a difference no more than ( Co5.0 ). The heater in each cylinder was provided with the required power from an AC source through a power stabilizer -with accuracy of 2% from the provided power- by using a voltage regulator type (SAT-2020). Also, a voltmeter type (K-1400) was used to measure the voltage difference generated on the heaters ends with reading range of (0 - 500 V) and accuracy (0.2 V). To measure the current passing through each heater, an ammeter type (ENERTEC) was used with reading range of (0 - 5 A) and accuracy (0.002 A). The temperature was measured by a digital thermometer type (Barnant-689) reads temperature with accuracy of ( Co5.0 ), where five calibrated thermocouples (T type) with diameter (0.27 mm) were used to measure the surface temperatures of each cylinder ( sT ). Three of these thermocouples were fixed after (200 mm) from one end of the cylinder and distributed evenly on the perimeter in the rate of a thermocouple each ( o120 ) while the two other thermocouples at a distance of (40 mm) from both sides of the cylinder to measure the temperature distribution on the cylinder circumference and its length, respectively. Also, five other thermocouples with the same specifications, dimensions and previous distribution method were used to measure the temperatures of the test section inner surface ( aT ).
The main purpose of the test is to demonstrate the influence of the rotation angle and spacing between the two cylinders on the natural convection heat transfer from these embedded cylinders in a porous medium. So in the beginning of any test, the rotation angle is determined at a certain spacing ratio where the two cylinders are fixed in their positions within the test section, then the test section is filled with the porous bed which is alumina granules (aluminum oxide), then the test section is closed. After that, the container is filled with water and ice and waiting, for few minutes, to ensure thermal homogeneity state within the test section is achieved. At the above case, the electric heater is supplied in each cylinder with equally elected electrical power by the voltage regulator and consequently obtained the amount of heat to be supplied to the two cylinders, and then the test rig is left for enough time until accessing the steady state. Then, the readings are taken for: the temperatures of the two cylinders surfaces, the internal surface temperatures of the test section (air temperature), and the amount of voltage and the current for each cylinder. After that, the amount of power supplied to the heaters is altered by another amount through the voltage regulator and the process is repeated again. These steps were repeated for several attempts (six attempts), which were sufficient to drawing the relationship between porous medium Rayleigh number and average Nusselt number. The results, obtained from the experimental work, were reduced in terms of average Nusselt number and medium Rayleigh number as a function of the rotation angle. The average heat transfer coefficient (h) had been calculated for each cylinder from the following equation:
)( ass
net
TTAQh
(1)
where netQ is the net natural convection heat transferred from the surface of each cylinder. It was found that the heat lost by conduction through the plugs did not exceed (4%) of the heat generated in the heater. The average Nusselt number (Nu) for each cylinder was calculated by the equation:
ek
hDNu (2)
where ek is the effective thermal conductivity for the porous medium at each cylinder, which can be calculated by Zehnder and Schlunder model (Kaviany, 1999) as follows:
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B
BBBB
BBk
kf
e
1
12
11ln)1(
)1(1
)1(2)1(1 2
(3)
where 910)1(25.1 B , pf kk , and pk is the thermal conductivity for alumina (35 W/m K). The bulk porosity of the porous medium ( ) was calculated by the volumetric method that used by (Nasr et al, 1994). It was shown that the bulk porosity of the bed used in the test equals (0.746). While calculating medium Rayleigh number (Ra) for each cylinder was from the equation:
ef
as KDTTgRa
)( ;
fT1
(4)
where K is the permeability through the porous medium, which was calculated from:
2
32
)1(150
pd
K (5)
and e is the effective thermal diffusivity for the porous medium at each cylinder. It was calculated by the equation:
fPf
ee C
k
(6)
All the physical properties of the air were taken as reported by (Incropera and DeWitt, 2003), at the film temperature ( fT ) which is:
2
asf
TTT (7)
Kline and McClintock method (Holman, 2012) was used for the experimental error analysis of the average heat transfer coefficient, medium Rayleigh number and average Nusselt number for each cylinder. The error was limited between (3% - 14%) for all experimental data. 3. Results and Discussion It has been illustrated the influence of the flow, resulting from the temperatures differences, on natural convection heat transfer from the two horizontal test cylinders in Figures (4, 5 and 6), which describes the relationship between the medium Rayleigh number (Ra) and the average heat transfer represented by Nusselt number (Nu). Every figure, within these figures, shows the behavior of Nusselt number with Rayleigh number for each of the two cylinders embedded in the porous medium at fixed spacing ratio between them and a certain rotation angle. From these figures, it is noticed that the rate of heat transfer from each cylinder increased with increase of Rayleigh number, i.e., increase the fluid flow around the hot surface. This is an expected result because increase the difference in temperatures between the surface and the fluid, resulting from increase the thermal energy supplied to the cylinder, leads to increase the change in density of the fluid adjacent to the surface, which increases the contact between the cold fluid stream and the hot surface per unit time. For all spacing ratios, it can be noticed that the heat transfer from the two cylinders is largely convergent and symmetric at rotation angle ( o0 ) compared with the other rotation angles because the shape of flow in each side of the vertical plane, passing through the center of the cylindrical enclosure, is symmetric at this rotation angle. Moreover, the heat transfer from the first cylinder ( ) in all the figures above is higher than the second cylinder ( ) because the hot fluid stream rising from the first cylinder leads to a decrease in heat transfer and an increase in the
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surface temperature of the second cylinder at the same heat flux. It is clear that the highest difference in heat transfer between the two cylinders is at rotation angle ( o90 ) as a result of the building completion and the formation of the rising thermal layer at this situation so that the upper cylinder (second cylinder) is completely buried in the thermal boundary layer of the lower cylinder (first cylinder). Also, it is obvious from these figures that they are not enough to study the effect of the other variables (rotation angle and the spacing between the two cylinders) on the heat transfer from the studied arrangement in this work. Therefore, it has been fitted the values of medium Rayleigh number and average Nusselt number to extract linking equations describes the relationship between them for the two embedded cylinders at fixed spacing ratios. From these equations, relations were extracted shows the effect of the rotation angle and spacing on the heat transfer at different Rayleigh numbers as shown in Figures (7, 8, 9 and 10). For these figures, it is evident that the heat transfer increased with increase of Rayleigh number. Also, the relationship between average Nusselt number with the rotation angle on one hand, and average Nusselt number with the spacing ratio, on the other hand, for both two cylinders is a relationship changing up and down depending on the change in the rate of heat transfer.
Figure (7), shows that the heat transfer from the first cylinder ( ) at spacing ratio (S/D=2) decreased with increase of the rotation angle, for all taken medium Rayleigh number values, because of being located in the nearest location to the second cylinder ( ) which its thermal boundary layer negatively affect, whenever the rotation angle has increased, by its intersection with the thermal boundary layer of the first cylinder. However, the heat transfer from the first cylinder ( ) at the spacing ratio (S/D=2.5) increased with increase of rotation angle, for all taken Rayleigh number values. As well as at the spacing ratio (S/D=3), the highest value of heat transfer in the present study occurs at rotation angle ( o45 ) for low medium Rayleigh numbers ( 3Ra ) and then returns to its same previous behavior at high medium Rayleigh numbers ( 9Ra ) where the highest level of heat transfer in the present study occurs at rotation angle ( o90 ), i.e., the cylinder is in the bottom of the cylindrical enclosure. The reason behind that whenever the cylinder moves away from the horizontal plane toward the bottom of the enclosure, its exposure to the cold convection stream has increased, which leads to increase the heat exchange between the cylinder and the surrounding fluid streams.
It is observed from Figure (8) that the heat transfer from the first cylinder ( ) at the two rotation angles ( o0 ) and ( o45 ) increased with increase of the spacing ratio between the two cylinders. While at the rotation angle ( o90 ), the behavior of the heat transfer will be almost different, where the highest value of heat transfer occurs at spacing ratio (S/D=2.5) for low Rayleigh numbers ( 3Ra ), besides the heat transfer increased with increase of the spacing ratio to reach its highest value in the present work at spacing ratio (S/D=3) for high Rayleigh numbers ( 9Ra ) because the vertical distance between the edge of the cylinder and the bottom of the cylindrical enclosure becomes in its lowest values at this spacing ratio in conjunction with the increase in Rayleigh number, which leads to increase the speed of relatively cold convection cells leading to increase the heat transfer at the above case.
Regarding the second cylinder ( ), it can be generally noticed from Figures (9 and 10) that the heat transfer decreased with increase of the rotation angle and spacing ratio. The increase in the rotation angle and spacing will raise the position of this cylinder within the enclosure to be more and more embedded in the warm convection streams, that rising upward, and the thermal boundary layer for the first cylinder, which necessarily leads to increase the difference in temperatures between the cylinder surface and the fluid streams within the enclosed enclosure (at fixed heat flux supplied to the cylinder), leading to decrease the heat transfer from this cylinder to reach its minimum value through the current study at the rotation angle ( o90 ) and spacing ratio (S/D=3) at low Rayleigh numbers ( 3Ra ) and spacing ratio (S/D=2) at high Rayleigh numbers ( 9Ra ), while it is noticed that the maximum value of heat transfer from it at the
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rotation angle ( o90 ) and spacing ratio (S/D=3) at Rayleigh number ( 18Ra ), and this is contrary to the above interpretation, which mostly may be due to the experimental error. To clarify the effectiveness of the porous medium in an improving natural convection heat transfer from the two cylinders of the test (the first and second cylinder), Figure (11) has been adopted, which represents the relationship between the improvement ratio of the heat transfer and the rotation angle of each cylinder, at a certain temperature difference and different spacing ratios, using results of the present study and results of the previous study that was conducted by (Al-Sammarraie, 2010). It is obvious that the improvement ratio in the heat transfer from each two cylinders is not less than (1.3). Also, the highest improvement ratio is (2.4), which means that the amount of the heat transfer from any embedded cylinder in the porous medium at any rotation angle and spacing ratio is almost higher twice than the amount of the heat transfer from any free cylinder at the same arrangement. Accordingly, it is clear that the granules of the porous medium operates as a catalyst or effective improver for the natural convection heat transfer as a result of the compound work for both the fluid and solid medium in transferring heat by conduction and convection, besides the continuous smashing, by the medium granules, for the viscous boundary layer which is trying to growth on the surfaces of the two cylinders, which leads to make the thermal boundary layer surrounding these cylinders thinner than the thickness of the thermal boundary layer surrounding the free cylinders, thereby increasing the dissipation of heat from the embedded cylinders in comparison with free cylinders at the same arrangement and the same temperature difference. An empirical formula was extracted for the experimental data of natural convection heat transfer with the effective parameters on this process for both two cylinders (the first and second cylinder). The least squares fitting method was used for this purpose. The average heat transfer from any cylinder was represented by average Nusselt number as a function of medium Rayleigh number and rotation angle, where the correlating equation can be expressed as:
cbRaaNu )cos1.(. (8)
where a, b, and c are the experimental constants. Using the form of the Equation (8), the experimental results for the heat transfer from the two cylinders and using the least squares fitting method, the experimental constants values were extracted for this equation for both two cylinders as shown in Table (1) for the first cylinder ( ) and Table (2) for the second cylinder ( ). A correlating equation was devised for natural convection heat transfer from each cylinder more general than the previous one. Where another variable was added to the two previous variables which is the spacing ratio between the two cylinders (S/D), so the average Nusselt number has become in terms of a set of non-dimensional variables as follows:
DSRafNu /,, (9)
As proposed in this formula, average Nusselt number has been linked versus the spacing ratio and rotation angle, as well as Rayleigh number for each cylinder as shown in the following equation:
dcb DSRaaNu )/.()cos1.(. (10)
Using the fitting method aforementioned and applying it on the experimental results, the experimental constants values of Equation (10), for the first and second cylinder, were extracted as follows:
1st Cylinder ( ): a = 0.877, b = 0.225, c = -0.256, d = 0.814 and:
2nd Cylinder ( ): a = 1.138, b = 0.233, c = 0.073, d = 0.219
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Therefore, the general form of the correlating equation of the natural convection heat transfer from the first cylinder ( ) inside a closed cylindrical enclosure filled with a porous medium at ( 1330/,976.4/ fpp kkdD ) is expressed as:
814.0256.0225.0 )/()cos1(877.0 DSRaNu , at
3/2900
DS
oo (11)
Also, the general form of the correlating equation of the natural convection heat transfer from the second cylinder ( ) inside a closed cylindrical enclosure filled with a porous medium at ( 1330/,976.4/ fpp kkdD ) is expressed as:
219.0073.0233.0 )/()cos1(138.1 DSRaNu , at
3/2900
DS
oo (12)
The results of heat transfer from the two cylinders (the first and second cylinder) were compared with correlating equations (11) and (12), respectively. As shown in Figure (12), a good agreement is noticed between the results and the equation curve for each cylinder. Also, the measured Nusselt number was compared with correlated Nusselt number for both two cylinders as shown in Figure (13). It is clear that, for fifty four reading points for each cylinder that (87%) of these points are located within the deviation range of ( %22 , %17 ) from their own two correlating equations, respectively. This shows how the compatibility was good between these equations and the experimental results. 4. Conclusions The present study has given a clear and fully vision about the impacts of rotation angle and spacing on the natural convection heat transfer from two embedded cylinders in a porous medium inside a closed cylindrical enclosure. These impacts can be summarized as follows: 1. The ability of dissipation heat from the two embedded cylinders is a function of the medium
Rayleigh number, rotation angle and spacing between them. This ability increases with increasing Rayleigh number, changing the rotation angle and widening the spacing between these two cylinders.
2. The heat transfer from the two embedded cylinders is almost identical at rotation angle ( o0 ) for any spacing ratio.
3. The lowest value for the heat transfer from the first cylinder ( ) is at rotation angle ( o90 ) and spacing ratio (S/D=2) for Rayleigh numbers that taken in the current study, and the highest value for the heat transfer from it is at spacing ratio (S/D=3) and rotation angle ( o45 ) at low Rayleigh numbers and ( o90 ) at high Rayleigh numbers.
4. The lowest value for the heat transfer from the second cylinder ( ) is at rotation angle ( o90 ) and spacing ratio (S/D=3) at low Rayleigh numbers and spacing ratio (S/D=2) at high Rayleigh numbers, and the highest value for the heat transfer from it is at spacing ratio (S/D=3) and rotation angle ( o0 ) for most Rayleigh numbers that taken in this study and rotation angle ( o90 ) at Rayleigh number ( 18Ra ).
5. The highest value of the improvement in heat transfer from the two cylinders as a result of burring or embedding them in the porous medium are about twice as much as the heat transfer from the same arrangement in the free case at the same temperature difference.
6. The two correlating equations, which were derived from the experimental results for the natural convection heat transfer from the two cylinders, are largely accepted to estimate the heat transfer from the studied arrangement within the range of their respective determinants.
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7. Natural convection heat transfer results have been compared with available literature and showed satisfactory agreement.
Acknowledgements The author would like to thank University of Tikrit for providing laboratory facilities and financial support.
References
Al-Thaher, M. A., & Yacoub, K. Y. (1992). Experimental investigation on the effect of side wall floor and side wall combination on convective radiative heat transfer from horizontal cylinder. Journal of Engineering and Technology, Iraq, 11(10), 9-24
Kuehn, T. H., & Goldstin, R. J. (1976). An experimental and theoretical study of natural convection in the annulus between horizontal concentric cylinders. Journal of Fluid Mechanics, 74(4), 695-719
Faruak, B., & Guceri, S. I. (1982). Laminar and turbulent natural convection in the annulus between horizontal concentric cylinders. Journal of Heat Transfer, 104(11), 631-636
Kuehn, T. H., & Goldstin, R. J. (1978). An experimental study of natural convection heat transfer in concentric and eccentric horizontal cylindrical annuli. Journal of Heat Transfer, 100(11), 635-640
Naylor, D., Badr, H. M., & Tarasuk, J. D. (1989). Experimental and numerical study of natural convection between two eccentric tubes. International Journal of Heat and Mass Transfer, 32(1), 171-181
Guj, G., & Stella, F. (1995). Natural convection in horizontal eccentric annuli: numerical study. Journal of Numerical Heat Transfer, 89-105
Al-Naddawi, A. S. D. (1994). Finding the distribution of shear stresses on the internal surfaces of two cylinders inside an external third cylinder as a result of the movement of the fluid confined between the surfaces of the three cylinders in an influence of natural convection. Scientific Journal of Tikrit University, Engineering Sciences Sector, Iraq, 1(1), 25-40
Ali, M. H. (2008). Numerical study of natural convection from two parallel horizontal cylinders enclosed by circular cylinder. Tikrit Journal of Engineering Sciences, Iraq, 15(1), 51-69
Al-Sammarraie, A. T. A. (2010). An experimental study on natural convection heat transfer from two parallel horizontal cylinders in horizontal cylindrical enclosure. Tikrit Journal of Engineering Sciences, Iraq, 17(1), 38-53
Kaviany, M. (1999). Principles of heat transfer in porous media. Corrected 2nd Edition, Springier-Verlag, Inc., New York, USA
Nasr, K., Ramadhyani, S., & Viskanta, R. (1994). An experimental investigation on forced convection heat transfer from a cylinder embedded in a packed bed. ASME Journal of Heat Transfer, 116(2), 73-80
Incropera, F. P., & DeWitt, D. P. (2003). Fundamentals of heat and mass transfer. 5th Edition, John Wiley & Sons Inc.
Holman, J. P. (2012). Experimental methods for engineers. 8th Edition, McGraw-Hill Book Company, New York, USA
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Nomenclature
Symbol Meaning A Area (m2). a Experimental constant in Equation (8).
B Variable in Equation (3). b Experimental constant in Equation (8).
PC Specific heat at constant pressure (J/kg K).
c Experimental constant in Equation (8). D Cylinder diameter (m). d Experimental constant in Equation (10).
pd Particle diameter (m).
g Gravity acceleration (m/s2). h Average heat transfer coefficient (W/m2 K).
K Permeability (m2). k Thermal conductivity (W/m K).
Nu Average Nusselt number.
netQ Net heat (W).
Ra Medium Rayleigh number. S Spacing (m). T Temperature (K).
Greek Symbols Thermal diffusivity (m2/s).
Volume coefficient of expansion (K-1).
Rotation angle (Degree).
Variable in Equation (3). Kinematic viscosity (m2/s). Density (kg/m3).
Bulk porosity (-).
Subscripts a Air. e Effective.
f Film.
p Porous medium granules. s Cylinder surface.
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Tables
Table 1. The experimental constants of Equation (8) for the first cylinder ( )
S/D a b c
2 1.299 0.215 0.208
2.5 1.999 0.239 -0.468
3 2.114 0.241 -0.349 Table 2. The experimental constants of Equation (8) for the second cylinder ( )
S/D a b c
2 1.389 0.192 0.170
2.5 1.406 0.257 -0.048
3 1.366 0.249 0.103 Figures
Figure 1. The physical problem and coordinate system
S D
Ts
Ts
Ta
Cyl. 1
Cyl. 2
Porous Medium
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Figure 2. The experimental rig
Figure 3. The test cylinder
/ / / / / / / / / / / / / / / / / / / / // / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / 1. Water container, 2. Test section, 3. Cylinder, 4. Teflon plug,
5. Iced water, 6. Digital thermometer, 7. Voltmeter, 8. Ammeter, 9. Voltage regulator, 10. Power stabilizer
10 9 7 6 8
1
3
2
5
4
1. Copper cylinder, 2. Sand, 3. Tube of Pyrex glass, 4. Electrical resistance, 5. Teflon plug, 6. Electric conductor
1 2 3 4 5 6
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2 3 4 5 6 7 8 9 21 10Ra
5
6
789
2
3
4
5
6
1
Nu
S/D=21st Cyl., th=0
2nd Cyl., th=0
1st Cyl., o0
2nd Cyl., o0
(4a)
2 3 4 5 6 7 8 9 21 10Ra
5
6
789
2
3
4
5
6
1
Nu
S/D=2
1st Cyl., th=-45
2nd Cyl., th=+45
1st Cyl., o45
2nd Cyl., o45
(4b)
2 3 4 5 6 7 8 9 21 10Ra
5
6
789
2
3
4
5
6
1
Nu
S/D=21st Cyl., th=-90
2nd Cyl., th=+90
1st Cyl., o90
2nd Cyl., o90
(4c)
2 3 4 5 6 7 8 9 21 10Ra
5
6
789
2
3
4
5
6
1
Nu
S/D=2.5
1st Cyl., th=0
2nd Cyl., th=0
1st Cyl., o0
2nd Cyl., o0
(5a)
2 3 4 5 6 7 8 9 21 10Ra
5
6
789
2
3
4
5
6
1
Nu
S/D=2.5
1st Cyl., th=-45
2nd Cyl., th=+45
1st Cyl., o45
2nd Cyl., o45
(5b)
2 3 4 5 6 7 8 9 21 10Ra
5
6
789
2
3
4
5
6
1
Nu
S/D=2.5
1st Cyl., th=-90
2nd Cyl., th=+90
1st Cyl., o90
2nd Cyl., o90
(5c)
Figure 4 (a, b, c). Variation of Nusselt number with Rayleigh number at spacing ratio (S/D=2) and a certain rotation angle for the two cylinders
Figure 5 (a, b, c). Variation of Nusselt number with Rayleigh number at spacing ratio (S/D=2.5) and a certain rotation angle for the two cylinders
Vol 20, No. 3;Mar 2013
Figure 6 (a, b, c). Variation of Nusselt number with Rayleigh number at spacing ratio (S/D=3) and a certain rotation angle for the two cylinders
Figure 7 (a, b, c). Variation of Nusselt number with the rotation angle of the first cylinder at a certain spacing ratio for different Rayleigh numbers
2 3 4 5 6 7 8 9 21 10Ra
5
6
789
2
3
4
5
6
1
Nu
S/D=3
1st Cyl., th=0
2nd Cyl., th=0
1st Cyl., o0
2nd Cyl., o0
(6a)
2 3 4 5 6 7 8 9 21 10Ra
5
6
789
2
3
4
5
6
1
Nu
S/D=31st Cyl., th=-45
2nd Cyl., th=+45
1st Cyl., o45
2nd Cyl., o45
(6b)
2 3 4 5 6 7 8 9 21 10Ra
5
6
789
2
3
4
5
6
1
Nu
S/D=3
1st Cyl., th=-90
2nd Cyl., th=+90
1st Cyl., o90
2nd Cyl., o90
(6c)
0 45 90
0
1
2
3
4
5
Nu
1st Cyl., S/D=2Ra=18
Ra=9
Ra=3
1st
)(Degree
(7a)
0 45 90
0
1
2
3
4
5
Nu
1st Cyl., S/D=2.5
Ra=18
Ra=9
Ra=3
1st
)(Degree
(7b)
0 45 90
0
1
2
3
4
5
6
Nu
1st Cyl., S/D=3Ra=18
Ra=9
Ra=3
1st
)(Degree
(7c)
Vol 20, No. 3;Mar 2013
Figure 8 (a, b, c). Variation of Nusselt number with the spacing ratio of the first cylinder at a certain rotation angle for different Rayleigh numbers
Figure 9 (a, b, c). Variation of Nusselt number with the rotation angle of the second cylinder at a certain spacing ratio for different Rayleigh numbers
(8a)
2.0 2.5 3.0S/D
0
1
2
3
4
5
Nu
1st Cyl., th=0Ra=18
Ra=9
Ra=3
1st Cyl., o0
2.0 2.5 3.0S/D
0
1
2
3
4
5
6
Nu
1st Cyl., th=-90Ra=18
Ra=9
Ra=3
1st Cyl., o90 (8c)
2.0 2.5 3.0S/D
0
1
2
3
4
5
Nu
1st Cyl., th=-45Ra=18
Ra=9
Ra=3
1st Cyl., o45 (8b)
0 45 90
0
1
2
3
4
5
Nu
2nd Cyl., S/D=2
Ra=18
Ra=9
Ra=3
2nd
)(Degree
(9a)
0 45 90
0
1
2
3
4
5
Nu
2nd Cyl., S/D=2.5Ra=18
Ra=9
Ra=3
2nd
)(Degree
(9b)
0 45 90
0
1
2
3
4
5
Nu
2nd Cyl., S/D=3
Ra=18
Ra=9
Ra=3
2nd
)(Degree
(9c)
Vol 20, No. 3;Mar 2013
Figure 10 (a, b, c). Variation of Nusselt number with the spacing ratio of the second cylinder at a certain rotation angle for different Rayleigh numbers
Figure 11 (a , b). The relationship between the improvement ratio of the heat transfer and the rotation angle of each cylinder at different spacing ratios
(10a)
2.0 2.5 3.0S/D
0
1
2
3
4
5
Nu
2nd Cyl., th=0
Ra=18
Ra=9
Ra=3
2nd Cyl., o0
2.0 2.5 3.0S/D
0
1
2
3
4
5
Nu
2nd Cyl., th=+45Ra=18
Ra=9
Ra=3
2nd Cyl., o45 (10b)
2.0 2.5 3.0S/D
0
1
2
3
4
5
Nu
2nd Cyl., th=+90Ra=18
Ra=9
Ra=3
2nd Cyl., o90 (10c)
(11a)
-135 -90 -45 0 45th (Degree)
0.5
1.5
2.5
0
1
2
3
hEm
b./h
Free
1st Cylinder
S/D=2
S/D=2.5
S/D=3
1st
)(Degree
h Em
b./h
Free
(11b)
-45 0 45 90 135th (Degree)
0.5
1.5
2.5
0
1
2
3
hEm
b./h
Free
2nd Cylinder
S/D=2
S/D=2.5
S/D=3
2nd
)(Degree
h Em
b./h
Free
Vol 20, No. 3;Mar 2013
Figure 12 (a , b). The relationship between heat transfer results from the two cylinders and their own correlating equations (11 and 12), respectively
Figure 13 (a , b). The comparison of measured Nusselt numbers with those correlated by Equations (11) and (12), respectively
(13a)
1.5 2.5 3.5 4.51 2 3 4Nu (Measured)
1.5
2.5
3.5
4.5
1
2
3
4
Nu
(Cor
rela
ted)
1st Cylinder1st
-22%
+22%
(13b)
1.5 2.5 3.5 4.51 2 3 4Nu (Measured)
1.5
2.5
3.5
4.5
1
2
3
4
Nu
(Cor
rela
ted)
2nd Cylinder2nd
-17%
+17%
(12a)
0.5 1.5 2.5 3.50 1 2 3ln (Ra)
0.5
1.5
2.5
3.5
0
1
2
3
ln (N
u)
1st Cylinder1st
(12b)
0.5 1.5 2.5 3.50 1 2 3ln (Ra)
0.5
1.5
2.5
3.5
0
1
2
3
ln (N
u)
2nd Cylinder2nd