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Analysis of natural convection heat transfer from a cylinder enclosed in a corner of two adiabatic walls
Experimental Thermal and Fluid Science 62 (2015) 9–20
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Analysis of natural convection heat transfer from a cylinder enclosed in a corner of two adiabatic walls M.H. Sedaghat a, M. Yaghoubi b,c, M.J. Maghrebi d,⇑ a
Department of Mechanical Engineering, Shahrood University, Shahrood, Iran Department of Mechanical Engineering, Shiraz University, Shiraz, Iran c Academy of Sciences of Islamic Republic of Iran, Tehran, Iran d Department of Mechanical Engineering, Ferdowsi University of Mashhad, Mashhad, Iran b
a r t i c l e
i n f o
Article history: Received 13 July 2014 Received in revised form 12 October 2014 Accepted 5 November 2014 Available online 25 November 2014 Keywords: Free convection Laminar flow Cold horizontal cylinder Adiabatic walls
a b s t r a c t A steady two-dimensional laminar natural convection heat transfer from a cold single horizontal isothermal cylinder enclosed between two adiabatic walls is studied both experimentally and numerically. Experiments are performed to study the effects of cylinder distance from horizontal and vertical walls for Rayleigh numbers of 3 105 and 6 105. Numerical computations are carried out by solving the governing equations of laminar natural convection using the finite volume technique. OpenFOAM which is an open source code software is used. Computations are made for a wide range of Rayleigh numbers from 104 to 106 and different ratios of vertical cylinder spacing (L/D) and that of horizontal cylinder spacing (S/D). Comparisons are made with the corresponding experimental measurements. Flow streamlines and isothermal lines around the cylinder so that by changing the value of L/D and S/D, due to damping effects of the adiabatic walls, the cold plume moves away from the cylinder at different angles. Results also indicate that local and average heat transfer coefficient is highly affected by L/D and Rayleigh number while S/D has a limited influence on heat transfer from the tube. Ó 2014 Elsevier Inc. All rights reserved.
1. Introduction The significance of natural convection heat transfer from cold pipes or tubes for designing refrigeration, air conditioning and freezing systems have continued interest among designers and manufacturers. Even though the study of natural convection heat transfer from pipes or an array of pipes and tubes in a medium with an infinite domain is regarded as a topic of wide variety of investigations [1–8], but natural convection heat transfer from a pipe with confining walls or pipe in a semi enclosure is not studied considerably. Some researchers analyzed the influences of adiabatic bounding walls on the natural convection heat transfer coefficient from a hot circular cylinder. Saito et al. [9] in an experimental investigation studied the effect of adiabatic plate over a horizontal heated cylinder for Grashof numbers between 2.1 106 and 3.2 106. They studied effects of the ratio of cylinder spacing from the floor to its diameter (L/D) on Nusselt number and showed that the rate of heat transfer would be minimized if this ratio is about 0.12. ⇑ Corresponding author. Tel.: +98 51 38805051; fax: +98 51 38763304. E-mail addresses: [email protected] (M.H. Sedaghat), [email protected] (M. Yaghoubi), [email protected] (M.J. Maghrebi). http://dx.doi.org/10.1016/j.expthermflusci.2014.11.003 0894-1777/Ó 2014 Elsevier Inc. All rights reserved.
Koizumi and Hosokawa [10] used flow visualization to study free convection heat transfer from a pipe under a ceiling. Their experiments also include the effects of changing pipe diameter, ceiling distance from the pipe and the type of ceiling (adiabatic or isothermal) on natural convection heat transfer for Rayleigh numbers from 4.8 104 to 107. Their results for an isothermal ceiling show that the fluid flow remains almost steady and two dimensional for all ranges of L/D for Rayleigh numbers over 105. When the ceiling is close to the pipe (L/D = 0.4) the fluid flow becomes unstable and three dimensional for Rayleigh number less than 105. For high values of L/D the plume oscillate laterally. They also reported local Nusselt number for both types of ceiling at Ra = 1.3 106 and L/D = 0.05, 0.2. Lawrence et al. [11] studied natural convection heat transfer from a hot pipe below an isothermal surface. They applied finite element method for their numerical solution and MachZehnder interferometer in their experiments for air flow; with Rayleigh numbers of 103–105 and 0.1 6 L/D 6 0.5. They showed that when the distance between ceiling and pipe is larger than the pipe diameter, ceiling does not influence the rate of heat transfer from the pipe. They also indicated that average Nusselt number increases as a result of conduction heat transfer between the pipe and the ceiling for spacing of about one fourth of cylinder diameter. However, when L/D > 0.25 the ceiling decreases heat transfer
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Nomenclature D g Gr h L Nu K P Pr q00 R Ra S T U U V W
cylinder diameter (m) gravitational acceleration (m/s2) Grashof number heat transfer coefficient (W/m2 K) vertical cylinder to floor distance (m) Nusselt number thermal conductivity (W/m K) pressure (Pa) Prandtl number heat flux (W/m2) radius (m) Rayleigh number horizontal cylinder to floor distance (m) temperature (°C, K) horizontal component of velocity (m/s) velocity vector (m/s) vertical component of velocity (m/s) floor length (m)
up to 10%. Ashjaee et al. [12] used a Mach-Zehnder interferometer to study natural convection heat transfer from a pipe under an adiabatic ceiling for Rayleigh numbers between 103 and 4 104. They showed that ceiling did not have a great effect on heat transfer from the cylinder when L/D > 1.5. They also showed that by decreasing the distance to L/D = 0.5 the Nusselt number reduced significantly. They reported that, by reducing this ratio and as a consequence the formation of vortex above the cylinder, both local Nusselt numbers at the top of the cylinder and the average Nusselt number decreased sharply. They also suggested the average Nusselt number as function of L/D and Rayleigh number. Similar investigations are devoted for natural convection heat transfer from circular pipe with horizontal [13–17] and vertical [18,19] confining walls. A few investigations are focused on natural convection heat transfer from a horizontal cylinder enclosed in a rectangular cavity. Cesini et al. [20] studied the influence of Rayleigh number and the geometry of the cavity on the natural convection heat transfer from a horizontal circular cylinder both numerically and experimentally. Hussain and Hussain [21] numerically investigated the two dimensional steady natural convection for a uniform heat source applied on the inner circular cylinder in a square in which all the boundaries are assumed to be isothermal (at a constant low temperature). They showed the effects of the vertical location of cylinder and Rayleigh numbers (from 103 to 106) on fluid flow and heat transfer from the pipe. They found that the location of cylinder at small Rayleigh numbers does not have much influence on the flow field while it has a considerable effect on the flow pattern at high Rayleigh numbers. According to the literature review, most of the studies are focused on natural convection from hot cylinders with horizontal and vertical confinements and a few analyses are reported from a cylinder or tube in a corner of two adiabatic walls. In this paper, a numerical and experimental analysis is performed for natural convection heat transfer from a single horizontal cooled cylinder located at the corner of two flat adiabatic walls. This configuration is specially used in power plants, chemical and refrigeration industries for transferring a fluid from one location to another place. Based on such simulation experiments and numerical computations are made for various cylinder space ratios and flow Rayleigh numbers.
Greek symbols b volumetric expansion coefficient (K1) a thermal diffusivity e emissivity h angle from top of cylinder (°) l dynamic viscosity (Pa s) m kinematic viscosity (m2/s) q density (kg/m3) r Stefan-Boltzmann constant (W/m2 K4) Subscripts ave average condition f fluid in inner layer of test section ins insulation out outer layer of test section s surface of cylinder 1 ambient condition h angle from top of the cylinder (°)
2. Experimental setup Fig. 1 shows the arrangement of horizontal tube and adjustment walls. A horizontal long cylinder placed at the corner of two flat adiabatic walls and surrounded by an ambient air. In this figure the surface temperature of the cylinder, Ts, is assumed to be lower than the ambient air temperature, T1, and higher than the ambient dew point temperature. The length of each floor (W) and width along the cylinder axial direction are considered to be sufficiently wide in order to reduce the end effects on heat transfer from the cylinder. In Fig. 2 the experimental system configuration is shown. The main components of the system contain: a circular cylinder as test section (36 cm in length and 8 cm in diameter), an adiabatic floor, a fluid cooling unit with R22 as a refrigerant, an air cooling unit, an air heating unit, a humidifier, a fan, valves, a data logger, T type thermocouples and a computer. The ends of the cylinder are kept insulated to reduce end heat losses and keep two dimensional natural convection. This figure shows that the temperature of cylinder surface is controlled by a fluid cooling unit which flows in a refrigeration cycle. The T-type thermocouples attached to the
Fig. 1. Cylinder and floors arrangement.
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Fig. 2. Experimental system and all equipments.
cylinder, measure temperature at various positions. All the measured data are recorded by a data logger which is connected to a computer. The test room consists of two blocks; test room and equipment room which are separated by an isolated wall. In the equipment room, air cooling unit, air heating unit and humidifier control the temperature and humidity of the test room. Fig. 2 also shows that a fan temporary blows air in the space between the roof and the suspended ceiling and passes through the holes with reduced air velocity in 10–15 min with in every two hours. This velocity at the outlet is less than 0.4 m/s only during fan operation. Free convection condition is maintained in the room for the rest of experimental duration time. Detailed explanations of the system operation is reported by Tahavvor and Yaghoubi [7]. 2.1. Apparatus and instrumentation
Surface temperature distribution indicates that for a steady state condition, heat conduction along the axis of the cylinder is negligible in comparison to heat transfer in radial direction. Hence, the axial heat transfer is neglected in this study. Fig. 3 also shows the relative position of thermocouple holes in the inner and outer layer of the test section. Since the thermal conductivity of insulation is much lower than the thermal conductivity of aluminum, the thermal resistance of the outer layer is negligible in comparison with the thermal resistance of the insulation layer. This leads to a considerable temperature gradient across the insulation, which provides an easy way to measure the radial heat flux. The radial heat flux (q00 ), average heat transfer coefficient (have) and average Nusselt number (Nuave) for the experimental measurements are determined by:
q00 ¼
The test section is shown in Fig. 3. This cylinder is made of aluminum, which has 36 cm long and 8 cm outside diameter. Sixteen T-type thermocouples are attached to the cylinder to measure its temperature. The thermocouples are first calibrated in a controlled water basin with 0.5 °C uncertainty. These thermocouples are placed in specific holes with a distance of 3 mm from the cold surface as shown in Fig. 3. The thermocouples are inserted in the both sides of the pipe (eight thermocouples in each side) to measure the average heat flux and the temperature distribution along the test section. The value of cold surface temperature is determined by averaging the temperature of four thermocouples at each side. As shown in Fig. 3, an insulated layer with a thermal conductivity of 0.17 (W/m K) is placed between two aluminum layers to obtain high temperature difference and more accurate heat flux.
2kins ðT out T in Þ Ln rrout D in
ð1Þ
q00 ðT sav e T 1 Þ
ð2Þ
h av e D kf
ð3Þ
hav e ¼
ðNuav e Þexp ¼
The vertical and horizontal adiabatic walls, which are placed beneath the test section is made of compact Polystyrene with rated thermal conductivity of 0.029–0.045 (W/m K). The adiabatic surface dimension is 150 100 cm2 with 4 cm thickness. The insulation is considered to be wide enough to reduce the end effects on heat transfer to the cylinder. Due to small temperature difference
Fig. 3. Test section and thermocouples position.
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and steady state condition, it is assumed that the insulated surface temperature is equal to the ambient temperature. So the radiation heat flux from the cylinder surface can be computed as follows:
q00rad ¼ eal r T 41 T 4sav e
ð4Þ
When r is the Stefan-Boltzmann constant and equal to 5.67 108 (W/m2 K4) and e is emissivity, which is 0.09 for an aluminum surface. For maximum thermal radiation at Ra = 104, the amount of q00rad is about 15 (W/m2) which is negligible in compared with the free convection heat flux of 745 (W/m2). The fluid cooling unit is TUV-BERLIN BRANDENBURG. It has the ability to cool the antifreeze solution to 30 °C. Ethylene-glycol/ water of 60/40 solution is used as refrigerant for controlling and keeping the test section surface temperature at a fixed value. Also the cooling unit used is TUV-BERLIN BRANDENBURG. It has the ability to cool the ambient air temperature to 0 °C. The air heating unit consists of three thermal elements, 2000 W in power. These elements have the ability to reach the test room air temperature to 50 °C. Two humidifiers are also used to humidify the test room ambient air. These humidifiers have a volume of 6 l approximately and have the ability to operate at least 20 h continuously. According to the volume of the test room these humidifiers are able to reach a relative humidity of the test room to 100% for about 4 h. The data acquisition units used are ADAM 4518 and 4520MOXA ioLogic. First unit has sixteen analogue input channels, able to measure voltage, resistance, current and frequency. It is used to measure the thermocouple output. The second unit is used to measure relative humidity and temperature of air in the test room. According to these data, command signals are relative humidity and temperature of the ambient air of the test room. These values are measured using four TMH-1 transmitters. These transmitters provide a reliable and economical method for the most relative humidity and temperature monitoring applications. Each unit is factory calibrated to provide excellent sensitivity, fast response and stability. The unit features a simple installation with a direct USB connection. Free friendly user software converts the signals into both humidity and temperature meter.
Fig. 4. Physical geometry and computational domain.
The governing equations for two-dimensional, laminar, natural convection around a horizontal cylinder with Boussinesq approximation are continuity, momentum and energy equations.
@u @ v þ ¼0 @x @y ! @u @u @P @2u @2u ¼ q u þv þl þ @x @y @x @x2 @y2
For experimental measurements, ambient air temperature is maintained at a uniform temperature of T1 = 30 ± 0.2 °C and relative humidity at 30 ± 0.5%. Experiments are performed for two surface temperatures of Ts = 13 ± 0.1 °C and Ts = 21 ± 0.1 °C which are well above the dew point of air temperature. In these conditions, Rayleigh numbers based on the cylinder diameter are 6 105 and 3 105, respectively. The experiments are performed for L/D = 0.1, 0.3, 0.5, 0.75, 1 and S/D = 0.1, 0.3, 0.5, 0.75, 1 with aforementioned Rayleigh numbers. During each experiment, cylinder ends are insulated with a thin polyethylene film so that heat transfers occur only in the radial direction. After reaching the surface temperature to a desired steady value, which takes about two hours, measurements started. Each test is started by saving temperature of all thermocouples in a data logger connected to a PC computer. Based on the recorded average temperature, radial heat flux is calculated using Eq. (1). 3. Numerical method 3.1. Computational modeling Physical model for CFD solution is shown in Fig. 4. The outer boundary has a pressure outlet with ambient temperature (T1) and zero pressure gradient.
ð6Þ
! @v @v @P @2v @2v ¼ q u þv þl þ þ qgbðT T 1 Þ @y @x @y @x2 @y2
qC u
@T @T þv @x @y
¼k
@2T @2T þ @x2 @y2
ð7Þ
! ð8Þ
The corresponding boundary conditions are prescribed in Table 1 and the non-dimensional parameters related to the governing equations are:
Ra ¼ 2.2. Experimental procedure and measurements
ð5Þ
gbD3 ðT s T 1 Þ
am
;
Nu ¼
hD Kf
ð9Þ
The governing equations are solved using OpenFOAM, an opensource CFD code. The OpenFOAM is an enormously powerful research CFD code, written in the C++ programming language. OpenFOAM uses finite volume scheme to solve systems of partial differential equations ascribed on any 3D unstructured mesh of polyhedral cells. The fluid flow solvers are developed within a robust, implicit, pressure-velocity, iterative solution framework, although alternative techniques are applied to other continuum mechanics solvers. Domain decomposition parallelism is fundamental to the design of OpenFOAM and integrated at a low level so that solvers can generally be developed without the need for any ’parallel-specific’ coding. In this study the incompressible Navier Stokes equations are solved by using the PISO algorithm. The pre-processing open-source software Gambit is used to create the grids of the computational model. The computational
Table 1 Boundary conditions for the domain of Fig. 4.
Surrounding Horizontal/vertical plate Cylinder
Thermal
Hydrodynamic
Pressure
Fixed value Zero gradient Fixed value
Pressure outlet Fixed value uniform (0 0 0) Fixed value uniform (0 0 0)
Fixed value Zero gradient Zero gradient
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Fig. 5. Generated grid for numerical solution (a) whole domain and (b) near the cylinder.
3.2. Numerical computations
Table 2 Various cases of numerical computations. Ra
D (m)
Ts (°C)
T1 (°C)
104 105 3 105 6 105 106
0.016 0.04 0.08 0.08 0.08
13 13 21.5 13 13
41.5 29.5 31.5 31.5 32.5
Numerical modeling is performed for different L/D = 0.1, 0.3, 0.5, 0.75,1 and S/D = 0.1, 0.3, 0.5, 0.75 and 1 at Rayleigh numbers ranging from 104 to 106 which correspond to different values of the cylinder diameter (D), cylinder temperature (Ts) and the ambient temperature (T1). Table 2 shows the specifications of various cases considered for the simulation. Energy balance at the cylinder surface yields the following expressions for the local and average Nusselt number
q00 ¼ kf
@T @r r¼D=2
ð10Þ
(a) Grid independency
(b) domain independency Fig. 6. Solution Independency based on average Nusselt number for Ra = 106.
domain is taken big enough to resemble the ambient boundary conditions. Fig. 5 shows the generated grid for the numerical solution.
Fig. 7. Comparison of Nuave/Nu1 for numerical and experimental measurements as a function of L/D for Ra = 6 105 for S/D = 0.1 and 0.5.
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Table 3 Percentage of relative error of average Nusselt number between present numerical solutions and experimental measurements.
hh ¼
q00 ðT s T 1 Þ
hav e ¼
1 2p
Nuh ¼
hh D kf
Z 2p
Ra
Pr
Churchill and Chu [8] (Eq. (15))
Present numerical solution
104 105 3 105 6 105 106
0.7143 0.7153 0.7144 0.7152 0.7151
4.3762 7.7847 10.4323 12.6184 14.5508
4.6457 7.7980 10.1583 11.8271 13.3505
hh dh
ð12Þ
0
ðNuav e Þnum ¼
Table 4 Comparison of average Nusselt number for a horizontal tube in an infinity media.
ð11Þ
ð13Þ hav e D kf
ð14Þ
For a horizontal cylinder in an infinity media, the following relation is defined to compute the average Nusselt number [8]:
8 92 > > < = 0:387Ra1=6 Nu1 ¼ 0:6 þ h i 8=27 > > : ; 1 þ ð0:559=PrÞ9=16
ð15Þ
4. Results and discussion The OpenFOAM code solves the governing equations unsteadily and the computations are continued iterated to reach a steady
Ra=10 4
Ra=10 5
Ra=10 6
L/D=0.1
L/D=0.5
L/D=1
L/D=1.5
Fig. 8. Streamlines around cylinder for various L/D and Rayleigh numbers for S/D = 0.1.
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state solution. For a typical calculation for a pipe in an infinite medium and for Ra = 104, the computations are performed till velocity residual becomes less than 106 and temperature residual reduces less than 105 after 100 s. Next the validity of the CFD solver and grid and domain independency of the solution method are explained. To obtain a grid independent solution, various meshes are considered. Fig. 6a shows the average Nusselt number obtained for different meshes for a cylinder in an infinite medium for Ra = 106. According to this figure, an average Nusselt number almost becomes constant for the mesh numbers larger than 7500. Fig. 6b shows variation of average Nusselt number with ratio of numerical domain to the pipe diameter (R/D) for a cylinder in an infinite medium for Ra = 106. This figure also illustrates that average Nusselt number become constant after R/D = 15. This means that the solution is domain-independent. Based on the above computations the domain size is selected 15D 15D and the number of grids for numerical computations is 120 120. To validate the present CFD solver, the numerical results with experimental measurements are compared for two Rayleigh numbers 3 105 and 6 105. Fig. 7 shows the numerical and experimental measurements comparison of Nuave/Nu1 as a function of L/D for Ra = 6 105 at two different S/D values. For making a better comparison Table 3 which lists the percentage of relative errors of
Ra=10 4
the average Nusselt number at different L/D and S/D is used. The relative error is calculated according to the following expression.
ðNuav e Þ ðNuav e Þ exp num Relative Error ¼ ðNuav e Þexp
ð16Þ
For experimental measurements and numerical computations the value of Nuave is determined from Eqs. (3) and (14), respectively. Table 3 as well as Fig. 7 show good agreements between the experimental measurements and the numerical computations in the range of Rayleigh number considered. The maximum difference is less than 11%, which is almost related to L/D = 0.1 in which the horizontal wall is nearest to the tube. The second validation test is carried out by comparing the average Nusselt number for the horizontal tube in an infinity media using Eq. (15). Table 4 shows that both results are in good agreement. The results of numerical computations for flow field and temperature field are plotted in Figs. 8–11 in detail. Fig. 8 indicates the streamlines around the cylinder for the Rayleigh numbers of 104, 105, and 106 and L/D ratios of 0.1, 0.5, 1 and 1.5, respectively at S/D = 0.1. All graphs indicate that the air plume from the cooled cylinder flows down and strikes to the horizontal plate and flows
Ra=105
Ra=106
L/D=0.1
L/D=0.5
L/D=1
L/D=1.5
Fig. 9. Streamlines around cylinder for various L/D and Rayleigh numbers for S/D = 0.5.
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Ra=10 4
Ra=10 5
Ra=10 6
L/D = 0.1
L/D = 0.5
L/D = 1
L/D =1.5
Fig. 10. Isothermal around cylinder for various L/D and Rayleigh numbers for S/D = 0.1.
opposite to the vertical wall. For low values of L/D (e.g. L/D = 0.1) cold air moves away about at an angle of 120° from the top of the cylinder due to damping effect of the floor. This angle increases by increasing L/D. Furthermore, cold plume is intensified by increasing L/D especially at higher Ra numbers. This figure also shows that by increasing Rayleigh number, the hydrodynamic boundary layer become thinner which is related to the increasing the buoyancy driven flow pattern. To investigate the effects of S/D on the flow field, aforementioned graphs are plotted for S/D = 0.5 in Fig. 9. This figure shows the pattern of different streamlines in comparison with previous case in Fig. 8. By increasing the value of S/D a rotating vortex is generated at the corner of the walls. This figure also shows that the size and the shape of these vortices change by increasing the value of L/D. By increasing Ra number, vortex size becomes wide and its velocity increases. Isothermal lines around the horizontal cylinder for various L/D and Ra = 104, 105, and 106 for S/D = 0.1 and S/D = 0.5 are indicated in Figs. 10 and 11 respectively. These figures represent that the cold plume flows downstream strikes to the floor and moves horizontally right from the cylinder over the floor. Fig. 10 shows the finite space between the cylinder and the floors has a great influence on the flow pattern. This figure also illustrates that by increasing L/D for each Rayleigh number and S/D, the thermal boundary layer becomes thinner. This results in a reduction of adiabatic floor influence and consequently increase the Nusselt number. Thinning the thermal boundary layer is also affected by increasing the Rayleigh number which enhances the buoyancy force in comparing to the viscous force as shown in Figs. 10 and 11. For low Rayleigh numbers, conduction heat transfer is dominant while at high Rayleigh numbers, the fluid flow adjacent to cylinder plays an important
role in convection heat transfer. These figures also show that for more closer of the cylinder to the vertical wall (Fig. 10) fluid motion reduces. Since the flow is free to move around the cylinder by increasing S/D, (Fig. 11) only the lower plate damps the cold plume. A more exhaustive investigation of heat transfer phenomena and a better comparison between the related factors of generating cold plume can be seen from the variation of local and average Nusselt number on the cylinder surface. The effects of horizontal and vertical adiabatic floor on local Nusselt number around the cylinder as a function of radial angle h at Ra = 104 are shown in Figs. 12 and 13, respectively. Fig. 12a shows that the horizontal adiabatic floor has a significant influence on the local Nusselt number and this effect is evident even at h = 0. This figure also shows that local Nusselt number has minimized at h ’ 240° approximately for all values of L/D in which the cylinder does not have enough space to convert heat to the surrounding air. The local Nusselt number is not symmetric, and even maximized at h ’ 30°. It also shows that the convection heat transfer on the wall side is less than on the free side. By increasing S/D as indicated in Fig. 12b, at very low L/D (L/D = 0.1) the heat transfer is almost negligible between h ’ 170° and the h ’ 190°. This range increases for lower S/D (as shown in Fig. 12a). It means that by reaching the adiabatic walls to the cylinder, the temperature gradient in radial direction (oT/or) converting to zero and so local Nusselt number and heat transfer can be disregarded due to damping effects of these walls at some angles (see Eqs. (10)–(13)). Fig. 12b shows that like the previous case the local Nusselt number maximized at h ’ 30°. Since the flow and heat transfer on the cylinder face side is almost identical, the variation of local Nusselt number become symmetric
M.H. Sedaghat et al. / Experimental Thermal and Fluid Science 62 (2015) 9–20
and the maximum of Nu at h ’ 30° disappeared by increasing L/D in Fig. 12b. Also minimum local Nusselt number shifted to the higher values of h. For example, it changed from 150° for L/D = 0.3 to 180° for L/D = 1.5. In other words, at large values of S/D, having enough space for heat transfer, only horizontal floor plays a significant role in the variation of local Nusselt number. Figs. 12b and c show that by increasing S/D from 0.5 to 1.5 the variation of local Nusselt numbers does not change considerably and this ratio does not have a great effect on heat transfer. Fig. 13 indicates the variation of local Nusselt number around the cylinder with h for Ra = 104 and for L/D = 0.1, 0.5 and 1.5 respectively at S/D = 0.1, 0.3, 0.5, 0.75, 1 and 1.5. Fig. 13a show that the variation of local Nusselt number is not symmetric, and heat transfer is almost negligible from h ’ 150° up to h ’ 210° and this range intensify as S/D decreases at low L/D values. This is because of the limited space between the cylinder and plate even for high S/D (e.g. S/D = 1.5). As mentioned before, this phenomena is also happened for very low L/D values (Fig. 11a). Fig. 13a also show that the local Nusselt number is maximized at h ’ 30°. Figs. 13b and c show the variation of a local Nusselt number of higher L/D values. These figures show that for L/D P 0.5 increasing S/D, does not affect the local Nusselt number greatly. For S/D 6 0.3 a variation of Nu is still non-symmetric and its variation is far from the cylinder in free space. Such a variation is even observed for L/D = 1.5 at S/D = 0.1. In other words, changing the position of vertical plane does not have great influence on the local Nusselt number of large space between the vertical plate and cylinder. Fig. 14 illustrates the variation of the average Nusselt number ratio (Nuave/Nu1) as functions of L/D and S/D at Ra = 104. This figure indicates that, the vertical displacement from the horizontal floor to cylinder (L/D) play a vital role for average Nusselt number ratio
Ra=104
17
and as L/D increases the Nusselt number ratio increases sharply. At a fixed value of L/D the Nusselt number ratio almost remains constant as S/D increases (for S/D P 0.3). To study the effect of the Rayleigh number on heat transfer rate, Figs. 15a and b show the variation of the Nusselt number ratio with L/D and Rayleigh numbers at S/D = 0. 1 and S/D = 0.5, respectively. Fig. 15a shows that L/D has a great effect on the Nusselt number ratio and this effect reduces by increasing the Rayleigh numbers. For example this figure shows that Nuave/Nu1 is about 0.55 for Ra = 105 at L/D = 0.1. This ratio reaches to 0.98 for L/D = 1.5 but the value of Nuave/Nu1 for Ra = 106 is about 0.8 at L/D = 0.1 and 1 for L/D = 1.5. Fig. 15b shows the similar effects of heat transfer rate. Due to higher space for heat transfer Nuave/Nu1 increases in the specific value of L/D and Rayleigh number. Fig. 15 also show that at a specific L/D, as Rayleigh number increases, the thermal boundary layer around the cylinder become thinner and dependency on S/D and L/D on heat transfer decreases. In other words the that Rayleigh number should be regarded as one of the significant factors which influence the ratio of Nuave/Nu1 on the cylinder when located at the corner of two adiabatic walls. 5. Uncertainty Analysis of uncertainty is required in order to evaluate the accuracy of measurements which is similar to the one described in the work of Batista et al. [22]. The total uncertainty U can be determined from uncertainties of the specified components which influence the experiment [23]. For a measurement M, whose results depend on uncorrelated input estimates, the standard uncertainty of the measurement is obtained by appropriately combining the standard uncertainties of these input estimates. The combined
Ra=105
Ra=106
L/D = 0.1
L/D = 0.5
L/D = 1
L/D = 1.5
Fig. 11. Isothermal around cylinder for various L/D and Rayleigh numbers for S/D = 0.5.
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(a)
(a)
S/D=0.1
L/D=0.1
(b)
(b)
L/D=0.5
S/D=0.5
(c)
(c)
L/D=1.5
S/D=1.5
Fig. 12. Variation of local Nusselt number around cylinder for Ra = 104 and for various S/D.
Fig. 13. Variation of local Nusselt number around cylinder for Ra = 104 and for various L/D.
standard uncertainty of the estimate M denoted by U is calculated from the following equations [24,25].
M ¼ f ðx1 ; x2 ; . . . xn Þ U 2 ðMÞ ¼
2 n X @f i¼1
@xi
U 2 ðxi Þ
ð17Þ ð18Þ
where f is the function of M in terms of input, estimates x1, x2, ... , xn, and each U(xi) is a standard uncertainty of any input. The expression for the standard uncertainty associated with have is developed from Eqs. (1) and (2), using the law of propagation of uncertainty. The resulting expression is:
" 2 2 2 @hav e @hav e @hav e dT 1 þ dT out þ dT in @T 1 @T out @T in 2 2 2 #12 @hav e @hav e @hav e dD þ dr out þ drin þ @D @r out @r in
dhav e ¼
ð19Þ
Fig. 14. Variation of average Nusselt number ratio with L/D and S/D for Ra = 104.
M.H. Sedaghat et al. / Experimental Thermal and Fluid Science 62 (2015) 9–20
19
6. Conclusion Natural convection heat transfer from a cold horizontal cylinder which is enclosed in a corner of two flat adiabatic walls is studied both numerically and experimentally. Results indicate that:
(a) S/D=0.1
(b) S/D=0.5 Fig. 15. Variation of average Nusselt number ratio with L/D and Rayleigh numbers for various S/D.
References
Table 5 Values for the parameters of Eq. (20). Parameter
xi
dxi
@hav e @xi
T1 (°C) Tout (°C) Tin (°C) D (mm) rout (mm) rin (mm)
31.5 13 9.96 80 25.9 22
0.2 0.1 0.1 0.02 0.02 0.02
0.0463 0.1408 0.1408 0.0011 0.0202 0.0238
so
dhav e ¼
1 ðT sav e T 1 Þ2
"
1 2kins ðT out T in Þ dT 1 ðT sav e T 1 Þ DLnðrout =r in Þ
dxi
2
2 2 2kins 2kins dT out þ dT in þ DLnðr out =r in Þ DLnðr out =r in Þ þ
þ
2kins ðT out T in Þ D2 Lnðr out =r in Þ 2kins ðT out T in Þ rin Dðr out =r in Þ2
1. Due to the damping effect of horizontal plate, streamlines and isothermal lines around the horizontal cylinder at low L/D (L/D = 0.1) separated and moved away about at an angle of 120° from the surface of the cylinder and this angle increases as L/D increased. 2. Owing to the enhancement of buoyancy force in comparison to the viscous force, as Rayleigh number increases, the hydrodynamic and thermal boundary layers become thinner, but fluid motion highly depends on the cylinder space from adiabatic walls. The higher is the value of Ra, the less is the effects of L/D and S/D on heat transfer from the horizontal cylinder. 3. In contrast to the vertical adiabatic floor, the horizontal adiabatic floor has a more significant influence on the local Nusselt number. This effect is evident even at the top of the cylinder (h = 0). 4. Due to the limitation of space between cylinder and plates, the variation of local Nusselt number is not symmetric, and heat transfer is almost negligible from h ’ 150° up to h ’ 210° at low L/D or S/D. Also the local Nusselt number has a maximum value at h ’ 30°. 5. The value of Rayleigh number and the ratio of vertical cylinder spacing of adiabatic floor (L/D) are regarded as two important factors which greatly influence the heat transfer from horizontal cylinder. In contrary the horizontal cylinder spacing of adiabatic floor (S/D) has less effect on the heat transfer rate. 6. More closer is the cylinder to the corner, more reduction in convection heat transfer takes place from the cylinder.
!2 dD
þ
!2 312 dr in 5
2kins ðT out T in Þ
!2
dr out r out DðLnðr out =rin ÞÞ2
ð20Þ
Lists of the experimental values of different parameters and associated uncertainties for Ra = 6 105 and L/D = 1 are prescribed in Table 5. The maximum uncertainty of the measurements is determined according to Table 5 and it is about 20%.
[1] V.T. Morgan, The overall convective heat transfer from smooth circular cylinders, Adv. Heat Transf. 11 (1975) 199–264. [2] T. Kuehn, R. Goldstein, Numerical solution to the Navier–Stokes equations for laminar natural convection about a horizontal isothermal circular cylinder, Int. J. Heat Mass Transf. 23 (1980) 971–979. [3] I. Tokura, H. Saito, K. Kishinami, K. Muramoto, An experimental study of free convection heat transfer from a horizontal cylinder in a vertical array set in free space between parallel walls, J. Heat Transf. 105 (1983) 102–107. [4] H. Badr, Heat transfer in transient buoyancy driven flow adjacent to a horizontal rod, Int. J. Heat Mass Transf. 30 (1987) 1997–2012. [5] J. Herraez, R. Belda, A study of free convection in air around horizontal cylinders of different diameters based on holographic interferometry. Temperature field equations and heat transfer coefficients, Int. J. Therm. Sci. 41 (2002) 261–267. [6] M. Corcione, Correlating equations for free convection heat transfer from horizontal isothermal cylinders set in a vertical array, Int. J. Heat Mass Transf. 48 (2005) 3660–3673. [7] A.R. Tahavvor, M. Yaghoubi, Natural cooling of horizontal cylinder using artificial neural network (ANN), Int. Commun. Heat Mass Transf. 35 (2008) 1196–1203. [8] S.W. Churchill, H.H. Chu, Correlating equations for laminar and turbulent free convection from a horizontal cylinder, Int. J. Heat Mass Transf. 18 (1975) 1049–1053. [9] T. Saito, R. Ishiguro, Y. Fujishima, Natural convection heat transfer around a horizontal cylinder (effect of a horizontal plate placed above the cylinder), in: 6th National Heat Transfer Symposium Japan, 1969. [10] H. Koizumi, I. Hosokawa, Chaotic behavior and heat transfer performance of the natural convection around a hot horizontal cylinder affected by a flat ceiling, Int. J. Heat Mass Transf. 39 (1996) 1081–1091. [11] G. Lawrence, G. Jardin, D. Naylor, A. Machin, Free convection from a horizontal heated cylinder located below a ceiling, Trans. Canad. Soc. Mech. Eng. 23 (1999) 19–35. [12] M. Ashjaee, A. Eshtiaghi, M. Yaghoubi, T. Yousefi, Experimental investigation on free convection from a horizontal cylinder beneath an adiabatic ceiling, Exp. Therm. Fluid Sci. 32 (2007) 614–623. [13] M. Correa, R. Parra, A. Vidal, J. Rodriguez, M. Aguilera, D. Gonzalez, Natural convection around a horizontal cylinder near an adiabatic cover wall, Proc. Fourth ICCHMT (2005) 336.
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[14] L. Ma, J.v.d. Zanden, J.v.d. Kooi, F.T. Nieuwstadt, Natural convection around a horizontal circular cylinder in infinite space and within confining plates: a finite element solution, Numer. Heat Transf. 25 (1994) 441–456. [15] M.S. Sadeghipour, Y.P. Razi, Natural convection from a confined horizontal cylinder: the optimum distance between the confining walls, Int. J. Heat Mass Transf. 44 (2000) 367–374. [16] B. Farouk, S.I. Güqeti, Natural and mixed convection heat transfer around a horizontal cylinder within confining walls, Numer. Heat Transf., Part A Appl. 5 (1982) 329–341. [17] B. Huynh, Cooling by free convection at high Rayleigh number of cylinders positioned above a plane, in: 16th Australasian Fluid Mechanics Conference, Gold Coast, Australia, 2007. [18] M.A. Atmane, V.S. Chan, D.B. Murray, Natural convection around a horizontal heated cylinder: the effects of vertical confinement, Int. J. Heat Mass Transf. 46 (2003) 3661–3672. [19] F. Karim, B. Farouk, I. Namer, Natural convection heat transfer from a horizontal cylinder between vertical confining adiabatic walls, J. Heat Transf. 108 (1986) 291–298.
[20] G. Cesini, M. Paroncini, G. Cortella, M. Manzan, Natural convection from a horizontal cylinder in a rectangular cavity, Int. J. Heat Mass Transf. 42 (1999) 1801–1811. [21] S.H. Hussain, A.K. Hussein, Numerical investigation of natural convection phenomena in a uniformly heated circular cylinder immersed in square enclosure filled with air at different vertical locations, Int. Commun. Heat Mass Transf. 37 (2010) 1115–1126. [22] E. Batista, L. Pinto, E. Filipe, A. Van der Veen, Calibration of micropipettes: test methods and uncertainty analysis, Measurement 40 (2007) 338–342. [23] P.L. Teoh, B. Shirinzadeh, C.W. Foong, G. Alici, The measurement uncertainties in the laser interferometry-based sensing and tracking technique, Measurement 32 (2002) 135–150. [24] X. Wang, E. Bibeau, G. Naterer, Experimental correlation of forced convection heat transfer from a NACA airfoil, Exp. Therm. Fluid Sci. 31 (2007) 1073–1082. [25] C.-C. Wang, R.-T. Huang, W.-J. Sheu, Y.-J. Chang, Some observations of the frost formation in free convection: with and without the presence of electric field, Int. J. Heat Mass Transf. 47 (2004) 3491–3505.
Contents lists available at ScienceDirect
Experimental Thermal and Fluid Science journal homepage: www.elsevier.com/locate/etfs
Analysis of natural convection heat transfer from a cylinder enclosed in a corner of two adiabatic walls M.H. Sedaghat a, M. Yaghoubi b,c, M.J. Maghrebi d,⇑ a
Department of Mechanical Engineering, Shahrood University, Shahrood, Iran Department of Mechanical Engineering, Shiraz University, Shiraz, Iran c Academy of Sciences of Islamic Republic of Iran, Tehran, Iran d Department of Mechanical Engineering, Ferdowsi University of Mashhad, Mashhad, Iran b
a r t i c l e
i n f o
Article history: Received 13 July 2014 Received in revised form 12 October 2014 Accepted 5 November 2014 Available online 25 November 2014 Keywords: Free convection Laminar flow Cold horizontal cylinder Adiabatic walls
a b s t r a c t A steady two-dimensional laminar natural convection heat transfer from a cold single horizontal isothermal cylinder enclosed between two adiabatic walls is studied both experimentally and numerically. Experiments are performed to study the effects of cylinder distance from horizontal and vertical walls for Rayleigh numbers of 3 105 and 6 105. Numerical computations are carried out by solving the governing equations of laminar natural convection using the finite volume technique. OpenFOAM which is an open source code software is used. Computations are made for a wide range of Rayleigh numbers from 104 to 106 and different ratios of vertical cylinder spacing (L/D) and that of horizontal cylinder spacing (S/D). Comparisons are made with the corresponding experimental measurements. Flow streamlines and isothermal lines around the cylinder so that by changing the value of L/D and S/D, due to damping effects of the adiabatic walls, the cold plume moves away from the cylinder at different angles. Results also indicate that local and average heat transfer coefficient is highly affected by L/D and Rayleigh number while S/D has a limited influence on heat transfer from the tube. Ó 2014 Elsevier Inc. All rights reserved.
1. Introduction The significance of natural convection heat transfer from cold pipes or tubes for designing refrigeration, air conditioning and freezing systems have continued interest among designers and manufacturers. Even though the study of natural convection heat transfer from pipes or an array of pipes and tubes in a medium with an infinite domain is regarded as a topic of wide variety of investigations [1–8], but natural convection heat transfer from a pipe with confining walls or pipe in a semi enclosure is not studied considerably. Some researchers analyzed the influences of adiabatic bounding walls on the natural convection heat transfer coefficient from a hot circular cylinder. Saito et al. [9] in an experimental investigation studied the effect of adiabatic plate over a horizontal heated cylinder for Grashof numbers between 2.1 106 and 3.2 106. They studied effects of the ratio of cylinder spacing from the floor to its diameter (L/D) on Nusselt number and showed that the rate of heat transfer would be minimized if this ratio is about 0.12. ⇑ Corresponding author. Tel.: +98 51 38805051; fax: +98 51 38763304. E-mail addresses: [email protected] (M.H. Sedaghat), [email protected] (M. Yaghoubi), [email protected] (M.J. Maghrebi). http://dx.doi.org/10.1016/j.expthermflusci.2014.11.003 0894-1777/Ó 2014 Elsevier Inc. All rights reserved.
Koizumi and Hosokawa [10] used flow visualization to study free convection heat transfer from a pipe under a ceiling. Their experiments also include the effects of changing pipe diameter, ceiling distance from the pipe and the type of ceiling (adiabatic or isothermal) on natural convection heat transfer for Rayleigh numbers from 4.8 104 to 107. Their results for an isothermal ceiling show that the fluid flow remains almost steady and two dimensional for all ranges of L/D for Rayleigh numbers over 105. When the ceiling is close to the pipe (L/D = 0.4) the fluid flow becomes unstable and three dimensional for Rayleigh number less than 105. For high values of L/D the plume oscillate laterally. They also reported local Nusselt number for both types of ceiling at Ra = 1.3 106 and L/D = 0.05, 0.2. Lawrence et al. [11] studied natural convection heat transfer from a hot pipe below an isothermal surface. They applied finite element method for their numerical solution and MachZehnder interferometer in their experiments for air flow; with Rayleigh numbers of 103–105 and 0.1 6 L/D 6 0.5. They showed that when the distance between ceiling and pipe is larger than the pipe diameter, ceiling does not influence the rate of heat transfer from the pipe. They also indicated that average Nusselt number increases as a result of conduction heat transfer between the pipe and the ceiling for spacing of about one fourth of cylinder diameter. However, when L/D > 0.25 the ceiling decreases heat transfer
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Nomenclature D g Gr h L Nu K P Pr q00 R Ra S T U U V W
cylinder diameter (m) gravitational acceleration (m/s2) Grashof number heat transfer coefficient (W/m2 K) vertical cylinder to floor distance (m) Nusselt number thermal conductivity (W/m K) pressure (Pa) Prandtl number heat flux (W/m2) radius (m) Rayleigh number horizontal cylinder to floor distance (m) temperature (°C, K) horizontal component of velocity (m/s) velocity vector (m/s) vertical component of velocity (m/s) floor length (m)
up to 10%. Ashjaee et al. [12] used a Mach-Zehnder interferometer to study natural convection heat transfer from a pipe under an adiabatic ceiling for Rayleigh numbers between 103 and 4 104. They showed that ceiling did not have a great effect on heat transfer from the cylinder when L/D > 1.5. They also showed that by decreasing the distance to L/D = 0.5 the Nusselt number reduced significantly. They reported that, by reducing this ratio and as a consequence the formation of vortex above the cylinder, both local Nusselt numbers at the top of the cylinder and the average Nusselt number decreased sharply. They also suggested the average Nusselt number as function of L/D and Rayleigh number. Similar investigations are devoted for natural convection heat transfer from circular pipe with horizontal [13–17] and vertical [18,19] confining walls. A few investigations are focused on natural convection heat transfer from a horizontal cylinder enclosed in a rectangular cavity. Cesini et al. [20] studied the influence of Rayleigh number and the geometry of the cavity on the natural convection heat transfer from a horizontal circular cylinder both numerically and experimentally. Hussain and Hussain [21] numerically investigated the two dimensional steady natural convection for a uniform heat source applied on the inner circular cylinder in a square in which all the boundaries are assumed to be isothermal (at a constant low temperature). They showed the effects of the vertical location of cylinder and Rayleigh numbers (from 103 to 106) on fluid flow and heat transfer from the pipe. They found that the location of cylinder at small Rayleigh numbers does not have much influence on the flow field while it has a considerable effect on the flow pattern at high Rayleigh numbers. According to the literature review, most of the studies are focused on natural convection from hot cylinders with horizontal and vertical confinements and a few analyses are reported from a cylinder or tube in a corner of two adiabatic walls. In this paper, a numerical and experimental analysis is performed for natural convection heat transfer from a single horizontal cooled cylinder located at the corner of two flat adiabatic walls. This configuration is specially used in power plants, chemical and refrigeration industries for transferring a fluid from one location to another place. Based on such simulation experiments and numerical computations are made for various cylinder space ratios and flow Rayleigh numbers.
Greek symbols b volumetric expansion coefficient (K1) a thermal diffusivity e emissivity h angle from top of cylinder (°) l dynamic viscosity (Pa s) m kinematic viscosity (m2/s) q density (kg/m3) r Stefan-Boltzmann constant (W/m2 K4) Subscripts ave average condition f fluid in inner layer of test section ins insulation out outer layer of test section s surface of cylinder 1 ambient condition h angle from top of the cylinder (°)
2. Experimental setup Fig. 1 shows the arrangement of horizontal tube and adjustment walls. A horizontal long cylinder placed at the corner of two flat adiabatic walls and surrounded by an ambient air. In this figure the surface temperature of the cylinder, Ts, is assumed to be lower than the ambient air temperature, T1, and higher than the ambient dew point temperature. The length of each floor (W) and width along the cylinder axial direction are considered to be sufficiently wide in order to reduce the end effects on heat transfer from the cylinder. In Fig. 2 the experimental system configuration is shown. The main components of the system contain: a circular cylinder as test section (36 cm in length and 8 cm in diameter), an adiabatic floor, a fluid cooling unit with R22 as a refrigerant, an air cooling unit, an air heating unit, a humidifier, a fan, valves, a data logger, T type thermocouples and a computer. The ends of the cylinder are kept insulated to reduce end heat losses and keep two dimensional natural convection. This figure shows that the temperature of cylinder surface is controlled by a fluid cooling unit which flows in a refrigeration cycle. The T-type thermocouples attached to the
Fig. 1. Cylinder and floors arrangement.
M.H. Sedaghat et al. / Experimental Thermal and Fluid Science 62 (2015) 9–20
11
Fig. 2. Experimental system and all equipments.
cylinder, measure temperature at various positions. All the measured data are recorded by a data logger which is connected to a computer. The test room consists of two blocks; test room and equipment room which are separated by an isolated wall. In the equipment room, air cooling unit, air heating unit and humidifier control the temperature and humidity of the test room. Fig. 2 also shows that a fan temporary blows air in the space between the roof and the suspended ceiling and passes through the holes with reduced air velocity in 10–15 min with in every two hours. This velocity at the outlet is less than 0.4 m/s only during fan operation. Free convection condition is maintained in the room for the rest of experimental duration time. Detailed explanations of the system operation is reported by Tahavvor and Yaghoubi [7]. 2.1. Apparatus and instrumentation
Surface temperature distribution indicates that for a steady state condition, heat conduction along the axis of the cylinder is negligible in comparison to heat transfer in radial direction. Hence, the axial heat transfer is neglected in this study. Fig. 3 also shows the relative position of thermocouple holes in the inner and outer layer of the test section. Since the thermal conductivity of insulation is much lower than the thermal conductivity of aluminum, the thermal resistance of the outer layer is negligible in comparison with the thermal resistance of the insulation layer. This leads to a considerable temperature gradient across the insulation, which provides an easy way to measure the radial heat flux. The radial heat flux (q00 ), average heat transfer coefficient (have) and average Nusselt number (Nuave) for the experimental measurements are determined by:
q00 ¼
The test section is shown in Fig. 3. This cylinder is made of aluminum, which has 36 cm long and 8 cm outside diameter. Sixteen T-type thermocouples are attached to the cylinder to measure its temperature. The thermocouples are first calibrated in a controlled water basin with 0.5 °C uncertainty. These thermocouples are placed in specific holes with a distance of 3 mm from the cold surface as shown in Fig. 3. The thermocouples are inserted in the both sides of the pipe (eight thermocouples in each side) to measure the average heat flux and the temperature distribution along the test section. The value of cold surface temperature is determined by averaging the temperature of four thermocouples at each side. As shown in Fig. 3, an insulated layer with a thermal conductivity of 0.17 (W/m K) is placed between two aluminum layers to obtain high temperature difference and more accurate heat flux.
2kins ðT out T in Þ Ln rrout D in
ð1Þ
q00 ðT sav e T 1 Þ
ð2Þ
h av e D kf
ð3Þ
hav e ¼
ðNuav e Þexp ¼
The vertical and horizontal adiabatic walls, which are placed beneath the test section is made of compact Polystyrene with rated thermal conductivity of 0.029–0.045 (W/m K). The adiabatic surface dimension is 150 100 cm2 with 4 cm thickness. The insulation is considered to be wide enough to reduce the end effects on heat transfer to the cylinder. Due to small temperature difference
Fig. 3. Test section and thermocouples position.
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and steady state condition, it is assumed that the insulated surface temperature is equal to the ambient temperature. So the radiation heat flux from the cylinder surface can be computed as follows:
q00rad ¼ eal r T 41 T 4sav e
ð4Þ
When r is the Stefan-Boltzmann constant and equal to 5.67 108 (W/m2 K4) and e is emissivity, which is 0.09 for an aluminum surface. For maximum thermal radiation at Ra = 104, the amount of q00rad is about 15 (W/m2) which is negligible in compared with the free convection heat flux of 745 (W/m2). The fluid cooling unit is TUV-BERLIN BRANDENBURG. It has the ability to cool the antifreeze solution to 30 °C. Ethylene-glycol/ water of 60/40 solution is used as refrigerant for controlling and keeping the test section surface temperature at a fixed value. Also the cooling unit used is TUV-BERLIN BRANDENBURG. It has the ability to cool the ambient air temperature to 0 °C. The air heating unit consists of three thermal elements, 2000 W in power. These elements have the ability to reach the test room air temperature to 50 °C. Two humidifiers are also used to humidify the test room ambient air. These humidifiers have a volume of 6 l approximately and have the ability to operate at least 20 h continuously. According to the volume of the test room these humidifiers are able to reach a relative humidity of the test room to 100% for about 4 h. The data acquisition units used are ADAM 4518 and 4520MOXA ioLogic. First unit has sixteen analogue input channels, able to measure voltage, resistance, current and frequency. It is used to measure the thermocouple output. The second unit is used to measure relative humidity and temperature of air in the test room. According to these data, command signals are relative humidity and temperature of the ambient air of the test room. These values are measured using four TMH-1 transmitters. These transmitters provide a reliable and economical method for the most relative humidity and temperature monitoring applications. Each unit is factory calibrated to provide excellent sensitivity, fast response and stability. The unit features a simple installation with a direct USB connection. Free friendly user software converts the signals into both humidity and temperature meter.
Fig. 4. Physical geometry and computational domain.
The governing equations for two-dimensional, laminar, natural convection around a horizontal cylinder with Boussinesq approximation are continuity, momentum and energy equations.
@u @ v þ ¼0 @x @y ! @u @u @P @2u @2u ¼ q u þv þl þ @x @y @x @x2 @y2
For experimental measurements, ambient air temperature is maintained at a uniform temperature of T1 = 30 ± 0.2 °C and relative humidity at 30 ± 0.5%. Experiments are performed for two surface temperatures of Ts = 13 ± 0.1 °C and Ts = 21 ± 0.1 °C which are well above the dew point of air temperature. In these conditions, Rayleigh numbers based on the cylinder diameter are 6 105 and 3 105, respectively. The experiments are performed for L/D = 0.1, 0.3, 0.5, 0.75, 1 and S/D = 0.1, 0.3, 0.5, 0.75, 1 with aforementioned Rayleigh numbers. During each experiment, cylinder ends are insulated with a thin polyethylene film so that heat transfers occur only in the radial direction. After reaching the surface temperature to a desired steady value, which takes about two hours, measurements started. Each test is started by saving temperature of all thermocouples in a data logger connected to a PC computer. Based on the recorded average temperature, radial heat flux is calculated using Eq. (1). 3. Numerical method 3.1. Computational modeling Physical model for CFD solution is shown in Fig. 4. The outer boundary has a pressure outlet with ambient temperature (T1) and zero pressure gradient.
ð6Þ
! @v @v @P @2v @2v ¼ q u þv þl þ þ qgbðT T 1 Þ @y @x @y @x2 @y2
qC u
@T @T þv @x @y
¼k
@2T @2T þ @x2 @y2
ð7Þ
! ð8Þ
The corresponding boundary conditions are prescribed in Table 1 and the non-dimensional parameters related to the governing equations are:
Ra ¼ 2.2. Experimental procedure and measurements
ð5Þ
gbD3 ðT s T 1 Þ
am
;
Nu ¼
hD Kf
ð9Þ
The governing equations are solved using OpenFOAM, an opensource CFD code. The OpenFOAM is an enormously powerful research CFD code, written in the C++ programming language. OpenFOAM uses finite volume scheme to solve systems of partial differential equations ascribed on any 3D unstructured mesh of polyhedral cells. The fluid flow solvers are developed within a robust, implicit, pressure-velocity, iterative solution framework, although alternative techniques are applied to other continuum mechanics solvers. Domain decomposition parallelism is fundamental to the design of OpenFOAM and integrated at a low level so that solvers can generally be developed without the need for any ’parallel-specific’ coding. In this study the incompressible Navier Stokes equations are solved by using the PISO algorithm. The pre-processing open-source software Gambit is used to create the grids of the computational model. The computational
Table 1 Boundary conditions for the domain of Fig. 4.
Surrounding Horizontal/vertical plate Cylinder
Thermal
Hydrodynamic
Pressure
Fixed value Zero gradient Fixed value
Pressure outlet Fixed value uniform (0 0 0) Fixed value uniform (0 0 0)
Fixed value Zero gradient Zero gradient
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13
Fig. 5. Generated grid for numerical solution (a) whole domain and (b) near the cylinder.
3.2. Numerical computations
Table 2 Various cases of numerical computations. Ra
D (m)
Ts (°C)
T1 (°C)
104 105 3 105 6 105 106
0.016 0.04 0.08 0.08 0.08
13 13 21.5 13 13
41.5 29.5 31.5 31.5 32.5
Numerical modeling is performed for different L/D = 0.1, 0.3, 0.5, 0.75,1 and S/D = 0.1, 0.3, 0.5, 0.75 and 1 at Rayleigh numbers ranging from 104 to 106 which correspond to different values of the cylinder diameter (D), cylinder temperature (Ts) and the ambient temperature (T1). Table 2 shows the specifications of various cases considered for the simulation. Energy balance at the cylinder surface yields the following expressions for the local and average Nusselt number
q00 ¼ kf
@T @r r¼D=2
ð10Þ
(a) Grid independency
(b) domain independency Fig. 6. Solution Independency based on average Nusselt number for Ra = 106.
domain is taken big enough to resemble the ambient boundary conditions. Fig. 5 shows the generated grid for the numerical solution.
Fig. 7. Comparison of Nuave/Nu1 for numerical and experimental measurements as a function of L/D for Ra = 6 105 for S/D = 0.1 and 0.5.
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Table 3 Percentage of relative error of average Nusselt number between present numerical solutions and experimental measurements.
hh ¼
q00 ðT s T 1 Þ
hav e ¼
1 2p
Nuh ¼
hh D kf
Z 2p
Ra
Pr
Churchill and Chu [8] (Eq. (15))
Present numerical solution
104 105 3 105 6 105 106
0.7143 0.7153 0.7144 0.7152 0.7151
4.3762 7.7847 10.4323 12.6184 14.5508
4.6457 7.7980 10.1583 11.8271 13.3505
hh dh
ð12Þ
0
ðNuav e Þnum ¼
Table 4 Comparison of average Nusselt number for a horizontal tube in an infinity media.
ð11Þ
ð13Þ hav e D kf
ð14Þ
For a horizontal cylinder in an infinity media, the following relation is defined to compute the average Nusselt number [8]:
8 92 > > < = 0:387Ra1=6 Nu1 ¼ 0:6 þ h i 8=27 > > : ; 1 þ ð0:559=PrÞ9=16
ð15Þ
4. Results and discussion The OpenFOAM code solves the governing equations unsteadily and the computations are continued iterated to reach a steady
Ra=10 4
Ra=10 5
Ra=10 6
L/D=0.1
L/D=0.5
L/D=1
L/D=1.5
Fig. 8. Streamlines around cylinder for various L/D and Rayleigh numbers for S/D = 0.1.
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M.H. Sedaghat et al. / Experimental Thermal and Fluid Science 62 (2015) 9–20
state solution. For a typical calculation for a pipe in an infinite medium and for Ra = 104, the computations are performed till velocity residual becomes less than 106 and temperature residual reduces less than 105 after 100 s. Next the validity of the CFD solver and grid and domain independency of the solution method are explained. To obtain a grid independent solution, various meshes are considered. Fig. 6a shows the average Nusselt number obtained for different meshes for a cylinder in an infinite medium for Ra = 106. According to this figure, an average Nusselt number almost becomes constant for the mesh numbers larger than 7500. Fig. 6b shows variation of average Nusselt number with ratio of numerical domain to the pipe diameter (R/D) for a cylinder in an infinite medium for Ra = 106. This figure also illustrates that average Nusselt number become constant after R/D = 15. This means that the solution is domain-independent. Based on the above computations the domain size is selected 15D 15D and the number of grids for numerical computations is 120 120. To validate the present CFD solver, the numerical results with experimental measurements are compared for two Rayleigh numbers 3 105 and 6 105. Fig. 7 shows the numerical and experimental measurements comparison of Nuave/Nu1 as a function of L/D for Ra = 6 105 at two different S/D values. For making a better comparison Table 3 which lists the percentage of relative errors of
Ra=10 4
the average Nusselt number at different L/D and S/D is used. The relative error is calculated according to the following expression.
ðNuav e Þ ðNuav e Þ exp num Relative Error ¼ ðNuav e Þexp
ð16Þ
For experimental measurements and numerical computations the value of Nuave is determined from Eqs. (3) and (14), respectively. Table 3 as well as Fig. 7 show good agreements between the experimental measurements and the numerical computations in the range of Rayleigh number considered. The maximum difference is less than 11%, which is almost related to L/D = 0.1 in which the horizontal wall is nearest to the tube. The second validation test is carried out by comparing the average Nusselt number for the horizontal tube in an infinity media using Eq. (15). Table 4 shows that both results are in good agreement. The results of numerical computations for flow field and temperature field are plotted in Figs. 8–11 in detail. Fig. 8 indicates the streamlines around the cylinder for the Rayleigh numbers of 104, 105, and 106 and L/D ratios of 0.1, 0.5, 1 and 1.5, respectively at S/D = 0.1. All graphs indicate that the air plume from the cooled cylinder flows down and strikes to the horizontal plate and flows
Ra=105
Ra=106
L/D=0.1
L/D=0.5
L/D=1
L/D=1.5
Fig. 9. Streamlines around cylinder for various L/D and Rayleigh numbers for S/D = 0.5.
16
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Ra=10 4
Ra=10 5
Ra=10 6
L/D = 0.1
L/D = 0.5
L/D = 1
L/D =1.5
Fig. 10. Isothermal around cylinder for various L/D and Rayleigh numbers for S/D = 0.1.
opposite to the vertical wall. For low values of L/D (e.g. L/D = 0.1) cold air moves away about at an angle of 120° from the top of the cylinder due to damping effect of the floor. This angle increases by increasing L/D. Furthermore, cold plume is intensified by increasing L/D especially at higher Ra numbers. This figure also shows that by increasing Rayleigh number, the hydrodynamic boundary layer become thinner which is related to the increasing the buoyancy driven flow pattern. To investigate the effects of S/D on the flow field, aforementioned graphs are plotted for S/D = 0.5 in Fig. 9. This figure shows the pattern of different streamlines in comparison with previous case in Fig. 8. By increasing the value of S/D a rotating vortex is generated at the corner of the walls. This figure also shows that the size and the shape of these vortices change by increasing the value of L/D. By increasing Ra number, vortex size becomes wide and its velocity increases. Isothermal lines around the horizontal cylinder for various L/D and Ra = 104, 105, and 106 for S/D = 0.1 and S/D = 0.5 are indicated in Figs. 10 and 11 respectively. These figures represent that the cold plume flows downstream strikes to the floor and moves horizontally right from the cylinder over the floor. Fig. 10 shows the finite space between the cylinder and the floors has a great influence on the flow pattern. This figure also illustrates that by increasing L/D for each Rayleigh number and S/D, the thermal boundary layer becomes thinner. This results in a reduction of adiabatic floor influence and consequently increase the Nusselt number. Thinning the thermal boundary layer is also affected by increasing the Rayleigh number which enhances the buoyancy force in comparing to the viscous force as shown in Figs. 10 and 11. For low Rayleigh numbers, conduction heat transfer is dominant while at high Rayleigh numbers, the fluid flow adjacent to cylinder plays an important
role in convection heat transfer. These figures also show that for more closer of the cylinder to the vertical wall (Fig. 10) fluid motion reduces. Since the flow is free to move around the cylinder by increasing S/D, (Fig. 11) only the lower plate damps the cold plume. A more exhaustive investigation of heat transfer phenomena and a better comparison between the related factors of generating cold plume can be seen from the variation of local and average Nusselt number on the cylinder surface. The effects of horizontal and vertical adiabatic floor on local Nusselt number around the cylinder as a function of radial angle h at Ra = 104 are shown in Figs. 12 and 13, respectively. Fig. 12a shows that the horizontal adiabatic floor has a significant influence on the local Nusselt number and this effect is evident even at h = 0. This figure also shows that local Nusselt number has minimized at h ’ 240° approximately for all values of L/D in which the cylinder does not have enough space to convert heat to the surrounding air. The local Nusselt number is not symmetric, and even maximized at h ’ 30°. It also shows that the convection heat transfer on the wall side is less than on the free side. By increasing S/D as indicated in Fig. 12b, at very low L/D (L/D = 0.1) the heat transfer is almost negligible between h ’ 170° and the h ’ 190°. This range increases for lower S/D (as shown in Fig. 12a). It means that by reaching the adiabatic walls to the cylinder, the temperature gradient in radial direction (oT/or) converting to zero and so local Nusselt number and heat transfer can be disregarded due to damping effects of these walls at some angles (see Eqs. (10)–(13)). Fig. 12b shows that like the previous case the local Nusselt number maximized at h ’ 30°. Since the flow and heat transfer on the cylinder face side is almost identical, the variation of local Nusselt number become symmetric
M.H. Sedaghat et al. / Experimental Thermal and Fluid Science 62 (2015) 9–20
and the maximum of Nu at h ’ 30° disappeared by increasing L/D in Fig. 12b. Also minimum local Nusselt number shifted to the higher values of h. For example, it changed from 150° for L/D = 0.3 to 180° for L/D = 1.5. In other words, at large values of S/D, having enough space for heat transfer, only horizontal floor plays a significant role in the variation of local Nusselt number. Figs. 12b and c show that by increasing S/D from 0.5 to 1.5 the variation of local Nusselt numbers does not change considerably and this ratio does not have a great effect on heat transfer. Fig. 13 indicates the variation of local Nusselt number around the cylinder with h for Ra = 104 and for L/D = 0.1, 0.5 and 1.5 respectively at S/D = 0.1, 0.3, 0.5, 0.75, 1 and 1.5. Fig. 13a show that the variation of local Nusselt number is not symmetric, and heat transfer is almost negligible from h ’ 150° up to h ’ 210° and this range intensify as S/D decreases at low L/D values. This is because of the limited space between the cylinder and plate even for high S/D (e.g. S/D = 1.5). As mentioned before, this phenomena is also happened for very low L/D values (Fig. 11a). Fig. 13a also show that the local Nusselt number is maximized at h ’ 30°. Figs. 13b and c show the variation of a local Nusselt number of higher L/D values. These figures show that for L/D P 0.5 increasing S/D, does not affect the local Nusselt number greatly. For S/D 6 0.3 a variation of Nu is still non-symmetric and its variation is far from the cylinder in free space. Such a variation is even observed for L/D = 1.5 at S/D = 0.1. In other words, changing the position of vertical plane does not have great influence on the local Nusselt number of large space between the vertical plate and cylinder. Fig. 14 illustrates the variation of the average Nusselt number ratio (Nuave/Nu1) as functions of L/D and S/D at Ra = 104. This figure indicates that, the vertical displacement from the horizontal floor to cylinder (L/D) play a vital role for average Nusselt number ratio
Ra=104
17
and as L/D increases the Nusselt number ratio increases sharply. At a fixed value of L/D the Nusselt number ratio almost remains constant as S/D increases (for S/D P 0.3). To study the effect of the Rayleigh number on heat transfer rate, Figs. 15a and b show the variation of the Nusselt number ratio with L/D and Rayleigh numbers at S/D = 0. 1 and S/D = 0.5, respectively. Fig. 15a shows that L/D has a great effect on the Nusselt number ratio and this effect reduces by increasing the Rayleigh numbers. For example this figure shows that Nuave/Nu1 is about 0.55 for Ra = 105 at L/D = 0.1. This ratio reaches to 0.98 for L/D = 1.5 but the value of Nuave/Nu1 for Ra = 106 is about 0.8 at L/D = 0.1 and 1 for L/D = 1.5. Fig. 15b shows the similar effects of heat transfer rate. Due to higher space for heat transfer Nuave/Nu1 increases in the specific value of L/D and Rayleigh number. Fig. 15 also show that at a specific L/D, as Rayleigh number increases, the thermal boundary layer around the cylinder become thinner and dependency on S/D and L/D on heat transfer decreases. In other words the that Rayleigh number should be regarded as one of the significant factors which influence the ratio of Nuave/Nu1 on the cylinder when located at the corner of two adiabatic walls. 5. Uncertainty Analysis of uncertainty is required in order to evaluate the accuracy of measurements which is similar to the one described in the work of Batista et al. [22]. The total uncertainty U can be determined from uncertainties of the specified components which influence the experiment [23]. For a measurement M, whose results depend on uncorrelated input estimates, the standard uncertainty of the measurement is obtained by appropriately combining the standard uncertainties of these input estimates. The combined
Ra=105
Ra=106
L/D = 0.1
L/D = 0.5
L/D = 1
L/D = 1.5
Fig. 11. Isothermal around cylinder for various L/D and Rayleigh numbers for S/D = 0.5.
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M.H. Sedaghat et al. / Experimental Thermal and Fluid Science 62 (2015) 9–20
(a)
(a)
S/D=0.1
L/D=0.1
(b)
(b)
L/D=0.5
S/D=0.5
(c)
(c)
L/D=1.5
S/D=1.5
Fig. 12. Variation of local Nusselt number around cylinder for Ra = 104 and for various S/D.
Fig. 13. Variation of local Nusselt number around cylinder for Ra = 104 and for various L/D.
standard uncertainty of the estimate M denoted by U is calculated from the following equations [24,25].
M ¼ f ðx1 ; x2 ; . . . xn Þ U 2 ðMÞ ¼
2 n X @f i¼1
@xi
U 2 ðxi Þ
ð17Þ ð18Þ
where f is the function of M in terms of input, estimates x1, x2, ... , xn, and each U(xi) is a standard uncertainty of any input. The expression for the standard uncertainty associated with have is developed from Eqs. (1) and (2), using the law of propagation of uncertainty. The resulting expression is:
" 2 2 2 @hav e @hav e @hav e dT 1 þ dT out þ dT in @T 1 @T out @T in 2 2 2 #12 @hav e @hav e @hav e dD þ dr out þ drin þ @D @r out @r in
dhav e ¼
ð19Þ
Fig. 14. Variation of average Nusselt number ratio with L/D and S/D for Ra = 104.
M.H. Sedaghat et al. / Experimental Thermal and Fluid Science 62 (2015) 9–20
19
6. Conclusion Natural convection heat transfer from a cold horizontal cylinder which is enclosed in a corner of two flat adiabatic walls is studied both numerically and experimentally. Results indicate that:
(a) S/D=0.1
(b) S/D=0.5 Fig. 15. Variation of average Nusselt number ratio with L/D and Rayleigh numbers for various S/D.
References
Table 5 Values for the parameters of Eq. (20). Parameter
xi
dxi
@hav e @xi
T1 (°C) Tout (°C) Tin (°C) D (mm) rout (mm) rin (mm)
31.5 13 9.96 80 25.9 22
0.2 0.1 0.1 0.02 0.02 0.02
0.0463 0.1408 0.1408 0.0011 0.0202 0.0238
so
dhav e ¼
1 ðT sav e T 1 Þ2
"
1 2kins ðT out T in Þ dT 1 ðT sav e T 1 Þ DLnðrout =r in Þ
dxi
2
2 2 2kins 2kins dT out þ dT in þ DLnðr out =r in Þ DLnðr out =r in Þ þ
þ
2kins ðT out T in Þ D2 Lnðr out =r in Þ 2kins ðT out T in Þ rin Dðr out =r in Þ2
1. Due to the damping effect of horizontal plate, streamlines and isothermal lines around the horizontal cylinder at low L/D (L/D = 0.1) separated and moved away about at an angle of 120° from the surface of the cylinder and this angle increases as L/D increased. 2. Owing to the enhancement of buoyancy force in comparison to the viscous force, as Rayleigh number increases, the hydrodynamic and thermal boundary layers become thinner, but fluid motion highly depends on the cylinder space from adiabatic walls. The higher is the value of Ra, the less is the effects of L/D and S/D on heat transfer from the horizontal cylinder. 3. In contrast to the vertical adiabatic floor, the horizontal adiabatic floor has a more significant influence on the local Nusselt number. This effect is evident even at the top of the cylinder (h = 0). 4. Due to the limitation of space between cylinder and plates, the variation of local Nusselt number is not symmetric, and heat transfer is almost negligible from h ’ 150° up to h ’ 210° at low L/D or S/D. Also the local Nusselt number has a maximum value at h ’ 30°. 5. The value of Rayleigh number and the ratio of vertical cylinder spacing of adiabatic floor (L/D) are regarded as two important factors which greatly influence the heat transfer from horizontal cylinder. In contrary the horizontal cylinder spacing of adiabatic floor (S/D) has less effect on the heat transfer rate. 6. More closer is the cylinder to the corner, more reduction in convection heat transfer takes place from the cylinder.
!2 dD
þ
!2 312 dr in 5
2kins ðT out T in Þ
!2
dr out r out DðLnðr out =rin ÞÞ2
ð20Þ
Lists of the experimental values of different parameters and associated uncertainties for Ra = 6 105 and L/D = 1 are prescribed in Table 5. The maximum uncertainty of the measurements is determined according to Table 5 and it is about 20%.
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