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Experimental investigation of natural convection heat transfer in narrow vertical rectangular channel heated from both sides
Experimental Thermal and Fluid Science 36 (2012) 72–77
Contents lists available at SciVerse ScienceDirect
Experimental Thermal and Fluid Science journal homepage: www.elsevier.com/locate/etfs
Experimental investigation of natural convection heat transfer in narrow vertical rectangular channel heated from both sides Salah El-Din El-Morshedy ⇑, Adel Alyan 1, Loula Shouman 2 Reactors Department, Nuclear Research Center, Atomic Energy Authority, 13759 Cairo, Egypt
a r t i c l e
i n f o
Article history: Received 1 August 2011 Received in revised form 23 August 2011 Accepted 23 August 2011 Available online 29 September 2011 Keywords: Natural convection Rectangular channels Thermal-hydraulics MTR reactors
a b s t r a c t The heat transfer characteristics of the natural convection regime through a vertical rectangular channel simulating a cooling channel of typical material testing reactor have been experimentally investigated. Experiments are performed on demineralized water as coolant passing under atmospheric pressure through narrow rectangular channel of 80 cm length, 7 cm width and 2.7 mm gap thickness under different heat fluxes ranging from 2.7 kW/m2 to 26.8 kW/m2 that covers almost all possible heat fluxes in the single-phase liquid. The measured local Nusselt number values are compared with the predictions of two correlations for both natural and combined natural and forced convection regimes respectively. The comparison shows the need for developing new correlation to represent the obtained experimental results and to be utilized for the prediction of the local Nusselt number in thermal–hydraulics and safety analysis of the reactor. The developed correlation fits the experimental data within 5.8% relative standard deviation. Ó 2011 Elsevier Inc. All rights reserved.
1. Introduction Natural convective flows through vertical rectangular channels are encountered in many engineering applications such as cooling of electronic packages, solar collectors, building heat transfer and cooling of Material Testing Reactors (MTRs). Therefore, a lot of both experimental and numerical works on this phenomenon were found in the literature. Many of the previous works were on air as a coolant fluid [1–5]. However, less works were conducted on water as a coolant. Sudo et al. [6] carried out experimental investigation of the forced convection, free convection and combined convection on water flows through vertical rectangular channels of gap sizes of 18, 6 and 2.5 mm, they defined the conditions when forced convective heat transfer is dominant and when free convection heat transfer is dominant as well as the region of combined convection. Many experiments and analyses have been so far reported on the combined convective heat transfer characteristics [7–9]. In thermal–hydraulics and safety of research nuclear reactors, the reactor could operated at low power with steady-state natural circulation mode. Besides, some abnormal operational transients and accidents are assumed as design basis events for safety evaluation. During these events, there is a case that the core
⇑ Corresponding author. Tel.: +20 1016576250. E-mail addresses: [email protected] (S.E. El-Morshedy), yahoo.com (A. Alyan), [email protected] (L. Shouman). 1 Tel.: +20 100 7530454. 2 Tel.: +20 114 1810929.
adelalyan@
0894-1777/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.expthermflusci.2011.08.006
flow decreases from a steady-state forced convective flow to natural convective flow induced between the core and the reactor pool in which the core is submerged. After loss of pumping power, the natural convective heat transfer becomes significant and important. This experimental study is therefore, carried out in order to understand the natural convective heat transfer characteristics in a narrow vertical rectangular channel heated from both sides simulates a coolant channel of a typical material testing reactor [10] and to obtain an empirical correlation for the prediction of the local heat transfer coefficient along its coolant channels under natural convective regime occurs due to the local temperature differences between the heated wall surface temperature and the bulk temperature of water flowing in the channel along the vertical direction in order to utilize the obtained correlation in thermal–hydraulic design and the safety analysis of the reactor core in which flatplate-type fuel is adopted. 2. Experimental setup and procedure The experimental setup is a closed loop for carrying out the heat transfer investigations at nearly atmospheric pressure, a schematic layout of the experimental setup is shown in Fig. 1. It was designed and constructed to investigate natural convection heat transfer of water in vertical rectangular channel. The loop consists of the rectangular test section of laterally heated stainless steel plates of 2 mm in width, and inlet and outlet plenums connected to a water storage tank through recirculation pipes. The water storage tank has a cooling copper coil in it to keep the water inlet temperature
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Nomenclature z
specific heat at constant pressure, J/kg °C equivalent hydraulic diameter, m acceleration of gravity, m/s2 mass flux, kg/m2 s 3 Grashof number ¼ gbðT w T b Þde =m2w 2 heat transfer coefficient, W/m °C current, ampere thermal conductivity, W/m °C channel length, m Nusselt number ¼ hde =k Prandtl number ¼ lCp=k Reynolds number ¼ Gde =l channel gap thickness, m temperature, °C coolant velocity, m/s voltage, volt channel width, m
Greek symbols b coefficient of thermal expansion, °C1 / surface heat flux, W/m2 l dynamic viscosity, kg/ms m kinematic viscosity, m2/s q density, kg/m3 Subscripts b bulk conditions, at bulk temperature c combined natural and forced convection f forced convection in inlet n natural convection w wall conditions, at wall temperature
constant during each run. The tank is open to the atmosphere and a make-up line with a control floating ball valve is fixed on it for automatic compensation of evaporated water to keep constant the water level inside the tank and so achieving equal hot and cold legs during experiments. The test section configuration is rectangular with 70 mm in width, 2.7 mm in gap and 800 mm in length simulating a coolant channel of a typical MTR. The test section plates are insulated from the environment by insulating layer of 5 mm thickness. A detailed cross section of the test section is shown in Fig. 2. Demineralized water was circulated through the test section by natural circulation depending on the density difference between the hot water in test section and cold water in the storage tank. The power was supplied to the test section by DC generator. The ends of the heating plates are tightly connected with two copper rods penetrating the insulation layer and are welded by copper to assure good electric contact. The two copper rods are 30 mm in diameter, and 300 mm in length. These copper rods serve as electric connectors; the large diameter of these connectors decreases their electric resistance, resulting in reducing heat generations in this part. The wall surface temperatures along the heating plates were measured by ten calibrated thermocouples (K-type) inserted into the blind holes drilled in each plate to measure the inner surface temperature where the average value of each two corresponding point on the two plates is considered for local cross section wall temperature. Also, the coolant inlet and outlet temperatures are measured by
storage tank
upper plenum
distance in axial direction, m
make-up
flow channel
stainless steel plates spacer
insulator 2.7mm
Cp de g G Grd h I k L Nu Pr Re S T u V W
70 mm Fig. 2. Test section details.
two thermocouples inserted in the lower and upper plenums respectively. Thermocouple end leads are connected to a temperature recorder with a resolution of 0.1 °C. Thermocouples are calibrated by direct comparison with a standard platinum resistance thermometer where the thermocouple junction and the thermometer are immersed in a calorimeter. Specified power is supplied to the heating plates and the loop is allowed to operate until steady state condition was ensured when the variation of the temperature reading was within ±0.1 °C. The inlet water temperature is kept constant by regulating the flow rate of the cooling copper coil in the water storage tank. Once steady state was reached, the readings of electric current, voltage, wall temperatures as well as the inlet and outlet temperatures were recorded. Bulk water temperatures along the channel were calculated by linear interpolation between the inlet and outlet temperatures.
lower plenum Fig. 1. Schematic layout of the experimental setup.
cooling coil
temperature recorder
test section
3. Experimental results and discussion Experimental runs are carried out for a wide range of heat fluxes covering the almost all possible heat fluxes for single-phase liquid regime starting from very low value (2679 W/m2) up to the value (26,786 W/m2) that leads the maximum wall surface temperature just below the saturation temperature. Typical measured temperature profiles of plate surface temperature and bulk water temperature are depicted in Fig. 3 for twelve different runs. With very low heat flux of run (1), the water velocity through the channel is 0.25 cm/s and this value increased by increasing the heat flux to reach 0.94 cm/s at heat flux of run (12), it means that, all experiments are performed under laminar flow regime. It is also noticed that there is an inlet transition region of about 20 cm where the slope of the wall-surface temperature profile is lower than that
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S.E. El-Morshedy et al. / Experimental Thermal and Fluid Science 36 (2012) 72–77
Wall temperature
Length, (cm)
Bulk Temperature 80
80
80
70
70
70
60
60
60
50
50
40 30
Run (1) φ = 2679
30 20
20
10
10
0
0
Length, (cm)
0 30 40 50 60 70 80 90 100
30 40 50 60 70 80 90 100
80
80
80
70
70
70
60
60
60
50
50
50
40
40
Run (4) φ = 9821
20
40 Run (5) φ = 11607
30 20
10
10
0
0 30 40 50 60 70 80 90 100
Length, (cm)
30
10
30
20 10 0
30 40 50 60 70 80 90 100
30 40 50 60 70 80 90 100
80
80
70
70
70
60
60
60
50
50
50
40
40 Run (7)
φ = 15625
20
10 0 30 40 50 60 70 80 90 100
70 60
30
φ = 17857
20
0 80
40 Run (8)
30
10
10 0 30 40 50 60 70 80 90 100
70 60
Run (9) φ = 20089
20
80 Run (10) φ = 22321
Run (6) φ = 13393
30
80
30
Run (3) φ = 8036
40
20
30 40 50 60 70 80 90 100
Length, (cm)
50 Run (2) φ = 5804
40
30 40 50 60 70 80 90 100 80
Run (11)
70
φ = 24554
60
50
50
50
40
40
40
30
30
30
20
20
20
10
10
10
0
0
Run (12) φ = 26786
0
30 40 50 60 70 80 90 100
30 40 50 60 70 80 90 100
30 40 50 60 70 80 90 100
Temperature (oC)
Temperature (oC)
Temperature (oC)
Fig. 3. Typical temperature profile along the test section.
for the rest of the channel where the flow becomes fully developed and this phenomenon appears clearer at higher heat fluxes. Typical experimental results of local Nusselt numbers are shown in Fig. 4 where the experimental Nusselt number values are given by:
Nue ¼
/de kðT w T b Þ
ð1Þ
where Tw and Tb are the local measured wall-surface and water bulk temperatures respectively and k is the water thermal conductivity calculated at local water bulk temperature. Higher local Nusselt number values are obtained at the channel inlet where the differences between wall-surface and bulk temperatures are lower and gradually decreases in the first 20 cm region of the channel inlet, then the Nusselt number values remains approximately unchanged in the rest of the channel, this is mainly due to the approximately constant temperature differences between wall-surface and bulk temperatures in this region as shown in Fig. 3. This result implies that, the distance z from the channel inlet is a key parameter for natural convection heat transfer characteristics in the channel inlet region.
The forced convective Nusselt number is also plotted in Fig. 4 to show how the obtained Nusselt number values are much higher than the corresponding forced convective values and so the natural convection heat transfer mode is much more dominant. The forced convective Nusselt number is given by Sieder and Tate correlation [11] for internal flow in tubes and is valid for forced laminar regime, Re 6 2100:
1 0:14 Re Pr 3 lb Nuf ¼ 1:86 L=de lw
ð2Þ
with Re and Pr are calculated at local water bulk temperature. Fig. 4 shows also plots of the Nusselt number along the channel for natural convection heat transfer mode calculated by using the Churchill correlation [12]:
Nun ¼
0:15ðGrd Pr w Þ1=3 ð1 þ ð0:437=Pr w Þ9=16 Þ16=27 3
ð3Þ
with Grd ¼ gbðT w T b Þde =m2w and Pr is calculated at local plate surface temperature.
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80
80
Length, (cm)
70
60
60
50
Nun
50
50
40
40
30
30
20
20
10
10
10
0
0
Nuc
40
Nup
30
Nue
20
2
4
6
8 10 12 14
Run (4)
2
4
6
8 10 12 14
0
Run (5)
60
50
50
50
40
40
40
30
30
30
20
20
20
10
10
10
0
0 4
6
8 10 12 14
80 Run (7)
2
4
6
8 10 12 14
Run (8)
50
50
50
40
40
40
30
30
30
20
20
20
10
10
10
0
0 6
8 10 12 14 80
70
70
Run (10)
60
2
4
6
8 10 12 14
Run (11)
60
50 40
30
30
30
20
20
20
10
10
10
0
0 8 10 12 14
6
8 10 12 14
Run (12)
60
40
6
4
70
50
4
2
80
40
2
8 10 12 14
Run (9)
0
50
0
6
0 0
80
4
70 60
4
2
80
70 60
2
8 10 12 14
Run (6)
0
60
0
6
0 0
80
70
4
70
60
2
2
80
70
60
0
Run (3)
0 0
80
70
Length, (cm)
70
Nuf
80
Length, (cm)
Run (2)
60
0
Length, (cm)
80
70
Run (1)
0 0
2
Nu
4
6
8 10 12 14
0
2
4
Nu
6
8 10 12 14
Nu
Fig. 4. Nusselt number variation along the test section.
It is noticed a large deviation from the experimental Nusselt number at the channel inlet and this deviation decreases sharply in the first 20 cm region of the test section and then the predicted natural convection Nusselt number is gradually close to the experimental data to obtain a very good agreement at the channel exit. The Nusselt number values for combined natural and forced convection heat transfer mode under the present experimental condition are depicted in Fig. 4 as well, it is calculated by Collier correlation [13]:
Nuc ¼ 0:17Re0:33 Pr0:43 f f
Pr f Prw
#0:1 0:25 " 3 gbde ðT w T b Þ
m
2 f
ð4Þ
where subscripts f and w indicate that physical properties are evaluated at film temperature and wall-surface temperature respectively. This relation is valid for heating in vertical upflow or cooling in vertical downflow for L/de > 50 and Re < 2000. It shows a much better agreement than Eq. (3) but still large deviation at the channel inlet. Therefore, there is a need for a new correlation to fit the experimental data and takes into account the axial distance from the channel inlet. The proposed correlation
to predict the local Nusselt number will taken the following dimensionless form: k4 k3 Nup ¼ k1 Grk2 d Pr ðde =zÞ
ð5Þ
By taking the logarithmic transformation of Eq. (5) and applying the least squares method, the constants k1, k2, k3 and k4 are evaluated as 1, 0.284, 0.12 and 0.25, respectively. So the developed correlation for prediction of local Nusselt number takes the following form:
Nup ¼ Gr0:284 Pr0:12 ðde =zÞ0:25 d
ð6Þ
with all water physical properties calculated at the local bulk temperature. The Nusselt number is evaluated by the present correlation and the other correlations for natural convection mode (Eq. (3)) and combined natural and forced convection mode (Eq. (4)) under the present experimental conditions. All the obtained results and experimental data are plotted in Fig. 5. The solid line is a reference with the slope of one is drawn on the plot to give the relation between the predicted and measured data. The present correlation shows a good agreement with the experimental data, it gives only
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S.E. El-Morshedy et al. / Experimental Thermal and Fluid Science 36 (2012) 72–77
for both correlations respectively. A new correlation of the form Nu ¼ Gr 0:284 Pr0:12 ðde =zÞ0:25 was developed to be utilized in the d thermal–hydraulics and safety analysis of the reactor. The developed correlation predicted the local Nusselt number distribution along the test section with 5.8% relative standard deviation from the experimental results.
14 Present correlation Churchill correlation Collier correlation
Predicted Nusselt number
12
10
Appendix A. Experimental uncertainty 8
The experimental measurements cannot be measured exactly. It can be only known with a certain range of uncertainty. The accuracy of experimental results depends on the accuracy of the individual measuring instruments. There are several methods for calculating the uncertainty of experimental parameters. Based on the square root method, the total uncertainty U comprises of uncertainties of many experimental parameters x1, x2, . . . , xn, the combined standard uncertainty of the Nu is calculated from the following equation:
6
4
2
UðNue Þ ¼
( n X @Nu i¼1
0 0
2
4
6
8
10
12
14
Measured Nusselt number
UðNuÞ ¼
Model
Standard deviation
Present correlation Churchill correlation Collier correlation
0.058 0.327 0.155
2 )1=2 ðA1Þ
Eq. (A1) shows that, Nu is function of /, de, Tw and Tb. Therefore, the uncertainty in Nu is determined as:
Fig. 5. Comparison of the correlation model with previous correlations.
Table 1 Relative standard deviation from experimental data for Nusselt number.
@xi
Uxi
( 2 2 2 2 )1=2 @Nu @Nu @Nu @Nu U/ þ Ude þ UT w þ UT b @/ @de @T w @T b ðA2Þ
The heat flux, / is a function of voltage, current and test section length and width as:
/¼
V I 2LW
ðA3Þ
So the uncertainty in heat flux, U/ is determined by:
Uð/Þ ¼
Table A1 Uncertainty values for the measured parameters. Parameter
Value
Uncertainty
V I L W S Tw Tb
1–3.75 V 300–800 A 80 cm 7 cm 2.7 mm 42–96 °C 32–75 °C
±0.1 V ±5 A ±0.1 mm ±0.1 mm ±0.1 mm ±0.1 ±0.1 °C
5.8% relative standard deviation from the experimental data while the others gives 32.7% and 15.5% for Churchill and Collier correlations, respectively as shown Table 1.
(
@/ UV @V
2
þ
@/ UI @I
2
þ
2 2 )1=2 @/ @/ UL þ UW @L @W ðA4Þ
The equivalent hydraulic diameter, de is calculated as:
de ¼
2W S W þS
ðA5Þ
where S is the channel clearance. So the uncertainty in the equivalent hydraulic diameter, de is determined by:
Uðde Þ ¼
(
@de UW @W
2 þ
2 )1=2 @de US @S
ðA6Þ
4. Conclusions The natural convection heat transfer of water in narrow rectangular vertical channel was studied experimentally. The water channel is represented by two adjacent stainless steel heating plates with a dimension of 800 mm active length, 70 mm in width and 2.7 mm in gap thickness simulating a coolant channel of a typical material testing reactor under atmospheric pressure. Experiments were carried out under heat fluxes ranged from 2.7 kW/m2 to 26.8 kW/m2 to cover all possible heat fluxes in the single-phase liquid regime. Wall surface temperature, coolant bulk temperature and local Nusselt number profiles were experimentally investigated. The measured Nusselt number values are compared with the predicted values of both Churchill correlation for natural convection mode and Collier correlation for combined natural and forced convection with 32.7% and 15.5% relative standard deviation
Table A1 lists the values and uncertainties in the measured parameters. Thus, the uncertainty of the average Nu correlated in Eq. (1) can be deduced to be ranges from 10.7% for 2679 W/m2 heat flux to 3.1% for 26,786 W/m2. References [1] W. Elenbass, Heat dissipation of parallel plates by free convection, Physica 9 (1942) 1–28. [2] W. Elenbass, The dissipation of heat by free convection: the inner surface of vertical tubes of different shapes of cross-section, Physica 9 (1942) 865–874. [3] A. Bar-Cohen, W.M. Rohsenow, Thermally optimum spacing of vertical, natural convection cooled, parallel plates, ASME, Journal of Heat Transfer 106 (February) (1984) 116–123. [4] K.T. Lee, W.M. Yan, Laminar natural convection between partially heated vertical parallel plates, Warme- und Stoffubertragung 29 (1994) 145–151.
S.E. El-Morshedy et al. / Experimental Thermal and Fluid Science 36 (2012) 72–77 [5] Qing Lu, Suizheng Qiu, Guanghui Su, Wenxi Tian, Zhonghao Ye, Experimental research on heat transfer of natural convection in vertical rectangular channels with large aspect ratio, Experimental Thermal and Fluid Science 34 (2010) 73– 80. [6] Yukio Sudo, Tohru Usui, Masanori Kaminaga, Heat transfer characteristics in narrow vertical rectangular channels heated from both sides, JASME International Journal, Series II 33 (4) (1990) 743–748. [7] Yukio Sudo, Masanori Kaminaga, Hiromasa Ikawa, Combined forced and free convection heat transfer characteristics in narrow vertical rectangular channels heated from both sides, Journal of Nuclear Science and Technology 24 (5) (1987) 355–364. [8] Donald D. Joye, Comparison of aiding and opposing mixed convection heat transfer in a vertical tube with Grashof number variation, International Journal of Heat and Fluid Flow 17 (1996) 96–101.
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[9] D.D. Joye, M.J. Wojnovich, Aiding and opposing mixed-convection heat transfer in a vertical tube: loss of boundary condition at different Grashof numbers, International Journal of Heat and Fluid Flow 17 (1996) 468–473. [10] Salah El-Din El-Morshedy, Prediction, analysis and solution of the flow inversion phenomenon in a typical MTR-reactor with upward core cooling, Nuclear Engineering and Design 241 (1) (2011) 226–235. [11] E.N. Sieder, G.E. Tate, Heat transfer and pressure drop of liquids in tubes, Industrial and Engineering Chemistry 28 (1936) 1429. [12] S.W. Churchill, Combined free and forced convection in channels, in: G.F. Hewitt (Ed.), Handbook of Heat Exchanger Design, Begell-House, 1992. [13] J.G. Collier, Convective Boiling and Condensation, second ed., Mc Graw-Hill International Book Company, 1981.
Contents lists available at SciVerse ScienceDirect
Experimental Thermal and Fluid Science journal homepage: www.elsevier.com/locate/etfs
Experimental investigation of natural convection heat transfer in narrow vertical rectangular channel heated from both sides Salah El-Din El-Morshedy ⇑, Adel Alyan 1, Loula Shouman 2 Reactors Department, Nuclear Research Center, Atomic Energy Authority, 13759 Cairo, Egypt
a r t i c l e
i n f o
Article history: Received 1 August 2011 Received in revised form 23 August 2011 Accepted 23 August 2011 Available online 29 September 2011 Keywords: Natural convection Rectangular channels Thermal-hydraulics MTR reactors
a b s t r a c t The heat transfer characteristics of the natural convection regime through a vertical rectangular channel simulating a cooling channel of typical material testing reactor have been experimentally investigated. Experiments are performed on demineralized water as coolant passing under atmospheric pressure through narrow rectangular channel of 80 cm length, 7 cm width and 2.7 mm gap thickness under different heat fluxes ranging from 2.7 kW/m2 to 26.8 kW/m2 that covers almost all possible heat fluxes in the single-phase liquid. The measured local Nusselt number values are compared with the predictions of two correlations for both natural and combined natural and forced convection regimes respectively. The comparison shows the need for developing new correlation to represent the obtained experimental results and to be utilized for the prediction of the local Nusselt number in thermal–hydraulics and safety analysis of the reactor. The developed correlation fits the experimental data within 5.8% relative standard deviation. Ó 2011 Elsevier Inc. All rights reserved.
1. Introduction Natural convective flows through vertical rectangular channels are encountered in many engineering applications such as cooling of electronic packages, solar collectors, building heat transfer and cooling of Material Testing Reactors (MTRs). Therefore, a lot of both experimental and numerical works on this phenomenon were found in the literature. Many of the previous works were on air as a coolant fluid [1–5]. However, less works were conducted on water as a coolant. Sudo et al. [6] carried out experimental investigation of the forced convection, free convection and combined convection on water flows through vertical rectangular channels of gap sizes of 18, 6 and 2.5 mm, they defined the conditions when forced convective heat transfer is dominant and when free convection heat transfer is dominant as well as the region of combined convection. Many experiments and analyses have been so far reported on the combined convective heat transfer characteristics [7–9]. In thermal–hydraulics and safety of research nuclear reactors, the reactor could operated at low power with steady-state natural circulation mode. Besides, some abnormal operational transients and accidents are assumed as design basis events for safety evaluation. During these events, there is a case that the core
⇑ Corresponding author. Tel.: +20 1016576250. E-mail addresses: [email protected] (S.E. El-Morshedy), yahoo.com (A. Alyan), [email protected] (L. Shouman). 1 Tel.: +20 100 7530454. 2 Tel.: +20 114 1810929.
adelalyan@
0894-1777/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.expthermflusci.2011.08.006
flow decreases from a steady-state forced convective flow to natural convective flow induced between the core and the reactor pool in which the core is submerged. After loss of pumping power, the natural convective heat transfer becomes significant and important. This experimental study is therefore, carried out in order to understand the natural convective heat transfer characteristics in a narrow vertical rectangular channel heated from both sides simulates a coolant channel of a typical material testing reactor [10] and to obtain an empirical correlation for the prediction of the local heat transfer coefficient along its coolant channels under natural convective regime occurs due to the local temperature differences between the heated wall surface temperature and the bulk temperature of water flowing in the channel along the vertical direction in order to utilize the obtained correlation in thermal–hydraulic design and the safety analysis of the reactor core in which flatplate-type fuel is adopted. 2. Experimental setup and procedure The experimental setup is a closed loop for carrying out the heat transfer investigations at nearly atmospheric pressure, a schematic layout of the experimental setup is shown in Fig. 1. It was designed and constructed to investigate natural convection heat transfer of water in vertical rectangular channel. The loop consists of the rectangular test section of laterally heated stainless steel plates of 2 mm in width, and inlet and outlet plenums connected to a water storage tank through recirculation pipes. The water storage tank has a cooling copper coil in it to keep the water inlet temperature
73
S.E. El-Morshedy et al. / Experimental Thermal and Fluid Science 36 (2012) 72–77
Nomenclature z
specific heat at constant pressure, J/kg °C equivalent hydraulic diameter, m acceleration of gravity, m/s2 mass flux, kg/m2 s 3 Grashof number ¼ gbðT w T b Þde =m2w 2 heat transfer coefficient, W/m °C current, ampere thermal conductivity, W/m °C channel length, m Nusselt number ¼ hde =k Prandtl number ¼ lCp=k Reynolds number ¼ Gde =l channel gap thickness, m temperature, °C coolant velocity, m/s voltage, volt channel width, m
Greek symbols b coefficient of thermal expansion, °C1 / surface heat flux, W/m2 l dynamic viscosity, kg/ms m kinematic viscosity, m2/s q density, kg/m3 Subscripts b bulk conditions, at bulk temperature c combined natural and forced convection f forced convection in inlet n natural convection w wall conditions, at wall temperature
constant during each run. The tank is open to the atmosphere and a make-up line with a control floating ball valve is fixed on it for automatic compensation of evaporated water to keep constant the water level inside the tank and so achieving equal hot and cold legs during experiments. The test section configuration is rectangular with 70 mm in width, 2.7 mm in gap and 800 mm in length simulating a coolant channel of a typical MTR. The test section plates are insulated from the environment by insulating layer of 5 mm thickness. A detailed cross section of the test section is shown in Fig. 2. Demineralized water was circulated through the test section by natural circulation depending on the density difference between the hot water in test section and cold water in the storage tank. The power was supplied to the test section by DC generator. The ends of the heating plates are tightly connected with two copper rods penetrating the insulation layer and are welded by copper to assure good electric contact. The two copper rods are 30 mm in diameter, and 300 mm in length. These copper rods serve as electric connectors; the large diameter of these connectors decreases their electric resistance, resulting in reducing heat generations in this part. The wall surface temperatures along the heating plates were measured by ten calibrated thermocouples (K-type) inserted into the blind holes drilled in each plate to measure the inner surface temperature where the average value of each two corresponding point on the two plates is considered for local cross section wall temperature. Also, the coolant inlet and outlet temperatures are measured by
storage tank
upper plenum
distance in axial direction, m
make-up
flow channel
stainless steel plates spacer
insulator 2.7mm
Cp de g G Grd h I k L Nu Pr Re S T u V W
70 mm Fig. 2. Test section details.
two thermocouples inserted in the lower and upper plenums respectively. Thermocouple end leads are connected to a temperature recorder with a resolution of 0.1 °C. Thermocouples are calibrated by direct comparison with a standard platinum resistance thermometer where the thermocouple junction and the thermometer are immersed in a calorimeter. Specified power is supplied to the heating plates and the loop is allowed to operate until steady state condition was ensured when the variation of the temperature reading was within ±0.1 °C. The inlet water temperature is kept constant by regulating the flow rate of the cooling copper coil in the water storage tank. Once steady state was reached, the readings of electric current, voltage, wall temperatures as well as the inlet and outlet temperatures were recorded. Bulk water temperatures along the channel were calculated by linear interpolation between the inlet and outlet temperatures.
lower plenum Fig. 1. Schematic layout of the experimental setup.
cooling coil
temperature recorder
test section
3. Experimental results and discussion Experimental runs are carried out for a wide range of heat fluxes covering the almost all possible heat fluxes for single-phase liquid regime starting from very low value (2679 W/m2) up to the value (26,786 W/m2) that leads the maximum wall surface temperature just below the saturation temperature. Typical measured temperature profiles of plate surface temperature and bulk water temperature are depicted in Fig. 3 for twelve different runs. With very low heat flux of run (1), the water velocity through the channel is 0.25 cm/s and this value increased by increasing the heat flux to reach 0.94 cm/s at heat flux of run (12), it means that, all experiments are performed under laminar flow regime. It is also noticed that there is an inlet transition region of about 20 cm where the slope of the wall-surface temperature profile is lower than that
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Wall temperature
Length, (cm)
Bulk Temperature 80
80
80
70
70
70
60
60
60
50
50
40 30
Run (1) φ = 2679
30 20
20
10
10
0
0
Length, (cm)
0 30 40 50 60 70 80 90 100
30 40 50 60 70 80 90 100
80
80
80
70
70
70
60
60
60
50
50
50
40
40
Run (4) φ = 9821
20
40 Run (5) φ = 11607
30 20
10
10
0
0 30 40 50 60 70 80 90 100
Length, (cm)
30
10
30
20 10 0
30 40 50 60 70 80 90 100
30 40 50 60 70 80 90 100
80
80
70
70
70
60
60
60
50
50
50
40
40 Run (7)
φ = 15625
20
10 0 30 40 50 60 70 80 90 100
70 60
30
φ = 17857
20
0 80
40 Run (8)
30
10
10 0 30 40 50 60 70 80 90 100
70 60
Run (9) φ = 20089
20
80 Run (10) φ = 22321
Run (6) φ = 13393
30
80
30
Run (3) φ = 8036
40
20
30 40 50 60 70 80 90 100
Length, (cm)
50 Run (2) φ = 5804
40
30 40 50 60 70 80 90 100 80
Run (11)
70
φ = 24554
60
50
50
50
40
40
40
30
30
30
20
20
20
10
10
10
0
0
Run (12) φ = 26786
0
30 40 50 60 70 80 90 100
30 40 50 60 70 80 90 100
30 40 50 60 70 80 90 100
Temperature (oC)
Temperature (oC)
Temperature (oC)
Fig. 3. Typical temperature profile along the test section.
for the rest of the channel where the flow becomes fully developed and this phenomenon appears clearer at higher heat fluxes. Typical experimental results of local Nusselt numbers are shown in Fig. 4 where the experimental Nusselt number values are given by:
Nue ¼
/de kðT w T b Þ
ð1Þ
where Tw and Tb are the local measured wall-surface and water bulk temperatures respectively and k is the water thermal conductivity calculated at local water bulk temperature. Higher local Nusselt number values are obtained at the channel inlet where the differences between wall-surface and bulk temperatures are lower and gradually decreases in the first 20 cm region of the channel inlet, then the Nusselt number values remains approximately unchanged in the rest of the channel, this is mainly due to the approximately constant temperature differences between wall-surface and bulk temperatures in this region as shown in Fig. 3. This result implies that, the distance z from the channel inlet is a key parameter for natural convection heat transfer characteristics in the channel inlet region.
The forced convective Nusselt number is also plotted in Fig. 4 to show how the obtained Nusselt number values are much higher than the corresponding forced convective values and so the natural convection heat transfer mode is much more dominant. The forced convective Nusselt number is given by Sieder and Tate correlation [11] for internal flow in tubes and is valid for forced laminar regime, Re 6 2100:
1 0:14 Re Pr 3 lb Nuf ¼ 1:86 L=de lw
ð2Þ
with Re and Pr are calculated at local water bulk temperature. Fig. 4 shows also plots of the Nusselt number along the channel for natural convection heat transfer mode calculated by using the Churchill correlation [12]:
Nun ¼
0:15ðGrd Pr w Þ1=3 ð1 þ ð0:437=Pr w Þ9=16 Þ16=27 3
ð3Þ
with Grd ¼ gbðT w T b Þde =m2w and Pr is calculated at local plate surface temperature.
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80
80
Length, (cm)
70
60
60
50
Nun
50
50
40
40
30
30
20
20
10
10
10
0
0
Nuc
40
Nup
30
Nue
20
2
4
6
8 10 12 14
Run (4)
2
4
6
8 10 12 14
0
Run (5)
60
50
50
50
40
40
40
30
30
30
20
20
20
10
10
10
0
0 4
6
8 10 12 14
80 Run (7)
2
4
6
8 10 12 14
Run (8)
50
50
50
40
40
40
30
30
30
20
20
20
10
10
10
0
0 6
8 10 12 14 80
70
70
Run (10)
60
2
4
6
8 10 12 14
Run (11)
60
50 40
30
30
30
20
20
20
10
10
10
0
0 8 10 12 14
6
8 10 12 14
Run (12)
60
40
6
4
70
50
4
2
80
40
2
8 10 12 14
Run (9)
0
50
0
6
0 0
80
4
70 60
4
2
80
70 60
2
8 10 12 14
Run (6)
0
60
0
6
0 0
80
70
4
70
60
2
2
80
70
60
0
Run (3)
0 0
80
70
Length, (cm)
70
Nuf
80
Length, (cm)
Run (2)
60
0
Length, (cm)
80
70
Run (1)
0 0
2
Nu
4
6
8 10 12 14
0
2
4
Nu
6
8 10 12 14
Nu
Fig. 4. Nusselt number variation along the test section.
It is noticed a large deviation from the experimental Nusselt number at the channel inlet and this deviation decreases sharply in the first 20 cm region of the test section and then the predicted natural convection Nusselt number is gradually close to the experimental data to obtain a very good agreement at the channel exit. The Nusselt number values for combined natural and forced convection heat transfer mode under the present experimental condition are depicted in Fig. 4 as well, it is calculated by Collier correlation [13]:
Nuc ¼ 0:17Re0:33 Pr0:43 f f
Pr f Prw
#0:1 0:25 " 3 gbde ðT w T b Þ
m
2 f
ð4Þ
where subscripts f and w indicate that physical properties are evaluated at film temperature and wall-surface temperature respectively. This relation is valid for heating in vertical upflow or cooling in vertical downflow for L/de > 50 and Re < 2000. It shows a much better agreement than Eq. (3) but still large deviation at the channel inlet. Therefore, there is a need for a new correlation to fit the experimental data and takes into account the axial distance from the channel inlet. The proposed correlation
to predict the local Nusselt number will taken the following dimensionless form: k4 k3 Nup ¼ k1 Grk2 d Pr ðde =zÞ
ð5Þ
By taking the logarithmic transformation of Eq. (5) and applying the least squares method, the constants k1, k2, k3 and k4 are evaluated as 1, 0.284, 0.12 and 0.25, respectively. So the developed correlation for prediction of local Nusselt number takes the following form:
Nup ¼ Gr0:284 Pr0:12 ðde =zÞ0:25 d
ð6Þ
with all water physical properties calculated at the local bulk temperature. The Nusselt number is evaluated by the present correlation and the other correlations for natural convection mode (Eq. (3)) and combined natural and forced convection mode (Eq. (4)) under the present experimental conditions. All the obtained results and experimental data are plotted in Fig. 5. The solid line is a reference with the slope of one is drawn on the plot to give the relation between the predicted and measured data. The present correlation shows a good agreement with the experimental data, it gives only
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for both correlations respectively. A new correlation of the form Nu ¼ Gr 0:284 Pr0:12 ðde =zÞ0:25 was developed to be utilized in the d thermal–hydraulics and safety analysis of the reactor. The developed correlation predicted the local Nusselt number distribution along the test section with 5.8% relative standard deviation from the experimental results.
14 Present correlation Churchill correlation Collier correlation
Predicted Nusselt number
12
10
Appendix A. Experimental uncertainty 8
The experimental measurements cannot be measured exactly. It can be only known with a certain range of uncertainty. The accuracy of experimental results depends on the accuracy of the individual measuring instruments. There are several methods for calculating the uncertainty of experimental parameters. Based on the square root method, the total uncertainty U comprises of uncertainties of many experimental parameters x1, x2, . . . , xn, the combined standard uncertainty of the Nu is calculated from the following equation:
6
4
2
UðNue Þ ¼
( n X @Nu i¼1
0 0
2
4
6
8
10
12
14
Measured Nusselt number
UðNuÞ ¼
Model
Standard deviation
Present correlation Churchill correlation Collier correlation
0.058 0.327 0.155
2 )1=2 ðA1Þ
Eq. (A1) shows that, Nu is function of /, de, Tw and Tb. Therefore, the uncertainty in Nu is determined as:
Fig. 5. Comparison of the correlation model with previous correlations.
Table 1 Relative standard deviation from experimental data for Nusselt number.
@xi
Uxi
( 2 2 2 2 )1=2 @Nu @Nu @Nu @Nu U/ þ Ude þ UT w þ UT b @/ @de @T w @T b ðA2Þ
The heat flux, / is a function of voltage, current and test section length and width as:
/¼
V I 2LW
ðA3Þ
So the uncertainty in heat flux, U/ is determined by:
Uð/Þ ¼
Table A1 Uncertainty values for the measured parameters. Parameter
Value
Uncertainty
V I L W S Tw Tb
1–3.75 V 300–800 A 80 cm 7 cm 2.7 mm 42–96 °C 32–75 °C
±0.1 V ±5 A ±0.1 mm ±0.1 mm ±0.1 mm ±0.1 ±0.1 °C
5.8% relative standard deviation from the experimental data while the others gives 32.7% and 15.5% for Churchill and Collier correlations, respectively as shown Table 1.
(
@/ UV @V
2
þ
@/ UI @I
2
þ
2 2 )1=2 @/ @/ UL þ UW @L @W ðA4Þ
The equivalent hydraulic diameter, de is calculated as:
de ¼
2W S W þS
ðA5Þ
where S is the channel clearance. So the uncertainty in the equivalent hydraulic diameter, de is determined by:
Uðde Þ ¼
(
@de UW @W
2 þ
2 )1=2 @de US @S
ðA6Þ
4. Conclusions The natural convection heat transfer of water in narrow rectangular vertical channel was studied experimentally. The water channel is represented by two adjacent stainless steel heating plates with a dimension of 800 mm active length, 70 mm in width and 2.7 mm in gap thickness simulating a coolant channel of a typical material testing reactor under atmospheric pressure. Experiments were carried out under heat fluxes ranged from 2.7 kW/m2 to 26.8 kW/m2 to cover all possible heat fluxes in the single-phase liquid regime. Wall surface temperature, coolant bulk temperature and local Nusselt number profiles were experimentally investigated. The measured Nusselt number values are compared with the predicted values of both Churchill correlation for natural convection mode and Collier correlation for combined natural and forced convection with 32.7% and 15.5% relative standard deviation
Table A1 lists the values and uncertainties in the measured parameters. Thus, the uncertainty of the average Nu correlated in Eq. (1) can be deduced to be ranges from 10.7% for 2679 W/m2 heat flux to 3.1% for 26,786 W/m2. References [1] W. Elenbass, Heat dissipation of parallel plates by free convection, Physica 9 (1942) 1–28. [2] W. Elenbass, The dissipation of heat by free convection: the inner surface of vertical tubes of different shapes of cross-section, Physica 9 (1942) 865–874. [3] A. Bar-Cohen, W.M. Rohsenow, Thermally optimum spacing of vertical, natural convection cooled, parallel plates, ASME, Journal of Heat Transfer 106 (February) (1984) 116–123. [4] K.T. Lee, W.M. Yan, Laminar natural convection between partially heated vertical parallel plates, Warme- und Stoffubertragung 29 (1994) 145–151.
S.E. El-Morshedy et al. / Experimental Thermal and Fluid Science 36 (2012) 72–77 [5] Qing Lu, Suizheng Qiu, Guanghui Su, Wenxi Tian, Zhonghao Ye, Experimental research on heat transfer of natural convection in vertical rectangular channels with large aspect ratio, Experimental Thermal and Fluid Science 34 (2010) 73– 80. [6] Yukio Sudo, Tohru Usui, Masanori Kaminaga, Heat transfer characteristics in narrow vertical rectangular channels heated from both sides, JASME International Journal, Series II 33 (4) (1990) 743–748. [7] Yukio Sudo, Masanori Kaminaga, Hiromasa Ikawa, Combined forced and free convection heat transfer characteristics in narrow vertical rectangular channels heated from both sides, Journal of Nuclear Science and Technology 24 (5) (1987) 355–364. [8] Donald D. Joye, Comparison of aiding and opposing mixed convection heat transfer in a vertical tube with Grashof number variation, International Journal of Heat and Fluid Flow 17 (1996) 96–101.
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[9] D.D. Joye, M.J. Wojnovich, Aiding and opposing mixed-convection heat transfer in a vertical tube: loss of boundary condition at different Grashof numbers, International Journal of Heat and Fluid Flow 17 (1996) 468–473. [10] Salah El-Din El-Morshedy, Prediction, analysis and solution of the flow inversion phenomenon in a typical MTR-reactor with upward core cooling, Nuclear Engineering and Design 241 (1) (2011) 226–235. [11] E.N. Sieder, G.E. Tate, Heat transfer and pressure drop of liquids in tubes, Industrial and Engineering Chemistry 28 (1936) 1429. [12] S.W. Churchill, Combined free and forced convection in channels, in: G.F. Hewitt (Ed.), Handbook of Heat Exchanger Design, Begell-House, 1992. [13] J.G. Collier, Convective Boiling and Condensation, second ed., Mc Graw-Hill International Book Company, 1981.