- Email: [email protected]
Natural convection heat transfer characteristics in vertical cavities with active and inactive top and bottom disks
International Journal of Heat and Mass Transfer 87 (2015) 390–398
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Natural convection heat transfer characteristics in vertical cavities with active and inactive top and bottom disks Gyeong-Uk Kang a, Bum-Jin Chung b,⇑ a b
Research & Development Institute, Korea Radioactive Waste Agency, 989-111 Daedeok-daero, Yuseong-gu, Daejeon 305-353, Republic of Korea Department of Nuclear Engineering, Kyung Hee University, 1732 Deokyoung-daero, Yongin-si, Gyonggi-do 446-701, Republic of Korea
a r t i c l e
i n f o
Article history: Received 15 December 2014 Received in revised form 8 April 2015 Accepted 8 April 2015
Keywords: Analogy concept Hydrodynamic interaction Natural convection Heat and mass transfer Vertical cavity
a b s t r a c t Natural convection heat transfer was investigated in vertical cavities where either all surfaces were active, or only the vertical surface was active for four different geometries, which were varied by placing a disk at the top and/or bottom of the cavity. A cupric acid–copper sulfate electroplating system was employed for mass transfer experiments exploiting the analogy with heat transfer. The Rayleigh number was varied in the range 4.55 109 6 RaLw 6 3.79 1013. Preliminary tests for a vertical pipe, upwardand downward-facing horizontal disks showed good agreement with existing correlations. The measured Nusselt numbers in the vertical cavities with all surfaces active were always greater than those with only the vertical surface active, which is attributed to greater hydrodynamic interaction of the flows generated by different surfaces. When all surfaces were active, the bottom-closed cavity exhibited the largest heat transfer rates, followed by both-closed, top-closed, and both ends open cavities; this trend was observed with laminar and turbulent flows. With only the vertical surface active, similar trends were observed except that the heat transfer rates were almost identical for both-open and top-closed cavities, which is attributed to the weak influence of the top disk on the heat transfer characteristics. Using these results, empirical correlations were derived for both laminar and turbulent flow conditions. Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction Natural convective flows in a vertical cavity are relevant to many practical applications in the design of both heat and mass transfer devices, especially the passive safety system design of nuclear power plants under hypothetical accident conditions. Much data are available for natural convection in a vertical cavity where both ends are open (i.e., a vertical heated pipe) over a wide range of Rayleigh numbers [1–3]; however, most practical problems related to engineering fields concern vertical cavities with arrangements other than a simple vertical heated pipe. The vertical cavity consists of a vertical cylindrical wall and horizontal circular disks. The horizontal circular disks seal the cavity at the top and/or bottom, and may be either active or inactive, where the term active corresponds to a heated wall and inactive to an adiabatic wall. The geometry of the vertical cavities may be classified into four cases: both ends open, bottom-closed (with an open top), top-closed (with an open bottom), and with both ends closed. There have been some reports of natural convection in vertical cavities [4–6], whereby the authors used the cavities with either ⇑ Corresponding author. Tel.: +82 31 201 3893; fax: +82 31 204 8114. E-mail address: [email protected] (B.-J. Chung). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2015.04.022 0017-9310/Ó 2015 Elsevier Ltd. All rights reserved.
all surfaces active (i.e., both the vertical walls and horizontal disks were heated), or where only the vertical surfaces were active. These works were restricted mainly to laminar flows and the bottom-closed and top-closed cavity geometries. The heat transfer behavior of the cavity geometry with both ends closed has been less well studied. Furthermore, the available data in the literature lack consistency in terms of the size of the vertical cavities, and the authors did not provide detailed phenomenological explanations of the flow interactions in the vertical cavities. For these reasons, detailed investigations are required to understand the heat transfer behavior of vertical cavities by exploring further geometrical arrangements with consistent sizes of cavity with both laminar and turbulent flows. This study investigated natural convection heat transfer due to hydrodynamic interactions in vertical cavities with various geometrical arrangements and with Rayleigh numbers in the range 4.55 109 to 3.79 1013. These ranges are sufficient to cover laminar and turbulent flows. The Nusselt number was measured for the four geometries of vertical cavities with all surfaces were active, and only the vertical surfaces were active. Exploiting the analogy between heat and mass transfer, we used a sulfuric acid-copper sulfate (H2SO4ACuSO4) electroplating system with the limitingcurrent technique to characterize the heat transfer coefficients.
391
G.-U. Kang, B.-J. Chung / International Journal of Heat and Mass Transfer 87 (2015) 390–398
Nomenclature Cb Cp d Dm Do F g GrH H hh hm I Ilim L LW M k n Nud NuH
concentration in the bulk [mol/m3] specific heat at constant pressure [J/kg K] diameter [m] mass diffusivity [m2/s] cylinder diameter [m] Faraday constant, 96,485 [C/mol] gravitational acceleration, 9.8 [m/s2] Grashof number [gbDTH3/m], [(gH3/m2)(Dq/q)] height of cathode [m] heat transfer coefficient [W/m2 K] mass transfer coefficient [m/s] electric current [A] limiting current density [A/m2] length of horizontal surface [m] surface area / perimeter projected onto a horizontal plane (H + d/4) [m] molarity [mol/l] thermal conductivity [W/m K] number of electrons in charge transfer reaction Nusselt number [hhd/k] Nusselt number [hhH/k]
Phenomenological explanations of the effects of the geometry on the flow interactions and heat transfer rates are provided, and empirical correlations based upon the results are derived for the four arrangements with laminar and turbulent flow conditions.
2. Natural convection heat transfer in a vertical cavity Table 1 lists a summary of previous studies on natural convection heat transfer in vertical cavities. For the case with both ends open, the thermal boundary layer that developed along the hot wall was thinner than the cylinder radius, and the heat transfer phenomena were similar to those for a vertical plate [1]. With this geometry, the Nusselt number can be calculated using correlations for vertical plates. Kang and Chung [7,8] showed this result experimentally and reported that a transition to turbulent flow occurred with a Grashof number of approximately GrH = 109, corresponding to a Rayleigh number of RaH = 1012 when a Prandtl number was
NuLw Pr RaH Sc ShH Shd ShLw tn U
Nusselt number [hhLW /k] Prandtl number [m/a] Rayleigh number [gbDTH3/m], [(gH3/Dm)(Dq/q)] Schmidt number [m/Dm] Sherwood number [hmH/Dm] Sherwood number [hmd/Dm] Sherwood number [hmLW /Dm] transference number uncertainty
Greek symbols a thermal diffusivity [m2/s] b volume expansion coefficient [1/K] c dispersion coefficient l viscosity [kg/m s] m kinematic viscosity [m2/s] q density [kg/m3] d velocity boundary layer thickness [m] dT thermal boundary layer thickness [m]
Pr = 2000. Their results were in good agreement with Le Fevre’s correlation for laminar flow [9] and Fouad’s correlation for turbulent flow [10]. There have been several reports of the heat transfer characteristics in bottom-closed and top-closed cavities. These cavities consist of a vertical wall and a horizontal disk either at the bottom or top. Some of the previous studies have used mass transfer experiments exploiting the analogy between heat and mass transfer systems. Somerscales and Kassemi [4] measured the natural convection heattransfer in bottom-closed cavities with all surfaces active for Rayleigh numbers in the range 7.1 107 6 Rad 6 6.9 109, where the diameter was used as the characteristic length. Krysa et al. [5] carried out experiments using bottomclosed cavities for Rayleigh numbers in the range 2 107 6 Rad 6 1.2 1010 and visualized the flows emerging from the cavity openings for a short cavity. Two types of bottom-closed cavities were formed with all surfaces active and with only the vertical surface active. Two characteristic lengths were used depending
Table 1 Previous studies for the vertical cavities. Authors
H (m)
d (m)
Range of Ra
Arrangement
Surface condition
Correlations
Le Fevre [9] Fouad [10]
– –
– –
GrH 6 109 GrH P 109
Both ends open Both ends open
– –
NuH = 0.67(GrHPr)0.25 NuH = 0.31(GrHPr)0.28
Somerscales and Kassemi [4]
0.00635, 0.0127, 0.0254 0.0127, 0.0254, 0.0508 0.019, 0.0381, 0.0762
0.0127
7.1 107 6 Rad 6 6.9 109
Bottom-closed (open top)
All surfaces active
Nud ¼ 0:232ðd=HÞ
2 107 6 RaLw 6 1.2 1010
Bottom-closed (open top)
All surfaces active Only vertical surface active
NuLW ¼ 0:559RaL0:265 W
(4)
0:265 NuH ¼ 0:480RaH
(5)
All surfaces active
NuLW ¼ 0:257RaL0:333 W
(6)
NuLW ¼ 0:187RaL0:297 W
(7)
Krysa et al. [5]
Sedahmed et al. [6]
0.003–0.0381
0.005, 0.01, 0.015, 0.025, 0.03 0.02
0:191
(1) (2) Sc0:056 Ra0:28 d
(3)
0.0254 0.0381 0.0135
0.0172 0.02, 0.0296, 0.032, 0.041
1 108 6 RaLw 6 5.02 109
Bottom-closed (open top) Top-closed (open bottom)
392
G.-U. Kang, B.-J. Chung / International Journal of Heat and Mass Transfer 87 (2015) 390–398
Table 2 Natural convection correlations for horizontal surfaces. Scholars
Orientation
Range
Correlation 9
9
Al-Arabi and El-Riedy [16]
Upward
4 10 6 RaL 6 10
Ishiguro et al. [17]
Upward
3 105 6 RaL 6 1010
Yousef et al. [18]
Upward
4 107 6 RaL 6 1.7 108
NuL ¼
Wragg et al. [19]
Downward
7 103 6 Rad 6 1 1011
Nud ¼
Fishenden and Sauders [20]
Downward
1 106 6 Rad 6 1 1011
Nud ¼
Loomba [21]
Downward
2 107 6 Rad 6 1 1011
Nud ¼
Patrick et al. [22]
Downward
–
Nud ¼
(8)
1=3
NuL ¼ 0:155RaL NuL ¼
(9)
1=3 0:20RaL 1=3 0:162RaL 2:08Ra0:178 d 0:28Ra0:25 d 0:64Ra0:22 d 0:289Rad0:25
(10) (11) (12) (13) (14)
on cavity surface conditions: Lw for all surfaces active and H with only the vertical surface active, where Lw = H + d/4, and where H is the height of the cavity and d is the diameter [11]. They reported that the Nusselt number was larger with all surfaces active than with only the vertical surface active, which was attributed to the flow at the base of the cavity, which aided the development of a boundary layer along the vertical surface. In the above two studies, the anode was located outside the cavity so that the cupric ions produced from the anode could not reach the entire cathode surfaces easily [7]. To overcome this limitation, the anode should be located at the center of the cavity to improve the current distribution. Lim and Chung [12] investigated the influence of a center anode numerically as a cold surface in a heat transfer system, and reported that a central anode in vertical pipes did not affect the cupric ion concentration near the cathode. Sedahmed et al. [6] performed experiments with bottom-closed and top-closed cavities for Rayleigh numbers in the range 1 108 6 RaLw 6 5.02 109, with all cavity surfaces active. In contrast to the two studies discussed above, the anode was located inside the cavity. They reported that the Nusselt number was larger for the bottom-closed cavity than for the top-closed cavity, which was attributed to the presence of oscillatory flow. The top three studies listed in Table 1 were carried out for a narrow range of Rayleigh numbers, which resulted from the relatively short cavities. Furthermore, the cavity sizes were not consistent between these sets of experiments, which complicates a comparison, and the reported correlations were based on different characteristic lengths (i.e., d or H or Lw). Recently, Kang and Chung [13– 15] investigated the effects of the geometrical arrangement on the heat transfer rate in a vertical cylinder. They verified the results in Refs. [4–6], and reported phenomenological explanations of the flow interactions in vertical cavities. Table 2 lists a summary of previous studies for heat transfer on horizontal planar surfaces [16–22]. It is well known that natural convection heat transfer at an upward-facing planar surface is more effective than that on a downward-facing surface because of the buoyancy of the hot fluid, which is replaced by descending cold fluid. Wragg and Loomba [23] investigated the flow patterns on upward-facing planar heated surfaces and reported that turbulent flow occurred for Rad > 3 107. The present work supplemented some of the shortage and weak points as discovered in the existing literatures by exploring a more diverse set of geometrical arrangements with a wide range of RaLw for vertical cavities with all surfaces active and with only the vertical surfaces active.
form, and contain the same classes of boundary and initial conditions [1]. The technique is attractive for achieving a high Rayleigh number with relatively small apparatus, and provides benefits of faster and more accurate measurements compared with heat transfer measurements. The dimensionless mass transfer is described by Sherwood number Sh and the Schmidt number Sc, which correspond to the Nusselt number Nu and Prandtl number Pr in a heat transfer system, respectively. The mass transfer system investigated here consisted of an H2SO4ACuSO4 electroplating system. Mass transfer results obtained using the electroplating technique must be carefully analyzed because of the large Prandtl number. The limiting electrolysis currents were measured by applying a voltage to the cell to determine the current–voltage relationship. These mass transfer measurements were based on the technique described in Refs. [24–26]. In this study, empirical relations in Eqs. (15)–(22) were used, which were originally reported buy Fenech and Tobias [27], to determine the values of the physical properties for the required dimensionless numbers. These data were accurate within an error bound of 0.5% at 22 °C. Several studies have applied this method to convection heat transfer phenomena [12–15,28–31].
c¼
C CuSO4 ; C CuSO4 þ C H2 SO4
ð21Þ
3. Experimental method and apparatus
bj ¼
1 @q : q @C j T;Ck –j
ð22Þ
q ðkg=m3 Þ ¼ 0:9978 þ 0:06406MH2 SO4 0:00167M2H2 SO4 þ 0:12755M CuSO4 þ 0:018202CuSO4 103 ;
ð15Þ
l ðC p Þ ¼ 0:974 þ 0:1235MH2 SO4 þ 0:0556M2H2 SO4 þ 0:5344M CuSO4 þ 0:53562CuSO4 ;
ð16Þ
lDm ðm2 =sÞ ¼ 0:7633 þ 0:00511MH2 SO4 þ 0:02044MCuSO4 10; ð17Þ
tCu2þ ¼ 0:2633 0:1020C H2 SO4 C CuSO4 ; DC H2 SO4 ; ¼ C CuSO4 bCuSO4 bH2 SO4 q DC CuSO4
Dq
ð18Þ ð19Þ
DC H2 SO4 ¼ 0:000215 þ 0:113075c1=3 þ 0:85576c2=3 0:50496c; DC CuSO4 ð20Þ
3.1. The analogy between heat and mass transfer 3.2. Apparatus and test matrix Heat transfer problems in a vertical cavity can be investigated experimentally using mass transfer experiments. The mathematical framework for heat and mass transfer processes are of the same
Fig. 1 shows a schematic diagram of the test facility and the electric circuit used. The test section was formed of cylindrical
393
G.-U. Kang, B.-J. Chung / International Journal of Heat and Mass Transfer 87 (2015) 390–398 Table 3 Test matrix for the cavity arrangements.
(a) Test apparatus.
d (m)
H (m)
RaLw
Pr
Cavity arrangements
Cavity surface condition
0.032
0.03–0.6
4.55 109– 3.79 1013
2094
Both ends open, Bottom-closed (open top), Top-closed (open bottom), Both ends closed
All surfaces active Only vertical surface active
temperature of the fluid and the ambient room was measured before and after each test to ensure temperature was maintained constant. The electric potential was applied using a power supply (VüPower IPS18B10) and the current was measured using a dualdisplay multi-meter (Fluke 45). Table 3 lists the test matrix for the vertical cavities. Two types of vertical cavity were used: all surfaces were active and only the vertical surface was active. The cavity height was varied in the range 0.03–0.6 m to investigate laminar and turbulent flows, which corresponds to a Rayleigh number in the range 4.55 109 6 RaLw 6 3.79 1013. The diameter of the horizontal copper disks (active) or acryl (inactive) disks sealing the top and/ or bottom of the cavity were 0.032 m, corresponding to a Rayleigh number of Rad = 5.53 109. The measurements were carried out with four different geometries: both ends open, bottomclosed (with an open top), top-closed (with an open bottom), and with both ends closed. The definition of Lw is commonly applied to cavities regardless of the arrangement. The concentration of copper sulfate was 0.1 M, and that of sulfuric acid was 1.5 M. Sulfuric acid was added to suppress electrical migration. The Schmidt number was 2094, as determined by the empirical relations in Eqs. (15)–(22). 4. Results and discussion 4.1. Verification of the experimental method
(b) Schematic diagram of electric circuit. Fig. 1. Test apparatus and schematic diagram of electric circuit.
acryl pipes to prevent corrosion by the H2SO4 and CuSO4. It was separated into upper, middle and lower sections, in which only the middle section was lined with copper to form the cathode; the height of this cathode was 0.03–0.6 m and the diameter was 0.032 m. With the electroplating system, the cathode simulates the hot wall in heat transfer system because of the buoyancy induced by decreased fluid density with the reduction of copper ions at the cathode surface. A small groove was cut at the inside or outside edges of the cathode so that the cavity height can be varied and the cathodes can adhere to each other without gaps. Acryl flanges were attached to the top and bottom of each section, which acted as a link between the test sections and the remainder of the vertical cavity. A rubber O-ring was attached between the flanges to prevent leakage of fluid. A copper anode of 2-mm-diameter was inserted from the top of the test section, which was longer than the cathode height and acted as a cupric ion source. The
In order to verify the experimental methods, preliminary tests were carried out with three geometrical arrangements: both ends open, upward- and downward-facing horizontal disks. Fig. 2 shows a comparison of the measured Nusselt numbers with existing correlations developed for these three geometries. The Nusselt numbers measured in the cavity with both ends open were in good agreement with the laminar and turbulent natural convection heat transfer correlations for a vertical plate in Eqs. (1) and (2). Those for upward- and downward-facing disks were in satisfactory agreement with the existing correlations for horizontal flat surfaces shown in Eqs. (8)–(14), regardless of the characteristic length d or L. The Nusselt numbers were larger for upward-facing disk than for the downward-facing disk due to more effective natural convection flow. These comparisons show that the mass transfer measurements described here are appropriate for the vertical cavity geometries. 4.2. Natural convection heat transfer characteristics for various geometries Fig. 3 shows the Nusselt numbers for the vertical cavities with all surfaces active and with only the vertical surface active. The results were compared with the existing laminar and turbulent natural convection correlations for a vertical plate in Eqs. (1) and (2). The triangle symbols show the results in for the bottom-closed cavity, the inverse triangle symbols show the results for the
394
G.-U. Kang, B.-J. Chung / International Journal of Heat and Mass Transfer 87 (2015) 390–398
Fig. 2. Comparison of the preliminary test results with the heat transfer correlations.
sides of symbols denote a closed face. The bottom-closed cavity exhibited the largest Nusselt number, followed by the both-closed, top-closed, both ends open cavities, both in laminar and turbulent flow conditions. Similar trends can be seen in the data shown in Fig. 3(b), except that the results with both ends open and the top-closed cavities were similar each other and to the existing correlations in Eqs. (1) and (2), indicating a the weaker influence of the inactive top plate. These trends were observed regardless of the magnitude of RaLw. In particular, the larger Nusselt number for the bottom-closed cavity than the top-closed cavity is consistent with the observations reported by Sedahmed et al. [6]. Fig. 4 shows the axial variation of convective mass transfer coefficients. In the same manner, the trends are similar to those shown in Fig. 3. In all cases, the axial variations were similar to the well-known heat transfer behavior for the internal flow in a vertical pipe. The coefficients were largest at the lowest height of 0.03 m, and decreased rapidly with increasing height until 0.3 m, after which they remained constant. It follows that, at heights in the range 0.03–0.3 m, the flow develops, and heights greater than 0.3 m correspond to a hydrodynamic fully developed flow. Because of the large Prandtl number of 2094, the thickness of a velocity boundary layer was larger than that of a thermal boundary
(a) All surfaces active
(a) All surfaces active
(b) Only vertical surface active Fig. 3. Comparison of the test results for various arrangements in a vertical cavity.
top-closed cavity, the square symbols show results for both ends closed, and the diamond symbols show results for the both ends open. The symbols were determined so that the horizontal flat
(b) Only vertical surface active Fig. 4. Axial variations of convective mass transfer coefficients.
G.-U. Kang, B.-J. Chung / International Journal of Heat and Mass Transfer 87 (2015) 390–398
layer. Thus, a hydrodynamic fully developed region could be achieved in tall cavities. Fig. 5 shows a comparison of the results for active and inactive disks. In all cases, the Nusselt number was always higher with all surfaces active than for only the vertical surface active. This is
395
attributed to greater hydrodynamic interactions of the flows that developed from the vertical walls and the horizontal active disk. 4.3. Hydrodynamic interactions of flows in vertical cavities Based on results described in Section 4.2, in this section we discuss phenomenological explanations for the influence of the cavity geometry on flow interactions and heat transfer rates.
(a) Bottom-closed cavity
(b) Top-closed cavity
(c) Both ends closed cavity Fig. 5. Comparison of all surfaces active and only vertical surface active.
4.3.1. (Case 1) bottom-closed cavity The heat transfer behavior with the bottom-closed geometry depended on the length of cavity. For a short cavity with all surfaces active, the flows were generated from the vertical wall and the active bottom disk. The flow from the vertical wall was upwards along the inner wall of the cavity. The flow developed from the active bottom disk moved upward in the middle of the cavity, which resulted in turbulent flow with a Rayleigh number of Rad = 5.53 109 [23]. In this case, the fresh fluid from outside could flow into the cavity due to negative pressure gradients, which resulted from the two upward flows, leading to secondary flows. These secondary flows interacted with the upward flows, exhibiting oscillatory behavior, which led to an enhancement of the heat exchange between the hot walls and the fresh fluids. Furthermore, the secondary flows, which were in the opposing direction to the primary flow, led to effective mixing of the fluid, increasing the turbulence by supplying a shear force. For a short cavity with only the vertical surface active, there was no flow at the inactive bottom disk. The flow was generated only by the vertical wall, and the secondary flow entering into the cavity was obstructed by the inactive disk, so that the secondary flow moved to the vertical wall. The obstructed flow added to the flow from the vertical wall and a buoyancy-aided flow of mixed convection heat transfer formed. For a long cavity, secondary flow generated during the initial stage only during the development of the boundary layer, which no longer generated due to the establishment of fully developed flow conditions. The flow at the active bottom disk consistently supplied large initial velocities to the flows along the vertical wall, and the obstructed flows from the inactive disk contributed small velocities to the flows along the vertical wall. These phenomena led to the development of a boundary layer, incurring the buoyancy-aided mixed convection flow. 4.3.2. (Case 2) top-closed cavity For a cavity with all surfaces active, the flow generated at the vertical walls moved up and was blocked by the active top disk. The flow from the active top disk moved horizontally to the top corners. The two flows in opposite directions tended to accumulate near top corner, forming light and stable layer, which led to low heat transfer rates. For a cavity with only the vertical surface active, the flow upward along the vertical wall was obstructed by the inactive top disk, and moved horizontally toward the top center. The flow then descended, forming a natural convective flow. In this way, the combination of the two convective flows formed axisymmetric motion. As shown in Fig. 4(b), the Nusselt numbers for the top-closed cavity were smaller than those with both ends open when the cavities were short, as the top disk blocked the flow and the buoyancy forces were weak, leading to small heat transfer rates. For longer cavities, however, the heat transfer rates were almost identical to those for the cavity with both ends open. It follows that the inactive top disk had little impact on heat transfer in longer cavities. 4.3.3. (Case 3) both ends closed For a cavity with all surfaces active, the three flows from the vertical wall and active top and bottom disks converged, leading
396
G.-U. Kang, B.-J. Chung / International Journal of Heat and Mass Transfer 87 (2015) 390–398
to stagnation near the top corner. The flows moving in different directions mixed, which increased the turbulence by supplying a shear force. For the cavity with an active vertical surface only, the flow from the vertical wall was obstructed by the top and bottom disks. The circulation of ascending and descending fluids resulted in the formation of cells due to the interaction of the three walls, leading to enhanced heat transfer rates. To investigate the influence of the interaction of the flows on the heat transfer rate, we compared the measured total heat transfer rates with the calculated heat transfer rates obtained using heat transfer correlations in Eqs. (1), (2), (8) and (11) for horizontal and
(a) All surfaces active
vertical surfaces with the same area. Fig. 6 shows a comparison of the heat transfer rates for three arrangements with all surfaces active, and with only the vertical surface active. The closed symbols correspond to measured heat transfer rates and the open symbols correspond to the calculated data. For short cavities (with lengths in the range 0.03–0.07 m), the gradients of the two curves were very similar; however, as the height increased, the difference between the measured and calculated data increased as a function of the cavity height for all geometries. The gradient for the closed symbols become steeper from specified heights than that for open symbols. As shown in Fig. 6, for all cases
(b) Only vertical surface active
Fig. 6. Comparison of the results with the calculated values from heat transfer correlations.
(32)
(34)
0:271 NuLw ¼ 0:45RaLw
0:281 NuLw ¼ 0:31RaLw
Max. (6.4%) Min. (0.4%) Max. (3.9%) Min. (0.7%) Max. (10.1%) Min. (0.4%) (30)
(31)
(33)
0:23 NuLw ¼ 1:35RaLw
NuLw ¼ 0:61Ra0:25 Lw
(24)
Max. (9.1%) Min. NuLw ¼ (1.8%) 0:296 (25) Max. (7.1%) Min. NuLw ¼ 2:25Ra0:221 NuLw ¼ 0:24RaLw Lw (0.6%) 0:281 (27) Max. (4.9%) Min. NuLw ¼ 1:31Ra0:231 NuLw ¼ 0:33RaLw Lw (0.2%) Same as the existing laminar and turbulent correlations (1) and (2) for
Fig. 7. Derived empirical correlations for laminar and turbulent flows.
4.4. Uncertainty analysis
Both ends open
Both ends closed Top-closed
Bottom-closed
(23)
other than the top-closed cavity with only the vertical surface active, the measured heat transfer rates were always greater than the sum of the calculated heat transfer rates for the vertical wall and horizontal surfaces. These results demonstrate the importance of the interactions of the flows. Among the three geometries, the bottom-closed cavity exhibited the largest difference between the measured and calculated heat transfer rates, followed by the cavity with both ends closed and the top-closed geometry. It follows that the bottom-closed geometry had the strongest effect on the heat transfer behavior. The data plotted in Fig. 6 reveal that obtaining heat transfer rates for such geometries by summing the contributions for individual surfaces is not accurate due to the interactions between the different flows generated by the heated surfaces. Based on these experimental results, empirical correlations were derived for the different cavities with laminar and turbulent flows, as listed in Table 4. Fig. 7 shows plots of these correlations.
NuLw ¼
0:296 0:26RaLw
(b) Only vertical surface active
2:41Ra0:221 Lw
Turbulent Error Laminar
All surfaces active Arrangements
Table 4 Derived empirical correlations for laminar and turbulent flows.
397
(a) All surfaces active
Max. (13.2%) Min. (1.0%) (26) Max. (8.2%) Min. (1.2%) (28) Max. (4.0%) Min. (0.2%) a vertical pipe
NuLw ¼
(29)
Max. (6.1%) Min.(0.3%) Max. (6.4%) Min. (1.2%) Max. (4.1%) Min. (0.1%)
NuLw ¼
0:271 0:48RaLw
Turbulent Error Laminar
Vertical surface active only
Error
0:23 1:51RaLw
Error
G.-U. Kang, B.-J. Chung / International Journal of Heat and Mass Transfer 87 (2015) 390–398
The uncertainties involved in natural convection measurements using the H2SO4ACuSO4 electroplating system were analyzed using a conventional data reduction approach [32]. Because the Sherwood number was the final dependent variable, the data reduction and uncertaintycan be expressed as follows:
398
G.-U. Kang, B.-J. Chung / International Journal of Heat and Mass Transfer 87 (2015) 390–398
ShH ¼
hm H ) ShH ¼ f ðhm ; H; Dm Þ Dm
ð35Þ
and
U 2ShH ¼
@Sh Uh @hm m
2 þ
2 2 @Sh @Sh UH þ : @H @Dm
ð36Þ
The calculated average uncertainties for the Nusselt number were about 3.63% and the average fractional uncertainties of these values were roughly 1.82%. 5. Conclusions Natural convection heat transfer were investigated experimentally in vertical cavities with/without top and bottom lids with all surfaces active and only vertical surfaces active. The diameter of the cavity was 0.032 m, and the height of the cathode was varied in the range 0.03–0.6 m, which corresponds to Rayleigh numbers in the range 4.55 109 6 RaLw 6 3.79 1013 with a fixed Prandtl number of 2094. The measurements were carried out using an H2SO4ACuSO4 electroplating system. Preliminary results show that the Nusselt numbers measured for simple geometries were in good agreement with existing correlations for natural convection developed for a vertical pipe, as well as upward- and downward-facing horizontal planar heated surfaces. Empirical heat transfer correlations were derived for each geometry and set of surface conditions based on our measured data. The main findings of this study are summarized as follows: The interactions of flows generated from different surfaces enhanced the natural convection heat transfer. The natural convection heat transfer rates for all surfaces active were always larger than those for only the vertical surface active. With all surfaces active, the bottom-closed cavity exhibited the highest heat transfer rates, followed the cavity with both ends closed, the top-closed cavity, and the cavity with both ends open. When only the vertical surface was active, similar trends appeared except for the fact that both ends open and for the top-closed cavities show similar heat transfer rates. In both cavities, the bottom surface enhanced the heat transfer greatly, while the top surface only enhanced the heat transfer only when it is active. Conflict of interest None declared. Acknowledgments This study was supported by MOTIE (The Ministry of Trade, Industry and Energy), Korea, under the radioactive Waste Management Technology Development Project (2014171020173A). References [1] A. Bejan, Convection Heat Transfer, third ed., Wiley, New York, 2003. [2] F.P. Incropera, D.P. DeWitt, Fundamentals of Heat and Mass Transfer, fifth ed., Wiley, New York, 2003.
[3] J.P. Holman, Heat Transfer, eighth ed., McGraw-Hill, New York, 1997. [4] E.F.C. Somerscales, M. Kassemi, Electrochemical mass transfer studies in open cavities, J. Appl. Electrochem. 15 (1985) 405–413. [5] J. Krysa, A.A. Wragg, D.M. Thomas, M.A. Patrick, Free convective mass transfer in open upward-facing cylindrical cavities, Chem. Eng. J. 79 (2000) 179–186. [6] G.H. Sedahmed, A.M. Ahmed, M.L. El-Rafey, A.Y. Hosney, E.A. Ayob, Natural convection mass transfer inside cylindrical cavities of different orientation, J. Appl. Electrochem. 25 (1995) 677–681. [7] K.U. Kang, B.J. Chung, The effects of the anode size and position on the limiting currents of natural convection mass transfer experiments in a vertical pipe, Trans. KSME B 34 (1) (2010) 1–8. [8] G.U. Kang, B.J. Chung, The experimental study on transition criteria of natural convection inside a vertical pipe, Int. Commun. Heat Mass Transfer 37 (8) (2010) 1057–1063. [9] E.J. Le Fevre, A.J. Ede, Laminar free convection from the outer surface of a vertical circular cylinder, in: Proceedings of the Ninth International Congress Applied Mechanics, Brussels, vol. 4, 1956, pp. 175–183. [10] M.G. Fouad, N. Ibl, Natural convection mass transfer at vertical electrodes under turbulent flow conditions, Electrochim. Acta 3 (1960) 233–243. [11] M.E. Weber, P. Austraukas, S. Petsalis, Natural convection mass transfer to nonspherical objects at high Rayleigh number, Can. J. Chem. Eng. 62 (1984) 68–72. [12] C.K. Lim, B.J. Chung, Influence of a center anode in analogy experiments of long flow duct, Int. Commun. Heat Mass Transfer 56 (2014) 174–180. [13] B.J. Chung, J.H. Heo, M.H. Kim, G.U. Kang, The effect of top and bottom lids on natural convection inside a vertical cylinder, Int. J. Heat Mass Transfer 54 (2011) 135–141. [14] G.U. Kang, B.J. Chung, Effects of the dimensions of upward- and downwardfacing vertical cavities on natural convection, Int. Commun. Heat Mass Transfer 39 (2012) 399–404. [15] G.U. Kang, B.J. Chung, The effects of top and bottom lids on the natural convection heat transfer inside vertical cylinders, J. Energy Eng. 20 (3) (2011) 242–251. [16] M. Al-Arabi, M.K. El-Riedy, Natural convection heat transfer from isothermal horizontal plates of different shapes, Int. J. Heat Mass Transfer 19 (1976) 1399–1404. [17] R. Ishiguro, T. Abe, H. Nagase, Natural convection over heated horizontal plates, Trans. Jpn. Soc. Mech. Eng. Ser. B 43 (366) (1977) 638–645 (in Japanese). [18] W.W. Yousef, J.D. Tarasuk, W.J. Mckeen, Free convection heat transfer from upward-facing isothermal horizontal surfaces, Trans. Am. Soc. Mech. Eng. J. Heat Transfer 104 (1982) 493–500. [19] A.A. Wragg, G. Batting, J. Krysa, Free convective mass transfer at down-facing horizontal surfaces with free or collared edges, Int. Commun. Heat Mass Transfer 25 (2) (1998) 175–182. [20] M. Fishenden, O.A. Saunders, An Introduction to Heat Transfer, Oxford Univ. Press, London, 1950. pp. 102–104. [21] L.P. Loomba, Free convection mass transfer at horizontal surfaces (M.Sc. thesis), University of Manchester, 1969. [22] M.A. Patrick, A.A. Wragg, D.M. Pargeter, Mass transfer by free convection during electrolysis at inclined electrodes, Can. J. Chem. Eng. 55 (4) (1977) 432– 438. [23] A.A. Wragg, R.P. Loomba, Free convection flow patterns at horizontal surfaces with ionic mass transfer, Int. J. Heat Mass Transfer 13 (1970) 439. [24] V.G. Levich, Physicochemical Hydro-dynamics, Prentice-Hall, Englewood Cliffs, NJ, 1962. [25] J.N. Agar, Diffusion and Convection at Electrodes, Discuss. Faraday Soc. 1 (1947) 27–37. [26] J.R. Selman, C.W. Tobias, Mass transfer measurement by the limiting current technique, Adv. Chem. Eng. 10 (1978) 211–318. [27] E.J. Fenech, C.W. Tobias, Mass transfer by free convection at horizontal electrodes, Electrochim. Acta 2 (1960) 311–325. [28] S.H. Ko, D.W. Moon, B.J. Chung, Applications of electroplating method for heat transfer studies using analogy concept, Nucl. Eng. Technol. 38 (2006) 251–258. [29] M.S. Chae, B.J. Chung, Laminar mixed-convection experiments in horizontal pipes and derivation of a semi-empirical buoyancy coefficient, Int. J. Therm. Sci. 84 (2014) 335–346. [30] J.E. Moon, B.J. Chung, Time-dependent Rayleigh–Bennard convection: cell formation and Nusselt number, Nucl. Eng. Des. 274 (2014) 146–153. [31] J.H. Heo, M.S. Chae, B.J. Chung, Influences of vertical and horizontal pitches on the natural convection of two staggered cylinders, Int. J. Heat Mass Transfer 57 (2013) 1–8. [32] H.W. Coleman et al., Experimental and Uncertainty Analysis for Engineers, second ed., John Wiley Son, 1999.
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Natural convection heat transfer characteristics in vertical cavities with active and inactive top and bottom disks Gyeong-Uk Kang a, Bum-Jin Chung b,⇑ a b
Research & Development Institute, Korea Radioactive Waste Agency, 989-111 Daedeok-daero, Yuseong-gu, Daejeon 305-353, Republic of Korea Department of Nuclear Engineering, Kyung Hee University, 1732 Deokyoung-daero, Yongin-si, Gyonggi-do 446-701, Republic of Korea
a r t i c l e
i n f o
Article history: Received 15 December 2014 Received in revised form 8 April 2015 Accepted 8 April 2015
Keywords: Analogy concept Hydrodynamic interaction Natural convection Heat and mass transfer Vertical cavity
a b s t r a c t Natural convection heat transfer was investigated in vertical cavities where either all surfaces were active, or only the vertical surface was active for four different geometries, which were varied by placing a disk at the top and/or bottom of the cavity. A cupric acid–copper sulfate electroplating system was employed for mass transfer experiments exploiting the analogy with heat transfer. The Rayleigh number was varied in the range 4.55 109 6 RaLw 6 3.79 1013. Preliminary tests for a vertical pipe, upwardand downward-facing horizontal disks showed good agreement with existing correlations. The measured Nusselt numbers in the vertical cavities with all surfaces active were always greater than those with only the vertical surface active, which is attributed to greater hydrodynamic interaction of the flows generated by different surfaces. When all surfaces were active, the bottom-closed cavity exhibited the largest heat transfer rates, followed by both-closed, top-closed, and both ends open cavities; this trend was observed with laminar and turbulent flows. With only the vertical surface active, similar trends were observed except that the heat transfer rates were almost identical for both-open and top-closed cavities, which is attributed to the weak influence of the top disk on the heat transfer characteristics. Using these results, empirical correlations were derived for both laminar and turbulent flow conditions. Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction Natural convective flows in a vertical cavity are relevant to many practical applications in the design of both heat and mass transfer devices, especially the passive safety system design of nuclear power plants under hypothetical accident conditions. Much data are available for natural convection in a vertical cavity where both ends are open (i.e., a vertical heated pipe) over a wide range of Rayleigh numbers [1–3]; however, most practical problems related to engineering fields concern vertical cavities with arrangements other than a simple vertical heated pipe. The vertical cavity consists of a vertical cylindrical wall and horizontal circular disks. The horizontal circular disks seal the cavity at the top and/or bottom, and may be either active or inactive, where the term active corresponds to a heated wall and inactive to an adiabatic wall. The geometry of the vertical cavities may be classified into four cases: both ends open, bottom-closed (with an open top), top-closed (with an open bottom), and with both ends closed. There have been some reports of natural convection in vertical cavities [4–6], whereby the authors used the cavities with either ⇑ Corresponding author. Tel.: +82 31 201 3893; fax: +82 31 204 8114. E-mail address: [email protected] (B.-J. Chung). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2015.04.022 0017-9310/Ó 2015 Elsevier Ltd. All rights reserved.
all surfaces active (i.e., both the vertical walls and horizontal disks were heated), or where only the vertical surfaces were active. These works were restricted mainly to laminar flows and the bottom-closed and top-closed cavity geometries. The heat transfer behavior of the cavity geometry with both ends closed has been less well studied. Furthermore, the available data in the literature lack consistency in terms of the size of the vertical cavities, and the authors did not provide detailed phenomenological explanations of the flow interactions in the vertical cavities. For these reasons, detailed investigations are required to understand the heat transfer behavior of vertical cavities by exploring further geometrical arrangements with consistent sizes of cavity with both laminar and turbulent flows. This study investigated natural convection heat transfer due to hydrodynamic interactions in vertical cavities with various geometrical arrangements and with Rayleigh numbers in the range 4.55 109 to 3.79 1013. These ranges are sufficient to cover laminar and turbulent flows. The Nusselt number was measured for the four geometries of vertical cavities with all surfaces were active, and only the vertical surfaces were active. Exploiting the analogy between heat and mass transfer, we used a sulfuric acid-copper sulfate (H2SO4ACuSO4) electroplating system with the limitingcurrent technique to characterize the heat transfer coefficients.
391
G.-U. Kang, B.-J. Chung / International Journal of Heat and Mass Transfer 87 (2015) 390–398
Nomenclature Cb Cp d Dm Do F g GrH H hh hm I Ilim L LW M k n Nud NuH
concentration in the bulk [mol/m3] specific heat at constant pressure [J/kg K] diameter [m] mass diffusivity [m2/s] cylinder diameter [m] Faraday constant, 96,485 [C/mol] gravitational acceleration, 9.8 [m/s2] Grashof number [gbDTH3/m], [(gH3/m2)(Dq/q)] height of cathode [m] heat transfer coefficient [W/m2 K] mass transfer coefficient [m/s] electric current [A] limiting current density [A/m2] length of horizontal surface [m] surface area / perimeter projected onto a horizontal plane (H + d/4) [m] molarity [mol/l] thermal conductivity [W/m K] number of electrons in charge transfer reaction Nusselt number [hhd/k] Nusselt number [hhH/k]
Phenomenological explanations of the effects of the geometry on the flow interactions and heat transfer rates are provided, and empirical correlations based upon the results are derived for the four arrangements with laminar and turbulent flow conditions.
2. Natural convection heat transfer in a vertical cavity Table 1 lists a summary of previous studies on natural convection heat transfer in vertical cavities. For the case with both ends open, the thermal boundary layer that developed along the hot wall was thinner than the cylinder radius, and the heat transfer phenomena were similar to those for a vertical plate [1]. With this geometry, the Nusselt number can be calculated using correlations for vertical plates. Kang and Chung [7,8] showed this result experimentally and reported that a transition to turbulent flow occurred with a Grashof number of approximately GrH = 109, corresponding to a Rayleigh number of RaH = 1012 when a Prandtl number was
NuLw Pr RaH Sc ShH Shd ShLw tn U
Nusselt number [hhLW /k] Prandtl number [m/a] Rayleigh number [gbDTH3/m], [(gH3/Dm)(Dq/q)] Schmidt number [m/Dm] Sherwood number [hmH/Dm] Sherwood number [hmd/Dm] Sherwood number [hmLW /Dm] transference number uncertainty
Greek symbols a thermal diffusivity [m2/s] b volume expansion coefficient [1/K] c dispersion coefficient l viscosity [kg/m s] m kinematic viscosity [m2/s] q density [kg/m3] d velocity boundary layer thickness [m] dT thermal boundary layer thickness [m]
Pr = 2000. Their results were in good agreement with Le Fevre’s correlation for laminar flow [9] and Fouad’s correlation for turbulent flow [10]. There have been several reports of the heat transfer characteristics in bottom-closed and top-closed cavities. These cavities consist of a vertical wall and a horizontal disk either at the bottom or top. Some of the previous studies have used mass transfer experiments exploiting the analogy between heat and mass transfer systems. Somerscales and Kassemi [4] measured the natural convection heattransfer in bottom-closed cavities with all surfaces active for Rayleigh numbers in the range 7.1 107 6 Rad 6 6.9 109, where the diameter was used as the characteristic length. Krysa et al. [5] carried out experiments using bottomclosed cavities for Rayleigh numbers in the range 2 107 6 Rad 6 1.2 1010 and visualized the flows emerging from the cavity openings for a short cavity. Two types of bottom-closed cavities were formed with all surfaces active and with only the vertical surface active. Two characteristic lengths were used depending
Table 1 Previous studies for the vertical cavities. Authors
H (m)
d (m)
Range of Ra
Arrangement
Surface condition
Correlations
Le Fevre [9] Fouad [10]
– –
– –
GrH 6 109 GrH P 109
Both ends open Both ends open
– –
NuH = 0.67(GrHPr)0.25 NuH = 0.31(GrHPr)0.28
Somerscales and Kassemi [4]
0.00635, 0.0127, 0.0254 0.0127, 0.0254, 0.0508 0.019, 0.0381, 0.0762
0.0127
7.1 107 6 Rad 6 6.9 109
Bottom-closed (open top)
All surfaces active
Nud ¼ 0:232ðd=HÞ
2 107 6 RaLw 6 1.2 1010
Bottom-closed (open top)
All surfaces active Only vertical surface active
NuLW ¼ 0:559RaL0:265 W
(4)
0:265 NuH ¼ 0:480RaH
(5)
All surfaces active
NuLW ¼ 0:257RaL0:333 W
(6)
NuLW ¼ 0:187RaL0:297 W
(7)
Krysa et al. [5]
Sedahmed et al. [6]
0.003–0.0381
0.005, 0.01, 0.015, 0.025, 0.03 0.02
0:191
(1) (2) Sc0:056 Ra0:28 d
(3)
0.0254 0.0381 0.0135
0.0172 0.02, 0.0296, 0.032, 0.041
1 108 6 RaLw 6 5.02 109
Bottom-closed (open top) Top-closed (open bottom)
392
G.-U. Kang, B.-J. Chung / International Journal of Heat and Mass Transfer 87 (2015) 390–398
Table 2 Natural convection correlations for horizontal surfaces. Scholars
Orientation
Range
Correlation 9
9
Al-Arabi and El-Riedy [16]
Upward
4 10 6 RaL 6 10
Ishiguro et al. [17]
Upward
3 105 6 RaL 6 1010
Yousef et al. [18]
Upward
4 107 6 RaL 6 1.7 108
NuL ¼
Wragg et al. [19]
Downward
7 103 6 Rad 6 1 1011
Nud ¼
Fishenden and Sauders [20]
Downward
1 106 6 Rad 6 1 1011
Nud ¼
Loomba [21]
Downward
2 107 6 Rad 6 1 1011
Nud ¼
Patrick et al. [22]
Downward
–
Nud ¼
(8)
1=3
NuL ¼ 0:155RaL NuL ¼
(9)
1=3 0:20RaL 1=3 0:162RaL 2:08Ra0:178 d 0:28Ra0:25 d 0:64Ra0:22 d 0:289Rad0:25
(10) (11) (12) (13) (14)
on cavity surface conditions: Lw for all surfaces active and H with only the vertical surface active, where Lw = H + d/4, and where H is the height of the cavity and d is the diameter [11]. They reported that the Nusselt number was larger with all surfaces active than with only the vertical surface active, which was attributed to the flow at the base of the cavity, which aided the development of a boundary layer along the vertical surface. In the above two studies, the anode was located outside the cavity so that the cupric ions produced from the anode could not reach the entire cathode surfaces easily [7]. To overcome this limitation, the anode should be located at the center of the cavity to improve the current distribution. Lim and Chung [12] investigated the influence of a center anode numerically as a cold surface in a heat transfer system, and reported that a central anode in vertical pipes did not affect the cupric ion concentration near the cathode. Sedahmed et al. [6] performed experiments with bottom-closed and top-closed cavities for Rayleigh numbers in the range 1 108 6 RaLw 6 5.02 109, with all cavity surfaces active. In contrast to the two studies discussed above, the anode was located inside the cavity. They reported that the Nusselt number was larger for the bottom-closed cavity than for the top-closed cavity, which was attributed to the presence of oscillatory flow. The top three studies listed in Table 1 were carried out for a narrow range of Rayleigh numbers, which resulted from the relatively short cavities. Furthermore, the cavity sizes were not consistent between these sets of experiments, which complicates a comparison, and the reported correlations were based on different characteristic lengths (i.e., d or H or Lw). Recently, Kang and Chung [13– 15] investigated the effects of the geometrical arrangement on the heat transfer rate in a vertical cylinder. They verified the results in Refs. [4–6], and reported phenomenological explanations of the flow interactions in vertical cavities. Table 2 lists a summary of previous studies for heat transfer on horizontal planar surfaces [16–22]. It is well known that natural convection heat transfer at an upward-facing planar surface is more effective than that on a downward-facing surface because of the buoyancy of the hot fluid, which is replaced by descending cold fluid. Wragg and Loomba [23] investigated the flow patterns on upward-facing planar heated surfaces and reported that turbulent flow occurred for Rad > 3 107. The present work supplemented some of the shortage and weak points as discovered in the existing literatures by exploring a more diverse set of geometrical arrangements with a wide range of RaLw for vertical cavities with all surfaces active and with only the vertical surfaces active.
form, and contain the same classes of boundary and initial conditions [1]. The technique is attractive for achieving a high Rayleigh number with relatively small apparatus, and provides benefits of faster and more accurate measurements compared with heat transfer measurements. The dimensionless mass transfer is described by Sherwood number Sh and the Schmidt number Sc, which correspond to the Nusselt number Nu and Prandtl number Pr in a heat transfer system, respectively. The mass transfer system investigated here consisted of an H2SO4ACuSO4 electroplating system. Mass transfer results obtained using the electroplating technique must be carefully analyzed because of the large Prandtl number. The limiting electrolysis currents were measured by applying a voltage to the cell to determine the current–voltage relationship. These mass transfer measurements were based on the technique described in Refs. [24–26]. In this study, empirical relations in Eqs. (15)–(22) were used, which were originally reported buy Fenech and Tobias [27], to determine the values of the physical properties for the required dimensionless numbers. These data were accurate within an error bound of 0.5% at 22 °C. Several studies have applied this method to convection heat transfer phenomena [12–15,28–31].
c¼
C CuSO4 ; C CuSO4 þ C H2 SO4
ð21Þ
3. Experimental method and apparatus
bj ¼
1 @q : q @C j T;Ck –j
ð22Þ
q ðkg=m3 Þ ¼ 0:9978 þ 0:06406MH2 SO4 0:00167M2H2 SO4 þ 0:12755M CuSO4 þ 0:018202CuSO4 103 ;
ð15Þ
l ðC p Þ ¼ 0:974 þ 0:1235MH2 SO4 þ 0:0556M2H2 SO4 þ 0:5344M CuSO4 þ 0:53562CuSO4 ;
ð16Þ
lDm ðm2 =sÞ ¼ 0:7633 þ 0:00511MH2 SO4 þ 0:02044MCuSO4 10; ð17Þ
tCu2þ ¼ 0:2633 0:1020C H2 SO4 C CuSO4 ; DC H2 SO4 ; ¼ C CuSO4 bCuSO4 bH2 SO4 q DC CuSO4
Dq
ð18Þ ð19Þ
DC H2 SO4 ¼ 0:000215 þ 0:113075c1=3 þ 0:85576c2=3 0:50496c; DC CuSO4 ð20Þ
3.1. The analogy between heat and mass transfer 3.2. Apparatus and test matrix Heat transfer problems in a vertical cavity can be investigated experimentally using mass transfer experiments. The mathematical framework for heat and mass transfer processes are of the same
Fig. 1 shows a schematic diagram of the test facility and the electric circuit used. The test section was formed of cylindrical
393
G.-U. Kang, B.-J. Chung / International Journal of Heat and Mass Transfer 87 (2015) 390–398 Table 3 Test matrix for the cavity arrangements.
(a) Test apparatus.
d (m)
H (m)
RaLw
Pr
Cavity arrangements
Cavity surface condition
0.032
0.03–0.6
4.55 109– 3.79 1013
2094
Both ends open, Bottom-closed (open top), Top-closed (open bottom), Both ends closed
All surfaces active Only vertical surface active
temperature of the fluid and the ambient room was measured before and after each test to ensure temperature was maintained constant. The electric potential was applied using a power supply (VüPower IPS18B10) and the current was measured using a dualdisplay multi-meter (Fluke 45). Table 3 lists the test matrix for the vertical cavities. Two types of vertical cavity were used: all surfaces were active and only the vertical surface was active. The cavity height was varied in the range 0.03–0.6 m to investigate laminar and turbulent flows, which corresponds to a Rayleigh number in the range 4.55 109 6 RaLw 6 3.79 1013. The diameter of the horizontal copper disks (active) or acryl (inactive) disks sealing the top and/ or bottom of the cavity were 0.032 m, corresponding to a Rayleigh number of Rad = 5.53 109. The measurements were carried out with four different geometries: both ends open, bottomclosed (with an open top), top-closed (with an open bottom), and with both ends closed. The definition of Lw is commonly applied to cavities regardless of the arrangement. The concentration of copper sulfate was 0.1 M, and that of sulfuric acid was 1.5 M. Sulfuric acid was added to suppress electrical migration. The Schmidt number was 2094, as determined by the empirical relations in Eqs. (15)–(22). 4. Results and discussion 4.1. Verification of the experimental method
(b) Schematic diagram of electric circuit. Fig. 1. Test apparatus and schematic diagram of electric circuit.
acryl pipes to prevent corrosion by the H2SO4 and CuSO4. It was separated into upper, middle and lower sections, in which only the middle section was lined with copper to form the cathode; the height of this cathode was 0.03–0.6 m and the diameter was 0.032 m. With the electroplating system, the cathode simulates the hot wall in heat transfer system because of the buoyancy induced by decreased fluid density with the reduction of copper ions at the cathode surface. A small groove was cut at the inside or outside edges of the cathode so that the cavity height can be varied and the cathodes can adhere to each other without gaps. Acryl flanges were attached to the top and bottom of each section, which acted as a link between the test sections and the remainder of the vertical cavity. A rubber O-ring was attached between the flanges to prevent leakage of fluid. A copper anode of 2-mm-diameter was inserted from the top of the test section, which was longer than the cathode height and acted as a cupric ion source. The
In order to verify the experimental methods, preliminary tests were carried out with three geometrical arrangements: both ends open, upward- and downward-facing horizontal disks. Fig. 2 shows a comparison of the measured Nusselt numbers with existing correlations developed for these three geometries. The Nusselt numbers measured in the cavity with both ends open were in good agreement with the laminar and turbulent natural convection heat transfer correlations for a vertical plate in Eqs. (1) and (2). Those for upward- and downward-facing disks were in satisfactory agreement with the existing correlations for horizontal flat surfaces shown in Eqs. (8)–(14), regardless of the characteristic length d or L. The Nusselt numbers were larger for upward-facing disk than for the downward-facing disk due to more effective natural convection flow. These comparisons show that the mass transfer measurements described here are appropriate for the vertical cavity geometries. 4.2. Natural convection heat transfer characteristics for various geometries Fig. 3 shows the Nusselt numbers for the vertical cavities with all surfaces active and with only the vertical surface active. The results were compared with the existing laminar and turbulent natural convection correlations for a vertical plate in Eqs. (1) and (2). The triangle symbols show the results in for the bottom-closed cavity, the inverse triangle symbols show the results for the
394
G.-U. Kang, B.-J. Chung / International Journal of Heat and Mass Transfer 87 (2015) 390–398
Fig. 2. Comparison of the preliminary test results with the heat transfer correlations.
sides of symbols denote a closed face. The bottom-closed cavity exhibited the largest Nusselt number, followed by the both-closed, top-closed, both ends open cavities, both in laminar and turbulent flow conditions. Similar trends can be seen in the data shown in Fig. 3(b), except that the results with both ends open and the top-closed cavities were similar each other and to the existing correlations in Eqs. (1) and (2), indicating a the weaker influence of the inactive top plate. These trends were observed regardless of the magnitude of RaLw. In particular, the larger Nusselt number for the bottom-closed cavity than the top-closed cavity is consistent with the observations reported by Sedahmed et al. [6]. Fig. 4 shows the axial variation of convective mass transfer coefficients. In the same manner, the trends are similar to those shown in Fig. 3. In all cases, the axial variations were similar to the well-known heat transfer behavior for the internal flow in a vertical pipe. The coefficients were largest at the lowest height of 0.03 m, and decreased rapidly with increasing height until 0.3 m, after which they remained constant. It follows that, at heights in the range 0.03–0.3 m, the flow develops, and heights greater than 0.3 m correspond to a hydrodynamic fully developed flow. Because of the large Prandtl number of 2094, the thickness of a velocity boundary layer was larger than that of a thermal boundary
(a) All surfaces active
(a) All surfaces active
(b) Only vertical surface active Fig. 3. Comparison of the test results for various arrangements in a vertical cavity.
top-closed cavity, the square symbols show results for both ends closed, and the diamond symbols show results for the both ends open. The symbols were determined so that the horizontal flat
(b) Only vertical surface active Fig. 4. Axial variations of convective mass transfer coefficients.
G.-U. Kang, B.-J. Chung / International Journal of Heat and Mass Transfer 87 (2015) 390–398
layer. Thus, a hydrodynamic fully developed region could be achieved in tall cavities. Fig. 5 shows a comparison of the results for active and inactive disks. In all cases, the Nusselt number was always higher with all surfaces active than for only the vertical surface active. This is
395
attributed to greater hydrodynamic interactions of the flows that developed from the vertical walls and the horizontal active disk. 4.3. Hydrodynamic interactions of flows in vertical cavities Based on results described in Section 4.2, in this section we discuss phenomenological explanations for the influence of the cavity geometry on flow interactions and heat transfer rates.
(a) Bottom-closed cavity
(b) Top-closed cavity
(c) Both ends closed cavity Fig. 5. Comparison of all surfaces active and only vertical surface active.
4.3.1. (Case 1) bottom-closed cavity The heat transfer behavior with the bottom-closed geometry depended on the length of cavity. For a short cavity with all surfaces active, the flows were generated from the vertical wall and the active bottom disk. The flow from the vertical wall was upwards along the inner wall of the cavity. The flow developed from the active bottom disk moved upward in the middle of the cavity, which resulted in turbulent flow with a Rayleigh number of Rad = 5.53 109 [23]. In this case, the fresh fluid from outside could flow into the cavity due to negative pressure gradients, which resulted from the two upward flows, leading to secondary flows. These secondary flows interacted with the upward flows, exhibiting oscillatory behavior, which led to an enhancement of the heat exchange between the hot walls and the fresh fluids. Furthermore, the secondary flows, which were in the opposing direction to the primary flow, led to effective mixing of the fluid, increasing the turbulence by supplying a shear force. For a short cavity with only the vertical surface active, there was no flow at the inactive bottom disk. The flow was generated only by the vertical wall, and the secondary flow entering into the cavity was obstructed by the inactive disk, so that the secondary flow moved to the vertical wall. The obstructed flow added to the flow from the vertical wall and a buoyancy-aided flow of mixed convection heat transfer formed. For a long cavity, secondary flow generated during the initial stage only during the development of the boundary layer, which no longer generated due to the establishment of fully developed flow conditions. The flow at the active bottom disk consistently supplied large initial velocities to the flows along the vertical wall, and the obstructed flows from the inactive disk contributed small velocities to the flows along the vertical wall. These phenomena led to the development of a boundary layer, incurring the buoyancy-aided mixed convection flow. 4.3.2. (Case 2) top-closed cavity For a cavity with all surfaces active, the flow generated at the vertical walls moved up and was blocked by the active top disk. The flow from the active top disk moved horizontally to the top corners. The two flows in opposite directions tended to accumulate near top corner, forming light and stable layer, which led to low heat transfer rates. For a cavity with only the vertical surface active, the flow upward along the vertical wall was obstructed by the inactive top disk, and moved horizontally toward the top center. The flow then descended, forming a natural convective flow. In this way, the combination of the two convective flows formed axisymmetric motion. As shown in Fig. 4(b), the Nusselt numbers for the top-closed cavity were smaller than those with both ends open when the cavities were short, as the top disk blocked the flow and the buoyancy forces were weak, leading to small heat transfer rates. For longer cavities, however, the heat transfer rates were almost identical to those for the cavity with both ends open. It follows that the inactive top disk had little impact on heat transfer in longer cavities. 4.3.3. (Case 3) both ends closed For a cavity with all surfaces active, the three flows from the vertical wall and active top and bottom disks converged, leading
396
G.-U. Kang, B.-J. Chung / International Journal of Heat and Mass Transfer 87 (2015) 390–398
to stagnation near the top corner. The flows moving in different directions mixed, which increased the turbulence by supplying a shear force. For the cavity with an active vertical surface only, the flow from the vertical wall was obstructed by the top and bottom disks. The circulation of ascending and descending fluids resulted in the formation of cells due to the interaction of the three walls, leading to enhanced heat transfer rates. To investigate the influence of the interaction of the flows on the heat transfer rate, we compared the measured total heat transfer rates with the calculated heat transfer rates obtained using heat transfer correlations in Eqs. (1), (2), (8) and (11) for horizontal and
(a) All surfaces active
vertical surfaces with the same area. Fig. 6 shows a comparison of the heat transfer rates for three arrangements with all surfaces active, and with only the vertical surface active. The closed symbols correspond to measured heat transfer rates and the open symbols correspond to the calculated data. For short cavities (with lengths in the range 0.03–0.07 m), the gradients of the two curves were very similar; however, as the height increased, the difference between the measured and calculated data increased as a function of the cavity height for all geometries. The gradient for the closed symbols become steeper from specified heights than that for open symbols. As shown in Fig. 6, for all cases
(b) Only vertical surface active
Fig. 6. Comparison of the results with the calculated values from heat transfer correlations.
(32)
(34)
0:271 NuLw ¼ 0:45RaLw
0:281 NuLw ¼ 0:31RaLw
Max. (6.4%) Min. (0.4%) Max. (3.9%) Min. (0.7%) Max. (10.1%) Min. (0.4%) (30)
(31)
(33)
0:23 NuLw ¼ 1:35RaLw
NuLw ¼ 0:61Ra0:25 Lw
(24)
Max. (9.1%) Min. NuLw ¼ (1.8%) 0:296 (25) Max. (7.1%) Min. NuLw ¼ 2:25Ra0:221 NuLw ¼ 0:24RaLw Lw (0.6%) 0:281 (27) Max. (4.9%) Min. NuLw ¼ 1:31Ra0:231 NuLw ¼ 0:33RaLw Lw (0.2%) Same as the existing laminar and turbulent correlations (1) and (2) for
Fig. 7. Derived empirical correlations for laminar and turbulent flows.
4.4. Uncertainty analysis
Both ends open
Both ends closed Top-closed
Bottom-closed
(23)
other than the top-closed cavity with only the vertical surface active, the measured heat transfer rates were always greater than the sum of the calculated heat transfer rates for the vertical wall and horizontal surfaces. These results demonstrate the importance of the interactions of the flows. Among the three geometries, the bottom-closed cavity exhibited the largest difference between the measured and calculated heat transfer rates, followed by the cavity with both ends closed and the top-closed geometry. It follows that the bottom-closed geometry had the strongest effect on the heat transfer behavior. The data plotted in Fig. 6 reveal that obtaining heat transfer rates for such geometries by summing the contributions for individual surfaces is not accurate due to the interactions between the different flows generated by the heated surfaces. Based on these experimental results, empirical correlations were derived for the different cavities with laminar and turbulent flows, as listed in Table 4. Fig. 7 shows plots of these correlations.
NuLw ¼
0:296 0:26RaLw
(b) Only vertical surface active
2:41Ra0:221 Lw
Turbulent Error Laminar
All surfaces active Arrangements
Table 4 Derived empirical correlations for laminar and turbulent flows.
397
(a) All surfaces active
Max. (13.2%) Min. (1.0%) (26) Max. (8.2%) Min. (1.2%) (28) Max. (4.0%) Min. (0.2%) a vertical pipe
NuLw ¼
(29)
Max. (6.1%) Min.(0.3%) Max. (6.4%) Min. (1.2%) Max. (4.1%) Min. (0.1%)
NuLw ¼
0:271 0:48RaLw
Turbulent Error Laminar
Vertical surface active only
Error
0:23 1:51RaLw
Error
G.-U. Kang, B.-J. Chung / International Journal of Heat and Mass Transfer 87 (2015) 390–398
The uncertainties involved in natural convection measurements using the H2SO4ACuSO4 electroplating system were analyzed using a conventional data reduction approach [32]. Because the Sherwood number was the final dependent variable, the data reduction and uncertaintycan be expressed as follows:
398
G.-U. Kang, B.-J. Chung / International Journal of Heat and Mass Transfer 87 (2015) 390–398
ShH ¼
hm H ) ShH ¼ f ðhm ; H; Dm Þ Dm
ð35Þ
and
U 2ShH ¼
@Sh Uh @hm m
2 þ
2 2 @Sh @Sh UH þ : @H @Dm
ð36Þ
The calculated average uncertainties for the Nusselt number were about 3.63% and the average fractional uncertainties of these values were roughly 1.82%. 5. Conclusions Natural convection heat transfer were investigated experimentally in vertical cavities with/without top and bottom lids with all surfaces active and only vertical surfaces active. The diameter of the cavity was 0.032 m, and the height of the cathode was varied in the range 0.03–0.6 m, which corresponds to Rayleigh numbers in the range 4.55 109 6 RaLw 6 3.79 1013 with a fixed Prandtl number of 2094. The measurements were carried out using an H2SO4ACuSO4 electroplating system. Preliminary results show that the Nusselt numbers measured for simple geometries were in good agreement with existing correlations for natural convection developed for a vertical pipe, as well as upward- and downward-facing horizontal planar heated surfaces. Empirical heat transfer correlations were derived for each geometry and set of surface conditions based on our measured data. The main findings of this study are summarized as follows: The interactions of flows generated from different surfaces enhanced the natural convection heat transfer. The natural convection heat transfer rates for all surfaces active were always larger than those for only the vertical surface active. With all surfaces active, the bottom-closed cavity exhibited the highest heat transfer rates, followed the cavity with both ends closed, the top-closed cavity, and the cavity with both ends open. When only the vertical surface was active, similar trends appeared except for the fact that both ends open and for the top-closed cavities show similar heat transfer rates. In both cavities, the bottom surface enhanced the heat transfer greatly, while the top surface only enhanced the heat transfer only when it is active. Conflict of interest None declared. Acknowledgments This study was supported by MOTIE (The Ministry of Trade, Industry and Energy), Korea, under the radioactive Waste Management Technology Development Project (2014171020173A). References [1] A. Bejan, Convection Heat Transfer, third ed., Wiley, New York, 2003. [2] F.P. Incropera, D.P. DeWitt, Fundamentals of Heat and Mass Transfer, fifth ed., Wiley, New York, 2003.
[3] J.P. Holman, Heat Transfer, eighth ed., McGraw-Hill, New York, 1997. [4] E.F.C. Somerscales, M. Kassemi, Electrochemical mass transfer studies in open cavities, J. Appl. Electrochem. 15 (1985) 405–413. [5] J. Krysa, A.A. Wragg, D.M. Thomas, M.A. Patrick, Free convective mass transfer in open upward-facing cylindrical cavities, Chem. Eng. J. 79 (2000) 179–186. [6] G.H. Sedahmed, A.M. Ahmed, M.L. El-Rafey, A.Y. Hosney, E.A. Ayob, Natural convection mass transfer inside cylindrical cavities of different orientation, J. Appl. Electrochem. 25 (1995) 677–681. [7] K.U. Kang, B.J. Chung, The effects of the anode size and position on the limiting currents of natural convection mass transfer experiments in a vertical pipe, Trans. KSME B 34 (1) (2010) 1–8. [8] G.U. Kang, B.J. Chung, The experimental study on transition criteria of natural convection inside a vertical pipe, Int. Commun. Heat Mass Transfer 37 (8) (2010) 1057–1063. [9] E.J. Le Fevre, A.J. Ede, Laminar free convection from the outer surface of a vertical circular cylinder, in: Proceedings of the Ninth International Congress Applied Mechanics, Brussels, vol. 4, 1956, pp. 175–183. [10] M.G. Fouad, N. Ibl, Natural convection mass transfer at vertical electrodes under turbulent flow conditions, Electrochim. Acta 3 (1960) 233–243. [11] M.E. Weber, P. Austraukas, S. Petsalis, Natural convection mass transfer to nonspherical objects at high Rayleigh number, Can. J. Chem. Eng. 62 (1984) 68–72. [12] C.K. Lim, B.J. Chung, Influence of a center anode in analogy experiments of long flow duct, Int. Commun. Heat Mass Transfer 56 (2014) 174–180. [13] B.J. Chung, J.H. Heo, M.H. Kim, G.U. Kang, The effect of top and bottom lids on natural convection inside a vertical cylinder, Int. J. Heat Mass Transfer 54 (2011) 135–141. [14] G.U. Kang, B.J. Chung, Effects of the dimensions of upward- and downwardfacing vertical cavities on natural convection, Int. Commun. Heat Mass Transfer 39 (2012) 399–404. [15] G.U. Kang, B.J. Chung, The effects of top and bottom lids on the natural convection heat transfer inside vertical cylinders, J. Energy Eng. 20 (3) (2011) 242–251. [16] M. Al-Arabi, M.K. El-Riedy, Natural convection heat transfer from isothermal horizontal plates of different shapes, Int. J. Heat Mass Transfer 19 (1976) 1399–1404. [17] R. Ishiguro, T. Abe, H. Nagase, Natural convection over heated horizontal plates, Trans. Jpn. Soc. Mech. Eng. Ser. B 43 (366) (1977) 638–645 (in Japanese). [18] W.W. Yousef, J.D. Tarasuk, W.J. Mckeen, Free convection heat transfer from upward-facing isothermal horizontal surfaces, Trans. Am. Soc. Mech. Eng. J. Heat Transfer 104 (1982) 493–500. [19] A.A. Wragg, G. Batting, J. Krysa, Free convective mass transfer at down-facing horizontal surfaces with free or collared edges, Int. Commun. Heat Mass Transfer 25 (2) (1998) 175–182. [20] M. Fishenden, O.A. Saunders, An Introduction to Heat Transfer, Oxford Univ. Press, London, 1950. pp. 102–104. [21] L.P. Loomba, Free convection mass transfer at horizontal surfaces (M.Sc. thesis), University of Manchester, 1969. [22] M.A. Patrick, A.A. Wragg, D.M. Pargeter, Mass transfer by free convection during electrolysis at inclined electrodes, Can. J. Chem. Eng. 55 (4) (1977) 432– 438. [23] A.A. Wragg, R.P. Loomba, Free convection flow patterns at horizontal surfaces with ionic mass transfer, Int. J. Heat Mass Transfer 13 (1970) 439. [24] V.G. Levich, Physicochemical Hydro-dynamics, Prentice-Hall, Englewood Cliffs, NJ, 1962. [25] J.N. Agar, Diffusion and Convection at Electrodes, Discuss. Faraday Soc. 1 (1947) 27–37. [26] J.R. Selman, C.W. Tobias, Mass transfer measurement by the limiting current technique, Adv. Chem. Eng. 10 (1978) 211–318. [27] E.J. Fenech, C.W. Tobias, Mass transfer by free convection at horizontal electrodes, Electrochim. Acta 2 (1960) 311–325. [28] S.H. Ko, D.W. Moon, B.J. Chung, Applications of electroplating method for heat transfer studies using analogy concept, Nucl. Eng. Technol. 38 (2006) 251–258. [29] M.S. Chae, B.J. Chung, Laminar mixed-convection experiments in horizontal pipes and derivation of a semi-empirical buoyancy coefficient, Int. J. Therm. Sci. 84 (2014) 335–346. [30] J.E. Moon, B.J. Chung, Time-dependent Rayleigh–Bennard convection: cell formation and Nusselt number, Nucl. Eng. Des. 274 (2014) 146–153. [31] J.H. Heo, M.S. Chae, B.J. Chung, Influences of vertical and horizontal pitches on the natural convection of two staggered cylinders, Int. J. Heat Mass Transfer 57 (2013) 1–8. [32] H.W. Coleman et al., Experimental and Uncertainty Analysis for Engineers, second ed., John Wiley Son, 1999.