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Natural Convective Mass-Transfer Behavior of Horizontal and Vertical Perforated Surfaces Magdy Zaki,† Inderjit Nirdosh,*,† and Gomma Sedahmed‡ Department of Chemical Engineering, Lakehead University, Thunder Bay, Ontario, Canada P7B 5E1, and Chemical Engineering Department, Faculty of Engineering, Alexandria University, Alexandria, Egypt
Rates of free convective mass transfer at horizontal and vertical perforated plates containing circular holes were measured by determining the limiting current of the cathodic deposition of copper from acidified copper sulfate solutions. Variables studied were the vertical plate height or horizontal disk diameter, the hole diameter, and the physical properties of solution. The data at the vertical perforated plate were correlated by the equation Sh ) 0.1(Sc × Gr)0.33, while the data at the horizontal perforated plate were correlated by the equation Sh ) 0.062(Sc × Gr)0.349. A comparison of the rate of mass transfer at perforated surfaces with that at unperforated surfaces shows that the rate of mass transfer at vertical perforated surfaces is almost equal to that at unperforated vertical surfaces, while for horizontal perforated surfaces the rate of mass transfer at horizontal perforated surfaces is lower than that at unperforated surfaces. Industrial implication of the present results for the electronic industry where perforated metallic casings are used to house electronic equipments is highlighted. Introduction Perforated metallic sheets are used widely in industry to fabricate solar collectors, laundry equipment, food technology equipment such as sugar mills, flour mills, and rice mills, equipment used in the filtration and separation of liquids, solids, and gases, and kitchen water drains. The most important application of perforated metallic sheets is in the manufacture of electronic equipment. Thermal control of electronic equipment is necessary to maintain a constant component temperature below the manufacture’s maximum specified service temperatures which are typically between 80 and 100 °C. An increase of temperature beyond this range reduces the reliability of the system. The trend toward size minimization of electronic equipment through the use of semiconductor chips has made the problem of heat generation and accumulation more acute.1 One of the techniques used to alleviate the problem of heat accumulation inside the electronic equipment is to fabricate the outer equipment casing of a perforated metallic sheet. The presence of perforations in the outer casing allows heat to be transferred from inside to outside by natural convection where hot air escapes from the perforations, giving way to cold air. Perforated metallic sheets are subjected during their fabrication to diffusion-controlled metal-finishing processes such as electroplating, electroless plating (chemical or autocatalytic plating), electropolishing, and etching. To predict the rate of such processes, natural convection mass transfer at perforated sheets should be studied. Natural convection dominates the kinetics of diffusion-controlled metal-finishing processes carried out in unstirred solutions and contributes a good deal to the kinetics of these processes if the solution is mildly stirred.2,3 In view of the analogy between heat and mass transfer, it is hoped that the present study would also assist in predicting natural convection heat transfer at perforated plates.4 † ‡
Lakehead University. Alexandria University.
The present study was carried out by measuring the limiting current of the cathodic deposition of copper from acidified copper sulfate5 using horizontal and vertical copper cathodes. Experimental Section The apparatus (Figure 1a) consisted of a cell and an electrical circuit. The cell used for vertical plates was a plexiglass rectangular container of the dimensions 4.5 cm × 10 cm for the base and 60 cm height. The cathode was a copper-plated perforated steel plate of 10 cm width and 60 cm height; the cathode was placed between two copper anodes of similar dimensions. The cathodeanode separation was 2.2 cm; this distance is sufficient to eliminate the effect of the cathode-anode separation on the rate of mass transfer at the cathode.6-8 The cathode and the two anodes were fixed in the cell by inserting them through side grooves machined in the cell walls. Four different cathodes with different numbers of holes per square meter and different hole diameters were used. The relevant details are given in Table 1 and Figure 1b. Active cathode heights were 5, 10, 15, 20, 30, 40, and 50 cm; the active electrode height was controlled by isolating the upper part of the electrode with an epoxy resin. The cell used for measuring the rate of mass transfer at a perforated horizontal disk electrode was made of a cylindrical Plexiglas column of 15 cm diameter and 20 cm height. The anode was a cylindrical copper sheet lining the column wall. The cathode was a perforated copper-plated steel disk of diameters 4, 6, 8, 10, and 12 cm. Each cathode was brazed at its center to a 2 mm diameter insulated copper wire which acted as a cathode holder and a current feeder. The electrical circuit consisted of a 10 V dc power supply with a voltage regulator and a multirange ammeter connected in series with the cell. The limiting current of the cathodic deposition of copper from acidified copper sulfate was measured by increasing the current stepwise and measuring the steady-state cath-
10.1021/ie0109556 CCC: $22.00 © 2002 American Chemical Society Published on Web 06/04/2002
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Figure 2. Typical polarization curves at a vertical perforated plate at different cathode heights.
Figure 1. (a) Apparatus. (b) Perforated-plate geometry used in the present work.
Figure 3. Variation of K with the vertical cathode height at different CuSO4 concentrations.
Table 1. Geometric Characteristics of the Perforated Plates Used in the Present Work
Table 2. Physical Properties of the Solutions
geometry no. hole diameter dh, cm pitch, cm open area, % no. of holes/m2 thickness, cm
1
2
3
4
0.9 1.25 47.01 7390 0.15
0.6 0.9 40.31 14256 0.15
0.45 0.6 51.01 32075 0.15
0.3 0.5 32.65 46188 0.07
ode potential against a Cu/CuSO4 reference electrode by means of a high-impedance voltmeter. The reference electrode was placed in the cup of a Luggin tube filled with the cell solution; the tip of the Luggin tube was positioned 0.5-1 mm from the middle of the cathode surface. Four CuSO4 concentrations were used: 0.050, 0.097, 0.189 and 0.240 M. In all cases 1.5 M H2SO4 was used as a supporting electrolyte to eliminate electrical migration. The CuSO4 concentration was checked with iodometry.9 All solutions were prepared using distilled water and analytical reagant chemicals. Before each run, the cathode was cleaned as mentioned elsewhere.7 Solution physical properties (F, µ, and D) required to correlate the present data as well as the density difference ∆F between the solution bulk and the interfacial solution were taken from the literature.10,11 The solution temperature was kept at 23 ( 1 °C by placing the cell in a thermostated water bath. Solution properties are given in Table 2.
CuSO4 concn, M
µ, P
F, g/cm3
D × 105, cm2/s
∆F, g/cm3
Sc
0.050 0.097 0.189 0.240
1.26 1.263 1.3 1.345
1.0933 1.0968 1.1034 1.1046
0.593 0.594 0.58 0.557
0.00675 0.0133 0.0248 0.0319
1943 1939 2031 2186
Results and Discussion Figure 2 shows typical polarization curves at different electrode heights; for relatively short electrodes, curves with well-defined limiting current plateaus were obtained. As the electrode height increases, the plateau becomes inclined; in this case the limiting current was taken at the middle of the inclined plateau. The limiting current obtained from these curves was used to calculate the mass-transfer coefficient according to the equation4,5
K ) I/ZFAC
(1)
Figures 3 and 4 show the effect of vertical perforated cathode height on the mass-transfer coefficient at different copper sulfate concentrations and different hole diameters; the mass-transfer coefficient shows a negligible dependence on the electrode height. Figure 4 shows that the mass-transfer coefficient tends to decrease slightly with increasing hole diameter. An overall mass-transfer correlation was envisaged in terms of the dimensionless groups Sh, Sc, and Gr usually used in correlating natural convection masstransfer data. Figure 5 shows that the present data for
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Figure 4. Variation of K with the vertical cathode height at different hole diameters.
Figure 5. Overall mass-transfer correlation at a vertical perforated metal plate.
the conditions 1 × 1010 < Sc × Gr < 5 × 1013 and 1939 < Sc < 2186 fit the equation
Sh ) 0.1(Sc × Gr)0.33
(2)
with a correlation coefficient of 0.97 and an average deviation of (10%; the electrode height was used as a characteristic length in calculating Sh and Gr. The exponent 0.33 as well as the independence of the masstransfer coefficient on the electrode height testify to the turbulent nature of the free convection stream at the vertical perforated plate.5 To throw some light on the role of perforations on the rate of mass transfer, it would be instructive to compare the present data with the data obtained using unperforated vertical plates. Wilke et al.10,12 correlated their vertical plate laminar flow data for the conditions 107 < Sc × Gr < 1012 by the equation
Sh ) 0.67(Sc × Gr)0.25
(3)
Fouad and Ibl13 correlated their vertical plate transitional flow data for the conditions 4 × 1013 < Sc × Gr < 1015 by the equation
Sh ) 0.31(Sc × Gr)0.28
(4)
A comparison of eq 2, obtained in the present case, with eqs 3 and 4 obtained for unperforated vertical plates under laminar and transitional flow, respectively, shows that the presence of perforation enhances the degree of turbulence as indicated by the exponent 0.33 of eq 2 compared to the values 0.25 and 0.28 of eqs 3 and 4, respectively. The enhanced degree of turbulence at the perforated plate may be attributed to the repeated buildup and disruption of the hydrodynamic boundary layer as the buoyancy-driven solution moves upward
Figure 6. Effect of the horizontal cathode diameter on the masstransfer coefficient at different concentrations.
Figure 7. Variation of K with the horizontal cathode diameter at different hole diameters.
from the plate leading edge to the upper edge past the holes. A hydrodynamic boundary layer starts to develop at the leading edge of the perforated plate; when the rising stream reaches a hole, boundary layer separation with probable eddy formation takes place. Despite the turbulent nature of the natural convection at perforated plates, the mass-transfer coefficient is almost equal to the value predicted from eqs 3 and 4 for a given Sc × Gr within the present range of conditions. However, the turbulent flow at the perforated plate leads to a more uniform mass-transfer coefficient all over the plate than given by the laminar (eq 3) or transition flow (eq 4).14 The uniform mass-transfer coefficient is a welcome advantage in electroplating because it produces an electrode of a uniform thickness. Figures 6 and 7 show the effect of the diameter of a horizontal perforated disk with both the upward and downward facing sides active on the mass-transfer coefficient. The plots indicate that the mass-transfer coefficient shows a slight decrease with increasing disk diameter. Figure 7 shows that the hole diameter has little effect on the mass-transfer coefficient at a perforated horizontal disk. The increase in the mass-transfer coefficient (K ) D/δ) with increasing CuSO4 concentration, as shown in Figures 3 and 6, may be attributed to the increase in the buoyancy force and Gr,10 with a consequent increase in the upward solution velocity which decreases the diffusion layer thickness (δ). This increases the masstransfer coefficient by an amount which outweighs the retarding effects of the increase in viscosity with increasing CuSO4 concentration such as the decrease in diffusivity and the increase in the force resisting the flow. Figure 8 shows that the mass-transfer data at perforated horizontal surfaces for the conditions 6 × 109
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Figure 10. Approximate pattern of natural convection flow at a downward facing surface with a hole.
Figure 8. Overall mass-transfer correlation at a horizontal perforated metal disk.
A comparison of the average mass-transfer coefficient of a horizontal unperforated disk calculated from eqs 7 and 8 with the value calculated for the perforated disk using eq 5 reveals that the average mass-transfer coefficient at the unperforated disk is higher than that at the perforated disk. The low average mass-transfer coefficient at the perforated disk compared to the average value at the unperforated disk may be attributed to the pronounced interaction of natural convection at the lower side with that at the upper side. As indicated in Figure 10, at the downward facing surface of the disk, the depleted natural convection stream moves radially parallel to the disk,15,16 and when the horizontally moving depleted solution meets a hole, it rises vertically in the hole to the upper surface where it dilutes the solution contacting the surface with a consequent decrease in the rate of mass transfer at the upward facing surface. Equations 2 and 5, which correlate the present data for vertical and horizontal perforated surfaces, respectively, may be extended to other Schmidt and Prandtl numbers through the function of Churchill and Churchill17
f(Sc) ) [1 + (0.5/Sc)9/16]-16/9 Figure 9. Overall mass-transfer correlation at a horizontal perforated metal disk.
Combining eqs 2, 5, and 9 yields the equations
< Sc × Gr < 1012 and 1943 < Sc < 2338 fit the equation
Sh ) 0.062(Sc × Gr)0349
(5)
with a correlation coefficient of 0.9 and an average deviation of (14%. The disk diameter was used as a characteristic length in calculating Sh and Gr. The slight effect of the hole diameter on the rate of mass transfer is taken into consideration, and as shown in Figure 9, the data can be correlated by the equation
Sh ) 0.059(Sc × Gr)0.35(dh/d)0.014
(6)
with a correlation coefficient of 0.9 and an average deviation of (15%. To appreciate the role of perforation in the masstransfer behavior of a horizontal perforated surface, with both the upper and lower surfaces active, the present data should be compared with previous data obtained using unperforated horizontal surfaces. Sedahmed and Nirdosh8 correlated their natural convection mass-transfer data at an upward facing horizontal disk (lower surface inactive) by the equation
Sh ) 0.16(Sc × Gr)0.33
(7)
For a downward facing horizontal disk (upper surface inactive), the data of Selman and Tavakoli-Atar15 and those of Wragg et al.16 were correlated by the equation
Sh ) 2.08(Sc × Gr)0.178
(8)
(9)
Sh ) 0.1[f(Sc) Gr]0.33
(10)
Sh ) 0.062[f(Sc) Gr]0.349
(11)
and
Recently,18 some questions have arisen concerning the quantitative comparability of heat and mass transfer in the turbulent regime because the largest Prandtl number is 100 as compared to much higher values of Sc herein. In view of this, rates of heat transfer at perforated plates predicted from eqs 10 and 11 are approximate values. Conclusions 1. Despite the turbulent natural convection masstransfer mechanism at vertical perforated plates, the rate of natural convection mass transfer at a vertical perforated plate for a given Sc × Gr is almost equal to the value at a vertical unperforated plate; the turbulent nature of the flow at the perforated plate makes the distribution of the rate of mass transfer more uniform all over the plate than in the case of unperforated plates where the rate of mass transfer decreases with increasing plate height within the present range of Sc × Gr owing to the laminar or transitional nature of the flow. 2. For a given Sc × Gr, the rate of natural convection mass transfer at a perforated horizontal surface is less than that at an unperforated horizontal surface with both the upward and downward facing surfaces active. The dimensionless equations obtained in the present work can be used to predict the rate of diffusion-
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controlled metal finishing processes which might be used in fabricating perforated metals. Because the present preliminary study was limited to perforated plates with circular holes, future studies should include other hole geometries. Also the effect of the number of holes per unit area for a given hole diameter needs to be studied to test the possibility that as the number of holes is reduced the behavior of the perforated plate would approach that of an unperforated plate. Nomenclature A ) true area of the perforated surface allowing for the presence of holes, m2 C ) bulk concentration of copper sulfate, mol/m3 d ) horizontal disk diameter, m dh ) hole diameter, m D ) diffusivity of Cu2+, m2/s F ) Faraday’s constant, C/mol g ) acceleration due to gravity, m/s2 I ) limiting current, A K ) mass-transfer coefficient, m/s L ) vertical plate height, m Z ) number of electrons involved in the reaction Gr ) Grashof number (gL3∆F/Fν2) Sc ) Schmidt number (ν/D) Sh ) Sherwood number (KL/D) for the vertical plate and (Kd/D) for the horizontal disk µ ) solution viscosity, kg/m‚s ν ) solution kinematic viscosity, m2/s F ) solution density, kg/m3 ∆F ) density difference between the solution bulk and the interfacial solution, kg/m3
Literature Cited (1) Peterson, G. P.; Ortega, A. Thermal Control of Electronic Equipments and Devices. Adv. Heat Transfer 1990, 20, 181. (2) Tobias, C. W.; Hickman, R. G. Ionic Mass Transfer by Combined Free and Forced Convection. Z. Phys. Chem. 1965, 145, 229. (3) White, F. W. Heat and Mass Transfer; Addison-Wesley Publishing Co.: New York, 1988.
(4) Chiang, H. D.; Goldstein, R. J. Application of the Electrochemical Mass Transfer Technique to the Study of Buoyancydriven Flows. In Transport Phenomena in Heat and Mass Transfer; Reizes, J. A., Ed.; Elsevier Science Publisher: New York, 1992. (5) Selman, J. R.; Tobias, C. W. Mass Transfer Measurement by the Limiting Current Technique. Adv. Chem. Eng. 1978, 10, 211. (6) Bohm, U.; Ibl, N. About the decay of natural convection with electrolysis in a tight enclosed space (electrode-electrode system) [Ueber Das Abklingen Der Naturalichen Konvektion Bei Der Elecktrolyse in Engen Raumen (System Elektrode-Elektrode)]. Electrochim. Acta 1968, 13, 891. (7) Sedahmed, G. H.; Shemilt, L. W. Free Convection Mass Transfer in Vertical Annuli. Chem. Eng. Commun. 1982, 14, 307. (8) Sedahmed, G. H.; Nirdosh, I. Natural Convection Mass Transfer at an Enclosure Between Two Horizontal Discs. Chem. Eng. Commun. 1991, 101, 93. (9) Vogel, A. I. A Text Book of Quantitative Analysis, 3rd ed.; Longmans: London, 1961. (10) Wilke, C. R.; Eisenberg, M.; Tobias, C. W. Correlation of Limiting Current under Free Convection Conditions. J. Electrochem. Soc. 1953, 100, 513. (11) Eisenberg, M.; Tobias, C. W.; Wilke, C. R. Selected Physical Properties of Ternary Electrolytes Employed in Ionic Mass Transfer Studies. J. Electrochem. Soc. 1956, 103, 413. (12) Wilke, C. R.; Tobias, C. W.; Eisenberg, M. Free Convection Mass Transfer at Vertical Plates. Chem. Eng. Prog. 1953, 49, 663. (13) Fouad, M. G.; Ibl, N. Natural Convection Mass Transfer at Vertical Electrodes under Turbulent Flow Conditions. Electrochim. Acta 1960, 3, 233. (14) Ibl, N.; Dossenbach, O. Convective Mass Transport. Compr. Treatise Electrochem. 1983, 6, 133. (15) Selman, J. R.; Tavakoli-Atar, J. Free Convective Mass Transfer to a Rod Shaped Vertical Electrode. J. Electrochem. Soc. 1980, 127, 1049. (16) Wragg, A. A.; Batting, G.; Krysa, J. Free Convective Mass Transfer at Down Facing Horizontal Surfaces with Free or Collared Edges. Int. Commun. Heat Mass Transfer 1998, 25, 175. (17) Churchill, S. W.; Churchill, R. U. Comprehensive Correlating Equation for Heat and Component Transfer by Free Convection. AIChE J. 1975, 21, 604. (18) Churchill, S. W. Turbulent Flow and Convection. Adv. Heat Transfer 2001, 34, 325.
Received for review November 27, 2001 Revised manuscript received April 11, 2002 Accepted April 18, 2002 IE0109556
