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I n d . Eng. Chem. Res. 1990, 29, 1728-1731
lution. Chem. Eng. Sci. 1983, 38, 1411-1429. Bosch, H.; Versteeg, G . F.; van Swaaij, W. P. M. Gas-liquid mass transfer with parallel reversible reactions-I. Absorption of COz into solutions of sterically hindered amines. Chem. Eng. Sci. 1989, 44, 2723-2734. Chakraborty, A. K.; Astarita, G.; Bischoff, K. B. COz absorption in aqueous solutions of hindered amines. Chem. Eng. Sci. 1986,41, 997-1003. Danckwerts, P. V. The reactions of COz with ethanolamines. Chem. Eng. Sci. 1979, 34, 443-445. Hikita, H.; Asai, S.; Ishikawa, H.; Honda, M. The kinetics of reactions of carbon dioxide with MEA, DEA and TEA by a rapid mixing method. Chem. Eng. J. 1977, 13, 7-12. Marquardt, D. W. An algorithm for least squares determination of nonlinear parameters. J.SOC.Ind. Appl. Math. 1963, 11, No. 2. Sartori, G.; Savage, D. W. Sterically hindered amines for COz removal from gases. Ind. Eng. Chem. Fundam. 1983,22,239-249.
Sartori, G . ; Ho, W. S.; Savage, D. W.; Chludzinski, G . R.; Wiechert, S. Sterically-hindered amines for acid-gas absorption. Sep. Purif. Methods 1987, 16(2), 171-200. Sharma, M. M. Kinetics of reactions of carbonyl sulphide and carbon dioxide with amines. Trans. Faraday SOC. 1965, 61, 681-688. Versteeg, G. F.; van Swaaij, W. P. M. On the kinetics between C 0 2 and alkanolamines both in aqueous and non aqueous solutions11. Tertiary amines. Chem. Eng. Sci. 1988, 43, 587-591. Yih, S. M.; Shen, K. P. Kinetics of carbon dioxide reaction with sterically hindered AMP aqueous solutions. Znd. Eng. Chem. Res. 1988, 27, 2237-2241. Zioudas, A. P.; Dadach, Z. Absorption of COz and HzS in sterically hindered amines. Chem. Eng. Sci. 1986, 41, 405-408.
Received for review August 4, 1989 Revised manuscript received March 15, 1990 Accepted March 23, 1990
Natural Convection Mass Transfer in Slender Conical Cavities Gomaa H. Sedahmed*tt and Abdel-Moneum M. Ahmed* Chemical Engineering Department, Faculty of Engineering, and Chemistry Department, Faculty of Science, Alexandria University, Alexandria, Egypt
The rates of free convection mass transfer inside a conical cavity were studied by measuring the limiting current of the cathodic deposition of copper from acidified copper sulfate solution. T h e variables studied were the physical properties of the solution and the slant height of the cavity. The data were correlated in the range 1.46 X 1O'O < ScCr < 8.3 X 10" with eq 3. Comparison with previous mass-transfer studies on hemispherical, rectangular, and cylindrical cavities shows that cavity geometry plays a significant role in determining the nature of flow and the rate of mass transfer inside the cavity.
Introduction Although much work has been done on natural convection heat transfer in enclosures (Ostrach, 19721, little has been done on natural convection mass transfer in these geometries despite their practical importance. Previous studies of natural convection mass transfer in enclosures include geometries such as horizontal (Sedahmed and Shemilt, 1981) and vertical annuli (Sedahmed and Shemilt, 1982), horizontal tube (Sedahmed and Shemilt, 1983), a vertical narrow gap (Bohm et al., 1966), and cylindrical (Somerscales and Kassemi, 1985) and rectangular cavities (Kamotani et al., 1985). The present work deals with natural convection mass transfer in conical cavities. Conical cavities are encountered frequently in chemical engineering practice either alone or as a part of a more complex geometry; for instance, in the design of storage tanks, chemical reactors, and crystallizers,a conical bottom is preferred to other geometries because it allows complete drainage of the solution. Other equipment such as cyclones, centrifuges, and hoppers also use conical cavities. The present study assists in predicting the rate of diffusion-controlled reactions that might take place in conical cavities under free convection or weak forced convection with free convection contributing, e.g., electroplating,
f
Chemical Engineering Department. Chemistry Department.
electropolishing, corrosion, electroless plating, electroforming, and electrochemical machining. In view of the analogy between heat and mass transfer, it is hoped that the present work may be useful to heat transfer since only heat transfer to cylindrical cavities was studied (Japkise, 1973). The present case corresponds to a cavity with a constant-temperature heated wall where the fluid density at the wall is less than that in the bulk. Cavities filled with liquids are used in cooling equipment such as nuclear reactors, transformer cores, turbine blades, and internal combustion engines under the name thermosyphon (Japkise, 1973). The rates of mass transfer were measured by the well-known electrochemical technique that involves measuring the limiting current of the cathodic deposition of copper from acidified copper sulfate solution (Selman and Tobias, 1978).
Experimental Technique The apparatus (Figure 1) consisted of a cell and electrical circuit. The cell consisted of a vertical conical cavity cathode machined in a copper block. The cavity was filled with electrolyte to different heights. Six cavities were machined with apex angles 32,36,42,48,56, and 68'. The slant height ranged from 5 to 9.1 cm. A 2-mm-diameter vertical copper wire was placed in the center of the cavity to act as the anode. The advantage of placing the anode inside the cavity is that the primary current distribution
0888-5885/90/2629-1728$02.50/0 @Z1990 American Chemical Society
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Ind. Eng. Chem. Res., Vol. 29, No. 8, 1990 1729
Potentiometer
I
1
1
slant height z 8 . 5 c m apex angle f
Luggin tube with a reference electrode
0.47M
2 m m diameter copper wire anode
J
LI
Electrolyte level
N
A conical cavity machined i n a copper blocklcathodel
5
3
i V
T
x
c C
6 volt d.c power supply
Figure 1. Cell and electrical circuit.
inside the cavity is uniform. Previous studies (Somerscales and Kassemi, 1985; Sedahmed and Shemilt, 1986) where the counter electrode was placed outside the cavity revealed that current distribution was not uniform inside the cavity. The electrical circuit consisted of a 6-V dc power supply with a voltage regulator and a multirange ammeter connected in series with the cell. Polarization curves were obtained by increasing the cell current stepwise and measuring the steady-state anode potential against a reference electrode. The reference electrode consisted of a copper wire immersed in the cup of a Luggin tube filled with the cell solution; the tip of the Luggin tube was placed 0.5-1 mm from the cavity wall. The potential difference between the reference electrode and the cathode was measured by means of a high impedance potentiometer. Six copper sulfate solution were used, namely, 0.05, 0.1, 0.189, 0.247,0.376, and 0.47 M; in all cases, 1.5 M H2S04 was used as a supporting electrolyte. All solutions were prepared from AR grade chemicals. Before each run, the cathode surface was treated as mentioned elsewhere (Wilke et al., 1953). Each experiment was repeated twice with a fresh soluton; the temperature was 23 f 1 O C .
*
0
E2 k U
1
O.1M
0
0.05 M
0
I
-300
-500 cothode potential
I
,m V
-700
Figure 2. Typical current-potential curves obtained a t different copper sulfate concentrations: (A)0.05 M, (0) 0.1 M, ( 0 )0.189 M, (A)0.47 M, (0) 0.376 M, ( X ) 0.47 M. Apexongle
iL B O
Copper sulphate c o n t X O.2L9
M
0.376
Results and Discussion Figure 2 shows typical polarization curves from which the limiting current was determined and used to calculate the mass-transfer coefficient from the equation
1.7c
..
0.8
Figure 3 shows the effect of the slant height of the cavity on the mass-transfer coefficient; the mass-transfer coefficient decreases with increasing slant height. This decrease in the mass-transfer coefficient indicates a laminar flow mechanism where the hydrodynamic boundary layer and the diffusion layer thickness increase with slant height. The data agree fairly well with the prediction of the hydrodynamic boundary layer, namely, K c1: L4.25 (2) An overall mass-transfer correlation was envisaged in terms of the dimensionless groups Sh, Gr, and Sc usually used in correlating free convection mass-transfer data. The physical properties used in calculating these dimensionless groups were taken from the literature (Wilke et al., 1953; Eisenberg et al., 1956). Following previous studies on heat and mass transfer at cones (Stewart, 1971; Shenoy, 1983; Alamgir, 1979; Ahmed and Sedahmed, 1988; Patric and
0.9 Log
L
1
Figure 3. Effect of the slant height on the mass-transfer coefficient: ( X ) 0.249 M, ( 0 )0.376 M.
Wragg, 1984), the slant height was taken as the characteristic length in calculating Sh and Gr. Figure 4 shows that for the conditions 1.46 X 1O'O < ScGr < 8.3 X 10" and 32" < 8 < 68" the data fit the equation Sh = 0.89(S~Gr)O.~~ (3) with an average deviation of f 5 % . An attempt to correlate the data using g cos (8/2) instead of g to calculate Gr did not improve upon using eq 2. The exponent -0.25 in eq 2 confirms the fact that under the present conditions mass is transferred inside conical cavities by laminar flow. It is noteworthy that eq 2 is in excellent agreement with the natural convection mass-transfer equation obtained for vertical cones with downward pointing apexes. Patric and
1730 Ind. Eng. Chem. Res., Vol. 29, No. 8, 1990
slender cavities and should not be applied to cavities with larger apex angles where the boundary layer separation may take place. It would be of interest to compare the present data with the data obtained by using other cavity geometries to shed some light on the role of cavity geometry in the rate of mass or heat transfer. Somerscales and Kassemi (1985) studied natural convection mass transfer in cylindrical cavities using the electrochemical technique. These authors correlated their data for vertical cavities with the equation
Sh = 0 . 2 3 2 ( G r S ~ ) ~ ~ ~ ~ S ~ ~ ~(9)~ ~ ~ ( d / h 1 5
I,
I J
t o g I sc
I2
12
Gr I
Figure 4. Overall mass-transfer correlation: (A)0.05 M, (0) 0.1 M, (A)0.189 M, (0) 0.376 M, ( X ) 0.376 M, ( 0 )0.478 M
Wragg (1984) correlated their data for the range 2 C ScGr < 1 X 10" by the equation
Sh
X
lo8
= 0.87(S~Gr)O~~
(4) while Ahmed and Sedahmed (1988) correlated their data for the range 4.9 X 1O1O C ScGr C 9 X 10" by the equation
Sh = 0 . 8 7 7 ( S ~ G r ) O ~ ~ (5) The agreement between eq 2 and eq 3 and 4 may suggest that secondary flow, which may arise from recycled natural convection streams in the bulk of the cavity, has a little effect on the rate of mass transfer inside the conical cavity. Theoretical analysis by Shenoy (1983) of natural convection heat transfer between isothermal vertical conical surfaces with downward pointing apexes and Newtonian liquids of high Pr has led to equation Nu = 0 . 7 1 8 ( G ~ P r ) O ~ ~ (6) The above equation was obtained without considering the effect of curvature. When curvature was considered in the analysis, it was found that the rate of heat transfer increases above the value predicted from eq 6 by an amount depending on curvature, E , which was defined by the equation 2 € = (7) Gr0.25tan (0/2) The discrepancy between the coefficient of eq 5 and that of eq 2 may be ascribed to the neglect of the curvature effect and the simplifying assumption, made in solving the hydrodynamic boundary layer equations, that the thermal boundary layer and the hydrodynamic boundary layer have equal thicknesses. The agreement between mass transfer in conical cavities and in downward pointing cones seems surprising in view of the fact that in the case of conical cavity the natural convection stream flows past an upward facing inclined surface while in the case of downward pointing cones the natural convection stream flows past a downward facing inclined surface. This finding is consistent with the finding of Patric and Wragg (1984), who studied natural convection mass transfer at inclined plates using electrochemical and optical techniques. According to these authors, mass transfer a t upward facing and downward facing plates where the angle of inclination is relatively small can be represented by the equation Sh = 0 . 6 8 ( S ~ G r ) ~ , ~ ~ (8) Gr was calculated by using the plate height as a characteristic length and g cos a. Patric and Wragg found that by increasing a the boundary layer separation takes place from the upward facing plate and eq 8 becomes no longer valid. In view of this, it should be emphasized that eq 2 applies only to
Kamotani et al. (1985), who studied the combined natural convection heat and mass transfer in a rectangular cavity using the electrochemical technique, correlated their data with the equation
Sh = 0 . 6 7 ( S ~ G r ) ~ . ~ ~
(10) Sedahmed and Nirdosh (1989), who studied free convection mass transfer in a horizontal hemispherical cavity using the electrochemical technique, correlated their data with the equation
Sh = 0 . 1 4 3 ( S ~ C r ) ~ . ~ ~
(11)
Equations 3 and 9-11 show that cavity geometry has a profound effect on the rate of natural convection mass transfer; while slender conical cavities and rectangular cavities exhibit laminar flow mass-transfer behavior, cylindrical cavities and hemispherical cavities show unstable and turbulent flow mass-transfer behavior, respectively. This result could be of importance for the design of thermosyphons by virtue of the analogy between heat and mass transfer.
Nomenclature C = specific heat
c'=bulk concentration of copper sulfate
D
= diffusivity of copper sulfate F = Faraday's constant g = acceleration due to gravity h = heat-transfer coefficient I = limiting current density K = mass-transfer coefficient k = thermal conductivity L = slant height of the cavity Z = number of electrons involved in the reaction Gr = Grashof number, gL3Ap/v2pi Sc = Schmidt number, u/pD Pr = Prandtl number, C p u / k Pr = Prandtl number, C p u / k Sh = Sherwood number, KLID Nu = Nusselt number, hllk u = viscosity of the solution
Greek Letters v = kinematic viscosity of the solution Pp = density difference between the bulk solution interfacial
solution = density of the interfacial solution and the bulk solution, respectively 8 = apex angle of the conical cavity LY = angle between the plate and vertical position (eq 8) pi,p
Literature Cited Ahmed, A. M.; Sedahmed, G. H. Natural convection mass transfer at conical surfaces. J . Appl. Electrochem. 1988, 18, 196-199. Alamgir, Md. Over-all heat transfer from vertical cones in laminar free convection: an approximate method. Trans ASME 1979, 101, 174-176.
Ind. Eng. C h e m . Res. 1990,29, 1731-1734 Bohm, U.; Ibl, N.; Frei, A. M. Zur Kenntnis der naturlichen Knvektion bei der Elektrolyse in engen raumen. Electrochim. Acta 1966,11,421-434. Eisenberg, M.; Tobias, C. W.; Wilke, C. R. Selected physical properties of ternary electrolytes employed in ionic mass transfer 1956,103,413-416. studies. J . Electrochem. SOC. Japkise, D. Advances in thermosyphon technology. Adv. Heat Transfer 1973,9,1-111. Kamotani, Y.; Wang, L. W.; Ostrach, S.; Jiang, H. D. Experimental study of natural convection in shallow enclosures with horizontal temperature and concentration gradients. Znt. J. Heat Mass Transfer 1985,28,165-173. Ostrach, S. Natural Convection in enclosures. Adv. Heat Transfer 1972,8,161-226. Patric, M. A.; Wragg, A. A. Steady and transient natural convection a t inclined planes and cones. Phys. Chem. Hydrodyn. 1984,5, 299-305. Sedahmed, G. H.; Nirdosh, I. Free convection mass transfer in hemispherical cavities. Chem. Eng. Res. Des. 1989,in press. Sedahmed, G. H.; Shemilt, L. W. Natural convection mass transfer in horizontal annuli. Lett. Heat Mass Transfer 1981,8,515-523. Sedahmed, G.H.; Shemilt, L. W. Free Convection mass transfer in
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vertical annuli. Chem. Eng. Commun. 1982,14,307-316. Sedahmed, G.H.;Shemilt, L. W. Free convection mass transfer in horizontal tubes. Chem. Eng. Commun. 1983,23,1-10. Sedahmed, G. H.;Shemilt, L. W. Electropolishing of cylindrical cavities under natural convection conditions. Surf. Coat. Technol. 1986,27,279-285. Selman, J. R., Tobias, C. W. Mass transfer measurement by the limiting current technique. Adu. Chem. Eng. 1978,10,211-318. Shenoy, A. V. Laminar natural convection heat transfer from a slender vertical cone to a power-law fluid. Can. J. Chem. Eng. 1983,61,869-872. Stewart, W. E. Asymptotic calculation of free convection in laminar three-dimensional systems. Znt. J. Heat Mass Transfer 1971,14, 1031. Somerscales, E. F. C.; Kassemi, M. Electrochemical mass transfer studies in open cavities. J . Appl. Electrochem. 1985,15,405-413. Wilke, C . R.;Eisenberg, M.; Tobias, C. W. Correlation of limiting currents under free convection conditions. 1953,100,513-523.
Received for review J u n e 21, 1989 Revised manuscript received February 22, 1990 Accepted April 25, 1990
COMMUNICATIONS Theoretical Equations for Agar-Diffusion Bioassay On the basis of two-dimensional diffusion, theoretical equations for cup-plate and paper-disk methods of bioassay have been established. With cycloheximide as the test substance, samples were taken at various times and radial distances and the theoretical equations have been verified. Integrating with the concepts of critical concentration and critical time, the possibility of their application for practical uses has been tested, and the results were found to be in conformity with that of the Cooper equation. The method of agar-diffusion bioassay has long been used for estimating the potency of antibiotics, yet its theory has not been sufficiently developed. The most popular theory is that of Cooper and Woodman (1946), starting from linear diffusion. Later, Mitchison and Spicer (19491, Vesterdal(1947), and Humphrey and Lightbrown (1952) put forth similar equations. All of them share the disadvantages that they are based on the diffusion of antibiotics through the homogeneous agar medium, without taking into consideration their partition between the solution and agar phases. We have developed the mathematical models of cupplate and paper-disk methods of bioassay. The latter may also be used for the cylinder-plate method with similar assumptions. Their brief derivations are shown below.
Mathematical Models 1. Cup-Plate Method. Suppose a petri dish of radius b is filled with agar medium and a well of radius a is punched out at the center. An antibiotic solution of known concentration is put into the well. As the antibiotic diffuses through the inoculated medium, the inhibition zone is formed. The depth of the agar medium is rather small compared with the diameter of the petri dish, and the diffusion could be regarded as two dimensional: %I($+;$) at (a
