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Ind. Eng. Chem. Res. 2002, 41, 882-884
RESEARCH NOTES Heat- and Mass-Transfer Analogy in a Bubble Column Ashok K. Verma† Department of Chemical Engineering and Technology, Institute of Technology, Banaras Hindu University, Varanasi 221 005, India
Heat- and mass-transfer analogy based on heat, mass, and momentum analogy in pipe flow proposed by Lin et al. (Lin, C. S.; Moulton, R. W.; Putnam, G. L. Mass Transfer between Solid Wall and Fluid Stream. Ind. Eng. Chem. 1953, 45, 636) has been presented for bubble columns. The ratio of constants for correlations based on the Chilton and Colburn analogy in the case of mass transfer to that for heat transfer has been predicted. It depends on the range of gas velocity and values of Pr and Sc. Introduction Bubble-column reactors have been used as gas-liquid contactors for a variety of reactions because of their good mixing ability and high transfer coefficient. Heat transfer in the bubble column has been studied by many investigators. Deckwer2 reviewed empirical correlations proposed by many investigators (Kolbel et al.,3 Kast,4 Burkel,5 Hart,6 and Steiff and Weinspach7). It was observed that the correlations proposed by these investigators almost completely correspond to the correlation originally proposed by Kast.4 Hence, a model based on Higbie’s surface renewal model and Kolmogoroff’s theory of isotropic turbulence was proposed. The following equation in terms of dimensionless number was obtained. 2 -0.25
Sth ) 0.1(Re × Fr × Pr )
for mass transfer, the constant had to be adjusted using experimental data on mass transfer. In the present paper, a model is proposed to determine the ratio of proportionality constants in heat- and mass-transfer correlations. Model Heat transfer in bubble columns is assumed to be analogous to heat transfer in pipe flow. The liquidcirculation velocity is taken in place of the fluid velocity. Now the Lin et al. analogy1 is assumed to hold well. The mass-momentum analogy in pipe flow proposed by them is given as
Sh )
(1)
1 f Re × Sc φD 2
()
(3)
where Joshi et al.8 have used an analogy between bubble column and flow through circular pipes to predict the heat-transfer coefficient in bubble columns. The liquidcirculation velocity was used in place of the fluid velocity. Zehner9 used the expression for heat transfer over a flat plate in a model based on the boundary layer concept. Patil and Sharma10 studied the solid-liquid masstransfer coefficient in bubble columns by dissolution of copper in an acidic solution containing potassium dichromate, with column diameters of 0.146, 0.380, and 1.00 m. The following equation analogous to eq 1, with proportionality constant 0.1 replaced by 0.052, was obtained.
Stm ) 0.052(Re × Fr × Sc2)-0.25
(2)
φD ) 1 +
Sc x2f [14.5 3
2/3
FD + 5 ln
]
1 + 5.64Sc - 4.77 6.64(1 + 0.041Sc) (4)
in which 5 Sc ] [1 + 14.5 5 5 1Sc Sc + ( 14.5 14.5) 1/3 2
1 FD ) ln 2
2
1/3
-1
x3 tan
Rai11
used electrochemical technique (a ferro-ferricyanide system) to study the mass-transfer coefficient in a bubble column of diameter 0.0515 m. The constant 0.052 in eq 2 was replaced by 0.047. Though the Chilton-Colburn analogy between heat and mass transfer was helpful in getting the correlation †
Fax: 091-0542-368428.
+ 2/3
10 Sc1/3 - 1 14.5 πx3 + (5) 6 x3
In the case of heat transfer, analogous equations may be written as
Nu )
1 f Re × Sc φH 2
()
(6)
in which φH is obtained by substituting Pr in place of 10.1021/ie0009280 CCC: $22.00 © 2002 American Chemical Society Published on Web 01/24/2002
Ind. Eng. Chem. Res., Vol. 41, No. 4, 2002 883 Table 1. Range of Parameters CD/CH investigator
U, m
Patil and Sharma10 Rai11
s-1
0.01-0.35 0.0033-0.0330
φH
/Pr0.5
1.103-1.248 0.935-1.008
Sc in eqs 4 and 5. In the case of bubble columns, fluid velocity is replaced by circulation velocity. The Stantan number for heat transfer in a bubble column may be expressed as
Sth )
( )( )( )
hdb µ λ h Nu (7) ) ) UFlCp λ dbUFl Cpµ Re × Pr
φD/Sc0.5
calcd
exptl
0.565-0.649 0.383-0.441
0.512-0.544 (ave ) 0.528) 0.410-0.436 (ave ) 0.423)
0.52 0.47
and Pr increase from 0.5 to 10 000, values of φD and φH increase monotonically from 0.813 to 261.9. Because for a fluid Pr and Sc are different, the ratio φD/φH is not 1 but depends on the fluid being used. To determine the ratio φD/φH, the liquid-circulation velocity, Vc, given by Zehner9 was used.
Vc )
Hence, eq 1 can be written as
Nu ) 0.1Re-0.25Fr-0.25Pr-0.5 Re × Pr
(8)
which on rearrangement after eliminating Fr gives
( )
µ 1 Nu ) 0.1 0.5 d Pr bFl
U0.25 Re × Pr (gD)0.25
0.25
()
( )
1 f 1 µ ) CH 0.5 φH 2 d Pr b Fl
1 U0.25 (gD)0.25
(10)
1 U0.25 (gD)0.25
(11)
0.25
Similarly for mass transfer
()
( )
1 µ 1 f ) CD 0.5 φD 2 d Sc bFl
0.25
Dividing eq 10 by eq 11
CH Pr0.5/φH ) CD Sc0.5/φ
(12)
D
Equation 12 thus relates the constants in correlations obtained by the Chilton-Colburn analogy for heat- and mass-transfer coefficients. Because CH and CD are constants, their ratio should also be constant. The quantities φD and φH may be obtained from eqs 4 and 5 using Sc and Pr, respectively. It may be seen that the ratio does not depend on the definition of Re. However, Re is required for calculation of the friction factor, f. Verification of the Model Variation of φD and φH as a function of Sc and Pr, respectively, is presented in Figure 1. As values of Sc
Figure 1. Variation of φD and φH as a function of Sc and Pr.
1/3
(13)
Values of the friction factor, f, were then obtained using the following equation12 valid in a wide range of Rec (3000 e Rec e 3 × 106). The Reynolds number Rec, based on Vc, was used.
(9)
Comparing eq 9 with an equation analogous to eq 3 for pipe flow
[ ( ) ] 1 Fl - Fg gDU 2.5 Fl
f ) 0.0014 +
0.125 Rec0.32
(14)
Values of φD and φH can now be calculated using eqs 4 and 5, respectively. Calculated values of φD/Sc0.5 and φH/Pr0.5 and their ratio CD/CH along with the range of superficial gas velocities are compared with data of Patil and Sharma10 and Rai11 in Table 1. It may be seen that Sc0.5/φD and Pr0.5/φH and their ratio CD/CH are almost constant, though these are different for the two studies. The values of Sc0.5/φD and Pr0.5/φH increase with an increase in U. Therefore, these along with their ratio CD/CH are different in the two studies. The average value of CD/ CH was calculated by taking an average of the extreme values. These are comparable with the values reported in the literature. Thus, the constant in correlation should depend on the range of superficial gas velocities. Conclusion Though the Colburn-Chilton analogy for heat and mass transfer is valid only for Sc ) Pr ) 1, it results in experimental determination of the proportionality constant. However, using the Lin et al. analogy,1 the ratio of constants for heat- and mass-transfer correlations may be determined in the case of bubble columns. This ratio depends on the range of gas velocities and the values of Pr and Sc. Nomenclature CD ) constant for mass transfer in eq 11 CH ) constant for heat transfer in eq 10 Cp ) heat capacity of liquid, W s kg-1 K-1 db ) bubble diameter, m D ) column diameter, m f ) friction factor in circular pipes FD ) parameter for mass transfer defined by eq 5 FH ) parameter for heat transfer obtained by replacing Sc by Pr in eq 5 Fr ) Froude number ()U2/gD) g ) acceleration due to gravity, m s-2 h ) heat-transfer coefficient, kW m-2 K-1 k ) mass-transfer coefficient, m s-1 Nu ) Nusset number ()hD/λ) Pr ) Prandtl number ()Cpµ/λ)
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Ind. Eng. Chem. Res., Vol. 41, No. 4, 2002
Re ) Reynolds number ()UFlD/µ) Rec ) Reynolds number based on the liquid-circulation velocity ()VcFlD/µ) Sc ) Schmidt number ()µ/FlD) Sh ) Sherwood number Sth ) Stantan number for heat transfer ()h/FlCpU) Stm ) Stantan number for mass transfer ()k/U) U ) superficial gas velocity, m s-1 Vc ) liquid-circulation velocity, m s-1 Greek Letters λ ) thermal conductivity of liquid, W m-1 K-1 µ ) viscosity of liquid, kg m-1 s-1 φD ) parameter for mass transfer defined in eq 4 φH ) parameter for heat transfer defined in eq 6 Fg ) density of gas, kg m-3 Fl ) density of liquid, kg m-3
Literature Cited (1) Lin, C. S.; Moulton, R. W.; Putnam, G. L. Mass Transfer Between Solid Wall and Fluid Stream. Ind. Eng. Chem. 1953, 45, 636. (2) Deckwer, W. D. On the Mechanism of Heat Transfer in Bubble Column Reactors. Chem. Eng. Sci. 1980, 35, 1341. (3) Kolbel, H.; Siemes, W.; Mass, R.; Muller, K. Warmeubergang an Blasensaulen. Chem. Eng. Tech. 1958, 30, 400.
(4) Kast, W. Analyse Des Warmeubergang in Blasensaulen. Int. J. Heat Mass Transfer 1962, 5, 329. (5) Burkel, W. Der Warmeubergang an Heiz- und Kuhlflachen in Begasten Flussigkeiten. Chem. Eng. Tech. 1972, 44, 265. (6) Hart, W. F. Heat Transfer in Bubble-Agitated SystemssA General Correlation. Ind. Eng. Chem. Process Des. Dev. 1976, 15, 109. (7) Steiff, A.; Weinspach, P. M. Heat Transfer in Stirred and Non-Stirred Gas-Liquid Reactors. Ger. Chem. Eng. 1978, 1, 150. (8) Joshi, J. B.; Sharma, M. M.; Shah, Y. T.; Singh, C. P. P.; Ally, M.; Klinzing, G. E. Heat Transfer in Multiphase Reactors. Chem. Eng. Commun. 1980, 6, 257. (9) Zehner, P. Momentum, Mass and Heat Transfer in Bubble Columns. Part 2. Axial Blending and Heat Transfer. Int. Chem. Eng. 1986, 26, 29. (10) Patil, V. K.; Sharma. M. M. Solid-Liquid Mass Transfer Coefficient in Bubble Columns up to One Metre Diameter. Chem. Eng. Res. Des. 1983, 61, 21. (11) Rai, S. Studies on Mass Transfer with Immersed Surface using Electrochemical Technique. Ph.D. Thesis, BHU, Varanasi, India, 1997. (12) McCabe W. L.; Smith, J. C.; Harriot, P. Unit Operations of Chemical Engineering, 5th ed.; McGraw-Hill: New York, 1993; p 100.
Received for review October 30, 2000 Revised manuscript received May 1, 2001 Accepted November 27, 2001 IE0009280
