Ind. Eng. Chem. Res. 1999, 38, 2503-2504
2503
RESEARCH NOTES Effect of Turbulent Schmidt Number on Mass-Transfer Rates to Falling Liquid Films Aleksandar Dudukovic´ * and Rada Pjanovic´ Department of Chemical Engineering, Faculty of Technology and Metallurgy, Karnegijeva 4, Belgrade, Yugoslavia
The purpose of this paper is to provide a rational explanation for the Sherwood number correlations for the gas-side mass-transfer coefficient in columns with falling liquid films. Specifically we want to show that proper physical representation can provide the exponent for the Reynolds number for the above correlations which is consistent with available data. The typical form of the Sherwood number correlation used for pipes and falling liquid films is of the form: b
c
Sh ) aRe Sc
Sct ) a2Re0.159
(1)
In our recent paper (Dudukovic´ et al., 1996) it was shown, based on both theoretical and experimental evidence, that for mass transfer in the wetted wall columns the exponent on the Schmidt number is 1/2 and not as often quoted 0.44. The origin of eq 1 can be found in Levich’s (1959) three-sublayer representation of the boundary layer resulting in the following form:
Sh ) a1Rexf(Sc/Sct)1/2
increases the turbulent intensity but also moves the turbulent spectrum toward higher frequencies, which in turn have a smaller effect on mass than on momentum transfer rates (Van Shaw and Hanratty, 1964; Sirkar and Hanratty, 1970; Shaw and Hanratty, 1977). As a result, the Sct number increases with the Reynolds number. For the flow of fluids in circular tubes it was found (Dudukovic´, 1985) that
(2)
To verify our approach for falling liquid films, we assumed that the same relation (4) will hold for the gas core in the column. We now combine eqs 2 and 4 with the correlated friction factor data of Gilliland and Sherwood (1934), given in eq 5, which is similar to the
f ) 0.0779Re-0.254
Sh ) a3Re0.794Sc1/2 (3)
The turbulent Schmidt number, Sct, is usually assumed to be constant. However, it is precisely this assumption of Sct ) constant that leads to misrepresentation of the Reynolds number dependence, as we have shown (Dudukovic´, 1985) in the treatment of mass transfer to tube walls. When one takes into account the dependence of the friction factor on Reynolds number (f ) f(Re)), the resulting exponent on the Reynolds number turns out to be higher than the experimentally found value of about 0.8. This was often rationalized as the effect of waves on the gas-liquid surface, without any further evidence (Skelland, 1974). We will show here that one reaches a different conclusion regarding the exponent on the Reynolds number if one considers the change in the turbulent Schmidt number with the Reynolds number. We have previously shown (Dudukovic´, 1985, 1995) that the turbulent Schmidt number is not a constant but changes with the changes in turbulent spectra, which shift to higher frequencies with increases in the Reynolds number. This is a consequence of different contributions of high- and low-frequency turbulence pulsations to mass (or heat) and momentum transfer rates. An increase in the Reynolds number not only
(5)
Blasius equation for smooth pipes. The resulting equation is
where
Sct ) M/D
(4)
(6)
The correlation for the gas-side mass-transfer coefficient to the falling liquid film, based on data of Gilliland and Sherwood (1934) and Barnet and Kobe (1941), is
Sh ) a4Re0.790Sc1/2
(7)
An excellent agreement between eqs 6 and 7 is evident. This provides further support to our claim that changes in the turbulent Schmidt number with the Reynolds number should be accounted for. If that is not done, an overestimate of the dependence of the Sh number on the Re number is obtained, leading to possible overestimates of mass transfer for highly soluble gases. Further applications of these results could be in the use of turbulence promoters (Perry and Green, 1984). One should design turbulent promoters so that they increase mostly the lower part of the turbulence spectrum, which affects mass transfer more and in that way saves energy and has greater efficiency. Notation a, a1 ) coefficients (eqs 1, 2, 4, 6, and 7) b ) exponent on Reynolds number (eq 1)
10.1021/ie980501h CCC: $18.00 © 1999 American Chemical Society Published on Web 05/12/1999
2504 Ind. Eng. Chem. Res., Vol. 38, No. 6, 1999 c ) exponent on Schmidt number (eq 1) f ) fanning friction factor Re ) Reynolds number Sc ) Schmidt number Sct ) turbulent Schmidt number (defined in eq 3) Sh ) Sherwood number M ) eddy viscosity D ) eddy diffusivity
Literature Cited Barnet, W. I.; Kobe, K. A. Heat and Vapor Transfer in a WettedWall Tower. Ind. Eng. Chem. 1941, 33, 436. Dudukovic´, A. Influence of the Turbulent Schmidt Number on Mass Transfer Rates Between Turbulent Fluid Stream and Solid Surface. AIChE J. 1985, 31, 1919. Dudukovic´, A. Momentum, Heat and Mass Transfer Analogy for Drag-Reducing Solutions. Ind. Eng. Chem. Res. 1995, 34, 3538. Dudukovic´, A.; Milosˇevic´, V.; Pjanovic´, R. Gas-Solid and GasLiquid Mass Transfer Coefficients. AIChE J. 1996, 42, 269. Gilliland, E. R.; Sherwood, T. K. Diffusion on Vapors into Air Streams. Ind. Eng. Chem. 1939, 26, 516.
Levich, V. G. Fiziko-Himicheskaya Gidrodinamika, 2nd ed.; Izdv. Akademii nauk SSSR: Moscow, 1959. Perry, R. D.; Green, D. Perry’s Chemical Engineer’s Handbook; McGraw-Hill: New York, 1984; pp 18-43. Shaw, D. A.; Hanratty, T. J. Influence of the Schmidt Number onthe Fluctuations of Turbulent Mass Transfer to a Wall. AIChE J. 1977, 23, 160. Sirkar, K. K.; Hanratty, T. J. Relation of Turbulent Mass Transfer to a Wall at High Schmidt Numbers to the Velocity Field. J. Fluid Mech. 1970, 44, 589. Skelland, A. H. P. Diffusional Mass Transfer; Wiley: New York, 1974; p 266. Van Shaw, P.; Hanratty, T. J. Fluctuations in the Local Rate of Turbulent Mass Transfer to a Pipe Wall. AIChE J. 1964, 10, 475.
Received for review July 30, 1998 Revised manuscript received March 11, 1999 Accepted March 24, 1999 IE980501H
