3538
Znd. Eng. Chem. Res. 1996,34, 3538-3541
Momentum, Heat, and Mass Transfer Analogy for Drag-Reducing Solutions Aleksandar Dudukovi6 Department of Chemical Engineering, Faculty of Technology and Metallurgy, Karnegtjeva 4, 11 000 Belgrade, Serbia, Yugoslavia
The new, generalized form of the analogy among momentum, heat, and mass transfer for both Newtonian and drag-reducing solutions is presented. This work rests on the understanding that the shift in turbulent spectra due to the presence of additives must be taken into account because different portions of the turbulent spectra unequally affect the rates of momentum, heat, and mass transfer. These effects are accounted for through the turbulent Schmidt (Sct) or turbulent Prandtl (PrJ number in a manner analogous to the way that Sc (or Pr) number accounts for contributions of molecular transport. The proposed approach was verified by (1) comparing the qualitative prediction for Sct number and the calculated values from mass transfer data and (2) comparing predicted and experimental data for heat transfer.
Introduction The addition of minute quantities of certain high molecular weight polymers to turbulent flow can dramatically reduce the frictional drag. This was first discovered (independently) by Mysels (1949)and Toms (1949),but many investigators since then have confirmed this effect. Drag reductions of up to 80% were found with polymer concentrations of up to 50 ppm. Drag reduction is accompanied by a reduction in heat and mass transfer. A number of authors presented experimental results for heat transfer in drag-reducing solutions and a few of them for mass transfer. Their results covered both the maximum drag reduction conditions and the “polymeric regime” (moderate drag reduction). Great differences between data were found, and much greater reductions in heat than in mass transfer (Cho and Hartnett, 1980,1981). Several theoretical and semiempirical models of heat and mass transfer in drag-reducing flows have been developed. In general, the proposed models resemble classical heat (mass) transfer models which had been used successfully for predicting heat or mas transfer in various turbulent flows of Newtonian character. All the models indicate that drag reduction is associated with a reduction in heat and mass transfer, but large differences exist between some of the models. Models that correlate satisfactorily some sets of data usually fail to agree with other sets. For mass or heat transfer at maximum drag reduction few models have been proposed Wirk and Suraiya, 1977; Cho and Harnett, 1981;Kawase and Ulbrecht, 1982). They contain adjustable parameters and are not applicable for moderate drag reduction. Ghajar and Tiederman (1977)used an eddy diffisivity distribution due to Cess (19581,together with experimental data on frictional drag reduction, to determine their numerical evaluations of the Lyon equation for heat transfer. Dimant and Poreh (1976)presented a phenomenological model for calculating heat transfer in flows with drag reduction. The model was based on Van Driest’s mixing length expression with a variable damping parameter. Hanna and Sandal (1981)used a modification of the drag-reducing eddy diffisivity model developed by Dimant and Poreh. It was used together with the analytical relationships for heat and mass transfer,
developed for drag-reducing conditions in a similar manner as for zero drag reduction. The resulting equations contain an adjustable parameter and a semiempirical form for the mixing length. Kawase (1983)derived a model based on Levich‘s three-zone concept, applicable t o moderate drag reduction. As proposed by Seyer and Metzner (19691,the extent of drag reduction was taken into account through the value of the Deborah number. The model contains an empirical correlation for the laminar sublayer thickness. On the basis of the previous work of Friend and Metzner (1958)and Seyer and Metzner (1969),starting from the analysis of Reichardt (19571,Kale (1977) derived a model for prediction of heat transfer data from the friction factor results. Starting from similar equations, Wells (1968) derived another model for the prediction of heat transfer rates. These two models take into account the extent of drag reduction either through the value of the Deborah number (Kale) or through the value of the velocity a t the edge of the viscous sublayer (Wells). Both parameters could be determined from the friction factor data. However, these two models are inconsistent, especially at high Prandtl or Schmidt numbers. Sedahmed and Griskey (1972)used the Wells equation to correlate their pioneering results for mass transfer in drag-reducing fluids. They presented the equation for short mass transfer entrance lengths a t maximum drag reduction. Some authors (Kwack and Hartnett, 1982, 1983;Yoon and Ghajar, 1984, 1986) have used the Weissenberg number (the ratio of the fluid time scale to the flow time scale) to take into account the effect of drag reduction. Discussing the great variation among the data and between proposed correlations for heat and mass transfer in drag-reducing solutions, and taking into account the differences in the extents of maximum drag reduction and maximum heat and mass transfer, it was concluded (Cho and Harnett, 1980,1981;Kwack et al., 1982)that there is no analogy among momentum, heat, and mass transfer in drag-reducing solutions. The goal of this paper is not only to prove the existence of these analogies but also t o show that such general analogies must take the turbulent Schmidt (Prandtl) number into
0 1995 American Chemical Society 0888-5885/95/2634-3538$09.00/0
Ind. Eng. Chem. Res., Vol. 34, No. 10, 1995 3639 account, i.e., to be of the form
St = F(f/2,Sc,Sct)
(la)
or
where St = Fcf/2,Pr,Prt)
(Ib)
Scope The analogy among momentum, heat, and mass transfer rates in Newtonian solutions is based on the fact that all of these three quantities are proportional t o the intensity of turbulence. The role of Sc or Pr number in the analogies is to take into account different contributions of molecular transport near the interface (wall) t o the three transport rates. The usual mathematical form is
St = Fv12,Sc) = F(f/2,Pr)
(2)
However, the eddies of different sizes have different contributions to the transport of momentum, heat, and mass. It was experimentally shown (Van-Shaw and Hanratty, 1964; Sirkar and Hanratty, 1970; Shaw and Hanratty, 1977) that high-frequency turbulent pulsations have little effect on mass transfer rates. Guided by these experimental findings, Dudukovik (1985) developed a model to illustrate the influence of the turbulent Schmidt number on mass transfer rates even in Newtonian solutions, and to show that the effect predicted by theory is small, and, hence, more of theoretical than practical interest. On the other hand, it was found experimentally (Fortuna and Hanratty, 1972; Makarenkov et al., 1973; Taylor and Middleman, 1974; Berman, 1978) that drag-reducing polymers not only decrease the turbulent intensity but also shift the turbulent spectra toward lower fequencies;i.e., the highfrequency portion of turbulent spectra is much more suppressed. As a consequence, in drag-reducing solutions both eddy viscosity and eddy mass diffisivity are reduced due to decrease in turbulent intensity. However, we expect eddy mass diffisivity to be reduced less than eddy viscosity because mostly high-frequency pulsations are suppressed and they had little contribution to mass transfer. As a result, the turbulent Schmidt number, Sct, should decrease with the extent of drag reduction (Dudukovik, 1986). The scope of this paper is (1) to show the decrease of turbulent Schmidt number with the extent of drag reduction, DR, by comparison of theory and experimental results for mass transfer, and (2) to predict heat transfer results based on the relationship Sct-RDR) found in step 1. Accomplishing the above should demonstrate the existence of the general form of the analogy among momentum, heat, and mass transfer for both Newtonian and drag-reducing solutions, and should indicate that the effect of the shift in turbulent spectra can be taken into account through the turbulent Schmidt or Prandtl number.
Theory A mathematical form of the analogy between momentum and mass transfer is developed starting from the analysis of Reichardt (1957) which, applied to mass transfer, may be represented in the form
and s c * = SCISC,
(5)
The assumptions involved are minor and not central to the present discussion. Several models were derived (Metzner and Friend, 1959; Friend and Metzner, 1938; Kale, 1977) or coud be derived starting from Reichardt's approach, but the usual assumption was Prt or Sct = 1. The ratio of the mean to maximum velocity r$ is nearly constant a t a value of about 111.2 under turbulent flow conditions (Friend and Metzner, 1958; Metzner and Friend, 1959). The analogous temperature difference parameter 8, increases rapidly with both increasing Reynolds number and Prandtl number, approaching a value of unity asymptotically for high Prandtl numbers (Metzner and Friend, 1959; Friend and Metzner, 1958). It is appropriate to assume 8 = 1 (Kale, 1977). Eq 3 is solved numerically for the function b. The radial variation of Sct number and its dependence on Reynolds number were neglected. Deissler (1954) proposed the following semitheoretical expression t o represent the conditions in the vicinity of the wall: Elv = n 2u+y + [l - exp(-n 2u+y + )I
(6)
with the value n = 0.124 for Newtonian fluids. The corresponding velocity profile was used for the wall region: u+ =
dy
+
(7)
Farther from the wall the logarithmic profile was assumed:
Sayer and Metzner (1969) correlated the parameter B with viscoelastic properties of the fluid through a Deborah number, and suggested for A the value of 2.5, the same as for Newtonian fluids. The integral eq 4 was solved for different values of n and the generalized Schmidt number Sc*. Taking into account the appropriate relations among n, B , and De number the computed results were approximated with the following equation:
b = [9.2
+ 1.2(De - 0.17De2)1S~*-0.255
(9)
Eqs 3 and 9 combined are a new form of the analogy between momentum and mass transfer. The extent of drag reduction is taken into account through the value of Deborah number, which can be determined directly from friction factor data, as shown by Sayer and
3640 Ind. Eng. Chem. Res., Vol. 34, No. 10,1995
Smith a l all (19691 Smith ai all (19691 0 Mc Nally (19681 0 Mc Naily 119681
0
A
+ I
i '
I
/
.-
1 00
Debrule and S a k n k v (lY74) Debrule and Sabrrskv (19711
02
04
Ofi
08
IO
Figure 1. Turbulent Schmidt number as a function of the extent of drag reduction.
Metzner (1969). The turbulent Schmidt number takes into account the shift in turbulent spectra.
Mass Transfer Eqs 3 and 9 were compared with experimental data for friction and mass transfer in drag-reducing solutions t o check our qualitative prediction that the turbulent Schmidt number, Sct, will decrease with the extent of drag reduction. Data of McConaghy and Hanratty (1977),for moderate drag reduction ("polymer regime"), and those of Virk and Suraiya (1977),for maximum drag reduction, were used. The results of Sedahmed and Griskey (1972) and Tenget et al. (1979) were not used because they were restricted t o the entry region. The results of Smith and Edwards (1982) were not used because they found a considerable effect of the electrolyte on their results. Comparison of experimental data and the values from eqs 3 and 9 show that the turbulent Schmidt number, Sct, really decreases with the extent of drag reduction (Figure 1). To compare these results, it was necessary to take into account the dependence of Sct on the Schmidt number. From the work of Shaw and Hanratty (19771, where the influence of the Schmidt number on the fluctuations of turbulent mass transfer to a wall was studied, it was concluded that the similarity parameter should be S C O . ~Therefore, ~. the calculated Sct number from the comparison of exerimental results with eqs 3 and 9 is presented in Figure 1 as a product (SC&CO.~~) versus D€UD&,, the relative extent of drag reduction.
Heat Transfer When Sc and Sct are replaced with Pr and Prt numbers, then eqs 3, 5, and 9 and Figure 1 could be used to predict the values for heat transfer rates. These predictions are compared with the experimental results of McNally (1968),Smith et al. (19691, and Debrulle and Sabersky (1974) and are presented in Figure 2. Some scatter of the data is present, but one should keep in mind that we are dealing with experiments of heat transfer in polymer solutions, where large scatter among results is commonly found. Furthermore, the predicted values are calculated on the basis of equivalence of Prt and Sct numbers, and Sct numbers were calculated from the mass transfer data, where considerable scatter is also present (see Figure 1). Altogether,
IO
Figure 2. Heat transfer in drag-reducing solutions. Comparison of the analogy and the experiments.
it can be concluded that Figure 2 supports and proves the proposed analogy. Conclusion The new, generalized form of the analogy among momentum, heat and mass transfer (eqs 3 and 9) can be used for turbulent flow of both Newtonian and dragreducing solutions. The relation between f12 and St numbers contains two parameters: De and Sct (or Prt) numbers. The Deborah number takes into account the degree of drag reduction and is determined directly from the friction factor data using the results of Sayer and Metzner (1969). This approach is based on the understanding that the relation between f12 and St numbers depends not only on the degree of turbulence but also on the changes in turbulent spectra, which are to be taken into account through the value of the Sct number. This was established in two ways: (1)by demonstrating the qualitative prediction of the decrease of Sct number with the extent of drag reduction, and (2) through the prediction of heat transfer data. Nomenclature A, B = parameters in eq 8 b = integral defined in eq 4
De = Deborah number DR = extent of drag reduction DFt,- = extent of drag reduction, corresponding to maximum drag reduction f = Fanning friction factor n = parameter in eqs 6 and 7 Nu = Nusselt number Pr = Prandtl number Pr* = generalized Prandtl number (PrlPrt) Prt = turbulent Prandtl number (€/€H) Re = Reynolds number Sc = Schmidt number Sc* = generalized Schmidt number (Sc/Sct)
Ind. Eng. Chem. Res., Vol. 34,No. 10, 1995 3541 Sct = turbulent Schmidt number ( E / E D ) St = Stanton number u = velocity U + = dimensionless velocity (u/w*)
w * = friction velocity ((tW/e)O9 y + = dimensionless radial position (yw*/v)
Greek Symbols E
= eddy viscosity
mass diffisivity = eddy thermal diffisivity v = kinematic viscosity 0 = ratio of mean t o maximum temperature difference Q = density tw = shear stress at the wall 4 = ratio of maximum to average velocity ED = eddy EH
Literature Cited Berman, N. S.; Griswold, S. T.; Elihu, S.;Yuen, J. An Observation of the Effect of Integral Scale on Drag Reduction. AIChE J. 1978,24, 124. Cess, R. D. A Survey of the Literature on Heat Transfer in Turbulent Tube Flows; Westinghouse Research Laboratories, Rep. 8-0529-R24, 1958. Cho, Y. I.; Hartnett, J. P. Analogy for Viscoelastic FluidsMomentum, Heat and Mass Transfer in Turbulent Pipe Flow. Lett. Heat Mass Transfer 1980, 7, 339. Cho, Y. I.; Hartnett, J. P. Mass Transfer in Turbulent Pipe Flow of Viscoelastic Fluids. Znt. J. Heat Mass Transfer 1981,24,945. Debrule, P. M.; Sabersky, R. H. Heat Transfer and Friction Coefficients in Smooth and Rough Tubes with Dilute Polymer Solutions. Int. J. Heat Mass Transfer 1974, 17,529. Deissler, R. G. Analysis of Heat Transfer, Mass Transfer and Friction in Smooth Tubes at High Prandtl and Schmidt Numbers; N.A.C.A. TN-3145, 1954. Dimant, Y.; Poreh, M. Heat Transfer in Flows with Drag Reduction. In Advances in Heat Transfer;Academic Press: New York, 1976; Vol. 12. Dudukovib, A. Analogies Between Momentum, Heat and Mass Transfer in Dilute Polymer Solutions. Encyclopedia of Fluid Mechanics: Cheremisinoff,. N. P., Ed.; Gulf h b l . Co.: Houston, 1988; Vol. 'VI, Ch. 12. Fortuna, G.; Hanratty, T. J. The Influence of Drag-Reducing Polvmers on Turbulence in the Viscous Sublayer. J.Fluid Mech. 19f2, 53 (31, 575. Friend, W. L.; Metzner, A. B. Turbulent Heat Transfer Inside Tubes and the Analogy Amount Heat, Mass, and Momentum Transfer. AlChE J. 1968,4, 393. Hanna, 0. T.; Sandal, 0. C.; Mazet, P. R. Heat and Mass Transfer in Turbulent Flow Under Conditions of Drag Reduction. MChE J. 1981,27, 693. Kale, D. D. An Analysis of Heat Transfer in Turbulent Flow of Drag-reducing Fluids. Znt. J. Heat Mass Transfer 1977, 20, 1077. Kawase, Y. Turbulent Heat and Mass Transfer in Flows with Moderate Drag Reduction. Phys.-Chem.Hydrodyn. 1983,4,29. Kawase, Y.; Ulbrecht, J. J. Turbulent Heat and Mass Transfer in Dilute Polymer Solutions. Chem. Eng. Sci. 1982,37, 1039. Kwack, E. Y.; Hartnett, J. P. Effect of Solvent Chemistry on Critical Weissenberg Numbers. Znt. J.Heat Mass Transfer 1982, 25, 1445. Kwack, E. Y.; Hartnett, J. P. Empirical Correlations of Turbulent Friction Factors and Heat Transfer Coefficients for Viscoelastic Fluids. Znt. Commun. Heat Mass Transfer 1983, 10, 451. Kwack, E. Y.; Hartnett, J. P.; Cho, Y. I. Turbulent Heat Transfer in Circular Tube Flows of Viscoelastic Fluids. Waeme-Stoffuebertrag. 1982, 16, 35.
Makarenkov, A. P.; Ninogradniy, G. P.; Skripanchev, U. V.; Kanarskiy, M. V. The Effect of Polymer Additives on the Pressure Drop Pulsations in the Boundary Layer (in Russian). Znzh. Fiz. Zh. 1973,25 (61, 1006. McConaghy, G. A.; Hanratty, T. J. Influence of Dragreducing Polymers on Turbulent Mass Transfer to a Pipe Wall. MChE J. 1977,23, 493. McNally, W. A. Heat and Momentum Transport in Dilute Polyethylene Oxide Solutions. Ph.D. Thesis, University of Rhode Island, 1968. Metzner, A. B.; Friend, W. L. Theoretical Analogies Between Heat, Mass and Momentum Transfer and Modifications for Fluids of High Prandtl or Schmidt Numbers. Can. J. Chem. Eng. 1968, 235. Metzner, A. B.; Firend, P. S. Heat Transfer to Turbulent NonNewtonian Fluids. Znd. Eng. 1959, 51 (71, 879. Mysels, K. J. Flow of Thickened FGluids. VS Patent, 2,492,173, Dec 27, 1949. Reichardt, I. I. The Principles of Turbulent Heat Transfer. Trans. Arch., Gas W m k c h . 1961, NO.6/7,129; NACATM-1408,1957. Sayer, F. A.; Metzner, A. B. Turbulence Phenomena in DragReducing Systems. AIChE J. 1969, 15, 426. Sedahmed, G. I. L.; Griskey, R. G. Mass Transfer in Drag-Reducing Fluid Systems. AIChE J. 1972, 18, 138. Shaw, D. A.; Hanratty, T. J. Influence of Schmidt Number on the 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 (3), 589. Smith, K. A.; Keuroghlian, G. H.; Virk, P. S.; Merrill, E. W. Heat Transfer to DragReducing Polymer Solutions. AZChE J. 1969, 15, 294. Smith, R.; Edwards, M. F. Pressure Drop and Mass Transfer in Dilute Polymer Solutions in Turbulent Drag-Reducing Pipe Flow. Znt. J. Heat Mass Transfer 1982,25, 1869. Taylor, A. R.; Middleman, S. Turbulent Dispersion in DragReducing Fluids. AZChE J. 1974,20, 454. Teng, I. T.; Greif, R.; Cornet, T.; Smith, R. Study of Heat and Mass Transfer in Pipe Flows with Non-Newtonian Fluid. Znt. J. Heat Mass Transfir 1979,22,493. Toms, B. A. Some Observations on the Flow of Linear Polymer Solutions Through Straight Tubes at Large Reynolds Numbers. Proceedings of the First International Rheology Congress; 1949; Part 2, 135. Van-Shaw, P.; Hanratty, T. J. Fluctuations in the Local Rate of Turbulent Mass Transfer to a Pipe Wall. AZChE J. 1964, 10, 475. Virk, P. S.; Suraiya, T. Mass Transfer a t Maximum Drag Reduction. Proceedings of the Second International Conferenceon Drag Reduction; G3-41;BHRA Fluid Eng.: Cranfield, England, 1977. Wells, C. S. Turbulent Heat Transfer in Drag-reducing Fluids. AZChE J. 1968,14,406. Yoon, H. K.; Ghajar, A. J. An Analysis of the Heat Transfer to Drag Reducing Turbulent Pipe Flows. ASME J.Heat Transfer 1984,106,898. Yoon, H. K.; Ghajar, A. J. A New Eddy Diffusivity eq for Calculation of Heat Transfer to Drag-reducing Turbulent Pipe Flow. Znt. Commun. Heat Mass Transfer 1986,13,449. Received for review January 19, 1995 Revised manuscript received June 20, 1995 Accepted J u n e 22, 1995" IE950060R
* Abstract published in Advance ACS Abstracts, August 15, 1995.
