International Journal of Heat and Mass Transfer 43 (2000) 4267±4274 www.elsevier.com/locate/ijhmt
Buoyancy eects on heat transfer and temperature pro®les in horizontal pipe ¯ow of drag-reducing ¯uids
K. Gasljevic, G. Aguilar, E.F. Matthys
Department of Mechanical and Environmental Engineering, University of California, Santa Barbara, CA 93106, USA Received 12 November 1999; received in revised form 7 February 2000
Abstract
We have studied the extent to which buoyancy eects in horizontal pipe ¯ows of drag-reducing viscoelastic ¯uids cause distortions to both laminar and turbulent temperature pro®les. In the case of laminar ¯ows, these distortions may lead to variations in Nusselt numbers that are larger than those seen for Newtonian pipe ¯ows under similar conditions. In the case of turbulent drag-reducing ¯ows, the eects of buoyancy can also be large and may in turn result in large errors in estimated Nusselt numbers if not properly accounted for. These errors are quanti®ed and recommendations are made on how to reduce them. 7 2000 Elsevier Science Ltd. All rights reserved.
1. Introduction on experimental data, Morcos and Bergles [3] provided
averaged Nusselt number (Nuavg) correlations which In heated ¯ows in horizontal pipes, gravity-induced take into account the eect of circumferential heat ¯ux body forces may result from density variations within variations for the problem of mixed convection on the ¯uid. Mori et al. [1,2] have shown ¯ow visualiza- laminar ¯ow of Newtonian ¯uids in horizontal tubes. tion results for mixed convection in laminar ¯ows of Metais and Eckert [4] proposed practical charts in Newtonian ¯uids in horizontal pipes, demonstrating which the regions of forced and mixed convection for that the local variations in the ¯uid density lead to Newtonian laminar and turbulent ¯ows in horizontal counteracting transverse vortices (or secondary ¯ow pipes can be clearly identi®ed in terms of the Re, Gr, patterns), which are superimposed to the main axial and Pr. ¯ow. More recently, viscoelastic drag-reducing ¯uids, They also showed that the dierences between their and particularly surfactant solutions, have attracted local measurements of temperature and velocity, and the attention of researchers because of their poten- the theoretical laminar pro®les with no buoyancy tial for energy savings applications. The buoyancy eects, could be as high as 50%, and that Ð for heat- eects on laminar and turbulent ¯ows for these ing of the ¯uid Ð the point of lowest temperature was ¯uids is not only interesting from a theoretical displaced from the center towards the lower portion of point of view, but also for the proper design of ex- the pipe. All these experiments were conducted in the perimental procedures. Shenoy and Ulbrecht [5] stu- range of Re  Ra =2 104±1.6  105. On the other died the eect of natural convection on a laminar hand, their measurements of turbulent temperature ¯ow next to a vertical ¯at plate for various sol- and velocity pro®les in the range of Re  Ra between utions of a viscoelastic ¯uid, and they found that 3.87  105 and 4.7  105 showed a negligible dierence the local convective heat transfer coecients (h ) with respect to those of pure forced convection. Based were systematically higher for elastic ¯uids than for
0017-9310/00/$ - see front matter 7 2000 Elsevier Science Ltd. All rights reserved. PII: S0017-9310(00)00065-X 4268 K. Gasljevic et al. / Int. J. Heat Mass Transfer 43 (2000) 4267±4274
Nomenclature
2 à 0:5 Cf 2tw=rV friction coecient u tw=r friction or shear vel- Cf; water friction coecient for ocity 1:58  ln Re 3:28 2 turbulent newtonian V bulk velocity (m/s) ¯ow (Filomenko) y yuÃ=n dimensionless distance D pipe diameter (m) from the wall DR 1 Cf =Cf, waterÂ100 drag reduction level (%) Greek symbols 3 2 Gr gbD DTw; avg-b=n Grashof number a thermal diusivity based on D (m2/s) 00 h q =DTw-b convective heat trans- b thermal expansion fer coecient (W/m2 coecient (1/K) K) DTw-b inner wall-bulk tem- kf thermal heat conduc- perature dierence (K) tivity (W/m K) DTw, avg-b average inner wall- 00 Nu q D=DTw-bkf Nusselt number based bulk temperature on D dierence for the three 00 Nuavg q D=DTw; avg-bkf average Nusselt num- wall locations (K) ber (based on the eM momentum eddy diu- average temperature sivity (m2/s) of the three wall sen- eH heat eddy diusivity sors) (m2/s) Pr n=a Prandtl number n kinematic viscosity 2 Prt eM=eH turbulent Prandtl (m /s) number r ¯uid density (kg/m3) 4 Ra gbDTw; avg-b=na Rayleigh number tw wall shear stress (N/ Re VD=n Reynolds number m2) based on D q0 heat ¯ux at the wall Subscripts (W/m2) up refers to top portion 00 Tb Ti q =rCpV bulk temperature (8C) of the pipe à 00 T Tw T u rCp=q dimensionless wall-to- dn refers to bottom por- ¯uid temperature tion of the pipe dierence
inelastic ones at similar heat and ¯ow rate con- of drag-reducing ¯uids in horizontal pipes, however, ditions, being in some cases higher by 40%. Other although one might guess that buoyancy can induce researchers have studied, for viscoelastic ¯uids, the secondary ¯ows perpendicular to the main free-stream eect of buoyancy on laminar ¯ows over vertical direction, presumably similar to those seen in laminar and horizontal plates, as well as around the stag- ¯ows. The only reference to the eects of buoyancy in nant region of a heated cylinder [6±8]. In all cases, channel ¯ows of drag-reducing surfactant solutions there is an increase in the convective heat transfer that we know is the one by Kawaguchi et al. [10]. coecient with increasing ¯uid viscoelasticity. They measured temperature pro®les in the cross sec- Regarding internal ¯ow, Ref. [9] appears to be the tion of a square channel heated at the bottom wall, only study that has addressed the problem of mixed and found a region of very high diusivity in the vis- convection of viscoelastic drag-reducing ¯uids on cous sublayer. They also measured an increase in Nus- vertical pipe ¯ow under turbulent conditions. For selt number (Nu ) of 20% due to a twofold increase in this particular case, the turbulence generation is the heat ¯ux, with all other conditions remaining con- aected by the redistribution of the shear stress stant. The increased diusivity and Nu were attributed across the pipe, which is in turn aected by the to buoyancy eects, although they also considered the buoyancy-driven ¯ow moving in the ¯ow direction. possibility of thermal destruction of the micelles. In Little work has been done on the mixed convection our recent studies on the heat transfer and temperature K. Gasljevic et al. / Int. J. Heat Mass Transfer 43 (2000) 4267±4274 4269 pro®le measurements of various drag-reducing polymer shape with a mean diameter of about 0.007±0.008 in. and surfactant solutions, we have found that buoyancy (0.18 mm). This sensor is displaced across the pipe by eects are present in laminar as well as in turbulent an external mechanism that allows it to move in incre- ¯ows. The aim of this paper is to draw attention to the ments of 0.001 in. (0.025 mm). This pro®le temperature eect of buoyancy in drag-reducing turbulent pipe sensor was located at approximately the same axial lo- ¯ows, which may aect experimental results if there is cation as the RTD temperature sensors, so that the a lack of awareness of its presence. values of h measured by the RTDs, and those calcu- lated by integration of the temperature pro®le should be about the same for symmetric pro®les. 2. Experimental installation
The experimental setup consists of a stainless steel 3. Results and discussion tube of 19.95 mm (20 mm nominal) inner diameter and 680 diameters length, a centrifugal pump or a 3.1. Laminar pro®les pressurized tank used for ¯uid circulation, and various pressure taps installed along the pipe for drag re- Fig. 1 shows the results of temperature pro®le duction measurements (DR). A more detailed descrip- measurements for a 1500 ppm polyacrylamide tion of the setup is given elsewhere [11]. (Separan AP-273) solution in deionized water. The Two types of heat transfer measurements were con- span of our device covers only approximately 8 mm, ducted: overall heat transfer coecients (h ), and local and the two halves of the pro®le had to be measured temperature pro®les across the pipe. A DC Joule heat- by turning the pipe 1808, leaving a small gap in data at ing source provided a good approximation of a con- the center. A theoretical no-buoyancy laminar pro®le stant and uniform heat ¯ux condition. We used four is shown for comparison, and there are two eects of temperature sensors: three sensors (miniature RTDs 10 buoyancy which one can readily see: a distortion of  2 mm, 100 O), which were cemented with RTD the temperature pro®le (or circumferential variation of epoxy adhesive (Omega OB-101-2) on the outer wall at local convective heat transfer), and a change in the the top, side, and bottom of the pipe at 675 diameters average Nu compared to the ¯ow without buoyancy. downstream of the entrance to detect the possibility of The pro®le looks similar to that measured for a New- circumferential asymmetry of h; and one temperature tonian ¯uid [1], and shows also the coldest point sensor, a shielded type RTD, which was inserted across shifted towards the bottom wall, presumably due to the pipe at the inlet in order to measure the ¯uid inlet the action of secondary ¯ows generated by buoyancy, temperature Ti). The local ¯uid bulk temperature Tb as in the case of laminar ¯ow. However, the eect is at the location of the three RTD temperature sensors larger than for Newtonian ¯uids at the same Re and is calculated through the inlet temperature, the ¯ow Pr numbers. The Nuavg calculated by averaging the rate, and the heat ¯ux measurements. These four tem- three wall-to-bulk temperature dierences DTw, avg-b perature sensors were connected to a precision multi- corresponding to each of the RTD sensors, is increased meter (Keithley DMM/Scanner) for data acquisition. by buoyancy by 53% over its theoretical value with no Altogether, the uncertainty of our data for circum- buoyancy eects (Nu = 4.36), whereas only a 12% ferentially-averaged Nuavg for water is around 212± dierence was expected based on the correlations pro- 15%, which is indeed about the usual uncertainty for posed by Morcos and Bergles [3] for horizontal lami- most of the Nu correlations available for Newtonian nar pipe ¯ows of Newtonian ¯uids. This is also turbulent ¯ows. For the case of the drag-reducing qualitatively consistent with previous observations for ¯uids, the uncertainty is reduced given the increase in vertical ¯at plates [5]; and indirectly with the results the temperature dierences (details of this analysis can for ¯ows around various geometries [6±8], for which be found in [12]). On the other hand, there is an ad- ¯uid elasticity enhances the buoyancy-generated heat ditional error of up to 10%, due to variations in the transfer. Note also that because of the eects of buoy- radial heat ¯ux, which is another consequence of buoy- ancy, the Nusselt number calculated with the top wall ancy eects (as explained below). temperature measurement Nuup 8:2 and that with The measurement of the temperature pro®les across the bottom one Nudn 3:9), dier by a factor of more the main ¯ow direction is a challenging task, but it than 2. provided us with much new information. For this pur- Circumferential variations of the convective heat pose, we have built a temperature sensor [13] which is transfer may also induce a circumferential heat ¯ux in moved perpendicularly to the main ¯ow stream. The the wall, which in turn may cause variations in the sensor is a home-made type E thermocouple (Chromel- outer wall temperature measurements around the pipe. Constantan). Each lead is 0.003 in. (0.08 mm) thick, In this case, the assumption of the constant radial heat and the welded bead is of an approximately spherical ¯ux may not be exactly valid, and the Nu also loses its 4270 K. Gasljevic et al. / Int. J. Heat Mass Transfer 43 (2000) 4267±4274
Fig. 1. Temperature pro®les measured in the vertical plane of a laminar pipe ¯ow of a drag-reducing ¯uid (water-based 1500 ppm polyacrylamide Ð Separan AP273 Ð solution). The top/bottom average temperature is 2.68C higher than the bulk temperature.
strict meaning since the DTw-b is no longer unique. eects, an eect that is generally considered to be neg- However, the local Nu parameter is a good indicator ligible for turbulent ¯ows of Newtonian ¯uids. Conse- of how how the local heat transfer varies around the quently, we had to determine under which conditions pipe. According to our analysis based on the tempera- the buoyancy could aect signi®cantly our results, and ture dierences between the top and bottom wall sen- then determine under which range it was possible to sors (1.88C for the case illustrated in Fig. 1), and on conduct the experiments to avoid these eects. For the pipe heat conductivity (kp), it was estimated that that purpose, we carried out tests for various ¯uid con- the radial heat ¯ux could vary by up to 10% between centrations, heat ¯uxes, and bulk velocities. Fig. 2 the top and bottom for average test conditions. Con- shows the results of wall-to-bulk temperature dierence sidering that under turbulent ¯ow conditions (the measurements for the three locations around the pipe: regime of greatest interest for these ¯uids, see below) top, side, and bottom. These measurements were con- the buoyancy eects are relatively weaker because of ducted for a 1500 ppm cationic surfactant solution increased forced convection mixing, it was concluded (Ethoquad T13/27) plus 1300 ppm of NaSal as coun- that it is acceptable to use the constant heat ¯ux ap- terion, diluted in tap water. On the ordinate are shown proximation in our analyses. each of the three DTw-b, normalized by DTw, avg-b; and on the abscissa, Gr Pr/Re, a combination of par- ameters which compares the eects of buoyancy (Gr 4. Turbulent pro®les Pr ) to the ¯ow intensity (Re ). The Reynolds number is used in this correlation under the assumption that During our measurements of temperature pro®les in the ¯ow is essentially turbulent, despite the fact that turbulent ¯ow of drag-reducing ¯uids, we noticed that for this particular ¯uid the turbulence is highly these pro®les, as well as the outer wall temperatures damped since it showed asymptotic DR over the whole used for the overall heat transfer coecient calcu- range of Re. lations, were also signi®cantly distorted by buoyancy One can see at high Gr Pr/Re (e.g. around 10) that K. Gasljevic et al. / Int. J. Heat Mass Transfer 43 (2000) 4267±4274 4271
if the Nu were calculated using the top wall tempera- be important, and the scatter of the data around Nuavg ture measurement (which is larger than the average looks to be about the normal uncertainty for h wall temperature), it would be as much as 50% smaller measurements (210%). Although not seen in this than the Nu calculated from the average wall tempera- ®gure, for Gr Pr/Re values less than 3 we have ture, whereas the one calculated from the bottom wall DTw; avg-bR1:88C, a value one should, therefore, strive temperature would be about 50% higher than the aver- not to exceed in this case if the eects of buoyancy are aged one. On the other hand, and very usefully so, the to be avoided. One should keep in mind, however, that sensor placed at mid-elevation shows approximately in the relationship shown in Fig. 2, the viscoelastic the average value between top and bottom throughout properties are not considered, and only one single ¯uid the whole range of Gr Pr/Re. As mentioned before, in is used, which exhibit asymptotic drag and heat trans- the case of turbulent ¯ows, the uniformity of the radial fer reductions. Consequently, this criterion may have heat ¯ux is less aected than it is for laminar ones (i.e. some limitations but it may well be valid for all below 10%), and therefore, the ratio between the wall- asymptotic ¯uids. to-bulk temperature dierences measured at the top To verify the eect of buoyancy on the temperature and bottom sections of the pipe, i.e. pro®les, we measured a couple of temperature pro®les DTw; up-b=DTw; dn-b, can be considered, in ®rst approxi- under the conditions where we expected the eects of mation inverse to the ratio of the h (or local Nu ). buoyancy to be negligible, and also where they would Interestingly, the dierence between DTw, up-b and be more signi®cant, i.e. at low and high heat ¯ux con- DTw, dn-b at higher values of Gr Pr/Re in turbulent ¯ow ditions, respectively. Fig. 3 shows the eect of an is comparable to that measured in laminar ¯ow (Fig. 1) increase in the Gr for a 400 ppm cationic surfactant which correspond to a DTw; up-b=DTw; avg-b 1 1:4, and, solution (Ethoquad T13/27 by Akzo) diluted in tap DTw; dn-b=DTw; avg-b 10:7, respectively, indicating a water, which provides slightly less than asymptotic strong eect of buoyancy. For values of Gr Pr/Re less DR. Two temperature pro®les measured at the bottom than about 3, the eects of buoyancy do not seem to half of the pipe (where the eects of buoyancy are
Fig. 2. Eect of buoyancy on the wall temperature measurements located at the top, side, and bottom of a circular pipe for turbu- lent pipe ¯ow of a drag-reducing ¯uid (1500 ppm cationic surfactant solution Ð Ethoquad T13/27). 4272 K. Gasljevic et al. / Int. J. Heat Mass Transfer 43 (2000) 4267±4274 believed to be highest), are plotted in the usual dimen- top and bottom wall RTD sensors did not show a sig- sionless coordinates T + vs. y +. For pro®le T10 (low ni®cant temperature dierence, whereas in the case of 00 2 heat ¯ux: q 755 W/m ), the DTw, avg-b is calculated higher heat ¯ux this dierence was about 10% (T11 by integration of the pro®le to be about 1.68C, and the was intentionally measured at much larger heat ¯ux corresponding Gr Pr/Re to be about 3.1; for pro®le than for the average tests). 00 2 T11 (high heat ¯ux: q 2100 W/m ), the DTw, avg-b is The shift of the coldest point in Fig. 3 is not dra- 5.28C and the corresponding Gr Pr/Re is about 8.1. matic, and in order to see it more clearly, Fig. 4 shows For temperature pro®le T10, the dierence between a temperature pro®le measured at the bottom half of Nuup 15:1 and Nudn 15:5 is less than 2%, whereas the pipe for the same ¯uid as in Fig. 2, with even for T11, the dierence between Nuup 12:0 and higher heat ¯ux than the one imposed for the exper- Nudn 13:2 is already about 10%. For the lower heat iments shown in Fig. 3. Although the level of DR is ¯ux, the temperature pro®le does not show a shift of similar in both cases, the pro®le presented in Fig. 4 the coldest point, which is usually a good indication of re¯ects a stronger distortion of the pro®le (shift of the the top/bottom asymmetry of the temperature pro®le coldest point towards the bottom half of the pipe) due to buoyancy. However, a slight shift of the coldest than the one shown in Fig. 3 (open symbols), as well point towards the bottom wall is apparent at the as a larger dierence between the DTw, up-b and higher heat ¯ux. More importantly, even though the DTw, dn-b measured. In this case the values of Nuup DTw, avg-b for the pro®le T11 is larger, when normal- 10:5 and Nudn 26:5 vary by a factor of 2.5, which ized by the wall heat ¯ux, the region closer to the pipe corresponds to a wall temperature dierence between center shows a somewhat smaller T +, indication of the top and the bottom of almost 38C. The ¯uid used the enhanced heat transfer due to stronger buoyancy in the tests presented in Fig. 4 (Ethoquad 1500 ppm) (note that both pro®les were measured along the bot- exhibits, however, signi®cantly higher elasticity (nor- tom half of the pipe). Also, for the lower heat ¯ux, the mal stress dierences) than the ¯uid in Fig. 3. It is
Fig. 3. Eect of Gr (i.e. heat ¯ux) on two turbulent pro®les measured along the vertical plane in the bottom half of the pipe for a drag-reducing ¯uid (400 ppm Ethoquad solution). K. Gasljevic et al. / Int. J. Heat Mass Transfer 43 (2000) 4267±4274 4273
possible that higher elastic properties aect local tur- shown in Figs. 3 and 4, the Nuavg is only three to four bulence, as is the case for turbulent ¯ows in vertical times higher than in the laminar ¯ow despite the high pipes [9]. If so, this should be seen through direct Re associated to those ¯ow conditions. measurements of the turbulent Pr, for which we have An important practical issue is worth noting here. recently estimated an average value of about 5 for var- Although the details of other experimental setups used ious drag-reducing ¯uids without buoyancy eects [13]. for heat transfer measurements in turbulent ¯ows of The reason for this particularly high value of the tur- drag-reducing ¯uids are not known to us, it is likely bulent Prandtl number (Prt), may be related to an that in many cases the sensors were located for con- unexplained eect of elasticity on the correlation venience on the upper surface of the pipes (as we did between temperature and velocity ¯uctuations. In this ourselves initially), thus providing possibly greatly respect, buoyancy in turbulent ¯ow of drag-reducing underestimated measurements of the Nu. To the best ¯uids deserves a deeper analysis that we cannot present in this short communication. of our knowledge, very few researchers in this ®eld Note that in Fig. 4 there is a very strong shift of the have reported being aware of the problems that buoy- point of coldest temperature towards the bottom wall, ancy eects could have on their heat transfer measure- comparable to the one showed in Fig. 1 for the lami- ments and analyses. It is indeed very easy to take it for nar ¯ow. This similarity of the temperature pro®les granted that the eects of buoyancy may be neglected aected by buoyancy in the laminar and drag-reducing for turbulent ¯ows of drag-reducing solutions as well, turbulent ¯ows, may suggest that in drag-reducing ¯ow as is indeed often done for Newtonian ¯uids. This we also have secondary ¯ows, just as in laminar ¯ow. would clearly be a serious mistake in some cases for This would not be surprising, because in the cases drag-reducing ¯uids.
Fig. 4. Temperature pro®le pro®les measured under high heat ¯ux along the vertical plane in the bottom half of the pipe for a drag-reducing ¯uid (1500 ppm Ethoquad solution). 4274 K. Gasljevic et al. / Int. J. Heat Mass Transfer 43 (2000) 4267±4274
5. Summary and conclusions References
The temperature pro®les of drag-reducing ¯uids are [1] Y. Mori, k. Futagami, S. Tokuda, M. Nakamura, signi®cantly aected by the action of buoyancy under Forced convective heat transfer in uniformly heated laminar ¯ow conditions. These buoyancy eects are horizontal tubes: experimental study of the eect of larger at the same Gr Pr than is generally observed buoyancy, Int. J. of Heat and Mass Transfer 9 (1966) with Newtonian ¯uids. In terms of Nu, buoyancy may 453±463. increase the Nu for drag-reducing viscoelastic ¯uids to [2] Y. Mori, K. Futagami, Forced convective heat transfer a level about 50% greater than that of the expected in uniformly heated horizontal tubes: theoretical study, theoretical value with no buoyancy eects, whereas Int. J. of Heat and Mass Transfer 10 (1966) 1801±1813. only a 12% dierence is expected under the same con- [3] S.M. Morcos, A.E. Bergles, Experimental investigation ditions for Newtonian ¯uids. of combined forced and free laminar convection in hori- Even in turbulent ¯ows, the eects of buoyancy are zontal tubes, J. of Heat Transfer 97 (2) (1975) 212±219. [4] B. Metais, E.R.G. Eckert, Forced, mixed and free con- still very noticeable for DR ¯uids. Given the distortion vection regimes, J. of Heat Transfer May (1964) 295± of the temperature pro®les, it appears likely that the 296. eect of buoyancy in turbulent ¯ows of DR ¯uids is of [5] A.V. Shenoy, J.J. Ulbrecht, Temperature pro®les for the same nature as in laminar ¯ows, i.e. caused by sec- laminar natural convection ¯ow of dilute polymer sol- ondary ¯ows. This is not surprising because even at utions past an isothermal vertical ¯at plate, Chemical relatively high Re (say up to 30,000), the Nuavg are Engineering Communications 3 (1979) 303±324. only three to four times higher than for laminar ¯ows [6] K.J. Ropke, P. Schummer, Natural convection of a vis- in the case of asymptotic DR, indicating very low tur- coelastic ¯uid, Rheologica Acta 21 (1982) 540±542. bulence. As a result of the asymmetry of the pro®les, [7] S.F. Liang, A. Acrivos, Experiments on buoyancy the usual heat transfer measurements in drag-reducing driven convection in non-Newtonian ¯uid, Rheologica ¯ows may show a large dierence between the Nu cal- Acta 9 (3) (1970) 447±455. culated with the top and the bottom wall temperatures, [8] A.V. Shenoy, Combined laminar forced and free con- in our case amounting to a ratio of over 2 between the vection heat transfer to viscoelastic ¯uids, AIChE Journal 26 (1980) 683±686. two. It is therefore very important that experimental- [9] A.V. Shenoy, Eects of buoyancy on heat transfer ists be aware of this issue, and use relatively low heat during turbulent ¯ow of drag reducing ¯uids in vertical ¯uxes (in our case corresponding to Gr Pr/Re of less pipes, Warms und Stoubertragung 21 (1987) 15±19. than 3) to reduce the eects of buoyancy on the [10] Y. Kawaguchi, H. Daisaka, A. Yabe, K. Hishida, M. measurements. If that is not convenient or feasible, a Maeda, Existence of double diusivity layers and heat temperature sensor placed on the side of the tube at transfer characteristics in drag reducing channel ¯ow, mid-elevation should be used, since it will give a good in: Proc. 2nd Int. Symp. Turbulence, Heat and Mass approximation of the average Nu between top and bot- Transfer, Delft, June, 1997. tom measurements. [11] K. Gasljevic, G. Aguilar, E.F. Matthys, An improved diameter scaling correlation for turbulent ¯ow of drag- reducing polymer solutions, J. of Non-Newtonian Fluid Acknowledgements Mechanics 84 (2/3) (1999) 131±148. [12] K. Gasljevic, An experimental investigation of the drag reduction by surfactant solutions and of its implemen- We acknowledge gratefully the ®nancial support of tation in hydronic systems, Ph.D. Thesis, University of the California Energy Commission (contract No. California at Santa Barbara, Santa Barbara, CA, 1995. 490260 to EFM). GA also wishes to acknowledge the [13] G. Aguilar, An experimental study of drag and heat Universidad Nacional Autonoma de Mexico, and es- transfer reductions in turbulent pipe ¯ows for polymer pecially the DGAPA and the IIM for support granted and surfactant solutions, Ph.D. Thesis, University of through their scholarship program. California at Santa Barbara, Santa Barbara, CA, 1999.