741
Ind. Eng. Chem. Fundam. 1986, 2 5 , 741-745 Patterson, J. M.; Haidar, N. F.: Shiue. C.; Smith, W. T., Jr. J . Ors. Chem. 1973, 38, 3052. Ratcliffe, C. T.; AbdeCBaset, M. Ger. Patent 3 007 257, 1980. Raymont, M. E. D. Ph.D. Thesis, Department of Chemistry, University of Calgary, Alberta, Canada, 1974. Ruberto, R . G.; Cronauer, D. C.; Jewell, D. M.; Seshadri. K. S. Fuel 1977, 56, 17, 25. Satterfield, C. N.; Model, M. U S . N T I S , PB Rep. 1975, 248101. Sondreal, E. A.; Wilson, W. G.; Stenberg, V. I . Fuel 1982, 67,925. Stenberg, V. I.; Baltisberger, R. J.; Ogawa. T.; Raman, K.; Woolsey, N. F. Prepr. Pap.-Am. Chem. SOC.Div. FuelChem. 1982, 2 7 , 22. Szladow, A. J.; Given, P. H. Prepr. Pap.-Am. Chem. Soc.,Div. Fuel Chem. 1978, 23, 161.
Takegami, Y.; Kajiyama, S.; Yokokawa. C. Fuel 1963, 4 7 , 291. Takemura, Y.; Itoh, H.; Ouchi, K. Fuel 1981, 60, 379. Takeya, G. Pure Appl. Chem. 1978, 5 0 , 1099. Van Buren. R . L. Ph.D. Dissertation, Department of Chemistry, University of North Dakota, Grand Forks, ND, 1981. Van Buren, R. L.; Stenberg, V. I . Chem. Ind. (London) 1980, 7 4 , 569. Wachowska, H.; Pawlak, W. Fuel 1977, 5 6 , 422. Wiewiorski, T. K. Endeavour 1970, 2 9 , 9. Yao, T.; Kamiya, Y. Bull. Chem. SOC.Jpn. 1979, 5 2 , 492.
Received for reuieu October 29, 1984 Accepted February 14, 1986
Drag Reduction in a Longitudinally Grooved Flow Channel John J. J. Chen' Chemical and Materials Engineering Department, The University of Auckland, Auckland, New Zealand
Yiu-Cheong Leung and Norman W. M. KO Mechanical Engineering Department, University of Hong Kong, Hong Kong
A survey of literature relevant to drag reduction using grooved surfaces is presented. I t was found that, for external air flows, longitudinal grooves were capable of reducing drag when their heights and spacings expressed in terms of the law-of-the-wall coordinates as h + and S+ were less than 25 and 100, respectively. Experiments were performed to confirm that, within this range of h+ and S+,drag reduction is also possible for internal turbulent flow of water in grooved channels.
Introduction Various methods have been explored with varying degrees of success to reduce the drag arising from the relative motion between a fluid and a solid boundary in both internal and external flows. These methods include polymer additives, a compliant surface, and the introduction of a second phase. A survey of the various methods may be found in Hefner and Bushnell (1977). In this article, a review of the literature pertinent to drag reduction using a grooved surface is presented. Experimental work on internal flow has been carried out t o determine whether the range of conditions already found to be possible for reduced drag in external flows is also applicable to internal flows. Literature Survey The experimental and theoretical study of flow through ducts with sharp-cornered cross sections had been carried out, giving anomalous results, by various workers including Eckert and Irvine (1956, 1957, 1958, 19601, Carlson and Irvine (1961), Aly et al. (1978), Hodge (19611, Wilkins (1965), Sparrow and Haji-Sheikh (1965), Gunn and Darling (1963), and Rehme (1973). Generally, it was found that for triangular channels the transition to turbulence starts a t a lower Reynolds number than for a circular pipe, but the range of transitions is wider and the "hump" that is a characteristic of the circular pipe flow is diminished. Irvine and Eckert (1960) suggested that the converging walls tend to exert a stabilizing influence on the flow field and that it may be possible to make use of wedge-shaped grooves to delay the laminar-to-turbulent transition. In a number of articles related to the augmentation of heat transfer, using finned tubes where the hydraulic diameter concept was employed, a lower friction factor had
been observed (Bergles e t al. (1971), Carnavos (1979a, 1979b), Scott and Webb (1981)). Carnavos (1979a) experimented with both straight fins and spiral fins and gave an empirical equation, eq 1, based on the hydraulic diameter, which correlates his results within &7%. However, deviations up to 21% were found with the rest of the data.
Scott and Webb (1981) also presented results based on the hydraulic diameter, and they noted that the friction factors of the finned channels were greater than those for smooth channels for some cases and smaller in other cases. They also pointed out that the use of the hydraulic diameter assumes that the velocity distribution is unaffected by the presence of the fins. Such correlations would therefore be inadequate because the velocity is reduced in the interfin regions relative to that in a smooth pipe. They developed a model which takes into account the difference in velocity in regions between the fins and in the core. The results were generally unsatisfactory, but for those fin configurations of interest for heat transfer, the predictions were &lo%. For the finned circular channels they gave fH
=
and for the finned rectangular channels fH
-( ")
= 0.566
ReH0.25 DH
0.26
(3)
Liu, Kline, and Johnstone (1966) and Kennedy et al. (1973)
0196-4313/86/1025-0741$01.50/00 1986 American Chemical Society
742
Ind. Eng. Chem. Fundam., Vol. 25, No. 4, 1986
Table I. Dimensions of Channels Used in the Experiments 2
3
4
5
model no. 6
8
9
10
11
0.29 0.61
0.50 0.35
0.50 0.13
0.50 0.6,j
0.50 1.04
0.70 0.63
0.70 1.45
1.00 2.06
0.50 2.08
0.50 2.08 1.00
90 134.5 25.29 19.02
30 295.8 25.50 8.79
45 251.7 25.50 10.33
60 177.6 25.50 14.64
90 136.0 25.50 19.13
45 240.5 26.70 10.98
90 136.5 25.70 19.35
90 137.3 26.00 19.69
90 119.9 25.24 81.25
__-
groove height h , mm groove spacing s, min groove width, mm groove apex angle. deg actual wetted perimeter, mm Dbl,
mnl
Dk$. inn,
1
100.0 25.00 25.00
also carried out work on flow through longitudinally grooved channels, but their results were inconclusive. I-Iefner and Bushnell (1977) suggested that turbulence seems to react to a Reynolds number based upon the local distance between the adjacent walls and that this behavior may be harnessed by employing a very small longitudinal V-groove on flow surfaces to confine the turbulent wall bursts to their initial burst regions and to regions of small transverse extent, thus altering the local turbulence production. This was studied further by Walsh and Weinstein (1979), who performed tests on longitudinally ribbed surfaces. They suggested that the presence of longitudinal fins tends to confine the turbulent bursts to regions of small transverse extent, thus altering the local turbulence production. In terms of the law-of-the-wall coordinates, grooves of height on the order of 25 and 15-20 showed drag reduction on the order of 2-4%. The repeatability was within * 3 % . This is consistent with the belief that the regions in which turbulent bursts are born have depths of about yt = 30. Separation between bursts in the direction transverse to the mainstream is about 100 in terms of the law-of-the-wall coordinates. This is somewhat in line with the observation of Ornatskii (1970), that transition to laminar flow occurs in the interfin channels only when the ratio of the fin width to fin height is less than or equal to 0.2 and the Reynolds number is less than 50 X lo3. Walsh (1980) pointed out that Liu, Kline, and Johnstone (1966) used h+ = 45-111 and St= 190-373 in their experiments, while Kennedy et al. (1973) had their rectangular fins at h+ = 70-150 and S' = 500-1100. Walsh (1980) substantiated his earlier work (Walsh and Weinstein, 1979) by showing t h a t , for certain V-groove riblets with heights h+ < 25 and S+ < 100. net drag reduction of up to 7'70 was possible. Walsh (1980) also pointed out that, even in those situations without drag reduction, there are relatively small drag increases considering the large wetted surface area increases (6.17-fold). He suggested that such surfaces might have potential for compact heat exchangers. From our review of the articles presented, it appears that grooved surfaces are able to confine turbulent bursts and their propagation downstream. This results in minimizing the generation of turbulence and a delay of the laminarto-turbulent transition, effecting lowered drag. This paper examines experimentally the possibility of using this concept for drag reduction for internal flow in closed conduits, which will be relevant in applications such as reticulation of process fluids, in design of nozzles, and in variuus internal flow situations. It is not the intent of this work to cover the entire range of h+ and S+ values but rather to determine whether drag reduction is possible in the region h t < 25 and S+ < 100 for turbulent internal channel flow Experimental Equipment A schematic diagram of our experimental equipment is L>hownin Figure I . It consists of a closed-circuit water
n
I
148.0 25.48 17 54
CARBON 'ETRACHLORIDE MANOMETER
CALMING SECT ON
TEST SECT ON
L2 r.4
t,-.-
Figure 1. Schematic diagram of the experimental setup.
loop. Town water was used and was changed after each series of tests. A constant-head tank at a height of 3 m above floor level supplied water through a control valve into the horizontal test section, preceded by a 0.7-m-long calming section which was found to be adequate for the development of the pressure profile. After the test section, the water was discharged into a calibrated measurement tank which was connected to a reservoir. The reservoir supplied water via a pump to the constant-head tank. A bypass was provided on the high-pressure side of the pump in order to regulate the amount of water being pumped up to the constant-head tank. To ensure that the head is constant, there must be an overflow of water from the constant-head tank into the reservoir. A valve was provided between the measurement tank and the reservoir to facilitate measurement of the flow rate, using a stopwatch. The test section may be easily removed and substituted by any one of the eleven sections fabricated for testing. Care was exercised to ensure alignment each time this was carried out. The models were fabricated with the characteristic dimensions listed in Table I. The process of manufacturing involved first milling longitudinal grooves on four pieces of 6-mm-thick flat PVC sheets of nominal dimensions, 700 mm X 30 mm. After machining, the sheets were cut to size and assembled by gluing at the corners to form the square test channels. The size of the grooves and the length of the test sections were therefore dictated by the size of the cutting tool and the length of the stroke of the milling machine, respectively (Table 11). Pressure tap points of 3-mm diameter were provided at the side and bottom of the test section, a distance 0.4 m apart. Each of these pressure tap holes may span more than one groove, depending on the width of the grooves. U-tube manometers using carbon tetrachloride (specific gravity 1.59) were used. The readings obtained from the side taps were identical with those from the taps at the
Ind. Eng. Chem. Fundam., Vol. 25, No. 4, 1986
743
Table 11. Profiles of Grooves in Channel Walls model no.
description
2-9
E
w
m
t L
10 11
7
z
F-
m lii IO’
I 2
3
L
I
5678910‘
2
-
3
4
5 6 78910’
Re,
Figure 3. Data for models 3 (0) and 10 ( X ) plotted as fM against ReM.
IiZ.
* -:&,*e
i
-‘
“k
O.8OC
060k
0‘01 10’
2
3
4
I 5 678910‘
-
1
2
3
4
5
6 7 8910’
Re,
Figure 4. Pumping power ratio, R, plotted against ReMfor models 3 (0) and 10 (x).
tively slightly. With a decrease in DH, the value of ReH shifted to the left and fH shifted downward in approximately equal amounts, provided that the changes in U and dP/dl were relatively small. Since the negative slope of the smooth-pipe line is not great, this caused a false impression of a large reduction in pressure drop, particularly for channels with grooves similar to model 3 which has a DH value approximately one-third that of the ungrooved channel. Alternatively, one may express the friction factors and Reynolds numbers in terms of a hydraulic diameter based on the average height of the grooves, DM, such that
and ReM = PuDM/P
(7)
Data for models 3 and 10 are plotted in Figure 3, based on eq 6 and 7. In these plots, the data were higher than the smooth-pipe line but crossed it and came down below the smooth-pipe line in the region of Re from lo4 to 2 X lo4. A graph based on eq 6 and 7 is also unsatisfactory since it is not possible to determine from it whether drag reduction occurred. If drag reduction occurs, a larger quantity of fluid can be transported for a fixed value of pressure drop or, alternatively, less pumping power is required to transport the same flow. This suggests a comparison of the pumping power required at the same flow rate. Thus, the pumping power ratios are plotted against the Reynolds number as shown in Figure 4 for models 3 and 10. The power ratio,
Ind. Eng. Chem. Fundam., Vol. 25, No. 4, 1986
744
Table 111. Range of Values of h + , S', and Re at R = 1.0 model no.
I
8 9 10 11
sc+
Re, X lo-* 1.45-1.68 1.95-2.15 1.65-2.00 1.67-1.82 1.70-1.80 1.38-1.58 1.40-1.50 1.50-1.72 1.30-1.55
11-13 18-21 25-30 37-42 23-24 46-52 62-66 75-83 58-62
'&
Id'l
'0'
hc+ 16-19 21-25 19-23 18-20 26-27 22-25 30-32 18-20 14-15
2
3
1
5 6 7 8910'
--
2
3
1
5 6 7 e510i i20xlO'
Re,
Figure 5. Comparison of the results for model 6 with the prediction from eq 3 of Scott and Webb (1981) (---). The solid line represents
+"
40-
i
1-
L
15.10'
eq 8.
i
R, is the pumping power for the channel in question, based on the measured pressure gradient, divided by the pumping power of a smooth channel for which the friction factor is based on eq 8. Equation 8 is preferred for smooth
f = 1/[1.8 log (Re/7)I2 channels over the Blasius equation since it is more accurate and is superior to the Prandtl-Karman equation in that it is explicit in the friction factor. The Reynolds number used is ReMsince it represents very nearly the same discharge in both grooved and ungrooved channels. It is justifiable to use eq 8 in this comparison since it provides a conservative estimate of the drag reduction. If we had used the experimental data for the smooth-walled channel or the transition equation of Colebrook and White (1939), which includes the roughness effects, not only would drag reduction have been predicted to occur a t a higher value of ReM,but an even smaller power ratio would have resulted where drag reduction occurred. Due to the limitations of the flow rig, accuracies of the very few data with ReM< 4000 become progressively worse as ReM decreased. No data were obtained for the smooth-walled channel for ReM 4000, and all the data obtained above ReM = 4000 were characteristic of flow in the turbulent region. Detailed data for the smooth-walled channel are also given in the supplementary material. Except for model 2, which exhibited a rather wide range of scattering, all the models showed that the power ratio increased with the Reynolds number (or the discharge). The ratio was less than unity when the Reynolds number was less than about lo4 t o 2 X lo4, indicating that drag reduction occurred. In fact, Figure 4 bears a direct correspondence to Figure 3. It therefore appears that the use of D, is more appropriate in this case than DH. Drag reduction is therefore possible in the turbulent flow regime to a Reynolds number of around lo4 to 2 X lo4. It is interesting to note that an inspection of Figure 3 shows that, before the data cross over the smooth-pipe line, they appear to be part of a gradual and extended transition from laminar to turbulent flow, similar to those observed by Gunn and Darling (1963) and Bergles et al. (1971). However, the apparatus we employed was not suitable for work in the low Reynolds number range. Beyond the crossover point, the data appear to follow lines which are roughly parallel to the smooth-pipe line. The region of increased drag is believed to be that for which the correlations of Carnavos (1979a) and Scott and Webb (1981) are expected to apply. However, these correlations are based on DH,and therefore for comparison purposes data must be presented as in Figure 2. It is interesting to note that for models 2, 6, and 8-10, which have values of DH reasonably close to the DH of the un-
0 0
05
-
h
IO mm
Figure 6. Values of h,+ and ReMcplotted against h for models 6, 8, and 9 which have grooves with apex angle of 90°.
grooved channel, the results approach those predicted by eq 3 when the Reynolds number is high. Equation 3 is inaccuarate for the remaining models which have significantly lower D,. Equation 1 of Carnavos (1979a) did not agree with any of the results obtained in this work. The comparison of eq 3 with the results obtained from model 6 is shown in Figure 5 . T o facilitate comparisons with the findings of Walsh (1979, 1980), the values of h+ and S+ for the crossover points have been calculated and listed in Table 111. It becomes obvious that drag reduction occurred at h+values s m d e r than about 25, which is consistent with the findings of Walsh (1979,1980). All the S+ values in this work were less than 100. The frictional drag increased beyond h,+, consistent with the work of Kennedy et al. (1973) which covered the range of h+ = 70-150. Some comparisons may be made using the data given in Tables 1-111. Referring to models 6, 10, and 11,which have the same groove heights of 0.5 mm but differently shaped grooves, the crossover Reynolds number appeared to decrease in going from model 6, to 10, to 11,indicating that drag reduction may be obtained a t higher Reynolds numbers by using model 6 rather than models 10 or 11. Figure 6 shows the values of h,+ and Re, plotted against the physical height, h, for models 6,8, and 9. The grooves in these channels had the same apex angle of 90°. It appears that the lower the physical height of a groove, the higher the flow rate necessary to maintain reduced drag. Unfortunately, as pointed out earlier, the height, h, is limited by the cutting tool, and some other techniques such as etching will have to be employed to obtain smaller h. However, as h gets smaller and smaller, a hydraulically smooth surface must be approached; the surface would become that of a smooth channel. Thus, it is expected that there is an optimum value of h which gives the best drag reduction behavior. Figure 7 shows the variation of h,+ and Re, for models 3-6, which have grooves of varying apex angles but equal physical height, h. This also gives the variations with respect to the fin spacing. The data indicate that an op-
Ind. Eng. Chem. Fundam., Vol. 25, No. 4, 1986
f fH fM
h h+
1 P R ReH ReM S S+ 0
10 20 30 LO 50 60 70 80 90
-GRGROOVE
APEX
ANGLE
Figure 7.
Values of h,+ a n d ReMc for models 3-6 w h i c h have t h e same values of h but have grooves of varying apex angle.
t U
745
Darcy friction factor given in eq 8 Darcy friction factor defined in eq 4 Darcy friction factor defined in eq 6 groove height groove height expressed in terms of the law-ofthe-wall coordinates = (hU/u)(fM/8)1/2 length pressure pumping power ratio Reynolds number, defined in eq 5 Reynolds number, defined in eq 7 groove width groove width expressed in terms of the law-ofthe-wall coordinates = ( SU / U) (fM/8)1/2 fin thickness, appears in eq 2 mean velocity of flow
Greek Symbols spiral fin tube helix angle in degrees; appears in eq 1 t equivalent sand roughness U kinematic viscosity P density a
timum range of groove apex angle exists in the range of 40-50'.
Conclusions A review of the literature relevant to the use of grooved surfaces for drag reduction has been presented. It appears that, for external flows, frictional drag may be reduced if surfaces with wedge-shaped longitudinal grooves of suitable dimensions were employed. A delay in the laminar-toturbulent transition may be effected, resulting in drag reduction, provided that the depth and width of the grooves are less than 25 and 100, respectively, when expressed in terms of the law-of-the-wall coordinates. This concept was tested for internal flow in square PVC channels. A friction factor smaller than the values for a smooth pipe were obtained when the data were presented by using the hydraulic diameter based on the total wetted perimeter. Comparisons of the ratio of the power required for a grooved channel to that required for an ungrooved channel were therefore made a t various discharge rates. This showed that drag reduction is possible if the groove heights are below about 25 in terms of the law-of-the-wall coordinates. Grooving of the flow surface is suggested as being a possible method of passive drag reduction in internal flow. Work is in progress to cover the lower Reynolds number range, as well as the use of smaller groove heights, which will provide values of h+ < 25 at higher Reynolds numbers. Acknowledgment We are grateful to Professor A. E. Bergles for clarifying a number of points in the manuscript. Nomenclature Afa free flow area nominal flow areas as if -grooves were not present; A .fn.. thus, A,, > Afa hydraulic diameter based on the total wetted perDH imeter diameter to the base of the fins in a finned tube Di hydraulic diameter based on the square section DM formed by the mean heights of the grooves ~~
Subscript C
signifies the point at which R = 1.0
L i t e r a t u r e Cited Aly, A. M. M.; Trupp, A. C.; Gerrard, A. D. J. Fluid Mech. 1978, 8 5 , 57-83. Bergles, A. E.; Brown, G. S.; Snider, W. D. Presented at the ASME Meeting, New York, 1971; paper 71-HT-31. Carlson, L. W.; Irvine, T. F. J. Heat Transfer 1961, 8 3 , 441-444. Carnavos, T. C. Heat Transfer Eng. 1979a, 1 , 41-46. Carnavos T. C. Natl. Heat Transfer Conf., 18th I979b, 61-67. Colebrook, C. F. J. Inst. Civ. Eng. 1939, 1 1 , 133-156. Eckert, E. R. G.; Irvine, T. F. Trans. Am. SOC. Mech. Eng. 1958, 78, 709-718. Eckert, E. R. G.: Irvine, T. F. Midwest Conf. Fluid Mech., 5th 1957, 122-145. Eckert, E. R. G.; Irvine, T. F. J. Heat Transfer 1960, 82,125-138. Eckert, E. R. G.; Irvine, T. F.; Yen, J. T. Trans. Am. SOC.Mech. Eng. 1958, 80, 1433-1438. Gunn, D. J.; Darling, C. W. W. Trans. Inst. Chem. Eng. 1963, 4 1 , 163-173. Hefner, J. N.; Bushnell, D. M. "Special Course on Concepts for Drag Reduction", Report 654; NATO Advisory Group for Aerospace Research 8, Development: Neuilly Sur Seine, France, 1977. Hodge, R. I.J. Heat Transfer 1981, 8 3 , 384-385. Irvine, T. F.; Eckert, E. R. G. Trans. ASME 1960, 325-332. Kennedy, J. F.; Hsu, S. T.; Lin, J. T. Proc. Am. SOC.Civ. Eng., J. Hyd. Div. 1973, 99, 605-616. Leung, Y. C. Ph.D. Thesis, University of Hong Kong. in preparation. Liu, S. K.; Kline, S. J.; Johnstone, J. P. Report MD-15; Stanford University: Stanford, CA, 1966. Ornatskii, A. P.; Shcherbakov, V. K.; Semena, M. G. Therm. Eng. (Engl. Trans/.) 1970, 17, 108-111. Rehme, K. Int. J. Heat Mass Transfer 1973, 16, 933-950. Schlichting, H. Boundary Layer Theory, 7th ed.; McGraw-Hill: New York, 1979; pp 612-615. Scott, M. J.; Webb, R . L. J. Heat Transfer 1981, 103, 423-428. Sparrow, E. M.; Haji-Sheikh, A. J. Heat Transfer 1965, 87C, 426-427. Walsh, M. J. Prog. Prog. Astronaut. Aeronaut. 1980, 7 2 , 168-184. Walsh, M. J.: Weinstein, L. M. AIAA J. 1979, 17, 770-771. Wilkins, J. E. J . Heat Transfer 1985, 87C, 427-428.
Received for review October 29, 1984 Revised manuscript received November 11, 1985 Accepted
F e b r u a r y 13,
1986
Supplementary Material Available: Tables of drag reduction data and plots of fM vs. ReM, fH vs. ReH,and R vs. ReM for models 1-11 (41 pages). Ordering information is given on any current masthead page.
