Heat transfer in rotating narrow rectangular pin-fin ducts

Heat transfer in rotating narrow rectangular pin-fin ducts

Experimental Thermal and Fluid Science 25 (2002) 573±582 www.elsevier.com/locate/etfs Heat transfer in rotating narrow rectangular pin-®n ducts Fred ...

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Experimental Thermal and Fluid Science 25 (2002) 573±582 www.elsevier.com/locate/etfs

Heat transfer in rotating narrow rectangular pin-®n ducts Fred T. Willett b


, Arthur E. Bergles


a Power Technology Incorporated, 2204 Maxon Road, Schenectady, NY 12308, USA Department of Mechanical, Aerospace and Nuclear Engineering, Rensselaer Polytechnic Institute, Troy, NY 12180-3590, USA

Received 2 May 2001; received in revised form 31 August 2001; accepted 2 October 2001

Abstract Rotational e€ects on heat transfer in radial-out¯ow ducts have implications for the design of gas turbine blades. The turbine blade trailing edge convective cavity generally has a narrow cross-section and often utilizes pin-®ns for heat transfer enhancement. Previous research on rotational e€ects considered cavity shapes quite di€erent from those of typical trailing edge cavities. In this research, experiments were conducted to determine the e€ect of rotation on heat transfer in pin-®n ducts of narrow cross-section (height-to-width ratio of 1:10), oriented with the heated sides at 60° to the r±z plane. In the experiment, a high-molecular-weight gas (Refrigerant-134A) at ambient pressure and temperature conditions was used to match the dimensionless parameters at engine conditions. The pin-®ns were arranged in a staggered array. Thin foil heaters were used to produce a constant heat ¯ux at the long sides of the duct; the narrow sides were unheated. Duct Reynolds numbers were varied up to 17,200; rotation numbers were varied up to 0.082. The test results show the e€ect of rotation on duct leading and trailing side heat transfer. In addition, the results show the variation in heat transfer coecient with transverse location in the duct. Results are compared to data obtained for a smooth duct of identical dimensions. Ó 2002 Elsevier Science Inc. All rights reserved.

1. Introduction The demand for more power, whether it be thrust from an aircraft engine or shaft horsepower from an industrial gas turbine, provides incentive for gas turbine manufactures to continually seek improvements in gas turbine performance. Turbine performance can be increased by raising the turbine inlet temperature. It can also be increased by decreasing the amount of air from the compressor that is dedicated to turbine cooling. In general, turbine blades, in order to survive in higher performance turbines, must either be made from superior materials or have superior cooling designs, or both. Insuciently cooled blades are subject to oxidation, creep rupture, and, in extreme cases, melting. A common design solution for the problems of trailing-edge cooling uses arrays of cast pin-®ns inside the trailing edge cavity to enhance heat transfer. The pin-®ns are provide structural support for the thin walls of the air*

Corresponding author. Tel.: +1-518-347-0271; fax: +1-518-3470273. E-mail address: [email protected] (F.T. Willett).

foil and eliminate the concern over panel-mode vibration of the airfoil walls. Passage size and manufacturing limitations dictate that the pin height-to-diameter ratio be small, often close to 1. The body of heat transfer data that exists for ¯ow across tube banks or arrays of long pin-®ns, then, is of limited usefulness, because the e€ect of the endwalls on ¯uid ¯ow and heat transfer is much di€erent. Brown et al. [1] concluded as much in an early study of turbine blade trailing edge heat transfer. For the short pin-®n case, endwall e€ects are very important. The heat transfer in a rotating heated channel is complex. The action of rotation on the coolant results in a non-uniform distribution of the heat transfer coecient at any cross-section. Numerical solutions are not yet reliable or accurate enough for use in design. Experimental data are required to help gain a better understanding of the e€ects of rotation on heat transfer in a heated channel. Fig. 1 describes the rotation-induced forces acting on radially outward-¯owing coolant in a heated channel. The buoyancy force is governed by both the centrifugal force acting on the ¯uid and the density gradient in the ¯uid. The density gradient is a function

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F.T. Willett et al. / Experimental Thermal and Fluid Science 25 (2002) 573±582

Nomenclature A heater surface area (mm2 ) Buo buoyancy number (Gr=Re2 ) (dimensionless) dH duct hydraulic diameter (mm) D pin diameter (mm) Gr Grashof number (X2 Rmean b…Tw Tb †dH3 =m2 ) h k Lp Nu Nu0 Pr q Re

(dimensionless) convective heat transfer coecient (W=m2 K) thermal conductivity (W/mK) pin length (mm) Nusselt number (hdH =k† (dimensionless) predicted pin-®n duct Nusselt number as correlated by Metzger (0:135Re0:69 …x=D† 0:34 ) (dimensionless) Prandtl number (m=a) (dimensionless) heat ¯ow (W) Reynolds number …VdH =m† (dimensionless)

Rmean Ro x V

mean duct radius (mm) Rotation number …XdH =V † (dimensionless) distance between pin-®ns (mm) axial velocity (m/s)

Greek a b m X

thermal di€usivity (m2 =s) volume expansion coecient (K 1 ) kinematic viscosity (m2 =s) rotational speed (rpm)


Subscripts b bulk in total supplied into heaters loss lost to surroundings w wall

Fig. 1. Outward ¯owing channel in rotation.

rectangular ducts, and concluded that aspect ratio is a critical parameter. They observed that the greatest Coriolis e€ect occurs in square ducts, i.e., an aspect ratio of 1:1. Willett and Bergles [7] investigated the e€ect of rotation and orientation on heat transfer in a 1:10 aspect ratio duct and concluded that the normalized Nusselt number is a strong function of transverse location and duct orientation. They also found that normalized Nusselt number at the far-aft end of the trail side of the duct, which represents the very trailing edge in an actual turbine blade, is a strong function of rotation number and buoyancy number. Fig. 2 shows a cross-section of the test duct superimposed on a turbine blade crosssection. All of the work on pin-®ns published in the open literature was done on stationary systems, i.e., not subject to rotation as in an actual turbine blade. Van Fossen [8] demonstrated experimentally that the average overall heat transfer coecient for an array of short pin®ns was lower than that for long pin-®ns in a similar array. Another important conclusion from Van Fossen's work was that existing correlations for long pin-®ns are

of the di€erence in temperatures of the ¯uid near the heated walls and the ¯uid in the core of the passage. Mori et al. [2] did an investigation of the Coriolis e€ect, studying ¯ow in round tubes both experimentally and analytically. Wagner et al. [3] investigated both Coriolis and buoyancy e€ects in their work, which used square passage and non-dimensional ¯ow parameters typical of gas turbine blades. Later work by Johnson et al. [4] studied the e€ects of rotation on inward ¯ow as well as outward ¯ow. Soong et al. [5], and later Kuo and Hwang [6] expanded the investigation of secondary ¯ow e€ects due to rotation by considering smooth-walled

Fig. 2. Test duct orientation.

F.T. Willett et al. / Experimental Thermal and Fluid Science 25 (2002) 573±582

not applicable for the shorter pin-®n geometry, even if the correlation attempts to account for pin-®n length. Metzger et al. [9] followed the initial work of Van Fossen, and studied developing heat transfer in stationary arrays of short pin-®ns. They concluded that the trend observed by Van Fossen of lower heat transfer coecient for short pin-®ns than for long pin-®ns was also true for developing heat transfer. They also observed a rise in heat transfer coecient over the ®rst three to ®ve rows of pins in the array, followed by a gradual decline over the remaining rows. Simoneau and Van Fossen [10] investigated the e€ect of location within an array on local heat transfer coef®cient. Their results expanded upon the work of Metzger et al. [9], who concentrated on the ®rst row of pin-®ns. Simoneau and Van Fossen showed that the number of upstream rows was unimportant for the case of in-line pin-®ns, but a€ected heat transfer coecient if the pin-®ns were in a staggered array; the number of downstream rows of pin-®ns was unimportant for both array patterns. Their results showed that heat transfer coecients peaked at the third row of pin-®ns. Brigham and Van Fossen [11] studied the e€ect of length-to-diameter ratio (Lp =D) on short pin-®n heat transfer, and concluded that Lp =D is the dominant factor in pin-®n heat transfer for Lp =D P 2. For values of Lp =D < 2, Nusselt number, Nu, is a function of Reynolds number, Re, only, and is independent of Lp =D. A comprehensive review of the literature through 1988 was made by Armstrong and Winstanley [12]; their summary covers the work described above, as well as the evolution of pin-®n research from the earlier studies of cross-¯ow over tube arrays. Lau et al. [13] and Kumaran et al. [14] studied the more complicated, but also more realistic, case of pin-®n channels with additional trailing-edge ejection. This work closely models the conventional use of pin-®ns in an array at the trailing edge of a blade using open-circuit convective cooling. They de®ned ejection ratio, the fraction of the total ¯ow that is exhausted through the blade trailing edge, and correlated the heat transfer enhancement and pressure loss with ejection ratio. Ejection ratio was found to have a strong e€ect on friction factor. Overall heat transfer decreases with increased trailing-edge ejection, due to the decrease in straightchannel mass ¯ow as ¯ow is exhausted through the trailing edge. Ejection hole arrangement was not shown to a€ect either pressure loss or heat transfer. The ¯uid ¯ow over a pin-®n has some relation to the classic case of ¯ow over a single, submerged cylinder. Given the velocity, u…x†, where x is the arc length measured from the stagnation point, there are several analytical techniques that can be used to predict the point of separation, and most of them match well with experimental observation. However, the wake caused by the separation has itself a ®rst-order e€ect on the velocity,


u…x†, and prevents accurate prediction of the ¯ow around a cylinder. Fortunately, the cylinders in this case are pin-®ns, which are small enough that the interest is not at the microscopic level, i.e., not concerned with ¯ow and heat transfer as a function of position on the cylinder. The interest here is at the macroscopic level, i.e., the e€ect of pin-®ns on the system. The Nusselt number averaged over the entire pin-®n or group of pin®ns is of great interest, but the variation in Nu as a function of location on the pin-®n is not. For a cylinder in cross-¯ow, the correlation Nu ˆ 0:26Re0:6 Pr0:3


was developed by McAdams [15] for air, and over a Reynolds number range of 1000±50,000. The same Re0:6 dependence is observed for banks of tubes in cross-¯ow, as well as for long pin-®ns. Armstrong and Winstanley [12] make the connection and postulate that the physical mechanism driving heat transfer is the ¯ow around the pins. As the pins become shorter, endwall e€ects become more signi®cant and the Reynolds number dependence changes from a 0.6 power dependence to a dependence approaching 0.7 power. The limiting case would be complete dominance by endwall e€ects, and the Re number dependence would be expected to be a power of 0.8, the Reynolds number dependence for turbulent duct ¯ow. Metzger et al. [16] proposed an array-averaged heat transfer correlation for pin-®ns ranging in height from 0.5 to 3.0 diameters:  x  0:34 Nu ˆ 0:135Re0:69 ; …2† D where x is the streamwise distance between pin-®ns. Armstrong and Winstanley [12] compared the Metzger correlation with published results for the same Reynolds number range, height-to-diameter ratio, and transverse spacing-to-diameter ratio and concluded that the correlation is adequate for design use. This correlation is, of course, developed for stationary pin-®ns arrays. It will be used as the basis for normalized Nusselt number for the case of rotating ducts with arrays of pin-®ns, just as the Dittus and Boelter/McAdams correlation is used for smooth ducts. Study of row-by-row variation in pin-®n heat transfer has been more limited than array-averaged heat transfer, with the most signi®cant work done by Metzger et al. [9] and Van Fossen [8]. At the local level, there are two e€ects. One is the entrance e€ect, similar to the entrance e€ect observed in duct ¯ow. The other e€ect is due to the broad wakes coming o€ the upstream cylinders, which raise the turbulence intensity levels downstream. This causes an increase in heat transfer from one row to the next, with peak values observed at the third of fourth row. The heat transfer decrease after the third or fourth row and reaches an asymptotic value. The maximum combination of wake e€ects and entrance e€ects is seen,


F.T. Willett et al. / Experimental Thermal and Fluid Science 25 (2002) 573±582

then, at the third or fourth row, with the entrance e€ects dying out after the fourth row. Unlike array-averaged heat transfer, there does not exist an accepted correlation for Nusselt number as a function of row number. 2. Experimental apparatus and method 2.1. Rotational test rig The test facility consisted of a rotor and stator enclosed in a closed loop system using R-134a as the test ¯uid. The refrigerant was supplied as a gas to the test sections to be evaluated. The use of a high-molecularweight gas, e.g., a refrigerant, means that the dimensionless parameters relevant to gas turbine operating conditions can be matched at ambient pressure and relatively low rotational speeds. A diagram describing the test rig is given in Fig. 3. The test sections, or ducts, were installed in a 622 mm diameter rotor. The rotor was constructed of mahogany and framed by an aluminum disk. The duct coolant was fed through the rotor via a rotating union and a radial supply passage in the disk. An inlet plenum was used to connect the supply tube to the rotor with an O-ring seal. Two inlet screens were used in the plenum to provide a uniform velocity at the inlet to the test duct. The inlet plenum was made of aluminum and was ®tted into the supply tube at the inlet to the test duct. The inlet gas temperature was measured between the two inlet screens. The duct inlet was located at a radius of 168 mm. Fig. 4 shows the rotor and test section before installation into the rig. 2.2. Test sections The duct was a two-piece construction and was made of mahogany. The ¯ow passage in the test section was created by removing material from one half of the section, then cementing the two halves together. The external dimensions of the test section are 26:7 mm 24:1 mm, which allow it to ®t in the inlet plenum. Continuous thin (0.013 mm) ®lm nickel-based

Fig. 4. Rotor disk with test duct ready for installation.

alloy heaters were attached to the wide sides of the duct. The heaters were backed with a very thin polymer ®lm, which prevents the foil from tearing easily, but more importantly, electrically insulates the foil heater from the thermocouples. A specially formulated epoxy cement was used to fasten the polymer backing of the heaters to the walls of the duct. This very thin layer of epoxy provided additional electrical insulation for the thermocouples. The narrow sides of the duct were not heated. The heated length of the duct was 114.3 mm. The two pieces of the duct were cemented together with epoxy cement and wrapped with tape to prevent leaks. The ¯ow area of the duct was 2:54 mm 25:4 mm. The pin-®ns were made of carbon steel and were cut from a 2.54 mm diameter drill rod. The pin-®ns had a height-to-diameter ratio of 1. The pins were attached to the thin foil heaters with an epoxy cement that had been impregnated with aluminum oxide. The aluminum oxide increases the thermal conductivity of the cement without having a large e€ect on the electrical conductivity. A resistance check of the assembly was performed prior to testing to con®rm that the pins were electrically insulated from the heaters. The pins were fastened at both ends to accurately represent pin-®ns in an actual turbine blade. The pins were con®gured in a staggered array as described in Fig. 5. 2.3. Instrumentation and data-acquisition

Fig. 3. Rotating test rig.

The duct was instrumented with Type-K thermocouples. The insulated thermocouple beads were in contact with the back of the heater. The thermocouple wires were of 0.25 mm diameter and were covered with 0.15 mm thick insulation. The thermocouples were fed through the duct wall, perpendicular to the heater surface. The thermocouples were connected to a slip ring, which allowed transmission of data to the data-acquisition system. One side of the duct was well instrumented; the other side had only three thermocouples used

F.T. Willett et al. / Experimental Thermal and Fluid Science 25 (2002) 573±582


Fig. 5. Test duct and thermocouple locations.

for measuring the heater temperature and regulating the power to the heaters. The total number of thermocouples was limited by the slip ring capacity. A total of 25 thermocouples was used on the well-instrumented side, 11 in the developing region and 14 in the fully developed region. An additional thermocouple was placed on the outer wall of the test section. Because of the limited number of slip ring channels available, lead-side and trail-side data could not be obtained simultaneously. The test ring was operated in both forward and reverse rotations to obtain data for both sides. Fig. 5 describes the test duct. A program written using commercially available software enabled continuous readout of the data. The data were displayed on the computer screen in a virtual instrument format con®gured especially for the experiment. The user could write the data from the virtual instrument to a ®le command. Flow levels were set manually using a ¯ow gage and monitored by the program. Rotating speed was measured with a tachometer and was set by the test operator using a variable-speed electric motor. Current to the heaters was controlled by the user via a DC power supply and displayed along with the other data. Heat input to the test section was entirely from resistance heating; the same current was supplied to both heated sides of the duct. The uniform-heat-¯ux boundary condition was applied on two sides of the duct. Uniform heat ¯ux is a reasonable ®rst-order representation of the heat transfer in an actual turbine blade. The nickel-based alloy heaters have a very low temperature coecient of resistance (TCR), so the resistance is relatively constant over the range of operations, however, a correction was made to the resistance for the calculation of heat input to the duct. The known or measured quantities for this experiment were: (1) the heat into the test section, (2) the R134a ¯ow rate, (3) the wall temperature, (4) the R-134a

inlet temperature, (5) the ¯uid properties, and (6) the duct geometry. The heat transfer coecient is de®ned as qin qloss hˆ : …3† A…Tw Tb † In the experiment described here, q, Tw , and A are known. Tb can be calculated knowing the inlet temperature and the heat supplied to the heaters. With the calculated value of Tbulk , the local heat transfer coecient can be determined at any thermocouple location. qloss , the heat lost from the heaters by conduction out of the test section, was determined using temperature measured at the outside wall of the test section and a 2-D ®nite-element model of the test section. Because the test section is made of non-conducting material, this loss is small, between 5% and 10% of the total heat supplied by the heaters, depending on test conditions. The duct-averaged heat transfer coecient can also be determined for each heated wall. When determining the duct-average heat transfer coecient, only the region aft of the fourth row of pin-®ns is considered. The duct-average heat transfer coecient is based on average temperature, i.e., qin qloss : …4† hˆ A…T w T b † The average Nusselt number is calculated as hdh : …5† k The Nusselt number is normalized to the Nusselt number for turbulent ¯ow in a pin-®n duct as correlated by Metzger, Eq. (2). A series of tests were conducted, over the range of Re anticipated for the rotating tests, with the duct in a static, i.e., non-rotating, condition to validate the use of this correlation for the 1:10 aspect ratio pin-®n duct. The stationary duct results were compared to the values

Nu ˆ


F.T. Willett et al. / Experimental Thermal and Fluid Science 25 (2002) 573±582

predicted by the Metzger correlation and were found to correlate well; the value of R2 calculated was 99.6%. From the results it can be concluded that the Metzger correlation can be used to normalize Nusselt number for the pin-®n duct data from this research. An uncertainty analysis was performed using the method described by Kline and McClintock [17]; the experimental uncertainty in normalized Nusselt number was calculated to be 10%. The largest sources of uncertainty were the temperature measurement and the measurement of heater area. Willett [18] describes the apparatus, procedure, and data reduction in greater detail. The duct-averaged normalized Nusselt number is calculated as Nu Nu ˆ 1 Pn ; Nu0 n iˆ1 …Nu0 †i


where n is the number of thermocouples in the section of the duct over which the average is taken and …Nu0 †i is the calculated smooth, non-rotating, duct Nusselt number corresponding to the thermocouple location. The variation in Nu0 with axial distance is quite small, and is due to the dependence of viscosity and Prandtl number on bulk ¯uid temperature. Normalized Nusselt number was studied as a function of both rotation number and buoyancy number. Rotation number was controlled by adjusting the mass ¯ow through the duct. Buoyancy number was controlled by adjusting mass ¯ow and wall temperature. The rotational speed was held constant at 2400 rpm. The Coriolis forces due to rotation generate secondary ¯ows and produce a complicated three-dimensional ¯ow and heat transfer distribution. The Coriolis force acts perpendicular to the ¯ow direction. Secondary ¯ows, produced by interaction of viscous forces and Coriolis forces, develop in the plane perpendicular to the ¯ow direction. The Coriolis force also has an e€ect on velocity pro®le, with the core ¯ow shifting in the direction of the Coriolis force. In radially outward ¯ow, this shift is towards the trail side of the duct. The work of Moore, as reported in [19], clearly demonstrated this shift in a rotating square duct. An excellent discussion of the ¯ow behavior in a rotating radial-out¯ow channel was published by Wagner et al. [3] and is pertinent to the present study. On the lead side, the boundary layer thickness as a result of the Coriolis force. The trail-side boundary layer is thinner and turbulence is increased, contributing to higher heat transfer. Moreover, the secondary ¯ows in the transverse direction bring cooler ¯uid from the core towards the trail-side wall and increase heat transfer. The lead-side heat transfer is reduced, as the thicker boundary layer acts to stabilize the ¯ow, so turbulent mixing is reduced.

Fig. 6. Transverse locations within the test duct.

Also, the secondary ¯ow patterns are such that warmed ¯uid is returned to the lead side, reducing heat transfer. Flow in a rotating channel is further complicated by the buoyancy e€ect. In a radially outward ¯owing duct, the buoyancy force acts counter to the ¯ow direction. These opposing forces create a strong shear gradient and generate additional turbulence, which results in increased heat transfer. When the Coriolis e€ect is taken into account, it follows that the buoyancy e€ect will be greater on the trail side. The density gradient is steeper on the trail side than on the lead side due to the movement of cooler core ¯ow towards the trail side and away from the lead side. With the duct rotated out of the r±z plane, i.e., heated sides not perpendicular to the h-direction, the lead-side and trail-side designations are no longer sucient. It is necessary now to add forward and aft designations. The transverse location in the duct is referenced to the forward and aft positions, with aft corresponding to the end at the turbine blade trailing edge. Data were collected at ®ve equally spaced transverse locations, one at the center and two each on either sides of the duct center line. The ®ve positions are designated as: far forward, center forward, center line, center aft, and far aft; these are shown in Fig. 6. By separating lead- and trail-side data and by segregating data as described above, it is possible to gain a more thorough understanding of the e€ects on heat transfer, not only of rotation and duct orientation, but also of local position within the duct. 3. Results and discussion 3.1. Lead-side data Fig. 7 shows averaged data for the lead and trail sides of the duct. Three sets of data are shown for the lead

F.T. Willett et al. / Experimental Thermal and Fluid Science 25 (2002) 573±582


Fig. 7. Average Nu=Nu0 vs. Buo.

side and ®ve sets for the trail side, corresponding to di€erent values of rotation number. The normalized Nusselt number is relatively constant with buoyancy number, but shows a slight increase with increasing rotation number. Both buoyancy number and rotation number are calculated on a smooth duct basis. In calculating the rotation number, the hydraulic diameter of the smooth duct, i.e., no pin-®ns, is used. The velocity term in the denominator is the smooth duct velocity, and not the maximum velocity used to calculate Reynolds number. Similarly, in the calculation of the buoyancy number, the smooth duct hydraulic diameter is used. This choice of calculation allows for a clear and meaningful comparison of the pin-®n results with similar results obtained using a smooth duct, reported by Willett and Bergles [7]. The in¯uence of rotation number and buoyancy number on normalized Nusselt is apparent from the duct average data and can also be seen from data for the far forward, center line, and far aft locations. These data are presented for the lead side in Fig. 8 and show a slight increase in Nu=Nu0 with increasing buoyancy number. Consistent with the observations of McMillin and Lau [20] for stationary pin-®n ducts, the heat transfer coef®cient near the side walls is higher than that at the center row of pin-®ns. The di€erence can be observed by inspection of the plots. In this case, the di€erence in normalized Nusselt number from center to sidewalls was about 15±20%. The di€erence between aft end Nu=Nu0 and center line Nu=Nu0 ranged from 17% to 25% while the di€erence between forward end and center line Nu=Nu0 ranged from 10% to 17%. The data are tightly grouped and show only a small in¯uence of transverse location of normalized Nusselt number. There is a consistent increase in Nu=Nu0 with increasing buoyancy number, and it is unclear whether it is due to increased rotation (Coriolis) e€ects or density gradient. In a square duct, previous researchers have observed a transition rotation number, marking the large-scale development of Coriolis-induced secondary ¯ow cells

Fig. 8. Lead-side Nu=Nu0 vs. Buo.

and increased lead-side heat transfer. Wagner et al. [3] reported a transition rotation number of 0.2. At the lead side of the square duct, for values of rotation number less than the transition value, normalized Nusselt number decreases with increasing Ro. Willett and Bergles [7] observed similar behavior at the lead-side farforward end of a smooth duct identical in size, shape, and orientation to the pin-®n duct of the present study. No such decrease in heat transfer coecient at the lead-side forward end was seen in the pin-®n duct. This suggests that the secondary ¯ow behavior due to the Coriolis e€ect is mitigated by the presence of pin-®ns, but it is unclear whether the in¯uence of Buo on Nu=Nu0 is due to density gradient or Coriolis-induced secondary ¯ow cells. The e€ect of density ratio is examined in Fig. 9, which presents normalized Nusselt as a function of density ratio and rotation number for the far-forward end. At this location (as at all transverse locations in the duct), the slope of the data is nearly zero, suggesting that density ratio is not important at the lead side in the pin®n ducts. There is a slight increase in Nu=Nu0 with increasing rotation number; in all three plots, the Nu=Nu0 data are strati®ed by rotation number. This suggests that any in¯uence on normalized Nusselt number at the lead side in pin-®n ducts comes from the Coriolis e€ect, and not from the buoyancy e€ect. 3.2. Trail-side data The slight increase in Nu=Nu0 with increasing buoyancy number observed in the duct average data can also be seen from data for the far forward, center line, and far aft locations. Five sets of data are shown in Fig. 10,


F.T. Willett et al. / Experimental Thermal and Fluid Science 25 (2002) 573±582

Fig. 9. Lead-side Nu=Nu0 vs. density ratio at the far-forward end of the duct.

values of Ro and Buo. At the forward end, there was a slight, but statistically signi®cant, di€erence; a paired t test of mean di€erences was done for veri®cation. The range of di€erences at the forward end was from 0% to 3%. The trail-side data also show the e€ect of transverse location on heat transfer in the pin-®n duct. The transverse variation in heat transfer coecient increases with buoyancy number, as was the case for the smooth duct results reported by Willett and Bergles [7]. The range of variations, however, is much less than observed for the smooth duct. As with the smooth duct, the highest heat transfer coecients are observed at the aft end. Unlike the smooth duct, however, the lowest heat transfer coecients are observed at the center of the duct, consistent with previous work on stationary pin-®n ducts as discussed earlier. The increasing heat transfer coecients at the aft end with increasing rotation number suggest that, even though the pin-®ns act to minimize the Coriolis e€ect, it is not entirely eliminated. It is still evident at the aft end of the duct, where, as was observed for the smooth duct, it is particularly strong. The e€ect of density ratio was studied to assess the relative importance of Coriolis and buoyancy e€ects. The results at the far-forward end and center line are very similar to the results obtained at the lead side, where a slight in¯uence of rotation number on Nu=Nu0 can be observed. Data are shown in Fig. 11 for the faraft end. At the far-aft end, a similar trend is observed for the in¯uence of density ratio; the slope of the data increases with increasing rotation number. The in¯uence of rotation number, however, is much stronger than at other transverse locations, consistent with the observations noted above. Also, unlike the results elsewhere in the duct, the results at the far-aft end of the trail side show an in¯uence of increasing density ratio at high Ro, showing the presence of the buoyancy e€ect.

Fig. 10. Trail-side Nu=Nu0 vs. Buo.

corresponding to di€erent values of rotation number. The normalized Nusselt number shows a slight increase with increasing buoyancy number at constant rotation number. As with the lead side, and for all pin-®n-duct data analysis, both buoyancy number and rotation number are calculated on a smooth duct basis. The data also show the trend of increasing Nu=Nu0 with decreasing rotation number at constant buoyancy number. The e€ect of buoyancy number is more pronounced at the aft end, as can be seen by the slopes of the data in Fig. 10. As with the lead side, the normalized Nusselt number near the side walls was higher than that at the center row of pin-®ns. At the aft end, the di€erence ranged from 14% to 37%, and was higher for higher

Fig. 11. Trail-side Nu=Nu0 vs. density ratio at the far-aft end of the duct.

F.T. Willett et al. / Experimental Thermal and Fluid Science 25 (2002) 573±582


An explanation of the e€ect of rotation on Nu=Nu0 at the far-aft end of the pin-®n duct is o€ered. The Coriolis e€ect changes the direction of the ¯ow vector and the ¯ow is channeled between the pin-®ns towards the aft end of the duct. Previous researchers have observed this type of behavior in stationary pin-®n channels with trailing edge ejection. 3.3. Forward and aft end data The Coriolis force in the duct oriented 60° to the r±z plane has two components relative to the heated walls. The parallel component forces ¯ow towards the aft end of the duct, and were shown by Willett and Bergles [7] to have a signi®cant e€ect on the heat transfer in the smooth duct. In the smooth duct, the e€ects were most pronounced at the forward and aft ends: Nu=Nu0 at the far-forward end of the lead side was as low as 0.7 for Buo ˆ 0:10; at the far-aft end of the trail side, it was greater than 3.0 for Buo ˆ 0:20. Fig. 12 compares the lead- and trail-side normalized Nusselt numbers at the far-aft end of the duct containing pin-®ns. The enhancement at the trail side is higher than that at the lead side, and the normalized Nusselt number at both sides is greater than 1.0 over the entire range of Buo. The leadside data follow close to the same trend as the trail-side data, but have a ¯atter slope. The enhancement due to rotation and buoyancy is much less than that observed for the smooth duct, showing that the pin-®ns mitigate, but do not eliminate, the rotation e€ects. Data for the far forward end of the duct are shown in Fig. 13. At buoyancy numbers below 0.1, there is not much di€erence between the heat transfer coecients for lead and trail sides. This is very di€erent from the smooth duct results, where the lead-side data showed a decrease in Nu=Nu0 similar to those observed by previous researchers, reaching a minimum value below 1.0. The trail-side heat transfer behavior is much di€erent, also. In the smooth duct, Nu=Nu0 at the trail side was

Fig. 13. Comparison of lead- and trail-side far-forward-end results.

much higher than at the lead side, but decreased sharply with increasing buoyancy number, approaching an asymptotic value of 1.0 at Buo ˆ 0:15. There are similarities with the smooth duct data at buoyancy numbers above 0.1. In both the smooth and pin-®n ducts, the normalized Nusselt number at the lead side was higher than that at the trail side. 3.4. Practical signi®cance of work The results of this investigation can be used by designers of turbine blades to predict more accurately the trailing edge metal temperatures. The results indicate that the overall heat transfer enhancement in the pin-®n duct is higher in a rotating system than in a static system; design engineers can use this information to reduce cooling ¯ow requirements or increase part life estimates. Engineers seeking a better understanding of the cause of turbine blade distress will be aided by the results showing the transverse and side-to-side variations in heat transfer coecient. 4. Conclusions

Fig. 12. Comparison of lead- and trail-side far-aft end results.

This research has expanded the study of rotational e€ects to include pin-®n ducts, and is unique in the ®eld of convective heat transfer in heated ducts. The passage aspect ratio and addition of pin-®ns make this data set unique. The results of this research are useful in their present form, and can also be used for the validation of computational models. Summarizing the conclusions: (a) The presence of the pin-®ns mitigates, but does not completely eliminate the Coriolis e€ect on heat transfer. Transverse variation in normalized Nusselt number is more apparent at the trail side of the duct because of the strong Coriolis e€ect at the far-aft end of the duct.


F.T. Willett et al. / Experimental Thermal and Fluid Science 25 (2002) 573±582

(b) Lead-side and trail-side values of Nu=Nu0 increase steadily with buoyancy number and were greater than 1.0 at all transverse locations for the range of Buo studied. (c) Local heat transfer near the side walls was greater than in the center of the duct. This was observed on both lead and trail sides and is consistent with previous results from stationary pin-®n ducts. The e€ect was greater on the lead side than on the trail side. (d) Lead-side and trail-side normalized Nusselt numbers at the mid-wall are less in¯uenced by buoyancy number than heat transfer coecients at the forward and aft ends. As buoyancy number increases, the difference between lead-side and trail-side Nu=Nu0 becomes insigni®cant. 4.1. Recommendations and future research needs Additional work is required to develop a more complete understanding of the e€ect of orientation angle, as only one angle, 30°, was evaluated here. Trailing edge ejection is commonly combined with the pin-®n cooling design, and an investigation of the combined interaction of pin-®ns, rotation, orientation angle, and trailing edge ejection is a suggested topic for future research. Finally, as this experiment was conducted with a uniform heat ¯ux boundary condition, a similar set of experiments with uniform wall temperature boundary condition is recommended. Acknowledgements This e€ort has bene®ted greatly from the technical discussions with Mr. Fred Staub of GE's Corporate Research and Development Center. The authors also gratefully acknowledge the support of Mr. Karl Hardcastle, also of GE CR&D, during the data-gathering phase of this research. The rotating test rig used for this research is located at GE's Corporate Research and Development Center. Dr. Norm Shilling, Dr. Kent Cueman and Mr. Gene Kimura graciously made the rig available for this work; their interest and encouragement is much appreciated. References [1] A. Brown, B. Mandjikas, J.M. Mudyiwa, Blade trailing edge heat transfer, ASME Paper No. 80-GT-45, 1980.

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