Detailed Heat Transfer Distribution on the Endwall of a Trapezoidal Duct with an In-Line Pin Array

This study presents the measurements of detailed endwall heat transfer and pressure drop in a pin-®nned trapezoidal duct. The trapezoidal duct is inserted with an in-line pin array of ®ve rows and ®ve columns, with the same spacings between the pins in streamwise and spanwise directions of Sx/d5Sy/d5 2.5. The eects of the lateral̄ow ejection (0<  e<  1.0) and ̄ow Reynolds number (6000< Re  40000) are examined. Transient liquid crystal technique is employed to measure the detailed heat transfer on the endwall of a pin®nned trapezoidal duct. Results indicate that the pin-®nned trapezoidal duct of a lateral̄ow rate about e5 0.3 has a local minimum endwall-averaged Nusselt number. In addition, the trapezoidal duct of lateral outlet ̄ow only (e5 1.0) has a highest endwall heat transfer as well as pressure drop. Finally, the correlation of the endwall-averaged heat transfer for the pin-®nned trapezoidal duct is developed by taking account of the lateral̄ow rate and the ̄ow Reynolds number.

Augmentation of heat transfer inside airfoil internal channels is an important issue for the gas turbine industry.As turbine inlet temperatures are increased, there is a greater need for more ecient cooling.Many cooling strategies have been developed over the years.The most familiar internal-cooling method as shown in Figure 1 is augmented forced convection in various-shaped channels by using rib turbulators and/or pin ®ns.This study focuses on the internal cooling near the trailing edge of a turbine blade, in which pin ®ns spanning between the suction and pressure surfaces to promote turbulence, and thus internal heat transfer.
Some relevant studies about pin ®n heat transfer are brie¯y surveyed below.VanFossen (1982) measured the overall heat transfer coecients in rectangular ducts with staggered arrays of short pin ®ns (0.5 < l/d < 2.0).All arrays had four rows of pins in the streamwise direction.Heat transfer rates were averaged over the four-row array.It was found that the overall heat transfer coecients for short pins were lower than those for long pin ®ns (l/d 8.0).Metzger et al. (1982) studied experimentally the local heat transfer variation in rectangular ducts with a staggered pin array for S x /d 1.5, and S y /d 2.5, and l/d 1.0.Results showed that heat transfer increased in the ®rst few rows, reached a peak value and then slowly decreased to a fully developed value.Later, Metzger et al. (1984) further studied the eects of using ¯attened pins and of varying the orientation of the pin ®n array with respect to the main ¯ow direction on the heat transfer and pressure drops in pin ®nned channels.They observed that, by varying the orientation of the pin-®n array, it was possible to increase the heat transfer and, at the same time, reduce the pressure drop.They also found that the use of ¯attened pins increased the heat transfer slightly but doubled the pressure loss.Lau et al. (1987) conducted a naphthalenesublimation experiment to determine the distributions of the local endwall heat transfer coecient in rectangular channels with in-line and staggered pin-®n arrays.The eects of varying the Reynolds number and the pin con®guration on the local endwall heat transfer coecient distribution were examined.Overall, and row-averaged Nusselt numbers were found to be in a good agreement with those in Metzger et al. (1982).Lau et al. (1989) further measured the overall heat transfer and friction factor in pin-®nned rectangular channels with lateral ejection.It was shown that the overall heat transfer for a pin-®nned rectangular channel with lateral ejection holes was lower than that for rectangular channel with no ejection holes.Chyu (1990) used a mass transfer sublimating system to evaluate the eects of ®llets at the cylinder-endwall junction and array geometry on the endwall heat transfer in rectangular ducts.Then, Chyu et al. (1998) examined the pin-shape eect on the heat/mass transfer characteristics in a narrow rectangular channel.The staggered cube array was recommended since it produced a signi®cant heat transfer enhancement and a moderate pressure penalty.Chyu et al. (1999) further conducted experiments to compare the contribution of heat transfer enhancement by the endwall and that by the pins in a pin-®n rectangular duct.Results revealed that the pin has about 10±20 percent higher heat transfer than the endwall.However, such a dierence in heat transfer coecient imposed very insigni®cant in¯uence on the overall array-averaged heat transfer since the wetted area of the uncovered endwalls was much greater than that of the pins.All aforementioned studies are for ¯ow through rectangular channels with a uniform pin length.However, as shown in Figure 1, the blade pro®le has a small wedge angle near the trailing-edge region that results in a narrow trapezoidal cooling cavity.In this circumstance, the pin pins spanning between two principal walls of the trapezoidal duct should be of dierent in length.A more realistic geometry of the pin-®nned trapezoidal cooling cavity in a typical turbine blade has not been studied until Hwang et al. (1999), in which the log-mean heat transfer and overall pressure drop in a trapezoidal pin-®nned duct are measured.However, the local heat transfer characteristics, which are highly desired in the turbine-blade cooling design, were not determined.
In this study, we continue our previous work (Hwang et al., 1999) to measure the detailed heat transfer coecient on the endwall of a pin-®nned trapezoidal duct by using a transient liquid-crystal technique (Hwang and Chen, 1999).Therefore, the eects of lateral-¯ow injection and the ¯ow Reynolds number on the endwall heat transfer and pressure drop in a trapezoidal duct with an in-line pin array are examined.In addition, an empirical correlation for the endwall heat transfer in a pin-®nned trapezoidal duct is developed by considering the eects of the lateral-¯ow rate and the ¯ow Reynolds number.The detailed distributions of heat transfer coecient provided by the present work could aord a better understanding of the endwall heat transfer enhancement by pin ®ns in a trapezoidal duct with lateral-¯ow injection.They could also provide a reference of computational-¯uid-dynamic-based studies relating to the pin-®n-duct heat transfer.

Apparatus
A schematic layout of the present experimental apparatus is shown in Figure 2. Air from the laboratory room is sucked into the ¯ow circuit by a 15-hp centrifugal blower.Before entering the test section, the air is ducted into an electric heater to be heated to a required temperature, subsequently traverses a noise-reduction chamber, and an entrance section.Finally, the air exits from the straight and/or lateral outlets.A ¯ow meter together with a valve situated at each outlet measures and controls the volume ¯ow rate of the air.As shown in Figure 3, the trapezoidalshaped test section and the pins are made of Plexiglas 1 .
The bottom endwall of a square area of 120 mm by 120 mm (L by W ) is sprayed with liquid crystals, representing the heat transfer active surface.Twenty-®ve pins spanning the distance between two principal duct walls are arranged by an in-line fashion.The diameter (d ) of the circular pin is 9 mm.All pins are screwed in place from the bottom endwall, and thus stand vertically on the endwall.They have dierent lengths from l/d 2.5 to 4.6, depending on the location within the trapezoidal duct.The pin spacings along the longitude and transverse directions are the same, i.e., S x S y 22.5 mm.The heights of the long (H 1 ) and short (H 2 ) side walls are 40.0mm and 18.0 mm, respectively, forming a wedge angle about 10.4 deg.As shown in Figure 2, two thermocouple rakes (each has ®ve beads) are located at the test-section entrance and exit (straight or lateral), respectively, to measure the mainstream temperatures.A real-time hybrid recorder (YOKOGAWA, AR 1100A) records the time-dependent mainstream temperatures.In addition, nine pressure taps are installed at the entrance and two exits of the test section (each has three taps) for the static-pressure measurement.They are connected to a transmitter (MODUS) to display the pressuredrop signals.
The image-processing system includes a digital color camcorder (SONY DCR-TRV7), a frame grabber card, and a Pentium II personal computer.The camcorder is focused on the liquid-crystal-coated surfaces to view and record their color change during the transient test.The frame grabber interface is programmed to analyze the color change using an image-processing software.The software  analyzes the picture frame-by-frame and simultaneously records the time lapse of the liquid crystals from colorless to green during the transient test.

Experimental Procedure
The test section is assembled after spraying the liquid crystals (Hallcrest, BW/R38C5W/C17-10) uniformly on the endwall surface.The camcorder is set up and focused on the endwall surface.Each test run is thermal transient, initiated by suddenly exposing the hot air to the test section, which results in a color change of the surface coatings.The liquid crystals are colorless at room temperature.It then changes to red, green, blue and ®nally colorless again during the heating process.The temperatures for color changed to red, green and blue are 38.2C, 39.0 C and 43.5 C, respectively (Hwang and Chen, 1999).Before the test run, the hot air bypasses the test section so that the endwall remains at the laboratory ambient temperature.The valve keeps in the diverted position until a required mainstream temperature (typically about 60±70 C) has been achieved in the diversion of the ¯ow loop.Then, the valve turns to route the hot air into the test section and, simultaneously, the recorder is switched on to record the mainstream temperature history.The image processing system records the transition time for the color change to green, and transfers the data into a matrix of time of the color change over the entire surface.The time and temperature data are entered into a computer program to obtain the local heat transfer coecient.

Heat Transfer Theory
The local heat transfer coecient over the test surface can be obtained by assuming one-dimensional transient conduction over a semi-in®nite solid.The one-dimensional transient conduction, the initial condition, and boundary conditions on the liquid crystal coated surface are k @ 2 T @Z 2 c p @T @t as t 0; T T i ; as t > 0; k @T @Z hT w ÿ T m at z 0; T T i ; as z ! 1 1 The surface temperature response to the equation above is shown as: The heat transfer coecient h can be calculated from the above equation, by knowing the wall temperature (T w ), the initial surface temperature (T i ), the oncoming mainstream temperature (T m ), and the corresponding time (t) required to change the coated-surface color to green at any location.The time required for the color change in a typical run is about 15 to 90 seconds depending on the location, mainstream temperature, and through¯ow rate.This testing time is so short that the heat ¯ow can hardly penetrate the depth of the acrylic.Therefore, the assumption of the semi-in®nite solid on the test surface is valid.Noteworthy that in the region where the heat transfer coecient varies signi®cantly in spatiality, the heat transfer coecient measured may be somewhat averaged by the axial conduction in the Plexiglas 1 plate (i.e., two-or three-dimensional eect).For checking this eect, two thermocouples are cemented into small holes drilled into the Plexiglas 1 plate approximately 1.0-mm in depth to measure the time-dependent axial conduction.The locations of these two thermocouples are behind the pin and ahead of the next pin, respectively, where the heat transfer is expected to be highly localized.The results show that the maximum axial conduction is less than 5% of the convection heat transfer from the ¯uid to the surface in the test duration.
Since the mainstream temperature is time-dependent, the solution in Eq. [2] should be modi®ed.First, the mainstream temperature history is simulated as a series of time step changes.Then, the time step changes of the mainstream temperature are included in the solution for the heat transfer coecient using Duhamel's superposition theorem.The solution for the heat transfer coecient at every location is therefore represented as where 1T m(j,j ÿ 1) and j are the temperature and time step changes obtained from the recorder output.

Data Reduction
The nondimensional heat transfer coecient on the endwall of the trapezoidal duct is represented by the Nussult number as

Nu hDe=k f 4
The Reynolds number used herein is based on the mean through¯ow velocity at the duct entrance and the equivalent hydraulic diameter of the trapezoidal duct, i.e., Re G 1 De= 5 Note that the above reductions of Nu and Re are similar to those in VanFossen (1982), but are dierent from those in Metzger et al. (1982), in which Nusselt number and the Reynolds number are based on the pin diameter.This is because Metzger et al. (1982) were devoted to pin-®n heat transfer, while the endwall-heat-transfer enhancement due to the pin ®ns in a trapezoidal duct is interested in the present study and VanFossen (1982).
The pumping power required to drive a ®xed volumetric ¯ow rate _ V in the trapezoidal duct with both straight and lateral outlets can be written as where _ V 1 and _ V 2 are the volumetric ¯ow rates, and 1P 1 and 1P 2 the measured pressure drops for ¯ow through the straight exit and the lateral exit, respectively.If the density of the ¯owing air does not vary signi®cantly in the teat section, the pressure drop across the ®nite-length duct of trapezoidal cross section with both straight and lateral outlets, can be made dimensionless as follows Eu 21P=G 2 = 7 The pressure-drop coecient obtained is based on adiabatic conditions (i.e., test with ambient-temperature mainstream).
By using the estimation method of Kline and McClintock (1953), the maximum uncertainties of the investigated nondimensional parameters are as follows: Re, 6.4%; Nu, 8.5% and Eu, 7.7%.The individual contributions to the uncertainties of the nondimensional parameters for each of the measured physical properties are summarized in Table I.

Detailed Heat Transfer Coecient Distributions
Typical examples showing the eects of lateral-¯ow rate on the detailed Nusselt number distributions on the endwall surfaces of a trapezoidal duct with pin ®ns are given in Figure 4.The trapezoidal duct is inserted with an in-line pin array at a ®xed Reynolds number of Re 40000.The lateral-¯ow rate varies from " 0 to 1.0.
For the duct of a straight outlet ¯ow only (Figure 4(a), " 0), the local heat transfer coecient distributions are slightly asymmetric due to the dierence in the sidewall eects.In addition, high and low heat transfer coecients are observed ahead of and behind each pin, respectively.
When the lateral exit is open with " 0.2 (Figure 4(b)), only a small portion of ¯uids turns laterally and most ¯uids still traverses the trapezoidal duct and exits through the straight exit.Therefore, the pattern of heat transfer distribution on a large portion of endwall shown in Figure 4(b) is not changed too much as compared to Figure 4(a).The notable dierences are a slightly reduction in Nu value and the de¯ection of wake shedding behind the pins adjacent to the lateral exit for Figure 4(b).It is seen that the wake shedding behind the pins adjacent to the lateral exit, except for the last row, is slightly de¯ected toward the lateral exit by the lateral ¯ow.Interestedly, the pin around the corner formed by tow opens of the duct (i.e., the last-row pin adjacent to the lateral exit) has a shedding wake directed from the lateral exit toward the straight exit.This means that there are some ¯uids enter the duct from the lateral opening and exit from the straight opening.Such a complex ¯ow transportation will be illustrated later in Figure 5.As the lateral ¯ow rate increases further (Figures 4(c) (d ) and (e)), the de¯ection of wake shedding behind the pins becomes signi®cant, and the heat transfer in the region near the lateral exit has been enhanced by increasing ".Meanwhile, the heat transfer coecient near the longer sidewall is evidently degraded.The former is attributed to the eect of the accelerating ¯ow through the convergent lateral exit and, partly, the ¯ow turning eect.The latter is because of the less in forced convection along the straight (X ) direction of the trapezoidal duct.In contrast to the results of small lateral ¯ow injection shown in Figure 4(b), the direction of the wake shedding by the pin around the corner formed by tow openings is from the straight exit to the lateral exit for the high lateral injection of " 0.8, indicating that the ¯ow across this pin is from the straight open to the lateral open.
When the straight exit is blocked (Figure 4( f ), " 1.0), the total main ¯ow has to turn laterally, and thus enhances the most heat transfer near the lateral exit.However, due to the ¯ow recirculation, a heat transfer deterioration is observed around the remote corner formed by the longer sidewall and the blocked straight exit.
From the heat transfer traces above, the ¯ow direction across the trapezoidal duct under various values of ", which strongly aects the local heat transfer distribution on the endwall surfaces, can be sketched in Figure 5.For 0 < " 0.4, due to a strong vacuum at the straight exit, the total straight outlet ¯ow is induced not only from the duct entrance but also from the lateral opening (Figure 5(a)).
For the duct of 0.6 " < 1.0 (Figure 5 Endwall Averaged Heat Transfer Figure 6 shows the eect of the lateral-¯ow rate on the endwall-averaged heat transfer in a trapezoidal duct at Re 40,000.The symbols are actual experiments and the solid line passing through these symbols is a curve-®tting result.As given in this ®gure, the endwall-averaged Nusselt number starts a decrease from " 0, reaches a local minimum at about " 0.3 ÿ 0.4, and then increases with increasing ".The ®rst decrease is because the apparent cross-sectional area for the through¯ow becomes large when the lateral exit is open.An increase in the apparent cross section area reduces the average through¯ow velocity; hence endwall-averaged heat transfer.Further increasing the lateral-¯ow rate augments the convective heat transfer by accelerating ¯uid through the convergent lateral exit and by enhancing the ¯ow turning eect.The more the lateral ¯ow is, the stronger the ¯uid acceleration becomes.This is the reason why the endwall-averaged heat transfer increases with the increase in the lateral-¯ow rate at high lateral-¯ow conditions.Noteworthy that the previous results by Lau et al. (1989) showed that the overall heat transfer is decreased monotonically with increasing the lateral-¯ow rate from " 0 to 1.0 in the pin-®nned rectangular duct.The disagreement in the "-dependence of the overall heat transfer between the present and previous results may be due to the dierence in the cross section of the test duct investigated.
The dependence of the endwall-averaged Nusselt number on the Reynolds number and the lateral-¯ow rate can be correlated by the equation of the form Nu 0:226Re 0:649 1 ÿ 0:21" 0:32" 2 8 FIGURE 4 Eect of the lateral-¯ow rate on the detailed heat transfer coecient distribution on the endwall of the trapezoidal duct at Re 40,000 (See Colour Plate at back of issue.).
The maximum deviations between the equations above and the experimental data are less than 6 7 percent.The combined eects of lateral-¯ow rate and Reynolds number on the endwall-averaged heat transfer of the trapezoidal duct with an array of various-shaped pins are shown in Figure 7.For simplicity, the eect of lateral-¯ow rate is expressed explicitly in the ordinate of the ®gure.As shown in Figure 7, the data for various " gather closely at a ®xed Reynolds number, meaning that the dependence of " on the Nu in Eq.
[8] is appropriate.From Figure 7, in general, the endwallaveraged Nusselt number of the trapezoidal pin-®nned duct increases with increasing the Reynolds number.In comparison of the smooth duct results of the straight ¯ow (Dittus and Boelter, 1930), the enhancement in the endwall averaged heat transfer is about 130±210% for various lateral ¯ow rates.

Pressure Drops
The results of the pressure-drop experiments are presented in Figure 8.In this ®gure, the Euler number, de®ned in Eq.
[7], is plotted as a function of the ¯ow Reynolds number for various lateral-¯ow rates.For all lateral-¯ow rates, the Euler number depends mildly on the ¯ow Reynolds number over the range of Reynolds number studied.
As for the eect of lateral-¯ow rate, similar to the heat transfer results, the Euler number is ®rst decreased and then increased with increasing ".The minimum Eu occurs at about " 0.4.Increasing the lateral-¯ow rate from " 0 to 0.4 has two counteracting eects that in¯uence the overall pressure drop across the trapezoidal duct.The ®rst is the reduction in the volumetric ¯ow rate or the air in the duct due to an increase in the apparent exit area, which reduces the pressure drop across the trapezoidal duct.The second eect is an enhancement in the ¯ow turning eect, which increases the pressure drop.Obviously, the former eect is more signi®cant than the latter eect for " 0.4.As " increases further, the ¯ow-turning eect is graduately crucial.After " 0.8, the Eu is higher that of " 0 (solid circles), meaning that the ¯ow-turning eect has overcame the area-increment eect.At " 1.0, the Euler number has a maximum value and is about 100 ± 130 percent higher than that of straight outlet ¯ow.The turning of the ¯ow through 90 deg angle and the recirculation of trapped air at the corner between the longer sidewall and the blocked straight exit for " 1.0 cause a signi®cant pressure drop in the pin-®nned channel.

CONCLUDING REMARKS
Measurements of endwall heat transfer and pressure drop in a trapezoidal duct with an in-line pin-®n array, simulating the trailing edge cooling cavity of a turbine blade, have been conducted.Lateral ¯ow rate and ¯ow Reynolds number are of the ranges 0 < " < 1.0 and 6000 < Re < 40000, respectively.New information of detailed heat transfer distributions on the endwall surfaces of a pin-®nned trapezoidal channel with various lateral ¯ow rates has been provided in this study.Increasing " enhances the local heat transfer in the region near the lateral exit but degrades the local heat transfer in the region adjacent the longer sidewall.The former is attributed to the eect of the accelerating ¯ow through the convergent lateral exit together with the ¯ow-turning eect.The latter is because of the less in forced convection along the straight (X ) direction of the trapezoidal duct.As for the endwallaveraged heat transfer, it decreases with increasing " from zero to a small lateral-¯ow rate (say, " 0.3 ÿ 0.4) due to an increase in the duct through¯ow area, which reduces the forced convection in the trapezoidal duct.For " 0.4, however, the endwall-averaged heat transfer increases with increasing " due to the signi®cance of ¯ow-turning eect.
Consequently, a local minimum in Nu occurs at about " 0.3 for the pin-®nned trapezoidal duct.Similarly, as the lateral-¯ow rate increases from " 0 to 1.0, the pressure drop across the trapezoidal duct decreases ®rst, reached a local minimum at about " 0.4 and then increases to a maximum at " 1.0.A new correlation for endwallaveraged heat transfer in the trapezoidal duct with an array of the pin ®ns is developed in terms of the lateral ¯ow rate and the ¯ow Reynolds number for the design purpose.

FIGURE 1
FIGURE 1 Typical modern internally cooled turbine blade and the pin-®nned trapezoidal duct.

FIGURE 2
FIGURE 2 Sketch of the ¯ow loop and experimental apparatus.

FIGURE 3
FIGURE 3 Dimensions and coordinate system of the test section.
(c)), contrarily, the ¯uids exiting from the lateral exit are provided by the duct entrance as well as the straight open.As for 0.4 < " < 0.6, the straight and lateral out¯ows are even almost (Figure 5(b)) due to the vacuum balance in the two exits.

FIGURE 5
FIGURE 5 Relation of the lateral-¯ow rate and the main ¯ow direction the trapezoidal duct.

FIGURE 7
FIGURE 7 Endwall-averaged Nusselt number as a function of the Reynolds number.
longer side wall of the trapezoidal duct, see Figure3 of air at the duct inlet _ V volumetric ¯ow rate of air W spanwise distance of the heated plate, see Figure3X, Y, Z coordinates as de®ned in Figure3

FIGURE 8
FIGURE 8 Reynolds-number dependence of Euler number for the pin-®nned trapezoidal duct under various lateral ¯ow rates.

TABLE I
Typical nondimensional interval for the relevant variables