Heat transfer enhancement by skewed wavy sidewall for two-pass ribbed channels with different aspect ratios

International Journal of Heat and Mass Transfer 73 (2014) 217–230

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International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt

Heat transfer enhancement by skewed wavy sidewall for two-pass ribbed channels with different aspect ratios Shyy Woei Chang a,⇑, Jing Yan Gao b a b

Thermal Fluids Laboratory, National Kaohsiung Marine University, No. 142, Haijhuan Road, Nanzih District, Kaohsiung City 81143, Taiwan, ROC Department of Marine Engineering, National Kaohsiung Marine University, No. 142, Haijhuan Road, Nanzih District, Kaohsiung City 81143, Taiwan, ROC

a r t i c l e

i n f o

Article history: Received 26 November 2013 Received in revised form 6 February 2014 Accepted 6 February 2014 Available online 4 March 2014 Keywords: Compound HTE Wavy ribbed two-pass channel Turbine blade cooling

a b s t r a c t Thermal performances of the newly devised compound heat transfer enhancement (HTE) method by deploying the in-line 45° ribs and skewed waves along the two opposite pairs of channel endwalls and sidewalls, respectively, for three two-pass sharp-bend channels with aspect ratios (AR) of 0.5, 1 and 2 are studied. For each test channel at Reynolds number (Re) between 5000 and 20,000, the full-field Nusselt number (Nu) distributions over the ribbed endwall and the channel-averaged pressure drop coefficients (f) are measured to determine the thermal performance factors (TPF) as the efficiency indices for heat transmissions. A set of Nu, f and TPF data obtained from the three test channels is selected to illustrate the HTE properties and the associated f augmentations. With present orientations for the skewed sidewall waves and the 45° endwall ribs to trip the co-current axial swirls, the local and areaaveraged endwall Nusselt numbers (NuA ) are considerably raised by elevating the HTE benefits over the mid-rib regions. The NuA levels over the ribbed endwalls for present test channels with AR = 0.5, 1 and 2 are respectively raised to 4.74–3.83, 6.25–4.94 and 7.43–6.09 times of the Dittus–Boelter references. With the accompanying f augmentations to 24.54–12.44, 17.35–10.42 and 26.71–19.82 times of the Blassius levels, the TPF values for present test channels of AR = 0.5, 1 and 2 fall in the ranges of 1.74–1.62, 2.42–2.24 and 2.51–2.28 respectively. The averaged Nusselt number correlations over inlet/ outlet legs, turning region and entire endwall, as well as the f correlations, for present test channels are generated to assist the design activities. Ó 2014 Elsevier Ltd. All rights reserved.

1. Introduction With cooling applications to gas turbine blades, the internal coolant channels are shaped to fit the blade profile for transferring the heat fluxes conducted through the suction and pressure walls of the blade. The serpentine coolant channel is now widely in use for reducing the coolant consumptions from those consumed by the multi parallel channels. Limited by the blade geometry, the radii of sharp bends connecting the serpentine coolant channels with radially outward and inward flows are much less than the channel hydraulic diameter, leading to the typical flow separation and reattachment downstream the 180° sharp bend. With the secondary vortical flows induced in the sharp turn, the heat/mass convection in the smooth multi-pass channels connected by 180° sharp bends is further complicated [1,2]. For acquiring HTE benefits, several early studies [3–7] explored the forced convection in the ribbed ⇑ Corresponding author. Tel.: +886 7 6126256; fax: +886 7 3629500. E-mail addresses: [email protected] (S.W. Chang), 1011532113@stu. nkmu.edu.tw (J.Y. Gao). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2014.02.015 0017-9310/Ó 2014 Elsevier Ltd. All rights reserved.

multi-pass channels with emphases on the detailed distributions of heat/mass transfer rates corresponding to the flow phenomena induced by the various types of surface ribs and the sharp bend. With various flow structures developed in the multi-pass straight legs enhanced by the 45°, 60° and 90° ribs [3–7], the combined effects of rib angle, rib orientation and 180° sharp bend considerably affect the Nu distributions over the ribbed endwalls [5]. Considering the typical variations of blade profiles from hub to tip section, the heat transfer performances in the tapered two-pass channels with the convergent-inlet and the divergent-outlet legs were studied [8]. Due to the flow acceleration in the convergent inlet leg and the combined effects of flow deceleration and downstream swirls after the sharp bend in the divergent outlet leg, the heat transfer performances were considerably improved in the tapered straight smooth channels. With the additional convective capacities immediate downstream the sharp bend, the Nu levels over the after-turn region were not predictable using the straight channel heat transfer correlations with local Re evaluated from local channel hydraulic diameters; while the heat transfer properties in the smooth tapered inlet leg are generally correlative with

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Nomenclature English symbols A, B correlation coefficients AR channel width-to-height (aspect) ratio = W/H a wave amplitude of sidewall (m) BR channel blockage ratio at ribbed section Cp specific heat of fluid (J kg1 K1) d hydraulic diameter of test channel (m) e rib height (m) f pressure drop coefficient of test channel = DP/ (0.5qW 2m )(d/4L) f1 pressure drop coefficient evaluating from Balssius equation = 0.079Re0.25 H channel height (m) k thermal conductivity of fluid (W m1 K1) L channel length (m) Nu endwall Nusselt number of test channel = qd/[k(Tw  Tb)] Nu1 Dittus–Boelter Nusselt number level Nu area-averaged Nusselt number P rib pitch (m) Pr Prandtl number = lCp/k q convective heat flux (W m2) Re Reynolds number = qWmd/l

local Re [8]. With the rib-induced flow phenomena, the differences in the heat transfer properties over the after-turn region between the straight and tapered ribbed channels were reduced from the smooth-wall scenarios [8]. As the internal coolant network of a gas turbine blade is often formulated by the sharp-turn connected multi-pass channels with various aspect ratios (AR), several previous works investigated the heat/mass transfer characteristics of straight and sharp-turn connected two-pass channels with different AR between 1 and 8 [9–11]. The combined effects of AR and channel-taper or divider wall-to-tip distance by adjusting the inclination/length of the divider wall on the thermal performances of the two-pass channels were comparatively examined [10,11]. The most salient AR impacts on the distributions of Sherwood number (Sh) emerge over the flow regions where the maximum and minimum mass/heat transfer rates were developed [10]. In general, the averaged endwall HTE ratios decreased against AR for any smoothwalled tapered channel; while the improved heat transfer performances were concluded for the channels with relatively low AR and a divider wall that was parallel or slightly inclined toward the converging outlet-leg sidewall [10]. Regarding the effect of distance between divider-tip and top-endwall of a 180° sharp bend, the smaller tip-to-wall distances increased the heat transfer rates on top endwall and outlet channel but at the cost of increased pressure losses [11]. In addition to the flow complexities induced by the 180° sharp bend for a multi-pass channel, the ever mounting turbine entry temperature urges the consistent pursuits for effective HTE measures. While the surface ribs and pin–fins are widely deployed along such internal coolant channels, the various HTE elements, such as dimples, wall-waves and the compound HTE measures were proposed. Sorting by the channel width-to-height (W/H) aspect ratio, Table 1 summarizes the Nusselt number ratios (Nu/ Nu1) and/or the accompanying f augmentations (f/f1) from the plain-tube references (Nu1, f1) for the various HTE measures with cooling applications to gas turbine blades [12–32]. Regardless the effects of channel geometry on the thermal impacts by the various HTE elements, the ranges of endwall Nu/Nu1 generated by the

S St SL TPF Tb Tw W Wm X, Y x, y

rib-wise coordinate (m) Stanton number = Nu/(RePr) rib length (m) thermal performance factor = Nu/Nu1/(f/f1)1/3 fluid bulk temperature (K) wall temperature of endwall (K) channel width (m) mean through flow velocity (m s1) dimensionless coordinates = x/d, y/d spanwise and streamwise coordinates (m)

Greek symbols k wave pitch of channel sidewall (m) l fluid dynamic viscosity (kg s1 m1) q fluid density (kg m3) Subscripts A refers IL refers OL refers TR refers

to to to to

entire endwall area inlet leg outlet leg turning region

continuous ribs, broken ribs, pin–fins, dimples and the undulant endwall fall in the respective ranges of 4.83–2.48, 4.04–3.1, 4.7–2.25, 2.55–1.44, 4.12–3.54; whereas the accompanying f/f1 ratios are in the ranges of 61.84–5.3, 7.7–10.24, 11.42–9.76, 4.19–5.48, 7.72–10.17, Table 1. For the passive HTE elements collected in Table 1 with the various geometries, the Nu/Nu1 ratios decrease with the increase of Re; while the f/f1 ratios are generally increased as Re increases, leading to the general Re-driven TPF reduction for these HTE elements. While most of the HTE treatments are focused on the endwalls of the internal coolant channels, few previous attempts deployed surface ribs along channel sidewall(s) [33,34]. Inheriting from the characteristic heat transfer and pressure drop properties over a rib floor, both Nu and f values were increased as the number of enhanced walls by 90° ribs was increased [33]. With the criterion of constant pumping power consumption, the TPF values decrease as Re and the number of ribbed wall increase; which fall in the respective ranges of 1.78–1.17, 1.62–1.04, 1.55–1.00, and 1.50–0.95 with one, two, three and four ribbed walls [33]. The varying trends of Nu, f and TPF against Re for the square channels enhanced by 90° ribs [33] are similarly followed by the channels fitted with 45° ribs [34]. To elevate the HTE impacts for a typical two-pass channel with 180° sharp bend, this study proposes a newly devised compound HTE measure combining the undulant sidewalls with skewed sinusoidal wall-waves and the ribbed endwalls with 45° ribs. The HTE mechanisms for present compound HTE measure are attributed to the interactive flow phenomena tripped by the wavy sidewalls and the ribbed endwalls; which include the swirling flows tripped by the 45° sidewall-waves and endwall-ribs, the turbulent augmentations by the surface ribs and the enhanced fluid mixings by the decelerating/accelerating flows through present furrowed channels [35]. This experimental study investigates the AR impacts on endwall heat-transfer and channel pressure-drop properties by comparatively examining the detailed and area-averaged endwall Nu distributions, the channel averaged pressure drop coefficients and the thermal performance factors detected from three two-pass test channels constructed by two wavy sidewalls and two ribbed

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S.W. Chang, J.Y. Gao / International Journal of Heat and Mass Transfer 73 (2014) 217–230 Table 1 Thermal performance of typical HTE elements for turbine blade cooling. AR

HTE configuration

1/4 0.5

[12] [13]

1

[14] [15] [16] [17]

Re/1000

Nu/Nu1

f/f1

TPF

10–20

3.54–2.48

10–60

2.55–1.44

4.19–5.48

1.58–0.82

Hemispherical dimple array X/D = 5.2, S/D = 3

7.5–24

1.99–4.08

61.84–28.65

2.34–0.95

45° V-rib, staggered p/e = 10, e/d = 0.083–0.167

6–60

4.3–2.38

11–48

3.37–3.86

129.6–44.96

2.27–1.07

45° Rib, staggered p/e = 5–10, e/D = 0.133–0.25

12

2.1–2.8

3.78–7.6

1.07–1.8

45° V-rib, one ribbed endwall p/e = 10, e/D = 0.12

7–25

1.89–1.71

90° Rib, in-line p/e = 10, e/D = 0.11

10–30

2.8–3.7

Saw-tooth 90° rib p/e = 7–12, e/d = 0.0625–0.15

5–30

2.22–2.68

45° Ribs, staggered p/e = 10, e/d = 0.1

40–16

4.83–3.49

45° Rib, staggered p/e = 10, e/D = 0.1

10–25

4.04–3.1

7.7–10.24

1.76–1.43

60° rib, broken-V, in-line p/e = 0.625, e/d = 0.0125–0.0625

4–16

1.2–1.34

1.83–2.62

0.98–0.97

90° Rib, half-size 4–5th ribs, p/e = 20, e/D = 0.1

50–200

3.1–1.96

11.6–30

1.37–0.63

60° Rib, staggered p/e = 15–20, e/H = 0.1–0.15

5–30

2.32–2.6

10–30

2.97–2.6

4.7–6.74

1.8–1.35

45° Scattering V-rib, staggered e/D = 0.04, P/e = 10

10–30

3.66–3.1

7.6–9.36

1.87–1.43

60° V-rib, staggered e/D = 0.04, P/e = 10

5–25

3.4–2.52

10–20

3.26–2.75

7.9–10.3

1.63–1.27

45° V-rib, in-line p/e = 10, e/D = 0.078

10–20

3.76–2.9

7.2–8.3

1.97–1.42

45° V-rib, discrete, in-line p/e = 10, e/D = 0.078

9–76

3.7–1.8

4.8–5.3

2.2–1.1

45° Rib, in-line p/e = 10, e/d = 0.078

5–30

4.12–4.03

7.72–10.5

3.5–4.2

45° Wave a/H = 0.178, a/k = 0.2

5–30

3.8–3.54

5.36–10.17

1.96–1.59

90° wave a/H = 0.178, a/k = 0.2

5–30

2.47–2.68

9–35.5

2.55–1.82

2.6–14.3

1.85–0.65

45° Rib, in-line p/e = 10–20, e/d = 0.09

10–30

2.87–2.55

7.37–10.46

1.47–1.13

45° Scattering V-rib, staggered e/D = 0.06, P/e = 10

10–30

3.5–3.02

11.63–16.68

1.54–1.19

60° V-rib, staggered e/D = 0.06, P/e = 10

10–30

2.87–2.49

9.2–13.25

1.34–1.04

45° Scattering V-rib, staggered e/D = 0.08, P/e = 10

10–30

3.3–3.07

15.53–21.57

1.32–1.097

60° V-rib, staggered e/D = 0.08, P/e = 10

45° Rib, in-line p/e = 10, e/D = 0.078–0.156

90° Rib, one ribbed endwall p/e = 10, e/d = 0.125

[18]

[19] [20]

[21]

[22] [23] 2 [24]

[20] 3

45° Ribs, staggered p/e = 10, e/d = 0.1

[25]

[25] 4

45° Rib, in-line p/e = 10, e/D = 0.125

[26] [27] [27] [28] [29]

[29] 45° Rib, staggered p/e = 10, e/d = 0.1

[20] 5 [30]

[25]

[25] 6.8 [25]

[25]

(continued on next page)

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Table 1 (continued) AR

HTE configuration

10

Re/1000

Nu/Nu1

f/f1

TPF

5–20

2.42–4.01

10.64–11.05

1.1–1.8

45° Rib, staggered p/e = 10, e/d = 0.14

5–22

3.6–1.39

2.05–43.9

1.3–0.915

60° Wedge-roughened channel p/e = 6.65, e/H = 0.2

7.5–24

4.7–2.25

11.42–9.76

3.4–1.45

Pin fin array S1/d = 2, S2/d = 1.73

[31]

[32] 64 [33]

endwalls with different AR of 0.5, 1 and 2. While present compound HTE measure enriches the HTE varieties for general engineering applications, the comparative thermal performances in terms of Nu=Nu1 , f/f1 and TPF between present test channels and the typical HTE measures for gas turbine blade cooling can enlighten the favorable HTE elements with the higher degrees of HTE benefits and/or TPF at the less expense of f augmentations. To assist the design activities using present HTE measure, the Nu and f correlations are generated with Re and AR as the controlling parameters. 2. Experimental facilities and data processing Fig. 1 depicts (a) constructional details of present twin-pass test channel with the two opposite sidewalls configured by skewed sinusoidal wall-waves and another two opposite endwalls roughed by 45° ribs. The two pairs of skewed sidewall waves and 45° endwall ribs are both in-line arranged. The details of the endwall rib floor and the undulant sidewall are illustrated by Fig. 1(b). The three test channels have the same channel width (W) of 40 mm but different channel heights (H) of 20, 40 and 80 mm, giving rise three channel width-to-height (Aspect, AR) ratios of 0.5, 1 and 2. The corresponding channel hydraulic diameters (d) for the three test channels of AR = 0.5, 1 and 2 are 47, 36 and 25 mm

respectively, which are selected as the length scales to define Re, Nu and f. To control the rib floor at the identical geometries for present three two-pass test channels, the distance between the tip of divider and the top wall of 180° bend is kept at one channel width for all three test channels; whereas the variation of channel aspect ratio is achieved by adjusting the channel height. As a result, the rib-height (e) to channel-height (H) ratios and the channel hydraulic diameters for present test channels are respectively increased and decreased as AR increases. Each of present twin-pass ribbed channels consists of the square/rectangular sectioned inlet and outlet ribbed legs connected by a 180° bend, Fig. 1(a). The nominal width of the undulant divider for all three test channels is 20 mm. Six pairs of in-line 45° ribs over the two opposite ribbed endwalls along the inlet and outlet legs are subject to the basically uniform heat fluxes. These in-line square sectioned ribs are parallel with a regular pitch (P) of 10 rib heights (e), giving P/e ratio of 10 for all present test channels. The origin of present x–y coordinate system locates at the entry corner of the inlet leg; whereas another set of channel-wise (S-wise) coordinate system locates its origin at x/W = 0.5 and y = 0 along the centerline of the ribbed endwall, Fig. 1(b). Regions IL, TR and OL respectively cover the areas of Inlet Leg, Turning Region and Outlet Leg. While the same attack angle of 45° is selected for both the skewed wall waves and ribs over the channel sidewalls and endwalls, the sidewall waves and the

Fig. 1. (a) Two-pass test channel with ribbed endwalls and wavy sidewalls (b) conceptual flow structures induced by endwall ribs, sidewall waves and centrifugal forces through bend.

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endwall ribs are both tilted toward the downstream direction along the inlet and outlet legs, Fig. 1(b). The sectional twin-vortical flow cells tripped by the two pairs of ribbed endwalls and wavy sidewalls are orthogonal as indicated by the conceptual flow structure shown by Fig. 1(b). Geometric specifications for the ribbed endwall and the wavy sidewall of each twin-pass test channel are characterized by five dimensionless parameters, namely width (W) to channel height (H) ratio of 0.5, 1 or 2, wall-wave amplitude (a) to channel width (W) ratio of 2.4 mm/40 mm = 0.06, wall-wave amplitude (a) to wave-pitch (k) ratio of 2.4/17 mm = 0.14, rib height (e) to rib pitch (P) ratio of 4 mm/40 mm = 10 and rib height (e) to channel hydraulic diameter (d) ratios of 4 mm/ 47 mm = 0.085 (AR = 0.5), 4 mm/0.036 mm = 1.11 (AR = 1) and 4 mm/0.025 mm = 1.6 (AR = 2). The two opposite rib floors (1) and (2) are forged from stainless steel foils into the specified geometry with 0.1 mm thick and 40 mm wide. The basically uniform heat flux over each ribbed endwall is generated by feeding electrical current through the two stainless steel ribbed foils (1) and (2) which are connected in series. As the width-to-thickness ratio of each stainless steel foil is about 400, the one-dimensional wall conduction with negligible conductive heat flux through the foil thickness is emulated. The ribbed endwall (1) over which the full-field distributions of wall temperature (Tw) are scanned by the infrared camera is sandwiched between Teflon end plate (3) and (4), Teflon frame (5) and Teflon divider (6). Another heating foil (2) is attached on Teflon back plate (7). The two Teflon sidewalls (8), two opposite ribbed heating foils (1) and (2), the undulant central divider (6) and the Teflon end plate of the rectangular bend (3) construct the present two-pass test channel. The length and nominal width of Teflon central divider (6) is 209.7 mm and 20 mm, respectively, with the circular top edge of radius 10 mm. The distance between divider-tip and the outer edge of the rectangular bend is one channel width of 40 mm for all three test channels. A series of axial bolts and four draw bolts (9) between the outer wall of the rectangular bend and the bottom Teflon end plate (4) are used to tighten the complete test assembly. Prior to the experimental tests, the test section is pressurized to the pressure level required for reaching the maximum Re. With these pressurized tests, both airflow and heat flux are supplied and the scale-plate is used to detect the deformation over the ribbed foil. With present Re range, the detected deformations over the ribbed foil are negligible and, in particular, any deformation over the heated rib floor will affect the infrared signal emission, which can be readily identified from the full-field wall temperature scan by present IR thermo-graphic system. To simulate the abrupt flow entry condition, a cylindrical air plenum chamber (10) with a segregation plate to separate the flow streams into the inlet leg and out of the outlet leg is installed. The sectional area ratio of present flow entrance is 3.53:1. The two ends of each stainless steel heating foil (1) and (2) are clamped between two pairs of copper plates (11) and (12) which connect with the controllable heater power supply. A type K thermocouple (13) is installed inside the entry plenum chamber (10) at the location immediate upstream the entry plane of the inlet leg to measure the inlet fluid temperature. Reynolds number at the flow entry is _ with the determined from the measured air mass flow rate (m) fluid properties evaluated at the entry fluid temperature. Variations of fluid properties due to different fluid entry temperatures at different heating levels consequently affect Re. Fine adjustments _ are frequently performed to compensate such Re deviations of m from the targeting values due to the temperature-driven fluid_ is measured by a mass property variations. The airflow rate (m) flow meter installed upstream the entry plenum chamber (10). Eight type K thermocouples (14) are installed with equal intervals over the exit plane of the outlet leg to measure fluid exit temperatures. The average of these eight thermocouple readings (14) is

221

treated as the measured exit fluid bulk temperature (Tb). Local Tb along the two-pass channel in streamwise direction is calculated using the enthalpy balance method. The Tb increase between two successive axial spots where Tw are measured is estimated as _ Cp) where qf, dA and Cp respectively stand for the local (qfdA)/(m convective heat flux, segmental heating area and the constant pressure specific heat of the test coolant. Such calculation procedure starts from the measured fluid entry temperature with the calculated and the measured exit fluid Tb compared. Raw data are collected when the differences between the calculated and measured exit fluid Tb are less than 10%. With the confirmation by the measured exit fluid Tb, the calculated local Tb is selected as the reference temperature to determine local Nu and all the temperature dependent fluid properties required for data processing. Nevertheless, with present Tb range of 296.32–358.24 K, the maximum variation in Prandtl number (Pr) is about 1.7% so that the Pr effect is exclusive from present investigation. The present IR thermo-graphic system detects the full-field Tw distributions at the scan rate of 60 Hz with the maximum percentage of uncertainties within +2%. Each Tw distribution over the ribbed heating foil (1) is scanned by the infrared radiometer which takes 60 frames of 239  255 matrix in one second. To minimize the background reflection during each Tw scan and the external heat loss from the heat transfer test module, a shield envelops the infrared camera and the heat transfer test module which is warped by thermal insulation fiber. With present optical IR setup, the infrared emission from the heating foil is normal to the plane of the focal lens. Prior to heat transfer measurements, the emissivity over the temperature range of 290–400 K is determined as 0.88 via the calibrating tests for present test configurations to ensure the maximum discrepancies between the calibrated thermocouple and the IR measurements are less than +0.3 K. The commercialized on-line monitoring program, which enables the control for scanning, collecting and transmitting IR signals for on-line monitoring or post data processing, is installed in the PC at the control console. Pressure drops across the entire twin-pass channel for f accountancies are measured at the isothermal conditions by a digital micro-manometer (15) with the precision of 0.01 mm-H2O. The micro-manometer (15) connects to two pressure taps installed at the flow entrance and exit of each twin-pass test channel to detect the pressure drops for f evaluations at all the tested Re. A series of heat transfer tests are performed at Re = 5000, 7500, 10,000, 12,500, 15,000, 17,500 and 20,000 with the heater powers adjusted to raise the highest Tw on the scanned ribbed wall at 120 °C. All the Tw scans for Nu evaluations are collected at the steady states which are assumed when several successive Tw scans at the monitoring spots over the heating foil are less than +0.3 °C. It generally takes about 30–45 min to reach the steady state after _ and/or heater power is adjusted. The parametric analysis is folm lowed to devise a set of heat transfer and f correlations with the HTE, pressure drop augmentation and TPF performances examined comparatively. The local Nu is experimentally evaluated as Nu = qfd/{kf(Tw  Tb)} in which d is the hydraulic diameter for each test channel. The local convective heat flux (qf) is calculated by subtracting the heat loss flux from the total heat flux supplied. The heating area selected to account for the heat flux includes the surface areas of all the angled ribs. The characteristic of heat loss flux is determined from the heat-loss calibration test results. With heat loss calibration tests, the channel passage is blocked and filled with the thermal insulation fiber. Driven by the temperature differences between the heated test section and the heat sink surrounding the test assembly, the heat loss flux can be expressed as the function of wall-to-ambient temperature difference. Having reached the steady state for each heat loss test run, the heater power fed to

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the test section is balanced with the heat loss via the conductive and convective pathways at the specific wall-to-ambient temperature difference. For each set of heat loss tests, ten heater powers are used to raise the wall temperatures up to 400 K for each test channel. As the heat convection in each test channel is diminished during each heat loss test, the Tw distribution over the heating foil scanned by present IR system is basically uniform. The variation of the external heat loss flux against the mean wall-to-ambient temperature difference exhibits the ascending trend which is well correlated as the function of wall-to-ambient temperature difference for each test channel. These heat-loss correlations are included in the post data processing program to enable the accountancy for local qf distributions. Nevertheless, due to the spatially varied forced convective capabilities in present test channels, the Tw distribution over the ribbed endwall for each heat transfer test is not uniform. Accordingly, the local heat loss flux varies over the heated endwall so that the qf distribution over the heated endwall is not perfectly uniform. The review of entire qf distributions generated by this study reveals that the maximum non-uniformity for qf distribution is about 14%. The streamwise Tb variation is subsequently calculated once the qf distribution over the ribbed endwall is determined. The full-filed endwall Nu distribution at each tested Re for each test channel is defined with the thermal conductivity of coolant evaluated at local Tb. Present pressure drop coefficient is defined by the Fanning friction factor (f) as ½DP=ð0:5qW 2m Þ=ðd=4LÞ where DP is the pressure drop detected from the channel endwall or sidewall with corresponding centerline length L at the mean flow velocity Wm. To enlighten the HTE performances and the associated pressure drop augmentations, the referenced Nusselt number (Nu1) and Fanning friction factor (f1) selected to normalize the present Nu and f data are respectively derived from the Dittus–Boelter correlation and the Blassius equation for the developed turbulent flow in smooth tube. The thermal performance factor (TPF) under the criterion of constant pumping power consumption was previously derived as (St/St1)/(f/f1)1/3 [36]. When both Re and Pr are controlled at the same values for the comparative groups with various HTE elements, the theoretical TPF in terms of (St/St1)/(f/f1)1/3 is reduced 1=3 to ðNu=Nu1 Þ=ðf =f1 Þ , which is adopted by present study to compare the relative efficiencies of heat transmissions among the collective HTE elements summarized in Table 1. The estimation of present experimental uncertainties follows the policy recommended by the editorial board of ASME J. Heat Transfer [37]. The sources of uncertainties for Re, f and Nu are mainly attributed to the measurements of temperature, flow rate, heater power and pressure drop. The major sources attributing to the experimental uncertainties for Nu and f are the temperature and DP measurements as the fluids properties are calculated from local Tb. Thus the maximum uncertainties of Nu are reduced by raising Tw and Tb due to the extended data range with the similar precisions of instrumentations; whereas the f uncertainties are reduced at high Re with the larger DP. The estimated maximum uncertainties for fluid density, viscosity, conductivity and thermal expansion coefficient are +0.6%, +0.7, +0.35% and +0.23%, respectively. With Tw  Tb between 15.9 and 63.06 K and DP in the range of 16.67–946.67 Pa, the experimental uncertainties for Nu, f and Re are about 9.8%, 4.5%, and 4.3% respectively. 3. Results and discussion 3.1. Heat transfer results The detailed endwall Nu distributions over three ribbed twopass channels with wavy sidewalls of AR = (a) 0.5 (b) 1 (c) 2 are typified by the Nu imprints collected at Re = 20,000, Fig. 2. For keeping the ratio between the endwall rib-height and the sidewall

wave-amplitude, as well as the ratio of endwall rib-pitch to sidewall wave-pitch at the same values, the variations of channel width-to-height (AR = W/H) ratio are achieved by varying the height of the wavy sidewall. As a result, the respective rib-height (e) to channel-height (H) ratios for present two-pass channels of AR = 0.5, 1 and 2 are e/H = 0.05, 0.1 and 0.2. As e/H increases, the overall endwall heat transfer levels and the later examined pressure drop coefficients (f) are accordingly increased with the emerging heat transfer characteristics attributed to the flow phenomena tripped by angled ribs, Fig. 2. In this regard, the high Nu stripe along the top face of each skewed rib shown by Fig. 2(c) is clearly visible over the ribbed endwall for present two-pass channel of AR = 2, e/H = 0.2. As compared by Fig. 2(c)–(b)–(a), such high Nu stripes along the rib-tops over the ribbed endwalls are systematically faded as AR and/or e/H decreases. Other heat transfer signatures induced by the angled ribs include the general rib-wise Nu decays from the obtuse edge to the acute edge of inlet/outlet leg and the high Nu regions initiated at the obtuse corner behind the angle ribs, Fig. 2. Adding the flow complexities tripped by the wavy sidewalls for present two-pass test channels, the high Nu zones behind the angled ribs over the inlet-leg/outlet-leg endwall are extended from those developed in a typical ribbed two-pass channel with flat sidewalls [21]. Moreover, acting by the complex vortical flow interactions between the Dean-type vortices, the rib induced swirls and the vortices tripped by the skewed sidewall waves in present sharp bends of three test channels, the considerable heat transfer elevations prevail over the most of turning regions as shown by the three endwall Nu imprints in Fig. 2. While the low Nu region attached along the inner edge downstream the tip of the central divider is typically induced by the separation ‘‘air bubble’’ attached on the central divider in the outlet leg of a two-pass channel with a sharp bend and flat sidewalls, there is no clear sign of such low Nu imprint to feature the re-circulating air bubble attached downstream the tip of the central divider for present two-pass channels with wavy sidewalls, Fig. 2. However, due to the limitation for unifying the Nu scales for the three endwall Nu imprints collected in Fig. 2, the rib-wise Nu variations along the last/first rib of the inlet/outlet leg over the endwall of the turning region for each of present two-pass test channels are not shown by Fig. 2 but to be later revealed by the sectioned Nu profiles in the rib-wise (S-wise) direction. To exemplify the differential endwall heat transfer properties between the two-pass channels with wavy and flat sidewalls, the centerline Nu profiles between present test channels of AR = 0.5, 1, 2 and the two-pass ribbed square channel with flat sidewalls [21] are compared by Fig. 3. Unlike the present test channels with skewed ribs installed on two endwall corners of the sharp bend, there is no rib on the bend endwall for the two-pass ribbed square channel with flat sidewalls [21]. As compared by Fig. 3, the present centerline Nu profiles sectioned through the ribbed endwalls of the three test channels consistently decrease following the order of AR(e/H) = 2(0.2) ? 1(0.1) ? 0.5(0.05) and are all elevated from the Nu references detected from the two-pass ribbed square channel with flat sidewalls [21]. With the identical AR(e/H) of 1(0.1) for present square test channel and the comparative channel with flat sidewalls [21], the centerline endwall Nu over present inlet and outlet legs are raised from the comparative Nu references [21], Fig. 3. While the overall axial Nu increases are followed by all the two-pass channels with present wavy and flat [21] sidewalls along the inlet legs, the elevations and extensions of the mid-rib high Nu bumps between two successive ribs for present test channels from the comparative counterparts [21] are evident, Fig. 3. With the coswirls tripped by the skewed ribs and the sidewall waves on the two endwall corners of present two-pass test channels, the twinpeak endwall centerline Nu profiles are consistently developed over the bend region for all present test channels; regardless the

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Fig. 2. Detailed Nu distributions over ribbed endwalls of two-pass channels with skewed wavy sidewalls of (a) AR = 0.5, e/H = 0.05 (b) AR = 1, e/H = 0.1 (c) AR = 2, e/H = 0.2 at Re = 20000, P/e = 10.

Fig. 3. Centerline endwall Nu profiles along present test channels of AR(e/H) = 0.5(0.05), 1(0.1), 2(0.2) and two-pass ribbed square channel of AR(e/H) = 1(0.1) with flat channel sidewalls [21] at Re = 12500.

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differential AR and e/H. Clearly, the orientations of present sidewall waves and angled ribs considerably boost the HTE performances in the sharp bend, Fig. 3. Over the outlet-leg endwall near the immediate downstream locations of the sharp bend for all the two-pass channels with present wavy and flat [21] sidewalls, the local centerline Nu are raised by the remaining swirls formulated through the sharp bends, after which the axial Nu decreases are noticeably followed over the first 1–2 rib pitches along the outlet legs of these two-pass channels, Fig. 3. Similar to the centerline endwall Nu profiles along the inlet leg, the mid-rib Nu bumps along the outlet-leg centerline are amplified via the vortical flow interactions between the swirls tripped by the angled ribs and by the skewed sidewall waves for present two-pass ribbed channels, Fig. 3. The peak values for these mid-rib endwall Nu bumps developed along the centerlines of the inlet/outlet legs of present test channels are even higher than the Nu peaks on the rib-tops, suggesting the considerable HTE benefits over the mid-rib regions for the particular channel configurations with present ribbed endwalls and wavy sidewalls. With the vortices induced by the two opposite wavy sidewalls, the near-wall flow structures along the obtuse and acute edges of present two-pass ribbed channels are considerably modified from those developed in the similar ribbed channels with flat sidewalls. In this respect, the additional undulant spaces constructed by present wavy sidewalls assist the relaxation of near-wall pressure gradients and reduce the near-wall shear strains around the sectional corners of the inlet and outlet legs. The degrees of f augmentations are accordingly moderated from the typical results detected from the ribbed two-pass channels with flat sidewalls, which will be later demonstrated. Also referring to the conceptual vortical flow structures depicted in Fig. 1, the two pairs of perpendicular vortical flow cells tripped by present endwall-ribs and sidewall-waves interact respectively in co- and counter-acting directions over the obtuse and acute regions along the inlet and outlet legs, leading to the further regional endwall HTE elevations along the obtuse side of each inlet and outlet ribbed endwalls. The rib-wise (S-wise) endwall Nu profiles are responsive to the synergistic effects of such bulk vortical flow interactions and the near-wall flow structures induced by the particular wavy-rib geometries around the sectional corners along the obtuse/acute edges in the inlet and outlet legs of present test channels. This is typified by Fig. 4 in which the rib-wise endwall Nu profiles sectioned through rib 4–5 (a)–(d) and rib 5–6 (e)–(h) of inlet leg and through rib 1–2 (i)–(l) and rib 5–6 (m)–(p) of outlet leg for present test channels with wavy sidewalls are compared with the S-wise endwall Nu profiles detected from the square two-pass channel with flat sidewalls [21]. Over the rib pitch 4–5 for the inlet leg as shown by Fig. 4(a)–(d), the Nu levels and the degrees of the typical obtuse-to-acute Nu decays caused by the angled ribs are systematically weakened as e/H(AR) decreases for present three test channels of AR = 2, 1 and 0.5. The flow phenomena attributed to the wavy sidewalls for present test channel of AR = 1 further elevate the HTE impact to raise present Nu levels from the comparative Nu references for the square two-pass ribbed channel with flat sidewalls [21]. Further downstream rib 5 into the region between rib 5–6, the sixth pair of present in-line ribs on two opposite rib-floors of the inlet leg extend toward the corner of the sharp bend to trip the turning bulk stream. On the ribbed endwall of the inlet legs for present test channels of AR = 0.5, 1 and 2, the obtuse-to-acute Nu decreases in S-wise direction are generally followed, Fig. 4(a)–(d). As indicated by Fig. 4(b) at mid-rib location, the Nu levels adjacent to the obtuse edge on each inlet-leg endwall are considerably elevated due to the emergence of the mid-rib high Nu bump extended from the obtuse edge along the inlet leg as seen in Fig. 3. Due to the localized flow acceleration and the separated flows tripped at the leading edge of each 45° rib, the major S-wise Nu peak develops along the rib-top as typified by Fig. 4(d). Along

the S-wise span between 0.5 < S/SL < 1 on the endwall of present sharp bend as indicated by Fig. 4(g), the obtuse-to-acute Nu decrease in the span of 0 < S/SL < 0.5 is reversed to the obtuse-toacute Nu increase after entering the turning region. The obtuseto-acute Nu increase emerging over the bend endwall along rib 6 shown by Fig. 4(h) suggests that the sweeping flow over the turning endwall by the swirls through present sharp bend is directed from the outer top wall toward the inner curved bend. As the near-wall flows driven by such Dean-type vortical flows through the bend act in the same direction with those tripped by the angled ribs deployed over the outlet-leg endwall, the obtuse-to-acute Nu decreases consistently emerge all over the outlet-leg endwall, Fig. 4(i)–(l) and (m)–(p). As the S-wise Nu variations over present sharp bend endwall follow the consistent decreasing trend from the outer wall toward the inner edge of the central divider, the swirls induced through the sharp bend for each of present test channels are dominantly generated by the centrifugal forces. Over rib 5–6 on the endwall of the outlet leg, the typical S-wise Nu profiles subject to the consistent obtuse-to-acute decreases are recovered, Fig. 4(m)–(p). In search for endwall heat transfer correlations for present twopass test channels with wavy sidewalls, the regionally averaged Nusselt numbers over the inlet leg (NuIL ), turning region (NuTR ), outlet leg (NuOL ) and entire endwall (NuA ) are evaluated. The variations of (a) NuIL , (b) NuTR , (c) NuOL and (d) NuA against Re for present test channels of AR(e/H) = 0.5(0.05), 1(0.1), 2(0.2) are shown by Fig. 5 in which the heat transfer results collected from the comparable two-pass ribbed square channel with flat sidewalls of AR(e/ H) = 1(0.1) [21] are also compared. As depicted by Fig. 5(a)–(d), all the regionally averaged endwall Nusselt numbers increase with the increase of Re. While all these regionally averaged Nusselt numbers obtained at each Re increase as AR increases mainly due to the accompanying e/H increase, the NuIL , NuTR , NuOL and NuA detected from present test channel of AR(e/H) = 1(0.1) are consistently raised from the comparative counterparts reported in [21]. Such AR(e/H) driven NuIL differences between present three test channels shown by Fig. 5(a) are amplified over the turning region and the outlet leg. With the amplified AR(e/H) effects in sharp bend and outlet leg of present test channel, the higher degrees of HTE benefits attributed from the flow interactions between the Den-type vortices in the bend and the vortical flows tripped by the endwall-ribs and the sidewall-waves in the respective turning region and outlet leg are expected. In particular, the combination of present sidewall waves and sharp bend results in significant NuTR elevations from the comparative counterparts with flat sidewalls [21], Fig. 5(b). This result enlightens the potential for further HTE elevations by deploying the ribbed endwalls and wavy sidewalls in planar spiral channels in which the centrifugal forces prevail. Justified by the consistent Re-driven data trends collected in Fig. 5, all the regionally averaged Nusselt numbers, namely NuIL , NuTR , NuOL and NuA , can be well correlated by A  ReB in which the coefficient A and exponent B vary with AR and e/H. Table 2 summarizes the A coefficient and B exponent in NuIL , NuTR , NuOL and NuA correlations for each of present test channels. As indicated by Table 2, the A coefficient and B exponent vary with AR and e/H in each heat transfer correlation. Clearly, the vortical flow structures induced by the endwall ribs, the sidewall waves and the sharp bend are AR dependent, leading to the AR impacts on endwall Nu. However, with a fixed e/H, the degree of ‘‘channel blockage’’ by the endwall ribs varies with AR, leading to the inter-correlative dependency of Nu on AR and e/H. Alternatively, the channel blockage ratio (BR) is introduced as the area-ratio between the flow pathway through the ribbed section and the entire channel section to reflect the relative rib height for any AR. By definition, BR features the degree of local flow acceleration through the ribbed section. The range of BR from the fully blocked ribbed channel with no flow to the smooth-walled channel with no rib is

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Fig. 4. S-wise endwall Nu profiles along present test channels of AR(e/H) = 0.5(0.05), 1(0.1), 2(0.2) and two-pass ribbed square channel of AR(e/H) = 1(0.1) with flat channel sidewalls [21] at Re = 15000.

Fig. 5. Variations of regionally (a)–(c) and endwall (d) averaged Nusselt numbers against Re and variations of correlative (e) A coefficients (f) B exponents with AR and BR for present two-pass channels with wavy sidewalls and square two-pass channel with flat sidewalls [21].

Table 2 A coefficient and B exponent in NuIL , NuTR , NuOL , NuA correlations. Present ribbed test channels with wavy sidewalls AR, e/H, BR ðNu ¼ A  ReB Þ

0.5, 0.05, 0.9

1, 0.1, 0.8

2, 0.2, 0.6

A

B

A

B

A

NuIL

0.201

0.67

0.24

0.71

0.33

0.68

NuTR

1.334

0.54

1.91

0.53

0.89

0.64

NuOL

0.39

0.65

0.44

0.66

0.71

0.63

NuA

0.31

0.68

0.61

0.62

0.64

0.65

B

0–1 for all types of ribbed channels with different AR and rib configurations. In Fig. 5(e) and (f), the A, B variations against AR and BR are respectively depicted. As indicated in Fig. 5(e) and (f), the zero A value at BR = 0 with no flow and the typical A coefficient and B exponent in Dittus–Boelter correlation are selected to typify the A, B values at the limiting conditions of BR = 0 and BR = 1. Within present BR range tested, the A coefficients collected in Fig. 5(e) consistently increase as BR decreases from unity (plain channel condition) by increasing the relative rib height. However, another physical constraint at the limiting condition of BR = 0, which features the full blockage condition, calls for the vanished

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A value to respond the no flow scenario. As a result, the BR-driven data trend of A coefficient in NuIL , NuOL or NuA correlation suggested by Fig. 5(e) for each ribbed passage deems to have the optimal BR between 0 and 1 to provide the highest A value. In view of the BR-driven B variations shown by Fig. 5(f), the B exponents consistently decrease from 0.8 at the plain straight channel condition (BR = 1) when BR decreases from unity toward 0. It is interesting to note that, even with NuTR , which properties are centrifuge dominant in the sharp bend, the typical trend of BR-driven B variation for the ribbed passages is similarly followed, Fig. 5(f). Adding the additional physical constraints for A coefficients at the limiting conditions of BR = 0 and BR = 1 and B = 0.8 at BR = 1, four sets of A and B functions collected in Table 3 for NuIL , NuTR , NuOL and NuA correlations are generated using BR as the controlling variable. Substitutions of A coefficients and B exponents into the general equation of Nu ¼ A  ReB give rise to the correlated NuIL , NuTR , NuOL , NuA values which show that the maximum discrepancies from the entire set of experimental data are less than +20%. To assess the relative HTE performances for present test channels, the regionally averaged Nusselt numbers are normalized by Dittus–Boelter (plain-tube) heat transfer references (Nu1) and compared with the heat transfer data reported for a variety of ribbed channels. As shown by Fig. 6 in which the variations of (a) NuIL =Nu1 (b) NuTR =Nu1 (c) NuOL =Nu1 (d) NuA =Nu1 against Re are depicted, all these normalized Nusselt numbers for present test channels decrease with the increase of Re. With 5000 < Re < 20,000, the ranges of present NuIL =Nu1 , NuTR =Nu1 , NuOL =Nu1 and NuA =Nu1 are collected in Table 4. Due to the flow mechanisms tripped by present wavy sidewalls, which appear compatible with the rib-induced HTE flow mechanics, the present NuIL =Nu1 range is considerably raised about 21–71% from the single-pass comparative group shown by Fig. 6(a). While present NuTR =Nu1 are significantly elevated from present inlet-leg and outlet-leg counterparts, the NuOL =Nu1 ratios are consistently higher than the NuIL =Nu1 ratios at all Re tested, Fig. 6(a)–(c). Clearly, the remaining swirls immediate downstream present sharp bend in the outlet leg generate the higher degrees of HTE benefits than those emerged at the entry region of inlet leg at present abrupt entry condition. Above all, the NuA =Nu1 ratios over the entire endwall for each of present two-pass test channels are considerably raised about 42–72% from the comparative two-pass channels with various rib configurations collected in Fig. 6(d). 3.2. f and TPF factors The f factors reported by preset study are the dimensionless pressured drops evaluated as Fanning pressure drop coefficients (f). The main focus to detect these pressure drops through present two-pass ribbed channels with wavy sidewalls is to comparatively examine the additional pumping power required for gaining the additional HTE benefits generated by the wavy sidewalls. Referring to the pressure-tap locations selected for measuring the pressure drops as indicated in Fig. 1, the f factors detected by present study indicate the channel-averaged pressure drop properties in respect to the specific locations where the pressure taps are installed. The

Table 3 A {BR}, B {BR} functions in NuIL , NuTR , NuOL , NuA correlations. Nu ¼ A  ReB A{BR} function

B {BR} function

NuIL

1.36  BR-1.34  BR2

1.04–1.11  BR + 0.88  BR2

NuTR

4.88  BR-4.84  BR2

1.55–3.2  BR + 2.46  BR2

NuOL

2.84  BR-2.82  BR2

1.44–2.4  BR + 1.76  BR2

2

2

NuA

2.58  BR-2.56  BR

1.4–2.22  BR + 1.63  BR

corresponding changes for these f factors to AR(e/H) with wavy sidewalls are comparatively examined to study their general impacts on f augmentations. As an attempt to develop the empirical f correlations, the depiction of (a) f (b) f/f1 variations by adjusting AR(e/H) and Re for present test channels is shown by Fig. 7. The f1 reference selected to normalized present f data is evaluated from Blassius equation as 0.079Re0.25. Also included in Fig. 7 are the f and f/f1 data detected from the two-pass ribbed square channels with V-ribs [17] and 45° ribs [38,39]. For a typical two-pass ribbed channel, the turning motions of bulk coolant stream through a 180° sharp bend incurs significant pressure drops due to flow momentum changes. While the HTE benefits developed in present test channels are consistently increased as AR increases, mainly due to the increased e/H, the f factors detected from present test channels of AR = 0.5 and 1 as compared in Fig. 7(a) are in close agreements, suggesting the pressure drops aroused through the sharp bends of these test channels (AR = 0.5 and 1) are dominant for their channel-averaged f. By way of further e/H increase, the f factors for present test channel of AR = 2 are noticeably raised from the f data bands obtained with AR = 0.5 and AR = 1, Fig. 7(a). The rib-induced form/frictional drags at AR(e/H) = 2(0.2) for present ribbed-wavy two-pass channel become compatible with the pressure drops incurred through the sharp bend. For present ribbed-wavy two-pass channels of AR(e/ H) = 0.5(0.05), 1(0.1) and 2(0.2), the f/f1 ratios are in the respective ranges of 24.54–12.44, 17.35–10.42 and 26.71–19.82 with 5000 < Re < 20000. Accompanying with the additional HTE benefits attributed to present wavy sidewalls as demonstrated by Fig. 6(d), the f/f1 performances for present test channels relative to those obtained from various ribbed two-pass channels with flat sidewalls are compared by Fig. 7(b). As compared by Fig. 7(b), the data trends of f/f1 for present test channels of AR(e/H) = 0.5(0.05) and 1(0.1) are similar to the test results reported in [17,38] for the square two-pass channels enhanced respectively by the one-leg V-ribs with forward flows [17] and the two-leg 45° ribs [38]; but are less than those reported in [17,39] for the two-pass channels fitted with the one-leg V-ribs at backward flows [17] and the two-leg 45° ribs [39]. Even with the additional vortical flow interactions between the vortices tripped by 45° endwall ribs and the skewed sidewall waves in present test channels, which are expected to add the drags among the coolant bulk stream, the additional undulant spaces constructed by present skewed sidewall-waves at the corner junctions that connect the endwall-ribs and the sidewallwaves assist the relaxation of near-wall pressure gradients and reduce the near-wall shear strains around the sectional corners along the entire ribbed channel. As a result, present f/f1 ratios fall close to those detected from the two-pass ribbed channels [17] and are less than the two-pass ribbed channel [39] with flat sidewalls. In the attempt to devise f correlations for present two-pass ribbed channels with wavy sidewalls, it is noticed that the momentum change of coolant bulk stream through the 180° sharp bend inevitably incurs the corresponding pressure drop even if the boundary layer thickness tends to be diminished as a result of Re elevation. Thus the typical mathematic form of f correlation for boundary-layer type ducted flows, such as the Blassius equation in terms of 0.079Re0.25, is replaced by the exponential decay function for present two-pass channels, which approaches a finite asymptotic level to account for the additional pressure drop through the sharp bend. Justified by the consistent Re-driven f data trends collected in Fig. 7(a) for present two-pass ribbed-wavy channels, the f correlations for present test channels of AR(e/ H) = 0.5(0.05), 1(0.1), 2(0.2) are respectively devised as Eq. (1)–(3).

f ¼ 0:079 þ 0:44  eð0:00022ReÞ

ARðe=HÞ ¼ 0:5ð0:05Þ

ð1Þ

f ¼ 0:042 þ 0:19  eð0:0001ReÞ

ARðe=HÞ ¼ 1ð0:1Þ

ð2Þ

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227

Fig. 6. Variations of (a) NuIL =Nu1 (b) NuTR =Nu1 (c) NuOL =Nu1 (d) NuA =Nu1 against Re.

Table 4 NuIL =Nu1 , NuTR =Nu1 , NuOL =Nu1 and NuA =Nu1 ranges for present test channels. 5000 < Re < 20000 AR, e/H, BR

0.5, 0.05, 0.9

NuIL =Nu1

3.79–3.31

4.94–4.32

5.26–4.49

NuTR =Nu1

6.4–4.23

8.08–5.78

9.91–8.05

NuOL =Nu1

4.02–3.94

5.74–4.72

7.13–5.72

NuA =Nu1

4.74–3.83

6.25–4.94

7.43–6.09

f ¼ 0:102 þ 0:26  eð0:00012ReÞ

1, 0.1, 0.8

ARðe=HÞ ¼ 2ð0:2Þ

2, 0.2, 0.6

ð3Þ

The comparison of entire experimental f data against the correlation results calculated by Eqs. (1)–(3) indicates that the maximum discrepancies are less than +7%. The advantage of using present sidewall waves to enhance the endwall HTE properties for typical two-pass ribbed channels as demonstrated by Fig. 6(d) is accompanied with the noticeably relaxations of corner drags in each present test channel, leading to the reduced or equivalent f/f1 ratios compared to those reported for the two-pass ribbed channels with flat sidewalls [17,38,39], Fig. 7(b). Having revealed the Nu and f properties for present two-pass ribbed-wavy channels, it is interesting to examine the correlative relationships between the heat-transfer and pressure-drop performances for this type of sharp-bend ribbed-wavy channels with large changes in the gradients of field variables, including both velocity and temperature, by plotting Stanton numbers (St) against f coefficients obtained at the same Re. As depicted by Fig. 8(a), the St data detected from all the tested Re for each of present two-pass

ribbed-wavy channels converge into a tight f-driven linear data trend. As well as a reconfirmation, the linear increases of St against f for present two-pass ribbed-wavy channels shown by Fig. 8(a) reconfirm the applicability of Reynolds’ analogy [40] for this type of two-pass ribbed-wavy channels. Based on the endwall HTE properties and the accompanying f augmentations from the f1 references for present test channels, the overall efficiency of heat transmission using present type of ribbed-wavy two-pass channels is assessed by comparing the TPF variations against Re in Fig. 8(b). The TPF value at each tested Re for each of present test channels is 1=3 evaluated as (NuA =Nu1 Þ=ðf =f1 Þ at constant pumping power consumption. Although the two-pass ribbed channel is typical as a part of the cooling network in a gas turbine blade, the relevant TPF values reported in the open literature are very rare. Only the limited TPF data collected from the two-pass ribbed square [17] and parallelogram (AR = 4) [41] channels with flat sidewalls are included in Fig. 8(b) to compare with present TPF results. As shown by Fig. 8(b), all the TPF values evaluated from present test channels are above than unity, confirming the efficient heat transmission using this type of two-pass ribbed-wavy channels. As an overall assessment, the Re-driven TPF variation for each of present test channels follows the similar decreasing trend in Fig. 8(b). Due to the elevated HTE performances by increasing AR(e/H) as compared in Fig. 6, the TPF values evaluated for present test channels generally increase with the increase of AR(e/H) at each tested Re. However, with the higher degree of counteracting f increase for the test channel of AR(e/H) = 2(0.2), the differential TPF values between present test channels of AR(e/H) = 2(0.2) and 1(0.1) shown by Fig. 8(b) are not as considerable as their differential HTE

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Fig. 7. Variations of (a) f (b) f/f1 against Re for present test channels and various two-pass ribbed channels [17,38,39].

respective ranges of 1.74–1.62, 2.42–2.24 and 2.51–2.28, suggesting that the undulant channel sidewalls with skewed sinusoidal waves appear as an effective and efficient HTE measure to improve the thermal performances of a two-pass ribbed channel with 180° sharp bend for turbine blade cooling applications. 4. Conclusions This experimental study explores the endwall heat transfer properties and the channel averaged pressure drop performances for the newly devised compound HTE measure aimed at improving the thermal performances of the internal cooling networks of gas turbine blades. Full-field endwall Nu distributions and channelaveraged f coefficients are measured at Reynolds number (Re) between 5000 and 20,000 for three two-pass sharp-bend channels of AR(e/H) = 0.5(0.05), 1(0.1) and 2(0.2) with 45° in-line endwall ribs and skewed sidewall waves. Several salient concluding remarks are summarized as follows.

Fig. 8. Variations of (a) St against f (b) TPF against Re for present test channels and two-pass ribbed square [17] and parallelogram channels [41] with flat channel sidewalls.

properties exhibited in Fig. 6(d). Nevertheless, the TPF values for present two-pass ribbed channels with wavy sidewalls of AR(e/ H) = 1(0.1) and 2(0.2) are considerably raised from the typical TPF values obtained from the two-pass ribbed channels with flat sidewalls [17,41]. Even for present test channel with low rib-height to channel width (e/H) ratio of 0.05, the TPF values are similar to those detected from the two-pass parallelogram channel enhanced by 45° ribs [41], Fig. 8(b). Clearly, the further improved endwall HTE properties by present wavy sidewalls, which also assist the relaxation of corner shearing actions at rib-sidewall junctions, act together to considerably elevate present TPF values from the comparing counterparts collected in Fig. 8(b) for the two-pass ribbed channels with flat sidewalls. The TPF values for present test channels of AR(e/H) = 0.5(0.05), 1(0.1) and 2(0.2) are raised to the

1. While the typical heat transfer signatures relevant to the angled ribs emerge clearly over present ribbed endwalls, the present high endwall Nu zones behind the angled ribs are extended from those developed in ribbed two-pass channels with flat sidewalls. With the co-swirls tripped by present 45° endwall ribs and skewed sidewall waves, the twin-peak endwall centerline Nu profiles are induced over the bend region. As the Den-type vortices through sharp bend interact with the vortical flows tripped by present endwall-ribs and sidewall-waves to further boost the HTE impacts, the significant NuTR elevations from the comparative counterparts obtained from the two-pass ribbed channels with flat sidewalls are observed. 2. Four sets of regionally averaged Nusselt number correlations for NuIL , NuTR , NuOL and NuA are generalized as the BR functions to account for the channel blockage effects by the angled ribs for each of present test channels. Justify by the BR-driven A variations in NuIL , NuOL and NuA correlations, an optimal BR between 0 and 1, which offers the highest A coefficient, along with the consistent decrease of B exponent from 0.8 at BR = 1 toward the full-blockage condition at BR = 0 by reducing BR are observed to characterize the heat transfer performances for present type of two-pass ribbed-wavy channels.

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3. In view of NuIL =Nu1 , NuTR =Nu1 , NuOL =Nu1 and NuA =Nu1 ranges collected in Table 4 with 5000 < Re < 20,000, the present compound HTE effectiveness for the two-pass ribbed channels are considerably raised from those reported for a variety of ribbed channels as demonstrated by Fig. 6. The ratios of NuA =Nu1 in the Re range of 5000–20,000 for present test channels of AR = 0.5, 1 and 2 are respectively raised to 4.74–3.83, 6.25–4.94 and 7.43–6.09. 4. With present wavy sidewalls to generate the additional HTE benefits, present f/f1 ratios are similar to or reduced from those detected from the various two-pass ribbed channels with flat sidewalls. The ratios of f/f1 for present ribbed-wavy two-pass channels of AR(e/H) = 0.5(0.05), 1(0.1) and 2(0.2) fall in the respective ranges of 24.54– 12.44, 17.35–10.42 and 26.71–19.82 with 5000 < Re < 20,000. Even with the complex flow interactions between the vortices tripped by present 45° endwall ribs and skewed sidewall waves, the additional undulant spaces constructed by present sinusoidal sidewall waves at the corner junctions, where the endwall ribs and wavy sidewalls are connected, assist the relaxation of near-wall pressure gradients and reduce the near-wall shear strains to moderate the f augmentations. Three sets of f correlations evaluating the channel averaged dimensionless pressure drops for present test channels are accordingly devised. 5. For present type of sharp-bend ribbed-wavy channels with large changes in the gradients of fluid velocity and temperature, the linear increase of St against f ensures the applicability of Reynolds’ analogy. Acting by the flow phenomena relevant to present wavy sidewalls, the TPF values for present test channels of AR(e/H) = 0.5(0.05), 1(0.1) and 2(0.2) are generally raised above the TPF references collected from the two-pass ribbed channels with flat sidewalls to the respective ranges of 1.74–1.62, 2.42–2.24 and 2.51–2.28. The present undulant channel sidewalls with skewed sinusoidal waves are demonstrated as an effective and efficient HTE measure to boost the thermal performances for a twopass ribbed channel with 180° sharp bends.

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