Numerical investigation of heat transfer and fluid flow ...

Applied Thermal Engineering 129 (2018) 1369–1381

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Applied Thermal Engineering journal homepage: www.elsevier.com/locate/apthermeng

Research Paper

Numerical investigation of heat transfer and fluid flow in a rotating rectangular channel with variously-shaped discrete ribs Jinsheng Wang a,b, Jian Liu b, Lei Wang b, Bengt Sundén b,⇑, Songtao Wang a,⇑ a b

School of Energy Science and Engineering, Harbin Institute of Technology, Harbin, China Division of Heat Transfer, Department of Energy Sciences, Lund University, Lund, Sweden

h i g h l i g h t s Effects of rib configuration is revealed by CFD. Coriolis force and channel orientation effects are taken into account. Effects of Rib streamwise/widthwise distance and inner-half-rib angle are examined. Best overall performance by discrete ribs with normalized streamwise rib gap 0.2.

a r t i c l e

i n f o

Article history: Received 7 June 2017 Revised 27 August 2017 Accepted 18 September 2017 Available online 24 October 2017 Keywords: Numerical simulation Discrete rib Channel orientation Coriolis force Heat transfer

a b s t r a c t A numerical study is performed to investigate the effects of various discrete rib configurations on the heat transfer and fluid flow characteristics of a rotating rectangular straight channel (aspect ratio of 2:1) with 45° orientation. Three parameters of the discrete rib configuration - rib streamwise distance, rib widthwise distance, and inner-half-rib angle - are examined based on a continuous inclined rib. The blockage ratio (e/Dh) is 0.1 and the pitch ratio (P/e) is 10. The Reynolds number is fixed at 10,000 in each case, while the rotation number is varied from 0 to 0.7. Details of the turbulent flow structure, turbulence kinetic energies, temperature fields, normalized Nusselt numbers, friction penalties and thermal performance factors were obtained by Computation of Fluid Dynamics (CFD) with the k-x SST turbulence model. The results show that a small streamwise rib gap can effectively enhance the leading wall heat transfer. The heat transfer decreases gradually as the streamwise rib gap is enlarged. The inner-half-rib angle b provides the most conspicuous effects on heat transfer and friction loss, and the best heat transfer appears at b = 60° or 75° for different rotating conditions. The case with a rib streamwise gap normalized distance of 0.2 and inner-half-rib angle of 45° provides best thermal performance. A widthwise rib gap is favorable in reducing pressure drop, but its heat transfer augmentation is limited. Ó 2017 Elsevier Ltd. All rights reserved.

1. Introduction To improve thermal efficiency, advanced gas turbines are designed to operate at high temperature. Correspondingly, the rotor inlet temperature may exceed the allowable metal temperature of blades and lead to thermal stresses on the blades. In order to guarantee that the blades operate reliably, several improved cooling techniques have been applied to enhance the heat transfer characteristics in turbine blades. Rib turbulators are widely used in the mid portion of the blade to augment the heat transfer in internal cooling channels. As the turbine blades are rotating, the extra ⇑ Corresponding authors. E-mail addresses: [email protected] (B. Sundén), [email protected] (S. Wang). https://doi.org/10.1016/j.applthermaleng.2017.09.142 1359-4311/Ó 2017 Elsevier Ltd. All rights reserved.

forces shift the flow field in the channel compared with stationary conditions, i.e., the Coriolis force pushes the coolant to the trailing wall for a radial outward flow with rotation, and the buoyancy force modifies the flow field by the density variation between core fluid and near-wall fluid. The extra forces along with the rib turbulators make the flow mechanism in a rotating channel much more complicated. Therefore, the rotation has a considerable impact on the heat transfer and pressure drop. The application of rib turbulators enhances the heat transfer by redeveloping the boundary layer after the ribs and induces rib secondary flow. There have been a number of experimental and numerical studies on the impact of rib shapes, rib configurations and channel geometries on the flow pattern and heat transfer distributions in single or two-pass channels. Comprehensive

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J. Wang et al. / Applied Thermal Engineering 129 (2018) 1369–1381

Nomenclature

English symbols specific heat (J kg1 K1) cp channel hydraulic diameter (m) Dh e rib height (m) f friction factor f0 friction factor from Blasius equation g rib widthwise gap (m) H channel height (m) h heat transfer coefficient (W m2 K1) L channel length (m) m rib streamwise gap (m) Nu average Nusselt number Nu Nusselt number Nu0 from Dittus-Boelter Nusselt number correlation P pressure (N m2) P rib pitch (m) Pr Prandtl number q heat flux (W m2) R rotating radius (m) Re Reynolds number Ro rotation number T temperature (K) Tb air bulk temperature (K) Tw wall temperature (K) U mean velocity (m s1) ui orthogonal velocities (m s1)

reviews of turbine internal cooling including rib-turbulated cooling can be found in [1–3]. Earlier studies on internal rib-turbulated cooling were based on single-pass or U-bend stationary models to study the configuration parameters such as channel aspect ratio, pitch ratio, blockage ratio, and rib angle of attack. Rib-turbulated heat transfer in rectangular channels with different aspect ratio, rib height, rib pitch, and rib angle of attack were studied by Han et al. [4]. It was observed that these configuration parameters significantly affect the heat transfer distribution and pressure drop in the channel. Different pitch ratios in a square channel on rib-turbulated cooling were tested through detailed aerodynamic and heat transfer measurements by Rau et al. [5]. The local heat transfer enhancement distribution on the symmetry line showed a slightly better performance for a pitch ratio of 12. The average overall floor data gave in contrast a maximum for a pitch ratio of 9. The heat transfer characteristics of a two-pass channel featuring ribs-alone and combination of ribs and dimples were investigated by Singh et al. [6]. It showed that the heat transfer enhancement by the V-compound two-pass channel is higher than V rib case followed by the dimpled configuration. Studied flow characteristics and thermal efficiency in corrugated ducts. The researches by Tokgoz et al. [7] found that heat transfer rate increases with the rise of the aspect ratio and the aspect ratio of 0.3 corresponds to the maximum value. The corrugated channel promotes the early generation of vortices which force the fluid to move between the core and wake flow regions resulting in a higher Nusselt number. Meanwhile, modified rib configurations, such as discrete rib, also were studied. It was shown that the application of discrete ribs slightly weakens heat transfer but observably reduces pressure drop penalty. Wang and Sundén [8] experimentally studied the local heat transfer in a square duct with truncated ribs. The experiments showed that the horseshoe vortices lead to higher Nusselt number downstream of the rib, with two peaks of enhancement

v W x y z

widthwise velocity (m s1) channel width (m) streamwise coordinate widthwise coordinate span-wise coordinate

Greek symbols a outer-half-rib angle (°) b inner-half-rib angle (°) g thermal performance factor h orientation angle (°) k thermal conductivity of air (W m1 K1) l air dynamic viscosity (kg m1 s1) lt turbulent viscosity (kg m1 s1) qb air bulk density (kg m3) qw near wall air density (kg m3) X angular velocity of rotational channel (rad s1) Subscripts in inlet of the heated section out outlet of the heated section Abbreviations CFD computational fluid dynamics TKE turbulence kinetic energy

occurring symmetrically around the centerline. However, hot spots still exist at the corner just behind the upstream rib. The experiments in [9] about heat transfer characteristics and friction loss of discrete ribs reported that the heat transfer enhancement with discrete ribs is remarkable. Kumar et al. [10] applied multi discrete V-blockages in a channel and Nusselt number and friction factor correlations were established. The experiment indicated that the secondary fluid streams are generated due to discretization of V block blockage pattern, which enhances the heat transfer. Several researchers studied the heat transfer and friction penalty by numerical methods. The simulations of parallelogram channels roughened by 45° ribs without/with in-line auxiliary fins [11] indicated angled ribs and auxiliary fins is applicable for improving thermal performances of turbulent channel flow. The flow structure and heat transfer in a square channel with offset midtruncated ribs were studied [12,13]. It showed that the cases with truncated ribs significantly reduced the pressure drop and friction drag compared to cases of continuous ribs, and staggered arrangement of mid-truncated ribs enhanced heat transfer at low Reynolds numbers. The heat transfer in a channel with inline V-shaped discrete ribs was numerically studied by Promvonge et al. [14]. The results showed that an increase of the blockage ratio values leads to an increase in heat transfer and friction loss. The research works above focused on heat transfer and flow structure based on stationary model and neglected the Coriolis and buoyancy forces caused by rotation, which alter the flow field and temperature distributions. The experimental study about square and triangular ribbed channels with different channel orientations by Dutta et al. [15] showed that the Nusselt number ratios of stabilized ribbed sides show a mixed dependence on the rotation number and the effects of rotation are mostly reduced for nonorthogonal alignment of the heat transfer surfaces with respect to the rotation plane. Griffith et al. [16] studied the heat transfer in a rectangular rotating channel with angled ribs and


found span-wise heat transfer differences up to 50–75% for the rotating ribbed channel across the leading and trailing walls. The heat transfer in a rotating ribbed two-pass channel with enginesimilar cross section was studied by Tao et al. [17]. The research showed that channel orientation influences the heat transfer more significantly at a larger Rotation number. Xu et al. [18] investigated the effect of rib spacing on heat transfer and friction in a rotating two-pass square channel. The results showed that the rib spacing effect is more pronounced in the first radially outward flow passage than the second passage and rotation reduces rib spacing effect and pitch ratio of 10 has the best surface average heat transfer enhancement. The experiment of a two-pass rectangular channel with different discrete ribs in rotating condition [19] showed that the discrete V-shaped ribs have a better overall thermal performance than other tested models in both rotating and nonrotating channels. Liu et al. [20] studied heat transfer augmentation in a rotating channel with discrete and V-shaped ribs. It was found that V-shaped and discrete V ribs both have the highest heat transfer enhancement in rotating condition. Numerical research works about heat transfer and pressure loss in rotating channels were also carried out. The combination of ribs, dimples and protrusions on heat transfer in a U-bend rotating channel was numerically studied by Shen et al. [21]. It was found that rib-protruded channel seems to possess better performance with obviously higher heat transfer rate and comparable friction factor especially for rotational condition. Wang et al. [22] experimentally and numerically studied the heat transfer performance in a two-pass rotating channel. The research showed that the heat transfer in ribbed trailing wall cases was not enhanced compared to stationary cases due to the fact that the spanwise velocity has been weakened by the interaction of rotation induced secondary flow and rib induced secondary flow. Dowd et al. [23] studied the flow and heat transfer in a rotating two-pass duct geometry with staggered ribs using large eddy simulations and found that Coriolis forces have no impact on the augmentation of heat transfer on the leading wall until the second half of the passage. The previous experimental and numerical investigations showed that the application of discrete ribs in the channel of a blade has greater advantage than the continuous ribs on heat transfer augmentation and especially friction penalty reduction. However, investigations concerning the effect of discrete ribs on heat transfer and fluid flow under rotating condition were very few, especially for cases with skew channel orientations. In particular, the recovery of the heat transfer reduction on the leading side due to rotation should be considered. In the present study, several

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discrete rib configurations and a continuous rib configuration are examined and the thermal/hydraulic performances are evaluated by utilizing a three-dimensional numerical method. 2. Description of the problem 2.1. Geometry & configuration The geometric configuration of the single-pass rectangular channel is explicitly presented in Fig. 1. The channel has a width (W) of 45 mm and a height (H) of 22.5 mm, resulting in an aspect ratio (AR) of 2:1 and a hydraulic diameter (Dh) of 30 mm. The total length (L) of the channels including the inlet section and outlet section is 685 mm and the length of the heated section is 300 mm. The ratio of the mean rotating arm radius-to-channel hydraulic diameter (R/Dh) is 12.7. The air flows radially outward from the axis of rotation through the test section. The cross-section of the ribs in all cases is a square with a rib height/width (e) of 3 mm. The pitch ratio (P/e) is 10 and the blockage ratio (e/Dh ) is 0.1. The top view of rib configuration is shown in Fig. 2, and the four parameters m, g, a,

Fig. 2. Top view of the rib configurations.

Fig. 1. Schematic diagram of the single-pass channel.

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inner-half-ribs and outer-half-ribs along the y-axis is increased from 0 to 0.1.

Table 1 The parameters of each case.

Case Case Case Case Case Case Case Case Case Case Case

O m1 m2 m3 m4 b30 b60 b75 b90 Og m1g

m/P

g/P

b (°)

0 0.2 0.4 0.6 0.8 0.2 0.2 0.2 0.2 0 0.2

0 0 0 0 0 0 0 0 0 0.1 0.1

45 45 45 45 45 30 60 75 90 45 45

and b represent rib streamwise gap, rib widthwise gap, outer-halfrib angle, and inner-half-rib angle, respectively. The parameter values in each case to describe the rib configurations are shown in Table 1. The outer-half-rib angle a is a constant. The ribs for each case are symmetrical on the leading and trailing walls. All the cases in Table 1 can be divided into three groups to study the fluid flow and heat transfer of the discrete rib configurations: Group 1: rib streamwise gap normalized distance m/P. Effect on fluid flow and heat transfer Case O is a traditional rib-turbulated configuration with 45° inclined continuous ribs. Case m1, Case m2, Case m3, and Case m4 are discrete rib cases based on Case O, i.e., the inner-half-ribs propagate along the x-axis with different rib streamwise distance m. Group 2: the angle of inner-half-ribs b. Effect on fluid flow and heat transfer Case b30, Case b60, Case b75, and Case b90 are based on Case m1, i.e., the angle b is gradually increased from 30° to 90°, while the values of m/P and g/P are fixed at 0.2 and 0, respectively. Group 3: rib widthwise gap normalized distance g/P. Effect on fluid flow and heat transfer Case Og and Case m1g are based on Case O and Case m1, respectively, i.e., the rib widthwise gap normalized distance g/P between

(a) Sketch of typical cooling techniques for modern gas turbine blades

2.2. Boundary conditions In this study, the research object was the ribbed channel of a turbine blade shown in Fig. 3(a). The channel orientation of 45° with respect to the direction of rotation was considered, as presented in Fig. 3(b). The Reynolds number was fixed at 10,000 based on channel hydraulic diameter. The rotation number was varied from 0 to 0.7. Constant mass flow rate of air with a temperature T b of 293.15 K and a turbulence intensity of 5% at the inlet were considered. The flow direction was radially outwards along the channel. An average static pressure of 1 atm was imposed at the outlet. Impermeable boundary and no-slip wall conditions were implemented over the four channel walls as well as on the ribs. A thermal boundary condition was supplied on the leading and trailing walls (including the ribs) with a constant temperature T w of 337.15 K rather than a uniform heat flux. The density ratio ðqb qw Þ=qb = 2ðT w T b Þ=ðT w þ T b ) [24] was fixed at 0.140 resulting in a relatively constant effect of buoyancy force. Meanwhile, the inner and outer walls of test section and the walls of the inlet and outlet sections were adiabatic.

3. Data reduction The Reynolds number is defined as

Re ¼

qU in Dh l

ð1Þ

where U in is the mean velocity at the inlet of the heated section. The rotation number is defined as

Ro ¼

XDh U in

where X is the angular velocity of the rotating channel. The local heat transfer coefficient h is defined as

(b) Schematic of the channel cross-section

Fig. 3. Research object and extracted model.

ð2Þ

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q Tw Tb

ð3Þ

4.0

where q is the wall heat flux. The Nusselt number is defined as

hDh k

3.5

ð4Þ

where k is the thermal conductivity of air. The Nusselt number is normalized by the Dittus-Boelter correlation (see, e.g., Sundén [25]).

Nu0 ¼ 0:023Re0:8 Pr 0:4

Nu/Nu

Nu ¼

ð5Þ

2 4L qU in Dh 2

2

3

Segment

4

5

3.8

14 Nu/Nu (experimental)

b)

Nu/Nu (numerical)

3.6

f/f (experimental)

ð7Þ

Based on the heat transfer enhancement (Nu=Nu0 ) and the frictional loss penalty (f =f 0 ), the thermal performance factor (g), of each rib configuration at all Re and Ro can be evaluated as

g ¼ ðNu=Nu0 Þ=ðf =f 0 Þ1=3

leading (experimental) tailing (experimental) leading (numercial) tailing (numercial)

1

ð6Þ

The Blasius equation (see, e.g., Sundén [25]) is used as the basis of comparison, i.e.,

f 0 ¼ 0:079Re0:25

2.5

1.5

ð8Þ

f/f (numerical)

3.4

Nu/Nu

ðP in Pout Þ

3.0

2.0

where the Prandtl number (Pr) is 0.7. The frictional losses in the channel are defined as the pressure drop from the inlet to the outlet of the heated section in the channel. The friction factor equation is shown below

f ¼

a)

13 12 11 10

3.2

9

f/f

h¼

8 3.0

7 6

2.8

5 4

2.6 10000

20000

30000

40000

Re Fig. 4. Comparison between the numerical and experimental results: (a) Nu=Nu0 on each segment; (b) Nu=Nu0 and f =f 0 in the channel.

4. Numerical simulation method 4.1. Selection of turbulence model

Momentum equation:

In this study, the k-x SST model by Menter [26] was employed. Walker et al. [27] predicted heat transfer in a ribbed two-pass channel by utilizing the commercial solver ANSYS CFX [28] and indicated that k-x SST model is an adequate turbulence model to predict the Nusselt numbers. The research of Schüler et al. [29] illustrated that the simulations by the k-x SST model corresponded well with experimental results of the heat transfer performance in a channel with rib turbulators for both stationary and rotating conditions. In order to make the numerical simulations more reliable, an additional validation was conducted by comparison with the results of Wright et al. [30] before the formal simulations were performed. The comparisons of Nu numbers and friction factors between experiments and the k-x SST model are shown in Fig. 4. On the leading and trailing walls, the numerical results are slightly higher than in the experiment. The maximum Nu=Nu0 deviations on the leading and trailing sides are 7.43% and 6.73%, respectively, and both appear at the first segment. The values of the average Nu=Nu0 and f =f 0 from the simulation and experiment match each other well with a maximum deviation of 5.08% and 7.74%, respectively. Therefore, the k-x SST turbulence model is considered suitable for the present study.

i @ u j i u j Þ @ðqu @P @ @u k 2qeijk Xj u ¼ þ ð l þ lt Þ þ @xj @xi @xj @xj @xi qeilm emjk Xl Xj Rk where R is the rotation radius. Energy equation for fluids:

i T @ qu @ @T cp ¼ k @xi @xi @xi

!

@ lt @T cp @xi Prt @xi

ð10Þ

! ð11Þ

The k-x SST turbulence model employed in this paper is according to [26]. The equation of the turbulent kinetic energy k reads

@ @uj @ @k ðqkui Þ ¼ sij b qxk þ ðl þ rk lt Þ @xi @xi @xi @xi

ð12Þ

The equation of the dissipation rate x reads

@ c @uj @ @x ðqxui Þ ¼ sij bqx2 þ ðl þ rx lt Þ @xi @xi v t @xi @xi qrx2 @k @ x þ 2ð1 F 1 Þ x @xi @xi

ð13Þ

where

4.2. Governing equations

b ¼ F 1 b1 þ ð1 F 1 Þb2

ð14Þ

The governing equations at low Mach number are described as follows [31]: Continuity equation:

rk ¼ F 1 rk1 þ ð1 F 1 Þrk2

ð15Þ

rx ¼ F 1 rx1 þ ð1 F 1 Þrx2

ð16Þ

c ¼ F 1 c1 þ ð1 F 1 Þc2

ð17Þ

i Þ @ðqu ¼0 @xi

ð9Þ

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Table 2 The normalized Nusselt numbers and friction factors for different mesh systems.

15M 10M 7.5M 4.8M 2.9M 1.5M

Nu=Nu0 on the leading wall

Nu=Nu0 on the trailing wall

f =f 0

2.927 2.923 2.881 2.856 2.836 2.848

3.122 3.116 3.092 3.078 3.063 3.079

9.132 9.114 8.977 8.839 8.652 8.494

Baseline 0.14% 1.57% 2.43% 3.11% 2.70%

Baseline 0.19% 0.96% 1.41% 1.89% 1.38%

Baseline 0.20% 1.70% 3.21% 5.26% 6.99%

Bold means the selected mesh system.

Fig. 5. Schematic of a mesh system.

c1 ¼

pffiffiffiffiffi b1 2 b rx1 j = b

ð18Þ

c2 ¼

pffiffiffiffiffi b2 rx2 j2 = b b

ð19Þ

vt ¼

a1 k maxða1 x; XF 2 Þ

ð20Þ

sij ¼ lt 2Sij

2 @uk dij 3 @xk

2 qkdij 3

ð21Þ

Sij ¼

1 @uj @ui þ 2 @xi @xj

ð22Þ

lt ¼

qa1 k maxða1 x; XF 2 Þ

ð23Þ

F 1 ¼ tanhðarg 41 Þ "

F2 ¼

# 4qrx2 k

pffiffiffi k 500v ; ; 2 b xd d2 x CDkx d 1 @k @w ¼ max qrx2 ; 1020 x @xi @xi

arg 1 ¼ min max

CDkx

ð24Þ !

tanhðarg 22 Þ

! pffiffiffi k 500v arg 2 ¼ max 2 ; 2 b xd d x

ð25Þ

The constants in the above equations have the following values:

rk1 = 0.85, rk2 = 1, rx1 = 0.65, rx2 = 0.856, b1 = 0.075, b2 = 0.0828, b = 0.09, a1 = 0.31, j = 0.41. 4.3. Grid independence A mesh generation code based on Matlab is used to generate high quality structured meshes. Structured meshes with grid adoption of yþ < 1 are applied at all solid boundary walls for all cases. The thickness of the first layer nearest a wall grid is 5.5e6 m. The qualities of all the mesh element in each case is above 0.45 based on ANSYS ICEM. The accuracy of the numerical results depends on the mesh cell number and the quality. Six mesh systems with different cell numbers have been tested (see Table 2) at a Reynolds number of 10,000 and a Rotation number of 0.3, namely 1.5–15M for channels with inclined continuous ribs. Using the Case 15M as the baseline, it is found that the deviations of the normalized Nusselt numbers on the leading and trailing sides, and friction factor in Case 10M mesh system are 0.14%, 0.19% and 0.20%, respectively. Thus, the grids of 10M cells are chosen for this study to ensure prediction accuracy. The generated mesh is shown in Fig. 5. 5. Result & discussion

ð26Þ 5.1. Effect of discrete ribs on the leading wall heat transfer

ð27Þ ð28Þ

Due to the presence of Coriolis force, the trailing wall usually has higher heat transfer coefficients than the leading wall. Therefore, the heat transfer characteristics on the leading wall which is enhanced by various-shaped ribs are of particular interest. For the sake of convenient comparison, Fig. 6(a) presents the Nusselt

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4.0

Inner

Case O Case m1

3.5 3.0

Nu/Nu

2.5 2.0 1.5 1.0 0.5 0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x/Dh Fig. 7. Streamwise Nu number distribution at Re = 10,000; Ro = 0.3.

Outer

(a) Case O Inner

Outer

(b) Case m1

Fig. 6. Nu number distributions of continuous and discrete rib cases on the leading side at Re = 10,000; Ro = 0.3.

number contours on the leading wall which is modified by inclined continuous ribs, where b = 45°, Re = 10,000, and Ro = 0.3. Close to the outer wall, the high heat transfer region indicates that the flow reattaches on the floor after it separates from the rib. However, close to the inner wall, the high heat transfer region disappears and is replaced by a low heat transfer band. A similar phenomenon

Inner

was also observed in a ribbed U-band channel where the centrifugal force plays a part in affecting the local heat transfer [32]. For inclined ribs, due to the pressure difference between the two ends, a secondary flow is generated along the rib-wise direction. Close to the inner wall, the rib-wise secondary flow may be strong enough to deflect the mainstream flow. In this case, the core flow does not reattach on the floor. Because the ‘jet-like’ reattached flow is attenuated by the secondary flow, the heat transfer is significantly deteriorated in the region close to the inner wall. In contrast, Fig. 6(b) demonstrates the Nusselt number contours on the leading wall modified by discrete ribs, where the rib streamwise gap m/P = 0.2. Close to the outer wall, the heat transfer patterns are nearly unchanged. However, close to the inner wall, another reattachment region downstream of the inner-half rib is clearly noticed, even though the heat transfer magnitudes are lower than in the corresponding region close to the outer wall. Moreover, in the vicinity of the inner wall, the low heat transfer band is significantly improved. This can be attributed to the streamwise rib gap which reduces the rib-wise pressure gradient and suppresses the secondary flow close to the inner wall. To further clarify the heat transfer pattern, Fig. 7 shows the Nusselt number profiles along the streamwise direction between consecutive ribs (i.e., the red line in Fig. 6(a) and (b)). For the discrete ribs (Case m1), the peak Nusselt number at about x/Dh = 0.25 indicates the flow reattachment point. For the continuous ribs (Case O), however, a plateau is observed, suggesting that the flow does not reattach on the floor. Due to the absence of flow reattachment, the magnitudes of heat transfer are significantly reduced.

Inner

Outer

Outer

(a) Case O

(b) Case m1

Fig. 8. Streamlines comparisons of Cases O and m1 at Re = 10,000; Ro = 0.3.

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Inner

C1

C2

Inner

C3

D1

C4

Outer

D2

D3

D4

Outer

(a) Case O

(b) Case m1

Fig. 9. Normalized rib-wise velocity contours comparisons of Cases O and m1 at Re = 10,000; Ro = 0.3.

Inner Wall

Symmetry Plane

Inner Wall

Symmetry Plane

Leading wall

Leading wall

(b) D1

(a) C1

Inner Wall

Symmetry Plane

Inner Wall

Symmetry Plane Leading wall

Leading wall

(d) D2

(c) C2

Inner Wall

Symmetry Plane

Inner Wall

Symmetry Plane

Leading wall

Leading wall

(f) D3

(e) C3

Inner Wall

(g) C4

Symmetry Plane

Inner Wall

Symmetry Plane

Leading wall

Leading wall

(h) D4 Fig. 10. Streamlines comparisons on four surfaces of Case O and m1.

Fig. 8 shows the streamlines as the flow is approaching the inclined ribs on the leading wall. For both cases, it is found that the streamlines reattach on the floor close to the outer wall. Due to the large pressure gradient, a rib-wise secondary flow is generated towards the inner wall. For the discrete ribs, however, the presence of the streamwise gap leads to intense mixing between the secondary flow and main stream, which is represented by the high turbulent kinetic energy in the gap region. The mixing of the two streams can significantly reduce the strength of the ribwise secondary flow and thus enhance the local heat transfer close to the inner wall via flow reattachment. Fig. 9 displays the rib-wise velocity contours at various planes between the third and fourth ribs. The planes are parallel to the rib direction and are selected to represent the flow separation and reattachment regions. Except the plane 1, the magnitudes of the rib-wise velocity are lower for the discrete ribs than for the

continuous ribs, indicating that the secondary flow is suppressed by the presence of the rib gap. The corresponding streamlines at the various planes are seen in Fig. 10. For the continuous ribs, a strong secondary flow is seen. Moreover, a large corner vortex is observed between the inner wall and the leading wall. The corner vortex has a low rotation speed, therefore, the local heat transfer is dominated by heat conduction, which results in lower heat transfer. This is consistent with the results shown in Fig. 6(a). Compared to the continuous ribs, Fig. 10(b) shows that the streamlines at the corresponding planes where the corner vortex significantly shrunk by the discrete ribs. As a result, the local heat transfer is improved, as shown in Fig. 6(b). Fig. 11 shows the streamlines at different planes for the two cases. The arrow of the core flow exhibits the flow direction of the main flow in the channel. Based on the flow structures at planes normal to the ribs, the separation lines and reattachment

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J. Wang et al. / Applied Thermal Engineering 129 (2018) 1369–1381 Inner

Inner

S2

S2 S1

Core flow

S3 Core flow

R2

R1

S1 R1

Rib secondary flow

Rib secondary flow Outer

Outer

(a) Case O Inner

(b) Case m1 Inner

Outer

Outer

(c) Case O

(d) Case m1

Fig. 11. Streamlines of continuous and discrete rib cases on the leading side at Re = 10,000; Ro = 0.3.

lines are identified, as shown in Fig. 11(a) and (b). For the continuous ribs, the separation lines S1 and S2 are observed in the vicinity of the rib. Between the separation lines is the reattachment line R1. For discrete ribs, a distinct feature is that an additional reattachment line R2 and a new separation line S1 exist. In the planes normal to the widthwise direction, as shown in Fig. 11(c) and (d), it is found the flow separates and reattaches on the floor close to the outer wall. However, close to the inner wall, the streamlines for the continuous ribs are nearly parallel to each other, indicating that the flow reattachment disappears. For the discrete ribs, the phenomena of flow separation and reattachment are still visible.

5.2. Effect of rib streamwise gap In what follows, the effects of the rib streamwise gap, the innerhalf-rib angle, and rib widthwise gap on the heat transfer and pressure drop are considered. Fig. 12 presents the Nu number distributions on the leading walls between the 3rd and 4th streamwise row of ribs for various rib streamwise gap. In this configuration, m/P = 0.2, 0.4, 0.6, and 0.8, respectively. While other parameters are as: the widthwise gap g/P = 0, and the inner half rib angle b = 45° at Re = 10,000 and Ro = 0.3. Heat transfer peaks appear at the region close to the outer side where flow reattachment occurs. The shape of the heat transfer peak is hardly affected by the rib streamwise gap size. However, the Nu number distribution at the region near the inner side is obviously influenced by the rib configuration. It can be seen that

the smaller gap, such as in Case m1 and Case m2, shows a stronger heat transfer enhancement at the region behind the inner-half-rib. The low heat transfer bands close to the inner wall in Case m1 and Case m2 are slightly narrower than for the other two cases due to the moderate rib-wise pressure gradient and diminished momentum of the rib secondary flow. The average normalized Nu number on the leading wall and friction factor for five cases at Re = 10,000 and Ro number ranging from 0 to 0.7 are given in Fig. 13(a) and (b). The average Nu number and friction loss both increase as the Ro number increasing for the five cases. It is found that the heat transfer of Case m1 (m/P = 0.2) is obviously higher than the other four cases particularly at higher Ro number. The trend is that the average Nu number on the leading wall gradually decreases as the rib streamwise gap is increased. However, Case m1 (m/P = 0.2) produces higher friction factors than the remaining four cases. Generally speaking, a smaller rib streamwise gap contributes to a higher heat transfer performance. 5.3. The effect of inner-half-rib angle The rib angle, which significantly affects the direction of the rib secondary flow, is also an important parameter to influence the fluid flow and heat transfer characteristics. Moreover, considering that in a discrete rib case the rib secondary flows induced by the inner-half-rib and the outer-half-rib may interact with each other. Thus, it is necessary to study the effects of different rib angles on heat transfer and fluid flow in a rotating channel. In this part, Case

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Inner

Inner

Outer

Outer

(a) Case m1

(b) Case m2

Inner

Inner

Outer

Outer

(c) Case m3

(d) Case m4

Fig. 12. Normalized Nu number distributions of Cases m1, m2, m3, and m4 on the leading wall at Re = 10,000; Ro = 0.3.

Case O Case m1 Case m2 Case m3 Case m4

12

3.4

11 10

f/f

Nu/Nu

3.2

3.0

Case O Case m1 Case m2 Case m3 Case m4

2.8

9 8 7

0

0.1

0.2

0.3

0.4

0.5

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Ro

(a) Average normalized Nu number on the leading wall

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Ro

(b) Normalized friction losses

Fig. 13. Local Nu number distributions, average normalized Nu number and normalized friction losses for Cases m1, m2, m3, and m4 at Re = 10,000.

m1 is the original model. A fixed outer-half-rib angle (a = 45°) is chosen while the inner-half-rib angle b is varied, i.e., b = 30°, 60°, 75° and 90°, respectively. The normalized Nu number contours on the leading wall for Cases b30, b60, b75 and b90 are presented in Fig. 14. The Nu number contours for b = 45° are seen in Fig. 12(a). It is found that the heat transfer patterns close to the inner wall are significantly affected by the inner-half-rib angles. For b = 60° and 75°, a large reattachment region is found between consecutive inner ribs. This significantly enhances the local heat transfer. For b = 30°, the inclined angle is shallow and the rib effect caused by flow separa-

tion and reattachment is attenuated. For b = 90°, the rib is normal to the mainstream so that the rib-wise secondary flow is suppressed. In this case, the heat transfer in the inner corner region is deteriorated. The average normalized Nu numbers on the leading wall and friction losses for the five cases are shown in Fig. 15. It is found that the heat transfer effect of inner-half-rib angle is more obvious than the rib distance along the streamwise or widthwise direction. However, the pressure penalty increases tremendously as the angle b increases. Cases b60 and b75 give the best heat transfer performance, while the pressure loss in Case b75 is largest.

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Inner

Inner

Outer

Outer

(a)

(b)

Inner

Inner

Outer

Outer

(c)

(d)

Fig. 14. Normalized Nu number distributions for Cases b30, b60, b75 and b90 on the leading wall at Re = 10,000; Ro = 0.3.

17

3.6

Case β30 Case A2 Case β60 Case β75 Case β90

16 15

3.4

13

3.2

12

f/f

Nu/Nu

14

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Case β30 Case A2 Case β60 Case β75 Case β90

2.8

11 10 9 8 7

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6 0

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0.4

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0.7

0

0.1

0.2

0.3

0.4

0.5

Ro

Ro

(a) Average normalized Nu numbers on the leading wall


0.6

0.7

Fig. 15. Average normalized Nu number and normalized friction losses for Cases b30, m1, b60, b75, and b90 at Re = 10,000.

5.4. Effect of rib widthwise gap The rib widthwise distance g is another parameter to control the shape and size of the rib gap, which influences the flow pattern around the gap and the downstream region. In this part, a fixed rib widthwise gap normalized distance (g/P = 0.1) is employed for the Case O and m1 to form the Case Og and m1g, respectively. The Nu number distributions of Case Og and m1g are shown in Fig. 16. It is found that for Case Og, a high Nu number area appears at the rib gap and it extends towards the inner side due to the impact of the rib secondary flow. Comparing the results in

Fig. 16(a) and Fig. 6(a), it shows that the low heat transfer band is slightly reduced, indicating that the separation near the inner wall is suppressed. The magnitudes and positions of high/low Nu number regions between the adjacent inner-half-rib in Case m1g are similar to those in Case m1 (Fig. 12(a)), except that the high heat transfer region beginning at the gap is slightly expanded towards the downstream. The cases with larger rib widthwise gap (g/P > 0.1) are also tested. The results show that the heat transfer dramatically decreases as the rib widthwise gap is enlarged. This reversed phenomenon indicated that the ratio of the truncation length to the total width length increases when the widthwise

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Inner

Inner

Outer

Outer

(a) Case Og

(b) Case m1g

Fig. 16. Normalized Nu number distributions of Cases Og and m1g on the leading wall at Re = 10,000; Ro = 0.3.

12

3.4

11 10

f/f

Nu/Nu

3.2

9

3.0

Case O Case Og Case m1 Case m1g

2.8

Case O Case Og Case m1 Case m1g

8 7

0

0.1

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0.6

0.7

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Ro

Ro


(a) Average normalized Nu number on the leading wall

Fig. 17. Local Nu number distributions, average normalized Nu number and normalized friction losses for Cases O, Og, m1, and m1g at Re = 10,000.

1.55

1.55

1.50

1.50

1.45

1.45

1.40

1.40

1.35

1.35

1.30

1.30 Wright et al. [30] Chang et al. [33]

1.25

Case O Case m1 Case m2 Case m3 Case m4 Case Og

1.25

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Case m1g Case 30 Case 60 Case 75 Case 90

1.20 0

0.1

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Ro

(a) On the leading wall

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Ro

(b) On the trailing wall

Fig. 18. Thermal performance on the leading and trailing walls for all tested cases at Re = 10,000.

gap is increased. Because the gaps turn to smooth-channel-like regions, the effect of the rib is impaired. However, the intensity of turbulence in the flow in this region does not increase as the gap is enlarged. Therefore, the reversed phenomenon is found when enlarging the rib widthwise gap. The average normalized Nu number on the leading wall and friction factors for these four cases are given in Fig. 17(a) and (b) at Re = 10,000 and Ro number ranging from 0 to 0.7. It is noted that

Case Og provides better heat transfer than Case O in the range of 0.1 < Ro < 0.5, indicating that widthwise gap can enhance heat transfer in these rotating conditions. However, the reverse is true for Cases m1 and m1g, which means that the rib gap is oversized. In addition, the case with streamwise gap (Case m1) gives better heat transfer performance than that with widthwise gap (Case Og), indicating that the streamwise gap is more advantageous than the widthwise gap. The comparison of Case m1 and Case Og shows


that Case m1g provides smaller pressure loss and similar results can be found in Case O and Case Og, which illustrates that the cases with widthwise gap have smaller pressure drop in most situations. 5.5. Overall performance The overall thermal performances of various rib configurations on the leading wall and trailing wall for Rotation numbers ranging from 0 to 0.7 and Reynolds number fixed at 10,000 are compared in Fig. 18. The experimental results reported by Wright et al. [30] and Chang et al. [33] are exhibited as well. It can be seen that the differences between leading wall and trailing wall is conspicuous and thermal performances on the trailing wall are always better for all the cases in rotating condition. On the leading wall, the overall thermal performances of most rib configurations vary between 1.4 and 1.45. Case m1 provides the best thermal performance at low Ro number but is exceeded by Case b30 as Ro > 0.5 due to the low flow resistance characteristics. Meanwhile, on the trailing wall, the thermal performances of the majority of the test cases vary between 1.45 and 1.5. Case m1 performs better than others for all the studied rotating conditions and peaks at 1.53 as Ro = 0.5. 6. Conclusions In this study, the flow field and heat transfer characteristics of a straight rectangular rotating channel with channel orientation of 45° and various discrete rib configurations were numerically studied for a fixed Re number of 10,000 and a range of Ro numbers from 0 to 0.7. Compared with continuous rib case, the discrete rib case was found to obviously enhance heat transfer on the leading wall. Three parameters of the discrete rib configuration - rib streamwise distance, rib widthwise distance, and inner-half-rib angle - were considered. The study found that a small streamwise rib gap was beneficial to improve heat transfer. However, as the streamwise rib gap is enlarged, the heat transfer decreased consistently. The influence of inner-half-rib angle on heat transfer was most conspicuous among these three parameters. As the inner-half-rib angle increased from 30° to 90°, the heat transfer on the leading wall and flow resistance endured a trend from rise to decline, and Cases b60 and b75 gave the best heat transfer performance. The heat transfer augmentation of the widthwise rib gap was not as remarkable as the streamwise rib gap, but it offered lower flow resistance. Among all the studied rib configurations, Case m1 provided best thermal performance both on the leading and trailing walls. Acknowledgments The authors acknowledge the financial support provided by the Swedish Scientific Council, and the China Scholarship Council (CSC). References [1] J.-C. Han, Recent studies in turbine blade cooling, Int. J. Rotating Mach. 10 (2004) 443–457. [2] J.-C. Han, M. Huh, Recent studies in turbine blade internal cooling, Heat Transf. Res. 41 (2010). [3] J.-C. Han, S. Dutta, S. Ekkad, Gas Turbine Heat Transfer and Cooling Technology, 2nd ed., CRC Press, 2012. [4] J.C. Han, J.S. Park, Developing heat transfer in rectangular channels with rib turbulators, Int. J. Heat Mass Transf. 31 (1988) 183–195. [5] G. Rau, M. Cakan, D. Moeller, T. Arts, The effect of periodic ribs on the local aerodynamic and heat transfer performance of a straight cooling channel, in: ASME 1996 International Gas Turbine and Aeroengine Congress and Exhibition, American Society of Mechanical Engineers, 1996, pp. V004T009A061–V004T009A061.

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