Heat transfer enhancement of wedge-shaped channels by replacing pin fins with Kagome lattice structures

International Journal of Heat and Mass Transfer 141 (2019) 88–101

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Heat transfer enhancement of wedge-shaped channels by replacing pin fins with Kagome lattice structures Beibei Shen a, Yang Li b, Hongbin Yan a,⇑, Sandra K.S. Boetcher c,⇑⇑, Gongnan Xie a,d,⇑ a

School of Marine Science and Technology, Northwestern Polytechnical University, Xi’an 710072, PR China School of Mechanical Engineering, Northwestern Polytechnical University, Xi’an 710072, PR China c Department of Mechanical Engineering, Embry-Riddle Aeronautical University, Daytona Beach, FL 32114, United States d Research & Development Institute of Northwestern Polytechnical University in Shenzhen, Shenzhen 518057, PR China b

a r t i c l e

i n f o

Article history: Received 19 December 2018 Received in revised form 22 May 2019 Accepted 15 June 2019 Available online 24 June 2019 Keywords: Trailing edge Kagome cores Pin fins Flow characteristics Heat transfer enhancement

a b s t r a c t This study introduces Kagome lattice structures to replace pin fins or ribs in a wedge-shaped channel, which represents a turbine-blade trailing edge, for heat transfer enhancement. Present simulation methods are verified against available experimental data. The local and overall thermo-fluidic characteristics of the four designed channels, at five Reynolds numbers and a given porosity, are investigated numerically and compared. The results reveal that the proposed structures exhibit 6–71% higher overall Nusselt numbers relative to the original structure but with similar pressure loss. The heat transfer enhancement is attributed to the fact that the first array of Kagome cores separates more high momentum fluid to the tip region, which results in the reduction of recirculation and increases convective heat transfer. Ó 2019 Elsevier Ltd. All rights reserved.

1. Introduction In order to improve the thermal efficiency and power output of gas turbine engines, the temperature at the inlet of the turbine needs to be as high as possible; recent developments show inlet temperatures as high as 2000 K [1], which is much higher than the allowable metal temperature of the turbine blades [2]. To improve operation safety and prolong lifespan of the blades in such a severe environment, advanced cooling technologies must be developed along with exploring advanced high-heat-resistant materials. Since cooling technologies are developing faster than heat-resistant materials, turbine blade cooling, the focus of the present study, is important to the development of highperformance gas turbine engines. In order to minimize aerodynamic losses as a result of the wake flow downstream the trailing edge of a blade, the transverse cross-section of the trailing edge decreases gradually along the flow direction. Therefore, the space

⇑ Corresponding authors at: School of Marine Science and Technology, Northwestern Polytechnical University, P.O. Box 24, Xi’an 710072, China. ⇑⇑ Corresponding author at: Department of Mechanical Engineering, Embry-Riddle Aeronautical University, 600 S. Clyde Morris Blvd., Daytona Beach, FL, 32114, United States. E-mail addresses: [email protected] (B. Shen), [email protected]. cn (Y. Li), [email protected] (H. Yan), [email protected] (S.K.S. Boetcher), [email protected] (G. Xie). https://doi.org/10.1016/j.ijheatmasstransfer.2019.06.059 0017-9310/Ó 2019 Elsevier Ltd. All rights reserved.

for mounting the trailing edge cooling elements is very limited. Additionally, its mechanical strength must be reinforced in consideration of the thin walls. Such a situation makes the effective cooling of trailing edges challenging; hence heat transfer enhancement in this region becomes very significant. In the past, various elements such as ribs, pin fins, etc., have been added to the smooth channel of the trailing edge to enhance heat transfer. Among them, pin-fin arrays sandwiched between the two sidewalls of the trailing edge are supposed to be an excellent design which can enhance heat transfer by intensifying flow mixing and enlarging the heat transfer area. They also act as the supporting structure to improve the mechanical strength and stiffness. In particular, circular pin fins are commonly used and are easy to implement in gas turbine applications [3]. Hwang and Lui [4] compared the fluid flow and heat transfer characteristics between wedge-shaped channels incorporating in-line and staggered pin fins; they found that the turned channel with staggered pin-fin arrays produced the highest pressure drop but resulted in a high temperature near the corner. Wedge-shaped channels with pentagonal pin fins provide less uniform heat transfer distribution compared to the staggered configuration [5]. To further improve the cooling performance, modifications of the turbulators in the cooling passage were conducted. Triangular-shaped pin fins were proposed by Katharine et al. [6] but produced comparable thermal performances to the traditional cylinder pin fin. Apart from pin fins, the wedge-shaped channel with

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Nomenclature cp dP1, dP2, dk1, dk2, Dh Et f h H1 H2 H3 hr k L0 L1 l1 lr M Nu Dp Pd Ph Pr Px Py r q00 Re Tf

specific heat (J/(kg∙K)) dP3 diameter of the pin fin (m) dk3 diameter of the Kagome lattice (m) hydraulic diameter of the wedge-shaped channel (m) turbulent kinetic energy (J/kg) friction factor defined in Eq. (8) overall heat transfer coefficient (W/(m2∙K)) defined in Eq. (6) height of the inlet channel with trapezoidal cross section (m) height of the inlet channel with trapezoidal cross section (m) height of the outlet channel (m) height of the rib (m) thermal conductivity (W/(m∙K)) length of the inlet channel (m) length of the wedge-shaped channel (m) length between the center of the horseshoe pin fin and back wall (m) length of the rib (m) Sutherland constant overall Nusselt number defined in Eq. (5) pressure drop (Pa) local pressure minus 101,000 (Pa) pitch of the horseshoe pin fin (m) pitch of the rib (m) pitch of cores along x-axis direction (m) pitch of cores along y-axis direction (m) radius of the horseshoe pin fin (m) heat flux (W/m2) Reynolds number defined in Eq. (2) bulk mean fluid temperature in Eq. (11) (K)

the +60° inclined ribs provided superior cooling efficiency [7]. To improve the cooling performance and reduce pressure drop concurrently, a combination of diverse pin fins and ribs/dimples were introduced and investigated experimentally and numerically [8–10]. It was reported that channels with different combinations of turbulators led to various heat removal capability and showed higher cooling performance in contrast to the channel only with pin fins. Further, intensified turbulence intensity and enlarged cross-section flow area were found to contribute to the improved performance. In addition, the effect of the exit direction of the fluid from wedge-shaped channels with V-shaped ribs [11], tapered ribs [12], ribs and pin fins [13] has been explored by experimental methods. The results suggest that due to the majority of the fluid being discharged through the lateral outlets, the inner-top corner of the channel receives less fluid flow which lowers the thermal performance. Subsequently, Li at el. [14,15] examined two entrances at the bottom and the top of the wedge-shaped channel and concluded that the heat transfer of the upstream-half of the channel was reduced and the overall thermal performances of the channel hadn’t been increased. Moreover, compared to a wedge-shaped channel with a closed-tip condition, the channel with an opentip condition provided a smaller area of fluid recirculation at the tip region and higher heat transfer capability for the outlet region near the tip [16]. Based on the aforementioned literature, optimization of the turbulators to enhance cooling performance and provide mechanical reinforcement is an important aspect pertaining to the design of a trailing edge. Truss-type periodic cellular metals (PCMs) with

Tin Tw Uin V W1 W2 W3 wc wh wr x, y, z y+

inlet fluid temperature (K) wall temperature (K) inlet velocity (m/s) velocity magnitude (m/s) width of the wedge-shaped channel (m) width of the trapezoidal cross section of the inlet channel (m) width of the outlet channel (m) width between first array of lattice in middle region and side wall (m) width of the horseshoe pin fin (m) width of the rib (m) Cartesian coordinates (m) dimensionless wall distance

Greek symbols a inclined angle (°) e porosity of lattice structure l dynamic viscosity (Pa∙s) q density (kg/m3) Abbreviations (capital) and subscripts (lowercase) PCM periodic cellular material WBK wire-woven bulk Kagome c core structure f fluid h horseshoe pin fin in inlet k Kagome lattice r rib s solid w wall

excellent specific strength/stiffness and heat transfer performance seem to be promising candidates for trailing edge cooling [17]. Since 2000, PCMs with various topologies, such as pyramid, tetrahedron, textile and Kagome [18–23], have been devised. In particular, the specific topology of the Kagome lattice endows itself with superior aerodynamic isotropy. It was found that the Kagome lattice has better resistance to plastic buckling under compression or shear load for a given relative density compared to the tetrahedral lattice [22,23]. Thermally, many experimental and numerical studies have been conducted on the Kagome lattice. Hoffman [24] carried out an experimental study to find the effect of flow orientation, core porosity and material properties on fluidic and thermal characteristics in a casted Kagome-cored sandwich panel. Krishnan et al. [25] compared the thermo-fluidic characteristics of tetrahedral, Kagome and pyramidal sandwich panels by numerical simulation. Later, Shen et al. [26] presented a numerical comparison of the fluid flow and heat transfer characteristics between the single-layered Kagome-cored sandwich panel and the singlelayer wire-woven bulk Kagome (WBK) cored sandwich panel. Results revealed that the Kagome-cored sandwich panel provided about 30% higher thermal performance relative to the WBK sandwich panel for a given pumping power and porosity. Many factors, including the configuration, the arrangement and the combination mode of turbulators, have an impact on the thermal performances of wedge-shaped channels. Due to their excellent thermal and mechanical characteristics, replacing conventional pin fins with Kagome lattices at the trailing edge may improve thermal and mechanical performances concurrently. Therefore, this study proposes a new concept of replacing conven-

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tional pin fins with Kagome lattice structures. Especially, for the wedge-shaped channel with lateral fluid exits, tip regions with low-momentum flow recirculation are a non-negligible factor for the heat transfer enhancement of the channel. To further enhance the heat transfer of the tip region, pin fins combined with ribs are distributed in the inner region of the wedged channel to guide more fluids towards the tip region. For comparison of the shunting effects of pin fins versus Kagome lattices, Kagome lattices combined with ribs are also placed in the inner region. In the present study, different L-shaped channels with different turbulators including pin fins, Kagome lattice structures and ribs are designed and numerically investigated for five Reynolds number values ranging from 10,000 to 25,000. Comparison of overall and local flow and heat transfer characteristics were carried out to understand the mechanisms. 2. Physical models According to the geometrical features of the trailing edge of a gas turbine blade as described in Fig. 1, the trailing edge is modeled as an L-shaped channel equipped with Kagome lattice structures, ribs, and pin fins (Fig. 2). All L-shaped channels consist of an inlet channel, a wedge-shaped channel, and extended channel. The inlet and extended channels are used to ensure numerical stability. The dimensions of the computational domain are taken from [27]. For clarity, the governing parameters are depicted and summarized in Table 1. The channel, with a trapezoidal cross section of H1 = 24.2 mm, H2 = 17.3 mm, and W2 = 25 mm, has a length (L0)

of 15 mm. The extended channel has a width (W3) of 47.4 mm and a height (H3) of 4.5 mm. The turbulators, which include pin fins, horseshoe pin fins, and the Kagome lattice, are arranged in a pattern of eight-cell units along the streamwise direction (i.e. along the y-axis), corresponding to a width (W1) of 204 mm. In addition, there are two arrays of ribs mounted on the top wall and bottom wall, respectively. The casted Kagome lattice has an inclined angle of a = 60° and is characterized by dimensions dk1 = 3.52 mm, dk2 = 3.28 mm and dk3 = 2.43 mm. To ensure a systematic comparison, the four channels have the same porosity (e = 0.983). The porosity is defined as the fraction of volume void divided by the total volume of the wedged-shaped channel. Correspondingly, the diameter (dp1) of the first array of pin fins is 5.8 mm, the diameter (dp2) of the second array of pin-fins is 4.25 mm, and the diameter (dp3) of the last array of pin-fins is 2.5 mm. Lastly, all the L-shaped channels have the same horseshoe pin fins with a radius (r) of 3.6 mm near the exit position. To simplify the numerical simulation, the thickness of the heating wall is ignored and only the solid domain of pin fins and the Kagome lattices are considered. Particularly, the turbulators are all made of ASTM type 310 stainless steel [28] (0Cr25Ni20) with a specific heat (cps) of 502 J/kgK and a density (qs) of 8030 kg/ m3. The thermal conductivity (ks) in W/mK varies linearly with temperature (T) and is correlated as

ks ðTÞ ¼ 0:0115T þ 9:9105

ð1Þ

Geometric details and configurations of the pin fins and the Kagome lattices are shown in Fig. 2. The Kagome unit cell is composed of three straight circular ligaments intersecting with one another, forming two identical tetrahedrons. The substrate acts as the tetrahedron base. In other words, six identical ligaments of identical length are needed to construct the unit cell. More detailed descriptions about the morphological features of Kagome lattices can be found in [23]. For convenience, the four L-shaped channels are separately denoted as Model 1, Model 2, Model 3, and Model 4 as presented in Fig. 2. All four models have the same arrangement of horseshoe pin fins near the outlets and two arrays of core structures placed in the middle position of the channel. The core structures are cylindrical pin fins in Models 1 and 3, while the core structures in Models 2 and 4 are Kagome lattices. In Models 1 and 2, only the ribs are acting as turbulators near the side wall, and in Models 3 and 4, ribs as well as pin fins/Kagome lattices act to help distribute the flow. 3. Details of numerical simulation 3.1. Computational domain and boundary conditions The detailed computational domains are presented in Fig. 2. Extended channels are placed downstream of each L-shaped channel to avoid backflow. Fully developed, isothermal velocity profiles are imposed at the inlet. In order to obtain these velocity profiles, fully developed isothermal turbulent airflow is simulated separately in the short channel as shown in Fig. 2 by applying a translational periodic boundary condition with zero pressure drop to the periodic faces, while the other walls are set to be no-slip adiabatic walls [29]. This separate simulation to obtain the inlet velocity profile is based on a Reynolds number of 15,000. Herein, Reynolds number (Re) is defined as

Re ¼

Fig. 1. Physical model of the trailing edge of a gas turbine blade cooled by Kagome lattices.

qU in Dh l

ð2Þ

where Dh is the hydraulic diameter based on the inlet and is equal to 22.5 mm, q and l are separately the density and viscosity of air. Uin is the mean flow velocity at the inlet.

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Fig. 2. Schematic description of wedge-shaped channels representative of a turbine blade trailing edge: (Model 1) with ribs and two arrays of cylinder pin fin; (Model 2) with ribs and two arrays of Kagome core lattices; (Model 3) with ribs and three arrays of cylinder pin fin; (Model 4) with ribs and three arrays of Kagome core lattices.

Table 1 Geometrical parameters of all models in the present study. Parameter

Value

Parameter

Value

Parameter

Value

dp1 dp2 dp3 dk1 dk2 dk3 H1 H2 H3

5.8 mm 4.25 mm 2.5 mm 3.52 mm 3.28 mm 2.43 mm 24.2 mm 17.3 mm 4.5 mm

hr L0 L1 lr l1 Ph Pr Px Py

3.0 mm 15 mm 204 mm 3.6 mm 20 mm 24 mm 24 mm 13.5 mm 24 mm

r W1 W2 W3 wc wp wr

3.6 mm 77.4 mm 25 mm 47.4 mm 43.2 mm 6.3 mm 30 mm 60° 0.983

a e

suitable to discretize the main flow region and solid domain. Ten layers of prism elements are generated near all the no-slip walls to resolve the fluid and thermal boundary layers. The scalable wall functions are used for all turbulence models based on the e–equation to establish correlation between the momentum (flow variables) and heat transfer features in near-wall region and fully developed turbulent flow of mainstream. For an accurate prediction of turbulence flow, the height of the first layer elements adjacent to the solid walls is set to be small enough to exhibit a dimensionless wall distance (y+) <1.0 [30]. In addition, nonconformal interfaces are adopted to couple the fluid and solid domains. The fluid is assumed to be compressible with temperaturedependent thermo-physical properties. The air has a constant specific heat capacity (cpf = 1006.4 J/kg/K), and the thermal conductivity (kf) of air in W/m K is a function of temperature

A uniform inlet temperature (Tin) of 300 K is specified for fluid at the entrance. A pressure outlet boundary condition with ambient gauge pressure of 0 is set at the outlets of the computational domain. The walls of the inlet region and the extended walls at the outlet region are assumed to be adiabatic. The conservative interface flux condition and no-slip condition are applied to all interfaces and walls, respectively. All other walls are heated with a uniform heat flux (q00 ) of 1500 W/m2.

Dynamic viscosity of the air is calculated based on Sutherlands formula

3.2. Numerical methods

lðTÞ ¼ l0 ð 0 Þ

ANSYS ICEM 15.0 is used to generate a hybrid mesh incorporating both tetrahedron and prism elements to discretize the solid and fluid domains as detailed in Fig. 3. Tetrahedron elements are

where l0 is the reference viscosity and is equal to 1.7894  105 Pas, T0 is the reference temperature with a constant value of 273.11 K, and M is the Sutherland constant with a value of 110.56 K.

kf ðTÞ ¼ 0:024  0:00000002  ðT  273:15Þ2 þ 0:00008  ðT  273:15Þ

T T

3=2

T0 þ M T þM

ð3Þ

ð4Þ

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Fig. 3. The representative meshes adopted in the numerical simulations: the enlarged views describe the mesh for the ribs, cylinder pin fin in Model 1, Kagome core in Model 2, horse-shoe pin fin and outlet region.

The turbulence was modeled using the RNG k-epsilon model since it exhibits the capability of precisely predicting separated flows [31]. This turbulence model presents reasonable agreement between the experimental and numerical results for the channel with ribs and helical wind turbine [32]. Therefore, the RNG kepsilon model is used in this study for all numerical simulations. The advection terms in the governing equations are discretized by a high-resolution scheme while the diffusion terms are discretized by the central-difference scheme. The computations are conducted by the simulation software ANSYS CFX 15.0, which is based on the finite-volume method. 3.3. Mesh independence test and model validation To assess grid independence and to validate the numerical model, several dimensionless parameters are defined first. The overall Nusselt number (Nu) is used to evaluate the overall heat transfer of the L-shaped channel is defined as

Nu ¼

hDh kf

ð5Þ

where Nu is the overall Nusselt number and kf is the thermal conductivity of fluid. In Eq. (5), the overall heat transfer coefficient (h) is defined as

h¼

q00 T w  Tb

ð6Þ

where q00 is the heat flux applied on the heated walls, Tw is the average temperature of all the heated walls. According to the references [33,34], an overall averaged bulk temperature, Tb, is employed to calculate the overall Nusselt number as

Tb ¼

T in þ T out 2

ð7Þ

where Tin is the inlet fluid temperature and Tout is the outlet fluid temperature. The dimensionless friction factor (f) is used to evaluate pressure drop and defined as

f ¼

  Dp Dh L qU 2in =2

ð8Þ

where Dp is the pressure drop from the inlet to outlet and L is the length of the L-shaped channel.

Model 1 is chosen for the mesh-independence study and three sets of meshes with 8.3 million, 11.0 million and 14.0 million elements are evaluated to check the mesh sensitivity. Table 2 presents the predicted average temperature of all the heated walls and the friction factor for each of these three mesh cases. It is noted that the maximum percentage change in the value of average temperature of all the heated walls and friction factor from the last two meshes are <3%. To ensure the computational accuracy and save computational cost, meshes with 11.0 million elements are employed in subsequent simulations for Model 1. Meshes of other models are generated along with similar mesh setup to the relevant parameters of Model 1. For verification of the numerical model, the comparison of the heat transfer coefficient (h) between the present numerical data and the experimental results of the L-shaped channel with pin fins in [27] is performed, as shown in Fig. 4. In the present study, a heat flux of 1500 W/m2 is used as the boundary condition for the heated walls. However, since the investigators in [27] used a wall temperature boundary condition of 335 K, in order to verify the current simulation with [27] a separate simulation was conducted with a wall temperature of 335 K. It can be seen that the predicted heat transfer coefficient (h) displays roughly the same trend with the experimental data as the unit number increase. The maximum deviations between the current simulation and the experiment in [27] are 11% and 55% for the heat transfer coefficient of the walls with pin fins and the bottom wall with ribs, separately. It is difficult to ascertain the reason behind the 55% deviation since the investigators in [27] did not present experimental uncertainties. However, the good agreement in the experimental data for the case of the pin fins and a maximum deviation of 13% between the numerical data of [27] and the present numerical simulation was deemed acceptable. Therefore, it is believed that the present numerical model is reliable to explore the relative merits of the convective heat transfer and the underlying flow mechanisms in the following simulations. Table 2 Nusselt numbers and friction factors under three different sets of meshes for Model 1 (Re = 15,000). Total elements

Tw

f

8.3 million 11.0 million 14.0 million

333.7911 K 332.8325 K 332.5047 K

0.311 0.316 0.317

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200 Experimental results in [27] Numerical results in [27] Numerical results in this study

2

h/[W/m /K]

150

100

50

0 0

2 4 6 8 Plate number of walls with pin fins

(a) Fig. 5. Comparison of overall Nusselt number for the four models in this study at Re = 15,000.

200 Experimental results in [27] Numerical results in [27] Numerical results in this study

h/[W/m2/K]

150

100

50

0 0

2 4 6 8 Plate number of bottom walls with ribs

(b) Fig. 4. Validation of the numerical model via comparison between the present numerical results and the experimental data and numerical data in [27]: (a) heat transfer coefficient of walls with pin fins; (b) heat transfer coefficient of bottom wall with ribs.

4. Results 4.1. Overall heat transfer Fig. 5 shows the overall thermal performances characterized by the overall Nusselt number (Eq. (5)) for four models at the same Reynolds number (Re = 15,000). Although the four models have the same porosity, they exhibit different heat transfer behavior. The overall Nusselt number of the Model 4 is superior to other models, approximately 23.9% and 35.4% higher heat removal capability than that of the cylindrical pin-fin models (Models 1 and 3). The models with two rows of tabulators (Models 1 and 2) exhibit similar thermal performance. However, the models with three rows of turbulators (Models 3 and 4) show significant difference. The following sections present an investigation and discussion on the reasons for the thermal performance of each model as it relates to flow patterns and local heat transfer mechanisms. 4.2. Fluid flow and local heat transfer characteristics 4.2.1. Analysis of bulk flow mechanisms To further understand the local fluid behavior, Fig. 6 presents three-dimensional streamlines for all of the models. It is noticed that in all of the models, the fluid crosses the inlet and then turns towards the outlets, which leads to low-momentum recirculation at the tip region. Particularly, the recirculation flow of Model 4

exhibits less vortices than that of other models. In order to recognize the local flow features clearly, the enlarged views in Fig. 6 describe the secondary flows around the turbulators. As shown in the enlarged view of Model 1, the direction of the fluid is changed by the blockage of the upstream of ribs, in addition, there are more recirculating flows at the downstream of ribs. It can be concluded that the ribs evidently promote the fluid mixing and redistribute the fluid. As illustrated in the enlarged views of Models 3 and 4, fluid moves along the direction of the y-axis and the velocity of the fluid decreases gradually. It may be inferred that the formation of recirculation zones is attributed to the fact that less fluid flows into the tip region and the adverse pressure gradients push the lowmomentum fluid to become vortices. Specifically, as the fluid flows around the pin fins of Model 3, more fluid is oriented toward the outlets, which results in fluid with low velocity magnitude entering the tip regions as shown in the enlarged view of Model 3. According to the enlarged view of Model 4, the crossed ligaments of the Kagome lattice make more fluid move forward along the streamwise direction and flow into the downstream channel, therefore, causing the fluid at the tip to have more momentum to resist the adverse pressure gradient, which leads to less vortices generating at the tip region. For Models 1 and 2, the size of recirculation zones are similar sized due to the identical effect of the ribs at the tip regions. In order to describe the corresponding relationship between pressure distribution and flow characteristics of all the models, Fig. 7 presents the surface streamlines and pressure distributions projected onto the mid-planes (Pd is the local pressure minus 101,000 Pa). The location of the adverse pressure gradient can be determined from the pressure distribution of Model 1 in Fig. 7. In conjunction with the three-dimensional velocity streamlines in Fig. 6, it is clearly seen that that the adverse pressure gradient along the x-direction in Model 1 is caused by the local high pressure generated by high momentum fluid hitting the back wall. At the same position in Model 2, the high momentum fluid is obstructed by the Kagome lattice ligaments which separates the fluid into other directions, eliminating the local high pressure. Under the same configurations, the models with Kagome lattices can effectively resist the adverse pressure gradient. 4.2.2. Local fluid characteristics In order to describe the internal flow characteristics of all models, Fig. 8(a) and (b) present the surface streamlines projected onto the mid-planes and upper planes, respectively. From Fig. 8(a), it is

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Fig. 6. Overall and local flow patterns (colored by velocity magnitude) around ribs and pin fins as characterized by streamlines for all models at Re = 15,000.

Fig. 7. Surface streamlines and pressure distributions (Pd is the local pressure minus 101,000 Pa) on mid-planes for all models at Re = 15,000.

obvious that the overall fluid features of Models 1 and 2 are similar on mid-planes; however, Model 2 with the Kagome lattices has the capability of splitting more fluid at the tip region which compresses the recirculation region. In Models 3 and 4, the pin fins and Kagome lattices replace partial ribs in the inner regions of the channels. It is evident that the first array of pin fins/Kagome lattices, closest to the side wall, divides the high flow rate fluid gradually along the direction of the y-axis. In particular, the Kagome lattices closest to the side wall separate more fluid from

the primary fluid into to the region nearby as demonstrated in Fig. 8(a). The surface streamlines projected onto the upper planes (Fig. 8(b)) show similar flow characteristics as the streamlines projected onto the mid-planes. Showing in both the surface streamlines on the mid-plane and upper plane, Model 4 with three arrays of Kagome lattices has a better flow distribution compared to the other three models. As a result, relative to pin fins, the Kagome lattices are more capable of reducing the recirculation flows downstream of the inlet

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Fig. 8. Surface streamlines and velocity distributions on (a) mid-planes and (b) upper planes for all models at Re = 15,000.

channel and facilitating the uniform distribution of flow in the L-shaped channel. This is due to the major disturbances that the six ligaments (shown in enlarged view of Model 4 in Fig. 6) of the Kagome lattices provide. The volume of the pin fin is identical to that of the Kagome lattice, while the Kagome lattice with crossed ligaments could induce more fluid flowing along the ydirection compared with the pin fin, as shown in enlarged views of Model 3 and Model 4 in Fig. 6. Therefore, the tip region of the Model 4 has more high momentum fluids compared with the Model 3. Furthermore, due to the sharp corner, vortices are generated near the inlet for all models. By comparison, Model 1 has the largest vortex structure at the tip region and Model 4 exhibits the smallest one; therefore, the L-shaped channel in Model 4 provides more opportunity for high-momentum fluid to enhance heat transfer compared with other models. It can be seen from the streamlines of Model 3, that as soon as the primary fluid enters the inlet region, the fluid flows around the pin-fin array closest to the side wall and then moves towards the spanwise direction. It is clear that the fluid in Model 3 tends to primarily exit from the sixth, seventh, and eighth outlets from the inlet (Fig. 8), which leads to low-momentum fluid at the tip region of Model 3. The most uniform distribution of fluid is seen

in Model 4, which is observed via the homogeneous outlet flow fields. In terms of the recirculation flows in the tip regions nearby the side walls, Fig. 9 presents the local fluid flow behaviors in the vicinity of the side wall for all models to demonstrate the different effect of the ribs and cores. First, the staggered ribs of all models produce constant vortices to improve turbulence, which is illustrated in the enlarged view of Model 1 in Fig. 6. Particularly, the same arrangement of ribs are responsible for the little difference in flow patterns between Models 1 and 2. For Models 3 and 4, the pin fins and the Kagome lattices break up the development of vortices along the streamwise direction. Further, directing attention to Model 4, the vortices formed downstream of each Kagome lattice intersection, also observed in [26], feed fluid into the top and bottom wall which gives rise to the increase of velocity gradient near the wall. Finally, the tip region of Model 4 has less lowmomentum vortices compared with other models. Relatively, for Model 3, the size of the recirculation flow near the side wall is larger and the velocity magnitude is lower. Fig. 10 shows the local turbulent kinetic energy in order to evaluate the mixing features of the flow. The turbulators, including ribs, pin fins, Kagome lattice and horseshoe pin fins, promote the mixing

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Fig. 9. Surface streamlines and velocity contours on vertical cross sections at the middle positions of ribs for all models at Re = 15,000.

Fig. 10. Local turbulent kinetic energy distributions on five vertical cross-sections of x/W = 0.013, 0.25, 0.47, 0.67, 0.9 for all models at Re = 15,000.

of fluid which results in high turbulent kinetic energy. The region with high turbulent kinetic energy is consistent with the region with high velocity as shown in Fig. 8. From Model 1 to Model 4, the turbulent kinetic energy of the mainstream fluid decreases

gradually. Moreover, due to Kagome lattices, the turbulent kinetic energy of Model 2 outperforms that of Model 1. In comparison with other models, Model 4 exhibits relatively strong fluid mixing in the tip region which corresponds to strong flow structures.

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4.2.3. Local heat transfer Figs. 11 and 12 have been prepared to summarize the temperature distributions both qualitatively and quantitatively. Fig. 11 shows contour diagrams of the local temperature on the heated outside walls. Fig. 12 shows plots of spanwise-averaged temperature profiles along the y-direction for the top walls and for the side walls. It can be seen from Fig. 11 that the temperature distribution is sensitive to the vortices and the value of velocity, since the higher temperature regions are near the tip regions. Compared to the other models, the contour diagram for Model 4 shows relatively low temperatures due to the increased flow velocity and fluid mixing at the tip region as shown in Figs. 6–10. For Model 3, there are some hot spots downstream the channel near the side wall. The reason for this is because the pin fins in the ribs section stall the fluid in this region (see enlarged view of Model 3 in Fig. 6). As shown in Fig. 12, the average temperature increases gradually along the y-direction as the cooling capabilities of the fluid decrease due to the decreasing temperature difference between the fluid and the hot walls. In particular, when y < 0.14 m for the top walls and y < 0.12 m for the side walls, the temperature differences among all models are negligible. Due to the similar flow characteristics seen in Models 1 and 2, the temperature distributions of the top wall for Model 1 is nearly identical to that of Model 2. Further, the average temperatures of the downstream top wall on Model 3 is the largest among the four models as a result of the heat transfer deterioration induced by low-momentum fluid distribution in the tip region. Average temperatures of the top wall and side wall for Model 4 are lower along the y direction, which is attributed to the absence of a large recirculation zone as shown in Fig. 6. Furthermore, the high velocity and turbulent kinetic energy of the fluid near the side wall and top wall for Model 4 reduces the thermal and flow boundary layers. Therefore, the fluid can more effectively transfer heat on the Kagome lattice surfaces and surrounding walls, which is responsible for the lower temperature at the tip region compared with the other models. Local Nusselt numbers are also used to assess the influence of the arrangement of the turbulators on thermal performance. The local Nusselt number based on the local heat flux, the wall temperature and the bulk mean fluid temperature on the wall of all channels is

Fig. 12. Profiles of spanwise averaged temperature along y direction: (a) top wall; (b) side wall at Re = 15,000.

hðyÞ ¼

hðyÞDH NuðyÞ ¼ kf

ð9Þ

where the local heat transfer coefficient (h(y)), is defined as

97

q00 T w ðyÞ  Tf ðyÞ

ð10Þ

where Tw (y) and Tf (y) are the wall temperature and the local bulk mean fluid temperature, respectively. The bulk mean fluid temperature as a function of y is defined as

Fig. 11. Temperature contours of the heated walls for all models at Re = 15,000.

98

T f ðyÞ ¼ Tin þ

B. Shen et al. / International Journal of Heat and Mass Transfer 141 (2019) 88–101

yq00

qDH Uin cp

ð11Þ

where cp is the specific heat of air. Local Nusselt number contours of all models are presented in Fig. 13. It can be seen from the figure that the areas with high Nusselt numbers are concentrated mainly in the lower half of the channel due to the high momentum of the fluid in this section. It can also be seen that the low-momentum recirculation zones near the tip region reduce heat transfer. The effects of the small recirculation zone at the tip for Model 4 can be seen in Fig. 13 as higher

Nusselt numbers in this region. Furthermore, high Nusselt numbers can be seen around the turbulators, due to the formation of high shear induced by high-momentum fluids around each turbulator. For quantitative analysis of the thermal performances of all models, the variation of spanwise averaged local Nusselt numbers along the y-direction on the top wall (including inner region, midspan region and outer region), and the side wall are plotted in Fig. 14. Fig. 14(a) presents the spanwise averaged Nusselt number on the inner regions of the top wall. Compared with Models 1 and 2, Models 3 and 4 provide about 38% higher heat removal capabil-

Fig. 13. Nusselt number contours of the heated walls for the considered models at Re = 15,000.

Fig. 14. Spanwise averaged Nusselt number value along inlet flow direction (y-direction): (a) inner region of top wall; (b) mid-span region of top wall; (c) outer region of top wall; (d) side wall at Re = 15,000.

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ities at the upstream of the channel due to improved fluid mixing by the first arrays of pin fins and Kagome lattices in the inner region. Meanwhile, the Nusselt number of the inner region in Model 4 is about 75% higher than that of Models 1 and 3 and 44% higher than that of Model 2 at downstream of the channel. This is attributed to the fact that Model 4 has less of a recirculation zone at the tip region relative to other models (Fig. 8). Fig. 14(b) displays the thermal performances of the mid-span region of the top wall. For y > 0.1 m, the spanwise averaged Nusselt number of Model 4 is about 26% lower than that of Models 1 and 2 because more fluid is divided into the inner region leading to the lowermomentum fluid in the mid-span region. Fig. 14(c) shows the heat dissipation behavior of the outer region of the top wall. It can be seen that the main differences in heat transfer is mainly located in the downstream channels where the recirculation zone is located. The thermal performance of Model 4 is slightly lower than that of the other models in this area due to the lower turbulent kinetic energy in this region. Although Model 4 has lower kinetic energy in outer region as shown in Fig. 10, the heat transfer difference of the outer region of Model 4 compared with the other models is within 3.6–7%. This is attributed to the fact that the narrow space in the outer region reduces the difference in kinetic energy of the fluid for different model. Similar to the temperature distribution, as described in Fig. 14 (d), the region of 0.1 m < y < 0.19 m of side wall for the Model 4 has about 20–309% higher cooling performance than that of the other models due to the high fluid velocity and turbulent kinetic energy. This also suggests that the turbulator configuration in Model 4 is more advantageous to heat transfer enhancement compared to the other models. In general, the superior thermal performance of the inner region and side wall (as shown in Fig. 13 for Model 4) causes better overall thermal performance, as displayed in Fig. 5. This demonstrates that the heat transfer performance of the tip region and side wall play a vital role in the overall heat transfer enhancement for wedgedshaped channels. However, Models 1 and 2 have the same arrangement of ribs in the inner region; therefore, the tip region of both models have similar fluidic and thermal characteristics, as shown in Figs. 5–14, leading to similar overall thermal performances. Moreover, the first array of pin fins of Model 3 causes less flow in the tip region (as shown in Figs. 8 and 9) compared with Model 1, thus the heat transfer of Model 1 outperforms that of Model 3. All the results suggest that turbulators of different geometries and arrangements play an important role in thermal performances. Fig. 15 presents a comparison of the area-averaged local Nusselt number for the four models in this study at Re = 15,000. The area-

averaged local Nusselt number of Model 4 is superior to other models, approximately 4.9% and 2.8% higher than that of the cylindrical pin-fin models (Models 1 and 3) and 5.3% higher than that of the Kagome lattice models (Model 2). These facts indicate that the area-averaged local heat transfer coefficient of Model 4 is the highest. In conjunction with the analysis of Fig. 5, the reason that Model 4 shows the best cooling performance is due to the increase in heat transfer area and area-averaged local heat transfer coefficient. Although the area-averaged local Nusselt number of Model 3 is larger than that of Model 1, the minimum heat transfer area of Model 3 results in the smallest overall Nusselt number among the four models. Similar overall Nusselt numbers are seen for Model 1 because Model 2 has a bigger heat transfer area, but a lower area-averaged local heat transfer coefficient compared with that of Model 1. The heat transfer area and the area-averaged local heat transfer coefficient play important roles in the heat transfer process.

Fig. 15. Comparison of area-averaged local Nusselt number for the four models in this study at Re = 15,000.

Fig. 16. Overall Nusselt number for the four models at Reynolds numbers of 5,000– 25,000.

4.2.4. Pressure drop The pressure drop and friction factor for each model are presented in Table 3. The four models provide similar friction factors at a given Reynolds number and porosity. This suggests that the configurations and arrangement of turbulators considered in this study have very little influence on the pressure drop. This is expected since the majority of the pressure drop is due to the channel shape and ribs, while the small pin fins and Kagome lattices contribute less to the overall pressure drop. This information can inspire further investigations on replacing traditional pin fins with lightweight and strengthened structures that can provide higher heat transfer enhancement with minimal pressure drop penalties. 4.3. Reynolds number effects To investigate the relationship between heat transfer characteristic and Reynolds number, Fig. 16 has been prepared to show the comparison of the overall Nusselt number for each of the four Table 3 Comparison of pressure drop and friction factor for each model (Re = 15,000). Model #

DP

f

1 2 3 4

115 Pa 116 Pa 108 Pa 113 Pa

0.31679 0.31954 0.2975 0.31128

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models. As expected, with increasing Reynolds number, the Nusselt numbers of four models ascend gradually. At a given Reynolds number within the range (5000–25,000), the heat transfer capability of Model 4 is 9%–28% superior to that of Model 1, and Model 4 exhibits 16%–71% more heat removal compared with Model 3. The results indicate that the heat transfer advantage of Model 4 decreases gradually with increasing Reynolds number. Furthermore, the thermal performances of Models 1 and 2 are nearly identical within the considered range of Reynolds number.

Acknowledgements

5. Conclusions

References

In this study, Kagome lattices are inserted into a wedge-shaped channel to further enhance the thermal performance of the trailing edge of a turbine blade. The cooling performance of the L-shaped channel with different turbulators, such as pin fins and Kagome lattices, are investigated systematically by numerical simulations. For a fixed Reynolds number and porosity, the effects of the geometry and configuration of the core structures on the heat transfer enhancement are compared comprehensively. Conclusions drawn in this study are summarized as follows.

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(1) For the given porosity and Reynolds number within the range of 5000–25,000, the overall Nusselt number of the model with three arrays of Kagome lattices is about 9%– 28%, 6%–25%, and 16%–71% higher than that of the model with two arrays of pin fins, the model with two arrays of Kagome lattices and the model three arrays of pin fins, respectively. The model with three arrays of pin fins provides inferior thermal performance. The model with two arrays of pin fins and the model with two arrays of Kagome lattices have almost the same heat removal capability. (2) Under identical situations, all models exhibited similar average temperatures and Nusselt number distributions as y < 0.1 m. The Nusselt number of the inner region in Model 4 is about 75% higher than those in Models 1 and 3, and 44% higher than that in Model 2 downstream of the channel. Within the region of 0.1 m < y < 0.19 m, the side wall of Model 4 has about 20–309% superior cooling performances to other models due to the reduction of the recirculation flows at tip region. (3) For a given Reynolds number and porosity, all models present similar overall fluid flow patterns in the L-shaped channel. Model 1 with two arrays of pin fins and Model 2 with two arrays of Kagome lattices have similar recirculation flows at the tip region due to the same rib structures, which shows that the core turbulator configurations have little effect on the mainstream flow due to the large porosity. (4) The tip region of Model 4 with three arrays of Kagome lattices has the smallest recirculation as a result of better shunting effects of the Kagome lattices, which feeds high velocity fluid to the tip region and improves thermal performance. (5) All four models in this study produce similar pressure loss at the same Reynolds number and porosity. (6) The scope of the present study is to determine the effect of Kagome lattices on the heat transfer rate of the trailing edge of a counter-vane, first in stationary position. The rotational effect on the trailing edge of the channel will be examined in further investigations.

Declaration of Competing Interest The authors declared that there is no conflict of interest.

This research was supported by the National Natural Science Foundation of China (51676163 and 51806176), the National 111 Project (B18041), the Natural Science Basic Research Plan in Shaanxi Province of China (2018JQ5159), the Fundamental Research Funds for the Central Universities (3102018zy004), the Fundamental Research Fund of Shenzhen City of China (JCYJ20170306155153048).

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