Multiconfiguration Shape Optimization of Internal ...

Bingxu Wang Engineering Simulation and Aerospace Computing (ESAC), Northwestern Polytechnical University, P.O. Box 552, Xi’an, Shaanxi 710072, China

Weihong Zhang1 Engineering Simulation and Aerospace Computing (ESAC), Northwestern Polytechnical University, P.O. Box 552, Xi’an, Shaanxi 710072, China e-mail: [email protected]

Gongnan Xie Engineering Simulation and Aerospace Computing (ESAC), Northwestern Polytechnical University, P.O. Box 552, Xi’an, Shaanxi 710072, China e-mail: [email protected]

Yingjie Xu Engineering Simulation and Aerospace Computing (ESAC), Northwestern Polytechnical University, P.O. Box 552, Xi’an, Shaanxi 710072, China

Manyu Xiao Engineering Simulation and Aerospace Computing (ESAC), Northwestern Polytechnical University, P.O. Box 552, Xi’an, Shaanxi 710072, China

1

Multiconfiguration Shape Optimization of Internal Cooling Systems of a Turbine Guide Vane Based on Thermomechanical and Conjugate Heat Transfer Analysis This study concerns optimization of shapes, locations, and dimensions of internal cooling passages within a turbine vane under severe environments. The basic aim is to achieve a design that minimizes the average temperature and ensures the structural strength. Considering the prohibitive computational cost of 3D models, numerical optimization process is performed based on 2D cross-sectional models with available experimental temperature data as boundary conditions of thermomechanical analysis. To model the cooling channels, three kinds of shape configurations, i.e., circle, superellipse, and near-surface holes, are taken into account and compared. Optimization results of 2D models are obtained by using a globally convergent method of moving asymptotes (GCMMA). Furthermore, full conjugate heat transfer (CHT) analyses are made to obtain temperature distributions of 3D models extruded from 2D ones by means of shear stress transport (SST) k-x turbulence model. It is shown that optimization of cooling passages effectively improves the thermomechanical performances of turbine vanes in comparison with those of initial C3X vane. The maximum temperature of optimized vane could be reduced up to 50 K without degrading mechanical strength. [DOI: 10.1115/1.4029852] Keywords: shape optimization, thermomechanical analysis, conjugate heat transfer, globally convergent method of moving asymptotes, internal cooling passage, superellipse, turbine vane

Introduction

Turbine inlet temperature recognized as an important factor and pursued objective dominates the thermal efficiency and thrustweight ratio of aircraft engines. However, the desire to increase inlet temperature would bring forward many crucial demands to the design of hot engine components. Bunker has provided some basic evaluation criteria concerning heat-resistant effect and mechanical strength [1]. For these criteria, temperature and stress should be taken into account simultaneously. Apart from the development of high temperature resistant materials, an effective way is to introduce an associated cooling system. In 1983, NASA first made an experiment [2] of turbine vane with internal cooling passages and provided exhaustive data of heat transfer test results. A literature survey indicates that CHT analysis plays an important role in simulating temperature distributions for the heat management. Bohn and Heuer [3] and Facchini et al. [4] applied CHT analysis to study a nozzle vane and numerical results were validated experimentally. CHT analysis was also used to study the behavior of transitional boundary layer. Takahashi et al. [5] used CHT analysis to predict the temperature distribution of a cooled blade with rib-roughened passages.

1 Corresponding author. Manuscript received March 27, 2014; final manuscript received August 1, 2014; published online March 17, 2015. Assoc. Editor: Giulio Lorenzini.

Journal of Heat Transfer

Finite element method (FEM) provides an effective tool in stress analysis of a cooling system as the temperature distribution is known. To reduce computational cost, this method was used by Chmielniak et al. [6] to deal with the heat transfer problem of a blade. In this paper, temperatures and stresses are predicted by using ANSYS and FLUENT, respectively. In general, optimization of a cooling system is a challenging issue involving strong fluid–structure interactions and strict restrictions of temperature and stress distributions. Numerical methods make it possible to effectively optimize cooling systems. Different problems concerning flow, thermal and structural criteria of convective cooling passages were studied in Refs. [7–10] for gas turbine blade/vane optimizations. Muller et al. [11] attempted to minimize the coolant supply and keep the cooling performance by finding proper locations and the number of cooling rows. Nowak et al. [12–15] explored optimization problems of cooling passages of different shapes and indicated that the combination of 3D model with CHT analysis is the research trend. Besides, Mangani et al. [16] designed film cooling within the C3X vane and the comparison is made between numerical results and experimental ones. Bohn et al. [17] investigated film cooling effects of different hole shapes. Favaretto and Funazaki [18] optimized impingement cooling system by means of pins and airdischarging holes. As an important part in cooling passage optimization, some efforts were devoted to explore better heat transfer effects in changing the geometrical shape of passage cross sections or their locations. For instance, Talya et al. [19] described

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the passage cross section using a set of Bezier–Bernstein polynomials, Nowak and Nowak [14] modeled each cooling passage with four Bezier curves which joined together to make a closed contour, although circular configuration is still the main research object. Dennis et al. [20] located a set of column cooling channels near the blade surface instead of the traditional arrangement. Furthermore, Haasenritter and Weigand [21] equipped the cooling passages with rib structures to increase turbulence and flow mixture. Kim and Lee [22] tried to use V-shape ribs for the enhancement of turbulent heat transfer effect. For these optimization problems, shape, location, and number optimizations of cooling passages [7–15,23–25] in convection cooling are conducted by using GA (genetic algorithm). In this paper, superellipse is first introduced to model the cooling passage, and two other configurations, i.e., circular configuration and near-surface hole configuration, are also discussed. Optimization is based on 2D thermomechanical analysis to achieve the maximum drop of average temperature subjected to thermal stress constraint. Once the optimized solution has been obtained from the 2D vane model, 3D models are extruded correspondingly and then confirmed by a full CHT analysis. As is known, GA was used in many existing optimization studies, however it is usually cost-ineffective. To remedy this, a gradientbased optimization algorithm named GCMMA [26] is implemented here.

2 Optimization Model of a Turbine Vane Cooling System 2.1 Optimization Model. The considered optimization problem could be expressed as the following mathematical programming model: 8 find X ¼ ðx1 ; x2 ;    xn Þ > > > > < min Taver ðXÞ (1) > s:t: rj ðXÞ  r; j ¼ 1; 2;    ; m > > > : xi  xi  xi ; i ¼ 1; 2;    ; n

     k  X @g xk  k  ui  xki c¼g x  @xi þ;i  k   X  2 @g x   xki xki  lki þ @xi ;i

(5)

Superscript k represents the kth iteration, uik and lik represent the upper and lower moving asymptotes of xi, respectively. qk stands for an internal global convergence control parameter. By choosing a proper value of qk, the convergence of the above optimization problem could be guaranteed. Sensitivity analysis can be carried out to obtain the partial derivative @g(xk)/@xi involved in Eqs. (3)–(5) by means of semianalytical or finite difference method. The latter corresponds to       @g Xk g xki þ Dxi  g xki  ði ¼ 1; 2;    nÞ (6) Dxi @xki Dxi means the perturbation of xik. For a problem of n design variables, this scheme requires n þ 1 times FEAs. 2.3 Description of Original C3X Vane Model. C3X is the well-known vane widely investigated by NASA. It is often considered as the benchmark with available experiment data in the open literature [2]. The C3X vane has a constant cross section which can be modeled by pieces of curve segments through polynomial fitting of feature points. Meanwhile, the 3D model can easily be extruded from cross section (Fig. 1).

3 Multiconfiguration Shape Representation and Thermomechanical Analysis 3.1 Superellipse Description. Recently, various novel geometrical shapes have been brought forward into blade cooling system optimization. In this study, superellipse is applied as the basic geometry to describe the passage cross section

X stands for the vector of design variables characterizing geometric shapes of cooling channels, locations, and dimensions. The objective function is to minimize the average temperature as much as possible. The constraints concern the thermal stress r being limited by the allowable value r. n and m are numbers of design variables and constraints, respectively. 2.2 Optimization Algorithm. The adopted GCMMA algorithm is a first-order algorithm, which is efficient in solving nonmonotonic function optimization problem with a fast convergence. In general, as temperature and stress could not be expressed explicitly in terms of design variables, approximations should be used. For a function g(X), it could be approximated in the following way [26]:

gð XÞ  c þ

X þ;i

X qi pi  ðui  xi Þ ;i ðxi  li Þ

(2)

with  pi ¼

       k  @g xk @g xk qk  k k k 2 þ ui  li ; if >0 u  li @xi @xi 2 i

(3)

    k   k  @g x @g xk qk  k k k 2 þ li  xi ; if < 0 (4) u  li qi ¼  @xi @xi 2 i

Fig. 1

C3X guide vane

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x m yn       þ  ¼ 1 a b

(7)

In the extreme case of m ¼ n ¼ 2 and a ¼ b, the geometry becomes a circle. Assume m ¼ n in this study. Clearly, the geometry almost degenerates into a rectangle as n tends to be infinite. 3.2 Multiconfiguration Setup and Related Design Variables. Suppose that internal cooling system within a turbine vane comprises ten cooling passages that are all straight for coolant supply. Passages have to keep a certain distance without overlap and be away from the border of the vane. Corresponding criteria are listed in Table 1 [15]. In this section, three configurations are taken into account. Case 1: Circular Configuration. Circular configuration is widely used in cooling system of turbine vane. The C3X vane is a typical example of this kind. Suppose circular passage i has three design variables: xi, yi (location coordinate), and di (diameter). The total number of design variables is 30. Case 2: Superellipse Configuration. Each passage cross section is defined as a superellipse. Six parameters including xi, yi, ai, bi, ni, and hi are needed to define the geometry. Notice that hi is the rotation angle between the symmetrical axis and global coordinate system. Therefore, the total number of design variables is 60. Case 3: Near-Surface Hole Configuration. Dennis et al. [20] made such a design optimization where a large number of coolant passages were located near the outer surface of turbine blade. In our work, suppose 19 cooling holes are distributed inside the annular region near the surface with a constant distance of only 2 mm. Two cases are considered.

3.4 Thermomechanical Analysis. Generally, thermomechanical analysis is separated into two parts: temperature field computing and thermal stress computing. And thermomechanical stress analysis can thus be represented by Eq. (11). rij ¼ Eijkl ekl  bij  ðT  T0 Þ

(9)

where r is stress tensor, e is strain tensor, E is elasticity tensor, and b is thermal elasticity tensor. T0 represents initial temperature distribution within a structure. Due to the fact that the hot flow around the vane is steady and approximately constant and uniform, fixed thermal boundary condition which is based on the experimental data is applied. In this case, experimental temperature data from literature [2] is regarded as the constant thermal loads. For cooling passages, continuous coolants are provided to transfer heat from surfaces. Considering that the locations and diameters of passages are changing during the optimization, their heat transfer coefficients differ from each other. To deal with the problem, there are three assumptions for coolant. • • •

Assume the flow phenomenon is steady and its thermal physical property is constant. Assume mass flow rate of each passage is uniform and constant. Assume temperature of each passage is uniform and constant.

The convective heat transfer coefficient is estimated on the basis of Nusselt number

(a) Circular holes are uniquely used with design variables being locations of holes while hole diameters are all fixed to the same value. (b) Circles and superellipses are used simultaneously. Design variables then include locations of holes and longitudinal dimensions of the superellipse while the diameters of circular holes and the width dimensions of superellipse are all kept to constants. 3.3 Design Domain and Coordinate Transformation. As illustrated in Fig. 2, the cooling holes are separated into several regular boxes. To limit the shape variation of each cooling passage inside each box, optimization is carried out by imposing constraints on the center coordinates and diameters of all holes. To do this, transformation is made from global coordinates to local coordinates   0    0      cos h  sin h x x0 x x x0 þ  0 ) ¼TþR 0 ¼ sin h cos h y0 y y0 y y (8) where x and y are global coordinates and x’ and y’ are local coordinates. T and R represent the translation matrix and the rotation matrix, respectively. As a result, the upper and lower bounds of design variables could be specified as the constant values that are suitable to the GCMMA algorithm.

Table 1 Basic domain restriction [15] Parameters Min radius Max radius Min distance between passages Min distance of passage from border

Journal of Heat Transfer

Values (mm) 0.8 6 2 1

Fig. 2 Coordinate transformation from global coordinate to local coordinate

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h¼

kNu d

(10)

where k is thermal conductivity, d is hydraulic diameter, and Nu is defined in experiential way according to the forced convection Nusselt number of tube [27] Nu ¼ 0:023 Re0:8 Pr 0:4

(11)

where Pr and Re stand for Prandtl and Reynolds number, respectively, and Re can be described by Eq. (14) Re ¼

qvd l

(12)

where q and l are the density and viscosity of the coolant, respectively. v is the speed of fluid. Based on the combined equations of (12)–(14), the heat transfer coefficient can be calculated as follows: h ¼ 0:0242  Cðk; M; lÞd1:8  0:8 M Cðk; M; lÞ ¼ k  l

(13) (14)

occurs. In detail, the average temperature declines from 504 K to 485 K, and the maximum thermal stress also drops from 325.58 Mpa to 299.082 Mpa. Case 2: The optimized superellipse configuration is shown in Fig. 3(c). The passage cross sections near the tail end become narrowed and elongated. The high temperature region thus diminishes drastically compared to the initial one. In addition, first four passages are staggered to increase the cooling effect as much as possible. As a result, average temperature decreases to 468 K and thermal stress has a significant improvement of dropping to 297 Mpa. Case 3(a): The optimized circle configuration is illustrated in Fig. 3(d). Average temperature drops by 35 K compared to Fig. 3(a) while thermal stress increases. Case 3(b): As shown in Fig. 3(e), suppose two circular holes are initially defined at both sides of vane head where the curvature of the vane is comparatively large and other holes are superellipses. Here, the locations of cooling holes change slightly after optimization. From the results, it can be found that the temperature improvement is similar to that in Case 3(a). However, the maximum thermal stress reaches 349 Mpa. Above results indicate that the temperature decrease is obvious because the cooling passages effectively take away a lot of surface heat, protect from local high temperature and thermal failure. In conclusion: •

C stands for the coefficient related to each specific cooling passage. It is constant for each passage due to the fact that the coolant mass flow rates and the fluid property are assumed to be constant. As a result, the heat transfer coefficient h just varies in terms of passage diameter. The mass flow rate M and temperature of the coolant Tin are listed in Table 2. Based on loading and boundary conditions, Fig. 3(a) shows initial temperature and thermal stress distributions on C3X midspan. Units of temperature and stress are K and Mpa, respectively.

• • •

The cooling passages at vane head are much closer to the border. The last passage approaches the end of vane tail as much as possible. The optimized solutions in case 1 and case 2 result in a decrease of thermal stress about 25 Mpa. The maximum thermal stress obtained in case 3 is larger than the initial one, since the temperature gradient grows when the cooling flow is located near the hot flow.

5 Numerical Simulation of 3D Extruded Models of C3X Vane 4

Optimization Tests

The whole optimization procedure is based on the optimization platform of Boss-Quattro [28] and ANSYS [29]. FEM analysis process including geometry setup, mesh generation, thermomechanical analysis, postprocessing, and sensitivity calculation is implemented by APDL (Ansys Parametric Design Language) script. The optimization process of updating variables, iteration control, and the GCMMA algorithm are integrated into BossQuattro optimization platform. When analysis completed, the calculation results and sensitivities are exported into Boss-Quattro, and the variable values change according to GCMMA algorithm, then the analysis of next iteration will be carried out until the optimization procedure tends to convergency. Here, three configurations are optimized and discussed. Case 1: Optimized configuration is shown in Fig. 3(b). Diameters of most passages increase and the last cooling hole moves toward the end of the vane tail where the highest temperature Table 2 Cooling channel dimensions and coolant mass flow rates Num

d/mm

Mass flow rate /kg/s

Tin/K

1 2 3 4 5 6 7 8 9 10

6.3 6.3 6.3 6.3 6.3 6.3 6.3 3.1 3.1 1.98

0.0222 0.0221 0.0218 0.0228 0.0225 0.0225 0.0216 0.00744 0.00477 0.00256

342 344 335 336 330 355 336 350 377 387

To demonstrate the benefice of different optimized configurations, 3D extruded models of C3X vane are correspondingly realized. Temperature distributions are evaluated by CHT analyses using FLUENT [29] and compared with experiment data of initial 3D C3X vane. 5.1 Properties of C3X Vane. In our computing model, periodic condition is imposed on vane side surfaces to simulate physical arrangement. As shown in Fig. 4, the whole computing domain of C3X vane is divided into three subdomains: hot flow, vane, and coolant. In hot flow domain, the top and bottom planes are adiabatic. Periodic side surfaces are spaced apart by 117.73 mm. The vane material is ASTM 310 stainless steel [30] with constant density of qs ¼ 8030 kg/m3 and specific heat capacity of Cs ¼ 502 J/kg/K. Its thermal conductivity varies in terms of temperature in the following way: Ks ¼ 0:0115T þ 9:9105 ðW=m=KÞ

(15)

For the hot gas and coolant, their viscosity, thermal conductivity and specific heat capacity are functions of temperature. Viscosity l and thermal conductivity Kf are defined by Sutherland formula  3=2 T T0 þ S T0 TþS  3=2 T T0 þ S Kf ðT Þ ¼ k0 T0 TþS lðT Þ ¼ l0

(16) (17)

where l0 ¼ 1.7894  105 Pa  s, T0 ¼ 273.11 K, k0 ¼ 0.0261 W/(m  K), and S ¼ 110.56.

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Fig. 3

Temperature and stress contours for different cases (up: temperature distribution down: thermal stress distribution) Table 3 Flow conditions in simulations Pi /Pa

Ti/K

Po/Pa

Tu/%

Mo

413,286

818

254,172

8.3

0.89

Fig. 4 C3X vane work condition consisting of three computing domains: hot flow, coolant, and vane

Fig. 6 Predicted and measured curves of different turbulence models at the vane midspan

The specific heat capacity Cf is approximated by polynomial fitting Cf ¼ a0 þ a1 T þ a2 T 2 þ a3 T 3 þ a4 T 4

Fig. 5 Computing mesh of entire domains with local refinement

Journal of Heat Transfer

(18)

5.2 Mesh and Boundary Conditions. The entire domain is discretized by a structured mesh to ensure the mesh quality and computing precision. O-type grids are located along the internal and external walls of cooling channels and vane, as shown in JUNE 2015, Vol. 137 / 061004-5

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each cooling passage are listed in Table 2. Such flow conditions are estimated based on temperature distribution of vane and the flow is assumed to be fully developed. The selection of turbulence models also has an important influence on the simulation precision. Inappropriate models might make large divergences. For exploring the right one, four turbulence models, i.e., standard k-e, k–kl–x, standard k-x and SST kx, are tested to fit the real work conditions. Standard k-e model is the most popular model applied in industry. Standard k-x model predicts low Reynolds number, compressibility and shear flow transmission more precisely, and SST k-x model is beneficial in improving the computing accuracy.

Fig. 7 Contours of velocity magnitude on the midspan plane

Fig. 5. The mesh has 50 layers in vertical direction and contains in total 2.1  106 cells, among which about 1.4  106 cells are for the hot flow domain, 0.3  106 cells for vane, and 0.4  106 cells for ten cooling passages. The arithmetic average volume of elements is 2.55 mm3 and no negative volume exists. The elements whose qualities are between 0.9 and 1 take up 89.32% and all the qualities are larger than 0.5, therefore the elements satisfy the calculation requirement and accuracy. The flow condition corresponds to “run 157” taken from Ref. [2]. Total pressure Pi and total temperature Ti are specified at inlet. Static pressure Po and match number Mo are specified at outlet. According to the experimental data, the turbulence intensity is evaluated as Tu, and viscosity ratio at inlet is specified to be 30. The specific values of hot flow boundary conditions are listed in Table 3. The setup for mass flow rates and inlet temperature of

5.3 Simulation Results. Through CHT computing, the distributions of temperature and pressure on vane midspan are compared with the original experimental data to find the proper turbulence model. Figure 6 shows the predicted and measured distribution curves of pressure and temperature. x/l is the axial chord representing the profile of 2D midspan, t/811 and p/p* denote the ratios of evaluation items t and p to their initial values (811 is the inlet temperature of main flow and p* is the static pressure at inlet). Meanwhile, the solid circle represents experimental data. Star, square, triangle, and sphere represent computing results related to SST k-x, standard k-x, k–kl–x, and k-e models. Comparatively, the k–kl–x model results in the temperature fluctuating largely at x/l ¼ 0.4 or so. The k-e model seems to have the most overestimation over the experimental temperature value and the maximum temperature error reaches about 9% at x/l ¼ 0.2. As for SST k-x and standard k-x models, well-fitted temperature results are obtained by both of them. The maximum error is about 3% which is small enough to be ignored. Besides, Fig. 6 shows that obtained pressure distributions fit well experiment data except for the range of x/l ¼ 0.1–0.5. This is illustrated in Fig. 7 where the shock (red region) occurs at vane head inflection point on suction side. Due to the high local speed, pressure has a sharp decline and Reynolds number changes at the same time. Numerical simulation seems to be difficult in estimating the pressure distribution, though the whole trend of results is consistent with the experiments. According to the above comparison, it is found that SST k-x model much fits the real physical condition. This assessment is similar to the recent study in Ref. [31]. Thus, the SST k-x model is used in the present CHT analyses.

Fig. 8 Predicted temperature contours for different cases (up: midspan, down: 3D)

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Table 4 Maximum configurations Case Temperature drop (K)

Fig. 9 Predicted and measured temperature curves at the vane midspan of optimized configurations

Fig. 10 Stress representations of different optimized cases by ANSYS and FLUENT

Figure 8 shows temperature distributions obtained by using in different optimized cases. These results are comparable with those obtained by ANSYS shown in Fig. 3. In detail, the comparison is shown along the circumference of midspan in Fig. 9. FLUENT

temperature

drops

for

different

Case 1

Case 2

Case 3(a)

Case 3(b)

46.4

46.4

50.9

50.2

The dotted line represents temperatures related to initial C3X vane. It is obvious that the maximum temperature reduction at the vane tail end is up to about 10% after optimization in all cases. In other places, the temperature distributions change slightly in case 1 and case 2 after optimization. However, the surface temperature becomes much lower than the initial one in case 3 because cooling holes are rearranged close to the surfaces, especially case 3(a) fluctuates more considerably. Since stress cannot be evaluated by FLUENT, temperature variation is selected as an indirect measure of thermal stress. As illustrated in Fig. 10, both ANSYS related to 2D models and FLUENT related to 3D models yield similar trends. In this sense, it is appropriate to use temperature variation in the representation of thermal stress. Temperature variations in case 1 and case 2 are small in comparison with that of the original configuration while case 3 results in large temperature variations. In Fig. 11, maximum and average temperatures obtained by FLUENT (noted as Max Temperature and Aver Temperature) and average temperature by ANSYS (noted as Opt Objective, optimization objective function in Sec. 2) of different cases are recorded. It is obvious that the general tendencies of two sets of average temperatures are different, because their boundary conditions in analyses are different. However, the trends of terms Opt Objective (ANSYS) and Max Temperature (FLUENT) are similar, so the mode using average temperature under fixed boundary condition as objective function in 2D optimization is effective to evaluate the maximum temperature which is the main optimization objective in 3D real work condition. The temperature improvement data are listed in Table 4. Maximum temperatures of optimized configurations are dropped by 48.5 K averagely in CHT analyses. Therefore, the optimization turns out to be effective.

6

Remarks and Conclusions

Three configurations of cooling passages, i.e., circle, superellipse, and near-surface hole, are optimized. Thermomechanical optimizations are performed to reduce high temperature and guarantee the structure strength simultaneously. The GCMMA algorithm is employed to search optimized solutions of 2D vane models. Full 3D CHT analyses are then carried out for verification. Four turbulence models are investigated based on the NASA C3X vane experiment. It is shown that the SST k-x model provides the most reasonable predictions and is then adopted in the full CHT analyses. Numerical results of CHT analyses show that the 2D optimized models and the 3D extruded models are globally in good consistence. Thus, the present model simplification is effective for optimization procedure and the methodology could reflect the true conditions to some extent. The maximum temperature of the vane can approximately be reduced by 50 K after thermomechanical optimizations. In cases of circle and superellipse configurations, stress and temperature intensities can be improved simultaneously. In case of the nearsurface configuration, the temperature can greatly be reduced while the stress intensity intends to increase.

Acknowledgment Fig. 11 Temperature variations of different optimized cases by ANSYS and FLUENT

Journal of Heat Transfer

The author appreciates funding from National Natural Science Foundation of China No. (51275424, 11202164) and NPU Graduate Research Fellowship. JUNE 2015, Vol. 137 / 061004-7

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Nomenclature ai, bi ¼ Cf ¼ Cs ¼ d¼ h¼ Kf ¼ Ks ¼ l¼ n¼ Nu ¼ p¼ p* ¼ Pr ¼ R¼ Re ¼ t¼ T¼ xik ¼ x, y ¼ x’, y’ ¼ xi , y i ¼

semimajor axis and semiminor axis of the ith superellipse specific heat capacity of fluid specific heat capacity of stainless steel hydraulic diameter convective heat transfer coefficient thermal conductivity of air thermal conductivity of stainless steel axial chord length number Nusselt number pressure distribution at midspan of vane static pressure at inlet Prandtl number rotation matrix Reynolds number temperature distribution at midspan of vane translation matrix the ith variable in the kth iteration global coordinates local coordinates center coordinates of the ith cooling hole

Greek Symbols h ¼ rotation angle between local coordinate and global coordinate k ¼ thermal conductivity l ¼ fluid viscosity  ¼ speed of fluid q ¼ density of air qs ¼ density of stainless steel qk ¼ global convergence control parameter in the kth iteration

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