Middle East Technical University

Aerospace Engineering&

Metallurgical and Materials Engineering

Jet Engine Design Competition-Teknofest-2020

Design of a Nozzle Guide Vane Cooling System

Team Name:

Ka-Fa-1500

Team ID:

T3-12873-154

Team Leader:

Olcay Nurtac Deniz

Team Members:

Cansu Yıldırım

Muhammed Mustafa Balcılar

Academic Advisor:

Prof. Dr. Yusuf Ozyoruk

Contents

List of Figures iii

List of Tables v

1 Introduction 11.1 Geometry of the Stator Blades . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Design Requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Flow Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Importance of the Turbine Blade Cooling 4

3 External Flow Boundary Conditions of the Blade 53.1 Overall Temperature Distribution Factor (OTDF) . . . . . . . . . . . . . . . 53.2 External Flow Mach Number Calculation . . . . . . . . . . . . . . . . . . . . 53.3 Heat Transfer Coefficient with Flat Plate Assumption . . . . . . . . . . . . . 73.4 Heat Transfer Coefficient with CFD . . . . . . . . . . . . . . . . . . . . . . . 9

3.4.1 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.4.2 Mesh and Boundary Conditions . . . . . . . . . . . . . . . . . . . . . 103.4.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

4 Cooling Techniques and Decision Matrix 144.1 Internal Cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

4.1.1 Internal Flow Holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.1.2 Convection Cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.1.3 Rib-Turbulated Convection Cooling . . . . . . . . . . . . . . . . . . . 16

4.2 Impingement Cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.3 Pin-Fin Cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.4 External - Film Cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

4.4.1 Transpiration Cooling . . . . . . . . . . . . . . . . . . . . . . . . . . 204.5 Decision Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

5 Cooled Blade Geometry 32

6 Computational Fluid Dynamics Analysis (CFD) 356.1 Flow Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356.2 Mesh Operations and Boundary Conditions . . . . . . . . . . . . . . . . . . 35

6.2.1 Mesh Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356.2.2 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

6.3 Fluent Solver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396.3.1 Turbulence Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396.3.2 Other Solver Parameters . . . . . . . . . . . . . . . . . . . . . . . . . 40

i

7 One-Dimensional Internal Flow 417.1 Comparison of Hand Calculations with CFD . . . . . . . . . . . . . . . . . . 467.2 Back Flow Margin (BFM) Calculations . . . . . . . . . . . . . . . . . . . . . 48

8 Heat Transfer Results 508.1 Hand Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 508.2 CFD Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 628.3 Cooling Effectiveness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

9 Cold Flow Test Results 68

10 Life-Cycle Calculations 69

11 Conclusion 73

References 74

Appendix 77

ii

List of Figures

1 3D Geometry of the Blade . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 Meridional Plane View of the Blade, Dimensions in [mm] . . . . . . . . . . . 23 Increasing Turbine Inlet Temperature Over the Years [26] . . . . . . . . . . . 44 Improvement of Cooling Techniques Over the Years [15] . . . . . . . . . . . . 45 Combustor Exit Temperature Profile – OTDF [32] . . . . . . . . . . . . . . . 56 Annulus Area of the Turbine Inlet . . . . . . . . . . . . . . . . . . . . . . . . 67 External Flow Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 External Flow Domain of the Real Blade Profile . . . . . . . . . . . . . . . 109 Mesh of the Flow Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1110 Velocity Contour of the Flow Domain . . . . . . . . . . . . . . . . . . . . . . 1211 Heat Transfer Coefficient Distribution Over the 2-D Blade Surface-Top View 1312 Heat Transfer Coefficient Distribution Over the 2-D Blade Surface-Isometric

View . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1313 Effect of Aspect-Ratio on the Nusselt Number [15] . . . . . . . . . . . . . . . 1514 Convection Cooling System [17] . . . . . . . . . . . . . . . . . . . . . . . . . 1515 Serpentine Passages on a Turbine Blade [3] . . . . . . . . . . . . . . . . . . . 1616 Rib-Turbulated Cooling Schematic [15] . . . . . . . . . . . . . . . . . . . . . 1717 Effects of Ribbed Walls and Rib Spacing on the Nusselt Number [15] . . . . 1818 Angled Ribs and Created Swirl [15] . . . . . . . . . . . . . . . . . . . . . . . 1819 Impingement Cooling at the Leading Edge [26] . . . . . . . . . . . . . . . . . 1920 Different ejection hole configurations [15] . . . . . . . . . . . . . . . . . . . . 2021 Control Volume for Mass ans Energy Balances, [16] . . . . . . . . . . . . . . 2222 Schematic of the Film Cooling, [15] . . . . . . . . . . . . . . . . . . . . . . . 2323 Geometric Parameters of the Film Cooling, [7] . . . . . . . . . . . . . . . . . 2524 Three different blowing ratio and local and mean film effectiveness, [24] . . . 2725 Three different blowing ratio and local and mean Nusselt Number, [25] . . . 2826 Three different blowing ratio and local and heat loading, [25] . . . . . . . . . 2827 Effect of the blowing ratio on the Nusselt Number distributions for air and

CO2, [29] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2928 Effect of the film hole location on the average film effectiveness, [18] . . . . . 3029 Isometric View of the Cooled Blade . . . . . . . . . . . . . . . . . . . . . . . 3230 Backward View of the Cooled Blade . . . . . . . . . . . . . . . . . . . . . . . 3231 Areas of the Coolant-1 and Coolant-2 Holes . . . . . . . . . . . . . . . . . . 3332 Areas of the TE Exits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3333 Dimensions of the Film Holes . . . . . . . . . . . . . . . . . . . . . . . . . . 3434 Passage Shape Flow Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . 3535 Location of the Data Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3636 Mesh Convergence Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3637 Mesh of the both fluid domain and solid body . . . . . . . . . . . . . . . . . 3738 Grids on the Solid Blade . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3739 Inflation Layers on the Walls . . . . . . . . . . . . . . . . . . . . . . . . . . . 3840 Names of the Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . 3841 Effect of Turbulence Model on the Analysis . . . . . . . . . . . . . . . . . . . 39

iii

42 Subdivision in the chord direction . . . . . . . . . . . . . . . . . . . . . . . . 4143 Basic Physical Model of the Internal Flow . . . . . . . . . . . . . . . . . . . 4244 Thermal circuit model of the chord-wise heat transfer, 1D model . . . . . . 4245 Discharge Coefficient vs velocity head for different orientation of the holes, [22] 4446 Flow network and discharging masses for channel 1 & 2 . . . . . . . . . . . . 4547 Flow Properties at the Coolant-1 . . . . . . . . . . . . . . . . . . . . . . . . 4648 Flow Properties at the Coolant-2 . . . . . . . . . . . . . . . . . . . . . . . . 4749 Model Simplification of the Blade Cooling Design, [2] . . . . . . . . . . . . . 5050 Separate parts of the blade . . . . . . . . . . . . . . . . . . . . . . . . . . . 5151 Flat plate model, [27] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5352 Trailing-edge Model, [15] . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5353 Mach Distribution on the plate . . . . . . . . . . . . . . . . . . . . . . . . . 5454 1st Part of the Blades both top view & 3-D graph . . . . . . . . . . . . . . . 5555 2st Part of the Blades both top view & 3-D graph . . . . . . . . . . . . . . . 5656 3rd Part of the Blades both top view & 3-D graph . . . . . . . . . . . . . . . 5757 4th Part of the Blades both top view & 3-D graph . . . . . . . . . . . . . . . 5858 5th Part of the Blades both top view & 3-D graph . . . . . . . . . . . . . . . 5959 6th Part of the Blades both top view & 3-D graph . . . . . . . . . . . . . . . 6060 7th Part of the Blades both top view & 3-D graph . . . . . . . . . . . . . . . 6161 Temperature Distribution on the Pressure Side . . . . . . . . . . . . . . . . . 6362 Temperature Distribution on the Suction Side . . . . . . . . . . . . . . . . . 6363 Temperature Distribution from the Top View . . . . . . . . . . . . . . . . . 6464 Temperature Distribution from the Bottom View . . . . . . . . . . . . . . . 6465 Temperature Distribution on the 20% of the Blade from Root . . . . . . . . 6566 Temperature Distribution on the 40% of the Blade from Root . . . . . . . . 6567 Temperature Distribution on the 60% of the Blade from Root . . . . . . . . 6668 Temperature Distribution on the 80% of the Blade from Root . . . . . . . . 6669 Streamlines at the Suction side of the Blade . . . . . . . . . . . . . . . . . . 6770 Larson-Miller Parameter Curves for CM 247 LC DS . . . . . . . . . . . . . . 6971 Larson-Miller Parameters for CM 247 LC DS at a stress level of 111 MPa . . 71

iv

List of Tables

1 Design Requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 Flow Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 Calculated Boundary Conditions at the External Flow . . . . . . . . . . . . 74 Different Coolant Systems and Properties . . . . . . . . . . . . . . . . . . . . 315 Properties of Coolant Holes . . . . . . . . . . . . . . . . . . . . . . . . . . . 336 Areas of the Trailing Edge Exit Slots . . . . . . . . . . . . . . . . . . . . . . 347 Number of Elements With Respect to the Mesh Quality . . . . . . . . . . . . 368 Values for Inlet Boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399 Fluid and Solid Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 4010 Area, Perimeter and Hydraulic Diameter of the channels . . . . . . . . . . . 4111 CD for the film holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4412 Back Flow Margin Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4913 Metal Temperature Results and Coolant Effectiveness . . . . . . . . . . . . . 6714 Cold Flow Test Results from CFD . . . . . . . . . . . . . . . . . . . . . . . . 6815 Obtained Results After Calculations . . . . . . . . . . . . . . . . . . . . . . 73

v

Abstract

In this project, a cooling system is designed and analyzed for the first stage sta-tor blade of the high pressure turbine for the Jet Motor Design Competition of theTEKNOFEST-2020. To design the cooling system a literature survey is done and dif-ferent heat transfer methods are used for analysis at that report. Different coolingmethods (film, impingement, pin-fin, etc.) are investigated and based on their ad-vantages/disadvantages, blade geometry is finalized as conventional internal coolingcombined with film cooling. The geometry is then analyzed with empirical methodsand simulation tools. The result of these analyses achieve the requirements given inthe technical specification, within a reasonable tolerance. The finalized geometry hasa temperature distribution ranging from 1021 to 1495 K, with a mean of 1231 K. Thegeometry is predicted to have a life cycle of approximately 6900 hours.

1 Introduction

In this project, a coolant system is designed for the given geometry which is presentedbelow. First of all, the importance of the coolant systems are mentioned. It is well knownthe metals are able to endure in a limited range of temperatures, and to increase the engineperformance that limit must be exceed with increasing the turbine inlet temperature. Thusturbine blades must be cooled to withstand higher temperatures. In the design procedure,firstly, external flow boundary conditions and heat transfer coefficients are determined. Withthese calculated coefficients, the required internal cooling system is designed. For that, ascheme is created according to the literature survey. Then, with channel scheme internal flowparameters are calculated. These parameters also guide the coolant system design. Thenefficiency of the coolant system is calculated. In all calculations and analysis, radiationeffects are neglected. Finally, life-cycle analysis of the blade is done according to the givenLarson-Miller parameter.

1.1 Geometry of the Stator Blades

Isometric and meridional plane views of the blades are showed on the Figure 1 and 2.That blade is a nozzle guide vane or first stage stator blade of a high pressure turbine. Inthe calculation and design procedure that geometry is used.

1

Figure 1: 3D Geometry of the Blade

Figure 2: Meridional Plane View of the Blade, Dimensions in [mm]

1.2 Design Requirements

For this design, the requirements which are listed on the Table 1, will be used. The require-ments are given by the competition organizers, namely T3 Foundation and Turkish EngineIndustries. The design of the coolant system is done according to that given parameters.

2

Table 1: Design Requirements

1.3 Flow Conditions

For the design and the analysis of the cooling system, flow conditions will be used givenon the Table 2. With these values external flow boundary conditions are determined. Alsosame methodology is used for the calculation of the internal flow parameters.

Table 2: Flow Conditions

3

2 Importance of the Turbine Blade Cooling

In the aircraft industry the speed of the aircraft is extremely important. To achieve thehigh number of speeds, a few ways can be followed. One of these ways is the increasingthe thrust which is produced by engines. To increase the engine thrust, the turbine inlettemperature has to be increased. However, turbine inlet temperature has a limit due to somematerial and structural limitations. Proper material selection allows a significant amountof increase in turbine inlet temperature but that is not sufficient for increasing thrust, thusadditional cooling systems have to be used to get higher turbine inlet temperatures. FromFigures 3 and 4 historical improvement of the turbine inlet temperature and the coolingsystems can be seen.

Figure 3: Increasing Turbine Inlet Temperature Over the Years [26]

Figure 4: Improvement of Cooling Techniques Over the Years [15]

4

3 External Flow Boundary Conditions of the Blade

Some of the important parameters which are used in the calculation of the flow, is givenby the Table 2. These are; total temperature, total pressure, mass flow rate and OTDF. Thevalues are 1600[K], 1180[kPa], 2.72[kg/s] and 0.2 respectively.

3.1 Overall Temperature Distribution Factor (OTDF)

After the combustion chamber, the temperature distribution occurs like in the Figure 5.

Figure 5: Combustor Exit Temperature Profile – OTDF [32]

And the given temperature at the Table 2 is the total average turbine inlet temperature.To find the maximum value of the turbine inlet temperature the equation 1 is taken fromreference [32].

OTDF =Tmax − TavgTavg − T3

(1)

Where the T3 is the combustor inlet total temperature. So, from given parameters, thetotal temperature of the coolant is equal to 700[K]. In addition to that, the coolant flowis sucked from the compressor. So, the combustor inlet temperature is assumed to be thesame as the coolant temperature. Then T3 equal to 700 [K]. Also Tavg and OTDF values areknown from Table 2. As a result, maximum turbine inlet total temperature is equal to;

Tmax = 1780[K]

3.2 External Flow Mach Number Calculation

From the OTDF value the turbine inlet temperature is calculated. According to the inlettemperature and given total pressure value, the total density is found as ρt = 2.3094[kg/m3]from reference [1]. With using the isentropic relations and the conservation of mass, theequation 9 is obtained. Isentropic relations and conservation of mass relation are representedbelow step by step.

5

ρ = ρt ·(

1 +γ − 1

2M2

)( −1γ−1

)

(2)

T = Tt ·(

1 +γ − 1

2M2

)(−1)

(3)

P = Pt ·(

1 +γ − 1

2M2

)( −γγ−1

)

(4)

M =v√γRT

(5)

m = ρvA (6)

v = m/ρA (7)

M =v√γRTt

·√

1 +γ − 1

2M2 (8)

M =m

Aa·

(1 + γ−1

2M2

)( 1γ−1

)

ρt· 1√

γRTt·√

1 +γ − 1

2M2 (9)

The specific heat ratio of the air at the turbine inlet is assumed to be γ = 1.3. In addition,gas constant of the air is taken as R = 287[J/kgK]. Also, mass flow rate is given on theTable 2 as m = 2.72[kg/s]. The only unknown in the equation 9 is the annulus area of theturbine inlet Aa. The front view of the turbine entrance is presented on the Figure 6.

Figure 6: Annulus Area of the Turbine Inlet

6

To find the annulus area equation 10 is used. Where the R2 is equal to 98.98 [mm] andthe R1 is equal to 76 [mm]. From that equation, annulus area is found as Aa = 0.01263[m2].

Aa = π(R22 −R2

1) (10)

At the end, all parameters are substituted in equation 9, and the turbine inlet machnumber calculated as;

M = 0.1153

Then, from isentropic relations which are the equations 2, 3, 5 below values of the floware calculated as;

ρ = 2.2941[kg/m3]

T = 1776.5[K]

v = 93.868[m/s]

Also static pressure of the air is again found from isentropic relations which is presentedon the equation 11. (

P

Pt

)=

(T

Tt

)( γγ−1

)

(11)

P = 1169.9[kPa]

Table 3: Calculated Boundary Conditions at the External Flow

Parameter ValueMach 0.1153

v [m/s] 93.868Aa [m2] 0.01263ρt [kg/m3] 2.3094ρ [kg/m3] 2.2941Tt [K] 1780T [K] 1776.5Pt [kPa] 1180P [kPa] 1169.9

3.3 Heat Transfer Coefficient with Flat Plate Assumption

In the previous part mach number of the external flow is calculated. With using machnumber formula external flow velocity is calculated as 93.868 [m/s]. Before introducing theflat plate solution, firstly flow domain is presented at the Figure 7.

7

Figure 7: External Flow Domain

As it is known, the nozzle guide vanes which is the cooled blade in this project, turnsflow through the flow axis, hence their position on the turbine assumed like Figure 7. Also,nozzle guide vanes create a nozzle-like passage, hence, a domain given above is created. Theangle of the blades are taken from the given geometry as nearly 49-50 degree from the flowaxis in clockwise sense. In addition to that the distance between the blades is calculated atthe mid span of the blade. From Figure 6, the annulus diameter is found at the mid spanand with simple geometric calculations, distance is calculated as 27.1716 [mm].

The average convection coefficient for flat plate assumption, assuming the flow is turbulent,can be calculated by using following relation [10]:

NuL = 0.037(ReL)0.8(Pr)1/3 (12)

where:

NuL is the average Nusselt number over the entire length of the flat plate NuL = h ·L/k

h is the average convection coefficient over the entire length of the flat plate [W/m2K]

k is the thermal conductivity of air [W/mK]

L is the length of the plate [m]

ReL is the average Reynolds number over the entire length of the flat plateReL = ρ · V · L/µ

ρ is the density of air [kg/m3]

V is the velocity of air [m/s]

µ is the dynamic viscosity of air [kg/ms]

8

Pr is the Prandtl number Pr = Cp · µ/k

Cp is the specific heat capacity of air at constant pressure [J/kgK]

The total temperature, pressure, and, density for air are taken as 1780K and 1180 kPa, and,2.3094 kg/m3 respectively, and isentropic relations given in equations 2, 3, and 4 are usedto calculate the static properties of air. The static properties of air are:

T = 1776.4 K

P = 1169.8 kPa

ρ = 2.2942 kg/m3

The velocity of air is found by using equations 5 and 9. Eq 9 is iteratively solved tofind the Mach number of the air. The configuration given in Figure 7 forces the flow in anarrower section, causing an increase in velocity. Since the Mach numbers in both freestreamand above and below the blade are below 1, the flow can be assumed as incompressible andhence as the cross-sectional area is constricted, the velocity is increased, given by the relation:

V1A1 = V2A2

Other relevant properties of air are found by using equations of state and ideal gas approx-imations. The chord length is assumed to be the entire flat plate length. A MATLAB® codeis given in the Appendix section that is used or the calculation of the parameters mentionedabove and the dimensionless numbers for the external flow. The convection coefficient forthe external flow with flat plate assumptions is found as:

h = 3171 W/m2K

In addition to solving simple external flow flat plate model, different correlations are alsosolved for cross flows, such as Zukauskas and Churchill–Bernstein. The convection coefficientsare the same order of magnitude and the values deviate about 10%.

3.4 Heat Transfer Coefficient with CFD

In the previous part, the heat transfer calculation is done with flat plate assumption.However, the blade has an airfoil shape which is very dominant on the flow physics. So dueto airfoil shape, the shape of passage become a nozzle shape, hence flow accelerates on thepassage. In addition to that, as we know heat transfer coefficient is related with the flowvelocity. So, to get a more accurate result, a computational fluid dynamics (CFD) analysisof the 2-D shape blade is done to fin the heat transfer coefficient. The airfoil profile is takenat the mid-span of the given blade geometry. Same fluid domain is created as Figure 7 andshowed on the Figure 8.

9

Figure 8: External Flow Domain of the Real Blade Profile

3.4.1 Geometry

CFD solution of the domain is obtained with ANSYS-Fluent. The airfoil shape of theblade is taken from the mid-span of the given blade geometry, and its copied to the top andthe bottom of the blade. With this way a flow passage is obtained. The distance betweenthe blades are calculated again from the mid span of the blade and its value showed at theFigure 7.

3.4.2 Mesh and Boundary Conditions

In this design only 2-D flow analysis is done, hence the number of the mesh elementsare not too high and do not take to much time to solve, thus a mesh convergence studyis not done on that analysis. A finer mesh is used, and inflation layers are added to thetop-bottom walls and the surface of the blade which is showed at the Figure 9. Inflationlayers are crated with the first layer thickness method and 15 layers are obtained. In total,nearly 55000 element are created and the solution takes a few minutes.

10

Figure 9: Mesh of the Flow Domain

The left hand side of the domain is taken as inlet and right hand side as outlet, thus thefluid flows in the positive x direction. External flow parameters are calculated at the previoussections. Same values are used in the CFD calculation. Thus inlet of the domain is takenas pressure-far-field and the mach number and pressure is given as; 0.1153 and 1169.9[kPa]respectively. In addition to that the thermal solver is opened, and the temperature of the inletis taken as 1776.5[K]. Also, the flow is coming with high turbulence levels due to combustionchamber. As known, to get better burning efficiency, combustion chamber increases theturbulence level. Thus, at the inlet, the turbulence intensity is taken as 10 which is also thesuggestion of the competition committee. Also at the inlet flow is assumed as continuous.

The k-omega SST turbulence model is used in the turbulence calculations and also thesecond order scheme is used. The top and bottom blades are determined as wall and constantsurface temperature is defined as 1400 [K] which is the allowable maximum temperaturevalue. Also, the blade at the mid, is defined as wall and again constant temperature valueis taken as 1400 [K]. Other, side walls are kept as default values. In addition the outlet istaken as pressure outlet and the same pressure value is defined as inlet pressure value.

For fluid materials, air is used as an ideal gas. Its viscosity is calculated from Sutherlandlaw on the reference [1]. Also other values like density and thermal conductivity is calculatedat the inlet temperature 1776.5 [K]. For solid materials, nickel base superalloy CM247LC DS(equivalent: MAR-M-247) is used and its properties are taken from the reference [13]. Theproperties of the nickel are:

ρ = 8540[kg/m3]

Cp = 600[J/kgK]

k = 30[W/mK]

where:

11

ρ is the density of MAR-M-247

k is the thermal conductivity of MAR-M-247 [W/mK]

Cp is the specific heat capacity of MAR-M-247 at constant pressure [J/kgK]

3.4.3 Results

Figure 10: Velocity Contour of the Flow Domain

As mentioned earlier, the airfoil shape has an huge impact on the velocity distribution ofthe flow domain. So, Figure 10, shows that impact, at the quarter chord region of the blade,the flow velocity is achieved at nearly 250 [m/s]. So according to that velocity distribution,wall heat transfer coefficient obtained at the blade surface and results are showed on theFigure 11. Also average heat transfer coefficient is found as h = 3583.7[W/m2K]. In additionto that average heat transfer coefficient of the flat plate assumption is close enough to theCFD results.

As seen from the Figure 11, the maximum heat transfer coefficient is happening at thenearly leading edge region. The values of heat transfer coefficient is achieving to the h =16000[W/m2K] at leading edge. Because of too much heat transfer coefficient that locationgetting hotter. To cool that portion also too much heat transfer coefficient is needed atthe inside from basic convection equation. However, increasing the internal heat transfercoefficient too much is not possible, thus in that project other way is used which is thedecreasing the external heat transfer coefficient with applying film cooling.

12

Figure 11: Heat Transfer Coefficient Distribution Over the 2-D Blade Surface-Top View

Figure 12: Heat Transfer Coefficient Distribution Over the 2-D Blade Surface-Isometric View

13

4 Cooling Techniques and Decision Matrix

For cooling a turbine blade several methods can be applied, and they can be classifiedunder two major categories as internal cooling and external cooling [21].

• Internal Cooling

– Convection cooling

– Impingement cooling

– Pin-fin cooling

• External Cooling

– Transpiration cooling

– Local film cooling

At the previous section, external heat transfer coefficient is calculated with using theanalytical and computational methods in 2-D domain. Result shows that, heat transfercoefficient is too high at the leading edge of the blade. Thus, to cool down the blade eitheran internal channel can be designed or heat transfer coefficient at the external flow canbe decreased with using film cooling. To find the cooling system which has the maximumefficiency, in that section various coolant systems will be introduced from the literature andtheir advantages will be discussed. Then a few coolant systems will be considered, importantcriteria will be introduced and a Pro/Con table will be given to these systems on a decisionmatrix. As a result of the decision matrix, optimum cooling system will be chosen andanalyzed.

4.1 Internal Cooling

4.1.1 Internal Flow Holes

To increase the efficiency of the internal cooling, internal flow holes play an importantrole. Dimensions of these holes and the Reynolds number at the inlet of the hole, affect theheat transfer coefficient. A study explained on the reference [15], shows the effects of theshape of internal cooling holes on the Figure 13. As seen from the figure, narrow aspectratio (width/height) channels has the greater Nusselt number at different internal channelconditions. So, that means greater heat transfer coefficient. Thus narrow shape holes aresuggested.

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Figure 13: Effect of Aspect-Ratio on the Nusselt Number [15]

4.1.2 Convection Cooling

Convection heat transfer can be explained like that, convection is the energy transferbetween solid and a moving fluid on the solid [10]. Convection cooling method is one of theearliest method on the blade cooling because it is simplest way of the cooling [26]. In thiscooling system, heat is transferred from hot blade to the cold coolant flow which is showedon the Figure 14. Convection cooling is simple way, however it can not be efficient alone,thus some additional vortex generators can be applied inside of the channels. In addition tothat, shape of the channels are also important.

Figure 14: Convection Cooling System [17]

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Serpentine Cooling Passages: Coolant passages can be either separate or serpentinetype. Separate passages are opened at the different location of the blade at different shapes,and each passage need a coolant mass flow rate, so that increases the coolant mass flow ratewhich is not desirable. On the other hand, serpentine passages have a continues structure andonly one coolant hole is sufficient. However, there are some disadvantages on the serpentinepassages.

Figure 15: Serpentine Passages on a Turbine Blade [3]

As seen from the Figure 15, coolant flow turns 180o inside of the serpentine passages. Thiscauses decrease in the pressure of the coolant, in addition, air supply pressure is limitedand at low pressure values effectiveness of the coolant also decreases [26]. Another problemis that, on the serpentine passages, temperature of coolant air is changing continuously,thus density and other properties of the coolant will also change [21]. So, this again causesinefficient cooling through the end of the serpentine passages which is placed nearly at thetrailing edge region. Also, serpentine passages increases the complexity, thus it causes somemanufacturing problems.

4.1.3 Rib-Turbulated Convection Cooling

As mentioned on the above paragraphs, heat transfer coefficient of the coolant air shouldbe increased at the inside of the coolant holes, otherwise a higher temperature gradienthappens on the blade, so this causes higher thermal stresses. To increase the heat transfercoefficient at the inside of the coolant holes some additional structures like rib-turbulators,may be applied, because additional turbulence increases the heat transfer coefficient.

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In fact, rib-turbulators increase the heat transfer coefficient, however they causes a reduc-tion on the coolant flow pressure. Though the end of the coolant hole, pressure has lowestvalues. So, as mentioned on the previous paragraphs, low pressure values causes the lowefficiencies.

Another disadvantage of the rib-turbulators is the manufacturing problems. Ribs are thevery small structures compared to the whole blade. Thus, building them inside of a coolantholes requires additional manufacturing techniques, so this causes increase in the cost.

Figure 16: Rib-Turbulated Cooling Schematic [15]

Some factors affects the performance of the rib-turbulated cooling. These are; channelaspect ratio which is mentioned on the above paragraphs, rib configuration and Reynoldsnumber.

Effects of Rib Layouts: Several studies about rib turbulated cooling done by Han’s(1988)is explained on the reference [15]. Results of the studies shows that; increasing the Reynoldsnumber, decreases the Nusselt number, hence decreases the heat transfer coefficient.

In the Figure 16, spacing between the ribs is presented. So, smaller spacing between theribs have better performance than the greater spacing. Because, wider rib spacing createsthe thicker boundary layer. In addition to that, Nusselt number at the ribbed wall is nearlytwo times higher than Nusselt number of the the smooth wall. Both spacing and ribbed walleffect on the Nusselt number is presented on the Figure 17.

17

Other important parameter is the placement angles of the ribs on the walls, because ofthat, angled ribs are more effective than the orthogonal ribs to the coolant flow. Because,angled ribs creates additional swirl on the flow which increases the turbulence. Also, withchanging the orientation of the angle, created swirl can be directed to significant point.Angled ribs and swirls on the flow are showed on the Figure 18.

Figure 17: Effects of Ribbed Walls and Rib Spacing on the Nusselt Number [15]

Figure 18: Angled Ribs and Created Swirl [15]

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4.2 Impingement Cooling

Heat transfer coefficient should be increased at the inside of the coolant holes as mentionedon the above sections. Thus, impingement cooling provides highest possible convective heattransfer coefficient augmentation factors compared with all other modes of single-phase cool-ing [6]. Impingement jets focus only small portion of the region and it is effective only on thisregion. Also, impingement cooling, weakens the structural integrity, because of that usedonly regions which have very high thermal loads [15]. So, this region generally is the stagna-tion point of the blade at the leading edge where, highest external heat transfer coefficientis done.

Impingement cooling works like that, with using designed holes, coolant flow is acceleratedand the resulting jet directed normal to the target surface [23]. As seen from the Figure 19,impingement cooling requires an additional channel and holes. So this, increases the com-plexity of the coolant system. Also, manufacturing these small holes used on the impingingthe jets needs additional manufacturing techniques.

Figure 19: Impingement Cooling at the Leading Edge [26]

4.3 Pin-Fin Cooling

As mentioned previous paragraphs, internal of the blade can be cooled with using convec-tion or impinging jets. However, through the trailing edge of the blade, shape of the bladebecome more narrow. Thus, cooling techniques on this region are limited. In addition tothat, due to passage shape, through the trailing edge velocity of the flow increases thus heattransfer coefficient increases, then more heat transfers to blade. To satisfies enough amountof cooling on this region only convection can be used due to lack of place. However, simple

19

convection may not satisfies the enough amount of cooling, thus additional structures areneeded to increase the heat transfer area and the turbulence, like pin-fins [15].

Pin-fins can be oriented with two ways; inline and staggered. In addition to that, height-to-diameter ratio of the pin-fins affects the efficiency of the cooling system. According toreference [15], height-to-diameter ratio can vary between 1/2 to 4. Also, closer spaced arraysshow higher heat transfer coefficient [15].The flow in the array is also affected by the eachelements of the structure, thus first two rows affect the third row, so heat transfer coefficientis maximum at the third row [15]. In addition to the array configuration, Reynolds numberis also affects the heat transfer coefficient. With increasing the Reynolds number, Nusseltnumber increases, hence heat transfer coefficient also increases.

Generally, coolant air used on the pin-fin regions ejected to main flow with the help of ejec-tion holes. Design of the ejection holes affects both thermal and aerodynamic performanceof the blade. With different hole shapes, aerodynamic performance may increase however,this may cause decrease on the thermal performance. Thus, design of ejection holes is atrade-off study [31].

Figure 20: Different ejection hole configurations [15]

4.4 External - Film Cooling

4.4.1 Transpiration Cooling

Transpiration or full coverage film cooling is one of the coolant method for external cooling.Coolant flow is ejected to main flow with film holes or porous structure of the blade. Thismethod is relatively new and not well studied on the literature. Hence, there will not bemore information about transpiration cooling.

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Fundamentals of Film Cooling

Cooling of the turbine blades achieved by cooling from outside and inside. External coolinginvolves allowing the coolant from inside the blade to eject out onto the hot-gas path-sidesurface through discrete holes. This method is called film cooling [15].

The film cooling is explained more accurately, by Goldstein [16]. He examined externalcooling method with a name of “Secondary Flow” which are ablation, transpiration (or sweator mass transfer), and film cooling. Ablation cooling and transpiration cooling have seriousdisadvantages. For ablation, it is not generally renewable so it is short duration solutionsuch as reentering the vehicles to atmosphere. On the other hand, transpiration coolingis suffered from the low strength properties. It generally does not use at mechanical partdesign. In contrast to these, Goldstein [16] continues to explanation of the advantages of thefilm cooling:

“Although a secondary fluid is also added to the boundary layer in film cooling thereare considerable differences in operation and even in goals as compared with ablation andtranspiration cooling. A key difference is that film cooling is not primarily intended asprotection of the surface just at the location of coolant addition, but rather the protectionof the region downstream of the injection location. Film cooling is thus the introductionof a secondary fluid (coolant or injected fluid) at one or more discrete locations along asurface exposed to a high temperature environment to protect that surface not only in theimmediate region of injection but also in the downstream region.” Other advantage of thefilm cooling is reducing the radiation between hos gases environment and blades.

Theoretical Approaches

General Remarks about Theoretical Approaches: In this section, generally RichardJ. Goldstein’s Film Cooling article, [16] is taken as reference. Since; for turbine coolingturbulent boundary layer is analyzed, some of the theoretical calculation still uses empiricalformulas. This theories generally based on the simple heat sink model. Secondary flow whichis injected onto the blades are thinking like heat sinks. Furthermore, all of the analysis basedon some assumptions and this causes some advantage and disadvantages.

Two-Dimensional Incompressible Flow Film Cooling-Heat Sink Model: In thismethod, purpose is developing to kernel to predict the heat transfer and temperature distri-bition along the non-isothermal surfaces.The strength of the line heat source is determinedby the net entalphy flow of the secondary fluid which is ρ2U2Cp2(T2 − T∞).

Then, Tribus and Klein [30] found the efficiency, by assuming that air is injected main airstream.

η = 4.62Re20.2(x/Ms)

−0.8 (13)

21

where, M:blowing ratio and x:the distance x is measured downstream from the point ofinjection.

Advantages and Disadvantages: However, this method is calculate the higher efficiencydue to the assumption of the method which is injected gas does not affect the velocityboundary layer of the main stream. As injected gas and main stream flow mixed heat, heattransfer coefficient increases due to turbulence and this causes the decreasing the efficiency.In addition to this, this article has importance because of the applicability of the all of theheat source models and it is useful for the low blowing ratios.

Energy Balance in the Boundary Layer: This model based on the continuity equationand assumed main stream and secondary flow is well-mixed at the boundary layer.

m = m2 + m∞ =

∫ δ

0

ρUdy (14)

Then, by using the definition of the average temperature (T ) on the boundary layer andspecific heat constant (Cp), equation 14 is modified with assumption of that flow is adiabatic.In the figure 21, mass and energy balance control volumes are showed.

Figure 21: Control Volume for Mass ans Energy Balances, [16]

From, energy balance at the control volume,

(m2 + m∞)CpT = m2Cp2T2 + m∞Cp∞T∞ (15)

Rearrange term and by taken as mean temperature on the wall as an adiabatic temperature.Adiabatic wall efficiency (η) is obtained. And by using some turbulent boundary layer

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analytic relationship, η is:

η =1.9Pr1/3

1 + 0.329Cp∞Cp2

ξ0.8β(16)

where, ξ = (x/Ms)[(µ2/µ∞)Re2]−0.25 and β = 1 + 1.5× 10−4Re2(µ2W∞/µ∞W∞)sinα and

α is injection angle according to wall.

Advantages and Disadvantages: Since this method assumes that main flow and sec-ondary flow is well-mixed, it will be suffer at secondary flow injection point.

Injection Through Discrete Holes - Three Dimensional Film Cooling: There aresome studies on the this subject; however, there will be a lot of effect on the process.Therefore, there is little hope of getting a relatively exact analytic description of the velocityand temperature distributions. Since, analytic methods of 3D flow is not mentioned in here.

Experimental Approaches

In this section two methods which have been used to determine heat load at surface areintroduced.

Figure 22: Schematic of the Film Cooling, [15]

At the constant property fluids, temperature fields are independent from the velocity field.Therefore, heat flux at no film cooling condition is described by Newton’s Law of Cooling.

q′′0 = h0(Tg − Tw) (17)

where,h0:heat transfer coefficient of the fluid at no film coolingTg:oncoming gas temperatureTw:wall temperature

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When the fluid is injected through the surface,

q′′ = h(Tdatum − Tw) (18)

According to Goldstein [16], datum temperature is the reference temperature for the cal-culation. However, how is determined?

For subsonic flow is used as adiabatic wall temperature. However, this fluid driven tem-perature changes for hypersonic flow condition due to the difference of the hypersonic tem-perature boundary layer. At this location, adiabatic wall temperature is taken as Tf whichis the film temperature, more precisely, this is the temperature of the where hot combustionand coolant mixed. Also, in here new term is introduced which is adiabatic efficiency (η):

“The adiabatic wall temperature is not only a function of the geometry and the primaryand secondary flow fields but also the temperatures of the two gas streams. To eliminate thistemperature dependence a dimensionless adiabatic wall temperature, q, called the film cool-ing effectiveness is used. For low speed, constant property flow the film cooling effectivenessis given by [16] ”

η =Tg − TfTg − Tc

(19)

This adiabatic efficiency (η), is used for the understanding of the film cooling efficiencywhich is described by

q′′

q′′0=

h(Tf − Tw)

h0(Tg − Tw)=

h

h0

[1− η Tg − Tc

Tg − Tw

](20)

At described at the beginning of the section, two methods are used for understanding ofthe heat load which is described mathematically at the equation 20. This methods are calledas the superposition approach and adiabatic wall effectiveness approach. These are bothdescribed at the reference [15].

“In the superposition approach, actual surface temperatures and gas temperatures aremeasured to calculate the heat transfer. The effects of injection are all accounted for by thevalue of the heat-transfer coefficient. Experiments are run for cases where the surface hasno film cooling and cases where film cooling is introduced onto the surface. The ratio of gastemperature to coolant temperature (Tg/Tc) is similar to values that exist in the real engineconditions. The heat-transfer coefficient depends on the injection temperature as well ason the injection rate.” This method is explained in details by Moffat at ’The superpositionapproach to film-cooling’. In addition, again according to reference [15], the heat-transfercoefficient is based on the local wall temperature and injection rate. A single test will provideall the heat transfer information for this approach. Typically, the result is presented as aparameter called Stanton number reduction (SNR), defined as

SNR = 1− StfilmStno−film

(21)

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where, St:Stanton Number.

On the other hand, in the adiabatic wall effectiveness approach adiabatic wall tempera-ture is determined by in the experiment both function of the position and injection rate.Then, when adiabatic wall temperature is determined adiabatic wall effectiveness is definedautomatically. In this approach, heat-transfer coefficient is also reported at the other exper-iment. Although, there are two approaches according to reference [15], both approaches arewell established and are conceptually equivalent. Data can be converted from one approachto another without any corrections.

Effect of the Film Cooling on the Turbine Blade Region

As an introduction of this section, some importance parameter for turbine cooling is listed.These parameters are strongly affected cooling situation of the turbine blades.

Some Importance Parameters in the Turbine Blade Cooling: Below parametershas a importance on the turbine blade cooling and they strongly affects the efficiency of thecooling.

Figure 23: Geometric Parameters of the Film Cooling, [7]

• Flow Properties

– Velocity ratio (uc/u∞)

– Density ratio (ρc/ρ∞)

– Blowing Ratio (M = ρcucρ∞u∞

)

– Momentum Flux Ratio (I = ρcu2cρ∞u2∞

)

– Pressure Gradient (∂P∂x

)

– Main stream Reynolds Number (ReD)

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– Boundary Layer Thickness (δ)

– Main stream turbulence intensity (Tu)

– Coolant Reynolds number (Rec)

– Surface Roughness

– Compressibility

• Geometric Properties (shown in the figure 23)

– Hole inclination

– Hole spacing to diameter ratio (s/D)

– Hole shape

– Surface curvature

– Number and spacing of rows and hole arrangement

Effect of the Parameters on the Leading Edge

Blowing ratio (M) Blowing ratio is important parameter for efficiency (η). Mick andMarley is studied at this subject. They studied at three different blowing ratio from M =0.38 to M = 0.97. As seen in the figure 24, blowing ratio M = 0.38 gives highest filmcooling effectiveness (η). This is due to fact that, at highest blowing ratios main stream andsecondary flow boundary layer is mixed and this cause turbulence and turbulence increasethe heat transfer coefficient and causes decreasing at the effectiveness. Another interestingresult in here is that blowing ratio high coolant effect is more longer at downstream of theflow. This is clearly seen at figure 24, spanwise averaged effectiveness.

Free- Stream Turbulence (Tu) This effect is important for the film cooling. However,importance of the this subject is realized very late. Since main stream flow is comingfrom the combustion chamber, its turbulence intensity factor is very high. At this subjectMehendale and Han [25] are studied. Again this study’s experimental set-up has somesimilarity between Mick and Marley’s one. At high blowing ratios Nusselt Number (Nu)increases due to turbulence and this means that high heat transfer coefficient (h) at figure25. Therefore, a seen in the figure 26 heat loading increases.

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Figure 24: Three different blowing ratio and local and mean film effectiveness, [24]

27

Figure 25: Three different blowing ratio and local and mean Nusselt Number, [25]

Figure 26: Three different blowing ratio and local and heat loading, [25]

Unsteady Wake Funazaki et al [14]. is studied the Unsteady wake and turbulence (Tu)relationship and when Turbulence (Tu) intensity factor increases, unsteady wake effect de-creases.

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Density Ratio Density ratio is an important parameter when occurring of the considerabletemperature difference occur between main stream flow and secondary flow. Because at thiscondition attainment of the adiabatic wall temperature is very difficult and mass transferanalogous will be important at this cases. Ekkad et al. used the two different injection typewhich are air and CO2 and three different blowing ratios which are M = 0.4, M = 0.8 andM = 1.2. In the same blowing ratio, type of the coolant is not big effect. However, in thedifferent blowing ratio. As indicated before [24], for low blowing ratio (M = 0.4) film coolingeffectiveness is higher for air. However, this result true for CO2 at M = 0.8. This can beseen at the figure 27 clearly.

Figure 27: Effect of the blowing ratio on the Nusselt Number distributions for air andCO2, [29]

Film Hole Geometry This is another important factor for leading edge film cooling.Hole location, type and angle of injection has significant on the this parameter.

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Figure 28: Effect of the film hole location on the average film effectiveness, [18]

As seen in the figure 28, Karni and Goldstein [18] used also Mehendale and Han [25]results. For low blowing ratios two row injections give good result. However, for high blowingratios, two row injection shows some coincidence between one row injection. Also, 40 degreeinjection rate gives better result for all blowing ratios. These results can be explained withmass-transfer phenomena.

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4.5 Decision Matrix

Table 4: Different Coolant Systems and Properties

Coolant System PropertiesConventional Internal Cooling, Through Hole

Simplest geometry, easiest to manufacturePressure drop is the lowest but most inefficient in terms of cooling

Conventional Internal Cooling, Several HolesMore complex geometry but more efficient cooling compared to first case

Cooling still not adequate for severe conditionsDifficult to manufacture

Internal Cooling, Several Holes, with Film CoolingFilm cooling effect is present

Simpler to manufacture compared to latter casesEfficient cooling

Pressure drop is not severe

Internal Cooling, Several Holes, with Film CoolingComplex geometry

Difficult to manufactureDifficult to model empirically

More efficient coolingbut pressure drop is higher compared to previous cases

Internal Cooling, Several Holes, with Film CoolingComplex geometry

Difficult to manufactureDifficult to model empirically

Most efficient due to turbulence generated inside the ductsbut pressure drop is most severe

As the result of desicion matrix, blade design is chosen with two coolant holes and filmcooling at the leading edge.

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5 Cooled Blade Geometry

Coolant systems are designed with using CATIA CAD program on the given blade geom-etry. As decided on the decision matrix, two coolant holes are drilled from the root of theblade. Wall thickness is taken as 1 [mm], then holes are designed. Also, holes are drilledtill the 0.25 [mm] above of the tip of the blade. The coolant hole at the leading edge partis called as Coolant-1 and, the coolant hole at the mid section is called as the Coolant-2.In addition to that, two film holes are opened on the both suction and pressure side of theblade. The coolant gas is ejected on the trailing edge with using rectangular slots, which arecalled as Ex1-Ex4 from root to tip.

Figure 29: Isometric View of the Cooled Blade

Figure 30: Backward View of the Cooled Blade

Areas of the coolant holes are very important on the analysis. They are presented on theTable 5 and Figure 31.

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Table 5: Properties of Coolant Holes

Area (m2) Aspect Ratio (W/H)Coolant-1 2.461e-5 1.52Coolant-2 3.927e-5 4.63

Figure 31: Areas of the Coolant-1 and Coolant-2 Holes

Figure 32: Areas of the TE Exits

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Table 6: Areas of the Trailing Edge Exit Slots

Area (m2)Ex1 2.55e-6Ex2 1.82e-6Ex3 1.4e-6Ex4 1.02e-6

Area of the slots which are used to eject the coolant air at the trailing edge is showed onthe Figure 32 and Table 6.

For the external cooling two rows of film holes are drilled on the both pressure and thesuction side of the blade. There are 18 holes are on each rows in total 36 film holes are opened.The distance between on each hole is the half of the diameter of the blade. As, showed onthe Figure 33, diameter of the film holes are equal to 0.8 [mm]. On the decision matrix forthe coolant holes simple convection method is chosen. Thus, that diameter is the allowablelimit for this configuration, because more turbulence is needed to decrease the diameter. Inaddition to that, a jet engine can be worked on the different kind of environments like desertor dirty mediums. The coolant flow is coming from the compressor, so as explained on thereferences [12] and [5] some little particles or dust can be sucked by the compressor. Thenthis particles blocks the film holes. As a result, diameter of the film holes are chosen as 0.8[mm] and area of the holes is equal to 5e-7 [m2]. Also the distance between holes is taken as0.4 [mm].

In addition to that, film holes have the 30 degree angles between coolant flow direction.Also, the film holes at the suction side has nearly 75 degree angle between main flow.

Figure 33: Dimensions of the Film Holes

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6 Computational Fluid Dynamics Analysis (CFD)

6.1 Flow Domain

Internal flow and metal temperature analysis are done with using hand calculations. Inaddition to these calculations a CFD analysis is done. On this section, details of the analysiswill be explained and the results will be showed at the related sections. The cooled bladegeometry is explained at the above paragraphs. To make analysis more accurate, flow domainmust be a passage shape which is showed on the Figure 34. On the passage, upper and lowerblades are placed with a 27.172 [mm] distance from the blade which is showed on the Figure7. Also, to decrease the number of elements inlet and outlet of the domain is placed the 30[mm] front and the back of the blade.

Figure 34: Passage Shape Flow Domain

6.2 Mesh Operations and Boundary Conditions

On this analysis, both fluid and the solid domains are solved together. Also, geometry isa 3-D geometry, any dimension are very small. Thus, number of elements get to very highlevels. So, to find the optimum number of elements a mesh convergence study is done.

6.2.1 Mesh Convergence

To measure the effect of the mesh on the analysis, this process is done. For comparisonof the results a location is determined on the flow domain. Data obtained on a Line-1 atthe middle of the Coolant-1 hole which is showed on the Figure 35. The length of the line isnearly 20 [mm] and 140 samples obtained from this line. For comparison, pressure level ischosen on the Coolant-1 hole.

Three different mesh quality is determined and created. Number of elements are showedon the Table 7.

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Figure 35: Location of the Data Lines

Table 7: Number of Elements With Respect to the Mesh Quality

Mesh Number of Elements [millions]Coarse 1.21

Medium 1.63Fine 1.87

On the Figure 36, results of pressure values at the Line-1 is presented for three differentmesh. On the analysis K-Omega turbulence model is used. As seen from the figure, throughthe tip of the coolant hole, pressure values are nearly equaly to each other for finer andmedium quality meshes. On the other hand, coarse mesh is far from others. As a result,quality of the medium is nearly equal to finer mesh. To decreases the computational cost,medium mesh quality is chosen.

Figure 36: Mesh Convergence Results

36

With using medium mesh quality, below 3-D elements are crated on the both fluid domainand the solid body. In addition to that, to capture turbulence better inflation layers addedon the walls showed at the Figure 39, with using first layer thickness method. First layerthickness is taken as 1e-5 [m] and 10 layers are added with 1.2 growth rate.

Figure 37: Mesh of the both fluid domain and solid body

Figure 38: Grids on the Solid Blade

37

Figure 39: Inflation Layers on the Walls

6.2.2 Boundary Conditions

Names of the boundary conditions are presented on the Figure 40. On the analysis twodomain is used, these are the fluid and solid domains. Main stream inlet called as Inlet anddefined as pressure inlet. For pressure inlet boundary conditions, total pressure is given as1800 [kPa] and the static pressure as 1169.9 [kPa] and total temperature 1776.5 which arecalculated as the above sections. Outlet of the main flow is defined as the pressure outlet.Top and the bottom of the domain defined as symmetry. For more accurate analysis, 1400[K] temperature is defined for the blades which are placed on upper and the bottom ofthe domain and they called as Blade-Wall. Coolant inlets are called as Inlet-1 and Inlet-2.Again they are defined as pressure inlet. For the coolant holes mass flow rate defined as5.6e-3 [kg/s] for each holes. With using this value a procedure is applied which is explainedon the section 3, on the Inlet-1 and Inlet-2. Results are presented on the Table 8.

Figure 40: Names of the Boundary Conditions

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Table 8: Values for Inlet Boundaries

Name of the BC Total Pressure [kPa] Static Pressure [kPa] Total Temperature [K]Inlet 1180 1169.9 1776.5

Inlet-1 1200 1195.6 700Inlet-2 1200 1198 700

6.3 Fluent Solver

6.3.1 Turbulence Models

On the solver, energy equation is opened and different turbulence models are used. Toshow the effects of the turbulence model on the analysis, K-Omega, K-Epsilon and Spalart-Allmaras models are used. Results are again obtained on the Line-1 which is showed on theFigure 35. Again pressure data on the Line-1 is compared for the medium mesh quality.

Figure 41: Effect of Turbulence Model on the Analysis

As seen from the Figure 41, at the root of the coolant hole, K-omega and Spalart-Allmarasmodels are nearly same. Through the tip of the coolant hole all models are getting closer toeach others. Hence, on the analysis K-Omega SST turbulence model is used. In addition tothat, on the all of the inlets turbulence intensity is taken as 10 % as the suggestion of thecompetition committee.

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6.3.2 Other Solver Parameters

On the analysis, a fluid domain and solid domain is used. For the fluid domain air is usedand properties of the air is calculated at the average temperature 1250 [K] and kept constantfor the hole domain. For the solid, properties of the blade material is used. Both air andthe solid parameters are presented on the Table 9.

Table 9: Fluid and Solid Parameters

Domain Properties Values

Fluid

Viscosity 4.74e-5 [kg/ms]Cp 1178.26 [J/kgK]

Thermal Conductivity 0.08 [W/mK]Density 3.316 [kg/m3]

SolidCp 600 [J/kgK]

Thermal Conductivity 30 [W/mK]Density 8540 [kg/m3]

To solve the conjugate heat transfer a coupled interface is defined between solid and thefluid domain. In additio to that, for pressure and velocity coupling, coupled scheme is used.And for the turbulence and energy solvers second order upwind scheme is defined.

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7 One-Dimensional Internal Flow

Calculation Methodology of the Internal Flow In this section calculation methodol-ogy of the internal flow is explained by itemizing the process. In addition to this two coolingchannel is used at the design so examination of the flow at this channels are consideredseparately. At the figure 42 this channels are numbered and taken as control volume forthe heat and flow calculations. Design details of the channels are given at table 10. In theprocess of the division of the channels, aspect ratio facts is considered at table 31, accordingto reference [15]. Furthermore, as mentioned before in the design two shower-head rows areused for the cooling of the leading edge and; for the trailing edge slots are used. Thereforedescribed methodology at the below part is appropriate both channel 1 and 2, at figure 42.In addition assumption is made in this section itemize in the below part.

Figure 42: Subdivision in the chord direction

Table 10: Area, Perimeter and Hydraulic Diameter of the channels

Area [m2] Perimeter [m] Hydraulic Diameter [m]Channel 1 0.00002461 19.978×10−3 0.0049Channel 2 0.00003927 37.285×10−3 0.0039

Assumptions

• Air is ideal gas.

• Smooth duct.

• Velocities and temperatures in the duct is mean.

• Heat transfer steady-state.

• No internal heat generation.

• Thermophysical properties of the metal is constant. Also, there is no property changechord-wise direction.

• Flow is fully-developed in the duct.

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• Negligible flow work.

• Negligible radiation.

1. Construction of model This item describes construction 1D physical model for theinternal flow. In the design part two rows of film cooling holes is used. Each oneof the rows are composed of the seven holes. As a result of this, internal coolingmodel is constructed as shown in figure 43. In fact that this as a consequence of thisconfiguration model is divided seven slides. In addition for our model r = 0.8[mm].

Figure 43: Basic Physical Model of the Internal Flow

2. Solution of the each slide This item describes the solution approach of the eachslide which means that there is no mass discharge from the film cooling holes. Sinceas flow keep going through the duct both heat transfer and fluid dynamics laws mustbe considered. Therefore, in the code, first external temperature distribution is foundwith known initial temperature of the coolant at the hub which is 700K for bothchannels. Also, Taw distribution is obtained by film cooling efficiencies, details of thisis explained in the external flow part. Then, in order to find the heat flux thermalcircuit is constructed as like figure 44.

Figure 44: Thermal circuit model of the chord-wise heat transfer, 1D model

After the initializing the external temperature distribution, according to thistemperature and by the help of the thermal circuit total heat fluxes are calculatedseparately for both channel 1 and channel 2. Then this flux are multiplied by the sidearea of the channels which is Perimeter ∗ delta. In this formulation delta is representthe iteration value for the span direction. In the calculation this taken asdelta = 0.1e− 3[m]. Then, by the helping of the equation 22 new coolanttemperature value is obtained. After all, according to new cooling temperatureexternal temperature distribution is obtained. Process is loop is taken like this. In

42

addition to this, for internal heat transfer coefficient calculation, empiric equation 23is used. Since this empiric relation is turbulent and fully-developed, it represent wellthe physics of the internal cooling passages. Moreover, in the model mean velocityand mean temperature is used. At the calculation of the velocity, one workingbetween continuity equation at 24.

Q = mCp∆T (22)

NuD = 0.023Re4/5D Pr0.4 (23)

m = ρumA (24)

Now, fluid dynamics calculations are described. The pressure drop in the smoothduct is obtained by the Darcy–Weisbach equation at 25 where ∆x = 0.1e− 03[m]. Inorder to calculate skin friction factor Petukhov correlation is used. This is shown atequation 26. After this new pressure values are obtained for the loop.

P2− P1 = fρum2Dh

∆x (25)

f = (0.790lnReD − 1.64)−2 (26)

In addition to this at every step thermophysical property of the fluids (Mach,viscosity, mean velocity, conductivity etc.) is calculated at new temperature andpressure value.

43

3. Solution for the film cooling discharge In this section film cooling air dischargingprocess is described. Film cooling hole are modeled as orifice. Therefore, orifice flowcalculations are done.Main problem of this section is calculation of the mass flow discharge from the holes.For the purpose of calculate this mass flow rate, equation 27. In fact that at equation27 CD : discharge coefficient is used. For finding discharge coefficients for the 30◦ hole,reference [22] is used. At figure 45, used data is given. Actually, discharge coefficientsis function of the velocity head. Therefore, Interp1 functions are used in order to takingdifferent discharge coefficients for the film holes which is given in table 3. Dischargingmass flows details are given in the flow network part.

W

CDPtiA=[PsoPti

]γ+1γ

√2γ

(γ − 1)RTti

√PtiPso

γγ−1

− 1 (27)

Figure 45: Discharge Coefficient vs velocity head for different orientation of the holes, [22]

Table 11: CD for the film holes

Channel 1 & 2Film Hole # 1 0.74Film Hole # 2 0.72Film Hole # 3 0.72Film Hole # 4 0.71Film Hole # 5 0.67Film Hole # 6 0.62Film Hole # 7 0.56

44

1D flow network and Mass flow rate Results In this section flow network is shownand results are given. As described at figure 46 both channels are divided seven parts anddischarging mass flow rates given separately. In addition to this for both channels equalamount of the cooling air is used which is calculated as W2 ∗ 0.07/2. Also, since there isno bend, pin-fin array or rib in the cooling design, this are not shown. However, in orderto model trailing edge slots, at the last 10[mm] of the blade, film cooling is modelled. Andagain diameters are taken 0.8[mm]. These parts detail explained at the external flow part.

Figure 46: Flow network and discharging masses for channel 1 & 2

45

7.1 Comparison of Hand Calculations with CFD

In this section both hand calculations and CFD results are shown for the internal flow.

Figure 47: Flow Properties at the Coolant-1

46

Figure 48: Flow Properties at the Coolant-2

• Hand Calculation Result

– At both channel pressure drop is very small. One of the reason of this in thedesign, there is no serpentine, rib or bend staff. Therefore, pressure drop is verylittle. Also, as indicated in the assumptions pipe is taken like smooth. However,in reality this is not the case.

– For both channels, temperature increase observed very sharp. Reason of this inthe calculation of the temperature rising, equation is used, Q = mCp∆T . In factthat main coolant system for our design is film cooling and as air is discharged mterm decrease along the passages. So, sharp temperature increasing main reason isthis. At second channel this increasing value goes to 1600K. However, at passage 1

47

maximum is 1200K. Reason of this at the first channel, there are two film coolingholes and this cause the decreasing of the conduct the hot air.

– Velocity calculations are expected result for both channel. Up to discharging air,velocities are increases due to temperature increases, after the discharging airvelocity is decreases.

– For both passages MFR are decreases when the discharging air. However, for thesecond passage at the end of the channel, there is a little amount of air. Reasonfor this modelling the trailing edge slots like film cooling holes. However, this canbe updated very easily.

• CFD Result

– On the both Coolant-1 and Coolant-2 holes pressure drop is occurred. However,at the Coolant-1 drop is much greater than the Coolant-2 due to film coolingholes.

– Temperature variations are similar on the both holes, and the temperature valueis nearly kept constant about 700-800 [K]

– As explained on the CFD section, coolant inlets are defined as the pressure inletboundary condition. In addition to that, properties of the flow domain keptconstant for the 1250 [K]. Thus, inlet velocities of the Coolant-1 and Coolant-2holes are different than the emprical calculations.

– Also, on both channel velocity values are decreased. However, through the end ofthe coolant holes, a little increase on the velocity levels is observed. This may behappen due to vortices at the end of the coolant hole.

– On the above, velocity calculation is explained. So mass flow rate of the coolantholes are find with using the equation 6. Thus similar fashion is also observed onthe mass flow rate results.

7.2 Back Flow Margin (BFM) Calculations

Back flow margin is calculated with using CFD results. On the coolant domain 4 differentsection is created to obtain data. Sections are parallel to the symmetry boundary condition.And their locations are the 20%, 40%, 60%, 80% of the blade from the root of the blade.

To find the back flow margin equation 28 is used and the obtained results are presented onthe Table 12. Data are obtained at the exit of the pressure side film holes on the presentedsections.

BFM =

(PtcPinlet

− 1

)× 100 (28)

48

Table 12: Back Flow Margin Results

Inlet Pressure 1169900 [Pa]Total Pressure After Cooling [Pa] Back Flow Margin (BFM) [%] Average BFM [%]

Section 1 1184727 1.267

1.183Section 2 1184212 1.223Section 3 1183275 1.14Section 4 1182808 1.10

49

8 Heat Transfer Results

8.1 Hand Calculations

Calculation Methodology of the Heat Transfer In this section methodology of theheat transfer calculations are explained. Also, assumptions for this section is listed.

Assumptions

• Air is ideal gas.

• For the chord-wise direction 1-D heat transfer.

• For the span-wise calculation 1-D heat transfer.

• Property of the metal is constant.

• No radiation.

• Steady-state heat transfer.

• Pressure and Suction side are both modelled as flat plate at same property. Differentphysical conditions for this sides neglected.

1. Literature Search In this item data and modelling approaches explained which aresearched from literature. When looking the literature, simplifying model of the coolingconfiguration is described at figure 49. Therefore flat plate assumption is done for the1-D configuration. In fact that 1-D assumptions solutions are also dividing. One of thesolution is solving the boundary layer equations. At cooling flow part, some of themis mentioned according to reference [16]. As an other solution method is described atLefebvre & Ballal at reference [8]. Details of the process is given the other parts.

Figure 49: Model Simplification of the Blade Cooling Design, [2]

50

2. Construction of the Model In this item simplified heat transfer model is explained.First of all, blade is divided seven part according to film cooling holes diameter andthis holes are at end of the each passages. This supply the mass flow charging. Thisis shown at figure 50. These all seven parts are solved individually. In fact that forfirst part of the blade (before 1st mass discharge) 20 loop is solved in order to gettrue result of the blade temperatures at the 1st part. This makes the iteration stepsize delta = 0.1e − 03[m]. Then, while reaching the film cooling holes procedure isagain applied. However, there is an problem about the initializing the loop. For thesolving this, hub coolant temperature is used. Since it is known fact at 700K. Then,by doing this external temperature is obtained at chord-wise. Again, by using thisdistribution next coolant temperature is got and going like this. Each of the span-wiseand chord-wise temperature distribution is are described detail below part.

Figure 50: Separate parts of the blade

As seen figure 50, blade is divided seven part. Each part has the length of 2[mm] anddiameter of the film holes taken as 0.8[mm]. In addition to this height of the bladetaken as havg = (rtip + rhub)/2. Therefore, remaining length of the blade which isR = havg − (2 ∗ 7 + 7 ∗ 0.8)[mm] giving like residual (R) for both hub and tip section.Also, since the film cooling holes diameter are relatively big and distance betweentwo holes are short, for all chord-wise temperature distribution solving film coolingeffect is taken. Also, related blowing ratios are calculated corresponding slide notonly in the film holes section.

• Chord-wise In order to model the heat transfer circuits, two control volume isused as described figure 42. After this thermal circuit analogy is used and circuitis created, this is shown at figure 44. In the circuit thermal barrier coating also

51

modelled. However, in the calculation taking off. In thermal circuit radiation andfin between the the control volumes are neglected. Total resistance of the circuitgiven below:

R′′

= 1/hin + 1/hout + L/kmaterial + Ltbc/ktbc (29)

According to this R′′

value at each chord location q′′

is calculated and wherekmaterial = 30W/(m.K) for, CM 247 LC DS(MAR-M-247).

Outlet HTC correlations For the external HTC Lefebvre & Ballal correla-tions are used which is given in the reference [8]. There are two empiric datawhich are separated according to blowing ratio (m). Actually, Lefebvre & Bal-lal gives this equations for the cooling of the combustion liner and combustionand turbine environment is relatively different from each other such as for the airvelocity. However, when detail search is done, this equations again used for theturbine cooling simulation. For example, one of the new NASA report is writtenaccording to this correlation at reference [27]. At the other part detail of thisreport is explained. Moreover, since these are slot equations equivalence conver-gence to hole diameter to slot height is done.For m between 0.5 and 1.3:

hout = 0.069(Resx/s)0.7kg/x (30)

For m between 1.3 and 4:

hout = 0.10Re0.8s kg(x/s)0.44 (31)

where Res = ρcUcs/µc , s: slot height and kg : conductivity of the gas.

Internal HTC At the internal flow part this correlation is given which is equa-tion 26.

Film Cooling Efficiencies These are used for the finding of the adiabatic walltemperature (Taw) at equation. Again, reference [8] and [27] is used.

Taw = Ttg − η(Ttg − Ttc); (32)

– 0.5 < m < 1.3

∗ 0 < x/s < 150, ηx = 0.6( xms

)−0.3(Res

mµcµg

)0.15∗ x/s > 150, ηx = 3.68( x

ms)−0.8

(Res

µcµg

)0.2– 4 > m > 1.3

∗ 0 < x/s < 150 and x/ms < 8, ηx = 1.0

∗ 0 < x/s < 150 and 8 < x/ms < 11, ηx =(0.6 + 0.05 x

ms

)−1∗ 0 < x/s < 150 and x/ms > 11, ηx = 0.7x

s−0.3(Res µcµg )0.15m−0.2

∗ x/s > 150, ηx = 3.68( xms

)−0.8(Res

µcµg

)0.252

Model of the Film Cooling Basic flat plate model which is used at NASAarticle, [27] is given in the figure 51.

– Shower-heads As seen in the figure 51 shower heads are simulated like usconfiguration.

– Trailing edge slots At the beginning of the code, trailing edge slots are notsimulated. However, as CFD results are got, this are shown for the trailingedge cooling these are very effective. For the modelling of this slots literaturesearch is done; however equivalent configuration is not found. However, atDutta’ s book [15] is examined an design is found at figure 52 and trailing edgeslots are taken like film at constant η = 0.75 by doing engineering judgementand beginning last 10[mm] of the chord.

Figure 51: Flat plate model, [27]

Figure 52: Trailing-edge Model, [15]

• Span-wise For the span-wise calculations both external and internal coolingsolved simultaneously for this Matlab code is used which is given in the appendixpart. However, as mentioned again procedure is followed at the code:

53

– Initialize code by the help of the 700K which is coolant initial temperature.Beginning of the 1st part of the blade as described figure 50.

– According to this temperature external cooling code is start, by using thermalcircuit analogy.

– Then, again internal cooling function start and calculate the heating of thecoolant at span-wise delta which is 0.1e− 03[mm].

– This loop continues while coming the first film hole, first one is at first 2[mm].

– After the coolant discharge, end of the first part and beginning of the secondpart, loop again continue as explained before items.

3. Results In this Matlab code suction pressure side of the blade is not separately modelsince design is symmetric according to chord. All of the figures are taken as contourplot from the Matlab and x represents the chord direction and y represents the spandirection. In addition to this figures are taken for the seven parts separately.

Figure 53: Mach Distribution on the plate

At figure 53 Mach number distribution of the blade is shown for both suction andpressure sides.

54

Figure 54: 1st Part of the Blades both top view & 3-D graph

55

Figure 55: 2st Part of the Blades both top view & 3-D graph

56

Figure 56: 3rd Part of the Blades both top view & 3-D graph

57

Figure 57: 4th Part of the Blades both top view & 3-D graph

58

Figure 58: 5th Part of the Blades both top view & 3-D graph

59

Figure 59: 6th Part of the Blades both top view & 3-D graph

60

Figure 60: 7th Part of the Blades both top view & 3-D graph

61

In the figure 54, 55, 56, 57, 58, 59 and 60 metal temperatures of the blades is given.Therefore, 3D blade analysis of the blade is done. In addition film cooling effectivenessgiven at table 13 for both CFD and hand calculations. However, there are relatively muchassumptions. As a result,

• At the figure 54, in the beginning of the span-wise calculations, temperatures doingsharp pick. This is due to fact that initializing of the code begin at this point.

• As flow going through the chord-wise, film effectiveness is decreasing. Therefore metaltemperature is increasing.

• As flow going through the span-wise direction coolant temperature is increasing aswell. Therefore, temperature values are relatively higher previous chord iteration.Also, again first item is appropriate.

• For part number is increasing this means that flow is discharged from the film holes,temperature increase observed very clear. Because, in the internal flow temperaturecalculation Q = mCp∆T is used. When the coolant amount is decreasing, coolantheated. Therefore, there is no cold, relatively beginning.

• At the graphs sharp increasing and decreasing observed, this is due to fact that, twocoolant hole models separately after the 81th iteration second cooling channel is activein the Matlab code. Also, in this channel sometimes sudden decreasing observed thisis the effect of the trailing edge film cooling configuration.

• According to inlet and outlet wall temperature average value is Tavg = 1492.77[K].This is the above of the given temperature limit. However, at CFD calculation 1250Kis obtained. Because, in the hand calculation there a lot of assumptions. Also, for thechord-wise solution temperature is taken only chord-wise. However, in reality this isnot the case. Same thing valid for the span-wise calculations. Also, blades pressureand suction sides are not modelled separately; and this is one of the missing physicalpoint.

8.2 CFD Results

On the above sections, methodology of the CFD is explained. On this section metaltemperature on the surface of the blade is showed. In addition to that, metal temperatureat the different sections also showed. Four different sections are created parallel to thesymmetry boundary of the domain. Obtained results are presented on the Figures from 61to 68.

62

Figure 61: Temperature Distribution on the Pressure Side

Figure 62: Temperature Distribution on the Suction Side

63

Figure 63: Temperature Distribution from the Top View

Figure 64: Temperature Distribution from the Bottom View

64

Figure 65: Temperature Distribution on the 20% of the Blade from Root

Figure 66: Temperature Distribution on the 40% of the Blade from Root

65

Figure 67: Temperature Distribution on the 60% of the Blade from Root

Figure 68: Temperature Distribution on the 80% of the Blade from Root

As seen from the above figures, at the some regions metal temperature is passing to 1400[K] requirements especially, at the tip of suction side. To understand the reason for thisincrease on the metal temperature, streamlines are showed on the Figure 69.

66

Figure 69: Streamlines at the Suction side of the Blade

As previously explained, CFD analysis is done on a passage. As known due to passageshape, some secondary flows are developed. These secondary flows creates vortices like horse-shoe vortex. In addition to that, at the tip of the blade tip vortices are also created. Asa result, these vortex structure cause the separation of the film coolant from the surface.Thus, metal temperature is passing the limitations at suction side of the tip region and thisalso affects the metal temperature distribution on the bottom of the blade which is showedon the Figure 64. to prevent this kind of separations, additional structures like squealer tips,may be applied to the blade.

8.3 Cooling Effectiveness

For the efficiency calculation of the coolant system below formula is taken from the refer-ence [23].

ηc = (Tg − Tm)/(Tg − Tc) (33)

Where; [Tg] :Temperature of the hot stream is equal to 1776.5 [K], [Tm] :Mean temperature,[Tc] :Temperature of the cold stream is equal to 700 [K].

Table 13: Metal Temperature Results and Coolant Effectiveness

CFD Hand Calculation RequirementsTmax [K] 1495.78 1656.5 1400Tavg [K] 1231.78 1492.78 1250Tmin [K] 1021.2 700 -

ηc 0.51 0.26 -

67

9 Cold Flow Test Results

Table 14: Cold Flow Test Results from CFD

Test - 1 / Tamb = 298.15 [K] P1 = 96900 [Pa], P2 = 92300 [Pa], PR = 1.05MFR [g/s] Cd Average Cd

Section 1 9.2×10−2 0.86

0.73Section 2 7.8×10−2 0.79Section 3 6.59×10−2 0.71Section 4 4.96×10−2 0.57Test - 2 / Tamb = 298.15 [K] P1 = 101500 [Pa], P2 = 92300 [Pa], PR = 1.10

MFR [g/s] Cd Average CdSection 1 1.1×10−1 0.99

0.97Section 2 1×10−1 0.95Section 3 9.2×10−2 0.94Section 4 8.2×10−2 0.98Test - 3 / Tamb = 298.15 [K] P1 = 106100 [Pa], P2 = 92300 [Pa], PR = 1.15

MFR [g/s] Cd Average CdSection 1 1.15×10−1 0.96

0.89Section 2 1.04×10−1 0.91Section 3 9.2×10−2 0.87Section 4 8.2×10−2 0.83Test - 4 / Tamb = 298.15 [K] P1 = 110800 [Pa], P2 = 92300 [Pa], PR = 1.20

MFR [g/s] Cd Average CdSection 1 1.19×10−1 0.91

0.84Section 2 1.04×10−1 0.84Section 3 9.2×10−2 0.80Section 4 8.1×10−2 0.80Test - 5 / Tamb = 298.15 [K] P1 = 115400 [Pa], P2 = 92300 [Pa], PR = 1.25

MFR [g/s] Cd Average CdSection 1 1.3×10−1 0.87

0.76Section 2 1.04×10−1 0.73Section 3 9.2×10−2 0.74Section 4 8.3×10−2 0.69

Cold flow test is done on the CFD analysis with given pressure values. Results are obtainedat sections which are created at 20%, 40%, 60%, 80% of the blade from the root which isexplained on the previous sections. Also, parameters are found at the exit of the filmholes, and calculations are done with using the below equation which is obtained from thereference [4]. On the equation, γ = 1.4, R=287 [J/kgK] and A=5×10−7 m2. Obtainedresults are presented on the Table 14.

68

10 Life-Cycle Calculations

The turbine blade life cycle calculations are primarily based on creep or stress-rupturebehavior since the materials used in jet engines are in service under elevated temperatures(above 40% of the melting point) and static mechanical and/or thermal stresses.

The study of creep behavior of materials is generally experimental since the equations topredict the creep behavior contains constants that require to be determined experimentally.The creep or stress-rupture test are performed under constant stress and temperature andduration of the test is recorded to plot what is known as Larson-Miller Curve. [11] This curvehelps to extrapolate the life of the turbine blade under a constant stress and temperaturewhich would be almost impossible to determine experimentally. For the design of this NGV,the Larson-Miller plots are given in Figure 70.

Notice that there are two curves. This is due to the anistropic nature of directionallysolidified (DS) superalloys. The blue curve is for the configuration where loading axis isaligned with solidification direction, and the grey curve is for the transverse loading withrespect to solidification direction.

Figure 70: Larson-Miller Parameter Curves for CM 247 LC DS

In Figure 70, the Larson-Miller Parameter on the plot is defined as

LMP = (T (°R)× [20 + log(tr)])× 10−3

69

where:

T is temperature in Rankine scale (1.8 R = 1 K)

tr is time until rupture

In order to determine the load which the NGV will operate under, a preliminary approx-imation must be made for the thermal stresses. The best approach is to assume the bladehas fixed end (root) support, the temperature gradient from tip to root in chord directionvaries linearly [9], and approximate the thermal stresses in 1-D form as [9]:

σthermal = Eα∆Tave (34)

where:

E is the Elastic Modulus in MPa of the CM 247 LC DS alloy

α is the mean coefficient of thermal expansion in K−1 for CM 247 LC DS alloy

∆Tave is the average temperature gradient in K on the blade (0.5×(Tmax − Tmin)

The study in Sehitoglu (2000) [28] gives the elastic modulus for the CM 247 LC DS alloyas a linear function of temperature (in Celsius) as:

E = a− b× Twhere: a and b are constants that have the value of a = 161000 MPa and b = 78.5MPa/◦C. The mean coefficient of thermal expansion for CM 247 LC DS is calculated fromthe Thermal Strain vs Temperature curve given in Sehitoglu (2000) [28] as

α = 16.3× 10−6[1/K]

The temperature values on the blade are obtained from the 3D thermal analysis given inCFD and Heat Transfer Analysis and average thermal gradient is calculated as:

∆Tave = 0.5× (Tmax − Tmin)

= 0.5× (1495.77− 1021.20)

= 237.29K

The elastic modulus is calculated by taking the maximum of the temperature distributiongiven in CFD and Heat Transfer Analysis and substituting it into the expression given above.The result is

E = 161000− 78.5× (1495.77− 273.15)

= 65024.33MPa

The elastic modulus for the system is calculated from the maximum temperature since themajor loss of stiffness comes from the region with the highest temperature. This assumptionis for the worst case scenario for the life calculation. Substituting E, α, and ∆Tave intoEquation 34 gives the thermal stress as

σthermal = 111.58MPa

70

If this stress value is found on Figure 70 then LMP values are found to be 51.5 and 54.8,for transverse and longitudinal loadings, respectively. The LMP relation in the curves givenare in terms of Rankine, which is 1.8 times the Kelvin scale. The time to rupture has theunits of hours. The temperature value substituted in the LMP expression is the averagetemperature distribution from the CFD and Heat Transfer Analysis results. If the LMPrelation is used to find the life of the blade, then the rupture life for transverse loading atan average temperature of K turns out to be

LMP =T (◦R)× (20 + log(tr))

1000

tr = 101000∗LMP1.8×T (◦K)

−20

tr = 101000∗51.5

1.8×1231.74−20

Which is tr = 6959.73 hrs = 290 daysFor longitudinal loading

tr = 101000∗54.8

1.8×1231.74)−20

Which is tr = 234623 hrs = 9776 days = 26.8 years.

Figure 71: Larson-Miller Parameters for CM 247 LC DS at a stress level of 111 MPa

From empirical relations, we have the following:

∆Tave = 0.5× (Tmax − Tmin)

= 0.5× (1656.5− 700)

= 478.25K

E = 161000− 78.5× (1656.5− 273.15)

= 52407.025MPa

71

Thus σ = E · α ·∆T = (52407.025)× (16.3× 10−6) ∗ 478.25 = 408.54MPaFor this stress level, the LMP for transverse and longitudinal loadings are 44 and 47,

respectively. If these values are used with an average temperature of 1492.78 K, the rupturelife values are 2.35 hours and 57.45 hours, which are obviously incorrect and inadequate.

The calculations above are concerned with pure creep only. However, the thermo-mechanical fatigue is also of great importance in high temperature turbine blade appli-cations. The main model to predict the life under thermo-mechanical fatigue is the wellknown Miner’s rule.

j∑i=1

niNi

= 1 (35)

This model assumes that at different stages of flight or operation, the damage to the bladewill accumulate and when the ratio reaches 1, failure will occur. The simplicity of the modelallows one to easily integrate varying operational conditions and given when failure is to beexpected after each operation cycle. But in this design, the operation is given at one modeonly at a certain stress level.

72

11 Conclusion

Table 15: Obtained Results After Calculations

Obtained ResultsParameter CFD Hand Caclulations Requirements

MFR [%W2] 7 7 7BFM [%] 1.183 - 1.1Tmax[K] 1495.78 1656.5 1400Tavg[K] 1231.78 1492.78 1250Tmin[K] 1021.2 700 -Life [hr] 6959.73 2.35 5000

Effectiveness 0.51 0.26 -

In conclusion, a coolant system is designed and analyzed in that project. First of all,external flow properties of the blade is calculated. With given pressure and temperaturevalues, flow parameters are found, then by using these parameters heat transfer coefficient iscalculated with two different ways which are the flat plate assumption and 2-D CFD. Theiraverage heat transfer are close to each other. Also, with CFD solution, distribution of theheat transfer coefficients are represented.

To find the both internal flow parameters and metal temperature distribution, 3-D conju-gate heat transfer analysis is done with using ANSYS-Fluent. In addition to that, internalflow parameters and metal temperature results are obtained from the empirical formulas.Obtained results are presented on the Table 15. As seen from the table, maximum tempera-ture is above than the requirements at the some local regions of the blade. However, averagemetal temperature requirement is satisfied on the CFD analysis.

Finally, life-cycle calculation are done with given Larson-Miller chart. The temperaturedistributions obtained from ANSYS CFD and Heat Transfer analyses are used to find ther-mal stresses. Several studies in literature are used to find different material properties. Afirst order approximation is employed to calculate the thermal stresses arising from largetemperature gradients, geometrical and structural constraints. The stresses calculated arethen used in Larson-Miller curves to find the LMP for each condition, which in turn de-termines the rupture life for the blade. Obtained result shows the life-cycle of the blade ishigher than the 5000 hours with this cooling configuration.

Designed geometry shared with ”iletisim@teknofestistanbul.org” address by email.

73

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[16] Irvine, T. F., and (Eds.), J. P. H. Advances in Heat Transfer 7. Academic Press,1971.

74

[17] J. Mattingly, H. v. O. Elements of Gas Turbine Propulsion, illustrated edition ed.Aiaa Education Series. AIAA (American Institute of Aeronautics & Ast, 2006.

[18] Karni, J., and Goldstein, R. Surface injection effect on mass transfer from acylinder in crossflow: A simulation of film cooling in the leading edge region of a turbineblade. ASME Journal of Turbomachinery 112 (1990), 418–427.

[19] Karni, J., and Goldstein, R. Surface injection effect on mass transfer from acylinder in crossflow: A simulation of film cooling in the leading edge region of a turbineblade. ASME Journal of Turbomachinery 112 (1990), 418–427.

[20] Karni, J., and Goldstein, R. Surface injection effect on mass transfer from acylinder in crossflow: A simulation of film cooling in the leading edge region of a turbineblade. ASME Journal of Turbomachinery 112 (1990), 418–427.

[21] Lakshminarayana, B. Fluid Dynamic and Heat Transfer of Turbomachinery, sec-ond ed. John Wiley and Sons, 1996.

[22] Lampard, N. H. D. Discharge coefficient of turbine cooling holes: A review. ASMEJournal of Turbomachinery 14 (1990).

[23] Logan, E., and Roy, R. Handbook of Turbomachinery, second ed. Marcel Dekker,2003.

[24] Mayle, W. J. M. R. E. Stagnation film cooling and heat transfer, including its effectwithin the hole pattern. Journal of Turbomachinery 110, 1 (1993), 66–72.

[25] Mehendale, A., and Han, J. Influence of high mainstream turbulence on leadingedge film cooling heat transfer. ASME Journal of Turbomachinery 114 (1992), 707–715.

[26] Oates, G. C. Aerothermodynamics of Aircraft Engine Components. AIAA EDUCA-TION SERIES, 1985.

[27] Schneider, S. J. Analysis of turbine blade relative cooling flow factor used in thesubroutine coolit based onfilm cooling correlations. NASA Glenn Research Center -(2015), –.

[28] Sehitoglu, H., Maier, H., on Fatigue, A. C. E.-., and Fracture. Thermo-mechanical Fatigue Behavior of Materials: Third volume. No. 3. c. in ASTM STP 1371.ASTM, 2000.

[29] S.V., Han, J., and Du, H. Detailed film cooling measurements on a cylindricalleading edge model: Effect of free-stream turbulence and coolant density. ASME Journalof Turbomachinery 120 (1998), 799–807.

[30] Tribus, M., and Klein, J. Heat transfer.

[31] Tuncel, T., and Kahveci, H. Effects of pin fin shape and size on turbine bladetrailing edge flow and heat transfer. J. of Thermal Science and Technology (2019),191–207.

75

[32] Walsh, P. P., and Fletcher, P. Gas Turbine Performance, second ed. BlackwellScience, 2004.

76

Appendix

Listing 1: Code is used for the getting the initial condition of the coolant

1 function [Ps_c1 ,Ps_c2 ,Ts_c1 , Ts_c2 ,Pr_c1 ,Pr_c2 ,mu_c1 ,mu_c2 ,k_c1

,k_c2 ,...

2 a_c1 ,a_c2 ,rhos_c1 ,rhos_c2 ,U_c1 ,U_c2 ,...

3 mdot_c1 ,mdot_c2 ,Cp_c ,R_c ,gama_c ,M_c1 ,M_c2] = init_mdot(

percent_coolant ,Ac1 ,Ac2)

4 %%Initial Thermophysical Properties of the Coolant

5 W_2 =3.20;

6 mdot_c = W_2 *0.07/20;%kg/sec .%%% Coolant amount for one blade.

7 Tt_c = 700;%K

8 Pt_c = 1200*10^3;%Pa

9 gama_c = 1.4;%%%As an approximate number , taken from

10 Cp_c = 1004;%J/(kg.K)%%% Elements of Propulsion: Gas Turbines

and Rockets , Jack D. Mattingly.

11 R_c = (gama_c -1)/gama_c*Cp_c;

12 rho_t = Pt_c/(R_c*Tt_c);

13 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% COOLANT AMOUNT FOR HOLES

14 mdot_c1 = mdot_c *( percent_coolant);%%% Coolant Amount for first

hole.

15 mdot_c2 = mdot_c *(1- percent_coolant);%%% Coolant Amount for

second hole.

16 %%%Newton -Raphson Method.

17 Cc1 = mdot_c1 / (Ac1*rho_t*sqrt(gama_c*R_c*Tt_c));

18 Cc2 = mdot_c2 / (Ac2*rho_t*sqrt(gama_c*R_c*Tt_c));

19 itmax =1000;

20 k1 = 0;

21 M_c1 = 0;

22 while (k1<itmax)

23 M_c1 = M_c1 - (Cc1 * (1 + ((gama_c -1)/2)*M_c1 ^2)^(( gama_c

+1) /(2*( gama_c -1))) - M_c1)/...

24 (Cc1*( gama_c +1) /2*(1 + ((gama_c -1) /2)*M_c1 ^2)^((3- gama_c)

/(2*( gama_c -1)))*M_c1 -1);

25 k1=k1+1;

26 end

27 k2 = 0;

28 M_c2 = 0;

29 while (k2<itmax)

30 M_c2 = M_c2 - (Cc2 * (1 + ((gama_c -1)/2)*M_c2 ^2)^(( gama_c

+1) /(2*( gama_c -1))) - M_c2)/...

31 (Cc2*( gama_c +1) /2*(1 + ((gama_c -1) /2)*M_c2 ^2)^((3- gama_c)

/(2*( gama_c -1)))*M_c2 -1);

32 k2=k2+1;

77

33 end

34 Ps_c1= Pt_c /((1+( gama_c -1) /2.* M_c1 ^2))^( gama_c /(gama_c -1));%Pa.

35 Ps_c2= Pt_c /((1+( gama_c -1) /2.* M_c2 ^2))^( gama_c /(gama_c -1));%Pa.

36 Ts_c1 = Tt_c /(1+( gama_c -1)/2* M_c1 ^2);%K.

37 Ts_c2 = Tt_c /(1+( gama_c -1)/2* M_c2 ^2);%K.

38 Pr_c =0.695;

39 Pr_c1 = Pr_c;

40 Pr_c2 = Pr_c;

41 mu_c =338.8*10^ -7;%N.s/m^2

42 mu_c1 = mu_c;

43 mu_c2 = mu_c;

44 k_c =52.4*10^ -3;%W/(m.K)

45 k_c1 = k_c;

46 k_c2 = k_c;

47 a_c1=( gama_c*R_c*Ts_c1)^0.5;

48 a_c2=( gama_c*R_c*Ts_c2)^0.5;

49 rhos_c1=Ps_c1/(R_c*Ts_c1);

50 rhos_c2=Ps_c2/(R_c*Ts_c2);

51 U_c1 = M_c1 * a_c1;

52 U_c2 = M_c2 * a_c2;

53 end

Listing 2: Code used in the calculation of the chord-wise heat transfer

1 function [Tw1 ,Tw2 ,Taw ,q1,q2,eta ,hout ,Ps_g ,x] = lateral(chord ,

delta ,Ts_c1 ,Ts_c2 ,break_point1 ,...

2 break_point2 ,d_extra ,d, rho_c1 ,rho_c2 , mu_c1 ,...

3 mu_c2 ,U_c1 , U_c2 ,hin1 , hin2)

4 it_x = chord / delta + 1;%%% iteration value chordwise

5 it_change = break_point1 / delta +1;

6 it_extra = break_point2 /delta +1;

7 itmax = 1000; %%%for Newton -Raphson.

8 %%%%%%%%%%%%%%%%%%%%%

9 rt1 = 98.98*10^ -3;%m

10 rt2 = 94.01*10^ -3;%m

11 Annular_Area = pi*(rt1^2-rt2 ^2);%m^2.

12 W_2 = 3.20;%kg/sec.

13 OTDF = 0.2;

14 T_avg = 1600;

15 T3 = 700;

16 Tt_max = OTDF*(T_avg - T3) + T_avg;

17 Tt_g = Tt_max;%%% Combustion air maximum total temp.%K

18 Pt_g = 1180*10^3;%Pa

19 gama_g = 1.3;%%%After the combustion.

20 Cp_g = 1239;%J/(kg.K)

78

21 R_g = (gama_g -1)/gama_g*Cp_g;

22 rhot_g = Pt_g/(R_g*Tt_g);%kg/m^3.

23 mdot_g = (1 -0.07)*W_2;%kg.

24 k2 = 0;

25 M_g = 0;

26 Cg = mdot_g /( Annular_Area*rhot_g*sqrt(gama_g*R_g*Tt_g));

27 while (k2<itmax)

28 M_g = M_g - (Cg * (1 + ((gama_g -1) /2)*M_g^2) ^(( gama_g +1)

/(2*( gama_g -1))) - M_g)/...

29 (Cg*( gama_g +1) /2*(1 + ((gama_g -1) /2)*M_g^2)^((3- gama_g)

/(2*( gama_g -1)))*M_g -1);

30 k2=k2+1;

31 end

32 Ps_g = Pt_g /((1+( gama_g -1) /2.* M_g^2))^( gama_g /(gama_g -1));%Pa.

33 Ts_g = Tt_g /(1+( gama_g -1) /2*M_g^2);%K.

34 a_g=( gama_g*Ts_g*R_g)^0.5;

35 rhos_g=Ps_g/(R_g*Ts_g);

36 U_g = a_g*M_g;%m/s.

37 mu_g =637*10^ -7;%N.s/m^2.

38 Pr_g =0.683;

39 kg =120*10^ -3;%W/(m.K).

40 %%%%%%%%%%%%%%%%%%%%%%%%

41 %%%%%% Properties for conduction resistances

42 L_b =1* 10^-3;%%%Duvar et kalinligi

43 k_mat =30;%W/(m.K), CM 247 LC DS(MAR -M-247)

44 L_tb = 0;%%% themal barier length

45 k_tb = 2;%%% conductivity of thermal barier coating

46 %%%%%%%%%%%%%%%%%%%%%%%%

47 x(1)=0;

48 for i = 1:it_x

49 if ( i <= it_change) %%%work with first channel

50 Ts_c=Ts_c1;

51 rho_c=rho_c1;

52 U_c = U_c1;

53 hin = hin1;

54 mu_c = mu_c1;

55 else

56 Ts_c=Ts_c2; %%%work with second channel

57 rho_c=rho_c2;

58 U_c = U_c2;

59 hin = hin2;

60 mu_c = mu_c2;

61 end

62 x(i) = delta*i;

63 m = rho_c*U_c / (rhos_g*U_g);

79

64 % s = pi* (d)^2/4;%%% slot%m

65 s = d;

66 Re_s = rho_c*U_c*s / mu_c;

67 if ( m < 0.5)

68 m = 0.5;

69 end

70 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

71 %%%Film Cooling Eff. Definitions

72 x(i) = delta*i;

73 if ( i > it_extra)

74 s = 0.8*10^ -3;

75 eta(i) = 0.75;

76 Re_s = rho_c*U_c*s / mu_c;

77 end

78 if (m <= 1.3) &&(m >= 0.5)

79 if ((x(i) / s) <= 150)

80 eta(i) = 0.6* ( x(i) / (m *s ))^( -0.3) * (Re_s*

m *mu_c / mu_g)^0.15;

81 end

82 if ((x(i) / s) > 150)

83 eta(i) = 3.68 * (x(i) / (m *s))^( -0.8) * (Re_s*

mu_c / mu_g)^0.2;

84 end

85 if (eta(i) > 1)

86 eta(i) = 1;

87 end

88 end

899091 if (m > 1.3) && (m < 4)

92 if ((x(i) / (m *s)) < 8) && ((x(i) / s) < 150)

93 eta(i) = 1.0;

94 end

9596 if ((x(i) / (m *s )) > 8) && ((x(i) / (m*s))

<11) && ((x(i) / s) < 150)

97 eta(i)=(0.6 + 0.05* x(i) / (m*s)) ^-1;

98 end

99100 if ((x(i) / (m*s)) > 11) && ((x(i) / s) < 150)

101 eta(i)= 0.7* ( x(i) / s)^ (-0.3)* (Re_s* mu_g /

mu_g)^ 0.15* m^( -0.2);

102 end

103104 if ((x(i) / s) > 150)

80

105 eta(i) = 3.68 * (x(i) / (m *s))^( -0.8) * (Re_s*

mu_c / mu_g )^0.2;

106 end

107108 if (eta(i) > 1)

109 eta(i) = 1;

110 end

111 end

112113114 Taw(i) = Tt_g - eta(i) * ( Tt_g - Ts_c);

115 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

116 %%% External and Internal HTC

117 if ( m <= 1.3) && ( m >= 0.5)

118 hout(i) = kg* 0.069* (Re_s* x(i) / s)^0.7 / x(i);

119 end

120121 if ( m > 1.3) && (m < 4)

122 hout(i) = kg* 0.10 * Re_s ^0.8* (x(i) /s)^ 0.44;

123 end

124125 % Regx(i) = rhos_g*U_g*x(i) / mu_g;

126 % hin(i) = kg * 0.0292 * Regx(i) ^0.8 * Pr_g ^(1/3);

127128 %%% Beginning of the thermal circuit

129 R_conv1(i) = 1 / hout(i); %%% External convection resistance

130 R_cond1 = L_tb / k_tb; %%% TBT conduction resistance

131 R_cond2 = L_b / k_mat; %%% Conduction resistance of material

132 R_conv2(i) = 1 / hin; %%% Internal convection resistance

133 R_total(i) = R_conv1(i) + R_cond1 + R_cond2 + R_conv2(i);%%%

Total resistance

134135 q(i)= (Taw(i)- Ts_c) / R_total(i);%%% Heat current

136 if ( i <= it_change) %%%work with first channel

137 q1(i) = q(i);

138 else if ( i > it_change)

139 q2(i-it_change) = q(i); %%%work with second channel

140 end

141 end

142 Tw2(i) = q(i) * R_conv2(i) + Ts_c ;%%% Inner Wall Temp.

143 Tw1(i)= Taw(i) - q(i) * R_conv1(i);%%% Outer Wall Temp.

144 end

145 end

Listing 3: Code used in the calculation of the span-wise heat transfer

81

1 function [Ps_c_new ,Ts_c_new ,del_temp ,del_pre , hin , Re_D ,f] =

long(Ts_c ,Ps_c ,rho_c ,mdot_c ,Cp_c ,R_c , mu_c , U_c , q, A,...

2 Perimeter , delta , Pr_c ,k_c ,break_point)

3 D = 4*A/Perimeter;%m

4 q = 2* sum(q);%%% Total heat value , multiplication with 2.%W.

5 q = break_point*q*delta;%%%Heat array

6 %%% Definition of coefficients

7 C1 = R_c * ( mdot_c / A ) ^ 2 /( Ps_c * ( 1 + ( mdot_c / A )

^ 2 * R_c ^ 2 * Ts_c /...

8 (Ps_c ^2 * Cp_c)));

9 C2 = 1- (mdot_c / A)^2 * R_c * Ts_c / Ps_c + C1 / ( Ps_c * Cp_c

)*( mdot_c /...

10 A *R_c * Ts_c / Ps_c )^2;

11 CT = 1+ (mdot_c / A)^2 * R_c ^2* Ts_c / (Ps_c ^2* Cp_c );

1213 %%% Definition of skin fricition factor

14 Re_D = rho_c * U_c * D / mu_c;

15 f = ( 0.790 * log (Re_D) -1.64 )^-2;

16 %%% HTC inlet corr.

17 hin = (0.023 * Re_D ^(4/5) * Pr_c ^0.4* k_c) / D;

1819 %%% Definition of pressure and temp. changes

20 % dp_dx = 1 / C2 * (-4* f* R_c* Ts_c * (mdot_c / A)^2 / (2* Ps_c

*D)- 4*q * C1 /...

21 % (mdot_c* Cp_c * D));%%% Pressure Drop

2223 del_pre = f * rho_c * U_c ^2 * delta / ( 2* D )* delta;

2425 del_temp = q / (mdot_c * Cp_c);

2627 Ps_c_new = Ps_c - del_pre;

28 Ts_c_new = Ts_c + del_temp;

29 end

Listing 4: Main Code

1 %%%%%% MAIN FUCTION

2 clc;clear all;

3 %%%%%%%%%%%%%%%%%%%%%

4 load('discharge.mat');%%% Discharge coeff. according to velocity

head.

56 %%%%% Thermophysical Properties of air

7 load ('table.dat'); %%% Heat and Mass Transfer , Incropera. %

Table A.4

82

8 % 1st: temp (K), 2nd: rho (kg / m^3), 3rd: cp (kJ / kg.K), 4th:

mu*10^7 (N

9 % *s / m^2), 5th: nu*10^6 (m^2 / s), 6th: k*10^3 (W / m*K), 7th

: alpha (m^2

10 % / s), 8th: Pr

1112 %Geometric Constraints

13 %%%%% Geometric Inputs of the lateral function.

14 d = 0.8*10^ -3;%m %%% diameter of the film holes

15 d_extra = 0.8* 10^-3; %%% extra film cooling hole at the

trailing edge

16 delta =0.1*10^ -3;%%% general iteration value

17 % chord = 74.126*10^ -3 / 2;

18 chord = 20.39*10^ -3;

19 break_point1 = 8*10^ -3;%m%contstricition of the 2nd cooling

hole.

20 break_point2 = chord - 10*10^ -3; %m%for trailing edge slot.

21 %%%%%% Geometric Inputs of the cooling holes

22 rt1 = 98.98*10^ -3;%m

23 rt2 = 94.01*10^ -3;%m

24 rt_avg = (rt1+rt2)/2;%m

25 rh = 76*10^ -3;%m;

26 h_total = rt_avg - rh;%m %%% average span of the blade.

27 %%%%%%%%%% number of the film holes

28 distance_hub = 1;%mm

29 distance_tip = 1;%mm

30 film_parameter = h_total *10^3 - distance_hub - distance_tip;

31 distanceOfFilmHoles = 2;%mm

32 diameterOfFilmHoles = 0.8;%mm

33 numberOfFilmHoles = (film_parameter + distanceOfFilmHoles) / (

diameterOfFilmHoles +...

34 distanceOfFilmHoles);

35 N = numberOfFilmHoles;

36 %%%%%%%%%%

37 Perimeter_1 = 19.978* 10^-3;%m

38 Perimeter_2 = 37.285* 10^-3;%m

39 Ac1 = 0.00002461;%m^2

40 Ac2 = 0.00003656;%m^2

41 D1 = 4*Ac1/Perimeter_1;%m

42 D2 = 4*Ac2/Perimeter_2;%m

43 %%%%%%%%%%%%%%%%%%%%%%%

44 %%Initial Thermophysical Properties of the Coolant

45 percent_coolant = 0.5;%%% Initialization of the coolant amount

for 1st hole

46 [Ps_c1 ,Ps_c2 ,Ts_c1 , Ts_c2 ,Pr_c1 ,Pr_c2 ,mu_c1 ,mu_c2 ,k_c1 ,k_c2 ,

83

a_c1 ,a_c2 ,rhos_c1 ,...

47 rhos_c2 ,U_c1 ,U_c2 , mdot_c1 ,mdot_c2 ,Cp_c ,R_c ,gama_c ,M_c1 ,M_c2]

...

48 = init_mdot(percent_coolant ,Ac1 ,Ac2);

49 hin1 = 10;

50 hin2 = 10;

51 mdot_c2_initial = mdot_c2;

52 %%%%%%%%%%%%%%%%%%%%%%

53 %%Initializing of the Lateral function

54 for j=1:N%%% number of slice

55 for i=1:( distanceOfFilmHoles *10^ -3 / delta)

56 % Iteration value spanwise

57 y(i) = delta * i;

58 yy = y(i);

59 [Tw1 ,Tw2 ,Taw ,q1 ,q2,eta ,hout ,Ps_g ,x] = lateral(chord ,delta ,Ts_c1

,Ts_c2 ,break_point1 ,break_point2 ,...

60 d_extra ,d,rhos_c1 ,rhos_c2 , mu_c1 ,mu_c2 , U_c1 , U_c2 ,hin1 , hin2);

61 [Ps_c_new1 ,Ts_c_new1 ,del_temp1 ,dp_dx1 ,hin_new1 , Re_D ,f] = long(

Ts_c1 ,Ps_c1 ,rhos_c1 ,mdot_c1 ,...

62 Cp_c ,R_c , mu_c1 , U_c1 , q1, Ac1 ,Perimeter_1 , delta , Pr_c1 ,k_c1 ,

break_point1);

63 [Ps_c_new2 ,Ts_c_new2 ,del_temp2 ,dp_dx2 ,hin_new2] = long(Ts_c2 ,

Ps_c2 ,rhos_c2 ,mdot_c2 ,...

64 Cp_c ,R_c , mu_c2 , U_c2 , q2, Ac2 ,Perimeter_2 , delta , Pr_c2 ,k_c2 ,

...

65 (chord -break_point1));

66 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

67 %%% Interpoletion of Thermophysical Prop. and others

68 rhos_c1 = Ps_c1 / (R_c*Ts_c1);

69 rhos_c2 = Ps_c2 / (R_c*Ts_c2);

70 mu_c1 = interp1 ( table (:,1) , table (:,4), Ts_c1) * 10 ^ (-7);

71 mu_c2 = interp1 ( table (:,1) , table (:,4), Ts_c2) * 10 ^ (-7);

72 Pr_c1 = interp1 ( table (:,1) , table (:,8), Ts_c1);

73 Pr_c2 = interp1 ( table (:,1) , table (:,8), Ts_c2);

74 k_c1 = interp1 ( table (:,1) , table (:,6), Ts_c1) * 10 ^ (-3);

75 k_c2 = interp1 ( table (:,1) , table (:,6), Ts_c2) * 10 ^ (-3);

76 a_c1 = (gama_c* R_c* Ts_c1) ^ 0.5;

77 a_c2 = (gama_c* R_c* Ts_c2) ^ 0.5;

78 U_c1 = mdot_c1 / (rhos_c1*Ac1);

79 U_c2 = mdot_c2 / (rhos_c2*Ac2);

80 M_c1 = U_c1 / a_c1;

81 M_c2 = U_c2 / a_c2;

82 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

83 Ps_c1star(i,j) = Ps_c_new1;

84 Ts_c1star(i,j) = Ts_c_new1;

84

85 Ps_c2star(i,j) = Ps_c_new2;

86 Ts_c2star(i,j) = Ts_c_new2;

87 Tw1star(:,i,j) = Tw1;

88 Tw2star(:,i,j) = Tw2;

89 Tawstar(:,i,j) = Taw;

90 etastar(:,i,j) = eta;

91 U_c1star(i,j) = U_c1;

92 U_c2star(i,j) = U_c2;

93 Ps_c1 = Ps_c_new1;

94 Ts_c1 = Ts_c_new1;

95 Ps_c2 = Ps_c_new2;

96 Ts_c2 = Ts_c_new2;

97 hin1 = hin_new1;

98 hin2 = hin_new2;

99 hin1star(i) = hin_new1;

100 hin2star(i) = hin_new2;

101 fstar(i) = f;

102 houtstar(:,i) = hout;

103 end

104 Tw1starSS(:,j) = Tw1;

105 %%% Discharge for film air

106 Ps_o = Ps_g;

107 A_film1 = pi*d^2/4;

108 A_film2 = pi* d_extra ^2 / 4;

109 g = 9.81;

110 vel_head = U_c1^2 / (2 * g);

111 if ( vel_head > 95)

112 CD = 0.8;

113 else

114 CD = interp1 (discharge (:,1),discharge (:,2),vel_head ,'spline ');115 end

116 CDstar(j) = CD;

117 Pt_i1(j) = Ps_c1* ( 1 + (gama_c -1) / 2 * M_c1 ^2) ^ (( gama_c) /

(gama_c -1));

118 Tt_i1(j) = Ts_c1* ( 1 + (gama_c -1) / 2 * M_c1 ^2);

119 W1(j) = ( Ps_o / Pt_i1(j) )^ (( gama_c +1) /gama_c) * ( 2* gama_c

/...

120 ( (gama_c -1)* R_c* Tt_i1(j) )) ^0.5 * (( Pt_i1(j) / Ps_o )^ ((

gama_c -1) /gama_c)...

121 -1) ^0.5;

122 W1(j) = W1(j) * (CD *A_film1* Pt_i1(j));

123 a1(j) = 2* W1(j);

124 mdot_c1star(j) = mdot_c1 - 2*W1(j);

125 back_flow1(j) = (Ps_c1 / Ps_o -1) *100;

126 %%% Property calc. after the discaherge.

85

127 mdot_c1 = mdot_c1star(j);

128 U_c1 = mdot_c1 / (rhos_c1*Ac1);

129 M_c1 = U_c1 / a_c1;

130 %%Discharge for trailing edge air

131 % Continious discharge is assumed.

132 Pt_i2(j) = Ps_c2* ( 1 + (gama_c -1) / 2 * M_c2 ^2) ^ (( gama_c) /

(gama_c -1));

133 Tt_i2(j) = Ts_c2* ( 1 + (gama_c -1) / 2 * M_c2 ^2);

134 W2(j) = ( Ps_o / Pt_i2(j) )^ (( gama_c +1) /gama_c) * ( 2* gama_c

/...

135 ( (gama_c -1)* R_c* Tt_i2(j) )) ^0.5 * (( Pt_i2(j) / Ps_o )^ ((

gama_c -1) /gama_c)...

136 -1) ^0.5;

137 W2(j) = W2(j) * (CD *A_film2* Pt_i2(j));

138 a2(j) = 2*W2(j);

139 mdot_c2star(j) = mdot_c2 - 2*W2(j);

140 back_flow2(j) = (Ps_c2 / Ps_o -1)* 100;

141 %%% Property calc. after the discaherge.

142 mdot_c2 = mdot_c2star(j);

143 U_c2 = mdot_c2 / (rhos_c2*Ac2);

144 M_c2 = U_c2 / a_c2;

145 end

146 %%%mean calculations

147 for j=size(Tw1star ,2)

148 for h= 1:size(Tw1star ,3)

149 A1(:,:) = mean(Tw1star (:,:,h));

150 B1(:) = mean(A1(:,j));

151 C1 = mean(B1);

152 A2(:,:) = mean(Tw2star (:,:,h));

153 B2(:) = mean(A2(:,j));

154 C2 = mean(B2);

155 end

156 end

157 mean = (C1 + C2)/2;

Listing 5: Code used in the calculation of the flat plate solution

1 clc;

2 clear;

3 %Flat plate convection coefficient calculator

4 %{

5 mdot_in: inlet mass flow rate in kg/s

6 r_in: inside radius in m

7 r_out: outside radius in m

8 A: Annular area in m^2

86

9 R: gas constant in J/kg.K

10 To: total temperature in K

11 Po: total pressure in Pa

12 rho_o: total density in kg/m^3

13 Cp: specific heat capacity of air in J/kg.K

14 gamma: specific heat ratio of air

15 M: Mach number

16 r: Isentropic ratio

17 T, P, rho: static properties of air

18 a: speed of sound

19 V: velocity in m/s

20 Veff: effective velocity in m/s

21 k: thermal conductivity of air in W/mK

22 mu: dynamic viscosity of air in kg/m.s

23 L: chord length in m

24 Re: Reynolds number

25 Pr: Prandtl number

26 Nu: Nusselt number

27 %}

28 r_out = 98.98E-3;

29 r_in = 76E-3;

30 mdot_in = 2.72;

3132 R = 287;

33 To = 1780;

34 Po = 1180E3;

35 rho_o = 2.3094;

36 Cp = 1235.23;

37 Cv = 948.18;

38 gamma = Cp/Cv;

3940 M = 0; %initial guess for Mach number in external flow

41 A = pi*( r_out^2-r_in ^2);

4243 %Newton -Raphson method

44 itmax = 1000;

45 k=0;

46 C = mdot_in /(A*rho_o*sqrt(gamma*R*To));

47 while (k<itmax)

48 M = M - (C * (1 + ((gamma -1) /2)*M^2)^(( gamma +1) /(2*( gamma

-1))) - M)/(C*(gamma +1) /2*(1 + ((gamma -1)/2)*M^2)^((3-

gamma)/(2*( gamma -1)))*M-1);

49 k=k+1;

50 end

51

87

52 r = 1 + ((gamma -1)/2)*M^2;

53 T = To/r;

54 P = Po / (r)^( (gamma)/(gamma -1) );

55 rho = rho_o / (r) ^ ( 1/(gamma -1) );

56 a = sqrt(gamma*R*T);

57 V = M*a;

58 Veff = 3*V;

59 k = 0.101920;

60 mu = 5.792E-5;

61 L = 20.39E-3;

62 %L = 0.8*20.39E-3;

63 x = 0 : 0.01E-3 : 20.39E-3;

64 Re = rho*Veff*L/mu;

65 Re_x = rho*Veff*x/mu;

66 Pr = Cp*mu/k;

67 %Fully turbulent flow

68 Nu_turb_avg = 0.037*( Re)^0.8* Pr ^(1/3);

69 h_turb = Nu_turb_avg*k/L;

70 Nu_turb_local = 0.0296*( Re_x).^(0.8)*Pr ^(1/3);

71 h_turb_local = Nu_turb_local*k/L;

72 fprintf('The convection coefficient for fully turbulent flow is

% 7.5f W/m^2.K \n',h_turb)73 %Fully laminar flow

74 Nu_lam_avg = 0.664*( Re)^0.5* Pr ^(1/3);

75 h_lam = Nu_lam_avg*k/L;

76 Nu_lam_local = 0.664*( Re_x).^(0.5)*Pr ^(1/3);

77 h_lam_local = Nu_lam_local*k/L;

78 fprintf('The convection coefficient for fully laminar flow is %

7.5f W/m^2.K \n',h_lam)79 %Cylinder in cross flow

80 Nu_circle = 0.027*( Re)^0.805* Pr ^(1/3);

81 h_circle = Nu_circle*k/L;

82 fprintf('The convection coefficient for cylinder in cross flow

is % 7.5f W/m^2.K \n',h_circle)83 %Zukauskas correlation

84 Cp_s = 1121.49; mu_s = 3.896E-5; k_s = 0.062708; %Air

properties in contact with surface assumed to be at 900K

85 Pr_s = Cp_s*mu_s/k_s;

86 Nu_circle_2 = 0.076* Re ^(0.7)*Pr ^(0.37) *(Pr/Pr_s)^0.25;

87 h_circle_2 = Nu_circle_2*k/L;

88 fprintf('The convection coefficient for cylinder in cross flow

-Zukauskas correlation - is % 7.5f W/m^2.K \n',h_circle_2)89 %Churchill -Bernstein equation

90 Nu_circle_3 = 0.3 + ((0.62* Re^0.5*Pr ^(1/3))/(1+(0.4/ Pr)^(2/3))

^0.25) *(1+(Re /282000) ^(5/8))^0.8;

88

91 h_circle_3 = Nu_circle_3*k/L;

92 fprintf('The convection coefficient for cylinder in cross flow

-Churchill -Bernstein - is % 7.5f W/m^2.K \n',h_circle_3)939495 figure

96 plot(x/L,Nu_turb_local ,'-r','LineWidth ' ,2)97 title('Local Nusselt number (Nu_x) vs Normalized Chord length (

x/c) for fully turbulent flow. Chord Length c = 20.39mm')98 xlabel('Normalized Chord length (x/c)')99 ylabel('Local Nusselt number (Nu_x)')

100 grid on

101 grid minor

102 set(gca ,'FontSize ' ,13)103104 figure

105 plot(x/L,h_turb_local ,'-r','LineWidth ' ,2)106 title('Local Heat Transfer Coefficient (h_x) vs Normalized

Chord length (x/c) for fully turbulent flow. Chord Length c

= 20.39mm')107 xlabel('Normalized Chord length (x/c)')108 ylabel('Local Convection Coefficient (h_x) [W/m^2.K]')109 grid on

110 grid minor

111 set(gca ,'FontSize ' ,13)112113 figure

114 plot(x/L,Nu_turb_local ,'-r','LineWidth ' ,2)115 title('Local Nusselt number (Nu_x) vs Normalized Chord length (

x/c) for fully laminar flow. Chord Length c = 20.39 mm')116 xlabel('Normalized Chord length (x/c)')117 ylabel('Local Nusselt number (Nu_x)')118 grid on

119 grid minor

120 set(gca ,'FontSize ' ,13)121122 figure

123 plot(x/L,h_lam_local ,'-r','LineWidth ' ,2)124 title('Local Heat Transfer Coefficient (h_x) vs Normalized

Chord length (x/c) for fully laminar flow. Chord Length c =

20.39mm')125 xlabel('Normalized Chord length (x/c)')126 ylabel('Local Convection Coefficient (h_x) [W/m^2.K]')127 grid on

128 grid minor

89

129 set(gca ,'FontSize ' ,13)

Listing 6: Code used in the calculation on the heat transfer coefficient distributions withoutfilm cooling

1 clear;

2 clc;

3 load('duzeltme.mat')4 % x and y vectors are airfoil profile coordinates

5 % h is convection coefficients obtained from ANSYS heat

transfer thermal analysis

6 z=18000* ones(length(x) ,1);

7 dt = delaunayTriangulation(x,y) ; % delaunay triangulation

8 points = dt.Points ; % points

9 tri = dt.ConnectivityList ; % nodal connectivity

10 xx = points (:,1) ;

11 yy = points (:,2) ;

12 F = scatteredInterpolant(x,y,h); % interpolates h values ,

linear interpolation

13 zz = F(xx,yy);

1415 figure

16 trisurf(tri ,xx,yy ,zz)

17 view(0, -90)

18 shading interp

19 c1 = colorbar;

20 c1.Label.String = 'Convection Coefficient [W/m^2K]';21 hold on

22 scatter3(x,y,z,'filled ','r')23 colormap(jet)

24 xlabel('x [m]')25 ylabel('y [m]')26 zlabel('Convection Coefficient [W/m^2K]')2728 figure

29 stem3(x,y,h)

30 hold on

31 scatter3(x,y,h,60,h,'filled ')32 view(0, -90)

33 c2 = colorbar;

34 c2.Label.String = 'Convection Coefficient [W/m^2K]';35 colormap(jet)

36 xlabel('x [m]')37 ylabel('y [m]')38 zlabel('Convection Coefficient [W/m^2K]')

90

3940 figure

41 scatter3(x,y,h,60,h,'filled ')42 view(0, -90)

43 c3 = colorbar;

44 c3.Label.String = 'Convection Coefficient [W/m^2K]';45 colormap(jet)

46 xlabel('x [m]')47 ylabel('y [m]')48 zlabel('Convection Coefficient [W/m^2K]')4950 figure

51 scatter3(x,y,h,60,h,'filled ')52 view(-180, 90)

53 c4 = colorbar;

54 c4.Label.String = 'Convection Coefficient [W/m^2K]';55 colormap(jet)

56 xlabel('x [m]')57 ylabel('y [m]')58 zlabel('Convection Coefficient [W/m^2K]')

Listing 7: Code used in the calculation of the heat transfer coefficient distributions with filmcooling

1 clear;

2 clc;

3 load('htc_film.mat')4 x = x_film;

5 y = y_film;

6 h = h_film;

7 % x and y vectors are airfoil profile coordinates

8 % h is convection coefficients obtained from ANSYS heat

transfer thermal analysis

9 z=18000* ones(length(x) ,1);

10 dt = delaunayTriangulation(x,y) ; % delaunay triangulation

11 points = dt.Points ; % points

12 tri = dt.ConnectivityList ; % nodal connectivity

13 xx = points (:,1) ;

14 yy = points (:,2) ;

15 F = scatteredInterpolant(x,y,h); % interpolates h values ,

linear interpolation

16 zz = F(xx,yy);

1718 figure

19 trisurf(tri ,xx,yy ,zz)

91

20 view(0, -90)

21 shading interp

22 c1 = colorbar;

23 c1.Label.String = 'Convection Coefficient [W/m^2K]';24 hold on

25 scatter3(x,y,z,'filled ','r')26 colormap(jet)

27 xlabel('x [m]')28 ylabel('y [m]')29 zlabel('Convection Coefficient [W/m^2K]')3031 figure

32 stem3(x,y,h)

33 hold on

34 scatter3(x,y,h,60,h,'filled ')35 view(0, -90)

36 c2 = colorbar;

37 c2.Label.String = 'Convection Coefficient [W/m^2K]';38 colormap(jet)

39 xlabel('x [m]')40 ylabel('y [m]')41 zlabel('Convection Coefficient [W/m^2K]')4243 figure

44 scatter3(x,y,h,60,h,'filled ')45 view(0, -90)

46 c3 = colorbar;

47 c3.Label.String = 'Convection Coefficient [W/m^2K]';48 colormap(jet)

49 xlabel('x [m]')50 ylabel('y [m]')51 zlabel('Convection Coefficient [W/m^2K]')5253 figure

54 scatter3(x,y,h,60,h,'filled ')55 view(-180, 90)

56 c4 = colorbar;

57 c4.Label.String = 'Convection Coefficient [W/m^2K]';58 colormap(jet)

59 xlabel('x [m]')60 ylabel('y [m]')61 zlabel('Convection Coefficient [W/m^2K]')

Listing 8: Code used in the calculation of thermal stress on the turbine blade

92

1 %Thermal Stress calculations for CM 247 LC DS (equivalent: MAR -

M-247)

2 clc;

3 clear;

45 %Elastic Modulus is dependent on temperature by a linear

variation with

6 %constants a and b

7 %E[MPa] = a - b*T

8 a = 1.61E5; %[MPa]

9 b = 78.5; %[MPa/Celsius]

1011 %Thermal expansion coefficient (\alpha) is necessary to

calculate thermal stresses

12 %by the relation \sigma = E*(\ alpha)*(\ Delta T)

13 %where \Delta T or dT is the temperature difference on the

blade

14 %rising from contact between hot and cold gases

1516 alpha = 16.3E-6; %[/K] %Taken from Sehitoglu

1718 %Temperatures with film cooling

19 %Coolant temperature 700 K

20 Tmin = 1021.20;%[K] %748.05 C

21 Tavg = 1231.74;%[K] %958.59 C

22 Tmax = 1495.77;%[K] %1222.62 C

2324 %Elastic modulus based on max temperature with regular temp

diff

25 E = a - b*(Tmax -273.15);

2627 %Temperature gradient for linear temperature variation

28 DT = 0.5*( Tavg - Tmin);

2930 %Thermal stress calculated with S(sigma) = E*alpha*DeltaT

31 S = E*alpha*DT;

32 fprintf('Thermal stress is %7.5f \n',S)

93