PROCEEDINGS OF ECOS 2016 - THE 29TH INTERNATIONAL CONFERENCE ON

EFFICIENCY, COST, OPTIMIZATION, SIMULATION AND ENVIRONMENTAL IMPACT OF ENERGY SYSTEMS

JUNE 19-23, 2016, PORTOROŽ, SLOVENIA

A lumped thermodynamic model of gas turbine blade cooling: prediction of first-stage blades temperature

and cooling flow rates

Roberta Mascia, Enrico Sciubbab

a Dept. of Mechanical and Aerospace Engineering, University of Rome “Sapienza”, Rome, Italy,

roberta.masci1@gmail.com (CA) b Dept. of Mechanical and Aerospace Engineering, University of Rome “Sapienza”, Rome, Italy,

enrico.sciubba@uniroma1.it

Abstract

Turbine Inlet Temperatures of 1500-2000K have become a sort of standard for most modern advanced

applications. First-stage blades are obviously the most exposed components to such hot gases, and

thus they need proper cooling. In the preliminary design of the blades and their cooling system,

designers must rely on simple models that can be further refined at a later stage, in order to have an

approximate but valuable set of guidelines and to reach a feasible first-order configuration. In this

paper, a simple lumped thermodynamic model of blade cooling is proposed. It is based on

mass/energy balances and heat transfer correlations and it predicts a one-dimensional temperature

profile on the blade external surface along the chord for a given gas temperature profile, as well as

the required cooling air flow rates to prevent blade material from creep. The greatest advantage of

the model is that it can be easily adapted to any operating condition, process parameter and blade

geometry, which makes it well suited to the last technological trends, namely the investigation of new

cooling methods and alternative coolants instead of air. Therefore, the proposed model is expected to

be a useful tool in the field of innovative gas turbine cycle analysis, replacing more computationally

intensive and very time-consuming models.

Keywords

Gas Turbine, Blade Cooling, Heat Transfer, Thermodynamics.

1 Introduction

Both thermal efficiency and power output of a gas turbine are known to increase with Turbine Inlet

Temperature (TIT). Therefore, over the past decades, aircraft and power generation gas turbine

designers have focused on increasing TIT until reaching the current temperature levels of 1500-

2000K in most advanced applications. Such high operating temperatures by far exceed the maximum

allowable blade material temperatures, so that an efficient cooling system is essential in order to

maintain the material temperatures under a defined threshold value. Turbine cooling is obtained by

the extraction of a small portion of air from the high-pressure compressor stages, which bypasses the

combustor and is reintroduced directly in the turbine to cool its most sensitive components. While on

the one hand such relatively cold air used as cooling medium induces a great benefit to the

components operational life, on the other hand, its extraction represents a direct loss of engine thermal

efficiency and power output. Therefore, maximum cooling with minimum cooling air flow rate is the

design goal. The best cooling is one that provides an as uniform as possible material temperature in

the blade (so to reduce thermal stresses) and with a peak value lower than the maximum allowable

for the blade material to avoid creep. Many improvements in blade cooling system design have been

made thanks to the industrial relevance of the problem and to the constant development of

Computational Fluid Dynamics (CFD) and of metallurgical methods over the last years.

2

In the preliminary design of gas turbine first-stage blading (both statoric and rotoric) and of their

cooling system, sophisticated or computationally intensive models are not required; they would result

exceedingly resource-consuming, due to the excessive computational burden required to attain a

satisfactory accuracy on a very complex geometry and the need for a less computationally demanding

solution is felt at this project stage. It is hence important for designers to rely on simpler models that

can be further refined later in the engineering process, mostly based on bulk quantities and one-

dimensional simplified equations, in order to arrive at approximate yet meaningful estimates of

temperatures and flow requirements.

El-Masri proposed a now classical, simple lumped model for expansion in cooled turbines in 1986

[5]. His aim was to quantify the turbine-cooling losses for different types of cooling methods, as a

detailed prediction of them was feasible for specific designs and operating conditions only. Many

models of blade cooling were published since [1, 4, 11, 12, 13]. Some of them arose as natural

developments of the El-Masri model, like those proposed by De Paepe [4] and by Bolland and Stadaas

[1], others as different approaches to the same problem, but with the same goals. In particular,

important contributions were made by Jordal [13] and Horlock [11] in their studies on effects on gas

turbine cycle performance induced by turbine cooling.

In this paper, a simple lumped thermodynamic model is adopted for the cooling of the first stage

(stator and rotor) blading. The aim is to predict the one-dimensional surface temperature along the

chord (streamwise) of a gas turbine first stage vane and blade, for a given gas temperature profile

along the stator and rotor channel mid-streamline. Then, once the blade metal temperatures are

known, a proper Ni-based superalloy is chosen in order to prevent blade material from creep at that

stress level. Finally, the required cooling air flow rates needed to guarantee such material

temperatures are estimated both for the vane and for the blade.

2 Turbine blade cooling: state-of-the-art The most common blade cooling methods currently implemented in the turbine industry belong to

two main categories: internal and external cooling.

Internal cooling is realized by making the coolant flow through properly shaped channels in the

interior of the blade, removing by internal convection a portion of the heat transferred to the blade by

the hot external mainstream, thus reducing the blade metal temperature. To make cooling systems

more efficient and reduce compressed air extraction, different internal cooling techniques have been

proposed [6, 7, 8, 9, 10]. Most popular are: jet impingement, rib turbulated cooling and pin-fin

cooling. Their names originate from the different features, commonly known as turbulence promoters,

adopted to enhance the heat transfer coefficient by augmentation of the flow turbulence and/or surface

area.

External cooling consists in ejecting the coolant that passes in the blade internal cooling channels

through small holes drilled into the airfoil external shell. The resulting ejected air jets are promptly

squeezed towards the wall by the main flow, and create a thin, cooler, and insulating film along the

external surface of the turbine blade, whence the name “film cooling”. This film provides blade

protection from the hot combustion gases, thus keeping the blade temperature lower than that of an

internally cooled blade, which helps increasing blade operational life. On the other hand, the non-

neglegible interference between the injected coolant and the main flow makes the positive effect of

the film cooling technique global rather than local and thus more complicated to predict.

The latest major improvement to turbine blade material technology was the development of Thermal

Barrier Coatings (TBC), which increased turbine blade temperature capability thanks to their very

low thermal conductivity λ [0.5-2 W/(mK)], improving blade lifetime as a consequence, and they

improved corrosion and oxidation resistance as well, both of great concern as temperatures increases.

A detail representation of how the different cooling methods are currently combined in stator vanes

and rotor blades, to adequately cool them, is given in Fig. 1.

3

3 The proposed lumped thermodynamic model The proposed model is totally based on prime principles

(mass and energy balances) and heat transfer correlations.

The main difference with respect to the other models that can

be found in the literature (see section 1.2) is in the treatment

of the blade energy balance. In particular, the heat transfer

process between the blade and the two flows (hot gas and

coolant) is performed by a local thermal balance on an

infinitesimal control volume, shown in Fig 2. This

elementary volume embeds the blade, spanning its entire

height l and extending on both the pressure and suction sides

for a length t/2 equal to half of the blade spacing. Its

meridional (streamwise) length is equal to dx, where x is a

curvilinear adimensional coordinate measured along the

blade chord starting from the leading edge (x=0) and ending

in the trailing edge (x=1).

A distinction is made between the use of internal convection cooling alone and combined with film

cooling, so that two different procedures are applied to both the stationary vane and the rotating blade.

The aim is to estimate the blade surface temperature along the chord and the coolant mass flow rates.

3.1 Internal Convection Cooling

The first goal of the model is to calculate the metal temperature profile along the chord on the blade

external surface Tbe(x), which are the maximum temperatures experienced by the material. A local

energy balance is applied to the elemental control volume by equalling the infinitesimal heat dqgas(x)

lost by the hot gas during its expansion along dx to the heat transferred to the blade by external

convection dqconv(x):

𝑑𝑞𝑔𝑎𝑠(𝑥) = 𝑐𝑝 [𝑇𝑔(𝑥) − (𝑇𝑔(𝑥) +𝑑𝑇𝑔(𝑥)

𝑑𝑥𝑑𝑥)] = −𝑐𝑝

𝑑𝑇𝑔(𝑥)

𝑑𝑥𝑑𝑥

𝑑𝑞𝑐𝑜𝑛𝑣(𝑥) = ℎ𝑒𝑥𝑡(𝑥) (𝑇𝑔(𝑥) − 𝑇𝑏𝑒(𝑥)) 𝑑𝐴𝑏 = ℎ𝑒𝑥𝑡 (𝑇𝑔(𝑥) − 𝑇𝑏𝑒(𝑥)) 2𝑙𝑑𝑥

𝑑𝑞𝑔𝑎𝑠(𝑥) = 𝑑𝑞𝑐𝑜𝑛𝑣(𝑥)

Combining the above expressions with equations 3.1 and 3.2:

−𝑐𝑝

𝑑𝑇𝑔(𝑥)

𝑑𝑥𝑑𝑥 = ℎ𝑒𝑥𝑡(𝑥) (𝑇𝑔(𝑥) − 𝑇𝑏𝑒(𝑥)) 2𝑙𝑑𝑥

(3.1)

(3.2)

(3.3)

Figure 2. The elemental control

volume [14]

Figure 1. Schematic view of a modern gas turbine vane (a) and

blade (b) with common cooling techniques [10]

(a) (b)

4

From eq. 3.3 we obtain:

𝑇𝑏𝑒(𝑥) = 𝑇𝑔(𝑥) +𝑐𝑝

2 ∗ 𝑙 ∗ ℎ𝑒𝑥𝑡(𝑥)∗

𝑑𝑇𝑔(𝑥)

𝑑𝑥

In the assumption of knowing the gas temperature profile along the channel Tg(x) and the values of

the external heat transfer coefficient hext(x), and assuming a constant cp, Tbe(x) can be computed from

eq. 3.4. First-stage turbine rotating blades are the limiting components of the gas turbine, since they

are the most thermally and mechanically loaded airfoils. In particular, for such airfoils long-term

creep resistance is the most relevant life-limiting factor. Therefore, a creep test on the rotor blade

must be performed in order to choose the proper superalloy able to withstand such hot gas

temperatures, without failure risks, for a prescribed number of operating hours. The Larson-Miller

Parameter (LMP) relation is used for this purpose:

LMP = T (C + log t)

where C is a material specific constant often approximated as 20, t is the stress-rupture time in hours

and T is the temperature in Kelvin.

The heat transferred to the blade surface is computed from eq. 3.2. The same heat is transferred within

the blade by thermal conduction, and Tbi(x) can be computed from eq. 3.6:

𝑑𝑞𝑏𝑙𝑎𝑑𝑒(𝑥) =𝜆

𝑠(𝑥)(𝑇𝑏𝑒(𝑥) − 𝑇𝑏𝑖(𝑥))2𝑙𝑑𝑥 = 𝑑𝑞𝑐𝑜𝑛𝑣(𝑥)

In the assumption that the heat flux dqblade(x) is totally absorbed by the coolant (no heat transfer to

the disk and to the casing), the following relation holds:

𝑑𝑞𝑖𝑛𝑡(𝑥) = ℎ𝑖𝑛𝑡(𝑥) 𝑟 (𝑇𝑏𝑖(𝑥) − 𝑇𝑐(𝑥))𝑑𝐴𝑐 = 𝑑𝑞𝑏𝑙𝑎𝑑𝑒(𝑥) (3.7)

Where r is a geometry factor expressly introduced to represent

the different heat transfer enhancements that are currently

adopted. In particular, according to their effectiveness, whose

schematic representation is given Fig. 3, they can be expressed

by means of factor r as:

- r=1 Smooth surface

- r=1.2 Jet impingement

- r=1.4 Turbulence promoters

- r=1.6 Pin fins

In the calculations presented here below, only the smooth-

channel configuration (r=1) will be analyzed, so that the worst

heat transfer conditions are tested. Finally, the heat transferred per unit of length q(x) can be analytically computed, since it turns out to

be equal to any of these different expressions:

𝑞(𝑥) = −𝑐𝑝

𝑑𝑇𝑔(𝑥)

𝑑𝑥= ℎ𝑒𝑥𝑡(𝑥) (𝑇𝑔(𝑥) − 𝑇𝑏𝑒(𝑥)) 2𝑙 =

𝜆

𝑠(𝑥)(𝑇𝑏𝑒(𝑥) − 𝑇𝑏𝑖(𝑥))2𝑙

= ℎ𝑖𝑛𝑡(𝑥) 𝑟 (𝑇𝑏𝑖(𝑥) − 𝑇𝑐(𝑥))2𝑙 [𝑊

𝑚]

Approximating the cooling channels to cylinders, the heat q(x) is then integrated for each portion of

the blade chord related to a certain cooling cylinder. Thus, for a number j of cylinders a correspondent

number of Δqj is obtained, whose sum is the whole heat transferred to the blade from the hot gases.

∆𝑞𝑗 = ∫ 𝑞(𝑥)𝑑𝑥𝑥𝑗

0

In the assumption that each of these Δqj is absorbed only by the coolant that flows into the

correspondent cylinder, the following relation can be written for the coolant in each cylinder:

(3.4)

(3.6)

(3.5)

(3.9)

(3.8)

Figure 3. Schematic view of the

enhance systems effectiveness [4]

5

𝛥𝑞𝑗 = 𝑚𝑐𝑐𝑝𝑐𝛥𝑇𝑐𝑗= 𝑚𝑐𝑐𝑝𝑐 (𝑇𝑐𝑜𝑗

− 𝑇𝑐𝑖𝑗)

Expressing the coolant mass flow rate as a fraction i of the gas mass flow rate:

𝑖 =𝑚𝑐

𝑚𝑔

→ 𝑚𝑐 = 𝑖 𝑚𝑔

Equation 3.10 can be rewritten as:

𝛥𝑞𝑗 = 𝑚𝑐𝑐𝑝𝑐𝛥𝑇𝑐𝑗= 𝑖 𝑚𝑔𝑐𝑝𝑐𝛥𝑇𝑐𝑗

The coolant temperature Tcj progressively increases along the blade height, from Tcij to Tcoj, as the

coolant heats up while flowing through the channels. A growing trend with blade height should hence

be given to Tcj to simulate such phenomenon. However, since in this work a two-dimensional analysis

of an airfoil section is performed, no 3-D radial phenomena can be included. Therefore, the Logarithm

Mean Temperature Difference (LMTDj) that drives the heat transfer between the blade and the coolant

in each cylinder is defined as:

𝐿𝑀𝑇𝐷𝑗 =∆𝑇𝑐𝑜𝑗

− ∆𝑇𝑐𝑖𝑗

𝑙𝑛 (∆𝑇𝑐𝑜𝑗

∆𝑇𝑐𝑖𝑗

)

=(𝑇𝑏𝑖𝑗

− 𝑇𝑐𝑜𝑗) − (𝑇𝑏𝑖𝑗

− 𝑇𝑐𝑖𝑗)

𝑙𝑛 ((𝑇𝑏𝑖𝑗

− 𝑇𝑐𝑜𝑗)

(𝑇𝑏𝑖𝑗− 𝑇𝑐𝑖𝑗

))

Knowing the internal heat transfer coefficient hint and the finite heat transfer area Ac of each cylinder,

relation 3.7 can be rewritten for the j-th cylinder as:

𝛥𝑞𝑗 = ℎ𝑖𝑛𝑡 𝑔 𝐿𝑀𝑇𝐷𝑗 𝐴𝑐

From which the LMTDj can be computed:

𝐿𝑀𝑇𝐷𝑗 =∆𝑞𝑗

ℎ𝑖𝑛𝑡 𝑔 𝐴𝑐

For a given coolant inlet temperature in the cooling system, the coolant temperature difference

between inlet and outlet in each cylinder ΔTcj, as well as Tcij and Tcoj, for different values of i, can be

estimated from the following relation, derived from eq. 3.12:

𝛥𝑇𝑐𝑗=

𝛥𝑞𝑗

𝑚𝑐 𝑐𝑝𝑐

=𝛥𝑞𝑗

𝑖 𝑚𝑔 𝑐𝑝𝑐

The internal blade temperatures Tbij on the cylinders surfaces can now be computed for each cooling

ratio. The following formula derived from the LMTD definition (3.13) is used:

𝑇𝑏𝑖𝑗=

𝑇𝑐𝑖𝑗− ( 𝑇𝑐𝑜𝑗

𝑒

(𝑇𝑐𝑜𝑗−𝑇𝑐𝑖𝑗

)

𝐿𝑀𝑇𝐷𝑗 )

1 − 𝑒

(𝑇𝑐𝑜𝑗−𝑇𝑐𝑖𝑗

)

𝐿𝑀𝑇𝐷𝑗

By joining all the Tbij temperatures, different profiles for the blade internal surface temperature Tbint(x)

can be obtained for each cooling mass flow rate. However, the more realistic temperature profile for

Tbi(x) is that derived from eq. 3.6, since it is related to the blade external surface temperature Tbe(x)

and to the thermal conductivity λ of the blade material. Therefore, in order to assess the coolant mass

flow rate required to guarantee such blade internal temperature Tbi(x), a comparison between it and

the various temperature profiles Tbint(x) is made. To this extent, one must consider that the selected

coolant mass flow rate will be the one for which the two profiles most closely match. As criterion of

choice the standard deviation is used, calculated as follows for each value of i:

𝜎𝑖 = √∫ |(𝑇𝑏𝑖𝑛(𝑥) − 𝑇𝑏𝑖(𝑥))2| 𝑑𝑥1

0

(3.14)

(3.15)

(3.16)

(3.17)

(3.18)

(3.11)

(3.12)

(3.13)

(3.10)

6

The value of i for which the standard deviation is lowest will be the needed cooling mass ratio, from

which then the absolute value of the required coolant mass flow rate can be assessed.

3.2 Film cooling

A different method is adopted to assess the coolant mass flow rates when film cooling is also applied.

The internal cooling channels are always approximated by cylinders, with the addition of the cooling

holes. Firstly, q(x) is computed from eq. 3.8, then, applying the same procedure as internal cooling,

the Δqj are computed from eq. 3.9, while the ΔTcj, Tcij and Tcoj are calculated from eq. 3.16 for each i-

th cooling mass ratio. A film cooling efficiency is introduced to compute the film temperature Tf(x),

defined as:

𝜀𝑓 =(𝑇𝑔(𝑥𝑓) − 𝑇𝑓)

(𝑇𝑔(𝑥𝑓) − 𝑇𝑐𝑜𝑗)

Once a value for the efficiency is assigned, such formula is applied to each cooling hole, in the

assumption that the film temperature remains constant between one hole and the following one.

Although this is certainly a strong assumption, it can be considered acceptable within the present

context. Indeed, in a physical sense this is the only reasonable assumption. The temperature Tgas is

considered as the gas temperature at each hole position xf along the chord. The value for Tcoj, which

is the temperature at which cooling air is discharged into the main flow, is assigned according to the

cylinder to which the hole refers.

𝑇𝑓(𝑥𝑓) = 𝑇𝑔(𝑥𝑓) − 𝜀𝑓 (𝑇𝑔(𝑥𝑓) − 𝑇𝑐𝑜𝑗)

A film temperature profile Tf(x) for each value of i is then obtained by fitting such Tf values. The heat

transfer is now between the cooling layer and the blade surface, so a new heat transfer equation can

be written:

𝑞(𝑥) = ℎ𝑓(𝑥) (𝑇𝑓(𝑥) − 𝑇𝑏𝑒(𝑥)) 2𝑙

Where hf is the heat transfer coefficient under film cooling conditions, which is known to be higher

than hext. Such increase in the heat transfer coefficient is accounted for by introducing an enhancement

factor u, taken equal to 1.5, so that:

ℎ𝑓(𝑥) = 𝑢 ℎ𝑒𝑥𝑡(𝑥) = 1.5 ℎ𝑒𝑥𝑡(𝑥)

Eq. 3.21 is then used to calculate the blade external surface temperature Tbe(x) for each cooling

percentage i, with the resulting following equation:

𝑇𝑏𝑒(𝑥) = 𝑇𝑓(𝑥) −𝑞(𝑥)

1.5 ℎ𝑒𝑥𝑡(𝑥)2𝑙

The first value of i (coolant mass flow rate) that provides a temperature profile Tbe(x) that has a

maximum temperature below the maximum allowable by the material is hence chosen.

4 Cooling analysis of a representative gas turbine first stage The proposed model was implemented in MATHEMATICA© in order to carry out the calculations.

Two separate codes were written to account for the different set of equations needed by internal

convection and film cooling analysis. In particular, for each cooling method, two additional sub-codes

were written to apply the model to both the turbine first-stage stationary vanes and rotating blades.

The first input for the MATHEMATICA© code is the hot gas temperature profile along the mid-

streamline of both stator and rotor channels. Since neither reliable nor realistic gas temperature

profiles have been found in the literature, and industrial data are unavailable as well, it was decided

to obtain the needed profiles from CFD simulations. Therefore, numerical 2-D simulations of the flow

on the midspan section of a gas turbine first stage vane (internally cooled by convection) and of a

(3.23)

(3.19)

(3.20)

(3.21)

(3.22)

7

rotor blade were performed. The midspan section was chosen for the sake of generality, so that the

calculations could then be equally adapted to any airfoil section (e.g. hub and tip).

The simulations were carried out with the commercial solver ANSYS Fluent 15.0. It is important to

remark in advance that all the successive results strongly depend on the particular choice of the gas

turbine, whose first-stage configuration parameters are listed in Table 1, while the geometry

parameters for the NGV and rotor blade are listed in Table 2 and their midspan sections are shown in

Fig.4. The assigned inlet boundary conditions values are all listed in Table 3. The static temperature

maps in the stator and rotor channel obtained from the simulations are plotted in Fig. 5.

Configuration parameters

Power output, P [MW] 4

Diameter, D [m] 0.5

Rotational speed, ω [s-1] 1200

Velocity, U [m/s] 300

Blades, z 44

Gas mass flow rate, mg [kg/s] 45

Euler work, weul [KJ/kg] 90

Degree of reaction, Rρ 0.5

ψ = V2t/U 1

Geometry parameters NGV Rotor

Blade pitch, t [m] 0.045 0.033

Blade height, l [m] 0.04 0.04

Blade chord, c [m] 0.04 0.03

Blade maximum spanwise

thickness, smax [m] 0.008 0.008

Gas Turbine Inlet

Boundary Conditions

Turbine Inlet Temperature [K] 1672

Gauge pressure [bar] 18

Mass flow-rate [kg/s] 0.65

Flow velocity [m/s] 136

Table 2. Selected gas turbine first stage

NGV and rotor blade geometry parameters

Table 1. Selected gas turbine first

stage configuration parameters

Table 3. Inlet boundary conditions

assigned for the simulations

Figure 4. NGV and rotor blade midspan section

Figure 5. Contours of Static Temperature in the stator and rotor channels and representation

of the channel mid-streamlines (blue lines) on which temperature values are extrapolated

8

The temperature values on the stator and rotor channel mid-streamline section corresponding to the

real gas expansion (blue lines in Fig. 5) are extrapolated and imported in the MATHEMATICA© code

for the cooling analysis. The obtained expansion gas temperature profiles Tg(x) are plotted in Fig. 6

as a function of a non-dimensional chord length, going from 0 to 1, on the x-axis.

4.1 Internal Convection Cooling Results

Blade Temperature Calculation

According to the set of equations presented in section 3.1, knowing the gas temperature profiles Tg(x),

a trend for the external convective heat transfer coefficient hext(x) is needed to compute the blade

external surface temperature Tbe(x) from eq. 3.4. Many typical trends can be found in the literature,

most of them of experimental origin. The following trends for the NGV and the rotor blade, given by

Han [6], were selected for our calculations. The obtained metal temperature profiles on the NGV and

rotor external surface Tbe(x) are plotted in Fig. 8, while their values are listed in table 4.

x 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

NGV Tbe(x) [K] 1229 1199 1123 1130 1075 1010 994 960 990 1092 1215

Blade Tbe(x) [K] 1233 1183 1192 1104 1021 1192 1052 1043 1061 1210 1222

Figure 6. Gas temperature profiles along stator (a) and rotor (b) channel mid-streamline

0.0 0.2 0.4 0.6 0.8 1.0

1640

1650

1660

1670

Tg(x)

0.2 0.4 0.6 0.8 1.0

1610

1620

1630

1640

Tg(x)

(a) (b)

Figure 7. External convective heat transfer coefficient trend along the NGV (a) and blade (b) chord

0.0 0.2 0.4 0.6 0.8 1.0

500

600

700

800

900

1000

1100

hext

0.0 0.2 0.4 0.6 0.8 1.0

400

600

800

1000

hext

(a) (b)

Table 4. External surface temperature values along the non-dimensional chord x

T [K] T [K]

x x

x x

h [W/m2K]

h [W/m2K]

9

Thermal gradients within the NGV and blade

material along the chord (red line in Fig. 9),

were computed by eq. 3.6, with a material

thermal conductivity equal to 22 W/(mK)

(common value for Ni-based superalloys). As a

consequence, knowing Tbe(x), the temperature

profiles Tbi(x) along the chord on NGV and

blade internal surfaces (Fig.10) can be

computed by subtracting such thermal gradients

from Tbe(x).

Creep test on rotor blade

Considering the most advanced technology for the selected turbine, a third-generation Single Crystal

(SC) superalloy, which is currently used for gas turbine blades, is chosen for the test: CMSX-10.

Rotor blades are subjected to both mechanical stresses induced by centrifugal force and thermal

stresses due to thermal gradients within blade material. Therefore, the maximum total stress can be

expressed by relation 4.1, and according to the selected turbine parameters (Table 1), the computed

maximum thermal gradient value within the blade material (ΔTmax=23 K) and CMSX-10 density

equal to 9.05 g/cm3, we obtain:

𝜎𝑡𝑜𝑡 = 𝜎𝑐 + 𝜎𝑡 =𝜌𝜔2

2(𝑟𝑒𝑥𝑡 − 𝑟) ∗ 10−6 + 𝐸 𝛼 ∆𝑇𝑚𝑎𝑥 = 130 𝑀𝑝𝑎

0.2 0.4 0.6 0.8 1.0

1000

1050

1100

1150

1200

1250 Tbe(x)

(a) (b)

0.0 0.2 0.4 0.6 0.8 1.0

1000

1100

1200

1300

Figure 8. NGV (a) and blade (b) external surface temperature profiles

Tbe(x)

Figure 9. Internal paths representing NGV (a)

and blade (b) chord

(a) (b)

0.2 0.4 0.6 0.8 1.0

900

1000

1100

1200 Tbi

(x)

0.2 0.4 0.6 0.8 1.0

900

1000

1100

1200

1300

Tbi

(x)

(a) (b)

Figure 10. NGV (a) and blade (b) internal surface temperature profiles

(4.1)

T [K] T [K]

x x

x x

T [K] T [K]

10

The resulting Stress/Density ratio is equal to 14.4

MPa/g/cm3, which, from the CMSX-10 Larson-Miller

curve, corresponds to a LMP value equal to 30.2 (Fig.

11). Applying the LMP relation (eq. 3.5) for a blade

lifetime expected to be equal to 24000 operating hours,

and assuming material constant C=20, the maximum

allowable temperature is given by:

𝑇𝑏𝑒𝑚𝑎𝑥=

𝐿𝑀𝑃 ∗ 103

(𝐶 + log 𝑡𝑟) = 1235 𝐾

Such value is higher than the maximum Tbe(x) values at

the leading edge of both the rotor blade (Tbemax=1233 K),

and the NGV (Tbemax=1229K). Therefore, such superalloy

is proper to be used as it prevents the blade from creep for

the estimated time.

Preliminary assessment of cooling air flow rates

The cooling system must be designed according to Tbi(x) trend. In particular, the cooling air has to be

injected in the cylinders that are in the coldest portion of the blade. A common practice is to split the

total amount of injected coolant into two or three portions, each of them following a specific winding

path through the internal channels. It was decided to split the coolant into two equal mass flows in

the NGV and into three mass flows in the blade, following the paths shown in Fig. 12. For a coolant

inlet temperature equal to 750 K, the coolant temperatures along each cylinder and thus the blade

internal temperature profiles Tbint(x), shown in Fig.13, can be computed as explained in section 3.1

for different cooling mass flow rates, i.e. for i going from 0.05 to 0.09.

Figure 11. Larson-Miller stress-rupture

curves for third generation SC Ni-based

superalloy CMSX-10 [3]

Figure 12. Coolant paths in NGV (a) ad rotor blade (b) cooling cylinders

mc1 mc2 mc1 mc2 mc3

(a) (b)

0.0 0.2 0.4 0.6 0.8 1.0

1000

1050

1100

1150

1200

1250

Tbint(x)

0.0 0.2 0.4 0.6 0.8 1.0

950

1000

1050

1100

1150

1200

1250

Tbint(x)

(a) (b)

Figure 13. Internal temperature profiles along NGV (a) and rotor blade (b) chord for each value of

i (0.06-0.09)

x x

T [K] T [K]

11

The cooling air flow rates were

assessed according to the

procedure presented in section

3.1. Plots of the resulting

standard deviation values for

each i are given in Fig. 14 for

both the NGV and rotor blade.

The values of i for which the

standard deviation is lower are

respectively 0.075 and 0.06.

4.2 Film Cooling Results

Blade Temperature Calculation

A film cooling efficiency equal to 0.4 was introduced to compute the film temperature values Tf for

each hole from eq. 3.20. A film temperature profile Tf (x) for each value of i going from 0.05 to 0.1

(Fig. 15) was hence obtained by joining consecutively such Tf values. The NGV and rotor blade

external surface temperature profiles Tbe(x) were computed for each value of i going from 0.05 to 0.1

(Fig. 16) according to the procedure presented in section 3.2.

Preliminary assessment of cooling air flow rates

The first value of i that provides a temperature profile Tbe(x) that has a maximum temperature below

the maximum allowable by the material (1235K) resulted to be equal to 0.06 for the NGV and to

0.055 for the rotor blade.

Figure 14. Standard deviation values for each i in the NGV (a)

and rotor blade (b)

0 2 4 6 8

50

60

70

80

90

i = 0.075

0 2 4 6 8 10 12

60

80

100

120

i = 0.06

(a) (b)

0.0 0.2 0.4 0.6 0.8 1.0

1380

1400

1420

1440

1460

1480 Tf(x)

0.0 0.2 0.4 0.6 0.8 1.0

1340

1360

1380

1400

1420

1440

1460

1480

Tf(x)

(a) (b)

Figure 15. Film temperature profiles for each i (0.05-0.1) in the NGV (a) and rotor blade (b)

0.0 0.2 0.4 0.6 0.8 1.0

1000

1050

1100

1150

1200

1250

1300

Tbe(x)

0.2 0.4 0.6 0.8 1.0

1000

1050

1100

1150

1200

1250 Tbe(x)

Figure 16. NGV (a) and rotor blade (b) external surface temperature profiles for each i (0.05-0.1)

x x

x x

T [K] T [K]

T [K] T [K]

12

i 0.05 0.055 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 0.1

Tbemax

[K](x=0)

NGV 1253 1239 1227 1220 1214 1209 1205 1201 1197 1194 1190

Blade 1248 1228 1218 1207 1197 1288 1280 1274 1268 1263 1258

5 Conclusions All the coolant mass flow rates estimated by the model are in accordance with some well-known

considerations, which confirms the reliability of the model. When internal cooling alone is used, much

more cooling air is required, while in both cooling methods less coolant is needed to cool the rotor

blades with respect to the stator vanes. Therefore, despite of its limitations, the method outlined in

this thesis is a useful design tool when the goal is to obtain a sufficiently accurate first estimate of the

cooling requirements in a quick and reliable way, with which a preliminary cycle calculation aiming

at efficiency prediction can be done. Furthermore, the greatest advantage of the model is that it can

be easily adapted to any operating condition as well as different process parameters, cooling methods,

fluid properties and blade geometries. Such versatility is particularly well suited for the last

technological trends, namely the investigation of new cooling methods (like porous and transpiration

cooling) and alternative coolants instead of air.

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Table 5. External surface temperature maximum values at the NGV and rotor blade leading

edge (x=0) for each value of i (0.05-0.1)

13

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