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CHANNEL FLOW MODELING OF IMPINGEMENT

COOLING OF A ROTATING TURBINE BLADE

by

JaHye Jenny Koo

GT&PDL Report No. 181

GAS TURBINE & PLASMA DYNAMICS LABORATORY

^MASSACHUSETTS INSTITUTE OF TECHNOLOGY

CAMBRIDGE, MASSACHUSETTS

/
CHANNEL PLOW MODELING OF IMPINGEMENT

COOLING OF A ROTATING TURBINE BLADE

by

JaHye Jenny Koo

GTSPDL Report No. 181 December 1984

This research was supported by the NASA Lewis Research Center

under Grant No. NAG3-335.
ABSTRACT

Local heat transfer distributions in impingement cooling have been

measured by Kreatsoulas [1] and Preiser [2] for a range of conditions which

model those in actual turbine blades, including the effects of rotation.

These data were reported as local Nusselt numbers, but referred to coolant

supply conditions. By means of a channel flow modeling of the flow in the

supply and impingement passages, the same data are here presented in terms

of local Nusselt number distributions such as are used in design. The

results in this form are compared to the nonrotating impingement results of

Chupp [3] and to the rotating but non-impingement results of Morris [4],

Rotation reduces the mean Nusselt numbers from these found by Chupp by

about 30 percent, and introduces important radial variations which are

sensitive to rotation and to leading edge stagger angle.

KJ^SS^CT »oon-

^MULL PCO^ ^OCL

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ACKNOWLEDGEMENTS

I would like to express my deepest appreciation to my advisor, teacher

and friend, Professor Jack L. Kerrebrock. It was his enthusiasm, optimism

and endless encouragement, as much as his extensive knowledge and hard work

into this project throughout the year, which led to the completion of this

work.

Special thanks to Professor Alan H. Epstein, Dr. Robert Norton, and

Dr. Hyoun-Woo Shin for their comments and suggestions in the initial stage

of the program.

My sincere thanks to Messrs. Bob Haimes, Mike Giles, and Mark Drela

for their patience in answering my numerous questions. Their personal

warmth and invaluable assistance throughout the program helped me survive

my first year at MIT.
TABLE OF CONTENTS

Page

Abstract

 2

Acknowledgements

 3

Table of Contents

 4

Nomenclature 5

List of Tables and Figures

 7

I. Introduction

 8

II. Experimental Apparatus

 11

III. Formulation of the Model

 13

IV. Data Analysis

 20

V. Results and Discussions

 24

VI. Conclusion

 28

References

 29

Tables

 30

Figures

 32
NOMENCLATURE

English Symbols

A cross-sectional area (sq. m)

Cp specific heat under constant (J/Kg K)

d impingement jet diameter (m)

D leading edge diameter (m)

de equivalent diameter (A/s) (m)

f friction factor

h heat transfer coefficient W/sq.m K)

k thermal conductivity (W/m K)

M Mach number

Nu Nusselt number

p pressure (N/sq.m)

P jet hole pitch (m)

r position vector

r radius (m)

Re Reynolds number

s perimeter (m)

S surface area (sq.m)

S jet span (m)

St Stanton number

Ts average surface temperature .- (K)

Tf average fluid temperature (K)

T£ total fluid temperature (K)
Lw effective wall temperature

u velocity vector

u velocity component in r direction

Vj impingement jet velocity

z wall to jet distance

Greek Symbols

Y Cp/Cv = specific heat ratio

Y stagger angle

\i viscosity

p density

n angular velocity

Subscripts

1 impingement channel

2 supply channel

(K)

(m/sec)

(m/sec)

(ra)

(rad)

(Nsec/sq.m)

(kg/cu.m)

(rad/sec)
LIST OF TABLES AND FIGURES

Tables

Table 1:

Table 2:

Table 3:

Test section geometry

Summary of -30 degree stagger angle tests

Summary of 0 degree stagger angle tests

Figures

Figure 1:

Figure 2:

Figure 3:

Figure 4:

Figure 5:

Figures 6-22:

Figures 23-34:

Experimental apparatus schematic drawing

Blade model cross-section (side view)

Flow in radial coolant passage

Elementary radial volume

Test section geometry (top view)

Local flow distribution and Nu for 30 degree

stagger angle tests

Local flow distribution and Nu for 0 degree

stagger angle tests

Page

30

31

31

32

33

34

34

36

70
I. INTRODUCTION

One of the most challenging tasks for advancing gas turbine technology

is to improve gas turbine efficiency by Increased turbine inlet temperature,

while maintaining low enough metal temperature levels to achieve acceptable

component life. To maintain acceptable temperatures of turbine components,

it is necessary to develop effective cooling methods, and the importance of

using coolant air in airfoils has been repeatedly emphasized. One of the

most effective cooling methods for local internal cooling of turbine blades

is jet impingement, which is achieved by blowing cooling air through a

series of holes located inside the blades. The effectiveness of this

internal cooling technique is measured by the heat transfer coefficient

which gives the heat flux per temperature difference between the airfoil

and the coolant.

The accurate prediction of the local convection heat transfer coefficient is essential in order to optimize the use of the coolant supply and

minimize thermal stresses. Although numerous impingement heat transfer

studies have been conducted and reported, most of the available data have

been measured in stationary systems, so they do not address the rotational

effects in impingement cooling systems. Recently, an extensive set of data

on impingement cooling with rotational effects has been obtained by Kreatsoulas [1]. These experiments have revealed large effects due to rotation,

on both the mean Nusselt number (averaged on the leading edge surface) and

on the detailed heat transfer distribution, which is created by the impinging jets. All of Kreatsoulas1 experiments were conducted with a -30 degree

stagger angle, typical of the inlet of a high work rotor at the mean radius.
10

modeled as channel flows, using friction coefficients and heat transfer

coefficients which represent averages around the circumference of the

passages, and are functions of the radial coordinate along each passage.

Because of the complexity of the flow, it has been possible to describe

the effectiveness of impingement cooling only with reference to specific

geometries. Thus, Ref. [3] gives correlations in terms of the jet Reynolds

number, jet diameter, and the geometric parameters. We describe the

effects of rotation relative to this correlation.

Some experiments have been conducted in simple rotating cylindrical

passages which show the effects of rotation in such simple geometries and,

again, we have compared the data of Kreatsoulas and Preiser with this data,

to show the effect of the impingement jets on the heat transfer in the

rotating passage. Within the channel flow approximation adopted here, the

effects of rotation which are encompassed in the empirical correlations, at

least in first approximation, are: 1) the radial pressure gradient in the

supply and impingement passages, and its dependence on the heat addition in

each passage, which is important in that it controls the flow rates in the

impingement orifices, 2) the effect of buoyancy on the heat transfer, and

3) the averaged effect of Coriolis induced cross flow in the passages.
11

II. EXPERIMENTAL APPARATUS

The experimental apparatus for study of impingement cooling in a

rotating system was mainly designed and developed at the MIT Gas Turbine

Laboratory and is described in detail by Kreatsoulas [1]. A brief description of the apparatus will be discussed in this section in order to present

the modeling of the flow in the impingement region.

The apparatus consists of the rotor, the supporting structure, heat

exchangers, instrumentation, calibrating body and blade model, as illustrated

in Figure [1]. The blade model, which simulates the leading edge geometry

of a real turbine blade, rotates in a vacuum chamber. The vacuum chamber is

used to reduce heat transfer by convection from the outer wall to the

environment, as well as to eliminate any absorbing medium between radiometer

and measurement spot. The infrared radiometer measures the external

surface temperature of the foil with a spatial resolution of about 1mm.

The local heat transfer coefficients throughout the leading edge region of

the blade are then determined from the measured electrical input power and

the skin temperature distribution.

Figure [2] shows a geometric description of the flow system of the

test model. A thin stainless steel foil (Kanthai A-1), a high resistivity

and low thermal coefficient of resistivity material, is electrically heated

by dissipation in the skin while being cooled from one side by impinging

jets of refrigerant gas (Freon 12). The coolant enters into the supply

plenum, flows through the jet holes to the impingement section, and

exhausts radially to the exhaust channel. Inside the cooling passage of

the model, type E thermocouples and pressure probes are placed to measure
13

III. FORMULATION OF THE MODEL

Figure [3] illustrates a tube rotating about an axis with a constant

angular velocity. Fluid flows in the tube and the motion of this fluid is

referred to as a reference coordinate system (r,9,z). The acceleration

vector of a fluid particle, which has a position vector r and a velocity

vector u, relative to the rotating coordinate system is

*

 = Dt + ^^ + B*^**) (1

 )

Du/Dt, 2flxu, and ftx(flxr) refer respectively to the total derivative of the

velocity vector, Coriolis acceleration, and the centrifugal acceleration.

The Navier-Stokes equation may then be approximated as

P 5£

 + 2nxu + nx(nxr) - -vp + uV u (2)

where p, \i, and p are the density, viscosity, and pressure of the fluid

respectively. For a steady flow in the radial direction, Eq. (2) may be

simplified as

Equation (3), however, does not include the effects of the incoming

jets from the supply channel, which we shall assume enter the impingement

passage with zero radial momentum. If an elementary radial volume, as

shown In Fig. 4, is considered, the momentum balance becomes

dp du . d , 4

— —*- o Du + U —•— (ou)

dr dr dr

From continuity,

dA
15

The energy equation can be derived in a similar manner as the equation

of motion by taking a radial element. The heat transfer rate across the

surface dS is written in terms of the heat transfer coefficient and the

temperature difference as

dq - h(T - T)dS (14)

As a first approximation, the heat transfer rate is estimated by

taking the temperature difference between the average temperature of the

fluid T and the effective wall temperature Tw, which is assumed to be

constant along the passage. Tw is defined as

„ Vs * Vf T

w ° A, + A2

 (15)

where

AI = impingement channel cross-sectional area

A2 = supply channel cross-sectional area

T8 = average surface temperature

Tf = average fluid temperature

In the final heat transfer coefficient calculation, Tw is replaced

with the measured local wall temperature, and T is replaced with the local

fluid temperature. This is to be described in detail in IV.

The average heat transfer coefficient, as a first approximation, is

estimated by applying the following analogies. The Reynolds analogy, which

assumes complete similarity of momentum and heat transfer, provides the

following equation.

St-J=5 H_

Pr Re pc u
17

where the total temperatures

2

T

t, ~

 T1

 +

1 Ic~

 P

and (22)

T - T + 2

t 2 2c 2 p

The final six differential equations describing the flow characteristics

derived for the regions of interest are presented in non-dimensional form

as follows.

Impingement region,

1-M2 du ,rd r T p V UYM

2 Tt dA

2

1+yM, dA. . t.

A, dr P.U..C T .2

1

 M1 1 p t. dr

dTt ,d r T PV, Tt dA

(23)

-1..2 p.u. M c T .2

1 1 p t dr

dV

r 1
19

where M = Mach number.

The measured total temperatures and the static pressures at the passage

hub of both the impingement and supply channels are prescribed as initial

conditions for integration of these equations. The velocities at the

passage hub of the regions are prescribed as ui = 0 and U2 = &/P2A2

 wnere

m is the total mass flow rate entering the supply region.
20

IV. DATA ANALYSIS

The solutions of the differential equations derived in III enable one

to obtain the local flow distribution, which further gives the circumferentially averaged flux to the coolant along the radius. In this section, a

method of determining the local flow distribution and heat transfer

distribution is outlined.

From the measured heat fluxes Kreatsoulas and Preiser calculated, the

local heat transfer coefficient h of the leading edge surface using the

definition

h = q/(T - T ) (30)

c c

where Tc is the measured inlet coolant temperature, and q and T are the

measured local heat flux and skin temperature, respectively. The value of

hc, calculated using Eq. (30), is based on coolant inlet temperature to

the blade, so it includes a number of complex effects, such as heating of

the fluid in the supply passage and variations in jet Reynolds number due

to differential pressure drops in the supply and impingement passages. A

more useful correlation of the local heat transfer distribution may be

obtained by basing h on the local fluid temperature Tf in the jets, and

by correlating in terms of the local jet Reynolds number. Thus,

hf = q/(T - Tf) (31)

For the channel flow analysis, the heat transfer coefficient is averaged

around the circumference of the passage along the radius, and Eq. (30)

becomes,
22

Chupp, et al. (Eq. 34) which is expressed as a function of geometric

parameters and the jet Reynolds number.

Nu = 0.63 Re"' P - exp[-1.27 -3 - - ~] (34)

avg d VPJ VDJ VdJ\Pj W

Nu is an arithmetic average of the Nusselt number for strips located in the

leading edge region. Re3 is the jet Reynolds number which is defined as

Re = Vjp2d/ii where d is the hole diameter. The friction factor is also

recalculated as a function of local Reynolds number.

f=^|i (35)

An approximate flow distribution is determined from the calculated

average Nusselt number and friction factor based on Eqs. (34) and (35),

respectively, with the prescribed initial conditions.

The calculated and measured values of the pressure ratio between

impingement and supply regions at the inlet are compared. Then an iteration

method is applied to satisfy the continuity and zero mass flow at the tip

of the supply region.

In the next and final iteration, the Stanton number defined in Eq. (16)

is calculated from the average heat transfer, calculated by using Eq. (32).

Subsequently, the new local heat transfer averaged in the theta direction,

which is based on the temperature difference between the local skin

temperature and fluid temperature, is calculated as defined in Eq. (33).

This analysis is applied to all the test data obtained by Kreatsoulas

and Preiser. The new calculated heat transfer distribution is then compared

with the correlations obtained by other investigators to ensure that the
24

V. RESULTS AND DISCUSSION

The independent parameters of primary interest in the present study

are Reynolds number, rotational speed, heat input, and stagger angle.

Tables 2 and 3 summarize the series of parametric studies performed by

Kreatsoulas and Preiser which are considered in this analysis. Figures 6A

to 34C are the graphical representation of the calculated local flow

distribution for all the cases. In Figures 6A to 22C, the velocity,

pressure, and temperature of the fluid in both the Impingement and supply

channels are plotted for each test conducted at -30 degree stagger angle.

The results of the local flow distribution calculated for 0 degree stagger

angle are similarly illustrated in Figs. 23A to 34C. On the temperature vs

radius graphs, the temperatures measured by themocouples at the passage hub

and tip are also plotted.

In the impingement passage, the initial condition was applied that Ttl

should be equal to the value given by the thermocouple at the base of the

impingement passage, and in the region below the first jet, conduction in

the radial direction was included in the energy equation. This is why the

temperature shows an initial drop before rising due to convective heating.

The outer thermocouple gave fluid temperatures quite different from

those computed, and quite inconsistent with the measured heat transfer

values. We infer that such measurements are unreliable in the complex,

thermally stratified flow found in the impingement passage.

Figures 6D to 22D and 23D to 34D present the Nusselt numbers averaged

around the circumference of the passage along the radius for the 30 and 0

degree stagger angle tests. Rotation introduces important radial variations
26

is subject to minimal cross flow and buoyancy effects, it is dominated by

the tangential velocity effect and deflects toward the hub. The third jet,

under the influence of increasing buoyancy and cross flow, is deflected

toward the tip and hence creates a large thermal gradient.

The effect of Reynolds number on heat transfer can be examined by

comparing Figs. 13D, 15D, and 17D. In all three cases, the tests were

conducted at medium rotational speed and high temperature ratio at 30

degree stagger angle. As the average jet Reynolds number varies from 17000

to 74000, the heat transfer rate increases about three times, the average

values ranging from 100 to 280. Similar trends are shown for the low and

high speed cases but, at the high rotational speed, the thermal gradients

between the jet holes along the passage are more pronounced. For the 0

stagger cases, Figs. 26D and 3ID show the increase in heat transfer rate by

a factor of 2 as the Reynolds number increases from low to medium. In all

the cases, the temperature ratio between the average skin temperature and

the coolant temperature seems to have almost no effect on Nusselt number as

shown in Figs. 21D and 22D.

The stagger angle effect on impingement-cooled rotating blades is

examined by comparing the data obtained by Kreatsoulas and Preiser, which

are conducted at 30 and 0 stagger angles, respectively, holding all other

nominal conditions the same. Figures 22O and 34D show that the heat

transfer rate are lower at zero degrees by approximately 30%, and the high

thermal gradient effect due to the dominant tangential velocity effect

between the second and third jets is not seen at this angle.

Also indicated in Figs. 6D to 34D are the comparable Nusselt number
28

VI. CONCLUSION

A channel flow model of flow in the impingement cooling passage of a

rotating turbine blade leading edge is used to correct measured heat

transfer coefficents to local fluid conditions. The flows in the supply

and impingement regions are both modeled as steady state channel flows,

using friction and heat transfer coefficents. The local flow properties

are obtained by solving the momentum, mass and energy equations with the

prescribed initial conditions, and using the local heat transfer rates

calculated from the data of Kreatsoulas and Preiser. The local Nusselt

number, based on calculated local fluid temperature, is re-evaluated by

circumferentially averaging the heat transfer rate. When this result is

compared with the Nusselt number, based on the inlet coolant temperature,

the maximum difference between them is determined to be about 17 percent.

By comparing the calculated Nusselt numbers with the correlations obtained

by other investigators, it is concluded that the experimental and modeling

formulations are satisfactory. The modeling and the analysis illustrated

here, therefore, offer a means for more accurate prediction of impingement

cooling heat transfer rates.
29

REFERENCES

1. Kreatsoulas, J.C., "Experimental Study of Impingement Cooling in

Rotating Turbine Blades," Ph.D Thesis, MIT Department of Aeronautics

and Astronautics, September 1983.

2. Preiser, Uriel Z., "Stagger Angle Effects on Impingement Cooling of a

Rotating Turbine Blade," M.S. Thesis, MIT Department of Aeronautics and

Astronautics, May 1984.

3. Chupp, R.E., Helms, H.E., McFadden, P.W. and Brown, T.R., "Evaluation

of Internal Heat Transfer Coefficients for Impingement Cooled Turbine

Airfoils," Journal of Aircraft, Vol. 6, 1969, pp. 203-208.

4. Morris, W.D. and Ayhan, T., "Observations on the Influence of Rotation

on Heat Transfer in the Coolant Channels of Gas Turbine Rotor Blades,"

Proceedings of the Institute of Mechanical Engineers, Vol. 193, 1979,

pp. 303-311.
30

TABLE 1: TEST SECTION GEOMETRY

Span

Hub radius

Tip radius

Leading edge diameter

Stagger angle (from axial)

Impingement hole diameter

Impingement hole pitch

Wall to jet distance

Impingement insert span

S

Rh

Rt

D

Y

d

P

z

Si

101.6 mm

406.4 mm

508.0 mm

12.7 mm

0.523 rad

2.1 mm

6.1 mm

4.1 mm

76.2 mm

4.000 in

16.000 in

20.000 in

0.500 in

30.0 deg

0.081 in

0.240 in

0.162 in

3.000 in
31

TABLE 2: -30 degree stagger angle tests

TEST //

43

. 42

39

44

40

41

57

56

55

. 54

59

58

63

65

64

66

67

ROTATIONAL SPEED

low

low

low

low

low

low

medium

medium

medium

medium

medium

medium

high

high

high

high

high

RE //

low

low

medium

medium

high

high

low

low

medium

medium

high

high

low

med ium

medium

high

high

TW/TC

low

high

low

high

low

high

low

high

low

high

low

high

low

low

high

low

high

TABLE 3: 0 degree stagger angle tests

TEST //

113

114

115

123

124

122

119

120

121

128

126

127

ROTATIONAL SPEED

low

low

low

medium

medium

medium

medium

medium

medium

high

high

high

RE //

low

medium

medium

low

low

medium

medium

medium

medium

med ium

medium

high

TW/TC

high

high

high

high

hifeh

low

high

high

high

low

high

high
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33 ORIGINAL PA@E-8§

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SUPPORTING STRUCTURE

IMPING EMENT INSERT

?ig. 2: Blade Model Cross Section (Side 7iew)

1. T • Thermocouple

2. P • Pressure Tap

Thermocouple and pressure tap locations

1.

2.

3.

4.

0.3905 m

0.4747 m

0.4008 m

0.4921 m

measured from the center of the shaft
34

r,u

m,

Figure 3: Flow in radial coolant passage

r.u

1 >*£"

dr <pu) dr

t P.pu1

Figure 4: Elementary radial volume
35

ROTATION

25.4 mm

1 in.

Fig 5: Test Section Geometry ( Top View)

1. Supply Plenum

2. Jet Hole

3. Impingement Space

4. Izooingenent Insert

5. Rubber Seal

S. Cover

7. Resistive Wall
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