Turbine Design Manual

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CHAPTER 4

PART 2 AXIAL TURBINE DESIGN MANUAL

Dr K W RAMSDEN

DIRECTOR – GAS TURBINE TECHNOLOGY PROGRAMMES DEPARTMENT OF POWER AND PROPULSION

SCHOOL OF ENGINEERING CRANFIELD UNIVERSITY CRANFIELD, BEDFORD MK43 0AL

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DISCLAIMER

SCHOOL OF ENGINEERING

DEPARTMENT OF POWER AND PROPULSION

These notes have been prepared by Cranfield University for the personal use

of course delegates. Accordingly, they may not be communicated to a third

party without the express permission of the author.

The notes are intended to support the course in which they are to be

presented as defined by the lecture programme. However the content may

be more comprehensive than the presentations they are supporting. In

addition, the notes may cover topics which are not presented in the

presentations.

Some of the data contained in the notes may have been obtained from public

literature. However, in such cases, the corresponding manufacturers or

originators are in no way responsible for the accuracy of such material.

All the information provided has been judged in good faith as appropriate for

the course. However, Cranfield University accepts no liability resulting from

the use of such information.

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SUMMARY

This document facilitates the aerodynamic design of both a low and high pressure

turbine allowing the user to work step by step through the calculation procedure.

The turbines are matched to a two spool compressor having an overall pressure ratio

of 16.

One of two alternative turbine entry temperatures may be chosen, namely, 1250K or

1650K representative of industrial and aeronautical technology, respectively.

The HP turbine RPM is chosen at 15000 whilst that of the LP is estimated by limiting

the LP compressor stage one rotor tip relative Mach number to 1.15.

In both cases, the turbines have a mean diameter of 0.45m.

The inlet Mach number to the HP turbine is 0.30 and the corresponding axial velocity

is maintained constant throughout.

A critical assessment is carried out in terms of likely performance and, where

appropriate, suggestions made for modifications taking into account the prescribed

application.

The results calculated by the user can be directely compared with the values

appended.

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AXIAL TURBINE DESIGN MANUAL

CONTENTS

PAGE

BACKGROUND NOTES

NOTATION AND UNITS 1

1.0 INTRODUCTION 2A

TWO SHAFT ARRANGEMENT 2B

2.0 SPECIFICATION

2.1 THE COMPRESSOR SYSTEM 3

2.2 THE HP TURBINE SYSTEM 4

3.0 HP TURBINE DESIGN CONSTRAINTS 5

4.0 HP TURBINE ANNULUS DIAGRAM 5

5.0 HP TURBINE DESIGN TABULATION

5.1 OVERALL SPECIFICATION 6

5.2 INLET ANNULUS GEOMETRY 6

5.3 EFFICIENCY PREDICTION 6

5.4 OUTLET ANNULUS GEOMETRY 7

6.0 HP TURBINE FREE VORTEX DESIGN

6.1A DESIGN TABULATION - TET = 1250K 8A

6.1B VELOCITY TRIANGLES - TET = 1250K 8B

7.0 HP TURBINE DESIGN ASSESSMENT

7.1A DESIGN SUMMARY - TET = 1250K 10A

7.1B RECOMMENDATIONS - TET = 1250K 10B

8.0 HP TURBINE DESIGN ASSESSMENT

8.1B RECOMMENDATIONS TET = 1650 K 11B

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CONTENTS ( CONTINUED )

PAGE 9.0 LOW PRESSURE TURBINE DESIGN

9.1 LP COMPRESSOR SPECIFICATION 12

9.2 LP COMPRESSOR DESIGN CONSTRAINTS 12

9.3 ESTIMATION OF LP COMPRESSOR ( LP TURBINE ) RPM 13 10.0 LP TURBINE OVERALL DESIGN

10.1 OVERALL SPECIFICATION 14

10.2 HP TURBINE EXIT ANNULUS GEOMOETRY 14

10.3 INTER-TURBINE ANNULUS GEOMETRY ESTIMATION 15

10.4 LP TURBINE EFFICIENCY PREDICTION 16

10.5 LP TURBINE OUTLET ANNULUS GEOMETRY 17

11.0 LP TURBINE FREE VORTEX DESIGN

11.1B VELOCITY TRIANGLES - TET =1250K 18B

12.0 LP TURBINE DESIGN ASSESMENT

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AXIAL TURBINE DESIGN MANUAL CONTENTS (CONTINUED) ANNEXES ANNEX A PAGE SUMMARY OF CONTENTS A1

A 1.O HP TURBINE DESIGN TABULATION

A 1.1 OVERALL SPECIFICATION A2

A 1.2 INLET ANNULUS GEOMETRY A2

A 1.3 EFFICIENCY PREDICTION A2

A 1.4 OUTLET ANNULUS GEOMETRY A3

A 2.0 HP TURBINE FREE VORTEX DESIGN

A 2.11 DESIGN TABULATION - TET = 1250K A4A

A 2.1B VELOCITY TRIANGLES-TET = 1250K A4B

A 2.2A DESIGN TABULATION - TET = 1650K A5A

A 2.2B VELOCITY TRIANGLES- TET = 1650K A5B

A 3.0 HP TURBINE DESIGN ASSESSMENT

A3.1A DESIGN SUMMARY - TET = 1250K A6A

A 3.1B DESIGN SUMMARY - TET 1650K A6B

ANNEX B

B 1.0 GUIDNACE NOTES FOR CALCULATIONS B1

ANNEX C

GAMMA = 1.40 C1 AND C2

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CONTENTS (CONTINUED) ANNEXES

ANNEX D

PAGE D 1.0 SMITH'S EFFICIENCY CORRELATION D1

ANNEX E

E1.0 LOW PRESSURE TURBINE DESIGN TABULATION

E1.1 ESTIMATION OF LP COMPRESSOR (LP TURBINE) RPM E1

E1.2 LP TURBINE INLET ANNULUS GEOMETRY E2

E1.3 LP TURBINE EFFICIENCY PREDICTION E2

E1.4 LP TURBINE OUTLET ANNULUS GEOMETRY E3

E2.0 LOW PRESSURE TURBINE FREE VORTEX DESIGN

E2.1A DESIGN TABULATION - TET = 1250K E4A

E2.1B DESIGN TABULATION - TET = 1650K E4B

E3.0 LOW PRESSURE TURBINE FREE VORTEX DESIGN

E3.1A DESIGN TABULATION - TET = 1250K E5A

E3.1B DESIGN TABULATION - TET = 1650K E5B

E4.0 LOW PRESSURE TURBINE DESIGN ASSESSMENT

E4.1A DESIGN SUMMARY - TET = 1250K E6A

E4.1B DESIGN SUMMARY - TET = 1650K E6B

ANNEX F

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-1-NOTATION AND UNITS

SYMBOLS UNITS

A Cross sectional area m2

Cp Specific heat at constant pressure Joules / kg.K

D Diameter m

h Annulus height m

H Stagnation enthalpy Joules / kg

M Mach number

N Revs per minute min. -1

p Static pressure n/m2

P Stagnation pressure n/m2

q Mass flow function (WT /Ap ) 1/( Joules kg/K )

Q Mass flow function (WT /AP ) 1/( Joules kg/K )

R Gas constant Joules/kg.K

Rc Compressor pressure ratio Rov Overall pressure ratio

t Static temperature K

T Stagnation temperature K

U Blade speed m/sec

V Velocity m/sec

W Mass flow kg/sec

 Gas angle degrees

 Ratio of specific heats



Change in:

 Work done factor

ABBREVIATIONS SUFFICES

BMH Blade mid height a Axial



isent Isentropic efficiency ann Annulus



poly Polytropic efficiency in Stage inlet

FAR Fuel air ratio mean At mid height

HP High pressure out outlet

LP Low pressure R (or H) At the root (or hub)

NGV Nozzle guide vane T At the tip or casing

stoi. Stoichiometric w Whirl direction

TET Turbine entry temperature 0 Nozzle outlet (abs)

1 Rotor inlet (rel) 2 Rotor outlet (rel) 3 Rotor outlet (abs)

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-2A-1.0 INTRODUCTION

This Document facilitates the aerodynamic design of both a low and high pressure turbine allowing the user to work step by step through the calculation procedure.

The turbines are matched to a two spool compressor having an overall pressure ratio of 16. One of two alternative turbine entry temperatures may be chosen, namely 1250K or 1650K, representative of industrial and aeronautical technology, respectively.

The HP turbine RPM is chosen at 15000 whilst that of the LP is estimated by limiting the LP compressor (stage one) rotor tip relative Mach number to 1.15.

In both cases, the turbines have a mean diameter of 0.45m.

The inlet Mach number to the HP turbine is 0.3 and the corresponding axial volocity is maintained constant throughout.

A critical assessment is carried out in terms of likely performance and where appropriate, suggestions made for improvements taking into account the prescribed application.

The results estimated by the user may be compared with values appended. The following design constraints are imposed

:-Constant axial velocity

Constant mean diameter = 0.45m

RPM = 15000

50% reaction at blade mid height

Free vortex flow distribution

Axial HP inlet flow with a Mach number of 0.3

Straight sided annulus walls

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2B

LPC

HPC

HPT

LPT

TWO SHAFT TURBOJET (OR TURBOFAN CORE ENGINE)

FIGURE 1

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-3-2.0 SPECIFICATION 2.1 THE COMPRESSOR SYSTEM.

The compressor system has the following specification :

Inlet temperature _(T1) 300

Inlet pressure _{(P1 )} 101325

Overall pressure ratio (Rov) 16.0

LP pressure ratio (Rc) 3.56

HP pressure ratio (Rc) 4.494

HP RPM _(Nhp) 15000

Polytropic efficiency (



_poly) ( both spools ) 0.90

Mass flow (W) 40.0

With these data and the formulae below, the following can be calculated :

LP COMPRESSOR HP COMPRESSOR Pressure ratio 3.560 4.494



isent 0.882 0.879 Inlet temperature 300 449 Temperature rise T 149 274 Outlet temperature 449 723 Power = W. Cp.T (megawatts) 5.99 11.03 NOTE :

1 R

poly 1 c γ 1 γ c isent







                1 R T T 1 -c isent 1 and

1 R

Cp





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-4-2.0 SPECIFICATION 2.2 THE HP TURBINE SYSTEM

The hp turbine is required to supply only the hp compressor power since it is assumed that there are no mechanical losses.

The turbine mass flow is the compressor flow plus the fuel flow. The latter is obtained by calculating the fuel flow and hence the fuel/air ratio (FAR) required to raise the compressor outlet temperature to the specified TET. This is calculated based on an enthalpy balance. The corresponding values of FAR are shown in the table below assuming a combustor efficiency of 100%.

The mean specific heat is calculated from values of Cp for both air as well as for the combustion products. See for example Walsh and Fletcher.

Cp air_{= ao + a1 X+ a2X}2 + a_3X3 + a_4X4... Where X = (T/1000) Cp kerosene =Cp f= _{bo + b1 X+ b2X}2 + b_3X3 + b_4X4... Cp comb_gas =Cp air+(FAR/(1+FAR))* Cp f R=287.05-0.0099FAR+1e-7(FAR2) A0 0.992313 B0 -0.71887 A1 0.236688 B1 8.747481 A2 -1.852150 B2 -15.8632 A3 6.083152 B3 17.2541 A4 -8.89393 B4 -10.2338 A5 7.097112 B5 3.081778 A6 -3.23473 B6 -0.36111 A7 0.794571 B7 -0.00392 A8 -0.08187 A8 -0.71887

Based on a similar, but slightly different, approach the following values are used here:

Compressor outlet temperature (K) 723 723

Turbine entry temperature (K) 1250 1650

Combustor temperature rise (K) 526.7 927

Fuel / Air Ratio (FAR) 0.0159 0.0289

Mass Flow (air +fuel) (Kg/s) 40.64 41.16

HP Turbine Power (megawatts)

(To drive hp compressor)

11.03 11.03

Mean specific heat - Cp (joules/Kg.K) 1184 1275.5

Inlet stagnation pressure - Pin (n/m2) (Assumes 5% Combustor pressure loss)

1540140 1540140

Ratio of specific heats,

 = 1/(1-R/Cp)

1.32

1.29

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-5-3.0 HP TURBINE DESIGN CONSTRAINTS.

The following design constraints are imposed

:-Axial inlet flow with a Mach number of 0.3 Constant axial velocity

Constant mean diameter RPM = 15000

50% reaction at blade mid height Free vortex flow distribution Straight sided annulus walls Constant mean diameter = 0.45m

The assumption of constant axial velocity would require an iteration on NGV exit gas angle,



o, so that mass flow continuity is satisfied.

The annulus area distribution would then be an automatic outcome of the calculations.

For simplicity, however, it is assumed that the annulus is straight sided (see the diagram below). This introduces only a small error.

Additionally, it is assumed that the exit plane of the NGV is half way along the annulus. This implies that the axial chord of the NGV is greater than that of the rotor which allows a reasonable spacing between the blade rows.

4.0 HP TURBINE ANNULUS DIAGRAM. The following general annulus configuration is used

:-AXIS h out h in L / 2 L NGV BLADE D mean

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-6-5.0 HP TURBINE DESIGN TABULATION. 5.1 OVERALL SPECIFICATION.

TET 1250 1650

Mass flow W (Kg / s) 40.64 41.16

Power (megawatts) 11.03 11.03

Specific Heat Cp (and



) 1184 (1.32) 1275.7 (1.290)

5.2 INLET ANNULUS GEOMETRY. P = 16 x 101325 x 0.95

Inlet Mach Number 0.30 0.30

Q = W.T / A.P

(See Tables - ANNEX C )

A = W.T / Q.P

h = A / (



.Dmean)

Dtip = Dmean + h

Dhub = Dmean - h

Hub/Tip Ratio = Dhub / Dtip

5.3 EFFICIENCY PREDICTION - (MEAN HEIGHT) Temperature Drop T = Power / W.Cp

Umean = U = RPM.



Dmean / 60

H/U2 = CpT /U2

Va /Tin

( for Min = 0.3, See ANNEX C - use appropiate



) Va

Va / U



isent (Smith's Chart value minus 2 %) (See Annex D)

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-7-5.0 HP TURBINE DESIGN TABULATION ( CONT. )

5.4 OUTLET ANNULUS GEOMETRY.

TET 1250 1650

Va

T3 = Tin -T

Work done factor  0.98 0.98

Vw = (H/U2) . U/ Vw3 = (Vw-Umean) /2 (50 % Reaction)



3 = tan-1 (Vw3/Va) V3 = Va/Cos3 V3/T3

M3 (See ANNEX C, use appropiate )

Q3 (See ANNEX C)

R = (1-T/ (



_{isent. Tin))} /(-1) P3 = Pin x Rov (See note below)

A3 = W.T3/ P3.Q3

Aann = A3 / Cos3

h = Aann / (



Dmean)

Dtip = Dmean + h

Dhub = Dmean - h

Hub/Tip Ratio = Dhub/Dtip

NOTE: _{P3 = Pout (In the direction of V3)}

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-8A-6.0 HP TURBINE-FREE VORTEX DESIGN 6.1A DESIGN TABULATION - TET = 1250K

ROOT BMH TIP

D (NGV exit) = (Din + Dout) /2 D (Rotor exit) (See Table 5.4 - page 7) Va (Constant radially) Vw3mean (See Table 5.4 - Page 7) Vwomean = (Vw-Vw3) mean (See Table 5.4)

Vwo = Vwomean x Dmean/D (D at NGV exit)



o = tan-1 (Vwo / Va) Vw3 = Vw3mean . Dmean/D (D at rotor exit)



3 = tan-1 (Vw3 / Va) U (For exit velocity triangles) = Umean . D/Dmean (D at rotor exit)

Vo = Va / Coso

Nozzle Acceleration, Vo / Vin (= Vo / Va)

V1 =(Va2+(Vwo-U)2)



1 = Cos-1 (Va / V1) V2 =(Va2+(U+Vw3)2)



2 = Cos-1 (Va / V2) Rotor Acceleration, V2 / V1

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-8B-6.0 HP TURBINE-FREE VORTEX DESIGN (CONT) 6.1B VELOCITY TRIANGLES - TET = 1250 K

From the data provided on Page A4A, draw below the velocity triangles appropriate to the stage at Root, Blade Mid Height and Tip.

NOTE: USE A SCALE OF 1cm = 100m/s

TIP

BMH

ROOT

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-9A-6.0 HP TURBINE-FREE VORTEX DESIGN 6.2A DESIGN TABULATION - TET = 1650K

D (NGV exit) = (Din+Dout)/2 D (rotor exit) (See Table 5.4 - page7)

Va (Constant radially)

Vw3mean (See Table 5.4 - page 7) Vwomean = (Vw-Vw3)mean (See Table 5.4) Vwo = Vwomean x Dmean/D (D at NGV exit)



o = tan-1 (Vwo/Va) Vw3 = Vw3mean x Dmean/D (D at rotor exit)  3 = tan-1 (Vw3/Va)

U (For exit velocity triangles) = Umean x D/Dmean (D at rotor exit)

Vo = Va/Coso

Nozzle Acceleration, Vo/Vin = Vo/Va

V1 =(Va2+(Vwo-U)2)  1 = Cos-1 (Va/V1) V2 =(Va2+(U+Vw3)2)  2 = Cos-1 (Va/V2) Rotor Acceleration, V2/V1

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-9B-6.0 HP TURBINE-FREE VORTEX DESIGN (CONT) 6.2b VELOCITY TRIANGLES - TET = 1650K

From the data provided on Page A5A, draw below the velocity triangles appropriate to the stage at Root, Blade Mid Height and Tip.

NOTE: USE A SCALE OF 1cm = 100m/s

TIP

BMH

ROOT

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-10A-7.0 HP TURBINE DESIGN ASSESSMENT. 7.1A DESIGN SUMMARY - TET = 1250K NOTE: See ANNEX B for method of calculation.

AT BLADE MID HEIGHT NGV EXIT BLADE EXIT Static temperature

Speed of sound Absolute Mach number Axial Mach number

DATA FROM PAGE A4A

HUB TO CASING ROOT BMH TIP

NGV Exit Gas Angle o Nozzle Deflection, o+in Rotor Deflection, ₁₊₂

Nozzle Acceleration Vo / Vin Rotor Acceleration _{V2 / V1}

Exit swirl, 3

Reaction

STAGE OVERALL DATA

Inlet hub/tip ratio (See Page A2) Outlet hub/tip ratio (See Page A3)

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-10B-7.0 HP TURBINE DESIGN ASSESSMENT 7.1B RECOMMENDATIONS - TET = 1250 K

(SEE PAGE A6A for data)

(A) ARE THE AXIAL MACH NUMBERS OK ?

(B) IS THE NGV LEAVING GAS ANGLE ACCEPTABLE ?

(C) IS THE ROTOR EXIT SWIRL ACCEPTABLE ?

(D) ARE THE GAS DEFLECTIONS OK ?

(E) IS THE ROTOR ROOT ACCELERATION OK ?

(F) IS THE NGV TIP ACCELERATION OK ?

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-11A-8.0 HP TURBINE DESIGN ASSESSMENT. 8.1A DESIGN SUMMARY - TET = 1650K NOTE: See ANNEX B for method of calculation.

AT BLADE MID HEIGHT

NGV EXIT

BLADE EXIT

Static temperature Speed of sound Absolute Mach number Axial Mach number

DATA FROM PAGE A5A

NGV Exit Gas Angle

o

Nozzle Deflection



o+



in Rotor Deflection



1+



2

Nozzle Acceleration Vo / Vin Rotor Acceleration _{V2 / V1}

Exit Swirl

3

Reaction

STAGE OVERALL

DATA Inlet hub/tip ratio

(See Page A2) Outlet hub/tip ratio (See Page A3)

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-11B-8.0 HP TURBINE DESIGN ASSESSMENT 8.1B RECOMMENDATIONS - TET = 1650 K

(SEE Page A6B for data)

(D) ARE THE GAS DEFLECTIONS OK?

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-12-9.0 LOW PRESSURE TURBINE DESIGN

9.1 LOW PRESSURE COMPRESSOR SPECIFICATION

The low pressure compressor has the following specification (See Page 3)

Inlet temperature Tin 300

Inlet pressure Pin 101325

Mass flow W 40

Polytropic efficiency

poly

0.90 Isentropic efficiency

isent

0.88

Compressor power 5.99 megawatts

9.2 LOW PRESSURE COMPRESSOR DESIGN CONSTRAINTS The following design assumptions are

made:-Axial inlet flow (no inlet guide vanes)

Inlet axial Mach number Ma = 0.5 Rotor tip relative Mach number _{M1 = 1.15}

Mean diameter Dmean = 0.45

The compressor RPM is limited to that value corresponding to a maximum rotor relative tip Mach number of 1.15. Accordingly, the following velocity triangle applies at the tip of the first stage

rotor:-M

U tip

Ma = 0.5

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-13-9.3 ESTIMATION OF LP COMPRESSOR (LP TURBINE) RPM

The following tabulation gives the sequence of calculations to estimate blade tip speed and RPM.

(See also velocity triangle at the rotor tip shown on page 12).

Ma 0.5

Va /Tin ( See ANNEX C, for = 1.4 ) Va

Qin = W.Tin / Pin.Ain Ain

hin = Ain/(

.

Dmean )

Dtip = Dmean + hin

Dhub = Dmean - hin

Hub/Tip Ratio = Dhub / Dtip

Tin/tin (See ANNEX C, for



= 1.4) t in

V1 _{= M1}



(



_{R tin )}

Utip _{= (V12 - Va2)}

RPM = 60.Utip/(



Dtip )

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-14-10.0 LP TURBINE OVERALL DESIGN 10.1 OVERALL SPECIFICATION.

LP TET 1021 1440

Mass flow 40.64 41.16

Power (megawatts) 5.99 5.99

Specific heat, Cp (and) 1184 (1.32) 1275.7 (1.290)

RPM 10980 10980

Blade mid height reaction 50% 50%

10.2 HP TURBINE EXIT ANNULUS GEOMETRY (SEE PAGE A3)

TET 1250 1650 Dmean 0.45 0.45 Dtip = Dmean + h 0.529 0.524 Dhub = Dmean - h 0.371 0.376 h = (Dtip-Dhub)/2 0.079 0.074 A =.Dmean.h 0.112 0.105

Hub/Tip Ratio = Dhub / Dtip 0.702 0.718

Va 205.1 233.0

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-15-10.3 INTER-TURBINE ANNULUS GEOMETRY ESTIMATION

The factors concerning selection of inter-turbine axial space and annulus flare angle are considered in ANNEX F. Accordingly, an annulus flare of 300_{( included angle ) is selected}

with an axial space of 0.00635m. This is an example estimate for a closely spaced blade rows. For your own designs select spacings based on the values of local upstream chord as discussed in the lectures (e.g. St≈0.25Cax)

The lp inlet annulus area is then estimated using the hp exit values of Table 10.2 and the inter-turbine data in table 10.3 below.

The inter-turbine geometry is shown diagramatically below

:-y y 15 0.00635 D mean AXIS HP EXIT LP INLET o

TABLE 10.3 LP TURBINE INLET ANNULUS GEOMETRY

LP TET. 1021 1440

LP Turbine inlet pressure ( See Table A1.4 ) 583713 768530

Dmean 0.45 0.45

Dtip _{= Dtip (hp exit) + 2y ( See ANNEX F )}

Dhub _{= Dhub (hp exit) - 2y}

h = (Dtip- Dhub)/2

A =



.Dmean . h

Hub / Tip Ratio = Dhub / Dtip

Va = Va(hp exit) x h(hp exit) / h(lp entry)

Vw in (mean) (As for HP exit) 215.4 210.5

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-16-10.4 LP TURBINE EFFICIENCY PREDICTION (SINGLE STAGE AT MID HEIGHT)

LP TET 1021 1440

Temperature Drop = Power / (W.Cp) Blade Speed, Umean = U = RPM.



. D / 60

H/U2 = CpT / U2 (Single Stage)

Va (See Table 10.3 - Page 15)

Va / U



isent (Smith's Chart Value minus 2 %)

NOTE : SEE PAGE E2 FOR SOLUTIONS

THE ABOVE EFFICIENCY PREDICTION IS VALID FOR A SINGLE STAGE TURBINE. THE DESIGNER CAN NOW SELECT A SINGLE OR TWO STAGE DESIGN.

For the low TET ( industrial ) case, a two stage design would probably be preferred to give a high overall efficiency in favour of low weight. If then, the work is split equally, each stage would have aH/U2 of 1.1015 and an efficiency of of approximately 91.5% (see Smith's Chart - ANNEX D ).

It is probable that an equal work split would be chosen since both stages would discharge at near axial leaving velocity.

IMPORTANT NOTE

THE PRELIMINARY DESIGN NOW CONTINUES ASSUMING A SINGLE STAGE LP TURBINE IS FEASIBLE FOR BOTH TET CASES CONSIDERED.

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-17-10.5 LP TURBINE OUTLET ANNULUS GEOMETRY.

LP TET 1021 1440

Va

T3 = Tin -T

Work Done Factor  0.97 0.97

Vw = (H/U2) . U/ Vw = (Vw - Umean )/2 ( 50% Reaction ) 3 = tan -1 (Vw3/Va) V3 = Va / Cos3 V3/T3

M3 (See ANNEX C, use Appropriate )

Q3 ( See Tables-ANNEX C )

Pressure Ratio R = ( 1 -T/ (



_{isent Tin ))} / (-1)

P3 = Pin x R (See note below)

A3 = WT3 / P3 Q3

Aann _{= A3 / Cos3}

h = Aann / (



Dmean)

Dtip = Dmean + h

Dhub = Dmean - h

Hub / Tip Ratio = Dhub/Dtip

NOTE : P3 = Pout ( In the direction of V3 )

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-18A-11.0 LP TURBINE-FREE VOTEX DESIGN 11.1A DESIGN TABULATION - TET = 1250K

D (NGV Exit) = (Din + Dout)/2

D (Rotor exit) (See Table 10.5 - Page 17 or Page E3) Va (Table 10.3, Constant Radially) Vw3mean (See Table 10.5 Page 17 or Page E3)

Vwomean = (Vw - Vw3)mean

(See Table 10.5 Page 17 or Page E3)

Vwo = Vwomean x Dmean / D

(D at NGV exit)

0 = tan -1 (Vw0/Va)

Vw3 = Vw3mean x Dmean / D

(D at Rotor exit)

3 = tan -1 (Vw3/Va)

U for exit velocity triangles = Umean x D/Dmean (D at Rotor exit, Umean Table 10.4)

V0 = Va / Cos0

in _{= tan -1 (Vw3hp. out / Valpin)} (Vw3hp. out - Table 6.1A, Page 8A)

Vin _{= Valpin / Cos in)}

Nozzle Acceleration _{= V0/Vin}

V1 =(Va2+(Vwo-U)2)

1 = Cos -1 (Va/V1)

V2 =(Va2+(U+Vw3)2)

2 = Cos -1 (Va/V2)

Rotor Acceleration _{= V2/V1}

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-18B-11.0 LOW PRESSURE TURBINE - FREE VORTEX DESIGN 11.1B VELOCITY TRIANGLES - TET = 1250 K

(MID HEIGHT REACTION = 50%)

From the data provided in Page E4A, draw below the velocity triangles appropriate to the stage at Root, Blade Mid Height and Tip.

NOTE: USE A SCALE OF 1cm = 100m/s

TIP

BMH

ROOT

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-19A-11.0 LP TURBINE-FREE VORTEX DESIGN 11.2A DESIGN TABULATION - TET = 1650K

D (NGV exit) = (Din + Dout)/2

D (Rotor exit) (See Table 10.5 - Page 17 or Page E3) Va (Table 10.3, Constant Radially) Vw3mean (See Table 10.5 - Page 17 or Page E3) Vwomean = (Vw - Vw3)mean

(See Table 10.5 - Page 17 or Page E3)

Vwo = Vwomean x Dmean / D

(D at NGV exit)

0 = tan -1 (Vw0/Va)

Vw3 = Vw3mean x Dmean/D

(D at Rotor exit)

3 = tan -1 (Vw3/Va)

U (for exit velocity triangles) = Umean x D/Dmean (D at Rotor exit, Umean Table 10.4)

V0 = Va / Cos0

in = tan -1 (Vw3hp. out / Valp.in) (Vw3hp out - Table 6.2A, Page 9A)

Vin _{= Valp. in /Cos in)}

Nozzle Acceleration. V0/Vin

V1 =(Va2+[Vwo-U]2)

1 = Cos -1 (Va/V1)

V2 =(Va2+[U+Vw3]2)

2 = Cos -1 (Va/V2)

Rotor Acceleration. V2/V1

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-19B-11.0 LOW PRESSURE TURBINE - FREE VORTEX DESIGN 11.2B VELOCITY TRIANGLES - TET = 1650 K

(MID HEIGHT REACTION = 50%)

From the data provided on Page E5A, draw below the velocity triangles appropriate to the stage at root, blade mid height and tip.

USE A SCALE OF 1cm = 100m/s

TIP

BMH

ROOT

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-20A-12.0 LP TURBINE DESIGN ASSESSMENT. 12.1A DESIGN SUMMARY - TET = 1250 K NOTE: See ANNEX B for method of calculation.

Speed of sound Absolute Mach number Axial Mach number

DATA FROM PAGE E4A

in

NGV Exit Gas Angle 0 Nozzle Deflection 0+in Rotor Deflection 1+2 Nozzle Accel. Vo/Vin Rotor Accel. _V2/V1 Exit swirl 3 Reaction

STAGE OVERALL DATA Inlet hub/tip ratio

Outlet hub/tip ratio

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-20B-12.0 LP TURBINE DESIGN ASSESSMENT 12.1B RECOMMENDATIONS - TET = 1250 K

(SEE PAGE E6A - DESIGN SUMMARY)

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-21A-12.0 LP TURBINE DESIGN ASSESSMENT. 12.2A DESIGN SUMMARY - TET = 1650K NOTE : see ANNEX B for method of calculation.

Speed of sound Absolute Mach number Axial Mach Number

DATA FROM PAGE E6B

 in

NGV Exit Gas Angle 0 Nozzle Deflection 0+in Rotor Deflection 1+2 Nozzle Acceleration _V0/Vin Rotor Acceleration _V2/V1

Exit Swirl 3

Reaction

STAGE OVERALL DATA Inlet hub/tip ratio

Outlet hub/tip ratio

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-21B-12.0 LP TURBINE DESIGN ASSESSMENT 12.2B RECOMMENDATIONS - TET = 1650 K

(SEE PAGE E6B- DESIGN SUMMARY)

(C) IS THE ROTOR EXIT ACCEPTABLE ?

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ANNEX A

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-A1-APPENDICES.

SUMMARY OF CONTENTS

ANNEX A

Presents the results of the high pressure turbine design.

Design tabulations and velocity triangles are included for free vortex flow distribution. A critical assessment of the alternative designs is included.

ANNEX B

Presents additional guidance notes for calculations. ANNEX C

Contains tables for the compressible flow of air for the three appropriate values of



. ANNEX D

Smith's Efficiency Prediction. ANNEX E

Presents the results of the low pressure turbine design.

Design tabulations and velocity triangles are included for free vortex flow distribution. A critical assessment of the alternative designs is included.

ANNEX F

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ANNEX B

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-B1 and B2-ANNEX B

B 1.0 GUIDANCE NOTES FOR CALCULATIONS.

These notes will assist in the calculations for tables 7.1A, 7.1B, (HP) and 12.1A, 12.1B (LP) of the turbine design assessment.

V

Vw

V

1

0

2

3

0 Vw

a



The above diagram shows the velocity triangles for a stage. The following calculation procedures are

recommended:-AXIAL MACH NUMBER AT NGV EXIT, Ma Ma = Va / (  R to )

Where to = To - (Vo2 / 2Cp) NOTE: To = Tin and from the geometry of the velocity triangles

above:-Vo2 = Va2 + Vwo2

AXIAL MACH NUMBER AT ROTOR EXIT, Ma Ma = Va /( R tout)

Where: tout = t3 = T3 - (V32 / 2 Cp) NOTE: T3= Tin -TStage and from the geometry of the velocity triangles

above:-V32 = Vw32 + Va2

ABSOLUTE MACH NUMBER AT NGV EXIT, Mo Mo = Vo / ( R to)

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-B3 and

B4-ABSOLUTE MACH NUMBER AT ROTOR EXIT, M₃ M₃ from Table 5.4 (HP Turbine) from Table 10.5(LP Turbine) NGV ACCELERATION, Vo / Vin

Vo as above

Vin = Va at inlet to the HP turbine. Vin = V3hp exit at inlet to the LP turbine.

ROTOR ACCELERATION, V₂/ V₁

Where from the velocity triangles above:-V₂= Va / Cos₂

V1 = Va / Cos1

DEFLECTIONS:

Rotor deflection =₁+₂ Where: _

         Va Vw U tan 1 3 2 and:           Va U Vw tan 1 0 1

NGV deflection =o + in Where: in = 0 for HP turbine and: in = ₃_{hp exit} for LP turbine

STAGE REACTION. stage rotor stage rotor

T

t

H

h

0 0

Reaction,











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ANNEX C

COMPRESSIBLE FLOW TABLES

GAMMA = 1.40

PAGE C1 AND C2

GAMMA = 1.32

PAGE C3 AND C4

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(51)

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ANNEX D

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-D1-ANNEX D

D1.0 EFFICIENCY CORRELATION (SINGLE STAGE TURBINES)

REFERENCE: SMITH S F., "A SIMPLE CORRELATION OF TURBINE EFFICIENCY" (Journal of The Royal Aeronautical Society. 69 (1969) 467)

(55)

ANNEX F

(56)

F1 ANNEX F

F1.0 INTER-TURBINE ANNULUS GEOMETRY ESTIMATION This note explains the calculations necessary to complete Table 10.3, page 15.

TURBINE OVERALL ANNULUS GEOMETRY

A X IS H P N O Z Z L E H P B L A DE L P N O Z Z L E L P B L A DE X  

A finite distance, x, is required between the HP exit and LP entry. The value of x is, typically, approximately 25% of the previous blade row axial chord or 1/4 inch. (whichever is larger).

The value of annulus flare angle,



, usually limited to 30o (included), will depend on the magnitude of axial chords chosen for each of the blade rows.

In any event, inter-turbine annulus flare will result in a reduction in the axial velocity between HP turbine exit and LP turbine inlet.

The whirl component of velocity, Vw3, at HP exit will, however, remain unchanged in the inter-turbine space since angular momentum will be conserved.

Since blading considerations are not covered in this design study, the axial distance, x, is assumed to be 1/4 inch. (0.000635m) and annulus flare angle is taken to be 30o (included).

If the annulus height increase between HP exit and LP inlet is 2y, the reduction of axial velocity can be estimated, as

follows:-y = 0.00635 tan (/2) h lp entry = h hp exit + 2y

Where:- _{h hp exit is the annulus height at hp exit.} (See Table 5.4 page 7 or Table A1.4 page A3) Va lp inlet = Va hp outlet. hhp exit / h lp inlet NOTE: _{Vw3hp exit = VWin lp inlet}

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1

Dr. David MacManus , Dr. Ken Ramsden, Dr. Anthony Jackson Gas Turbine Technology Programmes

DEPARTMENT OF POWER AND PROPULSION SCHOOL OF ENGINEERING

CRANFIELD UNIVERSITY

AXIAL TURBINE DESIGN AND PERFORMANCE

Presentation slides v2013-v1.1

1

Turbines - General Bibliography

1. Japikse, D., “Introduction to turbomachinery”, Oxford University Press, 1997.

2. Cohen, H., Rogers, G., and Saravanamuttoo, H., “Gas turbine theory”, Longman Scientific and Technical, 3rd_{Edition, 1987.}

3. “The jet engine”, Rolls-Royce plc, 5th_{Edition, 1996.}

4. Cumpsty, N., “Jet propulsion”, Cambridge University Press, 1997.

5. Dixon, S., ”Fluid mechanics and thermodynamics of turbomachinery”, Butterworth-Heinemann, 4th Edition, 1998.

6. Turton, R., “Principals of turbomachinery”, E.&F.N. Spon, 1984.

7. Lakshminarayana, B., “Fluid dynamics and heat transfer of turbomachinery”, John Wiley and Sons, 1996.

8. Van Wylen, G., Sonntag, R., “Fundamentals of classical thermodynamics”, John Wiley and Sons, 1985.

9. Wilson, D., Korakianitis, T., “The design of high-efficiency turbomachinery and gas turbines”, 2nd Edition, Prentice Hall, 1998.

11. Mattingley, J., et al.”Aircraft engine design”, AIAA education Series, 1987. 12. Kerrebrock, J., “Aircraft engines and gas turbines”, MIT Press, 1992.

13. Oates, G., “Aerothermodynamics of aircraft engine components”, AIAA education Series, 1985. 14. Aungier, R., “Turbine aerodynamics”, ASME Press, New York, 2006

15. Sieverding, C., “Secondary and tip-clearance flows in axial turbines”, Von Karman Institute, LS1997-1 16. Arts, T., “Turbine blade tip design and tip clearance treatment”, Von Karman Institute, LS2004-2 17. Booth, T., “Tip clearance effects in axial turbo-machines “, Von Karman Institute, LS1985-5

18. Sunden, B., Xie, G., “Gas Turbine Blade Tip Heat Transfer and Cooling: A Literature Survey”, Heat Transfer

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2

DISCLAIMER

SCHOOL OF ENGINEERING

DEPARTMENT OF POWER AND PROPULSION

These notes/slides have been prepared by Cranfield University or its agents for the personal use of course attendees. Accordingly, they may not be communicated to a third party without the express permission of the author.

The notes/slides are intended to support the course in which they are to be presented as defined by the lecture programme. However the content may be more comprehensive than the presentations they are supporting. In addition, the notes may cover topics which are not included in the presentations.

Some of the data contained in the notes/slides may have been obtained from public literature. However, in such cases, the corresponding manufacturers or originators are in no way responsible for the accuracy of such material.

All the information provided has been judged in good faith as appropriate for the course. However, Cranfield University accepts no liability resulting from the use of such information.

3

Turbine aerodynamics - programme

Part A: Turbine aerodynamics

• Introduction to aero design

• Arrangements, architectures, characteristics

• Work

• Frame of reference and parameters

• Introduction to turbine aerodynamic features

• Introduction to turbine aerodynamic design

• Turbine annulus design

• Turbine stage aerodynamics

• Loading, flow, coefficient, specific work and reaction

• Designing for high power

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3

• Turbine efficiency

• Turbine blading

• Three-dimensional aerodynamics

• Streamline curvature and secondary flows

• Unsteady aerodynamics

• Introduction to cooling

Part B: Axial turbine design exercise

• HP and LP designs

• Specification, constraints

• Effect of TET

• Design summary , assessment and recommendations

5

Preliminary design

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4

Gas turbine applications

This image cannot currently be displayed. Industrial Power generation

Siemens 340 megawatts (MW) SGT5-8000H gas turbine. Marine e.g. MT30 marinized version of an aero GT. 40MW range

Oil and gas

7

http://www.siemens.co.in/

Rolls-royce.com

7

Gas turbine applications

Propulsion 8 airbus.com Boeing.com Lockheed.com 8

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5

• Preliminary design stage considerations

• How much do you need to know…..and when?

• What is the application?

• Propulsion or power

• Civil

• Military

• Short duration? Disposable?

• How does this affect the design approach ?

• Time to market

• Market size and duration

• Preliminary design fidelity

• Evolution or revolution

₉

Turbine design drivers

• What are the design aspects for consideration ?

• Specific fuel consumption (and/or block fuel burn) • Temperature • Pressure • BPR • Component efficiency

• Emissions

• Weight

• Size

• Embedded configurations (civil or military)

• Life

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6 Turbine design drivers

• Reliability

• Risk/benefit trade off

• E.g. tip gap, TBC, cooling strategy, stress margins • Noise

• Turbine noise

• Effect of LPC noise on turbine design • Time to manufacture

• Robustness

• Change in operations • Change in future processes • Growth potential

• Cost

• Manufacture • Ownership • Replacement parts

• Power/thrust supply (risk ownership) • Maintanence

11

Turbine design disciplines

• Aerodynamics

• Cooling and thermal management

• Mechanical design

• Stress

• Lifing

• Costs

• Weights

• Manufacturing

• Logistics

• Purchasing

12

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7

• Number of stages

• Work split for multi-stage turbines

• Aerodynamic conditions

• Annulus shape and dimensions

• Blade and vane aspect ratio

• Blade and vane space/chord ratio

• Blade and vane airfoil numbers

• Radial work distribution

• Inter-row axial spacing

13

Design process and considerations

Stage 0

Preliminary evaluations

Stage 1

Preliminary design

Stage 2

Full concept definition

Stage 3 Product realisation Stage 4 Development and production Stage 5 In service Stage 6 Disposal

Main focus for turbine aerodynamic design work 2,3 and 4D aerodynamic design 1D, 2D and maybe 3D aerodynamic design 1D and maybe 2D aerodynamic design 14

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8 The importance of preliminary design

Jones 2002

Knowledge of the design

15

16

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9

         t (molecular activity) p C W (kinetic) 2 2 W.V

COMPRESSOR INLET TURBINE INLET

TEMPERATURE K 300 1600

SOUND SPEED m/s 350 780

MACH NUMBER 0.5 0.5

VELOCITY m/s 175 390

ENERGY

TEMPERATURE t , T (static – molecular; total –plus kinetic)

PRESSURE p , P (static - molecular bombardment; total - adds kinetic term) POWER W Cp ΔT (total energy change per second; molecular plus kinetic) SPEED OF SOUND a Rt (sound transmitted by molecular collision) MACH NUMBER

a V

M (better to use than velocity)

EXAMPLE: 17 18 COMPRESSOR POWER TURBINE POWER COMBUSTOR ENERGY INPUT USEFUL POWER ENERGY ENTROPY P2 P₁

= Turbine Power – Compressor Power

THERMAL

EFFICIENCY _Combustion_Energy_Input Power Useful 

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10

19

DESIGNER’S SOLUTIONS FOR HIGHEST USEFUL POWER DESIGN FOR HIGH TURBINE INLET TEMPERATURE

RedMINUSblue (PT-PC) equalsoutput power Largest when

Highest pressure ratio

and or Highest TET T S PT PC 20 T s

EFFICIENCY OF GAS TURBINE ENGINES

IDEAL COMPRESSOR WORK ACTUAL COMPRESSOR WORK COMBUSTOR ENERGY INPUT IDEAL TURBINE WORK ACTUAL TURBINE WORK 1 2 3 4’ 4 2’ P1 P2

Compressor Isentropic Efficiency

) T T ( ) T ' T ( c 1 2 1 2     W.Cp.(T’2-T1) = ideal compressor work W.Cp.(T2-T1) = actual compressor work ) ' T T ( ) T T ( T 4 3 4 3   



W.Cp.(T3- T4) = actual turbine work W.Cp.(T3–T’4) = ideal turbine work

Turbine Isentropic Efficiency

) T T ( ) T T ( THERMAL 2 3 1 4 1      Thermal Efficiency = (Useful Work/Combustor Energy Input) Where:

Useful work = turbine work - compressor work = W.Cp.(T3-T4) - W.Cp.(T2-T1)

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11

Basic arrangements

21

22

Engine architectures and gas path

This image cannot currently be displayed.

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12

23

Single spool axial flow turbojet

Images from Rolls Royce

Engine architectures and gas path

23

24

Engine architectures and gas path

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13

25

Images from Rolls Royce TEMPERATURE

VELOCITY

PRESSURE

25

IMAGE COURTESY ROLLS ROYCE

GAS GENERATOR TURBINES

POWER TURBINE

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14

TRENT AERO ENGINE – IMAGE COURTESY OF ROLLS-ROYCE

TURBINES 27 Rolls Royce T900 Specifications: BPR 8 OPR 41 Stages 1LPC, 8IPC, 6HPC, 1HPT, 1IPT, 5 LPT Fan diameter 116 inches

Thrust 76,500lb

Aircraft A380

A MILITARY LOW BYPASS RATIO TURBOFAN

Specifications:

BPR 0.4

OPR 25

Stages 3LPC, 5HPC

1HPT, 1LPT Fan diameter 29 inches

Thrust 20,000lb

Aircraft Typhoon

IMAGE COURTESY ROLLS ROYCE

EJ200

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15

29

Shrouded HP turbine

Unshrouded HP turbine

HIGH BYPASS RATIO TURBOFANS

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16

HIGH BYPASS RATIO TURBOFANS

IMAGE COURTESY ROLLS ROYCE 31

T800

~GE90

TURBINE TECHNOLOGY IMPROVEMENTS HISTORY

1950 NOW TIP SPEED m/s 250 350 + STAGE TEMPERATURE DROP, K 150 250 + EXPANSION RATIO 2 2.5 + STAGE POLYTROPIC EFFICIENCY % 86 92 + TURBINE ENTRY TEMPERATURE K 1200K 1800K + 32

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17

TERMINOLOGY

FUEL / AIR RATIO FAR

STOICHIOMETRIC ALL OXYGEN USED (COMPLETE COMBUSTION) OUTLET TEMPERATURE PROFILE TTQ

PRESSURE LOSS FACTOR _V2

2 1

P



LINER

FLAME TUBE DILUTION HOLES (BURNER)

SWIRLER

FUEL SPRAY NOZZLE

PRIMARY ZONE SECONDARY ZONE SECONDARY AIR LINER

TURBINE NEEDS GOOD TEMPERATURE TRAVERSE QUALITY

33

Combustor exit profile

He 2004 _{He 2004}

Povey 2009 T/Tmean

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18

35

Conventional multi-stage turbine

U1 U2 U₂ U1 U₁ Relative Absolute

Typical conventional arrangement

Vanes turn and accelerate flow for next blade row. Controlled work split between the HP and IP systems

36

Contra-rotation multi-stage turbine

U₁ U2 U1 U₁ U₂ Relative Absolute

Reversal of the HP shaft rotation relative to the IP (LP) shaft

IP NGV required to get the correct flow angle and velocity into the IP rotor Reduced turning on and reduced secondary flows on the IP NGV Increased IP NGV efficiency

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19

37 U₁ U2 U₁ U₁ U₂ Relative IP Absolute Relative HP IP NGV is removed. Reduced length, weight, cost Eliminated IP NGV loss Closely coupled HP-IP rotors can result in unsteady interactions -> reduced efficiency and possible vibration.

The inlet conditions to the IP rotor are limited by the exit conditions from the HP rotor. i.e. the absence of the IP NGV means that the flow cannot be pre-conditioned as in a conventional arrangement. The HP rotor exit swirl is limited by the HP rotor turning and the whirl velocity is limited by the rotor exit Mach number.

A consequence of this is that the work split is uneven. The HP stage typically has a much higher work level than the IP (LP).

Euler’s work equation

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20 Steady Flow Energy Equation

• For each kilogram of fluid entering the control

volume at position 1, the total energy is:

• Similarly at point 2:

• Q is the heat addition (positive into the system) and W is the work (positive when done by the fluid)

• The energy balance equation then becomes:

z₁ z₂ h1,, V1 h_2,, V₂ Q W System g z V h Etot 1 2 1 1 1  2 g z V h Etot 2 2 2 2 2  2



h V zg

 

h V zg



W Q 1 2 1 1 2 2 2 2 2   2  

This is known as the Steady Flow Energy Equation.

• For an axial turbomachine it reduces to: • For an ideal gas h = Cpt and the total enthalpy is

• Also recall, 2 2 2 0 0 CT Ct V H  p  p  2 2 2 1 1 , 1 that Remember . 2 1 M t T a V M Rt a and R C tC V t T p p             



 



01 02 0 2 2 2 2 1 1 V 2 h V 2 H H CT h W        _p 39

Compressible Form of Bernoulli’s Equation

If there is no heat transfer to or from the gas the flow is ADIABATIC. Hence conservation of energy tells us that the Total Energy (usually called the Total

Enthalpy) is conserved i.e. ho = constant.

Considering a perfect gas:

p = ρRT Equation of State

h= specific enthalpy= CpT Calorifically perfect gas

ho=specific total enthalpy = CpT0

The specific* enthalpy is defined as h = e + P/ρ and the specific internal energy e = CvT.

*The word specific means per unit mass flow and is often omitted.

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21

The energy equation for an adiabatic, steady flow is given by:

Therefore: Pressure Energy Internal Energy Kinetic Energy

all per unit mass flow

2 2 2 2 2 2 2 2 1 1 1 1 V p e V p e       

Recall that specific enthalpy is defined as h =e+p = e+p/ ( is specific volume). enthalpy) (total tan 0 2 2 2 2 1 1 2 2 cons t h V h V h     

For a calorifically perfect gas h=CpT and similarly h0=CpT0

e) temperatur total is ( tan ₀ ₀ 2 2 2 2 1 1 2 2 cons t CT T V T C V T Cp   p    p T 1 0 2 0 2 2 2 T T C V T C V T C p p p      Eqn 1.7 41

Compressible Form of Bernoulli’s Equation(continued)

Recalling: ,       _          2 2 0 2 1 1 M a a T T_o 

So far the ONLY assumptions have been a Perfect gas and ADIABATIC FLOW. If the flow is also ISENTROPIC (i.e. the entropy is constant – no shock present and outside viscous layers like the boundary layer) then:

p = k_{=RT and hence} RT V a V M   









2 2 2 0 2 1 1 2 1 1 2 RT M RT M T C V T p              1 T 1    R Cp and ₀2 2 2 ₀ 2 1 RT V a a    1 2 1 o o o _M 2 1 1 T T p p             _                  42

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22

A-A x r x q Vx V_q RotationW Rotor Streamtube r2 r₁ Centreline A-A Figure 1.1

Euler’s work equation

43

• One of the most fundamental aspects of turbomachinery aerodynamics is the process of work input (compressors) and the work extraction (turbines) processes. The same model is adopted for both compressors and turbines as outlined below.

• The work extraction and addition process is performed by rotation. It is the rotating components that transfer work. The fixed components, or stators, are not explicity involved.

• Figure 1.1 shows the flow field through a generic rotor passage for an axial-type machine but including a change in mean radius. Consider the flow along a streamtube that enters at radius r1and exits at radius r2.

• The shaft is rotating with an angular velocityW (rad/s) and is producing a torque T. • Torque is the rate of change of angular momentum and if the massflow is steady, then

the change in angular momentum in a timeDt is give by:

Euler’s work equation

) (1 1 2 2   V r V r m T rV t m T t mrV T           Rotation Rotor Streamtube r₂ r1 Centreline A-A Rotation Rotor Streamtube r₂ r1 Centreline A-A 44

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23

• The rate of change of angular momentum equals the torque:

• Power is defined as

• Work per unit mass of flow therefore is:

• Rotor blade speed at radius r is defined as U=Wr • Therefore.

• This is known as Euler’s work equation.

• It applies to all types of turbomachines. It shows that all transfer of work processes (either in or out) are reflected in a change in angular momentum via a rotating blade row. This is principally done using the pressure forces which act in the circumerential direction upon rows of rotating aerofoils.

• Recall:



1 1 2 2





1 1 2 2





V

r

V

r

m

V

r

V

r

t

m

T













r1V1 r2V2



m T P   



1 1 2 2



/

,

W

P

m

r

V



r

V



Work

k











2 2 1 1

V



U

V



U

W

_k





45



UV



T C W T C m P p k p       0 0 ork, Specific w Power, 

Frame of reference

46

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24

U

NGV

ROTOR

TURBINE STAGE

Turbine stage aerodynamics

47

Frame of reference

• For an axial machine the following co-ordinate system is defined:

x is axial r is radial q is circumferential x RotationW q V x Vr V_q r Please note:

This nomenclature is for this section only which applies to compressors and turbines alike. Subsequent sections use individual notation for turbines based on axial station

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25

• The absolute and relative frame of reference velocities are therefore

• (please note the changes in nomenclature from this section)

• Both axial and radial velocities are independent of frame of reference i.e. V_x =Wxand Vr=Wr. For the tangential velocities: Vq= Wq+Wr = Wq+U

• Notice that V_qand W_qare positive in the direction of rotation. U is Blade speed. Absolute Relative Vx Axial velocity Wx Vr Radial velocity Wr Vq Circumferential velocity (tangential, whirl or swirl

velocity) Wq Total velocity 2 2 2  V V V V x  r  2 2 2  W W W W  x  r  49

Frame of reference

• An important concept is the distinction between absolute and relative frames of reference. For the rotor shown, the inlet stationary frame velocity is V. It has two components and an absolute swirl angle of a1. By subtracting the

blade speed term, U, the relative velocity vector is obtained.

• This is the effective velocity seen by the rotor. A similar analysis at the exit plane transforms from the relative to absolute frame of reference. Conventional turbomachinery notation uses positive velocities and angles in the direction of rotation.

• Blade speed =Wr, where W is rotational speed and r is the local radius.

• Axial velocity is independent of frame of reference and relative whirl velocity is obtained from W_q= V_q-U

• For example, from a given inlet absolute velocity, flow angle and blade speed, all other vectors can be determined.

x q Rotors Relative whirl velocity Wq Absolute whirl velocity Vq Axial velocity Vx= Wx a1 50 Blade speed U

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26 Frame of reference

a1 51 b1 a1 b1 Relative velocity W b1 a1

Effect of NGV exit angle (fixed Va)

Effect of blade speed (fixed Va) Effect of NGV exit velocity

Blade speed U

Static, stagnation and relative properties

• Following on from the absolute and relative velocities there are also the equivalent relative and absolute stagnation (or total) properties.

• For example, for an incompressible flow, the absolute total pressure is:

• However, in the rotating frame of reference, the total pressure seen by the rotor is:

• Static quantities are unchanged by frame of reference.

• Stagnation properties are dependent on the frame of reference. • For compressible flows:

2 2 1 0

p

V

P







2 2 1 0

p

W

P

_{_}_REL







reference of frame Relative 2 reference of frame Absolute 2 1 0 0 2 0 1 0 0 2 0                         T T P P and C W T T T T P P and C V T T rel rel p rel p 52

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27

• For a rotor the Euler work equation applies:

• For a compressor work is done on the fluid (Wkis negative) so stagnation enthalpy rises (h02 > h01).

• For a turbine work is done by the fluid (W_kis positive) so stagnation enthalpy decreases.

• By rearranging this equation:

• Which states that h₀-UV_qis constant across a rotor blade row. This quantity is referred to a ROTHAPLY and is denoted by I.

02 01 h h Wk   02 01 2 2 1 1 2 2 1 1

h

V

U

V

U

W

V

U

V

U

W

k k













    2 2 02 1 1 01

U

V



h

U

V



h







2 2 02 1 1 01

U

V



h

U

V



h

I









53

Rothalpy and Frame of Reference

• Rothalpy in the absolute frame of reference is defined as :

• Looking at the change of reference frame:

• Therefore rothalpy in the rotating frame is given by:  

h

V

UV

H

I



_o







2



2

1 



2 2 2 2 2 2 2 2 2 2 2 2 2 2

2

2 U

UV

W

V

U

UW

W

V

U

W

V

r x r x



















    2 2

2

1

2

1 U

W

h

I







U W V W V W Vx x , r r ,    54

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28 Rothalpy and Frame of Reference

• Total enthalpy in absolute frame (absolute total enthalpy):

• Total enthalpy in relative frame of reference (relative total enthalpy):

• Rothalpy can be expressed as:

2 0 2 1 V h h   2 0 0 2 1 U h I UV h I rel    

• Rothalpy along a streamline is conserved across any blade row either moving or stationary. It applies along an arbitrary streamline for an adiabatic flow and in the absence of gravity and it is invariant. For axial machines with no change in radius the U2_{term cancels and changes in} relative stagnation enthalpy and rothalpy are the same.

2 0 2 1 W h h_rel  55

Rotary stagnation temperature

2 0 0 2 1 U h I UV h I rel    _

Rothalpy along a streamline is conserved across any blade row

Where T_0is the rotary stagnation temperature.





p rel p p p p p p C r T T r W C t C I T C U C W C H C I T 2 2 1 2 2 2 2 0 0 2 2 2 0 2 2 0 0                 0 0 0 T C I T C p p   Rothalpy, H , Enthalpy Total p p p p C rV T C UV C H C I T      0



 0 0 Relative Absolute

For axial machines with constant radius the changes in relative stagnation temperature and rotary stagnation temperature are the same.

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29

Relative Stagnation

p0rel, Torel

“What a rotor mounted probe sees”

Rotary Stagnation

p0w, Tow

“Equivalent of stagnation in a rotor”

Stagnation State

p0, To

“What a stationary probe sees”

1 0 0 2 0 2             T T p p C V T T p 1 0 0 2 0 2             T T p p C W T T r r p r Static State p, T “what the gas sees”





1 0 0 0 0 2 2 0 0 2              T T p p C V W T T r r p r





1 0 0 2 2 2 0 2                  T T p p C r W T T p 1 0 0 0 0 2 2 0 0 2                 r r p r T T p p C r T T 1 0 0 0 0 0 0                  T T p p C rV T T p 57 I Rothalpy = CpT0w M. Rose - 1998

Frame of Reference - notes

• Rothalpy, I = CpT0w, is conserved along a streamline.

• For isentropic flow the rotary stagnation pressure, p_0w, is also conserved along a streamline.

• For an adiabatic rotor and with a thermally perfect gas the rotary stagnation temperature is constant. This is true even for a change in radius, viscosity and effects of friction. If the flow is also reversible, then the rotary stagnation pressure (P_ow) is also constant.

• All relationships between the different states are isentropic compressible flow.

Nomenclature (for this section only) Subscripts

I Rothalpy = CpT0w r relative state

P pressure w rotary state

r radius 0 stagnation state

T temperature q whirl component

V absolute velocity w rotational speed W relative velocity

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30 Frame of Reference

• Relative total pressure is defined as

• Absolute and relative Mach numbers:

1 0 0 















 

T

p

_r _r 1 2 0 2 0

2

1

2

1















_











 



M

P

M

T

1 2 0 2 0

2

1

2

1















_











 



rel r rel r

M

P

M

T

Absolute Relative 59

Introduction to turbines

60