Turbomachinery for LREseitzman.gatech.edu/classes/ae6450/turbines.pdfMechanics and Thermodynamics of Propulsion, Hill and Peterson • To let largest power per unit mass flow rate

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Turbines - 1

Copyright © 2012 ,2017, 2019 by Jerry M. Seitzman. All rights reserved. AE6450 Rocket Propulsion

Turbomachinery for LRE

Turbines

Turbines - 2


Axial Turbine Analysis

• From Euler turbomachinery (conservation) equations need to understand change in tangential velocity to relate to forces on blades and power

• Analyze flow in a plane normal to rotational axis (cascade plane) to find c

ie

rcrcm

ie

ucucmW

Nozzle Rotor

2

Turbines - 3


Cascade Analysis

• You may have previously analyzed

flow over a “blade” (airfoil)

– but in blade’s reference frame

• Here there are moving

(e.g. rotor) and stationary

blades

– e.g., for turbine

12 nozzle (stator)

23 rotor

• Use velocity triangles to switch

between frames Mechanics and Thermodynamics of

Propulsion, Hill and Peterson

z

Nozzle Rotor

12

3

z

r

Turbines - 4


Velocity Triangles

• Two reference frames to use for fluid velocity

– engine’s

– blade’s

• Difference due to blade motion

• In 2-d (z,) “plane”

– u is in direction

– define angles (,)for each ref. frame

c

w

uwc

z

c

u

w

u

c

w

u

braytonenergy.net

u

3

Turbines - 5


Rotor Velocity Triangles

• Blade moves in direction, and in

(z,) plane, for fixed r, let ui=U

• Also have general geometric relations

– e.g.,

• Therefore

uwc

z

1ii zz cw 2Ucw

ii

izii iiccc tansin

3tan

2,1

33

33

zcUc

wUc

2tan22

zcc

Mechanics and Thermodynamics of

Propulsion, Hill and Peterson

c

w

U

Nozzle Rotor

Turbines - 6


Single-Stage Characteristics

(Axial Turbine)

• Goal - determine how turbine performance, e.g., PrT, affected by changes in operating conditions

• Start by analyzing single-stage turbine

• Rotor (23)

– Euler

– at fixed radial location

• Nozzle (12)

• So for stage

2323 23 ooR hhmcucumW

2323 ooR hhmccUmW

3,23,2 ohcU

120 ooS hhW

3,23,23,1 cUhh oo

while no work, there is still

torque on stationary blades

4

Turbines - 7


Axial Turbine Stage

• Combining

results

– assuming constant axial velocity

3tan33

zcUc 2tan

22 zcc

3,23,23,1 cUhh oo

23, tantan23

zzstageo ccUUh

322

,tantan1

U

c

U

hzstageo

, stage loading

coeff., flow coefficient

2

3

2 related to nozzle trailing edge angle

3 related to rotor trailing edge angle

IF flow attached (no separation)

1tantan 32

2,

U

cUc

A

Wz

zinlet

inlet

producedT

High output power:

1) high flow (cz)

2) high U (rpm, radius)

3) high 2 (max <90)

4) high 2 (large rev. turn)

Turbines - 8


Turbine Stage Pressure Ratio

• For adiabatic turbine with TPG/CPG

1

1

13

03

1 11

o

oo

st

oT

T

TT

p

pPr

1

2

1

23,111

1U

h

RT

U o

ost

1

1

2

3

1 3,211

U

c

RT

U

p

p

osto

o

U

c

U

ho 3,23,1

2

• Stage pressure ratio depends on

1. = f(U= r, c 2,3)

2. blade Mach number Mblade=f(r, To1)

3. st

>1 as written

<0 for turbine

M2blade

5

Turbines - 9


Turbine Characteristics

• For given Mb, blade design, , T

• As increase flowrate through

turbine (at fixed rpm), larger

pressure drop (more expansion) is

produced

– more work extracted per unit mass

1

21U

cCCPr z

T

TPr

U

cz

Higher Mb

1

32

2 tantan11

1U

cMPr z

b

st

T

Turbines - 10


Axial Turbine Maps

• Typically presented as separate curves for each rpm (Mb)

• x-axis - replace flow coefficient with corrected mass flow rate, recall

– at high corrected mass flowrate, nozzle becomes choked

• Peak efficiency around design point

Mechanics and Thermodynamics of Propulsion, Hill and Peterson

oo RTpAm

PrT

T

1/PrT

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Turbines - 11


Blade Design: Degree of Reaction

• We have TWO blade

parameters to design

– rotor trailing edge (match 3)

– nozzle trailing edge (match 2)

• How to do this?

1.Degree of reaction, R

2.Stage exit condition

constraint (3)

32 tantan13,2

U

c

U

cz

2tan22

zccUw

3tan3

zcc

2tan2

zcc

3tan33

zccUw 3,23,2 wc


Turbines - 12


Degree of Reaction

• Recall

– allows us to distribute load (static pressure

change) between rotor and nozzle (or stator)

– how to relate static enthalpy change to

azimuthal velocity changes?

• KE !!

– for stationary blade, no work done

• e.g., nozzle blade

stagerotor hhR

KEhho 0

2v2 hho

22 2222

12 2221 cccchh zz 222

21 cc

if cz constant, and negligible cr

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Turbines - 13


Degree of Reaction (Turbine)

• Rotor blades??– are “stationary” in rotor’s

reference frame

• Reaction

123 zzz ccc

222

23 32 wwhh

13

23

hh

hhR

23

32

2

22

ccU

ww

2

11

2

33

23

chch

hh

oo

if c1 c3

13

23

oo hh

hh

U

wwR

2

32

R relates design blade angles

to azimuthal KE change

23 tantan3,2

zcUc

323232

22

wwwwww

3,23,2 wc

3,232 cww

Turbines - 14


Impulse Turbine


U

cU

U

w

U

cz 2tan23,23,2

13

23

h

hR

• R = 0

– all the pressure change occurs across the

nozzle, or the nozzle

creates high KE

23

23 ww

2tan123,2

U

c

U

cz

23 tantan zz cc

23,22 ww

U

ww

2

32

8

Turbines - 15


Impulse Turbine


• So for impulse turbine,blade loading coeff.

• Relates blade loading to nozzle exit angle

• From previous & velocity triangles,rotor angles given by

22

tan1223

U

c

U

c

U

hzstageo

222

1tanU

h

c

U stageo

z

zc

U 223 tantantan

<0

Turbines - 16


Impulse Turbine


• To let largest power per unit mass flow rate large 2

– tends to produce high velocities and po losses

– practical limit, ~70-75

• Further possible constraint

– no exit swirl (c3=0)

233,2 ccc 23,2 cc

Uc2

U

c

U

c2323 12

22

U

hstageo

zcU2tan 2

2tan1223

U

c

U

cz

zcU3tan,

9

Turbines - 17


50% Reaction Turbine

• R 0.5

– balanced p drop across stage

– if no exit swirl

32 wwU

22

tan213,2

U

c

U

c

U

hzstageo

12

U

hstageo

23 tantan

23 tantan13,2

U

c

U

cz

23,2 cc

zcU1

22 tan

half loading of impulse: less power/stage Mechanics and Thermodynamics of Propulsion, Hill and Peterson

32 tantan zz ccU

less convergence in nozzle

vs impulse turbineU

wwR

2

32

Turbines - 18


Rocket Turbines

• Can combine results for no exit swirl condition to show

– as reaction decreases, power per stage increases

• To minimize size/weight, rocket turbopumps often employ impulse or low reaction turbines

– but efficiencies typically lower (<70%) for impulse turbines compared to higher reaction turbines (~90%)

• Can improve efficiency by decreasing flow coefficient (cz/U)

– for given flowrate, requires higher blade speed, RPM

– higher RPM = higher stresses = heavier, and larger gear ratio if geared to pump

RUhstageo 122

10

Turbines - 19


Velocity-Compound Impulse Turbine

• Can increase stage power even more using velocity-compounding• multiple nozzle/rotors in series

• Example, two-row compounded impulse turbine

– all p in 1st nozzle

– 1st rotor exits with highswirl (so large 2 allowed)

– 2nd nozzle redirects flow without p

– 2nd rotor extracts more work and reduces swirl

– stage loading is 4x that of single-row impulse stage

From Sutton

Turbines - 20


Highly-Loaded Turbine Efficiencies

• Can provide lower or improved efficiency

improvement over single row impulse stage

– still lower than high reaction turbines

From Hill and Peterson

0.1

From Sutton

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Turbines - 21


Turbine Inlet Temperature Limitswww.virginia.edu/ms/research/wadley/high-temp.html• Maximum inlet temp.

limited by blade stresses

• Advances

– higher T materials (superalloys)

– coatings (TBC)and blade cooling, not typical for rockets

• Rocket turbine Tmax

historically limited to 900-1100K with blade tip speeds of 400-700 m/s

– potential for increases to 1400-1500 K with better materials

Ni superalloys

single crystal super alloys

1200K

1400K

1500K

1100K

Turbines - 22


Turbine Design Example

• Consider preliminary design requirements for gas-generator cycle LRE turbine

– power/flow: 19.4 MW, 41.8 kg/s

– gas properties: =1.15, MW=27.7

– inlet: 1000K, 44 bar

– outlet: ?

• Constraints

– max tip speed 550 m/s

– assume geared so rpm not fixed by pump rpm

– assume zero swirl at exit, constant axial vel.

p

inoeocm

WTT

,,

would be more realistic to constrain blade-root stress

790K

12

Turbines - 23


0.1


• Step 1: Turbine type

– estimate U/co, co=theoretical gas spouting vel.

– U/co 0.5 single impulse stage, much higher

than 2 row compounded, less stages than

reaction turbine

• Step 2: blade angles

– use max 2=70

smmWhhc esoio 100022

75.0 T

9.53tantan 3221

3

11

1oi

oioe

st

TT

TTPr

3, 6.1,8.3m

kgbarp eeo

oeoee RTp

suggests nozzle will be supersonic

2=-53.9

Power 19.4MW

Flowrate 41.8kg/s

1.15

MW 27.7

To,in 1000K

po,in 44bar

To,exit 790K

Turbines - 24



• Step 3: sizing

728.0tan

2

2

mz Uc

smskgMW

Um 4872

8.4119

smcz 354

2

3075.0

35458.1

8.41m

smmkg

skg

c

mA

ze

e

13.1487

550

m

tip

m

tip

U

U

r

r

m

Ar e

m 207.0113.12 2

85.0m

root

r

r

s

rad

r

U

m

2350 rpmN 400,22

flow coefficient ()typical turbine values 0.5-1.5

root-tip ratio of 0.75

Me ~ 0.6-0.7

5.8cm

17.6cm

structure is mostly disk

702

for zero-swirlimpulse turbine

Power 19.4MW

Flowrate 41.8kg/s

1.15

MW 27.7

To,in 1000K

po,in 44bar

To,exit 790K

2

2

stageo

m

hU

2

22

2 roottip

m

rrr

222 2 tipmroot rrr

12

2222

222

22

mtipm

tipmtip

roottipe

rrr

rrr

rrA

12 22

2

mtip

em

rr

Ar

jerrys

Polygonal Line

Turbomachinery for LREseitzman.gatech.edu/classes/ae6450/turbines.pdfMechanics and Thermodynamics of Propulsion, Hill and Peterson • To let largest power per unit mass flow rate

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