Turbine design

NASA SP-290

/N-.o7

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TIIItI|INI{

I)KSll;l

a,id AI'I'I, II:AI'IIiN

(NASA-SP-290) APPLICATION

TURBINE (NASA)

DESIGN 390

N95-22341

AND

p Unclas

HlI07

NATIONAL

AERONAUTICS

AND

SPACE

0041715

ADMINISTRATION

NASA

SP-290

'lrlllel|lNl IDK Iq ,N annalAIDIDI,Iq;A"I' IqlPm

Edited by Arthur J. Glassman Lewis Research Center

Scientific and Technical Information Program--1994 NATIONAL AERONAUTICS AND SPACE ADMINISTRATION Washington,

DC

Available NASA

from the Center

800 Elkridge

for AeroSpace Landing

Linthicum Heights, Price Code: A17 Library

of Congress

Information

Road MD 21090-2934 Catalog

Card Number:

94-67487

PREFACE NASA and

has

space

an interest

turboshaft

spacecraft.

fluids,

interest

for turbine

trains,

etc.)

In view a

Program.

The

1972-73.

Various

entitled

course and

velocity

cooling, The

mechanical notes written

edited

for

are the

U.S.

sets

of units

units

and

after

Such

set

of

will

satisfy

units

consistent symbol

customary

units.

by including defining

a

Research

Application"

was

Graduate again

were

Study

presented

covered

can

serve

means

for

in

including

fundamental

aerodynamic

course,

power.

at Lewis

and

a publication

buses,

turbine design,

blade

revised

and

as a foundation self-study,

or

topics.

used the

and

blade

of

trucks,

In-House

concepts,

losses,

turbine

consistent

Design revised

providing

electrical

efforts

technology

gases,

applications

(cars,

and the

auxiliary

for

Other

and l%o-

inert

design, operation, and performance. and used for the course have been

for selected

given

somewhat of turbine

diagrams,

publication.

commonly

of

fluid-dynamic

introductory

Any

part

using

ground-based

interest as

was

aspects

concepts,

Two

and

aircraft,

studied

land-vehicle

"Turbine

1968-69

thermodynamic

an

power

been

jet

and

engines

spacecraft.

include

for

propulsion

have

for

to aeronautics provide

power

turbine

fluids

power

turbine-system

during

reference

rocket

engines

course

presented

engines

provide

propulsion

of the

primarily

turbine

as auxiliary

metal

electric

related

well

Closed-cycle

and

long-duration

Center,

as

turbines

for

organic

for

Airbreathing

propulsion,

pellant-driven power

in turbines

applications.

as unity

sets

of

definitions. A

single

all constants those

not

,°°

111

the units

These set

of

required required

equations and are

presented.

constant the

SI

equations for the

values units

covers U.S.

and both

customary

for the SI units. ARTHUR J. GLASSMAN

a

CONTENTS CHAPTER

PAGE

PREFACE

°

• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • . . . . . . . . . . .

THERMODYNAMIC CONCEPTS

AND

by Arthur

FL UID-D

YNAMI

J. Glassman

C

....................

BASIC CONCEPTS AND RELATIONS ....................... APPLICATION TO FLOW WITH VARYING AREA REFERENCES ...................................... SYMBOLS .........................................

,

BASIC

TURBINE

CONCEPTS

by Arthur

1 1 14 19 20

..............

J. Glassman

....

TURBINE FLOW AND ENERGY TRANSFER .................. DIMENSIONLESS PARAMETERS ......................... REFERENCES ...................................... SYMBOLS ......................................... GLOSSARY .........................................

.

VELOCITY DIAGRAMS by Warren Warner L. Stewart ...................................

J. Whitney

BLADE Arthur

DESIGN

by Warner

J. Glassman

21 45 62 63 65

and

L. Stewart

STUDIES

.....

CHANNEL

FLOW

ANALYSIS

lOl

.

INTRODUCTION by William

102 118 124 125

by Theodore

STREAMAND POTENTIAL-FUNCTION VELOCITY-GRADIENT ANALYSIS REFERENCES ...................................... SYMBOLS .........................................

ANALYSES ........................

TO BOUNDARY-LAYER

D. McNally

70 84 95 96 98

and

..................................

SOLIDITY .......................................... BLADE-PROFILE DESIGN .............................. REFERENCES ...................................... SYMBOLS .........................................

,

21

69

MEAN-SECTION DIAGRAMS ............................ RADIAL VARIATION OF DIAGRAMS ....................... COMPUTER PROGRAMS FOR VELOCITY-DIAGRAM REFERENCES ...................................... SYMBOLS .........................................

,

...

111

Katsanis ............

......

127 130 147 154 155

THEORY

...............................

157

NATURE OF BOUNDARY LAYER ......................... DERIVATION OF BOUNDARY-LAYER EQUATIONS SOLUTION OF BOUNDARY-LAYER EQUATIONS CONCLUDING REMARKS .............................. REFERENCES ...................................... SYMBOLS .........................................

............ ..............

157 160 172 188 188 191

V

iI

PAG'E__

I_T_NTtON_!.Ly BLANg

CHAPTER

o

PAGE

BOUNDARY-LAYER

LOSSES

by Herman

W. Prust,

Jr ....

BOUNDARY-LAYER PARAMETERS ........................ BLADE-ROW LOSS COEFFICIENTS ....................... BLADE-ROW LOSS CHARACTERISTICS .................... REFERENCES ....................................... SYMBOLS .........................................

o

MISCELLANEOUS

LOSSES

by Richard

195 201 217 221 223

J. Roelke

........

225

TIP-CLEARANCE LOSS ................................ DISK-FRICTION LOSS ................................. PARTIAL-ADMISSION LOSSES ........................... INCIDENCE LOSS .................................... REFERENCES ....................................... SYMBOLS ..........................................

o

SUPERSONIC

TURBINES

by Louis

225 231 238 243 246 247

J. Goldman

.........

METHOD OF CHARACTERISTICS ......................... DESIGN OF SUPERSONIC STATOR BLADES ................. DESIGN OF SUPERSONIC ROTOR BLADES .................. OPERATING CHARACTERISTICS OF SUPERSONIC TURBINES REFERENCES ....................................... SYMBOLS .........................................

10.

RADIAL-INFLOW

TURBINES

by Harold

E. Rohlik

OVERALL DESIGN CHARACTERISTICS .................... BLADE DESIGN ..................................... OFF-DESIGN PERFORMANCE ........................... REFERENCES ....................................... SYMBOLS .........................................

11.

TURBINE

COOLING

by Raymond

S. Colladay

EXPERIMENTAL DYNAMIC and Harold

DETERMINATION PERFORMANCE by Edward J. Schum ................................

....

.......

250 263 266 272 277 278

279

...........

307 307 314 328 330 332 340 345 347

OF AEROM. Szanca

TEST FACILITY AND MEASUREMENTS .................... TURBINE PERFORMANCE .............................. REFERENCES ....................................... SYMBOLS .........................................

vi

249

284 295 302 305 306

GENERAL DESCRIPTION .............................. HEAT TRANSFER FROM HOT GAS TO BLADE ................ CONDUCTION WITHIN THE BLADE WALL .................. COOLANT-SIDE CONVECTION ........................... FILM AND TRANSPIRATION COOLING ..................... SIMILARITY ........................................ REFERENCES ....................................... SYMBOLS .........................................

12.

193

351 352 374 387 388

CHAPTER 1

Thermodynamic andFluid-Dynamic Concepts ByArthur J. Glassman This

chapter

cepts are

of the

is intended

thermodynamics concepts

energy-transfer treatment textbooks. the

U.S.

needed

to

processes

purposes

given

and

of this

consistent commonly after

units

by

units

and

of

analyze

including

units

will

all

sets

must

know

generalizations of behavior,

The

flow

more

concerning are

far

how

not

and

complete

of

with

their

covers the

for

U.S.

the

values

are

and

the

units both

sets

of

customary

SI units.

RELATIONS

kind

of calculation

volume,

and

resulted

in

behavior. to

SI

State

any has

presented.

constant the

for

required

AND

gases

referred

and

of equations

pressure,

of

equations

are

required

those

very

study

gases

of units

set

CONCEPTS

get

we

These

the A

the

These

constants

as unity

we can

interrelated.

turbine.

satisfy

Equation

gases,

mechanics.

understand

a

con-

can be found in reference 1 and in many to be steady and one-dimensional for

A single

BASIC

Before

fundamental

fluid

and in

definitions.

units.

defining

compressible

consistent

symbol

customary

of the

chapter. set

used

the

some

occurring

of these subjects Flow is assumed

Any Two

to review

as being

In

temperature certain

discussing

either

involving

ideal

laws these or real.

are and laws The

TURBINE

DESIGN

ideal

is only

the

gas real

The

gas

AND

hypothetical

can

ideal

APPLICATION

only

gas

and

approach

equation

obeys

under

of state

various certain

simplified

laws

that

conditions.

is R*

T

pv=--_-

(1-1)

where P

absolute

pressure,

V

specific

R*

universal gas constant, (lb mole) (°R)

volume,

molecular

lb/ft 2

m3/kg;

weight,

absolute The

N/mS;

fta/lb 8314

kg/(kg

temperature,

quantity

R*/M,_

J/(kg

mole); K;

is often

1545

mole)(K);

lb/(Ib

(ft) (lb)/

mole)

°R used

as

a single

quafftity

such

that

R $

R-----_-_ where

R is the

Density law.

gas

constant,

is often

used

(1-2)

in J/(kg)(K)

instead

or

of specific

(ft)(Ib)/(lb)(°R). volume

in

the

ideal

gas

Thus, 1 p =-

where

p is density,

In

general,

sures

or

within

high

as 50 atmospheres, deviations

of several None

applicable

only

pressure. to use

similarity (ratio

pressure

2

between

above their critical to within 5 percent for

and

gases

may

below

appear

ideal

been the

cannot

their

free

over

have

the as

pressure.

to express

in the the

p-v-T

satisfactory, range

of these

justified

are

tempera-

resulted

a limited

useful

space

molecules

critical

universally

gas

pres-

temperatures, up to pressures

of state

most be

the

low

at 1 atmosphere

behavicr

found

to a single Even

in behavior

and

reduced

behavior.

which

forces

at

unless

and of tem-

equations

a high

are

degree

of

is required.

temperature forms

under

equations

have

behavior

attractive

from

hundred

of these

and

cumbersome The

gases

ideal

conditions the

while

of real

relation.

accuracy

and

which are be accurate

proposal are

(1-3)

approximate

of 2 to 3 percent

Deviations

perature

will

temperatures,

For gases gas law may

most

gas

gas is large

small. ideal tures,

RT = pRT

in kg/m 3 or lb/ft 3.

a real

high

the

V

the

of

of substances

temperature, (ratio

basis

of a relatively

The

method

of

T,

of pressure, simple general

at equal to

critical

p,

method correlation

values

of reduced

temperature,

to critical for

estimating is

to

To)

pressure, real

incorporate

pc) gas a

THERMODYNAMIC

correction law:

term,

called

the

AND

FLUID-DYNAMIC

compressibility

factor,

CONCEPTS

into

the

ideal

gas

p=zpRT where

z is the

The and

compressibility

compressibility

reduced

of the

gas.

Values

temperature other as

figure

with

for

This

may

and

pressures

be

are

The

molal

averages

of

calculations

do

never

quick

not

determination

approximate

error

of

the

associated

we

the

are

use

of the

pres-

using

the

law

ideal

in our

However, is always

factor

can

gas

of

in a large with

region. gas

region

result

concerned

this ideal

by

would

compressibility with

and

is a large

law

within

that

temperatures

components.

there

gas

that

fall

granted

average correla-

approximated

that

ideal

and here

agreement

temperatures

of the

shows

usually

for

an

if pseudocritical are

conditions

from

in rigorous

reduced

properties

of the

texts

compressibility-factor

mixtures

1-1

use

of reduced

in many

is not

The

nature

2 is reproduced

is derived

properties

figure

temperature of the

as a function

presented

and

to calculate

the

take

gas.

critical

where

Fortunately,

should

one

reduced

reference

correlation

to gas

used

are from

of gases

any

of the

conditions

error.

of

pseudocritical

Examination

A

for

extended

sures.

state

type

of

to be independent factor

charts

number

data

tion

a function

pressure

of the

a large

all the

is

is assumed

reduced

One

1-1.

of data

and

of compressibility

and

sources.

factor.

factor

pressure,

(1-4)

we valid.

show

the

I 9,0

F 10.0

law.

Reduced temperature, 1.20 T/Tc_ 1.10 1.00

_

_--

_._

--

1. 1_50

______

N

•90 .80 • 70 t_

.60

--

1.20

.50

-

I \ -1.1o

.30! _-'LO0 1.0

I 2,0

I 3,0

1 4.0

I 5.0

f 6.0

I 7.0

I 8.0

Reduced pressure, P/Pc

FIGURE

1-1.--Effect

of

reduced

pressibility

pressure factor.

(Curves

and

reduced from

ref.

temperature

on

com-

2.)

3

TURBINE

DESIGN

AND

Relation In

a flow

is the and

of

process,

enthalpy

position,

APPLICATION

Energy the

h. For

enthalpy

Change

energy

term

a one-phase

can

be

to

State

Conditions

associated

system

expressed

as

with

work

of constant a

and

chemical

function

of

heat com-

temperature

pressure: h=fcn(T,

where

h is specific

enthalpy

can

enthalpy,

in J/kg

be expressed

partial

derivatives

properties

as follows.

can By

(1-5)

or Btu/lb.

A differential

change

v dT+(00-_h)rp be

expressed

dp in

(1-6) terms

of

determinable

definition, Oh Cv=(-_'_),

where

cp is heat

(lb) (°R).

One

capacity

of the

at

basic

constant

differential

dh=

where s is specific entropy, conversion constant, 1 or derivative

with

determined

respect

from

of the

Maxwell

pressure,

in J/(kg)(K)

equations

of thermodynamics

pressure

or Btu/

vdp

at

is

(1-8)

in J/(kg)(K) or Btu/(lb)(°R), 778 (ft)(lb)/Btu. Therefore,

to

equation

(1-7)

Tds + j

constant

and J is a the partial

temprature

is,

as

(1-8), Oh'_

One

in

as

Oh dh:(-O-T) The

p)

relations

T los\

states

1

that

(1-10)

Substituting

equations

(1-7),

(1-9),

and

(1-10)

j

Ov T (_-_)p_

into

equation

(1-6)

yields dh=c,dT+

Equation change between 4

(1-11)

in

terms

two

states

is the of

the

rigorous state

is calculated

[v--

equation conditions, rigorously

for and as

dp

(1-11)

a differential the

enthalpy

enthalpy change

THERMODYNAMIC

Ts

AND

1

FLUID-DYNAMIC

P*

CONCEPTS

_V (1-12)

If

we

law,

now

assume

we can

that

the

gas

behaves

according

to

the

ideal

gas

set RT v------P

(1-13)

and

( Ov' By

using

these

pressure

on

last

two

enthalpy

equations

change

Empirical

equations

and

in

for most

gases

of equation

Although

one

might

calculations, If and

there

it can

be

/'2, then

not

bT+cT

assumption

gases,

there

of some

average

be within

usually loss,

value

a few

Relation In

is an

there

cv will

of the

true

Conditions

the

loss

be is no

heat

assumed change

in hand-

example,

(1-17)

type it for

of expression computer

between

for hand calculations.

temperatures

T_

T2-- T1) one

of State

a turbine, can

percent

available

If, for

(_--T_)

this

for

variation

for

remains

(1-16)

ca is constant

excellent

is a significant

of

becomes _=c.(

This

there

2

to avoid

that (1-15)

effect

yields

to use

reason

assumed

equation

and

of T are

of interest.

(1-15)

want

is no

the

(1-15)

T2-- T1) +_ b (_--_)+3

Ah=a(

(1-12),

to zero,

for ca as a function

textbooks

integration

equation

frTcz, dT

cp=a-tthen

(1-14)

is reduced

Ah:

books

R

to be

monatomic

in c_ with give

an

gases;

for

T. However,

approximation

other

the

that

use

should

value. for

Constant

is normally adiabatic.

in entropy.

(1-18)

small, For

Therefore,

Entropy

Process

and

flow

the

adiabatic the

flow

process with

no

constant-entropy

(isentropic) process is the ideal process for flow in the various parts of the turbine (inlet manifold, stator, rotor, and exit diffuser) as well as for the overall turbine. Actual conditions within and across the

TURBINE

DESIGN

AND

APPLICATION

turbine are usually determined in coniunction necessary For can

with

to be able a one-phase

some

state

conditions

of constant

as a function

a differential

From

change

equations

Substituting

(1-8)

(1-23)

we get

(1-21)

and

relating

temperature

If we assume into

equation

and

pressure

ideal-gas-law (1-23)

and

but

using

a relation

such

equation

(1-17),

a computer calculation With the additional

not

behavior the

as equation

peratures 6

T_ and

equation

equation

(1-25)

(1-24)

yields

(1-22)

particularly

and

useful,

expression

for an isentropie substitute

integration,

process.

equation

(1-14)

we get

Rln_

(1-24)

(1-16),

integration

yields

(T_--T_)

also

is more

becomes

(1-25)

suitable

than in a hand calculation. assumption that cp is constant

T2, equation

(1-20)

dp

1Rln_=aln_--kb(T2--T_)+2 J p_ Like

(1-20)

and

, _dT----

By

into

conditions

perform

as

(tp

0v (0-T),

ds=0

rigorous,

entropy

pressure (1-19)

(1-10)

dT--j

process,

is the

composition, and

be expressed

and

a constant-entropy

Equation

can

dT÷(_)r

equations

process.

(T, p)

ds----(_),

ds=_

For

chemical

in entropy

(1-7),

It is, therefore,

for an isentropic

of temperature s----fcn

and

isentropic process calculations

efficiency or loss term.

to relate system

be expressed

from

for use

between

in

tem-

THERMODYNAMIC

AND

Jc_ ln T2--R T1--

FLUID-DYNAMIC

CONCEPTS

p2 ln- pl

(1-26)

and p2=('%'_ p, \T1/

Jc,/R (1-27)

But Jcp R where

3" is the

capacity equation

ratio

of heat

at constant (1-27) yields

3' 3"-- 1

capacity

(1-28) at

constant

volume. Substitution of the more familiar form

pressure

equation

to

heat

(1-28)

into

p, _-_----\_]

Where should

specific heat ratio give a reasonable

(1-29)

3" is not constant, approxlmation. Conservation

The

rate

of mass

flow

through

the

use

of an average

value

of Mass an area

A can

be expressed

as

w=pAV

(1-30)

where w A

rate flow

of mass flow, kg/sec; area, m2; ft 2

V

fluid

velocity,

For across across

a steady any any

m/see; flow

lb/sec

ft/see

(and

nonnuclear)

section of the flow other section. That

path is,

process, must

equal

the

rate

of mass

flow

the

rate

of mass

flow

plA, VI = p2A2 V2 This

expresses

(1-31)

the

is referred

principle to as the

of conservation continuity

Newton's

are

All conservation

equations,

consequences

of Newton's

that an unbalanced in the direction force is proportional the body.

Second

Law

Second

product

of mass,

and

equation

equation.

theorems,

force that ac_s of the unbalanced to the

(1-31)

of etc.,

Law

Motion dealing of

with

Motion,

momentum which

states

on a body will cause it to accelerate force in such a manner that the of the

mass

and

acceleration

of

7

TURBINE

DESIGN

AND

APPLICATION

Thus, F= m a g

(1-32)

where F

unbalanced

force,

m

mass,

a

acceleration,

g

conversion

kg;

N;

lbf

lbm m/sec2

; ft/sec

constant,

2

1 ; 32.17

(lbm)

(ft)/(lbf)

(sec 2)

But dV a =--d- 7 where

t is time,

(1-32)

in seconds.

Substituting

the

mass

is constant,

into

equation

g Equation

(1-34b)

is equal mass

(1-34a)

can

to

specifies the

(1-34a)

equation

F=I

Since

(1-33)

dV dt

g

fluid

equation

yields F=m

Since

(1-33)

rate

(1-34a)

the

of

also

be written

as

be

written

time

F =w

as

(1-34b)

unbalanced

of change

increment

also

d(mV__) dt

that

per

can

force

of momentum is the

mass

acting

(mV) flow

on

with

rate,

the

time.

equation

dV

(1-35)

g

be

A useful derived

fluid

as indicated

ligible. of

relation, sometimes called the from second-law considerations.

A fIictional

fluid

forces

in figure resistance

is subjected acting

in

Expanding,

Gravitational (force)

downstream in

+(p+

simplifying,

and direction

the upstream direction is

of are

assumed

can of neg-

element

boundary-surface-pressure and

fluid-pressure Therefore,

(A+dA)--dRt

second-order

dropping

motion, an element

as R s. The

direction.

d-_)dA--(p+dp)

and

forces

is indicated

to fluid-pressure the

friction forces acting force in the downstream

F=pA

1-2.

equation Consider

and the

net

(1-36)

differentials

yields F= 8

-- Adp--

dR I

(1-37)

THERMODYNAMIC

The

mass

of the

element

AND

FLUID-DYNAMIC

is m:pAdx

Substituting

CONCEPTS

equation

(1-38)

into F_

(1-38)

equation

pAdx g

(1-34)

yields

dV dt

(1-39)

Since

(1-40) equation

(1-39)

can

be written

in the

form

F_pAV

dv

(1--41)

g Equating

(1-37)

with

(1-41)

now

yields

F=--Adp--dRs=

pA VdV g

(1-42)

and dp + VdV+

dRs

0

p-F--i-A--

(1-43)

p+dp 2

A+dA f f

% ,%

Flow

p+dp

p dx

V+dV

V

dRf FIGURE

1-2.--Forces

on

an

element

of

fluid.

TURBINE

DESIGN

If we now

AND

APPLICATION

let (1-44)

where

ql is heat

produced

by

friction,

in J/kg

dpp __V_VdgV+Jdq1= For

isentropic

flow,

a steady-flow

a system system

or or

electrical

(and

part

of of that

energy,

etc.,

(1-45)

0

P lvl

equal

If

we still

"Jr- V12

Energy process,

must

system.

kinetic energy work I¥,. Thus,

u,+--j-

of

nannuclear)

a system

part

energy pv, mechanical

we have

dqs----0. Conservation

For

or Btu/lb,

we

have

energy Z,

heat

J-J--

that energy, u, flow q,

and

(1-46)

V2" 2 .4 Z_2_4_W Tr

j-2g

entering

leaving

chemical

internal energy

p2l'2_l__ ,

-j+q=u:+

energy

energy

neglect

potential

_j[_Z1

_gj

the

can

to consider

V2/2g,

the

s

where u

specific

internal

Z

specific

potential

q W,

heat added mechanical For

by

energy,

J/kg;

energy,

to system, work (lone

a gas system,

the

Btu/lb

J/kg;

(ft)

(lbf)/lbm

J/kg; Btu/lb by system, J/kg;

potential

energy

can

Btu/lb be neglected.

In addition,

definition pv

(1-47)

h=u + -j Thus,

equation

(1-46)

reduces

to V

"{72

2

(1-48) gd Equation

(1-48)

as we will be using

is the

basic

g form

The

sum

of the

problems,

steady-flow

energy

balance

it. Total

in flow

of the

enthalpy and

and

Conditions the

kinetic

it is convenient

energy

to use

is always

it as a single

appearing quantity.

Thus, V2 h ' _-h + 2gJ 10

(1-49)

THERMODYNAMIC

where

h' is total

The

enthalpy,

concept

of

temperature. that

corresponds

cept

is most

capacity

to

the

leads can

total

when

us

be

enthalpy.

In

that

to

the

defined The

ideal-gas-law

be assumed.

FLUID-DYNAMIC

CONCEPTS

or Btu/lb.

enthalpy

temperature

useful

can

in J/kg

total

Total

AND

as

the

of

temperature

and

according

con-

constant

heat

to equation

(1-15),

h'--h=cp(T'--T) where

T'

with

is total

equation

temperature,

(1-49)

(1-50)

in K or °R.

Combining

equation

(1-50)

yields V2

T' = T+ 2gJcp The

total

attained

temperature

when

to rest The

total,

pressure

T'

can

at static

these

total

two

brought

equation

of

T and

can

between

as

the

temperature

velocity

is also

used

V is brought

called

stagnation

interchangeably. be regarded

isentropically

relation

(1-29)

are

pressure

to rest the

thought

temperature

terms

or stagnation,

p. Since

be

(1-51)

temperature

Thus,

and

of a fluid use

a gas

adiabatically.

temperature,

total

total-temperature

behavior

case,

concept

from p'

as the

a velocity

and

pressure

V and

p is isentropic,

static we can

to write

(1-52)

where

p'

With

is total

pressure,

regard

to the

in N/m 2 or lb/ft _. above-defined

total

conditions,

certain

points

should its use

be emphasized. The concept of total enthalpy is general, involves no assumptions other than those associated with

energy

balance

be seen,

as we have

is a very

tion,

but

heat

capacity.

useful

it is rigorous For

change,

the

use

pressure,

in addition

ture, involves conditions. Flow Let occurs work. each

us now, with This part

in

neither process

of the

only

for

systems

of an

considered

it.

convenience

total

temperature

to the

assumptions

is

With

terms

of total

conditions,

transfer

(adiabatic

heat

turbine

one

No

that

(including

not

between

Heat

occurs the

rotor,

and

of calculaand

reaction

constant

or

a

recommended.

associated

path

as will

burden

behavior

chemical

Process

is the

temperature, the

ideal-gas-law

involving

isentropic

Total

for easing

with

static

No

Work

examine process) (neglecting at constant

phase Total

total

the

and the

temperaand

total

a process

that

nor

mechanical

heat

losses)

radius,

in

when 11

TURBINE

the

DESIGN

velocities

AND

are

expressed

Substitution

of

rangement

APPLICATION

relative

equation

to the

moving

into

equation

(1-49)

blade). and

(1-48)

rear-

yields h2' -- hi' = q--W_

The

energy

dynamics

balance for

If we set

now

a flow

q and

looks

something

process,

W_

equal

as we

to zero,

(1-53)

like

were

the

first

First

Law

exposed

of Thermo-

to it in college.

we get

h_' =-h,' Therefore,

for

constant. that

adiabatic

Further,

total

flow

from

with

no

equations

temperature

also

(1-54) work,

(1-18)

remains

total

and

enthalpy

(1-50),

remains

it can

be shown

constant.

TJ =T/ Note

that

the

enthalpy Total and

process

and

total

pressure

(1-55)

tions,

not

have

the

matter.

ideal-gas-law

be shown

to be isentropic

to remain

is another

and

it can

does temperature

that

(1-55)

From

and

for

in order

for total

(1-22),

(1-52),

constant. equations

constant-heat-capacity

adiabatic

flow

with

assump-

no work,

eJd' l _= P_' p2' Only

for isentropic

constant.

For

flow

flow

(ds=0),

with

(1-56)

therefore,

loss

(ds>0),

does there

total is

pressure

remain

a decrease

in

total

pressure. Speed An wave

important

of

Sound

and

characteristic

propagation

or,

as

of

otherwise

small-pressure-disturbance

Velocity gases

is

called,

the

Ratios the

speed

speed

of

of sound.

From

a is speed the

ideal

of sound, gas

From

theory

a=_/g(-_p), where

pressure-

law

in m/sec and

(1-57) or ft/sec.

isentropic

process

relations,

this

reduces

to a= The factor called 12

ratio

of fluid

velocity

in determining the flow the Mach number M:

V

_gRT to sound characteristics

(1-58) velocit_y

a is an important

of a gas.

This

ratio

is

THERMODYNAMIC

AND

FLUID-DYNAMIC

CONCEPT_

M= --V

(1-59)

a

Mach

number

is a useful

behavior regimes, but expressions. Consider temperature, (1-59),

given

and

parameter

also the

with

(1-51).

equation

ratio

for

Combining

(1-51)

T'

Another velocity critical velocity

only

identifying

for simplifying and generalizing relation of total temperature

in equation

(1-28)

not

to

equations

certain static (1-58),

yields

l_I_7__

often

flow-

(1-60)

M 2

used

is the

ratio

of fluid

velocity

V V Vc,--acr where

V.

sound

at

is critical critical

is equal

to the

condition (1-60),

at

velocity,

velocity

(1-61)

in m/sec

condition,

in m/sec

of sound

or or

at the

is that condition where the critical condition

to

ft/sec, ft/sec.

critical

M----1.

and The

a.

is speed

critical

condition.

The

Consequently,

of

velocity critical

from

equation

2 T.=3"+ and

substitution

of equation

1 T'

(1-62)

into

(1-62) equation

(1-58)

yields

(1-63)

acr=_/_lgRT' Thus,

in any

no work),

flow process

the

value

of the

for the

entire

process,

as

the

static

temperature

The

ratio

the

critical

because while perature The

the Mach

constant

critical

whilethe

of fluid velocity

with

velocity value

velocity ratio.

to

Its

(Vc,=a,)

of the

critical

speed

velocity

use is often

velocity

ratio

number

is not

(since

the

temperature

(no heat remains

of sound

and

constant (a) changes

changes.

critical

in

total

preferred

is directly there

is sometimes over

proportional

is a square

root

Mach

called number

to velocity, of static

tem-

denominator).

relation

between

critical

velocity

ratio

(1-28),

and

static results

and from

total

temperature

combining

equations

in terms (1-61),

of the (1-63),

(1-51). TT '-1

3,--1(V) 3"-t-1 _

2

(1-64) 13

TURBINE

DESIGN

AND

APPLICATION

APPLICATION The the

equations

flow

(flow the

TO

already

through

in the turbine.

there

to learn

varying-area

passages

We

are

going

change,

ously

to

and

presented

examine

number.

equations

yields

to analyze

are

losses

in a turbine,

we can

about

the

of the

rotor,

are

behavior

and

exit

no

losses

diffuser)

use flow

of the

Regime

relations

a,mon_

Prot)er the

there

completely

that

Flow

the

Much

AREA

provided

(stator, of

VARYING

sufficient

something

Effect

area

are

passages,

Although

process

WITH

presented

turbine

is isentropic). loss-free

FLOW

pressure,

manipulation

following

velocity,

of the

equation

previ-

for isentropic

flo_v:

_(I_M_

Equation

(1-65)

velocity

is opposite

the

changes

whether let

flow),

static

to the

Much

Let

Much

with than

the

various

flow

(M0).

and

area

decreases

(dA_0).

and

area

(tecreases

area

increases

(dA_0).

diffuser. (dV>0)

supersonic

increasing

:

(dp0): (dVI):

supersonic

Decreasing

Sonic

of

depend

(dp0):

pressure

Supersonic

2.

change

in area

is a varying-area

Velocity increases (dV>0) This is the subsonic nozzle.

1.

(2) the

1 (subsonic

Velocity decreases (dV0).

nozzle.

1) : (dp>0) must

condition section

and

equal can

zero

occur

decreasing

(dp P2 > P3 E b--

I Entropy, s

FIGURE

34

2-6.--Typical

temperature-entropy

diagram.

con-

expansion

means of a temperature-entropy diagram is a plot of temperature and

of

is

be proven

in this

being

of constant

temperature

terms

assumptions, we have changes can be represented

ideal

by

in

Process

a given can

changes.

the

processes in a turbine The temperature-entropy entropy

for

we will not

a little

represent

increasing

to condiand

definition than

energy)

This

constant-heat-capacity energies and energy and

pressure,

kinetic

is isentropic. (but

the

processes,

of mechanical

graphically

peratures

2 refer stator,

is defined

V rather

Expansion

of

and that

of the

that

(i.e.,

expansion

equations

fact

0, 1, and

This

except

squared

(development process

(2-41)

reaction

energies.

Turbine For

W22

downstream

blade-row

instead

formation

W12

-

respectively.

literature,

power

l

subscripts

stator,

velocities first

W22

with from

BASIC

the

discussion

chapter

1,

example by

entropy,

the

For

the

These The the

coastant-entropy-1)rocess

figure

pressure

purposes four

four

the

A

At

diverge;

is also

diagrams

the

with will

like

process

values

of

therefore,

at

difference

in

looks

the

is repre-

temperature

and

increasing

values

between

any

two

increasing.

of clarity, steps,

diagram

constant-entropy

temperature

CONCEPTS

thermodynamics T-s,

increasing

curves

curves

into

or

2-6.

line.

entropy,

pressure

divided

in

a vertical

constant

given

the

temperature-entropy,

shown

sented of

of a

TURBINE

each

then

be

turbine shown

expansion

process

in a separate

combined

into

T-s

a single

four diagrams represent the stator expansion process relation between absolute and relative conditions

will

be

diagram. diagram.

(fig. 2-7 (a)), at the stator

P()"P'Lid

_,,_Jc-T_= T_

Ti

TO

Tj'

------7

vl p?

T1 Tl, kJ

T1

NJCp 0 (a)

(b)

i.--

p_

P_' T_'

/

/

Ti' : T_'

T_ 2gJcp

P2

>2gJCp

hl--_

T2

CPl

T2 ....

(c)

(d)

l Entropy, s

(a)

Expansion

process

across

stator.

(b)

Relation relative

(c)

Expansion

process

across

rotor.

(d)

Relation absolute

FIGURE

2-7.--Temperature-entropy flow

diagrams turbine.

for

between conditions

absolute at

stator

between conditions

flow-process

and exit.

relative at steps

and

rotor of

exit. an

axial-

35

TURBINE

exit

DESIGN

(fig.

blades

2-7(b)),

(fig. at

Figure

APPLICATION

the

rotor

2-7(c)),

conditions four

AND

and

the

rotor

2-7(a)

shows

before

is represented

point

and

the

the

expansion

indicated state

by

process

by

the

small

by

the

1 with

arrows. actual

It

related

gies

and

The represent If the

static

For

and

were

rotor

exit

process

The

is shown

four

state

as

the

enthalpy (p_,

figure

2-7(c),

it is on

for

the

T2._d), state refers

figure

2-8,

the

entire

stage

36

figure

the

and

by

the

the

which the

ideal

rotor

in

kinetic

in figure

en-

2-7(c)

in

curves

before

and

so

that

would

be that

and

after

the

T;'=

T_'.

indicated

are

is on

the point

ideal 2,id and

absolute

now

are

right state

!

line.

rotor).

The

the

one by

In

figure.

figure

ideal

l_,_t_fing

the

stage

diagram total

the

being, Note

arrows

ignore

the

that

the

line 2-8, the

rotor

alone.

of the

across subscript

in

where

2-7(c),

expansion

the isen-

relative

in figure

across

to the

at

constant-entropy

as in(licated

expansion refers

time

of the

at

into and

indicated

the

ideal

related

energies

total,

an

states

are

combined

For

developed by

total

states

kinetic

expansion on the

energy

developed

These

points.

same

kinetic be

absolute

0 constant-entropy the

relative

absolute

2-7

indicated

stator

relative

assumed, state

would

static,

turbine

subscript

final

2-7(d).

state

to

(both

is

relative

The

the

is not

2,id

by

developed

constant-pressure

flow

the

appropriate

script

that from

figure.

is shown four

pressures

than

of figure

2-8.

rotor

the

that

relative

differences

point

be

proceeds energy

and

total

is less

in

diagrams figure

processes

through

would

If

The actual process proceeds from state 1 to the small arrows, with an increase in entropy.

between

tropically, and the exit are indicated. shown

state

(1-51).

through

in the

The

axial

be noted

it can

process. The relation

static

developed

flow

at each

as indicated

kinetic

absolute

the

isentropic,

Here

actual

the

relative

2,id. by

again

state

process

be

analyze

across

simplicity,

expansion

the

would

conditions.

by the subscript state 2, as indicated by

the

in entropy,

are indicated

process relative

the

expansion.

and

temperatures

of the

than

The total

energy

equation

final

the

stator.

conditions. Figure 2-7(b) shows the relation between relative total states at the stator exit. These states

expansion

terms

with

the

that

we

isentropica]ly, total

absolute

absolute

between

actual

increase

be noted

the

and

kinetic

distance

The

a small

previously,

terms of relative the absolute and

static

The

isentropic,

is less

moving

and

across

the

in accordance

1,id.

can

process

process. As mentioned

are

vertical

were

to the

relative

process

expansion.

point

subscript

0 to state

the

the

state

between

represent

the

relative

2-7(d)).

expansion

after

total

process

relation (fig.

curves and

state

the

the exit

constant-pressure

pressures

expansion

subIn the 2,id

BASIC

TURBINE

CONCEPTS

T_" T_ i1

ii

P2

=__Po

To

Ah'

(h6-

T_? '_ T_' p-

Ahid

\ lh

T1

--

--j,,J

T_ Ca

E

I

T' 2, id

Ahid (h 6 " h2, id )

T2 TZ id

Entropy, FIGURE

is,

2-8.--Temperature-entropy

therefore,

diagram

ambiguous

obvious

from

process

(as represented

be obtained To-- T;. ,d).

figure

but

2-8 an

for

the

work

T o-

ideal

T_)

Since

turbine

used actual blade

for

blade

to express this

rows

is less

purpose

in both from

the

the

work

than

process

operate

performance.

is blade-row

exit kinetic energy row. For the stator,

axial-flow

(as

turbine.

senses. real

It

that

could

represented

by

isentropically, One

efficiency,

divided

by the ideal

common

which exit

we

need

is defined

kinetic

_

is stator

indicated in figure (1-55), we get

efficiency. 2-7(a).

The By

as the

energy

of the

For

the

(2-42)

relation

applying

between

equations

V_ (1-51),

and

where

is

and

(2-43)

rotor

Wi _'°=W_, ,d Vro is rotor

is indicated

2 Vl._a

(1-52),

\pod

L

a

parameter

V_I

where

is

turbine

Efficiency

do not

blade-row

of an

used

obtained

turbine

Blade-Row

parameter

a stage

is commonly

that by

from

s

in figure

efficiency. 2-7(c).

The For

(2-44)

relation

purely

axial

between

W_

and

W_._d

flow,

37

TURBINE

D]_SIGN

AND

APPLICATION

W_,,d=2gJc_,T;' Thus,

with

inlet

it is possible

conditions

and

to calculate

Blade-row

in

as a loss rather

known

for a given

for a specified

terms

than

(2-45)

j

efficiency

exit velocity

performance

expressed

1-\-_,/

of

blade

exit static

kinetic

energy

as an efficiency,

is

sometimes

as

e=l--n where

e is the

kinetic-energy

Blade-row total

differing by dimensionless. actual

also

Several the

(2-46)

loss coefficient.

performance

pressure.

can

be expressed

coefficients

of this

normalizing parameter Inlet total pressure, exit

dynamic

head

have

all

been

in terms

type

used ideal

used

have

this

Axial

I

/I

f

f

!

Y,

Y',

ff

!

between

the

pressure

Y"

Y"o

stated,

and

they

are

total-pressure

can

be derived.

loss

involve

I¢

t!

p2

process

or stage

energy

is isentropic.

Since

parameter that

for expressing

we use

ratio

is the

of actual

This

Overall stage or

stage

to

turbine

we

are

It is the the

ideal

or stage

discussing

process

inlet

transfer condition

aerodynamic

when

simply

which

refers energy

energy

based to

efficiency

the

exit and

transfer. The

definition

are

overall

transferred on

turbine

in the

isentropic pressure. are

as the

efficiency.

above

to the

a

parameter

is defined

or adiabatic the

expansion we need

The

(isentropic)

apply

the

isentropic,

performance.

isentropic

efficiency energy

total-

are not

dependency.

efficiency,

of actual

various

relations

is never

or stage

that we can to follow. ratio

the

Relations

Efficiencies

to ideal

as the

coefficients.

is maximum

or stage

e_ciency.--Overall

process.

the

38

the

transfer

ways sections

Stage

transfer

turbine

is known

several different discussed in the

and

turbine

energy

efficiency

number

(2-47c)

--P2

and

These

(2-47b)

P_

loss

coefficient

a Mach

Turbine Turbine

_f

P2

kinetic-energy

loss coefficients

(2-47a)

,, yT o__P, p_ --P2 --p_

"Po--Pl

and

It

--P2,,

'

• 't--P--_--pl

where

as follows:

PI

Po--p___._ "--P;--Pl

V"

each

the coefficient head, and exit

purpose

YTo-_P-'

Po

v'

used,

in

rotor:

t

po--pl,

of a loss

been

to make dynamic

for

Stator:

y,

row,

pressure.

not,

or

turbine

flow

from

Note

that

at

present,

BASIC

considering seal friction.

mechanical

We will turbine.

define

This

however,

Actual

absolute

total

in figure

2-8.

Now

work

energy

transfer

energy

for conversion conditions are used.

If

plenum, energy

turbine

then

the

could

shaft for

the

work the

in the

down

If we were the

kinetic

loss.

The

are

carried

shaft

to the

exit

ideal

case,

energy kinetic

work.

the

cases

where

pose,

the

entire

from

the

exit

case

is the

high

velocity

leaving

and

effi('iency exit

total

before turbine exit

based by

the exit total kinetic energy)

is always

between

the

two

the

ideal is

situation,

obvious must

as a

wasted,

but

of its exit

to static

of their

serves

of ideal

a useful work

exit pur-

computed

example

of this

be expanded

therefore,

In exit

converted

basis

most and,

not be

basis the

basis

engine, ideal

ideal

enthall)y

2-8.

It

can

must based

higher increasing

than

work

available

is culled

work

called

con(tition than that

efficien('y

are

the

gas

to

the

above

may

the

state

be considered

energy

the

equal

in tile

on

static energy.

would

The

Here

exit

to

exit. kinetic

stages on

converted

a high

to

a

velocity

a waste.

on

on the the

on

the

kinetic

desirable

they

rated

kinetic

('.on(litions

conditions in figure

are

conditions.

based static

the

as in a

wasted

been

would

other

is rated

the

zero

where

stages

leaving

inlet,

is available

be

state

stage

the

turbine.

have

turbine

stage,

stage

is not

the

This

it would

with

last

is rated

state

available

is dissipated,

we use

total

turbine-exit

turbine total

rel)resented in(licated

last

the

efficiency

total

next

other

jet-engine

the

The

the

the

leaving

to the

Thus,

state

a multistage

energies

over

energy At

in

is indicated

energy

wasted.

because

exit

only

the

decrease

this

ideal

energy

a case,

work

the

from

by

l)lus exit kinetic

and

kinetic

if it could

such

static

considering

the

is just

1)ut to use In

done

occasionally,

is the

conditions.

kinetic

of ideal

work

and

At the turbine or stage exit., static and total conditions are sometimes

energy

turbine.

condition, while total conditions. In

been

as bearing

work

or stage,

inlet

CONCEPTS

people;

herein

or total

work. used

shaft

as shaft

to define the

such

most

defined

turbine

exhaust-flow

have

this desirable static state.

the

exit kinetic

computation

expand

as

because

to shaft sometimes

by

is defined

of static

used

as the

used

whether

basis

is ahvays

to items

transfer

across

consider

due

one

transfer

enthalpy

on the

state

energy is the

we must

to do total

actual

definition

actual

energy.

inefficiencies

TURBINE

available tile be

the

between

the

inlet

efficiency.

The

decrease

for each

of these

that

the

be less (as long on the exit static static

efficiency, increasing

ideal

with

inlet The

total

and

conditions

work

as there condition. exit.

the

efficiency.

total seen

with

between

static

cases

are

based

on

is some Thus, the

kinetic

exit total

difference energy. 39

TURBINE

DESIGN

Overall

AND

turbine

APPLICATION

efficiency

_ and

similar equations. The inlet and exit conditions,

subscripts instead

stage.

static

Overall

turbine

stage

efficiency

_tg

are

defined

by

in and ex are used to denote turbine of the subscripts 0 and 2 used for the

efficiency

can

be expressed

as

(2-48a)

For the ideal-gas-law reduces to

and

constant-heat-capacity

assumptions,

this

T',.--T',_ V

/ p,, \(_-l)/_l

Overall

turbine

total

efficiency

(2-48b)

J

T;. is expressed

as

hh' h;.--h',_ _' = .-_-7-, =h' _'

For the ideal-gas-law reduces to

and

constant-heat-capacity

._, _

total

and

appropriate Relation

of turbine

efficiencies

are

(2-49b)

similarly

defined

but

with

the

as

However,

to stage

a measure

it is not

efficiency.--The

of the

a true

overall

indication

overall

turbine

performance

of the

of

efficiency

the

of the

comprising the turbine. There is an inherent thermodynamic hidden in the overall turbine efficiency expression. If equation

(2-48b)

or (2-49b)

a given

stage

which

were

pressure

for a stage

ture

would

gas

entering

2-8,

the

losses

perature stage though

for a stage,

and

stage

the of one

For

stage

appear

the

following

is then

capable

of

delivering

all the

individual

overall

turbine

the

number

of stages.

effect such

can as

stages

efficiency be shown

figure

2-9.

by The

the

a turbine, in the

stage have

del)ends

means solid

to

form

on

the

same the

line

transfer, teml)era-

of a higher

pressure

O-2,id

tem-

following

Therefore, stage

of a temt)erature-entropy verticnl

for

be __een from This

work. the

that

energy

as can

(T_T2,_,).

additional may

still

be seen

is prol)ortional

stage.

entering

it could

efficiency,

be (To--T_),

gas

the

This

written ratio

of the

figure

4O

[1 --\p_--_-/ (P-_= _('-'>"]

e3_ciency

is useful

turbine.

gram,

this

subscripts.

efficiency stages effect

static

assumptions,

T'_.--T',_ T_

Stage

(2-49a)

even

efficiency, ratio

and dia-

represents

BASIC

TURBINE

CONCEPTS

Pk k

_ \\

1

i '

b---

E

\\ A,'i .,tg

\

¢D

E

_\

¢D b--

Ahld:stg

\I:

\\

Ahld, stg

2,id

Entropy, s 2-9.--Temperature-entropy

FIGURE

isentropic

expansion

dashed

line

taking

place

The

0-2

from

obtained

ideal

work

difference

of temperature

increasing

p_),

the

that

represented

greater

by

constant E-F

stage

is greater

ing

the

the

sum

which and

by

virtue

ideal

Thus,

the

than work turbine

the

the

of

actual

work

B-2,id.

With

lines

three

stages,

it can ideal

be seen work

that

the

0-A,

work

where the

increases

stage

(p_

is greater for

this

stage

be greater.

C-D,

and

and

to

than stage

an(l,

is for

Similarly,

E-F

represent-

2; Ahld. stt representing

Z Ah;dstg

represented

second C-D

previous will

_' _'sto.

previously, pressure

the

line

The

efficiency

is ,7_tg Ah_,.stt,

isentropic

for

the

for the

p_.

efficiency

mentioned

of constant

by

inefficiency

efficiency,

of these,

is the

A-B.

of

pressure

stage

stage

Hence,

represented

in a multistage

turbine

same

As

lines

of entropy.

work

each

a stage.

between

values

isentropic

the

effect

exit

of overall

having

from for

reheat

Po to

process

each

work

the

showing

pressure

the

stages,

Ah_d. stg is with

inlet

represents

in three

actual

diagram turbine.

by

is greater the

sum

than of 0-A,

A-h_, A-B,

B-2,id. 41

TURBINE

DESIGN

The

total

to P'2 can these

two

AND

actual

APPLICATION

turbine

be

represented

values

must

work

obtained

by

either

be equal.

from

7'

the

expansion

Ah',d or

_tg

from

Z Ah'_.,t,,

Po and

Thus,

-'Ah'

'

(2-50)

z ah,_.,,, Ah'_

(2-51)

or

, _-_

Since

2_ Ah'_d,s,g

greater This

than effect

be confused between

_Ah_d,

with

the

stages,

The

of

efficiency

turbine

process

which

equation

stages

the

overall

isentropic

efficiency

the stage isentropic efficiencies, or _' _tg. in turbines is called the "reheat" effect. is also

for

constant

of adding called

calculating stage

heat

from

This

an

is

must

external

not

source

"reheat".

overall

pressure

turbine

ratio

efficiency

p_/p_

and

for

several

constant

stage

_t_ is

I--

-,

1--_,t_

J)

1--\_-£]

_1 --

(2-52)

( p i t nt(.y_ l)l_, ]

1 --\_oo/ where n is the number be found in reference The

fact

on the

that

stage

pressure

ratio,

of turbine

efficiencies

not

comparison

a true

higher

pressure

sirable

to be able

In

order

ciency

ratio

of stages. efficiency raises

from

machines

of their

to express

temperature

(T--dT),

by

a true

all reheat

the

consideration.

can

this

a gas where

efficiency

from

is expanded d T is the

we

It

have

ratios

is

as the

one

would

be

of de-

for a turbine. to be the

effi-

stage.

of isentropic

definition,

A comparison

effect. would

depending

pressure

behavior,

reheat

aerodynamic effect,

small

suppose stage

equation

efficiency,

of different

eff_ciency.--Starting

T,

of this

turbine

aerodynamic

is helped

Infinitesimal-stage

tropic-efficiency

derivation

an important

of an infinitesimally

infinitesimal

differs

of two

to eliminate

temperature an

The

1.

pressure

to pressure

increment

efficiency

p

(p--dp)

of temperature

vp. By

using

the

and and for isen-

write

d T = _pT E l --(ppdp

) (_- ' ) /_]

(2-53)

and dTT_ _1o[1--(I 42

__)(_-

z,l_]

(2-54)

BASIC

These

equations

efficiency

the be

the

the

be

than

static

rigorously

authors

the

in accord

ignore

proportional

the

to the

that

stage,

there

temperature

fact

that

the

is used

isentropicactual

in

the

work

differenOther

in kinetic

d T'----d T. However,

that

the

differential.

is no change

so that

with

CONCEPTS

total-temperature

static-temperature

assumption

infinitesimal

quite

Some

should

rather

make

not

definition.

differential tial

are

TURBINE

authors

energy

it ahvays

across seems

to

infinitesimal-efficiency

expression. Using

the

evaluation

series

expansion

of equation

approximation

(2-54)

(1-#x)

n= 1 +nx

dT 7_1 dp T --_ 7 I' Integrating

between

the

for

yields

turbine

inlet

(2-55)

and

exit

yields

Tfn

In _(2-56) z/P--7--

1 In p_" "Y

Equation

(2-56)

The

can

infinitesimal-stage

dynamic

efficiency,

efficiency

is also

from

method

the

constant, called

be written

where

we get

for the

as

efficiency exclusive

known

as the

n is called l)olytropic

the

effect

l)olytropic an

the

process.

_p is SUl)posedly

of

of expressing

a l)olytropic

Pex

of

the

efficiency.

irreversible

polytropic

for

This

process

exl)onent

Substituting

true

aero-

ratio.

This

name

arises

pressure

v from

path

as pv"=

, and

the

process

the

ideal

gas

process

{p,n'_

T,n

(n-l)/n

(2-5s)

\p,,! Equations process relate

(2-57) were

l)olytropic

and

(2-58)

to be expressed efficiency

are as

and

the

n

inlet

and

exit,

very

similar,

a ])olytroi)ic

n--1

If we neglect

is law,

polytrol)ic

and

if

the

process,

then

exponent

as

-),--1

turbine we

couhl

(2-59)

?

kinetic

energies

for

the

overall

turbine 43

TURBINE

DESIGN

process,

we

ciency.

Actual

AND

can

APPLICATION

relate

turbine

temperature

overall

efficiency

could

be expressed

drop

T_ _Te_='_T_,

to

E1 --\_-_,/ (Pe_ y"-l)/_

(_-e_)

1J

_'{(_-

1)/_]

L -\_/ Equating

(2-60)

with

(2-61)

then

(2-60)

k

(2-61)

J

yields

_=1--\_/

This proach

relation

is illustrated

each

other

unity.

However,

ciency

levels,

as at

the

two

in

(2-62)

figure

pressure

higher

effi-

as

or _1

polytropic

2-10.

ratio

and

pressure

efficiencies

can

The

two

efficiency

ratios,

especially

differ

signiticantly.

efficiencies each at

ap-

approach lower

effi-

.9-Turbine pressure

¢-

_.,,_

r_tio /.,///

09 ----

.8

B

b---

.5

1

I

.7 Turbine

FIGURE

2-10.--Relation

between Specific

44

I

,8 poly_ropic efficiency,

turbine heat

overall ratio

I

.9

7,

ffp

and 1.4.

1.0

polytropic

efficiencies.

BASIC

DIMENSIONLESS Dimensionless turbine

number introduced

of

the and

dimensionless

serve

geometry,

and

to correlate

Dimensional

analysis

variables the

or

minimum

more

the

analysis

that some

of the

procedure

variables for

and

obtaining

flow

yields

The

dimensions, term

resultant

ratios

implies

the

actual

insight

that

into

shape

and linear

]'his

based

on

on

is an

various

These viscous

a basic

ideal attributes

include

effects;

an elasticity

number),

which

number, pressing are

the The

matic, similitude.

which real

parameter

effects,

significant concept and

fluid,

ratios

which which for

for gas groups leads conditions

the

Reynolds

and

based

relations.

the

effect

of

surface-tension to the

and Of

analysis

ideal

reduces

effects;

change of the

an

expresses

a gas

than

groups,

expresses

effects. the

of

modify

of

geometrical rather

dimensionless

that

gravitational

quantities operating

physical

is a con trolling

characteristic other

in general,

of dimensionless two

are

(which

parameters

dynamic If

by itself,

compressibility

expresses

fluid

dimensions),

number,

expresses

formal

of fluid

basic

The

of linear

of

1, which

represent

dimension

number,

Weber

of the

of a

pertinent

problem

of velocities.

parameter

There

Reynolds

the

nature

The the

the ratio of the force due to the inertia force due to the motion

of a real

the

forces;

flow

fluid.

general

states

powers

reference

terms

ratios

(as a ratio

of each

factor. Another term expresses of pressure in the fluid to the fluid.

the

dimensionless

of forces,

magnitude

to the

and

form

of

group.

including

analysis

which

from

be

of dimen-

in the

groups

con-

once

basis

product

a dimensionless

the will

at least

7r-Theorem, a

some

each

groups

The

be expressed

in many texts, this discussion.

considerable

relations.

may

groups,

them.

is the

throw

for grouping

variables

dimensionless

of dimensional

are of

use

of variables

they

of such

representing

forming the

variables is presented served as the basis for Application

all the between

equation term

a group

so that

number

1)rocedure

each

allows

of dimensionless The

relation

physical

A

parameters for the

It is a procedure

to include

as a formal terms,

that

to be arranged

number

physical

a complete of

performance.

Analysis

relation.

variables.

necessary

to represent

number

of the

a smaller

two

sional

relation

nature

into

taining

diagrams,

analysis.

is a procedure

a physical

on the

velocity

turbine

is dimensional Dimensional

light

classify

more commonly used dimensionless discussed in this section. The basis

parameters

comprising

CONCEPTS

PARAMETERS

parameters

classify

TURBINE

the

these

Mach Froude

terms

Mach

ex-

numbers

flow. as ratios _o are

the such

of geometric, idea that

of

kine-

similarity all

the

or

dimen45

TURBINE

DESIGN

sionless

terms

of the

separate

obtained.

AND

have

APPLICATION

the

same

variables,

Complete

value,

then

physical

similarity

is ever

be approached

of similarity

the

to the

operation

rather

full-size

at some

Another fluid

design

Turbomachine Application flow

in are rows

the

use

in conjunction

_4th

size,

following variables tant relations : Volume

flow

Head,

to the

rate,

P,

Rotative

speed,

set

to ideal and

Q, m3/sec

or ft3/sec

parameters.

These

and

of the of the

power

fluid.

more

The

impor-

W or Btu/sec N,

rad/sec

or rev/min

Fluid

viscosity,

u, (N) (sec)/m

Fluid

elasticity,

E,

dimension,

variables,

the dimensional five

capacity, 3, which

D, nt or ft

_-or lbm/(ft)

N/m 2 or lbf/ft five

dimensionless

or flow is called

_2_' rate, the

groups

constants groups

can

in order

can

be formed.

to ease

be expressed

p-N3D 5' is exl)ressed capacity

(sec)

2

dimensionless

conversion

_fcn

Q/ND

of fluid

flow rate,

some

p, kg/m 3 or lb/ft 3

The

work),

to demonstrate

density,

the

problem

operational characterin the relation of head

used

Fluid

lation,

of

properties

linear

drop

involves conditions

or (ft)(lbf)/lbm

speed,

these

results

detailed examination of flow within In addition, dimensional analysis

Characteristic

From

46

relates

so that

the

ambient

general

the

are

H, J/kg

Power,

this

scale

with

Parameters

has great utility in the analysis of the overall istics. For any turbomachine, we are interested flow,

use

of similarity

at or near

mentioned

important for the of turbomachines.

(for compressible

it

One

condition.

analysis

previously

linear

the ratios

purposes

utility.

performed

Operational

of dimensional

results

parameters the blade

be

with

the

similar-

the ratios complete

practical

of smaller

can

are

are everywhere the velocity

for most

machine.

severe

(1) geometric

to be of great

of models

of machines

than

but

experiments

values

conditions

which means that is doubtful whether

closely

operation

inexpensive

applicable

attained,

sufficiently

is the

relatively

implies

similarity, same. It

individual

physical

dimension ratios which means that

are the same; and (3) dynamic of the different forces are the can

of the

similar

similarity

ity, which means that the linear same; (2) kinematic similarity,

physical

regardless

exactly

u

the

manipu-

as

(2-63)

' pN2D2/ il_ (limensionless

coefficient.

If we

It

can

form be

further

by

BASIC

rel)resented

TURBINE

CONCEPTS

as Q

VA

VI) 2

V

V

(2-64)

Thus, the capacity coefficient is equivalent to V/U, and a given value of Q/ND 3 implies a l)articular relation of fluid velocity Io blade speed or, in kinematic terms, similar velocity diagrams. The head is expressed in dimensionless form I)v H/N21) ', whi(,h is called the head coefficient. This can be rel)rcscnted as H H N 2D,_a: -_

(2-65)

Thus, a given value of t./N2D 2 iml)lies a particular rclatiol_ of hea_l to rotor kinetic energy, or dynamic similarity. The term P/pN3D 5 is _t power coefficient. It represents the actual power And thus is related to the capacity and hea(t coefficients, as well as to the efficiency. The term pND2/u is the Reynolds number, or viscous effect coefficient. Its effect on overall turbine 1)erformance, while still iml)ortant, can be regarded AS secondary. Ti_e Reynolds number effect will be discussed separately later in this chapter. The term E/aN2D 2 is the compressibility coefficient. Its effect depends on the level of .XIach number. At low NIach number, where the gas is relatively incoml)ressible, the effect is negligible or very secondary. As NIach number increases, the compressibility effect becomes increasingly significant. Velocity-Diagram

Parameters

We have seen that the ratio of fluid velocity to blade velocity and the ratio of fluid energy to blade energy are inq)ortant factors required for achieving similarity in turbomachines. Since completely similar machines shouhl perform similarly, 'these factors become iml)ortanl as a means for correlating performance. Since the fa('tor._ I_/'U aud H/U z are rclatc(I to the velocity (liagrams, factors of this type ar(, refcrrc(I to as velocity-diagram I)aramcters. Several velocity-(liagram parameters are co,mnonly us(,_l ill t_,rt)illc work. NIost of these arc ,ise(l I)rim'trily with rcsl)e('t to axial-ttox_ turbines. One of these parameters is the speed-work parameter X::-

U 2

(2-66

gJ/W 'Fh(_ re(:ipro('al

of the sl)ce(l-work

parameter

is also oflcn

use_l, aml

it 47

TURBINE

DESIGN

is referred

For

AND

to as the

an axial-flow

APPLICATION

loading

turbine,

factor

or loading

1 ¢=X--

g JAb' U2

we can

coefficient

(2-67)

write

(2-68)

Therefore,

equations

(2-66)

and

(2-67)

X=

Another

l)arameter

often

used

1

can

be expressed

as

U

(2-69)

¢--AV_ is the

blade-jet

speed

ratio

U p_

where

V_ is the

or spouting,

jet,

or spouting,

velocity

is defined

ideal

expansion

stage

or turbine.

from

inlet

That

(2-70)

"_fj

velocity, as the

total

in m/see velocity

to exit

or ft/sec.

The

corresponding

st, atie

(:onditions

across

of equation

(2-71)

back

into

(2-71) equation

(2-70)

yields

U _= -7-= _ 2gJAh,d A relation

between

parameter the

static

can

the

be obtained

efficiency

the

is, Vj2= 2gJ,_h,,_

Substitution

jet,

to the

blade-jet by

(2-72)

speed

ratio

use of equations

and

the

(.2-66)

and

speed-work (2-72)

and

definition zXh'

The

resultant

This

shows

must

also

is directh speed

ratio

Another 48

relation

that

if efficiency

be a function related

onh

n =Sh-_

(2-73)

v= _/____ _

(2-74)

is

is a funclion

of the

other.

to the

is related

to the

frequently

used

actual

velocity

of ()ne of these

While

the

veloc)tv diagram

velocity-diagram

parameters

speed-work diagram, and

to the

parameter

it

parameter the

blade-jet

efficiency. Is

the

flow

BASIC

factor,

or flow

TURBINE

CONCEPTS

coefficient

(2-75) The

flow

coefficient

can

be related

to the

loading

coefficient

as follows:

v, / v, By

using

equation

(2-69)

and

the

velocity-diagram

_=#

cot

(V,,. ,'_ al kAVu /

The term Vu. I/AV,, cannot be completely specific types of velocity diagrams, such next

chapter,

(a different each can

this

term

function

of the

becomes

for each

different

be expressed

types

of the

(2-77)

of loading

of velocity

of velocity

in terms

we get

generalized. However, for as will be discussed in the

a function type

geometry,

coefficient

diagram).

diagrams,

Therefore,

the

loadingcoefficient

alone

flow

and

for

coefficient

the

stator

exit

angle. It

is thus

related

seen

to each This

next

and

chapter. one

Where

of

the

efficiency. or the two flow

We

related

that

we have

with

constant

stage

of this this

at

of

For

are other

type

use

only loss, therefore, definition is

stage is

the

in figure exit

kinetic

of equation

(2-68)

into

in the

for

only

correlating parameter

efficiency

correlation,

ease

how static speed

impulse Further

(total energy.

equation

the

(W_:

Assume W2)

stage

diagram assume

efficiency The

efficiency

ratio.

A velocity

2-11.

be

Parameters

blade-jet U_)

the next

is specified,

static

h'o-h'2 Ah' ,7-- h£--h2. ,_ --_h,_ Substitution

in

One parameter must the loading coefficient.

(U_=

is isentropic

to these

speed-work

general

specific the

are

case

case

diagram

(V_._=V_,2).

is shown

turbine

real

Velocity-Diagram

to

axial-flow

be related specific

is required

a more

for an idealized

velocity

can

of velocity

required. is usually to

parameters

idealized general

generally

ratio.

a single

an

more

mathematically axial

efficiency

parameters

Efficiency

show

velocity-diagram

for

type

Lewis

parameters and the

will now

through

addition, shown

a somewhat

speed

of these coefficient,

be

for

In be

a particular

blade-jet

We

will

four

velocity-diagram

Relation

can

these

other.

parameters. section

that

for

that _'=

a

flow

1).

The

efficiency

(2-78) (2-78)

yichts 49

TURBINE

DESIGN

AND

APPLICATION

a1

Vu,1 j/ V2

Wu, 2

FIGURE

2-11.--Velocity-vector

diagram

for

an

axial-flow,

impulse

stage.

UAV,,

(2-79)

= gJAh,.,t The

change

in fluid

tangential

velocity

is (2-80)

zxV,,=V_,,-V_,z From

the

assumptions

convention

(Wl---- W2)

and

(W,._=

and

W,.2)

the

we adopted, (2-81)

W_. 2= --W_. 1 From

sign

equations V,.2=W,.5+

(2-6),

(2-81),

and

(2-80),

U=-W,._+U=-(V,.

we get 1- U) + U=-

V_.I+2U (2-82)

and AV.=V..,I--V.,2=V..1--(--V..,+2U)--2V..I--2U From

the

velocity-diagram

(2-83)

geometry (2-84)

V_, 1= V1 sin m Since

flow

is isentropic

and

the

turbine

stage

is of the

impluse

type

(h2, _d=h_=hl), Vl=_ Substitution 50

of equations

(2-84)

2gJAh_e and

(2-85) (2-85)

into

equation

(2-83)

BASIC

TURBINE

CONCEPTS

yields AVe=2 Substitution

of

sin al_/2gJAhia-2U

equation

(2-86)

back

into

4U sin al

definition

Equation stator only. an

The

(2-88) angle, with

is reached speed

tion

(2-88)

shows

ratio and

that

static

variation

example

0.88 jet

exit

speed

for this and

mathematically

ratio

derivative

of axial-flow,

case

and

of 0.47.

The

by

for of

optimum

bladeequa-

(2-89)

I

I

I

.6

.8

stage.

2-12

to zero:

.4

speed

speed ratio,

ratio Stator

on

ratio

efficiency

differentiating

I

impulse

constant

speed

in figure

.2

blade-jet

any

of blade-jet

sin al 2

Blade-jet

2-12.--Effect

(2-72) (2-88)

equal

Vo_t--

FIGURE

equation

of 70 °. A maximum

speed

the

from

is illustrated

angle

can

setting

ratio

particular

at a blade-jet be found

(2-87)

is a function

is parabolic exit

yields

sin a,--4_3

efficiency

a stator

(2-79)

2gJAh_

of blade-jet n=4z,

equation

4U _

_--4'2gJAh_a Now using the finally yields

(2-86)

static exit

1.0

v

efficiency angle,

of

an

isentropic,

70 °.

51

TURBINE

Since the

DESIGN

the

AND

stator

sine

APPLICATION

exit

of the

angle

angle

is normally

does

not

in the

vary

greatly,

speed ratio for most cases of interest be in the range of 0.4 to 0.5. Equation specific.

(2-88) While

ideal

case,

indeed, very

the

the

figure

2-12

levels

and

values

basic

it does. good

and

We

parabolic find

correlating

Likewise,

for

other

tion

(2-63).

This

does

would

of all sizes.

Such groups.

apply

would

however,

to

apply

be

parameter is found

exit

used

for

or turbine

blade-jet static

would

idealized the

from

same;

speed

and

and

ratio

total

Ds and

parameter

efficiency.

on the variables relating to parameters shown in equathe

number

A parameter not because values of

case,

by

having of the

geometrically not

this

of dimension-

rotative

values

of

at D

similar

having

two

turbospeed

the

all rotative

combining

excludes

the linear remaining

Q

remaining speeds.

of the

is known

\l/2/N2D2\3/4

the

volume

excludes

as

previous

the

specific

NQI/2

flow

rate

is taken

at the

the

Commonly, are 52

quoted

stage

N

(2-91)

is known

as the

specific

diameter

as ( H D_=\-_VD2]

With

N

Thus,

that

is found

is a

parameters.

NE),/2

The

the and,

as

a turbine, exit.

type

will differ

case,

exhaust

found

that

case remain

to a turbomachine

can

very

should both

range in

/

When

of course,

a parameter

because,

parameters N8 and

a

Also,

desirable

The

speed

of this

blade-iet

Parameters

are possible. be desirable

machines variables

not,

that would

variables be

a turbine

of dimensional analysis led to the dimensionless

less parameters dimension D

would

optimum

for a real

for

of 60 ° to 80 °, where

the

velocity-diagram Design

The operation turbomachines

are,

a real

parameter

so are the

with

trend

that

range

volume

but with

flow

not rotative

rate

,,_1/4 (V)

1/2-

taken

at

D,--

DH1/4 Qi/_x

exclusively,

the

speed

N

the

in

DH1/, Q1/2

stage

exit

(2-92) or

turbine

exit, (2-93)

values revolutions

for

these per

parameters minute,

exit

BASIC

volume flow rate H in foot-pounds of units,

Qe_ in cubic per pound,

specific

less because

speed

the

to be the

and

units

are

total-to-total

it is specified Specific

as the

speed

presented

not

diameter

consistent. (5h;_),

total-to-static diagram

parameters.

dimensional

The

ratio

of total

differing eters. all

efficiency

definitions Some

cases, The

thus

eliminating

parameter

speed-work (2-96)

to the

parameter

by

to the

blade

speed

previously

is

or 60 sec/min).

The

(n)-_

(2-95)

and

(2-72)

efficiency

used

use

the can

appears the work

ideal

ratio

from

expressed equation

equations

because

in defining

be

substituting

with

same

efficiency

interrelation

taken

convenience,

(2-94)

(2-93),

work

prefer

for

(27r rad/rev

to static

of ideal

authors

H is usually

be related

The

constant

equations (2-91), (2-95) yields

dimension-

(hh_).

can

H=JAh_d=J_d Combining (2-94) and

head

truly

or head, this set

_ND K

U-where K is the head is

not

sometimes,

value

diameter

are

The but

and specific

velocity

CONCEPTS

feet per second, ideal work, and diameter D in feet. With

specific

value

TURBINE

of the

various

param-

definition

equation in

(2-74)

(2-96).

terms

of

into

equation

NsD8 ' :-- K -v/g_'X -

speed The

and specific exit volume

diameter flow rate

can is

also

be related

to the

Qe_=A_V_ where (2-91),

Since diagram

A_x

is the

(2-93),

specific

flow

area,

(2-94),

and

(2-75)

with

NsD,

3=-- KD2 _'_A_

parameters,

and

specific

which

can

in m 2 or

diameter be used

flow

(2-98)

exit

speed

the

(2-97)

"/l"

Specific coefficient.

in

ft 2. Combining

equation

(2-98)

equations yields

(2-99)

are to

related correlate

to the

velocity-

efficiency,

then 53

TURBINE

DESIGN

specific

AND

speed

and

APPLICATION

specific

diameter

can

also

be

used

to

correlate

efficiency. Specific

speed

and

velocity-diagram flow rate, and (2-99),

diameter

are

sometimes

often

dictate

diameter

that

imply

sometimes referred the

shape. to

type

of design

these for

are

the

equal

applied

thus

overall

following

are

entire

can

that

be

might the

we

have

turbine.

and the

been

When represent

used

some

the

and volume appearing in

speed since

the

specific They

are

shape

will

Parameters

turbine,

the type of design that rapid means for estimating

that

parameters.

parameters,

parameters

and

specific shape

that

or to the

similarity

values to the

The

variables

to be selected.

parameters

to a stage

as

to as design

dimensionless

be applied

Thus,

referred

Overall The

contain

parameters do not. These are diameter their use leads to terms, such as D2/Aex

equation also

specific

most

to a stage,

similar

conditions

efficiency.

parameters

be most appropriate number of required commonly

can

applied

to correlate

of these

discussing

help

and stages.

When identify

serve

encountered

as

a

overall

parameters: Overall

specific

speed __

No1/2

N.----Overall

specific

(2-100)

D"_H1/'

(2-101)

diameter "_

Overall

___e, H3/4

speed-work

parameter (2-102) gj_'_

Overall

blade-jet

speed

ratio

;=

The

subscript

script

(--)

Of nificant

av refers refers

these

overall

considerations erally for that

its on the

specific

contribute

U..

some

value

the

nature speed

to the

value

for

perhaps

determined the

the

super-

evolved

geometry. to show specific

speed.

is most by

other

be restated of overall

and

turbine.

speed

always

Q,_=wv,_ 54

condition,

entire

values

of the can

the

specific

is almost

while

(2-103)

average

for

parameters, value

only,

depend overall

to

to the

because

,

parameters Equation the

sig-

application gen(2-100)

considerations Let (2-104)

BASIC

where

ve_ is specific

flow

rate

volume

be expressed

at exit,

in m3/sec

TURBINE

or ft3/lb.

CONCEPTS

Also,

let

mass

as JP w----_

(2-105)

_'H

Then, (2-100)

substitution yields

of

equations

-_,=1_.7 Thus, three

the

overall

terms.

The

be reasonably

estimated.

gas

thermodynamic

and

the

useful choice

for evaluating is available)

by the

application.

in other

cases,

of N and The

P,

second

both

product

by

in which

as the only

conditions.

the

and

than

of

which

can

on the specified

This

speed

rather

product

performance,

depends

rotative the

equation

oo)

be expressed

term

N_/J__,

into

,=\

expected

cycle

is established

manner

(2-105)

effects that different fluids on the turbine. The third

Often, the

can

reflects

The

the have

vex kll21

speed

term

and

t

/

specific first

(2-104)

second

term

is

(in cases where term is dictated

power

the

are

specified

individual

;

values

application.

overall

specific

speed

influences

the

tur-

axial flow

axial flow One-stage, _'_

A

flow

[

I

I

I

I

I

I

I

I

I

I

.2

.3

.4

.5

.6

.7

.8

.9

1.0

1.1

1.2

I 1.3

Specific speed, Ns, dimensionless

I

I

I

I

I

20

30

40

50

60

I 70

I

I

I

I

I

I

I

I

I

80

90

100

110

120

DO

140

150

160

112)(ibf3/4) Specific speed, Ns, ( ft3/4 )(Ibm314)/(min)(sec t FIGURE

2-13.--Effect

of specific

speed

on turbine-blade

a

shape.

55

TURBINE

DESIGN

AND

bine

passage

shape

bine

and

one-

ratio

for

of hub

speed.

For

decreases

is illustrated and

radius the

APPLICATION

axial-flow,

tip

decreases

to

radius

radius turbines,

ratio.

of the

Dividing

equation

for stage

specific

number (2-100)

speed

for

the

a radial-flow

tur-

turbines.

The

example with

increasing

Thus,

application indicates the type The values of some of the proximation

2-13

two-stage,

axial-flow

the

in figure

increasing

the

overall

specific

or types of design overall parameters

specific

number

of stages

speed

for

any

that will be required. give us a rapid ap-

of stages

required

for

for overall

specific

speed

a given

application.

by equation

(2-91)

yields N_

Qe_

1/2 (2-107)

If we neglect change

per

the

reheat

stage,

effect,

we can

which

is small,

and

assume

equal

H=nH Further,

if the

expansion

compressibility these

effect

last

two

head

write

ratio

and

is not

assume

conditions

(2-108)

into

too

that

large,

we

can

Qe_:-Q_:._t_.

equation

(2-107)

neglect

the

Substitution

and

of

rearrangement

yields n:(NsY/3

(2-109)

\NJ Since

stage

specific

experience assume

can

number

type

overall

speed-work Often,

basis

of stress

suming

2, where

by

requires

value

of blade

equal

Dividing

equation work

Or,

equation

(2-102) blade

per

blade

speed

on

this

can

speed

for

the

for is

is presented.

for

obtained the

of be

the

for

for stage (U2=U_v),

the

overall blade on the

varied

_peed-work

turbine

the

estimate

be selected

may

(2-66) overall

from

value

a value

to

stage

estimate

is often

a knowledge

for

para-

speed-work parameter, and

as-

stage,

ah' =n h' 56

an

an assumed

speed

speed with

known us

correction

of

however,

Thus,

of stages and

Knowledge

efficiency,

specific

speed gives

number

parameter

considerations.

a constant

specific

of compressibility

parameter.

if desired.

of stage

(2-109)

for

for

of efficiency.

a compressibility

speed-work

a reasonable

parameter assuming

effect

parameter,

speed.

overall

of estimate

speed-work

level

equation

The

in reference

metrically

and

parameter

value

a given

assumed

A similar the

a reasonable

requirement, of stages.

discussed

stage

us

is a correlating

to achieve

speed

application

from

tell

in order

specific

speed

(2-110)

BASIC

TURBINE

CONCEPTS

yields n==-

Equations studies

(2-109)

and

associated

(2-111)

with

are particularly

preliminary

Performance The

turbomachinery

perfectly

correct

variables, rate

Q because

considerably Change

for

the

ideal

is the

work

ratio,

depends

we include

number

power work

on the on

as

D

gas #. For

Now,

operating

still R,

simplicity with

heat

ratio

.

some complicated, and speed terms. Let

us operate

we can by using tionality

get

out

but on

of them.

f

not

of the

The

ratio

initial

and to

of

Since

pressure

Since

Mach

temperature

N and

fluid

is

a character-

properties

a molecular heat

H,

express

Ah'.

as on the

variable. speed

specific

T'_,,

are

weight,

ratio

inand

3' is assumed

Ah'

ND

D, R, #)

(2-112)

p,..

w

, -_ ) _/ RT,,,'_T"--7 'P_

been

assumed terms

above

mass-flow

equation,

N,

following:

second-order,

some

the continuity AocD _, so that

the

-- Icn_R-_n'

had

to

variables

produces

-, P; D_

specific

implies the

p'_,, p_,,

w_/RT,.

If the

which

constant.

term

as well

flow

preferred

enthalpy

introduction

here, the

preferred

The

of

Q changes

remains is

in total

Rotative

volume

expansion, w

as another

of interest.

the

pressure

temperature

elasticity.

are

analysis

the

temperature,

w=fcn(_h', dimensional

P,

are

choice

nondimensional to

ratio

on both

temperature

constant

viscosity constant.

while

or drop

initial

of

(2-63)

Another

expressing

pressure

depends

of

to introducing

dimension

turbine,

specific initial

depends

equivalent eluded

flow

Instead

work

for degree

as

parametric

in equation

machines.

w is preferred

significant

expressed

for compressible

actual

presented

rate

for

Parameters

flow

preferred

flow

any

pressure

temperature.

istic

mass

throughout

of

which

is often

The

useful

analyses.

Specification

compressible

however,

performance.

system

parameters for

(2-111)

),

the

(2-113)

constant,

there

modifying

terms parameter ideal-gas

to see

the what

may law,

would

flow,

be

work,

significance

be transformed and

RT .

the

propor-

(2-114) 57

TURBINE

DESIGN

Substitution (2-113)

of this

APPLICATION

relation

into

the

mass

flow

parameter

of equation

yields

Thus,

the

mass

of actual the

AND

critical, The

flow

mass speed

flow

rate rate

is represented to the

or sonic,

velocity.

parameter

may

nondimensionally

mass

flow rate

when

be transformed

by

the

the

velocity

ratio equals

as

ND

U U oc ,-7-_,_, oc-_]RT't,, _RT,_ ac, Thus,

the

rotative

of rotor-blade Mach

speed

velocity

number.

parameter fluid

have

certain

with

V/U,

Mach

number,

as for

temperature

a given

(2-113)

with

can

in order

be

the

for

rotor

critical the flow,

the

effect

only

velocity.

For speed

becomes

of varying

speed

be inlet

presented

The

must

also

must

parameters

of rotor the

must

but

ratio

similarity.

rotative

All variables

dimensionless

expressed

not

to the

fluid.

by

of

the

the

is a kind

parameter

condition but

incompressible

by

similarity,

therefore,

of the

gas,

for

respect

dimensions,

dimensionless form to be correlated. For

kinematic

a certain velocity

which

mass-flow

is that

variable the

the

nondimensionally velocity,

the

analysis

of fixed

singular

of

of this

fixed

machine

to critical

Division

gives

implication

is represented

(2-116)

the

have

a

a given is not

a

associated expressed

in

temperature as equation

as

(2-117) p_---_

For

a given

gas

-_-Icn

in a given

_T-'

turbine,

_=fcn

Depending tions

on

the

(2-113),

particular

(2-117),

or

the

_-r,

case,

--7'

the

(2-118)

parameters

further

be

to

(2-118)

-r-,

parameters

can

reduce

presented

used

to

in equa-

express

turbine

performance. Equivalent It

is very

useful

of temperature and 58

specific

to report

and heat

pressure

ratio.

This

Conditions

performance and

under

sometimes

is done

in order

standard

of fluid that

conditions

molecular results

weight

obtained

at

BASIC

different easily

conditions used

following sure,

may

are

101

the

325

as

of these Let constant.

With

subscript can then

are

parameters

the

known

with

basis

diameter

conditions similarity

and

the

conditions

(2-119)

!

Ah'eq l_,d

N

(2-120)

N_q

/

these

The

the

Pst_

ah' RT_.--R_,.

of

air. on

but

the

or are

conditions.

standard

P

P_.

Rearrangement conditions

standard

expressed

(2-113)

conditions,

K

1.4. These

as equivalent

std denoting

equivalent as

pres-

288.2

ratio,

NACA

speed

also The

atmospheric

heat

or and

and

we desire.

temperature,

of equation

subscript

eq denoting be expressed

used:

specific

work,

CONCEPTS

compared

condition

psia;

and

conditions,

of flow,

conditions

the

readily

usually

14.696 29.0;

standard

variables

us use

or

weight,

standard

and

for any

conditions

N/m _ abs

NACA

performance

directly

performance

standard

518.7 ° R; molecular known

be

to determine

TURBINE

,

(2-121)

equations

then

yields

for

the

equivalent

p',. w,q.=w

taT' ta

,

Ah_q-----,_h

(2-122)

, RT',_

(2-123)

N_q= N 4R--_ta / T:,a ,

by

As you may recall, assuming constant

always and

the

case,

fluid.

Let

since

The

used

yield

do not

(sonic)

velocity.

specific

heat

and

have

specific add

similarity and

a very

commonly used are expressed as

heat

ratio

change

corrections

under the

Mach small

can

a specific-heat-ratio

specific-heat-ratio However,

ratio

only

we started off the discussion of these specific heat ratio for all conditions.

us now

parameters.

(2-124)

all

terms number,

effect

specific-heat-ratio

conditions, that are

are

the

temperature

effect

into

the

that

are

commonly

but

only

out

depend

cumbersome

on equivalent terms,

left

with

parameters This is not

at

critical on both

to work

conditions. equivalent

above

with,

With

the

conditions

59

TURBINE

DESIGN

AND

APPLICATION

(2-125)

V_.v_,,.J \p,.:

8q--

----

t

(2-126)

_:,=t,h' ,,,_7 ) N_q_

Nj(V_'d.)

'

(2-127)

where

.>,,S. _2 V,;(,,,:,) "\%,,+1/

(2-128)

.d__Ly"

_

FIGURE 4-7.---Effe(:t

figure

I

__

4

-70

i" _

7.

Exit-flow angle, a2, deg

\ 8--

of reference

the there

in a manner

are that

at

this

experimany we do 115

TURBINE

DESIGN

AND

APPLICATION

Basedon analyticalresultsof fig. 4-_b) Basedon experimental resultsoffig. 4-7

------

Inlet-flow angle, al, deg - 2

-a 2

1-0 E -(]2

o

I

-30

FIGURE

not

yet

are

more

fully

mental

results, ¢,

such

recommended

of

Analytical

used

to

in reference

which

In the

past,

some

the

on the

modification

is suppressed The

to reduced

boundary

layer

increasing

the

the

Certain

Two

are

tandem

the

applying

alternate

blade

in references test

references The

15 and

practice

than

the

0.8

have

boundary-layer to stator

blades

blades

low-solidity

and

are presented tandem

removing

blades

jet-flap

on

marginal

better

potential

in figure as well

4--9. as the

are summarized

of low-solidity

in references

and

or

with

concepts tests

the

of turbulators

illustrated

rotor

Cascade

is one

by blowing,

perhaps,

are

separation

layer explored

treatment

10, respectively.

that

solidities,

include

by use

been have,

separation

lower

of separation

could layer

that

been

such

boundary

boundary

which

plain,

11 to 14. Turbine

rotors

are

presented

in

16, respectively. operates

of suction-surface

diffusion

minated

the

116

the

blades,

blade

at about

4-5,

experi-

occur.

region

treatments

energizing of the

do not

in the

concepts

jet-flap

tandem

Such

})e utilized

blade

concepts

with

layer

has

To achieve

losses

jet-flap

the

9 and

and

results

must

high

concepts

and

design

higher

in solidity blade.

of these

alternate

Studies

tandem,

solidity.

by suction, turbulence

success.

concept

of the boundary

approach

blade.

of the

the associated

treatment

Current

are

Blading

to reductions

surface

in blade

and

4-7.

of figure

than

1.

limitation suction

as those solidity

is slightly

Ultralow-Solidity

occurring

I -80

such

in figure

1.0,

I -10

solidities.

optimum

shown

of 0.9

optimum

results,

to determine

as those

values

I

-50 -_ Exit-flow angle,(]p deg

4-8.--Comparison

understand.

frequently

is to use

I

-40

point

on the principle is utilized of separation.

that,

(perhaps The

2),

although the

remaining

a high value

front

foil is ter-

diffusion

then

Tandemblades

FIGURE

takes

place

on the

20 to 30 percent The trailing the own

lift.

foil with

blading

a clean

mainstream

point

around

In addition,

momentum.

the

the

Figure

edge,

delivers

4-10

layer

through

perhaps

and thc

slot.

a secondary air stream main stream. This jet

trailing

jet

with

concepts.

boundary

air going

jet-flap blade operates with edge perpendicular to the

stagnation

DESIGN

Jet-flapblades

4-9.--Low-solidity

rear

of the

BLADE

thereby

some

shows

force

jetting moves

substantially to the

experimental

blade

velocity

out the the rear increasing

througl_

its

distributions

1.2 (-1 Jeton (4 percentflow) 0 Jetoff .-._- -O..[:r

1.0

_,

Suctionsurface IJ

_e'-

6I I°

4

2

Pressure

FIGURE

f_

I

__ ,

0

surface

I 20

4-10.---Jet-flap

I

I

40 60 Axialchord,percent

experimental

velocity

I

I

80

100

distributions.

117

TURBINE

DESIGN

AND

around

one such

longer

a requirement

APPLICATION

blade

with

to be equal

at the blade

a rectangular

shape,

unity.

Also,

thus

the

for solidity

sidered

only

other

load

on the

tendency

reductions. such

The

loading

coefficient

suction

the

blade

the The

determined This

from

involves

jet

connecting

flap,

to

channel

and

exit must

thc

provide

has

The

stream.

the

concepts

however,

been

reduced,

offer

the

poten-

will probably air

flow

and

the blade and

and

be con-

is required

parts

profiles

flow turning

blade

for

must

be designed.

of the

and

blade

between the

minimum

the

must the

connecting

with

spacing

geometries

transition

surface

the

itself exit

exit

efficient

The

required

selected

inlet

inlet

a smooth,

free

approaching

DESIGN

of the

profiles.

provide

now approaches

closely

is substantially

a secondary

considerations,

determination

surface

designed

is no

surfaces

cooling.

length

solidity

pressure

diagram

tandem-blade

where

as blade

chord

jet on, there

and

_ more

surface

BLADE-PROFILE After

the

to separate.

and

for applications

purposes,

on. With

on the suction

edge.

the

the jet-flap-blade

off and

velocities

trailing

with

diffusion

suppressing

Both tial

the

the jet

for the

be

blade

inlet

and

edge,

the

loss.

Exit Consideration throat,

and

the

Trailing trailing

of the suction

edge.--In edge

blade surface

17, an increase loss.

This

discussion

in chapter

significant

effect

4-11,

which

of the shows

new

exit-velocity

just

within

trailing-edge station The

example diagram

the

blade

blockage

2, which

that

at 2a include

sections

have

region.

in a higher just

been

conservation

beyond used

with

continuity

in

turbine-loss also

has

with

the

2a, which

reduced

velocity

use of figure

area

at station

blade this

trailing-edge

used. due

to the

2a than

(4-26)

_)_

at

region.

"within-the-blade"

momentum:

(p

A

is located

:

or2

a

exit region. nomenclature

of tangential

S COS

118

an increase

of the

the

obtain

smallest in

at station The

edge.

shown

thickness

blade

V,,.2, = V,,,2 and

the As

causes

as part

the

to

trailing

considerations.

will be made

is constructed

results

the

trailing-edge

effect

blade

and

in the

trailing-edge

is located

equations

diagram

addition,

blockage

throat

further

on the flow blockage

Consideration

trailing

thickness

is discussed

7. In

the

it is wise to utilize

mechanical

in trailing-edge

effect

includes the

of turbines,

with

the

section

between

the design

consistent

reference blade

exit

(4-27)

BLADE

DESIGN

Station 1

0

$

FIQU_E 4-11.--Blade

where

t is the

trailing-edge

is determined between

from

stations

are usually

thickness,

equations

2a and

small)

Mach

preceding

at

and

flow 3Iach

blockage

can cause

station

design

whether flow rate

Throat.--Since,

the

throat

design

assumes

no

change

inside a turbine

use of the

up to the throat, One

also

number

high

in general, o (see

fig. 4-11)

technique

flow

(since

at the

blade

conditions

region,

used

area, a rather

successfully velocity and

exit

a straight

the large

(station

2)

trailing-edge important

will occur

row operates

or minimum becomes

from

as is often the

an

as at station

It is, therefore, row

flow

changes

to have

angle

angle

asa

the the

be designed

the

blade

blade

flow angle

assuming

determined

choked. the

The

be

subsonic

"inside-the-trailing-edge" in

by

must

Because

be obtained.

procedure.

can

2a to become

choking

opening

sion makes

blade

cannot

the flow accelerating of the

in the

(4-27)

incompressible

2a

the

to be

the

station assumptions.

specified

or feet.

a velocity-diagram

and

is often

to determine

and

The

to produce

number

(65 ° or greater)

(4-26)

or isentropic.

equations

in meters

2 to bc either

exit angle of a2a in order 2 outside the blade row. The

section and nomenclature.

such

that

as a nozzle,

with

the determination critical

aspect

to give this diagram. suction

of

dimenIf one surface 119

TURBINE

DESIGN

between

the

obtained

from

AND

APPLICATION

throat

and

the

station

velocity

2a,

then

diagram

the

throat

at station

dimension

2a by

using

can

the

be

following

equation:

o( =

where

o is the

throat

If it is assumed throat

and

the

1

opening, that

cos a:_ 8

cos

in meters

the

velocity

"free-stream"

(4-28)

(T2a

or feet.

and

station

loss do not

change

between

the

2, then

0 -= 8

When the

this

method

angle

(4-28)

of the

and

solidities

(4-29)

applies

flow the

sonic

exit.

For

exit

(throat) perhaps, achieved additional the

case

predicted at exit

blade

condition

at the

flow

back

the

channel

occurs

channel

exit

downstream dimension

condition

i\[ach

o would

then

for expansion choking

such that

from

(4-28)

velocity,

1.3, the

low supersonic

the

be

across

flow is subsonic.

to account

supersonic

about

could

occur

at the section

a convergent-

numbers

been found that satisfactory performance is still located at the exit of the channel,

expansion

required

For

to the than

within

exit

This 60 ° and

equation

to a supersonic

be modified

throat

would

from

blade-row

than

deviation

that

(cqs.

8 compares (4-29).

greater

determined

changes

methods

by equation angles

to 35 ° . This

expands

greater

Both Reference

gradients

the

must

is obtained.

1.3), it has if the throat

as

row

numbers

be located

passage

down

wherc

dimension

5[ach

must

divergent

the

throat

from

following

those

thickness

its length.

dimensions.

agreement

dimension to the

within

this computed

not

as well as larger

throat-opening

If the

but

with

close

(4-29)

of trailing-edge

throat

of up to 5 ° for exit angles

due to lower the throat. The

angles

q2

effect

position

indicates

deviations

the

give similar

exit-flow

comparison

case,

throat

(4-29))

measured

or

is used,

COS

throat.

be computed

(up

to,

can be and the In by

this the

equation:

(4-30) o = o_ \A _} where

058

throat

opening

supersonic

computed velocity,

Act

flow area

for sonic

A,,

flow area

for supersonic

120

flow,

from

equation

m; ft m2; ft 2 flow,

me; fC"

(4-28)

or

(4-29)

for

BLADE

DESIGN

1.0

¢3

.9--

1

Rt

I

1.1 Mach

FIGUttE 4-12.--Variati(m

This

area

exit,

correction,

is shown

Suction surface

surface

be made edge

the

from

throat

such

region,

assumed

isentropic

from

throat.--The

and

trailing

considerations

5[ach

suction-surface

number

flow Mach munl_er.

flow between

throat

and

4-12.

downstream

between

1

1.4

ill flow area with supersonic

with

in figure

I

number

diffusion

on the

as structural

level (D_),

selection

edge

and

and

in the

losses,

surface

area

type

surface

integrity

associated

blade

of the

suction

of

must

trailing-

desired

level

resulting

of

from

the

design. A "straight unity)

back"

are specified

or transonic the

uses tail

problems solidity ture ably at

this

to the

curw'd

and

ref. 8), if the is small. 5[ach

surface

suction-surface is improved of remaining

and integrity

are

great. 5Iach

regions.

between

the

velocity

distribution.

curvature

region.

This

permits

some

numbers design type

and

it adds

by introducing is used,

t)y figure

(greater

0.8, than

curvatures

trailing throat

edge the

effect

(which

the

curvature

0.8),

the

has an effect

to trailing

is

effect

be lower

selected

velocity

angle of this

4-13

should

of curvature

dif-

consider-

a wedge the

is less than

In general, from

flimsy. of curva-

number

decreases

structurally

low-

amount

blading

and

Principal

preclude

some

As indicated

throat

next

utilizes

blade

The

low.

low D, values

tail of the blade,

Therefore,

number

if the

can become

trailing-edge

(,xit Mach

severe.

edge

subsonic

in the

flow acceleration

losses

the

()f the

High

to prevent

associated

blading

on the

(approximately

discussion

that

loaded

exit-flow

At higher

on loss can become suction

the

on loss is not

effect

higher

and

If eonv(,ntionally

surface

by the

in order

keep

of D,

are permissible.

of surface

h)ading

structural

exit.

edges

long trailing

throat

low values

be indicated

surface

the

when

trailing

gas-turbine

the

from

the

blade

a straight

and additional

the

long

type

conventional

between

fusion

is used

as would

of the

with designs

5lost

and

blading,

paragraph, on

design

in

for the on the

distribution edge

instead

constant. 121

TURBINE

DESIGN

AND

APPLICATION

Mach number 1.0

__o .-

j

m

_\\ L°7

.2 .4 .6 .8 Ratio ofbladespacingto surfaceradiusofcurvature

FIauR_

4-13.--Variation

of

profile

between

h)ss

throat

with and

Math

exit

numl_er

(from

ref.

and

surface

curvature

_).

Inlet The

leading-edge

than

the

exit-region

leading-edge erally

geometry

diffusion

and

numbers,

inlet

the

must blade

for thc

lead

that

inlet.

the

.\lach

through

the

blade

blading

area

With

5[aeh

arc

usually

lead5 [ach high

high

inlet

5[ach

is not so severe

and

to det('rmin('

and

The high

of suction-surface

contraction (4-26)

row.

is gen-

excessively

values

losses.

large

number and

binding,

to high

Equations

can also be used

a relatively

the

i:_creased

flow angle

circular

could

edges

can

limit

leading the

region. result

pressure-surface ellipses,

(4-27),

which

as

were

a blade-inlet

opening

to check

for blade-

number

freedom The

of

velocity

peaks

leading

edge.

permit

of the

variations or eliminate

the

leading-

remaining

and

is t:o join

the velocity

them

with

is arbitrary

selection

associated

in curvature

trailing-edge

this

velocity-distribution

curvatures

Blade-Surface Once

specified,

large

portions

which

edges

in undesirable

be used to minimize

122

inlet,

less critical

choking.

leading-edge

task

blade because

of low-reaction can

be taken exit,

row is usually

for low-reaction

toward

at the

bIade

case

region

"within-the-blade"

Although and

the

inlet

a tendency

care

to choke and

In

in the

increases

concern

blade

the

be used,

and then

a serious

blading.

velocities

At

earl usually

low at the inlet

number

used

geometry.

radius

ing edge bccomes

of a turbine

with

on both Other

around

in

circular the

leading

suction-

geometries, the leading

the

such edge,

and as can

peaks.

Profile geometri('s a profih_ that,

have

been

yields

the

selected, required

the flow

BLADE

turning

and

desigu

a satisfactory

procedure

must

to an accuracy Two

sufficient

of the

major

Velocity

gradients

pressure

surface

turn

the

flow

occur

as a result

of these

factors

influence

The

channel

and

of the

theory

from

Pressure

_ Suction

surface7

_su rface

The rows

in the

figure

difference and,

considerations.

serves

to

the

required

to

therefore, Since

distribution,

as the basis

available

4-14.

suction

position

velocity

limits).

both

the design

of a quasi-three-dimensional

that

ncxt

blade.

the blade diffusion

illustrated

channel

programs

in the

(e.g.,

static-pressure

least

the

through

in streamlinc

be at

computer

are discussed

arc

the

of radial-equilibrium

should the

around

controls

the blade-surface

flow analysis

procedures putations

across

variations

velocity

used

design

considerations

occur

Radial

distribution

the flow conditions

to impose

as a result

flow.

procedures

velocity

describe

DESIGN

nature.

for these

to perform

the

design com-

chapter.

..-Pressure surface

Suction surface_..

Cross-channel

distance

(a)

__

, 'lEi.:_.i:!_._.:._::..::::::_i_!| Tip : _i_ Rotorf_ii!i|

-'Nl ,

"'- Tip

_

_I

°--

NN

¢";1

,-Hub

_:: /

Velocity (b)

(a)

Cross-channel (b)

FIGURE

4-14.--Turbine

Radial

variation. variation.

blade-row

velocity

variations.

123

TURBINE

DESIGN

AND

APPLICATION

REFERENCES 1.

O.:

ZWEIFEL,

Angular 2.

LIEBLEIN, sion

The

STEWART,

Blade

Paper

Effective

RM

E56B29, Y.;

for

a Series

Four 6.

HELLER,

E56F21, A.;

Investigation

8.

AINLEY,

l).

G.;

BETTNER,

RM

Loaded

10.

H.

cepts

Designed

2, Apr. 11.

STABE,

ROY

NOSEK, tion

13.

NOSEK,

STABS,

U.;

AND

Gt.

E55BOS,

of

Turbine-Blade-

WARREN

WHITNEY,

by

Parameter

Cascade

and

for

1955.

Changes

in

J.:

Blade

RICHARD

Turbine

at

R.:

An

Analysis

Geometry.

H.:

Experimental

Four

Rotor-Blade

Examination

of

Turbines.

Rep.

Axial-Flow Britain,

the

Flow

R&M

and 2891,

1955. M.

:

Summary

of Tests

in Three-Dimensional

R.

J.:

Turbine

Some

on

Cascade

Experimental

Blade

Two

Highly

Sector.

Paper

Results

Loading.

J. Eng.

Ratio

Cascade

to Spacing

KLINE,

JOHN

Blade

Blade. and

with

Two-Dimensional Chord

of Two

Power,

NASA

JOHN NASA

of 0.5.

F.:

Chord

of

TM

vol.

Con92,

Turbine

no.

X-1836,

X-2183,

Stator

X-1991,

Cascade TM

1970. Investiga-

1969.

Two-l)imensional

TM

Two-Dimensional of Axial

Test

NASA

F.:Two-Dimensional

Design.

AND KLINE,

Rotor

l)_ign

Blade

of

1969.

and

Tandem M.;

ROY G.:

Use

Losses.

Diffusion

Two-Dimensional

AND

STANLEY

Nov.

of Axial

Turbine

Stator

C.

of

Concepts

])esign

Ratio

STANLEY

W.:

198-206.

of a Turbine

Jet-Flap 14.

pp.

STANLEY

JAMES

Rotor-Blade

with

AND CAVlCCHI,

G.

AND ROELKE,

G.:

with

Ex-

Diffusion

1952.

Rows

to Increase

1970,

Blade 12.

G.;

R.:

Using

Correlation

Designed

NOSEK,

ASME,

LUEDZRS,

L.;

L.;

Council,

Blade

69-WA/GT-5,

ROSE

Blade

; AND

Turbine

in

Affected

MATHIESON,

in

L.

L.:

RM

WARNER

E52C17,

Research

JAMES

MICHAEL

MISER,

Turbine

Thickness

NACA

Conservatively

AND

Aeronautical 9.

a

Losses

Diffu-

1956.

NACA

Pressure

AND

])escribing

Blades

Losses

WHITNEY,

of

Solidities.

L.:

in Axial-Flow-

1967.

J.;

WARNER

Rotors.

Viscous

RM

Large

436-444.

1953.

AND VANCO,

Nov.

Momentum

Turbine

STEWART,

JACK

pp.

ROBERT

Loadings

Characteristics

ASME,

on Wake

Turbine W.;

with

1945,

1956.

of Subsonic

JAMES

J.;

WARREN

AND STEWART,

Turbomachine

NACA 7.

Blade-Lolling

in

Blade

E531)O1,

Turbine

WHITNEY,

Based

Transonic

MISER,

of

Losses

RM

Especially 12, Dec.

AND BEODERICK,

ARTHUR

Thickness

ROBERT

Element

NACA

32, no.

Limiting

GLASSMAN,

Momentum

NACA WosG,

L.;

C.;

and

67-WA/GT-8,

WARNER

vol.

FRANCIS

Elements. L.;

Blading,

Rev.,

Losses

of Axial-Flow

Parameters.

5.

SCHWENK,

WARNER

STEWART,

Turbo-Machine

Boveri

for Estimating

amination

4.

of

Brown

SEYMOUR;

Factor

Compressor 3.

Spacing

Deflection.

Cascade

Test

of a

1971.

Cascade

Test

to Spacing

of a Jet-Flap

of 0.5.

NASA

TM

Turbine X-2426,

1971. 15.

16.

BETTNER,

JA_ES

Solidity

Tandem

BETTNER,

JAMES

Solidity, 17.

PRUST,

Jet

ing.

124

NASA

and TN

l)esign Rotor.

L.:

Flap

HERMAN

Geometry

L.:

l)esign Rotor.

W.,

Jm;

and

Experimental

NASA and

on

1)-6637.

1972.

Results

CR-1968,

AND the

HELON,

of a Highly

Loaded,

Low

of a Highly

Loaded,

Low

1971.

Experimental

NASA

Thickness

Results

CR-1803,

1972. RONALD

Performance

M.:

of Certain

Effect

of

Turbine

Trailing-Edge Stator

Bind-

BLADE

DESIGN

SYMBOLS A

flow area,

C

chord,

D

diffusion

F

force, N; lb conversion constant,

g K

ratio

m_; ft 2

m; ft parameter

of inlet

o

throat

P R

absolute reaction

8

blade

t

trailing-edge

V

absolute

X

axial

o_8

1; 32.17

to exit tangential

opening,

(Ibm)

(ft)/(lbf)

components

(sec 2) of velocity

m; ft

pressure, spacing,

N/m2;

lb/ft

_

m; ft thickness,

velocity,

distance,

m; ft

m/sec;

ft/scc

m; ft

fluid absolute

angle

from

axial

direction,

deg

blade

angle

from

axial direction,

deg

stagger

ratio of specific heat constant volume density,

( V_ ._/V,._)

kg/m'_;

lb/ft

at

constant

pressure

to specific

heat

at

a

solidity

_bz

loading

coefficient

defined

by equation

(4-6)

loading

coefficient

defined

by equation

(4-5)

Subscripts" cr

critical

inc max

incompressible maximum value

rain

minimum

opt

optimum

p s

pressure suction

ss u

supersonic tangential

x 1

axial component blade row inlet

2

blade

2a

within

value

surface surface

row

component

exit

trailing

edge

of blade

row

Superscript" '

absolute

total

state

125

CHAPTER 5

Channel FlowAnalysis ByTheodore Katsanis

The

design

of a proper

blade

profile,

chapter

4, requires

calculation

determine analysis

the velocities on the blade theory for several methods

of the

discusses associated computer Lewis Research Center. The

actual

cannot

velocity

be calculated viscous,

passages.

To calculate

simplifying simplified surfaces Similar mean

distribution

surfaces. This used for this

to flow

on or through

hub-to-shroud shown

directly,

but

stream in figure

provides

surface

extreme

to

at

the also

NASA

flow

field

complexity

geometrically therefore,

two-dimensional

of

complex certain

three-dimensional

5-1 (a),

(commonly

does

information

There

are

tribution

two

over

formulation

one

of the

mathematical we

velocity-gradient

parts

will

to a method

of these

flow

surfaces.

and

problem. discuss

the For

stream-

(stream-filament)

yield

The second

is

Such

and

is the

is the

mathematical The

surface which

yield

a velocity

dis-

mathematical

numerical

solution

formulation

potential-function

methods.

velocities

solutions,

part

part

meridional

blade-to-blade

to obtain

first

the

blade-surface

5-1 (c))

of analysis

the

called

for the

(fig.

surfaces.

problem,

not

required

(fig. 5-1 (b)) and orthogonal surface the desired blade-surface velocities.

problem,

in order

are illustrated in figure 5-1 for the case of a radial-inflow turbine. surfaces are used for an axial-flo(v turbine. A flow solution on the

surface),

of the

The

various

of

developed

distribution,

made.

field

section

a blade-row

of the

velocity be

last

chapter presents calculation and

were

flow through

a theoretical must

that

because

three-dimensional

assumptions

flow

throughout

time

in the

blade-row

programs

at this

nonsteady,

as indicated

stream-

of the

methods and

and

potential-

TURBINE

DESIGN

AND

APPLICATION

Blade-_,-blade

Hub-to-shroud stream surface

-_ |

surface] I

(a)

0_)

,- Ortho_nm

(c) (a) Hub-to-shroud

stream surface. (b) Blade-to-blade (c) Orthogonal surface across flow passage. FIOURE 5--1.--Surfaces used for velocity-distribution calculations.

128

surface.

CHANNEL

function

methods

will be described

solution.

A similar

type

The

of analysis

velocity-gradient

for solutions The

equation

on any of the

following

of analysis (1)

The

surface

flow

Thus,

if the

fixed

coordinate

(2)

The

blade-to-blade

surface

for the meridional is general

surface.

and can be used

are

made

in deriving

the

various

This

means

methods

herein:

at any

blade

can be made

ANALYSIS

surfaces.

is steady

velocity

to the

to be presented

assumptions

discussed

relative

FLOW

relative

given

point

is rotating,

the

to

the

blade.

on the blade flow

would

does not

not

that

vary

be steady

the

with

time.

relative

to a

system.

fluid obeys

the

ideal-gas

law

p=pRT

(5-1)

where p

absolute

pressure,

p

density,

kg/m_;

R

gas constant,

T

absolute

The

lb/ft

lb/ft

2

S

J/(kg)

(K);

temperature,

or is incompressible (3)

N/m_;

(ft) (lbf)/(lbm)

K; °R

(p = constant).

fluid is nonviscous.

A nonviscous

The blade-surface velocity is calculated, extends to the blade surface. (4)

The

fluid

(5)

The

flow is isentropic.

(6) inlet.

The

total

(7)

For

the

assumption

has

a constant

heat

temperature stream-

is made

and

and

that

Y is the

absolute

do

particles

total

vector.

not

change

their

absolute

change.

For

example,

particle

at

times

t and

t-t-At.

particle

changes

blade,

rotation the

is zero. particle

has rotated,

In

and

Of course,

layer.

free

stream

are

uniform

analyses,

across

the

irrotational.

the

shape

additional

because

(5-2)

Intuitively,

orientation figure

this with

5-2

means

time,

shows

absolute

frame instant

of time,

of reference

relative

the

frame

that

although

a hypothetical

at a later

in a frame

the

Therefore,

Y = VX V =0

may

net

no boundary as if the

pressure

flow is absolutely

velocity

its location

has

capacity.

shape

their

fluid therefore,

potential-function

the

curl where

(°R)

of reference,

of reference

the

but

the

to the

has rotated.

129

TURBINE

DESIGN

AND

APPLICATION

Direction of rotation

Time = t

r-l LJ

Time-t+At

I

I

(Absolute frame of reference)

Time=t + At (Relativeframe ofreference)

FIGURE

Some also

numerical

techniques

be discussed.

techniques excellent given

However,

for solving theoretical

in Chapter

STREAM-

5--2.--Absolutely

irrotational

for solving it must

the

simplest cascade 130

stream

IV of reference

AND

be emphasized

equations

that

there

can

be

of streamlines.

in figure

5-3.

only

will

are many a few. cascades

An is

1.

POTENTIAL-FUNCTION

function

is in terms as shown

mathematical

these equations, and we will discuss discussion of flow in two-dimensional

Stream-Function The

flow.

ANALYSES

Method

defined

several

Suppose

It is assumed

ways,

but

we consider that

there

two

perhaps blades

is two-dimensional

the of a

CHANNEL

FLOW

ANALYSIS

1 .8 Mass

.6

flow fractbn

.4

0

\\\\

FIGURE

axial

flow here,

5-3.--Streamlines

so that

the

there is no variation of the rotation about the centerline. Shown the

in figure

blades

passing line.

Thus,

the lower streamlines

the

radius

r from

flow

in the

5-3 are a number

is w. The

between

for a stator

number

the upper upper

surface have

surface

centerline

radial

streamline

of the lower

(which

The

is a streamline)

It will be recalled

There

mass

indicates blade

and has

and

may

be

flow between

the fraction the given the value

of w

stream0, and

of the upper blade has the value of 1, while the remaining values between 0 and 1. Note that a value can be asso-

ciated with any point in the passage. This value function value and can be used to define the stream (or uniform)

is constant

direction.

of streamlines.

by each surface

the

cascade.

that

mass

flow can be calculated

is called function.

the

stream-

for a one-dimensional

flow by w= pVA

(5-3) 131

TURBINE

DESIGN

AND

APPLICATION

where W

rate of mass

V

fluid absolute

A

flow area

This

flow,

kg/sec;

velocity,

normal

can be extended

lb/sec m/sec;

to the

ft/sec

direction

to a varying

of the velocity

flow by using

V, m2; ft _

an integral

expression:

w= f a pV dA Since

this stream-function

cascades relative

(blade velocity

analysis

rows),

the

W, which

velocity V. We will assume the mass flow wl._ between fig. 5-4)

can be calculated

fluid

applies velocity

(5-4) to both will

be

stationary

and

expressed

in terms

reduces

blade

to absolute

that any

has a uniform height b. Then, QI and Q2 in the passage (see

by QI

wl._ --/Q

132

5-4.--Arbitrary

(5-5)

_ pW,,b dq

-'_Q1

FIGURE

of

for a stationary our cascade two points

row

rotating

•

curve

joining

two

points

in

flow

passage.

CHANNEL

where

Wn is the

hand

normal

relative

of the line going

that

wl.t will be negative

The

integral

dependent With

velocity

is a line of path

from

for steady

function

in the direction

QI to Qt. This

if Qs is below integral

the use of equation

for the stream

component

the

flow relative (5-5),

points

of the right-

passing

means

through

Q_ and

Q2 and

Q1. is in-

to the cascade.

an analytical

u at a point

ANALYSIS

sign convention

a streamline

between

FLOW

expression

can be written

(x, y) :

/Q(_'Y) pWnb dq o

u(x,

(5-6)

y) = W

where integral in figure Since tively calculate is still

Qo is any

point

on the

is taken

along

any curve

upper

surface

between

of the

Qo and

lower

(x, y).

blade, This

and

the

is indicated

5--5. the

easy

integral

in equation

to calculate Ou/Ox

at the

the

partial

point

in the flow passage,

(5-6)

derivatives

(x, y).

as shown

Let

of path,

of u. For

Xo0.98 0 90 to 098 >0.80 to 0.90

E::3 >0 98 090 Io 098 0.80 to 0.90 >070 to 0.80

Pressure surface side Suction surface ,, side -

(b) Total-pressure ratio (blade exit to blade inlet)

Total-pressure ratio (blade exit to blade inlet} r_>

r---I >098 _zza 0,90 to 0.98 i_ 0.80 to 090 I_ 0.70 to 080 _lm >0.60 to 070

0.98 0 90 0.80 0.70 >060

to to to to

0.98 0,90 0.80 070

I I 4

L_j (d_

(c) (a)

Ideal ity

(c)

Ideal ity

FIGURE

after-mix

ratio,

critical

veloc-

critical

veloc-

(d)

0.823. of

obtained

total-pressure

Surface

profiles

Ideal

part

of the

that

yield

ratio

experimentally,

blading favorable

after-mix

ratio,

ity

12-25.--Contours

An important

Ideal ity

after-mix

ratio,

ance have been can be made.

surface

(b)

0.512.

a

critical

veloc-

critical

veloc-

0.671. after-mix

ratio,

0.859.

from

stator

turbine

annular

loss

surveys.

breakdown

Velocity design

is the

surface

selection velocity

of the

blade

distributions. 385

TURBINE

DE'SIGN

Analytical

AND

methods

chapter

5 (vol.

mine

achieved.

During

the

To

the

the

velocity

ments.

With

the

known,

the

velocity

static

pressure

can

v-f_'+i[-1

L_--1L-\p_-/

tribution

shown

Acceleration

on

smooth,

and

no

flow

large

(force

the

the

12-26(a)

suction

maximum is well

a blade

surface,

the

along

surfaces

the

in the

measureblade

from

the

surfaces relation

to

(12-15)

determined surface similar conditions.

is considered

surface

distributed

blade

jj

maximum

along

1.4

the

surface.

one.

velocity

is

There

are

subsonic.

on either

other hand, Flow on

velocity The dis-

to be a desirable

the

is maintained

(diffusions)

12-26(b), on the to be undesirable.

along

( P']'"-"'"7"lr"

velocity

decelerations

on blade)

Figure considered

in figure

to deteractually

be determined

Figure 12-26 shows the experimentally distributions for two stators tested under

in

were

on static-pressure

distribution

distribution

Vc,

along

section

discussed

velocities

distribution

in the

were

it is of interest

surface

are made

previously

velocities

program,

for"

measurements

discussed

surface test

"designed

obtain

static-pressure manner

for calculating

2).

whether

APPLICATION

The

loading

blade.

shows a velocity the suction surface

distribution accelerates

B

1.2

._

>_1.0

r- Suction surface

¢''_\ _ Suction surface

--

_

_.

i,/.....E,/ '

/

•2 _

.Z/

"0 f_).

0.. 3_. _

surface

i

.

I 0

.2

.4

surface

__Pressure .6

I

I_

I

'_ Pressure

I

I

.8 1.0 0 .2 Fraction of blade surface length

I

.6

(a)

(a)

Desirable FIGURE

386

I .8

(b)

distribution. 12-26.--Experimental

I

.4

(b) surface

Undesirable velocity

distribution.

distributions.

I 1.0

EXPERIMENTAL

to

a supersonic

deceleration thickening and face.

could

DETERMINATION

velocity

back

to a subsonic boundary

possibly

lead

A deceleration

result

in reattachment

distributions blades

with

are being

because sharp

and

velocity. layer

observed

of the

flow

of

any

the by

separated and

valleys

causes

increase

off the

pressure an

a rapid

a deceleration

associated

on

PERFORMANCE

undergoes

an

it is followed peaks

then

Such

with

to separation

is also

as critical,

AERODYNAIVIIC

(V/Vcr=l.2)

of the

is not

OF

suction

surface,

acceleration

flow. should

In

sur-

but that

general,

be avoided

a

in loss this

would velocity

when

the

designed.

REFERENCES 1.

DOEBELIN,

ERNEST 0.: Measurement Systems: Application and Design. McGraw-HiU Book Co., 1966. 2. PUTRAL, SAMUEL M.; KOFSKEY, MILTON; AND ROHLIK, HAROLD E.: Instrumentation Used to Define Performance of Small Size, Low Power Gas Turbines. Paper 69-GT-104, ASME, Mar. 1969. 3. MOFFITT, THOMAS P.; PRUST, HERMAN W.; AND SCHUM, HAROLD J.: Some Measurement Problems Encountered When Determining the Performance of Certain Turbine Stator Blades from Total Pressure Surveys. Paper 69-GT103, ASME, Mar. 1969. 4. WISLER, D. C.j AND MOSSEY, P. W.: Gas Velocity Measurements Within a

5.

Compressor Rotor Passage Using 72-WA]GT-2, ASME, Nov. 1972. ASME RESEARCH COMMITTEE ON Theory and Application, 5th ed., Engineers, 1959.

the

Laser

Doppler

FLUXD METERS: The American

Velocimeter.

Fluid Society

Paper

Meters, Their of Mechanical

387

TURBINE

DE,SIGN

AND

APPLICATION

SYMBOLS A

area,

C

discharge

D

diameter,

E

thermal

g hh_d

conversion ideal

m2; ft 2 coefficient m;

ft

expansion constant,

specific

pressure, turbine

J

conversion

K

conversion

M

approach

N

rotative

speed,

P R

absolute

pressure,

absolute

V

absolute change

AVu

work,

absolute

and exit,

kg/sec; angle,

torque,

5'

ratio of specific heat constant volume ratio

of

defined

ratio

from

to

kg/m

annulus

cr

critical

eq in

equivalent meter inlet

std

standard

t

meter

0

measuring

388

be-

or (12-6)

direction,

pressure

pressure

velocity

deg

critical

ratio, 8; lb/ft

defined

condition

by

sea-level

Mach

eq.

1)

condition

throat station

at turbine

standard

inlet

heat sea-level

by eq. (12-12) on

based

by eqs.

3

(at

to

velocity

defined

to specific

based

Subscripts" an

axial

defined

critical

temperature, speed

density,

velocity

ft/sec

at constant total

of

temperature blade-jet

absolute

by eq. (12-5)

pressure function of specific-heatratio,

sea-level

of

lb-ft

turbine-inlet

squared

ft/sec

m/sec;

measured

F

N-m;

(o R)

lb/sec

factor,

flow

m/sec;

m/sec; ft/sec component

inlet

flow rate,

(sec)

(12-4)

K; °R speed,

gas velocity, in tangential

compressibility

(min)/(rev) by eq.

rev/min

temperature,

mass

(rad)

defined

(K) ; (ft) (lbf)/(lbm)

mean-section

Y

to exit-static

N/m S; lb/ft 2

J/(kg)

W

of inlet-total

Btuflb

1 ; 9/30 factor,

rad/sec;

rotor

2)

1; 778 (ft)(lb)/Btu

constant, velocity

tween

on ratio

J/kg;

constant,

gas constant, blade

based

(lbm)(ft)/(lbf)(sec

Btuflb

specific

U

1;32.17

work J/kg;

hh'

T

factor

(12-10)

(12-14)

turbine-inlet on and

standard (12-13)

at

EXPERIMENTALDETERMINATIONOF 1

measuring

station

at stator

2

measuring

station

at turbine

AERODYNAMIC

PERFORMANCE

outlet outlet

Superscript: '

absolute

total

state

389

REPORT r_,_t?,.¢,r_

_r-_

, oll_ct4_f_ i;,,l_,_

,t t_l_;l_._,

,_._r,t,_,r_+,_

i T_,,

_r_f,_r_r,,_,,'_,r_. ,.

r.

_,j*_,_

!2C,t

]

_TI

_,,c,r_'._ ,,v!,r_

DOCUMENTATION

"_e'_. ,_,,_;

;_c,

_,

,in_ _,'_tq,_r,_

, :,

:c, rmpl_llnq _,.,r

r_ducln]

22202.,1_02,

_ncl

_6

_t,,_

r_.._,,._n,:

'_...

;,,_t_,_n

_r._,

:_f,,.,,

PAGE _h_ _'_

_f

2;

_h,rTrOn J'_h_n_on

_.t_n_,_ment

Of

,nl,_rmat_on

_teadquar ind

oMs _o. o7o4-o188 Form Approved S_'nd qr%

Budeet

Ser_

cP._.

` Paperwork

3. REPORT

I" AGENCY USEONLY(Leaveb/ank) 12" REPORT 1994 DATEjune 4. TITLE AND

I

comment_

regarding

Directorate

_ot

Reducteon

TYPE

SDecial

the%

Design

eslrmate

AND

DATES

and

Wa%hington.

.Jny

other

asDect

_epott_, DC

1215

Ot

thi%

]._fferson

2050].

COVERED

Publication 5. FUNDING

and

or

ODeratlon%

(0104-018B),

SUBTITLE

Turbine

burden

Information

Prolect

NUMBERS

Application

6. AUTHOR(S)

A.

J. Glassman

7. PERFORMINGORGANIZATION NAME(S) AND ADDRESS(ES) Lewis

Research

Cleveland,

OH

8. PERFORMING ORGANIZATION REPORT NUMBER

Center 44135 R-5666

9. SPONSORING/MONITORINGAGENCYNAME(S) AND AODRESS(ES) National Aeronautics and Washington, DC 20546

Space

10. SPONSORING / MONITORING AGENCY REPORT NUMBER

Administration NASA-SP-290

11. SUPPLEMENTARY NOTES Responsible 433-5890. 12a.

DISTRIBUTION

/ AVAILABILITY

Unclassified Subject

13, ABqTRACT

C.

Civinskas,

organization

code

12b.

STATEMENT

2760,

(216)

Ol S T"iB U TI--'0-_0

D E'

- Unlimited

Category

(Maxmnum

NASA turbine driven engines electric trucks,

Kestutis

person,

- 7

200 words)

has an interest in turbines relatedprJrnarily to aeronautics and space applications. Airbreathing engines provide jet and turboshaft propulsion, as well as auxiliary power for aircraft. Propellantturbines provide rocket propulsion and auxiliary power for space craft. Closed-cycle turbine using inert gases, organic fluids, and metal fluids have been studied forproviding long-duration power for spacecraft. Other applications of interest for turbine engines include land-vehicle (cars, buses, trains, etc.) propulsion power and ground-based electrical power.

In view of the turbine-s_stem interest and efforts at Lewis Research Center, a course entitled "Turbine Design and Application was presented during ]968-69 as part of the ]n-house Graduate Study Program. The course was somewhat rewsed and again presented in 1_72-73. Various aspects of turbine technology were covered including thermodynamic and fluid-dynamic concepts, fundamental turbine concepts, velocity diagrams, losses, blade aerodynamic design, blade cooling, mechanical design, operation, and performance. The notes written and used for the course have been revised and edited for publication. Such apublicaLion can serve as a foundation for an introductory turbine course, a means f-or self-study, or a reference for selected topics. Any consistent set of units will satisfy the equations presented. Two commonly used cons!stent sets of units and constant values are given after the symbol definitions. These are the ST units and the U.S. customary units. A single set of equations covers both sets of units by including all constants required for the U.S. customary units and defimn 8 as unity tho_o not required for the ST units. 14.

SUBJECT

TERMS

15. NUMBER

engine design, turbine engines, cooling 17.

SECURITY CLASSIFICATION OF REPORT

Unclassified NSN 7540-01-280-5500

lB.

engines,

aircraft

engines,

automobile

OF PA_GES

400 16. PRICE CODE

A17 SECURITY CLASSIFICATION OF THIS PAGE

Unclassified

19.

SECURITY CLASSIFICATION OF ABSTRACT

20. LIMITATION OF ABSTRACT

Unclassified

For sale bythe NASACenter for AeroSpace Information, 800 Elkridge Landing Road, Unthicum Heights, MD 21090-2934

Unlinlited Standard _rescrloed

298 !02

Form by

_INSI

29B _td

(Rev Z39-_8

2-B9)