Turbine design
Short Description
NASA Turbine design document...
Description
NASA SP290
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TIIItIINI{
I)KSll;l
a,id AI'I'I, II:AI'IIiN
(NASASP290) APPLICATION
TURBINE (NASA)
DESIGN 390
N9522341
AND
p Unclas
HlI07
NATIONAL
AERONAUTICS
AND
SPACE
0041715
ADMINISTRATION
NASA
SP290
'lrlllellNl IDK Iq ,N annalAIDIDI,Iq;A"I' IqlPm
Edited by Arthur J. Glassman Lewis Research Center
Scientific and Technical Information Program1994 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 210902934 Catalog
Card Number:
9467487
PREFACE NASA and
has
space
an interest
turboshaft
spacecraft.
fluids,
interest
for turbine
trains,
etc.)
In view a
Program.
The
197273.
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 selfstudy,
or
topics.
used the
and
blade
of
trucks,
InHouse
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
fluiddynamic
introductory
Any
part
using
groundbased
interest as
was
aspects
concepts,
Two
and
aircraft,
studied
landvehicle
"Turbine
196869
thermodynamic
an
power
been
jet
and
engines
spacecraft.
include
for
propulsion
have
for
to aeronautics provide
power
turbine
fluids
power
turbinesystem
during
reference
rocket
engines
course
presented
engines
provide
propulsion
of the
primarily
turbine
as auxiliary
metal
electric
related
well
Closedcycle
and
longduration
Center,
as
turbines
for
organic
for
Airbreathing
propulsion,
pellantdriven 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 UIDD
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 POTENTIALFUNCTION VELOCITYGRADIENT ANALYSIS REFERENCES ...................................... SYMBOLS .........................................
ANALYSES ........................
TO BOUNDARYLAYER
D. McNally
70 84 95 96 98
and
..................................
SOLIDITY .......................................... BLADEPROFILE DESIGN .............................. REFERENCES ...................................... SYMBOLS .........................................
,
21
69
MEANSECTION DIAGRAMS ............................ RADIAL VARIATION OF DIAGRAMS ....................... COMPUTER PROGRAMS FOR VELOCITYDIAGRAM REFERENCES ...................................... SYMBOLS .........................................
,
...
111
Katsanis ............
......
127 130 147 154 155
THEORY
...............................
157
NATURE OF BOUNDARY LAYER ......................... DERIVATION OF BOUNDARYLAYER EQUATIONS SOLUTION OF BOUNDARYLAYER EQUATIONS CONCLUDING REMARKS .............................. REFERENCES ...................................... SYMBOLS .........................................
............ ..............
157 160 172 188 188 191
V
iI
PAG'E__
I_T_NTtON_!.Ly BLANg
CHAPTER
o
PAGE
BOUNDARYLAYER
LOSSES
by Herman
W. Prust,
Jr ....
BOUNDARYLAYER PARAMETERS ........................ BLADEROW LOSS COEFFICIENTS ....................... BLADEROW LOSS CHARACTERISTICS .................... REFERENCES ....................................... SYMBOLS .........................................
o
MISCELLANEOUS
LOSSES
by Richard
195 201 217 221 223
J. Roelke
........
225
TIPCLEARANCE LOSS ................................ DISKFRICTION LOSS ................................. PARTIALADMISSION 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.
RADIALINFLOW
TURBINES
by Harold
E. Rohlik
OVERALL DESIGN CHARACTERISTICS .................... BLADE DESIGN ..................................... OFFDESIGN 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 .................. COOLANTSIDE CONVECTION ........................... FILM AND TRANSPIRATION COOLING ..................... SIMILARITY ........................................ REFERENCES ....................................... SYMBOLS .........................................
12.
193
351 352 374 387 388
CHAPTER 1
Thermodynamic andFluidDynamic Concepts ByArthur J. Glassman This
chapter
cepts are
of the
is intended
thermodynamics concepts
energytransfer 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 onedimensional 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=_
(11)
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
(12)
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
pvT
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
(13)
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
FLUIDDYNAMIC
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
compressibilityfactor
mixtures
11
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
11.
of data
and
of compressibility
and
sources.
factor.
factor
pressure,
(14)
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
11.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 onephase
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
(15)
or Btu/lb.
A differential
change
v dT+(00_h)rp be
expressed
dp in
(16) 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
(18)
in J/(kg)(K) or Btu/(lb)(°R), 778 (ft)(lb)/Btu. Therefore,
to
equation
(17)
Tds + j
constant
and J is a the partial
temprature
is,
as
(18), Oh'_
One
in
as
Oh dh:(OT) The
p)
relations
T los\
states
1
that
(110)
Substituting
equations
(17),
(19),
and
(110)
j
Ov T (__)p_
into
equation
(16)
yields dh=c,dT+
Equation change between 4
(111)
in
terms
two
states
is the of
the
rigorous state
is calculated
[v
equation conditions, rigorously
for and as
dp
(111)
a differential the
enthalpy
enthalpy change
THERMODYNAMIC
Ts
AND
1
FLUIDDYNAMIC
P*
CONCEPTS
_V (112)
If
we
law,
now
assume
we can
that
the
gas
behaves
according
to
the
ideal
gas
set RT vP
(113)
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,
(117)
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
(116)
ca is constant
excellent
is a significant
of
becomes _=c.(
This
there
2
to avoid
that (115)
effect
yields
to use
reason
assumed
equation
and
of T are
of interest.
(115)
want
is no
the
(115)
T2 T1) +_ b (__)+3
Ah=a(
(112),
to zero,
for ca as a function
textbooks
integration
equation
frTcz, dT
cp=atthen
(114)
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.
(118)
small, For
Therefore,
Entropy
Process
and
flow
the
adiabatic the
flow
process with
no
constantentropy
(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 onephase
some
state
conditions
of constant
as a function
a differential
From
change
equations
Substituting
(18)
(123)
we get
(121)
and
relating
temperature
If we assume into
equation
and
pressure
idealgaslaw (123)
and
but
using
a relation
such
equation
(117),
a computer calculation With the additional
not
behavior the
as equation
peratures 6
T_ and
equation
equation
(125)
(124)
yields
(122)
particularly
and
useful,
expression
for an isentropie substitute
integration,
process.
equation
(114)
we get
Rln_
(124)
(116),
integration
yields
(T_T_)
also
is more
becomes
(125)
suitable
than in a hand calculation. assumption that cp is constant
T2, equation
(120)
dp
1Rln_=aln_kb(T2T_)+2 J p_ Like
(120)
and
, _dT
By
into
conditions
perform
as
(tp
0v (0T),
ds=0
rigorous,
entropy
pressure (119)
(110)
dTj
process,
is the
composition, and
be expressed
and
a constantentropy
Equation
can
dT÷(_)r
equations
process.
(T, p)
ds(_),
ds=_
For
chemical
in entropy
(17),
It is, therefore,
for an isentropic
of temperature sfcn
and
isentropic process calculations
efficiency or loss term.
to relate system
be expressed
from
for use
between
in
tem
THERMODYNAMIC
AND
Jc_ ln T2R T1
FLUIDDYNAMIC
CONCEPTS
p2 ln pl
(126)
and p2=('%'_ p, \T1/
Jc,/R (127)
But Jcp R where
3" is the
capacity equation
ratio
of heat
at constant (127) yields
3' 3" 1
capacity
(128) at
constant
volume. Substitution of the more familiar form
pressure
equation
to
heat
(128)
into
p, __\_]
Where should
specific heat ratio give a reasonable
(129)
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
(130)
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
(131)
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
(131)
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
(132)
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,
(132)
in seconds.
Substituting
the
mass
is constant,
into
equation
g Equation
(134b)
is equal mass
(134a)
can
to
specifies the
(134a)
equation
F=I
Since
(133)
dV dt
g
fluid
equation
yields F=m
Since
(133)
rate
(134a)
the
of
also
be written
as
be
written
time
F =w
as
(134b)
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
(135)
g
be
A useful derived
fluid
as indicated
ligible. of
relation, sometimes called the from secondlaw 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
boundarysurfacepressure and
fluidpressure Therefore,
(A+dA)dRt
secondorder
dropping
motion, an element
as R s. The
direction.
d_)dA(p+dp)
and
forces
is indicated
to fluidpressure the
friction forces acting force in the downstream
F=pA
12.
equation Consider
and the
net
(136)
differentials
yields F= 8
 Adp
dR I
(137)
THERMODYNAMIC
The
mass
of the
element
AND
FLUIDDYNAMIC
is m:pAdx
Substituting
CONCEPTS
equation
(138)
into F_
(138)
equation
pAdx g
(134)
yields
dV dt
(139)
Since
(140) equation
(139)
can
be written
in the
form
F_pAV
dv
(141)
g Equating
(137)
with
(141)
now
yields
F=AdpdRs=
pA VdV g
(142)
and dp + VdV+
dRs
0
pFiA
(143)
p+dp 2
A+dA f f
% ,%
Flow
p+dp
p dx
V+dV
V
dRf FIGURE
12.Forces
on
an
element
of
fluid.
TURBINE
DESIGN
If we now
AND
APPLICATION
let (144)
where
ql is heat
produced
by
friction,
in J/kg
dpp __V_VdgV+Jdq1= For
isentropic
flow,
a steadyflow
a system system
or or
electrical
(and
part
of of that
energy,
etc.,
(145)
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
dqs0. Conservation
For
or Btu/lb,
we
have
energy Z,
heat
JJ
that energy, u, flow q,
and
(146)
V2" 2 .4 Z_2_4_W Tr
j2g
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
(147)
h=u + j Thus,
equation
(146)
reduces
to V
"{72
2
(148) gd Equation
(148)
as we will be using
is the
basic
g form
The
sum
of the
problems,
steadyflow
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
(149)
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
idealgaslaw
be assumed.
FLUIDDYNAMIC
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
(115),
h'h=cp(T'T) where
T'
with
is total
equation
temperature,
(149)
(150)
in K or °R.
Combining
equation
(150)
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
(129)
are
pressure
to rest the
thought
temperature
terms
or stagnation,
p. Since
be
(151)
temperature
Thus,
and
of a fluid use
a gas
adiabatically.
temperature,
total
totaltemperature
behavior
case,
concept
from p'
as the
a velocity
and
pressure
V and
p is isentropic,
static we can
to write
(152)
where
p'
With
is total
pressure,
regard
to the
in N/m 2 or lb/ft _. abovedefined
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
idealgaslaw
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
(149)
blade). and
(148)
rear
yields h2'  hi' = qW_
The
energy
dynamics
balance for
If we set
now
a flow
q and
looks
something
process,
W_
equal
as we
to zero,
(153)
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
(154) work,
(118)
remains
total
and
enthalpy
(150),
remains
it can
be shown
constant.
TJ =T/ Note
that
the
enthalpy Total and
process
and
total
pressure
(155)
tions,
not
have
the
matter.
idealgaslaw
be shown
to be isentropic
to remain
is another
and
it can
does temperature
that
(155)
From
and
for
in order
for total
(122),
(152),
constant. equations
constantheatcapacity
adiabatic
flow
with
assump
no work,
eJd' l _= P_' p2' Only
for isentropic
constant.
For
flow
flow
(ds=0),
with
(156)
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
smallpressuredisturbance
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
(157) 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
(158) velocit_y
a is an important
of a gas.
This
ratio
is
THERMODYNAMIC
AND
FLUIDDYNAMIC
CONCEPT_
M= V
(159)
a
Mach
number
is a useful
behavior regimes, but expressions. Consider temperature, (159),
given
and
parameter
also the
with
(151).
equation
ratio
for
Combining
(151)
T'
Another velocity critical velocity
only
identifying
for simplifying and generalizing relation of total temperature
in equation
(128)
not
to
equations
certain static (158),
yields
l_I_7__
often
flow
(160)
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 (160),
at
velocity,
velocity
(161)
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
M1.
and The
a.
is speed
critical
condition.
The
Consequently,
of
velocity critical
from
equation
2 T.=3"+ and
substitution
of equation
1 T'
(162)
into
(162) equation
(158)
yields
(163)
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
(128),
and
static results
and from
total
temperature
combining
equations
in terms (161),
of the (163),
(151). TT '1
3,1(V) 3"t1 _
2
(164) 13
TURBINE
DESIGN
AND
APPLICATION
APPLICATION The the
equations
flow
(flow the
TO
already
through
in the turbine.
there
to learn
varyingarea
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). lossfree
FLOW
pressure,
manipulation
following
velocity,
of the
equation
previ
for isentropic
flo_v:
_(I_M_
Equation
(165)
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 varyingarea
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
26.Typical
temperatureentropy
diagram.
con
expansion
means of a temperatureentropy 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 temperatureentropy entropy
for
we will not
a little
represent
increasing
to condiand
definition than
energy)
This
constantheatcapacity 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
(241)
reaction
energies.
Turbine For
W22
downstream
bladerow
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
coastantentropy1)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
constantentropy
temperature
CONCEPTS
thermodynamics Ts,
increasing
curves
curves
into
or
26.
line.
entropy,
pressure
divided
in
a vertical
constant
given
the
temperatureentropy,
shown
sented of
of a
TURBINE
each
then
be
turbine shown
expansion
process
in a separate
combined
into
Ts
a single
four diagrams represent the stator expansion process relation between absolute and relative conditions
will
be
diagram. diagram.
(fig. 27 (a)), at the stator
P()"P'Lid
_,,_JcT_= 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
27.Temperatureentropy flow
diagrams turbine.
for
between conditions
absolute at
stator
between conditions
flowprocess
and exit.
relative at steps
and
rotor of
exit. an
axial
35
TURBINE
exit
DESIGN
(fig.
blades
27(b)),
(fig. at
Figure
APPLICATION
the
rotor
27(c)),
conditions four
AND
and
the
rotor
27(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
27(c),
it is on
for
the
T2._d), state refers
figure
28,
the
entire
stage
36
figure
the
and
by
the
the
which the
ideal
rotor
in
kinetic
in figure
en
27(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 28, the
rotor
alone.
of the
across subscript
in
where
27(c),
expansion
the isen
relative
in figure
across
to the
at
constantentropy
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 constantentropy the
relative
absolute
27
indicated
stator
relative
assumed, state
would
static,
turbine
subscript
final
27(d).
state
to
(both
is
relative
The
the
is not
2,id
by
developed
constantpressure
flow
the
appropriate
script
that from
figure.
is shown four
pressures
than
of figure
28.
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
(151).
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 27(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
27(d)).
expansion
after
total
process
relation (fig.
curves and
state
the
the exit
constantpressure
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,
28.Temperatureentropy
therefore,
diagram
ambiguous
obvious
from
process
(as represented
be obtained To T;. ,d).
figure
but
28 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 bladerow
exit kinetic energy row. For the stator,
axialflow
(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 (155), we get
efficiency. 27(a).
The By
as the
energy
of the
For
the
(242)
relation
applying
between
equations
V_ (151),
and
where
is
and
(243)
rotor
Wi _'°=W_, ,d Vro is rotor
is indicated
2 Vl._a
(152),
\pod
L
a
parameter
V_I
where
is
turbine
Efficiency
do not
bladerow
of an
used
obtained
turbine
BladeRow
parameter
a stage
is commonly
that by
from
s
in figure
efficiency. 27(c).
The For
(244)
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
Bladerow
in
as a loss rather
known
for a given
for a specified
terms
than
(245)
j
efficiency
exit velocity
performance
expressed
1\_,/
of
blade
exit static
kinetic
energy
as an efficiency,
is
sometimes
as
e=ln where
e is the
kineticenergy
Bladerow total
differing by dimensionless. actual
also
Several the
(246)
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
totalpressure
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
(247c)
P2
and
These
(247b)
P_
loss
coefficient
a Mach
Turbine Turbine
_f
P2
kineticenergy
loss coefficients
(247a)
,, yT o__P, p_ P2 p_
"PoPl
and
It
P2,,
'
• 'tP_pl
where
as follows:
PI
Pop___._ "P;Pl
V"
each
the coefficient head, and exit
purpose
YTo_P'
Po
v'
used,
in
rotor:
t
popl,
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
28.
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
28.
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
turbineexit
turbine total
rel)resented in(licated
last
the
efficiency
total
next
other
jetengine
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
exhaustflow
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
(248a)
For the idealgaslaw reduces to
and
constantheatcapacity
assumptions,
this
T',.T',_ V
/ p,, \(_l)/_l
Overall
turbine
total
efficiency
(248b)
J
T;. is expressed
as
hh' h;.h',_ _' = ._7, =h' _'
For the idealgaslaw reduces to
and
constantheatcapacity
._, _
total
and
appropriate Relation
of turbine
efficiencies
are
(249b)
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
(248b)
or (249b)
a given
stage
which
were
pressure
for a stage
ture
would
gas
entering
28,
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
29.
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
O2,id
tem
following
Therefore, stage
of a temt)eratureentropy 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 (ToT_),
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
(249a)
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 29.Temperatureentropy
FIGURE
isentropic
expansion
dashed
line
taking
place
The
02
from
obtained
ideal
work
difference
of temperature
increasing
p_),
the
that
represented
greater
by
constant EF
stage
is greater
ing
the
the
sum
which and
by
virtue
ideal
Thus,
the
than work turbine
the
the
of
actual
work
B2,id.
With
lines
three
stages,
it can ideal
be seen work
that
the
0A,
work
where the
increases
stage
(p_
is greater for
this
stage
be greater.
CD,
and
and
to
than stage
an(l,
is for
Similarly,
EF
represent
2; Ahld. stt representing
Z Ah;dstg
represented
second CD
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
AB.
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 0A,
Ah_, AB,
B2,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'
'
(250)
z ah,_.,,, Ah'_
(251)
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 
(252)
( 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
(TdT),
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
Infinitesimalstage
tropicefficiency
derivation
an important
of an infinitesimally
infinitesimal
differs
of two
to eliminate
temperature an
The
1.
pressure
to pressure
increment
efficiency
p
(pdp)
of temperature
vp. By
using
the
and and for isen
write
d T = _pT E l (ppdp
) (_ ' ) /_]
(253)
and dTT_ _1o[1(I 42
__)(_
z,l_]
(254)
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
totaltemperature
statictemperature
assumption
infinitesimal
quite
Some
should
rather
make
not
definition.
differential tial
are
TURBINE
authors
energy
it ahvays
across seems
to
infinitesimalefficiency
expression. Using
the
evaluation
series
expansion
of equation
approximation
(254)
(1#x)
n= 1 +nx
dT 7_1 dp T _ 7 I' Integrating
between
the
for
yields
turbine
inlet
(255)
and
exit
yields
Tfn
In _(256) z/P7
1 In p_" "Y
Equation
(256)
The
can
infinitesimalstage
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
(nl)/n
(25s)
\p,,! Equations process relate
(257) were
l)olytropic
and
(258)
to be expressed efficiency
are as
and
the
n
inlet
and
exit,
very
similar,
a ])olytroi)ic
n1
If we neglect
is law,
polytrol)ic
and
if
the
process,
then
exponent
as
),1
turbine we
couhl
(259)
?
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
(260)
with
(261)
then
(260)
k
(261)
J
yields
_=1\_/
This proach
relation
is illustrated
each
other
unity.
However,
ciency
levels,
as at
the
two
in
(262)
figure
pressure
higher
effi
as
or _1
polytropic
210.
ratio
and
pressure
efficiencies
can
The
two
efficiency
ratios,
especially
differ
signiticantly.
efficiencies each at
ap
approach lower
effi
.9Turbine pressure
¢
_.,,_
r_tio /.,///
09 
.8
B
b
.5
1
I
.7 Turbine
FIGURE
210.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
surfacetension 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
7rTheorem, 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
fullsize
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
pN3D 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
(263)
' pN2D2/ il_ (limensionless
coefficient.
If we
It
can
form be
further
by
BASIC
rel)resented
TURBINE
CONCEPTS
as Q
VA
VI) 2
V
V
(264)
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: _
(265)
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. VelocityDiagram
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 velocitydiagram 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 axialttox_ turbines. One of these parameters is the speedwork parameter X::
U 2
(266
gJ/W 'Fh(_ re(:ipro('al
of the sl)ce(lwork
parameter
is also oflcn
use_l, aml
it 47
TURBINE
DESIGN
is referred
For
AND
to as the
an axialflow
APPLICATION
loading
turbine,
factor
or loading
1 ¢=X
g JAb' U2
we can
coefficient
(267)
write
(268)
Therefore,
equations
(266)
and
(267)
X=
Another
l)arameter
often
used
1
can
be expressed
as
U
(269)
¢AV_ is the
bladejet
speed
ratio
U p_
where
V_ is the
or spouting,
jet,
or spouting,
velocity
is defined
ideal
expansion
stage
or turbine.
from
inlet
That
(270)
"_fj
velocity, as the
total
in m/see velocity
to exit
or ft/sec.
The
corresponding
st, atie
(:onditions
across
of equation
(271)
back
into
(271) equation
(270)
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
bladejet by
(272)
speed
ratio
use of equations
and
the
(.266)
and
speedwork (272)
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_
(273)
v= _/____ _
(274)
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
velocitydiagram
parameters
speedwork diagram, and
to the
parameter
it
parameter the
bladejet
efficiency. Is
the
flow
BASIC
factor,
or flow
TURBINE
CONCEPTS
coefficient
(275) The
flow
coefficient
can
be related
to the
loading
coefficient
as follows:
v, / v, By
using
equation
(269)
and
the
velocitydiagram
_=#
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
(277)
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
(268)
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
211.
be
Parameters
bladejet U_)
the next
is specified,
static
h'oh'2 Ah' ,7 h£h2. ,_ _h,_ Substitution
in
One parameter must the loading coefficient.
(U_=
is isentropic
to these
speedwork
general
specific the
are
case
case
diagram
(V_._=V_,2).
is shown
turbine
real
VelocityDiagram
to
axialflow
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
velocitydiagram
for
type
Lewis
parameters and the
will now
through
addition, shown
a somewhat
speed
of these coefficient,
be
for
In be
a particular
bladejet
We
will
four
velocitydiagram
Relation
can
these
other.
parameters. section
that
for
that _'=
a
flow
1).
The
efficiency
(278) (278)
yichts 49
TURBINE
DESIGN
AND
APPLICATION
a1
Vu,1 j/ V2
Wu, 2
FIGURE
211.Velocityvector
diagram
for
an
axialflow,
impulse
stage.
UAV,,
(279)
= gJAh,.,t The
change
in fluid
tangential
velocity
is (280)
zxV,,=V_,,V_,z From
the
assumptions
convention
(Wl W2)
and
(W,._=
and
W,.2)
the
we adopted, (281)
W_. 2= W_. 1 From
sign
equations V,.2=W,.5+
(26),
(281),
and
(280),
U=W,._+U=(V,.
we get 1 U) + U=
V_.I+2U (282)
and AV.=V..,IV.,2=V..1(V..,+2U)2V..I2U From
the
velocitydiagram
(283)
geometry (284)
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
(284)
2gJAh_e and
(285) (285)
into
equation
(283)
BASIC
TURBINE
CONCEPTS
yields AVe=2 Substitution
of
sin al_/2gJAhia2U
equation
(286)
back
into
4U sin al
definition
Equation stator only. an
The
(288) angle, with
is reached speed
tion
(288)
shows
ratio and
that
static
variation
example
0.88 jet
exit
speed
for this and
mathematically
ratio
derivative
of axialflow,
case
and
of 0.47.
The
by
for of
optimum
bladeequa
(289)
I
I
I
.6
.8
stage.
212
to zero:
.4
speed
speed ratio,
ratio Stator
on
ratio
efficiency
differentiating
I
impulse
constant
speed
in figure
.2
bladejet
any
of bladejet
sin al 2
Bladejet
212.Effect
(272) (288)
equal
Vo_t
FIGURE
equation
of 70 °. A maximum
speed
the
from
is illustrated
angle
can
setting
ratio
particular
at a bladejet be found
(287)
is a function
is parabolic exit
yields
sin a,4_3
efficiency
a stator
(279)
2gJAh_
of bladejet n=4z,
equation
4U _
_4'2gJAh_a Now using the finally yields
(286)
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.
(288) While
ideal
case,
indeed, very
the
the
figure
212
levels
and
values
basic
it does. good
and
We
parabolic find
correlating
Likewise,
for
other
tion
(263).
This
does
would
of all sizes.
Such groups.
apply
would
however,
to
apply
be
parameter is found
exit
used
for
or turbine
bladejet 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
(291)
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
bladeiet
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
velocitydiagram 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
(292) or
turbine
exit, (293)
values revolutions
for
these per
parameters minute,
exit
BASIC
volume flow rate H in footpounds of units,
Qe_ in cubic per pound,
specific
less because
speed
the
to be the
and
units
are
totaltototal
it is specified Specific
as the
speed
presented
not
diameter
consistent. (5h;_),
totaltostatic diagram
parameters.
dimensional
The
ratio
of total
differing eters. all
efficiency
definitions Some
cases, The
thus
eliminating
parameter
speedwork (296)
to the
parameter
by
to the
blade
speed
previously
is
or 60 sec/min).
The
(n)_
(295)
and
(272)
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,
(294)
(293),
work
prefer
for
(27r rad/rev
to static
of ideal
authors
H is usually
be related
The
constant
equations (291), (295) yields
dimension
(hh_).
can
H=JAh_d=J_d Combining (294) and
head
truly
or head, this set
_ND K
Uwhere 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
(274)
(296).
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 (291),
Since diagram
A_x
is the
(293),
specific
flow
area,
(294),
and
(275)
with
NsD,
3= KD2 _'_A_
parameters,
and
specific
which
can
in m 2 or
diameter be used
flow
(298)
exit
speed
the
(297)
"/l"
Specific coefficient.
in
ft 2. Combining
equation
(298)
equations yields
(299)
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
velocitydiagram flow rate, and (299),
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
(2100)
D"_H1/'
(2101)
diameter "_
Overall
___e, H3/4
speedwork
parameter (2102) gj_'_
Overall
bladejet
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
(2103)
average
for
parameters, value
only,
depend overall
to
to the
because
,
parameters Equation the
sig
application gen(2100)
considerations Let (2104)
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_
(2105)
_'H
Then, (2100)
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
(2105)
effects that different fluids on the turbine. The third
Often, the
can
reflects
The
the have
vex kll21
speed
term
and
t
/
specific first
(2104)
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 Onestage, _'_
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
213.Effect
of specific
speed
on turbineblade
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
axialflow,
tip
decreases
to
radius
radius turbines,
ratio.
of the
Dividing
equation
for stage
specific
number (2100)
speed
for
the
a radialflow
tur
turbines.
The
example with
increasing
Thus,
application indicates the type The values of some of the proximation
213
twostage,
axialflow
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
(291)
yields N_
Qe_
1/2 (2107)
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
(2108)
into
too
that
large,
we
can
Qe_:Q_:._t_.
equation
(2107)
neglect
the
Substitution
and
of
rearrangement
yields n:(NsY/3
(2109)
\NJ Since
stage
specific
experience assume
can
number
type
overall
speedwork Often,
basis
of stress
suming
2, where
by
requires
value
of blade
equal
Dividing
equation work
Or,
equation
(2102) 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
_peedwork
turbine
the
estimate
be selected
may
(266) overall
from
value
a value
to
stage
estimate
is often
a knowledge
for
para
speedwork 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
(2109)
for
for
of efficiency.
a compressibility
speedwork
a reasonable
parameter assuming
effect
parameter,
speed.
overall
of estimate
speedwork
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
(2110)
BASIC
TURBINE
CONCEPTS
yields n==
Equations studies
(2109)
and
associated
(2111)
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, #)
(2112)
p,..
w
, _ ) _/ RT,,,'_T"7 'P_
been
assumed terms
above
massflow
equation,
N,
following:
secondorder,
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
(263)
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
(2111)
),
the
(2113)
constant,
there
modifying
terms parameter idealgas
to see
the what
may law,
would
flow,
be
work,
significance
be transformed and
RT .
the
propor
(2114) 57
TURBINE
DESIGN
Substitution (2113)
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 rotorblade Mach
speed
velocity
number.
parameter fluid
have
certain
with
V/U,
Mach
number,
as for
temperature
a given
(2113)
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
massflow
is that
variable the
the
nondimensionally velocity,
the
analysis
of fixed
singular
of
of this
fixed
machine
to critical
Division
gives
implication
is represented
(2116)
the
have
a
a given is not
a
associated expressed
in
temperature as equation
as
(2117) p__
For
a given
gas
_Icn
in a given
_T'
turbine,
_=fcn
Depending tions
on
the
(2113),
particular
(2117),
or
the
_r,
case,
7'
the
(2118)
parameters
further
be
to
(2118)
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
(2119)
!
Ah'eq l_,d
N
(2120)
N_q
/
these
The
the
Pst_
ah' RT_.R_,.
of
air. on
but
the
or are
conditions.
standard
P
P_.
Rearrangement conditions
standard
expressed
(2113)
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
,
(2121)
equations
then
yields
for
the
equivalent
p',. w,q.=w
taT' ta
,
Ah_q,_h
(2122)
, RT',_
(2123)
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 specificheatratio
specificheatratio However,
ratio
only
we started off the discussion of these specific heat ratio for all conditions.
us now
parameters.
(2124)
all
terms number,
effect
specificheatratio
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
(2125)
V_.v_,,.J \p,.:
8q

t
(2126)
_:,=t,h' ,,,_7 ) N_q_
Nj(V_'d.)
'
(2127)
where
.>,,S. _2 V,;(,,,:,) "\%,,+1/
(2128)
.d__Ly"
_
FIGURE 47.Effe(:t
figure
I
__
4
70
i" _
7.
Exitflow 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. 47

Inletflow angle, al, deg  2
a 2
10 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
boundarylayer to stator
blades
blades
lowsolidity
and
are presented tandem
removing
blades
jetflap
on
marginal
better
potential
in figure as well
49. as the
are summarized
of lowsolidity
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 suctionsurface
diffusion
minated
the
116
the
blades,
blade
at about
45,
experi
occur.
region
treatments
energizing of the
do not
in the
concepts
jetflap
tandem
Such
})e utilized
blade
concepts
with
layer
has
To achieve
losses
jetflap
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
47.
of figure
than
1.
limitation suction
as those solidity
is slightly
UltralowSolidity
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 _ Exitflow angle,(]p deg
48.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
410
layer
through
perhaps
and thc
slot.
a secondary air stream main stream. This jet
trailing
jet
with
concepts.
boundary
air going
jetflap blade operates with edge perpendicular to the
stagnation
DESIGN
Jetflapblades
49.Lowsolidity
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
410.Jetflap
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
tandemblade
where
as blade
chord
jet on, there
and
_ more
surface
BLADEPROFILE After
the
to separate.
and
for applications
purposes,
on. With
on the suction
edge.
the
the jetflapblade
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
411,
which
of the shows
new
exitvelocity
just
within
trailingedge 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
turbineloss also
has
with
the
2a, which
reduced
velocity
use of figure
area
at station
blade this
trailingedge
used. due
to the
2a than
(426)
_)_
at
region.
"withintheblade"
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
trailingedge
effect
blade
and
in the
trailingedge
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 trailingedge
effect
includes the
of turbines,
with
the
section
between
the design
consistent
reference blade
exit
(427)
BLADE
DESIGN
Station 1
0
$
FIQU_E 411.Blade
where
t is the
trailingedge
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. 411)
technique
flow
(since
at the
blade
conditions
region,
used
area, a rather
successfully velocity and
exit
a straight
the large
(station
2)
trailingedge 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
"insidethetrailingedge" in
by
must
Because
be obtained.
procedure.
can
2a to become
choking
opening
sion makes
blade
cannot
the flow accelerating of the
in the
(427)
incompressible
2a
the
to be
the
station assumptions.
specified
or feet.
a velocitydiagram
and
is often
to determine
and
The
to produce
number
(65 ° or greater)
(426)
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
"freestream"
(428)
(T2a
or feet.
and
station
loss do not
change
between
the
2, then
0 = 8
When the
this
method
angle
(428)
of the
and
solidities
(429)
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
(428)
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
bladerow
than
deviation
that
(cqs.
8 compares (429).
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
throatopening
If the
but
with
close
(429)
of trailingedge
throat
of up to 5 ° for exit angles
due to lower the throat. The
angles
q2
effect
position
indicates
deviations
the
give similar
exitflow
comparison
case,
throat
(429))
measured
or
is used,
COS
throat.
be computed
(up
to,
can be and the In by
this the
equation:
(430) 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"
(428)
or
(429)
for
BLADE
DESIGN
1.0
¢3
.9
1
Rt
I
1.1 Mach
FIGUttE 412.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
suctionsurface
number
flow Mach munl_er.
flow between
throat
and
412.
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
suctionsurface 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
413
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
trailingedge
(,xit Mach
severe.
edge
subsonic
in the
flow acceleration
losses
the
()f the
High
to prevent
associated
blading
on the
(approximately
discussion
that
loaded
exitflow
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
gasturbine
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_
413.Variation
of
profile
between
h)ss
throat
with and
Math
exit
numl_er
(from
ref.
and
surface
curvature
_).
Inlet The
leadingedge
than
the
exitregion
leadingedge 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 suctionsurface
contraction (426)
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
pressuresurface ellipses,
(427),
which
as
were
a bladeinlet
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
trailingedge
this
velocitydistribution
curvatures
BladeSurface Once
specified,
large
portions
which
edges
in undesirable
be used to minimize
122
inlet,
less critical
choking.
leadingedge
task
blade because
of lowreaction can
be taken exit,
row is usually
for lowreaction
toward
at the
bIade
case
region
"withintheblade"
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
414.
suction
position
velocity
limits).
both
the design
of a quasithreedimensional
that
ncxt
blade.
the blade diffusion
illustrated
channel
programs
in the
(e.g.,
staticpressure
least
the
through
in streamlinc
be at
computer
are discussed
arc
the
of radialequilibrium
should the
around
controls
the bladesurface
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_..
Crosschannel
distance
(a)
__
, 'lEi.:_.i:!_._.:._::..::::::_i_! Tip : _i_ Rotorf_ii!i
'Nl ,
"' Tip
_
_I
°
NN
¢";1
,Hub
_:: /
Velocity (b)
(a)
Crosschannel (b)
FIGURE
414.Turbine
Radial
variation. variation.
bladerow
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
TurbineBlade
WARREN
WHITNEY,
by
Parameter
Cascade
and
for
1955.
Changes
in
J.:
Blade
RICHARD
Turbine
at
R.:
An
Analysis
Geometry.
H.:
Experimental
Four
RotorBlade
Examination
of
Turbines.
Rep.
AxialFlow Britain,
the
Flow
R&M
and 2891,
1955. M.
:
Summary
of Tests
in ThreeDimensional
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
TwoDimensional Chord
of Two
Power,
NASA
JOHN NASA
of 0.5.
F.:
Chord
of
TM
vol.
Con92,
Turbine
no.
X1836,
X2183,
Stator
X1991,
Cascade TM
1970. Investiga
1969.
Twol)imensional
TM
TwoDimensional of Axial
Test
NASA
F.:TwoDimensional
Design.
AND KLINE,
Rotor
l)_ign
Blade
of
1969.
and
Tandem M.;
ROY G.:
Use
Losses.
Diffusion
TwoDimensional
AND
STANLEY
Nov.
of Axial
Turbine
Stator
C.
of
Concepts
])esign
Ratio
STANLEY
W.:
198206.
of a Turbine
JetFlap 14.
pp.
STANLEY
JAMES
RotorBlade
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
69WA/GT5,
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 AxialFlow
1967.
J.;
WARNER
Rotors.
Viscous
RM
Large
436444.
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.
BladeLolling
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
67WA/GT8,
WARNER
vol.
FRANCIS
Elements. L.;
Blading,
Rev.,
Losses
of AxialFlow
Parameters.
5.
SCHWENK,
WARNER
STEWART,
TurboMachine
Boveri
for Estimating
amination
4.
of
Brown
SEYMOUR;
Factor
Compressor 3.
Spacing
Deflection.
Cascade
Test
of a
1971.
Cascade
Test
to Spacing
of a JetFlap
of 0.5.
NASA
TM
Turbine X2426,
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
CR1968,
AND the
HELON,
of a Highly
Loaded,
Low
of a Highly
Loaded,
Low
1971.
Experimental
NASA
Thickness
Results
CR1803,
1972. RONALD
Performance
M.:
of Certain
Effect
of
Turbine
TrailingEdge 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
trailingedge
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
(46)
loading
coefficient
defined
by equation
(45)
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
hubtoshroud shown
directly,
but
stream in figure
provides
surface
extreme
to
at
the also
NASA
flow
field
complexity
geometrically therefore,
twodimensional
of
complex certain
threedimensional
51 (a),
(commonly
does
information
There
are
tribution
two
over
formulation
one
of the
mathematical we
velocitygradient
parts
will
to a method
of these
flow
surfaces.
and
problem. discuss
the For
stream
(streamfilament)
yield
The second
is
Such
and
is the
is the
mathematical The
surface which
yield
a velocity
dis
mathematical
numerical
solution
formulation
potentialfunction
methods.
velocities
solutions,
part
part
meridional
bladetoblade
to obtain
first
the
bladesurface
51 (c))
of analysis
the
called
for the
(fig.
surfaces.
problem,
not
required
(fig. 51 (b)) and orthogonal surface the desired bladesurface velocities.
problem,
in order
are illustrated in figure 51 for the case of a radialinflow turbine. surfaces are used for an axialflo(v turbine. A flow solution on the
surface),
of the
The
various
of
developed
distribution,
made.
field
section
a bladerow
of the
velocity be
last
chapter presents calculation and
were
flow through
a theoretical must
that
because
threedimensional
assumptions
flow
throughout
time
in the
bladerow
programs
at this
nonsteady,
as indicated
stream
of the
methods and
and
potential
TURBINE
DESIGN
AND
APPLICATION
Blade_,blade
Hubtoshroud stream surface
_ 
surface] I
(a)
0_)
, Ortho_nm
(c) (a) Hubtoshroud
stream surface. (b) Bladetoblade (c) Orthogonal surface across flow passage. FIOURE 51.Surfaces used for velocitydistribution calculations.
128
surface.
CHANNEL
function
methods
will be described
solution.
A similar
type
The
of analysis
velocitygradient
for solutions The
equation
on any of the
following
of analysis (1)
The
surface
flow
Thus,
if the
fixed
coordinate
(2)
The
bladetoblade
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
idealgas
law
p=pRT
(51)
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 bladesurface 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
ttAt.
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
(52)
Intuitively,
orientation figure
this with
52
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,
potentialfunction
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
rl LJ
Timet+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
52.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
53.
only
will
are many a few. cascades
An is
1.
POTENTIALFUNCTION
function
is in terms as shown
mathematical
these equations, and we will discuss discussion of flow in twodimensional
StreamFunction The
flow.
ANALYSES
Method
defined
several
Suppose
It is assumed
ways,
but
we consider that
there
two
perhaps blades
is twodimensional
the of a
CHANNEL
FLOW
ANALYSIS
1 .8 Mass
.6
flow fractbn
.4
0
\\\\
FIGURE
axial
flow here,
53.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
53 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 onedimensional
flow by w= pVA
(53) 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 streamfunction
cascades relative
(blade velocity
analysis
rows),
the
W, which
velocity V. We will assume the mass flow wl._ between fig. 54)
can be calculated
fluid
applies velocity
(54) 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
54.Arbitrary
(55)
_ 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 (55),
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,
(56)
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
55. the
easy
integral
in equation
to calculate Ou/Ox
at the
the
partial
point
in the flow passage,
(56)
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) Totalpressure ratio (blade exit to blade inlet)
Totalpressure ratio (blade exit to blade inlet} r_>
rI >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
aftermix
ratio,
critical
veloc
critical
veloc
(d)
0.823. of
obtained
totalpressure
Surface
profiles
Ideal
part
of the
that
yield
ratio
experimentally,
blading favorable
aftermix
ratio,
ity
1225.Contours
An important
Ideal ity
aftermix
ratio,
ance have been can be made.
surface
(b)
0.512.
a
critical
veloc
critical
veloc
0.671. aftermix
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
vf_'+i[1
L_1L\p_/
tribution
shown
Acceleration
on
smooth,
and
no
flow
large
(force
the
the
1226(a)
suction
maximum is well
a blade
surface,
the
along
surfaces
the
in the
measureblade
from
the
surfaces relation
to
(1215)
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)
1226(b), on the to be undesirable.
along
( P']'""'"7"lr"
velocity
decelerations
on blade)
Figure considered
in figure
to deteractually
be determined
Figure 1226 shows the experimentally distributions for two stators tested under
in
were
on staticpressure
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
staticpressure 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. 1226.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. McGrawHiU 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 69GT104, 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 69GT103, ASME, Mar. 1969. 4. WISLER, D. C.j AND MOSSEY, P. W.: Gas Velocity Measurements Within a
5.
Compressor Rotor Passage Using 72WA]GT2, 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 (126)
direction,
pressure
pressure
velocity
deg
critical
ratio, 8; lb/ft
defined
condition
by
sealevel
Mach
eq.
1)
condition
throat station
at turbine
standard
inlet
heat sealevel
by eq. (1212) 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 bladejet
absolute
by eq. (125)
pressure function of specificheatratio,
sealevel
of
lbft
turbineinlet
squared
ft/sec
m/sec;
measured
F
Nm;
(o R)
lb/sec
factor,
flow
m/sec;
m/sec; ft/sec component
inlet
flow rate,
(sec)
(124)
K; °R speed,
gas velocity, in tangential
compressibility
(min)/(rev) by eq.
rev/min
temperature,
mass
(rad)
defined
(K) ; (ft) (lbf)/(lbm)
meansection
Y
to exitstatic
N/m S; lb/ft 2
J/(kg)
W
of inlettotal
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
(1210)
(1214)
turbineinlet on and
standard (1213)
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. o7o4o188 Form Approved S_'nd qr%
Budeet
Ser_
cP._.
` Paperwork
3. REPORT
I" AGENCY USEONLY(Leaveb/ank) 12" REPORT 1994 DATEjune 4. TITLE AND
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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%
(0104018B),
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 R5666
9. SPONSORING/MONITORINGAGENCYNAME(S) AND AODRESS(ES) National Aeronautics and Washington, DC 20546
Space
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Administration NASASP290
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Ol S T"iB U TI'0_0
D E'
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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. Closedcycle turbine using inert gases, organic fluids, and metal fluids have been studied forproviding longduration power for spacecraft. Other applications of interest for turbine engines include landvehicle (cars, buses, trains, etc.) propulsion power and groundbased electrical power.
In view of the turbines_stem interest and efforts at Lewis Research Center, a course entitled "Turbine Design and Application was presented during ]96869 as part of the ]nhouse Graduate Study Program. The course was somewhat rewsed and again presented in 1_7273. Various aspects of turbine technology were covered including thermodynamic and fluiddynamic 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 for selfstudy, 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.
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