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IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
1-1
Module 3A
3 - Turbom
achinery P
rof How
ard Hodson
1 Introduction 1.1 D
efinition
A T
urboma
chine is a
stea
dy flow de
vice (non-positive
displace
me
nt) which
crea
tes/consum
es sha
ft-work by cha
nging the m
ome
nt of mom
entum
(angula
r m
ome
ntum) of a
fluid passing through a
rotating se
t of blade
s.
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
1-2
1.2 Exam
ples of Turbom
achines 1.2.1
Ve
ry La
rge M
achine
s
MH
I 501 single shaft Gas T
urbine
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
1-3
Three Low
Pressure R
otors from a large steam
turbine (approx 150 M
W per rotor)
Ma
nufacture
rs of
large
ga
s turbine
s a
nd stea
m turbine
s for industrial pow
er
gene
ration include
• Alstom
• Mitsubishi
• Sie
me
ns
• Ge
nera
l Ele
ctric
All use
axia
l flow turbom
achine
s
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
1-4
1.2.2 A
ero E
ngines &
Ae
ro De
rivative
s
Big 4 m
anufa
cturers:
• Ge
nera
l Ele
ctric
• Pra
tt & W
hitney
• Rolls-R
oyce
• SN
EC
MA
All use
axia
l flow com
pressors a
nd axia
l flow turbine
s exce
pt for the
sma
llest of e
ngines (e
g helicopte
rs a
nd UA
Vs) w
hen ra
dial flow
compre
ssors are
used
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
1-5
1.2.3 R
adia
l Turbom
achine
ry
Ma
ny types &
configurations
Most com
mon type
s
• centrifuga
l pump or com
pressor w
ith axia
l inflow a
nd radia
l outflow
• radia
l inflow-a
xial outflow
turbine
An industrial centrifugal com
pressor
A S
mall T
urbocharger
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
1-6
The K
aplan turbine has an radial flow
stator and an axial flow rotor
The F
rancis Turbine has an radial flow
stator and a radial-axial flow
rotor
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
1-7
1.3 Aim
s of the Course
This course
aim
s to provide a
n understa
nding of the principle
s that gove
rn the fluid dyna
mic ope
ration
of axia
l and ra
dial flow
turboma
chines.
At the
end of this course
, you should be a
ble to
• Identify a
nd understa
nd the ope
ration of diffe
rent type
s of turboma
chinery.
• Ana
lyse turbom
achine
ry perform
ance
.
• Unde
rstand the
cause
s of irreve
rsibilities w
ithin the bla
de pa
ssage
s
• Ana
lyse com
pressible
flow through turbom
achine
s.
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
1-8
1.4 What is in this course?
• 4 types of m
achine
s:
o A
xial com
pressors
o A
xial ga
s turbines a
nd axia
l stea
m turbine
s,
o C
entrifuga
l compre
ssors
o R
adia
l inflow-a
xial outflow
turbines
• Ana
lysis of the flow
in blade
rows a
nd stage
s (1D)
• Dyna
mic sca
ling, chara
cteristics of com
pressors a
nd turbine
• Com
pressible
Flow
Ma
chines
• Hub-T
ip varia
tions in flow prope
rties (2D
)
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
1-9
1.5 Laboratory Experim
ent E
valua
tion of pump pe
rforma
nce
• me
asure
perform
ance
para
me
ters
• study effe
cts of Re
ynolds numbe
r
• exa
mine
the e
ffects of a
nd visualise
cavita
tion
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
1-10
1.6 Recom
mended B
ooks A
uthor T
itle
Shelf M
ark
Dixon, S
L
Fluid
me
chanics,
The
rmodyna
mics
of T
urboma
chinery,
TN
24
Cohe
n, H.,
Roge
rs, G.F
.C., a
nd S
ara
vana
muttoo, H
.I.H.
Ga
s Turbine
The
ory V
K 33
Cum
psty, N.A
. C
ompre
ssor Ae
rodynam
ics V
S 16
Rolls-R
oyce
The
Jet E
ngine
VN
36
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
1-11
1.7 Notation
Ge
ome
tric & F
low P
ara
me
ters
Ax
Annulus a
rea
=
hR
RR
Ax
mean
2hu
b2casin
g2
)(
ππ
=−
=
Ax
Pa
ssage
are
a =
(blade
height) × (bla
de pitch)
A
Effe
ctive flow
are
a =
A = A
x cosα b
axia
l width of ra
dial im
pelle
r (i.e. bla
de spa
n) α
F
low a
ngle in a
bsolute co-ordina
te syste
m
F
low a
ngle in rota
ting co-ordinate
system
D
dia
me
ter (usua
lly me
an or tip)
ψ
stage
loading coe
fficient
h A
nnulus height, bla
de he
ight, span =
h
= r
casin
g – rh
ub
h e
nthalpy
m &
Ma
ss flow ra
te
Q
Volum
etric flow
rate
φ F
low coe
fficient =
UV
x (or
3D
QΩ
& in “sca
ling” applica
tions) Λ
re
action
r R
adius
s pitch (spa
cing) of blade
s σ
slip factor
U
Bla
de spe
ed (usua
lly at m
ea
n radius) =
U
= 1/
2(Uca
sing +
Uh
ub )
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
1-12
V
Flow
velocity
Vx
Axia
l velocity
Vθ
Ta
ngentia
l velocity
Vρ
Ra
dial ve
locity V
θ ,rel
Ta
ngentia
l velocity in rota
ting co-ordinate
system
V
rel
Ve
locity in rotating co-ordina
te syste
m
S
uffices:
0 sta
gnation
1 inle
t to 1st blade
row
2 ca
scade
exit or 1st bla
de row
exit / 2nd bla
de ro
w inle
t 3
stage
exit / 2nd bla
de row
exit / 3rd bla
de row
inle
t m
, me
an
value
at m
ea
n radius
r ra
dial
rel
rela
tive fra
me
of refe
rence
(rotating fra
me
) x
axia
l y, θ
tange
ntial
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
2-13
2 Basic C
oncepts 2.1 S
tagnation (total) and Static (T
rue Therm
odynamic) Q
uantities
wx
q
2 1
Co
ntrol
Volum
e
Assum
ing gravity ca
n be ne
glecte
d, applica
tion of the
SF
EE
to the a
bove give
s
+
−
+
=−
22
211
222
Vh
Vh
wq
x
Now
, the sta
gn
atio
n (total) spe
cific entha
lpy h0 is given by:
2
2 10
Vh
h+
≡
so the S
FE
E ca
n be w
ritten:
01
02h
hw
qx
−=
−
In a turbom
achine
, the w
ork excha
nge occurs be
cause
of change
s in mom
entum
(velocity) so the
im
portance
of the kine
tic ene
rgy in the S
FE
E ca
nnot be
ignored. IIA
Paper 3A
3 Fluid M
echanics II: Turbom
achinery/HP
H
2-14
The
refore
, espe
cially w
hen w
e a
re de
aling w
ith individua
l stage
s (i.e. single
rotor+sta
tor combina
tions),
we
must spe
cify if the p, T
and h tha
t we
are
using a
re the
• stag
na
tion (tota
l) pressure
, tem
pera
ture a
nd specific e
nthalp
y or
• the sta
tic (ie true
therm
odynam
ic) pressure
, tem
pera
ture a
nd spe
cific entha
lpy.
For a
perfe
ct gas, the
static a
nd stagna
tion tem
per
ature
s T a
nd T0 are
rela
ted to h a
nd h0 by
2
2 10
0)
(V
TT
ch
hp
=−
=−
It is T0 ra
ther tha
n h0 that is m
ea
sured e
xperim
enta
lly. This ca
n be done
by m
ounting a the
rmocouple
inside
some
thing like a
Pitot tube
.
By w
orking in term
s of stagna
tion (total) qua
ntities
• kinetic e
nergy e
ffects a
re a
utoma
tically ta
ken ca
re of,
• ana
lyses a
re e
asie
r (sta
gn
atio
n quantitie
s are
ea
sier to m
ea
sure tha
n sta
tic quantitie
s).
Note
that if the
type is not spe
cified or im
plied,
it is usually sa
fer to a
ssume
that the
p, T a
nd h re
prese
nt the
stag
na
tion pre
ssure, sta
gn
atio
n tem
pera
ture a
nd sta
gn
atio
n specific e
nthalpy.
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
2-15
2.2 The air-standard Joule (B
rayton) cycle T
he clo
sed a
ir-standa
rd Joule/B
rayton cycle
is the
• is the sim
plest m
odel of the
o
pe
n circuit gas turbine
• is the ba
sic standa
rd aga
inst which w
e a
ssess pra
ctica
l applica
tions
• is a ve
ry good mode
l of the a
ctual e
ngine
Assum
ptions:
• All proce
sses a
re re
versible
• cp a
nd γ are
constant a
round the cycle
• No pre
ssure cha
nge (i.e
. no losses) in the
hea
t exc
hange
rs
• In the com
pressors a
nd turbines, e
verything ha
ppens
so quickly that the
re is no tim
e for a
ny hea
t tra
nsfer, i.e
., they a
re
ad
iab
atic
In this course, w
e w
ill assum
e a
ll of the a
bove e
xce
pt that w
e w
ill often a
llow irre
versibilitie
s to occur.
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
2-16
TurbineS
haft
CO
NT
RO
L SU
RFA
CE
2
1
3
4
inout
.. CW
.
W =
W -W
x
..
CT
.
Closed C
ircuit Gas T
urbine
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
2-17
You should be
able
to show tha
t the e
fficiency is g
iven by
t
cycler 1
1−
=η
whe
re r
t is the ise
ntropic tem
pera
ture ra
tio
γ
γ)1
(
4 3
1 2−
==
=p
tr
T T
T Tr
and r
p is the pre
ssure ra
tio
4 3
1 2
p p
p prp
==
For the
idea
l cycle
• ηcycle de
pends only on the
pressure
ratio rp .
(for a rea
l gas turbine
, it also de
pends on the
ratio 13
TT
).
• ηcycle incre
ase
s monotonica
lly with incre
asing
rp .
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
2-18
2.3 Irreversible Turbom
achines: Isentropic efficiencies W
e a
re use
d to dea
ling with the
se in the
context of e
ntire turbine
s (gas or ste
am
) or entire
compre
ssors.
How
eve
r,
• The
sam
e de
finitions can a
lso be a
pplied to individ
ual rotor+
stator com
binations
(i.e. sta
ges)
The
isen
trop
ic efficie
ncies a
re de
fined a
s the ra
tio of the
• the a
ctual w
ork and
• the ise
ntropic work
that occur be
twe
en
• the spe
cified inle
t conditions and
• the spe
cified e
xit pressure
The
refore
, espe
cially w
hen w
e a
re de
aling w
ith individua
l stage
s, we
must spe
cify if the p, T
and h th
at
we
are
using are
the
stag
na
tion (tota
l) or the sta
tic value
s.
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
2-19
2.3.1 T
otal-T
otal Ise
ntropic Efficie
ncies
Although you m
ay not ha
ve re
alise
d it, in Pa
rt I you ha
ve be
en using sta
gnation (tota
l) quantitie
s to de
fine the
isentropic e
fficiencie
s. The
se a
re use
d w
hen
• the kine
tic ene
rgy of the flow
is very sm
all or
• whe
re the
kinetic e
nergy of the
flow le
aving one
com
ponent (e
g stage
) is not wa
sted by a
dow
nstrea
m com
ponent
Com
pressor
01Entropy s
02
h (or T)
02s
wis
w
0102
01s
02tt
hh
hh
work
actual work
ideal− −
=≡
η
Gas or S
team T
urbine
03
Entropy s
04
h (or T)
04s
w
wis
s04
03
0403
tth
hh
h w
orkideal w
orkactual
− −=
≡η
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
2-20
2.3.2 T
otal-S
tatic Ise
ntropic Efficie
ncies
We
use the
se de
finitions whe
n
• the kine
tic ene
rgy of the flow
lea
ving one com
ponen
t (eg sta
ge) is w
aste
d by a dow
nstrea
m
compone
nt
This m
ost often ha
ppens w
hen w
e w
aste
the e
xit kinetic e
nergy of a
n entire
turboma
chine, e
.g.
• in the e
xhaust duct of a
stea
m turbine
• whe
n a fa
n or pump e
xhausts dire
ctly into the a
tmos
phere
The
total-sta
tic efficie
ncy is alw
ays le
ss than the tota
l-total e
fficiency. T
he diffe
rence
is due to the
so-ca
lled le
avin
g lo
ss (i.e. the
exit K
E)
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
2-21
Com
pressor
w
w
is, no exit KE
actual
Exit KE
01Entropy s
022
h (or T)
02s2s
0102
01s
2ts
hh
hh
work
actual work
ideal− −
=≡
η
Gas or S
team T
urbine
03
Entropy s
04
h (or T)
04s
w
w
is, no exit KE
actual
44s
P03
P04
P4
Exit KE
s4
03
0403
tsh
hh
h w
orkideal w
orkactual
− −=
≡η
Total-S
tatic Isentropic Efficiencies
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
2-22
2.3.3 E
xam
ple
Th
e in
let a
nd
exit co
nd
ition
s to a
turb
ine
are
:
inle
t: T
03 =
1000 K
P
03 =
2.0 bar
e
xit: T
04 =
874 K
P
04 =
1.2 bar
P
4 = 1.17 ba
r
Ca
lculate
both the tota
l-to-total a
nd total-to-sta
tic ise
ntropic efficie
ncies. A
ssume
the flow
is a p
erfe
ct ga
s with γ =
1.4.
total-to-tota
l: γ
γ)1
(
03
0403
04
−
=P P
TT
s =
4.
1)4
.0(
0.2
2.1
1000
=
864.2 K
928.0
2.864
1000874
1000 w
orkide
al w
orka
ctual
0403
0403
0403
0403
=− −
=− −
=− −
=≡
ss
ttT
T
TT
hh
hh
η
total-to-sta
tic: γ
γ)1
(
03 403
4
−
=P P
TT
s =
4.
1)4
.0(
0.2 17.1
1000
=
857.97 K
887.0
97.
8571000
8741000
work
idea
l work
actua
l
403
0403
403
0403
=− −
=− −
=− −
=≡
ss
tsT
T
TT
hh
hh
η
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
2-23
2.4 The R
ankine Steam
Cycle
This is the
basis of a
lmost e
very pra
ctical ste
am
cycle
for large
scale
powe
r gene
ration.
4
3
2
1
Qin from
combu
stion gas
WP
WT
feed
pum
p stea
m turbine
steam
genera
tor
condenser
Qo
ut to cooling
water
.
..
.
03
s
T
04s 04
0201
You should a
lrea
dy know tha
t per unit m
ass of ste
am
circulating, the
fee
d pump w
ork input is given by
combining the
SF
EE
with Td
s = d
h − d
p/ρ a
nd assum
ing that the
wa
ter is incom
pressible
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
2-24
ρ
ηρ
ηη
pu
mp
s
pu
mp
pu
mp
sp
um
pp
pd
ph
hh
hw
0102
0201
0102
0102
1)
(−
≅=
−=
−=
∫
whe
re η
pu
mp is the
total-tota
l isentropic e
fficiency of the
fee
d pump a
nd ρ is the de
nsity of wa
ter.
1
The
hea
t input in the boile
r and he
at re
jecte
d in the
condense
r are
given by
02
03h
hq
in−
= a
nd 01
04h
hq
ou
t−
=
The
turbine w
ork output is given by
)
(04
0304
03s
isen
trop
icT
hh
hh
w−
=−
=η
whe
re η
isen
trop
ic is the tota
l-total ise
ntropic efficie
ncy of the w
hole
turbine.
1 Th
e final p
art of th
is expressio
n is m
uch
mo
re accu
rate and
con
venien
t to u
se than
interp
olatin
g fo
r liq
uid
enth
alpies
in th
e steam tab
les.
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
2-25
2.5 Sum
mary
We
must a
lwa
ys specify if the
p, T a
nd h tha
t we
are
using are
the
• stagnatio
n (total) pre
ssure, te
mpe
rature
and spe
cific entha
lpy or
• static (ie
true the
rmodyna
mic) pre
ssure, te
mpe
rature
and s
pecific e
nthalpy.
For com
pressors:
• 01
02
0102
work
actua
l work
idea
lh
h
hh
stt
− −=
≡η
01
02
012
work
actua
l work
idea
lh
h
hh
sts
− −=
≡η
For a
gas or ste
am
turbine:
• s
tth
h
hh
0403
0403
work
idea
l work
actua
l− −
=≡
η
sts
hh
hh
403
0403
work
idea
l work
actua
l− −
=≡
η
For incom
pressible
pumps, if pum
pw
is the a
ctual spe
cific work input
• ρ
ηp
um
ptt
w
pp
0102
−=
ρ
ηp
um
pts
w
pp
012
−=
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
3-26
3 Flow
Velocities and V
elocity Triangles
3.1 Basic C
oordinate System
s and Velocities
Ea
rlier, w
e de
fined a
turboma
chine a
s a ste
ady flow
device
which cre
ate
s/consume
s shaft-w
ork by cha
nging the m
ome
nt of mom
entum
of a fluid pa
ssing through a
rotating se
t of blade
s.
The
refore
, we
must conside
r
• the m
ome
nt of mom
entum
• rotation a
bout an a
xis
As a
result,
• we
use a
n x-r-θ coordinate
system
x = a
xial dire
ction
r = ra
dial dire
ction
θ = ta
ngentia
l/circumfe
rentia
l direction
• we
nee
d to work in the
sta
tionary (a
bso
lute) a
nd rota
ting (re
lative) fra
me
s of refe
rence
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
3-27
In the sta
tionary fra
me
, we
have
Vx =
axia
l velocity
Vr =
radia
l velocity
Vθ =
tange
ntial/circum
fere
ntial/sw
irl velocity
Vx
Vr
Vθ
V
xr
θ
Ω
We
note tha
t:
The
sign convention use
d throughout this course (a
nd by much of indust
ry) is that
tange
ntial/circum
fere
ntial/sw
irl velocitie
s are
positive
if they a
re in the
sam
e
direction a
s the rota
tion of the rotor.
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
3-28
The
ana
lysis of the flow
through rotating bla
de rows (rotors) ca
n be gre
atly sim
plified by w
orking in a
fra
me
of refe
rence
so that the
rotors appe
ar to be
a
t rest.
Vr
Vθ,rel
rΩ Ω
r
Vθ
Axial view
of the components of the absolute and rotor relative velocity vectors
We
first note tha
t in the both fra
me
s of refe
rence
, w
e ha
ve
V
x = a
xial ve
locity
V
r = ra
dial ve
locity
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
3-29
and tha
t the tw
o (stationa
ry/absolute
and rota
tional/rotor re
lative
) fram
es of re
fere
nce a
re re
late
d a
ccording to the ve
ctor expre
ssion
a
bsolute ve
locity = re
lative
velocity +
rotationa
l ve
locity
Since
Vx a
nd Vr a
re the
sam
e in both fra
me
s of refe
rence
, the only diffe
rence
betw
ee
n the a
bsolute a
nd re
lative
velocitie
s is due to the
ma
gnitude of the
circum
fere
ntial ve
locity.
In fact,
r
VV
rel,
Ω+
=θ
θ
whe
re
rel,
Vθ
= rotor re
lative ta
ngentia
l/circumfe
rentia
l/swirl ve
locity
and
=
=Ω
Ur
blade
spee
d
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
3-30
For a
xial m
achine
s:
V
x >>
Vr
For ra
dial m
achine
s, at the
outer ra
dius
V
x <<
Vr
and a
t the inne
r radius, de
pending on w
hethe
r the f
low is m
ainly a
xial or ra
dial,
V
x >>
Vr
or
V
x <<
Vr
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
3-31
3.2 Mean-line A
nalyses D
esign m
ethods for turbine
s, compre
ssors and pum
ps usua
lly involve a
numbe
r of sepa
rate
processe
s.
The
first step is to use
• 1-D ca
lculations a
long me
an ra
dius, i.e. m
ea
n-line
ana
lyses
to exa
mine
• me
an ra
dius velocity tria
ngles
before
and a
fter e
ach bla
derow
In doing so, we
assum
e tha
t
• the spa
n (hub-tip length) of the
blade
s is sma
ll in re
lation to the
me
an ra
dius so that
• the va
riation of the
flow in the
hub-tip direction
can be
negle
cted
• the m
ea
n radius =
r
mea
n = (r
casin
g + r
hu
b ) / 2
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
3-32
Axial F
low P
ump
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
3-33
3.3 Velocity T
riangles for an Axial T
urbine Stage (S
tator+R
otor) F
or simplicity, w
e w
ill assum
e tha
t
• the va
riation of the
flow in the
radia
l direction i
s sma
ll
• the ra
dial com
ponent of ve
locity is negligible
(Vr =
0)
• there
is no change
of radius (r) through the
stage
• the bla
de spe
ed (
Ur=
Ω) is consta
nt
• the va
riation of the
flow in the
circumfe
rentia
l dire
ction is sma
ll
• we
can e
xam
ine the
flow by looking a
n unwra
pped (ie de
velopm
ent of) cylindrica
l surface
of re
volution, i.e. by using the
ca
scade (x–y or x-rθ) pla
ne
We
reca
ll that
• flow a
ngles a
re positive
if they a
re in the
sam
e di
rection a
s the rota
tion of the rotor.
Now
,
• turbines use
stators to cre
ate
a m
ome
nt of mom
entum
which is the
n rem
oved in the
rotor.
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
3-34
V2
,rel
α2,re
l
α2
V2
U
U
STA
TO
RR
OT
OR
rθ
xV
θ2,rel
Vθ1
Vθ2
Vx2
α2
V2
V2
,rel
V1
α2
,rel
α1
Vx2
Vx1
Axial T
urbine Stator E
xit/Rotor Inlet V
elocity Triangle V
iewe
d Radially
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
3-35
At the
inlet of our sta
tor
axia
l velocity
11
1cos α V
Vx
=
tange
ntial ve
locity 1
11
sin α V V θ
=
At the
exit/outle
t of our stator
axia
l velocity
22
2cos α V
Vx
=
tange
ntial ve
locity 2
22
sin α V V θ
=
Fina
lly, we
note tha
t (see
late
r):
• the a
bsolute e
xit flow a
ngle
α of a
stator &
the re
lative
exit flow
angle
rel
α of a
rotor tend to be
inde
pende
nt of the ope
rating condition e
ven w
hen th
e inle
t flow a
ngle to the
sam
e bla
derow
or the
velocitie
s change
.
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
3-36
Now
, we
aga
in note tha
t the a
nalysis of the
flow th
rough rotating bla
de row
s (rotors) can be
grea
tly sim
plified by w
orking in a fra
me
of refe
rence
so tha
t the rotors a
ppea
r to be a
t rest.
We
reca
ll that the
axia
l velocity
2x
V is the
sam
e in both fra
me
s of refe
rence
and tha
t
U
Vr
VV
rel
rel
+=
Ω+
=,2
,22
θθ
θ
whe
re
Ur
=Ω
⇒
UV
Vre
l−
=2
,2θ
θ
The
refore
, the rotor re
lative
inlet flow
angle
is give
n by
2
2
2
2 ,22
,2ta
nta
nx
xx
rel
V U
V
UV
V
Vre
l−
=−
==
αα
θθ
We
now look a
t the rotor e
xit.
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
3-37
V3
,rel
V3
V2
,rel
α2,re
l
α3,re
l
α2
α3
V2
STA
TO
RR
OT
OR
rθ
x
Vθ1
V1
α1V
x1U
3
U2
Blade Speed
Velocity T
riangles for an Axial T
urbine Stage V
iewed R
adially
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
3-38
At rotor e
xit, we
note tha
t
U
VV
rel +
=,3
3θ
θ
and tha
t
3
3
,3
3 3,3
3ta
nta
nx
x rel
xV U
V
UV
V Vre
l +=
+=
=α
αθ
θ
If we
study the ve
locity triangle
s of the turbine
as w
e ha
ve dra
wn the
m, w
e should notice
that
• 1
2α
α>>
and
rel
rel
,2,3
αα
>>
- turbine bla
des m
ake
the flow
more
tange
ntial
• 3
21
xx
xV
VV
≈≈
- this is ve
ry comm
on
• 1
2V
V>>
and
rel
rel
VV
,2,3
>>
- turbine bla
des a
ccele
rate
the flow
- boundary la
yers thin a
nd losses in e
fficiency are
sma
ll
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
3-39
3.3.1 E
xam
ple
The
flow
leavin
g a
n a
xial tu
rbin
e sta
tor b
lade
row
has a
velo
city 700 m
s -1 at a
n a
ngle
of 7
0°. T
he
ro
tor h
as a
bla
de
spe
ed o
f 500 m
s-1. T
he
flow
leavin
g a
roto
r bla
de
row
also
has a
rel
ative
velo
city of
700 m
s -1 at a
rela
tive a
ngle
of -7
0°. N
egle
ct any ra
dia
l ve
locitie
s and a
ssum
e th
at th
e a
xial ve
locity is
consta
nt th
rough th
e sta
ge
Calcu
late
the
rela
tive flo
w a
ngle
at ro
tor in
let a
nd th
e a
bso
lute
flow
angle
at ro
tor e
xit.
The
stator e
xit/rotor inlet a
xial ve
locity is
2
22
cos α V V
x=
= 700 × cos70° =
239.4 ms -1
The
stator e
xit/rotor inlet a
bsolute ta
ngentia
l vel
ocity is
2
22
sin α V V θ
==
700 × sin70° = 657.8 m
s -1
The
stator e
xit/rotor inlet re
lative
tange
ntial ve
locity is
U
V
Vθ,rel θ
−=
22
= 657.8 – 500 =
157.8 ms
-1
The
stator e
xit/rotor inlet re
lative
flow a
ngle is
=
−
2 ,21
,2ta
nx
rel
rel
V
Vθα
=
°=
−
4.33
4.239
8.157
tan
1
(sign indica
tes sa
me
direction a
s blade
spee
d)
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
3-40
The
rotor exit a
xial ve
locity is
VV
xx
23
==
239.4 ms -1
The
rotor exit re
lative
tange
ntial ve
locity is
re
lx
rel
rel
rel
θαV
α V
V,3
3,3
,3,3
tan
sin=
==
239.4 × tan(-70°) =
-657.8 ms -1
(sign indica
tes opposite
direction a
s blade
spee
d)
The
rotor exit a
bsolute ta
ngentia
l velocity is
U
V
V
rel
θθ+
=,3
3=
-657.8 + 500 =
-157.8ms -1
(sign indica
tes opposite
direction a
s blade
spee
d)
The
rotor exit a
bsolute flow
angle
is
°
−=
−=
=
−−
4.33
4.239
8.157
tan
tan
1
3 31
3x θV V
α
(sign indica
tes opposite
direction a
s blade
spee
d)
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
3-41
3.4 Velocity T
riangles for an Axial C
ompressor S
tage (Rotor+
Stator)
For sim
plicity, we
will a
gain a
ssume
that
• the ra
dial com
ponent of ve
locity is negligible
(Vr =
0)
• the va
riation of the
flow in the
radia
l direction i
s sma
ll
• there
is no change
of radius (r) through the
stage
• the bla
de spe
ed (
Ur=
Ω) is consta
nt
• the va
riation of the
flow in the
circumfe
rentia
l dire
ction is sma
ll
• we
can e
xam
ine the
flow by looking a
n unwra
pped (ie de
velopm
ent of) cylindrica
l surface
of re
volution, i.e. by using the
casca
de (x–y or x-r
θ) plane
Now
:
• compre
ssors use rotors to cre
ate
a m
ome
nt of mom
ent
um w
hich is then re
move
d in the sta
tor to cre
ate
a furthe
r pressure
rise.
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
3-42
V1
,rel
V2
,rel
V1
V2
UU
RO
TO
RS
TAT
OR
UBlade Speed
θ
xα
2,re
l
α2
Vθ3
V3
α3V
x3
α1
α1,rel
Velocity T
riangles for an Axial C
ompressor S
tage View
ed Radially
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
3-43
If we
study the ve
locity triangle
s of the com
presso
r as w
e ha
ve dra
wn the
m, w
e notice
that
• 3
2α
α>>
and
rel
rel
,2,1
αα
>>
- compre
ssor blade
s ma
ke the
flow m
ore a
xial
• 3
21
xx
xV
VV
≈≈
- this is ve
ry comm
on
• 3
2V
V>
and
rel
rel
VV
,2,1
>
- compre
ssor blade
s dece
lera
te the
flow (by a
bout 30%
)
- static pre
ssure rise
s
- boundary la
yers thicke
n & se
para
tion is a big ri
sk
- losses in e
fficiency a
re highe
r than in turbine
s
- more
stage
s for sam
e pre
ssure cha
nge cf. turbine
s.
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
3-44
3.5 Sum
mary
We
use tw
o fram
es of re
fere
nce tha
t are
rela
ted a
ccording to the
vector e
xpression
a
bsolute ve
locity = re
lative
velocity +
rotationa
l ve
locity
⇒
UV
Vre
l +=
,22
θθ
whe
re
Ur
=Ω
In axia
l flow turbine
s (stator +
rotor):
• blade
s ma
ke the
flow m
ore ta
ngentia
l
• often
32
1x
xx
VV
V≈
≈
• flow a
ccele
rate
s (thin boundary la
yers) so good e
fficie
ncy.
In axia
l flow com
pressors (rotor +
stator):
• blade
s ma
ke the
flow m
ore a
xial
• often
32
1x
xx
VV
V≈
≈
• flow de
cele
rate
s (boundary la
yers thicke
n) so lowe
r e
fficiency
• more
stage
s nee
ded for sa
me
pressure
change
cf. turbine
s.
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
4-45
4 Mass F
low R
ates/Forces/W
ork/SF
EE
4.1 T
he calculation of mass flow
rate in axial turbomachines
The
ability to a
pply the la
w of conse
rvation of m
as
s to a turbom
achine
blade
row is funda
me
ntal to
ma
ny turboma
chine ca
lculations.
Co
ntrol
Volum
e
rel
1
scosα1rel
scosα2rel
s
rel2
Inlet and exit flow areas of an axial com
pressor rotor in x-rθθθ θ plane
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
4-46
First, w
e e
xam
ine the
flow a
t inlet to a
nd exit fro
m a
2D com
pressor rotor of bla
de spa
n h a
nd blade
pitch s in the
rela
tive fra
me
. We
assum
e tha
t
• the bla
de spa
n h
is sma
ll in rela
tion to the m
ea
n radius
• the ge
ome
try and flow
conditions (velocitie
s and a
ngle
s) are
constant a
cross the spa
n.
Conse
rvation of m
ass give
s for one bla
de pa
ssage
(
)(
),re
l,re
ls
hV
= ρs
hV
ρAV
ρm,re
l,re
l,re
l,re
lp
assa
ge
12
coscos
11
22
22
2α
α=
=&
or, more
gene
rally, if Ax =
hs w
hich is the cro
ss-sectio
na
l or fron
tal a
rea
of the pa
ssage
, then
(
)(
)consta
ntcos
cos=
==
αρ
αρ
xx
rel
pa
ssag
eA
VA
Vm
rel
&
whe
re
• the flow
are
a (
)α
cosx
A is a
lwa
ys me
asure
d perpe
ndicular to the
velocity ve
ctor
• failure
to observe
this importa
nt simple
rule ha
s se
rious conseque
nces w
hen de
aling w
ith com
pressible
flow (you ha
ve be
en w
arne
d!)
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
4-47
It also ha
ppens tha
t the a
bove ca
n be w
ritten a
s (he
nce the
previous w
arning)
(
)(
)x
xx
xre
lp
assa
ge
AV
AV
AV
mre
lρ
αρ
αρ
==
=co
sco
s&
Now
, if there are Z blades, then the total mass flow
rate through the com
pressor rotor is
(
)(
))
Z(cos
cosx
xx
xre
lp
assa
ge
com
pre
ssor
AV
AZ
VA
ZV
mZ
mre
lρ
αρ
αρ
==
==
&&
h
Rhub
Rcasing
RS
RS
Axial (r-θ)
θ) θ) θ) and M
eridional (x-r) views of a 1-stage com
pressor
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
4-48
For the com
pressor, the mean radius defined as
2
sinh
ub
gca
mea
nr
rr
+=
Now
m
ean
rZ
sπ2
=
so the area of the annulus is
(
)()
2hub2casing
hubcasing
hubcasing
2r
rr
rr
rh
r
Zsh
Am
ea
nx
ππ
ππ
−=
+−
==
=
Therefore, w
hether we exam
ine a complete bladerow
or just one blade passage:
xx
rel
xre
lx
AV
A
V
VA
AV
mρ
αρ
αρ
ρ=
==
=co
sco
s&
= const
where A
x is the annulus area or the passage area as appropriate.
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
4-49
This is w
hy in compressors:
• flow is turned to be m
ore axial
• Inlet flow area >
Exit flow
Area
• Flow
decelerates
• Static pressure rises in each bladerow
And in turbines:
• Flow
is turned to be more tangential
• Inlet flow area <
Exit flow
Area 2
• Flow
accelerates
• Static pressure falls in each bladerow
2 Th
is is gen
erally true - excep
t for th
e true im
pu
lse ro
tor w
here th
ere is no
chan
ge in
pressu
re and
co
nseq
uen
tly no
ch
ang
e in relative velo
city across th
e roto
r (see sectio
n 5
.3.2
)
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
4-50
4.1.1 E
xample
Th
e a
xial tu
rbin
e in
Exa
mp
le 3
.3.1
ha
s a co
nsta
nt m
ea
n ra
diu
s of 0
.5 m
an
d th
e b
lad
e sp
an
is con
stan
t a
nd
eq
ua
l to 0
.07
5 m
. Th
e in
let sta
gn
atio
n te
mp
era
tu
re to
the
stag
e is 1
80
0 K
an
d th
e in
let sta
gn
atio
n
pre
ssure
is 30
ba
r. Th
e flo
w is ise
ntro
pic. A
ssum
e
tha
t the
ga
s ha
s the
pro
pe
rties o
f air. N
eg
lect a
ny
ra
dia
l velo
cities.
Ca
lcula
te th
e m
ass flo
w ra
te o
f the
turb
ine
.
We already know
V2
= 700 m
s -1
Vx2
= 239.4 m
s -1
There is no w
ork done in the stator therefore, the S
FE
E
(
)() 21
2 11
222 1
2V
hV
hw
qx
+−
+=
−
can be written
01
02h
h=
where
22 1
0V
hh
+≡
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
4-51
Therefore
⇒
22 1
202
01V
Tc
Tc
Tc
pp
p+
==
The static tem
perature at stator exit is therefore
2.
15561005
7005.
01800
2
2=
×−
=T
K
The density can be obtained from
(
)(
)(
)029.4
1800 2.1556
18005.
287000,
000,3
14.
11
11
02 2
02
021
1
02 202
2=
×=
=
=
−−
−γ
γρ
ρT T
RT P
T Tkg/m
3
The turbine m
ass flow rate is therefore
(
)2
22
22
22
22
cosx
xx
AV
AV
AV
mre
lρ
αρ
ρ=
==
&
⇒
()
3.227
075.0
5.0
14159.3
24.
239029.4
22
2=
××
××
×=
=h
rV
ρmm
ean
xπ
& kg/s
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
4-52
4.2 Axial and T
angential Forces on a 2D
Blade
We exam
ine the flow at inlet to and exit from
a 2D
compressor blade of span
h and pitch s.
Note that
• The blades and control volum
e have a span h
• The upper and low
er boundaries are streamlines (thi
s is for convenience) and they are exactly one pitch s in the circum
ferential direction
o N
o mass crosses the upper and low
er boundaries
o N
o net pressure forces are exerted on the two bound
aries
• The forces show
n are those on the flow
• The force on each blade is equal and opposite to th
at on the flow in one passage
• The forces are (m
ainly) created by the pressure differences betw
een the suction and pressure surfaces of the blades
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
4-53
Control
Volum
eSuc tion S
urface
s
V1
V2
α2
α1
scosα1
scosα2P
ressure Surface
Fx
Fθ
Forces on an A
xial Com
pressor Stator
We start by recalling that
F
orce on flow =
rate of change of mom
entum of flow
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
4-54
Hence, the A
xial Mom
entum E
quation is
(
)12
21
xx
pa
ssag
ex
VV
mh
sp
hs
pF
−=
−+
&
⇒
()
()1
21
2x
xp
assa
ge
xx
VV
mA
pp
F−
+−
=&
where
hs
Ax
= and
xx
xx
pa
ssag
eA
VA
Vm
22
11
ρρ
==
&
We note that w
hen the axial velocity remains consta
nt through a bladerow (often true)
• axial force (thrust bearings) mainly a result of th
e inlet to exit pressure difference.
The T
angential Mom
entum E
quation is
(
)12
θθ
θV
Vm
Fp
assa
ge
−=
&
If the mean radius changes, w
e use the analogous Mo
ment of T
angential Mom
entum E
quation
T
orque on fluid = rate of change of m
oment of m
ome
ntum
⇒
()1
12
2θ
θVr
Vr
mT
To
rqu
ep
assa
ge
−=
=&
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
4-55
4.3 Euler's W
ork Equation
By com
bining the mom
ent of mom
entum equation (radiu
s×tangential mom
entum equation) w
ith the S
FE
E it is possible to derive E
uler's Work E
quation even for the case w
here there is a change of radius.
This is the m
ost important equation in the analysis
of turbomachinery.
r1
r2
Rotor
Control V
olume
τΩ
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
4-56
We consider a control volum
e which is form
ed by a narrow
streamtube and w
hich contains a row of
rotor blades that
• has a mass flow
rate m &
• has angular velocity Ω
• produces a torque τ
• has a flow w
hich enters at a mean radius
r1
• has a flow w
hich leaves at a mean radius
r2
We first observe that
T
orque exerted by flow on blade row
= shaft output
torque = τ
Therefore:
(
) fluid
of
mom
entum
of
mom
ent
of
change
of
rate
−=
τ
⇒
()1
12
2θ
θτ
VrV
rm
−−
=&
⇒
()Ω
−−
=Ω
=1
12
2θ
θτ
VrV
rm
Wx
&&
[eqn 1]
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
4-57
The S
FE
E is:
(
)(
)2,1
2 11
2,22 1
2re
lre
lx
Vh
mV
hm
WQ
+−
+=
−&
&&
&
For adiabatic flow
(and using stagnation enthalpy), S
FE
E becom
es
(
)01
02h
hm
Wx
−=
−&
&
[eqn 2]
Com
bining these two expressions for the shaft-pow
er gives:
(
)(
)Ω−
−−
11
22
0102
θθx
VrV
rm
=h
hm
=W
&&
&
Now
rΩ =
U =
mean radius blade speed. T
hus Euler's W
ork Equat
ion is:
()
()1
12
201
02θθ
xV
UV
Um
=h
hm
=W
−−
−&
&&
11
22
0102
θθx
VU
V=
Uh
=h
w−
−−
Which m
eans that
To transfer w
ork either from or to a turbom
achine, a change in the m
oment of
mom
entum of the flow
must occur through a rotating
bladerow
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
4-58
So • turbines use stators to create a m
oment of m
omentum w
hich is then removed in the rotor
• compressors use rotors to create a m
oment of m
oment
um w
hich is then removed in the stator to
create a further pressure rise
and that Euler's w
ork equation is valid for:
(1) steady flow
(or time average of a periodic
flow)
(2) adiabatic flow
(m
ust modify for turbine bl
ade cooling)
(3) com
pressible flow
(any Mach num
ber)
(4) changing stream
line radius ( r1 ≠ r
2 ) (radial or axial m
achines)
(5) viscous flow
in the rotor
(no viscous effect
s on stationary walls)
(6) stators
(no work because
02
01
0
hh
U=
⇒=
)
Note that the S
FE
E can also be w
ritten as:
Roth
alp
y =
=− θUVh0
constant along a streamline
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
4-59
4.3.1 E
xample
Calcu
late
the
rise in
stagnatio
n te
mpe
ratu
re a
cross
a ro
w o
f axia
l com
pre
ssor ro
tor b
lade
s give
n th
at
the
inle
t tange
ntia
l velo
city is 75 m
s-1, th
e e
xit tange
ntia
l velo
city is 175 m
s-1 a
nd th
e m
ean b
lade
spe
ed
is 250 m
s -1 at b
oth
inle
t and e
xit.
Also
de
term
ine
th
e
exit
stagnatio
n
pre
ssure
if
the
in
let
stagnatio
n
conditio
ns
are
1
bar
and
300 K
and th
e ro
tor is ise
ntro
pic.
Euler’s w
ork equation gives
1
12
201
02θθ
xV
UV
=U
h=
h-w
−−
= 250 × ( 175 – 75 ) =
25 kJ/kg
Now
, for air (a perfect gas)
⇒
()
()
==
−=
−100525000
0102
0102
pc
hh
TT
24.9 K
Finally, because the rotor is isentropic
(
)132190
300 9.324
105.
31
01
0201
025
=
=
=−
γγ
T TP
P P
a
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
4-60
4.3.2 E
xample
Usin
g E
ule
r’s work e
quatio
n, ca
lcula
te th
e w
ork d
on
e p
er kg
of m
ass flo
w a
nd th
e to
tal p
ow
er o
utp
ut o
f
the
axia
l flow
turb
ine
of o
ur p
revio
us e
xam
ple
(see
3.3
.1 a
nd 4
.1.1
)
We already know
that the stator exit/rotor inlet absolute tangential velocity is
2
θV=
657.8 ms -1
the rotor exit absolute tangential velocity is
3
θV =
-157.8 ms -1
and the mass flow
rate is
3.
227=
m & kgs
-1
So, using E
uler’s work equation
(
)(
)=+
×=
==
−−
8.157
8.657
5003
23
32
2θθ
θθx
VV
UV
UV
Uw
407,800 J/kg
and the power output is
=
×=
=800,
4073.
227x
xw
mW
&&
92.7 MW
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
4-61
V3
,rel
V3
V2
,rel
α2,re
l
α3,re
l
α2
α3
V2
STA
TO
RR
OT
OR
rθ
x
Vθ1
V1
α1
Vx1
U3
U2
Blade Speed
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
4-62
4.4 SF
EE
in a rotating frame
It is often easier to analyse the performance of ro
tor blade rows by w
orking in the relative frame.
• a rotor then appears at rest and "looks" very simil
ar to a stator row
Also, w
e know from
Part I therm
odynamics:
• the values of the true thermodynam
ic properties such as pressure, tem
perature and enthalpy are the sam
e in both the absolute and relative frames.
We now
define
• Absolute stagnation enthalpy
22 1
0V
hh
+=
• Relative stagnation enthalpy
22 1
,0re
lre
lV
hh
+=
This m
eans that
• the stagnation conditions are different in the absolute and relative fram
es.
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
4-63
This is alright because w
e already know from
Part I therm
odynamics
• the amount of w
ork (changes in stagnation enthalpy) perceived depends on the fram
e of reference of the observer.
Recall that E
uler's Work equation
1
12
201
02θθVUV
U=
hh
−−
can be written in term
s of the rothalpy:
R
oth
alp
y = θUVh−0
= constant along a stream
line
Therefore:
θ
θU
VV
hU
Vh
−−
+=
22 1
0 =
const
⇒
θr
xθ
UV
VV
Vh
UV
h−
−
+
++
=2
22
2 10
θ =
const
⇒
()
−
−+
++
=−
22
22
2 10
UU
VV
Vh
UV
hr
xθ
θ =
const
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
4-64
⇒
−+
++
=−
22,
22
2 10
UV
VV
hU
Vh
rel
rx
θθ
= const
⇒
22 1
22 1
0U
Vh
UV
hre
lθ
−−
+=
= const
co
nst
Uh
UV
hre
lθ
=−
=−
22 1
,00
So, the S
FE
E in stationary and rotating fram
es of reference for stators and rotors becom
es
2
2 1,0
0U
hU
Vh
Roth
alp
yre
lθ
−=
−=
= constant along stream
line
which for a perfect gas, can also be w
ritten as
co
nst
UT
cU
VT
cre
lp
θp=
−=
−2
2 1,0
0
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
4-65
We note that for stators
• U=
0
• co
nst
UV
h θ=
−0
⇒
const
h=
0 ⇒
no work
And for rotors,
• co
nst
Uh
rel
=−
22 1
,0
• If co
nst
r=
⇒
const
U=
⇒
con
sth
rel =
,0 (often true for axial m
achines).
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
4-66
4.4.1 E
xample
As in
our p
revio
us e
xam
ple
(see
3.3
.1 a
nd 4
.3.2
), the
flow
leavin
g a
turb
ine
stato
r bla
de
row
has a
ve
locity o
f 700 m
s -1 at a
n a
ngle
of 7
0°. T
he
roto
r has a
bla
de
spe
ed
of 5
00 m
s -1. The
flow
leavin
g th
e
roto
r bla
de
row
has a
rela
tive ve
locity 7
00 m
s-1 a
t a re
lative
angle
of -7
0°. N
egle
ct any ra
dia
l ve
locitie
s and a
ssum
e th
at th
e a
xial ve
locity is co
nsta
nt th
rough th
e sta
ge
The
inle
t stagnatio
n te
mpe
ratu
re to
the
stage
is 18
00 K
. Assu
me
that th
e g
as h
as th
e p
rope
rties o
f air
. C
alcu
late
the
roto
r rela
tive in
let a
nd e
xit stagnat
ion e
nth
alp
ies. A
lso ca
lcula
te th
e ro
tor a
bso
lute
exit
stagnatio
n e
nth
alp
y. Use
this va
lue
to d
ete
rmin
e th
e w
ork d
one
pe
r kg o
f mass flo
w.
We already know
that the stator exit/rotor inlet absolute tangential velocity is
2
θV=
657.8 ms -1
and the rotor exit absolute tangential velocity is
3
θV =
-157.8ms -1
and Euler’s w
ork equation gave
(
)(
)=+
×=
==
−=
−−
8.157
8.657
5003
23
32
201
02θθ
θθx
VV
UV
UV
Uh
hw
407,800 J/kg
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
4-67
V3
,rel
V3
V2
,rel
α2,re
l
α3,re
l
α2
α3
V2
STA
TO
RR
OT
OR
rθ
x
Vθ1
V1
α1
Vx1
U3
U2
Blade Speed
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
4-68
The rotor inlet absolute stagnation enthalpy is
0
0T
ch
p=
=1005 × 1800 =
1,809,000 J/kg
Now
,
co
nst
Uh
UV
hre
lθ
=−
=−
22 1
,00
So, the rotor inlet relative stagnation enthalpy is
2
2 12
02,
02U
UV
hh
θre
l+
=−
=
100,
605,1
5000.5
657.8500
-1,809,000
2=
×+
×J/kg
and the rotor exit relative stagnation enthalpy is
100,
605,1
,02
,03
==
rel
rel
hh
So, the rotor exit absolute stagnation enthalpy is
2
2 13
,03
03U
UV
hh
θrel
−+
= =
200,
401,1
5000.5
157.8500
1,605,1002
=×
−×
−J/kg
and the work done
800,
407200,
401,1
000,
809,1
0302
=−
=−
=h
hw
x J/kg
as before.
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
4-69
4.5 Sum
mary
The T
angential Mom
entum E
quation is
(
)(
)12
12
xx
pa
ssag
ex
xV
Vm
Ap
pF
−+
−=
&
The T
angential Mom
entum E
quation is
(
)12
θθ
θV
Vm
Fp
assa
ge
−=
&
The M
oment of T
angential Mom
entum E
quation
(
)11
22
θθ
VrV
rm
TT
orq
ue
pa
ssag
e−
==
&
Euler's W
ork Equation is:
(
)(
)11
22
0102
θθx
VU
VU
m=
hh
m=
W−
−−
&&
&
or 1
12
201
02θθ
xV
UV
=U
h=
hw
−−
−
Rothalpy is defined as
2
2 1,0
0U
hU
Vh
Roth
alp
yre
lθ
−=
−=
= constant along a stream
line
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
5-70
5 Turbom
achinery Design P
arameters
5.1 Flow
Coefficient
Defined as:
U V
x=
φ
it describes the “squareness” of the velocity triangles.
Axial turbines w
ith φφφ φ = 0.37 (solid line)
and φφφ φ = 0.53(dashed line)
where ΛΛΛ Λ
=0.5 &
ΨΨΨ Ψ=
1.0:
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
5-71
5.2 Stage Loading C
oefficient D
efined as:
(
)2
2 0
U UV
U
hθ
ψ∆
=∆
≡
It affects the “skew” of the velocity
triangles
V3,rel
V3
V2,rel
V2
U
U
STATOR
ROTO
R
Tw
o axial turbines with Λ
Λ
Λ
Λ
= 0.5 and φφφ φ =
0.5: S
olid line ΨΨΨ Ψ =
1.7; dashed line ΨΨΨ Ψ = 1.0
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
5-72
5.3 Reaction
Defined as
sta
ge
roto
r
h h
∆ ∆=
Λ
it affects the asymm
etry of velocity triangles and blade shapes
Most axial m
achines have relatively high efficiencies (typically, η >
90%)
so that
• ρ
ρd
pd
hd
pd
hT
ds
≈⇒
−=
and we see that reaction also
describes
changes in pressure across rotor com
pared to across the stage
V3
V2,rel
V2
U
V3,rel
U
V2,rel
Tw
o axial turbines with and φφφ φ =
0.37: S
olid line ΛΛΛ Λ =
0.5, Ψ
Ψ
Ψ
Ψ =
1.0; dashed line ΛΛΛ Λ
= 0.25, ΨΨΨ Ψ
= 1.5
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
5-73
5.3.1 E
xample: T
he 50% R
eaction Turbine
Eva
lua
te th
e d
eg
ree
of re
actio
n fo
r a tu
rbin
e th
at
ha
s symm
etric ve
locity tria
ng
les a
nd
a co
nsta
nt
rad
ius.
V3
,rel
V3
V2
,rel
α2
,rel
α3,re
l
α2
α3
V2
STA
TO
RR
OT
OR
rθ
x
Vθ1
V1
α1
Vx1
U3
U2
Blade Speed
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
5-74
Now
reaction is
(
)(
)(
)32
21
32
hh
hh
hh
hh
h
h h
roto
rsta
tor ro
tor
stag
e
roto
r
−+
−−
=∆
+∆
∆=
∆ ∆=
Λ
For the stator
(
)0
22 1
0=
+∆
=∆
Vh
hsta
tor
⇒
212 1
122
2 12
Vh
Vh
+=
+
⇒
212 1
222 1
21
VV
hh
−=
−
For the rotor (U
=const):
co
nst
Uh
rel
=−
22 1
,0
⇒
rel
rel
hh
,02
,03
=
⇒
2,22 1
22,3
2 13
rel
rel
Vh
Vh
+=
+
⇒
2,22 1
2,32 1
32
rel
rel
VV
hh
−=
−
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
5-75
Hence
(
)(
)()
2,22 1
2,32 1
222 1
212 1
2,22 1
2,32 1
rel
rel re
lre
l
VV
VV
VV
−+
−
−=
Λ
But, by sym
metry
a
nd
13
23
2V
V V
VV
,rel
,rel
==
=
⇒
()
()(
)5.
021
2221
22
2122
=−
+−
−=
ΛV
VV
V
VV
i.e. the turbine has 50% reaction
In fact:
All turbines and com
pressors with sym
metric velocit
y triangles have 50% reaction
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
5-76
5.3.2 E
xample: T
he impulse turbine
An
axia
l turb
ine
stag
e h
as a
roto
r in w
hich
the
inl
et a
nd
exit ve
locitie
s are
ide
ntica
l. Th
is is kno
wn
as
an
“imp
ulse” sta
ge
. Fin
d th
e d
eg
ree
of re
actio
n o
f such
a sta
ge
. Fo
r simp
licity, you
ma
y assu
me
tha
t th
e flo
w is a
xial a
t inle
t to a
nd
exit fro
m th
e stag
e.
Consider the follow
ing velocity diagram. W
e note that
re
lre
lV
V,2
,3=
We w
ill assume that there is no change of radius. S
o, from the rothalpy equation
co
nst
Uh
rel
=−
22 1
,0
⇒
rel
rel
hh
,02
,03
=
⇒
2,22 1
22,3
2 13
rel
rel
Vh
Vh
+=
+
⇒
02,2
2 12,3
2 13
2=
−=
−re
lre
lV
Vh
h
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
5-77
So
0)
(
)(
31
32
=− −
=∆ ∆
=Λ
hh
hh
h h
stag
e
roto
r
We also note that for an isentropic
rotor,
ρρ
dp
dh
dp
dh
Td
s≈
⇒−
=
So, m
any people refer (incorrectly) to an im
pulse rotor as one where there is
no static pressure change.
V3
,rel
V3
V2
,rel
V2
U
U
Velocity triangles for an im
pulse turbine with no
inlet or exit swirl
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
5-78
5.3.3 E
xample
Eva
lua
te th
e re
actio
n, th
e flo
w co
efficie
nt a
nd
the
stag
e lo
ad
ing
coe
fficien
t of th
e tu
rbin
e in
exa
mp
le
s 3
.3.1
, 4.1
.1,4
.3.2
an
d 4
.4.1
. No
te th
at th
e a
bso
lut
e flo
w a
ng
le a
t inle
t to th
e sta
tor is th
e sa
me
as
the
a
bso
lute
flow
an
gle
at ro
tor e
xit.
Since the turbine has sym
metric velocity triangles,
Λ
= 50%
Also,
479.0
500 4.239
==
=U V
xφ
and
63.1
500
407,800
22
2 0=
==
∆≡
U w
U
hx
ψ
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
5-79
5.4 Sum
mary
Sta
ge de
sign is about se
lecting
ψ, φ a
nd Λ a
t the de
sign point.
The
flow coe
fficient
U Vx
=φ
describe
s the “squa
rene
ss” of the ve
locity triangle
s
The
stage
loading coe
fficient
2 0
U
h∆
=ψ
define
s "skew
ness" of the
velocity tria
ngles
The
rea
ction sta
ge
roto
r
h h
∆ ∆=
Λde
scribes
• asym
me
try of velocity tria
ngles/bla
de sha
pes
• approx. cha
nges in pre
ssure a
cross rotor compa
red to the
sta
ge
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
6-80
6 The S
caling of Incompressible T
urbomachines
6.1 Introduction W
e sha
ll
• form non-dim
ensiona
l groups and
• invoke the
principles of ge
ome
tric and dyna
mic sim
ilari
ty
to: • repre
sent the
perform
ance
of turboma
chines in a
wa
y which
is convenie
nt and ra
tional
• describe
the ope
rating point of a
compre
ssor & turbine
• perform
scaling ca
lculations for a
llow for cha
nges in cond
itions or size
We
will de
al w
ith (low M
ach num
ber) incom
pressible
flow
ma
chines. E
xam
ples a
re:
• Industrial fa
ns
• Hydra
ulic pumps/turbine
s
• High pre
ssure ste
am
turbines
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
6-81
We
will not conside
r the e
ffects of:
• change
s in Re
ynolds numbe
r
• compre
ssibility
• cavita
tion
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
6-82
6.2 Turbine C
haracteristics E
uler’s w
ork equa
tion for an a
xial turbine
with consta
nt a
xial ve
locity and bla
de spe
ed (ra
dius)
(
)(
)0
VV
Uh
hh
32
0301
0>
−=
−=
∆θ
θ
ma
y be w
ritten a
s
(
)(
)U
tanV
tanV
Uh
rel,3
x2
x0
−α
−α
=∆
and the
stage
loading is
(
)1
tantan
U VU h
rel,3
2x
2 0−
α−
α=
∆=
ψ
⇒
()
1tan
tanrel
,32
−α
−α
φ=
ψ
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
6-83
V1
χ1
V2
α2
α1
7270
Incidence, i=-
αχ1
1
-300
+30
Exit Flow Angle α2
Cascade test results for an axial
flow turbine
V3
,rel
V3
V2
,rel
V2
U
U
STA
TO
R
RO
TO
R
Vx
Vx
α2
α3,rel
α3
V2
U
V3
,rel
V3
U
Effect of changing blade speed U
on velocity triangles
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
6-84
We
note from
the pre
vious page
that
• 0
a
nd
0,3
2<
>re
lα
α a
nd both are
approxim
ate
ly constant
and tha
t changing φ
• change
s ()
rel
,23
1
and
α
αα
= but not
rel
,32
a
nd
αα
so • ψ incre
ase
s with φ
• whe
n ψ =
0, no work is e
xtracte
d so ma
x. rpm
rea
ched for give
n ma
ss flow (
run
aw
ay
con
ditio
n )
ψ
φ0-1
Ideal Turbine C
haracteristic
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
6-85
6.3 Fan/P
ump C
haracteristics E
uler’s w
ork equa
tion for an a
xial pum
p with consta
nt ax
ial ve
locity and bla
de spe
ed (ra
dius)
(
)12
0θ
θV
VU
h−
=∆
ma
y be w
ritten a
s
(
)(
)1,2
0ta
nta
nα
αx
rel
xV
UV
Uh
−+
=∆
and the
stage
loading be
come
s
(
)(
)re
lre
lx
U V
U
h,2
11
,22 0
tan
tan
11
tan
tan
αα
φα
αψ
−−
=+
−=
∆=
Te
st results show
that
• the e
xit flow a
ngles
rel
,21
a
nd
αα
are
approxim
ate
ly constant
and the
velocity tria
ngles show
that
• changing φ cha
nges
2,1
a
nd
αα
rel
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
6-86
V1
χ1
V2
α2
α1
3230
Incidence, i=-
αχ1
1
-300
+30
Exit Flow Angle α2
34 36
Cascade test results for an axial flow
com
pressor
V2,rel
V1
V2
U
U
RO
TO
R
STA
TO
R
α1,rel
α2,rel
α1
α2
V1,rel
Effect of changing blade speed on axial
compressor velocity triangles
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
6-87
Now
,
• 0
a
nd
01
,2>
<α
αre
l
• ψ de
crea
ses w
ith φ
Now
, for incompre
ssible flow
,
d
p/ ρ dh
Td
s −
=
⇒
isen
hs
Th
P,0
00
∆=
∆−
∆=
∆ρ
ψ
φ0 1
actual ideal
so that
ηψ
==
⇒=
=2 0
2 0
0 0
0
0
U h
ηρU P
ρ h
P
h
h
η,ise
n
The
refore
• whe
n φ = 0, m
ax. pre
ssure rise
should occur
• in practice
this is limite
d by sepa
ration of the
bounda
ry laye
rs (stall).
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
6-88
6.4 More “usual” incom
pressible stage parameters
We
have
alre
ady se
en tha
t only one inde
pende
nt para
me
ter
UV
x=
φ de
term
ines
• the ope
rating point of a
g
iven sta
ge.
In the ca
se of a
compre
ssor3, this is be
cause
U
Vx
fixes
• the re
lative
flow a
ngle in to the
rotor
• the flow
patte
rn in the rotor
• the re
lative
flow a
ngle &
losses out of the
rotor
• the flow
angle
into the sta
tor
• the flow
patte
rn in the sta
tor
• the flow
angle
& losse
s out of the sta
tor
• the non-dim
ensiona
l opera
ting point of the sta
ge/m
ach
ine
3 Th
e argu
men
t for a tu
rbin
e is very similar
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
6-89
In context of ove
rall pe
rforma
nce, a
nd espe
cially during t
he initia
l design proce
ss, it is more
usual to
work w
ith a diffe
rent de
finition of the flow
coefficie
nt:
U V
AR
VA
D m
D Qx
xm
ea
n xx
33
αρ
ρα
ρφ
ΩΩ
=Ω
=&
as this a
lso define
s the ope
rating point of the
ma
chine
. Note
that Q is the
volume
tric flow ra
te.
Sim
ilarly, w
e pre
fer the
powe
r coefficie
nt ξ
to the sta
ge loa
ding coefficie
nt 2 0
U
h∆
whe
re
2 0
22
20
53
x
D
DD
D
wm
U
h
U Vh
VA
xx
x∆
ΩΩ
∆=
Ω=
αρ
ρρ
ξ&
Fina
lly, we
note tha
t for incompre
ssible flow
,
0 0
0 isen0,
h P
h
hp
um
p∆ ∆
=∆
∆=
ρη
whe
re the
efficie
ncy depe
nds on the R
eynolds num
ber.
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
6-90
The
refore
, we
often re
place
the pow
er coe
fficient by the
efficie
ncy and the
pressure
coefficie
nt
2 0
22
02
2 01
D
1
DU
hh
P∆
∝Ω ∆
=Ω ∆
=η
ηρ
ψ
The
se dim
ensionle
ss groups apply e
qually to a
xial
ma
chines a
nd radia
l flow m
achine
s (whe
re
D is
usually the
outer dia
me
ter of the
rotor)
0 1
D
22
0
Ωρ
P
3
Theoretical
Actual
Useful range
Typical characteristic for a centrifugal fan
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
6-91
6.5 Sum
mary
Cha
nging the flow
coefficie
nt φ =
Vx /U
change
s the incide
nce onto the
stator a
nd rotor blad
es.
In term
s of velocity tria
ngles, w
e se
e tha
t:
o S
tage
Loading C
oeffice
nt
()φ
ψf
U Vf
U
hx
=
=∆
=2 0
o E
fficiency
()φ
ηf
U Vf
x=
=
or o P
ressure
Rise
Coe
fficient
()φ
ρψ
fD Q
fP
= Ω
=Ω ∆
=3
22 0D
o E
fficiency
()φ
ηf
D Qf
= Ω
=3
o P
owe
r Coe
fficient
()φ
ρξ
fD Q
fw
mx
= Ω
=Ω
=3
53D
&
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
7-92
7 Radial F
low M
achines (Pum
ps, Com
pressors and Turbines)
7.1 Introduction M
any type
s & configura
tions
• Most com
mon turbom
achine
- the
centrifuga
l pump or co
mpre
ssor
• Less w
ell unde
rstood -
more
comple
x 3-D flow
s than in axia
ls
• Low cost
- ca
st, fabrica
ted, m
achine
d from solid
• Short de
velopm
ent cycle
-
optimize
d designs for low
volum
e production
• Me
chanica
lly robust -
low m
ainte
nance
/haza
rdous environm
ents
• Large
frontal a
rea
-
limits a
ero a
pplications to sm
all e
ngines
• Pre
ssure ra
tios -
typically 3:1 or gre
ate
r
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
7-93
To be
gin with, w
e note
that for ra
dial m
achine
s:
V
x <<
Vr a
t the oute
r radius
V
x >>
Vr or V
x <<
Vr a
t the inne
r radius
For e
xam
ple, the
turbocharge
r has a
turbine rotor (a
nd a co
mpre
ssor rotor) whe
re
V
x <<
Vr a
t the oute
r radius (i.e
. 0
Vx
≈)
But
V
x >>
Vr a
t the inne
r radius (i.e
. 0
Vr ≈
)
The
radia
l fan how
eve
r often ha
s
V
x <<
Vr a
t the oute
r radius (i.e
. 0
Vx
≈)
and
V
x <<
Vr a
t the inne
r radius (i.e
. 0
Vx
≈)
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
7-94
7.2 Application of E
uler’s Work E
quation to a Radial R
otor T
hough we
have
so far e
xam
ined E
uler’s w
ork equa
tion in the
context of a
xial m
achine
s, the proof w
as
not restricte
d to this type of m
achine
.
He
re, w
e re
call tha
t the flow
• ente
rs at a
me
an ra
dius r1
• lea
ves a
t a m
ea
n radius r
2
so that
• 2
1U
U≠
We
also note
that E
uler's W
ork Equa
tion is:
r1
r2
Rotor
Control V
olume
τΩ
(
)(
)11
22
0102
θθx
VU
VU
m=
hh
m=
W−
−−
&&
&
or
1
12
201
02θθ
xV
UV
U=
hh
w−
−=
−
and tha
t
co
nst
Uh
Uh
VU
hV
Uh
Ro
tha
lpy
rel
rel
θθ
=−
=−
=−
=−
=22
2 1,
0221
2 1,
012
202
11
01
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
7-95
7.3 Velocity T
riangles for a 90 degree Radial Inflow
Turbine
Scroll
StatorR
otor
12
3F
lowV
3
V2,rel
V2
Radial V
iewR
otor Exit
U2
U1
V3,rel
Axial V
iewR
otor Inlet
Velocity triangles for a radial inflow
turbine with stator vanes
Note
that
• the e
xit velocity tria
ngle looks ve
ry simila
r to that
from a
n axia
l flow turbine
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
7-96
From
velocity tria
ngles a
t design conditions
V
θ2 = U
2 &
Vθ3 =
0
∴ E
uler's W
ork Equa
tion reduce
s to
W
x = ∆
h0 =
UV
θ2 = U
2 2 ⇒
∆
h0
U2 2 =
1.0
In fact, it is usua
lly found that
• work done
is ma
inly a ƒ n of the
square
of the im
pelle
r tip spee
d U2 2
A sim
ilar a
nalysis show
s that
Λ
≈ 12
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
7-97
7.4 Velocity T
riangles for a 90 degree Centrifugal C
ompressor
Scroll
Diffuser
Rotor
1
2 3
Flow
V1
,rel
V1
V2
,rel
V2
U2
U1
Axial V
iewR
otor Exit
Radial V
iewR
otor Inlet
Velocity triangles for a centrifugal com
pressor with a “radial” rotor and stator vanes
Note
that
• the inle
t velocity tria
ngle is sim
ilar to tha
t from the 1
st stage
of an a
xial flow
compre
ssor
• exit re
lative
velocity doe
s not quite follow
blade
sha
pe – this is know
n as “slip”
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
7-98
7.5 Application of S
FE
E to a R
adial Rotor
We
can conside
r a com
pressor or turbine
. Whiche
ver w
e cho
se, the
Rotha
lpy equa
tion is
consta
nt
22 1
00
==
=U
hU
Vh
Ro
tha
lpy
-,re
l-
θ
If we
take
the rota
ting fram
e pa
rt, the a
bove ca
n be w
ritte
n as
consta
nt
22 1
0=
Uh
-,re
l
22 1
,0re
lre
lV
hh
+=
⇒
constant
2
2 12
2 1=
+U
-V
hre
l
22 1
0V
hh
−=
⇒
constant
2
2 12
2 12
2 10
=+
U-
V)
V-
(hre
l
(
) 22 1
22 1
22 1
0V
VU
hre
l +−
− =
constant
So,
hh
wx
0203
−=
− =
()
()2
22 1
22 1
22 1
32
2 12
2 12
2 1V
VU
VV
Ure
lre
l+
−−
+−
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
7-99
So for a
radia
l inflow turbine
:
)
V(V
)V
(V)
U(U
wre
lre
lx
2223
2 122
232 1
2223
2 1−
+−
−−
=−
< 0
⇒
U3 <
U2
⇒
Ra
dial IN
flow
M
inimise
2 1V2
,rel 2 ⇒
α
2,rel ≈ 0°
M
inimise
2 1V3 2
⇒
α3 ≈ 0°
And for com
pressors/pum
ps
)
V(V
)V
(V)
U(U
-wre
lre
lx
2122
2 121
222 1
2122
2 1−
+−
−−
= > 0
⇒
U2 >
U1
⇒
Ra
dial O
UT
flow
M
inimise
2 1V2
,rel 2 ⇒
α
2,rel ≈ 0°
M
inimise
2 1V1 2
⇒
α1 ≈ 0°
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
7-100
7.6 Exam
ple In
a ra
dia
l turb
ine
, the
flow
leavin
g th
e rin
g o
f stato
r bla
de
s has a
static te
mpe
ratu
re o
f 1000 K
an
d
velo
city 600 m
s -1 at a
n a
ngle
of 7
0° to
the
radia
l dire
ction. A
t e
ntry to
the
roto
r whe
el th
e b
lade
sp
ee
d is 5
00 m
s -1 whilst a
t flow
exit it is 1
00 m
s -1. Calcu
late
the
rela
tive sta
gnatio
n te
mpe
ratu
re a
t
entry a
nd e
xit of th
e ro
tor w
he
el
Rotor
2
3F
lowV
3
V2,rel
V2
Radial V
iewR
otor Exit
U2
U1
V3,rel
Axial V
iewR
otor Inlet
Stator
2
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
7-101
At e
ntry the flow
is radia
l-tange
ntial a
nd at e
xit the
flow is a
xial-ta
ngentia
l
1
12
20570
cos600
cos-
r m
s.
α V V
=°
==
1
18
56370
sin600
sin-
θ m
s.
α V V
=°
==
ms
..
UV
V-
,rel
11
18
63500
8563
=−
==
−θ
θ
Now
:
co
nst
Uh
,rel
=−
22 1
0
(
)()p
,rel
θrre
lc
/V
V
T
T2
2121
101
++
=
(
) K
./
..
Tre
l97
10221005
28
632
205(
1000)
22
01=
×+
+=
Re
arra
nging SF
EE
in rela
tive fra
me
:
(
)(
)pre
lp
,rel
cU
Tc
UT
22
22,
0221
01−
=−
(
)()p
,rel
rel
cU
UT
T2
2221
01,
02−
−=
K
/-
.T
rel
57.
903)
10052(
)500
100(
971022
22
,02
=×
+=
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
7-102
7.7 Sum
mary
In radia
l ma
chines
• Applica
tion of SF
EE
lea
ds to turbines w
here
most w
ork is obta
ined for:
⇒
Ra
dial Inflow
⇒
N
ea
r radia
l blade
s at rotor inle
t (α
2,rel ≈ 0°)
⇒
No e
xit swirl (α
3 ≈ 0°)
and for com
pressors/pum
ps:
⇒
Ra
dial O
utflow
⇒
Ne
ar ra
dial bla
des a
t rotor exit (α
2,rel ≈ 0°)
⇒
No e
xit swirl (α
1 ≈ 0°)
• Work e
xchange
is ma
inly a ƒ n of the
square
of the im
pelle
r tip spee
d
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
8-103
8 Losses In Turbom
achines 8.1 Introduction S
trictly, we
should define
the losse
s in term
s of the e
ntropy crea
ted but, w
e usua
lly dete
rmine
the losse
s from
stagna
tion pressure
me
asure
me
nts and, w
hen
co
nst
h=
0
we
find that
0
00
0ρ
dp
dh
ds
T−
=
⇒
00
0ρ
dp
ds
T−
=
and in the
particula
r case
of incompre
ssible flow
(con
st=
ρ)
ρ
00
ps
T∆
−=
∆
So viscous e
ffects (including those
due to shock w
ave
s) a
re usua
lly quantifie
d using
• Sta
gnatio
n P
ressu
re L
oss C
oe
fficients
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
8-104
Developm
ent of blade surface boundary layers and wakes in an axi
al compressor
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
8-105
8.2 The 2D
(Linear) Cascade
Linea
r or 2D ca
scade
s
• produced by de
velopm
ent of cylindrica
l surface
s
provide da
ta on
• me
an flow
angle
s
• losses
but only valid w
hen in the
turboma
chine
• radius cha
nge is sm
all from
inlet to e
xit of blade
row
• effe
cts of twist, le
an, sw
ee
p, rotation a
re sm
all
there
fore
• can only re
ally a
pply to axia
l ma
chines
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
8-106
Exam
ple of linear cascade wind tunnel
In a
ca
scade
e
xperim
ent,
at
exit,
the
pitchwise
(ie
ta
ngentia
l) va
riation
of ve
locity, flow
a
ngle,
stagna
tion pressure
and sta
tic pressure
are
usually m
ea
sure
d.
Dow
nstrea
m of the
blade
row the
stagna
tion pressure
is pitchw
ise non-uniform
and be
low the
isentropic
value
. The
ave
rage
exit sta
gnation pre
ssure ca
n be de
fined by:
∫
∫+−
+−=
2/2/2
2
2/2/02
22
02
)(
)(
)(
)(
)(ss
x
ssx
dy
yV
y
dy
yP
yV
y
P
ρ
ρ
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
8-107
Note
s:
• Flow
acce
lera
tes a
s e
xpecte
d for a turbine
• Be
twe
en the
wa
kes, flow
is ise
ntropic (no shock w
ave
s in this case
)
• Losses a
ppea
r only in the
wa
kes
• Exit flow
is alm
ost pa
ralle
l so static pre
ssure
is uniform
01
0.0
0.1
0.8
0.9
1.0
2.5
3.0
P-P
P- p
010 (y)
01 2
p-p
P- p
12 (y)
01 2
VV21
ys
01
2
01
2 2
Typical m
idspan wake traverse results
for a turbine cascade (4A3 C
ascade Experim
ent)
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
8-108
8.3 Stagnation pressure Loss C
oefficients T
he sta
gnatio
n p
ressu
re lo
ss coe
fficient
is define
d as
Pre
ssure
Dyna
mic
c)
(Isentropi
R
efe
rence
ilityirre
versib
to
due
pressure
S
tagna
tion
(rela
tive)
of
Loss
=
Yp
Since
the
sta
gnation
pressure
ca
n cha
nge
due
to adia
batic+
reve
rsible=
isentro
pic
change
s in
the
stagna
tion entha
lpy (or stagna
tion tem
pera
ture),
• the loss is e
valua
ted re
lative
to the ise
ntropic case
in
o the
absolute
fram
e of re
fere
nce for sta
tors/casca
des
o the
rotating fra
me
of refe
rence
for rotors
The
re a
re m
any diffe
rent de
finitions for loss coefficie
nt so ta
ke ca
re w
hen consulting te
xt books and
other publishe
d works.
The
loss coefficie
nt definitions use
d in this course a
re:
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
8-109
• For com
pressors
- w
e a
re inte
reste
d in how m
uch is costs to slow dow
n the
flow a
cross a bla
derow
- loss coe
fficients a
lwa
ys base
d on inlet
conditions
- m
ost comm
on is:
101
0201
101
02,
02
- pP
- PP
= - p
P
- PP
Yusu
ally
isen
p≡
• For turbine
s
- we
are
intere
sted in how
much is costs to spe
ed up the
flow a
cross a bla
derow
- alw
ays ba
sed on e
xit conditions
- most com
mon a
re
2
02
0201
202
02,
02
- pP
- PP
=- p
P
- PP
Yu
sua
llyise
np
≡ IIA
Paper 3A
3 Fluid M
echanics II: Turbom
achinery/HP
H
8-110
V2
α2
3230
0.08
0.04
0.00-300
+30
Incid
enc
e, i=
-α
χ11
-300
+30
Exit Flow Angle α2Profile Loss Coefficient Yp
34
36
Typical cascade test results for an axial
flow com
pressor
V1
ROTO
R
χ1
V2
α2
α1
7270
0.08
0.04
0.00-300
+30
Incid
enc
e, i=-
αχ1
1
-300
+30
Exit Flow Angle α2Profile Loss Coeffic ient Yp
Typical cascade test results for an axial flow
turbine
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
8-111
Note
that w
hen the
losses re
fer only to those
due to t
he bla
de surfa
ce bounda
ry laye
rs and w
ake
s (e.g.,
at the
mid-spa
n of a 2-D
casca
de) the
n we
often ca
ll the sta
gnation pre
ssure loss coe
fficient the
profile
loss coefficie
nt.
We
note tha
t changing 1
α
• results in a
change
of incidence
1
1χ
α−
=i
• does not cha
nge 2
α (until the
boundary la
yers se
para
te a
t high i)
• does not cha
nge the
losses (until the
boundary la
yers
sepa
rate
at high i)
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
8-112
8.3.1 E
xam
ple
At a
particu
lar o
pe
ratin
g p
oin
t, an a
xial tu
rbin
e r
oto
r bla
de
row
has re
lative
inle
t Mach
num
be
r M1
,rel
= 0
.6, a
rela
tive e
xit Mach
num
be
r M2,rel =
1.0
5 a
nd a
loss co
efficie
nt Yp =
0.0
5. If th
e re
lative
sta
gnatio
n p
ressu
re a
t inle
t to th
e ro
tor ro
w is 8
.0 b
ar, ca
lcula
te th
e re
lative
stagnatio
n p
ressu
re a
t exit.
The
loss co
efficie
nt is d
efin
ed a
s
Since
we
have
a rotor bla
de row
, we
will w
ork with re
lat
ive flow
quantitie
s. We
will a
ssume
that the
re
is no change
in radius
2,
02
,02
,01
2,
02
02,
,02
- PP
- PP
- PP
- PP
Yre
l
rel
rel
rel
isen
rel
p=
≡
⇒
1
1
,
022
,02
,01
rel
rel
rel
pP
P -
- P
P Y
=
⇒
1
1
,02 2
,02 ,01
+=
rel
pre
l
rel
P
P -
YP P
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
8-113
But
re
lP
P
,02 2
=
)1(
222
11
−−
−
+γ
γγ
rM
= ()
5.3
205.1
2.0
1−
×+
= 0.4979
⇒
()
025.1
4979.0
105.0
1,
02 ,01
=+
= -
P P
rel
rel
⇒
80.7
025.1
/8
025.1
/,
01,
02=
==
rel
rel
PP
bar
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
8-114
8.4 Sum
mary
The
crea
tion of entropy is usua
lly dete
rmine
d from the
losse
s of stagna
tion pressure
Linea
r or 2D ca
scade
s apply to a
xial m
achine
s only an
d provide da
ta on
• me
an flow
angle
s
• losses
The
stagnatio
n p
ressu
re lo
ss coe
fficient
is eva
luate
d rela
tive to the
ise
ntro
pic ca
se a
nd is define
d as
Pre
ssure
Dyna
mic
c)
(Isentropi
R
efe
rence
ilityirre
versib
to
due
pressure
S
tagna
tion
of
Loss
=
Yp
Com
pressor losse
s are
norma
lised by inle
t conditions
Turbine
losses a
re norm
alise
d by exit conditions
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
9-115
9 Com
pressible Flow
through Turbom
achines M
any turbom
achine
s involve com
pressibility e
ffects (M
ach
> 0.3).
To ca
lculate
the pe
rforma
nce of the
se m
achine
s, very simila
r me
thods are
used a
s those for ca
ses w
here
the
flow is conside
red incom
pressible
.
The
most re
leva
nt compre
ssible flow
rela
tions, which a
re a
ll tabula
ted a
s functions of Ma
ch numbe
r for γ=
1.4 and γ=
1.333, are
:
12
02
11
−
−+
=M
T Tγ
1
1
2
02
11
−−
−
+=
γγ
ρ ρM
1
2
02
11
−−
−
+=
γ γγ
Mp p
2 1
2
02
11
1−
−
+−
=M
MT
c Vp
γγ
− +−
−
+−
=1 1
2 1 2
0
0
2
11
1γ γ
γγ γ
MM
Ap
Tc
mp
&
2 1
20
2
11
1
−
+−
=M
MA
p Tc
mp
γγ γ
&
The
most im
portant is
0
0
Ap
Tc
mp
&
since w
e usua
lly know m & a
nd 0
p a
nd 0
T a
re ofte
n constant
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
9-116
Ma
ch N
umb
er
− +−
−
+−
=1 1
2 1
2
0
0
2
11
1
γ γγ
γ γM
MA
p
Tc
mp
&
12
02
11
−
−+
=M
T Tγ
1
1
2
02
11
−−
−
+=
γγ
ρ ρM
12
02
11
−−
−
+=
γ γγ
Mp p
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
9-117
9.1 Relative F
low Q
uantities W
e ha
ve a
lrea
dy observe
d that in the
absolute
fram
e of re
fere
nce tha
t
• 0
T T,
0p p
, 0
0
Ap
Tc
mp
&
, 0
Tc Vp
, …..
()
Mf
=
• T and p (in fa
ct all sta
tic quantitie
s) are
the sa
me
in both the
absolute
and re
lative
fram
es
The
refore
, in the re
lative
fram
e
rel
T
T,0,
rel
p
p,0,
rel re
lp
Ap
Tc
m
,0
,0&
, re
lp
rel
Tc V
,0, …
.. (
)re
lM
f=
So, w
e ca
n use the
sam
e ta
bles for both a
bsolute a
nd re
lative
flows providing
• we
use the
appropria
te sta
gnation qua
ntities (e
.g. 0
T or
rel
T,0
)
• we
use the
appropria
te M
ach num
bers(
M or
rel
M)
• A is the
effe
ctive flow
are
a m
ea
sured N
OR
MA
L to the a
ppropria
te flow
vector (V
or re
lV
)
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
9-118
9.2 Com
pressibility and Conservation of M
ass
Co
ntrol
Volum
e
rel
1
scosα1rel
scosα2rel
s
rel2
Inlet and exit flow areas of am
axial compressor rotor in x-rθθθ θ plane
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
9-119
Conse
rvation of m
ass give
s for one bla
de pa
ssage
(
)(
),re
l,re
lhs
V= ρ
hs
V ρm
,rel
,rel
pa
ssag
e1
2cos
cos1
12
2α
α=
&
or, more
gene
rally, if
xA
is the cross-se
ctional a
rea
(
)(
)α
ρα
ρcos
cosx
xre
lA
VA
Vm
rel
==
&
whe
re
• the e
ffective
flow a
rea
()
αcos
xA
A= is a
lwa
ys me
asure
d perpe
ndicular to the
velocity ve
ctor.
• failure
to observe
this importa
nt simple
rule ha
s serious conse
quence
s whe
n dea
ling with
compre
ssible flow
beca
use
00
Ap
Tc
mp
& de
pends on the
true flow
are
a
We
will e
xam
ine the
flow a
t inlet to a
nd exit from
a c
ompre
ssor rotor of pitch s a
nd constant spa
n h in
rela
tive fra
me
.
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
9-120
Conse
rvation
of m
ass
also
me
ans
that
in a
bsence
of
radi
us cha
nge
(re
lre
lT
T,
02,
01=
) a
nd loss
(re
lre
lP
P,
02,
01=
)
(
)()
rel
rel
rel
rel
rel
rel
rel
p
rel
rel
rel
p
Ph
s
Tc
m Ph
s
Tc
m
,1 ,2rel
2,,1 ,2
,02
,2
,02
rel1,
,01
,1
,01
cos
cos )
F(M
cos
cos
cos=
)F
(M =
cos
α αα α
α
α
=&
&
This is use
ful beca
use
• we
can find
rel
M,2
given
rel
M,1
,.re
l,1
α a
nd re
l,2
α
or • we
can find
rel
,2α
given
rel
M,1
,.re
l,1
α.a
nd re
lM
,2
or ……
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
9-121
9.3 Exam
ple: Flow
Through an A
xial Com
pressor Rotor
Fo
r the
hig
h M
ach
nu
mb
er co
mp
resso
r roto
r bla
de
de
scrib
ed
be
low
, find
the
static p
ressu
re ra
tio (p
2 /p1 )
acro
ss the
roto
r, the
ab
solu
te e
xit flow
an
gle
(α
2 ) an
d th
e e
xit Ma
ch n
um
be
r (M2 ).
Ge
ome
trical D
ata
:
m
ea
n radius (consta
nt) r
0.300 m
bla
de he
ight (span, consta
nt) h
0.050 m
⇒
annulus cross-se
ctional a
rea
A
x 0.0942 m
2
Ope
rating C
onditions:
bla
de spe
ed
U
250 ms -1
m
ass flow
rate
m .
16.0 kgs -1
inle
t stagna
tion pressure
(abs)
p0
1 1.4 ba
r
inle
t stagna
tion tem
pera
ture (a
bs) T
01
340 K
a
bsolute inle
t swirl
α1
10°
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
9-122
Bla
de P
erform
ance
(at a
bove ope
rating point):
rotor pre
ssure loss coe
fficient
Yp
0.034
rotor re
lative
exit a
ngle
α2
,rel -35.0°
Assum
e tha
t the w
orking fluid is air w
ith
γ=
1.4
R=
287 Jkg -1K-1
cp =
1005 Jkg -1K-1
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
9-123
Ste
p 1
First, w
e dra
w ve
locity triangle
s:
V1,rel
V2,rel
V1
V2
UU
RO
TO
R UBlade Speed
Note
tha
t the
ve
locities
(and
the
velocity
triangles)
ma
y be
conve
rted
to M
ach
numbe
rs (a
nd ge
ome
trically e
quivale
nt triangle
s) by dividing by the
local sound spe
ed γR
T .
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
9-124
We
note tha
t the flow
are
a norm
al to the
flow ve
ctor (this is ve
ry importa
nt) is given by
1
cosα
xA
A=
The
n, in the a
bsolute fra
me
, we
find the inle
t flow conditions using
01
1 01
01 01
cosp
A
Tc
m
Ap
Tc
m
x
pp
α&
&
==
5
104.
110
cos0942.0
3401005
16
××°
××
× =
0.720
Using the
table
s gives
M
1 = 0.350
and
V
1 /01
p Tc
= 0.219
⇒
V
1 = 128.0 m
s -1
T
1 /T0
1 = 0.976
⇒
T
1 = 331.8 K
P
1 /P0
1 = 0.919
⇒
P
1 = 1.287 ba
r
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
9-125
Ste
p 2
From
the tria
ngles, w
e obse
rve tha
t
Vx1 =
V1 cosα
1 = 126.1 m
s -1
Vθ1 =
V1 sinα
1 = 22.2 m
s -1
Vθ1
,rel = V
θ1 – U =
–227.8 ms -1
V1
,rel = 260.4 m
s -1
α1
,rel = -61.0°
M1
,rel = V
1 /1
p Tc
= 0.713
From
the T
able
s, we
find that:
T1 /T
01,rel =
0.908 ⇒
T
01,rel =
365.4 K
P1 /P
01,rel =
0.713 ⇒
P
01,rel =
1.805 bar
Note
that w
e could a
lso have
used
T0
1,rel = T
1 + (V
1,rel ) 2/2c
p = 365.5 K
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
9-126
Ste
p 3
He
re, w
e e
xam
ine the
rotor in the re
lative
fram
e, a
pplying the loss a
nd flow turning
Now
, a fixe
d radius com
bined w
ith the R
othalpy e
quation (
constant
22 1
,00
=−
=−
Uh
UV
hre
lθ
)
⇒
T0
1,rel =
T0
2,rel =
365.4 K
He
nce
P
02
,rel ,isen = P
01
,rel = 1.805 ba
r
Loss C
oefficie
nt (given)
Y
P =
1rel,
01
rel,
02rel,
01
PP
PP
− − =
0.034
⇒
P0
2,rel =
1.787 bar
Exit a
ngle (give
n)
α
2,rel =
-35°
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
9-127
Ste
p 4
He
re, w
e find the
rotor exit conditions in the
rela
tive fra
me
:
re
lre
lx
rel
p
PA
Tc
m
,0
2,2
,0
2
cosα&
=
510
787.1
)35
cos(0942.0
4.365
100516
××
°−
××
× =
0.703
Using the
table
s (γ=1.4) give
s: M2,rel =
0.340
The
other flow
propertie
s are
obtaine
d (from ta
bles
):
V
2,rel /
rel,
02p T
c =
0.213 ⇒
V
2,rel =
129.1 ms -1
T
2 /T0
2,rel =
0.977
⇒
T
2 = 357.0 K
P
2 /P0
2,rel =
0.923
⇒
P
2 = 1.649 ba
r
Note
that w
e could a
lso have
used
V2
,rel = M
2,rel ×
γRT
2 = 128.8 m
s -1
The
static pre
ssure ra
tio is then give
n by
P
2
P1 =
1.6491.287 =
1.28
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
9-128
Ste
p 5
He
re, w
e, conve
rt back to a
bsolute flow
conditions dow
nstrea
m of the
rotor
Vx2 =
V2
,rel cosα2
,rel = 105.8 m
s -1 < V
x1 due to com
pressibility
Vθ2
,rel = V
2,rel sinα
2,rel =
-74.0 ms -1
Vθ2 =
Vθ2
,rel + U
= 176.0 m
s -1
V2 =
205.4 ms -1
α2 =
59.0°
M2 =
V2 /
2p T
c=
0.542
From
the T
able
s (γ=1.4), w
e find tha
t:
T
2 /T0
2 = 0.945 ⇒
T
02 =
377.8 K
⇒
∆T
0 = 377.8 – 340 =
37.8 K
Using E
uler's W
ork Equa
tion, we
can che
ck this resu
lt:
h
02 – h
01 =
U(V
θ2 – Vθ1 ) =
38.45 kJkg -1 ⇒ ∆
T0 =
38.3 K
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
9-129
9.4 Sum
mary
• The
functions of Ma
ch numbe
r
0
T T,
0p p
, 0
0
Ap
Tc
mp
&,
0T
c Vp
can be
used in a
bsolute or re
lative
fram
es so long
as corre
ct value
s (absolute
or rela
tive) a
re use
d.
• Re
me
mbe
r:
E
ffective
flow a
rea
= A
=
αcos
xA
• Conse
rvation of m
ass m
ea
ns that if w
e know
02
0201
01
and
,
,P
TP
T, w
e ca
n use
(
)(
)1 2
01
02
02 012
1 2
01
02
02 01
022 02
101
1 01
cos
cos
cos
cos
coscos
α α P P
T T)
f(Mα α
P P T T
P
αA Tc
m) =
f(M
PαA T
cm
x
p
x
p=
=&
&
to find
o
rel
M,2
given
rel
M,1
,.re
l,1
α a
nd re
l,2
α
o
rel
,2α
given
rel
M,1
,.re
l,1
α.a
nd re
lM
,2 …
… e
tc
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
10-130
10 Hub-T
ip Variations
10.1 Introduction
So fa
r, we
have
ma
de significa
nt progress using
• 1-D m
ea
n-line a
nalyse
s turboma
chine
The
next sta
ge is to use
• Sim
ple R
adia
l Equilibrium
theory or
• Stre
am
line C
urvature
calcula
tions
to exa
mine
the hub-tip va
riations in, for e
xam
ple,
the circum
fere
ntial a
vera
ges of
• the ve
locities Vx , V
r and V
θ
• flow a
ngle α
at the
inlet a
nd exit of e
ach bla
derow
.
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
10-131
10.2 S
imple R
adial Equilibrium
W
e w
ill assum
e
• Ana
lysis applie
s in stationa
ry fram
e (i.e
. all ve
locitie
s are
absolute
, eve
n for a rotor)
• Axisym
me
tric flow (
0=
∂∂
θ)
• Curva
ture of the
strea
mline
s in the M
eridiona
l (x-r) pla
ne is ne
gligible (no a
ccele
rations norm
al to
the stre
am
surface
)
• Ra
dial ve
locity r
V is ne
gligible
• Isentropic F
low 4
rr
θ
x
4 Sim
ple R
adial E
qu
ilibriu
m th
eory d
oes n
ot req
uire
this b
ut th
is is a con
venien
t and
com
mo
n sim
plific
ation
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
10-132
r
dr
p+dp
p
Vθ
Unde
r these
circumsta
nces, w
e a
re de
aling w
ith the
equilibrium
of a sw
irling flow w
here
the pre
ssure
forces cre
ate
the ce
ntripeta
l acce
lera
tion:
r V
dr
dp
21
θρ
=
Now
, for isentropic flow
0
=−
=ρ dp
dh
Td
s
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
10-133
⇒
r V
dr
dp
dr
dh
21
θρ
==
Now
if 0
=r
V
2
2 10
Vh
h+
≡
⇒
22 1
22 1
0θ V
Vh
hx
++
=
⇒
dr
dV
Vd
r
dV
Vd
r
dh
dr
dh
xx
θθ
++
=0
Substituting for dh
/dr give
s
⇒
dr
dV
Vd
r
dV
Vr V
dr
dh
xx
θθ
θ+
+=
20
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
10-134
This ca
n be re
-arra
nged to give
the S
imple
Ra
dial E
quilibrium e
quation for ise
ntropic flow
(
)d
r rVd
r V
dr
dV
Vd
r
dh
xx
θθ
+=
0
For incom
pressible
flow, the
above
becom
es
(
)d
r rVd
r V
dr
dV
Vd
r
dp
xx
θθ
ρ+
=0
1
We
see
that for a
given ra
dial distribution of sta
gna
tion entha
lpy (or stagna
tion pressure
)
• θ
rV (w
hich results from
the ra
dial distribution of w
ork a
ccording to Eule
r’s equa
tion)
• x
V
• the flow
angle
(
)xV
Vθα
1ta
n −=
are
all de
pende
nt on ea
ch other
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
10-135
10.3 E
xample: F
ree Vortex C
ompressor R
otor “F
ree
vortex” m
ea
ns
θ
rV =
constant
The
ma
jority of axia
l ma
chines do not de
viate
far f
rom a
free
vortex de
sign.
Conside
r the ca
se w
here
the sta
gnation e
nthalpy is ra
dially uniform
at inle
t
r
x
RS
12
3
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
10-136
In this case
we
have
co
nst
h=
01
1
1C
con
strV
==
θ
2
2C
con
strV
==
θ
Now
, upstrea
m of the
rotor, the S
RE
gives
(
)d
r
rVd
r V
dr
dV
Vd
r
dh
xx
11
11
01θ
θ+
=
⇒
dr
dV
Vx
x1
10
=
⇒
con
stV
x=
1
The
specific w
ork at a
radius
r is given by E
uler’s w
ork equa
tion
(
)(
)(
)21
21
CC
rVrV
UV
wx
−Ω
=−
Ω=
∆=
θθ
θ =
constant
But
02
01h
hw
x−
=
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
10-137
⇒
con
sth
=02
The
refore
, by the sa
me
argum
ent a
s above
,
⇒
con
stV
x=
2
Conse
rvation of m
ass for incom
pressible
flow the
ref
ore give
s
⇒
con
stV
VV
xx
x=
==
21
Fina
lly, the ra
dial distributions of the
absolute
and re
lative
flow a
ngles a
re give
n by
=
=
−−
xx
V
rC
V V1
11
11
tan
tan
θα
Ω
−=
Ω
−=
=
−−
−
xx
x rel
rel
V
rr
C
V
rV
V
V1
11
1,1
1,1
tan
tan
tan
θθ
α
=
=
−−
xx
V
rC
V V2
12
12
tan
tan
θα
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
10-138
Ω
−=
Ω
−=
=
−−
−
xx
x rel
rel
V
rr
C
V
rV
V
V2
12
1,2
1,2
tan
tan
tan
θθ
α
V1
,rel
V2
,rel
V1
U1
U2
V2
RO
TO
R
Blade Speed
rθ
xα
2,re
l
α2
α1
α1,re
l
Mean line rotor velocity triangles
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
10-139
So, a
free
vortex de
sign has
• a uniform
work distribution a
cross the spa
n
• constant a
xial ve
locity
• varying bla
de sha
pe (inle
t and e
xit angle
s) along the
span
ME
AN
Hu
b
Me
an
Tip
TIP
HU
B
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
10-140
Exa
mple
:
=1
α0
and a
t me
an ra
dius:
408.0
2 0=
∆=
U
hψ
952.0
==
U Vx
φ
-80.00
-60.00
-40.00
-20.00
0.00
20.00
40.00
60.00
0.40.5
0.60.7
0.80.9
1r/rtip
angle
Abs Inlet
Abs E
xit
Rel Inlet
Rel E
xit
rmean
IIA P
aper 3A3 F
luid Mechanics II: T
urbomachinery/H
PH
10-141
10.4 S
umm
ary T
he S
imple
Ra
dial E
quilibrium e
quation for ise
ntropic flow
is
(
)d
r rVd
r V
dr
dV
Vd
r
dh
xx
θθ
+=
0
For incom
pressible
flow, the
above
becom
es
(
)d
r rVd
r V
dr
dV
Vd
r
dp
xx
θθ
ρ+
=0
1
So the
radia
l distributions of
• θ
rV
• x
V
• the flow
angle
s (
)rV
Vθα
1ta
n −=
cannot be
chosen inde
pende
ntly
As a
result of the
above
• blade
shape
s (i.e. inle
t and e
xit angle
s) vary a
long the
span.
