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IIA P

aper 3A 3 Fluid M

echanics II: T urbom

achinery/H P

H

1-1

**M
odule 3A
**

**3 - T
urbom
**

**achinery
P
**

**rof H
ow
**

**ard H
odson
**

**1 Introduction
1.1 D
**

**efinition
**

A T

urbom achine is a steady flow

device (non-positive displacem ent) w

hich creates/consum

es shaft-w ork by changing the m

om ent of m

om entum

(angular m

om entum

) of a fluid passing through a rotating set of blades.

IIA P

aper 3A 3 Fluid M

echanics II: T urbom

achinery/H P

H

1-2

**1.2 E
xam
**

**ples of T
urbom
**

**achines
**1.2.1

V ery L

arge M achines

**M
H
**

**I 501 single shaft G
as T
**

**urbine **

IIA P

aper 3A 3 Fluid M

echanics II: T urbom

achinery/H P

H

1-3

**T
hree L
**

**ow
P
**

**ressure R
otors from
**

** a large steam
**

**turbine (approx 150 M
W
**

** per rotor**)

M anufacturers

of large

gas turbines

and steam turbines for industrial pow

er generation include

• A lstom

• M itsubishi

• Siem ens

• G eneral E

lectric

A ll use axial flow

turbom achines

IIA P

aper 3A 3 Fluid M

echanics II: T urbom

achinery/H P

H

1-4

1.2.2 A

ero E ngines &

A ero D

erivatives

B ig 4 m

anufacturers:

• G eneral E

lectric

• Pratt & W

hitney

• R olls-R

oyce

• SN E

C M

A

A ll use axial flow

com pressors and axial flow

turbines except for the sm allest of engines (eg helicopters

and U A

V s) w

hen radial flow com

pressors are used

IIA P

aper 3A 3 Fluid M

echanics II: T urbom

achinery/H P

H

1-5

1.2.3 R

adial T urbom

achinery

M any types &

configurations

M ost com

m on types

• centrifugal pum p or com

pressor w ith axial inflow

and radial outflow

• radial inflow -axial outflow

turbine

**A
n industrial centrifugal com
**

**pressor
**

**A
Sm
**

**all T
urbocharger
**

IIA P

aper 3A 3 Fluid M

echanics II: T urbom

achinery/H P

H

1-6

**T
he K
**

**aplan turbine has an radial flow
**

**stator and an axial flow
rotor
**

**T
he F
**

**rancis T
urbine has an radial flow
**

** stator and
a radial-axial flow
**

** rotor **

IIA P

aper 3A 3 Fluid M

echanics II: T urbom

achinery/H P

H

1-7

**1.3 A
im
**

**s of the C
ourse
**

T his course aim

s to provide an understanding of the principles that govern the fluid dynam ic operation

of axial and radial flow turbom

achines.

A t the end of this course, you should be able to

• Identify and understand the operation of different types of turbom achinery.

• A nalyse turbom

achinery perform ance.

• U nderstand the causes of irreversibilities w

ithin the blade passages

• A nalyse com

pressible flow through turbom

achines.

IIA P

aper 3A 3 Fluid M

echanics II: T urbom

achinery/H P

H

1-8

**1.4 W
hat is in this course?
**

• 4 types of m achines:

o A

xial com pressors

o A

xial gas turbines and axial steam turbines,

o C

entrifugal com pressors

o R

adial inflow -axial outflow

turbines

• A nalysis of the flow

in bladerow s and stages (1D

)

• D ynam

ic scaling, characteristics of com pressors and turbine

• C om

pressible Flow M

achines

• H ub-T

ip variations in flow properties (2D

)

IIA P

aper 3A 3 Fluid M

echanics II: T urbom

achinery/H P

H

1-9

**1.5 L
aboratory E
**

**xperim
ent
**

E valuation of pum

p perform ance

• m easure perform

ance param eters

• study effects of R eynolds num

ber

• exam ine the effects of and visualise cavitation

IIA P

aper 3A 3 Fluid M

echanics II: T urbom

achinery/H P

H

1-10

**1.6 R
ecom
**

**m
ended B
**

**ooks
A
**

**uthor
T
**

**itle
Shelf M
**

**ark
**

D ixon, S L

Fluid

m echanics,

T herm

odynam ics

of T

urbom achinery,

T N

24

C ohen, H

., R

ogers, G .F.C

., and Saravanam

uttoo, H .I.H

.

G as T

urbine T heory

V K

33

C um

psty, N .A

. C

om pressor A

erodynam ics

V S 16

R olls-R

oyce T

he Jet E ngine

V N

36

IIA P

aper 3A 3 Fluid M

echanics II: T urbom

achinery/H P

H

1-11

**1.7 N
otation
**

G eom

etric & Flow

Param eters

A x

A nnulus area =

*h
*

*R
R
*

*R
A
*

*x
*m

ean 2hub

2casing 2

) (

π π

= −

=

A x

Passage area = (blade height) × (blade pitch)

A

E ffective flow

area =
*A
*

* =
A
*

*x * cosα
b

axial w idth of radial im

peller (i.e. blade span) α

Flow

angle in absolute co-ordinate system

Flow

angle in rotating co-ordinate system

D

diam eter (usually m

ean or tip) ψ

stage loading coefficient

h A

nnulus height, blade height, span =
*h =
*

* r
casing – r
*

*hub *
h

enthalpy m &

M

ass flow rate

Q

V olum

etric flow rate

φ Flow

coefficient =
*U
*

*V
x
*

(or 3

D Q

Ω &

in “scaling” applications) Λ

reaction

r R

adius s

pitch (spacing) of blades σ

slip factor U

B

lade speed (usually at m ean radius) =

*U
* =

1/
2(*U
*

*casing * +
*U
*

*hub *)

IIA P

aper 3A 3 Fluid M

echanics II: T urbom

achinery/H P

H

1-12

V

Flow velocity

V x

A xial velocity

V θ

T angential velocity

V ρ

R adial velocity

V θ

� ,rel T

angential velocity in rotating co-ordinate system

V rel

V elocity in rotating co-ordinate system

Suffices: 0

stagnation 1

inlet to 1st blade row

2 cascade exit or 1st blade row

exit / 2nd blade row inlet

3 stage exit / 2nd blade row

exit / 3rd blade row inlet

m , m

ean value at m

ean radius r

radial rel

relative fram e of reference (rotating fram

e) x

axial y, θ

tangential

IIA P

aper 3A 3 Fluid M

echanics II: T urbom

achinery/H P

H

2-13

**2 B
asic C
**

**oncepts
2.1 Stagnation (total) and Static (T
**

**rue T
herm
**

**odynam
ic) Q
**

**uantities
**

*w
x
*

*q
*

2 1

C o

ntro l

Vo lum

e

A ssum

ing gravity can be neglected, application of the SFE E

to the above gives

+

−

+

= −

2 2

21 1

22 2

*V
h
*

*V
h
*

*w
q
*

*x
*

N ow

, the *stagnation* (total) specific enthalpy h
0 is given by:

2

2 1 0

*V
h
*

*h
*+

≡

so the SFE E

can be w ritten:

01

02
*h
*

*h
w
*

*q
x
*

− =

−

In a turbom achine, the w

ork exchange occurs because of changes in m om

entum (velocity) so the

im portance of the kinetic energy in the SFE

E cannot be ignored.

IIA P

aper 3A 3 Fluid M

echanics II: T urbom

achinery/H P

H

2-14

T herefore, especially w

hen w e are dealing w

ith individual stages (i.e. single rotor+ stator com

binations), w

e m ust specify if the p, T

and h that w e are using are the

• *stagnation* (total) pressure, tem
perature and specific enthalpy or

• the *static *(ie true therm
odynam

ic) pressure, tem perature and specific enthalpy.

For a perfect gas, the static and stagnation tem peratures T

and T 0 are related to hand h

0 by

2

2 1 0

0 )

(
*V
*

*T
T
*

*c
h
*

*h
p
*

= −

= −

It is T 0 rather than h

0 that is m easured experim

entally. T his can be done by m

ounting a therm ocouple

inside som ething like a Pitot tube.

B y w

orking in term s of stagnation (total) quantities

• kinetic energy effects are autom atically taken care of,

• analyses are easier (*stagnation* quantities are easier to m
easure than *static* quantities).

N ote that if the type is not specified or im

plied, it is usually safer to assum e that the p, T

and h represent
the *stagnation* pressure, *stagnation* tem

perature and *stagnation* specific enthalpy.

IIA P

aper 3A 3 Fluid M

echanics II: T urbom

achinery/H P

H

2-15

**2.2 T
he air-standard Joule (B
**

**rayton) cycle
**T

he *closed* air-standard Joule/B
rayton cycle is the

• is the sim plest m

odel of the *open* circuit gas turbine

• is the basic standard against w hich w

e assess practical applications

• is a very good m odel of the actual engine

A ssum

ptions:

• A ll processes are reversible

• *c
p *and γ are constant around the cycle

• N o pressure change (i.e. no losses) in the heat exchangers

• In the com pressors and turbines, everything happens so quickly that there is no tim

e for any heat
transfer, i.e., they are *adiabatic*

In this course, w e w

ill assum e all of the above except that w

e w ill often allow

irreversibilities to occur.

IIA P

aper 3A 3 Fluid M

echanics II: T urbom

achinery/H P

H

2-16

Turbine S

haft

C O

N T

R O

L S U

R FA

C E

2

1

3

4

inout

.. C W

.

W =

W -W

x

. .

C T

.

**C
losed C
**

**ircuit G
as T
**

**urbine **

IIA P

aper 3A 3 Fluid M

echanics II: T urbom

achinery/H P

H

2-17

Y ou should be able to show

that the efficiency is given by

*t
*

*cycle
r *1

1 −

= η

w
here *r
*

*t * is the isentropic tem
perature ratio

γ

γ )1

(

4 3

1 2 −

= =

=
*p
*

*t
r
*

*T T
*

*T T
r
*

and *r
p * is the pressure ratio

4 3

1 2

*p p
*

*p p
rp
*

= =

For the ideal cycle

• η
*cycle * depends only on the pressure ratio rp .

(for a real gas turbine, it also depends on the ratio 1

3
*T
*

*T
*).

• η
*cycle * increases m

onotonically w
ith increasing *r
*

*p *.

IIA P

aper 3A 3 Fluid M

echanics II: T urbom

achinery/H P

H

2-18

**2.3 Irreversible T
urbom
**

**achines: Isentropic efficiencies
**W

e are used to dealing w ith these in the context of entire turbines (gas or steam

) or entire com pressors.

H ow

ever,

• T he sam

e definitions can also be applied to individual rotor+ stator com

binations (i.e. stages)

T
he *isentropic* efficiencies are defined as the ratio of the

• the actual w ork and

• the isentropic w ork

that occur betw een

• the specified inlet conditions and

• the specified exit pressure

T herefore, especially w

hen w e are dealing w

ith individual stages, w e m

ust specify if the p, T and h that

w
e are using are the *stagnation* (total) or the *static *values.

IIA P

aper 3A 3 Fluid M

echanics II: T urbom

achinery/H P

H

2-19

2.3.1 T

otal-T otal Isentropic E

fficiencies

A lthough you m

ay not have realised it, in Part I you have been using stagnation (total) quantities to define the isentropic efficiencies. T

hese are used w hen

• the kinetic energy of the flow is very sm

all or

• w here the kinetic energy of the flow

leaving one com ponent (eg stage) is not w

asted by a dow

nstream com

ponent

**C
om
**

**pressor
**

01Entropy s

02

h (or T)

02s

w is

w

01 02

01 s

02 tt

h h

h h

w ork

actual w ork

ideal − −

= ≡

η

**G
as or Steam
**

** T
urbine
**

03

E ntropy s

04

h (or T)

04s

w

w is

s 04

03

04 03

tt h

h h

h w

ork ideal w

ork actual

− − =

≡ η

IIA P

aper 3A 3 Fluid M

echanics II: T urbom

achinery/H P

H

2-20

2.3.2 T

otal-Static Isentropic E fficiencies

W e use these definitions w

hen

• the kinetic energy of the flow leaving one com

ponent (eg stage) is w asted by a dow

nstream

com ponent

T his m

ost often happens w hen w

e w aste the exit kinetic energy of an entire turbom

achine, e.g.

• in the exhaust duct of a steam turbine

• w hen a fan or pum

p exhausts directly into the atm osphere

T he total-static efficiency is alw

ays less than the total-total efficiency. T he difference is due to the so-

called *leaving loss* (i.e. the exit K
E

)

IIA P

aper 3A 3 Fluid M

echanics II: T urbom

achinery/H P

H

2-21

**C
om
**

**pressor
**

w

w

is, no exit KE

actual

Exit KE

01Entropy s

022

h (or T)

02s2s

01 02

01 s

2 ts

h h

h h

w ork

actual w ork

ideal − −

= ≡

η

**G
as or Steam
**

** T
urbine
**

03

E ntropy s

04

h (or T)

04s

w

w

is, no exit KE

actual

4 4s

P 03

P 04

P 4

Exit KE

s 4

03

04 03

ts h

h h

h w

ork ideal w

ork actual

− − =

≡ η

**T
otal-Static Isentropic E
**

**fficiencies
**

IIA P

aper 3A 3 Fluid M

echanics II: T urbom

achinery/H P

H

2-22

2.3.3 E

xam ple

*T
he inlet and exit conditions to a turbine are:
*

inlet:

*T
*03 =

1000 K

*P
*

03 = 2.0 bar

exit:

*T
*04 =

874 K

*P
*

04 = 1.2 bar

*P
*

4 = 1.17 bar

C alculate both the total-to-total and total-to-static isentropic efficiencies. A

ssum e the flow

is a perfect gas w

ith γ = 1.4.

total-to-total: γ

γ )1

(

03 04 03

04

−

=
*P P
*

*T
T
*

*s
* =

4.

1 )4

.0 (

0. 2

2. 1

1000

=

864.2 K

928 .0

2. 864

1000 874

1000 w

ork ideal w

ork actual

04 03

04 03

04 03

04 03

= − −

= − −

= − −

= ≡

*s
s
*

*tt
T
*

*T
*

*T
T
*

*h
h
*

*h
h
*

η

total-to-static: γ

γ )1

(

03 4 03

4

−

=
*P P
*

*T
T
*

*s
* =

4.

1 )4

.0 (

0. 2 17 .1

1000

=

857.97 K

887 .0

97 .

857 1000

874 1000

w ork

ideal w ork

actual

4 03

04 03

4 03

04 03

= − −

= − −

= − −

= ≡

*s
s
*

*ts
T
*

*T
*

*T
T
*

*h
h
*

*h
h
*

η

IIA P

aper 3A 3 Fluid M

echanics II: T urbom

achinery/H P

H

2-23

**2.4 T
he R
**

**ankine Steam
C
**

**ycle
**T

his is the basis of alm ost every practical steam

cycle for large scale pow er generation.

4

3

2

1

*Q
in * from

com bustion gas

*W
P
*

*W
T
*

feed pum

p steam

turbine

steam generator

condenser

*Q
out * to cooling w

ater

.

. .

.

03

s

T

04s 04

0201

Y ou should already know

that per unit m ass of steam

circulating, the feed pum p w

ork input is given by com

bining the SFE E

w
ith *T
*

*ds =
dh *−* dp/*ρ and assum

ing that the w ater is incom

pressible

IIA P

aper 3A 3 Fluid M

echanics II: T urbom

achinery/H P

H

2-24

ρ

η ρ

η η

*pum
p
*

*s
*

*pum
p
*

*pum
p
*

*s
pum
*

*p
p
*

*p
dp
*

*h
h
*

*h
h
*

*w
*01

02 0201

01 02

01 02

1 )

( −

≅ =

− =

− =

∫

w here η

*pum
p * is the total-total isentropic efficiency of the feed pum

p and ρ is the density of w ater. 1

T he heat input in the boiler and heat rejected in the condenser are given by

02

03
*h
*

*h
q
*

*in
*−

= and

01 04

*h
h
*

*q
out
*

− =

T he turbine w

ork output is given by

)

( 04

03 04

03
*s
*

*isentropic
T
*

*h
h
*

*h
h
*

*w
*−

= −

= η

w here η

*isentropic * is the total-total isentropic efficiency of the w
hole turbine.

1 T

he final part of this expression is m uch m

ore accurate and convenient to use than interpolating for liquid enthalpies in the steam

tables.

IIA P

aper 3A 3 Fluid M

echanics II: T urbom

achinery/H P

H

2-25

**2.5 Sum
m
**

**ary
**W

e m ust alw

ays specify if the p, T
and *h* that w

e are using are the

• *stagnation* (total) pressure, tem
perature and specific enthalpy or

• *static *(ie true therm
odynam

ic) pressure, tem perature and specific enthalpy.

For com pressors:

• 01

02

01 02

w ork

actual w ork

ideal
*h
*

*h
*

*h
h
*

*s
tt
*

− − =

≡ η

01

02

01 2

w ork

actual w ork

ideal
*h
*

*h
*

*h
h
*

*s
ts
*

− − =

≡ η

For a gas or steam turbine:

•
*s
*

*tt
h
*

*h
*

*h
h
*

04 03

04 03

w ork

ideal w ork

actual − −

= ≡

η

*s
ts
*

*h
h
*

*h
h
*

4 03

04 03

w ork

ideal w ork

actual − −

= ≡

η

For incom pressible pum

ps, if pum

p w

is the actual specific w ork input

• ρ

η
*pum
*

*p
tt
*

*w
*

*p
p
*

01 02

− =

ρ

η
*pum
*

*p
ts
*

*w
*

*p
p
*

01 2

− =

IIA P

aper 3A 3 Fluid M

echanics II: T urbom

achinery/H P

H

3-26

**3 F
low
**

** V
elocities and V
**

**elocity T
riangles
**

**3.1 B
asic C
**

**oordinate System
s and V
**

**elocities
**E

arlier, w e defined a turbom

achine as a steady flow device w

hich creates/consum es shaft-w

ork by changing the m

om ent of m

om entum

of a fluid passing through a rotating set of blades.

T herefore, w

e m ust consider

• the m om

ent of m om

entum

• rotation about an axis

A s a result,

• w e use an x-r-θ coordinate system

x = axial direction

r = radial direction

θ = tangential/circum

ferential direction

• w e need to w

ork in the *stationary* (*absolute*) and *rotating* (*relative*) fram
es of reference

IIA P

aper 3A 3 Fluid M

echanics II: T urbom

achinery/H P

H

3-27

In the stationary fram e, w

e have

V x =

axial velocity

V r =

radial velocity

V θ =

tangential/circum ferential/sw

irl velocity

V x

V r

V θ

V

x r

θ

Ω

W e note that:

T he sign convention used throughout this course (and by m

uch of industry) is that tangential/circum

ferential/sw irl velocities are positive if they are in the sam

e direction as the rotation of the rotor.

IIA P

aper 3A 3 Fluid M

echanics II: T urbom

achinery/H P

H

3-28

T he analysis of the flow

through rotating blade row s (rotors) can be greatly sim

plified by w orking in a

fram e of reference so that the rotors appear to be at rest.

V r

V θ,rel

r Ω Ω

r

V θ

**A
xial view
**

** of the com
ponents of the absolute and rotor relative velocity vectors
**

W e first note that in the both fram

es of reference, w e have

V

x = axial velocity

V

r = radial velocity

IIA P

aper 3A 3 Fluid M

echanics II: T urbom

achinery/H P

H

3-29

and that the tw o (stationary/absolute and rotational/rotor relative) fram

es of reference are related according to the vector expression

absolute velocity =

relative velocity + rotational velocity

Since V x andV

r are the sam e in both fram

es of reference, the only difference betw een the absolute and

relative velocities is due to the m agnitude of the circum

ferential velocity.

In fact,

r

V V

rel ,

Ω +

= θ

θ

w here

rel ,

V θ

= rotor relative tangential/circum

ferential/sw irl velocity

and

=

= Ω

U r

blade speed

IIA P

aper 3A 3 Fluid M

echanics II: T urbom

achinery/H P

H

3-30

For axial m achines:

*
V
*

*x >
>
*

* V
r *

For radial m achines, at the outer radius

*
V
*

*x <
<
*

* V
r*

and at the inner radius, depending on w hether the flow

is m ainly axial or radial,

*
V
*

*x >
>
*

* V
r *

or

* V
*

*x <
<
*

* V
r *

IIA P

aper 3A 3 Fluid M

echanics II: T urbom

achinery/H P

H

3-31

**3.2 M
ean-line A
**

**nalyses
**D

esign m ethods for turbines, com

pressors and pum ps usually involve a num

ber of separate processes.

T he first step is to use

• 1-D calculations along m

ean radius, i.e. m ean-line analyses

to exam ine

• m ean radius velocity triangles

before and after each bladerow

In doing so, w e assum

e that

• the span (hub-tip length) of the blades is sm all in relation to the m

ean radius so that

• the variation of the flow in the hub-tip direction can be neglected

• the m ean radius =

*r
m
*

*ean *=
(*r
*

*casing * +
*r
*

*hub *) / 2

IIA P

aper 3A 3 Fluid M

echanics II: T urbom

achinery/H P

H

3-32

**A
xial F
**

**low
P
**

**um
p **

IIA P

aper 3A 3 Fluid M

echanics II: T urbom

achinery/H P

H

3-33

**3.3 V
elocity T
**

**riangles for an A
xial T
**

**urbine Stage (Stator+R
otor)
**

For sim plicity, w

e w ill assum

e that

• the variation of the flow in the radial direction is sm

all

• the radial com ponent of velocityis negligible (V

r = 0)

• there is no change of radius (r) through the stage

• the blade speed ( U

r= Ω

) is constant

• the variation of the flow in the circum

ferential direction is sm all

• w e can exam

ine the flow by looking an unw

rapped (ie developm ent of) cylindrical surface of

revolution, i.e. by using the *cascade* (x–y or x-rθ) plane

W e recall that

• flow angles are positive if they are in the sam

e direction as the rotation of the rotor.

N ow

,

• turbines use stators to create a m om

ent of m om

entum w

hich is then rem oved in the rotor.

IIA P

aper 3A 3 Fluid M

echanics II: T urbom

achinery/H P

H

3-34

V 2

,re l

α 2,re

l

α 2

V 2

U

U

S TA

T O

R R

O T

O R

rθ

x V

θ2,re l

V θ1

V θ2

V x2

α 2

V 2

V 2

,re l

V 1

α 2

,re l

α 1

V x2

V x1

**A
xial T
**

**urbine Stator E
xit/R
**

**otor Inlet V
elocity T
**

**riangle V
iew
**

**ed R
adially **

IIA P

aper 3A 3 Fluid M

echanics II: T urbom

achinery/H P

H

3-35

A t the inlet of our stator

axial velocity 1

1 1

cos α

* V
*

*V
x
*

=

tangential velocity 1

1 1

sin α

* V
*

*V
*θ

=

A t the exit/outlet of our stator

axial velocity 2

2 2

cos α

* V
*

*V
x
*

=

tangential velocity 2

2 2

sin α

* V
*

*V
*θ

=

Finally, w e note that (see later):

• the absolute exit flow angle α

of a stator & the relative exit flow

angle
*rel
*

α of a rotor tend to be

independent of the operating condition even w hen the inlet flow

angle to the sam e bladerow

or the velocities change.

IIA P

aper 3A 3 Fluid M

echanics II: T urbom

achinery/H P

H

3-36

N ow

, w e again note that the analysis of the flow

through rotating blade row s (rotors) can be greatly

sim plified by w

orking in a fram e of reference so that the rotors appear to be at rest.

W e recall that the axial velocity

2 x

V is the sam

e in both fram es of reference and that

*U
*

*V
r
*

*V
V
*

*rel
rel
*

+ =

Ω +

= ,2

,2 2

θ θ

θ

w here

*U
r
*

= Ω

⇒

*U
V
*

*V
rel
*

− =

2 ,2

θ θ

T herefore, the rotor relative inlet flow

angle is given by

2

2

2

2 , 2

2 ,2

tan tan

*x
x
*

*x
*

*rel
*

*V U
*

*V
*

*U
V
*

*V
*

*V
rel
*

− =

− =

= α

α θ

θ

W e now

look at the rotor exit.

IIA P

aper 3A 3 Fluid M

echanics II: T urbom

achinery/H P

H

3-37

V 3

,rel

V 3

V 2

,re l

α 2,re

l

α 3,re

l

α 2

α 3

V 2

S TA

T O

R R

O T

O R

rθ

x

V θ1

V 1

α 1V

x1 U

3

U 2

Blade Speed

**V
elocity T
**

**riangles for an A
xial T
**

**urbine Stage V
iew
**

**ed R
adially
**

IIA P

aper 3A 3 Fluid M

echanics II: T urbom

achinery/H P

H

3-38

A t rotor exit, w

e note that

*U
*

*V
V
*

*rel *+
=

,3 3

θ θ

and that

3

3

,3

3 3 ,3

3 tan

tan
*x
*

*x rel
*

*x
V U
*

*V
*

*U
V
*

*V V
rel *+

= +

= =

α α

θ θ

If w e study the velocity triangles of the turbine as w

e have draw n them

, w e should notice that

• 1

2 α

α >>

and
*rel
*

*rel
*,2

,3 α

α >>

- turbine blades m

ake the flow m

ore tangential

• 3

2 1

*x
x
*

*x
V
*

*V
V
*

≈ ≈

- this is very com

m on

• 1

2
*V
*

*V
*>>

and
*rel
*

*rel
V
*

*V
*,2

,3 >>

- turbine blades accelerate the flow

- boundary layers thin and losses in efficiency are sm all

IIA P

aper 3A 3 Fluid M

echanics II: T urbom

achinery/H P

H

3-39

3.3.1 E

xam ple

*T
he flow
*

* leaving an axial turbine stator blade row
has a velocity 700 m
*

*s
-1 at an angle of 70°. T
*

*he
rotor has a blade speed of 500 m
*

*s
-1. T
*

*he flow
leaving a rotor blade row
*

* also has a relative velocity of
700 m
*

*s
-1 at a relative angle of -70°. N
*

*eglect any radial velocities and assum
e that the axial velocity is
*

*constant through the stage
*

*C
alculate the relative flow
*

* angle at rotor inlet and the absolute flow
angle at rotor exit.
*

T he stator exit/rotor inlet axial velocity is

2

2 2

cos α

* V
*

*V
x
*

= =

700 × cos70° = 239.4 m

s -1

T he stator exit/rotor inlet absolute tangential velocity is

2

2 2

sin α

* V
*

*V
*θ

= =

700 × sin70° = 657.8 m

s -1

T he stator exit/rotor inlet relative tangential velocity is

*U
*

* V
*

*V
*θ

*,rel
*θ

− =

2 2

= 657.8 – 500 =

157.8 m s

-1

T he stator exit/rotor inlet relative flow

angle is

=

−

2 , 2

1 ,2

tan
*x
*

*rel
rel
*

*V
*

*V*θ
α

=

° =

−

4. 33

4. 239

8. 157

tan 1

(sign indicates sam

e direction as blade speed)

IIA P

aper 3A 3 Fluid M

echanics II: T urbom

achinery/H P

H

3-40

T he rotor exit axial velocity is

*V
V
*

*x
x
*

2 3

= =

239.4 m s

-1

T he rotor exit relative tangential velocity is

*rel
*

*x
rel
*

*rel
rel
*

θ

α

*V
*α

* V
*

*V
*,3

3 ,3

,3 ,3

tan sin

= =

= 239.4 × tan(-70°) =

-657.8 m s

-1

(sign indicates opposite direction as blade speed)

T he rotor exit absolute tangential velocity is

*U
*

*
V
*

*
V
*

*rel
*θ

θ

+ =

,3 3

= -657.8 +

500 = -157.8m

s -1

(sign indicates opposite direction as blade speed)

T he rotor exit absolute flow

angle is

°

− =

− =

=

− −

4. 33

4. 239

8. 157

tan tan

1

3 3 1

3
*x *θ

*V V
*

α

(sign indicates opposite direction as blade speed)

IIA P

aper 3A 3 Fluid M

echanics II: T urbom

achinery/H P

H

3-41

**3.4 V
elocity T
**

**riangles for an A
xial C
**

**om
pressor Stage (R
**

**otor+Stator)
**For sim

plicity, w e w

ill again assum e that

• the radial com ponent of velocityis negligible (V

r = 0)

• the variation of the flow in the radial direction is sm

all

• there is no change of radius (r) through the stage

• the blade speed ( U

r= Ω

) is constant

• the variation of the flow in the circum

ferential direction is sm all

• w e can exam

ine the flow by looking an unw

rapped (ie developm ent of) cylindrical surface of

revolution, i.e. by using the cascade (x–y or x-rθ) plane

N ow

:

• com pressors use rotors to create a m

om ent of m

om entum

w hich is then rem

oved in the stator to create a further pressure rise.

IIA P

aper 3A 3 Fluid M

echanics II: T urbom

achinery/H P

H

3-42

V 1

,re l

V 2

,re l

V 1

V 2

U U

R O

T O

R S

TA T

O R

U Blade Speed

θ

x α

2 ,re

l

α 2

V θ3

V 3

α 3V

x3

α 1

α 1,rel

**V
elocity T
**

**riangles for an A
xial C
**

**om
pressor Stage V
**

**iew
ed R
**

**adially **

IIA P

aper 3A 3 Fluid M

echanics II: T urbom

achinery/H P

H

3-43

If w e study the velocity triangles of the com

pressor as w e have draw

n them , w

e notice that

• 3

2 α

α >>

and
*rel
*

*rel
*,2

,1 α

α >>

- com

pressor blades m ake the flow

m ore axial

• 3

2 1

*x
x
*

*x
V
*

*V
V
*

≈ ≈

- this is very com

m on

• 3

2
*V
*

*V
*>

and
*rel
*

*rel
V
*

*V
*,2

,1 >

- com

pressor blades decelerate the flow (by about 30%

)

- static pressure rises

- boundary layers thicken & separation is a big risk

- losses in efficiency are higher than in turbines

- m ore stages for sam

e pressure change cf. turbines.

IIA P

aper 3A 3 Fluid M

echanics II: T urbom

achinery/H P

H

3-44

**3.5 Sum
m
**

**ary
**W

e use tw o fram

es of reference that are related according to the vector expression

absolute velocity =

relative velocity + rotational velocity

⇒

*U
V
*

*V
rel *+

= ,2

2 θ

θ w

here
*U
*

*r
*=

Ω

In axial flow turbines (stator +

rotor):

• blades m ake the flow

m ore tangential

• often 3

2 1

*x
x
*

*x
V
*

*V
V
*

≈ ≈

• flow accelerates (thin boundary layers) so good efficiency.

In axial flow com

pressors (rotor + stator):

• blades m ake the flow

m ore axial

• often 3

2 1

*x
x
*

*x
V
*

*V
V
*

≈ ≈

• flow decelerates (boundary layers thicken) so low

er efficiency

• m ore stages needed for sam

e pressure change cf. turbines.

IIA P

aper 3A 3 Fluid M

echanics II: T urbom

achinery/H P

H

4-45

**4 M
ass F
**

**low
R
**

**ates/F
orces/W
**

**ork/SF
E
**

**E
**

**4.1 T
he calculation of m
**

**ass flow
rate in axial turbom
**

**achines
**T

he ability to apply the law of conservation of m

ass to a turbom achine blade row

is fundam ental to

m any turbom

achine calculations.

C o

ntro l

Vo lum

e

*rel
*1

scosα 1rel

scosα2rel

s

*rel
*2

**Inlet and exit flow
areas of an axial com
**

**pressor rotor in x-r**θθθ θ** plane
**

IIA P

aper 3A 3 Fluid M

echanics II: T urbom

achinery/H P

H

4-46

First, w e exam

ine the flow at inlet to and exit from

a 2D com

pressor rotor of blade span *h* and blade
pitch *s* in the relative fram

e. W e assum

e that

• the blade span *h *is sm
all in relation to the m

ean radius

• the geom etry and flow

conditions (velocities and angles) are constant across the span.

C onservation of m

ass gives for one blade passage

(

) (

)
*,rel
*

*,rel
s
*

*h
V
*

*=
*ρ

*s
h
*

*V
*ρ

*A
V
*

ρ

*m
,rel
*

*,rel
,rel
*

*,rel
passage
*

1 2

cos cos

1 1

2 2

2 2

2 α

α =

= &

or, m
ore generally, if *A
*

*x * =
*hs* w

hich is the *cross-sectional* or *frontal* area of the passage, then

(

) (

) constant

cos cos

= =

= α

ρ α

ρ
*x
*

*x
rel
*

*passage
A
*

*V
A
*

*V
m
*

*rel
*&

w here

• the flow area (

) α

cos
*x
*

*A
* is alw

ays m easured perpendicular to the velocity vector

• failure to observe this im portant sim

ple rule has serious consequences w hen dealing w

ith com

pressible flow (you have been w

arned!)

IIA P

aper 3A 3 Fluid M

echanics II: T urbom

achinery/H P

H

4-47

It also happens that the above can be w ritten as (hence the previous w

arning)

(

) (

)
*x
*

*x
x
*

*x
rel
*

*passage
A
*

*V
A
*

*V
A
*

*V
m
*

*rel
*ρ

α ρ

α ρ

= =

= cos

cos &

N ow

, if there are *Z* blades, then the total m
ass flow

rate through the com pressor rotor is

(

) (

) )

Z( cos

cos
*x
*

*x
x
*

*x
rel
*

*passage
com
*

*pressor
A
*

*V
A
*

*Z
V
*

*A
Z
*

*V
m
*

*Z
m
*

*rel
*ρ

α ρ

α ρ

= =

= =

& &

h

R hub

R casing

R S

R S

**A
xial (r-**θ)

θ)
θ)
θ) **and M
**

**eridional (x-r) view
s of a 1-stage com
**

**pressor
**

IIA P

aper 3A 3 Fluid M

echanics II: T urbom

achinery/H P

H

4-48

For the com pressor, the m

ean radius defined as

2

sin
*hub
*

*g
ca
*

*m
ean
*

*r
r
*

*r
*+

=

N ow

*m
*

*ean
r
*

*Z
s
*

π2 =

so the area of the annulus is

(

)( )

2hub 2casing

hub casing

hub casing

2
*r
*

*r
r
*

*r
r
*

*r
h
*

*r
Z
*

*sh
A
*

*m
ean
*

*x
*π

π π

π −

= +

− =

= =

T herefore, w

hether w e exam

ine a com plete bladerow

or just one blade passage:

*x
x
*

*rel
x
*

*rel
x
*

*A
V
*

*
A
*

*V
*

*V
A
*

*A
V
*

*m
*ρ

α ρ

α ρ

ρ =

= =

= cos

cos &

= const

w
here *A
*

*x * is the annulus area or the passage area as appropriate.

IIA P

aper 3A 3 Fluid M

echanics II: T urbom

achinery/H P

H

4-49

T his is w

hy in com pressors:

• flow is turned to be m

ore axial

• Inlet flow area >

E xit flow

A rea

• Flow decelerates

• Static pressure rises in each bladerow

A nd in turbines:

• Flow is turned to be m

ore tangential

• Inlet flow area <

E xit flow

A rea

2

• Flow accelerates

• Static pressure falls in each bladerow

2 T

his is generally true - except for the true im pulse rotor w

here there is no change in pressure and consequently no change in relative velocity across the rotor (see section 5.3.2)

IIA P

aper 3A 3 Fluid M

echanics II: T urbom

achinery/H P

H

4-50

4.1.1 E

xam ple

*T
he axial turbine in E
*

*xam
ple 3.3.1 has a constant m
*

*ean radius of 0.5 m
and the blade span is constant
*

*and equal to 0.075 m
. T
*

*he inlet stagnation tem
perature to the stage is 1800 K
*

* and the inlet stagnation
pressure is 30 bar. T
*

*he flow
is isentropic. A
*

*ssum
e that the gas has the properties of air. N
*

*eglect any
radial velocities.
*

*C
alculate the m
*

*ass flow
rate of the turbine.
*

W e already know

*V
*2

= 700 m

s -1

*V
x*2

= 239.4 m

s -1

T here is no w

ork done in the stator therefore, the SFE E

(

)( ) 21

2 1 1

22 2 1

2
*V
*

*h
V
*

*h
w
*

*q
x
*

+ −

+ =

−

can be w ritten

01

02
*h
*

*h
*=

w here

2 2 1

0
*V
*

*h
h
*

+ ≡