Turbomachinery, Lecture Notes- Physics, Study notes for Physics. University of Cambridge
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hawking7 September 2011

Turbomachinery, Lecture Notes- Physics, Study notes for Physics. University of Cambridge

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Definition, Example of turbomachinery, aero Engines and aero derivatives ,radial turbomachinery stagnation and static quantities, air standard joule cycle, Irresversible turbomachinery, rankine steam, cycle mean line ana...
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Microsoft Word - 3A3 Lecture Notes V5.doc

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

QQ

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

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

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

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

Vθ 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 =

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-rθθθ 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 x2

= 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

+ ≡

comments (1)
Covers axial turbo-machinery very well. Quite limited with radial turbo-machinery though...
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