Turbine Design Parameters - energy.kth.seLecture_n… · Turbomachinery Lecture Notes 4 2007-09-06...

Turbomachinery Lecture Notes 1 2007-09-06

Turbine Design Parameters

Damian Vogt Course MJ2429

Nomenclature

Symbol Denotation Unit c Absolute velocity m/s h Enthalpy J/kg m& Mass flow rate kg/s r Radius m u Tangential velocity m/s w Relative velocity m/s α Absolute flow angle deg β Relative flow angle deg ω Rotational speed rad/s C Normalized absolute

velocity ucC = -

W Normalized relative velocity uwW =

-

U Normalized tangential velocity 1== uuU

-

Subscripts

0 Total 1 Inlet stator 2 Outlet stator (inlet rotor) 3 Outlet rotor n Normal r Radial component x Axial component θ Tangential component

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Turbine Stage Denotations and Conventions

1 2 stator

3 rotor

Reference radius

Stage denotations

1 stator inlet 2 rotor inlet 3 rotor outlet

ω

30

20

10

0

10

20

30

−30

−20

−10

0

10

20

c1

c2

w2

u

u

w3

c3

rotor stator Stage velocity triangles u

Velocity triangles denotations and conventions

Relative flow angle

Axial direction

Origin

Circumferential speed

Absolute velocity

Relative velocity

u

wθ

cθ

β α

w

c

Absolute flow angle

Relative circumferential velocity component

Absolute circumferential velocity component

Axial velocity component (absolute and relative)

cx=wx

θ neg α, β neg

x pos

α, β pos

θ pos

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Stage Velocity Triangles The stage velocity triangles are commonly employed as graphical measure to represent averaged kinetics of the flow at a reference radial position throughout the stage. Usually one of the following reference radial positions is used:

• Mean radius

2sh

mrr

r+

= Eq. 1

• Euler radius ( radius that splits the annular cross section in half)

2

22sh

Err

r+

= Eq. 2

The absolute frame of reference is bound to the stator and is therefore non-rotating. The relative frame of reference is bound to the rotor and rotates with the circumferential speed of the rotor u at the reference radius obtained from ω⋅= refru Eq. 3 The relation between the velocities in the absolute frame of reference (denoted “absolute velocities”) and the ones in the relative frame of reference (respectively denoted “relative velocities”) is the following Eq. 4 xx cw = Eq. 5 ucw −= θθ where cx and cθ are the axial and circumferential components of the respective velocity as follows Eq. 6 222

θccc x +=

Eq. 7 222θwww x +=

The flow angles are defined as

xccθα =tan Eq. 8

xwwθβ =tan Eq. 9

Note:

• The relative velocity is the velocity that an observer sees while sitting on the rotor • The rotor blades thus see the relative flow velocities • The direction of the absolute flow velocity at stator outlet corresponds approximately

to the stator blade metal angle at the trailing edge • The direction of the relative flow velocity at rotor outlet corresponds approximately to

the rotor blade metal angle at the trailing edge

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First Design Parameter: Degree of Reaction The degree of reaction relates the change in enthalpy effectuated in the rotor to the change in enthalpy of the stage as follows

stage

rotorhh

RΔΔ

= Eq. 10

, which can be rewritten as

3221

32hhhh

hhhh

hhh

Rrotorstator

rotor

stage

rotor−+−

−=

Δ+ΔΔ

=ΔΔ

= Eq. 11

The change in enthalpies in stator and rotor respectively are related to the velocities as follows

In the stator the stagnation enthalpy 2

2

0chh += is constant, thus

22

22

2

21

1c

hc

h +=+ Eq. 12

leading to

( ) Eq. 13 21

2221 2

1 cchh −=−

In the rotor the rothalpy 22

22 uwhI −+= is constant, thus

2222

23

23

3

22

22

2uw

huw

h −+=−+ Eq. 14

leading to

( 22

23

22

2332 2

1 uuwwhh +−−=− ) Eq. 15

Substituting these expressions into the equation of stage reaction above leads to the following general expression

22

23

22

23

21

22

22

23

22

23

uuwwcc

uuwwR

+−−+−

+−−= Eq. 16

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For a normal repetition stage with the following restrictions 31 cc rr

= Eq. 17 constccc xxx === 3,2,1, Eq. 18 Eq. 19 32 uu = the expression of the degree of reaction can further be simplified. Firstly it can be noted that the circumferential speed u cancels out. Secondly the velocities shall be written in terms of their components as and respectively. This yields the following expression

222θccc x += 222

θwww x +=

22,

23,

21,

22,

22,

23,

θθθθ

θθ

wwcc

wwR

−+−

−= Eq. 20

The relative velocity components in the denominator shall be expressed by the absolute velocity components as leading to ucw −= θθ

2

2,22,

23,

23,

21,

22,

22,

23,

22 uuccuucccc

wwR

−+−+−+−

−=

θθθθθθ

θθ Eq. 21

After canceling out elements the expression can be rewritten as

( )3,2,

22,

23,

2 θθ

θθ

ccuww

R−⋅

−= Eq. 22

At this stage the enumerator shall be expressed as ( ) ( )2,3,2,3,

22,

23, θθθθθθ wwwwww +⋅−=− . The

absolute velocity components in the denominator shall be expressed in terms of relative velocities as leading to uwc += θθ

( ) ( )

( )33,22,

2,3,2,3,2 uwuwu

wwwwR

−−+⋅

+⋅−−=

θθ

θθθθ Eq. 23

Both the circumferential speeds in the denominator and the relative components

cancel out finally yielding 2,3, θθ ww −

( 2,3,21

θθ wwu

R +−= ) Eq. 24

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At this position a more intimate analysis of the degree of reaction is appropriate. For this purpose the relative circumferential velocity component at position 2 shall be expressed in the absolute frame of reference as yielding ucw −= θθ

( ) ( 2,3,2,3, 21

21

21

θθθθ cwu

ucwu

R +−=−+−= ) Eq. 25

By expressing the circumferential velocity components in terms of flow angles as xccθα =tan the

following expression is obtained for the degree of reaction

( 23 tantan22

1 αβ +−=uc

R x ) Eq. 26

In the above equation the degree of reaction is expressed in terms of axial velocity component, circumferential speed and stator and rotor outflow angles respectively, which are approximately equal to blade metal angles at trailing edge. According to the convention of velocity components depicted above flow angle β3 is negative whilst flow angle α2 is positive. This leads to the following observations

• An increase in flow angle α2 leads to a decrease in degree of reaction ( ), i.e. the contribution of enthalpy change in the stator to the total change in enthalpy in the stage gets larger

↓↑⇒ R2α

• An increase in flow angle β3 leads to an increase in degree of reaction ( ), i.e. the contribution of enthalpy change in the rotor to the total change in enthalpy in the stage gets larger

↑↑⇒ R3β

• For turbine stages the degree reaction usually lies in the range [0…1]

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Second Design Parameter: Loading Factor The loading factor relates the change in total enthalpy effectuated in the stage to the rotational speed as follows

20

u

hΔ=ψ Eq. 27

Under application of Euler’s turbine equation the change in enthalpy can be expressed as

leading to 3,32,20 θθ cucuh −=Δ

2

3,32,2

u

cucu θθψ−

= Eq. 28

For a normal repetition stage with the following restrictions 31 cc rr

= Eq. 29 constccc xxx === 3,2,1, Eq. 30 Eq. 31 32 uu = the expression of the loading factor can further be simplified to

ucc 3,2, θθψ

−= Eq. 32

Expressing the absolute flow velocities in the relative frame of reference as the loading factor can be expressed as

uwc += θθ

uww 3,2, θθψ

−= Eq. 33

An equivalent expression can be obtained by substituting the relative velocity component at position 2 in the absolute frame of reference as uwc += θθ yielding

uwc 3,2,1 θθψ

−+−= Eq. 34

, which also can be expressed in terms of flow angles α2 and β3 as follows

( 32 tantan1 βαψ −+−=ucx ) Eq. 35

According to the convention of velocity components depicted above flow angle β3 is negative whilst flow angle α2 is positive. This leads to the following observations:

• Increase in flow angles α2 and β3 lead to increase in loading factor ( ) ↑↑⇒ ψβα 32 ,• To obtain a loading factor smaller than one 3tan β must be greater than 2tanα

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Third Design Parameter: Flow Coefficient The flow coefficient relate the axial velocity component to the circumferential speed as follows

ucx=φ Eq. 36

The only observation to make for this coefficient is that the higher the axial velocity in the stage the higher the flow coefficient. As can be recognized below the flow coefficient stretches the velocity triangles in the axial direction.

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The Normalized Velocity Triangle At this position the normalized velocity triangle shall be introduced. The normalization consists therein that all velocity components are depicted with reference to the outlet circumferential velocity u3. The normalized velocity components are denoted by the respective capital letters and yield from

3ucC = Eq. 37

3uwW = Eq. 38

3uuU = Eq. 39

The special case of a normal repetition stage shall be regarded here for the sake of simplicity. The applied principle is however valid for all types of turbine stages. Conveniently the velocity triangle is drawn with a common origin for stator and rotor outlet. As a normal repetition stage with the condition constccc xxx === 3,2,1, is considered the height of the

triangle corresponds to φ==uc

C xx , i.e. the flow coefficient.

W3

C3 W2

C2

U=1 U=1

22,3, θθ WW +−

R

ψ

Φ

Note:

• The height of the velocity triangle corresponds to the flow coefficient Φ • The loading coefficient corresponds to the circumferential distance between C2 and

C3. In the case of a repetition stage this equals to the circumferential distance between W2 and W3.

• The degree of reaction equals to the distance between axial and half the midpoint between W2 and W3.

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Special Cases The special cases are here analyzed for the case of normal repetition stage. Similar analysis can be performed in a general manner for other types of stages. Degree of Reaction equal to Zero (R=0; Action Turbine) The expression of the degree of reaction yields the following

( ) 3,2,2,3,210 θθθθ wwwwu

R −=⇒+−== Eq. 40

Substituting this expression into the equation of loading coefficient yields

( 1222,

2,3,2, −= )⋅=⇒

−= θ

θθθ ψψ cuu

wuww

Eq. 41

Velocity triangle

W3

C3 W2

C2

U=1 U=1

22,3, θθ WW +−

ψ

Φ

Note:

• As and normal stage it follows that 3,2, θθ ww −= 32 ww = and consequently . The change in enthalpy in an action stage is thus entirely due to change

in enthalpy in the stator. 0=Δ rotorh

• The forces acting on the rotor are action forces, as the fluid is not accelerated through the rotor. This leads to the denotation of “action stage”

• The rotor only effectuates deflection of the fluid but not expansion as 32 ββ −= • As the fluid is not expanded throughout the rotor the pressure up- and downstream of

the rotor is (practically) unchanged. In reality a minimum pressure drop is necessary due to losses to drive the fluid, thus 23 pp ≈

• As a consequence there is little axial force on the rotor in an action turbine

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Degree of Reaction equal to one half (R=0.5; Reaction Turbine) The expression of the degree of reaction yields the following

( ) 3,2,2,3,21

21

θθθθ wuwwwu

R −=+⇒+−== Eq. 42

, which is equivalent to Substituting this expression into the equation of loading coefficient yields

3,2, θθ wc −=

12

12 2,2,3,2, −=+

⋅=⇒

−=

uc

uw

uww θθθθ ψψ Eq. 43

Velocity triangle

W3

C3 W2

C2

U=1 U=1

22,3, θθ WW +−

ψ

Φ

R Note:

• As and normal stage it follows that 3,2, θθ wc −= 32 wc = and with the assumption of repetition stage ( ) consequently 31 cc = statorrotor hh Δ=Δ . The change in enthalpy in a reaction stage is thus equally split on stator and rotor.

• The forces acting on the rotor are partially action and partially reaction forces, as the fluid is accelerated through the rotor. This leads to the denotation of “reaction stage”

• Both stator and rotor effectuate expansion of the fluid and thus and 12 pp < 23 pp < • As a consequence there is a considerable axial force on the rotor in a reaction

turbine. In most cases this force is too large to be submitted to an axial bearing and thus must be compensated for. Possible compensations are appropriate arrangement of components such as to cancel out axial forces or application of a thrust compensation devices (e.g. piston), see further below

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Zero Exit Swirl (cΘ,3=0) With the degree of reaction writes to uwuwc −=⇒+== 3,3,3, 0 θθθ

( )u

www

uR

221

21 2,

2,3,θ

θθ −=+−= Eq. 44

The loading coefficient yields from

12,3,2, +=−

=uw

uww θθθψ Eq. 45

Combining the two expressions leads after reformulation to a relationship between degree of reaction and loading coefficient as follows )1(2 R−⋅=ψ Eq. 46 Velocity triangle

Φ

W3

C3

W2

C2

U=1 U=1

R 2

2,3, θθ WW +−

ψ

Note:

• The flow exists the stage purely axial, i.e. there is no swirl at stage exit • For a zero exit swirl stage the degree of reaction and the loading factor are

dependent • Example for a zero exit swirl stage are for example last stages of jet engines thrust

maximized when jet flow purely axial

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Turbine Design Parameters - energy.kth.seLecture_n… · Turbomachinery Lecture Notes 4 2007-09-06...

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