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1

Axial Flow Turbines

3-D Flow theories

Following Three 3-D flow models in axial turbines are often used for design and analysis

1. Free Vortex flow

2. Constant nozzle exit angle, α 2

3. Arbitrary vortex case, C (^) w = rn

1. Free Vortex Flow model

C (^) w. r = constant, applied on the rotor flow which normally entails a few assumptions :

At turbine rotor entry , dH 02 /dr = 0 ; C (^) w2 .r = constant; C (^) a2 = const

Rotor specific work done :

H 02 – H 03 = U (C w2 + C w3 )= ω(r 2 .C w2 - r 3 .C w3 )

= constant With C (^) w3 .r = constant, it follows C (^) a3 = const

Constant Nozzle exit angle model

This model has been utilized for the practical purpose of creating stator-nozzle blades with zero twist. When stator-nozzles are facing very high inlet temperature elaborate cooling mechanism is embedded inside the blades; to facilitate efficient cooling of the blades, it is thought that such blades may not be twisted at all.

α 2 = constant

2 a2 w a2 w2 2

w

2 a

.cot^ α dr

dC dr

C C .cot^ α; which yields dC

const C

C

cot^ α

= =

= =

dH dC^ dC^ C^2

= C a^ +C w^ + w

dr a dr w dr r

Now invoking the radial equilibrium equation in energy eqn dH = 0 dr

and,

We get,

dC dC C^2

C a^ +C w^ + w

a dr w dr r = 0

sin^2 α

r

rm C 2 C2m

sin^2 α

r

rm Ca2 Ca2m

, C .r const

sin^2 α

r

rm C .r const ; Cw2 Cw2m

(^22)

(^22)

sin^ α a

sin^ α w

 

  

= 

 

  

= 

=

 

  

= = 

and finally in terms of absolute velocity

and then

alternate ly

and then

So, at the rotor inlet station one can say,

α 2 = constant

then,

r m

r C2m

C 2 Ca2m

Ca Cw2m

Cw2 (^) = = =

if

U (C (^) w2 + C (^) w3 )= ΔH (^0)

a) Constant Total Enthalpy at the outlet condition, if applied

ω

α

0

2 2 2

0 3

where,

tan

K^ H

C r

K C U

H C (^) w w a

=^ ∆

− = −

∆

And, whirl component of the velocity at rotor outlet is found from :

And, subsequently C (^) a3 may be also computed Both of which are computed from root to tip, using the variation shown in slide 10

b) Zero rotor exit whirl velocity α 3 = 0 this means, dH/dr = C (^) a3. d C (^) a3 / d r And , H 03 =H 02 – U C (^) w2 = H 02 -U.C (^) w2m(r (^) m /r)Sin (^2) α

sin^22

2

α

r

r

U C

dr

d

dr

dH m

w m

Which, produces the enthalpy distribution radially at exit :

3. Arbitrary vortex case, C (^) w = rn

Depending on the value of n there are four flow variation possibilities

i) n = -1 -- resolves to Free Vortex Model

ii) n= 0 ----- resolves to constant free vortex

iii) n= 1 ---- gives ‘solid body rotation’ model

iv) n= -2 --- produces ‘strong vortex flow’

All these flow models are considered for use in Turbine design and preliminary analysis