Useful Equations For Turbine Impulse Turbine: Cos U V r Q T

USEFUL EQUATIONS EQUATIONS FOR TURBINE IMPULSE TURBINE

It consists of one or more stationary inlet nozzles (Spear nozzles), a runner, and a casing. The runner has multiple buckets mounted on a rotating wheel. The pressure head upstream of the nozzle is converted into kinetic energy contained in the water et leaving the nozzle. !s the et strikes the rotating bucket, the kinetic energy is converted into a rotating tor"ue.

a) Tor"ue Tor"ue delivered delivered to to the wheel by the the li"uid li"uid et et  ρ Q r (V # U )(# Cosβ ) $here % is the discharge from all the ets and U  T 

=

−

(#)

−

=

ω  r   and friction is neglected.

 b) The et velocity &# V #

=

C v

(')

'  g  H T 

The velocity coefficient

C v

 accounts for the nozzle losses.

c) ower delivered delivered by the fluid fluid to the the turbine turbine runner  runner 

  shaft  W 

=

 ρ  Q U (V #

−

U )(#

−

Cosβ )

() d) *ondit *ondition ion for +aim +aimum um owe ower  r  U  = V # - '

e)

η T 

()

Turbine /fficiency

= ' C v 'φ (# − φ )(# − Cosβ )

(0)

REACTION TURBINE

1or &elocity triangle at inlet U #  2

3unner vane velocity 43 Tangential velocity of 3unner 43 vane peripheral velocity at inlet V   #  2 !bsolute velocity of water (leaving the guide vane) at inlet W  #  2 &elocity of water relative to runner vane (3elative velocity of water) at inlet α #  2 5uide vane angle V   # 2 Tangential component of the absolute velocity at inlet 43 &elocity of whirl at inlet 43 Swirl at inlet θ  

β #  2 3unner vane angle at inlet (&ane angle at inlet) V  r #  2

&elocity of flow (flow velocity) at inlet 43 3adial velocity at inlet

1or velocity triangle at outlet U '  2 3unner vane velocity 43 Tangential velocity of 3unner 43 vane peripheral velocity at outlet V  '

 2 !bsolute velocity of water at outlet W  '  2 &elocity of water relative to runner vane (3elative velocity of water) at outlet V   2 Tangential component of the absolute velocity at outlet 43 &elocity of whirl at outlet 43 Swirl at outlet β '  2 3unner vane angle at outlet (&ane angle at outlet) V  r  '  2 &elocity of flow (flow velocity) at outlet 43 3adial velocity at outlet θ   '

i) Q

6ischarge =

' π  r #b#V r #

' π  r 'V r '

=

(7)

ii)

Theoretical tor"ue delivered to the shaft

T  shaft 

=

 ρ Q (r # V θ #

−

r ' V θ  ' )

(8) iii)

ower delivered to the shaft

  shaft  W 

=

ω T  shaft 

=

 ρ  Q (U # V #Cosα #

−

U ' V ' Cosα ' )

(9) iv)

ower input to the turbine ($ater ower)

 water  power  W 

=

ρ  g  Q

H T 

(:)

where ;T is the actual head drop across the turbine. v) η T 

vi)

α #

4verall Turbine /fficiency

=

 shaft  W   water  power  W 

=

ω T  shaft   ρ  g Q H T 

 

(#<)

 

(##)

5uide vane angle

= Cot −# ('π  r #'b# ω - Q + Cot β # )

AXIAL FLOW TURBINE

= U ' = U  = ω ( Dt  +  Dh ) - ., and  V r # = V r ' = V r  = Q -  A, where U #

 A =

π  .

( Dt 

'

−  Dh

'

 

(#')

)

!t maimum efficiency V θ ' <  and &' 2 &f , it follows that the energy transferred by the fluid to the turbine per unit weight of the fluid =

 E 

=

U V θ #  g 

In which V θ # V   f  Cot θ  . Since / should be the same at the blade tip and at the hub, but = is grater at the tip, it follows that V   #  must be reduced. Similarly, the velocity of flow &f  should remain constant along the blade and, therefore, Cot θ   must be reduced towards the tip of the  blade. Thus, θ   has to be reduced and, conse"uently, the blade must be twisted so that it makes a greater angle with the ai at the tip than it does at the hub. =

θ  

GENERAL RELATIONSHIPS

vii)

>et ositive Suction ;ead

 NPSH  =

viii)

−  pv

 ρ  g 

− ∆ Z  + h L

−  pv ) - ρ  g  − ∆ Z  + h L

 H T 

i)

 

(#)

 

(#)

 

(#0)

 

(#7)

 

(#8)

 

(#:)

 

('<)

 

('#)

Thoma *avitation >umber  ( patm

σ  =

 patm

1low 3ate *oefficient

C Q

Q

=

ω  D 

) ower *oefficient C  P 

=

 W   ρ ω   D 0

i) C  H 

;ead *oefficient =

ii) C  NPSH 

 g  H 

ω '  D '

>S; *oefficient =

 g  NPSH 

ω '  D '

(#9)

iii)

Similarity 3ules

 H '

 ω      D'      =  '      g '  ω #       D#  

 H #

 g #

'

ω    D = '  ' 6IS*;!35/? Q# ω #   D# Q'

iv)

0

    D'         D         #  

 ρ   ω  4$/3?  = '  ' W #  ρ #  ω #

;/!6?



 W  '

'



       

S/*I1I* S//6 (Turbine)

#- '

 N  s v) φ 

=

C  P  C  H 

0- .

=

(

)

 - ρ  ω  W  T 

#- '

( g  H T  ) 0 - .

S//6 1!*T43   r ω 

=

' g  H T 

CLASSIFICATION OF PUMPS AND TURBINES

 

('')