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 

 

(#