USEFUL 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 jet leaving the nozzle. As the jet
strikes the rotating bucket, the kinetic energy is converted into a rotating torque.
a) Torque delivered to the wheel by the liquid jet
T   Q r (V1  U )(1  Cos )
Where Q is the discharge from all the jets and U   r and friction is neglected.
(1)
b) The jet velocity V1
V1  Cv 2 g HT
The velocity coefficient C v accounts for the nozzle losses.
(2)
c) Power delivered by the fluid to the turbine runner
W shaft   QU (V1  U )(1  Cos )
(3)
d) Condition for Maximum Power
U  V1 / 2
e)
(4)
Turbine Efficiency
T  2 Cv 2 (1   )(1  Cos )
(5)
REACTION TURBINE
For Velocity triangle at inlet
U 1 = Runner vane velocity OR Tangential velocity of Runner OR vane peripheral velocity at inlet
V1 = Absolute velocity of water (leaving the guide vane) at inlet
W1 = Velocity of water relative to runner vane (Relative velocity of water) at inlet
 1 = Guide vane angle
V 1 = Tangential component of the absolute velocity at inlet OR Velocity of whirl at inlet OR Swirl
at inlet
1 = Runner vane angle at inlet (Vane angle at inlet)
Vr1 = Velocity of flow (flow velocity) at inlet OR Radial velocity at inlet
For velocity triangle at outlet
U 2 = Runner vane velocity OR Tangential velocity of Runner OR vane peripheral velocity at outlet
V2 = Absolute velocity of water at outlet
W2 = Velocity of water relative to runner vane (Relative velocity of water) at outlet
V 2 = Tangential component of the absolute velocity at outlet OR Velocity of whirl at outlet OR
Swirl at outlet
 2 = Runner vane angle at outlet (Vane angle at outlet)
Vr 2 = Velocity of flow (flow velocity) at outlet OR Radial velocity at outlet
i)
Discharge
Q  2  r1b1Vr1  2  r2Vr 2
ii)
Theoretical torque delivered to the shaft
Tshaft   Q (r1 V 1  r2 V 2 )
iii)
(9)
Overall Turbine Efficiency
W shaft
 Tshaft
T  

Wwater power  g Q H T
vi)
(8)
Power input to the turbine (Water Power)
W water power   g Q H T
where HT is the actual head drop across the turbine.
v)
(7)
Power delivered to the shaft
W shaft   Tshaft   Q (U1 V1Cos1  U 2 V2Cos 2 )
iv)
(6)
(10)
Guide vane angle
1  Cot 1 (2 r12b1  / Q  Cot1 )
(11)
AXIAL FLOW TURBINE
U 1  U 2  U   ( Dt  Dh ) / 4, and
Vr1  Vr 2  Vr  Q / A, where
A

4
( Dt  Dh )
2
2
(12)
At maximum efficiency V 2  0 and V2 = Vf, it follows that the energy transferred by the fluid to
the turbine per unit weight of the fluid
U V 1
E
g
In which V 1  V f Cot . Since E should be the same at the blade tip and at the hub, but U is
grater at the tip, it follows that V 1 must be reduced. Similarly, the velocity of flow Vf should
remain constant along the blade and, therefore, Cot must be reduced towards the tip of the
blade. Thus,  has to be reduced and, consequently, the blade must be twisted so that it makes a
greater angle with the axix at the tip than it does at the hub.
GENERAL RELATIONSHIPS
vii)
Net Positive Suction Head
NPSH 
viii)
patm  pv
 Z  hL
g
(13)
Thoma Cavitation Number

( p atm  pv ) / g  Z  hL
HT
ix)
Flow Rate Coefficient
CQ 
(14)
Q
 D3
(15)
x) Power Coefficient
CP 
W
xi)
Head Coefficient
CH 
gH
 2 D2
xii)
(17)
NPSH Coefficient
C NPSH 
xiii)
(16)
  3 D5
g NPSH
 2 D2
(18)
Similarity Rules
3
5
W 2  2   2   D2 
  

POWER:  
W1 1  1   D1 
HEAD:
H 2 g1   2 
 

H 1 g 2  1 
2
 D2 


 D1 
(19)
2
(20)
Q
 D 
DISCHARGE: 2  2  2 
Q1 1  D1 
xiv)
Ns 
xv)

3
(21)
SPECIFIC SPEED (Turbine)
CP
1/ 2
CH
5/ 4
 WT /  
1/ 2

g H T 5 / 4
SPEED FACTOR
r
2 g HT
(22)
CLASSIFICATION OF PUMPS AND TURBINES