Turbomachinery – Euler’s Equation
ME 362
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Centrifugal Pump:
W2
V2
V2
2
2
Flow Direction
U2
Subscript:
W1
1

V1
1
1 - inlet
2 - exit
V1
U1
Velocity Triangle:
V
W

W
Vr


V
U
circumferential speed of impeller ( U  r )
velocity tangent to blade surface
  
absolute velocity ( V  U  W )
radial component of V
circumferential component of V
blade angle
U:
W:
V:
Vr:
V:
:
To construct a velocity triangle:



Draw U tangent to the rotor
Draw W tangent to the blade surface
Draw V
radial
circumferential
Turbomachinery – Euler’s Equation
ME 362
Page 2 of 4
Euler Turbomachine Equation:

Shaft torque:
Tshaft  Qr2V 2  r1V 1 

Brake horsepower:
bhp  QU 2V 2  U 1V 1 
Note:

Euler’s equation is valid for both pump and turbine

bhp is the power required to drive shaft of pump (bhp > 0)
or
the power required to deliver to shaft of turbine (bhp < 0)
Pump vs. Turbine:
2
Pump
1
U2 (exit) > U1 (inlet)
V2 (exit) > V1 (inlet)
bhp > 0
U2 (exit) < U1 (inlet)
V2 (exit) < V1 (inlet)
bhp < 0
1
Turbine
2
Pump Performance Characteristics:

Water horsepower:
Pw  gQH

Pump efficiency:

Pw
(1)
bhp
where




Pw:
H = hs - hL:
hs:
hL:
power delivered to fluid
available head
shaft work head
head loss
Turbomachinery – Euler’s Equation
ME 362
Pump in Ideal Flow Condition:

Ideal flow condition:  = 1 (no head loss) and V1 = 0 (maximum bhp)

Velocity triangle at the inlet:
W1
radial
1
V1
U1
circumferential
U1  r1
Vr1  V1  U1 tan 1
Q1  2r1bVr1 (b = blade height)

Velocity triangle at the exit:
W2
V2
Vr2
V2
radial
2
U2
circumferential
Vr2cot2
U 2  r2
Q2  2r2bVr 2  Q1
Vr 2 
Q2
2r2 b
V 2  U 2  Vr 2 cot  2
U 2V 2
g

Ideal head:
Hi 

Maximum power:
bhp  Pw  gQH i  QU 2V 2
( = 1)
Page 3 of 4
Turbomachinery – Euler’s Equation
ME 362
Page 4 of 4
Examples:
Problem 1 (Pump):
Given pump geometry and specifications on the figure below. Assume ideal flow
condition. Find the power required to drive it.
0.75”
55
 = 1.94 slugs/ft3
Q = 0.25 ft3/s
 = 960 rpm
V1
4” 12”
960 rpm
0.25 ft3/s
Problem 2 (Turbine):
A water turbine with radial flow has the dimensions shown below. The absolute entering
velocity is 50 ft/s, and it makes an angle of 30 with the tangent to the rotor. The absolute
exit velocity is directed radially inward. The angular speed of the rotor is 120 rpm. Find
the power delivered to the shaft of the turbine.
V1
30
b
r1
V2

r2
V1 = 50 ft/s
 = 120 rpm
r1 = 2 ft
r2 = 1 ft
b = 1 ft
 = 1.94 slug/ft3