Turbomachinary - Euler's Equation



Centrifugal Pump:

Velocity Triangle:

U: circumferential speed of impeller ([pic])

W: velocity tangent to blade surface

V: absolute velocity ([pic])

Vr: radial component of V

V(: circumferential component of V

(: blade angle

To construct a velocity triangle:

• Draw U tangent to the rotor

• Draw W tangent to the blade surface

• Draw V

Euler Turbomachine Equation:

• Shaft torque: [pic]

• Brake horsepower: [pic]

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:

|Pump | |U2 (exit) > U1 (inlet) |V(2 (exit) > V(1 (inlet) |bhp > 0 |

| | | | | |

| | | | | |

| | | | | |

| | | | | |

|Turbine | |U2 (exit) < U1 (inlet) |V(2 (exit) < V(1 (inlet) |bhp < 0 |

| | | | | |

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| | | | | |

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Pump Performance Characteristics:

• Water horsepower: [pic]

• Pump efficiency: [pic] ( ( 1 )

where

• Pw: power delivered to fluid

• H = hs - hL: available head

• hs: shaft work head

• hL: head loss

Pump in Ideal Flow Condition:

• Ideal flow condition: ( = 1 (no head loss) and V(1 = 0 (maximum bhp)

• Velocity triangle at the inlet:

[pic]

[pic]

[pic] (b = blade height)

• Velocity triangle at the exit:

[pic]

[pic]

[pic]

[pic]

• Ideal head: [pic]

• Maximum power: [pic]

(( = 1)

Examples:

Problem 1 (Pump):

Given pump geometry and specifications on the figure below. Assume ideal flow condition. Find the power required to drive it.

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.

-----------------------

circumferential

Flow Direction

V1

W1

α1

Vθ1

β1

U1

(

4”

V1

0.25 ft3/s

960 rpm

55(

radial

circumferential

Vr2cot(2

Vr2

V(2

U2

W2

(2

V2

Subscript:

1 - inlet

2 - exit

radial

0.75”

W1

Vθ2

U1

(

(1

U2

V2

W2

α2

β2

V1

(

(

V(

W

W

V

U

12”

ρ = 1.94 slugs/ft3

Q = 0.25 ft3/s

ω = 960 rpm

radial

circumferential

Vr

1

2

1

2

V2

V1

(

30(

r1

r2

b

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V1 = 50 ft/s

( = 120 rpm

r1 = 2 ft

r2 = 1 ft

b = 1 ft

( = 1.94 slug/ft3

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