Download relates P to V`s and A`s Other: P=VF F BETZ LIMIT

CATs
COHERENT APPLICATION
WindTHREADS
Turbines
Supplemental Learning Module
ME 303 Fluid Mechanics
Control Volume Analysis and the Betz
Limit
Don Wroblewski
Associate Professor, Mechanical Engineering
Associate Chair, Aerospace Engineering
Undergraduate Program
BETZ LIMIT SETUP
Streamtube
Wind Speed
downstream of
rotor
Wind Speed
through rotor
Undisturbed
Wind Speed
Force exerted
by rotor on
wind
F
V2
V
V1
A1
Rotor area
Power extracted at rotor
A
P
A2
BETZ LIMIT ASSUMPTIONS
•
•
•
•
Wind velocity is always axial (normal to flow areas)
Steady State, Uniform Flow
Incompressible, adiabatic flow
Ideal rotor
• Zero mass
• Zero drag
• No Wake
F
V2
V
V1
A1
A
P
A2
BETZ LIMIT ANALYSIS
•
•
•
•
Apply following to control volume
Conservation of mass: relates V’s to A’s
Momentum equation: relates F to V’s and A’s
Energy equation: relates P to V’s and A’s
Other: P=VF
Control volume
F
V2
V
V1
A1
A
P
A2
BETZ LIMIT RESULTS
Coefficient of Performance C P 
P
PAVAIL
P
 1
3

AV
2
1
Coefficient of Performance vs Velocity Ratio

V1
V
V2