Transcript Axial Momentum Theory for Turbines with Co

Axial Momentum Theory for
Turbines with Co-axial Counter
Rotating Rotors
By
Chawin Chantharasenawong
Banterng Suwantragul
Annop Ruangwiset
Department of Mechanical Engineering, KMUTT
Presented at the Commemorative International Conference on the Occasion of the 4 th Cycle Celebration of KMUTT
Sustainable Development to Save the Earth: Technologies and Strategies Vision 2050
Millennium Hilton Hotel, Bangkok, Thailand 7-9 April 2009
Co-axial Twin Rotor HAWT
Wind direction
Downstream rotor
Upstream rotor
Inspiration
[5] Jung S N, No T S and Ryu K W (2004) Aerodynamic performance prediction of a 30kW
counter-rotating wind turbine system, Renewable Energy, Vol. 30, pp.631-644
Existing Theory and Literature
Actuator Disc
Theory
Mass
conservation
This image is taken from www.esru.strath.ac.uk
The Betz Limit

Power coefficient
CP 
P
1
AV 3
2
•The Betz limit states that
CP
C P max [single rotor]
16

 59.3%
27
for a single rotor wind turbine
Existing Theory and Literature
C P max [two rotor discs]
16

 64%
25
[2] Newman B G (1983) Actuator-disc theory for vertical-axis wind turbine, Journal
of Wind Engineering and Industrial Aerodynamics, Vol.15, pp.347-355
Existing Theory and Literature
C P max [infinite no. of discs]
2
  66.7%
3
[3] Newman B G (1986) Multiple actuator-disc theory for wind turbine, Journal of
Wind Engineering and Industrial Aerodynamics, Vol.24, pp.215-225
Methodology & Assumptions
Rotor 1
Upstream
Rotor 2
Downstream
Flow Velocities and Pressure
Pressure profile in stream tube 1
Pressure profile in stream tube 2
Methodology

Axial loading on rotor
T  A   p1  p2 

Bernoulli’s equation
1
1
2
2
p1   U1   p2   U 2 
2
2

Axial flow momentum equation
 V 1  m
 V 2
T  m
Inner Section of Upstream Rotor
Pressure profile in stream tube 1



Axial loading on rotor
Bernoulli’s equation
Axial flow momentum
equation
Inner Section of Upstream Rotor

Axial loading on rotor
T1inner  A1inner


1 inner

1 inner
p

b  2a
Bernoulli’s equation
p0 
1
1
V 2  p1inner   1  a 2 V 2
2
2

1 inner
p

 p

1 inner
p
p1inner 

1
1
 1  a 2 V 2  p0   1  b 2V 2
2
2

1
2
  1  1  b  V 2
2
Axial flow momentum equation
T1 inner  A1 innerV 1  a V   A1 innerV 1  b V 
 A1 inner 1  a 1  1  b V 2
Mechanical Power
Inner section of
upstream rotor
Outer section of
upstream rotor
b  2a
d  2c


Downstream
rotor
f  2e


1
2
3
Mechanical
P


A
V
4
a
1

a
Rate
of change
of kinetic energy
1 inner
1 inner
Power
2
P1 outer


1
2
3
 A1 outerV 4e1  e 
2




1
m V 2
2
1
P2  A2V 3 1  b  c  2d  2bd  d 2
2

Power Coefficient


A1 inner
1
2 2
3
P1 innerCP1 inner
A1 innerV 4a4a11a a 
2
A
P1 outer
 


A1 outer
1
2 2
3
C P1 outer
A1 outer
 V 4e41e1 ee 
2
A




1
3A2
P2  CPA2 2V
 11 b 
b cc2d2d 
2bd
2bd 
d 2d 2
2
A
CP total  CP1 inner  CP1outer  CP 2
CP total  f (a, b, c, d , e)
Maximum Power Coefficient
CP total  f (a, b, c, d , e)
Function of 5 variables
1. Mass conservation
2. Betz limit implies that
CP total  f (a, c)
1
e
3
Function of 2 variables
Determine maximum
power coefficient
Optimisation Algorithm
solution
CP
a
c
Power coefficient of a turbine with two rotor discs
a=0
c = 0.418
C p max  0.814
Power coefficient of each rotor
Proposed design of a co-axial twin rotor
counter rotating wind turbine
a=0
c = 0.418
‘bladeless’ area in upstream
rotor (58% of rotor area)
Wind speed at downstream
rotor is 0.582V
Conclusions
1.
58% ‘Bladeless’ area in upstream rotor
2.
Wind speed at downstream rotor is 58.2% of free
stream velocity
3.
Wind speed at outer part of upstream rotor is 33.3%
of free stream velocity (Betz limit condition)
4.
Theoretical power coefficient increases to 0.814
Design
Single rotor disc
CPmax
0.593
Two rotor discs
0.640
Infinite rotor discs
0.667
Proposed design
0.814
Questions and Comments
Thank you for your attention
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