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Axial Momentum Theory for Turbines with Co-axial Counter
Rotating Rotors
By Chawin Chantharasenawong
Banterng SuwantragulAnnop Ruangwiset
Department of Mechanical Engineering, KMUTT
Presented at the Commemorative International Conference on the Occasion of the 4th 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
Upstream rotor
Downstream rotor
Wind direction
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
This image is taken from www.esru.strath.ac.uk
Actuator Disc Theory
Mass conservation
The Betz Limit
Power coefficient3
21AV
PCP
•The Betz limit states that
for a single rotor wind turbine
%3.5927
16rotor] [singlemax PC PC
Existing Theory and Literature
[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
%6425
16discs]rotor [twomax PC
Existing Theory and Literature
[3] Newman B G (1986) Multiple actuator-disc theory for wind turbine, Journal of Wind Engineering and Industrial Aerodynamics, Vol.24, pp.215-225
%7.663
2discs] of no. [infinitemax PC
Methodology & Assumptions
Rotor 1
Upstream
Rotor 2
Downstream
Flow Velocities and Pressure
Pressure profile in stream tube 1
Pressure profile in stream tube 2
Axial loading on rotor
Bernoulli’s equation
Axial flow momentum equation
Methodology
21 ppAT
222
211 2
1
2
1UpUp
21 VmVmT
Inner Section of Upstream Rotor
Pressure profile in stream tube 1
Axial loading on rotor Bernoulli’s equation Axial flow momentum
equation
Axial loading on rotor
Bernoulli’s equation
Axial flow momentum equation
Inner Section of Upstream Rotor
inner 1inner 1inner 1inner 1 ppAT
22inner 1
20 1
2
1
2
1VapVp
2
inner 1
inner 1inner 1inner 1
111
11
VbaA
VbVAVaVAT
220
22inner 1 1
2
11
2
1VbpVap
22inner 1inner 1 11
2
1Vbpp
ab 2
Mechanical Power
Inner section of upstream rotor
ab 2
Mechanical Power Rate of change of kinetic energy 23
inner 1inner 1 142
1aaVAP
23outer 1outer 1 14
2
1eeVAP
2322 221
2
1dbddcbVAP
Outer section of upstream rotor
cd 2
Downstream rotor
ef 2
2
2
1Vm
Power Coefficient
2outer 1outer 1 14 ee
A
ACP
222 221 dbddcb
A
ACP
2inner 1inner 1 14 aa
A
ACP 23
inner 1inner 1 142
1aaVAP
23outer 1outer 1 14
2
1eeVAP
2322 221
2
1dbddcbVAP
),,,,( total
21outerinner 1 total
edcbafC
CCCC
P
PPPP
Maximum Power Coefficient
),,,,( total edcbafCP
3
1e
Determine maximum power coefficient
),( total cafCP
1. Mass conservation
2. Betz limit implies that
Function of 5 variables
Function of 2 variables
Optimisation Algorithm
solution
PC
ac
Power coefficient of a turbine with two rotor discs
a = 0
c = 0.418
814.0max pC
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
Design CPmax
Single rotor disc 0.593
Two rotor discs 0.640
Infinite rotor discs 0.667
Proposed design 0.814
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
Questions and Comments
Thank you for your attentionchawin.cha@kmutt.ac.th
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