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