Fortsätt till huvudinnehåll

[VAWT] Relation between rotor radius and conduction losses

Since the last post a lot of effort has been dedicated to a presentation and mid-project report so little concrete work has been done in the project itself.

Today I however played around with some equations and got a new one of interest. I haven't double checked it yet though so full disclaimer. However, it seems that the steady state conduction losses in relation to it's produced shaft power in a DC machine connected to a Savonius turbine with a radius r can be expressed as

\$ P_{cond.loss,frac}=\frac{P_{cond.loss}}{P_{shaft}} = \frac{R \rho H V C_{p0}}{\lambda_0 k n}r^3 \$ (1)
where V is the wind speed, R is the DC machine series resistance, Roh is the air density H is the rotor height, Cp0 denotes the turbine effiency and lambda a desired tip-to-wind-speed-ratio, both assumed to be constant (a good approximation if a controller is used). Furthermore k is the ideal DC machine constant and n is a gearing ratio between the turbine and DC machine shaft assumed to have no losses.

All in SI units. Off course :)

So the fraction of the power that is lost increases linearly with for instance the rotor height and the generators internal resistance, but in cubic with the rotor radius. Could be worth considering when deciding the turbine proportions in relation to a generator with a given resistance. Though off course this must be put in the perspective that narrowing the turbine will decrease it's wind capturing area and in some cases probably impair it's aerodynamic properties.

 I've made a derivation below and tried LaTex for the first time. It's a bit time consuming to work with I think. Anyway, getting to business.

 The conduction losses in a DC machine can be expressed as

\$ P_{con.losses} = RI^2 = R { \left( \frac{T_{DCmachine} }{k} \right) }^2 \$ (2)

 See my previous blogpost for further explanations. The shaft output power on any wind turbine can be expressed by

\$P_{shaft} = P_{wind}C_p = \frac{2rH \rho V^3}{2} C_p \$ (3)

The shaft power can also be expressed in terms of the axis angular velocity omega and it's output torque T by

\$P_{shaft} = T_{turbine} \omega \$ (4)

The angular velocity can be related to the wind speed via the desired tip-to-wind speed ratio

\$ Desired TSR = \lambda _0 = \frac{ \omega r}{V} \$ (5)

that is assumed to be held constant, if a controller is used.

Setting (4) and (5) equal, assuming a constant effiency Cp0, also due to a controller, inserting an expression for the angular velocity from (6) and solving for the turbine torque gives

\$ T_{turbine} = \frac{ \rho r^2HV^2C_{p0}}{\lambda _0} \$ (6)

The torque acting on the generator during steady angular velocity (no angular acceleration) is simply

\$ T_{DCmachine} = \frac{T_{turbine}}{n} \$ (7)

Substituing (7) into (2)  generates

\$P_{cond.loss}= R { \left( \frac{\rho HV^2C_{p0} }{\lambda_0 k n } \right)}^2r^4 \$  (8)

Dividing (8) by (3) yields (1).

Comments and questions are as always very much appreciated.


Populära inlägg i den här bloggen

[VAWT] Affordable wind measurements

The shop Clas Ohlsson in Sweden sells a spare anemometer for the  WH-1080 weather station for a hobbyist friendly price around 15 euros. This anemometer can probably be found in other countries too.

The WH-1080 spare anemometer

Here  you can read about how to use this anemometer together with an Arduino or any other embedded system.  As we couldn't find any data on the characteristics of this sensor, we did our own calibration in a wind tunnel and the results are presented in the report above. This anemometer was used in a Bachelors Thesis project related to the devlopment of optimization of the Savonius Turbine, here tagged [VAWT].

[VAWT] Plausible improvements of the Savonius windmill

Cut a barrel in half, mount it on a shaft and you have a windturbine. If the barrel is used, then what else probably would end up on a scrap heap is instead used for converting renewable energy.

Simplicity and low cost makes this an attractive option especially for societies with limited economy and a malfunctioning or non existing electric grid. Small off-grid electrical networks can be built and people who perhaps most needs electricity get that. Isn't that neat?

Savonius (from

An overlooked potential?
The Savonius though has a widespread reputation of having low effiency and is often dismissed as a credible option around forums and in formal litterature. However, when looking at the graph below from a publication Wortman did 1983, the effiency can be realtively high provided that the TSR(Tip-to-Wind-Speed-Ratio) is held at a correct value and the windmill should work quite nicely. In practice this could probably be done by controlling the generato…

[RR] Automated rescue boat

We have a project course at our  master program System, Control and Mechatronics, were we could apply to one of around 30 projects available. I was really happy that I got my first choice, which is a project carried out as a joint effort between SSRS (Swedish Sea Rescue Society) and Chalmers. The goal is to develop  autonomous or/and semi-autonomous vehicles that can help out in rescue missions. The current vehicle, a large water scooter named Rescue Runner, is able to follow a larger leader boat.

However, the people involved want some better tracking capabilities and stability of the control system. We are currently a team of three persons hoping to achieve this. Stay tuned, more information will be posted when we finish. Or well, not if we sink though. Then I'll try to cover it up, and hope for dedicated readers to spam me with comments ;)

Here you can read some in Swedish about the project: