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[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.

Derivation
 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.


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