Introduction to Wind Turbine Technology - Small Wind Turbines: Analysis, Design, and Application - page 2

Fig. 1.1 Wind flow past a
circular disk representing the
blades

δ x

Elemental
volume of
length δ x and
area

U 0

U 0

A about
to cross disk

δ

Circular
disk of
area A

2 qDAdxU 0 . The time taken for this element to cross the blade disk,
dt, is given simply by dx = U 0 dt. The contribution of the element to the total
amount of KE that passes in dt is symbolized as DKE, and is given by

1

and its KE is

Þ¼ 1

2 qDAU 0 dtU 0

d DKE ð ð 1 : 1 Þ

Summing over all elements of area that make up the disk gives the KE passing the
disk as

d K ðÞ¼ 1

2 qAU 0 dt

ð 1 : 2 Þ

This equation can now be taken formally to the limit as dt ? 0, to give

P ¼ dK ð = dt ¼ 1

2 qAU 0

ð 1 : 3 Þ

where P is the power, the time rate change (derivative) of the energy. Equation 1.3
is extremely interesting because it suggests, as indeed is approximately the case,
that the output power of any turbine depends on the cube of the wind speed. 1 This
simple and fundamental fact must never be forgotten. If this cubic dependence
seems strange, remember that the wind speed determines both the amount of
energy, proportional to U 0 , and the mass of air carrying that energy through the
blade disk per unit time, which is proportional to U 0 . In practice the power output
is never as great as that suggested by Eq. 1.3 because extraction of all the available
KE would require the wind to be decelerated to rest. Furthermore a turbine cannot
capture all the wind that would otherwise pass through the disk, even if it could
decelerate this flow to rest, so that finding the KE in the absence of the blades will
over-estimate the actual energy capture. Including the finite efficiency of the