Blade Element Theory for Wind Turbines - Small Wind Turbines: Analysis, Design, and Application - page 43

Fig. 3.1 Annular streamtube
intersecting a blade element

dr 0

r ∞

r

r 0

upwind:
U 0 , W = 0

blade:
U 1 , W 2

far-wake:
U ∞ , W ∞

assumed that dr r typically. Any velocity in the radial direction is ignored,
but the circumferential or swirl velocity will be included in the analysis. The
conservation equations for mass, momentum, angular momentum, and energy are
now considered in turn.

3.3.1 Conservation of Mass

Dividing the conservation of mass Eq. 2.1 by the density and applying it to the
streamtube whose flow area is approximately 2prdr, gives

U 0 2pr 0 dr 0 ¼ U 1 2prdr ¼ U 1 2pr 1 dr 1

or, in a form analogous to ( 2.3 ),

U 0 r 0 dr 0 ¼ U 1 rdr ¼ U 1 r 1 dr 1 ð 3 : 1 Þ

3.3.2 Conservation of Momentum

Because the force of main interest is in the direction of the wind and the turbine's
axis, it is easier to revert to scalars. Using Eq. 3.1 , the contribution to the axial
thrust, T, from the streamtube is:

dT ¼ qU 0 U 0 2pr 0 dr 0 qU 1 U 1 2pr 1 dr 1 ¼ 2pqU 1 rU 0 U 1 ð Þ dr

Note that this is the total force acting on the N blade elements that intersect this
streamtube. Using Eqs. 2.15 and 2.17 , the equation can be rewritten as

dT ¼ 4prqU 0 a 1 a ð Þ dr ð 3 : 2 Þ

a is the axial interference factor, sometimes called the ''axial inflow factor'',
defined in Sect. 2.6 :