Starting and Low Wind Speed Performance - Small Wind Turbines: Analysis, Design, and Application - page 110

maximum when k = 0 because the drag does not influence Q when X and k are
zero. Q then reduces as the rotor accelerates from rest and the drag reduces the
torque. The initially high torque in the hub region decreases, and, generally, the
torque in the outer part of the blade increases as the blade accelerates. However,
this is not the whole story. For ( 6.17 ) with constant cr, dQ/dr goes through zero at
the radius where h p = 0:

2
3k p tan a max

r ð 6 : 18 Þ

Since a max & 6 typically, the most efficient blade will have regions of neg-
ative starting torque whenever k p C 7 approximately, which is the case for many
small wind turbines. If the more general assumption is made that C l and C d are
determined by the aerofoil's pressure distribution at high a, so that C l /C d =1/tan a
and using the result of Exercise 6.2, it can be shown that dQ/dr goes through zero
when

k r ¼ 1 = tan a ð 6 : 19 Þ
Positive blade element torque again requires positive h p . Both ( 6.18 ) and ( 6.19 ) are
approximate, and are to be taken only as indications that there may be regions of
negative aerodynamic torque on the outer part of starting blades. More aerofoil
data and measurements of starting rotors are required to resolve this issue. Finally,
the R 3 dependence indicates the difficulty in starting small rotors.
The integral in Eq. 6.17 is easily evaluated in blade element form, which
corresponds to using the midpoint rule for the quadrature.
An important consequence of the assumption that no power is extracted during
starting is that the rotor torque Q acts only to accelerate the blades. Thus at any
time during starting

dt ¼ RQ Q r

ð Þ

dk

ð 6 : 20 Þ

JU

where J is the total rotational inertia and Q r is the resistive torque. For most wind
turbines, J is dominated by the contribution from the blades as shown by the
discussion of Fig. 1.14 and Table 6.1 for the 500 W turbine. When Q r = 0,
N cancels when ( 6.17 ) is equated to ( 6.20 ). Thus: starting is independent of the
number of blades in the absence of resistive torque, unless the local solidity of the
blade is high enough to alter the lift and drag. The next section shows how
Eq. 6.20 can be evaluated using standard methods for solving ordinary differential
equations (ODEs).
The predictions using ( 6.17 ) are labelled as (c) in Figs. 6.4 and 6.8 and are not
as accurate as predictions (b) which were obtained by accounting for low aspect
ratio effects at high incidence as described by Clifton-Smith et al. [ 4 ]. However, as
noted earlier, ( 6.17 ) was found to be more accurate for the newer measurements on
the higher aspect ratio 2.5 m long blades.
Current knowledge of high-a aerodynamics at low Re is too rudimentary to
provide any firm conclusions on the appropriate formulation of lift and drag. As
discussed in Chap. 4 , the issues of aspect ratio are not understood and its effects