Wind Energy Handbook. Michael Barton Graham

Wind Energy Handbook - Michael Barton Graham


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a direction that is tangential to the rotation as well as an axial component; see Figure 3.4.

      The acquisition of the tangential component of velocity by the air means an increase in its kinetic energy that is compensated for by a fall in the static pressure of the air in the wake in addition to that which is described in the previous section.

Schematic illustration of the trajectory of an air particle passing through the rotor disc. Schematic illustration of the Tangential velocity grows across the disc thickness.

      An abrupt acquisition of tangential velocity cannot occur in practice and must be gradual. Figure 3.5 shows, for example, a sector of a rotor with multiple blades. The flow accelerates in the tangential direction through the ‘actuator disc’ as it is turned between the blades by the lift forces generated by their angle of attack to the incident flow.

      The tangential velocity will not be the same for all radial positions, and it may well also be that the axial induced velocity is not the same. To allow for variation of both induced velocity components, consider only an annular ring of the rotor disc that is of radius r and of radial width δr.

      The increment of rotor torque acting on the annular ring will be responsible for imparting the tangential velocity component to the air, whereas the axial force acting on the ring will be responsible for the reduction in axial velocity. The whole disc comprises a multiplicity of annular rings, and each ring is assumed to act independently in imparting momentum only to the air that actually passes through the ring.

      The torque on the ring will be equal to the rate of change of angular momentum of the air passing through the ring.

      Thus, torque = rate of change of angular momentum

      = mass flow rate through disc × change of tangential velocity × radius

      where δAD is taken as being the area of an annular ring.

delta upper P equals delta upper Q normal upper Omega delta upper P equals 2 rho delta upper A Subscript upper D Baseline upper U Subscript infinity Baseline Superscript 3 Baseline a left-parenthesis 1 minus a right-parenthesis squared

      Hence

2 rho delta upper A Subscript upper D Baseline upper U Subscript infinity Baseline Superscript 3 Baseline a left-parenthesis 1 minus a right-parenthesis squared equals rho delta upper A Subscript upper D Baseline upper U Subscript infinity Baseline left-parenthesis 1 minus a right-parenthesis 2 normal upper Omega squared a prime r squared

      and

upper U Subscript infinity Baseline Superscript 2 Baseline a left-parenthesis 1 minus a right-parenthesis equals normal upper Omega squared r squared a prime

      Ωr is the tangential velocity of the spinning annular ring, and so lamda Subscript r Baseline equals StartFraction r normal upper Omega Over upper U Subscript infinity Baseline EndFraction is called the local speed ratio. At the edge of the disc r = R and lamda equals StartFraction upper R normal upper Omega Over upper U Subscript infinity Baseline EndFraction is known as the tip speed ratio.

      Thus

delta upper P equals delta upper Q normal upper Omega equals left-parenthesis one half rho upper U Subscript infinity Baseline Superscript 3 Baseline Baseline 2 italic pi r delta r right-parenthesis 4 a prime left-parenthesis 1 minus a right-parenthesis lamda Subscript r Baseline Superscript 2

      The first term in brackets represents the power flux through the annulus in the absence of any rotor action; the term outside these brackets, therefore, is the efficiency of the blade element in capturing that power.

      Blade element efficiency is

      in terms of power coefficient

StartFraction italic d upper C Subscript upper P Baseline Over italic d r EndFraction equals StartFraction 4 pi rho upper U Subscript infinity Baseline Superscript 3 Baseline left-parenthesis 1 minus a right-parenthesis a prime lamda Subscript <hr><noindex><a href=Скачать книгу