Wind Energy Handbook. Michael Barton Graham
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Название: Wind Energy Handbook

Автор: Michael Barton Graham

Издательство: John Wiley & Sons Limited

Жанр: Физика

Серия:

isbn: 9781119451167

isbn:

СКАЧАТЬ infinity Baseline upper U Subscript infinity Superscript 2 Baseline plus p Subscript infinity Baseline plus rho Subscript infinity Baseline italic g h Subscript infinity Baseline equals one half rho Subscript upper D Baseline upper U Subscript upper D Superscript 2 Baseline plus p Subscript upper D Superscript plus Baseline plus rho Subscript upper D Baseline italic g h Subscript upper D"/>

      Assuming the flow speed to be at low Mach number M (typically M < 0.3 is sufficient), it may be treated as incompressible (ρ∞ = ρD) and to be independent of buoyancy effects (ρgh∞ = ρghD) then,

      (3.5c)one half rho upper U Subscript infinity Superscript 2 Baseline plus p Subscript infinity Baseline equals one half rho upper U Subscript upper D Superscript 2 Baseline plus p Subscript upper D Superscript plus

      Similarly, downstream,

      (3.5d)one half rho upper U Subscript upper W Superscript 2 Baseline plus p Subscript infinity Baseline equals one half rho upper U Subscript upper D Superscript 2 Baseline plus p Subscript upper D Superscript minus

      Subtracting these equations, we obtain

      (3.6)left-parenthesis p Subscript upper D Baseline Superscript plus Baseline minus p Subscript upper D Baseline Superscript minus Baseline right-parenthesis equals one half rho left-parenthesis upper U Subscript infinity Superscript 2 Baseline minus upper U Subscript upper W Superscript 2 Baseline right-parenthesis

      (3.7)one half rho left-parenthesis upper U Subscript infinity Superscript 2 Baseline minus upper U Subscript upper W Superscript 2 Baseline right-parenthesis upper A Subscript upper D Baseline equals left-parenthesis upper U Subscript infinity Baseline minus upper U Subscript upper W Baseline right-parenthesis rho upper A Subscript upper D Baseline upper U Subscript infinity Baseline left-parenthesis 1 minus a right-parenthesis

      and so,

      (3.8)upper U Subscript upper W Baseline equals left-parenthesis 1 minus 2 a right-parenthesis upper U Subscript infinity

      That is, half the axial speed loss in the streamtube takes place upstream of the actuator disc and half downstream.

      The force on the air becomes, from Eq. (3.4),

      As this force is concentrated at the actuator disc, the rate of work done by the force is TUD and hence the power extraction from the air is given by

      A power coefficient is then defined as

      (3.11)upper C Subscript upper P Baseline equals StartFraction italic Power Over one half rho upper U Subscript infinity Superscript 3 Baseline upper A Subscript upper D Baseline EndFraction

      where the denominator represents the power available in the air, in the absence of the actuator disc.Therefore,

      (3.12)upper C Subscript upper P Baseline equals 4 a left-parenthesis 1 minus a right-parenthesis squared

      3.2.3 The Betz limit

      The maximum value of CP occurs when

StartFraction d upper C Subscript upper P Baseline Over italic d a EndFraction equals 4 left-parenthesis 1 minus a right-parenthesis left-parenthesis 1 minus 3 a right-parenthesis equals 0

      that gives a value of a equals one third

      Hence,

      (3.13)upper C Subscript upper P max Baseline equals StartFraction 16 Over 27 EndFraction equals 0.593

      The efficiency of the rotor might more properly be defined as

      (3.14)StartFraction italic Power extracted Over italic Power available EndFraction equals StartStartFraction italic Power extracted OverOver StartFraction 16 Over 27 EndFraction period left-brace one half rho upper U Subscript infinity Superscript 3 Baseline upper A Subscript upper D Baseline right-brace EndEndFraction

      but note that CP is not the same as this efficiency.

      3.2.4 The thrust coefficient

      (3.15)upper C Subscript <a href=СКАЧАТЬ