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tan ϕ from the ratio of the non‐dimensional rate of change of angular momentum to the non‐dimensional rate of change of axial momentum, which is not changed because it deals with the average flow through the disc and so uses average values. If drag is ignored for the present, Eq. (3.62) becomes

      (3.87)tangent phi equals StartFraction lamda mu a prime left-parenthesis 1 minus a right-parenthesis Over a left-parenthesis 1 minus a right-parenthesis plus left-parenthesis a prime lamda mu right-parenthesis squared EndFraction

StartFraction left-parenthesis 1 minus a right-parenthesis lamda mu a prime Over a left-parenthesis 1 minus a right-parenthesis plus left-parenthesis a prime lamda mu right-parenthesis squared EndFraction equals StartStartFraction left-parenthesis 1 minus StartFraction a Over f EndFraction right-parenthesis OverOver lamda mu left-parenthesis 1 plus StartFraction a prime Over f EndFraction right-parenthesis EndEndFraction

      which becomes

      As before, Eq. (3.60) still applies, StartFraction italic d a Over italic d a prime EndFraction equals StartFraction 1 minus a Over a prime EndFraction

      Consequently,

left-parenthesis 1 minus a right-parenthesis left-parenthesis 1 minus 2 StartFraction a Over f EndFraction right-parenthesis equals lamda squared mu squared a prime

      which, combined with Eq. (3.89), gives

a squared minus two thirds left-parenthesis f plus 1 right-parenthesis a plus one third f equals 0

      so

Graph depicts the axial flow factor variation with radius for a three blade turbine optimised for a tip speed ratio of 6.

      Clearly, the required blade design for optimal operation would be a little different to that which corresponds to the Prandtl tip‐loss factor because ab = StartFraction a Over f EndFraction; the local flow factor does not fall to zero at the blade tip. The use of the Prandtl tip‐loss factor leads to an approximation, but that was recognised from the outset.

mu sigma Subscript r Baseline lamda upper C Subscript l Baseline equals StartFraction 4 lamda squared mu squared a prime Over StartRoot left-parenthesis 1 minus StartFraction a Over f EndFraction right-parenthesis squared plus left-bracket lamda mu left-parenthesis 1 plus StartFraction a prime Over f EndFraction right-parenthesis right-bracket squared EndRoot EndFraction left-parenthesis StartStartFraction 1 minus a OverOver 1 minus StartFraction a Over f EndFraction EndEndFraction right-parenthesis

      Introducing Eq. (3.89) gives

      (3.92) Скачать книгу