Downhole Turbine Design and Short Wind Tunnel Testing - Measurement While Drilling Signal Analysis, Optimization and Design - page 213

rotating conditions), but also experimentally observed when the angles are large,
that

T air = T s,air (1 - Z/Z NL,air ) (8.4)

where Z will vary from 0 to Z NL,air . In other words, the relationship between
dynamic torque and turbine rotation speed is linear. This is only true of turbines
and not generally applicable to sirens (again, sirens may not even move ! ). Now,
turbine power P air is simply the product of T air and Z air , or,

P air = T s,air Z air (1 - Z/Z NL,air ) (8.5)

That is, turbine power is a quadratic function of Z. It vanishes under stall
conditions (Z = 0) and no-load (Z = Z NL,air ) conditions. Whether we deal with
air or mud, for every desired power level P * in practice, two different values of
Z will give that power because Equation 8.5 is a quadratic equation in Z - the
rotation rate chosen in practice will depend on considerations other than turbine
aerodynamics. The two values of rotation speed needed to provide P * are given
by the solution to Equation 8.5 as a quadratic equation
Z 1,2 = [ T s r—{T s 2 - 4 T s P * /Z NL } ] / (2T s /Z NL ) (8.6)
In engineering design, the appropriate rotation rate may be dictated by the
possibility of shaft vibrations, mechanical packaging constraints, dynamic seal
performance, electrical alternator efficiency, and so on. The maximum power
possible from this turbine has the value

P max = T s Z NL /4 . . . at Z = ½ Z NL (8.7)
Figure 8.6 shows typical turbine properties. Also note from the plot of
speed U versus Z NL that this should be a linear relationship. This should be
demonstrated experimentally during any test. If it is not obtained, there are
measurement errors that must be corrected, e.g., excessive bearing friction,
errors in calculating U from manometer measurements, and so on.

Figure 8.6. Torque and power versus Z. Also, Z NL versus axial speed U.