Downhole Turbine Design and Short Wind Tunnel Testing - Measurement While Drilling Signal Analysis, Optimization and Design - page 209

In MWD applications, very high pitch angles are needed because all of the
power desired must come from a single stage. This, together with large blade-
to-blade distances means that the flow will separate. Clearances between rotor
tips and housing imply high three-dimensionality, inefficiencies in torque
creation, massive vortex shedding, and so on. Rotor-stator flow interactions
render fluid motions unsteady. Any one of these conditions means that analysis
is impossible. However, simple observations have led to accurate means for
MWD turbine design which provide almost perfect results: wind tunnel analysis.
Although wind tunnels have been used extensively in the petroleum
industry to model unsteady loads associated with vortex shedding from offshore
platforms in water, its application in downhole tool design was apparently not
well known until the publication of Gavignet, Bradbury and Quetier (1985)
which examined flow in tricone drillbits. These investigators, at the lead
author's suggestion, used “air as a flowing fluid,” noting that the “substitution of
fluids is justified by the highly turbulent nature of the flow.” In the past decade,
the lead author has developed the approach more extensively. As we will see,
the cited reason represents only part of the justification: air, which is convenient,
free, clean and providing of quick turn-around, is optimal for other reasons.
Fluid mechanics topics typically introduce the subject by developing ideas
in “dynamic similarity.” In the context of the aerodynamic design of turbines
(and mud sirens too), consider a fluid with density U, viscosity P and speed U,
and a geometry with a characteristic surface area S and a characteristic radius R.
The dimensional torque T will be a function of (i) the geometry or shape of the
turbine and the shape of the blades, (ii) the dimensionless Reynolds number Rey
= UUR/P, and finally, (iii) the dimensional quantity UU 2 SR. If two different
situations are such that (i) and (ii) are identical, then the two are physically
equivalent even if the torque values themselves are not.
Now consider the possibility of using a wind tunnel. This would mean
inexpensive and fast tests since the models can be made of, say, balsa wood,
constructed using simple wood-working tools. Hundreds can be tested in a
matter of days. Because downhole turbines (and sirens) are relatively small, we
will test them “full scale” with identical size and shape. Thus, condition (i) is
satisfied. Next, consider Reynolds number. Since the test fixture is small and
turbines do not substantially block the flow, we can test at the actual downhole
speed U using simple squirrel cage blowers.
For constant density, incompressible laminar flow, we need only require
that P/U, known as the “kinematic viscosity,” be identical. Let us consider the
kinematic viscosity versus temperature relationships in Figure 8.5 for various
fluids. Surprisingly, it turns out that the kinematic viscosity of a typical drilling
mud is not that of water, as one might surmise, but that of two gaseous fluids, air
and methane. Methane is dangerous because it is explosive. Air at room
temperature and pressure is free and abundant. By using air, our Reynolds
numbers are very close, satisfying condition (ii). If Rey is such that turbulent