Downhole Turbine Design and Short Wind Tunnel Testing - Measurement While Drilling Signal Analysis, Optimization and Design - page 212

Without loss of generality, we assume T = ½UU 2 SRC T . It is important to
remember that the torque coefficient is a function of geometry only and not fluid
properties, at least to first order. Thus, once we have tested a wood model in the
wind tunnel, it can be determined from the formula
C T = T air /(½U air U air 2 SR)

(8.1)

Now, consider a test under field conditions with a mud of density U mud and
downhole flow speed U mud . This yields a torque T mud . Its torque coefficient
would be C T = T mud /(½U mud U mud 2 SR). But since the two torque coefficients
must be the same, we have T mud /(½U mud U mud 2 SR) = T air /(½U air U air 2 SR) or
T mud = (U mud /U air )(U mud /U air ) 2 T air (8.2)
That is, the torque in mud is linearly proportional to the ratio of mud densities
and varies quadratically with the ratio of oncoming speeds (and hence, the
volume flow rates).
Next, focus on the wind tunnel test. Whether we perform a test using a
wind tunnel, a mud or water test loop, it is essential to use bearings that are low
in friction; otherwise, the torques needed to overcome bearing friction may
cause significant error in interpreting true fluid-dynamic properties. Because air
densities are typically 700-800 times smaller than those in mud, it is essential
that the highest quality low-friction bearings be used. Many of these are sealed
so that contaminants do not enter. It is also preferable to test at higher air speeds
in order to minimize bearing errors associated with torque measurement.
A simple method is available to determine if torques are measured
correctly. We assume that a manometer system has been set up to measure the
flow speed U. Since the equation
T = ½UU 2 SRC T

(8.3)

holds, one should measure torque at several flow speeds U, increasing U from
low to high speed. The equation shows a quadratic dependence on U. Thus, if
the plot of T versus U is not parabolic, measurements for U, T, or both, may be
incorrect. This provides a simple error-checking procedure.
We now turn to turbine performance and experimental details. If the
turbine is installed in a wind tunnel and held still so that it does not move while
wind of speed U air is blowing past it, the torque that is measured can be denoted
as the “stall torque, air,” that is, the air turbine “stalls” and does not move, and
the torque is given the symbol T s,air . This torque can be measured by drilling a
small hole through the wind tunnel wall and inserting a linear force gauge - the
torque is simply the product of the measured force and the moment arm. At the
opposite extreme, let the turbine turn freely at its maximum or “no-load rotation
speed” (when this occurs, there is no load across blade upper and lower surfaces
and the torque is zero). We denote this rotation speed by Z NL,air . Then, it is true
in linear theory (that is, for small flow angles relative to blade pitch under