Downhole Turbine Design and Short Wind Tunnel Testing - Measurement While Drilling Signal Analysis, Optimization and Design - page 215

For computing purposes, we can write Equation 8.12 in the form
P mud = (U mud /U air )(U mud /U air ) 3 T s,air Z air (1 - Z/Z NL )

(8.13)

This relationship states that, under the same dimensionless conditions, the
power in mud increases by the ratio of mass densities (or specific gravities) and
varies with the third power of velocity (or volume flow rate) ratio. Here, the
ratio Z/Z NL refers to rotation speeds for the oncoming speed U obtained for the
mud test. If U mud > U air , then the corresponding no-load speed is higher as
determined from Equation 8.9. If torques and powers are plotted versus rotation
rate, where rotation rate appears on the horizontal axis, the range of rotation
values for mud will be larger than that for air.

8.4 Software Reference
To understand physically what these equations imply, we perform
calculations using the software “turbine.exe,” which embodies all of the above
theory. Clicking on this filename brings up the application in Figure 8.7. The
numbers in the text input boxes are chosen for illustrative purposes and do not
represent any real test data. They are selected to illustrate relationships between
physical variables. Note, from Figure 8.7, that the two radii are needed so that
the program can calculate axial speed from volume flow rate and also determine
torque; these inner and outer radii are apparent from, say, Figure 8.17.
The wind tunnel data shown contains the no-load rpm, the stall torque, the
air specific gravity and the volume flow rate (it is also important, for
repeatability testing, to record temperature and humidity, which affect air
density). The computer program will plot the 'torque versus Z' and 'power
versus Z' curves for the wind tunnel first. Now, note in Figure 8.7 that we have,
for simplicity, assumed the same flow rate for the mud, but a density that is
2,000 times higher. We desire predictions for this situation. (Before running the
program the very first time, be sure to click “Install Graphics.”) Now click
“Simulate.” We obtain a sequence of four graphs, as shown below - each
graphics window must be closed before the next appears.
Note that the 1 in-lbf in Figure 8.7 is 0.08333 ft-lbf since there are 12
inches in one foot. The 1,000 rpm is 104.7 rad/sec since 1 rpm = 2S rad/60 sec,
where “rpm” is “revolutions per minute.” Next, the program calculates results
for the mud test. Note in Figure 8.8c that the 104.7 rotation speed value is
unchanged because in Figure 8.7 we kept the same volume flow rate. But
because the density has now increased 2,000 times, the stall torque for the mud
test becomes (2,000)(0.08333) or 166.7 ft-lbf as shown in Figure 8.8c. The peak
power for the wind tunnel test from Figure 8.8b is about “2.2.” For the mud test,
this also increases by a factor of 2,000. It is now (2,000)(2.2) or about “4,400”
as shown in Figure 8.8d.