Geology Reference
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radius R, the torque T, the fluid mass density U, the rotation rate Z, and the flow
rate Q. It is known from rotating flow turbomachinery analysis (e.g., refer to the
classic topics by Hawthorne (1964) and Oates (1978)) that these individual
parameters by themselves are unimportant; instead, the complete physical
problem depends on two parameters only, namely, the dimensionless groups
RT/UQ
2
and SZR
3
/Q. The first group measures a force-like quantity against the
dynamic head; the second, also known as the “tip speed ratio,” compares
azimuthal to axial velocities. Plots of one versus the other will generally yield
straight lines (Oates, 1982). For hundreds of data points obtained in a CNPC
mud flow loop, running at different flow rates with different mud densities and
viscosities, it was found that, for a given siren geometry, RT/UQ
2
is in fact a
straight line function of SZR
3
/Q. That this should have been expected is
anticipated from the wide body of empirical turbomachinery literature available.
An example from mud loop tests is shown in Figure 9.5, where different
flow rates, mud weights and viscosities were incorporated into the test program.
Again, RT/UQ
2
and SZR
3
/Q are dimensionless and have no physical units. The
straight line curve fit will take the form RT/UQ
2
= Slope
u
SZR
3
/Q + Vertical
Intercept. Using our data, Vertical Intercept = - 0.09 and Slope = (0.12 +
0.09)/1.0 = 0.21. Thus, RT/UQ
2
= 0.21 SZR
3
/Q - 0.09. Hence, the siren
torque acting at any fluid density U, flow rate Q and rotation rate Z is given by
the equation T = ( UQ
2
/R ) (0.21 SZR
3
/Q - 0.09). This result is, of course,
applicable only to the particular tapered-edge rotor design tested. But it is
important to observe that dynamic torque is not a simple quadratic function of Q
as in turbine flows. However, the equation applies generally to the hardware
class considered and can then be used with any set of physical units desired.
R Torque/ UQ
2
R Torque/ UQ
2
0.12
GPM (low)
GPM (medium)
GPM (high)
GPM (low)
GPM (medium)
GPM (high)
0.10
0.10
0.05
0.05
SZR
3
/ Q
SZR
3
/ Q
0.00
0.00
1.0
0.5
1.0
0.5
- 0.05
- 0.05
- 0.09
Figure 9.5
. Siren torque from numerous mud tests fit to single straight line
(results applicable to a research design never used commercially).
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