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Figure 9.10b. Time (left) and frequency domain (right) displays.
Test procedures for source strength are state-of-the-art. Determining signal
strength experimentally, that is, finding p s = p s (Q,Z) for a given siren, is very
challenging. Some researchers have assumed, following Bernoulli's equation,
that p s must vary like Q 2 , however, this equation describes constant density and
not compressible flows. On the other hand, from water hammer considerations,
p s might vary linearly with Q and Z. Montaron, Hache and Voisin (1993) and
Martin et al (1994) speculate that pressure pulse amplitudes created are roughly
independent of frequency, however, recent CNPC experiments indicate a
monotonic decrease with frequency for the same flow rate, e.g., as is evident
from experimental data in Figure 9.10c taken to about 60 Hz. This result is
expected physically: at higher rotation rates, rapid rotor movement does not
provide enough time for the fluid to come to a complete stop and recover.
Figure 9.10c. Experimental CNPC results, 'p versus frequency.
Additional discussion is available for positive pulsers. In one-dimensional
acoustics, a semi-infinite pipe with a piston at one end that is quickly struck will
create a propagating wave with strength p s = UVc. The piston model, an exact
solution attributed to the nineteenth century physicist Joukowsky, applies more
to positive pulsers than to sirens. Here, U is fluid density, c is sound speed and
V is the instantaneous piston speed. If we consider water, with U = 1.935 lbf
sec 2 /ft 4 and c = 5,000 ft/sec, and assume a flow rate of 500 gpm in a 6 inch
diameter pipe, we find that p s = 381 psi. This is a realistic downhole value for
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