Geology Reference
In-Depth Information
It is important to emphasize an additional property of turbines with respect
to no-load rotation speeds. It is known from aerodynamics, that under most
conditions, the no-load speed is linearly proportional to the speed of the
oncoming flow as seen in the far right diagram of Figure 8.6. That is,
Z
NL
= alpha u U (8.8)
where “alpha” is a constant of proportionality that does not depend on fluid
properties (it depends only on the geometry of the turbine - the same “alpha” is
obtained whether we test in air or in wind and at any speed U). The easiest and
most accurate way to determine it is to use a fast air speed in a wind tunnel (to
minimize bearing friction effects) and to measure the corresponding no-load
rotation speed. Note that we have only needed to obtain two other data points,
the stall torque and the no-load speed - as we will show, this all that is required
to determine all turbine performance for any mud flowing at any speed.
In summarizing, we have a wind tunnel setup with a free-standing turbine
installed
without
any drive motors. We measure the stall torque and the no-load
rotation speed. A high oncoming speed should be used to allow accurate
measurement of high stall torque T
s,air
since low torques may be degraded by
bearing friction effects - similar considerations apply to no-load speed. The
wind tunnel plots for Equations 8.4 and 8.5 are easily created and appear as
shown in Figure 8.6.
We now ask, how do we extrapolate these results to any mud of any
density flowing at any speed? To do this, we observe that we have assumed that
the same geometries (that is, flow patterns when the angle of the oncoming
flows are considered under rotating conditions) for both air and mud tests. This
requires that the ratios of the transverse velocity to axial velocities be identical,
that is, Z
air
/U
air
= Z
mud
/U
mud
so that
Z
mud
= (U
mud
/U
air
) Z
air
(8.9)
This simple relationship also follows from Equation 8.8. Since “alpha” can be
computed using air or mud conditions, it follows that we again have Z
air
/U
air
=
Z
mud
/U
mud
for which Equation 8.9 follows. Now, let us combine Equations 8.2
and 8.4 to give
T
mud
= (U
mud
/U
air
)(U
mud
/U
air
)
2
T
s,air
(1 - Z/Z
NL,air
)
(8.10)
We multiply Equation 8.10 by Equation 8.9 to give
T
mud
Z
mud
= (U
mud
/U
air
)(U
mud
/U
air
)
3
T
s,air
Z
air
(1 - Z/Z
NL,air
)
(8.11)
Now, the right side of Equation 8.11 can be simplified using Equation 8.5 which
states that P
air
= T
s,air
Z
air
(1 - Z/Z
NL,air
). The left side of Equation 8.11 is the
power in mud denoted by P
mud
. Thus,
P
mud
= (U
mud
/U
air
)(U
mud
/U
air
)
3
P
air
(8.12)
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