Page 64 - Mechanical Engineers' Handbook (Volume 2)

4 Operating Point of Static Systems 53

Figure 13 Chordal impedance matching.

2 2
Q CA P CA (P P ) (37)
d
o
o

d o
up down
where Q is the orifice flow, C is the discharge coefficient of the orifice, A is the orifice
o
d
o
area,
is the density of the fluid, and P P up P down is the pressure drop across the
orifice. If the upstream pressure is kept constant, then P down may be considered the output
pressure and Q may be considered the output flow from this orifice, a characteristic typical
o
of many hydraulic valves. Where is the maximum power point on this characteristic?
Hydraulic power P is the product Q P. It is a maximum when it is stationary with
respect to either Q or P. If we nondimensionalize Eq. (37) with respect to the maximum
flow through the orifice when P down 0, then the orifice equation becomes the following:*
Q P
Q* o 1 down 1 P* (38)
Q max P up
P
P* P*Q* P* 1 P* (39)
P max
dP* P*
1 P* 0 (40)
dP* 2 1 P*
for which P* –, and by substitution into (38), the flow at that point is Q* 1/ 3, and
2
3
the maximum power delivered is ( 2/3) P max Q max . The output admittance of the orifice is
the slope of the curve at any operating point:
3
dQ* 1
Z (41)
o
dP* 2 1 P* 2 P* 2/3
The load for which maximum power will be delivered, whether it has a linear or nonlinear
characteristic, must pass through the maximum power point. Its chordal admittance must be

*Where bold letters will be used to indicate the nondimensional forms.