Dynamax HPFM3, Installation And Operation Manual, Page: 12 / 86

Revision: JUNE 2017 12
V = Vo + P2/e (7)
Where: Vo is the initial volume of the system at P2 = 0 and e is the bulk modulus of elasticity. The time derivative of
Equation (7) gives the flow to cause the elastic volume change:
Fe = dV/dt = (1/e) dP2/dt (8)
If air bubbles are present anywhere in the system, they will be compressed according to the ideal gas law as the
pressure of the fluid around the bubble increases. If Vb is the volume of a bubble at absolute gas pressure Pb then
the ideal gas law gives:
PbVb = nRT = PiVi (9)
Where: n is the number of moles of gas in the bubble, R is the gas constant, T is the Kelvin temperature, and Pi is the
initial gas pressure and Vi is the initial volume. If we write Equation (9) as Vb = ViPi/Pb and take the derivative with
time we get the rate of volume change of the bubble:
dVb/dt = -(ViPi/Pb2)dPb/dt (10)
The negative sign of the derivative indicates that Vb decreases with increasing Pb. The flow of water to compress the
gas volume is the negative of Equation (10) so we have:
Fb = -dVb/dt = (ViPi/Pb2)dPb/dt (11)
In the case where all of the air bubbles are near the outlet side of the flow meter, we can equate Pb to P2 + Pi. Pi is
the initial absolute pressure of the bubble when P2 = 0. Pi approximately equals 0.1 MPa. The barometric pressure
will determine the Pi. There will also be a small contribution by the surface tension of the air-water interface to bubble
compression. In the special case where the air bubbles are near the outlet of the flow meter, the dynamic flow will be
given by substituting Equations (6, 8 and 11) into (5):
F = K P2 + (1/e) dP2/dt + ViPi/(P2 + 0.1)2 dP2/dt (12)
During normal operation of the HPFM, the inlet pressure P1 increases linearly with time, and after a short time delay
this causes P2 to increase linearly with time to make dP2/dt equal to a constant. This permits easier interpretation of
results because it makes the elastic contribution a constant offset to Fh. The contribution of the three terms in Equation
(12) is illustrated in Figure 2 for the case where P2 increases linearly with time from time = 0 (Figure 2 (A)).
The component flows (Fh, Fe, and Fb) and total flows are shown in Figure 2 (B).
A
B
C
D
P
2
F F
F
h
F
e
F
b
F
b
F
e
F
h
P
1
P
2
Figure 2
Transient Measurements Conductance of Roots
Flow → Pressure → Flow → Pressure →
Time → Time →
PT2 → PT2 →