Dynamax HPFM3, Installation And Operation Manual, Page: 13 / 86

Revision: JUNE 2017 13
It is not possible, in practice, to make P2 increase linearly with time from time 0. This is because an extra conductance
equal to the conductance of the capillary tubes of the HPFM (KCT) is interposed between the pressure transducers that
measure P1 and P2 (see Figure 1). When the full set of equations is derived (not shown) a short time lag is predicted
before P2 increases linearly with time (see Figure 2 (C)). The resulting component and total flows are shown in Figure
2 (D).
Air bubbles generally can be avoided within the flow meter tubing and apparatus, but sometimes air is in the vessels
and/or intercellular air spaces in the base of roots. Although the compression of air causes an increase in F at any given
P2 during a transient measurement, bubbles cause an underestimation of the root conductance, Kr. This is because
Fb decreases with increasing P2 causing a negative contribution to the slope used to calculate Kr. Fortunately, the
contribution of bubble compression to total flow diminishes with increasing pressure. The slope of the F versus P2 for
P2 > 0.25 MPa is a reasonable approximation of the hydraulic conductance of the root system being measured. It is
probably good practice to measure transient flows for P2 up to 0.5 MPa in order to reduce the underestimation of Kr
caused by compression of air bubbles.
3.3 Quasi-steady State Measurements of Hydraulic Resistance, QSS
During steady-state measurements of QSS, water flow and applied pressure are both constant and by definition the
water flow into the object measured equals the flow out of the object. In practice it is never possible to keep flow and
pressure perfectly constant, so it is best to refer to such measurements as quasi-steady state.
The capillary tubes (CT) on the HPFM have been calibrated, i.e., their conductance, KCT, or resistance RCT = 1/KCT,
has been measured. During flow measurements, the control program monitors the difference in pressure dP across
the CT. The program uses dP to calculate the flow, F, from:
F = KCT dP (1A)
or F = dP/RCT (1B)
Since the flow into the object closely equals the flow out, the unknown hydraulic resistance, RU, can be calculated
from:
RU = (P2 - PO) /F (2A)
or KU = F/(P2-PO) (2B)
Where: P2 is the pressure recorded by the pressure transducer at the outlet, PT2, and PO is the pressure of the water
where it emerges from the object. In many cases PO is known to be zero. When PO is unknown then RU or KU cannot
be computed. Examples will be given later of instances where PO is unknown. For the rest of this section we will
assume PO = 0 and can be omitted from the equations.
Another useful way of viewing Quasi-steady state measurements is through the Ohm’s law analogue for flow through
resistance in series. For any given resistance, R, in series Ohm’s law states that:
RF = dP (3)
So for the capillary tube in series with the unknown resistance we have:
RCT F = dP (4A)
and RU F = P2 (4B)
So dividing Equation (4B) by (4A) we have:
RU/RCT = P2/dP (4C)
or
RU = RCT P2/dP (4D)
or KU = KCT dP/P2 (4E)