Page 550 - Mechatronics with Experiments

JWST499-Cetinkunt
JWST499-c07
536 MECHATRONICS Printer: Yet to Come October 9, 2014 8:41 254mm×178mm
rate across the valves more accurately as a function of the pressure difference across them,
compared to the case where the pressure loss (drop) along the piping is neglected.
p = p − p loss−1−2 (7.434)
s
2
p = p − p loss−1−3 (7.435)
3
s
p = p + p loss−4−5 (7.436)
t
4
p = p + p loss−6−7 (7.437)
t
6
In short, the pressure drop can be considered as:
1. zero (neglected), or
2. estimated as constant throughout the whole simulated condition as a function of a
constant nominal viscosity, and flow rate, and geometric properties, or
3. can be estimated as a variable as a function of the variable flow rate and viscosity
(which varies with temperature).
For this example, the length and diameters of the straight pipes and connectors are as
follows,
l = l = l = l = l = l = 100 mm (7.438)
1 2 3 4 5 6
d = d = d = d = d = d = 10 mm (7.439)
1 2 3 4 6 7
The nominal flow rate from the pump is

Q (t) = D (t) ⋅ w shaft (t) (7.440)
p
p
The input shaft speed to the pump is constant for all the cases considered for simulation,
w shaft (t) = 2000 rpm (7.441)
The nominal pump flow rate is
Q (t) = D (t) ⋅ w shaft (t) (7.442)
p
p
= 5ml∕rev ⋅ 2000 rev∕min (7.443)
3
= 10 l∕min = 10∕60 ⋅ 10 −3 m ∕s (7.444)
Given this information, and the pressure drop data graphs provided by pipe-connector
suppliers, specific to the used pipes and fittings, we can estimate the nominal (constant)
pressure drops.
The displacement of the pump is different as a function of time for the different
simulation cases considered (Figure 7.92c). The relief valve reacts to the line pressure. If
the line pressure is below the maximum pressure setting, then the valve is fully closed.
When the line pressure reaches the maximum pressure value, the relief valve opens to
full orifice opening position within a small increment of the pressure above the maximum
pressure setting. The line pressure causes the relief valve spool position to change. Due to
the preset load in the spring, the spool does not move until the line pressure reaches the
maximum pressure setting (determined by the preset load on the spring). When the line
pressure reaches the maximum pressure and a little above, the relief valve spool moves
quickly to fully open the valve. Let us assume the relief valve spool position to orifice area
is a linear relationship plus saturation, as shown in the figure (Figure 7.92c, Case (3)). If we
model the relief valve position as being instantaneously determined by the line pressure, we
can model it as a static relationship between line pressure and relief valve spool position.
Notice that the valve orifice opening is zero until the line pressure reaches the maximum