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Pressure increase
(fill rate) may be important for many of the same reasons as the
rate of pressure decay. This might include pneumatic time delays,
plumbing capacity, the flow or Cv of Valves, Filters and
other components.
The fill rate, pump
up, time can also determine the capacity of a compressor and the
volume of an unknown system.
Previously we
explored the decay rate of compressed air. You may recall the
formula for pressure decay [P2=P1
x e^-t/Tk] and the Time Constant
concept sometimes referred to as Relaxation Time.
Now it gives me
great pleasure to point out the other side of the story or the rate
of pressure increase when filling a volume. The good news is that
when all things are equal the fill, or pump up, rate curve is a
mirror image of the decay curve.
Fortunately this
makes working with both applications easier to understand and to
use. Also the time constant structure is consistent in both cases.
The amount of time that would fill the volume to 63% of the inlet
pressure [P1 x (1- e^-1) is one time constant. A total of 5 time
constants will fill the volume to 99+%.
See Figure 1
Calculated Time Constant
(Tk)
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Chose a pressure and select a
reservoir. Convert the volume to cubic feet. Many tanks and
reservoirs are rated in gallons. To convert gallons to cubic
feet divide gallons by 7.48. A ten gallon reservoir is 10/7.48 =
1.34 cubic ft.
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Std. Cubic Feet (
SCF) = V(cubic feet)
x P1(psig) / 14.7) filled to P1.
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Select the orifice or device
to fill the reservoir for flow rate,
SCFM.
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Divide
SCF x 60 /
SCFM =
Tk
(Time Constant) in seconds. *
V
(CUFT)
X P1(PSIG)
X 60
/
[14.7 X SCFM] =Tk(sec.)Time
Constant
Empirical Time Constant (Tk)
Use this method if
the flow rate is known or a flowmeter is used.
FYI The initial
flow rate is gradually reduced with the changing pressure ratios.
The pressure in the volume at the end of the first time constant is
63% of P1. If the initial flow rate remained unchanged it would fill
the volume to 99+% of P1 in one time constant. In practice 5 time
constants fill the volume to P1. We can find the time constant ,
Tk, by simple observation as follows:
1. Note the reservoir
pressure (0 psig in most cases).
2. Note the pressure
to fill the reservoir.
3. Start the fill flow
and the clock.
4. The time to reach
63% of inlet pressure,
psig
[P1x (1-e^-1) or P1x .63] |
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By either method
this time constant (Tk) is unique to this specific volume
and the key to determine all the main facts. This specific amount of
time is the same for filling or discharging the pressure within it
by the law of organic growth or decay. For charging (filling) the
volume the formula is P2 = P1 x
(1 - e^-t/Tk)
Elapsed time “t”
(sec.) divided by time constant, Tk, is negative
exponent of e.
TIME DELAY [P2 =
P1 x (1-e^ - t/Tk)]
Enter the desired
time, t, in the formula to determine P2 at that elapsed time.
The desired time
may be any time equal to or less than 5 times
Tk.
The desired time may be selected and adjusted with a variable set
point (pressure switch), variable pressure setting (regulator top
change P1) and adjustable flow device (needle valve).
UNKNOWN VOLUME [
P1(psig), Q(scfm) known, Tk (Time to reach to .63 x P1)
A device, orifice,
flowmeter with a known flow rate establishes Q(scfm).
Q(scfm) x
Tk
/60 = SCF
CLOCK / OSCILLATOR
[Tick –Tock]
Fill and turn off
at high pressure. Decay and turn on at low pressure.
FIGURE 1 |
