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Someone said, “When
your only tool is a hammer every problem looks like a nail.” You
might throw up your hands and say, “Here he comes again with that
same darn formula.”
If you don’t, you may
need to be more critical and less accepting in your thinking.
An instruction about
the amount of flow that would produce a 10% pressure drop of
compressed air in 100 feet of schedule 40 pipe stated a common
equation based on based on the Darcy or Harris formula. The next
statement tripped the BS-amometer. In essence it said that for 200
feet double the pressure calculated for 100 feet. It follows that
starting with 100 PSIG and the flow that would cause a 10 PSIG
pressure drop in 100 feet that in 1,000 feet the pressure drop would
be 100%. If the reader interpreted this statement of double the
pressure drop for an additional 100 feet of pipe would the pressure
drop double at 200 feet and again at 300 feet and again at 400 feet
for an 80 PSI drop at 400 feet?
The best way to
illustrate the problem here is use a monetary example to compare
simple interest to compound interest. Some equations taught and used
in error for pneumatic calculations based on a linear or additive
approach. In the real world these changes are compounded. Resistance
to compressed air flow in plumbing causes a pressure drop. With
lower pressure the loss is aggravated. The loss in two feet is not
the same as the loss in one foot times two.
Back to our old
friend the decay formula or die away curve, P1 = P0 (e^-KL).
The constant of pressure loss, K would be found by experiments and L
represents the length. This concept and utilization of Newton’s law
of cooling formula is treated respectfully by Silvanus P. Thompson
in his book, “Calculus made easy” and as stated, “Is very important
in physical science.” |
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The formulas we have
proposed for flow and pressure decay are also somewhat simplistic.
At some point we must deal with the change in resistance to flow
when the velocity of the compressed air drops below 1200 feet per
minute and the flow become laminar. The sonic choke at pressure
ratios of 53% or less is another factor to consider.
For starters, the
formulas for die away and organic growth are technically correct and
significantly more accurate than linear, additive techniques. For
tubing it is simple to start with 100 feet of tubing and record the
time required to vacate a known volume. Then cut the tubing
progressively shorter. We have only tested ¼ “ OD Nylon. We used
100’, 75’, 50’, 25’, 20’, 15’, 10’, 5’, 4’, 3’, 2’, 1’, 6”, 4”, 2”,
and 1”. The number of test points may be reduced if a consistent
pattern emerges when testing other tube sizes. The entrance factor
is significant in addition to the friction resistance.
From the time in
seconds to decay the volume to 37% of the initial pressure we
established the flow with each length of tubing. By using the top
63% of the pressure we are operating in sonic, turbulent flow. These
results are consistent for the most common usage of flow through
tubing above the laminar threshold. With minor discrepancy, possibly
from the entrance factor the flow or resistance to flow for this
tubing follows the die away curve.
I’m off for Alaska.
If the bear wins you are on your own. |
