In this month's column, we'll cover how to use Boyle's Law to calculate the amount of usable water in a captive air tank.
The drawdown capacity of a captive air pressure tank is determined by a formula known as Boyle's Law. Simply put, Boyle's Law says as the volume of the air cushion in a tank decreases, the pressure of that air cushion increases. Conversely, as the volume of the air cushion increases, the pressure of that air decreases. Therefore, for any given tank, drawdown equals the volume of air at cut-in minus the volume of air at cut-out. Stated as a mathematical formula taking the total volume of a pressure tank into consideration, it looks like this:
Drawdown = P1V / P2 - P1V / P3 where,
P1 is the pre-charge pressure.
P2 is the cut-in pressure
P3 is the cut-out pressure
V is the total tank volume
Remember that all pressures must be stated in terms of absolute pressure. At sea level, add 14.7 psi to the gauge pressure to get absolute pressure. Table 1 shows the atmospheric pressures at different altitudes above sea level.
Let's try a couple of examples.
Example 1 -- for a 30/50 pressure switch:
P1 the pre-charge pressure is 28 psi
P2 the cut-in pressure is 30 psi
P3 the cut-out pressure is 50 psi
V the total tank volume is 85 gallons
Altitude is sea level
The drawdown formula then looks like this: Drawdown = (28 + 14.7) X 85 divided by (30 + 14.7) - (28 + 14.7) X 85 divided by (50 + 14.7). Or, (42.7 X 85 / 44.7) - (42.7 X 85 / 64.7)
Therefore drawdown = 81.2 - 56.1 or 25.1 gallons.
Example 2 -- the same tank conditions, but put the job up at Lake Tahoe in California at 6,000 feet above sea level:
This time drawdown = (28 + 11.8) X 85 / (30 + 11.8) - (28 + 11.8) X 85 / (50 + 11.8)
Now drawdown = 80.9 - 54.7 or 26.1 gallons.
Example 3 -- what happens if we ignore the difference between the pre-charge pressure and the cut-in pressure using 30 psi for both in our formula:
Drawdown = (30 + 14.7) X 85 / (30 + 14.7) - (30 + 14.7) X 85 / (50 + 14.7)
Drawdown = 85 - 58.7 or 26.3 gallons.
From this calculation, you can see that by ignoring the difference between the pre-charge and cut-in pressures, you overstate the drawdown by about 5 percent (Example 1 @ 25.1 vs. Example 3 @ 26.3).
Example 4 -- 30/50 pressure switch, but no pre-charge as could be the case with a conventional tank:
Drawdown = (0 + 14.7) X 85 / (30 + 14.7) - (0 + 14.7) X 85 / (50 + 14.7)
Drawdown = 28 - 19.3 or 8.7 gallons, about 10 percent of capacity.
Example 5 -- Instead of a 30/50 pressure switch, use a 40/60 pressure switch with a 38-psi pre-charge and the numbers look like this:
Drawdown = (38 + 14.7) X 85 divided by (40 + 14.7) minus - (38 + 14.7) X 85 divided by (60 + 14.7). Or, (52.7 X 85 / 54.7) - (52.7 X 85 / 74.7)
Therefore drawdown = 81.9 - 60 or 21.9 gallons.
Example 6 -- How the drawdown changes if we widened the differential between the cut-in and cut-out pressures from 30/50 to 30/60:
Drawdown = (28 + 14.7) X 85 / (30 + 14.7) - (28 + 14.7) X 85 / (60 + 14.7)
Drawdown = 81.2 - 48.6 or 32.6 gallons.
Compare this with the 26.1 from Example 1. A word of caution: Before expanding the cut-in/cut-out differential beyond 20 psi, check with your tank manufacturer to make sure that by doing so, you will not be over-expanding the water chamber.
Another term used in determining the drawdown of a captive air tank is "acceptance factor." This simply is the factor by which you multiply the total tank volume to get drawdown. In other words, drawdown equals the acceptance factor times the total tank volume.
If you would rather work with acceptance factors, take the "V" out of the basic Boyle's Law formula and it becomes:
Acceptance factor = P1/ P2 - P1/ P3
Taking it one step farther, if you consider the pre-charge to be the same as the pump cut-in pressure, P1/ P2 becomes 1, and the formula now:
Acceptance factor = 1 - P1/ P3
However, as pointed out in earlier articles, the pre-charge must be set a minimum of 2 psi below the pump cut-in pressure to avoid the possibility of having the system pressure drop abruptly to zero before the pump turns on. It is imperative in a captive air tank to have the pre-charge set at least 2 psi below the pump cut-in pressure. I strongly recommend using the proper version of Boyle's Law, which includes a separate factor for pre-charge in order to accurately represent the drawdown of your system.
That's a lot of formulas and numbers. I hope I haven't fried your brain.