Take a 4 cylinder engine. 3.875” piston, 4.25” stroke. With an intake at 1500 psig @ 60 deg. F, exhausting to 1000 psig. Back pressure under the piston is 14.7 psia. The exhaust stroke is at a constant 1000 psig resistance. Lets say we're using full pressure during the intake stroke.
4 cylinders with 1 intake and 1 exhaust per revolution, I get air volume moved as 234.04 cu ft/min of air @ 1500 psig, or the free air equivalent at 23,910 cu ft/min.
How much real world HP would it take to re-compress that 234.04 cu ft/min of air back up to 1500 psig?
On page 27 of Uncle Buddy's Compressed Air Power Secrets there is a chart by the great Herman Haupt which lists the hp required to compress 1 cfm of free air (which I'll call 1 cfmfa) to various pressures. To reach 1000 psi takes .364 hp, and to 1500 psi takes .385 hp.
So, to go from 1000 psi to 1500 psi would take .021 hp. Which doesn't sound like much.
Plug that into your scenario, though, and your 23,910 cfmfa will suck up 502 hp!
If I recall, Haupt also amended his figures to add 10% for mechanical inefficiency, so your real-world number becomes approx 550 hp.
You didn't include the hp produced by your system. I don't have the formulas here in front of me, but I think we can ballpark it.
I've got an air motor that uses about 20 cfmfa of 100 psi air to produce 1 hp. It's about average, maybe a little above. Let's say your motor would be of a similar efficiency.
If we're pushing 500 psi, I think this means that we'd need about 4 cfmfa for 1 hp when expanding fully and exhausting to the atmosphere. But since your exhaust is at 1000 psi (about 67x atm), we'd need 67x that amount, or about 270 cfmfa per hp.
This tells me that with 23,910 cfmfa available, your system would produce about 88 hp.
Unless my math is horribly wrong, which is entirely possible.
Is that chart for single stage compression? If it is those numbers could be reduced significantly, but still would be a loss. Mechanical re-compression and exhaust back-pressure are the air motor killers.
Wish I had a copy of Compressed Air Power Secrets, but as far as I can tell it's no longer available.
502 HP to re-compress is interesting. I came up with a little over 501 HP of work performed at the pistons in the motor. Estimating 15 to 20% net from that, or 17.5%, is right at 88 HP.
One of the best texts for air data is COMPRESSED AIR, Simons, 1914. A google search will bring it up, then choose download as pdf, it is FREE! The chapters are arranged very well. The text gives formulas but also explanation of the real-world efficiencies, and rules of thumb. The end contains lookup tables which are very valuable to avoid using math formulas, but need to read the text to get the background of when/where/how to apply the tables correctly. Here are the tables which should be printed on paper for future reference:Simons1914TABLESsmallsize.pdf (718.71 KB)
You can use this to calculate different in and out pressures, theoretical numbers, then apply the real world percentage from Simons, percentage for comp and for motors. The pre-made tables assume 1 atm reference, need to calculate for other pressures. AdiabaticFormulas.pdf (17.5 KB)AdiabaticFormulas.xls (100 KB)