Work of Breathing

Work-of-breathing

The CE standard for a regulator is 3J/L max.
That is three Joules of energy expended by the diver to get one litre of gas into their lungs and, I assume, out again.
What's that is real numbers? I need to get my head round it.
The best analogy I can think of is snorkelling. How far do I have to suck my one litre of gas down to expend 3 joules of work?
Well to compress a gas, the equivalent operation, you put in energy so we need a formula...

Consider a container with a moving piston of area A and current depth D
Hence the total volume is V=A*D
Let's keep it simple and think isothermal so PV=constant so the pressure is k/V
Start at pressure P as ambient so P=k/(A*D) so k=P*A*D
Right it is ambient so there is no force on the piston.

Now push the piston in a bit so 'D' is now d so the new volume is A*d and the pressure becomes p=P*A*D/(A*d) so p=P*D/d or d=P*D/p
This means the force on the piston is f = (p-P)*A
Now to push the piston in another δd we need f*-δd work
(Notice that pushing the piston in is reducing d so δd is negative here)
so δw = -(P*D/d-P)*A*δd
However what we want is a formula to go from pressure to work so take d=P*D/p and derive δd/δp = -P*D/p²
substitute δw = -(P*D/(P*D/p)-P)*A*(-P*D/p²)*δp
reshuffle δw = A*P*D*(p-P)/p²*δp
reshuffle again so A and D go and we know we are independent of geometry δw = P*V*(p-P)/p²*δp
Now integrate from p=P (w=0) to p=x (w=result)
(p-P)/p² is 1/p-P/p² so integrates to ln(p)+P/p
P*V*[ln(p)+P/p]
W = P*V*(ln(x)+P/x - ln(P)-P/P)
W = P*V*(P/x+ln(x/P)-1)

So if I want SI units that's pressure in Pascals and volume in cubic meters
However I like bar and litres so 1bar = 10^5 pascals, 1litre=10^-3cu meter
W = 100 * P*V*(P/x+ln(x/P)-1)

Finally stuff it into eXcell and get a graph. Depth in cm across the bottom and WOB in Joules up the side:

And the answer to my initial question is 29cms. Just under a foot in old money. That's probably just a bit deeper than the center of my chest when I'm in freediving trim and relaxed, flat in the water. Actually if I go from flat to upright it probably is the center of my chest without the snorkel.

by Nigel Hewitt