Dr Paul Thomas once bubbled...
My two cents,
1) An "empty" 15 litre cylinder at 30 metres will give an additional 15 litres of breathable gas when the external pressure is halved from 4 bar to 2 bar at 10 metres.
OR
2a) An "empty" 15 litre cylinder at 30 metres will give an additional five litres of accessible gas at 20 meters;- ((15 x 4/3) - 15 = 5) since the external pressure is reduced from 4 bar to 3 bar.
2b) When this 15 litre cylinder is emptied at 20 metres it will give an additional 7.5 litres of breathable gas at 10 metres when the external pressure is reduced from 3 bar to 2 bar;- ((15 x 3/2) - 15 = 7.5)
If you do not panic and so assume a tidal volume (TV) of under two litres you appear to have a choice of ascending from 30 metres directly to 10 metres where you can take about seven breaths or you could "stage" the ascent and take about three short breaths at 20 metres followed by about four more at 10 metres. (If you have a miniscule TV like J.P. even more breaths).
I started diving in the 1970's, when - although frowned upon - running out of air was not uncommon (and deco stops in recreational diving unheard of!) I remember not being too concerned about running dry because in the minute or so it took to ascend you could take a few breaths from the dead cylinder provided you did not forcibly exhale during the ascent to "empty" the lungs and simply blew bubbles to prevent barotrauma!
How times have changed!
Please note I am not suggesting running out of gas is a good idea or that the above should be practiced or taught. Far from it!! I am simply indicating that if an emergency swimming ascent is necessary there will indeed be a few breaths in the cylinder.
I am not so sure that voluntarily removing much needed buoyancy from your BC is at all wise or indeed necessary when you have nothing with which to replace it, particularly if this leads to the diver sinking!
As for spare air?
Please note I am no instructor and claim no credentials as such!
wait a second here. First of all a scuba tank is a rigid metal container. The outside pressure has no influence on the air inside the tank. You cant apply Boyle's law to a rigid container this way.
what do you consider an 'empty' cylinder?
Usually a cylinder at atmospheric pressure, i open the valve and nothing comes out, is considered empty correct?
Then, if i take an empty cylinder as defined above to ANY depth i will not be able to breathe because the pressure in the tank is not high enough.
suppose i am diving and run out of air at 30m. The reason i can take some breaths on the way up is because the cylinder is NOT empty as defined earlier, rather the cylinder pressure at that depth is too low to supply air to the regulator. Depending on regulator design the tank pressure has to be above a critical pressure, depending on the surrounding pressure the reg is in, for the regulator to work. When i ascend the tank pressure will at a certain shallower depth be above that critical pressure again allowing you to take a breath.
To find out exactly how much air you would get on the way up depends on regulator design.
Your calculation is incorrect.
Think about it.
You say you will get 5 L out of this tank at 20 m (coming from 30m), 5L at 20m (3 bar)corresponds to 15L of 1 bar (which would then per definition have to be the tank pressure, 15L tank holds 15L of gas at 1 bar)
Then at 10m you get an additional 7.5 L (2 bar) which again corresponds to 15L of 1 bar (tank pressure).
Doing this calculation from 20m to the surface would imply getting (15 x 3/1)-15 = 30 L which then implies a tank pressure of 2 bars!
You are getting many liters of gas out of the tank and the tankpressure INCREASES , this is simply impossible. As said, you cant apply Boyle's law this way.