I am a scuba diver, trying to understand some physics that occurs throughout a dive. Primarily I am trying to understand the surface air consumption (SAC) derivation.
For those unfamiliar, SAC is the change in pressure (PSI or bar) per unit time at the surface (1 atm). SAC is calculated by measuring the change in PSI at the dive depth and dividing it by the pressure at depth (expressed in atm).
$ SAC = \frac{\Delta PSI* P_{sea level}}{P_{depth}} $
Since the pressure at sea-level is 1 atmosphere the equation can be simplified by expressing the pressure at depth in atmospheres as $ P_{depth} = N_{atm}*1atm$. $N_{atm}$ is the number of atmospheres the pressure at depth is equivalent to.
In this case, SAC is expressed as
$SAC = \frac{\Delta PSI * 1atm}{N_{ATM}*1atm} = \frac{\Delta PSI}{N_{ATM}} $
I am trying to determine the assumptions/derivation to arrive at this equation.
So far, what I have done is assumed that
- The volume of air inhaled per breath is constant regardless of depth
- Breathing rate is constant regardless of depth
- Temperature of the gas is independent of depth
Using these assumptions and Boyle's law, it is easy to show that one breath at depth would be equivalent to $ N_{ATM} $ breaths at sea level.
I am not sure about what steps to take next. I need to figure out how much my tank's PSI will change for a breath, given the volume of 1 breath, the ambient pressure, and the tank's volume. I also need to show the PSI change per breath is independent of the PSI of the cylinder. I am rusty on my thermodynamics and looking for advice on how to derive/prove these two quantities.