SAC rates are used to predict gas consumption in both open and closed circuit diving. It is defined as the total psi used over a given time span at a depth expressed in ATA’s (Atmospheres Absolute). ATA’s are calculated by using the formula:
Since the Partial Pressure of a gas is equal to the Fraction of the gas multiplied by the pressure expressed in ATA’s we can use this formula to calculate partial pressures:
PPg is the Partial Pressure of the gas and Fg is the Fraction of the gas or its’ % in the mix and the + 1 is to account for the 1 atmosphere on the surface. As a function of depth, one atmosphere equals 14.7 psi so on the surface we are at 14.7 psi of pressure and at 33 feet we are at 29.4 psi. These numbers are linear with depth for our purposes.
Daltons Law
The total pressure exerted by a mixture of gasses equals the sum of their individual pressures. This is expressed as:
These individual pressures are referred to as partial pressures so the partial pressure of Oxygen in a nitrox mix of 21% oxygen and 79% nitrogen is .21 because the Partial Pressure of a gas is equal to the fraction of the gas multiplied by the pressure, in this instance, 1 atmosphere or ATA (Atmosphere Absolute). Because the surface is considered 1 atmosphere and in sea water every 33 feet of depth is considered another atmosphere of pressure, at 33 feet the Partial Pressure of that same .21% Oxygen mix is now .42 In simple terms, the partial pressure of a gas is equal to the fraction of the gas adjusted for pressure. The same goes for any other gasses in the mix: PPN2 on the surface or 1 ATA is .79 and at 33 feet or 2 ATA’s it is 1.58
Back to SAC Rates
SAC rates are mathematically calculated by taking the psi consumed and dividing it by time. That quotient needs to be adjusted for depth because we want surface consumption but the measurements are taken while submerged.
Simple enough, right? Well not really. The result from the above equation is in psi per minute which is meaningless unless we know the tank volume it came from..
1000 psi used over 20 minutes at 33 feet (2 ATA’s) gives us:
This is your calculated SAC rate which tells us nothing unless we know the tank it came out of. 1000 psi out of a LP 104 cu ft. tank is significantly more gas than 1000 psi out of an aluminum 40 cu ft. tank so the SAC rate of 25 psi is meaningless without the tank information.
To do this we must calculate the “Tank Factor” or the tank volume divided by the rated pressure. The actual fill pressure doesn’t matter.
So if the above 1000 psi were consumed from a LP 85 the tank factor would be:
To give meaning to the previously calculated SAC rate we multiply the SAC by the Tank Factor to get our RMV (Respiratory Minute Volume).
Were the tank a 40 cu ft. rated at 3000 psi the result would be quite different:
So, right now you are wondering what is RMV? It is Respiratory Minute Volume or the total volume consumed over a period of one minute at one atmosphere of pressure. We can use this number to predict gas consumption by multiplying it by ATA’s and dividing it into the cu ft. available. IE: at 2 atmospheres the Aluminum 40 would have 60 minutes in it:
Were we to go to 66 feet or 3 ATA’s the bottle would last us 40 minutes:
So, in summation, the SAC rate is in psi and is essentially meaningless without tank information. The RMV is the SAC rate adjusted for pressure (depth) and tank size. We need this information in order to predict gas usage for open circuit dives and rebreather bailout calculations. For the sake of simplicity, we will say that you should use maximum square profile depths and highest expected RMV’s for these calculations. No one ever drowned because they had too much gas.
Maximum Operating Depth or MOD
Because Oxygen becomes toxic at high partial pressures it is important to know the maximum operating depth of any given mix. This is calculated by dividing the desired PPO2 by the fraction of oxygen in the mix and subtracting 1 from the quotient. The answer is in ATA’s:
So using our 21% mix we could determine the MOD for a desired PPO2 of 1.4:
This is good information to have but is not particularly useful in terms of utility. What is more practical is a formula that tells us the best mix for a given dive plan.
Best Mix
How do we know which mix to use for a given dive plan? Like everything else in diving, we have a formula for it:
So if we want to plan a dive to 132 FSW or 5 ATA’s and we want a maximum PPO2 of 1.4 we would calculate:
In any form of technical diving our first concern is always the oxygen content so for nitrox dives the Best Mix formula is sufficient. As our diving evolves and we introduce a third gas into the mix we need a way to calculate the correct gas mixture to produce our desired result. The third gas used is Helium. It is used to mitigate narcosis which is the narcotic effect produced by breathing nitrogen at elevated partial pressures. You may have heard of “Martini’s Law” which states that for every atmosphere of depth the partial pressure of nitrogen in air has the net effect of drinking one Martini. Accurate? Probably not, but it makes the point.
So, how do we calculate this mixture? We do it by first deciding our maximum desired PPO2. Then we decide our END or Equivalent Narcotic Depth. Narcotic Depth and Nitrogen Depth are often used interchangeably. Next we select a depth on AIR that we are comfortable with. So, for our example let’s use a 70 foot END for a 150 foot (5.54 ATA’s) dive with a maximum PPO2 of 1.4. The END is determined by the partial pressure of the nitrogen at that depth so PPN2 of air at 70 feet is:
This tells us that the maximum PPN2 at maximum depth can be no greater than 2.46. Next we determine the maximum depth of the dive we are planning and use Best Mix formula to determine our maximum PPO2:
Now we need to know the fraction of Nitrogen that will give us a PPN2 of 2.46 at 150 feet or 5.54 ATA’s.
So now we have determined that our FO2 is 25% and our FN2 yielding a maximum END of 70feet is 44% so by process of elimination we can determine that our Helium percentage will be 31% because:
So now that we have figured out how to decide on what mix to use let’s approach things from a different perspective and say that we have a Trimix of 25/31/44 hereafter referred to as 25/31. This is because the standard protocol is the first two numbers will be the Oxygen and Helium content and the remainder will always be the nitrogen content. We want to know the limits of this gas mixture so we will need to know its MOD and its’ END. The MOD formula from above is:
So we know that our FO2 is .25 and our desired max PPO2 is 1.4
Next we need to know the END of our mix. The formula is:
Some divers believe that Oxygen is also narcotic and factor that into their equations. To do that simply calculate using the combined percentages of the Oxygen and the Nitrogen content in place of the Nitrogen only, which would actually be 1 minus the Helium content. For the above calculation for a 25/31 mix we would use .69 to factor the oxygen into the equation in place of the .44 so we would have:
You may have noticed in our END formula where we are only concerned with the nitrogen content we divide the fraction part of the equation by .79 This is because we are concerned with the equivalency of the Nitrogen content of air as a percent. Since air is 21% O2 / 79% N2 we divide by .79.
In the formula which considers Oxygen narcotic there is no need to divide by .79 because the nitrogen and oxygen are combined and therefore represent the total of the gasses not Helium. The formula you use is your choice depending on whether or not you believe Oxygen is narcotic.
Gas Matching
For open circuit cave divers dissimilar tanks can be a problem. The long established Rule of Thirds governs gas usage: one third in, one third out and one third for emergencies. This is fairly straight forward if the tanks are the same size. Simply calculate one third of your pressure and subtract it from your total pressure and that becomes your “Turn Pressure” or where you begin your exit.
As discussed previously in this document the same psi in different sized tanks represents a different volume of gas so the only way you can “Match Gas” is by volume. This would be done by using our “Tank Factor” formula and calculating the volume in cubic feet of one third of the gas in the smaller tanks and use that volume to determine the “Turn Pressure” for the larger tanks. This sounds far more complex than it really is.
So, let’s calculate turn pressures for 2 divers using Double 85’s and Double 104’s both filled to 3600 psi:
cu ft per one psi
We double it for 2 tanks =.0644
cu ft per one psi
We double it for 2 tanks = .0788 cu ft per one psi
Since both tanks are filled to 3600 psi we know that one third of the gas is 1200 psi. The dive is always controlled by the smaller tanks so we would use the double LP 85’s tank factor to determine the turn pressures for both divers. Obviously, the LP 85’s turn pressure is going to be 2400 psi but we need to know what that is in terms of volume so we can apply it to the larger tanks:
Next we need to calculate how many psi 77 cu ft are in the 104’s so we divide the 77 cu ft by .0788 (one cubic foot of gas per psi).
So, for this dive the turn pressure for the 104’s would be:
3600 – 977=2623 psi
The turn pressure for the LP 85’s would be:
3600-1200=2400 psi
Simply stated we have calculated the cu ft volume of 1/3 of the smaller (controlling) tanks and converted that to a volume equivalent psi for the larger tanks so we could subtract it from the total fill. Remember that even if the larger tanks were filled to a higher number, let’s say 4000 psi, the turn pressure would be:
4000-977=3023 psi
Typically these calculations are done per 100 psi so the respective “Tank Factors” would be 6.44 cu ft per 100 psi for the double LP 85’s and 7.88 cu ft per 100 psi for the double LP 104’s.
So, the process is as follows:
- Calculate or look up the appropriate tank factors.
- Calculate 1/3 of the smaller tanks psi.
- Multiply that psi by the tank factor to get the cu ft equivalent of thirds.
- Divide the cu ft of thirds in the smaller tanks by the tank factor of the larger ones to get the equivalent tank volume in psi for the larger tanks.
- Subtract that number from the total fill pressure of the larger tanks to get the turn pressure.