What is SAC Rate?

SAC (Surface Air Consumption) Rate is a measure of how much air a diver consumes per minute at the surface (1 atmosphere pressure). It's expressed in liters per minute (L/min).

The SAC rate allows divers to:

• Estimate gas consumption at different depths
• Plan dive durations based on available gas
• Compare breathing efficiency across different dives
• Calculate required cylinder sizes for planned dives

SAC Rate Calculation

The SAC rate is calculated from actual dive data using the following formula:

SAC = Total gas used / Absolute pressure / Dive time

Where:

• Total gas used = The amount of gas consumed during the dive, calculated from the difference between start and end pressures in the cylinder(s), normalized to surface pressure (liters at 1 atm)
• Absolute pressure = The absolute pressure at the average depth, calculated as (depth + 10) / 10 in bar (where depth is in meters)
• Dive time = The duration of the dive in minutes

The absolute pressure conversion accounts for the fact that at greater depths, each breath contains more gas molecules due to the increased ambient pressure. By dividing the total gas used by the absolute pressure, we normalize the consumption to what it would be at the surface.

For example, at 20 meters depth, the absolute pressure is 3 bar (1 bar atmospheric + 2 bar from 20m of water). A diver using a 12L cylinder consuming 150 bars of gas at this depth over 30 minutes would have a SAC rate of: (12 * 150) / 3 / 30 = 20 L/min

The values calculated this way can be used for dive planning in the minimum gas calculator.

Van der Waals Equation

This calculator uses the Van der Waals equation of state to calculate gas amounts at higher pressures, providing more accurate results than the ideal gas law.

At high pressures (above ± 230 bar for air), gases deviate significantly from ideal behavior. Using Van der Waals law ensures that gas consumption calculations are accurate, especially when working with scuba cylinders filled to 300 bar, where the ideal gas law would overestimate the actual gas content.

The Equation

The Van der Waals equation improves upon the ideal gas law (PV = nRT) by accounting for real gas behavior:

(P + a(n/V)²)(V - nb) = nRT

Where:

• P = pressure
• V = volume
• n = number of moles
• R = gas constant (0.08206 L·atm/(mol·K))
• T = temperature in Kelvin
• a = parameter accounting for intermolecular attractive forces (1.38 L²·atm/mol²)
• b = parameter accounting for finite molecular size (0.03767 L/mol)

The equation corrects for two non-ideal behaviors:

• Intermolecular attractive forces (a) - The term a(n/V)² is added to pressure, accounting for the reduction in effective pressure due to molecular attractions
• Finite molecular size (b) - The term nb is subtracted from volume, accounting for the volume occupied by the molecules themselves

Implementation

The calculator uses several functions to solve the Van der Waals equation:

You can view the implementation here.

Since the Van der Waals equation is nonlinear, it cannot be solved algebraically. The implementation uses the Newton-Raphson method to iteratively find the root numerically:

1. Start with an initial guess (using the ideal gas law approximation)
2. Calculate the function value and its derivative at the current guess
3. Update the guess using: x_new = x_old - f(x)/f'(x)
4. Apply physical constraints (e.g., n > 0, V > nb)
5. Repeat until convergence (when f(x) ≈ 0)

This method is used to first find the number of moles (n) in a cylinder at a given pressure and temperature. Then that value for n is used to find the volume that amount of gas would occupy at 1 atmosphere pressure.