Gas Concentration Over TimeBuild-up of gas in a ventilated compartment from a constant leak

What this calculates

The purpose of this sheet is to calculate the concentration of a gas in a compartment, given a constant incoming flow of gas (from an open pipe end or some other leak).

How to use it

This sheet requires the user to enter air changes per hour (ventilation), gas flow rate, and the volume of the space being filled up.

Enter the air changes per hour in cell C3; the gas flow rate in cubic meters per hour into cell C4, the volume of the space in cell C5 and a time step in cell C8.

The time step (C8) is not directly related to the calculation. Rather, it helps set the range for the graph. If you have a timeline of 4 hours or less use a value of 1. If you have a timeline of 8 hours, use 2, and so on.

The graph will display a curve of the concentration of gas (in %) over time (in hours). Generally you would want to look of the LEL and UEL of whatever gas you are dealing with to get a sense for when you could have had an explosive mixture.

Variables

C(t): Concentration of gas at time t [% by volume]
Q_a: Ventilation rate (air changes × volume) [m³/hr]
Q_g: Gas leakage rate [m³/hr]
V: Volume of the gas-filled space [m³]
t: Time since leak began [hr]

Equations

$$ C(t) = 100 \cdot \dfrac{ Q_{g} }{ Q_{a} + Q_{g} } \, \left( 1 - e^{- (Q_{a} + Q_{g}) \, t / V } \right) $$Concentration with constant leak and constant ventilation

Discussion

An air change per hour is the number of times that all of the air in the space will be replace each hour. Without pressure testing (which is usually not possible after an explosion) it is impossible to say exactly how many air changes per hour there might have been. Generally it is good practice to run the calculations using a number of possible air changes per hour from 0 to 3 or so to see what the impact may be.

The volume that you use for this calculation is not necessarily the total volume of the compartment, but the volume being filled up by the gas. For natural gas (or other gases lighter than air) you should consider the volume to be the space above wherever the leak was. For propane (and other gases heavier than air) you should consider the volume to be the space below wherever the leak was. You might also consider subtracting out some volume for any large items (furniture, machinery, etc.) that take up space in the compartment.

If you have a scenario with an open-ended pipe, you can calculate the gas flow rate using the “Open_Pipe” spreadsheet.

Worked example

Example

Two-step problem using the “Open_Pipe” and “Gas_Concentration” spreadsheets.

Problem: A warehouse 40’ x 100’ with a ceiling height of 30’ has natural gas (specific gravity of 0.65) supplied at an incoming pressure of 10 inches WC. The gas pipe is 2 inches (inner diameter 1.8 inches); somebody opened a fitting 50 feet from the regulator at a height of 3 feet off the floor.

The warehouse has a lot of overhead items that take about 10% of the volume out of the ceiling space.

It is a fairly tight building, with less than1 air change per hour.

Approximately how much time would elapse between opening the pipe, and reaching the LEL of 5% in the warehouse?

Answer: Go to the “Open_Pipe” spreadsheet. Enter the pressure of 10 inches WC, an inner pipe diameter of 1.8, a length of 50 and a specific gravity of 0.65 (for natural gas). The gas flow rate should be 169.53 cubic meters per hour, round up to 170.

Go to the “Gas_Concentration” spreadsheet. Enter an air change of 1, and gas flow rate of 170.

The volume of the space will be 40x100 (4000) times the height above the leak (30-3=27). 4000 x 27 = 108,000 cubic feet. The problem mentions 10% of the volume is taken up, so we subtract out 10,800 to get a volume of 97,200 cubic feet. Enter this into cell C5.

With the time step set to “1” we can see the LEL. If you wanted to see what the concentration looked at out to 8 hours, you could change it to “2”.

To find the time at which the concentration exceeds 5%, follow the blue line until it crosses 5% on the left axis. Straight down from there on the bottom axis is the time of about 1.8666 hours (just under 2 hours). You could also find that number by looking down column C for 5% (if you have the time step at 2 this will be cell C71). Look left to cell B71 for the time of 1.8666 hours.

So, the answer would be about 2 hours to reach the LEL of natural gas for the given scenario.

References

Harris, R. J. — The Investigation and Control of Gas Explosions in Buildings and Heat Plant, 1983.