t² Fire Growth CurveNFPA Slow / Medium / Fast / Ultrafast HRR-vs-time

What this calculates

This sheet estimates fire growth rate assuming it is proportional to time-squared.

How to use it

The chart at the top provides some examples of the growth rates classified by NFPA standards. To get an idea for what values of t1 or what growth categories might be appropriate for different fuels see Quintiere Table 6-6 and Figure 6-22.

Choose an appropriate t1 value for your fuel and enter it in the appropriate yellow cell below. Enter a value for the peak heat release rate in the next cell. This sets a cap on the growth phase, preventing it from getting any larger. If you don’t want a cap, just put some very large number here.

The “time step” in the last yellow cell is not related to the t-squared fire calculations. Rather, it determines how much time will be calculated out. If you have a very short fire and want to know what is going on every second, put in a value of 1. If you have a longer fire and want to know what is going on every minute, put in a value of 60.

The graph at the bottom is provided to help visualize the fire growth. The blue line is the t-squared fire created by using the spreadsheet.

To the right, the columns in red show the calculated heat release rate over time for the fire. If you want to extend the timeline out, increase the value of the timestep cell.

Variables

Q
Heat release rate [kW]
α
Fire growth coefficient = 1000 / t₁² [kW/s²]
t
Elapsed time since ignition [s]
t₁
Time to reach 1 MW [s]
Q_peak
Peak heat release rate (cap) [kW]

Equations

$$ Q(t) = \alpha \, t^{2}, \quad \alpha = \frac{1000}{t_{1}^{\,2}} $$Fire growth equation (Quintiere eq. 6-3)
$$ Q(t) = \min\bigl(\alpha t^{2}, \; Q_{peak}\bigr) $$Capped at peak

Discussion

The spreadsheet has been set up to automatically stop the growth rate at the user-specified peak heat release rate. There is no built-in function in the spreadsheet to show any kind of decay phase.

In general, it is a good idea to test a range of t1 values for the fuel of interest to determine a broader potential range of options. For instance, if you were looking at a stack of wood pallets 1.5 feet high, the recommendation in Table 6-6 of Quintiere would be to use 155-310.

Worked example

Example

This example will be a two-part problem that uses the “HRR” sheet as well as the t^2_fires sheet.

Problem: Fire department arrives on the scene of a warehouse fire. The main fuel that is burning is stacks of recycled polyethylene. Based on a picture from the dashboard cam on the fire truck, on their arrival there was about 200 m2 of polyethylene product burning. Assuming no accelerants were used, what was the approximate time of ignition of the fire?

Answer: Go to the HRR spreadsheet, choose polyethylene from the dropdown box and enter the estimated area of 200 m2. That will calculate a very large fire size of 225,160 kW (225 MW), but that makes sense because it is an enormous pile of plastic material burning.

Next go to the t2_fires spreadsheet. Enter 225,160 kW as the peak HRR. This doesn’t mean that the fire will really stop growing at this point in real life, it just makes it easier to look at the graph and determine the timeline.

Check table 6-6 (Quintiere) for something close to the polyethylene plastic. You will see there is a wide range of potential PE products. Most of them seem to be in the range of t1 = 53 to t1 = 82. There is one at 145, but that seems oddly outside the range of the others, so it might be best to consider that an outlier (but keep it in mind as you go forward with your case).

First, plug the value of 53 as t1. In order to get a timeline long enough to reach 255,160 kW we will also need to adjust the timestep cell to at least 30. Now, by looking at the graph we can see that the time at which 225,160 was reached was at about 810 seconds. We could also look at the red columns to the right, going down until we find at least 225,160, which is at 810 seconds. Keep in mind that it might be a little sooner than that because we are only checking every 30 seconds here.

Next, we will do the same thing for a t1 value of 82. This gives us an answer of 1230 seconds for the growth time. Just to experiment, try setting the timestep to 40 seconds, it’ll show more of the curve on the graph, but you’ll notice the growth time is now 1240 seconds. Just to demonstrate that further, try a timestep of 10 seconds, now you’ll see that we don’t get enough of the graph to reach the peak.

So, the range of times to grow from zero to the fire that was shown on the dash cam when the fire department arrived was between 810 and 1240 seconds, or about 13.5 to 20.7 minutes.

References

  1. Quintiere, J. G., Principles of Fire Behavior, Chapter 6, eq. 6-3; growth rates Table 6-6 and Fig. 6-22.
  2. Slow / Medium / Fast / Ultrafast classifications from NFPA standards (t₁ = 600, 300, 150, 75 s respectively).