How Uncertainty Transforms the Way we Quantify Fire

Posted in the name of Prof Jose Torero.
(related to the previous blog entry "Study or Gamble, but not both - 2nd annual Christmas tree fire test")



It is common practise to use experimental data for many purposes in the analysis of Fire Safety. It can be used as direct input (HRR, flame spread rates, ignition times, etc.) to models (analytical, semi empirical and CFD) as well as to obtain parameters that then can be used as input to other more fundamental models (heat of combustion, thermal properties, etc.). In many cases, due to the complexity of the tests, we rely of single data points to infer the values that we need. We can conduct a detailed analysis of the data and provide output values. In this particular case, the output values were the pHRR and the burn time of the tree. If I was to use this data for modelling, both parameters will be of critical importance and I could define a Q_dot=alpha x t^2 fire on the basis of both parameters. Furthermore, I could divide the HRR curve by the burning rate and obtain a heat of combustion that together with a flame spread model I could convert into another form of Q_dot. I could even use this data as part of a fundamental model that will attempt to predict all processes involved. Much of the research work we do tries to do two things, develop better models and try to make best use of the data we have. Thus this test is a fun example of what we are all about!

So, given the interest that this particular test has created I thought that it will be important to do a little exercise of uncertainty, not to question the winner, or to question the methodology used in defining this winner, simply to establish how important it is to look at these tests with caution and how difficult it is to use them in a manner that is truly representative of the event we are trying to describe via our engineering techniques. Furthermore, it is important to do this analysis to establish one of the key values of apriori estimations coupled with aposteriori explanations.

Apriori estimations have the distinct value of providing predictions that are only biased by the user’s knowledge or experience and not by the knowledge inferred through the observation of the test. Aposteriori estimations always carry the bias associated to having the knowledge of the results of the test. The aposteriori analysis of the apriori predictions reveals the effectiveness of the thought process associated to the apriori predictions. This analysis is extremely valuable in the sense that it can allow to separate the logic that is “user robust” from that that is purely a “guess.” It is also important because it allows establishing which of these “user robust” criteria have large experimental variability. Finally, it allows to identify common errors that can lead you to a “bad guess” but most important to a “good guess.” “User robust” logic with known “variability” is what we want to use to interpret test data and extract this information to introduce into our Fire Safety calculations.

I will do an aposteriori analysis of my estimates not to over-emphasize/or counteract the ridicule of being among the furthest away from the answer or to incontestably establish how my brain seems to have deteriorated with years doing fire research. The objective of this analysis is to encourage you to retrospect on how you achieve your estimate, post it, and let’s see what are the “user robust” criteria, which criteria is not robust, what were purely “guesses” and of these which ones are good or bad.

I know I am taking the risk of taking the joy out of a fun event, but given my role as an educator I find myself compelled to do this. The effort put on the tests and Guillermo’s fantastic statistical analysis encouraged me to do this. In any case, if you do not feel it is important, you will not participate and that is the end of the story!

900 kW and 20 seconds – How did I get there?

The way I reached my estimates, which I tried to qualify, but was told I could not (fair enough), was based on my experience of similar data published in the literature and the previous test conducted in Edinburgh.
When estimating the pHRR I made the following assumptions:

• The variability between tress in the literature was small.
• The pHRR was dominated by upward flame spread (VS) and time to burn out (tBO) of the leaves. Lateral flame spread is negligible compared to upward flame spread of a fuel of such low density, thus the effect of radial spread will happen after the pHRR.
• The base of the fuel burning (A) will be dominated by buoyancy not lateral spread, thus it should be the same for all tests.
• The tBO is very small and the base of the tree tends to have a higher density than the top, thus the pHRR will be generally attained before the flames reach the top of the tree.
• The HRR (given that this is a low density porous medium) will be proportional to the burning volume, so given a constant value of “A” it will be proportional to the height, thus to H=VS.t_b.
• The available data generally estimates a pHRR that ranges between 900 kW and 1100 kW.

So, given that the real height of the tree should not matter, then the pHRR should be similar to that of the literature. Because the tree was small, it was not so dry and it did not seem that dense I decided to opt for the lower bound value and estimate 900 kW.

Now, that being said, generally, literature values tend to be corrected by the time delay of the calorimeter. Our calorimeter has a time delay of about 10 seconds. What does that mean? Basically, it means that oxygen consumption measurements lag by 10 seconds the mass burning rate measurements. This generally makes no difference for events where things do not change within that period. If the event time scale is of the same order of magnitude of the time delay, then the measured value is somewhere between the measurement and that 10 seconds later. So, if I was to take the HRR curve measured by the calorimeter, then the value will be somewhere between what was measured 697 +/- 25 kW and 1000 kW.

An important lesson to learn is that the pHRR of a fast event (actually, even a slow event, but for different reasons) is a very difficult quantity to estimate precisely, thus the +/-25 kW stated as the error is truly only the direct measurement error. The true error will have to include the variability associated to the burn out time, the buoyantly driven upward spread, the global density and the comprehensive experimental error which is a parameter that is relevant in this case because the times are so short. So, any estimates within +/- 200 kW will probably have exactly the same value if the variable used is the pHRR. Thus 11/28 of you truly guessed the same answer.

If a different variable, such as the average HRR, or the Heat of Combustion was to be used as the “estimate,” then once all corrections due to time delay were made, would have probably delivered a smaller error bar.

The second variable to be estimated was the burning time. My estimate was 20 seconds and was based on a simple calculation of a typical upward flame spread rate of 10 cm a second. This had nothing to do with trees but with a fuel I know better (polyurethane foam). I estimated that the global value of “krhoC” is dominated by the density and I assumed that the density was more or less the same for both fuels. Thus I took that number. The tree was about 1.5 m, this gave about 15 seconds, time to burnout is so short that once the flame spread to the top, I could assume the fire was over.

Now, here is where I tried (unsuccessfully) to introduce a qualifier, I could not engage to estimate the initiation time (from the moment of ignition to the moment when the fire truly takes off). Furthermore, after the pHRR, what is left is lateral spread, then the branches and finally the trunk. The trunk will extinguish as soon as the branches die (bulk wood does not burn unless assisted!), but the lateral spread (being dominated by the shape of the tree) and the branches (being dominated by their individual shape and size) are impossible to predict. So at the end I gave up and simply estimated the time that it will take to achieve the pHRR from the moment the fire truly takes off. I reluctantly added a 5 second buffer for the slow initiation. While not a good estimate for what I was being asked, there is something to be said for the accuracy of the estimate! From the HRR curve we can establish that the primary burning will be somewhere between 10-20 sec (considering the instrument delay).

Now, what have I learnt, buoyancy is such a strong driving force that the estimate of the upward flame spread is a very robust one. The estimate of total burning time is one that carries a massive error bar, thus I will be reluctant to dismiss any of your estimates. From my perspective 28/28 gave estimates that I will consider within the expected error bars. Needless to say, last year’s Christmas tree was the proof to this point; the initial time could have been infinite if I did not decide to push the candle towards the denser part of the tree!

A final point, did I think of all of this in the 2 minutes that passed between the moment I learnt of the bet and the moment I provided my estimates? Obviously not! Most of this knowledge resides within your experience, and the estimate is an “educated guess.” Nevertheless, for the estimate to be adequate we need to carefully assess the question being asked (which I unfortunately decided to ignore) and the question needs to be posed correctly (meaning that what is being asked needs to have error bars that are smaller than the discrimination we are seeking). Otherwise, our guess will not be educated, nor it will be an estimate, it will just be a guess. If the error bars are small your chances of being the closest answer are very small (the educated estimate will have a much greater chance), but if the error bars are large you have as much of a chance to get it right as the most educated of estimates.

So citing Guillermo Rein: “while many stories can be told aposteriori,” and 3 hours of rationalizing my estimates can lead to this story, the stories need to be told and the discussion needs to follow. It is within the aposteriori 3 hours of introspection that I have truly managed to gain some insight into what happened not within the 2 minutes it took me to “guess.”

Congratulations to the winner!

Prof Jose Torero.