There was a formula on Wiki for Thermobaric overpressure calculation (Thermobaric=FAE). However, I wasn't sure enough to use it, as I did a calc based on a 75lb. FAE and got an answer that was something like 20 kilometers, so apparently I did something wrong
Oh, that was it!
Yeah I calculated it out for your value of 51,000lbs and got 4.9 million km
Although that is for 1 psi overpressure, which probably is not even worth considering... I wonder what the formula would be for 90 psi (approx guaranteed fatal).
Anyway... funnily enough NIST have already done what we're kind of trying to do.
In
this report they do a simple model to calculate how much fuel was burned off in the fireballs created when UA175 hit the South Tower.
I can't make much sense of it - as you'd expect it is INCREDIBLY technical - demonstrating the insanity of what Killtown is attempting here.
However, it is worth nothing that for an estimated burn of 1600kg of fuel (out of an estimated fuel load of 28,000kg) they produced (by my reading) a total spherical fireball radius of 70m after 2.5 seconds - or 140m across.
Now, if we're saying 51,000lbs of fuel on UA93 - that is 23,000kg.
Now, to compare this to our theoretical fireball we have to calculate spherical volume of the fireball first.
I just want to reiterate this is an appalingly shoddy calculation, given the enormous assumptions I am making, however given it is based very roughly on the NIST calculations, I figure it is better than anything Killtown has thrown up.
So for the UA175 fireball r = 70m
[latex]$$V = \frac{4}{3}\pi r^3$$[/latex]
[latex]$$V = \frac{4}{3}\pi 70^3$$[/latex]
[latex]$$V = 1,436,755.04m^3$$[/latex]
Working backwards...
If volume for 1600kg = 1.4 million m^3
Purely throwing out an even-ratio calculation, a fireball consuming Killtown's 75% (17,250kg) and 95% (21,850kg) will have volumes of about 15 million m^3 and 19 million m^3.
Someone let me know if this is wrong...
[latex]$$V = \frac{4}{3}\pi r^3$$[/latex]
[latex]$$V\div\frac{4}{3} = \pi r^3$$[/latex]
[latex]$$V\div\frac{4}{3}\div\pi = r^3$$[/latex]
So for the first example (75% burn)
[latex]$$15,000,000\div\frac{4}{3}\div\pi = r^3$$[/latex]
[latex]$$3,580,986 = r^3$$[/latex]
[latex]$$r = 152.99m$$[/latex]
So a fireball 306m across.
Or for 95% burn:
[latex]$$19,000,000\div\frac{4}{3}\div\pi = r^3$$[/latex]
[latex]$$4,535,915.878 = r^3$$[/latex]
[latex]$$r = 165.534m$$[/latex]
So a fireball 331m across.
Now it is important to note, in each of these cases, that the NIST calculation was only until the point where the fireballs began to rise. They continued to expand after this, and expanded even further as the fires died out and the mushroom cloud formed.
So could a 330m initial fireball produce a 600m across mushroom cloud?
Well, for comparison the
Tsar Bomba - the largest nuclear bomb ever detonated - had a fireball about 8-9km across, and produced a mushroom cloud 30 - 40km across. Bear in mind, despite the differing weapon, the basic physics involved in each is the same - all mushroom clouds form the same way.
Based on this very incredibly extremely rough example, you'd be looking at, easily, 1100 - 1650m across for a mushroom cloud from a 330m across fireball (and remember, based on UA175, the fireball would have reached BIGGER than 330m).
Now, I just want to stress, because it is incredibly important, that all of these calculations are INSANELY loose. A multitude of factors come into play at each step.
I have simply used a single example of a high speed airliner impact and resulting fireball, and applied the most loose interpretation of that data (and frankly, I could easily have interpreted it wrong!) scaled up to the percentage of fuel Killtown is proposing was burned in the crash of UA93.
This is not real science at all. But it at least has semi-stable foundations. I feel more comfortable with MY calculation than with Killtown's "looks about the right size".
-Andrew