#1. I'm not sure I agree that there is no cost to get the air moving. The air is not moving relative to it's compartment. I agree that it will have momentum and as the lower floor of the compartment is decellerated during impact there will be some pressure caused by the decelleration of the air as well but I think that this will be negligible compared to the pressure caused by the compaction of the compartment.
The reason there's no cost to get the air moving is because it too has gravitational potential -- on the order of 0.7 GJ if I didn't slip an order of magnitude. This is the same reason a sealed box full of air falls at the same rate as a box full of vacuum, corrected for mass, in a freestream.
As that air is expelled and "turns the corner" against the ground, there is a local felt pressure by the ground (dynamic pressure), but the pressure at the descending "piston" is the same. Incompressible flow. From the "piston"'s perspective, the streamlines haven't changed. It doesn't know about the air turning ahead of it.
What backpressure you get, again, will be primarily due to Venturi effect (the remainder being turbulence, etc.).
#2. In order to arrive at the expended energy during collapse. I am removing energy interatively in "jerks" at each floor impact such that the fall time becomes 6.5 sec. My interpretation of "very little structural resistance" is based on the air resistance accounting for all resistance rather than the comparatively low total energy dissipated during collapse which is expected in a bottom up collapse.
The "jerks" aren't quite the same thing as your spreadsheet -- you're summing up in a Riemann fashion rather than a continuous integral, which is perfectly adequate for this estimate. What I mean is that the
total work done against the air is much higher than it might seem because the backpressure applies over the entire height of collapse, whereas the structural elements apply only over a small fraction, equal to or less than the yield strain of the materials, or ~3%.
#3. Dr. Greening has pointed out previously that adiabatic heating can be a significant energy factor. Have you taken this into account in your more "pressure oriented" method?
I did not, simply because "adiabatic," among other things, means "reversible." The heating is a temporary cost. When the air escapes the structure, that heat will be returned as kinetic energy, minus some small correction of course since our system and our gas aren't ideal.
While the energy loss to heating will be very small, we can also bound it by computing the heating during pressurization. Suppose our structure withstands 1 PSI of overpressure before it starts breaking windows all over, so 1 PSI is our self-regulating backpressure. In adiabatic compression, we have the following equation of state:
p Vγ = constant (1)
where
γ = 1.4 for air. We also will need the Ideal Gas Law:
p V = n R T (2)
So let State 1 be the air in WTC 7 before compression, and State 2 be the air after compression. For sake of argument, assume State 1 is STP -- yes, I know the fire would have heated the air, but ignore that for now; it doesn't make that much difference here. We've already decided that
p2 = 16 PSI, or
p2 / p1 = 16/15. From Eq. (1), we can then calculate the relative volumes, and we find:
V2 / V1 = (p1 / p2) (1/γ) = 0.95.
Next, we find the resulting increase in temperature by writing the ideal gas law for both State 1 and State 2, and then dividing both equations, to get:
(p2 V2 / p1 V1) = T2 / T1 (3)
and thus
T2 / T1 = 1.02. Since above we assumed that
T1 = 300 K, we find that the compression causes a rise of about six Kelvins in the temperature.
Now we can estimate the energy content as a result using the specific heat of air. At this point we have to make a decision based on our method of estimation -- under the adiabatic model, there is no heat entering or leaving, so this equation doesn't strictly exist. We're highballing it. We can choose instead to use either the specific heat under constant pressure (
CP) or constant volume (
CV), but neither is strictly appropriate. I've chosen to go with
CV, modeling that the air will be expelled as it is and
only then expand, thus carrying away the maximum amount of energy as it does so.
Under this assumption, the energy content from heating is as follows:
ΔH = ρ V N CV ΔT (4)
where
ρ is the density of air (and hence
ρ V is the total mass of air);
N is the molar mass of air, approximately 29 g/mol; and
CV is the specific heat under constant volume, 20 J / mol K. I roughly estimate the air mass at 850 tonnes, assuming a footprint of 3900 m
2 and 180 m height. This gives us the maximum energy loss possible due to heating:
ΔH = 3.5 GJ
This is about 40% of the energy expended against the backpressure that I computed before. Since that estimate is sensitive to the actual backpressure seen in the building, which could be +/- 100%, this result is not particularly significant. Remember that this is the
worst case loss to adiabatic heating, and the expected value is closer to zero. I think this will be lost in the noise.
Again, this calculation supposes the building only withstands 1 PSI of overpressure before the air finds another way out. That number is going to be very, very hard to estimate, and the result is strongly dependent upon it.
#5. At this level of refinement, I would caution the truth movement not to jump up and down crying CD is proven. There are still a lot of factors that have not been taken into account.
The other big uncertainty in the problem is the actual energy dissipated during collapse. Since you've estimated a collapse time very close to "freefall," your calculation will be maximally sensitive to small errors in the timing. An underestimate of only 0.5 seconds, for instance, means about another 14% of the total gravitational potential was actually consumed, or about 13 GJ additional energy. As a result, your error bars on the energy available are going to be large -- I'd estimate +/- 100% -- unless you can estimate the collapse time
very accurately. I can't.
At the end of the day, if we could sum up all of the energies properly, I would expect to find the following:
Ebuilding = 14 GJ +/- 10
+ Eair = 7 GJ +/- 10
= Etotal = 20 GJ +/- 10
Something like that. The uncertainties are so high that I doubt we will ever conclude one way or the other.
What you can conclude, however, is the following:
- The energy budget is rather tight
- The energy expenditure against the air is much higher than most people would expect, possibly higher than the destruction energy of the structure during collapse
- The speed of collapse suggests, but does not definitively prove, that the structure suffered very heavy internal damage prior to total collapse
I'll be interested to see how the third conclusion tracks against NIST's final report. We know they'll propose that an internal failure triggered the collapse. This investigation suggests that the internal failure was very widespread -- this investigation seems to conflict with a finding that the failure was only on one floor. I'm expecting to see a nearly total core failure leading the collapse after having walked through this derivation.