Chemical energy of the jet fuel

Accoring to Wikipedia, which claims to quote NIST, either plane hitting the two WTC towers contained about 10,000 gallons of jet fuel, or 30,400kg:

According to the estimation in NIST's NCSTAR 1.5, page 58, AA11 carried 30,000kg of jet fuel, and UA175 carried 28,100 kg at the time of impact

http://en.wikipedia.org/wiki/American_Airlines_Flight_11
http://en.wikipedia.org/wiki/United_Airlines_Flight_175

Wikipedia further says about jet fuel:

Specific energy: >42.80 MJ/kg

So, AA11 carried with it fuel which, upon combustion, would this much energy as heat:
E_fuel = 30,000kg * 42.8MJ/kg = 1.284 * 10¹²J

UA175 carried with it fuel which, upon combustion, would this much energy as heat:
E_fuel = 28,100kg * 42.8MJ/kg = 1.203 * 10¹²J

At this time, I haven't found data about how much of that fuel burned outside of the towers and was not availabe to do damage inside.

In addition, each plane carried about 12,000kg of other combustibles (cargo, tires, mail, luggage, cabin materials). Humans are not included. "Other combustibles" generally having somewhere between 10 and 30 MJ/kg energy. For example: Paper about 15, many plastics 25-30. An assumption of 15MJ/kg seems not entirely unreasonable. So each plane put about 180GJ of chemical energy into the towers.

Kinetic energy of flight 175

The kinetic energy (KE) of a moving object is given by the formula

E = 1/2 m * v²

Where, in the case of flight UA 175:

• m is the mass of the plane: That is, empty mass plus fuel load plus passengers, their baggage, and additional cargo. AA 11 was a Boeing 767-222, which has an operating emtpy weight of 80,130kg and a maximum take-off weight of 142,880kg. Maxium fuel capacity is 63.216l (see Specs at ZAP16.com). Maximum range is 7,300 km. Flight UA171 had 168 passenger seats, but only 56 passengers (33% capacity), plus 9 crew members. 65 people, weighing on average 70kg, have a mass of 4,550kg. For each, luggage, food, drinks and other supplies of 20kg may be added, or 1,300kg. It was scheduled to fly from Boston to Los Angeles, a distance of about 46% of its maximum range. Wikipedia, quoting NIST, states that it carried about 38,000l of fuel - about 60% of maximum fuel capacity, by the time of impact. Jet fuel has a specific weight of about 0.8 kg/l, so 38,000l fuel is about 30,000kg. The total mass of the plabe was thus at least:
  m = 80,130kg + (4,550+1,300)kg + 30,000kg = 115,980kg
  (empty weight + passengers + fuel)
• v is the velocity of the plane at impact, which was estimated by NIST as 545mph or 243m/s

Using these estimates, the kinetic energy of flight AA11 works out as

E_kin = 1/2 * 115,980kg * (243m/s)² = 3.42 * 10⁹J

Kinetic energy of Flight 11

The kinetic energy (KE) of a moving object is given by the formula

E = 1/2 m * v²

Where, in the case of flight AA 11:

• m is the mass of the plane: That is, empty mass plus fuel load plus passengers, their baggage, and additional cargo. AA 11 was a Boeing 767-223ER, which has an operating emtpy weight of 82,380kg and a maximum take-off weight of 179,170kg. Maxium fuel capacity is 90,770l (see Boeing technical specs). Maximum range is 12,195 km. Flight AA11 had 158 passenger seats, but only 81 passengers (51% capacity), plus 11 crew members. 92 passengers, weighing on average 70kg, have a mass of 6,440kg. For each, luggage, food, drinks and other supplies of 20kg may be added, or 1,840kg. It was scheduled to fly from Boston to Los Angeles, a distance of about 30% of its maximum range. Wikipedia, quoting NIST, states that it carried about 38,000l of fuel - about 42% of maximum fuel capacity, by the time of impact. Jet fuel has a specific weight of about 0.8 kg/l, so 38,000l fuel is about 30,000kg. The total mass of the plabe was thus at least:
  m = 82,380kg + (6,440+1,840)kg + 30,000kg = 120,660kg
  (empty weight + passengers + fuel)
• v is the velocity of the plane at impact, which was estimated by NIST as 750km/h or 208m/s

Using these estimates, the kinetic energy of flight AA11 works out as

E_kin = 1/2 * 120,660kg * (208m/s)² = 2.61 * 10⁹J

There are, however, other numbers available for mass an velocity: NIST: 443mph = 198m/s, which would lower my estimate to 2.365 * 10⁹J

GPE of building WTC7

The mass of of building WTC7 had potential energy in the gravitational field (GPE) given by the formula

E = m * g * h

The values of m, g, and h are not all known, but can be estimated as follows, using measurements taken from Wikipedia and NIST's revised technical briefing of November 19, 2008:

• g: That value is known. It is about 9.805 m/s2 in New York
• h: Is the height of the center of mass of the building above ground level, minus the hight of the center of mass of the rubble after the collapse. The latter may have been up to 5m. The former would be somewhat lower than half the hight of the roof of the building, considering these factors: a) floors would have about equal masses from ground to top b) load-bearing structure (columns, mainly) would be much heavier in the lower part than the upper part. In my blog post "Potential energy of the mass of each twin tower" I showed that Gregory H. Urich, in his paper "Analysis of the Mass and Potential Energy of World Trade Center Tower 1", assumes that the center of mass would be about at 41% percent of the height of the building. Since WTC7 had a height of 186m, the center of mass might have been near a height of 76m. Deduct from this up to 5m to account for the height of the rubble pile, the total mass descended by a height of h= 71m.
  
• m: Is the total mass of WTC7. As I write this (september 9, 2010), I know no good source that gives that mass. In my post on the GPE of the twin towers, however, I found that a good estimate would be one sixth of the displacement of water. The volume of WTC7 can be computed by its height (186m), width of the south front (247ft = 75m), north front (329ft = 100m), and depth (140ft = 42.7m): V = 186m * (75+100)m/2 * 42.7m = 694942.5m3. Total mass would therefore be 694942.5/6 tons or about m = 1.16 * 10E8 kg

Therefore, I estimate the total GPE of WTC7 that was converted into other forms of energy to be about

E = 1.16*10E8kg * 9.805m/s2 * 71m = 80,753,980,000J
I will round that down to

GPE = 8 * 10E10J

Potential energy of the mass of each twin tower in the gravitational field (GPE)

When mass is lifted against the force of gravity, it acquires potential energy in the gravitational field (GPE) according to the formula

E = m * g * h

Where

• m is the mass, measured in g or kg.
• g is the (nearly) constant acceleration of earth's gravity - in New York, that value is about 9.805 m/s²
• h is the difference in hight, in meters, measured between the "ground" level from which the mass was lifted to where it was lifted. In the case of an object like a building, one would have to consider the height of its center of mass

What was the total mass m of a twin tower?

If you google this, you may find various quoted figures. One often given is 500,000 metric tons. This appears too round a number to be accurate.

One back-of-the-envelope estimate occurred to me the other day: Compare a Twin Tower to ship, and estimate its displacement. Looking at steel ships. I'd think that they displace about one sixth of their total volume. Surely, the WTC would float on water like a ship, if you assumed the windows would not break. The volume of the tower is 415m x 63m x 63m = 1647135m³. Displacement might be 1647135m³/6 = 274522.5m³, or 274,522.5 tons.

A very thorough estimate was given by Gregory H. Urich, B.S. Electrical and Computer Engineering, in his whitepaper "Analysis of the Mass and Potential Energy of World Trade Center Tower 1":

His result: WTC1 had a mass of 288,100 metric tons (in the Abstract, page 1). That's only about 5% off of my back-of-the-envelope, confirming my method wasn't so bad.

Urich also gives us an estimate of GPE in 5.1 Summary of Results (p. 23): 480,600 MJ, or 4.806 * 10¹¹J

With E = m * g * h <=> h = E / (m*g), this corresponds to an estimated hight of the Center of mass of
h = 4.806 * 10¹¹J / (288,100,000kg * 9.805m/s²) = 170m (rounded to full meters), or 41% of the height of WTC1. This seems to be reasonable, considering that...

• Most of the 110 floors are evenly spaced and nearly identical to each other, making the mass of them evenly spread
• The perimeter columns likewise are of the same dimensions from about 3rd floor to roof, making their mass evenly distributed along the total height
• The massive core columns, on the other hand, are much more massive in the lower floors and become lighter and lighter towards the top. Their center of mass, in isolation, would likely to be much lower than half the hight of the tower; maybe 25% or 30% of its height, or 104-125m

I note in passing that Urich, on page 4 (3.1 Popular numbers) quotes FEMA (Hamburger ed.al.) with an estimated GPE of "> 4 E+11 J" and NIST (Sunder ed.al.) with an estimated total mass of "> 250,000 tons", confirming that both operated with conservative, realistic numbers.

When that mass collapsed, a good portion of it formed a debris heap above ground level. Thus, not all the potential energy was realeased in the fall. It is difficult to estimate the height of the center of mass of the rubble after the fall. It certainly wasn't very much: Much of the rubble actually fell below ground level (into the basement. and very much more assembled  below 1m than beteween, say, 10 and 11 meters above ground. So I have reason to assume that the new center of mass was at most 5m above ground level, or under 3% of the original height of 170m. I will therefore go with Urich's original number, bearing in mind that a 3% deviation is easily possible, given the uncertainties we have about mass and mass distribution.

Conclusion:
I agree with G. Urich that each of the Twin Towers had a

• mass of about 2.88* 10⁸ kg
• GPE of about 4.8 * 10¹¹J

Energy considerations

The variously proposed mechanisms by which the WTC highrises have been turned from erect buildings to piles of smoldering rubble all seek to explain the forces necessary to initiate and propagate these destructions. Viewed with the eyes of a phycist, "force" was applied over a certian "time" along certain "distances", creating "heat" and doing "work". In short: "energy" was transformed.

Many Conspiracy Theorists (CTers) claim that plane impacts and fires would not be sufficient to explain collapse initiation. Some go even further and claim that obervations like "pulverisation of concrete and steel" or "molten steel" would require more energy, heat or force than was available in the common storyline.

So we need to look at the energy available, and the energy (supposedly) needed for various CT claims:

• Potential energy of the mass of each twin tower in the gravitational field (GPE)
• GPE of building WTC7
• Kinetic energy of Flight 11
• Kinetic energy of Flight 175
• Chemical energy of the jet fuel in either plane
• Chemical energy of the combustible office contents, per floor
• Energy release of high explosives, per mass unit
• Energy release of thermite, and derivatives, per mass unit
• Capabilities of Directed Energy Weapons (DEW)
• Energy required to heat, and melt, various relevant metals
• Energy required to break structural steel
• Energy required to "pulverise" concrete

If a CTer claims that the energy available as GPE, combustibles, and kinetic energy was insufficient, and proposes a different mechanism such as high explosives, thermitic incendiaries, or Directed Energy Weapons (DEW; for example lasers),  then it is clear that this method should add a substantial amount of energy in relation to the available energy. Certainly, adding 10% would not suffice, since that is within the margin of error of many of the considerations listed above. We don't know the mass of each building precisely, and we don't know precisely how much concrete was turned into pieces of what sizes to compute the fracture energy. We have no precise idea about how much metal was heated to what temperatures at what time. I suggest that any CTer claim that rests on the assumption that the energy contained in the buildings and planes alone was insufficient to explain the destrcution, must propose a method that adds, at least, 50% of that energy.

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