|This paper has been updated. Please see Version 4. The version on this page is archived for historical interest. See the Revision History.|
On September 11th, Both of the Twin Towers disintegrated into vast clouds of concrete and other materials, which blanketed Lower Manhattan. This paper shows that the energy required to produce the expansion of the dust cloud observed immediately following the collapse of 1 World Trade Center (the North Tower) was much greater than the gravitational energy available from its elevated mass. It uses only basic physics.
Vast amounts of energy were released during the collapse of each of the Twin Towers in Lower Manhattan on September 11th, 2001. The accepted source of this energy was the gravitational potential energy of the towers, which was far greater than the energy released by the fires that preceded the collapses. The gravitational energy of each tower was on the order of 400,000 KWH (kilowatt hours). Of the many identifiable energy sinks in the collapses, one of the only ones that has been subjected to quantitative analysis is the thorough pulverization of the concrete that constituted the floor slabs of the towers. Jerry Russell estimated that the energy required to convert a tower's 600,000 tons of concrete into dust of 60 micron particle diameter is about 900,000 KWH. (See http://www.911-strike.com/powder.htm.) The fact that this figure is almost twice that of the available gravitational energy does not prove the presence of other energy inputs, given the paucity of detailed data about the physical characteristics of the powder.
There was, however, another energy sink that was many times the magnitude of the gravitational energy: the energy needed to expand the dust clouds to several times the volume of each tower within 30 seconds of the onset of their collapses. Note that the contents of the dust clouds had to come from air and materials inside the building, modulo any mixing with outside air. Given that the Twin Towers' dust clouds behaved like pyroclastic flows, with distinct boundaries and rapidly expanding frontiers (averaging perhaps 35 feet/second on the ground for the first 30 seconds), it is doubtful that mixing with ambient air accounted for a significant fraction of their volume. Gases at constant pressure require the input of heat energy to expand. Given an observed ratio of expansion, a lower bound for the the required input of heat energy can be computed using the ideal gas law.
How much heat energy was involved in expanding the dust clouds? To calculate the energy we need to answer three questions:
Since I have better photographs for North Tower dust, I did the calculation for it.
To answer question 1, I made estimates based on photographs taken at approximately 30 seconds after the onset of the collapse. The photo in Figure 1 appears to have been taken around 30 seconds after the initiation of the collapse of the North Tower. The fact that the spire is visible directly behind Building 7 indicates the photo was not taken later than the 30 seconds, since video records show that the spire started to collapse at the around 29 seconds. In this photograph, as in other ones taken around that time, the dust clouds still have distinct boundaries.
|Figure 1. Photograph from Chapter 5 of FEMA's Building Performance Assessment Report.|
I used landmarks in this photo to make several approximate measurements of the frontier of the dust cloud. The following table lists some of them. Measurements are in feet. The first column lists heights above the street, and the second lists distances from the vertical axis of the North Tower.
|3||230||1011||west corner of 45 Park Place|
|5||228||729||top of south corner of building with stepped roof|
|6||204||658||east corner of Building 7, 30 stories below top|
|7||600||776||upwell towering over southeast end of Post Office|
|8||700||?||upwell slightly higher than the top of Building 7|
|11||190||870||top of west corner of 22 Cortland St tower|
|12||508||588||8 stories below top of face of WFC 3|
|13||498||517||3 stories below top of upper face of WFC 2|
To approximate the volume I used a cylinder, coaxial with the vertical axis of the North Tower, with a radius of 800 feet, and a height of 200 feet. All the above reference points lie outside of this volume. Although the cylinder does not lie entirely within the dust cloud, there are large parts of the cloud outside of it, such as the 700 foot high upwelling column south of Building 7. The cylinder has a volume of:
pi * (800 feet)^2 * 200 feet = 402,000,000 feet^3.I subtract about a quarter for volume occupied by other buildings, giving 300,000,000 feet^3.
To accurately answer question 2 would require detailed knowledge of the fluid dynamics involved. However it does appear that for at least a minute, the dust cloud behaved as a separate fluid from the ambient air, maintaining a distinct boundary. There are several pieces of evidence that support this:
Instinctively I lifted the camera up, and something took over that probably saved my life. And that was [an urge] to run rather than take pictures. I got down to the end of the block and turned the corner when a wave-- a hot, solid, black wave of heat threw me down the block. It literally picked me up off my feet and I wound up about a block away.
The answer to question 3 is easy. The volume of a tower, with it's 207 foot width and 1368 foot height, is:
1368 feet * 207 feet * 207 feet = 58,617,432 feet^3.
So the ratio of the expanded gasses and materials from the tower to the original volume of the tower is:
200,000,000 feet^3 / 58,617,432 feet^3 = 3.41.
The ideal gas law can be used to compute a lower bound for the amount of heat energy required to induce the observed expansion of the dust cloud. That law states that the product of the volume and pressure of a parcel of a gas is proportional to absolute temperature. It is written PV = RT, where:
P = pressure V = volume T = absolute temperature R = constantAbsolute temperature is expressed in Kelvin (K), which is Celsius + 273. Applied to the tower collapse, the equation holds that the ratio of volumes of gasses from the building before and after expansion is roughly equal to the ratio of temperatures of the gasses before and after heating. That allows us to compute the minimum energy needed to achieve a given expansion ratio knowing only the thermal mass of the gasses and their average temperature before the collapse.
I say that the ideal gas law allows the computation of only the lower bound of the required energy input due to the following four factors.
In this paper I examine only the fourth factor. Before considering its effect on energy requirements, I first consider the energy requirements of heating only the gasses in the clouds to the level needed to achieve the observed expansion.
According to the ideal gas law, expanding the gasses 3.4-fold requires raising their absolute temperature by the same ratio. If we assume the tower was at 300 degrees K before the collapse, then the target temperature would be 1020 degrees K, an increase of 720 degrees. Given a density of 36 g/foot^3 for air, the tower held about 2,000,000,000 g of air. Air has a specific heat of 0.24 (relative to 1 for water), so one calorie will raise one g of air 1 / 0.24 = 4.16 degrees. To raise 2,000,000,000 g by 720 degrees requires:
2,000,000,000 g * 720 degrees * 0.24 = 345,600,000,000 calories = 399,500 KWHTo evaluate the energy requirements of the fourth factor, it is necessary to consider the composition of the dust cloud. The cloud was a suspension of fine particles of concrete and other solids in gasses consisting mostly of air. Since concrete was the dominant solid, I will ignore the others, which included glass, gypsum, asbestos, and various hydrocarbons. The small size of the particles, being in the 10-60 micron range, would assure rapid equalization between their temperature and that of the embedding air. Therefore any heat source acting to raise the temperature of the air would have to raise the temperature of the suspended concrete by the same amount. Assuming all 500,000,000,000 g of concrete was raised 720 degrees (300 K to 1020 K), the necessary heat, given a specific heat of concrete of 0.15 is:
500,000,000,000 g * 720 degrees * 0.15 = 54,000,000,000,000 calories = 62,800,000 KWH.
It is clear that the dust cloud was not uniformly heated to 1020 K. Eyewitness reports suggest that the cloud's ground-level temperatures more than a few hundred feet away from its center were humanly survivable. Whereas a number of survivors reported encounters with the South Tower dust cloud within a minute of the start of the tower's collapse, it is not clear if any people survived encounters with the North Tower dust cloud so early in its development. It does appear, based on seismic and video evidence, that more energy was involved in the leveling of the North Tower. Video records, and particular the live CNN feed, suggest that a very large exothermic event coincided with the afterglow, which commenced around 17 seconds after the collapse onset. The CNN video shows large upwelling columns developing within a few hundred feet of the cloud's center, and growing to heights exceeding 600 feet seconds after the afterglow started. The previously gray cloud appears to glow with a reddish hue. Since the cloud was already over 500 feet in radius before the afterglow, and the exothermic event appeared to have a central locus, it would not be surprising that the periphery of the cloud advancing down the streets remained at a fraction of the temperature of the central portions.
Whatever the distribution of temperatures in the cloud, the average would have to be at least 1020 K to achieve the observed expansion ratio. Indeed the reddish glow indicates that the upper portions of the cloud were considerably hotter than that. Note that the image in Figure 1 does not show the color, since its color balance is highly skewed toward blue and is missing reds. Most other photographs show the cloud to have a reddish hue, and video recordings the cloud appears to glow incandescently. The color of the top surface of the cloud is consistent with a temperature well above 1000 K.
The distribution of concrete dust in the cloud, and its relationship to the temperature distribution in the cloud, would greatly affect the total energy requirements of the cloud's expansion. Less energy would be required if the hotter portions of the cloud had a lower density of dust. The density was probably greater toward the central portions of the cloud, which also seem to have experienced the most heating. On the other hand, some of the dust may have settled out by the 30 second mark. That a large fraction of the dust had settled by that time seems doubtful given the violent churning of the cloud, and the opaque appearance of its frontier.
The dominant energy source assumed to be in play during the leveling of each of the Twin Towers was the gravitational energy due to its elevated mass, whereas the energy sinks included the pulverization of it's concrete, and the heating of the concrete and air in the ensuing dust cloud. Estimates for these energies are:
|energy, KWH||source or sink|
|+ 400,000||falling of mass (6.8e8 g falling average of 207 m)|
|- 400,000||heating of gasses (2e9 g air from 300 to 1020 K)|
|- 900,000||crushing of concrete (0.5e12 g to 60 micron powder)|
|- 60,000,000||heating of suspended concrete (0.5e12 g from 300 to 1020 K)|
The imbalance between sources and sinks is striking. Moreover, it is very difficult to imagine how the gravitational energy released by falling mass could have contributed much to any of the sinks, since the vast majority of the tower's mass landed outside its footprint. The quantity for the crushing of concrete appears to be conservative since some reports indicate the average particle size was closer to 10 microns. The quantity for the heating of suspended concrete has a large amount of uncertainty. However, it is conservative in several respects.
The amount of energy required to expand the North Tower's dust cloud was many times the entire potential energy of the tower's elevated mass due to gravity. The over 100-fold disparity between this estimate and the gravitational energy is not easily dismissed as reflecting uncertainties in quantitative assessments.
The official explanation that the Twin Tower collapses were gravity-driven events appears insufficient to account for the documented energy flows.