|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 magnitude of that source cannot be determined with much precision thanks to the secrecy surrounding details of the towers' construction. However, FEMA's Building Performance Assessment Report gives an estimate: "Construction of WTC 1 resulted in the storage of more than 4 x 10^11 joules of potential energy over the 1,368-foot height of the structure." That is equal to about 111,000 KWH (kilowatt hours) per tower.
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 in the towers. It is well documented that nearly all of the non-metallic constituents of the towers were pulverized into fine powder. The largest of these constituents by weight was the concrete that constituted the floor slabs of the towers. Jerry Russell estimated that the amount of energy required to crush concrete to 60 micron powder is about 1.5 KWH/ton. (See http://www.911-strike.com/powder.htm.) That paper incorrectly assumes there were 600,000 tons of concrete in each tower, but Russell later provided a more accurate estimate of 90,000 tons of concrete per tower, based on FEMA's description of the towers' construction. That estimate implies the energy sink of concrete pulverization was on the order of 135,000 KWH per tower, which is already larger than the energy source of gravitational energy. However, the size of this sink is critically dependent on the fineness of the concrete powder, and on mechanical characteristics of the lightweight concrete thought to have been used in the towers. Available statistics about particle sizes of the dust, such as the study by Paul J. Lioy, et al., characterize particle sizes of aggregate dust samples, not of its constituents, such as concrete, fiberglass, hydrocarbon soot, etc. Based on diverse evidence, 60 microns would appear to be a high estimate for average concrete particle size, suggesting 135,000 KWH is a conservative estimate for the magnitude of the sink.
A second energy sink, that has apparently been overlooked, 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 building constituents -- gases and materials inside of or intrinsic to 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. Therefore the dust clouds' expansion must have been primarily due to an expansion of building constituents. Possible sources of expansion include:
How much energy was involved in expanding the dust cloud from either tower? To calculate an estimate we need to answer four 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 suspended materials from the tower to the original volume of the tower is:
200,000,000 feet^3 / 58,617,432 feet^3 = 3.41.
Above I identified two energy sinks that could have driven expansion of the dust cloud: thermodynamic expansion of gases, and vaporization of liquids and solids. Since most constituents and contents of the building other than water would require very high temperatures to vaporize, I consider only the vaporization of water in evaluating the second sink.
It is clearly not possible to determine with any precision the relative contributions of these two sinks to the expansion of the dust cloud. If the cloud remained uniform in temperature and density for the first 30 seconds, then the expansion would consist of three distinct phases:
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, assuming that the expansion was entirely due to thermodynamic expansion. 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 = nRT, where:
P = pressure V = volume T = absolute temperature n = molar quantity 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 90,000,000,000 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:
90,000,000,000 g * 720 degrees * 0.15 = 9,720,000,000,000 calories = 11,300,000 KWH.
If we assume that the water vaporization sink absorbed all available energy once temperatures reached water's boiling point, we can compute the size of the heat sink of thermodynamic expansion that was in play as temperatures rose from room temperature to 100 C, or from 300 K to 373 K:
2,000,000,000 g * 73 degrees * 0.24 = 35,040,000,000 calories = 40,744 KWHThe associated sink of heating the suspended solids to this temperature would be:
90,000,000,000 g * 73 degrees * 0.15 = 985,500,000,000 calories = 1,145,000 KWH.
At 100 C at sea-level, water expands by a factor of 1680 when converted to steam. Hence it is reasonable to expect that water in the building accounted for a significant part of the expansion. How much energy would be required to expand the volume of the cloud by the 3.41 ratio if water vaporization were entirely responsible for the expansion? Since water vaporization involves the introduction of volumes steam from comparatively negligible volumes of water, I assume that all the incremental volume was occupied by steam. The estimated 3.41 expansion ratio means that the incremental volume was:
200,000,000 feet^3 - 58,617,000 feet^3 = 141,383,000 feet^3 = 4,003,542,000 litersGiven the 1680 to 1 ratio between the volume steam and liquid water, 2,383,000 liters of water would have been required. The heat of vaporization of water is 540 calories/gram at 100 C. Therefore the heat energy required to produce the expansion is:
2,383,000,000 g * 540 = 1,286,820,000,000 calories = 1,496,000 KWH
Was there enough water in the building for this sink to be anywhere near this large? That is a matter of great uncertainty. Even well-cured concrete has a significant moisture content. Assuming that the estimated 90,000 tons of concrete in the tower was 1 percent water by weight, that would have provided 900 tons of water or about 900,000 liters -- well short of the 2,383,000 liter estimate above. However, there is a large amount of uncertainty in the water content of the concrete, which, like the rest of the remains of the disaster, was apparently disposed of with little or no examination. Moreover there were other sources of water in the building, such as the plumbing system, which could have accounted for tens of thousands of liters, and, gruesomely, people. The thousand victims never identified could have accounted for about 30,000 liters of water.
Both thermodynamic expansion and water vaporization have the capacity to produce vast expansion in gas volume given sufficient heat. Two major difference in the features of these sinks may help in understanding the relative contributions of each. First, thermodynamic expansion to the observed ratio requires very high temperatures, whereas vaporization-driven expansion occurs at a constant temperature of 100 C. Second, vaporization-driven expansion would be limited by the available supply of water.
If all the expansion was due to thermodynamic expansion, it would require that the dust cloud was heated to an average temperature of about 1020 K. Certainly the temperatures of the cloud near the ground were no-where near that high. Eyewitness reports show that the cloud's ground-level temperatures more than a few hundred feet away from its center were humanly survivable. Most of these reports are from the South Tower collapse, and it is unclear how similar the dust cloud temperatures following the two collapses were. Although serious fires raged in Buildings 4, 5, and 6, other nearby buildings that suffered extensive window breakage from the tower collapses, such as the Banker's Trust Building, and Word Financial Center Buildings 1, 2, and 3, did not experience fires. Digital photographs and videos show a bright afterglow with a locus near the center of the cloud, commencing around 17 seconds after the onset of the North Tower's collapse. Once the afterglow started, the cloud developed large upwelling columns towering to over 600 feet, and the previously gray cloud appeared to glow with a reddish hue. This suggests that at lest the upper and central regions of the North Tower cloud reached very high temperatures, but the evidence is insufficient to draw even general quantitative conclusions about the ranges and distributions of temperatures.
If enough water was present for vaporization to drive most of the expansion, temperatures in much of the cloud would have remained around 100 C until most of the water had vaporized. Thermodynamic expansion would occur in regions with liquid phase water until 100 C was reached, and again after the water was vaporized.
To the extent that thermodynamic expansion was the dominant factor driving the expansion, 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. 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, much of the dust may have settled out by the 30 second mark. The violent churning of the cloud, and the opaque appearance of its frontier, suggest that most of the dust had not settled that early.
The dominant energy source assumed to be in play during the leveling of each of the Twin Towers was the gravitational energy due to their elevated mass. The energy sinks included the thorough pulverization of each tower's concrete, the vaporization of water, and the heating of air and suspended concrete dust in the ensuing dust cloud. Estimates for these energies are:
|energy, KWH||source or sink|
|+ 111,000||falling of mass (1.97e11 g falling average of 207 m)|
|- 135,000||crushing of concrete (9e10 g to 60 micron powder)|
|ignoring water vaporization|
|- 400,000||heating of gasses (2e9 g air from 300 to 1020 K)|
|- 11,300,000||heating of suspended concrete (9e10 g from 300 to 1020 K)|
|assuming water vaporization sink was not supply-limited|
|- 1,496,000||vaporization of water (2.38e9 g water)|
|- 41,000||heating of gasses (2e9 g air from 300 to 373 K)|
|- 1,145,000||heating of suspended concrete (9e10 g from 300 to 373 K)|
The imbalance between sources and sinks is striking, no matter the relative shares of the thermodynamic and water vaporization sinks in accounting for the expansion. 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 than 60 microns. The quantity for the heating of suspended concrete has a large amount of uncertainty, but the energy imbalances remain huge even when it is ignored entirely. All of these energy sink estimates are 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 10-fold disparity between the most conservative 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.