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Estimates of the WTC dust cloud expansion are used to provide a lower bound for the quantity of explosives used to bring down the towers.

The North Tower's Dust Cloud

Analysis of Energy Requirements for the Expansion of the
Dust Cloud following the Collapse of 1 World Trade Center

by Jim Hoffman, October 16th, 2003
[Version 3.0]


This paper uses photographic evidence -- primarily a reference photograph taken from FEMA's report -- to estimate the volume of the dust cloud that grew from the collapse of the North Tower at about 30 seconds after the commencement of the collapse. The paper then estimates the thermal energy required to produce the observed expansion in the volume of the dust cloud, based on the assumption that most of the gasses and suspended solids in the cloud originated from within the building.

The most recent version of the paper identifies two major mechanisms for the expansion -- thermodynamic expansion of gasses due to increases in temperature, and expansion due to the vaporization of water. Both represent vast energy sinks. Whatever the relative contributions of these mechanisms to the expansion, the required energy inputs far exceeds the energy available in the form of the gravitational potential energy due to the tower's elevated mass.

Previous versions of the paper did not consider expansion due to water vaporization, and considered only thermodynamic expansion of gasses present in the building at the time of collapse. That required average dust cloud temperatures of around 1000 K, a feature several people found implausible. The addition of the heat of water vaporization to the analysis changes the picture dramatically. The heat energy requirements are similar, but the temperatures need not have been anywhere near 1000 K, since the phase change of water to steam occurs at 100 C.

The paper shows a large disparity between the energy required to produce the observed expansion of the dust cloud and that available from the conversion of all the tower's gravitational potential energy to heat. It does not consider the possible energy source of the unlikely rapid combustion of the tower's contents during its collapse, but even the energy available from consuming all of the oxygen in the tower to burn hydrocarbons is far short of the estimated size of the energy sink of dust cloud expansion.

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 [Ref. (1)]: "Construction of WTC 1 resulted in the storage of more than 4 x 1011 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. [Ref. (2)]. 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 [Ref. (3)], 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:

  • thermodynamic expansion of gases
  • vaporization of liquids and solids
  • chemical reactions resulting in a net increase in gaseous phase molecules That is explosives.
The evidence does not support the idea that chemical reactions in the dust cloud liberated vast quantities of gases.

Actually, the evidence does support the use of explosives to collapse the towers [Ref. (4)].

That leaves increases in gas temperatures and vaporization of solids and liquids, primarily water, to drive the expansion.

How much heat energy was involved in expanding the dust clouds? To calculate the energy we need to answer three questions:

  1. What was the volume of the dust clouds from a collapse at some time soon after it started (before the clouds began to diffuse)?
  2. How did the mixing of the dust cloud with ambient air contribute to its size, and how can this be factored out to obtain the volume occupied by gases and suspended materials originally inside the building?
  3. What is the ratio of that volume to the volume of the intact building?
  4. How much heat energy was required to produce that ratio of expansion?

Since I have better photographs for North Tower dust, I did the calculation for it.

1. Quantifying Dust Cloud Volume

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.

The WTC Dust Cloud

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.

32301011West corner of 45 Park Place
5228729Top of south corner of building with stepped roof
6204658East corner of Building 7, 30 stories below top
7600776Upwell towering over southeast end of Post Office
8700?Upwell slightly higher than the top of Building 7
11190870Top of west corner of 22 Cortland St tower
125085888 stories below top of face of WFC 3
134985173 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 x (800 feet)2 x 200 feet = 402,000,000 feet3.

I subtract about a quarter for volume occupied by other buildings, giving 300,000,000 feet3.

2. Factoring out Mixing and Diffusion

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:

  • The WTC dust clouds inexorably advanced down streets at around 25 MPH. This is far faster than can be explained by mixing and diffusion.
  • As the dust clouds advanced outward, features on their frontiers evolved relatively slowly compared to the clouds' rates of advance. This indicates that that clouds were expanding from within and that if surface turbulence was incorporating ambient air, it's contribution to expansion was minor.
  • The top surface of the clouds looked like the surface of a boiling viscous liquid - churning but not mixing with the air above. Sinking portions of the clouds were replaced by clear air, not a mixture of the cloud and air.
  • The dust clouds maintained distinct interfaces for well over a minute. Mixing and diffusion would have produced diffuse interfaces.
  • There are reports of people being picked up and carried distances by the South Tower dust cloud, which felt solid. New York Daily News photographer David Handschuh recalled:

    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.
Initially the dust clouds must have been much heavier than air, given the mass of the concrete they carried and the distances they transported it. As time went on the cloud became more diffuse, but all of the photographs that can be verified as being within the first minute show opaque clouds with distinct boundaries, indicating the dominant mode of growth was expansion, not mixing or diffusion. It seems reasonable to assume that mixing with ambient air did not account for a significant fraction of the expansion in the volume of the dust cloud by 30 seconds of the start of the North Tower collapse. Nevertheless, I reduce the estimate of the dust cloud volume of building origin to 200,000,000 feet3, imagining that a third of the growth may have been due to assimilation of ambient air.

3. Computing the Expansion Ratio

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 x 207 feet x 207 feet = 58,617,432 feet3.

So the ratio of the expanded gasses and suspended materials from the tower to the original volume of the tower is:

200,000,000 feet3 / 58,617,432 feet3 = 3.41.

4. Computing the Required Heat Input

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 temperature would increase to 100 C, accompanied by thermodynamic expansion.
  • The temperature would remain at 100 C until all of the water was vaporized.
  • The temperature would increase above 100 C, again accompanied by thermodynamic expansion.
Since such uniform conditions were not present, I will first treat the two energy sinks separately, and will compute the energy requirements for each if it alone were responsible for the expansion.

4.1. The Thermodynamic Expansion Sink

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 = constant

Absolute 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.

  • The finite size of molecules leads to a slight departure from the ideal gas law wherein the expansion of a parcel of gas leads to a decrease in its temperature. This means that slightly more heat energy is needed to achieve a given expansion ratio than is predicted by the ideal gas law.
  • The dust cloud at the time of the photograph used to estimate its volume had not finished expanding. Videos show that it continued to expand well after the 1 minute mark.
  • The suspended dust in the cloud had many times the mass of the gasses. This increased the energy needed to expand the dust cloud since it takes energy to lift and accelerate mass.
  • The suspended dust in the cloud had many times the thermal mass of the gasses. Increasing in temperature of the dust cloud to a level needed to induce the observed expansion entailed raising the temperature of the gasses and suspended solids by similar amounts. Since the solids had many times the thermal capacity of the gasses, this multiplied the energy requirements.

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.

Of course, this begs the question: What was the source of energy that heated the debris cloud from 298 K (25 C) to 1020 degrees K?

Given a density of 36 g/foot3 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 x 720 degrees x 0.24 = 345,600,000,000 calories = 399,500 KWH

To 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 x 720 degrees x 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 up to 100 C, or 373 K:

2,000,000,000 g x 73 degrees x 0.24 = 35,040,000,000 calories = 40,744 KWH

The associated sink of heating the suspended solids to this temperature would be:

90,000,000,000 g x 73 degrees x 0.15 = 985,500,000,000 calories = 1,145,000 KWH.

4.2. The Water Vaporization Sink

At 100 C at sea-level, water expands by a factor of 1680 when converted to steam.

Of course, this begs many questions, prominent being:

  1. What was the source of energy that heated the building to 100 degrees C?
  2. Was there enough time for the building to reach 100 degrees C before the collapses?
  3. How much of the water in the concrete slab is able to escape as the slab is heated to 100 degrees C?
Hence it is reasonable to expect that water in the building accounted for a significant part of the expansion.

This is only reasonable if the concrete has already been pulverized, and if this is so, begs the question: What pulverized it?

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 of 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 feet3 - 58,617,000 feet3 = 141,383,000 feet3 = 4,003,542,000 liters

Given the 1680 to 1 ratio between the volume steam and 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 x 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.

This is somewhat misleading. Well-cured concrete is indeed about 1 percent FREE water, but concrete is also about 7-20 percent chemically bound water. The free water evaporates from concrete at 100 - 150 C, whilst chemically bound water remains until temperatures of 450 C [Ref. (5)].

So it turns out that there is more than enough water to account for the expansion (as long as the concrete reaches temperatures of about 450 C).

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.

4.3. Which Energy Sink Was Dominant?

Both thermodynamic expansion and water vaporization have the capacity to produce vast expansion in gas volume given sufficient heat.

Explosives would produce vast expansions of gas on detonation.
Explosives would produce vast quantities of concrete particles.
Explosives would also produce vast quantities of heat.
Vast quantities of heat applied to the concrete particles might cause the release of some of the chemically bound water, which would contribute to a breakup the particles, reducing them to a fine dust.

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 least 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 Unexplored Option -- Explosives.

We will consider the explosive amatol which is a mixture of trinitrotoluene (TNT) and ammonium nitrate (AN).

Trinitrotoluene (left) has the chemical composition C7H5(NO2)3 and ammonium nitrate (right) has the chemical composition NH4NO3.

We choose the TNT to AN molar ratio in the amatol mixture to be 4 : 42.
This translates to a 4 x 227 : 42 x 80 = 908 : 3360 = 21 : 79 ratio by weight.

In this case the explosion proceeds according to the equation:

4 C7H5(NO2)3 + 42 NH4NO3 => 28 CO2 + 94 H2O + 48 N2

From the equation we see that 4 moles of TNT and 42 moles of AN produces 28 + 94 + 48 = 170 moles of gaseous product.

At standard temperature and pressure, 170 moles of gas occupies 170 x 22.4 = 3,808 liters. This is the volume that the explosion products would occupy if, after the explosion, they were cooled to 25 C (with the assumption that the water remains a vapor). In order to calculate the volume of the hot gaseous products generated by the explosion, we initially need to know the amount of heat released by the explosion of 4 moles of TNT and 42 moles of AN. We also need to know the amount of heat required to raise each of the gases, by one degree K. That is, we need to know the enthalpy of explosion ?explosionH and the heat capacities Cp (also know as specific heats) of each of the gases.

?explosionH = - [4 ?f H?solid(TNT) + 21 ?f H?solid(AN)] + [28 ?f H?gas(CO2) + 94 ?f H?gas(H2O) + 48 ?f H?gas(N2)]
= - [4(-60) + 21(-365.5)] + [28(-393.5) + 94(-242) + 48(0.00)]
=240 + 7,675.5 - 11,018 - 22,748
= - 25,850 kJ per 4 moles of TNT and 42 moles of AN.

Here we have used the following facts:

?f H?gas(H2O) = -242 kJ/mol
?f H?gas(CO) = -110.5 kJ/mol
?f H?gas(CO2) = -393.5 kJ/mol
?f H?solid(TNT) = -60 kJ/mol
?f H?solid(AN) = -365.5 kJ/mol

The heat capacities Cp for the gases involved are:

Cpgas (N2) = 28.87 J/mol*K
Cpgas (O2) = 28.91 J/mol*K
Cpgas (H2O) = 30.43 J/mol*K
Cpgas (CO2) = 37.12 J/mol*K

Since the heat capacities are all approximately 30 J/mol*K, we will assume this value for all the gases, i.e., we assume:

Cpgas (all relevant gases) = 30 J/mol*K

So, by assumption, 30 joules of energy will raise the temperature of one mole of the gaseous product (of the explosion) by 1 degree K.

Summarizing from earlier, we have that 4 moles of TNT and 42 moles of AN,
  1. produces 170 moles of gaseous product and
  2. liberates 25,850 kJ of energy.
We will assume that the original 170 moles of explosion products mix with N moles of air.

This 170 + N mole mixture of gases is initially assumed to be at the temperature T0 = 298 degrees K (25 C).
This 170 + N mole mixture of gases has an initial volume of V0 = 22.4 (170 + N) liters.

We calculate the increase in temperature ?T of this 170 + N moles of gases after being heated by the 25,850 kJ of energy released by the explosion of the 4 moles of TNT and 42 moles of AN.

Now 30 joules raises the temperature of one mole of the gases by 1 degree K.
Hence, 25,850,000 joules raises the temperature of the 170 + N moles by

?T = 25,850,000/(30 x (170 + N)).

We now calculate the volume V1 of this 170 + N moles of gas after being heated by the explosion to the temperature

T1 = T0 + ?T

From the ideal gas law we have that V1 / V0 = T1 / T0. Rearranging we obtain

V1 = V0 (T1 / T0) = V0 (T0 + ?T) / T0 = V0 + V0 ?T / T0. On substituting we obtain

V1 = V0 + 22.4 x (170 + N) x 25,850,000 / (30 x (170 + N) x 298) = V0 + 22.4 x 25,850,000 / (30 x 298) = V0 + 64,770 liters.

Hence, the increase in volume of the mixture of gases ?V = V1 - V0 = 64,770 liters.

So, summing up, the explosion of 4 moles of TNT and 42 moles of AN produce 64,770 + 22.4 x 170 = 64,770 + 3,808 = 68,578 liters of hot gases.
That is, the explosion of 4,268 grams of amatol produces 68,578 liters of hot gases.
That is, the explosion of one kilogram of amatol produces 68,578 x 1,000 / 4,268 = 16,068 liters of hot gaseous product.

Hence the 200,000,000 liter expansion calculated by Hoffman can be explained by the detonation of

200,000,000/16,068 = 12,447 kg = 12.5 tonnes (14 tons) of the high explosive amatol.


The 200,000,000 liter expansion calculated by Hoffman can be explained by the detonation of 12.5 tonnes (14 tons) of the high explosive amatol.

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, the vaporization of water, and the heating of the concrete and air in the ensuing dust cloud. Estimates for these energies are:

Energy,KWHSource or Sink
+111,000falling of mass (1.97 x 1011 g falling average of 207 m)
-135,000crushing of concrete (9 x 1010 g to 60 micron powder)
Ignoring Water Vaporization
-400,000heating of gasses (2 x 109 g air from 300 to 1020 K)
-11,300,000heating of suspended concrete (9 x 1010 g from 300 to 1020 K)
Assuming Water Vaporization Sink was not Supply-Limited
-1,496,000vaporization of water (2.389 g water)
-41,000heating of gasses (2 x 109 g air from 300 to 373 K)
-1,145,000heating of suspended concrete (9 x 1010 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. 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.
  • It is based on an estimate of dust cloud volume at a time long before the cloud stopped growing.
  • It uses a liberal estimate of the contribution of mixing to the volume.
  • It ignores thermal losses due to radiation.
The calculation also ignores the role the mass of the suspended materials in impeding the expansion of cloud and thereby increasing the required energy.


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.

However, the use of explosives explains all the observed facts, and is thus probably the correct explanation.


  1. http://members.fortunecity.com/911/wtc/WTC_ch1.htm
  2. http://www.911-strike.com/powder.htm
  3. http://ehpnet1.niehs.nih.gov/docs/2002/110p703-714lioy/abstract.html
  4. http://members.fortunecity.com/911/wtc/tower-explosions.htm
  5. http://members.fortunecity.com/911/fire/SLamont.htm

Revision History

The paper is now in its third version. A complete version history is archived here.

Version 2 adopts much smaller estimates of concrete and total building mass, and refines the argument that mixing could not have accounted for much of the expansion. Version 3 considers a source of expansion ignored in the earlier versions -- the vaporization of water.


I wish to thank Jerry Russell, proprietor of www.911-strike.com, for his work on the physics of the World Trade Center collapses, work which was invaluable in the development of my thermodynamics analysis.

This article by J. Hoffman is a deliberate attempt to divert your attention from the fact that explosives were used to bring down the WTC towers. By presenting a possible explanation for the debris cloud without considering explosives, he is implicitly stating that he, as an expert in the field, does not consider explosives an option, so why should you? He is deliberately pointing you in the wrong direction.

As for the web-site http://physics911.org. It is generally of poor quality, is full of misinformation, and has very few contributors. It is clearly a site designed to miss-direct and cover-up for the official media/government conspiracy theory.

"The best way to control the opposition is to lead it ourselves." V. I. Lenin.

The comment in red has been added to the original article.


you know... 10.Dec.2003 17:58


... You could always read the American Society of Civil Engineers' report on what happened at the Pentagon and World Trade Center (which I believe is what your photo is from).  http://www.asce.org/responds/

Structural engineering isn't my specialy, but flipping through the tons of great photos in the Pentagon report put to rest, for me at least, all those theories about how it wasn't actually a plane that hit the pentagon (e.g. it could have been a missle  http://www.rense.com/general26/penta.htm "Pentagon Video Evidence Shows Fraud Of War On Terror").

I mean, it's one thing to fool the media, but why or how could anyone possibly go and fool an engineering reanalysis team (i.e. by planting airplane parts in the pentagon after hitting it with a missle)?

No one will ever know for sure if there are cloaked men twiddling their handlebar mustaches while hatching their masterplan for world domination in smoky back rooms or if the world is incredibly boring, there is no god and when we die we just become wormfood... but I sense that there must be some kind of psychology associated with wanting to believe that there's a conspiracy associated with every catastrophic act. The kennedy assassination is probably a bad example, but I can imagine it's hard to accept that someone so good and loved and powerful could be taken down by a single mere piddling mortal. There could only be some equally evil and hated and sinister force behind such an act.... organizations that are intangible to the average person are always likely suspects.

I also wonder if there's an element of confirmation bias, where one tends to notice and look for what one agrees with or wants to believe and ignores or discounts the relevance of contradictory evidence. Here's what I would suggest... Make a concerted effort to disprove your theory. This may be hard to believe but providing contradictory evidence can lend credibility to the rest of your argument. Good luck!

Background articles on conspiracy analysis and conspiracy phobia 10.Dec.2003 18:48


An intelligent and thoughtful approach to analyzing corruption and clandestine policymaking in high places which avoids the simplistic and naive views at either extreme on one side, the classic pathological conspiracy theorist who sees cabals and control everywhere; on the other side, the dogmatic structuralist / institutional theorist who makes every possible effort to deny that conspiracies play a meaningful role at all;


Anticonspiratorial dogmatism has little to do with enlightened, progressive thinking, and instead bears its lineage from centrist liberal elitism and McCarthyism. Also, there is interesting historical background on the scapegoating of the grassroots right in the wake of the 1995 Oklahoma City bombing (in which Left anti-conspiracist John Foster "Chip" Berlet played a leading role).

The 9/11 anti-conspiracist campaign was similar to and involved some of the same media figures as the response to revelations of CIA drug smuggling several years ago, when the groundbreaking investigative reporting of Gary Webb was dismissed and attacked. It was later vindicated.There is a clear pattern of the same general group of "progressives" going on the attack on behalf of the government whenever critically damaging information about official high crimes makes it out of the woodwork.

Mind you -- you could also do some more reasearch for yourself. 10.Dec.2003 18:58

Try reading these, for example:




The official media/government conspiracy theory:

On conspiracy theories.

Do you believe the official media/government conspiracy theory, or the one's that actually make sense?


It is blatently anti-semitic and totally stretches the bounds of believability (Arabs are Semites).

It goes something like this:

There is a gigantic Arab/Semite conspiracy to destabilize the West.

There is a gigantic network of Arab/Semite spies and agents throughout the world, at the beck and call of a BIG Semite called bin Laden.

Controlling this gigantic network of Semite spies and agents from a cave in Afghanistan (while all the time under direct observation by the CIA and friends (who produced tapes of this master plotter conversing with his mother)) bin Laden managed to destroy the two largest commercial buildings in the USA, put a dent in the Pentagon and stand-down the United States Air Force while this destruction occurred.

Boggles the mind, doesn't it.

Only the maddest -- probably insane -- conspiracy theorists believe this dribble.

Seriously guys, people actually believe this crap.

reply 10.Dec.2003 20:17


Oh hey, you know, I believe "they" are dirty rotten scoundrels as much as the next person... and I never thought I'd find myself in a situation of being called a mccarthyist dupe for suggesting someone else is a conspiracy theorist.

This is just my humble opinion here, but... I know there's a long history of misunderstood geniuses out there who had wild theories that were right... I vaguely remember the story of a 19th century physicist who was driven out by the establishment, was fired from academia and killed himself in despair, only to have his theory proven correct a couple years later, and of course they now give awards in his name. But for every one of him there are probably 10 academics that were just looney and their theories wrong.

The problem is that we forget the folks that were wrong, but remember the eccentrics that were right and then end up saying "well because people hate my theory that means it must be right because look at all the other theories they hated that were right." The best strategy is probably to have a little of both, creative free thinkers and serious conservative researchers.

Of couse, if you used risk assessment, the choice is clear. What do you have to lose by putting forth a wild theory? If you're wrong, then people will forget about you. If you're right, then they'll heap praise on you for seeing what nobody else could see. The key is probably to have lots of extreme theories and eventually you'll get one right (wasn't that shirley maclaine's approach?). If the original author's reading these comments, would you care to put some kind of probability or confidence level that your theory is correct? Give all the evidence for and against, would you say there's 100% chance that there was a bomb in the world trade center? 75%? 10%? 5%? 1%?

no planes? 01.Jan.2004 12:47


does this mean that the planes I saw on TV was trick photography?

Air pressure 27.Apr.2004 14:15

Ben Apgar

This paper completely ignores the forces of air pressure on the dust cloud created by the compression of air inside the building, as well as falling debris. It makes a generalization of the volume of the cloud based on a single photograph. It ignores the air pressure created above the building which would draw in clean air. It assumes that the cloud is equally dense, when it is far more likely that the cloud is densest at the frontal edge of the cloud. The paper is based on the generalized assumption that the cloud is expanding only due to heat expansion, and just 30 percent is mixed with ambient air. This means that 70 percent of the cloud is debris and the air which was originally inside the building. Even if terrorists placed a bomb underneath the building, the bomb would only contribute to heat. The fact that his numbers are so huge does not disprove the theory that the towers collapsed due to the damage from the planes, as he implies, and goes more to prove that his generalisations of volume, density, and assumptions of cause and effect are erroneous.

Corrections to errors in Jim Hoffman's Dust Cloud analysis 22.Sep.2006 22:18

John Kimber, PhD johnkimber@sbcglobal.net

In a December 10, 2003 posting on the forum


an anonymous author using the name XXXX posted an annotated version of Jim Hoffman's "The North Tower's Dust Cloud", Version 3.0 of October 16, 2003. The posting by XXXX had the title, "It Took at Least 14 Tons of High Explosive to Bring Down One WTC Tower."

The annotations are in red, and include a two-page calculation wherein XXXX believed he had calculated that 14 tons of amatol (about 30% more powerful than TNT) would account for the visual evidence on the collapse of the WTC North Tower.
However there are two huge flaws in XXXX's calculation.

1. He used 200,000,000 liters as the expansion calculated by Hoffman, whereas in reality Hoffman's calculation was 200,000,000 - 58,617,432 = 141,382,568 cubic feet. Using 28 liters per cubic foot gives

141,382,568 x 28 = 3,958,711,904 liters

instead of the 200,000,000 liters expansion used by XXXX. The ratio of the correct number of liters of expansion divided by the 200,000,000 used is

3,958,711,904/200,000,000 = 19.8.

This 19.8 must be multiplied by the 14 tons calculated by XXXX to correct the error. So the corrected value of XXXX's calculation is 19.8 x 14 = 277 tons of amatol (instead of 14).

This figure is consistent with Hoffman's calculation of 400,000 KWH for heating of air (when water was ignored), because amatol is 30% more powerful than TNT, which has the energy output of 1,162 KWH per ton. (So 400,000 KWH is equivalent to 400,000/1,162 = 344 tons TNT, close to 277 tons amatol.)

2. This 277 tons of amatol is negligible compared to the 1,145,000 KWH that Hoffman calculates to have been necessary to heat suspended concrete, in the case where water was considered, and the 11,300,000 KWH for heating the concrete when water was not considered. Therefore neither the 14 tons of amatol calculated incorrectly by XXXX nor the corrected value of 277 tons is significant compared to other "energy sinks."


1. The erroneous 14 ton, or 16 ton, calculation has spread all over the Internet. It has even been appropriated by Hoffman, himself, as a figure he himself calculated, failing to acknowledge that it was calculated by XXXX and failing to mention the huge error in the calculation. For example, in a January, 2004 "Guns and Butter" interview with transcript available at Hoffman's web page,
he says, "To accomplish the expansion of the dust cloud that I calculate, you'd need about sixteen tons of high explosives like amatol." Here he kicked up XXX's 14 tons to 16 tons. But he failed to mention that this calculation was by an anonymous person, and that it had a huge error, and that his own calculations were that at least 1,145,000 KWH were necessary to account for the visual evidence on the North Tower alone, which comes to about 1,000 tons of TNT per tower, or 2,000 tons TNT for the Twin Towers. Or about 1,600 tons of amatol.
One reason Hoffman does not like to go around saying what the results of his calculations, 1,600 tons of amatol, is is that it would have been impossible to bring that much into the Twin Towers, carry it up to higher floors, and distribute it and wire it up for timed demolition in such a way that visual evidence would fail to show the huge amount of planted explosives: thirty-two truckloads assuming each load was fifty tons; or more than a hundred times as much explosive as went into the most powerful bomb ever used in war (other than nuclear bombs).

On the other hand, when Hoffman says he calculated 16 tons of amatol to have been sufficient it seems believable (even though he fails to mention that this figure was arrived at by an anonymous person by way of huge miscalculations.) My point here is not that Hoffman plagiarizes someone elses erroneous calculation, but that he himself doesn't believe that he made the calculation that he claims to have made. Also it is logically impossible for Hoffman to believe 1,600 tons of amatol to have been necessary to collapse the Twin Towers, while at the same time believing 16 tons to have been sufficient.

2. One thing that Hoffman and other 9/11 "truth researchers" should do after performing any calculation is to examine the result and see if it is plausible. If Hoffman had done this, he never would have published his four versions of his "Dust Cloud Analysis". On his website,


he referred to version 4.0, which is "in preparation." It's been "in preparation" for over two years and will never be completed because his earlier data and calculations are so wildly implausible that there is nothing he can do to salvage the paper.

3. One more thing. Some persons calculating energy to fit Hoffman's Dust Cloud data do the most obvious thing to calculate energy, which is to use

work (in ft-lbs) equals pressure (in lbs/sq ft) multiplied by volume change

This will always come up with a figure about one-fourth what a calculation of input energy gives. The reason is that the work achieved in an expansion with a volume ratio of 3.5 or so is only about one-fourth the input energy, just as with an old low-compression gasoline engine with efficiency of conversion of input energy to output work only about 25%.

1146 Birch Ave Spc 95, Seaside, CA 93955

Major errors in the 14 tons calculation. 23.Sep.2006 13:28

John Kimber johnkimber@sbcglobal.net

In "IT TOOK AT LEAST 14 TONS . . . " the anonymous author made an error in the sentence,

"Hence the 200,000,000 liter expansion calculated by Hoffman can be explained by the detonation of 200,000,000/16,068 = 12,447 kg = 12.5 tonnes (14 tons)of the high explosive amatol."

Actually Hoffman estimated the final volume of the expansion to be 200,000,000 cubic feet (NOT LITERS) and the initial volume to be 58,617,432 cubic feet. Subtraction yields a 141,382,568 cubic feet expansion. One cubic foot equals 28 liters, so the expansion in liters is 28 times the 141,382,568 cubic feet, or 3,958,711,904 liters.

This is 19.79 times as many liters as the 200,000,000 that XXXX used. Therefore the number of tonnes and tons of amatol needed is 19.79 times as much as the 12.5 tonnes and 14 tons that XXXX came up with: that is 19.79 x 14 = 277 tons of amatol. This is consistent with Hoffman's "heat sink" of 400,000 KWH for heating of air in the case where water is ignored.

It should be noted that this 400,000 KWH from the 277 tons of amatol is neglibible compared to the lowest estimate that Hoffman has for the heat-sink total, which is 2,952,000 KWH (from crushing of concrete, vaporization of water, heating of gasses, and heating of suspended concrete) in his table in the SUMMARY section. Neither 14 tons nor 277 tons of amatol comes close to the minimum energy needed in Hoffman's calculations. Subtracting the 111,000 source energy (from falling of mass) from the heat-sink total yields 2,841,000 KWH, or 1,967 tons of amatol for just the North Tower. For the twin towers Hoffman's minimal calculation would be twice this, or 3,394 tons of amatol. This would require 67 50-ton truckloads of explosives to be transported to the site, lifted to upper floors, and elaborately wired for controlled demolition. THIS WOULD BE IMPOSSIBLE.

1146 Birch Ave #95, Seaside, CA 93955