CALCULATIONS SAY AT LEAST 14 TONS OF EXPLOSIVE TO BRING DOWN EACH WTC TOWER.
The North Towers 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 FEMAs 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 towers 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 towers gravitational potential energy to heat. It does not consider the possible energy source of the unlikely rapid combustion of the towers 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, FEMAs Building Performance Assessment Report gives an estimate (Ref. (1)
): "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. (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 FEMAs 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:
1). thermodynamic expansion of gases
2). vaporization of liquids and solids
3). 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.
Figure 1. Photograph from Chapter 5 of FEMAs 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.
Label Height Distance Description
1011 West corner of 45 Park Place
729 Top of south corner of building with stepped roof
658 East corner of Building 7, 30 stories below top
776 Upwell towering over southeast end of Post Office
? Upwell slightly higher than the top of Building 7
870 Top of west corner of 22 Cortland St tower
588 8 stories below top of face of WFC 3
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 x (800 feet)^2 x 200 feet = 402,000,000 feet^3.
I subtract about a quarter for volume occupied by other buildings, giving 300,000,000 feet^3.
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:
1). The WTC dust clouds inexorably advanced down streets at around 25 MPH. This is far faster than can be explained by mixing and diffusion.
2). 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, its contribution to expansion was minor.
3). 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.
4). The dust clouds maintained distinct interfaces for well over a minute. Mixing and diffusion would have produced diffuse interfaces.
5). 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 feet^3, 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 its 207 foot width and 1368 foot height, is:
1368 feet x 207 feet x 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.
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:
1). The temperature would increase to 100 C, accompanied by thermodynamic expansion.
2). The temperature would remain at 100 C until all of the water was vaporized.
3). 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
= absolute temperature
= molar quantity
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.
1). 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.
2). 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.
3). 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.
4). 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/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 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 waters 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 feet^3 - 58,617,000 feet^3 = 141,383,000 feet^3 = 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 clouds 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 Bankers 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 Towers 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 H_explosion and the heat capacities C_p (also known as specific heats) of each of the gases.
H_explosion = - (4 H_solid(TNT) + 21 H_solid(AN)) + (28 H_gas(CO_2) + 94 H_gas(H_2O) + 48 H_gas(N_2))
= - (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:
H_gas(H2O) = -242 kJ/mol
H_gas(CO) = -110.5 kJ/mol
H_gas(CO2) = -393.5 kJ/mol
H_solid(TNT) = -60 kJ/mol
H_solid(AN) = -365.5 kJ/mol
The heat capacities C_p for the gases involved are:
C_p^gas (N2) = 28.87 J/mol*K
C_p^gas (O2) = 28.91 J/mol*K
C_p^gas (H2O) = 30.43 J/mol*K
C_p^gas (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:
C_p^gas (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 T_0 = 298 degrees K (25 C).
This 170 + N mole mixture of gases has an initial volume of V_0 = 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 V_1 of this 170 + N moles of gas after being heated by the explosion to the temperature
T_1 = T_0 + T
From the ideal gas law we have that V_1 / V_0 = T_1 / T_0. Rearranging we obtain
V_1 = V_0 (T_1 / T_0) = V_0 (T_0 + T) / T_0 = V_0 + V_0 T / T_0. On substituting we obtain
V_1 = V_0 + 22.4 x (170 + N) x 25,850,000 / (30 x (170 + N) x 298) = V_0 + 22.4 x 25,850,000 / (30 x 298) = V_0 + 64,770 liters.
Hence, the increase in volume of the mixture of gases V = V_1 - V_0 = 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, KWH Source or Sink
falling of mass (1.97 x 10^11 g falling average of 207 m)[/color]
crushing of concrete (9 x 10^10 g to 60 micron powder)[/color]
Ignoring Water Vaporization
heating of gasses (2 x 10^9 g air from 300 to 1020 K)
heating of suspended concrete (9 x 10^10 g from 300 to 1020 K)
Assuming Water Vaporization Sink was not Supply-Limited
vaporization of water (2.389 g water)
heating of gasses (2 x 10^9 g air from 300 to 373 K)
heating of suspended concrete (9 x 10^10 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 towers 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.
1). It is based on an estimate of dust cloud volume at a time long before the cloud stopped growing.
2). It uses a liberal estimate of the contribution of mixing to the volume.
3). 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 Towers dust cloud was many times the entire potential energy of the towers 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.
The paper is now in its third version. A complete version history is archived here.
1). Version 1
- released June 13, 2003
2). Version 2
- released July 23, 2003
3). Version 3
- released October 16, 2003
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 [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.