Archive for the ‘thermodynamics’ Category

Credit: Extinction Rebellion

On Tuesday 2nd April 2019, environmental activists from the Extinction Rebellion group staged a parliamentary protest in London by glueing their bottoms to the windows of the viewing gallery of the House of Commons during a Brexit debate.

They used this cheeky tactic to call on British politicians to act on the ‘Climate and Ecological Crisis’. On its website, Extinction Rebellion wrote that “Government must act now to halt biodiversity loss and reduce greenhouse gas emissions to net zero by 2025”.

There is an irony here in that the call to immediate action was delivered at the precise moment when the UK government was in a state of complete paralysis over Brexit. Then again, the demonstrators’ demands weren’t meant to be taken entirely at face value. Even the most optimistic environmental biologists and chemical engineers would shake their heads at what was proposed to be accomplished in the space of just six years.

What the demonstration did achieve was to focus attention on the task in hand and the current state of progress. And it prompted CarnotCycle to pen this post on one promising technology for carbon capture and conversion (CCC) in which UK researchers are playing a key role.

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It is a well-known fact, not to mention a subject of serious environmental concern, that cows burp methane as a result of anaerobic fermentation of the grass they eat. And being aerobic respirers they also exhale carbon dioxide. Both these substances are potent greenhouse gases associated with human activity whose atmospheric levels, according to the best available science, must at least be stabilized in order to stand a chance of keeping global warming within manageable limits.

The cow in our picture is asking us an interesting question. What if it were possible to react carbon dioxide and methane together to form products that are not greenhouse gases? Even better, what if the reaction products could be put to useful purposes? Just think how cute that would be!

Putting our physical chemistry hats on for a moment and looking at the above equation, we notice that the oxidation states of carbon in the two molecules are at opposite ends of the scale. Methane has the most reduced form of carbon (-4) while carbon dioxide has the most oxidized form of carbon (+4). A redox reaction between the two looks possible and indeed is possible, albeit at elevated temperatures:

This process – called dry reforming of methane or DRM – was first introduced by Germany’s dynamic duo, Franz Fischer and Hans Tropsch, in 1928 but extensive investigation only started in the 1990s when increasing concerns about the greenhouse effect were raised by the international scientific community.

Notice how DRM simultaneously converts two greenhouse gases into two non-greenhouse gases which together make valuable syngas (1:1), a key industrial intermediate in the production of chemicals and clean fuels.

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Well, sort of. The big difficulty in making DRM viable relates to side reactions. The DRM reaction proceeds above 918K but then so does the thermal decomposition of both methane and carbon monoxide which results in carbon being deposited on the catalyst, clogging up the pores and thereby deactivating it.

This problem can however be mitigated in a very neat way by combining DRM with another methane-reforming process, namely steam reforming (SRM). Coupling reactions in this way not only reutilizes the deposited carbon but also adds a product stream with an H2/CO ratio of 3:1 which enables the syngas ratio to be adjusted for the synthesis of methanol, ethanoic acid or dimethyl ether (DME), which has promise as a sulfur-free diesel fuel, or towards the (2n+1):n H2/CO ratio required for Fischer-Tropsch synthesis of alkane fuels

The other problem is that all these reactions are endothermic (heat requiring). This energy has to be obtained from somewhere, and now here comes the next neat idea. Adding oxygen to the reactant stream allows partial oxidation of methane (POM) and catalytic combustion of methane (CCM) to take place, which are exothermic reactions that can supply the necessary heat

Putting three reforming agents – carbon dioxide, water and oxygen – together in the reactant stream with a methane feedstock creates a sufficiently energy-efficient overall process known as ‘tri-reforming’.

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To complete the conceptual scheme, the tri-reforming process is integrated into a cycle where the syngas output is utilized in the tri-generation of fuels, industrial chemicals and electricity, with the flue gases from these processes being fed back after nitrogen purging to the tri-reforming reactor. Carbon dioxide can also be fed into the cycle from external sources such as power plants and cement works.

Note that in principle, carbon dioxide can be fed into the cycle from established carbon capture and storage (CCS) processes. In this way underground reservoirs of anthropogenic carbon dioxide can be utilized as a feedstock for additional tri-generation.

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Almost. I mentioned at the start that the UK was involved in CCC research so I should say a few words about that. Scientists at Oxford and Cambridge are working with the King Abdulaziz City for Science and Technology in Saudi Arabia and the National Natural Science Foundation of China on the tri-reforming/tri-generation technology detailed above.

Since China is the world’s largest CO2 emitter and Saudi Arabia is the world’s largest oil producer, the Anglo-Sino-Saudi initiative seems a sensible geoscientific cluster. The news was announced on 28 January 2018, more than a year before the Extinction Rebellion demonstrations in London.

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Suggested further reading

A mini-review on CO2 reforming of methane

Published in June 2018 this is a useful and easily readable grounder covering the thermodynamic, kinetic, catalysis and commercial aspects of the subject.

Turning carbon dioxide into fuel

A paper co-written by the Oxbridge scientists involved in the Anglo-Sino-Saudi initiative. It was published in 2010, which shows that these guys have been on the case a while. Climate activists take note, and read their stuff.

Turning carbon dioxide into fuel – a new UK-China-Saudi Arabia initiative

The January 2018 press release referred to above. I reckon my poly-alliterative Anglo-Sino-Saudi sounds better.

Tri-reforming: a new process for reducing CO2 emissions

A bedrock paper (January 2001) from Chunshan Song at Penn State. The process diagram featured above is taken from this paper. If you don’t read anything else, read this one.

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P Mander May 2019

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I dare say most of you will remember this classroom demonstration, in which water passes through a semi-permeable membrane and causes the liquid level to rise in the stem of the thistle funnel. The phenomenon is called osmosis, and at equilibrium the osmotic pressure is equal to the hydrostatic pressure.

Historical background

This experiment has its origins way back in the mid-18th century, when a French clergyman named Jean-Antoine Nollet tied a piece of pig’s bladder over the mouth of a jar containing alcohol and immersed the whole thing in a vat of water. What prompted him to do this is not known, but we do know the result of his experiment. The bladder swelled up and ultimately burst from the internal pressure.

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Jean-Antoine Nollet 1700-1770

Nollet published his findings in Recherches sur les causes de Bouillonement des Liquides (1748) in which he gave a correct interpretation of the phenomenon, which arises from the much more marked permeability of the bladder to water as compared with alcohol.

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Moritz Traube 1826-1894

The actual measurement of osmotic pressure had to wait for over a century, until the German chemist Moritz Traube showed in 1867 that artificial semipermeable membranes could be made using gelatin tannate or copper ferrocyanide. Traube’s compatriot Wilhelm Pfeffer, a botanist, succeded in depositing the latter in the walls of a porous jar, which when filled with a sugar solution, connected to a mercury manometer and then plunged into pure water, provided a means of measuring osmotic pressures.

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Wilhelm Pfeffer 1845-1920

Following Pfeffer’s osmotic pressure measurements using sucrose solutions, on which JH van ‘t Hoff based his famously flawed gaseous theory of solutions, there were two notable teams of experimentalists – one on each side of the Atlantic – which provided high quality osmotic pressure data to test the ideas of theoreticians. In the USA, Harmon Northrop Morse and Joseph Christie Whitney Frazer led a team at Johns Hopkins University, Baltimore, Maryland from 1901 to 1923. In Britain meanwhile, the aristocrat-turned-scientist Lord Berkeley and co-worker Ernald Hartley set up a private research laboratory near Oxford which operated (with gaps due to war service) from 1904 to 1928.

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Ernald Hartley (1875-1947) Besides being a research chemist, Hartley was an amateur clarinetist who played in the Oxford orchestra for many years. The photo dates from 1925.

While Morse and Frazer used the same principle as Pfeffer, albeit with a more advanced electrochemical method of depositing the membrane in the pores, Berkeley and Hartley reversed the arrangement of solvent and solution, applying measured pressure to the latter to attain equilibrium.

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Theoretical development in Europe

In Europe, the rigorous application of thermodynamics to the phenomenon of osmosis started in 1887 with Lord Rayleigh, who combined the use of the ideal gas law PV = nRT with the idea of a reversible isothermal cycle of operations in which the sum of work in the complete cycle is zero.

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Lord Rayleigh 1842-1919

Being essentially an attempt to provide hypothesis-free support to van ‘t Hoff’s troubled gaseous theory of solutions, the solute in Rayleigh’s cycle was a mole of ideal gas, which was first dissolved in the solution by applied pressure and then recovered from the solution by osmotic pressure to return the system to its original state.

Rayleigh’s approach, using a zero-sum cycle of operations, was thermodynamically sound and continued to form the basis of theoretical development in its next phase, which in Europe focused on vapor pressure following the influential papers of Alfred Porter in 1907 and Hugh Callendar in 1908.

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Alfred Porter 1863-1939

By 1928, the theoretical model in JAV Butler’s popular textbook The Fundamentals of Chemical Thermodynamics was close to the familiar classroom demonstration of osmosis shown at the head of this post, in which the hydrostatic pressure acting on the solution counteracts the tendency of the solvent to pass through the semi-permeable membrane. At equilibrium, the hydrostatic pressure P is equal to the osmotic pressure.

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JAV Butler 1899-1977

To obtain a thermodynamic relation for osmotic pressure in terms of vapor pressures, Butler uses Rayleigh’s idea of a reversible isothermal cycle of operations together with a semipermeable membrane in the form of a movable piston between the solution and the solvent:

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The diagram shows a solution under hydrostatic pressure P which is equal to the osmotic pressure. Below the semi-permeable piston is pure solvent. Butler then applies the following argumentation:

1] Vaporize 1 mole of the pure solvent at its vapor pressure p0, and expand it reversibly so that the vapor pressure falls to p equal to the partial pressure of the solvent in the solution (Butler assumes that p is not affected by P applied to the solution). Condense the vapor into the solution. Since the work of vaporization and condensation cancel out, the only work done is the work of expansion. Assuming the vapor obeys the ideal gas law, the work (w) done is given by the textbook isothermal expansion formula

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2] Now move the semi-permeable piston up against the pressure P until a quantity of solvent equivalent to 1 mole of vapor has passed through it. If the decrease in the volume of the solution is ΔV, the work done is PΔV.

The cycle is now complete and the system has returned to its original state. The total work done is zero and we may equate the two terms

osm12 (1)

where P is the osmotic pressure, ΔV is the partial molal volume of the solvent in the solution, p0 is the vapor pressure of the pure solvent and p is the vapor pressure of the solvent in the solution. This thermodynamically exact relation, which involves measured vapor pressures, is in good agreement with experimental determinations of osmotic pressure at all concentrations.

There is a great irony here, in that this equation is exactly the one that JH van ‘t Hoff found his way to in Studies in Chemical Dynamics (1884), before he abandoned his good work and went completely off-track with his gaseous theory of solutions.

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JH van ‘t Hoff 1852-1911

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Theoretical development in America

In the US, the theory of the semipermeable membrane and the ‘equilibrium of osmotic forces’ was the work of one supremely gifted man, Josiah Willard Gibbs, who more or less single-handedly laid the theoretical foundations of chemical thermodynamics in his milestone monograph On the Equilibrium of Heterogeneous Substances.

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J Willard Gibbs 1839-1903

But before delving into the powerful idea he introduced, let us return to the subject of equilibrium in a system subject to osmotic pressure with a set-up that is slightly different to that used by Butler. In the diagram below, the piston supplies pressure Psoln to the solution which is just enough to stop solvent passing through the membrane and bring about equilibrium at constant temperature; the osmotic pressure is defined as the excess pressure Psoln – p01.

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The question can now be asked: Does the condition of osmotic equilibrium coincide with equality of a thermodynamic variable on either side of the membrane? Clearly it cannot be pressure or volume, nor can it be temperature since constant temperature does not prevent osmotic disequilibrium.

The P, V, T variables do not provide an affirmative answer, but in his monumental masterwork, Gibbs supplied one of his own invention which did – the chemical potential, symbolized μ. It is an intensive variable which acts as a ‘generalized force’, driving a system from one state to another. In the present context the force drives chemical components, capable of passing through a membrane, from a state of higher potential to a state of lower potential.

So given a membrane dividing solution from solvent and permeable only to the latter, we can understand the osmotic force driving the solvent (designated by subscript 1) through the membrane into the solution in terms of movement to a region of lower potential since

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Now the difference in potential can be calculated according to the textbook formula

osm15 (2)

where x1 is the mole fraction (<1) of the solvent in the solution. To achieve equilibrium, the chemical potential of the solvent in the solution must be increased by the amount –RTlnx1 (a positive quantity since lnx1 is negative). This can be done by increasing the pressure on the solution since

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is always positive (V1 is the partial molar volume of the solvent in the solution).

The osmotic pressure is defined as the excess pressure Psoln – p01. As can be seen from the diagram below, this is the pressure required to raise the chemical potential of the solvent in the solution so that it becomes equal to the chemical potential of the pure solvent.

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Since the slope is V1, it follows that

osm17 (3)

Combining (2) and (3) and designating the osmotic pressure by P gives the desired equilibrium relation

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This is exactly equivalent to equation (1) derived by Butler, since by his terminology

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The two methods of proof are thus shown to be equivalent – we can regard osmotic pressure as the excess pressure required to increase either the chemical potential or the vapor pressure of the solvent in the solution. But Gibbs saw an advantage in using potentials, which he voiced in an 1897 letter to Nature entitled Semi-Permeable Films and Osmotic Pressure:

“The advantage of using such potentials in the theory of semi-permeable diaphragms consists … in the convenient form of the condition of equilibrium, the potential for any substance to which a diaphragm is freely permeable having the same value on both sides of the diaphragm.”

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Tottenham Court Road, London WC2 in 1880

In the study of chemical reactions, thermodynamics enables us to calculate changes in state functions such as enthalpy, entropy and free energy, and determine the direction in which a reaction is spontaneous. But it tells us nothing about the speed of reaction; that is the province of chemical kinetics. Thermodynamics and chemical kinetics can be viewed as complementary disciplines, which together provide the means by which the course of a reaction can be elucidated.

A classic case which exemplifies the dual application of thermodynamics and chemical kinetics is the Tottenham Court Road gas explosion which occurred in July 1880.

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The incident

It was a time of great expansion of the network for gas pipeline transport in London. Gas lighting of streets and buildings was well-established, but now the gas stove was about to become a commercial success, and new gas mains were being laid to supply the anticipated demand.

The Gas Light and Coke Company, which supplied coal gas from a number of gasworks in London, had laid a new 1.2 kilometer (0.75 mile) section of main from Bedford Square to Fitzroy Square, the pipeline crossing Tottenham Court Road at the junction with Bayley Street and running along Percy Street before turning north along the entire length of Charlotte Street.

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On the evening of Monday 5th July 1880, workmen were preparing to connect the new main to the existing network at Bayley Street. Unknown to them however, a faulty valve at the other end of the new main was leaking coal gas, which had mingled with the air in the pipe to form an explosive mixture. In a presumed act of carelessness by one of the workmen at Bayley Street, a flame or other ignition source came in close proximity to the pipe.

The gas mixture detonated and the explosion ripped through the entire length of the new 1.2 kilometer main. A number of people were killed and injured in the blast, and 400 houses were damaged by flying debris. The entire incident lasted about 12 seconds.

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The investigation

A singularly worrying feature of the Tottenham Court Road gas explosion was that it had ripped through over a kilometer of pipeline in a matter of seconds. How could this happen? And how easily could this happen again? For the safety of millions of Londoners, answers had to be found.

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Augustus Vernon Harcourt (1834-1919)

The authorities turned to one the country’s leading chemists, Augustus Vernon Harcourt, who was conducting a program of research in chemical kinetics at Oxford University. Together with his student Harold Baily Dixon (1852-1930), Harcourt began to investigate the rates of propagation of gaseous explosions.

In what sounds like a rather risky experiment, they set up long metal pipes under the Dining Hall of Balliol College Oxford to measure the speed with which explosion waves travel when a mixture of air and coal gas detonates.

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The Dining Hall of Balliol College, Oxford

Twenty three years earlier, the German chemist Robert Bunson (of Bunsen burner fame) had investigated the rate of propagation for the ignition of coal gas and oxygen and concluded that the flame front velocity was less than 1 meter per second. From the experiments at Balliol however, Harcourt and Dixon arrived at a very different answer. In a report to the Board of Trade on the Tottenham Court Road blast, Harcourt concluded that the velocity of a coal gas/air explosion wave exceeded 100 yards per second (91 meters per second).

From the safety point of view, Harcourt and Dixon had shown how absolutely essential it was to prevent air becoming mixed with coal gas in the gas pipeline network. But it would take decades before sufficient theoretical progress was made to allow a detailed understanding of what exactly happened in the great gas explosion of 1880.

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Branching chains

The development of chemical kinetics involved many different contributors in the decades after Harcourt and Dixon’s pioneering work at Oxford. Theories were advanced on several different aspects of the subject, but one piece of theoretical work had particular relevance to the study of explosions.

In 1921, a Danish physical chemist by the name of Jens Anton Christiansen (1888-1969) completed his PhD studies in reaction kinetics at Copenhagen University. In his thesis he incorporated an idea first suggested by Bodenstein in 1913 and introduced the term “kædereaktion”. This term, and the conceptual idea behind it, attracted considerable attention and the equivalent English expression “chain reaction” came into use. Two years later, Christiansen and the Dutch physicist Hendrick Anthony Kramers (1894-1952) published a paper in which they suggested the possibility of branching chains. Their idea was that a chain reaction could involve steps in which one chain carrier (an atom or radical) might not only regenerate itself but also produce an additional chain carrier. If such chain branching occurred, the number of chain carriers could increase extremely rapidly and result in an explosion.

The idea proved to be well-founded, and was further developed by Nikolai Semyonov (1896-1986) and Cyril Norman Hinshelwood (1897-1967). Their work also showed that chain carriers were removed at the walls of the reaction vessel. If the rate of removal of the chain carriers was fast enough to counteract the effect of chain branching, a steady reaction ensued. But if the removal rate could not keep pace with the chain branching rate, an explosion would result.

On the basis of their thinking, the reaction rate expression assumed the form

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where F is a function of the concentrations characteristic of the chain branching step, fa is a function determining the removal of chain carriers, and fb is a function expressing the branching nature of the chain reaction.

In steady reaction conditions, fa is sufficiently greater than fb. But if conditions change so that fa and fb converge, a point will be reached where the difference between them becomes vanishingly small. The reaction rate will soar towards infinity however small F may be, and the evolution of heat in the system will be so great as to cause an explosion.

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Semyonov and Hinshelwood were awarded the Nobel Prize in 1956 for their work on reaction rates

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Piecing the facts together

From the information contained in newspaper reports, and the application of kinetic theory and thermodynamics, it is possible to arrive at a likely explanation of why the great gas explosion of 1880 happened in the way it did.

It is known that coal gas leaked into the newly laid main at its northern end, and that detonation occurred at the other end in Bayley Street. From this it can be inferred that the entire pipeline between these two points contained coal gas admixed with the air that the pipe originally contained. On the assumption that the leaking valve was introducing coal gas at a modest and steady rate, it is likely that the partial pressures of the gases in the pipe were being brought into equilibrium as the coal gas seeped along the pipe.

Newspaper reports stated that the new main between Bayley Street and Fitzroy Square was a metal pipe of fixed (3 ft/0.91 m) diameter. The ratio of the surface area to the enclosed volume or, which is the same thing, the ratio of the circumference to the cross-sectional area

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was therefore constant along its length*.

*assuming the geometry of the bend had no effect on fa. This point is examined later.

At the moment of detonation at Bayley Street, it is a reasonable hypothesis that the function F in the Semyonov-Hinshelwood rate expression was not subject to large variations along the length of the new main. The same can be said of fb, and since the ratio of the circumference to the cross-sectional area of the pipeline was constant, the function fa determining the removal of chain carriers at the walls of the pipe was also constant. In short, the reaction rate expression applying at the end of the pipe – where detonation is known to have occurred – applied at every other point along its length.

At this juncture, it is convenient to recall the combustion reactions of the principal components of coal gas, namely hydrogen, methane and carbon monoxide:

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We observe that from a stoichiometric perspective, none of the reactions involves an increase in volume; in fact two of them result in a decrease. The overall entropy of reaction is negative, and this tells us that the conversion of reactants into products, however rapidly it took place, could not in itself have resulted in any pressure increase under the constant volume conditions of the pipe.

From an enthalpy of reaction perspective however, the situation is very different. The above reactions are all significantly exothermic processes – the calorific value of coal gas is typically around 20 megajoules per cubic meter. In the circumstances of detonation, the virtually instantaneous release of a large amount of heat would result in a similarly rapid rise in temperature, causing sudden compression of the adjacent volume element in the pipe and heating it to the point of detonation. This sequence would be repeated from one volume element to the next, with a wave of adiabatic compression intensifying the pressure as it traversed the pipe. A continuously propagating explosion would then follow the pressure wave along the course of the main as the pipe ruptured.

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The bend in the pipe

The junction of Percy Street with Charlotte Street was the only point along the entire length of the new main which deviated from a straight line. Here the pipeline executed a 90 degree turn, and it raises the question of how a detonation wave can go round corners. The exact construction of the bend is not recorded, but it is likely that an elbow joint was used.

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Geometrically, the bend itself is a quadrant of a torus, whose geometry is such that regardless of whether the elbow has a long or short major radius R, the ratio of the surface area to the enclosed volume is constant

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This is the same ratio as that of the straight pipe. The bend at the junction of Percy Street with Charlotte Street introduced no changes to the fa term in the Semyonov-Hinshelwood rate expression, and thus the conditions for detonation were met at every point of the bend.

So the 90 degree elbow made no difference to the detonation wave. It simply turned sharp right and carried on up to Fitzroy Square, at a velocity of almost 100 meters per second.

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Estimating the power of the explosion

It is known from the analysis of coal gas that one volume of coal gas requires approximately 10 volumes of air for its complete combustion. This means that an explosive mixture with air cannot be formed at coal gas concentrations much above 9%, since there would be insufficient oxygen to support the necessary rate of reaction. Below 7% coal gas concentration, the mixture is also non-explosive, for other reasons.

An average coal gas concentration of 8% throughout the pipeline is therefore a fair estimate, and seems plausible given that the new main contained air when laid and that coal gas was introduced at a modest rate from a leaking valve. We know that the new 1.2 kilometer main had a radius of 0,455 meters, giving a total volume of 780 cubic meters. At the moment of detonation, coal gas is estimated to have filled 8% of this volume i.e. 62 cubic meters. The calorific value of coal gas is typically 20 megajoules per cubic meter, so we can conclude that the Tottenham Court Road gas explosion released around 1,240 MJ in the 12 seconds it took to traverse the pipeline. The power of the explosion was therefore 1240/12 = 103 MW.

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The 3×2 flagstones used on London sidewalks weigh around 70 kg each. The energy released by the Great Gas Explosion of 1880 was sufficient to blast 59,000 flagstones to a height of 30 meters.

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Contemporary accounts

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Charlotte Street after the blast

Newspaper accounts remarked on the rapid progression of the explosion, with one commenting:

“[The main pipe at Bayley Street] burst with a terrific report, and sheets of flame issued suddenly from the earth. Instantly the report seemed to run along Percy Street, which was torn up for sixty or seventy yards (ca. 60 meters), the paving stones flying on each side against the houses.”

“At the corner of Charlotte Street the basements of two houses were shattered. The paving stones were here also sent into the air, falling on and through the roofs of the houses opposite. Further on, the pipe burst again, near the corner of Bennett Street, where there is a large gap in the roadway. Another burst-up occurred near the corner of Howland Street, and at the corner of London Street (now Maple Street) still further on…”

One eye-witness was in Percy Street when the explosion occurred. He experienced the effect of not only the pressure wave from the bursting pipe, but also the decompression wave which followed in its wake:

“I was walking down Percy Street, when I felt the ground shaking under my feet. I immediately saw the centre of the street rising in the air. A tremendous report followed, and then there was a shower of bricks and stones. I felt myself lifted from the ground, and the next moment I was lying among the debris at the bottom of a deep hole in the roadway.”

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P Mander December 2015

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I can’t think of a better introduction to this post than Ludwig Boltzmann gave in his Vorlesungen über Gastheorie (Lectures on Gas Theory, 1896):

“General thermodynamics proceeds from the fact that, as far as we can tell from our experiences up to now, all natural processes are irreversible. Hence according to the principles of phenomenology, the general thermodynamics of the second law is formulated in such a way that the unconditional irreversibility of all natural processes is asserted as a so-called axiom … [However] general thermodynamics (without prejudice to its unshakable importance) also requires the cultivation of mechanical models representing it, in order to deepen our knowledge of nature—not in spite of, but rather precisely because these models do not always cover the same ground as general thermodynamics, but instead offer a glimpse of a new viewpoint.”

Today, the work of Ludwig Boltzmann (1844-1906) is considered among the finest in physics. But in his own lifetime he faced considerable hostility from those of his contemporaries who did not believe in the atomic hypothesis. As late as 1900, the kinetic-molecular theory of heat developed by Maxwell and Boltzmann was being vigorously attacked by a school of scientists including Wilhelm Ostwald, who argued that since mechanical processes are reversible and heat conduction is not, thermal phenomena cannot be explained in terms of hidden, internal mechanical variables.

Boltzmann refuted this argument. Mechanical processes, he pointed out, are irreversible if the number of particles is sufficiently large. The spontaneous mixing of two gases is a case in point; it is known from experience that the process cannot spontaneously reverse – mixed gases don’t unmix. Today we regard this as self-evident, but in Boltzmann’s time his opponents did not believe in atoms or molecules; they considered matter to be continuous. So the attacks on Boltzmann’s theories continued.

Fortunately, this did not deter Boltzmann from pursuing his ideas, at least not to begin with. He saw that spontaneous processes could be explained in terms of probability, and that a system of many particles undergoing spontaneous change would assume – other things being equal – the most probable state, namely the one with the maximum number of arrangements. And this gave him a new way of viewing the equilibrium state.

One can see Boltzmann’s mind at work, thinking about particle systems in terms of permutations, in this quote from his Lectures on Gas Theory:

“From an urn, in which many black and an equal number of white but otherwise identical spheres are placed, let 20 purely random drawings be made. The case that only black spheres are drawn is not a hair less probable than the case that on the first draw one gets a black sphere, on the second a white, on the third a black, etc. The fact that one is more likely to get 10 black spheres and 10 white spheres in 20 drawings than one is to get 20 black spheres is due to the fact that the former event can come about in many more ways than the latter. The relative probability of the former event as compared to the latter is the number 20!/10!10!, which indicates how many permutations one can make of the terms in the series of 10 white and 10 black spheres, treating the different white spheres as identical, and the different black spheres as identical. Each one of these permutations represents an event that has the same probability as the event of all black spheres.”

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By analyzing the ways in which systems of particles distribute themselves, and the various constraints to which particle assemblies are subject, important links came to be established between the statistical properties of assemblies and their bulk thermodynamic properties.

Boltzmann’s contribution in this regard is famously commemorated in the formula inscribed on his tombstone: S = k log W. There is powerful new thinking in this equation. While the classical thermodynamic definition of entropy by Rankine and Clausius was expressed in terms of temperature and heat exchange, Boltzmann gave entropy – and its tendency to increase in natural processes – a new explanation in terms of probability. If a particle system is not in its most probable state then it will change until it is, and an equilibrium state is reached.

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P Mander April 2016

Torricelli using a lot more mercury than necessary to demonstrate the barometer principle

The scientific study of the atmosphere can be said to have begun in 1643 with the invention of the mercury barometer by Evangelista Torricelli (1608-1647). Although the phenomenon had been observed and discussed by others – including Galileo – in the preceding decade, it was Torricelli who provided the breakthrough in understanding.

The prevailing view at the time was that air was weightless and did not exert any pressure on the mercury in the bowl. Instead, it was thought that the vacuum above the liquid in the barometer tube exerted a force of attraction that held the liquid suspended in the tube.

Torricelli challenged this view by proposing the converse argument. He asserted that air did have weight, and that the atmosphere exerted pressure on the mercury in the bowl which balanced the pressure exerted by the column of mercury. The vacuum above the mercury in the closed tube, in Torricelli’s opinion, exerted no attractive force and had no role in supporting the column of mercury in the tube*.

The assertion that air had weight, Torricelli realized, could be tested. In elevated places like mountains the reduced weight of the overlying atmosphere would exert less pressure, so the corresponding height of the mercury column in the barometer tube should be lower. It seems that Torricelli did not have the opportunity in his short life to do this experiment, but in the year following his death the experiment was carried out in France at the behest of the scientific philosopher Blaise Pascal (1623-1662).

*CarnotCycle wonders if Torricelli tilted the barometer tube and observed the disappearance of the space above the mercury – see diagram below. This would have shown that something other than a vacuum held the liquid suspended in the tube.

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The Torricelli experiment

In 1644 the French salon theorist Marin Mersenne (1588-1648) travelled to Italy where he learned of Torricelli’s barometer experiment. He brought news of the experiment back with him to Paris, where the young Blaise Pascal was a regular attendant at Mersenne’s salon meetings.

Puy de Dôme in south-central France, close to Clermont-Ferrand

Pascal had moved to Paris from his childhood home of Clermont-Ferrand. The 1,465 meter high Puy de Dôme was a familiar feature in the landscape he knew as a youngster, and it provided an ideal means of testing Torricelli’s thesis. Pascal’s brother-in-law Florin Périer lived in Clermont-Ferrand, and after some friendly persuasion, Périer ascended Puy de Dôme with a Torricellian barometer, taking measurements as he climbed.

The Torricelli experiment was conducted by Florin Périer on Puy de Dôme on Saturday 19 September 1648

At the base of the mountain, Périer recorded a mercury column height of 26 inches and 3½ lines. He then asked a colleague to observe this barometer throughout the day to see if any change occurred, while he set off with another barometer to climb the mountain. At the summit he recorded a mercury column height of 23 inches and 2 lines, substantially less than the measurement taken 1,465 meters below, where the barometer had remained steady.

The Puy de Dôme experiment provided convincing evidence that it was the weight of air, and thus atmospheric pressure, that balanced the weight of the mercury column.

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Measuring pressure

A 14th century shopkeeper weighing out sugar cubes using hand-held scales

When Florin Périer conducted the Torricelli experiment on Puy de Dôme in 1648, the measurements he recorded were the heights of mercury columns in barometer tubes. From these measurements, Blaise Pascal inferred a comparison of atmospheric pressures at the top and bottom of the mountain.

This experiment took place, we should remind ourselves, when Isaac Newton was only 5 years old and had not yet formulated his famous laws which gave concepts like mass, weight, force and pressure a systematic, mathematical foundation. In the pre-Newtonian world of Torricelli and Pascal, their thinking was based on the balancing of weights in the familiar sense of a shopkeeper’s scales. The weight of the mercury column in the barometer tube, which acted on the mercury in the bowl, was balanced by the weight of the air acting on the mercury in the bowl. Since the height of the mercury column was directly proportional to its weight, it was valid to use a length scale marked on the barometer tube to compute the weight of the air acting on the mercury in the bowl.

It is instructive to compare the language of Robert Boyle (1627-1691) and Isaac Newton (1643-1727) when discussing the barometer in the decades which followed. In the second edition of Boyle’s New Experiments Physico-Mechanicall of 1662 – which contains the first statement of Boyle’s Law – the word pressure appears frequently and has a meaning synonymous with weight. In Isaac Newton’s Principia of 1687, pressure is regarded as a manifestation of force. Boyle and Newton are thus speaking in essentially the same terms since according to Newtonian principles, weight is a force.

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Newtonian principles applied

The crucial advance in atmospheric science that Newton supplied in his Principia was the second law, which gave mathematical expression to force, and thus to weight and pressure, through the famous formula

The weight of a mercury column of cross-sectional area A and height h is

where ρ is the mass density of mercury and g is the acceleration due to gravity. The pressure exerted by the mercury column, which balances the atmospheric pressure, is

Thus P is directly proportional to h.

For a column of mercury 1 mm in height in a standard gravitational field (g = 9.80665 ms-2) at 273K, P is equal to 133.322 pascals. This is a unit of pressure called the torr. Pascal and Torricelli are thus both commemorated in units of pressure.

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A question of balance

Torricelli, Pascal and Boyle were in agreement with the proposition that air has weight. According to Newton’s interpretation the atmosphere possesses mass which is subject to gravitational acceleration, resulting in a downward force. This raises the question – Why doesn’t the sky fall down?

Since the sky is observed to remain aloft, there must exist a counteracting upward force. The vital clue as to the nature of this force was obtained on Pascal’s behalf by Florin Périer on Puy de Dôme in 1648 – namely that pressure decreases with height in the atmosphere.

A difference in pressure produces a force. In this way a parcel of air in a vertical column of cross-sectional area A exerts a force in the opposite direction to the gravitational force, as shown in the diagram.

At equilibrium, the forces are equal. Thus

where ρ is the density of the air.

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The decrease of temperature with altitude

A snow-capped Puy de Dôme

The appearance of snow above a certain height in elevated places provides plain evidence that temperature decreases with altitude, at least in that part of the atmosphere into which our earthly landscape protrudes. No doubt Torricelli, Pascal and other scientific philosophers of their time noticed this phenomenon and pondered upon it. But the explanation had to wait for another two centuries until the industrial revolution began, ushering in the age of steam and the associated science of thermodynamics.

The air in the troposphere, the lowest layer of the atmosphere where almost all weather phenomena occur, exhibits convection currents which continually transport air from lower regions to higher ones, and from higher regions to lower ones. When air rises it expands as the pressure decreases and so does work on the air around it. Thermodynamic principles dictate that this work requires the expenditure of heat, which has to come from within since air is a poor conductor and very little heat is transferred from the surroundings. As a result, rising air cools.

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Atmospheric convection processes fall within the province of the first law of thermodynamics, which can be expressed mathematically (see Appendix I) as

This equation states an energy conservation principle that applies to processes involving heat, work and internal energy. The atmospheric convection process is adiabatic meaning that no heat flows into or out of the system i.e. dQ = 0. Applying this constraint and using the combined gas equation to eliminate pressure p the above equation becomes

Integration yields

Converting from logarithms to numbers gives

Since by Mayer’s relation R = CP – CV

where γ = CP / CV. Using the combined gas equation to substitute V, the above equation can be rendered (with the help of γ√) as

Applying logarithmic differentiation gives

Assuming hydrostatic equilibrium, dp can be substituted giving

Since ρ = m/(RT/p) the above becomes

This adiabatic convection equation gives the rate at which the temperature of dry air falls with increasing altitude. Taking the following values: γ = 1.4 (dimensionless) ; R = 8.314 kgm2s-2 K-1 mol-1 ; m = 0.0288 kg mol-1 ; g = 9.80665 ms-2 gives

At the top of Puy de Dôme (1465 meters), dry air will be 14°C cooler than at the base of the mountain. This explains why snow can appear on the summit while the grass is still growing on the lower slopes.

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Appendix I

Benoît Émile Clapeyron (1799-1864)

In 1834, more than a century after Newton’s death, the French physicist and engineer Émile Clapeyron wrote a monograph entitled Mémoire sur la Puissance Motrice de la Chaleur (Memoir on the Motive Power of Heat). It contains the first appearance in print of the ideal gas equation, which combines the gas law of Boyle-Mariotte (PV)T = k with that of Gay-Lussac (V/T)P = k. Clapeyron wrote it in the form

where R is a constant and the sum of the terms in parentheses can be regarded as the thermodynamic temperature.

Sixteen years later in 1850, the German physicist Rudolf Clausius wrote a monograph on the same subject entitled Ueber die bewegende Kraft der Wärme und die Gesetze, welche sich daraus fuer die Wärmelehre selbst ableiten lassen (On the Motive Power of Heat, and on the Laws which can be deduced from it for the Theory of Heat). Seeking an analytical expression of the principle that a certain amount of work necessitates the expenditure of a proportional quantity of heat, he arrived at the following differential equation in the case of an ideal gas

where Q is the heat expended, U is an arbitrary function of temperature and volume, and A is the mechanical equivalent of heat. Earlier in his paper Clausius had represented Clapeyron’s combined statement of the laws of Boyle-Mariotte and Gay-Lussac as pv = R(a + t) so he recognized the right-hand term as corresponding to pdv, the external work done during the change

We know this equation today as an expression of the first law of thermodynamics, where U is the internal energy of the system under consideration.

U is a function of T and V so we may write the partial differential equation

Since U for an ideal gas is independent of volume and dU/dT is the heat capacity at constant volume CV, the first law for an ideal gas takes on the form

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P Mander January 2018