Ever since the works of J. Willard Gibbs were first published, people have struggled to understand him. Back in 1892 Lord Rayleigh, by all accounts a capable physicist and mathematician, complained to Gibbs that his masterwork On the Equilibrium of Heterogeneous Substances was “too condensed and too difficult for most, I might say all, readers”.

In 1927 faculty members at Yale University where Gibbs taught decided to do something about it. A committee was appointed to oversee the creation of a work that would elucidate and facilitate understanding of Gibbs’ writings and the result of their labors was a two-volume Commentary published in 1936. The first volume was on Thermodynamics and the second on Theoretical Physics.

Volume 1 Thermodynamics is over 700 pages long and was written by ten authors of high standing including Edward Guggenheim whose influential textbook Modern Thermodynamics by the Methods of Willard Gibbs had been published by Methuen & Co., London in 1933.

The volume is organized in 13 sections, all but three of which refer to On the Equilibrium of Heterogeneous Substances, reflecting the exceptional difficulty of this milestone monograph:

A. Note on symbols and nomenclature, F.G. Donnan
B. Mathematical note, J. Rice
C. Papers I and II as illustrated by Gibbs’ lectures on Thermodynamics, E.B. Wilson
D. The general thermodynamic system of Gibbs, J.A.V. Butler
E. Osmotic and membrane equilibria, including electrochemical systems, E.A. Guggenheim
F. The quantities ψ, χ and ζ, and the criteria of equilibrium, E.A. Milne
G. The Phase Rule and heterogeneous equilibrium, G.W. Morey
H. The graphical representation of equilibria in binary systems by means of the Zeta (Free Energy) function, F.A.H. Schreinemakers
I. The conditions of equilibrium for heterogeneous masses under the influence of gravity (and centrifugal force), D.H. Andrews
J. The fundamental equations of ideal gases and gas mixtures, F.G. Keyes
K. The thermodynamics of strained elastic solids, J. Rice
L. The influence of surfaces of discontinuity upon the equilibrium of heterogeneous masses. The theory of capillarity, J. Rice
M. The general properties of a perfect electrochemical apparatus. Electrochemical thermodynamics, H.S. Harned

Sadly this book has long been out of print and today hardly anyone knows of its existence. A revival of interest is this amazingly useful Guide to Gibbs is greatly deserved and long overdue, so if you happen to read this post please check out the link below and spread the word in your scientific community.

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See the whole book on the Internet Archive


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P Mander January 2022


The above photograph shows the largest jet engine in commercial use at the time of writing (March 2023) – the General Electric GE9X high-bypass turbofan developed exclusively for the Boeing 777X. The fan is well over 3 meters in diameter and sucks in copious quantities of air, some of which enters the engine core to be pressurized and heated while a substantial amount does not enter the core but is channeled around it as bypass air.

The effect of this double air stream is to increase significantly the mass of air flowing through the engine per unit of time, which increases thrust (F) in accordance with the equation

where m dot is the mass flow rate (MT-1) and v the velocity (LT-1) at the exit and inlet respectively. F is the change in momentum per unit of time and has the dimensions of force (MLT-2).

This equation tells us some useful things. F is positive only if vexit > vinlet so the airplane cannot fly faster in level flight than the exhaust gas velocity. Increasing this velocity to gain extra thrust requires adding more kinetic energy which in turn means consuming more fuel. This is why commercial aviation has focused instead on sucking in more air. So long as vexit > vinlet an increase in m dot will produce higher thrust without much change to air velocity or fuel consumption.

High-bypass turbofans have thus brought significant improvements to the overall efficiency of the commercial jet engine. But big fans are not the only things determining the mass of air flowing through jet engines, as we shall see in the next section.

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Atmospherics and m dot

Inspection of the above equation shows that the thrust generated by a jet engine is directly proportional to the mass flow rate m dot. Now consider the steady-state condition where a unit volume of air enters the engine every second. The mass of air in that unit volume is proportional to its density, which is increased under high pressure, low temperature conditions and decreased under low pressure, high temperature conditions.

If we regard air as approximating an ideal gas, we can apply the ideal gas equation PV = nRT to calculate the effect of pressure and temperature on air density, taking as our starting point the density of dry air being 1.2752 kg/m^3 at STP (10^3 hPa, 273.15K).

Effect of pressure on dry air density at constant temperature (273.15K)

Effect of temperature on dry air density at constant pressure (10^3 hPa)

These figures demonstrate the considerable increase in thrust generated by a jet engine at take-off on a cold winter’s day with high atmospheric pressure compared to a hot summer’s day with low atmospheric pressure. Note that although the presence of water vapor makes air less dense, the density decrease is so small that it hardly makes any difference to jet engine performance.

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Jet engine performance at altitude

As the airplane climbs to cruising altitude, atmospheric temperature and pressure both decrease. While the former acts to increase air density the latter acts to decrease it. The two effects do not counterbalance each other however, since the proportional drop in pressure is considerably more than the proportional drop in temperature

Comparison of the two charts shows a 63% reduction in air density and 67% reduction in thrust during a climb to 30,000 feet, demonstrating a close correlation between these parameters. This is to be expected since thrust is a function of mass flow rate which in turn is a function of air density.

Reduction in thrust does not mean a corresponding reduction in velocity however, as drag forces on the airplane are also reduced due to the air becoming less dense. Newtonian mechanics reminds us that so long as the jet engines are producing thrust in level flight, the airplane will accelerate and velocity will increase until the thrust and drag forces are equal.

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Air density and the Brayton cycle

Since this is the CarnotCycle blog, it seems appropriate to end this blogpost – the last in a set of three on aeronautical themes – with a few words about the ideal thermodynamic cycle of the gas turbine, otherwise known as the Brayton cycle. George Brayton was a 19th century engineer from Rhode Island who invented heat engines operating on the same thermodynamic principles as the jet engine, long before the jet engine was even thought about.

Compared with the Carnot cycle, the Brayton cycle shares the adiabatic compression (1>2) and adiabatic expansion (3>4) steps. But while the Carnot cycle operates step (2>3) isothermally, the Brayton cycle operates this step under isobaric conditions. The temperature increases as heat is supplied, resulting in a greater volume increase as the pressure falls to atmospheric. This accounts for the extended volume axis.

Consider a unit mass M of air undergoing steps 1 to 4 of the Brayton cycle per unit of time. The density of the air at point 1 is M/V1, corresponding to the jet engine’s fresh air inlet. At point 4 the density is M/V4, corresponding to the exhaust outlet. Since V4 > V1, air density ρ decreases between inlet and outlet and exhaust velocity  v increases according to the relation M dot = ρvA. This acceleration of unit air mass between inlet and exhaust is the source of the jet engine’s thrust.

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P Mander March 2023

In all the years I have been getting on airplanes, I have never really thought about the jet engines that provide the propulsion to fly me from A to B. Until one day not so long ago when I happened to have a window seat looking out on the starboard engine of an Airbus A330 as it soared into the skies over New York. It was a moment to savor and I found myself wondering what Sadi Carnot would have thought if he were sitting in my seat. That got me thinking on his illustrious behalf about the thermodynamics of jet engines.

A distant memory from my college days reminded me that these engines produce thrust by sucking in air at the front, accelerating it and blowing it out the back. By Newton’s Third Law the airplane experiences an equal thrust in the opposite direction:

The thrust of a gas turbine engine is conventionally written

where m dot is the mass flow rate (MT^-1) of the working substance (air) and v is its velocity (LT^-1) at the exit and inlet respectively. F is the change in momentum per unit of time and has the dimensions of force (MLT^-2). Aeronautical engineers refer to the first term on the right as gross thrust and the subtracted second term as ram drag.

In order to ascertain the overall efficiency ɳ of the engine’s conversion of heat into work, the force pushing the airplane forwards needs to be multiplied by the distance travelled in a chosen unit of time and divided by the amount of heat from fuel combustion transferred to the air passing through the combustion chamber in the same unit of time. Hence

where mf dot is the fuel mass flow rate and q is the thermal energy per unit mass of fuel absorbed by the air. In dimensional terms ɳ is the ratio of propulsive power (MLT^-2 × LT^-1 = ML^2T^-3) to thermal power (MT^-1 × ML^2T^-2/M = ML^2T^-3). Dividing numerator and denominator by T, ɳ also expresses the classic thermodynamic efficiency ratio of work obtained from heat supplied.

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Jet fuel chemistry

Gas turbine-powered aircraft use Jet fuel, a mixture of liquid hydrocarbons whose exact composition varies depending on source. Performance specifications and physical properties (such as freezing point) define individual product requirements, but in general terms Jet fuel can be thought of as principally a mixture of alkanes and cycloalkanes, whose general formulas are CnH2n+2 and CnH2n respectively, with a carbon number distribution between about 8 and 15.

The carbon oxidation state of Jet fuel is thus a tad below –2 in the zone where one would expect a liquid state. In terms of bond energy storage, Jet fuel has excellent qualities with full combustion covering 2/3 of carbon’s energy release range. Energy density is 43 MJ/kg or better.

For use in commercial airplanes, it seems unlikely that Jet fuel will be much improved upon as a practical source of hydrocarbon combustion energy. Biosynthetic routes have obvious merit but in terms of energy density there would not appear to be much to go after. Famous last words.

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Increasing thrust

Inspection of the first equation shows there are two ways in which more thrust can be obtained: by increasing m dot or by increasing exit velocity in relation to inlet velocity. Commercial aviation has focused on m dot. So long as vexit > vinlet an increase in m dot will produce higher thrust. This is the theory behind high-bypass turbofan engines: big fans can suck in a lot of bypass air without much change to air velocity or fuel consumption. By increasing thrust in relation to fuel consumption, the overall efficiency of the engine is improved. High-bypass engines also have the advantage of quieter operation. Below is a photo of the GE9X high-bypass turbofan developed exclusively for the Boeing 777X. The fan is housed in a 3.4 meter (134 inch) diameter case.

Photo courtesy of GE Aviation

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Want to know more?

Jet propulsion and jet engine design are massive subjects to which this blogpost is just a very brief introduction. The following links are useful starting points for growing your knowledge.

How Jet Engines Work – 5 minute YouTube video

Brilliant animated graphics and clear explanation. A must see.

See Thru Jet Engine – 10 minute YouTube video

Get an inside look at a jet engine in operation. Amazing stuff.

General Thrust Equation, and more
NASA is the go-to place for the physics behind jet engines.

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P Mander February 2023

I am not a frequent flier, but I have spent enough time flying at altitude to start thinking about the difference in pressure between the air outside the window and the air inside the cabin. Except I haven’t, until now. I like looking at the screen tracking the progress of the flight and showing the altitude, temperature, groundspeed etc so I know how cold it gets at 37,000 feet. But it doesn’t show the pressure, and for whatever reason I suddenly became interested in finding out.
An internet search for ‘pressure at altitude’ yielded this formula:

P = pressure at altitude h [any unit so long as same as Pb]
Pb = pressure at sea level [any unit so long as same as P]
h = height above sea level [m]
hb = height at ground level [m]
Tb = temperature at sea level [K]
Lb = temperature lapse rate = -0.0065 Km-1
R = universal gas constant = 8.31432 kg.m2s-2K-1mol-1
g0 = gravitational acceleration constant = 9.80665 ms-2
M = molar mass of atmosphere = 0.0289644 kg.mol-1

Plugging in Pb = 1 atm, Tb = 293K, h = 11,277 m (37,000 ft) and taking hb as sea level gave me an answer I honestly didn’t expect. The outside pressure at 37,000 feet is only 0.22 atm – barely a fifth of its value at sea level!

That got me thinking about the structural stresses at altitude if the cabin is maintained at ground-level pressure, and I soon discovered that aeronautical engineers had been there long before me. In order to reduce these stresses, the cabin pressure is programed to reduce gradually during ascent from the airport of origin to a regulatory cabin altitude of 8,000 ft (2,438 m) and then increase gradually during descent until the cabin pressure matches the air pressure at the destination. So what is the pressure at 8,000 ft? I plugged h = 2,438 m into the equation and discovered that the cabin pressure at cruising altitude is 25% less than at sea level. A significant difference.

This in turn got me thinking about the reduction in oxygen availability, given that passengers do not appear to be distressed by it. Applying the ideal gas equation brought me to the conclusion that the reduction must be in proportion to the pressure difference, other things like cabin temperature and breathing rate and tidal volume being equal, since under these circumstances:
n1 = oxygen availability at ground level, n2 = oxygen availability at cruising altitude
P1 = cabin pressure at ground level, P2 = cabin pressure at cruising altitude
n1RT/P1V = n2RT/P2V
n2/n1 = P2/P1

In other words, the cabin air I am breathing at altitude contains 25% less oxygen per unit volume than it did at takeoff. I suspect the reason we don’t notice the change is that we are doing nothing more physically demanding than sitting in our seats. If we were all riding exercise bicycles we would probably notice soon enough.

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Checking cabin pressure at altitude


Watches like the SKMEI 1358 and Casio Pro Trek have a barometer function. I used one on a recent flight; at take-off the cabin pressure was 1007 hPa and at a cruising altitude of around 35,000 feet (10.7 km) the cabin pressure was 760 hPa, exactly 25% less than at take-off. Given the big fall in pressure it is remarkable how little we seem affected by it, although some online writers ascribe the tiredness and lassitude some passengers experience to depleted oxygen levels.

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Breathing in, breathing out

The type of airplane I usually fly in has around 270 passengers and crew. Each of us inhales an average 7½ liters of air per minute of which 20% is oxygen, so we inhale around 400 liters of oxygen per minute of which we utilize something like 5%. That corresponds to consuming 1.2 cubic meters of oxygen per hour.

We use this oxygen to sustain vital cellular processes, and biochemistry tells us that the oxygen we consume is replaced with equal amounts of exhaled carbon dioxide and water vapor. These gases cannot be allowed to build up in the cabin airspace since increased levels of carbon dioxide can have adverse physiological effects while water vapor carries a risk to the aircraft of condensation and corrosion. Again, aeronautical engineers have long known this and have solved both this and the oxygen depletion problem by continuously replacing the cabin air with air from outside which fortunately has the same nitrogen-oxygen composition as air at sea level. This air is compressed to the required cabin altitude using either bleed air from the jet engine compressor or purpose-built electrical compressor systems.

Water vapor is a different matter. At cruising altitude the outside temperature is -76°F (-60°C) or thereabouts. The vapor pressure of water at this temperature is extremely low so there can be only tiny quantities of water vapor in the air outside – this reference proves the point with a graph showing how the Mixing Ratio decreases exponentially with altitude.

Given this fact, it is a bit of a mystery to me where any replacement water vapor comes from if not from the breath of passengers and crew. In any event, cabin air has a reputation for dryness with relative humidity often as low as 20%. This explains why bottles of water are supplied as a courtesy service on longer-haul flights.

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“If the cabin air system should fail …”

When the safety video plays and it gets to the oxygen mask part, I often wonder if I will be able to breathe normally as the voice asks me to do once I have put the mask on as quickly as I can and tightened the elastic bands before helping others. I am not the excitable type but I think normal breathing is a tall order in this situation, especially since I know that the flow of oxygen is coming from a chemical generator usually designed to last not much more than 15 minutes. Enough time for the airplane to descend to a safe height I have read. However I have also read that the cockpit crew have the use of compressed oxygen cylinders which last somewhat longer, which makes me wonder what I would do in the meantime if they needed that extra capacity.

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When flying is a pain in the ear

I thought I would add a few words on this topic as I have experienced problems with this myself. If you have read this far, you will know that cabin air undergoes decompression on the way up to cruising altitude and compression on the way down. This can cause some passengers to experience ear pain – I have noticed that children seem more vulnerable to this than adults, especially during descent.

The problem centers on the middle ear cavity, which has the eardrum and ear canal on one side and a tube connected to the nasal canal on the other called the Eustachian tube. Its purpose is to ensure equal pressure on either side of the eardrum. Trouble is that the Eustachian tube is not always up to the job and when it doesn’t function properly, pain from pressure imbalance on the eardrum can result. This can be relieved by yawning or swallowing but if these techniques don’t work there are a number of proprietary products that might be worth trying, including pressure-controlling ear plugs and nasal balloons.

Nasal balloons encourage a closed Eustachian tube to open and allow pressure on both sides of the eardrum to equalize.

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P Mander December 2022, additions February 2023

One of the many statements of the Second Law of Thermodynamics is the following:

Spontaneous changes are those which, if carried out under the proper conditions,
can be made to do work

The problem with this statement is that there are examples of spontaneous change where it is not immediately obvious what the ‘proper conditions’ might be. The mixing of two perfect gases is a case in point. As the header diagram shows, two gases initially compartmentalized in a container will each expand to fill the space available to them when the partition is removed. The process is spontaneous but no work is done.

There is however another means of achieving the same mixing result if the partition is replaced with a piston equipped with a membrane permeable to gas A but not to gas B. This can be achieved in practice for example with palladium, which is easily permeable to hydrogen but not to other gases.

During this process the piston does PV work of 1 unit, displacing the original volume of the left compartment V = 1 at pressure P = 1. If the piston were connected to an external arrangement for extending a spring or raising a weight, the spontaneous mixing of gases could be made to do mechanical work.

Reversing the above process by doing mechanical work on the system would afford a method of separating the mixed gases. In either case, maximal efficiency would be achieved (see reference below).

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The remarkable diffusion properties of hydrogen

In Munich in 1883, Max Planck conducted an experiment in which a platinum tube originally containing hydrogen at atmospheric pressure was heated until the platinum became permeable to hydrogen, upon which it was found that almost the whole contents diffused out leaving a high vacuum (see reference below)!

Reference: G.H. Bryan, Thermodynamics, published by BG Teubner, Leipzig 1907, page 126

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