Posts Tagged ‘Temperature’

Sadi Carnot (1796-1832) One of the great original thinkers.

Sadi Carnot (1796-1832) One of the great original thinkers.

The milestone memoir Réflexions sur la Puissance Motrice du Feu (Reflections on the Motive Power of Fire), published by French engineer Sadi Carnot in 1824, marks the starting point of thermodynamics as a theory-based science. In this work, Carnot developed his powerful ideas with the aid of the caloric theory, which viewed heat as an effect caused by an all-pervading, invisible fluid called caloric. An important tenet of the theory was that caloric was considered to be conserved in all thermal processes.

The fact that Carnot published no other work during his short life led later theoreticians, notably Rudolf Clausius and James Clerk Maxwell, into the error of assuming that Carnot never questioned the validity of the caloric theory. But after Carnot’s death in 1834, a bundle of his papers was found whose contents reveal that he had not only questioned the caloric hypothesis, but had reached the point where he felt compelled to abandon it in favour of its eventual successor, the dynamical theory.

This cannot have been an easy decision for Carnot, since the rejection of the caloric theory in favour of the dynamical theory robbed him of the very principle he had employed in Réflexions sur la Puissance Motrice du Feu to reach his groundbreaking conclusions regarding the motive power of heat.

In this post, I will examine the dilemma that Carnot faced in contemplating this conceptual shift.

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When Carnot began thinking about how steam engines turned heat into work, the theory of machines that produced work through mechanical force supplied by men or animals, the wind or a waterfall was already well understood.

In his memoir, Carnot begins by saying that “a similar theory is evidently needed for heat-engines”, and commences his analysis by noting that the production of motive force in steam engines is always accompanied by the transportation of caloric from a body at elevated temperature to another whose temperature is lower.

It should be noted that in the early 19th century when Carnot was writing his memoir, steam engines had such low thermal efficiency that any consumption of caloric in its passage between the hot and cold reservoirs would barely have been perceptible. So it was quite reasonable for Carnot to think that caloric was conserved by a steam engine in motion.

But how did a steam engine produce motive power? With the aim in mind of ‘a similar theory’ to the mechanical engine, Carnot likened the passage of caloric to the passage of water from a higher reservoir to a lower reservoir as it drives a water wheel – there is no loss of water, and motive power depends on the quantity of water transported and the height of the waterfall.

In an equivalent way, Carnot saw the motive power of a steam engine arising from the quantity of caloric transported, and the ‘height of its fall‘, by which he meant the difference in temperature between the hot and cold bodies. And as with the water wheel analogy, the process involved no loss of caloric.

With this model-based principle in place, Carnot was ready to start finding answers to important questions.

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The first question Carnot poses is one that had been long asked by engineers: Is the motive power of heat invariable or does it vary with the working substance employed to realize it?

To furnish an answer, Carnot conducts a thought experiment involving an imaginary steam engine undergoing a cyclical sequence of four entirely reversible isothermal and adiabatic operations, the hot and cold reservoirs being maintained at given temperatures – a stroke of conceptual genius which we know today as the Carnot cycle.

In this steam engine, Carnot imagines the transfer of a quantity of caloric from a body A to a colder body B, both maintained at a given temperature. Assuming no loss of motive power of caloric, he shows that the motive power produced in a single Carnot cycle can be used to return the same quantity of caloric from body B to body A by operating the cycle in the reverse direction. He then concludes: “An indefinite number of alternative operations of this sort could be carried on without in the end having either produced motive power or transferred caloric from one body to the other.”

He then puts the question “Can there exist a working substance that makes better use of the heat than the steam employed in the cycle just described?” and uses the following argumentation to supply the answer:

If it were possible by any method whatever to make the caloric produce a quantity of motive power greater than we have made it produce by our first series of operations, it would suffice to divert a portion of this power in order by the method just indicated to make the caloric of the body B return to the body A from the refrigerator to the furnace, and thus be ready to commence again an operation precisely similar to the former, and so on: this would be not only perpetual motion, but an unlimited creation of motive power without the consumption either of caloric or of any agent whatever. Such a creation is entirely contrary to ideas now accepted, to the laws of mechanics and of sound physics. It is inadmissible.

To put it in modern parlance, in the first series of operations using steam as the working substance, a quantity C of caloric is transported from body A to body B, producing a quantity M of motive power. Then operating the cycle in reverse, the same quantity M of motive power is used to transport the same quantity C of caloric from body B back to body A, thus restoring the initial conditions. The net result is that no caloric is transferred and no motive power produced.

In the second series of operations using a working substance that is imagined to be more effective than steam, an identical quantity C of caloric is transported from body A to body B, but this time it produces a quantity M+m of motive power. Then operating the previously described cycle (using steam) in reverse, the quantity M of motive power is used, as before, to transport the quantity C of caloric from body B back to body A, thus restoring the initial conditions. The net result is that no caloric is transferred, but a quantity m of motive power is produced – an operation which could be endlessly repeated, producing an unlimited quantity of motive power from nothing. This violates the principle and disproves the initial assertion. Carnot thus concludes: “the maximum of motive power resulting from the employment of steam is also the maximum of motive power realizable by any means whatever“.

Carnot has proved that the  motive power of heat is independent of the working substance employed to realize it.

He then asks a second question – whether the motive power of heat is unbounded or subject to an assignable limit – and goes on to prove by further thought experiment that its quantity is fixed solely by the temperatures of the bodies between which the transfer of caloric is effected.

The important point to note is that Carnot arrives at these powerful results by applying the principle that  caloric is a conserved quantity in thermal processes.

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But even as his memoir was going to print (he published it at his own expense), it appears that doubts about the caloric theory were already forming in Carnot’s mind. A sentence in the manuscript which describes the theory as “beyond doubt” was changed at proof stage into less certain language: “the theory … does not appear to be of unquestionable solidity. New experiments alone can decide the question.

The kind of experiments Carnot had in mind are revealed in the bundle of papers found after his death. The 23 loose sheets, containing questions, speculations, fragments of essays, and proposed experiments almost identical to those Joule was later to conduct, chart Carnot’s increasing belief that heat and work are equivalent. He lists Rumford’s experiments on the boring of cannon and the friction of wheels on their spindles among the experimental facts undermining the caloric theory.

One of the sheets contains the following paragraph, in which Carnot identifies heat and work as interconvertible forms of a conserved quantity, and effectively states the first law of thermodynamics in relation to cyclical thermodynamic processes (ΔU = 0, Q – W = 0) :
Heat is simply motive power, or rather motion which has changed its form. It is a movement among the particles of bodies. Wherever there is destruction of motive power, there is at the same time production of heat in quantity exactly proportional to the quantity of motive power destroyed. Reciprocally, wherever there is destruction of heat, there is production of motive power.

On one of the last sheets, Carnot writes:
When a hypothesis no longer suffices to explain phenomena, it should be abandoned. This is the case with the hypothesis which regards caloric as matter, as a subtle fluid.

But as Carnot becomes increasingly convinced that heat and motive power are interconvertible, he is at the same time caught in a dilemma. Because whereas the caloric theory enabled him to prove an assertion by disproving the contrary assertion, the dynamical theory fails to do so.

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As Carnot contemplated the interconvertability of heat and work, he would no doubt have re-run the reversible steam engine thought experiment on the basis of the dynamical theory:

In the first series of operations using steam as the working substance, a quantity H of heat is taken from body A, a quantity M of motive power is produced, and a quantity H-M of heat is transferred to body B. Then operating the cycle in reverse, the quantity H-M of heat is taken from body B, a quantity M of motive power is consumed, and a quantity H of heat is transferred to body A, thus restoring the initial conditions. The net result is that no heat is transferred and no motive power produced.

In the second series of operations using a working substance that is imagined to be more effective than steam, an identical quantity H of heat is taken from body A, but this time a quantity M+m of motive power is produced. Taking h to be the quantity of heat equivalent to the quantity m of motive power, a quantity H-M-h of heat is transferred to body B. Then operating the previously described cycle (using steam) in reverse, the quantity H-M of heat is taken as before from body B, a quantity M of motive power is consumed, and a quantity H of heat is transferred to body A, thus restoring the initial conditions.

The net result is that a quantity h of heat has been taken from the colder body B and a quantity m of motive power has been produced, an operation which could be endlessly repeated. But since we have defined the equivalence relation h=m, there is no violation of principle and the dynamical theory therefore fails to disprove the absurd result of the limitless production of motive power solely by consuming the heat of a body.

This is the core of the dilemma Carnot faced.
He doubted the caloric theory, but it proved his assertions.
He favoured the dynamical theory, but it did not prove his assertions.

One of the loose sheets found after Carnot’s death reveals this difficulty in a passage where he convinces himself that heat can be converted into motive power and vice versa. But while he sees many advantages in this hypothesis, he notes “it would be difficult to explain why, in the development of motive power by heat, a cold body is necessary; why motion cannot be produced by consuming the heat of a body.

The explanation, as we now know, is provided by the second law of thermodynamics. Although Carnot had no time in his short life to grapple with this dilemma, it is not fanciful to suggest that he would have found the solution had he lived. More than twenty years before Clausius, Sadi Carnot had already effectively stated the first law, and as the last quote shows, he only needed to turn his question into an assertion to find the essential statement of the second law.

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I have a digital weather station with a wireless outdoor sensor. In the photo, the top right quadrant of the display shows temperature and relative humidity for outdoors (6.2°C/94%) and indoors (21.6°C/55%).

I find this indoor-outdoor thing fascinating for some reason and revel in looking at the numbers. But when I do, I always end up asking myself if the air outside has more or less water vapor in it than the air inside. Simple question, which is more than can be said for the answer. Using the ideal gas law, the calculation of absolute humidity from temperature and relative humidity requires an added algorithm that generates saturated vapor pressure as a function of temperature, which complicates things a bit.

Formula for calculating absolute humidity

In the formula below, temperature (T) is expressed in degrees Celsius, relative humidity (rh) is expressed in %, and e is the base of natural logarithms 2.71828 [raised to the power of the contents of the square brackets]:

Absolute Humidity (grams/m3) = 6.112 x e^[(17.67 x T)/(T+243.5)] x rh x 18.02
                                                                            (273.15+T) x 100 x 0.08314

which simplifies to

Absolute Humidity (grams/m3) = 6.112 x e^[(17.67 x T)/(T+243.5)] x rh x 2.1674

This formula is accurate to within 0.1% over the temperature range –30°C to +35°C

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Additional notes for students

Strategy for computing absolute humidity, defined as density in g/m^3 of water vapor, from temperature (T) and relative humidity (rh):

1. Water vapor is a gas whose behavior approximates that of an ideal gas at normally encountered atmospheric temperatures.

2. We can apply the ideal gas equation PV = nRT. The gas constant R and the variables T and V are known in this case (T is measured, V = 1 m3), but we need to calculate P before we can solve for n.

3. To obtain a value for P, we can use the following variant[REF, eq.10] of the Magnus-Tetens formula which generates saturated vapor pressure Psat (hectopascals) as a function of temperature T (Celsius):

Psat = 6.112 x e^[(17.67 x T)/(T+243.5)]

4. Psat is the pressure when the relative humidity is 100%. To compute the pressure P for any value of relative humidity expressed in %, we multiply the expression for Psat by the factor (rh/100):

P = 6.112 x e^[(17.67 x T)/(T+243.5)] x (rh/100)

5. We now know P, V, R, T and can solve for n, which is the amount of water vapor in moles. This value is then multiplied by 18.02 – the molecular weight of water ­– to give the answer in grams.

6. Summary:
The formula for absolute humidity is derived from the ideal gas equation. It gives a statement of n solely in terms of the variables temperature (T)  and relative humidity (rh). Pressure is computed as a function of both these variables; the volume is specified (1 m3) and the gas constant R is known.

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Author comment: Weather station manufacturers should consider a toggled RH/AH display


January 2017: Relative humidity (RH) is well understood by people trained in atmospheric science, but for ordinary folk who have an indoor-outdoor thermometer-hygrometer like the device pictured above, RH values can easily be misunderstood.

Consider the data illustrated. I would not blame anyone for thinking there is more water vapor in the air outside, since the RH outside (58%) is more than two-and-a-half times the RH inside (21%).

But the data is deceptive. Plug the numbers into my formula, which computes absolute humidity (AH) as water vapor density in g/m^3, and you will find that the AH inside is twice the AH outside. Could you have predicted that result from looking at the T and RH numbers? Nor could I.

Which is why I think that adding AH as an extra feature could be something for manufacturers of these devices to consider. People understand concepts like mass per unit volume much more easily than percentages of variable temperature-dependent maxima.

For manufacturers, adding AH to the display is a no-brainer. My formula computes AH directly from T and RH, so the RH displays could simply be toggled to show AH. Maybe visitors to this blogpost who are actively involved in microprocessor projects could consider pioneering a prototype?

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Igor uses my formula to keep his cellar dry


October 2016: I am impressed by this basement humidity control system developed by Igor and reported on forum.

Inside the short pipe is a fan equipped with a 3D-printed circumferential seal. The fan replaces basement air with outdoor air, and is activated when absolute humidity in the cellar is 0.5 g/m^3 higher than in the street, subject to the condition that the temperature of the outdoor air is lower. This ensures that water in the cellar walls is drawn into the vapor phase and pumped out; the reverse process cannot occur.


Full details (text in Russian) with lots of excellent photos → here

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Formula powers AH measurements from high-precision RH&T sensor


The SHT75 RH&T sensor from SENSIRION

April 2016: Prof. Antonietta Frani has made a miniature device for measuring absolute humidity, using my formula to power an Arduino Uno microcontroller board equipped with an SHT75 RH&T sensor which connects to a computer via a USB cable. Systems Integrator Roberto Valgolio has developed an interface to transfer the data to Excel spreadsheets with their associated graphical display functions.

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Formula powers online RH←→AH calculator


March 2016: German website is using my formula to power an online RH/AH conversion calculator.

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Formula cited in academic research paper


January 2016: A research article in Landscape Ecology (October 2015) exploring microclimatic patterns in urban environments across the United States has used my formula to compute absolute humidity from temperature and relative humidity data.

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Formula finds use in humidity control unit

August 2015: Open source software/hardware project Arduino is using my absolute humidity formula in a microcontroller designed to control humidity in basements:


“The whole idea is to measure the temperature and relative humidity in the basement and on the street, on the basis of temperature and relative humidity to calculate the absolute humidity and make a decision on the inclusion of the exhaust fan in the basement. The theory for the calculation is set forth here –” [на русском → здесь]

More photos on this link (text in Russian):

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AH computation procedure applied in calibration of NASA weather satellite

June 2015: My general procedure for computing AH from RH and T has been applied in the absolute calibration of NASA’s Cyclone Global Navigation Satellite System (CYGNSS), specifically in relation to the RH data provided by Climate Forecast System Reanalysis (CFSR). The only change to my formula is that Psat is calculated using the August-Roche-Magnus expression rather than the Bolton expression.

The CYGNSS system, comprising a network of eight satellites, is designed to improve hurricane intensity forecasts and was launched on 15 December 2016.

Reference: (Technical report “An Antenna Temperature Model for CYGNSS” June 2015)

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Formula cited in draft paper on air quality monitoring

May 2015: Metal oxide (MO) sensors are used for the measurement of air pollutants including nitrogen dioxide, carbon monoxide and ozone. A draft paper concerning the Air Quality Egg (AQE) which cites my formula in relation to MO sensors can be seen on this link:


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Formula used by US Department of Energy in Radiological Risk Assessment

June 2014: In its report on disused uranium mines, Legacy Management at DoE used my formula for computing absolute humidity as one of the meteorological parameters involved in modeling radiological risk assessment.

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