the journal of enginering

International Journal of Mechanical Engineering Education 39/1

Entropy pollution of the environment: a teaching approach to the Second LawJeffery LewinsCambridge University Engineering Department, Magdalene College, Cambridge CB3 0AG, UKE-mail: [email protected]

Abstract To introduce the concept of entropy to students, we propose building on what is currently a well aired concept, that of thermal pollution of the environment. This is then clarifi ed, refi ned and made quantitative by the addition of the quality of heat, the temperature. The student will be familiar with the idea that temperature differences drive heat fl ows. The new concept is that the individual temperatures are signifi cant, not just their difference. A high temperature drives a heat fl ow to a low temperature and in so doing can provide something useful, like work. We pollute our environment if we dissipate this opportunity. This approach leads naturally to the Second Law, to Carnot effi ciency and the Kelvin scale.

Keywords temperature scales; ideal Carnot engine; thermal effi ciency; ideal perfect gas thermometer

Fear no more the heat o’ the sunNor the furious winter’s rages(Cymbeline, W. Shakespeare)

Introduction

The concept of entropy is notoriously diffi cult to teach and, in my experience, fi rst-year engineers are not well prepared for this step. They need to have mastered the idea of work as the necessary effort to move against a resisting force. They should have studied heat transfer to calculate heat fl ows, analytically and perhaps with simple fi nite difference methods to establish the concept of a temperature fi eld. They should have made measurements of temperature with various devices and used practical scales of temperature. They will then be ready for the First Law in the form that heat and work are manifestations of the transfer of energy and that work can be wasted into heat at a constant, universal ratio1 that allows us to measure power fl ows, mechanical and thermal, in common units of watts. They will also need to be happy that heat fl ows naturally from ‘hot’ to ‘cold’ on any sensible scale of temperature; going the other way consumes work. But like us all, they are probably handicapped by the imprecise vocabulary that we inherit from the peculiar history of heat; a calorimeter measures heat quantities, a thermal insulator stops heat fl ow but a ther-mometer measures temperature. And why are there so many temperature scales that broadcasters have to use both Fahrenheit and Celsius?

1 A certain amount of work will always give you the same amount of heat, as Joule demonstrated in Manchester, allowing us to measure energy in joules (J) and power in watts (W), whether heat or work.

Entropy pollution 61


Students today will also be familiar with another concept, the thermal pollution of our environment. This is a good starting point to introduce the Second Law, the quantitative limit on turning heat fl ows to work fl ows.

Temperature as the quality of heat

I put forward therefore (Fig. 1) that it is not just the quantity of heat fl ows put in and taken out of the environment that matter, but also their quality. Temperature provides just such a quality of heat because a hot temperature has more capability than a cold temperature: it moves heat naturally from hot to cold. What we need is a temperature, θ, measured by some ideal thermometer on a numerical scale that has this desirable property. We use the symbol θ at this stage for temperature until we have made our choice.

Unlike the scales of Celsius, Centigrade, Fahrenheit, etc., the numbers should be always positive so we can multiply and divide by this quality and not change the direction of heat fl ows, destroying our sign convention for heat in and heat out. Just adding a large, positive number to these ‘practical’ scales is inadequate; how do we know it is big enough not to go negative at some cold temperature? But at least there is wide agreement that ‘hot’ should be represented by a larger, higher number than ‘cold’.2

The ideal thermal engine

Temperature as the quality of heat has more signifi cance than simply dictating the direction of natural heat fl ow. It must be a good thing in terms of pollution to take heat out of the atmosphere at a rate Q′

1 and at a temperature or quality θ1 and do something useful with it. Ultimately we may have to return heat at a temperature or quality θ2 and at a rate Q′

2 W. If we have done nothing with the heat fl ow while it is in our care then the heat rates are the same: Q′

2 = −Q′1, with a sign convention for

Q ′1

Q ′2

T1

T2

Fig. 1 Heat exchange with the environment.

2 Another of the historical quirks is that Celsius originally proposed 100 for the freezing point of water and zero for its boiling point. That one did not survive.

62 J. Lewins


in and out. Indeed, this is just what would happen in the environment if left to itself, with heat fl owing from the hot region to the cold.

But what we should have done is to turn some of that heat into useful work while it was in our care and thus decrease the heat rate returning to the environment and decrease the thermal pollution. We can represent this by introducing an entropy fl ow rate, S′ = Q′/θ W/?, where the ‘?’ in the units indicates that we have not yet defi ned the unit of temperature on our scale. So let us give the unit a name: the kelvin, symbol K, after William Thomson, Baron Kelvin, a great man in the history of thermodynamics, and defi ne it later.

Entropy as we have defi ned it, heat divided by its quality, is environmental pol-lution. If we do nothing with this heat (or leave it to degenerate in the environment)

then this pollution increases as ∆ ′ = −

′ = −

′ >S Q Q1 1

02 1

11 2

1 21θ θ

θ θθ θ

, necessarily

positive, an additional entropy pollution. We should have used the opportunity to turn some of the heat into work and thus reduce the rate of pollution we have gen-erated; the entropy generation rate is S′

gen = Q′2/θ2 − Q′

1/θ1. When this generation is weighted by the temperature at which the entropy is released to the environment, we have the dissipative power, the rate of thermal pollution that should have been turned to mechanical power, θ2S′

gen. So how shall we measure temperature as the quality of heat?

Temperature, scales and thermometers

What do we know about temperatures, before choosing a thermometer and a scale? This is the province of the so-called Zeroth Law. We are all familiar with the heat of a summer’s day and the cold of a winter’s night, the hot stove and the cold refrig-erator. ‘Hot’ and ‘cold’ are different temperatures. When hot and cold bodies are brought into contact, things tend to change; the hot body typically contracts and the cold body expands. Heat, a form of energy in transit, passes from hot to cold. Of course we can delay or even stop this by preventing the thermal contact with a thermal insulator. But supposing the contact remains good, and we do not supply heat to the hot body; the heat transfer continues, but increasingly slowly, until it stops and the two bodies are in thermal equilibrium. Then they have something in common: the same temperature.

Thermal expansion or a similar change with temperature can be the basis of a thermometer. For a thermometer to be useful, we need to be assured that when it reads the same temperature in two separate bodies, these two bodies indeed have the same temperature and would be in thermal equilibrium if brought into direct thermal contact. The Zeroth Law states therefore that if system A is in thermal equilibrium with system B and likewise system B with system C, then indeed C is in thermal equilibrium with A. All three have the same temperature as measured by a thermometer made from any one of the systems.3

3 This may seem obvious. Ask students for counter-examples in other fi elds such as ‘twinning’ towns or sports teams with different styles of play.



To establish a numerical scale of temperature, we must agree the value or number to be associated at two or more reproducible fi xed temperature points; the chosen thermometer then interpolates between these values. Popular practical choices are the freezing point of water and its boiling point. On the centigrade scale these are assigned the values 0 and 100 respectively.4 Now, some thermometer is chosen, such as mercury in glass for length or a thermocouple for voltage, and other temperatures are measured by linear interpolation or even extrapolation. There is no reason to expect, though, that ‘50’ on different thermometers should indicate the same tem-perature precisely.5 We need something better.

The ideal thermal cyclic engine of Carnot

Sadi de Carnot, a railway engineer concerned with the performance of steam engines, had the key to the problem. Instead of looking at the detail of the combustion process consuming coal (thermodynamics can do that later), he idealised the situation to a thermal engine that takes in heat at a single, constant temperature from a reservoir and similarly rejects heat.6 Model the engine as working in a closed cycle so that in steady operation everything returns cyclically to the original state – only heat has been exchanged with the environment. This is obviously valid for steam engines, where the steam is condensed and returned to the boiler, the Rankine cycle. Non-cyclic engines have to be idealised and modelled as, say, Otto or Diesel cycles, exchanging only heat with the environment.

This is the cyclic thermal engine that exchanges heat and does work in the form equivalent to raising a weight against gravity. Now for the ‘ideal’ thermal engine.

We know that an engine can be run backwards, taking in work and transferring heat from a cold to a hot reservoir: a refrigerator. Carnot’s great insight was to see that the ideal engine would be fully reversible. If all the original work was taken back, the engine left unchanged at the end of a cycle and all the heat restored, there would be no trace or evidence that the forward–backward process had ever hap-pened. Nothing would have changed overall and the engine would not just run backwards but would be fully reversible.

Three signifi cant things follow:

(1) All reversible thermal cyclic engines working between the same two tempera-tures have the same effi ciency, the fraction of heat taken and turned into work, ηrevc = 1 − Q1(θ1)/Q2(θ2), and therefore, what is more useful, the same ratio of heat taken in from the environment to the heat returned, Q1(θ1)/Q2(θ2)revc = const, dependent only on the temperatures.

4 These defi nitions are not precise because these ‘fi xed’ points vary with pressure. Skaters know that the pressure of the blade melts the ice beneath and mountaineers know that tea making is diffi cult because the lower pressure lowers the boiling point of water.5 The student could be asked to look at the detailed and different correction tables for different ther-mocouple pairs.6 For more than two reservoirs and temperatures, model a compound thermal engine from two-reservoir units.

64 J. Lewins


(2) No irreversible engine has a better effi ciency than a reversible engine operating between the same temperatures.

(3) There is an absolute zero temperature.

These claims are justifi ed by supposing we could indeed construct a device which contradicted the claim and then testing the claim against some common-sense facts that make up the Second Law of thermodynamics.7 We appeal to experience and common sense. Engineers might well regard the Second Law as a statement of the obvious; you do not get something useful for nothing.8

To prove the fi rst corollary, suppose there was indeed a more effi cient reversible engine; it produces more work from the same heat input and returns less heat. So use another reversible engine to drive this ‘worse’ engine backward, restoring heat to the hot reservoir using all the work from the ‘better’ engine. But by assumption, you are pumping more heat from the lower reservoir than it is receiving. Congratu-lations, you have invented a work-free refrigerator, allowing heat to fl ow naturally to higher temperature!

To prove the second, suppose there was a better irreversible engine. So drive a reversible engine backward needing less work to balance out the heat into the cold reservoir. Congratulations, you have net work left over simply taking heat from one reservoir and not returning any. You would become rich taking work from any constant-temperature ocean.

For the third corollary, surely we cannot have a thermal effi ciency greater than unity, taking heat from two reservoirs and producing only work? So the heat ratio cannot change sign.

The way forward is now simple: use the Carnot ratio in an ideal reversible thermal engine(the restriction to reversible is essential) as our thermometer and write:9

T

T

Q

Qrev

0 0

= − (1)

(The negative sign comes from our sign convention for the direction of heat fl ow.) We use the symbol T for such an absolute temperature and use t on a practical scale not having an absolute zero.

One point on our scale is the absolute zero, allowing us to call this the absolute scale. Here, T0 is a further single fi xed calibration number we agree to give to a convenient reproducible system. The internationally agreed Kelvin scale of tem-perature agrees that the triple point of pure water, where ice, liquid and steam coexist in equilibrium, shall be at 273.15 kelvin (K). This unlikely number was chosen in

7 The Second Law has many versions, all coming to the same thing: you cannot get something for nothing in the real world. It is like the various reports of a blindfolded committee on examining the fi rst elephant; one feels a leg, one a tail, one a trunk, etc. All report aspects of the same beast.8 Einstein described common sense as the layer of prejudice laid down by the age of 18. Appropriate perhaps in quantum physics but not, I suggest, in engineering.9 The student might reasonably ask, after the emphasis given to the Carnot effi ciency, why this does not appear in the defi nition of temperature. We use the reversible heat ratio because it is unique. We would have the same Carnot effi ciency from two different temperatures satisfying T1/T0 = T0/T2.



order that the gap between the freezing point and the boiling point of water might remain about 100 K and hence follow a familiar Celsius scale close to the centigrade scale, but where temperatures below freezing are negative and therefore useless for our present purposes. For the purist, the unit of temperature on the Celsius scale is the kelvin; there is no ‘degree’, even though we have to write the symbol as °C to avoid confusion with the coulomb.10

Since the reversible heat ratio cannot go negative, there is no temperature below zero on our scale. It is therefore called an absolute scale, since it has an absolute zero.11 The other agreed scaling point is arbitrary and not absolute, so on this scale we are not like length, needing to scratch marks to defi ne the unit of distance, but like mass, where we cannot conceive of negative mass but fi x the kilogram arbitrarily as the mass of a lump of matter kept at Sevres. The analogy is not an accidental coincidence. Following Einstein and special relativity, energy and mass are con-nected as E = mc2, so that energy, too, cannot be negative and has an absolute zero, needing only a single calibration point. Boltzmann studied the distribution of kinetic energy in simple molecules. In equilibrium, there is a distribution of speeds and energies, an average as the molecules collide with each other. In this view, the tem-perature on the absolute scale is simply an average energy per molecule at the

equilibrium temperature, 3

2kT , where k is Boltzmann’s constant, connected to the

universal molar gas constant, R, by Avogadro’s constant, N0, so that R = kN0.This identifi cation with the ideal gas laws brings a very welcome practicality to

our theoretical, ideal absolute scale of temperature. In the range of ideal gases, the product of absolute (not gauge) pressure and volume are proportional to the absolute temperature: PV = RT. Constant-pressure or constant-volume thermometers are therefore measuring temperature on the ideal Carnot scale.12

Entropy pollution of the world

We can now denote the rate of entropy pollution of the environment where heat is exchanged at different temperatures. The rate of entropy generation is:

10 There is a conceptual problem that cannot easily be resolved about the meaning of ‘temperature’ in a body not in thermal equilibrium. Our thermal reservoirs are large and separated, large enough for their temperature to remain unchanged during the process, an idealization allowing us to treat them as in thermal equilibrium and therefore at a precise temperature. But what is ‘temperature’ along a bar, say, held hot at one end and cold at the other? The bar may be at steady state but not in thermal equilibrium. The pragmatic answer is that if the active region of the thermometer is small enough we can suppose the temperature of the bar does not vary (much) and we accept a local average. But there are circumstances in quantum physics, say, where an intense non-equilibrium gradient makes identifi cation diffi cult. Most engineers accept the pragmatic view; to adapt the well known curse, may the instructor be blessed with more inquisitive students!11 This is where we have, as teachers, to tell a ‘white lie’; there are negative temperatures but they are hotter than infi nity, not colder than zero.12 A further white lie: although the simple ‘point’ molecules come to rest with zero kinetic energy and zero temperature, their electrons are governed by Fermi–Dirac statistics and retain a zero-point energy at zero temperature.

66 J. Lewins


∆SQ

T

Q

Tgen = − ≥1

1

2

2

0 (2)

with temperatures measured on our absolute scale (equation 1). Since the reversible heat ratio (Q1/Q2)rev is greater than for any real, irreversible process, the rate of generation of entropy is positive except for the limiting, reversible process, when it is zero. Thus entropy is a conserved property only in reversible processes, in contrast to energy, which is conserved in all processes if we include relativity.

The dissipation is given by:

dissipation T Sgen= ∆ (3)

If we use T1 in this expression, we have the work (or mechanical power) that should have been drawn from the hot reservoir. If we use T2 we have the amount of energy returned to the cold reservoir as heat that should have been taken as work.

These results led to Clausius putting forward yet another view of the Second Law: the energy of the universe is constant but its entropy is increasing. Those cosmolo-gists who favour continuous creation would disagree, but consider how we might interpret this at the world level. If our world, the Earth, were isolated, the dictum would apply on the terrestrial scale: taking heat out of the environment and return-ing it after utilizing, as far as possible, the temperature difference. But the Earth is not isolated: it receives a continuous supply of solar radiation, characterised by the Sun’s surface temperature, some 4 kK and re-radiates to space at some 300 K. The balance is complicated by the heat coming from the hot magma core and the radio-active decay of certain minerals. Even in steady state, therefore, we are contributing to the entropy pollution of the universe.

Logic then suggests that to reduce entropy pollution we should, as engineers, seek to extract work from sunlight, either directly or by taming the results of the great solar engine in the form of wave power and wind power, or, above all, perhaps, in the formation of biomass, as a store of entropy.

Conclusion

I hope that this approach to the Second Law in the context of environmental pollu-tion may be of interest to colleagues. Students often regard the whole concept of the ideal, reversible thermal engine as a highly impractical theoretical artifi ce.13 But we actually have a highly practical limit to the performance of real engines telling us how far away we are from our goal, the limit of ideal performance and whether, then, it is worthwhile to attempt improvements to our designs. This I suggest is the real practical value of studying thermodynamics.

13 Is it not of practical use that the limit 12

14

18 1+ + + =... even if we cannot carry out all the steps?



Sources

[1] J. D. Lewins, Engineering Thermodynamics: Frontiers and Foundations, (OECD Nuclear Energy Agency, Paris, 2009).

14

[2] J. D. Lewins, Teaching Thermodynamics (Plenum, London, 1985).[3] And see www.teachingthermodynamics.co.uk.

14 For a free copy on CD-ROM email [email protected] specifying Heritage Books, Lewins, Thermodynamics.

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