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We now move on to one of the two main underpinning areas of science - a quick introduction tobasic thermodynamics of systems and fluids.

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The First Law and Second laws of thermodynamics will be introduced for systems, and I willthen look at some of the classical results that follow from the First and Second laws.

Then I shall take the thermodynamics of fluids a little further and introduce the role of somekey functions and their differentials.

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The First lay of thermodynamics is otherwise known as the principle of conservation of energy.

It is usually stated in the following way. There is a property of a system, called the internalenergy, U, such that the change of the internal energy is equal to the sum of the work input tothe system and the heat input to the system.

By a property of the system we mean a function of certain measurable characteristics of thesystem – such as for instance the specific volume and the temperature.

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The Second law is considerably more subtle than the First. It can be stated in a variety ofdifferent ways. The most useful way though is probably to introduce a second property of thesystem, the entropy, S, and then state the Second law in the form of the Clausius-Duheminequality.

Thus states that the increase of entropy of the system is always greater that the entropy flux intothe system, where the entropy flux is the heat flow divided by the temperature.

The first and most obvious result is that an unchanged system exchanging heat with just onesource or sink can convert work into heat but not heat into work.

This is because, if the system is unchanger S_dot is zero. In the left hand diagram Q_dot / thetais negative, which is allowable, but in the right hand case it is positive, which is not allowable bythe Second law.

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Let us explore some other consequences of the Second law.

We consider another unchanged system, this time exchanging just heat with two reservoirs atdifferent temperatures.

We write the Second law in terms of the Clausius-Duhem inequality, and note that S_dot is zerofor the unchanged body.

The First law tells us that Qdot_2 = -Qdot_1, so we can re-arrange the equation.

And then if we choose Qdot_1 to be the positive quantity, and note that the temperature isalways positive, then we can cross-multiply and divide through by Qdot_1.

… and get the result that theta_1 must be greater than theta_2.

In other words “heat flows from a hotter body to a cooler body”.

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Now let us consider another unchanged system, which we shall call a heat engine. – it runs onheat and delivers work. We have already found out that this is impossible if the engineexchanges heat with only one reservoir, but what if it exchanges heat with two. What we shallexplore is what is the maximum possible efficiency of the heat engine, where the efficiency isthe amount of work delivered divided by the amount of heat supplied from one of thereservoirs.

For an unchanged bot we can write -W_dot = Q_dot, and it is convenient to divide through byQ_1 to get the efficiency –W_dot/Q_dot. (Note that the work output is minus W_dot).

As the system is unchanged S_dot is also zero, and we can write the Clausius-Duhem inequality,and then rearrange it in the form shown.

We can now combine the two laws, and with a little rearrangement we see that the efficiencycannot exceed the difference in temperature between the two heat reservoirs divided by thetemperature of the hotter one (which is the one that supplies the heat, Qdot_1 is positive,Qdot_2 is negative).

So to make an efficient engine we want as hot as possible a heat supply, and as cold as possibleheat sink to dump the rejected heat to.

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We shall now explore some special systems: reversible systems which can operate in eitherdirection.

When we reverse the heat flow, the sign of Qdot_i changes, as does the sign of Sdot, but the signof theta_i does not, so by simple inspection we can see that the only way we can satisfy theinequality in this case is if the equality holds.

The entropy change is simply equal to the entropy flux into the system (remembering that heatflux divided by temperature is entropy flux)

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Now let us look at the thermodynamics of a fluid treated as a continuum.

We restate the First law as follows.

The heat supply per unit mass plus the work input per unit mass is equal to the change ofinternal energy per unit mass. We call quantities “per unit mass” ,“ specific”, so the specificvolume is the volume per unit mass, i.e. 1 / density.

Working from the right hand side, udot is just the change of specific internal energy; the workper unit mass is just the pressure multiplied by the change of volumetric per unit mass. Theminus sign arises because pressure is compressive but change of volume is positive forexpansion.

Unfortunately it is easier to write the heat supply per unit volume, so to understand the firstterm rewrite the first equation by dividing through by the specific volume. It is easy to show thatthe heat supply per unit volume is the (negative) divergence of the heat flow vector – you shouldsatisfy yourselves about this by considering a small element, say in 2D.

In a similar way we can write the Second law per unit volume. s_dot / v is greater to or equalthan the divergence of the entropy flux (again with a negative sign, just as for the heat flow).

For a reversible process the equality sign holds, and we can expand the spatial differential.Again, we can observe that if we reverse the process s_dot changes sign, as does the first termon the right hand side, but not the second. We can deduce that the second term must be zerofor a reversible process – in other words a process is only reversible if the heat flux (and thermalgradient) are negligible.

Finally we can put the first and second laws together to eliminate the divergence of the heatflow , and arrive at the final equation, which is called the Gibbs equation.

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We now make a bold assumption, which is central to the Gibbsian approach to thermodynamics.

The internal energy is a function of state, and we choose to define the state in terms of thespecific volume and the entropy, so u = u(v, s).

If we differentiate we get the expression for u_dot.

… and subtracting from the Gibbs equation for the reversible material we get the fourthequation.

Now changes of volume and changes of entropy may be treated as independent processes, sothat this equation can only be true if both the brackets are themselves equal to zero.

… So we can identify p = (minus) du/dv and theta = du/ds.

This is an incredibly powerful statement. It means that if we know u as a function of v and s wecan derive p and theta also as functions of v and s – in other words we can deduce theconstitutive response!

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So how might this work in practice?

Consider a very simple function for u – a quadratic function of v and s

On differentiation the u_0 term just disappears – it represents the datum point for u

The differentials give us the pressure and temperature, and note that in fact the linear terms(u_1 and u_2) just set the datum point for pressure and temperature

Differentiate again, and write the resulting equation in matrix form to make the structure easier.

We can see that the u_3 term relates pressure to volume – in other words the volumetricstiffness

The u_5 term relates temperature change to entropy, and remember that the entropy increaseis simply related to the heat input, so the u_5 term represents the heat capacity of the fluid

And the u_4 term represents the coupling between temperature and volume change and alsobetween pressure and heat supply – in other words thermal expansion effects. The presence ofthe u_4 term in both places demonstrates that you can’t have one type of coupling without theother.

Another way of looking at this is that if we look at the cross term in the second derivative, wecan see that there is a relationship between differentials of temperature and pressure – this isone of the so-called Maxwell relations

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It turns out that there are different possible choices for the fundamental energy expression thatwe specify.

The internal energy is taken as a function of specific volume and entropy

… or a quantity that is called the Helmholtz free energy is a function of specific volume andtemperature, which some people find easier to understand. We can then derive the pressureand entropy by differentiation.

Further alternatives are the enthalpy and the Gibbs free energy

Importantly, each of these quantities is not independent, they are related by some simpleexpressions – once we know any one of them we can find any of the others.

… but I am effectively skipping ahead a little, because these relationships are what are calledLegendre Transforms, and we shall need to learn more about the maths of the transforms torealise why the expressions are written this way. This will come a little later, and LegendreTransforms will play an extremely important role in hyperplasticity.

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So in summary:

I have introduced the first and second laws of thermodynamics. The first law (conservation ofenergy) is best understood by introducing the property called internal energy, and the secondlaw, which is a little more subtle, is best expressed by the Clausius Duhem inequality andintroduction of the property called entropy.

The laws of thermodynamics are amongst the very few scientific laws that are almost universallyregarded as “true”, and not merely working hypotheses that might be disproven. Eddington putit rather eloquently.

We have then considered some simple thermodynamic systems and observed someconsequences – heat flows from hotter to cooler, a heat engine has a maximum efficiency whichit cannot exceed.

Finally we have reformulated the laws in terms of a fluid continuum, and seen that for reversiblesystems we can express the whole constitutive response through knowledge of the free energyfunction, and I have hinted at the idea that there are a series of alternative forms which usedifferent energy functions – we shall return to those later.