Page 78 - Advanced thermodynamics for engineers

4.2 AVAILABILITY 63

to follow the derivation using the specific case, but a more general result is obtained from the
arrangement shown in Fig. 4.2(a).
First consider that System A is a constant volume system which transfers heat with the surroundings
via a small reversible heat engine. Applying the First Law to the System A

dU ¼ dQ dW ¼ dQ dW s ; (4.5)
where dW s indicates the shaft work done (e.g. System A could contain a turbine).
Let System B be System A plus the heat engine, E R . Then applying the First Law to System B gives

dU ¼ SðdQ dWÞ¼ dQ 0 ðdW s þ dW R Þ (4.6)
As System A transfers energy with the surroundings, it undergoes a change of entropy defined by
dQ dQ 0
dS ¼ ¼ ; (4.7)
T T 0
because the heat engine transferring the heat to the surroundings is reversible and there is no change of
entropy across it.

Hence ðdW s þ dW R Þ¼ ðdU T 0 dSÞ (4.8)
As stated previously, (dW s þ dW R ) is the maximum work that can be obtained from a constant volume,
closed system when interacting with the surroundings. If the volume of the system was allowed to
change, as would have to happen in the case depicted in Fig. 4.2(b), then the work done against the
surroundings would be p 0 dV where p 0 is the pressure of the surroundings. This work, done against the
surroundings, reduces the maximum useful work from the system in which a change of volume takes
place to dW þ dW R p 0 dV, where dW is the sum of the shaft work and the displacement work.
Hence, the maximum useful work which can be achieved from a closed system is

dW þ dW R ¼ dU þ p 0 dV T 0 dS (4.9)
This work is given the symbol, dA. Since the surroundings are at fixed pressure and temperature
(i.e. p 0 and T 0 are constant) dA can be integrated to give

A ¼ U þ p 0 V T 0 S (4.10)
A is called the non-flow availability function. Although it is a combination of properties, A is not
itself a property because it is defined in relation to the arbitrary datum values of p 0 and T 0 . Hence it is
not possible to tabulate values of Awithout defining both these datum levels. The datum levels are what
differentiate A from Gibbs energy, G. Hence the maximum useful work achievable from a system
changing state from 1 to 2 is given by
W max ¼ DA ¼ ðA 2 A 1 Þ¼ A 1 A 2 (4.11)
The specific availability, a, i.e. the availability per unit mass is

a ¼ u þ p 0 v T 0 s (4.12a)
If the value of a were based on unit amount of substance (i.e. kmol) it would be referred to as the
molar availability.