Exergy Balance: Closed Systems

By Jai Schrem

Consider a simple compressible system inside this piston-cylinder device which is connected to a heat engine:

We will assume that we don't need to consider any energy due to mixing of substances or chemical reactions. Therefore, the only internal energy of the system under consideration are its sensible and latent energies - both of which are output as heat. The derivation for the exergy balance is based on considering how much of this heat can be converted into useful work (in other words exergy).

Any heat and/or work input to the system will be taken up by the system as internal energy. Taking heat and energy output by the system to be positive:

-dQ - dW = dU

Considering the Work Term

The only type of reversible work that can be done on a simple compressible system is boundary work (aka moving the piston), so 

dW = PdV. (NOTE: use absolute pressure for P). This can also be rewritten as:

dW = (P - Po)dV + PodV

The (P-Po)dV term represents the useful boundary work delivered by the piston. PodV represents energy lost due to changing the volume of the surroundings, with this taking place at a consistent ambient pressure.

Considering the Heat Term

To transfer heat reversibly between the system (at temperature T) and the surroundings (at temperature To) we use a reversible heat engine. We can then perform the following derivation:

Total Useful Work

The total useful work, in other words the exergy, comes from the useful parts of the work and heat terms:

dX = (P - Po)dV + dQ + TodS

Note that positive dQ denotes heat leaving the system. Due to the system storing heat as internal energy, this corresponds to a negative dU:

dX = (P - Po)dV - dU + TodS

Integrating from the system state (denoted by no subscript) to the dead/surroundings state (denoted by a 0 subscript) we obtain:

X = (P - Po)(Vo - V) - (Uo - U) + To(So - S)

Generalisation

More broadly, a closed system may possess kinetic and/or gravitational potential energies as well as its internal energy. Both these forms of energy can be fully converted into exergy, unlike heat. This results in the following expression for the exergy of a closed system:

X = (P - Po)(Vo - V) - (Uo - U) + To(So - S) + EK + EGP

The exergy change of a system during a process is equal to the difference between the net exergy transfer through the system boundary and the exergy destroyed within the system boundaries as a result of irreversibilities: