Available energy from a finite energy source

Consider hot gas of mass m_g at temperature T when the temperature of environment is T₀ (Fig 1.25). Let the gas be cooled at constant pressure to T₀ from state 1 and the heat given up by the gas Q₁ is utilized in heating up reversibly a working fluid of mass m_wf from state 3 to 1 along the same path so that the temperature difference between the gas and working fluid at any instant is zero and hence entropy increase of the universe is also zero.

The working fluid expands reversibly and adiabatically in an engine or turbine from state 1 to 2 doing work W_E and then rejects heat Q₂ reversibly and isothermally to return to the initial state 3 to complete the heat engine cycle.

Here,

Q1 = mgCpg(T - T0) = mwf Cpwf (T - T0) = Area 1-4-5-3-1

1.207

Fig. 1.25 Available and unavailable energy in finite processes

∴ mg Cpg = mwf Cpwf

1.208

Now

( negative as T0 < T)

1.209

1.210

∴ Δsuniv = Δsgas + Δswf = 0

1.211

1.212

∴ Available energy

1.213

Hence, exergy of a gas of mass m_g at temperature T is

1.214