Second Law Efficiency

A power plant receives two heat inputs, 25 kW at 825°C and 50 kW at 240°C, rejects heat to the environment at 20°C, and produces power of 12 kW. Calculate the second-law efficiency of the power plant.

from pint import UnitRegistry
ureg = UnitRegistry()
Q_ = ureg.Quantity

The second law efficiency is

\[\begin{split} \eta_2 = \frac{\text{exergy of desired output}}{\text{exergy supplied}} \\ \eta_2 = \frac{\dot{W}}{\dot{X}_{Q_{HT}} + \dot{X}_{Q_{MT}}} \;, \end{split}\]

where the exergy input due to heat transfer is

(26)\[\begin{equation} \dot{X}_{Q_i} = \dot{Q}_i \left( 1 - \frac{T_0}{T_i} \right) \end{equation}\]

heat_in_hot = Q_(25, 'kW')
temp_hot = Q_(825, 'degC').to('K')

heat_in_med = Q_(50, 'kW')
temp_med = Q_(240, 'degC').to('K')

temp_out = Q_(20, 'degC').to('K')

work_out = Q_(12, 'kW')

exergy_heat_hot = heat_in_hot * (1.0 - (temp_out / temp_hot))
exergy_heat_med = heat_in_med * (1.0 - (temp_out / temp_med))

eta_2 = work_out / (exergy_heat_hot + exergy_heat_med)

print(f'second law efficiency: {100*eta_2.magnitude: .1f}%')

second law efficiency:  30.2%

Let’s compare to the first-law efficiency:

\[ \eta = \frac{\dot{W}}{\dot{Q}_{HT} + \dot{Q}_{MT}} \]

eta = work_out / (heat_in_hot + heat_in_med)
print(f'first law efficiency: {100*eta.magnitude: .1f}%')

first law efficiency:  16.0%