Chambadal’s optimization of the steady-state power plant can be retraced by considering the model shown in Fig. P7.7. The power plant is driven by a stream of hot single-phase fluid of inlet...

Chambadal’s optimization of the steady-state power plant can be retraced by considering the model shown in Fig. P7.7. The power plant is driven by a stream of hot single-phase fluid of inlet temperature T₁
and constant specific heat c_P. The power plant model consists of two compartments. The lower compartment sandwiched between the surface of temperature T₂
and the ambient T₀
operates reversibly. The area of the heat transfer surface T₂
is infinite, and as a consequence, the outlet temperature of the stream is equal to T₂. Determine the optimal hot-end temperature T₂
such that the instantaneous power output Ẇ is maximized. Show that when the power is maximum, the efficiency 𝜂 = Ẇ
_max∕Q̇ is equal to 1 − (T₀∕T₁)
^1∕2.