The furnace of a particular power plant can be considered to consist of two
chambers: an adiabatic combustion chamber where the fuel is burned completely
and adiabatically, and a heat exchanger where heat is transferred to a Carnot
heat engine isothermally. The combustion gases in the heat exchanger are well
mixed so that the heat exchanger is at uniform temperature at all times that
is equal to the temperature of the exiting product gases, \(T_{p} .\) The work
output of the Carnot heat engine can be expressed as
$$ w=Q \eta_{c}=Q\left(1-\frac{T_{0}}{T_{p}}\right)$$
where \(Q\) is the magnitude of the heat transfer to the heat engine and \(T_{0}\)
is the temperature of the environment. The work output of the Carnot engine
will be zero either when \(T_{p}=T_{\mathrm{af}}\) (which means the product
gases will enter and exit the heat exchanger at the adiabatic flame
temperature \(T_{\mathrm{af}}\), and thus \(Q=0\) ) or when \(T_{p}=\) \(T_{0}\)
(which means the temperature of the product gases in the heat exchanger will
be \(T_{0}\), and thus \(\eta_{c}=0\) ), and will reach a maximum somewhere in
between. Treating the combustion products as ideal gases with constant
specific heats and assuming no change in their composition in the heat
exchanger, show that the work output of the Carnot heat engine will be maximum
when
$$T_{p}=\sqrt{T_{\mathrm{af}} T_{0}}$$
Also, show that the maximum work output of the Carnot engine in this case
becomes
$$W_{\max }=C
T_{\mathrm{af}}\left(1-\sqrt{\frac{T_{0}}{T_{\mathrm{af}}}}\right)^{2}$$
where \(C\) is a constant whose value depends on the composition of the product
gases and their specific heats.
