# Power produced by steam turbine

Why for the same turbine exit condition, if the pressure and temperature of the air vapor entering the turbine are increased, the power produced by the turbine is greater?

For approximately adiabatic reversible operation of the turbine, the variation of enthalpy per mole of gas passing through the turbine is given by $$dh=C_pdT=vdP=\frac{RT}{P}dP$$. This integrates to $$\frac{T_{out}}{T_{in}}=\left(\frac{P_{in}}{P_{out}}\right)^{\frac{(\gamma-1)}{\gamma}}$$From this it follows that the power delivered per mole of air passing through the turbine is given by $$\dot{p}=C_p(T_{in}-T_{out})=C_pT_{in}\left[1-\left(\frac{P_{out}}{P_{in}}\right)^{\frac{(\gamma-1)}{\gamma}}\right]$$So the higher $$T_{in}$$ and $$P_{in}$$, the higher the power delivered per mole of air.
Assuming an adiabatic steam turbine, the power output equals the enthalpy per unit time of the incoming steam, $$\dot h_i$$, minus the enthalpy per unit time of the exiting steam, $$\dot h_e$$. Increasing the temperature and pressure of the incoming steam increases $$\dot h_i$$. Therefore for the same $$\dot h_e$$ output power increases.