To solve this problem it is necessary to apply the concepts related to the adiabatic process that relate the temperature and pressure variables
Mathematically this can be determined as
[tex]\frac{T_2}{T_1} = (\frac{P_2}{P_1})^{(\frac{\gamma-1}{\gamma})}[/tex]
Where
[tex]T_1 =[/tex]Temperature at inlet of turbine
[tex]T_2 =[/tex] Temperature at exit of turbine
[tex]P_1 =[/tex] Pressure at exit of turbine
[tex]P_2 =[/tex]Pressure at exit of turbine
The steady flow Energy equation for an open system is given as follows:
[tex]m_i = m_0 = m[/tex]
[tex]m(h_i+\frac{V_i^2}{2}+gZ_i)+Q = m(h_0+\frac{V_0^2}{2}+gZ_0)+W[/tex]
Where,
m = mass
[tex]m_i[/tex] = mass at inlet
[tex]m_0[/tex]= Mass at outlet
[tex]h_i[/tex] = Enthalpy at inlet
[tex]h_0[/tex] = Enthalpy at outlet
W = Work done
Q = Heat transferred
[tex]V_i[/tex] = Velocity at inlet
[tex]V_0[/tex]= Velocity at outlet
[tex]Z_i[/tex]= Height at inlet
[tex]Z_0[/tex]= Height at outlet
For the insulated system with neglecting kinetic and potential energy effects
[tex]h_i = h_0 + W[/tex]
[tex]W = h_i -h_0[/tex]
Using the relation T-P we can find the final temperature:
[tex]\frac{T_2}{T_1} = (\frac{P_2}{P_1})^{(\frac{\gamma-1}{\gamma})}[/tex]
[tex]\frac{T_2}{1400K} = (\frac{0.8bar}{8nar})^{(\frac{1.4-1}{1.4})}[/tex]
[tex]T_2 = 725.126K[/tex]
From this point we can find the work done using the value of the specific heat of the air that is 1,005kJ / kgK
So:
[tex]W = h_i -h_0[/tex]
[tex]W = C_p (T_1-T_2)[/tex]
[tex]W = 1.005(1400-725.126)[/tex]
[tex]W = 678.248kJ/Kg[/tex]
Therefore the maximum theoretical work that could be developed by the turbine is 678.248kJ/kg
