Air flows steadily through an insulated constant area duct, where (1) denotes the inlet and (2) the outlet. Properties change along the duct as a result of friction.

(a) Beginning with the control volume form of the first law of thermodynamics, show that the equation can be reduced to

\[h_{1}+\frac{V_{1}^{2}}{2}=h_{2}+\frac{V_{2}^{2}}{2}=\text { constant }\]

(b) Denoting the constant by \(h_{0}\) (the stagnation enthalpy), show that for adiabatic flow of an ideal gas with friction

\[\frac{T_{0}}{T}=1+\frac{k-1}{2} M^{2}\]

(c) For this flow does \(T_{0_{1}}=T_{0_{2}}\) ? \(p_{0_{1}}=p_{0_{2}}\) ? Explain these results.