Air flows in a constant-area, insulated duct. The air enters the duct at \(520^{\circ} \mathrm{R}, 50\) psia, and \(\mathrm{Ma}=0.45 .\) At a downstream location, the Mach number is one. Find: (a) The pressure and temperature at the downstream location (b) The change in specific entropy (c) The frictional force if the duct is circular and \(1 \mathrm{ft}\) in diameter.

We first find the properties downstream where Mach number is \(1.0\) (sonic point). For a perfect gas undergoing an isentropic process, the ratio of pressure to its corresponding pressure when Mach number is \(1.0\) is given by \(P/P^*= ((1+(\gamma-1)/2*M^2)/((\gamma+1)/2))^(\gamma/(\gamma-1))\). Here, \(\gamma\) is the adiabatic index; for air, \(\gamma=1.4\). From this we find that \( P^*= 0.5283 * P\) and by using the given number for pressure, we find that \( P^* = 0.5283 * 50\). To find the temperature, we can use similar relation which is \( T^* = T/(1+(\gamma-1)/2)\) which would give us \(T^*= T/1.2\). Using the given value for temperature, \(T=520^{\circ} \mathrm{R}\), the temperature at sonic point is \(T^*=520/1.2\).

The change in specific entropy's formula for an isentropic process is given by \(\Delta s = c_p *ln(T_2/T_1) - R*ln(P_2/P_1)\), where \(\Delta s\) is the change in specific entropy, \(c_p\) is the specific heat at constant pressure, \(R\) is the gas constant, \(T_1\) and \(T_2\) are the initial and final temperatures, and \(P_1\) and \(P_2\) are the initial and final pressures. For air, \(c_p = 1.005 kJ/kgK\) and \(R = 0.287 kJ/kgK\). Assume initial state to be the inlet and final state to be the downstream via the entropy relation. Using the values from step 1 and given values, we can calculate \(\Delta s\).

The change in momentum flux across the duct influences the frictional force. But considering this as an isentropic flow, momentum flux change becomes zero. The frictional force encountered by the flow is balanced by the pressure change, hence frictional force is nothing but the pressure difference multiplied by the area. So, \(F = (P_1-P_2).A\), where \(F\) is the frictional force, \(A\) is the cross-sectional area of the duct. The diameter given for circular duct is 1ft. Hence, the area \(A = \pi*(d/2)^2\). Put these values in frictional force equation to get \(F\).