Combustor Performance: We can model the performance of a combustor using a simplified, constant$-$area, quasi$-$one$-$dimensional compressible flow, as shown in the figure below.

$($a$)$ In the first region $(335),$ heat is added to the flow via combustion; the heat transfer from the flame to the rest of the flow happens in a short enough distance that we neglect friction in this region. The combustor inlet stagnation temperature, $T_{03}=746K,$ and the heat released via combustion per unit air flow through the engine, $q=8\mathrm{.}37{10}^5\frac{J}{k}g,$ are known engine cycle analysis $($see Homework #$3,$ Problem $3).$ Assume the combustor inlet Mach number is $M_3=0\mathrm{.}2$ and the flow is an ideal gas with constant specific heat $c_p=1,110\frac{J}{k}\frac{g}{K}$ and $=1\mathrm{.}4.$

i$.$ Calculate the stagnation temperature at the end of the region of heating, $T_{035}.$

ii$.$ Calculate the Mach number at the end of the region of heating, $M_{35}.$

iii. Calculate the stagnation pressure ratio across the region of heating, $\frac{p_{035}}{p_{03}}.$

$($b$)$ The flow in the second region $(354)$ is adiabatic, but the combustor is long enough that we must account for the effects of friction. Assume the skin friction coefficient is $C_f=0\mathrm{.}01,$ the length of the second region is $x=0\mathrm{.}2m,$ and the hydraulic diameter of the channel is $D=1\mathrm{.}0m.$

i$.$ Calculate the stagnation temperature at the exit of the combustor, $T_{04}.$

ii$.$ Calculate the Mach number at the exit of the combustor $M_4.$

iii. Calculate the stagnation pressure ratio across the region of adiabatic flow with friction, $\frac{p_{04}}{p_{035}}.$

iv$.$ What is the stagnation pressure ratio of the combustor, $\frac{p_{04}}{p_{03}}?$ Would you say this represents good combustor performance?