$9\mathrm{.}30.$ Air flows at a steady rate through a $0\mathrm{.}08$ m diameter $1\mathrm{.}5$ m long pipe. A convergent$-$divergent nozzle expands the air to a Mach number of $2\mathrm{.}25$ and a pressure of $40$ kPa at the inlet to the pipe. The air from the pipe is discharged into a large chamber in which the pressure can be varied. Assuming a friction factor of $0\mathrm{.}003,$ find the pressure in this chamber if:

$1.$ There are no shock or expansion waves in the flow

$2.$ There is a normal shock wave on the exit plane of the pipe.

Answer is supposed to be: $(1)$ The back pressure is $68\mathrm{.}58$ Kpa. $(2)$ The back pressure is $177\mathrm{.}1$ Kpa.