Problem $3$: Air flowing steadily in a nozzle experiences a normal shock at a Mach number of $M=2\mathrm{.}5.$ The upstream static pressure and temperature of the air is $P_1=61\mathrm{.}64$kPa and $T_1=262\mathrm{.}15K,$ respectively. Calculate the following properties downstream of the shock at State $2$:

a$.$ the stagnation temperature $T_{0,2}(K)$ and explain the physical reason for the result;

b$.$ the static temperature $T_2(K)$ and pressure $P_2(kPa)$ using two different methods;

c$.$ the velocity of the flow $K_2(\frac{m}{s})$;

d$.$ the entropy change, $s_2-s_1(k\frac{J}{k}gK),$ using two different methods;

Re$-$calculate the properties for a flow of helium $($He$).$ For both gases, calculate the ratios $\frac{M_2}{M_1},\frac{T_2}{T_1},\frac{P_2}{P_1},\frac{P_{0,2}}{P_{0,1}},$ and $\frac{T_{0,2}}{T_{0,1}}$ and comment on the differences.

ANSWERS

Problem $3$: air

$($a$)T_{0,1}=T_{0,2}=590K$

$($b$)T_2=560K,P_2=439$kPa

$($c$)A{A}_2=244\frac{m}{s}$

$($d$)s_2-s_1=200\frac{J}{k}gK$

He

$($a$)T_{0,1}=T_{0,2}=811K$

$($b$)T_2=736K,P_2=467$kPa

$($c$){}_2=884\frac{m}{s}$

$($d$)s_2-s_1=1139\frac{J}{k}gK$