Air has a velocity of 1000 km/hr at a pressure of $9.81 KN/m^2$ vacuum and a temperature of $47^0C$. Compute the stagnation properties (Pressure, Temperature and Density). Take atm pressure $98. 1 KN/m^2, R =287 J/Kg^0K$ and $\gamma = 1.4$

Velocity of air,

$V_0\ =\ 1000\ km/hr\ =\ \frac{1000\ \times 1000}{60\ \times 60} \ = 277.78\ m/s$

Temperature of air,

$T_0\ =\ 47\ +\ 273\ =\ 320\ K$

Atmospheric pressure,

$p_{atm} = 98.1 \;kN/m^2$

Pressure of air (static),

$p_0 = 98.1 -9.81=88.29 \;kN/m^2 \\ R=287 \;J/kg \;K \\ γ = 1.4$

Sonic velocity,

$C_0 = \sqrt{γ RT_0} = \sqrt{1.4 \times 287 \times 320} = 358.6 \;m/s$

Mach number,

$M_0 = \frac{V_0}{C_0} = \frac{277.78}{358.6} = 0.7746$

Stagnation pressure, ps :

The stagnation pressure is given by,

$p_s=p_0[1 +(\frac{γ-1}{2}M_0^2)]^{γ/(γ-1)} \\ p_s = 88.29[1 +(\frac{1.4-1}{2} \times 0.7746^2)]^{1.4/(1.4-1)} \\ = 88.29(1.12)^{3.5} = 131.27 \;kN/m^2$

Stagnation temperature, Ts :

$T_s =T_0[1 +(\frac{γ-1}{2})M_0^2] \\ T_s = 320[1 +(\frac{1.4-1}{2}) \times 0.7746^2] = 358.4 \;K \;or \;85.4 \;°C$

Stagnation density, ρs :

$ρ_s = \frac{p_s}{RT_s} = \frac{131.27 \times 10^3}{287 \times 358.4} = 1.276 \;kg/m^3$