Derivation

Expression for Stagnation Pressure $(P_s)$

$$ \frac{P_{s}}{P_1}=\left[\left(1+\frac{k-1}{2} M_{1}^{2}\right)^{\frac{K}{K-1}}\right] ..(i)$$

Expression for Stagnation Density $\left(\rho_ {s} \right)$

For Adiabatic process

$$\begin{aligned} \left(\frac{\rho_{1}}{\rho_{s}}\right)=\left(\frac{p_{1}}{p_{s}}\right)^{\frac{1}{K}} \text{or}\left(\frac{\rho_{s}}{\rho_{1}}\right)=\left(\frac{p_{s}}{p_{1}}\right)^{\frac{1}{k}} \end{aligned}$$

(Taking reciprocal)

$$ \quad \rho_{s}=\rho_{1}\left[\frac{p_{s}}{p_{1}}\right]^{\frac{1}{K}} $$

Substituting the value of $\left(\frac{p_{s}}{p_{1}}\right)$ from equation $(i)$,

$$\begin{aligned} \rho_{s} &=\rho_{1}\left[\left(1+\frac{K-1}{2} M_{1}^{2}\right)^{\frac{K}{K-1}}\right]^{\frac{1}{K}} \\ \rho_{s} &=\rho_{1}\left[1+\frac{K-1}{2} M_{1}^{2}\right]^{\frac{1}{K-1}} ..(ii) \end{aligned} $$

Above equation gives value of Stagnation Density.

Expression for Stagnation Temperature $T_s$

as $\frac{P_s}{\rho_s}=R T$

For the stagnation point, we have equation of state as $\frac{P_{s}}{\rho_{s}}=R T_{s}$

$\therefore \quad T_{s}=\frac{1}{R} \frac{P_{s}}{\rho_{s}}$

Substituting the value of $P_{s}$ and $\rho_{s}$ from equations $(ii)$, we have

$$\begin {aligned} T_{s} & =\frac{1}{R} \frac{P_{1}\left[1+\left(\frac{K-1}{2}\right) M_{1}^{2}\right]^{\left(\frac{K}{K-1}\right)}}{\rho_{1}\left[1+\left(\frac{K-1}{2}\right) M_{1}^{2}\right]^{ \frac{1}{K-1}} }\\ &=\frac{1}{R} \frac{P_{1}}{\rho_{1}}\left[1+\left(\frac{K-1}{2}\right) M_{1}^{2}\right]^{\left(\frac{K}{K-1}\right)-\left(\frac{1}{K-1}\right)} \\ &=\frac{1}{R} \frac{P_{1}}{\rho_{1}}\left[1+\left(\frac{K-1}{2}\right) M_{1}^{2}\right]^{\left(\frac{K-1}{K-1}\right)}\\ &=\frac{P_{1}}{\rho_{1} R}\left[1+\left(\frac{K-1}{2}\right) M_{1}^{2}\right] \\ &=T_{1}\left[1+\left(\frac{K-1}{2}\right) M_{1}^{2}\right] \\ \end{aligned}$$

$$ \left(\because \frac{P_{1}}{\rho_{1}}=R T_{1}\right) $$

$$T_s=T_{1}\left[1+\left(\frac{K-1}{2}\right) M_{1}^{2}\right] $$

Above equation gives value of Stagnation Temperature.