Question

# Helium expands in a nozzle from $0.8\ \mathrm{MPa}, 500 \mathrm{~K}$, and negligible velocity to $0.1\ \mathrm{MPa}$. Find the throat and exit areas for a mass flow rate of $0.34 \mathrm{~kg} / \mathrm{s}$, assuming the nozzle is isentropic. Why must this nozzle be convergingdiverging?

Solution

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The stagnation temperature of helium at inlet is:

$\begin{equation*} T_{01}=T_1+\frac{V_1^2}{2c_p} \end{equation*}$

where $T_1$ is static temperature at inlet, $V_1$ is helium velocity at inlet and $c_p$ is constant-pressure heat capacity of the helium.

If the velocity at inlet is negligible ($V_1=0$), then we get:

$\begin{equation*} T_{01}=T_1=500\;\text{K} \end{equation*}$

The stagnation pressure at inlet is given by:

$\begin{equation*} P_{01}=P_1\bigg(\frac{T_{01}}{T_1}\bigg)^ {k/(k-1)} \end{equation*}$

With $P_1=0.8\;\text{MPa}$, $T_{01}=T_1=500\;\text{K}$ and $k=1.667$, it is obtained:

$\begin{equation*} P_{01}=0.8\;\text{MPa}\cdot\bigg(\frac{500\;\text{K}}{500\;\text{K}}\bigg)^{1.667/(1.667-1)}=0.8\;\text{MPa} \end{equation*}$