A tank fitted with a convergent nozzle contains air at a temperature of 20°C. The diameter at the outlet of the nozzle is 25 mm. Assuming adiabatic flow, find the mass rate of flow of air through the nozzle to the atmosphere when the pressure in the tank is :

(i) 140 kN/m2 (abs.),

(ii) 300 kN/m2

Solution:

Temperature of air in the tank, $ T_{1}=20+273=293 \mathrm{~K} \\ $

Diameter at the outlet of the nozzle, $ D_{2}=25 \mathrm{~mm}=0.025 \mathrm{~m} \\ $

Area, $\quad A_{2}=\pi / 4 \times 0.025^{2}=0.0004908 \mathrm{~m}^{2} \\ $

$ R=287 \mathrm{~J} / \mathrm{kg} \mathrm{K}, \gamma=1.4 \\ $

(i) Mass rate of flow of air when pressure in the tank is, $ 140 \mathrm{kN} / \mathrm{m}^{2}(abs.) \\ $ :

$$ \rho_{1}=\frac{p_{1}}{R T_{1}}=\frac{140 \times 10^{3}}{287 \times 293}=1.665 \mathrm{~kg} / \mathrm{m}^{3} \\ $$

$$ p_{1}=140 \mathrm{kN} / \mathrm{m}^{2} \text { (abs.) } \\ $$

Pressure at the nozzle, $\quad p_{2}=$, atmospheric pressure, $ =100 \mathrm{kN} / \mathrm{m}^{2} \\ $

Pressure ratio, $$ \quad \frac{p_{2}}{p_{1}}=\frac{100}{140}=0.7143 \\ $$

Since the pressure ratio is more than the critical value, flow in the nozzle will be subsonic, hence mass rate of flow of air is given by,

$$ m=A_{2} \sqrt{\frac{2 \gamma}{\gamma-1} p_{1} \rho_{1}\left[\left(\frac{p_{2}}{p_{1}}\right)^{\frac{2}{\gamma}}-\left(\frac{p_{2}}{p_{1}}\right)^{\frac{\gamma+1}{\gamma}}\right]} \\\\ $$

$$ =0.0004908 \sqrt{\left(\frac{2 \times 1.4}{1.4-1}\right) \times 140 \times 10^{3} \times 1.665\left[(0.7143)^{\frac{2}{1.4}}-(0.7143)^{\frac{1.4+1}{1.4}}\right]} \\\\ $$

$$ =0.0004908 \sqrt{1631700(0.7143)^{1.4285}-(0.7143)^{1.7142}} \\\\ $$

$$or$$

$$ m=0.0004908 \sqrt{1631700(0.6184-0.5617)}=0.1493 \mathrm{~kg} / \mathbf{s} \quad \text { (Ans.) } \\ $$