GATE 2021 AE – Question 54
Hot gases at 2100 K and 14 MPa expand ideally to 0.1 MPa through a convergent-divergent nozzle of throat area 0.1 m². Molecular mass is 22 kg/kmol, γ=1.32, universal gas constant 8314 J/(kmol K), and g=9.8 m/s². Specific impulse is ___ seconds (two decimals).
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Correct answer: 217.08 to 219.26
Explanation
**Specific impulse of a rocket with ideal expansion:**
$$I_{sp}=\frac{V_e}{g},\qquad V_e=\sqrt{\frac{2\gamma}{\gamma-1}\,R\,T_0\left[1-\left(\frac{p_e}{p_0}\right)^{\frac{\gamma-1}{\gamma}}\right]},$$
because the nozzle is ideally expanded to the ambient pressure, so there is no pressure thrust.
**Gas constant:** $R=\dfrac{8314}{22}=377.9$ J/(kg K).
**Numbers.**
- $\dfrac{2\gamma}{\gamma-1}=\dfrac{2.64}{0.32}=8.25$
- $\left(\dfrac{0.1}{14}\right)^{0.32/1.32}=(0.007143)^{0.2424}=0.3011$
- $V_e^2=8.25\times377.9\times2100\times(1-0.3011)=4.5757\times10^6$
$$V_e=2139\ \text{m/s},\qquad I_{sp}=\frac{2139}{9.8}=\mathbf{218.2\ s}.$$
(The throat area is not needed for the specific impulse.)