Question

Hot gases at 2100 K and 14 MPa expand ideally to 0.1 MPa through a rocket nozzle.
Molecular mass = 22 kg/kmol, heat-capacity ratio $\gamma = 1.32$,
Universal gas constant = 8314 J/kmol-K, \(g = 9.8\,\text{m/s}^2\).
Throat area = \(0.1\ \text{m}^2\). Find the specific impulse (round off to 2 decimals).

Hint:

Specific impulse is simply $I_{sp}=V_e/g$ once exit velocity is known.

Answer 1

Correct Answer: 216

Specific gas constant: \[ R = \frac{8314}{22} = 378\ \text{J/kg·K} \] Exit Mach for ideal expansion: \[ M_e = \sqrt{\frac{2}{\gamma - 1} \left[1 - \left(\frac{p_e}{p_0}\right)^{\frac{\gamma-1}{\gamma}}\right]} \] \[ = \sqrt{\frac{2}{0.32} \left[1 - (0.1/14)^{0.2424}\right]} \] \[ M_e \approx 3.14 \] Exit temperature: \[ T_e = \frac{T_0}{1 + \frac{\gamma - 1}{2} M_e^2} \approx \frac{2100}{1 + 0.16 \times 9.86} \approx 1120\ \text{K} \] Exit velocity: \[ V_e = M_e \sqrt{\gamma R T_e} = 3.14 \sqrt{1.32 \times 378 \times 1120} \] \[ \approx 3.14 \times 680 \approx 2135\ \text{m/s} \] Specific impulse: \[ I_{sp} = \frac{V_e}{g} = \frac{2135}{9.8} \approx 218\ \text{s} \]