Question

A monopropellant liquid rocket engine has 800 injectors of diameter 4 mm each, and with a discharge coefficient of 0.65. The liquid propellant of density 1000 kg/m³ flows through the injectors. There is a pressure difference of 10 bar across the injectors. The specific impulse of the rocket is 1500 m/s. The thrust generated by the rocket is __________ kN (rounded off to one decimal place).

Hint:

For monopropellant engines, the thrust can be calculated by determining the mass flow rate using the discharge coefficient and the pressure difference, then multiplying by the specific impulse.

Answer 1

Step 1: Given Data

• Number of injectors: \( n = 800 \)
• Diameter of each injector: \( d = 4\, \text{mm} = 0.004\, \text{m} \)
• Discharge coefficient: \( C_d = 0.65 \)
• Density of propellant: \( \rho = 1000\, \text{kg/m}^3 \)
• Pressure difference: \( \Delta P = 10\, \text{bar} = 10^6\, \text{Pa} \)
• Specific impulse: \( I_{sp} = 1500\, \text{m/s} \)

Step 2: Flow Rate Through One Injector

Cross-sectional area of one injector:
\[ A = \frac{\pi d^2}{4} = \frac{\pi (0.004)^2}{4} = 1.2566 \times 10^{-5}\, \text{m}^2 \]
Mass flow rate through one injector:
\[ \dot{m}_{\text{one}} = C_d \cdot A \cdot \sqrt{2 \rho \Delta P} = 0.65 \cdot 1.2566 \times 10^{-5} \cdot \sqrt{2 \cdot 1000 \cdot 10^6} \] \[ = 0.65 \cdot 1.2566 \times 10^{-5} \cdot 44721.4 \approx 0.365\, \text{kg/s} \]

Step 3: Total Mass Flow Rate
\[ \dot{m}_{\text{total}} = n \cdot \dot{m}_{\text{one}} = 800 \cdot 0.365 \approx 292\, \text{kg/s} \] Step 4: Thrust Calculation
\[ F = \dot{m} \cdot I_{sp} = 292 \cdot 1500 = 438000\, \text{N} = 438.0\, \text{kN} \] Final Answer:
\[ \boxed{438.0 \, \text{kN}} \]