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: $d = 4\, {mm} = 0.004\, {m}$ Discharge coefficient: $C_d = 0.65$ Density of propellant: $\rho = 1000\, {kg/m}^3$ Pressure difference: $\Delta P = 10\, {bar} = 10^6\, {Pa}$ Specific impulse: $I_{sp} = 1500\, {m/s}$ \end{itemize} Step 2: Flow rate through one injector \[ A = \frac{\pi d^2}{4} = \frac{\pi (0.004)^2}{4} = 1.2566 \times 10^{-5}\, {m}^2 \] \[ \dot{m}_{{one}} = C_d \cdot A \cdot \sqrt{2 \rho \Delta P} = 0.65 \cdot 1.2566 \times 10^{-5} \cdot \sqrt{2 \cdot 1000 \cdot 10^6} \] \[ = 0.65 \cdot 1.2566 \times 10^{-5} \cdot 44721.4 \approx 0.365\, {kg/s} \] Step 3: Total mass flow rate \[ \dot{m}_{{total}} = n \cdot \dot{m}_{{one}} = 800 \cdot 0.365 \approx 292\, {kg/s} \] Step 4: Thrust calculation \[ F = \dot{m} \cdot I_{sp} = 292 \cdot 1500 = 438000\, {N} = 438.0\, {kN} \]