Discussion
A bipropellant liquid rocket motor operates at a chamber pressure of 40 bar with a nozzle throat diameter of 50 mm. The characteristic velocity is 1540 m/s. If the fuel-oxidizer ratio of the propellant is 1.8, and the fuel density is 900 kg/m3, what should be the minimum fuel tank volume for a burn time of 8 minutes
Answer: B
The minimum fuel tank volume is 1.75 m\(^{3}\).
Total propellant flow from the chamber pressure, throat area and characteristic velocity:
\[ \begin{aligned} A_{t} &= \frac{\pi}{4}(0.05)^{2} = 1.963 \times 10^{-3}\ \text{m}^{2} \\ \dot{m} &= \frac{p_{c}A_{t}}{c^{*}} = \frac{4 \times 10^{6} \times 1.963 \times 10^{-3}}{1540} \\ &= 5.10\ \text{kg/s} \end{aligned} \]
With fuel-to-oxidizer ratio 1.8, fuel is \(1.8/2.8\) of the total:
\[ \begin{aligned} \dot{m}_{f} &= 5.10 \times \frac{1.8}{2.8} = 3.28\ \text{kg/s} \\ m_{f} &= 3.28 \times 480 = 1574\ \text{kg} \\ V_{f} &= 1574/900 = 1.75\ \text{m}^{3} \end{aligned} \]
The fuel share is the step to watch: taking the whole flow as fuel gives about 2.72 m\(^{3}\), and reading the ratio the wrong way round (1 part fuel in 2.8) about 0.97 m\(^{3}\). Only 1.8 parts in 2.8 gives 1.75 m\(^{3}\); 1.65, 1.85 and 1.95 m\(^{3}\) do not follow from the data.
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