Discussion
A solid propellant of density 1800 kg/m3 has a burning rate law r = 6.65 x 10-3p0.45 mm/s, where p is pressure in Pascals. It is used in a rocket motor with a tubular grain with an initial burning area of 0.314 m2. The characteristic velocity is 1450 m/s. What should be the nozzle throat diameter to achieve an equilibrium chamber pressure of 50 bar at the end of the ignition transient?
Answer: B
The throat diameter should be about 38 mm.
At equilibrium the gas generated by the burning grain equals the flow through the nozzle: \(\rho_{p} r A_{b} = p_{c}A_{t}/c^{*}\). First the burning rate at 50 bar (\(5 \times 10^{6}\) Pa):
\[ \begin{aligned} r &= 6.65 \times 10^{-3}(5 \times 10^{6})^{0.45} \\ &= 6.65 \times 10^{-3} \times 1033.6 \\ &= 6.87\ \text{mm/s} \end{aligned} \]
\[ \begin{aligned} \dot{m} &= 1800 \times 0.00687 \times 0.314 = 3.885\ \text{kg/s} \\ A_{t} &= \frac{\dot{m}c^{*}}{p_{c}} = \frac{3.885 \times 1450}{5 \times 10^{6}} \\ &= 1.127 \times 10^{-3}\ \text{m}^{2} \\ d_{t} &= \sqrt{4A_{t}/\pi} = 0.0379\ \text{m} \end{aligned} \]
That is 37.9 mm, so 38 mm. 35 mm would give too small a throat and a higher chamber pressure; 41 mm and 45 mm too large a throat and a lower pressure.
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