An ideal rocket has characteristic exhaust velocity of 1200 m/s, mass flow rate of 75 kg/s, thrust coefficient of 1.5, and nozzle throat area of 0.025 \(\text{m}^2\). The chamber pressure in kPa and the specific impulse due to gravity in seconds are ________, respectively. Assume that the acceleration due to gravity is 9.8 \(\text{m/s}^2\).
Show Hint
- 3600 and 183.67
- 4600 and 190.51
- 3600 and 175.23
- 3500 and 183.67
The Correct Option is A
Solution and Explanation
The characteristic velocity \(c^*\) links the chamber pressure \(p_c\), the throat area \(A_t\), and the mass flow rate:
\[ c^* = \frac{p_c A_t}{\dot{m}} \]
This number depends only on the propellant and chamber conditions, not on the nozzle expansion, so it is a good first quantity to use.
Step 2: Solve for the chamber pressure.
Rearranging,
\[ p_c = \frac{c^*\dot{m}}{A_t} = \frac{1200 \times 75}{0.025} = \frac{90000}{0.025} = 3600000 \text{ Pa} = 3600 \text{ kPa} \]
Step 3: Find the thrust using the thrust coefficient.
The thrust coefficient relates thrust to the chamber pressure and throat area:
\[ C_F = \frac{F}{p_c A_t} \implies F = C_F\, p_c A_t \]
\[ F = 1.5 \times 3600000 \times 0.025 = 1.5 \times 90000 = 135000 \text{ N} \]
Step 4: Get the effective exhaust velocity, then the specific impulse.
The effective exhaust velocity is the thrust divided by the mass flow rate:
\[ c = \frac{F}{\dot{m}} = \frac{135000}{75} = 1800 \text{ m/s} \]
The specific impulse due to gravity is this velocity divided by \(g\):
\[ I_{sp} = \frac{c}{g} = \frac{1800}{9.8} \approx 183.67 \text{ s} \]
Step 5: Check the other options.
Options (B) and (D) give a chamber pressure other than 3600 kPa, which does not match the direct \(c^*\) relation above. Option (C) keeps the correct chamber pressure but lands on a lower specific impulse, which does not come out of the correct thrust and mass-flow numbers.
Final Answer:
The chamber pressure is 3600 kPa and the specific impulse is 183.67 s. \[ \boxed{p_c = 3600 \text{ kPa}, \ I_{sp} \approx 183.67 \text{ s}} \]
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