Problem 1.11

A solid fueled rocket is fitted with a convergent-divergent nozzle with an exit plane diameter of 30 cm. The pressure and velocity on this nozzle exit plane are 75 kPa and 750 m/s, respectively, and the mass flow rate through the nozzle is 350 kg/s. Find the thrust developed by this engine when the ambient pressure is (a) 100 kPa and (b) 20 kPa.

Given: ${\dot{m}}_{e} = 350 kg/s$ , $V_{e} = 750 m/s$ , $p_{e} = 75 kPa$ , $D_{e} = ⌀ 0.3 m$ .
To calculate:
(a) Thrust when $p_{amb} = 100 kPa$ .
(b) Thrust when $p_{amb} = 20 kPa$ .

The schematic diagram of the problem description is shown in Fig. 1.

(a) Thrust when $p_{amb} = 100 kPa$ .

Applying the conservation of momentum on the control-volume around the rocket,

\begin{aligned} Thrust = & rate of momentum exiting - rate of momentum entering \\ + pressure force at exit - pressure force at inlet \\ Thrust = & {\dot{m}}_{e} V_{e} - 0 + (p_{e} - p_{amb}) A_{exit} \\ Thrust = & 350 \times 750 - 0 + (75 \times 10^{3} - 100 \times 10^{3}) \times \frac{π}{4} \times {0.3}^{2} \end{aligned}

when p_{amb} = 100 kPa ⟹ Thrust = 260732.85 N .

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(b) Thrust when $p_{amb} = 20 kPa$ .

Applying the conservation of momentum on the control-volume around the rocket,

\begin{aligned} Thrust = & rate of momentum exiting - rate of momentum entering \\ + pressure force at exit - pressure force at inlet \\ Thrust = & {\dot{m}}_{e} V_{e} - 0 + (p_{e} - p_{amb}) A_{exit} \\ Thrust = & 350 \times 750 - 0 + (75 \times 10^{3} - 20 \times 10^{3}) \times \frac{π}{4} \times {0.3}^{2} \end{aligned}

when p_{amb} = 20 kPa ⟹ Thrust = 266387.72 N .

Problem 1.11:

Solution: