What happens with thrust if I decrease the convergent divergent nozzle exit area?

Question

I am new to rocket design and I have a couple of questions.

1. If I have a convergent - divergent nozzle and I have a choked flow condition for the throat, what will happen a) with my exit pressure b) thrust if I decrease the nozzle exit area?

2. So I know that there is one point, where $$p_{ambient} = p_{exit}$$ , this is where the thrust becomes maximized. Is this also the point, where the exit velocity is at its peak?

Because if this would be the case, then according to $$ V_e = \sqrt{\frac{TR}{M} \cdot \frac{2\gamma}{\gamma-1} \cdot \Biggl( 1- \bigg(\frac{P_e}{P}\bigg)^{(\gamma-1)/\gamma} \Bigg)} $$ the lower the exit pressure is, the higher my exit velocity would be?

And if I want to lower my exit pressure, then according to

$$\frac{A_e}{A_t} = \frac{ (\frac{p_t}{p_e})^{\frac{k+1}{2 k}}}{\sqrt{\frac{2}{k-1}((\frac{p_t}{p_e})^{\frac{k-1}{k}}-1)}} * (\frac{k+1}{2})^{-\frac{k+1}{2(k-1)}}$$ I have to increas/decrease my exit area?

Oh I am a bit confused.... can someone help me?

Thank you very much !

Lucas

Answer 1

You are omitting an important part of the force exerted by a rocket engine. A better expression is $$F = \dot m v_e + (P_e-P_a)A_e$$ In this expression,

• $F$ is the force exerted by the engine on the rocket body,
• $\dot m$ is the mass flow rate of exhaust exiting the engine,
• $v_e$ is the velocity of the exhaust gas at the exit plane,
• $P_e$ is the pressure of the exhaust gas at the exit plane,
• $P_a$ is the ambient pressure of the atmosphere in the vicinity of the engine, and
• $A_e$ is the area of the rocket engine at the exit plane.

At face value, this suggests that having the exhaust pressure be greater than ambient pressure. This is not the case. A high exhaust pressure results from having a short engine bell, and that reduces the exit velocity. Exhaust pressure decrease as exit velocity increases, but in a rather non-linear way. The end effect is that it is best to have the exhaust pressure equal to ambient pressure.

That is not possible with a fixed-sized rocket engine attached to a rocket that uses its engines to make the rocket climbing through the Earth's atmosphere. This is because ambient pressure drops with increased altitude. With most launch vehicles, the rocket engines are designed to be over-expanded at launch ($P_e < P_a$), ideally expanded ($P_e = P_a$) at some altitude, and under-expanded ($P_e > P_a$) after that. The goal is to maximize the change in velocity experienced by the rocket rather than to maximize the force at any point in time generated by the rocket engine.