# Momentum Thrust of an Over-Expanded Rocket Nozzle

• BrandonBerchtold
In summary, the conversation discusses the calculation of momentum thrust in an over-expanded rocket nozzle. The exit velocity is determined by the maximum velocity attained before the normal shock, as the supersonic flow prevents any information from traveling upstream. There are two methods to study this, one involving forces on the nozzle and the other using conservation of total momentum.
BrandonBerchtold
TL;DR Summary
Momentum Thrust of Over-Expanded Rocket Nozzle
Hi All,
I'm trying to get a better understanding of the momentum thrust given by an over-expanded rocket nozzle (I realize this case voids the isentropic flow assumption used for the 1D isentropic gas expansion equations typically used for rocket engine design since the normal shock is not an isentropic process, but figuring out how momentum works in a supersonic flow is really messing with my brain right now).

Momentum thrust is given by the mass flow rate of propellants times the exit plane velocity, so when the flow mach number plot looks like the image below, what velocity is used as the "exit velocity"? Is the nozzle assumed to stop right before the normal shock, therefore using the maximum attained flow velocity before the shock as the "exit velocity", or is the exit velocity taken as the velocity across the exit plane downstream of the normal shock?

My reasoning is that since the flow is supersonic, no information may travel upstream, so if the flow slows down within the nozzle due to a shock, it can't really affect the nozzle walls upstream of the shock, therefore the exit velocity would be the max velocity before the normal shock. Is my reasoning incorrect?

View attachment 272099 [Source: https://www.mas.bg.ac.rs/_media/istrazivanje/fme/vol40/3/03_vzmijanovic.pdf]

Last edited by a moderator:
There are two ways to study this:

- by forces on the nozzle. Probably the more complicated way.
- by conservation of total momentum. Only the final exit velocity (after the nozzle) matters - and the speed of surrounding air if it gets accelerated backwards.

Hi there,

Your reasoning is correct. The exit velocity in this case would be the maximum velocity before the normal shock. This is because, as you mentioned, once the flow is supersonic, no information can travel upstream, so the shock cannot affect the nozzle walls upstream of it. Therefore, the exit velocity would be the maximum velocity attained before the shock.

I hope this helps clarify your understanding of momentum thrust in over-expanded rocket nozzles. Let me know if you have any further questions.

## What is momentum thrust?

Momentum thrust is the force generated by a rocket engine through the ejection of high-speed exhaust gases. It is a result of the conservation of momentum, where the mass of the exhaust gases multiplied by their velocity equals the mass of the rocket multiplied by its velocity.

## What is an over-expanded rocket nozzle?

An over-expanded rocket nozzle is a type of nozzle used in rocket engines where the expansion of the exhaust gases is greater than the design limit of the nozzle. This can occur when the nozzle is too large for the amount of exhaust gases being expelled, resulting in a decrease in thrust and efficiency.

## How does the momentum thrust of an over-expanded rocket nozzle differ from a properly expanded nozzle?

The momentum thrust of an over-expanded rocket nozzle is lower than that of a properly expanded nozzle. This is because the exhaust gases are not fully expanded and therefore do not have as high of a velocity, resulting in a decrease in momentum and thrust.

## What factors affect the momentum thrust of an over-expanded rocket nozzle?

The momentum thrust of an over-expanded rocket nozzle is affected by several factors, including the design and size of the nozzle, the amount and velocity of the exhaust gases, and the ambient pressure and temperature of the surrounding environment.

## Why is it important to consider the momentum thrust of an over-expanded rocket nozzle?

The momentum thrust of an over-expanded rocket nozzle is important to consider because it can significantly affect the performance and efficiency of a rocket engine. If the nozzle is over-expanded, it can result in a decrease in thrust and efficiency, which can impact the overall success of a rocket launch.

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