Nozzles on Vacuum Optimazed Rockets vs. Nozzles on Sea Level Engines

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Why are sizes of rocket nozzles in second or upper-stage engine bigger than those used by sea-level engines?

A sea-level optimized Raptor nozzle (on the left) stands next to a vacuum-optimized Raptor nozzle (on the right) at the SpaceX development facility in McGregor, Texas. Credit: SpaceX 

Well, the answer is quite simple. A vacuum-optimized rocket engine nozzle is bigger then a sea-level engine nozzle since it needs to match the significantly lower air density in the upper atmosphere and in space. Let's have a look in more detail.

For a rocket engine is important to operate at its most efficient level, which means to produce the maximum amount of thrust. It is a key for the hot gases exiting the engine nozzle to match the external air ambient pressure. The importance of this is clear when you realize that the air pressure at sea level is about 1000 millibars, or 100 kPa, and it decreases with increasing altitude. The value quickly decreases with altitude to 100 millibars, or 10 kPa, at 12 km and only 1 millibar, or 0.1 kPa, at 50 km. This makes it impossible for one single nozzle design to maintain maximum efficiency throughout a launch ascent.

The number of molecules in the atmosphere decreases with height. Credit: NOAA

Also to make things more complicated, the air pressure changes are caused by changes in air density, and air density is related to temperature. Warm air is less dense than cooler air because the gas molecules in warm air have a greater velocity and are farther apart than in cooler air.

How temperature affects the height of pressure. Credit: NOAA.


Nozzle shape:
The nozzle is designed in order to accelerate the high-pressure, high-temperature gases generated in the combustion chamber to a very high supersonic velocity. The nozzle typically consists of two sections. First section is convergent and and the second one is divergent, as you can see on the following Figure. 

Convergent/divergent nozzle; source Rocket Engines – Introduction to Aerospace Flight Vehicles

The design of the nozzle, meaning the length, the shape, and the used material of the divergent section, is crucial for achieving optimal rocket engine thrust. In general, the nozzle must be long enough, and the exit area must be large enough to ensure that the pressure at the exit is close to the ambient pressure. This is essential to maximize the efficiency of the propulsion system and achieve maximum thrust.


The expansion process:
Expansion is a process that converts the thermal energy of combustion into kinetic energy to move an object forward. It means that the hot gases, which are created by burning fuel inside a rocket engine, are exhausted through a nozzle to produce thrust. The shape of the nozzle is key to this expansion process. 
The nozzle expansion process. Credit: aerospacenotes.com



The principles of fluid dynamics order that the pressure of rocket exhaust gases decreases as the surface area of a nozzle increases without a loss in velocity. This means that the bigger a rocket nozzle is, the lower the gas pressure. As a result, the first-stage engine nozzles of a rocket are smaller than upper-stage nozzles, which are significantly bigger to allow the pressure of the gases exiting the nozzles to decrease enough to match ambient air pressures.

Diagram of the gas flow through a de Laval nozzle used to produce supersonic velocity of the exhaust gases from a rocket engine. Credit: Wiki.


These configurations allow the smaller first-stage nozzles to operate at maximum efficiency in the dense atmospheric air close to the surface, while the larger second-stage nozzles are deployed later in the upper atmosphere, where is less dense air, allowing to operate in optimal efficiency.

Note: Second-stage rocket booster typically deploys at a altitude of 50 to 80 kilometers above the Earth’s surface, where the air pressure is 0.1% or less than the air pressure at sea level. But it is important to conclude that rocket nozzle configuration is a compromise. At certain point it works in part of the atmosphere where the pressure of gases exiting the nozzle doesn't match the ambient air pressure. 

For example, in case of the Space Shuttle’s three main engines and the Delta IV rocket engine used nozzles that were overexpanded during launch. It resulted in certain pattern exhaust plume. In contrast, all engine nozzles, including vacuum-optimized nozzles, are under-expanded in the vacuum of space, in order to match basically non existing ambient pressure. 

Diagram showing the different effects an under-expanded, ambient, and overexpanded nozzle has on the way a rocket engine’s exhaust plumes interact with the external air pressure. Credit: headedforspace.com


As the diagram above illustrates, a nozzle can be in one of three states, depending on nozzle size and the external air pressure at a given altitude:

1. Under-Expanded Nozzle

An Under-Expanded Nozzle produces exhaust gases with a higher pressure than the surrounding air, which results in the exhaust plumes that are extending beyond the edge of the nozzle. Such a case reduces the efficiency. Nozzles optimized for vacuum usually exhibit this characteristic, but as you see in Figure during the launch of Saturn V. 

Underexpanded plumes; source aerospacenotes.com


2. Ambient Nozzle or the Ideal Expanded Nozzle

Unfortunately, this is a situation that can only occur at one specific atmospheric pressure on a nozzle with a specific geometry. As we already discussed earlier, pressure decreases with increasing altitude. Which is impossible situation. Nozzle designs work optimally only at one altitude. There will be always losses at lower and higher altitudes. 

AmbientNozzle; source aerospacenotes.com and Everyday Astronaut


3. Overexpanded Nozzle

The overexpanded nozzles of a Delta IV Heavy rocket firing liftoff, displaying the recompressing and decompressing shock waves in its exhaust plumes, source headedforspace.com

The Delta IV rocket engine used nozzles that were overexpanded during launch. 

This is a case, where the external pressure is higher than the exit pressure. As the over-expanded stream passes through the nozzle, the higher atmospheric pressure causes it to be pushed back in and separated from the walls of the nozzle. This “pinching” of the flow reduces efficiency because that extra nozzle wall is basically wasted. So, the particular solution would be that the nozzle should have been shorter. 

I have used many sources to learn about the nozzles. I will try to cite all of them:

Air Pressure | National Oceanic and Atmospheric Administration

Over-Under Expanded Nozzle - Propulsion 1 - Aerospace Notes

Why Nozzles On Vacuum Optimized Rocket Engines Are Bigger Than Those On Sea Level Engines - Headed For Space

Rocket engine nozzle - Wikipedia

Aerospaceweb.org | Ask Us - Nozzle Overexpansion & Underexpansion

Rocket Engines – Introduction to Aerospace Flight Vehicles




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