Have you noticed the rocket exhaust of Falcon 9 at different stages of its flight? Did you notice any differences in them? If not, then have a look at figures below—

To know the reason behind this, we need to first understand the importance of rocket nozzle. A rocket nozzle is basically used to expand and accelerate the combustion gases produced by burning the propellants so that the exhaust gases exit the nozzle at hypersonic velocities. Simply stating, the nozzle turns the static high pressure high temperature gas into rapidly moving gas at near-ambient pressure.

Now let’s refer above figure.

For brief explanation, let’s just focus on two terms: Ambient Pressure (Pa) and Exit Pressure of exhaust (Pe). Maximum efficiency (i.e. maximum thrust) is obtained when Pa = Pe. For rockets travelling from the Earth to orbit, a simple nozzle design is only optimal at one altitude (as Pa decreases with height).

Case (a): At sea level, Pa > Pe this is called as “over expanded” which reduces the efficiency.

Case (b): At optimum altitude, Pa = Pe this is called as “optimal expansion” which provides maximum efficiency.

Case (c): At higher altitude, Pa < Pe this is called as “under expansion” which reduces the efficiency.

Now let’s discuss the technicalities and the science behind it.

The rocket nozzle which is most frequently used in rockets is called as de Laval nozzle. It is named after its developer Gustaf de Laval, and first used in a rocket engine developed by Robert Goddard. Let’s consider a half-section of the nozzle (as it is symmetry around the longitudinal axis) as shown in Figure below.

Here —

P_{a} is the ambient pressure

P_{com. chamber }is the combustion chamber pressure

P_{e }is the exit pressure (of the exhaust gases)

A_{e }is the exit area.

As per the Newton’s third law of motion (to every action there is an equal and opposite reaction) P_{e} = P_{com. chamber}.

The thrust equation of the rocket nozzle is:

T = mU + A_{e} (p_{e} – p_{a} )

Where,

m is mass of rocket’s exhaust

U is the exhaust velocity of gases

While solving this equation to get maximum thrust (using differential equation, momentum and continuity equation) we get maximum thrust at optimum expansion i.e. p_{e} = p_{a}.

A nozzle longer than this point results in net force in negative thrust direction due to over-expansion (as the pressure acting on the inner walls is less than p_{a}). The nozzle shorter than the optimal expansion position would result in a net force in positive thrust direction due to under-expansion (as the pressure acting on the inner walls is higher than p_{a}). But it should be noted that the exiting momentum is not fully regained from p_{e} in case of shorter nozzle. Hence, nozzle design is only optimal at one altitude (as Pa decreases with height).

Magnetic nozzles have been proposed for some types of propulsion (for example VASIMR), in which the flow of plasma or ions are directed by magnetic fields instead of walls made of solid materials. These can be advantageous, since a magnetic field itself cannot melt, and the plasma temperatures can reach millions of kelvins. Currently they are in experimental and research phase.

To keep this newsletter brief, we provide a list of other designs proposed for altitude compensation:

- Aerospike engines
- Single Expansion Ramp Nozzle (SERN)
- Plug nozzle
- Expansion-deflection nozzle

Now some food for thought. Figure below shows the interior of a rocket nozzle. Why are some rocket nozzles have their interior grooved?