The Shape of Rocket Exhaust

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 the figures below—

To know the reason behind this, we need to first understand the importance of a 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 stated, 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 a 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 traveling 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 was first used in a rocket engine developed by Robert Goddard. Let’s consider a half-section of the nozzle (as it is symmetrical 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 Newton’s third law of motion (to every action there is an equal and opposite reaction) Pe = Pcom. chamber.

The thrust equation of the rocket nozzle is:
T = mU  + Ae (pe – pa )
Where,
              m is the mass of the rocket’s exhaust
              U is the exhaust velocity of gases

While solving this equation to get maximum thrust (using the differential equation, momentum, and continuity equation) we get maximum thrust at optimum expansion i.e. pe = pa.

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 pa). The nozzle is shorter than the optimal expansion position would result in a net force in a positive thrust direction due to under-expansion (as the pressure acting on the inner walls is higher than pa). But it should be noted that the exiting momentum is not fully regained from pe in case of a 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 is 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 the 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. The figure below shows the interior of a rocket nozzle. Why are some rocket nozzles have their interior grooved?