- #1

chyvak

- 6

- 0

I have problems understanding the prolific use of a certain term in the rocket thrust equation and its validity.

The general rocket thrust equation contains a term referred to as the pressure thrust. It is calculated as (p_e – p_a)Ae where p_e is the pressure in the nozzle exit plane, p_a ambient pressure and Ae the area of the nozzle exit plane. Both pressures assumed constant. Many textbooks give no satisfactory (to me) explanation of the origin of this term. However, the term naturally pops up in a careful analysis of a system whose outer boundary is defined by the rocket engine plus the nozzle exit plane. By accounting for the interchange of mass within this system it can be considered closed for infinitesimal periods of time. Then we can apply Newton’s 2. law to the system which tells us that the rate of change of linear momentum equals the sum of external forces acting on the system. The forces acting on the system are gravity and pressure forces. If the rocket engine is static integration of the pressure over the system boundary yields precisely the pressure thrust term (given the assumptions on constant pressure).

So far so good.

What I don’t understand is that this term is also included when the rocket engine is part of a rocket and what is sought is the motion of the rocket. In that case the system is no longer the rocket engine. The system boundary would most conveniently be defined as the outer surface of the rocket plus the nozzle exit plane. We can then go through the exact same steps as above and find that one of the external forces that act on the system is the integral of pressure over the system boundary. The pressure distribution on the system boundary is the result of the system (rocket) moving through the atmosphere. In addition to the pressure distribution there will also be shear stresses due to viscous effects. But the sum of integrating the pressure and shear stresses over the system boundary is the total aerodynamic force acting on the rocket. And this force is usually accounted for separately by drag and lift forces. There is no other pressure force to account for here, so the “pressure thrust” term completely drops out – it is implicitly included in the aerodynamic force terms. It should not be added explicitly. It would be tantamount to accounting for the pressure distribution in the nozzle exit plane twice.

So why is this term often included explicitly in the equations of motion of a rocket?

I suspect that the aerodynamic drag and lift forces are calculated based on data from wind tunnels or CFD simulations of rockets when the engine is not working. This will certainly give a different pressure distribution in the nozzle exit plane compared to when the engine is working. So if the drag and lift forces acting on the rocket during powered flight are calculated from data related to unpowered flight there needs to be a correction to the total aerodynamic force. That correction should subtract the integral of the invalid, unpowered-flight pressure distribution in the nozzle plane from the total aerodynamic force and add in the integral of the actual, powered-flight pressure distribution in the nozzle plane. If one (incorrectly) assumes that the former pressure distribution is constant p_a, and the latter is (incorrectly) assumed to be constant p_e, then, yes, the “pressure thrust” pops up again. But it is a correction arising from the nature of the data we are using, not a term that should just be added in an ad hoc manner.

NASA hosts a popularized page where the “pressure term” in a static rocket engine is motivated by the need to account for some additional linear momentum caused by pressure differences. This sounds dubious to me. Pressure _is_ momentum flux and we have already accounted for momentum flux in the other terms of the rocket thrust equation.