# Static forces on a rocket engine

I am trying to understand where and how the thrust forces act in a rocket engine. I had assumed that the net thrust could be "ball-parked" as the chamber pressure multiplied by the throat area, but this doesn't work out. Take for example the Rocketdyne F-1. It has a chamber pressure of about 1,015 psi, sea level thrust of 1,522,000 lbf, injector plate diameter of about 39 inches (largest interior diameter of the combustion chamber), and throat diameter of about 17 inches. Doing the math, it doesn't work out (using throat area, only about 230,000 lbf). Just looking at the total force acting on the injector plate doesn't work out either (only about 1,212,000 lbf). I've read that there is essentially no pressure acting on the walls of the nozzle (downstream of the throat), so where is the force coming from to act on the body of the engine as thrust?

I understand that thrust results as the reaction to accelerating the propellant, but somehow there must be a force acting against the interior surfaces of the combustion chamber, nozzle, etc., such that when you integrate up all the vertical components of pressures at various points, you get the actual engine thrust. Or am I misunderstanding something?

The nozzle is there to maximize thrust, which is the same thing as maximizing the backward-going momentum of the exhaust gas. To increase that momentum, Newton's 2nd and 3rd laws means that the gas has to be exerting a forward-going force on the nozzle, i.e. generating thrust.

If you have the numbers, you can directly calculate this part of the total thrust from the increase in momentum flow: The increase in backward exhaust velocity from the throat to the nozzle exit, times the mass flow.

How does the expanding part of the nozzle, a static element, increase that velocity? At the macro level, it's usually explained as "reducing the pressure and temperature, turning that energy into backwards velocity". But you can also see it at the micro level: An atom/molecule moving sideways due to temperature and pressure will hit the inclined nozzle surface and have some of that sideways velocity turned into backwards velocity due to the sloped nozzle surface. In return, there's a force on the nozzle that has a forward component, which becomes thrust (and an outward force attempting to expand the nozzle that has to be resisted by the strength of the material).

From Sutton, 4th edition, page 36

The axial thrust can be determined by integrating all the the pressures acting on areas that can be projected on a plane normal to the nozzle axis.

I think your injector area X chamber pressure is a rough approximation but you'd have to take the pressure on all the surfaces including the nozzle to get it exactly.

• So it would appear that, with reference to your attached image, (1) pressure in the nozzle downstream of the throat is not negligible, and (2) pressure acting against the walls of the nozzle downstream of the throat contributes more axial force than the axial force on the chamber walls upstream of the throat "takes away", at least in the case of the F-1. Nov 16, 2019 at 19:18
• Agreed, and consider that the converging portion of the nozzle (contributing negative thrust) is usually a lot shorter than the diverging portion (contributing positive thrust) despite what the schematic shows. Nov 16, 2019 at 19:27
• Since it is the axial component that matters, I would think that it is not length that is relevant here, but cross-sectional area. I also wonder if the effective pressure on the chamber walls upstream of the throat actually decreases due to Bernoulli effect. Nov 16, 2019 at 19:32
• Take a look at page 25 of the pdf (not page number, pdf page) here to see the relative dimensions of the converging / diverging parts of the nozzle large.stanford.edu/courses/2011/ph240/nguyen1/docs/… Nov 16, 2019 at 19:38
• The random sideways motion of the hot gas is converted into systematic axial motion by impacts with the expanding part of the nozzle. This delivers axial thrust via the nozzle and depends on flow rate and throat temperature more than chamber pressure. Nov 17, 2019 at 0:00