# Validity of rocket pressure thrust term

• chyvak
In summary, the pressure thrust term is included in the equation of motion of a rocket when the engine is part of a rocket system. It is a correction arising from the nature of the data used to calculate the aerodynamic forces acting on the rocket.
chyvak
My first post, and a very long one...

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.

Usually they are just accounting for the difference in thrust between operation at atmospheric pressure and operation in vacuum. I doubt there's much accounting for aerodynamic forces - too complicated. Even the exhaust plume modifies the flow around the vehicle.

That is exactly my point. The correct term to include in the equation of motion is the integral of the pressure and shear stresses over the system boundary. Since this is indeed very complicated when the engine is operating, that integral is replaced by the integral over an unpowered system which is precisely the aerodynamic forces acting on a non-operating rocket. The "pressure thrust" is a correction added during powered flight to account for this simplification. For a static system this correction is exact, in all other cases it is an approximation. What is confusing (to me) is when a textbook presents the equation of motion of a rocket with the “pressure thrust” and aerodynamic forces both present without explaining the link between them. My point is that if we actually could calculate this integral over the system boundary for a working rocket, there would be no explicit “pressure thrust” term. Agreed ?

chyvak said:
My first post, and a very long one...

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.

As you said, so far so good here (I notice you didn't mention the momentum thrust term at all above, but since it is irrelevant to your question, this doesn't matter).

chyvak said:
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?
Usually, the aerodynamic drag (and lift) term is considered to be a thrust-independent condition. As you mention, there are certainly plume effects caused by the rocket motor being in operation that change the overall aerodynamic force acting on the rocket, but by far the most convenient way to deal with this from a modeling point of view is to look at the rocket's aerodynamic characteristics independent of the rocket engine effects, and then consider the motor as applying a thrust to the system that includes the pressure thrust term. If you just care about a relatively simple model, this is frequently sufficient. If you want a more advanced model, the rocket is analyzed as a complete system, without breaking the forces down into completely discrete "drag" and "thrust" terms (which is really not possible, for the reasons you mentioned).

Besides, consider a rocket operating outside of the atmosphere. The correct equations of motion for such a rocket will include a pressure thrust term equal to Pexit*Aexit, but there will be no drag acting on the rocket. Are you suggesting that it is more reasonable to consider this as a negative drag force rather than as a thrust caused by the engine?

Finally, as far as mathematical convenience is concerned, this term really makes more sense as a thrust term than a drag term. In the flow regime in which pretty much all rockets operate, drag terms should scale linearly with the density of the medium and scale (approximately) with the square of the velocity through the medium, but the pressure thrust term is only very loosely dependent on the velocity (since it changes the pressure distribution behind the rocket), and it is dependent on ambient pressure, not on ambient density. On the other hand, it does scale perfectly with the engine chamber pressure, which is directly related to throttle position. This means that it is much more convenient to account for it as a thrust term.

chyvak said:
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.

This is all fairly true, though there are plenty of CFD simulations where the full engine operation is accounted for. However, in a simple model, the basic assumptions are all you can really do, since it dramatically complicates the calculations to include the effect of the engine in the overall pressure distribution around the rocket. Pre-CFD, not a lot better could be done, as far as I know (though you could improve on it a bit by measuring the pressure at the exit plane of your model's engines in the wind tunnel, so the drag contributed to your model by the low pressure region behind the rocket could be corrected for).

Also, why do you say that the powered flight pressure distribution in the nozzle plane will be different from what it would be on a test stand? As long as you didn't have flow separation from the nozzle walls when you were testing the engine, and as long as there are no differences in fuel feed rate or chamber pressure in flight vs on the test stand, there shouldn't be any difference in exit pressure distribution at all.

chyvak said:
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.

Why do you say that pressure is momentum flux? Maybe I'm just misunderstanding you here, but pressure does not require any momentum flux to be crossing the boundary of the system...

chyvak said:
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.
There most certainly are other pressure forces to account for here. There's buoyancy, for example (but buoyancy is typically ignored because it's so small). Another is that pressure term in the expression for rocket thrust. Like buoyancy, this term is ever present in the sense that unlike drag and lift, it doesn't depend on velocity relative to the air flow.

Thank you for your detailed reply. It was exactly what I was hoping for.

cjl said:
Are you suggesting that it is more reasonable to consider this as a negative drag force rather than as a thrust caused by the engine?

No, I am not trying to classify this term as either a thrust term or a drag term, merely to understand where it is coming from. In analyzing the forces acting on the system I could identify the above mentioned integral of pressure and shear stresses over the entire boundary as one of them. To me that took care of the external pressure whether the rocket is moving or not. I then got confused when I looked at the rocket equation of motion in various textbooks where they included both drag/lift and the pressure thrust, because my understanding was that the drag/lift was precisely that integral of pressure/shear over the entire rocket plus nozzle. So there should be no explicit pressure thrust term in the same equation. Re-reading my original post I can see why my labelling of this integral as ‘aerodynamic forces’ may have polluted the actual message I was trying to get through.

cjl said:
Also, why do you say that the powered flight pressure distribution in the nozzle plane will be different from what it would be on a test stand?

I am not. But it may be a language barrier (not native english speaker). I am saying that the pressure distribution in the nozzle exit plane during powered and unpowered flight will be very different. The point I was trying to make is that the integral of pressure/shear is hard to evaluate during powered flight so I assumed the evaluation is carried out on a non-powered vehicle. And this is then not directly applicable to powered flight so a correction accounting for the actual pressure/shear in the nozzle is needed. And that would be the pressure thrust term. This way I can explain to myself why that term is there at the same time as the drag/lift forces.

cjl said:
Why do you say that pressure is momentum flux? Maybe I'm just misunderstanding you here, but pressure does not require any momentum flux to be crossing the boundary of the system...

Agree. That was a very imprecise formulation on my part. What I meant to say was that when I analyze the interchange of linear momentum in the system, at no point does pressure show up. Thus I did not understand their statement suggesting that such an analysis had somehow missed an additional contribution to momentum exchange caused by a pressure difference across the system boundary. To me all the linear momentum interchange has already been considered in the analysis of the system, so this extra pressure-difference induced linear momentum they refer to was simply non-existing, or already taken care of.

D H said:
There most certainly are other pressure forces to account for here. There's buoyancy, for example (but buoyancy is typically ignored because it's so small). Another is that pressure term in the expression for rocket thrust. Like buoyancy, this term is ever present in the sense that unlike drag and lift, it doesn't depend on velocity relative to the air flow.

But is buoyancy really something extra that needs to be included in addition to the integral of the pressure/shear over the entire system boundary? I don’t believe so. If we were able to integrate the stress tensor (pressure and shear) over the entire system boundary then I don’t see what other forces apart from gravity that need accounting for. Pressure thrust, buoyancy and drag/lift – they are not distinct forces with separate origins. They all come from the integral of the stress tensor. Our inability to calculate that integral or desire to simplify things may require us to introduce another layer of modelling where this term is split up into more tractable parts, like a buoyancy term, drag/lift, pressure thrust. What confused me is that this additional layer of modelling is introduced in textbooks with no explanation. And with no explanation of what that model is they have difficulties explaining exactly what e.g. the pressure thrust is and what it's origins are.

chyvak said:
. Pressure thrust, buoyancy and drag/lift – they are not distinct forces with separate origins. They all come from the integral of the stress tensor.

This was a bit imprecise. I am well aware that external fluid flow around a rocket and internal fluid flow in a rocket are very different and have different origins. What I mean to say here is that from a system analysis perspective they are just stresses on the system boundary.

chyvak said:
But is buoyancy really something extra that needs to be included in addition to the integral of the pressure/shear over the entire system boundary? I don’t believe so. If we were able to integrate the stress tensor (pressure and shear) over the entire system boundary then I don’t see what other forces apart from gravity that need accounting for. Pressure thrust, buoyancy and drag/lift – they are not distinct forces with separate origins. They all come from the integral of the stress tensor. Our inability to calculate that integral or desire to simplify things may require us to introduce another layer of modelling where this term is split up into more tractable parts, like a buoyancy term, drag/lift, pressure thrust. What confused me is that this additional layer of modelling is introduced in textbooks with no explanation. And with no explanation of what that model is they have difficulties explaining exactly what e.g. the pressure thrust is and what it's origins are.

I agree with this - if you somehow got the pressure and shear at every point around the rocket, you could just integrate that to find the force applied, and then add in gravity and momentum flux to get the net force. Similarly, you could draw a very large box around the rocket that encompasses a fairly large fluid volume, and integrate the net momentum flux through the control volume's boundaries to find the force on the rocket. However, there is no simple way to obtain such detailed knowledge about the flow everywhere, hence the normal breakdown of the forces into separate terms.

cjl said:
I agree with this - if you somehow got the pressure and shear at every point around the rocket, you could just integrate that to find the force applied, and then add in gravity and momentum flux to get the net force. Similarly, you could draw a very large box around the rocket that encompasses a fairly large fluid volume, and integrate the net momentum flux through the control volume's boundaries to find the force on the rocket. However, there is no simple way to obtain such detailed knowledge about the flow everywhere, hence the normal breakdown of the forces into separate terms.

Thank you very much. I believe I have the full picture now.

chyvak said:
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.

Maybe the "additional linear momentum caused by pressure differences" is pointing to the difference between the speed of the exhaust particles and the ambient static air?

The rocket moving through the air is "working best" when the air speed of the rocket and the exhaust speed relative to the rocket are the same magnitude... so that the exhaust particles are being placed statically into the ambient air in a line behind the rocket.

At slower rocket speeds, a portion of kinetic energy is retained by the exhaust particles moving backwards with respect to the ambient air; and at faster rocket speeds, the exhaust particles retain some kinetic energy moving forward with respect to the ambient air... but at rocket speed where the particles are placed into the air at rest with respect to the air, the most kinetic energy is retained by the rocket itself.

Maybe the "pressure differences" is referring to the difference between exhaust speed relative to the rocket vs exhausted particle speed with respect to the ambient air? And maybe the "additional linear momentum" is pointing to rocket speeds where there is a difference between these two?

A static rocket test would always be in the case of the particles exhausting with a backwards kinetic energy with respect to the air. Without a wind tunnel to produce air flow speeds comparable to the exhaust speed, maybe this term is used to compensate for that?

bahamagreen: That's only because you're considering the air to be your frame of reference. You could pick a different frame of reference and get a different result for the energy split between rocket and exhaust, and it really makes no difference. With a jet, it does matter because the reaction mass is the air that it is traveling through, but for a rocket, the reaction mass is carried on board.

In fact, depending on how you define working "best", you could make a case that a rocket works best when the pressure thrust term goes to zero (that is, when the exit pressure of the rocket motor equals the ambient pressure), or you could make the case that the rocket works best when the ambient pressure is zero. However, I really don't see any justification for claiming that a rocket is working best at any particular speed, especially since for most rockets (at least the ones that are used on launch vehicles), the majority of their operation will be in a near-vacuum environment.

## 1. What is the rocket pressure thrust term in physics?

The rocket pressure thrust term refers to the force generated by the exhaust gases of a rocket as they are expelled from the nozzle. This force is what propels the rocket forward.

## 2. How is the rocket pressure thrust term calculated?

The rocket pressure thrust term is calculated using Newton's third law of motion, which states that for every action, there is an equal and opposite reaction. The force generated by the exhaust gases is equal to the rate of change of momentum, which is the mass flow rate of the exhaust gases multiplied by their velocity.

## 3. Why is the validity of the rocket pressure thrust term important?

The validity of the rocket pressure thrust term is important because it is a crucial factor in determining the performance and efficiency of a rocket. If the calculation of this term is inaccurate, it can lead to errors in the design and operation of the rocket.

## 4. What factors affect the validity of the rocket pressure thrust term?

The validity of the rocket pressure thrust term can be affected by various factors such as the type and design of the rocket nozzle, the composition and temperature of the exhaust gases, and the ambient pressure and temperature of the surrounding air.

## 5. How can the validity of the rocket pressure thrust term be improved?

The validity of the rocket pressure thrust term can be improved by using advanced computational models and simulations, conducting experiments and tests, and continuously refining and improving rocket design and technology.

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