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This post is from a private email....
The subject was/is kbd512's intuition that thrust of a rocket will increase as atmospheric pressure decreases...
No, Brian is right about the thrust.
The thrust you get in vacuum is KEefficiency*m*Ve + Pe*Ae, where m represents the total massflow coming through the nozzle and Pe is the expanded pressure at the exit plane, and Ve is the expanded velocity at the exit plane.
The thrust you get down in the atmosphere is KEefficiency*m*Ve + Pe*Ae - Pa*Ae, often written as KEefficiency*m*Ve + (Pe - Pa)*Ae, but which really should be written as vacuum thrust - Pa*Ae, where Pa is the surrounding atmospheric pressure (or "backpressure").
A lot of texts initially do not show the KEefficiency factor, because it is usually close to 1 in value with fixed expansion bell geometries.
That KEefficiency factor is quite simply the cosine factor of the integrated average streamline divergence angle, measured off of straight-aft axial.
Once you understand what it really is, then you understand why it applies to the exit momentum term, but NOT to either of the exit plane pressure terms.
If you do this in thrust coefficient form, F = CF Pc At, where CF = (Pe/Pc)*Ae/At)*(1 + gamma*KEefficiency*Me^2) - (Pa/Pc)*(Ae/At).
The first term is vacuum thrust / Pc*At, and the second term is the atmospheric backpressure force / Pc*At, which is Pa*Ae/Pc*At.
The advantage of this mathematical reformulation is that once you have a value of Me, you already have values for both Pe/Pc and Ae/At without iterations.
They're just complicated-looking power functions of gamma and Me.
All you need is a suitable ideal gas model specific heat ratio gamma, which is pretty near 1.2 for just about any rocket gases at all.
Primitive variables or thrust coefficient form, both say the very same thing.
The consensus appears to be that either LH2 or carbon will work for SSTO.
Either one requires more work than can be justified for economic value.
The only way an SSTO will ever be built and flown is if a wealthy individual human decides to take it on as an ego project.
We have many examples of such ego driven investment in human history.
(th)
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