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## On Pressure Fed Rockets and Scaling

Well, scaling seems to be my pet issue. I recently wrote something not entirely well reasoned in a comment at Paul Breed’s. (For some reason Chrome complains about blogrolling.com malware there so continue if you’re sure you’re safe.)
So let’s make it better. (A word of caution though, I’m quite sleep deprived now.)
For those who like to jump into conclusions, it’s going to be like this all over again.

Assume a pressure fed rocket first stage has a certain propellant chemistry, tank and thus chamber pressure and must operate in the atmosphere, hence has a certain exit pressure (0.1 MPa or 1 bar or 15 psi is the optimal). Then it has certain thrust per nozzle area.

Now, the rocket needs thrust to lift off. If we assume a constantly scalable shape, its mass will be base area times length times density.

Since the maximum fittable nozzle exit plane also depends on the base area, we find that for a certain area, the rocket can only have so much mass – or that the rocket has a maximum density times length parameter. If we assume the propellants have been picked early on, density is set and the rocket only has a height constraint.  Each pressure and propellant chemistry basically has a “characteristic length” that can’t be exceeded. Otherwise it can’t lift off.

The higher the exhaust velocity, the smaller the nozzle, so raising chamber pressure reduces the needed nozzle size per thrust and the rocket can be lengthened.

For small rockets, I’d hunch that they have little length and thus they don’t really have to worry about this. They can be as thin (and thus long) as practical, to try to avoid drag losses.

For upper stages, the thrust to weight needed is less and the weight even less so it’s even less of a problem – except that the expansion ratios can be huge since there’s no back pressure anymore. Still, with small rockets, pretty huge expansions might be possible without having much problems because the second stage is very small (=also short) anyway and thus there’s little mass per nozzle exit plane area.

On really tall rockets like Saturn V, the thrust per base area has to be huge, hence it had to have those base extensions for the corner engines (note how the N-1 had a conical shape with a wider base, the engines had a bit higher pressure but the upper stages were kerosene – these cancel out a bit but the base of the rocket had some empty space) . Similarly with STS, putting so much thurst on the tiny orbiter’s tail required high chamber pressures and some tail shaping

I don’t have any numbers handy, but if we assume a 10 m tall 1000 kg/m^3 density (water) rocket, then it has 10,000 kg per m^2 or the thrust required for a 20 m/s² acceleration is 200,000 N/m^2. This is easily achievable. With an exhaust velocity of 2000 m/s, the mass flow needs to be 100 kg/(s*m²) to produce that thrust. Again with the exhaust velocity that mass flow means a density of 0.05 kg/m^3. Air’s density is 1.2 kg/m^3 at 300 K, so that’s 20 times less dense which means hotter, the density is like hot air at 6000 K. Though the molecules might be mostly lighter OH instead of N2 and O2, making that rocket exhaust at 3000 K for the density. Rocket exhaust isn’t that hot – it’s cooler and denser and thus more thrust per unit area.

For a second stage we can look at the pressure fed AJ-10 from Delta 2: 1.7 meters diameter (certainly constrained), 40 kN of thrust. For a T/W of 1, density of 1000 kg/m^3, we get 4 tonnes and 1.7 meters of depth. Quite a stubby stage with a roughly spherical tank! Isp is 321 s. The real Delta II second stage weighs 7 tons and the payload is some too, but reusable rockets won’t have such high performance first stages (nevermind solids!), so they might need more T/W.

Oh, BTW, I assume three stages to orbit for pressure feds though I haven’t looked it that closely.  Mass ratios and ISP:s I’ve only hunched.

## Humans Fast, Machines Slow

Rand Simberg talks about impedance matching. So I’d like to make a post of my comment there (I’ve always wondered why this obvious alternative gets mentioned so little…)

## What to do when you arrive at Mars or Earth with your solar electric propelled vessel?

So, the problem with most low fuel demand velocity change schemes is that they only give slow accelerations. Low fuel high velocity change means solar or nuclear electric propulsion and aerocapture mainly.

High delta vee aerobraking is hard to do in one pass – it gets dangerous because of atmospheric variability and potentially other reasons.

Simple: detach a small capsule with the humans that goes directly to the surface (with only days of life support) and leave the untended craft to do multi-pass aerobraking. Hitting van Allen belts a few more times or taking a long time doesn’t matter that much with no humans onboard.

You could also potentially ultimately leave the long distance craft at some Lagrange point instead of LEO. (Cue some clever and complex maneuvers to save fuel – maneuvers that take long.)

Something similar could also be done when a long distance stack is assembled in LEO: send the humans there only after it’s through the belts. They can go with a smallish capsule again. Potentially at some Lagrange point, or in space without any fixed reference, just along the way. It could be dangerous though if the capsule doesn’t have much life support.

Many of these things have potential delta vee penalties as well as timing inflexibilities, but they could have enough other benefits that they should be considered.

I was not completely happy with the last post’s vagueness so I’m adding more formal treatment here. This long post still ends up lazily speculating around the advantages and disadvantages of air breathing propulsion in the end though. 🙂

Effective or apparent exhaust velocity (ISP to some, but that is an old-fashioned troublesome term so I won’t use it) is $v_{exeff}=\frac{F}{\dot{m}_{prop}}$. That means, thrust for a certain propellant flow. Higher is better.

For rockets, the thrust $F_R=\dot{m}_{propR}v_{exR}$ but for air breathers, $F_A=\dot{m}_{totA}v_{exA}$. Air breather is denoted by A and rocket by R.

Thus, for the rocket, the effective exhaust velocity is the real exhaust velocity (it exhausts only propellant products). $v_{exeffR} = v_{exR}$.

For the air breather it’s more complicated. Only fuel is used as propellant flow but a lot more is expelled:
$v_{exeffA} = v_{exA}\frac{\dot{m}_{totA}}{\dot{m}_{propA}}$

So, what is the ratio of fuel flows for a rocket and an air breather of same thrust? The inverse ratio of effective exhaust velocity of course. How much better is an air breather in this case?

$R_{breathe}= \frac{v_{exeffA}}{v_{exeffR}} = \frac{v_{exA}}{v_{exR}}\frac{\dot{m}_{totA}}{\dot{m}_{propA}}$ – the Air Breather’s advantage.

So, the rocket might benefit from a higher real exhaust velocity but the air breather benefits from the ratio of total exhaust mass versus propellant mass.

But that’s not all.