Rockets - in theory, does thrust scale linearly with mass?

haruspex

DH, I'm saying X implies Y; you're saying Y is false. Where's the problem?
Anyway, it wouldn't necessarily violate conservation of energy. You'd have to increase the power somehow, maybe by burning fuel faster but still funnelling it through the same exhaust aperture. I expect there's no reasonable way of doing that, but it does not violate the laws of the universe.

D H

Yes, it does violate the laws of the universe. Exhaust velocity is a function of the specific chemical potential energy of the fuel.

Consider LOX/LH₂. Burning oxygen and hydrogen at an 8:1 (stoichiometric) mixing ratio yields 242 kilojoules per mole of water vapor produced, or 1.344×10⁷ joules per kilogram of oxygen+hydrogen. Multiplying by two and taking the square root yields 5185 meters per second, or a specific impulse of 529 seconds. That's the highest possible exhaust velocity with this LOX/LH₂, and that assumes the exhaust is perfectly collimated and leaves the nozzle at absolute zero. In reality, the specific impulse will always be less than this upper limit.

Aside: LOX/LH₂ engines are almost always run fuel-rich, typically a 4:1 or 6:1 mixing ratio. This reduces exhaust velocity but has other benefits that outweigh this reduction in specific impulse.

haruspex

You're assuming all of the fuel is used to propel its own combustion product. Is there any inviolable reason why some could not be used in other ways, e.g. to superheat other fuel prior to combustion? As I keep saying, I don't claim that there is any practical way to increase exhaust speed, nor does it matter for my original post whether it would violate any laws of physics. But I'd like you to understand that my original post was not ridiculous.

Pkruse

Many good answers, but I'll make an attempt to distill them down into simple terms.

In a real solid rickety motor the thrust is a function of the case pressure and the exit cone design. The pressure is determined by the burn rate, which is often designed to change as the fuel burns, because more thrust is often desired at lift off when the rocket is heavier. Since the fuel only burns on the surface, burn rate is often controlled by designing the surface area to change as it burns. The fuel has a hole down the center. If that hole is star shaped, it will have more surface area. Then as it burns the star becomes a cylinder and the burn rate slows down. Also, the fuel is often poured non-homogeneously such that a hotter chemical mix burns first, and then a different chemical mix burns more slowly.

So doubling the fuel mass has nothing to do with thrust. If you change nothing else, you will have the same thrust but twice the burn time.

cephron

Thanks, Pkruse! Fuel isn't what I'm actually worried about, though. We can pretend, for the purposes of this question, that the fuel materializes out of nowhere at the injector. My concern is specifically about thrust capability as a function of engine mass, not fuel mass.


Anyways...if we assume exhaust velocity remains essentially unchanged, as D_H points out is the case with chemical rockets, then a rocket's thrust is directly proportional to fuel burn rate. So the question becomes, how does maximum fuel burn rate change as a function of engine (not engine+fuel) mass?

Here's some fictitious stats that highlight the question:

Rocket A (single unit)
-mass: 1 ton
-exhaust velocity: 3 km/s
-maximum fuel burn rate: 20 kg/s

Rocket A (x2)
-mass: 2 tons
-exhaust velocity: 3 km/s
-maximum fuel burn rate: 40 kg/s

Rocket B (single unit)
-mass: 2 tons
-exhaust velocity: 3 km/s
-maximum fuel burn rate: ???

Would rocket B's maximum fuel burn rate increase linearly with mass? (y = x)
Would it go up by a factor of the square root? (y = x√x)
Something else?
I know it's more complicated than this, but I'm looking for a general sort of principle, like how flow rate increases relative to pipe diameter.

D H

You assumed that scaling the rocket meant that exhaust velocity scales. That's a faulty assumption. Rhetorical question: Why didn't you scale the intermolecular / interatomic distances by that same length factor? The answer is doing so doesn't make sense. The distance between the iron atoms in the body of a little HotWheels car are exactly the same as in the HotWheels' full-size counterpart.


That's just wrong. The OP is scaling lengths. This of course changes mass, but it also changes a lot of other things. Increasing the diameter of the model rocket by some factor f and increasing the height by that same factor increases the quantity of fuel by a factor f³. It increases the area of the burning fuel by a factor of f². Thrust increases, but not by the same factor that mass increases. It's a cube-square law problem. The cube-square law pops up all over the place in the world of engineering. This is one of those places.

Pkruse

Looks like I misunderstood the discussion. My mind was stuck on solid fuel, but we are talking about liquid fuel. Very little of what I said would apply to liquid fuel motors.

D H

What you said doesn't apply to solid fuel rockets, either. They too are subject to cube-square law constraints.

haruspex

No, you still don't get what I said. I said that if you want to take a physical set-up and scale up the distances, and expect it to behave the same way, then, at the least, you have to scale all distances (and even that might not work). Since time is not scaled, that would imply scaling the exhaust velocity. This does not depend in any way on whether that would be feasible.

russ_watters

I agree with DH. "Exhaust velocity" is not a physical dimension of the rocket, read off of a blueprint. The question does not in any way imply scaling it. In fact, 'what happens to exhaust velocity' is one of the questions being asked - not a starting assumption of the question!

256bits

In that case the answer is as simple as DH proposed, if simple scaling of the blueprint is done. And I still don't get if the OP ( what does OP mean anyway ) means a whole rocket ship or just the engine scaling as the post seems to jump around.

Fact is, it is a cylindrical dimensional problem, not a volumetric. A surface area analysis would be more apt for a rocket whcih as far as I know are shaped like a tube, within which mostly all components are either tubes or hollow vessels.

The skin of the rocket ship does not have to be scaled up in thickness to have the same load carrying capacity. Same for the tubes in the plumbing. The pressure does not change as the system is scaled up in any proportion to length or mass but stays the same. Pumps do not have to be scaled up as r^2 but only as r^2. Again, combustion chamber volume would have to be scaled in proportion to length. One doe not need to cube the size of the combustion chamber if the size if the engine throat area is squared,

The only question I can see here for scaling is to a achieve a goal - altitude and acceleration. How much more or less fuel does one need to carry if a rocket ship is scaled up in linear dimenssions so that the same acceleration and burn time will carry the rocket to the same orbital altitude as before, and will the rocket ship be able to carry that amount of fuel within those linear dimensions, or can it carry less. In both these cases a redesign would be most beneficial for economics andf efficiency.