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

1. Jun 12, 2012

### cephron

An example of what I mean:
Suppose you had a blueprint for a chemical rocket.
You build one, and it has mass m and provides thrust x.
Suppose you scale the whole blueprint up by 1% and build another.
The volume (and therefore the mass) of each part in the rocket has increased by a factor of 1.01*1.01*1.01 = 1.030301, so the new rocket masses 1.030301m.
In theory, should it provide 1.030301x thrust?

General idea: provided the materials stay within the range of temperatures/pressures/etc in which they function properly--basically, ignoring the engineering problems that would no doubt arise--does doubling the mass of a rocket double its thrust?

2. Jun 12, 2012

### Andrew Mason

Thrust is determined entirely by the rate of change of momentum of the expelled rocket gases. So if the speed of those gases does not change, if you double the mass of the rocket you have to double the rate at which mass is expelled. If that is determined by the cross-sectional area of the rocket engine so that if that area doubles when you double the rocket mass and if the rate at which mass is expelled per unit area remains the same, the rocket thrust should double.

AM

3. Jun 12, 2012

### bahamagreen

Thrust is the force directed out the back end, which depends on the rate and mass of the material being expelled. The size of the fuel tank does not change the thrust...

If you look closer at thrust as the expulsion of matter out the back end at some fixed rate, you do get something interesting happening if the rocket is traveling through the atmosphere...

When the rocket first launches, the speed of the material out the back is fast and the speed of the rocket through the air is slow. So at a particular point "x" in the air where the rocket goes by in the positive "x" direction, you would observe that the material being expelled at "x" will have a velocity in the "-x" direction.

As the rocket increases speed, this velocity of the expelled material with respect to the rocket is constant, but with respect to the static air, the relative backward velocity of the expelled material is decreasing.

Eventually, if the constant rate of expelled material out the back of the rocket is, say, 2000m/s, then there comes a point where the rocket itself is traveling forward through the air at 2000m/s... so at that speed, the material being pushed out the back has a speed of 2000m/s with respect to the rocket, and the rocket is traveling forward at 2000m/s as well. The net result of this is that the material coming out the back of the rocket is being placed AT REST with respect to the still air the rocket is going through!

This speed of the rocket is the maximum efficiency of the rocket because all of the energy of forward motion of the fuel in the rocket is being used by placing the expelled material dead still in the dead still air.

If the rocket continues to increase speed, the subsequent material expelled will actually have a velocity with respect to the still air in the forward direction of the rocket, so some of the efficiency is now lost. So the rocket is most efficient when traveling through the air at the same rate as the material expelled is leaving the back of the rocket - so as to place that expelled material in the still air at rest.

But... for a normal rocket launch, the rocket needs to travel through the denser lower air up into the thinner higher air... the density of the air provides drag, and the combination of rocket speed and air density at a particular altitude provides what is called dynamic pressure.

At low altitudes the rocket is moving slower through denser air and too slow a fuel expulsion won't lift it, but as it gains speed at higher altitudes it is moving through thinner air. Dynamic pressure is included in the determination for how much thrust is need to be efficient. If you recall the shuttle launches, there is a point after about 90 seconds where the controllers do what they call "throttle back". The thrust is not constant throughout the launch; at the beginning the thrust is around "115%", later they take it back down to "100%"... that 100% is that thing mentioned above - where the thrust rate out the back equals the rocket speed through the air (kind simplified), to get maximum efficiency for most of the launch...

4. Jun 12, 2012

### D H

Staff Emeritus
Ignoring things such as change in specific impulse, scaling every linear dimension by a factor of 1.01 should provide 1.02 times the thrust. Note: Saying 1.0201 is incorrect because that would be lying about precision.

It's a cube-square law thing. Example: A 20 gram mouse that is 90 mm long (sans the tail) can jump 3 meters high, 33 times as high as its body is long. A 5000 kg elephant that is 6 m long (sans trunk, tusks, and tail) can't jump 200 meters high. Elephants can't jump, period.

Given the fuel, the two biggest factors that dictate the thrust from a rocket are the throat area and the exit area. Area scales with the square of a length scale. Volume (mass) scales with the cube of a length scale. Another way to put it: Length scales as the mass1/3, so (ignoring details such as change in Isp), thrust scales as mass2/3.

5. Jun 12, 2012

### bahamagreen

Seems to me that when the throat/exit area is increased the rate of flow will decrease resulting in diminished thrust...?

6. Jun 12, 2012

### cephron

@Andrew: Yes, I understand the basics of exhaust velocity and mass ejection rate. If I were to deal with these as givens, the answer would be something we could calculate. I have no idea how scaling up the size of a combustion chamber would affect these, though...this is kinda what I'm wondering.

@D_H: I see, you make some good points about cube vs square laws. But I don't think the thrust/mass ratio should decrease. Consider, instead of doubling the mass of the second engine, simply adding a second, identical engine. The two engines together make a system which is double the mass, and double the thrust. It also has double the fuel consumption and double the mass ejection rate, although the exhaust velocity remains unchanged. Could a single, double-mass engine perform better than the two identical smaller ones is kinda what I'm wondering. By "perform better", I guess I mean burn more fuel per second than the pair of smaller engines, but at the same efficiency?

Intuitively, it doesn't make sense to me that it should do worse...otherwise, space shuttles should contain huge arrays of smaller engines. It sounds like a naive sort of intuition, but I'm just wondering how the thrust/mass ratio behaves as mass is increased.

Last edited: Jun 12, 2012
7. Jun 12, 2012

### 256bits

Generally, all things being equal, to achieve twice the thrust the engine and componesnts would scale up by √2, so the extra mass for double the thrust engine would be 1.41 times as much as two engines. Structurally, two engines or one double sized engine would need pretty much the same load bearing members to attach to the rocket ship, so there is questionable difference there in what the mass gain/loss would be.

But the scaling up is most likely within a limited range. For example, as piping is made larger, pipe walls have to be made thicker to withstand the pressures, which adds to mass, so the 1.41 is just a figure to start with rather than what is achievable.

Perhaps that is one reason the space shuttle does not have one humungous engine,besides others, is that the plumbing might have been cumbersome with only one engine. A many engine design is most likely avoided for the simple reason that complexity increases and with it chances of failure, although redundancy could be a design topic also.

Efficiency of the combustion chamber and nozzle is one thing the engineers would have to tweak as the system scales up ( down).

8. Jun 12, 2012

### haruspex

You are right in your intuition that increasing all linear dimensions by the same percentage should produce the same answer. In fact, since that would include increasing the acceleration (distance scaled up but time the same), the thrust/mass ratio would increase. But what you're overlooking is that that would include increasing the exhaust velocity by the same percentage and increase the exhaust mass/second cubically. If the linear factor is r we have the following factors:
- rocket mass: r3
- exhaust velocity: r
- exhaust mass/sec: r3
- thrust: r4
- thrust/mass: r
If you keep the exhaust velocity the same and only scale up exhaust mass/sec by r2 (for the increased x-sectional area) then the thrust/mass ration will go down by r.

9. Jun 12, 2012

### D H

Staff Emeritus
Not at all. Exhaust velocity is first and foremost a function of the type of fuel being burned. This is what makes specific impulse a useful concept. It would not be useful if changing the size of a rocket engine changed the exhaust velocity by a significant fraction. Isp does change as an engine is scaled up, but the change is small, and it's not always an improvement.

10. Jun 12, 2012

### haruspex

You miss my point. I'm saying that the notion of scaling every linear dimension up equally would imply scaling the exhaust velocity by the same amount. Whether this is actually feasible is another matter.

11. Jun 12, 2012

### cephron

Ok. Thank you very much, haruspex, for your helpful analysis, and for clarifying your position concerning scaling exhaust velocity. I think, for the problem I'm facing, I will need to respect the specific impulse of the propellants involved. But is it possible that the exhaust ejection rate would still increase cubically?

12. Jun 12, 2012

### cephron

Actually!

Take a look at > this. <

I know I said chemical rocket in my first post, but this is actually a lot closer to what I'm thinking of.

13. Jun 12, 2012

### D H

Staff Emeritus
That's ridiculous. You still have to obey the laws of physics. Exhaust velocity doesn't scale. It would violate conservation of energy.

14. Jun 12, 2012

### cephron

Please don't start an argument, guys. Check out the link I posted--this sort of engine (more in the ion engine category than chemical rockets) can vary its exhaust velocity. Since I am ultimately interested in magnetic/ion based propulsion systems, perhaps increases in exhaust velocity are feasible. (Like I said before, apologies for starting the thread using chemical rockets as an example)

15. Jun 12, 2012

### D H

Staff Emeritus
We don't know how to make big ion propulsion engines. Make them a bit bigger and they work the more or less the same. Not much change in specific impulse, but a bit more thrust. Make them a lot bigger and they don't work.

16. Jun 12, 2012

### D H

Staff Emeritus
Here's a good example of what I'm getting at. The RL10B-2 (277 kg), the Space Shuttle Main Engine (3,526 kg) , and the RS-68 (6,600 kg) all burn liquid oxygen and liquid hydrogen. Their specific impulses are 464 seconds, 452.3 seconds, and 410 seconds, respectively. Exhaust velocity does not scale with size.

The low Isp for the RS-68 results from a design goal to make that engine relatively simple and inexpensive. It's used once and ditched in the ocean. The Shuttle Main Engine was reusable.

17. Jun 12, 2012

### haruspex

Perhaps. If the area only increases quadratically and the exhaust velocity (as dist/time) doesn't change then you'd need to increase the exhaust density somehow.

18. Jun 12, 2012

### 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.

19. Jun 13, 2012

### D H

Staff Emeritus
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.

20. Jun 13, 2012

### 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.

21. Jun 13, 2012

### 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.

22. Jun 13, 2012

### 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.

23. Jun 13, 2012

### D H

Staff Emeritus
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.

24. Jun 13, 2012

### 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.

25. Jun 13, 2012

### D H

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