Aerodynamic Heating: Equation for Rate of Heating?

[physwil90]

Is there an equation for the rate of heating on a surface at any time, and at any angle?

 

[rcgldr]

I'm not aware of anything simple that is accurate. The mathematical models are usually empirical (based on actual experiements) and complex. It's a combination of friction and compression. At hypersonic speeds, it's mostly due to compression.

 

[K^2]

No, there isn't. It all depends on the regime.

If you are still talking about your supersonic rocket, for the nose cone, you can make an estimate like so:

T_{eff} = \sqrt{T^2 + \left(\frac{m V^2}{k_B}\right)^2}

T is ambient temperature, T_eff is the effective temperature experienced by the nose cone, V is the velocity of the rocket relative to air, m is the "average" mass of air particle, and k_B is the Boltzman's constant.

Yes, they don't really add quite like that, but I don't think you need an exact number. For an estimate, this will do.

Keep in mind that only the very tip of your nose cone should be exposed to that.

I'm not aware of anything simple that is accurate. The mathematical models are usually empirical (based on actual experiements) and complex. It's a combination of friction and compression. At hypersonic speeds, it's mostly due to compression.

That's usually the worst in transonic regime. An estimate from simple ballistics is all he really needs. Not that it's going to account for nearly everything, but he's not building a jet fighter.

 

[physwil90]

To be a little more clear I am trying to get a good empirically value for how hot the back fins of a rocket I am constructing will get. I am assuming that the equation I am looking for will most likely have to be in the form of a rate of change rather than an exact value so that I can account for the fins heating up over time.

 

[K^2]

Like I told you in another thread, your fins are actually in a subsonic stream, and will not experience any significant heating.

 

[physwil90]

First off, I do not understand why you are saying that the fins will be in a subsonic airflow and how this affects it being cooler. And secondly, I would be inclined to think that the heating on the fins would be greater than the rest of the rocket because they are protruding from the rocket. This leading edge will contribute to high localized aerodynamic heating and so I think the heating will be significant...

 

[K^2]

That's the thing about the shock wave generated by a body going at supersonic speeds. The air behind the shock wave is moving with the body. The effect of this is that if the wings are sufficiently swept back, they are entirely within subsonic flow.

This effect is extremely important. Supersonic flow separates from a wing, and makes it effectively stalled. Look at any subsonic/transonic airplane. Wings are typically only somewhat swept back. Look at any supersonic aircraft. Note that the wings are entirely within the cone formed by the shock wave. Faster airplane means narrower cone. Means shorter wings. Think it's just for looks?

Anyways, the effect this has on heating is very direct. If the relative air stream is subsonic, the heating will be the same as that during subsonic flight. Have you ever heard of model rockets having problems with fin heating in subsonic flight? Typically, you have more trouble with heat from the engine. This should be no different.

So the only special challenge with the fins is making sure they are structurally strong enough to make it through transonic regime, where the drag will significantly increase, and airflow will be separating, which can cause all sorts of additional stress on the fins.

Just as an illustration, I attached a picture I grabbed off Wikipedia of F-22, and I added to it a cone extended from the nose. The aspect ratio of that cone is 1:1.8, because Mach 1.8 is the supercruise speed of F-22. Perfect fit. Engineered to be, no doubt.

Granted, F-22 can get up to Mach 2.2, but that's a different discussion. Afterburners and vectored thrust do wonderful things for an airplane.

So basically, don't worry about the fins. Worry about the nose cone and making sure that you have enough thrust to make it past transonic regime. If you can punch through transonic without shaking anything apart, you'd have a good chance of getting to Mach 2.

 

[rcgldr]

Supersonic flow depends on how long the rocket is. Generally you get secondary sets of shock waves every time the fuselage gets thicker on many aircraft. One at the nose and the next at the wing root is common, since the nose shock wave is fairly small and weak and doesn't accelerate the air much. Note how far aft the shockwave is on this F14 in the first part of this video:

http://rcgldr.net/real/f14flyby.wmv

Image of a f22's shockwave:
http://rcgldr.net/misc/f22.jpg

 

[physwil90]

I do not think in my case that the nose cone will disperse the air around the rocket like it does in the F-22 design.

Furthermore, I have a raytheon design guide on missile design and I am using as a reliable reference for this rocket. In this guide it states that "the highest heating rates on the missile are typically associated with stagnation heating...these include radome/nosecone tips (stagnation point), and control surface (fin) leading edges (attachment line, or stagnation line for control surface with no sweep)." (This is for all missiles, sub and supersonic) This statement leads me to believe that there is significant heating on the fins...

 

[K^2]

Image of a f22's shockwave:
http://rcgldr.net/misc/f22.jpg

That's a transonic shock. The plane forcing its way through air acts like a venturi, so the air speeds up every time the fuselage gets thicker. That means as you gradually build speed, shock waves will first form off the widest part of the plane. That's what they usually show off at air shows.

When in actual supersonic flight, you don't see the condensation cone. If you saw it, however, for F-22 in supercruise, you'd see that the cone completely contains air surfaces, with no secondaries off any of these.

Furthermore, I have a raytheon design guide on missile design and I am using as a reliable reference for this rocket. In this guide it states that "the highest heating rates on the missile are typically associated with stagnation heating...these include radome/nosecone tips (stagnation point), and control surface (fin) leading edges (attachment line, or stagnation line for control surface with no sweep)." (This is for all missiles, sub and supersonic) This statement leads me to believe that there is significant heating on the fins...

Are you building an interceptor SAM that's supposed to outrun planes going Mach 3.5? These things can go up to Mach 5. Yes, one of these will have supersonic flow behind primary shock wave. It will result in secondary shock wave around the fins. And that will generate huge amounts of heat.

If you build the rocket right, it might make it past transonic regime once, and get to supersonic flight. Rocketry enthusiasts have managed to achieve that. In order to get supersonic flow over fins, however, you have to get past transonic again, this time, transonic flow behind primary shock wave. Which is wave drag on fins and body in addition to shock wave on the nose cone. I can guarantee you that your rocket will not have enough thrust to do punch through that.

So your fins will never experience anything over Mach 1, and heating from Mach 1 isn't bad enough to damage fins if you make them from balsa. Not that balsa is a good material here, due to problems with transonic.

 

[rcgldr]

That's a transonic shock.

True, but the rocket will have to transition through this speed.

So your fins will never experience anything over Mach 1, and heating from Mach 1 isn't bad enough to damage fins if you make them from balsa. Not that balsa is a good material here, due to problems with transonic.

I would assume that balsa would self destruct at high speed due to flutter issues, probably well below mach 1. I thought high speed model rockets would be made of composite (fiberglass, carbon, kevlar) materials or metal.

 

[K^2]

True, but the rocket will have to transition through this speed.

My reply in the first thread he started:

In supersonic flight, fins will be in sub-sonic flow due to the shock produced by the nose. So you only have to worry about supersonic heating on the nose cone.

When punching through transonic, you might have supersonic flows forming around fins, but hopefully, your rocket will punch through transonic region fast enough for it not to be an issue.

I would assume that balsa would self destruct at high speed due to flutter issues, probably well below mach 1.

Not that balsa is a good material here, due to problems with transonic.

That's where the effect is the worst. Though, yes, it might become a problem even earlier.

I thought high speed model rockets would be made of composite (fiberglass, carbon, kevlar) materials or metal.

Composites, most likely. I'm not sure if use of metals is actually illegal anywhere, but it certainly frowned on, and violates NAR member rules. If the engine explodes, and everything is made of plastics, no big. If there are metal shards, it's a different matter.

 

[my_wan]

Is there an equation for the rate of heating on a surface at any time, and at any angle?

Not generally. K^2 tried to use plane wings to explain. I'll use use laminar verses turbulent flow.

First off, I do not understand why you are saying that the fins will be in a subsonic airflow and how this affects it being cooler. And secondly, I would be inclined to think that the heating on the fins would be greater than the rest of the rocket because they are protruding from the rocket. This leading edge will contribute to high localized aerodynamic heating and so I think the heating will be significant...

In general heating from a gas is the result of molecular collisions with a surface, but all gas motion is not equal. A laminar flow results when a large number of molecules are traveling in the same direction. Consider crowded kids running in the hall between classes at school. They will collide a lot more often than they would if they are all going the same direction. This directional flow is analogous to laminar flow, and can be a lot denser and faster with fewer heat producing collisions. A similar effect is taken advantage of using sound as a refrigerant.

This is why there's not a single general equation for all situations. There is a set of laws that give good answers in any situation you can effectively model. But such modeling is not so simple as an equation that can be applied in all situations.

 

[engee127slo]

Why don't you try looking at the aerodynamic heating at the stagnation point of the rocket...stagnation temperature: http://en.wikipedia.org/wiki/Stagnation_temperature