Re-entry Heating Model

[PakledHostage]

The guys at Squad may already have an idea for how they’re going to implement re-entry heating in the game, but the Kerbal Science: The atmosphere of Kerbin thread got me thinking about re-entry and how it could be implemented. I thought I’d take a stab at it myself. And while I realise that I may have gone off the deep end a bit with this little project, it turned out to be kind of fun. I’d be interested to hear other forum member’s feedback about it.

Accurate modelling of re-entry dynamics is extremely complex and would require (at a minimum) the use of computational fluid dynamics. That obviously isn’t going to be done in a game. Instead, I would think that a method would have to be found that would have low computational overhead, yet still maintain some physical significance.

Having looked through some papers on the subject and dusted off an old textbook or two, I came up with what I think would be a good model. It is based on the assumption that a re-entry vehicle forms a detached supersonic shock, and that the shock stays detached all the way down to trans-sonic speeds.

Example

I recorded telemetry data for a capsule re-entering Kerbin’s atmosphere and then used the equations in the 'Background' section below to estimate the air temperatures that the capsule would have experienced during re-entry. In my test, the 1.45 tonne chute/pod/sensor assembly was placed into a 1 km x 85 km orbit for re-entry. The 1 km periapsis resulted in a fairly shallow re-entry trajectory and a maximum g-loading of 4 g’s. Even so, the pod would have experienced an extreme environment during that re-entry.

Initial mach numbers during my spacecraft’s re-entry approached Mach 8. This high Mach number results in very high temperatures inside the spacecraft’s supersonic shock, but little heating would have occurred during this part of the re-entry because the air is still too thin to transfer much heat. Although shock layer temperatures outside the spacecraft approach 2500 K during the early part of the re-entry, the spacecraft’s temperature wouldn’t have started to rise significantly until 50 km altitude. Maximum heating occurs between 40 km and 30 km altitude, and the maximum spacecraft temperature of 1800 K occurs at 26 km altitude. Our intrepid Kerbals were still moving at about Mach 5.4 at that time. The spacecraft then rapidly cools to equal the static air temperature by the time it reaches about 8000 m.

In calculating the data plotted in this example, I picked values of K₁ and C_p that made the spacecraft temperature data “look rightâ€. These would have to be experimented with in the game to get the desired behaviour. As stated in the 'Background' section below, K₁ would be hard coded in the game, while C_p could be a parameter in each part’s part.cfg file that could be adjusted to account for whether or not a part is, for example, a heat shielded part. I’ve included some additional plots below.

Background

The spacecraft begins re-entry with both kinetic and potential energy which is dissipated during the descent. The orbital energy is dissipated in chemical energy, light energy, atmospheric kinetic energy, sound energy, heat energy, etc. Some of the energy also heats the spacecraft itself. Calculating the amount that it heats the spacecraft is what I’m focused on. That heating is dependent on the external air temperatures (shock layer temperatures) surrounding the spacecraft during re-entry.

Fortunately, there’s a relatively simple equation that could be used to simulate the temperatures that the spacecraft is exposed to during re-entry. It takes advantage of the fact that a detached supersonic shock (normal shock) forms ahead of the spacecraft as it re-enters. The temperature rise through a normal shock in an ideal gas is approximately given by:

(1)

Where M is the Mach number, T₁ is the temperature from the in-game atmospheric temperature profile and gamma is the heat capacity ratio of the ideal gas that the shock forms within.

If we assume that Kerbin’s atmosphere is an ideal gas¹ with heat capacity ratio gamma = 1.4, then the above equation reduces to:

(2)

Similarly, if Kerbin’s atmosphere is an ideal gas, then the speed of sound is just

(3)

And while Equation (2) is increasingly inaccurate at high Mach numbers because the heat capacity ratio gamma cannot be assumed to be constant, we’ve got to remember that this is just a game! If we take these liberties, then calculating Mach number and shock layer temperatures that the spacecraft is exposed to during re-entry is trivial. All that remains, then, is to approximate how much heating occurs as a result of the temperature that the re-entry vehicle is exposed to.

The spacecraft is heated during re-entry by forced convection. Heat transfer due to forced convection can be modelled according to equation (4) below.

(4)

Where h is the heat transfer coefficient, A is a representative area, T₂(t) is the current air temperature from equation (2) above, and T_craft is the current spacecraft surface temperature.

The heat transfer coefficient “h†is a function of atmospheric viscosity, density, heat capacity, thermal conductivity, as well as the spacecraft’s geometry and the airspeed. Of these variables, the one that experiences the greatest variation throughout the re-entry is density. In fact, density experiences several orders of magnitude more variation than all of the other parameters, even when considering conditions inside the detached supersonic shock. As a result, for the purposes of an in-game re-entry heating model, equation (5) could be used to estimate the convective heat transfer coefficient.

(5)

Equation (5) is derived from the definition of h for laminar forced convection. Well designed re-entry vehicles maintain laminar flow over large areas because this results in lower heat transfer rates. In the laminar flow case, h is a function of the square root of the Reynolds number. Reynolds number is, in turn, a function of the fluid density, viscosity, velocity and a characteristic length. Early in the re-entry, where densities are very low, Reynolds numbers are low enough that laminar flow is relatively easy to maintain.

K₁ in Equation (5) is a constant chosen to give desirable qualities to the game’s re-entry heating model. Equation (4) can be combined with Equation (5) to give Equation (6)

(6)

And the result from Equation (6) can be used in Equation (7) below to iteratively calculate the spacecraft’s temperature at any time during the re-entry (if we assume a uniform temperature distribution within the spacecraft part).

(7)

Where C_p is the spacecraft part’s specific heat and m is the spacecraft part’s mass. Both of these values would come from the part’s part.cfg file.

Summary

This method is based on the assumption that the re-entry vehicle forms a detached supersonic shock, and that the shock stays detached all the way down to trans-sonic speeds. This may not be a valid assumption for some configurations (i.e. space planes), but it may still be “good enoughâ€.

And while some of the equations above may 'look ugly', they really aren\'t that bad. I calculated the values plotted in this post using just an Excel spreadsheet. The method can also be 'tuned' to give desirable heating response by adjusting one or two parameters. As a result, it is also easily adjusted to yield different atmospheric heating characteristics for different planets.

Anyway, as I said above, the guys at Squad may already have an idea for how they’re going to implement re-entry heating in the game but I found putting this post together to be a fun exercise. I’d be interested to hear other forum member’s comments about it. I hope some of it is useful.

¹ Kerbin’s atmosphere cannot be an ideal gas given its current pressure, density and temperature distribution. This is very easy to fix however. The pressure distribution needs only to be re-defined as

Defining pressure in terms of this equation results in the “Kerbin Standard Atmosphere†shown in the plot below. The difference in pressure distribution is slight. Would this affect anything other than the altitude at which the chute opens?

[thorfinn]

Very nice resume on basic aerothermodynamics

Just three points:

1) The density of Kerbin\'s atmosphere seems to be much higher that 1,2 kg/m³ (See Closette\'s work for details.)

2) If the density is actually very high, a lower gamma would suit better an atmosphere with a high percentage of complex molecules.

3) You are completely neglecting radiative heat flux: but in Earth reentry conditions, it actually dominates. What\'s the temperature of your shock layer? Is it a reasonable assumption?

[VincentMcConnell]

Cool. When do you think you can implement a heatshield part that bursts into flames during reentry?

[Dobrodav]

Reentry heat is important, but that task is comparatively easer, than the ultimate goal, - applying different level of heat on different surfaces depending on their attitude. Actually, looks like the reentry heat mechanics should be extremely simplified, and parts should consist of few meshes (ie - low, upper, front) to be able use that simplified algorithm to each mech separately. Even for that, we need DPI - dynamic pressure variable to be in the game simulation. That all looks like massive rework of game engine needed. Hope i am wrong.

[PakledHostage]

Very nice resume on basic aerothermodynamics

Thanks!

1) The density of Kerbin\'s atmosphere seems to be much higher that 1,2 kg/m³ (See Closette\'s work for details.)

I\'m using the unmolested values out of the game engine. I retrieve them using:

dAtmosphericPressure = FlightGlobals.getStaticPressure();

dAtmophericDensity = FlightGlobals.getAtmDensity(dAtmosphericPressure);

3) You are completely neglecting radiative heat flux: but in Earth re-entry conditions, it actually dominates.

Good point. The temperature of the shock layer reached a peak of about 2500 K for this re-entry. I will look at the influence of radiation for temperatures in that range. That being said, I did some more reading this evening and it seems that radiative heat transfer only becomes important for Earth atmospheric entries when speeds exceed 15 km/s.

What\'s the temperature of your shock layer? Is it a reasonable assumption?

I read that a useful rule of thumb for estimating peak shock layer temperatures (in degrees Kelvin) during Earth re-entry is to use the magnitude of the re-entry speed in m/s. In this case, the re-entry speed is about 2200 m/s (relative to the surface), and the peak shock layer temperature is 2500 K. We\'re at least in the right ballpark...

Reentry heat is important, but that task is comparatively easer, than the ultimate goal, - applying different level of heat on different surfaces depending on their attitude.

One option may be to apply the re-entry heat to the forward most components in the stack during re-entry. If the forward most components (i.e. the ones oriented within some tolerance of the prograde direction) can handle the heat, then nothing happens. If they can\'t, then the spacecraft blows up. If it were done that way, then a heat-shield component could be added at the base of the pod, much like in Vostok\'s excellent suggestion over in the KSP development section. It should be fairly easy to determine whether the heat shield component is facing within some tolerance of prograde during re-entry.

[ferram4]

At the very beginning of reentry, radiation will play an important contribution simply because the temperature difference is large and radiation heat transfer goes with T^4.

Since most of the heated layer will be very close to the spacecraft, I\'d say that assuming the system can be modeled as a pair of infinite parallel plates wouldn\'t be too far off. That would make this equation work for radiative heat transfer:

I suspect that adding this into the calculations would bring the peak ship temperature above 2000K and at a higher altitude. I don\'t think adding this bit would be difficult at all.

Very nice work PakledHostage! I hope your work goes to good use.

[N2maniac]

Whether or not radiative heat transfer is important at 2km/s, it may be important at higher re-entry speeds or aero-capture. An aero-capture maneuver at high speed would avoid the highest conductive/convective heat flux deeper down, yet have a potentially high temperature initially (suppose it was 8,10,12, even 21km/s).

Not trying to nag or negatively criticize, just bringing up a point.

[PakledHostage]

Not trying to nag or negatively criticize, just bringing up a point.

It is a fair point and I appreciate the discussion about radiative heat transfer, but I wonder if we’re not barking up the wrong tree?

I did some more reading this evening and it seems that the black tiles on the bottom of the Space Shuttle were designed to maximise radiative heat transfer rather than minimise it. They have an emissivity of 0.8 because they are designed to dissipate heat by radiation during re-entry. Maybe someone with some expertise on the shuttle TPS would like to comment?

Also, I read this evening that radiative heating isn’t a significant factor for objects entering Earth’s atmosphere until speeds exceed about 15 000 m/sec. That would seem to validate your concerns about extremely high speed atmospheric entries, N2maniac, but I wonder if it is really a significant enough effect at typical KSP entry speeds that we need to ask that Squad add it to the game?

[thorfinn]

I did some more reading this evening and it seems that the black tiles on the bottom of the Space Shuttle were designed to maximise radiative heat transfer rather than minimise it. They have an emissivity of 0.8 because they are designed to dissipate heat by radiation during re-entry. Maybe someone with some expertise on the shuttle TPS would like to comment?

Remember that the Shuttle had wings

I\'m not an expert, but I\'ve seen radiative heating cited as an important factor in Mars entry, which is relatively mild... but you are also encountering a lot of molecules and radicals with excited vibrational states there. I don\'t know. Where did you get that 15 km/s figure?

Anyway, that was more for continuing conversation about the physics than for direct application to KSP: in the game, we will probably have to 'cheat' to make Kerbin reentry interesting (in the Chinese sense...) The energy involved is just so low....

Ah, I\'ve found a presentation that may be of interest to those who are reading this:

http://research.nianet.org/~grossman/Fundamentals/Hypersonic%20Aerodynamics/Section.5.pdf

[haltux]

Reentry heat is important, but that task is comparatively easer, than the ultimate goal, - applying different level of heat on different surfaces depending on their attitude. Actually, looks like the reentry heat mechanics should be extremely simplified, and parts should consist of few meshes (ie - low, upper, front) to be able use that simplified algorithm to each mech separately. Even for that, we need DPI - dynamic pressure variable to be in the game simulation. That all looks like massive rework of game engine needed. Hope i am wrong.

I don\'t think that this kind of real-time mesh-by-mesh modelling is the right approach. I could be wrong, but it sounds too ambitious and probably not very efficient.

My opinion is that reentry heating (with associated destruction threshold) models should be statically computed at rocket design validation. For each potentially independant part of the rocket there would be different models computed (for example for a 2 stage rocket there would be 3 models:stage 1+2, stage 1 and stage 2).

Each of this models, once computed, would be no more than simple functions altitude, attitude,speed -> heat. Of course that would not be completely realistic, but that would be good enough or a game. And that would be very cheap in terms of real-time computation.

Anyway, I think even the most basic heating/destruction model would be much better than nothing. Entering kerbin atmosphere at whatever speed and whatever angle with whatever rocket and just opening the parachute to be 100% sure to go back home safe is killing half of the game right now (the trip back part).

[PakledHostage]

I\'m not an expert, but I\'ve seen radiative heating cited as an important factor in Mars entry, which is relatively mild... but you are also encountering a lot of molecules and radicals with excited vibrational states there. I don\'t know. Where did you get that 15 km/s figure?

Thank you for posting the link to the NASA PowerPoint slides. Plasma physics, high temperature gas dynamics and even heat transfer are well outside my expertise. But as so often happens, KSP has lead me down the rabbit hole and I\'m enjoying learning something new.

The 15 km/s figure comes from a textbook chapter titled Returning from Space:Re-entry, but I don\'t know the title of the book and I can\'t find links to the rest of the chapters on the FAA website where I found this reference. Even so, there is some very good information there about re-entry heat and the physics of re-entry. Some of it would probably even be useful for refining a re-entry model so that it works well for both capsules and space planes. The textbook says on page 4.1.7-322:

'Convection is the primary means of heat transfer to a vehicle entering Earth’s atmosphere at speeds under about 15,000 m/s. (For a re-entry to Mars or some other planet with a different type of atmosphere, this speed will vary.)'

I don\'t know why the 15000 m/s value is significant and why it would be different on Mars, but that will be my next mystery to investigate.

[illectro]

Proper re-entry heating also needs aerodynamic lift at high speeds and altitude to work. Capsules are shaped with a flat bottom so that they can generate lift and keep the capsule in the thin upper atmosphere longer. Gagarin\'s spherical re-entry vehicle entered ballisticly and deceleration was 9-10g, modern capsules generate much lower forces due to the lift, unless the re-entry control system fails, in which case it\'s passivly stable but hard on the pilots.

[The_Duck]

This is really interesting, PakledHostage. I\'m somewhat tempted to try making a simple reentry heat plugin with these equations...

[PakledHostage]

I am no good at 3D modelling and I don’t know enough about Unity to make visual re-entry effects, but I did code a little module to compute the heating effects predicted by the method described in this thread. I have posted it over in the add-ons section of this forum.

As was discussed upthread, I didn’t include radiation heat transfer in this calculation because the purpose of this exercise was to create a re-entry heating model that has somewhat realistic behaviour, with minimal computational overhead. Also, at least in the case of Earth re-entry, radiation heat transfer actually cools the re-entry vehicle rather than heats it.

I’ve attached screenshots showing the raw numbers predicted by these calculations for a re-entry from low Kerbin orbit. As expected, re-entries from higher orbits (i.e. higher speed re-entries) and steeper re-entries result in more significant heating.

But they only result in increased heating to a point. I also tried a high speed straight down re-entry from a 70000 km apoapsis, and although the shock temperatures reached upwards of 4500 °C, the heat shield temperature didn’t rise above about 1200 °C. This may actually make sense though, because the capsule was only exposed to maximum heat for a very short period of time. Much like in the case of fried ice cream or waving your hand through a flame, you can expose something to high temperatures for brief periods without causing significant heating.

In the straight down re-entry case however, the resulting g-loading was off the scale. Maybe a proper re-entry model should also consider g-loading. A steeply re-entering spacecraft may not be destroyed by heat because it isn’t exposed long enough, but it would be destroyed by the extreme g forces.

The_Duck kindly provided me with links showing the work that Nova has already done to develop a re-entry heating plug-in (those links are broken in the new forum), but Nova doesn’t seem to have finished that work and I suspect that he’s got other priorities now that he’s working with Squad. If anyone reading this wants to collaborate on developing a re-entry heat module, please send me a PM. (Or feel free to use the library that I posted in the add-ons area in your own project, if you like what I’ve done here but prefer to work alone.)

PH.