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I'm building a simple spreadsheet model of reentry and I'm looking for any hints, advice, comments, pointers, etc.

I've read all over the place conflicting opinions on the question of 'can you reenter big lumps of naked metal in a safe and recoverable way'; responses range from 'no we'd have to place the material in reentry capsules' through 'sure just make a foam/fractal low density lump'. We find lumps of metal that have reentered but usually (always?) don't know what the original mass was.

I thought I'd explore the question for myself.

• The basic reentry heating math I'm getting from this FAA (really?) doc .
• I can play with entry velocity and angle.
• So far I've used a simple scale height / exponential atmosphere density model, but I could move to on of the more accurate standard models when it was worth it.
• I'm playing with a 1000 kg lump, sphere for simplicity, platinum ( \$\$, solid is about 40 cm in diameter )
• I can make it hollow with different wall thickness to play with ballistic coefficient
• I can kind of match the results the paper has, I plan to find some other sources to cross check later.
• I can compute W/m2 heat flux
• I'm assuming that it impinges all sides of the reentering object. The linked doc says for Earth atmosphere sub 15 km/s convection dominates heat transfer which makes me think it might be more accurate to use 'area facing the flow' to take up heat (i.e. less m2 => less total heat). what do you think?
• My current model cannot find a trajectory that won't melt the whole mass (and more - unless it impacts at like 7 km/s) but my heat may be incorrectly high (see above) and I need to account for
• radiation of heat from the hot surface
• heat conduction through the mass. If the total heat is not going to melt the whole ball then the question becomes can the surface transport heat fast enough to be below melting point.
• calcing radiation as if the surface was at melting point would get rid of a big proportion of the heat (and it would very quickly get that hot)
• I've found some heat transport equations. Working with hollow sphere looks 'simple enough' and if the material can transport heat away from the surface fast enough it looks like you might be able to arrange to impact a ball of hot but not melted metal. However if I should move to only receiving heat from the flow facing surface then the radiation becomes more complex, maybe:
• I can conservatively pretend only the flow facing surface radiates any heat
• I could pretend the lump is a flat disk for heat sink purposes (with some mass removed for 'back structure' for aerodynamic stability etc, patch of a sphere facing the flow and a conical back structure)

Anyway doing a full finite element analysis / computational fluid dynamics style treatment is more than I want to (could) do but it seems like a conservative approximation should be doable.

Maybe there is already some online tool that does this kind of thing?

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lol, most of the links when you google "reentry calculator" are for KSP mods!

You get a much better selection (for your purposes) if you include the term "-KSP" (without quotes) in your search.

Some choice picks:

https://what-if.xkcd.com/28/ (Theres an XKCD for everything)

NB: turns out reentry dynamics can be quite complex, so the further you look into it, the harder making a simple approximation is. But there is a lot of good info out there.

A not-completely-useless-but-VERY-simple approximation is just comparing total kinetic energy of your lump, with the heat capacity of the mass. If there isn't enough energy in the system to completely melt your lump, then you know part of it will probably reach the ground. Calculating just how much metal melts off quickly gets complex again, for example: the more metal that boils/melts off, the greater the insulating sheath of gas that surrounds your lump, lowering heating rates.

My gut says that "foaming" the metal is your best bet, all those little bubbles would make great insulation, and the low density means more slowdown in the thin upper atmosphere, decreasing peak heat flux.

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22 hours ago, p1t1o said:

lol, most of the links when you google "reentry calculator" are for KSP mods!

I know - funny. Thanks for the links I'll check em out.

For many entry angles I see enough heat to more than melt the lump => it will melt and 'run off' and be lost, for some angles it wont melt it all but the remainder impacts 'very fast'. I want to achieve a 'no melt off' non destructive to the lump impact velocity.

Re the foam it sounds right but:

• too steep an entry might pancake it => not foam any more and who knows what aero consequences - I can easy get 400g max deceleration - maybe metal foam laughs at that
• you'd have to worry that the heat is now impinging on a small mass of material that has to transport heat through a much smaller amount of material - so I worry it might make melt off worse if the heat is 'too high'. Modelling a foam sounds 'not simple' maybe someone has done the work and has some kind of approximation I can work with.

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The other problem you might get with a foam is the fact that it's not totally solid might mean it doesn't form a coherent shock out in front of it and so it just gets obliterated by aerodynamic forces/local heating of very thin bits of metal

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I'm unclear about your goal... do you want a "simple spreadsheet model" for KSP, or for reality? In either case, I have serious problems with your FAA doc's "Similar to a rock skipping off a pond, a vehicle that doesn’t slow down enough may literally bounce off the atmosphere and back into the cold reaches of space" line, straight out of NASA/Hollywood PR hype. You don't bounce off; you simply fail to shed enough kinetic energy. Given metals' thermal conductivity, I think you could make a reasonable guess by considering work done (average drag [lots of work to be done here] x distance in atmosphere [most significant input, periapsis]) = Q into the metal = some coefficient times delta T. If T > melting point, game over. That your metal lumps don't survive is probably why NASA doesn't use bare metal for reentry shields

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34 minutes ago, SlowThought said:

I have serious problems with your FAA doc

I don't like the tone, but I haven't found serious problems with the math. I think a high level energy approach, as I described above, gives you the easiest "back of the envelope" estimate. May do a political rant elsewhere on why the FAA is spending my dollars writing this stuff.

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2 hours ago, SlowThought said:

Similar to a rock skipping off a pond

Yeah, I just ignored that - as far as I can tell from KSP you just rip through the atmosphere but are left with a subsequent Pe in the atmosphere.

I think some warhead reentry vehicles were metal. Partly it a g-force, max, total heat tradeoff, people are squishy. Also if you are bringing your heat shield it has to be light, but in my case the heat shield is the cargo.

My goal is a real life conservative approximation - so it can be not too accurate as long as it's an overestimate of heat and underestimate of radiation & heat sink effect.

This has some links to Mars entry craft where front face got 2 MW/m2 but back face gets only 1 or 5 kW/m2  - so I guess it's pretty safe to set my back face flux to 0.25 to 1 % of front face. I could treat front and back as independent disks and see how that goes.

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Any sanity checks are welcome.

I couldn't find any magic formula in college/uni level thermo papers that would let me plug max flux, dimensions, and material properties and have it spit out a Kelvin over depth through a plate of material - except once it reached 'thermal equilibrium' with a temp on each side of the material. Spending a few seconds at 300 g while being blasted with 20 MW/m^2 is not equilibrium. So I went to 'first principles' and a spreadsheet. But if anyone knows a magic formula that would be great.

• treat the reentering slab as many wafer thin slices (1000 in 4.66cm 1kg per wafer of a 1000kg 1m2 slab)
• let flux W/m2 = J/(sm2) be dumped into wafer 0 every slice of time - so track the J in the wafer
• specific heat J/(K kg) can then compute a temperature K for the wafer
• chose time slice size so that temp is only a fraction of the melting temp (0.001 sec)
• the next time tick J will move out of wafer 0 into wafer 1
• best case the J in the two wafers will equalise
• but could be limited by: time s * thermal conductivity J/(smK) * temp difference K / distance m = J out of wafer 0 - this seems to be 10 times the above
• do all the J out calcs, then apply wafer n-1 J out as wafer n J in -
• now we know J in for wafer 1 at time slice 1
• repeat the logic for each wafer for each time

If I didn't mess up then the slab would start melting off after 0.16 seconds of peak heat and lose around 40 kg during peak heating, probably after than radiative cooling would stabilise things. I could have messed up:

• I misunderstood the heat transport from wafer to wafer
• I've only used a few wafers so far - makes it look artificially good
• time slice it too big and artificially limits heat transport - maybe I have to choose dt to be where thermal conductivity is coming into play

I guess I have to write a program if I'm going to slice time 10 times finer and compute all the wafers.

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4 hours ago, DBowman said:

Any sanity checks are welcome.

I couldn't find any magic formula in college/uni level thermo papers that would let me plug max flux, dimensions, and material properties and have it spit out a Kelvin over depth through a plate of material - except once it reached 'thermal equilibrium' with a temp on each side of the material. Spending a few seconds at 300 g while being blasted with 20 MW/m^2 is not equilibrium. So I went to 'first principles' and a spreadsheet. But if anyone knows a magic formula that would be great.

Regrettably I imagine that there is no magic formula, precisely because there's a lot of factors at play, all changing at different rates due to different things (as it looks like you're discovering). AFAIK the only way you're going to get an answer is numerical simulations (which is what it looks like you're putting together with your first principals stuff). Word of advice on numerical simulations, don;t try to do them on an Excel sheet, you'll just give you and your computer a headache!

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Posted (edited)

21 hours ago, Steel said:

Question - heat of fusion - after the metal reaches melting point you have to put another 'heat of fusion' Joules into it for it to liquify. So can I rely on this? having the Pt solid but with contained Joules between that required to melt it and to liquify it? Once it liquifies I can just remove that wafer from the simulation and apply the reentry heating to the next remaining wafer?

edit - forgot the most important part Fouriers law for thermal conduction has energy flow depending on temperature gradient - that would mean the energy flow out of the wafer would be constant as the joules contained by it increased over melting temperature - but it seems like if there are more joules there then they should flow more into the next wafer even though the temp is clamped. Is it that Fouriers law only works while its solid below melting point? or can one 'just' rewrite it as a Joule gradient rather than a temperature gradient?

Edited by DBowman

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Posted (edited)

4 hours ago, DBowman said:

Question - heat of fusion - after the metal reaches melting point you have to put another 'heat of fusion' Joules into it for it to liquify. So can I rely on this? having the Pt solid but with contained Joules between that required to melt it and to liquify it? Once it liquifies I can just remove that wafer from the simulation and apply the reentry heating to the next remaining wafer?

edit - forgot the most important part Fouriers law for thermal conduction has energy flow depending on temperature gradient - that would mean the energy flow out of the wafer would be constant as the joules contained by it increased over melting temperature - but it seems like if there are more joules there then they should flow more into the next wafer even though the temp is clamped. Is it that Fouriers law only works while its solid below melting point? or can one 'just' rewrite it as a Joule gradient rather than a temperature gradient?

You could rewrite it as an energy gradient (you can do whatever you like!) but it would be incorrect. The whole heat of fusion thing is accurate, more energy in doesn't change the temperature above the melting point, thus doesn't change the heat transfer to the next wafer until it melts.

Just to throw another major annoyance into the works, have you considered the fact that the heat capacity of materials changes with temperature? EDIT: not that you really need to, a reasonable average will get you close enough

Edited by Steel

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11 hours ago, Steel said:

You could rewrite it as an energy gradient (you can do whatever you like!) but it would be incorrect.

Ah - thanks - good to know (sad cause it makes it harder not to melt off wafers). After I get the basic scheme working I can go back and turn some of the 'constants' into lookups by temperature. Also I need to work out a starting temp, but that one should be K such that sun heating + Earth heating and radiative cooling are equal.