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A lot of discussions of SSTOs and the potential for making upper stages re-usable hinge on the fact that thermal protection systems for reentry from full orbital velocity are pretty heavy (and historically either expendable or so expensive to maintain to make one wish that they were expendable).

Is there a reasonably simple rule of thumb to estimate a minimum mass for a thermal protection system?

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    $\begingroup$ I think it's more complicated than just having a simple rule of thumb that suits most cases. It seems to be that historically the majority of expendable atmospheric entry heat shields were ablative. The shield mass would heavily depend on ablative properties, shape of the vehicle (blunt objects give best ablation results), entry profile (high angle->more peak heat, but the heat is gone quicker, low angle- less peak heat but for longer period), entry velocity (earth orbit or returning from Moon/Mars) etc. $\endgroup$ Jan 29, 2020 at 11:00
  • $\begingroup$ Yep, it's complicated. If the original plan for the Space Shuttle heat protection had worked - i.e. 8 or 10 large thermal pads instead of a few thousand, the cost would have been quite reasonable. (and very low mass) . But reentry from orbital will require careful control of the upper stage's attitude too. $\endgroup$ Jan 29, 2020 at 15:44
  • $\begingroup$ @LeoS definitely I would imagine so. Part of why I thought this might be tractable is that the vehicle is here shaped like a booster, has mass/density like a booster, and is coming along whatever is optimal from LEO. $\endgroup$
    – ikrase
    Jan 30, 2020 at 10:54

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Maybe not quite what you're after, but slide 85 of NASA's Aerothermodynamics Lecture has this figure:

TPS mass fraction

Of course this requires an estimate of instantaneous heat flux along the entry trajectory (integrated into total heat load). A good value for total heat load for a LEO reentry is ~$10^4$ to $2$x$10^4$ $\frac{J}{cm^2}$ which translates to ~10-15% of the entry vehicles mass as TPS (thermal protection system).

This is of course just a 1st (maybe even 0th) order design estimation tool, and specific to ablative entry systems which are generally not reusable.

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