What kind of heating would occur during a suborbital re-entry?

Question

What kinds of peak temperatures would a stage similar in proportions to the Space shuttle with a similar belly-first approach experience when re-entering from a low suborbital trajectory (similar to a first stage)? Would this necessitate heat shielding or would a sufficiently robust stage be able to re-enter intact without special equipment?

Answer 1

What kind of heating would occur during a suborbital re-entry

Convective Heat Transfer is the dominant source of heat for any highspeed object moving through atmosphere.

What kinds of peak temperatures

The temperature of the gas that rubs against the vehicle is called the recovery temperature

$$T_r = T_s(1+\frac{\gamma-1}{2}rM^2)$$

Here $T_r$ is the recovery temperature, $T_s$ is the static temperature of the atmosphere: this is the temperature you would feel if you were floating in a balloon near the passing vehicle. $\gamma$ is the ratio of specific heats, 1.4 for air, $M$ is the vehicle Mach number, and $r$ is the "recovery factor".

Notice that this equation is very similar to the equation for total temperature:

$$T_t = T_s(1+\frac{\gamma-1}{2}M^2)$$

The only difference is the recovery factor, which is a function of Prandtl number, and ranges from 0.8 to 0.95 for this class of flows. This means that the temperature the vehicle feels is only slightly less than the total temperature. The principle here is very similar to how air conditioners work: when you squeeze together many low temperature gas molecules, there is much more heat per volume and the temperature increases. The recovery factor tells us that this process is not perfectly efficient, some energy of the squeezed gas is lost to radiation or other mechanisms such that it doesn't all reach the vehicle.

I will make a big assumption that a suborbital re-entry will be at around Mach 5 for the majority of the atmosphere. In this case, you should expect temperatures 15 times the atmosphere's temperature. Another assumption is that most of the atmosphere is at ~-40 degrees Celsius, then the temperatures the vehicle feels are a whopping 3,500 degrees Kelvin (5,800F).

Would this necessitate heat shielding?

This depends on how quickly the vehicle manages to slow down. While the 3,500K temperatures seem wicked hot, what matters is how effectively the convection transfers the heat to the vehicle. This is characterized by the convective heat flux:

$$ \dot{q} = h(T_r-T_w)$$

Here $h$ is the convective heat transfer coefficient, and $T_w$ is the vehicle wall temperature. You'll want your wall temperature much lower than 3,500K, so the delta between the two is pretty significant. The heat transfer coefficient will determine if that large temperature difference is a problem. If the gas is rarified (very low density) then it will not transfer heat effectively, and you make be fine. The physics determining $h$ are complex, so engineer's developed correlations from various data sources to estimate this value.

The Nusselt number is used to compute $h$. Here is a correlation for Nusselt number from 'Cengel's Fundamentals of Thermal-Fluid Sciences'

$$Nu = 0.037Re^{0.8}Pr^{1/3} = \frac{hL}{k}$$

Here $Re$ is the Reynold's number, which scales with atmosphere density and vehicle velocity; $Pr$ is the previously mentioned Prandtl number, which ratios the heat motives of convective action to conductive action; $h$ is our convective heat transfer coefficient; $L$ is a characteristic length, e.g. the wingspan of the shuttle or the diameter of the booster etc.; and $k$ is the thermal conductivity of the fluid.

So basically, if the vehicle is moving very fast through very thick atmosphere, $h$ will be large. If the vehicle is moving through very thin atmosphere, $h$ will be very small.

This is why Falcon 9 does a reentry burn in addition to its boost-back burn. Slowing down from crazy high Mach numbers before hitting the thick parts of the atmosphere is critical to avoid regimes where both $h$ and $T_r$ are high. The Falcon 9 is also very heavy with a thin profile, giving it a very high ballistic coefficient, meaning that it won't slow down on its own nearly as much as a shuttle (very large drag) or lighter sounding rocket might.

To conclude: a suborbital trajectory provides a unique cooling challenge only for certain high-energy trajectories (like an orbital first stage, unlike a typical sounding rocket that just goes up then down) and for certain very low drag (in terms of deceleration) vehicles. The Falcon 9 first stage does use a thermal protective coating. I refuse to do the calculations I describe, so here is a source that cites peak heat loads of $70 kW/m^2$ for a reusable first stage. Wood auto-ignites with $30 kW/m^2$

Answer 2

The footage from various first stages suggest that there will be some aerodynamic heating, but the successful recoveries of Space Shuttle boosters and SpaceX and RocketLab first stages suggest it can be handled without a true heat shield.

We even have the recent falcon heavy fairings re-entering damaged but intact from close to orbital velocity.

The main factor would be the mass and area of the vehicle. If it was mostly empty and a decent lifting body it would be like the Falcon Heavy fairings above and just need careful choices of paint. If it is a payload carrying vehicle with substantial mass coming down then it will be looking more like fully fledged space shuttle thermal protection system because the heating experienced ties closely to kinetic energy at start of re-entry.

It also depends on the shape as careful design can dissipate most of that kinetic energy into the atmosphere rather than the vehicle skin and structure.