The problem is not so much propulsion as it is heat protection. The reason has to do with the enormous energies of high speed flight, and with steady-state and transient heat transfer. Any good rocket can push you to hypersonic speeds in the atmosphere. But it is unlikely that you will survive very long there!
The flow field around most supersonic and hypersonic objects looks somewhat like that in Figure 1. There is a bow shock caused by the object parting the oncoming air stream. Then, the flow re-expands back to near streamline direction along the side of the object. Then it over-expands around the aft edge, having to experience another shock wave to straighten-out its direction parallel to free stream again. This aft flow field usually also features a wake zone of one size or another, as shown.
The conditions along the lateral side of the object are not all that far from free stream, in terms of static pressures, flow velocities, and air static temperatures. One can compute skin heat transfer using those free-stream values as values at the edge of the local boundary layer, and be “in the ballpark”. That is what I do here, for illustrative and conceptual purposes.
Once flow is supersonic, the boundary layer behavior isn’t so simple any more. There is a phenomenon that derives from the very high kinetic energies that one simply does not see in subsonic flow: energy conservation. The value of that kinetic energy shows up as the air total temperature Tt, which is the upper bound for how hot things could be. Air captured on board by any means will be very close to Tt, if subsonic relative to the airframe after capture. This includes any “cooling air” one might use!
In addition, there is “viscous dissipation”, which has the effect of raising the actual (thermodynamic) temperature of the air in a max shearing zone within that boundary layer, to very high temperatures. The peak of this temperature increase is called the recovery temperature Tr. The difference between this recovery temperature and the local skin temperature Ts is what drives air friction heat transfer to the skin, not the difference between the air static temperature and the skin temperature, as is typical in subsonic flow. See Figure 2. The temperature rise from static to recovery is around 88 to 89% of the rise from static to total, in turbulent flow, which this almost always is.
Most heat transfer calculations for this kind of flow regime take the basic form and sequence illustrated in Figure 3. “How high and how fast” determines the conditions of flow, ultimately. Total and recovery temperatures may be computed from this, and total temperature is conserved throughout the flow field around the object, regardless of the shock and expansion processes. The flow alongside the lateral skin is not far from free-stream to first order, and that may be used to find out “what ballpark we are playing in”. Better local edge-of-boundary layer estimates must come from far more sophisticated analyses, such as computer fluid dynamics (CFD) codes.
In Figure 3, the process starts by determining recovery temperature. The velocity, density, and viscosity at the edge of the boundary layer won’t be vastly different from free stream, unless you are really hypersonic, or really blunt (detached bow shock). The various correlations account for this.
Using whatever dimension is appropriate for the selected heat transfer correlation, one computes Reynolds number Re. Low densities at high altitude lead to low values, and vice versa. High speeds lead to high values. Different correlations have the density and viscosity (and thermal conductivity) evaluated in different ways and at different reference temperatures. You simply follow the procedure for the correlation you selected. Sometimes this is neither simple, nor straightforward.
The complexity of these correlations varies. My favored lateral skin correlations use a T* for properties evaluation that is T* = mean film plus 22% of the stagnation rise above static. My favored slower than reentry stagnation zone correlation evaluates fluid properties at total conditions behind a normal shock. In the stagnation case, Reynolds number is based on the pre-shock freestream velocity.
The next step is the correlation for Nusselt number Nu. This nearly always takes the form of a power function of Re (plus some other nontrivial factors), usually with an exponent in the vicinity of 0.8 or so. Nusselt number is then converted to heat transfer coefficient h, using the appropriate dimension and the appropriately-evaluated thermal conductivity of the air, for the selected correlation.
The heat transfer rate is then as given in Figure 3, which shows the Tr – Ts temperature difference.
One should note that because both density (which is in Re) and thermal conductivity k (which is in h) are low at high altitudes, the computed values of h will be substantially smaller at high altitudes in the thin air. High speeds act to raise h, and to very dramatically raise Tr and Tt. That last effect is truly exponential.
Having the heat transfer rate is only part of the problem. One must also worry about transient vs steady-state effects. If the skin is completely uncooled in any way, it is then only a heat sink of finite capacity, with the convective input from Q/Aconv = h (Tr - Ts). One can use material masses and specific heats to estimate the heat that is sinkable as skin temperature rises. The highest it can reach is Tr = Ts, where it is fully “soaked out” to the recovery temperature. That zeroes heat transfer to the skin.
The time it takes to soak out can be very crudely estimated as 3 “time constants”, where one “time constant” is the heat energy absorbed to soak-out all the way from initial Ts to Tr, divided by the initial heat transfer rate when the skin is at the initial low Ts.
More complex steady-state situations must find the equilibriating Ts when there is convective input from air friction, conductive/convective heat transfer into the interior of the object (something not illustrated here), and re-radiation from the hot skin to the environment. In high speed entry, there is also a radiative input to the skin from the boundary layer itself, which is an incandescent plasma at such speeds, and this is very significant above about 10 km/s speeds.
Not covered here in the first two estimates are heat transfer correlations for nose tips and leading edges. Those heat transfer coefficients tend to be about an order of magnitude higher than the coefficients one would estimate for “typical” lateral skin. Stagnation soak-out temperatures should really be nearer Ttot than Tr, although those temperatures are really very little different.
Suffice it to say here that if one flies for hours instead of scant minutes or seconds with uncooled skins, they will soak out rather close to the recovery temperature Tr or total temperature Ttot. That brings up practical material temperature limits. See Figures 4 and 5.
For almost all organic composites, the matrix degrades to structural uselessness by the time it reaches around 200 F. The fiber might (or might not) be good for more, but without a matrix, it is useless. For most aluminum alloys, structural strength has degraded to under 25% of normal by the time it reaches about 300 F, which is why no supersonic aircraft made of aluminum flies faster than Mach 2 to 2.3 in the stratosphere, and slower still at sea level. Dash speeds higher are limited to several seconds.
Carbon steels and titaniums respond to temperature very similarly, it is a very serious mistake to think that titanium is a higher-temperature material than carbon steel! Titanium is only lighter than steel. And you “buy” that weight savings at the cost of far less formability potential with titanium. Both materials are pretty-much structurally “junk” beyond about 750 F. Various stainless alloys have max recommended use temperatures between 1200 and 1600 F. Inconel is similar to the higher end at about 1500 F. There are a very few “superalloys” that can be used to about 2000 F, give or take 100 F.
Figure 4 compares steady-state recovery (max soak-out) and total temperatures to material limitations on a standard day at sea level. Max speed for organic composites are barely over Mach 1, and just under Mach 2 with aluminum. Steel and titanium are only good to about Mach 2.5, unless cooled in some way. Stainless steels can get you to about Mach 3.5-to-4, the superalloys not much higher.
Figure 4 – Compare Tt and Tr to Material Limitations at Sea Level
One should note that stratospheric temperatures are only -69.7 F from about 36,000 feet altitude to about 66,000 feet altitude. Above 66,000 feet, air temperatures rise again, to values intermediate between these two figures! That lowers the speed limitation some, for altitudes above 66,000 feet.
This steady-state soak-out temperature comparison neatly explains why most ramjet missile designs (usually featuring shiny or white-painted bare alloy stainless steel skin) have been limited to about Mach 4 in the stratosphere, and around Mach 3.3 or so at sea level. Those limitations on speed are pretty close to the 1200 F isotherms of total or recovery temperature. Without re-radiation cooling, the skins soak out fairly quickly (the leading edges and nose tips extremely quickly).
To fly faster will require cooled skins, or one-shot ablatives, or else the briefest episodes (scant seconds) of transient flight. The nose-tip and leading edge problem is even worse! That means for long-duration / long-range flight, the skin must be cooled, or else coated with a thick, heavy, one-shot ablative. There are two (and only two) ways to do cooling: (1) backside heat removal, and (2) re-radiation to the environment. Or both!
Backside heat removal must address (1) conduction through the materials, (2) some means of removing the heat from the backside of the materials, and (3) some means of storing or disposing of all the collected heat (what usually gets forgotten). Liquid backside cooling using the fuel comes to mind, with the heat dumped in the fuel tank. However, there are two very severe limits: (1) the liquid cooling materials and media may not exceed the boiling temperature at tolerable pressures, and (2) the heat capacity of the fuel in the tank is very finite, and decreasing rapidly as the vehicle burns off its fuel load.
Re-radiation to the environment requires a very “black” (highly emissive) surface coating, and is further limited by the temperature of the environment to which the heat is radiated. These processes follow a form of the Stefan-Boltzmann Law, to wit: Q/A = σ εs (Ts4 – Te4), where σ is the Stefan-Boltzmann constant, and the εs is spectrally-averaged material emissivity at the corresponding temperature. Subscript s refers to the hot radiating skin panels, and subscript e refers to the environment.
While deep space is ~4 K, earth temperatures are nearer 300 K, and that is what most atmospheric vehicles usually “see”. The material absorptivity is its emissivity, which is why that value is also used for the radiation received from the environment. A truly “black” hot metal skin might have an emissivity near or above 0.8. This could be achieved in some cases by a metallurgical coating or treatment, in others by a suitable black paint (usually one of ceramic nature, and very high in carbon content).
One More Limitation to Consider
Once the boundary layer air is hot enough, it is no longer air, it is becoming an ionized plasma. The kinds of heat transfer calculations that I used here become increasingly inaccurate when that happens, and other correlations developed for entry from space need to be used instead. As a rough rule-of-thumb, that limit is about 5000 F air temperature.
If you look at Figure 4 (sea level), you hit the “not-air anymore” limitation starting around Mach 7. In figure 5 for coldest stratosphere, that limit gets exceeded starting around Mach 8. The only calculation methods that “work” reliably above these limits would be CFD codes, and even then, only if the correct models and correlations are built into the codes. That last is not a given! “Garbage-in, garbage-out”. That expression is no joke, it is quite real.
With Re-Radiation Cooling at Emissivity = 0.80
This applies only to lateral skins, not leading edges, because the heat transfer rates are an order of magnitude higher for leading edges. That effect alone changes the energy balance enormously.
But for lateral skins, the speed limitation occurs when the re-radiation heat flow equals the convective input to the skin. The complicating factor is that convective heat transfer is a strong function of altitude via the air density, while re-radiation is entirely independent of altitude air density. There are now more variables at work on the energy balance than just ambient air temperature.
That means two charts depicting the “typical” effects are entirely inadequate. We need a sense for how this changes with altitude air density. What follows is a selection of equilibrium re-radiating temperature versus speed plots, at various altitudes, in a US 1962 Standard Day atmosphere model. Material temperature capabilities are superposed, as before.
Figure 11 – Lateral Skin Radiational Equilibrium at 110,000 feet
Tough Design Problem
How exactly one achieves this re-radiation cooling is quite a difficult design problem. The skin itself will be very hot, in order to re-radiate effectively. Not only will it be very structurally weak, there will be heat leakage from it into the vehicle interior. This is inherent, but by careful design, can be limited to rather small (1-2%) values compared to the energy incident and re-radiated from the outer surface.
There must be a sufficient thickness of low density insulation between that skin and the interior, one capable of surviving at the skin temperature. This insulation must be some sort of mineral fiber wool. There are no simple glasses that survive at the temperatures of interest for hypersonic flight.
The mountings that hold the skin in place constitute metallic conduction paths into the interior. These must be made of serpentine shape, of length significantly greater than the insulation thickness, in order to effectively limit heat leakage by the metallic conduction path.
Finally, there is the issue of sealing the structure against throughflow induced by the surface pressure distribution relative to the pressure in the interior. Because it is much easier to design seals that survive cold, than seals that survive incandescently-hot, it seems likely that the surface skins must be vented, with the pressure distribution resisted by colder structures deeper within the airframe.
Two Sample Cases
The SR-71 and its variants featured a “black” titanium skin, cooled by re-radiation, but nothing else. The leading edges (at least very locally) would approach the soak-out temperature limits shown in Figures 4 and 5 above. Typical missions were flown at around 85,000 feet, with speeds up to, but not exceeding Mach 3.3. In the very slightly-colder air at 66,000 feet, that leading edge limit was Mach 3.5.
As figure 10 shows, the lateral skins had a higher speed limit nearer Mach 4. So we can safely draw the rough conclusion that the SR-71 airframe was likely limited by leading-edge heating to about Mach 3.5 or so, at something around 80,000 or 85,000 feet.
The X-15 featured skins of Inconel-X that were radiationally very “black”. About the max recommended material use temperature is 1500-1600 F. Leading edges might tend toward the local soak-out limit at about Mach 4 to 4.5, unless internally cooled by significant internal conduction toward the lateral surfaces of a solid piece, which these were. Thinner air “way up high” might help with that balance, by reducing both the stagnation, and lateral, heating rates.
As shown in figure 11, the re-radiation equilibrium limitation near 110,000 feet is closer to Mach 10 for the lateral skins, and higher still at higher altitudes, as the other figures indicate by their trends. The fastest flight had a white coating, which effectively killed radiational cooling. For that, the soak-out speed limit is closer to Mach 4.5 to 5.5, based upon figures 4 and 5.
Again, we might very crudely conclude the X-15 was limited by its leading edges to something between Mach 5 and Mach 10. The fastest flight actually flown reached Mach 6.7, without any evident wing leading edge or nose damage, excepting some shock impingement heating damage in the tail section.
Most of the outfits claiming they have vehicle designs that cruise steadily at Mach 8+ (high-hypersonic flight) have not done their thermal protection designs yet.
That lack inherently means they do not have feasible vehicle designs at all, since thermal protection is the enabling item for sustained high-hypersonic flight.
“Hypersonic cruise” (meaning steady state cruise above Mach 4 or 5 for extended ranges) is therefore nothing but a buzz word, without an advanced thermal protection system in place.
The faster the cruise speed, the more advanced this thermal protection must be, and the more unlikely there will be a metallic solution.
Blunt vehicles = hypersonic Mach 3+
Sharp vehicles = hypersonic Mach 5+
Formally, “hypersonic” is when the bow shock position relative to the vehicle surface contour becomes insensitive to flight speed.
A Better Leading Edge Model
That is entirely out of scope here. It might consist of one solid leading edge piece, to be assumed isothermal. It would have a very small percentage of its surface area calculated for stagnation heat transfer, with the remainder calculated as lateral skin heat transfer, except as modified for convexity into the flow near the leading edge. There would be no conduction or convection into the interior. All surfaces would re-radiate to cool.
The next best model is a finite-element approximation, which allows for temperature variations and internal conduction, within the leading edge piece. Adding conduction and convection paths into the interior is the next level of modeling fidelity. None of this is amenable to simple hand calculation.
Supersonic Inlet Structures
These are an even more difficult problem, as the inner surfaces are (1) blocked from viewing the external environment for radiational cooling, and (2) are exposed to edge-of-boundary layer conditions that are very far indeed from freestream conditions.