This article is a generic look at heat protection of lateral skins in sustained high-supersonic and low-hypersonic flight. No particular application is modeled. Heat protection of nose tips and aerosurface leading edges is not included! The flow and heat transfer models require use of ideal gas models, so that high-hypersonic flight cannot be extrapolated from this, although the trends are somewhat similar.
This work was done with simple calculator- or slide-rule-type analysis equations, embedded in a spreadsheet. For a copy of that spreadsheet, contact the author. Otherwise, this is easy enough for a knowledgeable, experienced aerothermal design analyst to do for himself or herself. (It might be beyond the ability of an amateur to get it right.)
Materials selections, insulation thicknesses, and cooling requirements for steady-state high-speed flight depend upon:
(1) where on the airframe you look,
(2) whether or not you can cool by re-radiating to the environment,
(3) how high you fly,
(4) how fast you fly, and
(5) how much conduction you can tolerate into the interior.
I explored this with a generic shape, simplified compressible flow aerodynamics, and common heat transfer models. These are restricted to conditions where the ideal gas equation of state applies. That means the air flow past the body cannot be undergoing significant ionization.
The generic shape is “sort-of a projectile” with an ogive-like nose, a cylindrical body, and a slight boat-tail taper toward a bluff rear. The nose tip approximates a 10 degree half-angle cone locally. The boat-tail taper is approximately 4 degrees half-angle. This object is nominally 10 feet long.
Flow Field About the Airframe
Simplified compressible flow analysis around this shape uses NACA 1135 cone shock charts for the nose tip surface conditions, and NASA 1135 supersonic tables, specifically the Prandtl-Meyer expansion, from those nose tip conditions to the lateral surface, and then on to the boat-tail surface. No attempt was made to look at stagnation point or aft base area conditions.
This flow field analysis was done at Mach 5 speed and 60 kft altitude on a US 1962 standard day. In addition to the 3 surface locations, a fictitious planar surface parallel to the free steam without shock waves was included for comparison. Results summarize in Figure 1 (all figures are located at the end).
If instead this were to represent a wing or other aerosurface, very similar methods would be used. Instead of an ogive nose, a two-dimensional wedge shape would be used. The wedge and boat-tail angles would likely be lower than 10 and 4 degrees, respectively. Instead of the cone shock charts, the wedge shock charts in NACA 1135 would be used. The same kind of Prandtl-Meyer expansion analysis to the other surface slopes would be used, starting from the initial wedge surface conditions. Again, this uses the supersonic flow tables in NACA 1135.
The projectile flow field results shown in Figure 1 clearly show higher pressures and density ratios-to-standard on surfaces more-or-less oriented into the oncoming free stream. Pressures are lower and velocities are higher on surfaces oriented more-or-less away from the oncoming wind. There are no surprises about that.
It is the lateral surface parallel to the oncoming wind that I wish to emphasize. Here, the velocity is almost exactly the same as the oncoming free stream velocity. The pressure and density ratio are a bit lower than free stream, but they are still remarkably close, considering that this air has passed through an oblique bow shock wave. This is the justification for using free stream conditions as representative of conditions on a parallel lateral surface, when doing exploratory “ballpark” heat transfer analyses.
Gas Properties and Convective Heat Transfer Model
Because this is generic “ballpark” work, I did not use exact high-temperature properties for air. Instead, for convenience, I used empirical generic gas properties correlation equations applicable to air or other gases, including combustion gases. Like the compressible flow analysis methods, these are restricted to ideal gas equation-of-state applicability (roughly 5500 R ~ 5000 F max total temperature).
Those models presume that an appropriate specific heat ratio and molecular weight are known as inputs. Two of the properties (Prandtl number and specific heat at constant pressure) are independent of temperature in these correlations. The rest depend upon temperature and molecular weight in various ways. These include thermal conductivity and absolute (not kinematic) viscosity. Density uses the ideal-gas equation of state. This is shown as part of Figure 2 below.
For the heat transfer model, it is presumed that velocity V, static pressure P, static temperature T, and total temperature Tt are known edge-of-boundary-layer conditions. In addition, the surface temperature Ts must be known (or usually guessed-for-iteration). This is also indicated in Figure 2.
The actual heat transfer correlation is for an average film coefficient over a “flat plate” surface parallel to the flow, of surface area A, with a dimension (in the streamwise direction) of L. In high-speed flow where compressibility and energy dissipation effects dominate, the best recommended textbook correlations evaluate properties at a reference temperature T*, instead of the usual average film temperature (T + Ts)/2. These also use the boundary layer recovery temperature Tr instead of bulk fluid static temperature T, as the driving temperature in the convective heat transfer equation. This is also indicated in Figure 2.
As with most convective heat transfer correlations, this one starts with an overall flat plate Reynolds number computed from edge-of-boundary layer bulk flow velocity V, plate streamwise length L, and with both density and viscosity evaluated at T*. This and the Prandtl number at T* are used to calculate the turbulent Nusselt number as shown in Figure 2. This is an empirical correlation, and every situation has its own empirical correlation. Nusselt number gets converted to film coefficient using the dimension L and the thermal conductivity evaluated at T*, as shown in Figure 2.
Then this film coefficient is an effective overall thermal conductance applied to the temperature difference to calculate heat flow per unit area Q/A. The temperature difference is that between Tr and Ts, as shown. For Tr > Ts, the resulting positive Q/A is heat transferred to the surface from the flow.
Heat Balance and Radiation and Conduction Models
The fundamental steady-state heat balance requires that heat added to the surface equals heat lost from the surface. At the conditions presumed here, there is no radiation to the surface, there is only convection from the hot air flow about the body to its surface. There is heat lost as thermal radiation to the environment, and there can be conduction inward into the interior, as indicated in Figure 3 below.
Those radiation and conduction models are simpler, and are also shown in Figure 3 below.
The radiation model requires inputs for surface thermal emissivity and for the effective temperature of the surroundings. The “view factor” here is unity, so the geometry cannot be complex. Typical reflective or “white” surfaces have a low thermal emissivity in the vicinity of 0.2. Typical highly-emissive or “black” surfaces have a high thermal emissivity in the vicinity of 0.8. Typical “Earth temperatures” for the surroundings are near 300 K ~ 540 R. These data fit Boltzmann’s equation as shown in the figure.
The conduction-inward model is even simpler. It presumes only two layers of different thermal conductivities and thicknesses, operating between the equilibrium surface temperature Ts and a constant cold sink temperature Tc inside the airframe. The individual layer thermal resistance is its thickness divided by its thermal conductivity, in appropriate units, as (t, ft)/(k, BTU/hr-ft-R).
The sum of the two layers’ thermal resistances is the overall thermal resistance. The temperature difference Ts – Tc divided by the overall thermal resistance is the conductive heat flow per unit area Q/A, as shown in Figure 3. Alternatively, the inverse of the overall thermal resistance is the effective thermal conductance, which multiplies Ts – Tc to produce Q/A. This was used in the spreadsheet.
Note that everything depends very fundamentally upon surface temperature Ts, something not known at the outset of analysis. In the spreadsheet, Ts is bounded, and all results computed vs Ts between those limits. The net heat flow to the surface is (Q/A convective) – (Q/A radiation) – (Q/A conduction). The steady-state equilibrium value of Ts is that which makes the net Q/A zero. The spreadsheet includes a row where trial Ts values can be input to make this net value as close to zero as desired.
To zero-out the conduction heat flow, input an extremely-large thickness for the layer of lower thermal conductivity. This makes its thermal resistance very large, in turn making the overall thermal resistance very large, without risking any division-by-zero problems. That makes the effective thermal conductance essentially zero, thus zeroing the conductive heat flow.
To zero-out the re-radiated heat to the environment, merely input a surface emissivity of e = 0. This makes the radiation heat flow zero without risking any division-by-zero problems.
You need not zero-out either of the heat flows from the surface, the spreadsheet uses both in the balance against convective input. You may zero out one or the other, as desired. You may not zero out both. You cannot zero-out the only heat input: convection from the hot air.
Spreadsheet inputs are highlighted yellow. For the air, these include MW = 28.97 and γ = 1.40. The edge-of-boundary layer conditions are input as local Mach M, static pressure P in psf, and static temperature T in degrees R. These are used to compute total and recovery temperatures Tt and Tr, constant regardless of surface temperature Ts, all degrees R. They are also used to compute speed of sound and local flow velocity V, both ft/sec.
The inputs for altitude and day type are visual reminders only. It is easy to forget what you are doing.
There are inputs for surface emissivity e and for the Earth (or surroundings) temperature (typically 540 R, equivalent to 300 K). To make radiative cooling zero, input a zero e value. But, e cannot exceed 1.
There is an input for the plate dimension L, ft. This needs to be large enough to make the problem qualify as turbulent, since the turbulent Nusselt correlation and recovery temperature are presumed.
There are 5 inputs for the conduction model. Each of the 2 layers has a thickness t, inch, and a thermal conductivity k, BTU/hr-ft-R. Subscript s refers to the surface layer, subscript b refers to the layer buried deeper within. The cold sink temperature Tc, R, is the 5th input. 15 C = 77 F = 536.7 R. Individual and summed thermal resistances are computed from these.
To model a metal skin with insulation underneath it, use a smaller ts and a larger ks, with a larger tb and a smaller kb. To model an ablative or refractory heat shield over an interior structure, use a thicker ts and lower ks, with a thinner tb and higher kb. This minimal 2-layer model actually is quite versatile.
The effective thermal conductance is also shown, and highlighted light blue for convenience when recovering data for plotting. To zero-out heat conduction into the interior, make the highlighted conductance near zero with a ridiculously-large tb or ts input, whichever has the lower k value.
Values of Ts are bounded by the fluid static temperature T and its total temperature Tt. That total temperature is highlighted reddish, for easy comparison to 5500 R as the limiting value for analysis applicability. Values of Ts are distributed evenly across 11 rows between those bounds, with a 12th row added at the bottom. The Ts in that 12th row is a yellow-highlighted input for iteration.
Columns across show the Ts value, then computed T*, then the density, thermal conductivity, and viscosity that are evaluated at T*. Then there are Reynolds number (needs to exceed ~500,000 to be turbulent), Nusselt number, film coefficient (h, BTU/hr-ft2-R), and convective heat flow per unit area (Q/A, BTU/hr-ft2). The next 3 columns are absolute radiation from the surface to the environment, absolute radiation received from the environment, and the net re-radiation to the environment.
The last two columns are the net heat flow per unit area to the surface, and the conductive heat flow from the surface to the cool sink within. These are out of order, because the conductive model was added as an afterthought. The net heat flux column is highlighted light green to call it out.
The bottom iteration row is largely highlighted light blue. There is a plot to the right to aid in selecting trial Ts values. If the net heat flux is positive, try a larger Ts. If negative, try a smaller Ts.
Most of the time, net heat fluxes will be order of magnitude 103 BTU/hr-ft2 or more. Once you are down to heat fluxes of order of magnitude 100 or lower, you are “close enough”. This usually happens at about the nearest quarter of a degree R, or thereabouts. I usually try to find the nearest 0.1 R.
A sort-of approximate half-interval search is the fastest way to do this. Don’t be obsessive-compulsive about being exactly halfway between two earlier values. Don’t worry about decimals until you are dealing with tenths of a degree. March until you see a sign change, then search between those two.
Flowfield Results at Mach 5, 60 kft, and Zeroed Conduction
These were run for the nose surface, the lateral surface, and the boat-tail surface, plus a fictitious surface out front, at freestream conditions. See Figure 4 below. This included low (“white”) and high (“black”) surface thermal emissivities. The “white” or reflective surface is the data across the upper part of the figure, while the “black” or highly-emissive data lies across the lower portion.
As depicted in the figure, the lateral surface film coefficients and equilibrium surface temperatures are “close enough” to the freestream model (less than 100 F different), to justify using that simplification for the other parts of this trend investigation, or for “ballpark” analyses in general.
Realism of the simplified flow-field calculations
As expected, the temperatures on the slightly more forward-facing nose surfaces are 150-200 F hotter than lateral, and the slightly more aft-facing boat-tail surfaces about 100 F colder than lateral. Thus, we may conclude this is a fairly realistic thermal analysis, despite the very simplified nature of the flow analysis calculations. Finite-element computer fluid dynamics (CFD) had no role in this.
Importance of surface radiation efficiency
Also as expected, the effects of surface radiation efficiency are quite important. This shows up as about a 250-300 F difference between equilibrium surface temperatures Ts for e = 0.20 vs e = 0.80. Both radiate (that being the only heat loss modeled here), but the “black” e = 0.80 surface radiates more easily, lowering the equilibrium surface temperature considerably. Most metal surfaces are highly reflective (low e), and most ceramics are “white” (low e). This result shows the crucial importance of a highly emissive (“black”, high-e) surface for high speed flight in the atmosphere.
Effects of Thermal Conduction Into the Interior
This was investigated using only the fictitious free stream surface for simplicity. It was done at Mach 5 60 kft conditions only, to conserve effort. The scope includes the zeroed thermal conductance (as already done), plus insulation thicknesses of 1 and 4 inches underneath a thin (0.160 inch) metal skin.
“Typical” hot values of metal ks = 15 BTU/hr-ft-R, and “warm” insulation kb = 0.2 BTU/hr-ft-R, were used. (The ridiculously-large value tb = 100,000 inches was how conduction was zeroed previously.)
Results are shown in Figure 5 below, for freestream plates of both e = 0.2 and 0.8. To understand the trends better, plots were made of equilibrium surface temperature Ts versus skin conductance values. These are shown in Figure 6 below.
Shedding heat by conduction into the interior clearly reduces Ts dramatically, regardless of the value of surface emissivity e. However, unlike re-radiation, there is a price to be paid for that conduction-lowered Ts. The heat that conducts through into the interior must be dealt with, either by direct heat sinking into some adjacent mass, or by active cooling of the inside surface. That conduction heat rate which must be dealt with is plotted in Figure 7.
Active cooling is really just a means of heat-sinking into a non-adjacent mass. Either way, it is still heat-sinking: you may only fly for a finite time before your heat sink is full. If all the cooling is re-radiative, there is no fundamental flight time limit. The distinction could not be more stark!
As shown in Figure 6, the effects upon external surface temperature Ts, of having an effective heat conductance path inward, are quite modest. This is simply because the inward conducted heat flux is 10 to 100 times smaller than the convective heat flux to the surface, and also the re-radiated heat flux from the surface.
It therefore makes very little difference to Ts (something like 50-100 F) to assume some inward conduction with some heat sink required. The surface temperatures are very little different from those cooled only by radiation. For purposes of selecting materials and flight limits, that radiation-only design analysis is pretty much “good enough”.
As shown in Figure 7, the effects of having an effective heat conductance path inward upon the quantity of heat to be dealt with, are not so modest. We assume water as the cooling fluid (with specific heat c = 1.0 BTU/lbm-R), and a max allowable coolant rise across any square foot of 5 F = 5 R. One divides the heat flux (Q/A) by the product of specific heat and temperature-rise (c ΔT), to obtain the “loading” of coolant flow rate per square foot (wc/A) that is required.
At something like 2700 BTU/hr-ft2 for about 1 inch of insulation, this is wc/A = 540 lbm/hr-ft2 = 0.15 lbm/sec-ft2. It would be roughly twice that, using jet fuel as the coolant, at c = 0.5 BTU/lbm-R, or near 0.30 lbm/sec-ft2. That would be 30 lbm/sec (at 60 kft) for 100 square feet of area to be cooled!
Whether that is “modest” depends upon how many square feet of surface there is to cool relative to how many square feet of propulsive cross section is needed, how much fuel is on-board, and how fast that fuel is being used for propulsion. Every design is different. But recycling flow to the tank seems far more likely than one-way through-the-cooling to the propulsion, for any reasonable size at all.
Practicality: some sort of active cooling is virtually certain, as no matter how the skins are mounted, there will be one or another kind of thermal conduction path into the interior. The only way to stop radiant heating of the interior by the skin is to include insulation just under the skin. Most flight vehicles will be short on internal space, precluding insulation thicknesses beyond about an inch or so.
That last situation will be true even for configurations with an ablative or refractory heat shield layer on top of a metal or composite substructure. The substructure temperatures will be lower, but the amount of heat flow to deal with depends directly and mostly on the heat shield thickness.
Effects of Speed and Altitude
To investigate the effects of altitude, speed was held constant at Mach 5, with conduction into the interior zeroed. Both “white” (e = 0.2) and “black” (e = 0.8) surface emissivities were used. Altitudes of 20 kft, 60 kft (reference point already done), and 100 kft were used. 60 kft is a stratospheric altitude at moderate pressure, with the coldest air. 20 kft is tropospheric, with high pressure and quite warm air. 100 kft has very low pressure, but air temperatures not much warmer than stratospheric. Warmer air raises the surface temperatures in the distribution; thinner air lowers the convective heating (and thus the surface temperature) by reducing the film coefficient values.
To investigate the effects of flight speed, altitude was held constant at the 60 kft already investigated, with conduction into the interior zeroed. Both “white” (e = 0.2) and “black” (e = 0.8) surface emissivities were used. Mach numbers of 6 and 7 were used, leading to higher total and recovery temperatures. These in turn raise equilibrium surface temperatures.
All these results are given in Figure 8. The altitudes are grouped, as are the speeds, taking advantage of the Mach 5 / 60 kft / no-conduction data already obtained. Again, the “white” surface data are across the upper part of the figure, with the “black” surface data across the lower part.
These results show that equilibrium surface temperatures are even more sensitive to altitude and Mach number variations than to surface emissivity. Temperatures get very large very quickly at 20 kft vs 60 kft, at only Mach 5. They get much smaller very quickly at 100 kft vs 60 kft, at Mach 5. The “thin air” effect is thus very strong. This is true regardless of emissivity, it’s just that surface temperature levels are substantially lower with the higher emissivity, since radiation is easier than convection in thin air.
At 60 kft, equilibrium surface temperatures get very much larger very quickly with increasing Mach number. This, too, is a very strong effect, and it’s true regardless of the surface emissivity, since the Tr is so much higher. It’s just that temperatures are a little lower with the larger emissivity.
If we assume that 1200 F is a “max survivable skin material temperature”, then just about Mach 5 is survivable at 60 kft, but only with the high emissivity. Flying faster at 60 kft, or even just Mach 5 lower down in the atmosphere, would seem to be quite infeasible in terms of that material temperature limit. The strength and direction of the sensitivities suggests that we might successfully fly faster in the far thinner air at 100 kft. This would seem to be even more feasible with the high emissivity only.
One Final Look
As confirmation, I ran a sweep of higher Mach numbers at 100 kft conditions, and only with the higher surface emissivity. I did this with inward conduction zeroed. Those results are listed in Figure 9, and plotted in Figure 10. This shows the surprisingly very mild nonlinearity of these results. Total temperature gets included in the data list of Figure 9, to verify whether the analysis technique assumptions get violated at the higher speeds in the warmer air up that high. Again, this is for the fictitious freestream panel, not the actual locations on the airframe.
Looking at the data in Figure 9, the approximate limit of 5500 R for air total temperature is violated at Mach 8. This means the analysis is becoming fundamentally inaccurate at Mach 8 / 100 kft conditions, due to significant ionization of the air into something that really isn’t air anymore. While this violation isn’t large, it does indicate that looking at higher speeds with these methods would be pointless.
Looking at the plot in Figure 10, and making the same material max temperature limit assumption of 1200 F as before, it is immediately apparent that speeds beyond about Mach 7 will definitely overheat even high-emissivity skins at 100 kft. Considering the variations with location around the airframe found earlier, that lateral skin speed limit at 100 kft is likely nearer Mach 6.8 or so. Compare this to just under Mach 5 at 60 kft. The thin air effect makes the skin Ts max speed limit a function of altitude.
Not examined here were stagnation points (noses) or lines (along aerosurface leading edges). These see higher air temperatures and higher local pressures, and the stagnation heating correlations are inherently different. For similar construction, surface temperatures at stagnation zones will be even higher, something long seen in actual practice.
However, construction approaches and material selections will be different in those stagnation zones. These are of limited extent, and will thus be far smaller fractions of the airframe weight. The lateral skins cover large portions of the airframe, and will thus be far larger fractions of the airframe weight.
A similar analysis of stagnation zones must be done to quantify those flight speed and altitude limits. Whichever result (lateral skins vs stagnation zones) governs, sets the actual airframe speed and altitude limits. You do Mach sweeps at multiple altitudes to accomplish any of these investigations.
A word of caution: airbreathing propulsion systems will feature air inlets and internal ducting. Because of locally-higher pressures, and two-sided convective heating, the materials making up inlet lips will be hotter still. Internal ducting cannot “see” the external environment to cool by re-radiation. Those features will inherently require active cooling of some sort. As thin as these structures need to be, self-heat sinking will simply not be feasible for long flight times.
Complicated shapes and faster conditions
The more complicated these shapes and flow situations, the less likely such a simplified flow analysis, as was used here, will be feasible. This is an area better addressed by finite element models used with CFD codes and thermal-structural codes, in computers.
Once the total temperature limit gets violated at the higher speeds, this kind of simplified analysis is no longer accurate enough to be feasible at all. That also requires CFD codes using finite element models that can also be fed through thermal-structural analyses. Trends in the answers will resemble those reported here; it’s just that the analyses must account for the ionization, much like entry problems.
Finally, the “max material limit” of 1200 F used here for realistic illustration is mostly arbitrary. It “sort-of applies” to 17-7 PH alloy steel, as long as the large hot strength reduction at 1200 F can be tolerated. This effect gets rapidly worse as the material heats up. Some stainless steels can go a little hotter without scaling, but they have even less hot strength, all are “soft butter” beyond about 1000 F.
Other steels like low-alloy D6ac and martensitic stainless 4130 tolerate lower max temperatures, nearer 900-1000 F. There are some non-steel superalloys that can go into the 1600-2000 F range, but they are quite weak that hot, and most of them very hard to work, and all are quite expensive.
Max temperature for mild carbon steels and for titanium are lower yet, at about 700-800 F, and they are quite weak that hot. The common assertion that titanium is a high-temperature material is just plain wrong. It has similar strength to mild steels while being lighter in weight. Many steels go hotter.
Most aluminum is “junk” at about 350 F, and the organic (epoxy) matrices of carbon (and other fiber) epoxy composites falls apart somewhere between 200 and 290 F. The same is true of vinyl ester and polyester matrices. Glass and Kevlar fibers get you more toughness and impact resistance at lower composite stiffness. Carbon fiber gets you high stiffness, but at the cost of extreme impact vulnerability, and with very high hidden-damage risks.
For any of these, if you need better strength, you simply must lower the max operating temperature. Period. Good sources for strength vs temperature data are manufacturer’s data sheets, and (for some of the more traditional aerospace materials) Mil Handbook 5.
I have said before (repeatedly) that the enabling factor for sustained hypersonic (or even just high-supersonic) flight in the atmosphere is heat protection. Most such applications feature long flight times, so this means the enabling factor is steady-state heat protection.
The steady-state heat protection solution is far more difficult to achieve than that for the brief transient of entry, even at planetary entry speeds. That is fundamentally the reason why we humans have been flying spacecraft back from space for over half a century, yet we still do not fly sustained missions down in the atmosphere at high supersonic speeds, much less hypersonic speeds.
Until that picture changes substantially, you can be rather sure of dismissing high speed vehicle concepts and proposals as hype, if they do not begin from a steady-state heat protection solution! This heat protection problem far outweighs any propulsion considerations as the key enabling factor. (One can always push something hypersonic, for at least a short time, with a big-enough rocket.)
Related Thermal-Structural Articles Posted on:
Heat Protection Is the Key to Hypersonic Flight posted 7-4-17 by GWJ
Shock Impingement Heating Is Very Dangerous posted 6-12-17 by GWJ
Why Air Is Hot When You Fly Fast posted 11-17-15 by GWJ
Commentary on Composite-Metal Joints posted 6-13-15 by GWJ
Building Conformal Propellant Tanks, Etc. posted 10-6-13 by GWJ
Entry Issues posted 8-4-13 by GWJ
Low-Density Non-Ablative Ceramic Heat Shields posted 3-18-13 by GWJ
Figure 8 – Results of Mach and Altitude Variation About the Reference Point, Conduction Zeroed
Figure 9 – “Final Look” Results for Mach Sweep at 100 kft, Conduction Zeroed, High Emissivity Only
Figure 10 – Plot of Surface Temperature vs Mach at 100 kft, Conduction Zeroed, High Emissivity Only