The following discussions define the various heating and cooling notions for entry stagnation heating, in terms of very simple models that are known to be well inside the ballpark. How to achieve the energy conservation balance among them is also addressed. This is more of an “understand how it works” article than it is a “how to actually go and do” article.
Convective Stagnation Heating
The stagnation point heating model is proportional to density/nose radius to the 0.5 power, and proportional to velocity to the 3.0 power. The equation used here is H. Julian Allen’s simplest empirical model from the early 1950’s, converted to metric units. It is:
qconv, W/sq.cm = 1.75 E-08 (rho/rn)^0.5 (1000*V)^3.0, where rho is kg/cu.m, rn is m, and V is km/s
The 1000 factor converts velocity to m/s. This is a very crude model, better correlations are available for various shapes and situations. However, this is very simple and easy to use, and it has been "well inside the ballpark" since about 1953. This is where you start. See Figure 1.
Plasma Radiation Stagnation Heating
There are all sorts of correlations for various shapes and situations. However, to get started, you just need a ballpark number. That comes from the widely-published notions that (1) radiational heating varies with the 6th power of velocity, and (2) radiation dominates over convection heating at entry speeds above 10 km/s. What that means is you can use a very simple radiational heating model, and "calibrate" it with your convection model:
qrad, W/sq.cm = C (1000*V)^6, where V is input as km/s
The 1000 factor converts speed to m/s. The resulting units of the constant C are W-s^6/sq.cm-m^6. You have to "calibrate" this by evaluating C with your convective heating result at 10 km/s, and a "typical" entry altitude density value, for a given nose radius for your shape:
find qconv per above at V = 10 km/s with "typical" rho and rn, then
C, W-s^6/sq.cm-m^6 = (qconv at 10 km/s)(10^-24)
This should get you into the ballpark with both convective and radiation heating. Figure both and then sum them for the total stagnation heating. Below 10 km/s speeds, the radiation term will be essentially zero. Above 10 km/s it should very quickly overwhelm the convective heating term.
As an example, I had data for an Apollo capsule returning from low Earth orbit. I chose to evaluate the peak stagnation heating point, which occurred about 56 km altitude, and about 6.637 km/s velocity. See Figure 2. Dividing that convective heating value of 55.72 W/sq.cm by the velocity cubed, and then multiplying by 10 km/s cubed, I was able to estimate stagnation convective heating at 10 km/s and 56 km altitude as 190.59 W/sq.cm.
Dividing that value by 10 km/s to the 6th power gave me a C value of 1.90588 x 10-4, to use directly with velocities measured in km/s, for estimating radiation heating from the plasma layer adjacent to the surface. The resulting trends of convective, radiation, and total stagnation heating versus velocity (at 56 km) are shown in Figure 3.
This is a form of Boltzmann's Law. The power you can radiate away varies as the 4th power of the surface temperature, but gets modified for an effective temperature of the surroundings receiving that radiation (because that gets radiated back, and emissivity is equal to absorptivity):
qrerad, BTU/hr-ft^2 = e sig (T^4 – TE^4) for T’s in deg R and sig = 0.1714 x 10-8 BTU/hr-ft^2-R^4
For this equation, T is the material temperature, TE is the Earthly environment temperature (near 540 R = 300 K), e is the spectrally-averaged emissivity (a number between 0 and 1), and sig is Boltzmann’s constant for these customary US units.
This radiation model presumes transparency of the medium between the radiating object and the surroundings. That assumption fails rapidly above 10 km/s speeds, as the radiating plasma in the boundary layer about the vehicle becomes more and more opaque to those wavelengths.
Therefore, do not attempt radiationally-cooled refractory heat protection designs for entry speeds exceeding about 10 km/s. They won't work well (or at all) in practice. Ablative protection becomes pretty much your only feasible and practical choice.
Heat Conduction Into The Interior
This is a cooling mechanism for the exposed surface, and a heating mechanism for the interior structure. In effect, you are conducting heat from the high surface temperature through multiple layers of varying thermal conductivity and thickness, to the interior at some suitable "sink" temperature.
The amount of heat flow conducted inward in steady state depends upon the temperature difference and the effective thermal resistance of the conduction path. The electrical analog is quite close, with current analogous to heat flow rate per unit area, voltage drop analogous to temperature difference, and resistance analogous to thermal resistance.
In the electrical analogy to 2-D heat transfer, conductance which is the inverse of resistance is analogous to a thermal conductance which is a thermal conductivity divided by a thickness.
Resistances in series sum to an overall effective resistance, so the effective thermal resistance is the sum of several inverted thermal conductances, one for each layer. Each resistance sees the same current, analogous to each thermal resistance layer seeing the same thermal flux, at least in the 2-D planar geometry.
Like voltage/effective resistance = current, heat flow per unit area (heat flux) is temperature drop divided by effective overall thermal resistance. (The geometry effect gets a bit more complicated than just thickness in cylindrical geometries.) In 2-D:
qcond = (Tsurf - Tsink)/effective overall thermal resistance
For this the effective overall thermal resistance is the sum of the individual layer resistances, each in turn inverted from its thermal conductance form k/t:
eff. th. resistance (2-D planar) = sum by layers of layer thickness/layer thermal conductivity
Using the electrical analogy, the current (heat flux) is the voltage pressure (temperature difference) divided by the net effective resistance (thermal resistance). The voltage drop (temperature drop) across any one resistive element (layer) is that element's resistance (layer thermal resistance) multiplied by the current (heat flux). See Figure 4.
What that says is that for a given layering with different thicknesses and thermal conductivities, there will be a calculable heat flux for a given overall temperature difference. Each layer has its own temperature drop once the heat flux is known, and the sum of these temperature drops for all layers is the overall temperature drop.
Any layer with a high thermal resistance will have a high temperature drop, and vice versa. High thermal resistance correlates with high thickness, and with low thermal conductivity. A high temperature drop over a short thickness (a high thermal gradient) requires a very low thermal conductivity indeed, essentially about like air itself.
On the other hand, any high-density material (like the monolithic ceramics) will have high thermal conductivity, and thus the thermal gradients it can support are inherently very modest. Such parts trend toward isothermal behavior. Their high meltpoint does you little practical good, if there is no way to hang onto the "cool" end of the part. In point of fact, there may not be much of a “cool” end.
Active Liquid Cooling
In effect, this is little different than the all-solid heat conduction into a fixed-temperature heat sink, as described just above. The heat sink temperature becomes the allowable coolant fluid temperature, and the last “layer” is the thermal boundary layer between the solid wall and the bulk coolant fluid. The thermal conductance of this thermal boundary layer is just its “film coefficient” (or “heat transfer coefficient”). The simple inverse of this film coefficient is the thermal resistance of that boundary layer. See Figure 5.
The main thing to worry about here is the total mass of coolant mcoolant recirculated, versus the time integral (for the complete entry event) of the heating load conducted into it. That heat is going to raise the temperature of the coolant mass and indirectly the pressure at which it must operate. That last is to prevent boiling of the coolant.
∫ qcond dt = mcoolant Cv (Tfinal – Tinitial) where Tfinal is the max allowable Tsink
Balancing the Heat Flows: Energy Conservation
A patch of heat shielding area sees convective heating from air friction, and may see significant radiation heating if the entry speed is high enough. That same patch can conduct into the interior, and it can radiate to the environment, if the adjacent stream isn't opaque to that radiation. If the heat shield is an ablative, some of the heating rate can go into the latent heat of ablation. See Figure 6.
The correlations for convective and radiation stagnation heating given above depend upon vehicle speed, not plasma temperature. The equation for conduction into the interior depends upon the surface and interior temperatures. Re-radiation to the environment depends upon the surface and environmental temperatures. Of these, both the environmental and heat sink temperatures are known fixed quantities.
If the heat shield is ablative, then the surface temperature is fixed at the temperature at which the material ablates; otherwise, surface temperature is free to "float" for refractory materials that cool by radiation.
The way to achieve energy conservation for refractories is to adjust the surface temperature until qconv + qrad - qcond - qrerad = 0. For ablatives, the surface temperature is set, and you just solve for the rate of material ablation (and the recession rate): qabl = qconv + qrad - qcond - qrerad.
The heating flux rate qabl that goes into ablation, divided by the latent heat of ablation Labl times virgin density ρ, can give you an estimate of the ablation surface recession rate r (you will want to convert to more convenient units):
(qabl BTU/ft^2-s)/(Labl BTU/lbm)(ρ lbm/ft^3) = ft^3/ft^2-s = r, ft/s
(qabl W/m^2)/(Labl W-s/kg)(ρ kg/m^3) = m^3/m^2-s = r, m/s
Those require the use of empirical correlations or actual test data to get accurate answers. However, to just get in the ballpark, any guess is better than no guess at all! For lateral windward-side heating, try about half the heat flux as exists at the stagnation point. For lee-side heating in the separated wake, try about 10-20% of the stagnation heating.
Bear in mind that these equations are for steady-state (thermal equilibrium) exposure. The conduction into the interior is the slowest to respond to changes. Transient behavior takes a finite-difference solution to analyze. There is no way around that situation.
But if that conduction effect is small compared to the applied heating terms because there is lots of re-radiation or there is lots of ablation to balance them, you can then approximate things by deleting the conduction-inward term. You simply cannot do that if there is no ablation or re-radiation. And your conduction effect will not be small compared to the heating, if you are doing active liquid cooling.
If you are entering from Earth orbit at speeds no more than 8 km/s, you can reasonably ignore the plasma radiation stagnation heating term. On the other hand, for entry speeds above 8 km/s, your re-radiation cooling term will rapidly zero as the plasma layer goes opaque to thermal radiation. Its transmissibility must necessarily zero, in order for its effective emissivity to become large. The zero transmissibility is what zeroes the re-radiation term.
If you choose to do a transient finite-difference thermal analysis, what you will find in the way of temperature distributions within the material layers has little to do with steady-state linear temperature gradients. Instead, there will be a “humped” temperature distribution that moves slowly like a wave through the material layers. This is called a “thermal wave”. It forms because heat is being dumped into the material faster than it can percolate through by conduction. See Figure 7.
Figure 7 – The Thermal Wave Is a Transient Effect
This humped wave of temperature will decrease in height and spread-out through the thickness, as it moves through the material. But, usually its peak (even at the backside of the heat shield) is in excess of the steady-state backside temperature estimates. It may take longer to reach the backside than the entire entry event duration, but it will certainly tend to overheat any bondlines or substrate materials.
Material properties such as thermal conductivity are actually functions of material temperature. These are usually input as tables of property versus temperature, into the finite-difference thermal analyses. Every material will have its own characteristics and property behavior. I did not include much material data in this article, in the way of typical data, for you to use. As I said at the beginning, this article is more about “understanding” than it is “how-to”.
The local heating away from the stagnation point is lower. There are many correlations for the various shapes to define this variation, but for purposes of finding out what “ballpark you are playing in”, you can simply guess that windward lateral surfaces that see slipstream scrubbing action will be subject to crudely half the stagnation heating rate. Lateral leeward surfaces that face a separated wake see no slipstream scrubbing action. Again, there are lots of different correlations for the various situations, but something between about a tenth to a fifth of stagnation heating would be “in the ballpark”.
Low density ceramics (like shuttle tile and the fabric-reinforced stuff I made long ago) are made of mineral flakes and fibers separated by considerable void space around and between them. The void space is how minerals with a high specific gravity can be made into bulk parts with a low specific gravity:
sgmineral * solid volume fraction = sgmineral *(1 – void volume fraction) = bulk effective sg
The low effective density confers a low thermal conductivity because the conduction paths are torturous, and of limited cross-section from particle to particle, through the material. Such thermal conductivity will be a lot closer to a mineral wool or even just sea level air, than to a firebrick material.
This low effective density also reduces the material strength, which must resist the wind pressures and shearing forces during entry. A material of porosity sufficient to insulate like “mineral wool-to-air” will thus be no stronger than a styrofoam.
So, typically, these low density ceramic materials are weak. And they are very brittle. The brittleness does not respond well to stress- or thermal expansion-induced deflections in the substrate, precisely because brittle materials have little strain capability. That fatal mismatch has to be made up in how the material is attached to the substrate. If bonded, considerable flexibility is required of the adhesive.
Related Thermal/Structural Articles On This Site
Not all of these relate directly to entry heat transfer. The most relevant items on the list are probably the high speed aerodynamics and heat transfer article, and the article taking a look at nosetips and leading edges.
The “trick” with Earth orbit entry using refractories instead of ablatives is to maximize bluntness in order to be able to use low density ceramics at the stagnation zone without overheating them. That leaves you dead-broadside to the slipstream, and thus inevitably ripping off your wings, unless you do something way “outside the box”.
The pivot wing spaceplane concept article is a typical “outside the box” study restricted to 8 km/s or less. The older folding wing spaceplane article is similar. In both studies, the wings are relocated out of the slipstream during entry, which is conducted dead broadside to the oncoming flow.
The reinforced low density ceramic material that I made long ago is described to some extent in the article near the bottom of the list (about low density non-ablative ceramic heat shields). It is like the original shuttle tile material that inspired it, but is instead a heavily reinforced composite analogous to fiberglass. This information was also presented as a paper at the 2013 Mars Society convention.
The fastest way to access any of these is to use the search tool left side of this page. Click on the year, then on the month, then on the title. It is really easy to copy this list to a txt or docx file, and print it.
1-2-20…On High Speed Aerodynamics and Heat Transfer
4-3-19…Pivot Wing Spaceplane Concept Feasibility
1-9-19…Subsonic Inlet Duct Investigation
1-6-19…A Look At Nosetips (Or Leading Edges)
1-2-19…Thermal Protection Trends for High Speed Atmospheric Flight
7-4-17…Heat Protection Is the Key to Hypersonic Flight
6-12-17…Shock Impingement Heating Is Very Dangerous
11-17-15…Why Air Is Hot When You Fly Fast
6-13-15…Commentary on Composite-Metal Joints
10-6-13…Building Conformal Propellant Tanks, Etc.
3-18-13…Low-Density Non-Ablative Ceramic Heat Shields
3-2-13…A Unique Folding-Wing Spaceplane Concept
1-21-13…BOE Entry Analysis of Apollo Returning From the Moon
1-21-13…BOE Entry Model User’s Guide
7-14-12…Back of the Envelope Entry Model
There are also several studies for reusable Mars landers that I did not put in the list. They are similar to the two spaceplane studies, but entry from Mars orbit is much easier than entry from Earth orbit. Simple capsule shapes work fine without stagnation zone overheat for low density ceramics, even for the large ballistic coefficients inherent with large vehicles.
I’ve been retired for some years now, and I have been retired out of aerospace work much longer than that. I’ve recently been helping a friend with his auto repair business, but that won’t last forever.
Not surprisingly, I am not so familiar with all the latest and greatest heat shield materials, or any of the fancy computer codes, and I have little beyond these paper-and-pencil-type estimating techniques to offer (which is exactly “how we did it” when I first entered the workforce long ago). Yet these simple methods are precisely what is needed to decide upon what to expend the effort of running computer codes! Today’s fresh-from-school graduates do not know these older methods. But I do.
Sustained high speed atmospheric flight is quite distinct from atmospheric entry from orbit (or faster), but if you looked at that high speed aerodynamics and heat transfer article cited in the list, then you already know that I can help in that area as well.
Regardless, it should be clear that I do know what to worry about as regards entry heat protection, and how to get into the ballpark (or not) with a given design concept or approach in either area (transient entry or sustained hypersonic atmospheric flight)! I can help you more quickly screen out the ideas that won’t work, from those that might.
And if you look around on this site, you will find out that I also know enough to consult in ramjet and solid rocket propulsion, among many other things. I’m pretty knowledgeable at alternative fuels in piston and turbine engines, too.
If I can help you, please do contact me. I do consult in these things, and more.
More? More: I also build and sell cactus eradication farm implements that really work easier, better, and cheaper than anything else known. I turned an accidental discovery into a very practical family of implements. It’s all “school of hard knocks” stuff.