Border “Crisis”? Nope

Update 5-5-19:  The continuing crisis has morphed somewhat.  There now seems to be a bigger population of migrants classifiable as refugees seeking asylum than guest workers looking for work. 

This is for reasons beyond our control:  the effect of at least 3 failed states in Central America.  This might be temporary,  or it might be permanent.  Statements by the various border control agencies do confirm this assessment,  whether worded that way or not.

The problem:  we are set up to pursue and deal with illegal migrant workers (a problem we created with artificially-low quotas for guest worker visas).  Most of these folks are Mexican residents,  not Central American refugees. 

We are not set up to deal humanely with refugee families from Central America.  THAT is what the border control agencies are really telling us they cannot deal with. 

I have already described what to do about the Mexican guest worker problem:  revise the visa quotas to reflect the demonstrable size of that particular job market. That is something for Congress to do,  and they have ignored this for ~7 decades now,  thus creating the 10-12 million strong illegal alien population in the US. 

My fellow citizens:  PLEASE HOLD YOUR CONGRESSIONAL REPRESENTATION ACCOUNTABLE FOR THAT FAILURE!  That applies to both the House and the Senate. 

If your current representation does not talk about fixing this,  then elect someone new.  You could NOT POSSIBLY do any worse!

I have also described the ONLY ethical way to handle the exploding refugee problem:  staff up and just deal with it!  More judges,  more personnel in general,  and more facilities appropriate to the task (and tent cities along the Texas and Arizona borders are NOT what I mean !!!). 

Several administrations in recent years have neglected this issue,  but the current Trump administration is quite egregiously failing to deal with this at all.  They are motivated by a self-evident "we want no brown-skinned immigrants at all" policy,  if you deign to call that a policy.

I do not:  this is racism far beyond the abhorrent "Jim Crow" type,   this is closer to that utter abomination demonstrated by the Nazi Germans during World War Two.

It manifests as who showed up as a lot of the pro-Trump supporters at Charlottesville,  VA,  for one. These included a lot of people wearing Nazi uniforms,  carrying torches and swastika flags,  and other Nazi regalia.  Despite what Trump said,  those are NOT "fine people". 

Face the truth:  there are no good Nazis!  There never were.

My fellow citizens:  PLEASE HOLD THE TRUMP ADMINISTRATION ACCOUNTABLE FOR THIS UTTER FAILURE TO CONFORM TO MINIMAL STANDARDS FOR HUMAN DECENCY !!!!   

I don't care whether this accounting is his non-re-election in 2020,  or his impeachment and conviction / removal from office sooner. 

Just do it!  Before he finishes destroying our country by artificially dividing it (with lies) for political gain (his playbook since entering the race in 2015).  His fact-checking rate is almost exactly 0%.

As for details about how to properly handle guest workers and refugees seeking asylum,  read on. I also deal with border smugglers (drugs or people). 

Update 2-3-19:  see below at end.
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lead-in

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This “crisis” is not about compromising on wall funding vs DACA,  it is about the hurtful practice of holding America hostage,  by means of damaging government shutdowns,  to get a political desire not otherwise obtainable.  This evil practice has to stop,  and this time around is as good a time as any,  to put a permanent stop to such behavior. 

The border “crisis” itself goes far beyond just walls and DACA.  “They” are lying to you when they cast it only in those terms!  Both sides in Congress,  and the administration,  chronically lie about this issue,  but the Trump administration has been (by far) the most egregious with its lying. 

Most of the so-called “news” about this is also a lie,  even if only lying-by-omission.  Be careful of your sources:  if you hear no divergent voices to your own opinions,  then you are in an echo chamber being fed propaganda.  “Propaganda” is just a long-winded way of spelling “lie”.

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article

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Asylum seekers,  guest workers,  and cross-border smugglers are different problems with different solutions.  Only false political arguments lump them together.

Government statistics prove that asylum seekers and guest workers are less likely to commit crimes than US citizens at large.  There is no real threat there,  despite all the “justification” claims by the Trump administration.

“Less likely” is not zero,  there are bad apples in any barrel.  Finding the bad ones at the border crossings merely requires adequate staffing to do the job.  A wall does not help that.

Cross-border smugglers are the drug cartel and gang members;  those are the real threat.

Asylum seekers

The number of refugees seeking asylum at our southern border is up in recent years.  The reasons why are not under our control. 

These people have the legal right to cross the border and ask an official for asylum.  There is no question about that aspect of federal law,  despite all the political denials. 

This legal right was illegally denied by the Trump administration,  by means of “criminalizing” any crossing not at a port-of-entry,  then claiming that “criminality” as an excuse not to hear the cases and separate children.  That law is settled:  it does not care where the asylum-seeker crossed. 

Trump administration officials have admitted in public that they intended to use the threat of separation of children as a deterrent to stop other asylum seekers.  This is not just immoral,  it is evil.

It is a fact that we have too few immigration judges to hear these asylum requests.  The backlog is unconscionably high,  and getting far worse with the shutdown.

Our holding facilities were designed to handle men,  not women and children.  This plus the backlog leads to cages and tent cities.  These are an immoral evil,  that no one can deny.

A whole-border wall “fixes” none of this.

Why not repair the asylum process and staff-up to do it right?  That would be far cheaper than any wall,  and it frees up many of our agents to chase the cross-border smugglers instead!

Guest workers

This is a very old problem,  the result of about 7 decades of neglect by Congress.

The quota limit of ~120,000 per year for legal workers is about 10-100 times too small.  This is where our ~10-11 million illegal immigrant population grew from,  over those same decades of neglect. 

The jobs are here,  the workers that will do them are from down there.  They have to come just to eat,  legal or illegal.  A guest worker visa itself is not a path to citizenship,  but it need not preclude the holder from seeking such. 

Our border agents,  tied up trying to stop so many illegal guest workers,  cannot also deal so effectively with the cross-border smugglers.

A whole-border wall “fixes” little to none of this,  because any wall can be defeated.

Why not just adjust the guest worker visa quotas to realistic levels (and staff up to track them properly,  not done now),  thus freeing a great many border agents to deal with the smugglers? 

Smugglers (of People or Drugs)

Government data clearly shows most drugs come through designated and manned ports of entry,  hidden among legal cargoes.  A whole-border wall does nothing to fix that problem.

Some drugs come by sea or air.  A whole-border wall does nothing to fix that problem.

Asylum seekers do resort to illegal smuggling,  because legal entry has been made so difficult.  And the recent Trump administration rules changes have worsened that entry difficulty,  further incentivizing their resort to smugglers. 

Our border agents are too tied up dealing with asylum seekers and illegal guest workers to deal adequately with cross-border smugglers. 

Why not just fix the two problems (asylum seekers and guest workers) that are sopping up all the agent manpower?  Turn them loose upon the smugglers. 

About the Whole-Border Wall

This was a campaign promise to wall-off the entire southern border.  Sounds great as a sound bite,  but it won’t fix the real problems.

Such a wall is ineffective because defeating a wall is easy:  ladders,  ropes,  gloves,  shovels,  bolt cutters,  and saws-alls are all very much cheaper than fences or walls of any type.

Border walls are ineffective against the majority of the drug smuggling,  because they cross at ports-of-entry,  not all along the border.

According to government statistics,  there are no terrorists at the southern border.  There have been a small handful apprehended at the northern border,  but by far the most were apprehended at airports.  Despite the false claims,  terrorists are no reason for a whole-border wall.

Building a whole-border wall requires the government taking private lands by eminent domain.  This is extremely unpopular in Texas,  especially among border region landowners.  This is true regardless of party affiliation,  according to the polls.  As well it should be. 

Repairs or additions to existing border fencing are fine,  but there is quite obviously no need for a whole-border wall. 

About the Shutdown

The government shutdown is merely a way to hold hostage some Americans,  the US economy,  and US public safety,  in order to fulfill a campaign promise in a highly-visible way.  This does increasing damage the longer it goes on.

The President cannot do this alone.  Key members of Congress must collude with him,  for this damaging grandstand play to be successful.  They do damage to the country for party advantage.

The House and Senate both already had funding bills that contained border security funding, including for barriers.  There are enough votes in the House and Senate,  to pass one of those existing bills,  and end this shutdown,  right now!  There are actually enough votes to override a Presidential veto.

Trump reneged on his promise to sign one of those existing bills,  because of bad publicity he got from the gadflies on Fox-and-Friends and talk-radio.  That is no justification for damaging the country.

In the Senate,  majority leader Mitch McConnell will not allow any of the Senate funding bills to come to a vote without pre-approval from Trump.  Since when does the Senate need approval from the President to do its business? 

This is McConnell prioritizing party advantage above providing for the good of the country.  Is that what any of you really want?  Damaging the country to score political “points”?

What Are the Right Things To Do?

The opposition in Congress cannot give in to hostage blackmail from the White House that is damaging and endangering the country.  Yielding only ensures this evil behavior will be repeated in the future.

The House and Senate need to pass one of the existing funding bills as fast as they can,  and end this travesty.  They should have done this the very first week. 

If necessary,  the House and Senate should quickly override a Trump veto.   The votes are there to do it.

If Mitch McConnell will not cease obstructing these votes,  then the Senate needs a new majority leader,  one who knows that the good of the country outweighs other considerations!   

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postscript
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For $5 billion,  you could hire more than 20,000 new immigration judges and their support staff,  border agents and their support staff,  and beaucoup other paperwork officers and clerks to track visas.  This assumes on-average about $200,000 each,  annually,  to cover salaries and benefits.  Some cost more,  some less,  but that’s a decent ballpark cost figure for estimates.  Over-20,000 is a whole lot more people than we have now,  working these problems.

That means you could staff up to take care of asylum seekers properly,  while cutting the processing delay to near zero,  and thus reducing the need for proper holding facilities.  It means you could staff up to issue a whole lot more guest worker visas,  and actually track them to ensure proper renewals and no overstays.  And,  the DACA problem goes away within a generation,  once these guest workers are legal. 

Doing those two items correctly frees up a whole lot of border agents to deal with the smugglers a whole lot more effectively!  And very likely with some money left over to upgrade or replace existing border fences,  and to add some more,  where such is actually needed. 

Together,  that solves all the problems,  and without doing an inherently-defeatable whole-border wall,  and taking people’s lands to build it (which takes years to accomplish)!  And so doing this right actually solves the problems quicker,  to boot!

Now,  the facts are quite different from the propaganda,  which is why I wrote this article.  The sane things to do are quite different from the campaign slogans and sound bite crap we are being fed.  

Why on God’s green Earth would anyone with two working brain cells to rub together,  actually believe we need hundreds more miles of tall wall along our southern border?  When we can do far more,  for less money,  and get a better result?

I recommend you apply “grassroots term limits” and vote all these corrupt incumbents out,  who have been damaging our country for nothing but political points scored.  That applies to both parties,  the current senate majority leader,  and the current occupant of the White House.  They all get corrupted by the big money that infects our system everywhere,  within about 1 term in office.  We don’t need that.  No more.

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Update 2-3-19:  The end of the shutdown came with a bipartisan passage of yet another short continuing resolution.  Trump signed,  I presume so as not to endure the bad publicity of a veto override.  Yet this is just another in a long series of very short-term continuing resolutions;  this one only 3 weeks!  What looms quickly is another shutdown,  or else serious abuse of the emergency powers law. That is entirely unacceptable!

One of Congress's mandated jobs is funding the government for the fiscal year,  not just temporarily!  This chronic continuing resolution process is nothing but abusing the process to play party politics instead of doing the people's business.  It damages the country.  It is clear evidence of mis-prioritizing party advantage above the public good!

In my opinion,  that is a crime against all Americans. It has gone unpunished for decades.

As I said above,  we voters already have the "grassroots term limits" power.  Vote these people out of office!  It really doesn't matter who,  or what party,  replaces them!  Just exactly how could you do worse than what you have now?

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Subsonic Inlet Duct Investigation

For a realistic estimate of ramjet subsonic duct thermal-structural conditions and construction approaches,  I looked at a generic engine/inlet combination,  sized at an arbitrary 1.00 square feet of combustor internal flow area.  Conditions inside that subsonic portion of the duct are more driven by the downstream combustor conditions than the upstream supersonic inlet characteristics.  That outcome is unlike the supersonic capture features and shockdown diffuser upstream.  Analyses here rely on standard (NACA 1135-type) compressible flow methods restricted to temperatures at which ideal gas assumptions are appropriate (under about 5000 F).

Inlet Duct Construction

I looked at arbitrary-but-realistic ramjet flight conditions of Mach 3.5 at 40 kft (on a US 1962 standard day) for “design”,  Mach 2.5 at sea level for a “low altitude minimum speed takeover”,  and Mach 5.5 at 80 kft for a “high-altitude / high-speed” point.  The best construction approach seemed to be a thin sheet metal pressure shell for the duct,  located on the outside of some thickness of magnesia insulation,  and a thin sheet metal liner shell that is perforated so as not to resist pressurization,  but does provide a smooth internal flow surface that is also impermeable to the injected fuel.  The fuel injection location is guessed as one duct ID upstream of the combustor entry,  so that there is time to achieve some vaporization.  The value for inlet duct ID d₂ is based on a dump area ratio A₂/A₄ of 0.50.

The pressure capability this inlet duct structure must resist is a max ramjet chamber pressure in the vicinity of 200 psig.  This would be experienced only in a transient max-speed terminal dive to sea level.  The combustor is not at issue here,  and its size is provided only for a practicality reference.  I used the typical strength and thermal conductivity values of stainless steel for estimating thickness and thermal behavior.  That would be k ~ 10 BTU/hr-ft-R and tensile stress allowables of ~1-2 ksi “hot”  and ~40 ksi “cold”.  I presumed the heat sink temperature maintained at the outer pressure shell was 100 F. 

This basic construction concept is depicted in Figure 1.  All figures are located at the end of this article. 

The insulation was presumed to be a fibrous magnesia,  similar to mineral wool,  but capable of withstanding higher temperatures.  Its “typical” thermal conductivity is about k ~ 0.0405 BTU/hr-ft-R.  That is actually a little higher than the conductivity of mineral wool,  but ordinary mineral wool is not rated to serve at temperatures at (or slightly exceeding) 2000 F. 

The outer pressure tube runs cool enough to be made of aluminum,  but there must be some sort of centering-connections between it and the inner tube,  which is very,  very hot.  Those standoffs or centering connections (not detailed here) must then survive hot,  so stainless steel is the better option.  In order to weld these to the outer tube,  it had to be stainless as well.  A simple SS 304L sheet metal tube will do nicely for the outer pressure tube,  with SS 316L standoff/centering devices that more-or-less resemble leaf springs.  This outer tube got sized at 18 gauge thickness (t = 0.0500 inch) to meet or exceed the pressure capability in “cool” conditions,  at the largest finished OD in the study. 

The inner tube need resist no pressure,  and may rest on the centering devices without solid attachment.  The study shows about 2200 F soakout temperature at the most demanding flight condition,  which is beyond the recommended no-scaling service limit of 1900 F for SS 309/310.  Some scaling will occur,  which roughens the inner surface,  but that may or may not actually be objectionable.  If scaling is objectionable ,  a non-ferrous superalloy must be used for this inner tube.  One of the alloys commonly used for afterburner parts would serve well.  The thickness I show for this part is the thinnest stainless sheet available,  30 gauge,  or t = 0.0125 inch.  The superalloy should be available in something comparable,  if it is needed.

Inner Surface Film Coefficient

Of the three flight conditions,  the highest film coefficient occurs at the lowest altitude,  while the largest possible driving temperature occurs at the highest speed,  which is at the highest altitude.  As it turns out,  film coefficient varies weakly with inner surface temperature,  while the heat transfer varies all the way down to zero if the surface temperature is fully equal to duct air temperature.  The net effect is that the largest heat transfer potential to be dealt with (average film coefficient multiplied by max driving temperature difference) occurs at that highest-speed condition.  A brief summary of those heating potential data is given in Figure 2 below. 

The mild film coefficient variation is shown in Figure 3 below (note scale break !!) for the high speed / high altitude condition.   Plot shapes at design and low takeover are similar,  but not shown.  The heat transferred at design is shown in Figure 4 below.  Plot shapes at design and low takeover are similar,  but these not shown here.

These were computed from a diameter-based Reynolds number ReD evaluated at bulk flow conditions (static temperature T₂ and pressure P₂,  and velocity V₂,  with density from the ideal gas equation of state).  For the other properties,  I used correlations as good for combustion gases as air,  instead of real tabulated air values.  This was as much for convenience as anything.  The Nusselt number correlation is:

                NuD = 0.027 ReD^0.8 Pr^1/3 (µ/µ_s)^0.14

for which h = NuD k / D,  with k also evaluated at bulk flow conditions.  The two viscosities sown in the equation are µ evaluated at bulk flow conditions T₂,  and µ_s evaluated at surface temperature T_S conditions.  The heat flux equation is Q/A =  h (T₂ – T_S).  Conditions are rather subsonic (well under M₂ = 0.7),  so compressibility and dissipation are simply not large-enough issues to warrant modeling. 

I used the average film coefficient h without any final-T_S correction in the subsequent cylindrical heat transfer model,  because the variation of h with T_S is so mild.  This is good enough to find out “what ballpark” we are playing in.  The cylindrical-geometry heat transfer model has fluid at bulk temperature T₂ transferring heat to the inner surface at T_S through that film coefficient .  That heat conducts through 3 concentric layers:  the inner layer is metallic and thin,  the middle layer is insulative and thicker,  and the outer layer is metallic and thin.  That outer layer’s outer surface is presumed to be held at a constant heat sink temperature T_sink,  in this case,  100 F.

This sort of 3 layer construction with insulation sandwiched between two metal layers is actually quite practical,  if the inner layer is vented so as not to hold duct pressure.  That makes it a structurally-unloaded piece of thin sheet metal,  serving only to be a smooth surface for the air flow,  and an impermeable surface for the injected fuel spray.  It being hot and weak is then irrelevant.  The cool outer layer is the actual pressure shell for the duct,  and because it is cool,  it is much stronger,  leading to a thinner,  lighter part.  The real variable to investigate is the insulation thickness:  we are trading off higher inner surface temperature for lower heat transfer to the sink at thicker insulation.  That insulation must be fibrous or at least open-cell,  so that pressure can instantaneously equalize right through it.

Layered Conduction Model

The heat transfer model is a textbook cylindrical geometry,  as shown in Figure 5 below.  It works by summing thermal resistance terms for the film coefficient and the three layers,  and is formulated to determine heat transfer rate Q per unit length of inlet duct L.  The cylindrical geometry shows up in the logarithmic variation in terms of layer radii,  and the 2 pi factor.  The individual thermal resistance terms can be used to determine the temperature drops through each layer,  including the thermal boundary layer represented by the film coefficient:

                Q/L, BTU/hr-ft  =  2 pi (T₂ – T_sink) / [denominator]

              where denominator = 1/R₂ h  + sum of {ln(ro/ri)/k} for the 3 layers

                and R₂ = 0.5 d₂;  with ro = ri + t for each layer

As shown in the figure,  the inner metal layer has ri = R₂,  with ro = ri + t for the metal.  That metal ro is the ri for the insulation layer,  with its ro equal to its ri + t.  The ro for the insulation is the ri for the outer metal layer.  Its ro is that ri plus t.  The OD for the layered inlet structure is then just twice the ro for that outer metal layer.

As a nod to the notion of heat-sinking that outer layer,  it is instructive to compute a Q for a representative duct length,  in this case L₂/d₂ = 1.00 to allow adequate length to spray and vaporize fuel. This answer is appropriate to heat-sinking into adjacent structure,  which would have to be in intimate contact over all of the duct outer surface.

Otherwise,  that outer duct surface would have to be liquid-cooled with some sort of jacket.  If one divides the heat flow by the product of liquid coolant heat capacity c and the allowed temperature rise dT,  one obtains the coolant flow rate wc.  Multiplying that by a suitable flight time gets the mass of coolant fluid required,  and by means of a density,  the volume of that coolant. 

I used c = 0.5 BTU/lbm-R as typical for a hydrocarbon fuel,  and dT = 20 F as “reasonable”.  Flight time was assumed 1000 sec as “typical”,  and liquid specific gravity is 0.8 for a typical kerosene-like hydrocarbon fuel.  These values are not-necessarily-at-all “right”,  but they are realistic enough to see informative trends in the answers.

Results

Figure 6 below shows the trends of heat rate to be dealt with (Q), coolant volume required (V),  and insulated duct OD,  all vs insulation layer thickness.  If you look at the OD trend:  at about 3 inches thick,  that finished duct size matches the combustor OD size for a 1.25 inch case-plus-ablative thickness allowance.  That sets the max feasible duct insulation thickness at 3 inches,  in a very real and practical sense.

Both Q and V decrease very rapidly with thickness from 0.5 inches to 1 inch,  then not so fast,  from 1 inch on thicker.  In a practical sense,  then,  1 inch insulation is about the thinnest insulation we should consider.  Thus,  one has the apparent design freedom to choose from about 1 to about 3 inches of insulation thickness,  in this 3-layer approach.  But,  bear in mind that heat rates and coolant requirements are actually a little larger,  due to the extra conduction paths afforded by the centering standoff structures that support the inner metal layer.

The thermal-structural design ranges are not so sensitive to insulation thickness,  as shown in Figure 7 below.  At any practical insulation thickness,  from half an inch on up,  the heat rate is reduced enough by the simple presence of insulation,  to limit the temperature drop across the thermal boundary layer to trivial values.  In effect,  to within just a few degrees,  the inner metal shell soaks out steady state to the inlet bulk air temperature T₂ = 2228 F,  which at subsonic velocities is,  in turn,  very close to the inlet total temperature T_t2 = 2803 R = 2343 F. 

Model results vary from T_S = 2188 F at half an inch,  to 2222 F at 4 inches.  In effect,  the lesson here is that one can use the inlet total temperature T_t2 as a good guide to suitable material selection for that inner shell,  and also just how hot the inner layer fibers of the insulation will get.  T_t2 is easy to compute from only flight Mach number and the outside air temperature at altitude.  Such is given in Figure 8 below.  The value of temperature for which these calculation methods fail (not being ideal gas anymore) is also shown.  One can derive speed limits from that,  outside of which predictions made by these methods will simply not be accurate. 

The speed limit for that “not air” aeroheating-analysis technique limitation is about Mach 8 in the cold stratosphere,  and about Mach 7 at sea level,  and 160 kft,  where the outside atmospheric air is about as warm as at sea level.  Good non-scaling max service temperatures would be 1200 F for SS 304/304L,  1600 F for SS 316/316L,  and 1900 F for SS309/310.  By way of comparison,  both titanium and plain carbon steel are listed as about 750 F max service.  There are several alloy steels capable of service to about 1400 F,  but only one also has very high cold strength:  17-7PH (that is what makes it suitable for ramjet cases that must also serve as integral boosters).  The nonferrous superalloys used for afterburner parts will go to ~2000-2500 F,  but also have low cold strength.

As for the inlet duct pressure capability,  this varies mildly with insulation thickness throughout the practical range of thicknesses.  It might be possible to reduce the sheet metal thickness to the next higher gauge number at lower insulation thickness nearer 1 inch,  but this has to be traded against the risk of cracking at joints during detail design.  The sharp shape changes at joints very effectively act as serious stress concentrators.  Rise factors can range from 1.5 to ~5.  These thicknesses will be larger at larger combustor sizes,  but the trend shapes will be similar.

Final Comments

First,  these results vary with combustor size.  The data shown here are for a generic combustor of 1 square foot flow cross section.  At larger sizes,  the sheet metal thicknesses for the inlet duct will need to be thicker,  particularly the outer pressure shell layer.  But the basic behavior trends will be the same.

Second,  for the Mach 5.5 at 80 kft flight condition analyzed here,  the inlet air temperature is really too hot at ~2200 F for repeated use of SS 309 or SS 310 construction as the inner tube.  Such parts would survive,  but would experience an ever-increasing surface oxidation scaling effect that roughens the inner surface.  One of the afterburner-part superalloys would be necessary for repeated operation.

Third,  we have NOT addressed here the rest of the supersonic inlet structures.  These include the compression spike or ramp structures,  the cowl lip structures that are heated on both sides,  the supersonic throat structures,  and the supersonic-to-subsonic shockdown divergence channel that connects to the subsonic duct analyzed here.  All of these will be more demanding problems to solve than this subsonic duct.

Fourth,  we have NOT considered here the use of smooth-surfaced non-porous ceramics as the inner tube sleeve for subsonic duct construction.  Such could be used to higher flight speeds than even the nonferrous afterburner-part superalloys.  However,  these would be brittle and shock-sensitive,  and of substantially-larger wall thicknesses.  Being dense,  they are highly thermally-conductive,  thus tending toward isothermal behavior.  Once hot enough,  there is no practical way to hang onto such a hot part!

Fifth,  I did a quick check of radiation cooling capability toward 70 F surrounding structures for a duct OD sink temperature of 100 F.  The results showed Qrad ~ 0.02 to 0.03 BTU/sec when otherwise Q ~ 1.2 to 0.2 BTU/sec.  At factor 6 to 60 too small a radiative heat flow,  there just isn’t any practical help there,  at nice,  low outer duct shell temperatures.  The outer tube would have to run significantly hotter (somewhere in the neighborhood of 500-600 F),  to reach “steady state” cooling to the structure that way,  at 2 or 3 inches of insulation.  And that adjacent structure would warm rapidly,  reducing its effectiveness as a radiation heat sink.  It’s a possibility,  but probably not that practical.

Related Articles by GWJ on http://exrocketman.blogspot.com:

“A Look at Nosetips (or Leading Edges)”,  1-6-19

“Thermal Protection Trends for High-Speed Atmospheric Flight”,  1-2-19

To navigate on that site,  look for the by-date-and-title navigation tool on the left of the web page.  Click on year,  then month,  then title (if more than one article was posted that month).  If you click on a figure,  you can see all the figures enlarged.  You “x-out” to return to the article itself. 

Figure 1 – Construction Approach Concept and Principal Study Dimensions

Figure 2 – Comparison Among the Flight Conditions

Figure 3 – Mild Variation of Film Coefficient vs Inner Surface Temperature (Scale Break!!)

Figure 4 – Strong Variation of Heat Transferred with Inner Surface Temperature

Figure 5 – Cylindrical Geometry Thermal Conduction Model

Figure 6 – Variation of Heating and Cooling Parameters with Insulation Thickness

Figure 7 -- Variation of Thermal-Structural Parameters with Insulation Thickness

Figure 8 – Inlet Air Total Temperatures vs Speed and Altitude,  for Selecting Inner Shell Materials

A Look at Nosetips (or Leading Edges)

This is a follow-up to the generic lateral skin panel investigation.  Flight speeds are supersonic to low hypersonic,  with analysis methods limited to ideal gas-based compressible flow.  This work presumes the same 10-ft long projectile shape as the lateral panel investigation. 

The area studied here is the nose tip,  presumed with a half-inch nose radius,  per Figure 1 below (all figures at end).  This article calls out 2 references (see list below),  one of which is the closely-related lateral skin investigation. 

That article has a list of several other relevant or related articles.  All of these are posted at http://exrocketman.blogspot.com,  and can be found with the navigation tool on the left of the web page.  Click on the year,  then the month,  then the title. 

The heat transfer correlation is based on stagnation conditions just behind the bow shock,  at normal shock conditions.  There are slightly different film coefficient estimates for a leading edge versus a nose tip,  reflected by the coefficient of the Nusselt-Reynolds number correlation equation. 

Reynolds number is figured from freestream velocity,  a diameter appropriate to the nose radius (so that only “several thousand” is turbulent),  and the density and viscosity calculated at stagnation conditions behind the normal shock wave.  The normal shock conditions are calculated from freestream conditions,  and the total pressure ratio equation from NACA 1135 (ref.1).  Free stream totals are computed by standard compressible flow relations,  also per NACA 1135.  These are limited to ideal gas conditions. 

The Nusselt number correlation is NuD = C ReD^0.5 Pr^0.4 (rhof/rhoinf)^0.25,  where rhoinf is the free stream air density,  and rhof is the density at stagnation conditions behind the normal shock.  The coefficient C is 0.95 for a cylindrical geometry appropriate to a leading edge,  and 1.28 for a spherical geometry appropriate to a nose tip.  The nose tip C = 1.28 was used for this study.

Properties versus temperature are correlations applicable to combustion gases as well as air,  just as in the lateral skin investigation (ref. 2).  These depend fundamentally upon inputs for molecular weight and specific heat ratio.   For the film coefficient h derived from NuD,  the thermal conductivity k is also evaluated at post-shock stagnation conditions. 

The actual convective heat transfer depends upon the recovery temperature Tr:  Q/A = h(Tr – Ts).  In this investigation,  recovery temperature is presumed turbulent so that r = Pr^1/3 in Tr = r(Tt – T) + T.  There is radiation from the hot surface to an environment that radiates back. 

Part of this environment is cool at earth temperatures (in this case,  520 R),  and part of it is hot at the recovery temperature Tr,  reflecting the presence of a hot-soaked shroud of some kind.  The fraction of this environment that is cool is Fsink,  which is a number between 0 and 1.  For this investigation,  Fsink = 1,  meaning all of the environment is cool.  That is appropriate for a nose tip not shrouded by anything.

Radiation from (and to) the surface is figured with Boltzmann’s equation at some surface emissivity,  in this case a “black” highly-emissive e = 0.80.   Based on the strength of the lateral skin investigation’s sensitivity to surface emissivity,  only the “black” highly-emissive e = 0.80 was investigated for the stagnation heating.  This presumes some sort of metallurgical coating or a black ceramic paint.

The lateral skin investigation (ref.2) ran sweeps of Mach at 60 kft and 100 kft on a US 1962 standard day,  plus a sweep of altitudes at Mach 5.  This stagnation investigation found very little altitude effect,  so that only sweeps of Mach number at 60 kft and 100 kft were made. 

Initially,  sweeps were run with no active cooling,  then adjusted to find the required active cooling for a desired max surface temperature of Ts = 1600 F,  as representative of a metallic nose tip construction. This produced curves of uncooled equilibrium temperature vs speed,  and then curves of required cooling rate per unit area vs speed to maintain only the desired temperature. 

The results at 60 kft on a standard day for uncooled nose tip equilibrium surface temperature Ts vs Mach are given in Figure 2 below.  If one presumes that 1600 F is feasible for uncooled metallic construction,  given a highly-emissive surface finish,  then the “speed limit” to avoid stead-state overheating is just about Mach 5.  This result is a bit different for different presumed acceptable temperatures.

That equilibrium Ts vs M result does not presume any particular material,  or any particular construction technique,  it is merely the physics of energy accounting!  Figure 2 also shows the max service temperature levels recommended for several materials.  Something like an Inconel X or a 316 stainless steel might serve.  Bear in mind that the higher-temperature alumino-silicate and zirconia-based low-density ceramics shown on the figure are experimental-only,  not yet ready-to-apply. 

Carbon-carbon composite ablative offers a higher speed limit nearer Mach 8,  but requires replacement every few flights,  precisely because it is an ablative,  albeit a slow one.  Titanium only gets you to about Mach 3.5 steady state:  titanium is very definitely NOT a high temperature material,  despite the commonly-held belief that it is.  That mistaken premise traces to titanium’s successful use as the skin material on the SR-71 and similar craft,  which had a max speed limit of Mach 3.3,  or else face damage.

The way around this quandary is possibly active cooling,  but it very quickly reaches ridiculously-infeasible levels with increasing speed,  as shown in Figure 3,  for a constant Ts = 1600 F,  allowing metallic construction.  At Mach 6,  the required cooling rate is near 700,000 BTU/hr-ft² = 194 BTU/sec-ft² = 221 W/cm²,  which is quite high. 

Dividing 194 BTU/sec-ft² by the specific heat of water (1.0 BTU/lbm-R) and a presumed 10-degree R temperature rise,  the coolant loading is 19.4 lbm/sec-ft²,  or about 0.135 lbm/sec water for a piece of about 1 square inch cross section.  For a 1000-sec flight at Mach 6,  that’s 135 lbm (2.16 US gallons) of water.  That’s why the cooling rate at Mach 6 is “high”.  At Mach 8,  the cooling rate is about 3.4 million BTU/hr-ft² = 944 BTU/sec-ft² = 1073 W/cm²,  which is utterly ridiculous. 

Quite unlike the lateral skin panels,  for stagnation zones,  one doesn’t get very much relief by flying higher in the thin air.  For the uncooled nose tip,  the Ts vs M trends at 100 kft is shown in Figure 4 below.  This plot is all but indistinguishable from the 60 kft plot in Figure 2.  For 1600 F equilibrium temperature,  it’s still just about a Mach 5 speed limit.  That would be for the same metallic construction with a highly-emissive surface finish. 

One can see the thin-air effect a little better in the required cooling data of Figure 5,  for the same high-emissivity material,  just thinner air at 100 kft.   At Mach 6,  the required cooling rate is about 300,000 BTU/hr-ft² at 100 kft,  vs 700,000 at 60 kft.  At Mach 8,  it’s about 1.4 million BTU/hr-ft² at 100 kft,  vs 3.4 million at 60 kft.  That’s only about a factor-2 reduction for freestream air at factor-6.5 lower pressure.  The water required for a 1000 sec flight at Mach 6 is down to about 1 gallon from 2 gallons.  That’s still quite “high”,  for only a 10-foot long projectile shape. 

Conclusions:

If space and weight for cooling systems are severely limited (as with most flight vehicles),  you are generally better off selecting a nose tip (or leading edge) material that can withstand the aeroheating uncooled,  except by radiation to the environment and conduction into the interior.  You will have to solve the design problem of how to “hang onto” a very hot part. 

Allowing for what materials are actually ready-to-apply,  this probably means limiting yourself to about Mach 5 with an Inconel or SS316 nose tip (or leading edge) at about 1600 F,  at any altitude.

Or,  you might limit yourself to about Mach 6 or 7 with a carbon-carbon ablative.  This approach will be more limited by needing to design the attachment to a 3000 F part,  than by ablation and erosion of a part that otherwise could approach 4000 F at Mach 8.  Again,  this is more independent than dependent,  upon altitude.  There will be places in the flight envelope that you could go with uncooled lateral skins,  that you simply cannot reach,  with uncooled non-ablative nose tips or leading edge pieces. 

For a highly conductive metal part,  extending it laterally where the heating is less than stagnation,  is actually a viable way to cool the part down some.  Once extended far enough,  the part is no longer isothermal,  perhaps allowing a practical attachment method.  That kind of thermal analysis cannot be done with methods this simple;  it inherently requires finite element analysis. 

The same extended-part approach might work with high-density ceramic radome materials,  which can go hotter than metallic materials.  There are several,  silicon nitride is but one.  Temperatures in the 2000+ F class can be tolerated easily.  However,  the part attachment design problem is aggravated by the higher temperature.  These parts will have to extend far back along toward the lateral skin panel locations,  to achieve a cool-enough temperature at the attachment point.  That is not analyzable with the simple methods here.  It will require finite-element thermal analysis.

Note that inlet structures for airbreathing propulsion have not yet been considered at all.  These will result in yet-different flight limits,  a separate topic from stagnation items (this article),  or from the lateral skin panel topic.  Radiation cooling of inlet structures will be less-feasible-to-infeasible,  depending upon the geometry,  which is never simple.  Some of these surfaces will inherently soak out near the recovery temperature Tr,  which zeroes net convection to the part.  Others will equilibriate differently.  These will likely be the source of the most-strict limits in the flight envelope. 

References:

#1. “Equations,  Tables,  and Charts for Compressible Flow”,  NACA report 1135,  by the Ames research staff,  1953.

#2. “Thermal Protection Trends for High-Speed Atmospheric Flight”,  by G.W. Johnson,  2 January,  2019 (this one has a list of relevant or related articles).

 Figure 1 – Geometry and Models for Stagnation Heating Investigation

 Figure 2 – Equilibrium Ts (Without Active Cooling) At 60 kft,  e = 0.80

 Figure 3 – Required Active Cooling for Desired Ts At 60 kft,  e = 0.80

 Figure 4 – Equilibrium Ts (Without Active Cooling) At 100 kft,  e = 0.80

Figure 5 – Required Active Cooling for Desired Ts At 100 kft,  e = 0.80

Thermal Protection Trends for High-Speed Atmospheric Flight

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.)

Article follows:

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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.

Spreadsheet Model

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 12^th row added at the bottom.  The Ts in that 12^th 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. 

                Free-stream assumption

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. 

Other Considerations

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. 

Materials Limits

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. 

Final Comments

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 http://exrocketman.blogspot.com:

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

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Figure 1 – Compressible Flow Analysis Results Around the Body

 Figure 2 – Models for Gas Properties and Compressible,  Dissipative Heat Transfer

 Figure 3 – Fundamental Heat Balance at Conditions Without Radiation to the Surface

 Figure 4 – Initial Results Around the Airframe with Conduction Inward Zeroed

 Figure 5 – Results for Conduction Study at the Mach 5 / 60 kft Reference Point

 Figure 6 – Effect of Inward Conductance on Surface Temperature at Reference Point

Figure 7 -- Effect of Inward Conductance on Interior Heat Flow at Reference Point

 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