Friday, April 24, 2020

Revised Hat

The image speaks for itself.  Sorry about the language,  but this is the most concise and precise way to say this.

Update 4-26-2020:  

After Mr. Trump's exposition of the notion of injecting or ingesting known-fatally-toxic disinfectants to "cure" Covid-19 disease,  the phone numbers for poison control around the country have fielded a lot of calls about this topic!  And,  the makers of those disinfectants have felt compelled to issue warnings NOT to ingest or inject these materials!

Science very clearly says DO NOT DO what Mr. Trump suggested!  In matters of science,  he is quite clearly far dumber,  than the average fence post.

Mr. Trump now claims he was being "sarcastic" when he suggested that,  but the video footage of him suggesting this to Dr. Birx very simply denies that claim.  Period!  End of issue!  It is quite clear he will not admit to making a very serious mistake.  Especially one that would be fatal to those who might follow it.

Add to that the recent absence of Dr. Fauci from the White House briefings.  (Update 5-2-2020 the White House has blocked Fauci from testifying before Congress!)  Fauci has contradicted Trump's ignorant lies on multiple occasions,  accompanied by shouts of "fire Fauci" from Trump's most ardent supporters.  Is anyone in their right mind now surprised by that absence?

OK,  I understand that Mr. Trump's most ardent supporters are also deniers of science in multiple arenas.  That much is quite clear from their positions on climate science and on the dangers of pollution,  versus what humans might really be able to do.  Perhaps that disparity is to the advantage of the country as a whole. 

Applying the epithet "Branch Covidians" to his most ardent supporters,  then perhaps letting them kill themselves,  by injecting or ingesting bleach and isopropyly alcohol,  or by defying quarantine,  is actually appropriate.  That way,  there are fewer of them left,  come November,  to vote for him.  Death cults do,  in fact,  lead to death.   And Mr. Trump's most ardent supporters do indeed qualify as a death cult.

If you want to know why I say that (meaning the danger of glorious-leader death cults),  then peruse the article "Beware of Leader Cults",  posted 2-13-2020,  on this same site.

Saturday, April 4, 2020

On the Covid-19 Pandemic

Update 4-10-2020:  see modeling analysis appended to the end of this article.

Update 4-11-2020:  see also recommendations for how and when to end the quarantines appended below.

Update 4-23-2020:  added second model;  see update appended at end below.

Final Update 5-2-2020:  evaluation of second model appended below.

Update 4-11-2020:  a version of this basic article appeared as a board-of-contributors article in the Waco "Tribune-Herald" today.

The current pandemic is a disease about which we know little,  for which we have no vaccine,  and for which we have no real treatments.  After this is over,  we will know more,  but for now,  the only thing we can do is to use the same thing we have used for centuries:  quarantining at one level or another,  to slow its spread.  Calling it "social distancing" makes no difference,  it is still a simple quarantine.

Here is what we do know,  as of this writing,  learned the hard way as the epidemic sickens and kills people.  It seems similar to,  but not the same as,  the 1918 "Spanish Flu" pandemic.  We have not seen this dangerous a disease since then.  It is a once-in-a-century event.

Covid-19 seems to be at least as contagious as,  and perhaps more contagious than,  the 1918 flu.  It seems to have a similar death rate (the number who die compared to the number thought to be infected),  which is somewhere around 10 to 20 times higher than ordinary influenzas.  Those are seriously-dangerous characteristics.

There seems to be another unusual characteristic that combines with the other two to make Covid-19 a truly dangerous threat.  It seems to be more generally spread by people showing no symptoms,  than by people who are just getting sick and beginning to run a fever. 

That makes all of us potential "Typhoid Mary" carriers of the disease.  It also makes taking temperature rather useless as a screening tool to determine who might be infected,  and who might not be. Without massively-available testing,  one must presume that all other persons are contagious,  which argues for using stricter levels of quarantine.

So far,  it is thought that the Covid-19 virus is spread within the moisture droplets ejected by sneezing or coughing,  or even by talking.  5 minutes talking spews the same droplet numbers and size distribution as one cough.  A sneeze just spews a lot more.  The Covid-19 virus does not seem to be able to remain airborne outside of those droplets,  the way a chickenpox or measles virus does.

Masks vary in their effectiveness against particle sizes.  It is hard to breathe through a mask that stops particles the size of a large bacterium.  No mask stops a virus particle.  But even a simple cloth bandana will stop most of the moisture droplets from coughing or sneezing,  as does about 6 feet of space (the droplets quickly fall to the floor).  See Figure 1 at end of article.  

What that means is that the new CDC recommendation to wear masks in public is not to prevent the infection of the mask wearer,  but to stop the mask wearer from infecting others.  It would protect the wearer only when someone got right in their face to sneeze,  cough,  or talk at very close range.  The 6 foot distance rule already stops that effect.

The recommendation to wear a mask is actually based on this uncomfortable reality:  that many seemingly-well people are actually infected,  just not showing symptoms,  and are walking around spreading the disease.  This "Typhoid Mary" effect is not common,  but may well be the case with this particular virus.

As already indicated,  a simple bandana will work.  Leave the real surgical masks for the health professionals.  They need them.  We ordinary citizens do not.  When you go to the store,  wear a bandana or a home-made mask.  That's all you need,  to protect others.  The 6 foot rule protects you.

And for Heaven's sake,  quit panic-buying toilet paper and other supplies!  There is plenty being made,  and plenty in the supply chain,  for everybody's needs.  The shelves are bare because so many folks panicked and took far more than their share (their "share" being what they really need).  Shame on you!

Predictions about this pandemic are still guesswork.  The CDC figures show a peak of 100,000 or more deaths in about another month.  Maybe a month or two after that,  it will be more-or-less over,  and we can safely re-open our lives and businesses.  But that's a guess,  and it will likely change.  See Figure 2 at end of article.  

Had we started with the quarantining a month or two sooner than we did,  the death totals would have been lower,  but the time to the end of this gets longer.  Time spent shut down costs all of us money and jobs. 

That is the inevitable tradeoff:  lives versus money. And it is quite the serious effect,  make no bones about that.  Job losses are already beginning to resemble those of the Great Depression of the 1930's.

But almost all of your mothers and churches taught you to value lives over money,  that valuing money over lives was evil!  Think about that,  when you vote.  Not just next time,  but from now on.

 Figure 1 -- Data on Particles Versus Filter Pore Sizes
Figure 2 -- How Quarantining Works,  and What It Does

Update 4-5-2020:                                                                       

The best numbers I have seen on Dr. Fauci’s curves and predictions,  as of end-of-March,  say that with “social distancing” quarantining in place,  US deaths may accumulate to 100,000 to 240,000 people lost.  That death rate trend should peak out somewhere in early May.  Without the quarantining measures,  something like 2 million deaths would be expected.  Maybe more.

Just to “calibrate” the threat of this thing,  the US lost 407,300 soldiers in WW2,  for a 1939 population of 131 million.  That’s 0.31% of the population dead from war

With Covid-19 at a population of 325 million today,  it is 0.03-0.07% of the population dead with quarantining,  and something like about 0.6% of the population dead without quarantining.  You don’t credibly compare this pandemic to yearly traffic deaths or the H1N1 epidemic.  You compare it to the casualties of a major world war.

Based on the numbers published in the newspaper,  the US death rate appears to be near 2% of known cases of infection.  For Dr. Fauci’s predicted death accumulation numbers,  that corresponds to something like 5 to 12 million accumulated known infections.  That’s about 1.5-3.7% of the US population infected,  and 0.03-0.07% of the population dying of it.  These numbers are clouded by uncertainty,  because without widespread testing,  we cannot know the real number of infections.

Using the rough-estimate 2 million deaths for no quarantining,  and the same 2% death rate of those infected,  the accumulated infections would be about 100 million,  which is 31% of the US population.  Quarantining is thus very,  very important,  by about a factor of 10 on the total infections,  and on total deaths.  So,  those who deny or ridicule the risk are dead wrong,  if you will forgive my choice of words.

According to Wikipedia,  the 1918 Spanish flu killed something like 1-6% of the world population.  The same article gives these statistics for the US:  about 28% of the population became infected,  and about 1.7% of those infected died of it. 

The death rate among those infected is quite comparable between Covid-19 and the 1918 flu.  The number of expected Covid-19 infections is lower,  probably because of our quarantining efforts,  despite our delay getting started.  The estimate of infections without quarantining is actually quite comparable to 1918. 

The Covid-19 pandemic really is an event comparable to the 1918 flu pandemic.  We have not seen such a thing in 102 years.

This is quite serious,  so I reiterate the recommendations I gave above:

#1. Stay away from crowds and gatherings,  and when you must go out,  stay at least 6 feet apart (which is what protects you from infection,  not any mask you might wear).

#2. If you must go out where 6 feet apart is not feasible,  wear a bandana or home-made mask to protect others in case you are unknowingly contagious (save the real masks for the health care folks who need them).

               Corollary:  if you are sick in any way,  DO NOT GO OUT.

#3. Stop panic-buying and hoarding supplies,  there is no need for that.

#4. Watch what your public leaders do (not what they say) to judge whether they values lives over money,  or not.  Then stop re-electing those with the wrong priorities. 


Update 4-10-2020:

Pulsed events like the daily infection rate for Covid-19 are actually well-modeled by the mathematics of something called the “logistic distribution”,  which is similar to,  but numerically a little different from,  the “normal distribution” in statistics.  The daily rate corresponds to a pulse function f,  and the accumulated total follows a S-curve shape corresponding to the F-function.  F is the integral of f (which means f is the derivative of F).  F is defined from 0 to 1,  so you have to scale it to apply it in the real world.   The following mathematics were obtained from Wikipedia under the article name “Logistic Distribution”.

Derivative (like a probability density):
f(x, µ, s) = exp(-(x - µ)/s) / s[1 + exp(-(x - µ)/s)]2

Cumulative function (S-shaped accumulation curve):
F(x, µ, s) = 1/[1 + exp(-(x - µ)/s)]

Variable definitions:
x is the independent variable,  usually time in applications,  a real number,  from – infinity to infinity
µ is the location variable,  center of the f-distribution,  and location of half the total accumulation
s is the scale variable > 0,  a measure of the distribution width;  a bigger s is a flatter and longer pulse
F varies from 0 to 1;  you have to scale it by the total accumulated T

The shape of the pulse function is shown in Figure I for multiple values of the 3 model parameters.  The smaller s is,  the “peakier” the pulse.  The location parameter µ merely moves the shape left or right on the graph.  The larger T is,  the taller the pulse,  as a direct scale factor.

 Figure I – Shape and Characteristics of the Unscaled Pulse Function

The shape of the unscaled S-curve accumulation function is shown in Figure II.  The smaller s is,  the steeper and shorter-in-time the S-curve shape is.  The location parameter µ merely shifts the shape left or right,  same as with the pulse function.  The T factor merely scales the shape from its 0-to-1 variation to whatever numbers your data are.  

 Figure II – Shape and Characteristics of the Unscaled Accumulation Function

To model a pulse of something that eventually totals to T instead of 1,  scale up both the cumulative and the derivative with the factor T.  Thus:
Derivative (pulse function)
Pulse rate vs time = T f(x, µ, s) = T exp(-(x - µ)/s) / s[1 + exp(-(x - µ)/s)]2 
Cumulative (S-curve function)
Accumulated total vs time = T F(x, µ, s) = T/[1 + exp(-(x - µ)/s)]

At the peak of the pulse in your real-world data,  there is a max rate with time,  and a location in time,  and in the accumulation function at that same location,  the total is exactly half the eventual total.   The model parameters can be calculated quite easily from those three pieces of data,  if you can confirm you have actually seen the max of the pulse.  Here is how you do that:

Where in x it peaks is µ
T = 2 times the accumulated total-at-peak
s = T/(4 times the peak rate)

I went on the CDC website 4-9-2020 and retrieved their posted Covid-19 Infections data as of 4-8-2020.  They had posted both accumulated cases,  which I used,  and a daily case rate vs time from a presumed infection date,  which did not share the same time scale.  I did not use the daily case data because of differing assumptions,  and because they said in no uncertain terms that the final week or so was clouded by as-yet unreported data.  Here in Figure A is the CDC’s own accumulated case data (as of 8 April 2020) vs time from 12 January 2020,  as plotted from the spreadsheet in which I put it.

 Figure A – CDC Data for U.S. Accumulated Cases of Covid-19 Infections as of 8 April 2020

The accumulated data are simply summed from one day to the next,  from the daily reported infections data.  So,  I simply recreated their daily case rate data by differencing the accumulated data from one day to the next.  That way everything shares the same sources and assumptions.  See Figure B.

 Figure B – Recreated CDC U.S. Daily Case Rate Data as of 8 April 2020

Early on,  the numbers are small,  and even a large-percentage inherent scatter is not significant.  Later,  as the numbers climb,  a large-percentage inherent scatter becomes very significant,  actually to the point of obscuring the trend.  There would seem to be a suggestion of the daily case rate bending over a peak value,  but picking a number for the peak point would be difficult indeed.  This some sort of averaging is needed to actually “see” the peak in the data well enough to quantify it.  I tried the moving-average technique,  with a 2-day average,  a 5-day average,  and a 3-day average,  as seen in Figure C.

 Figure C – Comparison of Raw U.S. Data Versus 3 Different Moving Averages

The 2-day moving average did not fully suppress the up-down scatter variation,  but showed very little “lag” in its trend behind the raw data.  The 5-day average suppressed the scatter,  but lags the data trend by about 4-5 days,  which is too much.  The 3-day average showed the peaking behavior the clearest,  and with about a 2-3 day lag behind the actual data trend.

The daily case rate seems to peak at about 32,400 cases per day at day 86 in the 3-day moving average data.  Its rise trend is seemingly 2 days too late in terms of the rise before peaking.  So revise the peak date to day 84,  keep the 32,400 cases,  and read the accumulated case data at day 84 (not 86) for about 331,000 accumulated cases at the peak point.  The corresponding T,  s,  and µ data for the model are T = 662,000 final total cases,  a scale parameter s = 5.108 days,  and a location parameter µ = 84 days (on the graph time scale). 

That model matches the accumulated case data quite well,  as shown in Figure D.   If you do not compensate for the lag of the moving average technique,  and use the wrong µ,  your model fails to match the initial upturn (near day 60 to 70).

The same choices of parameters do a good job matching the daily case rate data,  as long as you compare it to the moving average that reveals the peak.  Like with the accumulated data,  if you do not compensate for the lag of the moving average,  then the model fails to match the initial upturn in the data (again near day 60-70).  This is shown in Figure E.  

 Figure D – Comparison of Model and Data for U.S. Accumulated Cases

Figure E – Comparison of Model and Data for U.S. Daily Case Rates

This case study illustrates the inherent difficulty in choosing the “right” model parameters T,  s,  and µ unless you have already reached the peaking daily case rate in your data.  This is due to the mismatch around the initial upturn if you do not compensate for your moving average lag.  There are multiple combinations of the parameters that might match up the tail of the daily case rate distribution over time,  but most of these will not correctly predict the peak.  And the inherent scatter problem forces you to use a moving average to “see” the peak,  which inherently introduces the lag that causes the error,  if not compensated.

All that being said,  with a peak you can “see” and quantify,  this modeling technique becomes very accurate and very powerful for pulsed events like epidemics,  as the plots above indicate.  As a nationwide average,  the U.S. Covid-19 epidemic seems to have peaked just about 5 April 2020 (day 84 in the plots),  at about 32,400 cases per day reported,  and an expectation of being “over” by about 11 May at ~100 cases/day at the earliest,  or at worst about June 1 with 2 cases /day in this model,  and with about 661,000-to-662,000 total accumulated cases.

You do NOT release the quarantine restrictions until the event is actually effectively over!  Relaxing just after the peak pretty-much guarantees a second pulse of infections just about as bad,  and just as long,  as the first.  To propose doing so is a clear case of valuing money over lives,  instead of lives over money.  Valuing lives over money is what your mothers and your religious institutions taught to most of you readers!  I suggest that you use it as a criterion to judge your public officials.

A final note:  this same pulse model has been used to predict resource extraction and depletion.  The most notable example was geologist M. King Hubbert trying to predict “peak oil”.  One of his two best models came very close to predicting peak oil from data he had long before the peak actually occurred,  something very difficult at best.  Since then,  the model has diverged from reality and thus fallen into disrepute. 

Since that peak,  the development of then-unanticipated cheap fracking technology has not only made fracked oil and fracked gas available,  their simple availability has vastly increased the total recoverable resources available.  Those are very large and very fundamental changes in the assumptions underlying the formulation of any predictive model.  That’s analogous to the situation of a second wave of infections in the epidemic application. 

A new peak fracked oil/gas model would be the right thing to do to respond to these developments.  And if recovery technology improves much past the current 2-3% recovery rates,  yet a further new prediction would be warranted.  That’s just the nature of prediction models being sensitive to the assumptions underlying them.


Update 4-11-2020:

These predictive models cannot tell you when to lift the quarantine!  Period!  These exponential functions never,  ever predict when the daily case rate goes to zero.  Mathematically,  they cannot. 

Instead,  you have to watch (1) the daily infection rate field data   and (2) you have to determine from experience during the epidemic,  what the actual incubation time really is.  You CANNOT lift the quarantine,  until the daily infection rate has been zero for an interval longer than the observed incubation time.  There is NO WAY around that requirement!  To do otherwise is to value money over lives,  an evil according to the morality you were taught as a child.

If the observed incubation time is 7 days,  then 8 or 9 or 10 days of zero infections ought to do the trick.  If the observed incubation time is 10 days,  then 11 or 12 or 13 days of zero infections ought to do it.  If the observed incubation time is 14 days,  then something like 15 or 16 or 17 days ought to do.  There is simply no way around such a criterion,  if you intend to be moral and value lives above money!

A caveat:  this needs to be on a regional basis,  not nationwide.  That is because the infection pulses did not all start at the same time around the country.  They will not last the same interval,  nor end,  at the same time.  A national edict to end the quarantine by this or that date is just wrong technically,  and demonstrably immoral by the criterion I have offered.  Regions can de-quarantine,  but travel between them should stay restricted until the last region is past the crisis. 


Update 4-23-2020:

As time went by,  the first model I set up looks ever poorer.  It was clearly not “right” in the sense that the predicted ultimate total accumulated cases of infection were quite demonstrably wrong by 4-21-2020,  using the published CDC data for the US.  A lot of that can be attributed to the wild scatter in daily rates about the peak point,  making it hard to quantify that peak point.  This is shown in Figure I.

 Figure I – Increasingly Erroneous First Model

So,  I repeated the process as of 4-21-2020,  obtaining a second model.  It has about the same peak daily rate,  just a bit lower and later in time,  with a larger accumulated case number at that later time.  This led to a larger scale factor with a broader peak,  which matched the peak data quite well. 

However,  out in the initial “tail” the match is not as good,  as can be seen in both the accumulated and daily rate data in Figure II.  And,  the time interval is longer.  We will see as time goes by whether this is really significantly better than the first model.  I think it is,  but I’d also bet it will be “wrong”,  too.

Figure II – Second Model at its Inception

This just emphasizes the point I tried to make in the article:  that these models are quite uncertain even if you have peak data.  If you don’t,  this is even more uncertain.  Exact predictions from the model are not the point of doing this.  The trend shapes and behavior are the real goal. 

This modeling process gives you only a crude idea what the time interval will be from peak daily rate to no more new infections.  That would be half the width of the predicted pulse of daily rate data.  It’s only crude,  but it’s far better than nothing.  My first model’s pulse was about 70 days wide.  The second model’s pulse is about 130 days wide,  ending well into June.   

It would appear from my experience here that predicting the ultimate infection total (where the accumulated curve levels out) is even more problematical than modeling peak behavior.  That seems significantly more uncertain than identifying the peak in the daily rate curve. 

Clearly,  ultimately,  one must live through the epidemic event,  and just use the actual data after it is all over,  if one wants accurate statistics.  You won’t get that from this modeling activity.

But,  the other thing I want to point out is that the peak curve shape in the daily rate data is symmetrical.  There is a fall-off over time after the curve peaks,  it does NOT immediately crash to zero!

This is modeling a very real effect:  after the peak,  the daily infection rates are not zero for some time interval,   meaning there are still infectious people walking around out there and spreading the disease!  Ending the quarantine during this time guarantees a resurgence,  a second wave that starts everything again from scratch.  You have to start your quarantine all over again!  And the longer that goes on,  the more jobs and money everyone loses.  There is no way around that!

This model behavior,  which matches real-world experiences,  is exactly why I say you do not end the quarantine until your daily infection rate has zeroed,  and has been zero for longer than the microbe’s incubation time.   The health professionals would agree with me,  not the politicians who want to end it too soon,  just so that so very much money is not lost.

Once again,  I submit that you should use a simple criterion to judge whether public officials have your best interests in mind:  they either value lives over moneyor they don’t.  If they don’t,  then you don’t want them making decisions for you.  Simple as that.

Where did I get that?  From the moral teachings just about all of us got from our mothers and our churches.  It’s a question of moral fitness.  That has to take precedence.  And nearly every one of you readers knows that,  somewhere deep down.

Final Update 5-2-2020:

The CDC quit updating the national database that I was using as my data source.  They said modeling communities was more appropriate than the entire nation.  And I believe that may be correct,  as the totals for states show mostly steady or still-rising daily cases,  strongly at variance with each other.  As a result,  I am no longer able to update the 4-21-2020 model posted in the previous update.  The last reliable data I have are for 4-27-2020. 

However,  as can be seen in the figure,  the daily infection rate appears to be defying the previous interpretation of a peak in daily cases about 4-13-2020.   There was a bit of a downward trend,  but in the last few days it has trended sharply upward again.  Possibly this is related to some states ending quarantine measures,  trying to reopen for business,  especially without adequate testing and contact tracing.  Or it may just be the inherent variability in the data.  There is no way to know. 

From what I have read about the 1918 flu pandemic,  here in the US there was the initial pulse of infections,  followed by two resurgence peaks in daily rates,  for a total of three.  I added that qualitatively to the figure. 

This Covid-19 disease epidemic seems similar in many ways to that earlier epidemic.  It is both highly infectious (meaning easily transmitted),  and has a higher death rate than most other flus.  The most dangerous aspect seems to be “asymptomatic carriers”,  meaning “Typhoid Marys” who have the virus,  are contagious,  but do not know it because they are not sick.  For most of you,  the mask you are asked to wear in public is to protect the public from you!  You may well have the virus and not know it. 

Multiple pulses and “Typhoid Mary” transmission may be why some authorities are now warning that the pandemic may be with us for as much as 2 years yet.  We are going to have to figure out how to get back to business while at the same time interrupting the transmission of this disease.  It would appear that we do not yet know how to do that successfully!  All that we do know is that the pre-pandemic status quo is NOT it!  So for now,  wear your mask and keep your distance,  when in public. 

Wednesday, April 1, 2020

Entry Heating Estimates

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/ = 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. 

 Figure 1 – Old,  Simple Model for Entry Stagnation Convection Heating

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/ = 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/^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/^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/ 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/ 

Dividing that value by 10 km/s to the 6th power gave me a C value of 1.90588 x 10-4to 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.

 Figure 2 – Relevant Peak Convective Stagnation Heating Data for Apollo From LEO

 Figure 3 – Estimated Apollo Stagnation Heating Trends Versus Velocity at 56 km Altitude

Radiational Cooling

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.

 Figure 4 – Modeling the Thermal Resistances of Multiple Layers for Conduction

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.

 Figure 5 – Modifying the Thermal Conduction Model for Active Liquid Cooling

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.

 Figure 6 – The Energy Conservation Balance

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

Other Locations

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. 

Clarifying Remarks

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.             
8-4-13…Entry Issues                                                         
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.

Final Notes

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.