Fact Checking

Presented without comment.  It speaks for itself.

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Space-Based AI? Not Easy!

I see a lot of hoopla and speculation about why Elon Musk has “officially” changed his goal from Mars to the moon.  The answer is simple in the shorter term:  money talks! 

Musk’s companies SpaceX and Tesla are both serious government contractors.  SpaceX provides launch services to NASA and to DOD,  plus it is contracted to attempt to land humans on the moon for NASA’s Artemis program.  I am unsure whether the Tesla connection has to do with electric vehicles or the “Powerwall”,  but that does not really matter.

The point is,  much of the income for both SpaceX and Tesla come from their government contracts.  It is one thing to honestly try to fulfill a contract and fail.  It is quite another thing to default by not making the attempt. 

Musk is being paid by no one to go to Mars.  Why is this hard for anyone to understand?  He must focus SpaceX’s contracted efforts on the moon,  or else lose funding,  and worse,  all credibility as a government contractor.

There’s also been a lot of hoopla over recent Musk statements about AI data centers,  in space,  or maybe on the moon.  I see all sorts of speculation about this,  none of it based on any sort of facts. 

AI data centers involve enormous amounts of power.  All of that power gets eventually converted to waste heat,  which must be gotten rid of,  somehow.  Yes,  space is cold,  but getting rid of waste heat in space is just NOT that easy!

In space,  there is no heat loss capability due to either convection or conduction.   There is only thermal radiation to the cold background of deep space.  On the moon,  there could be conduction into the lunar surface,  but no convection,  because there is no air.  Mars is similar,  with “air” that is close to vacuum.

As for “in space”,  the cheapest destination is Earth orbit.  And from there,  the only way to shed waste heat,  is to radiate it into deep space.

I ran some numbers.  They do NOT look very good,  if one is limited to a coolant temperature compatible with cooling silicon electronics,  which is near the boiling point of water at normal atmospheric pressure.  Thermal radiation is bound by physics to be inefficient,  until the radiating surface is well above 1000 F.  The relative effects are given in the figure.

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The El Paso Airport Shutdown Fiasco

Update 2-17-2026:  A slightly toned-down version of this ran as a board-of-contributors column in the Waco Tribune-Herald,  Tuesday,  17 February,  2026.  

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A few days ago,  the FAA closed all airspace over El Paso,  Texas,  for 10 days,  catching everyone by surprise.  A few hours later,  they rescinded that order,  but a lot of damage to air travel had been done.  And,  during the closure,  even emergency services like air ambulances were grounded.  This caused a lot of problems,  and created the uproar we have seen in the news recently.

Lies and misinformation and wild speculations immediately flooded both the internet and the news.  There were mentions of laser weapons,  of drug cartel drones,  of military involvement,  and more,  all of it very confusing.  This initially was made even worse by lies about what happened,  from Trump administration figures.

As usual,  there were grains of truth in many of these speculations,  but how it all fits together is very different.  The drug cartels really do use drones to smuggle drugs across the border.  That really is the best way for them to do it.  Which brings up the next question:  how do you stop the drones?

Somehow,  Customs and Border Protection (CBP) obtained a military laser weapon intended to shoot down incoming drones.  How they obtained it,  and from what agency within the military,  is still unclear.  What they did with it,  is the start of a big problem.

Without notifying the city of El Paso,  the airport,  the FAA,  or anybody else,  CBP started trying to identify and shoot down some drug cartel drones.  They did this in close proximity to the airport,  which is quite close to the Rio Grande border,  actually. 

That was the first big mistake:  not telling anybody.  The second mistake was misidentifying a party balloon as a drone.   Given the small sizes as seen from a distance,  that is quite understandable.  They shot it down with the laser.

Exactly what happened next is unclear,  but somehow the FAA learned of lasers being used near the El Paso airport,  probably from their air traffic control employees in the tower,  and maybe the pilots flying near the airport.  Even low-power lasers near airports are already a forbidden and illegal hazard nationwide.  That is what prompted the FAA to close the airspace over El Paso entirely,  because they did not know what was going on.

It is not clear exactly who finally called whom and when,  but when the FAA learned it was CBP trying to shoot down drug smuggling drones,  and their activity had ceased,  FAA rescinded the shutdown.  About this same time,  the administration figures started claiming that their border agents shot down a drug smuggling drone with a laser weapon,  as if that would “justify” disrupting air travel around El Paso so badly.

Then,  the “party balloon” thing surfaced,  as it inevitably would,  if CBP had misidentified a balloon as a drone,  and shot it down.  Most likely,  that is why they stopped firing the laser weapon,  when they realized they had made a target identification mistake.  There are a lot of drones flying over the border,  so confusion and mistakes are easily understandable. 

Since the event,  I noticed that no administration or CBP people have admitted to what really happened.  The administration has ceased lying about “shooting down a drone”,  which indicates the veracity of the claim that CBP downed a party balloon by mistake. 

What this episode really indicates is the extreme incompetence of Trump appointees heading various agencies.  This was evidently an experiment to see if CBP could actually shoot down drug-smuggling drones with a military laser weapon.   If they could,  that would certainly be something we all can agree that they should be doing,  if they can do it “right”. 

But doing this dangerous experiment right next to a busy airport in a big city,  was stupid in the extreme,  because it endangered significant commercial air traffic!  Doing it without telling anyone they were going to do it,  was even stupider!  Not understanding that the main difficulty would be target identification,  plus ignoring the hazard of laser injury to innocent bystanders,  was egregiously stupid!

The proper place to have done this was many miles further downstream,  away from El Paso and its airport.  There would less drug drone traffic to shoot at,  but the risk to air travel would have been far lower,  and the local airspace closure to do it “right” would have affected far fewer people. 

So why did they not do it “right”? 

I already told you:  incompetent appointees.  Appointees willing to do anything Trump wants,  but demonstrably incompetent to do anything right.  Our government is rife with such.  We see it almost every day in the news.

You can fix that,  at the mid-term election,  by voting out of office all of Trump’s enablers in Congress.  You can,  unless he can successfully steal it with voter suppression and gerrymandering,  something all of you have also already seen going on in the news.

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Since I first wrote this,  I did find out that the laser weapon was Blue Halo’s LOCUST weapon,  pictured.  Supposedly,  this is a 20 KW laser system.   Blue Halo has been acquired by Aerovironment.  The weapon was seen at a conference mounted to an Army Strycker vehicle.  It was also seen near El Paso last August,  according to Axios. 

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More About Trump Treasons

This is essentially the text I sent to my federal representation, as of this date:

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These are but two headlines I saw on the PBS NewsHour Website on this date,  related to the weakening of NATO by Trump’s insults and threats toward our allies:

Strained U.S. ties loom over NATO leaders ahead of Munich Security Conference

By Nick Schifrin, Veronica Vela

'Are we still on the same team?' Ischinger asks as Trump fuels NATO tensions

By Nick Schifrin, Sonia Kopelev, Dan Sagalyn

Our NATO alliance helps to protect us all against Putin’s Russia.  Putin and his Soviet predecessors have all wanted to weaken or destroy NATO,  but could not!  Trump is doing it for Putin,  as the quoted headlines attest!  Putin’s Russia is the indisputable enemy of the US (and most of Europe).  Trump weakening the alliance is doing what Putin wants but cannot do himself.  And THAT is aid and comfort to the enemy,  per the Constitution!

The US and NATO have an ally and proxy in Ukraine,  to help hold back Putin’s ambitions in Europe.  (I think it was a mistake not to put Ukraine into NATO long ago,  long before even Putin’s illegal annexation of the Crimea.)  Nevertheless,  Ukraine has held back Putin with aid and weapons from the US and Europe,  for several years,  even past the invasion. 

Trump took office for his second term and almost immediately stopped US aid to Ukraine.  Within but days he removed the US sanctions that had impeded Russia in its invasion.  It has since taken about a year to deplete the weapons coming only from Europe,  so that Zelenskyy is forced to surrender and sue for peace,  on some or all of Putin’s terms,  which is so very evidently the entire point of Trump’s so-called “peace process”. 

The image dates to August 2025.  At one of Trump’s “peace process” meetings.  

How is forcing an ally to surrender to an enemy NOT providing aid and comfort to that enemy? 

That’s one count of aid and comfort treason for betraying Ukraine,  and multiple counts of aid and comfort treason for every ally in NATO that Trump has alienated.  Including especially Denmark and Greenland,  against which he threatened war!

DO SOMETHING ABOUT TRUMP’S TREASONS!  Or else be complicit in them!

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There will be a reckoning for these crimes;  it is coming! 

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Search code DDMMYYYY format:     13022026

Search keywords:                                     bad government,  idiocy in politics, treason

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Incompetents End the Fight Against Climate Change

Donald Trump and his appointed EPA Administrator Lee Zeldin have ended the EPA’s fight against climate change by repealing the EPA’s “endangerment finding” policy.  That policy was supported by very-strong,  evidence-based science that cannot rationally be disputed. 

These two men are entirely incompetent to judge the veracity of a scientific finding!  We already know that about Trump and the many lies he tells,  including the long-debunked “climate change is a hoax”.   He demonstrably does not value truth or fact,  and never has.

Zeldin is not competent to make that judgement either.  He got a law degree,  not a science degree,  and has only ever been an elected politician!   He knows nothing of science!  So why did Trump put him over the EPA?  Because he would do what Trump wanted him to do,  no matter how wrong.  The same as all his other appointees.

This further endangers the lives and livelihoods of Americans by “justifying” the repeal of all the pollution regulations,  making Trump’s giant corporate donors (especially but not exclusively in the fossil fuel industry) even richer!  They get richer at YOUR expense,  I might add!  Your air and water are about to get dirtier,  to the point of killing a lot of people.  We have seen this before,  and it took decades to fix. 

The EPA guy Zeldin is not the only incompetent Trump has appointed (nearly all of them are).  The other most glaring example is his HHS Secretary,  RFK Jr.  This idiot wants to take away from you all the immunizations that keep you and your children safe from things like polio,  smallpox,  diptheria,  tetanus,  yellow fever,  Covid,  and the flu.  And even things like mumps,  measles,  and chickenpox (which sometimes do kill,  especially measles). 

His anti-vaccine lies have already caused massive disease outbreaks.  But he got his kids vaccinated!  How hypocritical is that?  Does that tell you anything?  It should!

If you are tired of being treated like crap by these incompetent idiots,  then vote them all out of office!  That includes Trump,  who is not only an incompetent idiot,  but a convicted felon,  and a now-almost-successful wannabee dictator,  who has created a secret police force to intimidate you,  out of what used to be ICE.  Plus,  he is the most prolific egregious liar I have ever seen,  since Hitler’s propaganda minister Goebbels.  And all his underlings do the same lying,  about everything!

You are paying the price for his tariffs!  You are getting infected by the diseases he refuses to fight!  You are going to die fighting the wars he needlessly starts.  You will be killed for his desires!  You believed his lies and voted him in,  twice!  You should be ashamed!

You need to write your representation and demand that Trump and all his incompetent appointees be impeached and removed!   Go protest!  Do something!  I already have!

And stop believing the lies! 

Myself,  I never did believe any of them,  but then I was taught science and facts and critical thinking in the public schools,  long before these idiots dumbed those schools down,  with their stupid tests!

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Search keywords                                       bad government,  idiocy in politics

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The Trump Dictatorship Is Here

The Trump dictatorship is almost fully in place,  today!   The figure shows 7 of 10 hallmarks fully in place,  with the other 3 mostly completed.  Looking at it should scare the ever-lovin’ hell out of you!

And this dictator is also a traitor,  on at least 2 counts of the aid-and-comfort type of treason,  as defined in our Constitution.  He has betrayed one ally (Ukraine) to the enemy (Putin),  and he has weakened all our alliances by angering or threatening to attack allies!

We are already in a situation where we cannot trust the public statements of any high official in this Trump administration.  Not the leadership of the DOJ,  not the leadership of the FBI,  not Homeland Security,  not Health and Human Services,  not the FDA.  They all lie egregiously,  essentially every time they speak.   We have already seen it!  You dare believe NOTHING that any of them say!

All these appointed offices have been packed with dedicated Trump supporters as their leadership,  who will very apparently say anything Trump wants,  or do anything Trump says,  no matter how illegal or evil.  We’ve seen our immigration agencies turned into a secret police force,  deliberately being brutal to intimidate the population,  and killing people!  This has occurred in several opposition-led cities,  the latest being Minneapolis.

You can no longer believe even the Supreme Court!  There were 3 pre-existing far-right justices,  to which Trump added 3 dedicated supporters,  and THAT is where the current Supreme Court majority came from,  that has made such bad decisions recently!  They have been very busy over-ruling lower court decisions that were not in favor of Trump,   which are coming from those lower courts Trump has not yet packed with rabid supporters.  And,  they gave Trump immunity for any crimes he commits while in office!

Some lower courts are already packed with rabid Trump supporters.  This is why needless court delays forced Jack Smith to drop two slam-dunk-conviction criminal cases against Trump,  because they would not go to trial before the 2024 election,  which Trump might win (and he did).  Without those needless delays,  Trump would have been in jail and therefore un-electable,  by the 2024 election time.

Summarizing:  both the Executive and Judicial branches are essentially already taken over and neutralized as the sources of any opposition to Trump. 

As for Congress,  opposition there is already neutralized because (and as long as) the Trump personality cult that used to be the GOP holds the majority of seats in both houses.  Which is also why I believe there will be no laws made to retard any of this evil that is going on,  including that being done by the secret police force. 

Possible premature loss of control over Congress is why Trump is so desperate to steal the 2026 election with the gerrymandering he demanded!  If he gets past that election with his control of Congress still intact,  he will have time to finish purging the military of high-ranking officers who might oppose him.  Once that purge is achieved,  he no longer needs or cares about control of Congress!  Trump as a military-supported dictator will be a fait accompli. 

If Trump fails to steal the 2026 election,  he is in some trouble,  with an impeachment coming from the House.  It only takes a simple majority to do that,  and he has already publicly warned his supporters about that risk.  But,  it takes a 2/3 majority in the Senate to convict him!  I don't foresee a 2/3 majority of the opposition coming from the 2026 election in the Senate,  so a significant number of disaffected GOP members would have to vote with the opposition to convict.  I do NOT believe that to be a very probable outcome!  

Failure to convict in the Senate leaves him in place as dictator,  and if that happens,  he will convert the 2028 election to something rigged or a sham,  should he live that long.  If he does not live that long,  we are in for civil wars to see which cultist succeeds him as dictator.  That will likely have factions of the military fighting each other,  and killing us,  too.  

That failure-to-convict outcome leaves ONLY revolution in the streets as the lone remaining way to get rid of him,  before it is too late and we are destroyed by civil wars among competing dictator-wannabees trying to succeed him!  And,  he will be fully-supported by the surviving high-ranking military officers,  by somewhere around the 2026 election time,  or shortly thereafter.  “We the people” outnumber the military,  but they outgun us!  Revolution in the streets is a bloodbath at best!  

And the real problem with revolution is,  about half of "we the people" were suckered by the lies into becoming Trump cult believers and voting this dictator into office!  Twice!  That likely means there are fewer of us than we'd need,  by far,  to fight the revolution.  Why?   Because nothing changes the minds of fervent cult believers!  They will fight against the rest of us,  whether it is for Trump,  or one of his successors.  Why?  Because they prefer belief over facts and evidence!

Let's just say I am pessimistic about our future as a free people.  VERY pessimistic! 

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Search keywords                       bad government,  idiocy in politics,  treason

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Rocket Nozzles

This article is intended to acquaint the nontechnical or non-specialized person with the basics of compressible flow in rocket nozzles,  and how they are sized for rocket engines and rocket vehicles.  Scope here is limited to only conventional bell nozzles.

The author had two 20-year careers,  the first in aerospace/defense new product development engineering.  He is qualified!  The second was mostly teaching at all levels from high school to university,  but with some civil engineering and aviation work,  as well. 

Nontechnical and non-specialized people have difficulty with this topic,  because the behavior of supersonic compressible gases is quite foreign to their experience.  This paper attempts to address that,  as simply as is possible,  so that the behavior is not so foreign as to obscure what is needed to do the figuring. 

A familiarity with high school-level algebra is the only math required!  There are spreadsheet tools to do this kind of figuring,  but it really helps for the user to understand what the spreadsheet is actually doing for him or her.  That’s how to detect input problems.

The converging-diverging passage of a rocket nozzle is quite unlike the garden hose sprayer gun or spray nozzle that most people are familiar with!  On the subsonic side of the throat,  behavior is familiar,  because the flow accelerates in speed (and the pressure drops) as the passage narrows. 

It is the supersonic side,  downstream of the min-area throat,  where behavior is quite unlike common experience.  The supersonic flow continues to accelerate in speed (with further drop in pressure) as the passage grows larger!  Other than that,  the nozzle works to expel a fast jet that creates thrust,  by means of the large pressure drop from upstream of the nozzle out to ambient conditions.  That is grossly the same as the garden hose sprayer experience,  actually.  Just the supersonic-side details are different!

The gases cool off as they accelerate in speed,  because the sum of the heat energy and the flow kinetic energy,  anywhere in the nozzle,  is a constant that pretty much matches the heat energy of the almost-stationary gases upstream of the nozzle.  Energy conservation is not all that unfamiliar a concept,  even for nontechnical people.

Everyone is familiar with the reaction “thrust” of a garden hose,  or more especially that of a small fire hose.  That thrust is the momentum of the ejected stream of water.  The faster it moves,  the bigger the thrust.  The more water is ejected,  the bigger the thrust.  Simple!

In the compressible nozzle,  the momentum of the ejected stream of gas is also part of the reaction thrust,  but there are pressure forces that are also part of the thrust,  unlike the water hose.  The pressure of the gas just as it leaves the supersonic nozzle exit can be quite different from the surrounding atmospheric pressure!  The thrust of a rocket nozzle is the sum of the exit momentum and the pressure forces at the exit plane.  See Figure 1. 

Figure 1 – Fundamentals of Compressible Flow Nozzles

That difference in pressure between the exiting gas stream and the surrounding atmosphere leads to some behaviors of such nozzles,  that would otherwise look incomprehensible to the nontechnical or non-specialized person.  There is a force associated with the exiting gas pressure that adds to thrust,  and a force associated with the surrounding atmospheric pressure that subtracts from thrust.  Both of these pressures act upon the flow cross section area right at the exit plane. 

As shown in Figure 2,  when the exiting gas pressure is greater than atmospheric,  we say that the nozzle is “underexpanded”,  since expansion to just the right size larger exit area,  would reduce that exiting gas pressure to exactly atmospheric,  while at the same time increasing its speed still further.   This is the leftmost image in the figure,  corresponding to a high chamber-to-ambient pressure ratio.  The exiting plume actually spreads out wider after leaving the nozzle exit,  because its pressure really is higher than atmospheric.

For the same geometry,  the “perfect expansion” to atmospheric pressure,  at a slightly lower chamber-to-ambient pressure ratio,  is the second image in the figure.  That exiting plume neither spreads wider,  nor does it narrow,  after leaving the nozzle exit!  The pressure forces add to zero,  leaving only the momentum component of thrust.

The third image shows what happens when the exiting gas pressure is lower than atmospheric,  but not by too much.  We call this “overexpanded”,  because at this pressure ratio,  we would need less expansion of the nozzle passage than we have,  to bring the exit pressure back up to equal to atmospheric. The plume actually does contract some,  after leaving the nozzle exit!  Under certain circumstances,  the indicated oblique shock waves from the exit lip can actually be seen, often as the lead in a series of “shock diamonds”.

Figure 2 – Nozzle Behavior as Chamber to Ambient Pressure Ratio Reduces

Where one gets into trouble is illustrated in the 4^th and 5^th images in the figure,  where the chamber-to-ambient pressure ratio is too low for proper operation.  The oblique shocks first coalesce into a normal shock wave at the exit plane,  then move a bit upstream,  causing flow to separate-off of the inner wall of the nozzle!  The lower the chamber-to-ambient pressure ratio,  the further upstream this shock-separation phenomena moves!  Flow downstream of a normal shock is always subsonic (meaning very low speed),  so there is very little thrust,  once the shock is inside the nozzle and separating the flow from the wall.

The “trouble” one gets into is called “shock-impingement heating”.  Where the shock wave hits the nozzle wall (causing flow separation),  there is a large but very local amplification of the rate at which heat is transferred from the hot gas to the cool wall.  The nozzle can actually burn through and fail,  in a matter of only several seconds,  when this happens! 

The last (rightmost) image in the figure shows what happens when the atmospheric backpressure exceeds about 50-some percent of the chamber pressure.  The throat “unchokes” (goes subsonic),  and flow throughout the nozzle is subsonic.  There is no useful thrust when this happens.  There is almost no useful thrust even when choked,  if shock-induced flow separation occurs.  There is none when unchoked.

Most rocket engines have a set of turbopumps,  with pre-burners of one sort or another to create modestly-hot gases at high pressure,  which then get used to drive those turbopumps.  How this is done varies,  and is what we call the “cycle” of the engine.  

Those details do not matter to the functioning of the rocket nozzle!  All that stuff up to the chamber right before the nozzle entrance is just a “hot gas generator” that feeds the nozzle.  It is the nozzle that creates the thrust and its associated performance with that hot gas. 

The only effect of the engine “cycle” upon that nozzle behavior is whether-or-not any of the turbopump drive gas gets dumped overboard,  without going through that nozzle!  That does reduce the performance ,  even at the same thrust!  This is indicated in Figure 3,  among several other things.  

Figure 3 – How the Engine and Nozzle Work Together to Create Thrust

There are a couple of nozzle efficiency factors that depend upon the exact geometry going through the nozzle.  The effective flow area of the throat is slightly smaller than its geometric area,  because of “boundary layer displacement” effects.  This can be held to a minimal difference,  by using a smooth profile curve through that throat,  from ahead to downstream.  Effectively,  you just need the profile radius of curvature to be about the same as the throat diameter,  in order to get a good,  high discharge coefficient C_D. 

The shape of the supersonic expansion passage,  called the nozzle “bell”,  influences something called the nozzle kinetic energy efficiency factor η_KE.  Curved bells,  like that illustrated,  have half-angles that are large near the throat,  and smaller near the exit lip,  which need to be averaged.  Simple conical bells have only the one half-angle.  Curved bells require careful design using a computer program that does something called “method of characteristics” analysis.  Conical bells of half-angle equal to the average curved-bell half angle,  have exactly the same kinetic energy efficiency,  but are only somewhat longer than the curved bell.  They require no complicated analysis in order to lay out a design!

This η_KE factor measures the effect of having many of the exiting streamlines oriented not exactly aft.  There is a very simple empirical estimate of this efficiency,  computed with the bell average half angle,  as shown in the figure.  It applies to the momentum component of thrust,  but not to the pressure-forces component of thrust. 

One does need to address the subsonic contraction area ratio from chamber to throat!  If this is not large enough,  the flow Mach number at the nozzle entrance may be too high to use the measured chamber pressure as if it were the “total” or “stagnation” pressure for the nozzle flow.  There is a simple correction factor to increase measured chamber pressure slightly,  in order to have exactly the right “total” pressure for the nozzle thrust analysis.

Note in the figure that there is a nozzle massflow,  that depends upon both throat geometric area and its discharge coefficient.  That massflow may not be the massflow actually drawn from propellant tankage,  if there is dumped bleed from the turbopump drives!  It is the massflow drawn from tankage that affects rocket vehicle masses,  so for “rocket equation estimates” of vehicle performance,  the specific impulse needs to be computed from thrust using that total massflow,  not just the nozzle massflow! 

The other factor affecting calculation of the nozzle massflow is the “chamber characteristic velocity”,  usually denoted as “c*”.   That will be discussed below.  Just be aware that experimental values are far more reliable than theoretical thermochemical estimates.

In figuring all these things out,  one needs to be aware that there are two different design applications,  each with its own sizing methods.  Those are “vacuum design”,  for use outside the atmosphere,  and “atmospheric design”,  for use down in the atmosphere.  They are done differently using the same basic math,  just in a different sequence and with different constraints.  See Figure 4. 

We start by determining the “right” nozzle bell area expansion ratio.  For the vacuum case,  this number is assumed from the outset!  For the atmospheric design case,  this is determined by the pressure ratio at the exit,  in one fashion or another.  There are actually 3 distinct options to do atmospheric design sizing. 

Be aware that the very same math will analyze the chamber to throat contraction for us,  determining whether we need to factor-up the chamber pressure measurement. 

Figure 4 – Both Streamtube and Ratio Analyses Get Used First

Vacuum sizing is done to a presumed max expansion ratio, limited only by having the engine (or engines) actually fit behind the stage.  There is simply no such thing as a “vacuum-optimized” design!  Everything about it is constraint-driven,  and constrained even more if gimballing is needed for thrust vector control.  See Figure 5.

Figure 5 – Essentials of Vacuum Nozzle Sizing

Atmospheric nozzle sizing is done in one of three distinct ways,  starting with appropriate ratios of expanded pressure to chamber total.  These all use the same math,  just not in quite the same ways.  This is shown in Figure 6.  

Figure 6 – Three Options for Atmospheric Nozzle Sizing

The option on the left in the figure is “standard” sea level perfect-expansion sizing.  One knows a suitable chamber pressure.  One assumes the expanded exit plane pressure to be exactly equal to sea level atmospheric pressure.  That sets the ratio of expanded pressure to total.  From that comes the exit Mach number,  and from that,  the expansion area ratio.  These three items (and a nozzle kinetic energy efficiency) are needed to get a thrust coefficient,  in turn a way to book-keep where the thrust comes from.

Sea level nozzles have good thrust at sea level,  but their thrust does not increase much,  as you climb to higher altitudes.  Which in turn means the specific impulse does not increase very much with altitude.  They typically have rather low area expansion ratios. 

If you want to average a higher specific impulse as you climb to much higher altitude,  you can obtain it by sizing the expansion ratio to a higher altitude’s ambient pressure (top right),  or by sizing the nozzle to incipient separation at sea level (bottom right).  The penalty you pay for that higher average specific impulse during ascent,  is lower thrust right at sea level,  at liftoff,  when weight is largest!  So,  the design point selection is a tradeoff!  These do have somewhat larger area expansion ratios.

Clearly the compressible streamtube analysis math is crucial to running numbers for a nozzle.  This streamtube math is illustrated in Figure 7.  

Figure 7 – Compressible Streamtube Analysis

This is the analog to V₁A₁ = V₂A₂ in incompressible flow,  that many people have actually heard of,  or actually even used.  The equation is different,  but it is the same fundamental idea!  However,  everything is figured relative to the choked min area at the throat. 

One case of interest is finding the area ratio from a known Mach number.  That is a direct solution.  Just fill in the formula items,  starting with the gamma constants.

The other case of finding Mach number from a known area ratio has no direct solution,  because the equation is what they call “transcendental” in Mach number!  It is impossible to isolate Mach number in the equation,  because it appears in two places under very different mathematical circumstances (different exponents and functional forms). 

For that case,  there is only the iterative (trial-and-error) solution.  Keep guessing Mach numbers until the equation result is the area ratio you really want.  That is where spreadsheet-assist is so useful:  it makes such iteration very easy,  boiling down to just inputting the guesses in one cell and looking at the result in another cell.

The other piece of this math is the set of compressible flow ratios,  static vs total (or stagnation).  Those are shown in Figure 8.  These are “reversible” in the sense that a known Mach number gets you all the ratios,  and a known pressure ratio can be solved directly for a Mach number.  The basic math here is based on total/static ratios,  but their inverses are what we need for thrust coefficient and flow separation.  Those inverses are included.

Figure 8 – The Compressible Flow Ratios

We use the thrust coefficient form of this math to separate the variables,  allowing expansion ratio to be determined before actually sizing dimensions to meet a thrust requirement.  You cannot do that,  working directly in the primitive variables!  That thrust-sizing math based on thrust coefficient is shown in Figure 9. 

Thrust coefficient has two components,  the vacuum thrust coefficient,  and a correction term that reduces it somewhat to the thrust coefficient down in the atmosphere.  The vacuum thrust coefficient is actually independent of the specific value of chamber pressure!  The correction term depends directly upon chamber pressure and ambient pressure,  so that the down-in-the-atmosphere thrust coefficient is also dependent explicitly upon them,  as well.

Once you know the thrust coefficient,  you can use it,  your intended chamber total pressure,  and a thrust requirement,  to find the geometric throat area.  Knowing the expansion (and contraction) area ratios,  lets you define those chamber and exit areas from that throat area!  Very simple,  actually. 

Once you have a throat area,  you can compute nozzle massflow,  adjust it to total,  and compute specific impulse.  That is discussed below.  

Figure 9 – Thrust Coefficient Math Equations

The math for thrust-based sizing is a bit more complicated than simple performance of an already-sized configuration.  This is shown in Figure 10. 

Figure 10 – Thrust Requirement-Based Sizing of Dimensions and Flow Rates

By definition of the thrust coefficient,  thrust is the product of thrust coefficient,  chamber total pressure,  and geometric throat area!  Once you have a thrust coefficient defined,  you can size throat area from a required thrust value and your chamber pressure.  The contraction and expansion ratios then size those areas from your sized throat. 

You can use the sized throat area,  your chamber total pressure,  the c* model for your propellant at that pressure,  and your throat discharge coefficient,  to size the nozzle massflow.  That and the dumped bleed fraction define your total massflow drawn from tankage.  In turn,  that and your sizepoint thrust define your sizepoint specific impulse. 

Computing performance of an already-sized system is even easier,  as is also shown in the figure.  You compute the thrust from thrust coefficient at that altitude and your chamber pressure,  and you already know the total flowrate at that pressure.  Thrust at altitude divided by total massflow rate is specific impulse at that altitude.   Very simple indeed!

Be aware that all of these estimates are computed assuming there is no shock-separation going on in the nozzle bell!  So,  you must check for that!  If it occurs,  your calculated performance data are no good!  Do not use them!

The math predicting flow separation is an old correlation from designing tactical missile rocket nozzles.  It is slightly conservative.  The math is given in Figure 11.

Figure 11 – Math for Dealing With Flow Separation

The use of this empirical correlation is quite straightforward when computing performance vs altitude at any given throttle setting.  The nozzle expansion has a fixed ratio of exit plane pressure to chamber total pressure.  From that,  the correlation determines the ratio of separation backpressure to chamber total pressure.  That ratio and your operating chamber total pressure,  give you the value of backpressure that will risk inducing flow separation. 

If your ambient atmospheric pressure is less than,  or just equal to,  the separation pressure,  no separation occurs and your thrust and performance estimates are good.  If your ambient atmospheric pressure exceeds the separation pressure,  shock-separation will occur,  and your thrust and performance estimates are no good!  Simple as that!

When sizing an atmospheric nozzle for incipient separation at sea level (as discussed above),  you use the empirical correlation in reverse (which is also shown in the figure).  You know a suitable value of your chamber total pressure,  and you literally set the separation pressure equal to sea level atmospheric pressure.  Their ratio determines the expansion pressure ratio exit-to-chamber for your design process.  That gets you a Mach number,  and from that,  the expansion area ratio.  From them,   thrust coefficient is easy to find.

You have to think about your rocket vehicle and where it is flying,  to determine suitable thrust requirements.  Some items typical of launch vehicles are given in Figure 12.  Cases do vary,  though!

Figure 12 – Typical Considerations for Thrust Requirements

For launch vehicles,  you need to accelerate the vehicle at half a standard gee or more,  above the retarding effects of drag and the pathwise weight component.  Such would apply at vehicle masses appropriate to stage ignitions,  where vehicle weight is high.  The half-gee figure is only a rule-of-thumb minimum.  If you achieve lower,  you will definitely “dawdle around” at low speeds near the launch pad burning off lots of propellant,  without it actually buying you very much in the way of flight speed.  Higher gee is better,  but that requires more thrust,  and the engines might not fit behind the stage.  It’s a tradeoff!

Vehicle acceleration also has max values,  especially if crewed,  but a lot of potential payloads have similar acceleration limits.  Those limits might be roughly in the 4 to 6 gee range.  You can always turn off some engines while throttling others,  to stay within such limits.  They would occur when vehicle masses are low,  near stage burnout. 

The nozzle massflow equation uses characteristic velocity c* as the denominator.  You must have a suitable model for this value,  consistent with your propellant combination and design chamber pressure.  In the real world,  c* is weakly dependent upon chamber pressure as a power function.  This is shown in Figure 13.  

Figure 13 – About Chamber Characteristic Velocity (c*)

You must run a thermochemical code (computer program) on your propellant combination at your intended chamber pressure,  and your intended oxidizer-to-fuel ratio,  to determine the resulting combusted chamber temperature and gas properties.  These are theoretical values,  and from them a theoretical c* can be computed with the equation shown.  It will have a very weak power-function dependence upon chamber pressure. 

In the real world,  delivered test c* is always a little less than the theoretical value,  by a factor we call the “c* efficiency”.  This factor also typically has a weak power-law dependence on chamber pressure.  Therefore,  the actual experimental delivered c* is best modeled as a weak power-law dependence upon chamber pressure,  with an exponent that is usually crudely in the vicinity of 0.01. All of this is shown in the figure.

In Figure 14 is a table of values for PR = Pt/Pc vs Mach number,  including values for Ac/At,  created with the usual factors,  for gamma = 1.2 as “typical”.  Plotting Pt/Pc vs Ac/At reveals the importance of allowing for non-zero Mach number at station c (the aft “chamber”,  right before the nozzle entrance). 

Pc is always measured on real engines as a simple static pressure tap.  This is the total pressure fed to the nozzle only if the Mach number of the flow in the chamber is trivially close to zero.  For the recommended and often-observed Ac/At ratios,  this Mach number is simply not trivial,  so the Pt/Pc ratios are not trivially close to 1!  The error incurred by using Pc as Pt would seem to range from about 1% to 6%.  Pt = Pc * (Pt/Pc for the Ac/At ratio).

Figure 14 – Why Accounting for the Contraction Ratio Is Important

For an engine with a maximum nominal chamber pressure of around 3000-4000 psig in test,  one might select a pressure transducer of nominal 5000 psig capability at the very least,  which might have an accuracy of 0.25% of full scale.  That would be an expected error of 12.5 psi.  That is the inherent uncertainty,  within which one simply cannot distinguish measurements. 

For 3000 psig Pc,  that 12.5 psi is 0.42% error,  and for 4000 psig,  it is 0.31% error.  Most of the Pt/Pc error factors in the figure are very much larger than that,  so correcting for Pt/Pc before doing a nozzle analysis,  really is crucial for getting accurate results!

To Sum Up

Everything shown here is basically pencil-and-paper calculation stuff,  using the algebra equations given in the figures.  However,  this is better done with spreadsheet software,  to make iteration much easier!  In particular,  the computation of exit Mach number from the nozzle area expansion ratio is inherently iterative. 

The latest and best version of my own spreadsheet for this is the Excel spreadsheet file “liquid rockets.xlsx”.  It has 3 worksheets,  one the nozzle-sizing work space,  one has a compressible flow streamtube tool for relating Mach number and expansion ratio,  and there is one that is a propellant data library.  There is a “.PNG” file that goes with it,  as the template for your results report.  You just copy-and-paste your results into a copy of it. 

Figure 15 is an overall view of the nozzle-sizing worksheet,  where it is too small to read things in this view.  The main working area is top left across to the top center,  the results to be copied are top right,  and there are tables and plots of altitude performance across the bottom. 

Figure 16 shows the compressible flow tool worksheet that supports this.  You just iterate your Mach number until you get the desired area ratio,  then copy the pressure ratio for pasting into the main working space.   Figure 17 shows the propellant data library worksheet.  You just copy what you need,  and paste it in where it goes,  in the main workspace space. 

Figure 18 shows just that portion of the nozzle sizing worksheet where you actually do your sizing.  It is large enough to read easily.  Figure 19 shows just that portion of the nozzle sizing worksheet where your results are summarized.  This is what you copy,  and then paste it into the results report. 

Figure 20 shows an image of the “.PNG” file that you make a copy of,  and then paste your results into it,  and finally do some minor final edits as needed.  This “.PNG” file was drawn in the old 2-D Windows “Paint” software,  which is where I do my copying and pasting and editing. 

Figure 15 – Overall View of the Main Nozzle-Sizing Worksheet in the Spreadsheet File

Figure 16 – Image of the Compressible Flow Tool Worksheet

Figure 17 – Image of the Propellant Library Worksheet

Figure 18 – Image of the Main Working Area of the Nozzle-Sizing Worksheet

Figure 19 – Image of the Results Summary Block on the Nozzle-Sizing Worksheet

Figure 20 – Image of the “.PNG file” Results Report Into Which Results Get Pasted

Such spreadsheet tools already exist and are freely available to interested persons.  In particular,  a good spreadsheet embodying the rocket nozzle math calculations,  is available as part of the course materials included with the “orbits+” course materials on the Mars Society’s “New Mars forums” website.  These are available to anyone for free download.  That rocket spreadsheet is same “liquid rockets.xlsx” that was just described.

You go to the forums website newmars.com/forums/.  Go to the “interplanetary transportation” topic and select the “orbital mechanics traditional” thread.  The links to all sorts of lessons and multiple spreadsheets are in those postings.  These go way beyond just rocket nozzle sizing and performance,  to include orbital mechanics,  and even entry,  descent,  and landing.  All these materials are located in an online dropbox accessed by those links. 

The author has other materials and courses available directly from him.  Contacting him by email is preferred at gwj5886@gmail.com.  He has a blog site with all sorts of stuff posted,  much of it technical.  That is http://exrocketman.blogspot.com.  You may copy anything you like from that blog site.  He also has a presence on LinkedIn,  and another on Youtube under the name “exrocketman1”.  

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Appendix – Where the Thrust Coefficient Comes From

The boundary layer displacement factors are all very close to 1 and so divide-out of all the Ae/At ratios.  It appears explicitly only in the nozzle massflow equation as C_D,  not in C_F.