Presenter at Workshop

I attended the American Carbon Society Symposium and Workshop on thermal management,  held at North Carolina State University,  March 18 and 19,  2024.  I was an invited speaker at that conference,  and I took 3 presentations,  ready to present.  Only one was “live”,  we turned the other two into poster presentations on-site. 

The live presentation (photo) had to do with old-style by-hand methods of design analysis being used up-front for concept screening,  to enable efficient use of the concept brainstorming process to increase chances of project success.  That enables concentrating the real design efforts and heavy-duty design analyses with software packages,  to be reserved for only the one or two best concepts,  thus using resources and schedule time efficiently.   Analysts who can do this sort of by-hand analysis can also more readily-recognize “garbage-in,  garbage-out” problems with software packages!

The specific example used for this presentation was the old H. Julian Allen by-hand simplified re-entry analysis,  used for warhead design about 1953,  and declassified in the late 1950’s.  I have taken that analysis and re-implemented it in the form of an Excel spreadsheet file,  with worksheets representing the atmospheres of Earth,  Mars,  and Titan.  Those models came from the Justus and Braun paper regarding entry,  descent,  and landing,  presented several years ago. 

There are only 4 easily-estimated pieces of data required to represent the entering object:  its speed at entry interface,  its trajectory angle below horizontal at entry interface,  its hypersonic ballistic coefficient,  and its effective nose radius which determines how bad the stagnation heating will be.  The altitude at entry interface is part of the atmosphere model.  My spreadsheet creates plots,  the most useful of which tell you the peak heating rate,  followed closely in time by the peak deceleration gees. 

Those lead easily and immediately to the peak pressure on the heat shield material,  and (by way of a thermal balance) the surface temperature that must be withstood.  Those in turn constrain your material selection.

The other two presentations were about a unique ceramic composite heat shield material I created out of essentially hardware-store materials,  decades ago,  and about the ramjet combustor ablative materials I tested decades ago,  in a particularly-productive direct-connect test series.

The spreadsheet entry analysis,  and others about orbital mechanics,  compressible flow,  high speed heat transfer,  and rocket engine performance,  are all things I can make available.  Contact me.

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Up to this point,  I was able to post the same remarks on LinkedIn and stay within a 400 word limit.  Here on “exrocketman” I can say more and provide more informative detail. 

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The entry spreadsheet uses worksheets with the atmosphere models already set up for each of three worlds:  Earth,  Mars,  and Titan.  All use the same stagnation heating model.  There are only 4 inputs needed to model an entering vehicle. It generates plots automatically,  but you need to make sure the altitude data in the worksheet do not go past something very close to the Mach 3 point.  Or else you would have to recreate the plots from scratch,  limiting what data you select for plotting to the Mach 3 point,  in order to prevent extreme scale distortion.   This is what the Mars entry worksheet looks like:

This old model is 2-D Cartesian (you have to “wrap” its results around the planet).  The trajectory has a constant angle Ɵ with respect to horizontal,  making it a simple straight line (in the real world,  it will “droop” significantly after the peak deceleration pulse).  It uses a very simple scale-height type of exponential model for density variation with altitude:  ρ = ρ₀ exp[h/h_scale],  where ρ₀ and h_scale are merely the curve fit constants for modeling density in the altitude range of interest.  It presumes a constant hypersonic ballistic coefficient β = M_entry/(C_D A_block),  which for blunt shapes means the entry analysis math assumptions are violated below local Mach 3.  Allen came up with a simple closed-form double-exponential equation modeling speed versus altitude,  under these particular assumptions: 

V = V_atm exp{-C exp[h/h_scale]}, 

where V_atm is the object’s speed at entry interface,  C = 1000*ρ₀*h_scale/(2*β*sin Ɵ),  and the analysis starts downward from h = h_atm,  the altitude for entry interface (a property of the atmosphere model along with ρ₀ and h_scale).  The factor of 1000 converts the customary km units of h_scale to m.  While the equations create results at speeds under local Mach 3,  they are in error for not being hypersonic (β is no longer constant),  and those points should not be included in any reported results or plots.  

Allen used a stagnation convective heating correlation that is surprisingly accurate,  even today.  It is q = Q/A = 1.75 x 10^-8 (ρ/R_N)^0.5 (1000*V, km/s)³,  where ρ is measured in kg/m³,  and R_N is the effective nose radius in meters.  The value of Q/A = q  is measured in Watts/cm².  Its integral with time is in the spreadsheet.  This is convective heating only,  one would have to add a model for plasma sheath radiation heating,  for speeds at entry exceeding about 9 km/s.  That is currently not in the spreadsheet,  but is considered to be negligible at entry interface speeds of 8 km/s and less.  The analysis is summarized in this figure:

Where Do-It-Yourselfers Can Obtain Such Materials

At least the entry spreadsheet,  the orbital mechanics spreadsheet,  and one version of the rocket engine performance spreadsheet,  can downloaded for free,  using links that are on the Mars Society’s “New Mars” forums site:  newmars.com/forums/ 

These are located on that forums site in the “Acheron Labs” section,  under the topic “Interplanetary transportation”.  Scroll down a page or two,  to the thread titled “orbital mechanics class traditional”.  The list of available lessons is in the first posting there.  Subsequent posts have the links to all the lessons,  which are actually located in a drop box on-line.  All three named-above spreadsheets are available from that drop box,  as part of the supplied class materials for this course.  

The course comprises multiple lessons that acquaint the student with classical 2-body orbital mechanics of elliptic orbits,  to include interplanetary transfers,  adds in empirical corrections for losses during launch and when 3 bodies are involved,  acquaints you with entry,  descent,  and landing issues,  then takes up rocket vehicle performance estimation (and the rocket engine performance estimation methods to support it).

Be aware that I have two other courses not available from this New Mars forums site,  but instead directly from me.  One is about compressible flow,  to include flow with losses and with heat addition,  as well as shock waves and expansion fans,  plus the same rocket engine performance estimations as are in one of the orbits course lessons.  The other has to do with high-speed heat transfer,  complete with recommended models for various situations.  Both of these courses are associated with spreadsheets as part of the class materials. 

All these class materials include pdf documents that are essentially texts from which to teach yourself how to do these things.  They include demonstration problems with solutions,  and assigned problems to be worked,  plus solutions to those assigned problems,  for comparison afterwards.   For the already-adept,  there are also slide shows from which you can teach others.

GW’s Ramjet Book

Also be aware that I have offered my ramjet book “A Practical Guide to Ramjet Propulsion” as a self-published item.  Just contact me by email,  it currently comes as a series of pdf files,  which I email to you upon receipt of payment.  I hope to soon have a fully automated site,  with a final single download file for the book.  This is not an academic work,  it is a real “how-to” guide written from my direct experiences doing that kind of work in the aerospace/defense industry long ago.  It deals with plain subsonic-combustion ramjets,  to include integral boosters,  but not ejector ramjets,  combined cycles,  or supersonic combustion.  If that interests you,  please contact me (email preferred).

Other Technical Articles Posted On “Exrocketman”

There are many technical articles on a variety of topics posted here on “exrocketman”,  along with a few things posted on youtube under the channel name “exrocketman1”.  Here on “exrocketman” the blog site,  there is a catalog article posted,  that I try to keep current,  which has these things as a list for each one of multiple topic areas.  This is the article “Lists of Some Articles by Topic Area”,  posted 21 October 2021. 

All you need are the posting date and title of the article you seek,  to find anything quickly on this site,  using the blog archive tool,  left side of page.  Click on the year,  then the month,  then the title if need be (such as if multiple articles were posted that month).  Just peruse the lists and jot down the dates and titles you want to see,  then use the archive tool.

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Thoughts on Multiple Bad Things

There’s lots of bad things going on the world,  notably in the middle east and Ukraine.  And there’s no clear “good guys vs bad guys” about the middle east,  despite so many claims otherwise on both sides.  There’s plenty of blame to go around there.  But not so much in Ukraine.

One thing has finally begun to become clear in the news reporting:  most of the death and destruction in the middle east is being done by proxy terrorist armies that are funded,  supplied,  and given marching orders by Iran.  These would include Hamas and Hezbollah,  the Houthis in Yemen,  plus some others in Iraq that have injured and killed US troops.  Here is the sense of a list of Iranian proxies that I got from the US State Department:

To this we need to add the following:  (1) the current direct Iranian attack upon Israel,  and (2) its Revolutionary Guard,  which has also directly attacked international shipping,  as well as being the private army that keeps the ruling mullahs in power in Iran.

The common thread here is Iran:  they have been ordering all this death and destruction,  and until recently,  have let their proxies do it for them.  The western nations have been striking back at the proxies,  but without any success at stopping the death and destruction,  not in about 4 decades! 

Iran itself has suffered no punishment whatsoever for doing all these evils,  mostly because the western nations do not want a general war in the middle east with Iran.  That has apparently been a policy mistake,  and needs to change.  And the mass attack upon Israel shows that the Iranians have finally been emboldened into acting directly,  instead of through their proxies,  likely precisely because they have never been punished.

Now,  don’t get me wrong:  the Iranian people are good people,  and have been our friends before,  and could be again.  They live in a sham democracy,  where the ruling mullahs can over-rule anything the elected people decide.  Those ruling mullahs are propped-up by a private army that ruthlessly suppresses dissent.  The mullahs themselves masquerade as religious leaders,  when they are in fact extremist terrorists who misuse and abuse religion to “justify” the evils they do. 

In point of fact,  there almost was a revolution in Iran about a decade ago,  but it failed for lack of outside support from us.  They had no chance against the private army (the Revolutionary Guard).

Now what Israel does is not under US control,  let that be perfectly clear!  They are a valued ally and friend in the middle east,  and one of the few democracies in the region that is not a sham democracy,  at least not yet.  And the US is sworn to help defend them when they are attacked,  that is also clear.  And we just did that,  helping to thwart the Iranian attack.

But,  the current elected government of Israel is a re-elected right-wing prime minister (Netanyahu) who this term leads a far-right coalition that is demonstrably a bit extreme.  This shows in the policies and strategies Israel has used in its Gaza war against Hamas,  that has led to incredibly-massive civilian casualties,  plus the approaching mass death-by-famine there. 

One might conclude that the overall goal here is that there be no Palestinian state,  because there are no Palestinians left to populate it.  And,  the current attack from Iran came about because (intentionally or not) the Israelis bombed an Iranian embassy in Syria. 

That being said,  Hamas is far worse!  That is a known terrorist organization that became the government in the Gaza strip,  in an armed takeover.  They give the people of Gaza just enough food to make them think that Hamas is their friend,  but then created a network of tunnels under every populated place in Gaza!  They hide behind a human shield at all opportunities,  and in fact prevented those civilians who were shielding them,  from evacuating before announced Israeli attacks!  No real government would do that,  but evil terrorists certainly would!  And did!

Now,  bear in mind another thing that must be made perfectly clear:  when you decide to attack an evildoer who hides behind a human shield,  the only way to strike him is through that shield!  There will be high civilian casualties when you do that,  it is inherent!  Using a human shield violates international law and anybody’s rules of war,  and that is exactly what the Hamas terrorists did. 

But the civilian casualties among the Palestinians in Gaza have been much higher than they needed to be,  and that’s the result of the policies of the right-wing extremist government of Israel.  So,  there really is plenty of blame to go around!  Israel needs to change-out its government,  that much is for sure!  Extremism is the true evil here;  right wing,  left wing,  makes no difference.  Both end up doing the same needless death,  destruction,  and oppression.  They both look the same to me!

 I think the wish to avoid a general war with Iran is the right thing,  but no consequences for Iran is the wrong thing! 

Here is an out-of-the-box idea:  what if Iran were to be struck,  but NOT invaded?  Struck in such a way as to enable the Iranian people to rise up and overthrow the mullahs?  The mullahs that oppress them,  and who committed all this death and destruction all these decades?  What if the strike were to attempt two things:  (1) kill most or all of the mullahs,  decapitating that government,  and (2) destroying as many combat assets as possible,  of the Revolutionary Guard?  But without any invasion at all!  None!  Simply enable the people of Iran to do their own revolution.

I would predict that the terrorist proxies would slowly go away over time,  should we be successful.  They are currently funded and supplied massively by Iran,  and if Iran no longer did that,  their means to cause death and destruction would dry up. 

Iran has been supplying Putin’s Russia with weapons for its Ukraine war, too.  If that were to stop,  Ukraine would have a better chance to defend itself.  You need to understand,  Ukraine is the west’s proxy in the war against Putin’s Russia,  who will next attack NATO members in eastern Europe,  if he succeeds in Ukraine.  It was that way with Hitler in the 1930’s,  it is that way with Putin today.  History says all these dictators are pretty much alike. 

Ukraine has run short of weapons and ammunition,  which is partly stalled by the GOP in the US House of Representatives.  We’ve all seen that on the news.  They are now losing to Russia as a result. We have all seen that,  too.

If Ukraine loses,  we lose,  and thereby will eventually face World War 3 in Europe,  when an emboldened Putin tries to rebuild the old Soviet empire in eastern Europe,  out of countries now NATO members.  Plus,  that other evil dictator in China,  Xi Jinping,  will be emboldened to invade Taiwan and start World War 3 in the Pacific,  if Russia is successful in Ukraine!  So,  Ukraine cannot lose!  Period!  End of issue!

One GOP representative in the US House has publicly said he has heard colleagues spouting Russian propaganda on the floor of the US House.  That propaganda is aimed at stopping aid to Ukraine,  so that Putin can win.  I submit to you that voting against Ukraine aid,  is giving aid and comfort to Putin’s Russia,  which is clearly America’s enemy here!

It says in the US Constitution that one definition of treason is “aid and comfort to the enemy”,  and the standard for conviction requires only two witnesses in court to the act.  I wish the Justice Department would do something about what looks to me,  to be an awful lot like treason,  going on in the US House!

Defeating Putin’s Russia in Ukraine might possibly lead to one other good outcome for the US and the world:  he might be overthrown!  There’s likely a lot of Putin clones waiting in the wings,  but if the Russian people can get rid of them,  they might once again set foot on the path to a not-sham democracy.  They are good folk,  and could also be our friends once again.

Tornado Precautions?

My wife found this on her Facebook.  We both thought it was hilarious,  primarily because it is close to the truth. 

Many years ago,  before we went to Minnesota for my first formal teaching job,  we did just about what is pictured for Texans.  Except,  we were sitting in lawn chairs,  eating and drinking,  just watching the show.  That show was multiple little F1’s spun up out of nothing,  about a thousand feet away or so,  out here on the farm.  They got bigger as they moved downwind.   

The nearest town is 4 miles away line-of-sight,  but there were no tornadoes over there,  so there were no sirens to ignore.

Ascent Compromise Design Trade Study

Update 4-8-2024:  Should any readers want to learn how to do what I do (estimating performance of launch rockets or other space vehicles),   be aware that I have created a series of short courses in how to go about these analyses,  complete with effective tools for actually carrying it out.  These course materials are available for free from a drop box that can be accessed from the Mars Society’s “New Mars” forums,  located at http://newmars.com/forums/,  in the “Acheron labs” section,  “interplanetary transportation” topic,  and conversation thread titled “orbital mechanics class traditional”.  You may have scroll down past all the “sticky notes”. 

The first posting in that thread has a list of the classes available,  and these go far beyond just the two-body elementary orbital mechanics of ellipses.  There are the empirical corrections for losses to be covered,  approaches to use for estimating entry descent and landing on bodies with atmospheres,  and spreadsheet-based tools for estimating the performance of rocket engines and rocket vehicles.  The same thread has links to all the materials in the drop box. 

The New Mars forums would also welcome your participation.  Send an email to newmarsmember@gmail.com to find out how to join up.

A lot of the same information from those short courses is available scattered among the postings here.  There is a sort of “technical catalog” article that I try to main current.  It is titled “Lists of Some Articles by Topic Area”,  posted 21 October 2021.  There are categories for ramjet and closely-related,  aerothermodynamics and heat transfer,  rocket ballistics and rocket vehicle performance articles (of specific interest here),  asteroid defense articles,  space suits and atmospheres articles,  radiation hazard articles,  pulsejet articles,  articles about ethanol and ethanol blends in vehicles,  automotive care articles,  articles related to cactus eradication,  and articles related to towed decoys.  All of these are things that I really did. 

To access quickly any article on this site,  use the blog archive tool on the left.  All you need is the posting date and the title.  Click on the year,  then click on the month,  then click on the title if need be (such as if multiple articles were posted that month).  Visit the catalog article and just jot down those you want to go see.

Within any article,  you can see the figures enlarged,  by the expedient of just clicking on a figure.  You can scroll through all the figures at greatest resolution in an article that way,  although the figure numbers and titles are lacking.  There is an “X-out” top right that takes you right back to the article itself. 

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I updated the “compressible.xlsx” spreadsheet file as “liquid rockets.xlsx”,  and deleted the extraneous worksheets.  I added a convenient block of relevant outputs that requires no new inputs other than a name for the propellant combination.  I developed a “Paintbrush” file “engine sizing report.png” on which to copy and paste the convenient outputs block in one fell swoop.  You need only adjust the name text above the engine diagram.  See Figure 1 for what this looks like.  

Figure 1 – Image of What the “Engine Sizing Report” Format Looks Like

This is all you really need,  to understand what the engine can do,  except that you must look at the green-highlighted separation limits data,  and understand that the design shown is unseparated at sea level for full throttle,  and part-throttle settings (in this case 80% Pc).  This example has a backpressure-induced flow separation in the bell at min throttle,  below very near 12.6 kft.

You still have the two performance vs altitude plots already made by the “r noz alt” worksheet.   If you want to use them,  I  recommend you copy and paste them to a “Paintbrush” png file,  then annotate them for separation.  Such is illustrated in Figure 2 below.  These are located just to the right of the altitude performance calculation block. 

When evaluating flow separation in any of the calculation blocks,  note that the pressure in the standard atmosphere was modeled,  for purposes of quick and easy estimates of the altitudes below which to expect separation.  That model was reversed to altitude as a function of Pa,  plotted,  and a 4^th-degree polynomial trend line developed with the spreadsheet software.  The quality of the fit was excellent.  But,  because of the nature of the fitted curve shape,  using this on pressures above sea level standard 14.696 psia produces nonsensical results.  See Figure 3 below for why.

Figure 2 – Example Plots,  Showing How to Annotate for Flow Separation

Figure 3 – When and How to Use the Estimates of Separation Altitude

               Using the spreadsheet

To obtain such results quickly and conveniently,  I added an automated determination of the expansion-design value of Pe appropriate to compromise-ascent design,  once the design Pc has been selected and input.  Just copy and paste-123 the design Pe value into the indicated input cell for it.  I am recommending that you use 80% Pc for this purpose,  and that you size for thrust at sea level,  using the sea level C_F for that input.

If you are doing a traditional sea level-optimized design,  I recommend you use max (100%) Pc,  and the sea level standard 14.696 psia as your design Pe.  Again,  you can just copy and paste-123 the values into the cells quickly.  I recommend that you use the sea level C_F for sizing to your thrust. 

If you are doing a vacuum design,  there is some known expansion area ratio A/A* to which you are designing.  I have retained the “compr flow” worksheet for this purpose.  Go to it,  and make sure you have the correct specific heat ratio selected.  Then in the indicated input cell,  iteratively adjust exit Mach number Me until you hit exactly the desired value of A/A*.  Read the pressure ratio PR at that Mach,  and go back to the “r noz alt” worksheet and input that PR value, almost top right. 

Input PR in “r noz alt” where shown,  and the appropriate design Pe will appear,  to be used with your input expansion design Pc.  Copy and paste-123 that design Pe value into the appropriate input cell.  I also recommend that you copy and paste-123 the vacuum C_Fvac to size vacuum thrust for a vacuum design,  which in all probability cannot be unseparated at sea level,  even at full throttle.

               Trade Study for Throttle Setting to Use for Ascent Compromise Design

For the purpose of determining what throttle-setting Pc-value to use for ascent-compromise designs,  I ran a trade study.  I ran ascent-compromise designs at Pc values from 60% to 95% max Pc,  by increments of 5%.  This was for a liquid oxygen-liquid methane (LOX-LCH4) propellant combination,  in an engine technology characterizable as “low-tech”. 

Max Pc was assumed to be only 2000 psia.  I ran a constant pressure turndown ratio (P-TDR) of 2,  which set the min Pc to 50% of max,  or 1000 psia.  The intermediate throttle setting is what I varied.  I ran this with an 18^o-8^o curved bell,  a throat discharge coefficient C_D = 0.995,  and a dumped bleed fraction BF = 0.05.  All of this assumes a gas specific heat ratio of 1.20. 

For comparison,  I also ran a traditional sea level-optimized design,  and two vacuum designs at A/A* = 100 and 300.  The results were graphed in 4 different plots,  presented as annotated in Figure 4 below. 

Top-left of the figure,  the plot shows how sea level,  ascent-averaged,  and vacuum specific impulse (Isp) vary versus the range of throttle settings investigated.  The ascent-averaged Isp trend is not linear,  and I show a sort of “aft-tangent”  determination of the rather weak knee in this curve near 80% throttle setting. 

Top-right in the figure is the same basic plot,  but to a different scale,  showing the ascent-compromise trends and the bounds represented by the sea level and the two vacuum designs.  All of the ascent-compromise ascent-average Isp values beat the sea level design’s ascent-averaged Isp value.  They even beat the sea level design’s vacuum Isp value!  They are not significantly far below the vacuum Isp value for the A/A* = 100 vacuum design,  which in turn is not far below the vacuum design for A/A* = 300!  

Figure 4 – Trade Study Plots

Bottom left is a plot of estimated expansion bell lengths (L_bell) vs the throttle setting.  These are crudely estimated as L_bell = (De – Dt)/(2*tan(avg a)),  where avg a = 0.5*(a1 + a2).  The point here is twofold:  (1) the 80% or maybe 85% values for L_bell are halfway between the sea level and the vac-100 (A/A*=100) designs,  and (2) the trend is flat enough that none of the compromise choices are far from that “halfway-between” point.

Bottom-right is a plot of A/A* vs throttle setting,  quite similar to the L_bell plot.  In this case,  the 80% point is about a quarter of the way up between the sea level and vacuum A/A*=100 designs.  The trend is flat enough that no power setting investigated is far from that point.

               Recommendations for sizing engines

For the “traditional sea level” designs,  size the expansion between max Pc and Pe = Pa = 14.696 psia (1 standard atmosphere). That produces a sea level thrust coefficient C_F,  which you use with a sea level thrust requirement to size dimensions and flow rates.

For the “vacuum designs”,  there is some max expansion ratio,  allowable in terms of fitting the engine into the available space for it,  aboard the vehicle.  Determining that fit may well be iterative!  Size the expansion area ratio to that max area ratio A/A*,  at max Pc,  which produces a vacuum thrust coefficient C_Fvac,  once you translate A/A* into a design Pe value in the spreadsheet.   Use that C_Fvac and max Pc to size the flow rates and dimensions to a vacuum thrust requirement.

For the ascent compromise designs,  determine separately what throttle setting (percent of max Pc) will be used to size the expansion.  The recommendation here is 80%,  although variations of 5% up or down from that make very little difference.  I would not recommend less than 75%,  nor any more than about 85%,  though. 

Higher setting is higher vacuum Isp but lower sea level Isp.  Lower setting is lower vacuum Isp but higher sea level Isp.  The increase in ascent-averaged Isp with increased setting is almost negligible,  because of the offsetting effects on vacuum and sea level Isp.  But using near-80% setting gives you more “room for error” in a sea level open-air nozzle test,  where you need to ignite at a low setting,  and then throttle-up rather quickly above the separation-point setting,  before any damage is done.

               What this spreadsheet does not do

This spreadsheet is for calculating good estimates of performance for liquid rocket engines of fixed nozzle geometry.  It does not do variable geometry notions such as bell extensions,  excepting as separate estimates for the two geometries as if they were fixed.  It does not do free-expansion designs at all.  Those would include both coaxial and linear aerospike geometries,  expansion-defection designs,  or any exit stream with both a free surface and contact with a physical surface.

While it provides very good performance estimates of fixed bell geometry designs,  it does not model the “cycle” that powers the turbopumps.  One does not need to do that,  to model thrust and specific impulse,  as long as one has a good estimate of the dumped bleed gas fraction representing the cycle.

               Availability of the spreadsheet

I would be happy to share this spreadsheet.  Simply contact me to make the request. There is no user’s manual (see #1 Update 4-4-2024),  although its basic operation is described in this article and another on this site. User inputs are highlighted yellow.  Significant results are highlighted blue.  Things you need to check or to iterate are highlighted green.  The Excel “copy” command,  and the “paste-123” command are the best way to transfer numerical data from one cell to another.

#1 Update 4-4-2024:  This document was originally written during 12 through 16 March,  2024.  There is now a user’s manual,  available as a pdf document,  along with the spreadsheet file “liquid rockets.xls”.   The “Paintbrush” file “engine sizing report.png” is also available as a convenient tool for reporting results.  Open it in “Paintbrush”,  cut the block of data out of it,  and copy and paste the new block from the spreadsheet into it.

               Miscellaneous things to know about

Be aware that off to the right on the “r noz alt” worksheet are some other data and plots that I used deciding how to correlate Pa vs altitude for purposes of determining the altitudes where separation might occur.  These would be of little use to any user.  Just ignore them.

That does bring up the separation backpressure estimate,  which is entirely empirical,  and was developed originally for the straight conical nozzles seen in missile solid rocket motors.  It is slightly conservative for curved bells.  It takes this form:

               Psep/Pc = (1.5*Pe/Pc)^0.8333

The ratio Pe/Psep is a simple function of the nozzle area expansion ratio.  Psep is thus an easily-computed constant times whatever your operating Pc might be.  Whenever the ambient atmospheric pressure Pa equals Psep,  separation is likely.  When Pa exceeds Psep to any noticeable extent,  separation is certain! 

There are shocks that touch the inside bell surface when separation occurs.  These greatly amplify the localized heating at the impingement location,  leading to burn-throughs and destruction in only several seconds.  That is why separation is to be avoided.  See Figure 5.

Figure 5 – Sketch of Separation Phenomena in a Bell Nozzle

#2 Update 4-4-2024:  There are two very closely related articles on this site that this document and its partial throttle setting trade study supports.  They are

“Bounding Calculations for SSTO Concepts”,  dated 4-2-2024

“Bounding Calculations for TSTO”,  dated 4-3-2024 

Both of these have reference lists of other closely-related earlier articles on this site,  including two where I investigated free-expansion nozzle design approaches,  among other things. 

In the performance of the trade study,  I sized multiple engines with the “r noz alt” worksheet,  and reported those results (rather easily and quickly) using the “engine sizing report.png” Paintbrush file.  Those sized engine results follow,  as a collection of unnumbered and untitled figures.  

Bounding Analyses for TSTO

Update 4-8-2024:  Should any readers want to learn how to do what I do (estimating performance of launch rockets or other space vehicles),   be aware that I have created a series of short courses in how to go about these analyses,  complete with effective tools for actually carrying it out.  These course materials are available for free from a drop box that can be accessed from the Mars Society’s “New Mars” forums,  located at http://newmars.com/forums/,  in the “Acheron labs” section,  “interplanetary transportation” topic,  and conversation thread titled “orbital mechanics class traditional”.  You may have scroll down past all the “sticky notes”. 

The first posting in that thread has a list of the classes available,  and these go far beyond just the two-body elementary orbital mechanics of ellipses.  There are the empirical corrections for losses to be covered,  approaches to use for estimating entry descent and landing on bodies with atmospheres,  and spreadsheet-based tools for estimating the performance of rocket engines and rocket vehicles.  The same thread has links to all the materials in the drop box. 

The New Mars forums would also welcome your participation.  Send an email to newmarsmember@gmail.com to find out how to join up.

A lot of the same information from those short courses is available scattered among the postings here.  There is a sort of “technical catalog” article that I try to main current.  It is titled “Lists of Some Articles by Topic Area”,  posted 21 October 2021.  There are categories for ramjet and closely-related,  aerothermodynamics and heat transfer,  rocket ballistics and rocket vehicle performance articles (of specific interest here),  asteroid defense articles,  space suits and atmospheres articles,  radiation hazard articles,  pulsejet articles,  articles about ethanol and ethanol blends in vehicles,  automotive care articles,  articles related to cactus eradication,  and articles related to towed decoys.  All of these are things that I really did. 

To access quickly any article on this site,  use the blog archive tool on the left.  All you need is the posting date and the title.  Click on the year,  then click on the month,  then click on the title if need be (such as if multiple articles were posted that month).  Visit the catalog article and just jot down those you want to go see.

Within any article,  you can see the figures enlarged,  by the expedient of just clicking on a figure.  You can scroll through all the figures at greatest resolution in an article that way,  although the figure numbers and titles are lacking.  There is an “X-out” top right that takes you right back to the article itself. 

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I also want to keep this one fairly short.  The ascent-averaged and vacuum specific impulses for the propulsion presumed here,  came from earlier analyses of various engines contained in other articles posted on this site.   What I did here was to first size an all-expendable two-stage to orbit (TSTO) vehicle as a baseline,  then afterwards,  look at two different ways to make the first stage recoverable and reusable.  Then I compared those to previous single stage to orbit (SSTO) results.

The basic mission is surface launch to low Earth orbit (LEO) at about 300 km altitude,  eastward at low inclination,  as illustrated in Figure 1.    The surface circular orbit speed 7.9 km/s is the “ideal speed” to reach,  which covers the effects of both the kinetic energy of the speed at that altitude,  and the potential energy of being at that altitude. 

There is a staging point that is just barely exo-atmospheric in terms of the rather modest ascent speeds reached at that point (and which is not at all “exo-atmospheric” in terms of orbital speeds at only around 50-60 km altitude).  This stage point for the expendable baseline was presumed to be at 2 km/s,  and very nearly horizontal locally.  I estimated 5.6% each, for gravity and drag losses,  at 0.45 km/s each,  to be covered by adding to required delta-vee (dV).  That makes the min mass ratio-effective dV to “barely reach LEO” 8.8 km/s.   To which must be added a presumed rendezvous dV budget of 0.6 km/s,  and a deorbit dV of 0.1 km/s,  for a total expendable mission dV = 9.5 km/s.

Figure 1 – The Basic Mission Parameters and Possible Design Approaches

In the expendable baseline,  the first stage must accelerate from zero to the staging velocity Vstg,  and shoulder essentially 100% of the drag loss,  and a presumed 80% of the gravity loss.  It does this while carrying the fully-loaded second stage as its payload,  complete with the payload shroud protecting the actual payload during atmospheric ascent.  Note that the second stage shoulders a presumed 20% of the gravity loss,  and none of the drag loss;  not “right”,  but well in the ballpark.

In only the recoverable/reusable first stage scenarios,  the first stage coasts downrange toward an atmospheric entry at essentially the staging speed (approximately Mach 7).  To be recovered as an exposed aluminum structure item,  this stage must decelerate to an acceptable entry speed,  presumed to be 2.5 Mach,  and deploy grid fins to keep it below Mach 3 in the thin air,  and a terminal speed of only about Mach 2.5 in the thicker air close to the surface.   To land retro-propulsively,  it must “kill” the 2.5 Mach terminal speed of about 0.75 km/s,  factored up by 1.5 to cover any losses plus a budget for hovering-and-diverting clear of obstacles.

There would seem to be two ways to “redesign” the expendable baseline for first stage recovery and reuse.  First,  one might offload some payload from an otherwise fully-loaded second stage,  which lightens the second stage and increases its share of the mission dV,  while reducing the ascent dV requirement on the first stage,  due to both the decrease in payload mass,  and the decrease in staging velocity.   Second stage recovery was not considered,  since it must be a real entry vehicle.

Second,  one keeps the payload and second stage exactly the same as the baseline,  and just increases the size of the first stage,  so that it also has enough propellant to land.  Going into this,  my preconceived notion was the larger first stage would be slightly more favorable in terms of overall payload fraction,  because I thought we would have to offload around half the payload.

More details of exactly how I went about about the baseline design,  and the two re-designs for first stage recovery,  are shown in Figure 2.  I used a fixed 10 metric tons of baseline delivered payload. 

Figure 2 – The Best Design Analysis Notions and Baseline Data Going Into This Study

The baseline design was done using simple rocket equation calculations in a spreadsheet,  to include kinematic acceleration requirements on both stages to determine overall stage thrust requirements.  Being only a bounding calculation,  I did not size numbers of engines and their individual thrusts and turndown ratios.   Modest propulsion technologies being presumed,  pressure turndown ratios might be in the 2 to 2.5 class.  Thrust turndown ratios would be similar in vacuum,  and a bit more at sea level. 

The baseline analysis used in both stages a presumed inert/ignition mass fraction of 4% (0.04),  pretty much representative of bare aluminum alloy tankage exposed as the stage airframes.  The dV requirements and effective exhaust velocities Vex determine mass ratio MR = exp(dV/Vex) for each stage,  from which the propellant/ignition mass fraction can be computed as 1 – 1/MR. 

The payload/ignition mass fraction is then 1 – inert fraction – propellant fraction.  Then the payload mass and payload fraction determine the ignition mass,  from which then everything else in the weight statement scales.  Vex scales from input Isp as Vex = g_c * Isp / 1000. 

This calculation,  done for the two stages with the second stage as the payload for the first,  is quite simple as illustrated in Figure 3.  Note that the payload shroud is jettisoned at staging.  The “overall payload fraction” is the actual delivered payload mass divided by the complete launch vehicle ignition mass.  Note also that gees estimated at stage burnout masses are acceptable without throttling-down or shutting-down engines,  being under 4 gees.   This thing sized out near 138 metric tons at launch,  for a 10-ton payload delivery,  about half the size of a Falcon-9.  Under these analysis conditions,  it shows just over 7% delivered payload,  relative to launch mass. 

Figure 3 – The Baseline Expendable Vehicle Results

The re-design calculations are a bit more complex,  but are still done the same way with the rocket equation and some kinematic acceleration constraints for the stages,  that bound the thrust levels.    For those,  the first thing to do was to copy the stage weight statements and thrust results into new worksheets,  and then copy the dV items for a reanalysis of dV requirements,  that includes those for stage 1 recovery.  To this I added a revision analysis for the lower stage inerts,  to account for adding grid fins and landing legs.  Values for these are simply presumed,  and added to the expendable stage inert mass value.   Also,  one has to iterate the payload reduction to produce the same total propellant mass in the first stage,  but with an increased stage inert mass.

There are also added kinematic requirements on the first stage during its unladen descent.  The min thrust needs to be a bit less than the dry-tanks burnout weight,  so that it can gently set down.  That also enables hover at thrust equal to weight.  The max thrust relates to how soon the landing burn begins.  I simply arbitrarily set it to 2 gees,  for an effective 1 gee kinematic rate over gravity.

These worksheets include a revised dV analysis that includes the descent dV for stage 1,  located on the right of the worksheet.  For the reduced-payload approach,  I had to let the staging velocity shift downwards,  responding to the greater dV capability of the second stage at reduced payload.  This reduced the first stage ascent dV requirement,  accomplished on the laden weight statement.  The descent dV requirement has to be met unladen,  which determines the descent propellant requirement,  that must be carried as if it were payload during the ascent.   See Figure 3.   What surprised me was how little payload I had to offload,  which then had very little effect on the overall payload fraction,  still being about 7%.  The launch weight only increased by about a ton. 

Figure 4 – The Reduced-Payload Approach Results

For the larger-first-stage approach,  the payload and second stage are entirely unchanged,  and the first stage need only be large enough to accommodate both the ascent dV laden,  and the descent dV unladen.  See Figure 5.  Again,  you do the first stage descent first to determine the descent propellant,  in turn carried as if it were added payload during the first stage ascent.  I actually did this larger first stage analysis first,  developing the basic format.  Then I copied it to the reduced-payload worksheet and modified it into that analysis.  

I was surprised to see the smaller overall payload fraction nearer 6%,  but in retrospect,  the larger first stage increases the launch mass significantly (nearer 164 tons compared to the original 138 tons),  so the payload fraction reduction should not have been such a surprise.   There is no iteration in this approach,  we are simply up-sizing the first stage to a larger propellant mass and inert mass. 

Figure 5 – The Bigger First Stage Approach Results

Results and Recommendations

Because the overall propellant fraction fell more when upsizing the first stage,  than it did reducing payload mass,  I have to recommend reducing payload mass as the better design approach to making an expendable first stage recoverable and reusable!  This is in fact exactly what SpaceX did with their Falcons,  when they made the first stage cores recoverable and reusable,  although their recovery path proceeds back up-range,  while what I analyzed here leads to recovery far downrange.  They have to offload more payload to make that work,  because the first stage descent dV requirements are substantially higher when reversing flight direction.

To Summarize: 

If designing all-expendable vehicles,  I can get up to about 5% payload fraction out of a LOX-LH2 SSTO with modest engine technology and a good compromise-bell conventional nozzle (based on the previous SSTO article).  I can get about 7% payload fraction out of a TSTO that uses LOX-LH2 modest technology vacuum engines in the second stage,  and LOX-RP1 modest technology compromise-bell engines in the first stage.  The expendable TSTO and SSTO should be more-or-less competitive with each other,  but not with any renewable technologies.  The costs of thrown-away hardware will be too high to compete.

Of the possibilities to make the first stage of the TSTO recoverable and reusable,  I can offload just a tad of payload and recover the first stage far downrange for re-use.  The payload fraction actually stays near 7% doing that (it would drop if I recovered up-range)!  It drops to about 6% if I up-size the first stage instead (and would also drop further if recovering up-range).  The expendable SSTO just cannot compete with either one of those,  having both a lower payload fraction and much more hardware thrown away (first stages and SSTO’s have many engines,  second stages have few).

As the previous SSTO article listed among the references below indicates,  there is no possibility of a reusable SSTO that is chemically powered,  with more than about 1% payload,  if even that. 

References

All of these listed below are earlier articles posted on this site that relate to this topic in some way.  Use the blog archive on the left side of this page as a fast navigation tool to find them quickly.  All you need is the article title and its posting date.  Click on the year,  then on the month,  then on the title if more than one article was posted that month.  The 2-4-23 article (bold italic) has a lot of the earlier propulsion work used here.  The other 11-12-18 article (also bold italic) was the first to explore free expansion designs as an alternative to conventional bells.  The bold italic 4-2-24 article at the top of the list,  is the previous SSTO bounding study. 

4-2-24                 Bounding Calculations for SSTO Concepts (compared to TSTO in this article)

3-3-24                 Launch to Low Earth Orbit:  1 or 2 Stages?

3-4-24                 Launch to Low Earth Orbit: Fixed Geometry Options

6-20-23               TSTO Launch Fundamentals

2-4-23                 Rocket Nozzle Types (bells and aerospikes)

10-1-22               Rocket Engine Calculations

2-9-21                 Rocket Vehicle Performance Spreadsheet

2-16-20               Solid Rocket Analysis (solid ballistics and more)

11-12-18             How Propulsion Nozzles Work

 

Bounding Calculations for SSTO Concepts

Update 4-8-2024:  Should any readers want to learn how to do what I do (estimating performance of launch rockets or other space vehicles),   be aware that I have created a series of short courses in how to go about these analyses,  complete with effective tools for actually carrying it out.  These course materials are available for free from a drop box that can be accessed from the Mars Society’s “New Mars” forums,  located at http://newmars.com/forums/,  in the “Acheron labs” section,  “interplanetary transportation” topic,  and conversation thread titled “orbital mechanics class traditional”.  You may have scroll down past all the “sticky notes”. 

The first posting in that thread has a list of the classes available,  and these go far beyond just the two-body elementary orbital mechanics of ellipses.  There are the empirical corrections for losses to be covered,  approaches to use for estimating entry descent and landing on bodies with atmospheres,  and spreadsheet-based tools for estimating the performance of rocket engines and rocket vehicles.  The same thread has links to all the materials in the drop box. 

The New Mars forums would also welcome your participation.  Send an email to newmarsmember@gmail.com to find out how to join up.

A lot of the same information from those short courses is available scattered among the postings here.  There is a sort of “technical catalog” article that I try to main current.  It is titled “Lists of Some Articles by Topic Area”,  posted 21 October 2021.  There are categories for ramjet and closely-related,  aerothermodynamics and heat transfer,  rocket ballistics and rocket vehicle performance articles (of specific interest here),  asteroid defense articles,  space suits and atmospheres articles,  radiation hazard articles,  pulsejet articles,  articles about ethanol and ethanol blends in vehicles,  automotive care articles,  articles related to cactus eradication,  and articles related to towed decoys.  All of these are things that I really did. 

To access quickly any article on this site,  use the blog archive tool on the left.  All you need is the posting date and the title.  Click on the year,  then click on the month,  then click on the title if need be (such as if multiple articles were posted that month).  Visit the catalog article and just jot down those you want to go see.

Within any article,  you can see the figures enlarged,  by the expedient of just clicking on a figure.  You can scroll through all the figures at greatest resolution in an article that way,  although the figure numbers and titles are lacking.  There is an “X-out” top right that takes you right back to the article itself. 

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I want to keep this short and sweet.  It makes use of things computed in other posted articles,  as well as what I did for this one with a simple spreadsheet.  I often run into correspondents who advocate single-stage-to-orbit (SSTO) as the better means to send payload to low Earth orbit.  I disagree that it is “better” than two-stage-to-orbit (TSTO),  but it is actually possible to do SSTO with chemical propulsion!  However,  it is possible technologically  only if it is done as an expendable,  which simply cannot be inexpensive!  Equal payload but more expensive is not better!  See Figure 1.

Figure 1 – Overall Results for SSTO Bounding Calculations

There is a lot contained in that figure.  Upper left is a plot of SSTO vehicle inert mass fraction versus its propulsion’s ascent-averaged specific impulse (Isp).  This is for a constant 1% payload mass fraction,  which is quite low.  I indicated which ranges of Isp could be attained with chemical propulsion,  versus nuclear.  This is very simple rocket equation-based stuff.

The equations used to compute these things are given underneath the sketch of the vehicle concept lower left.  The flat lines in the plot are the levels of vehicle inert mass fraction that might be “credible” under different circumstances.  The arrows indicate where there might be payload fraction added to the baseline 1%.  5% inert is only credible for expendable stages! 

10% inert is almost not credible at all for a reusable vehicle,  due to the airframe items required as a heat shield,  and retro-propulsion to effect some sort of landing,  which also increases the dV requirement further!  15-20% inert are more credible as the inert mass fractions of some sort of lifting body vehicle with internal tanks,  heat protection,  and landing gear for a glide landing.

Most bomber and transport aircraft are nearer 40% inert,  and so was the X-15 rocket plane!  None of those has (or had) the heat protection for full orbital entry!  Naval carrier aircraft are closer to 50% inert,  because of the hard knocks they must take. 

Upper right is a depiction of the mission and the resulting delta-vee (dV) requirements.  The mission is eastward low Earth orbit at 300 km altitude,  at low inclination.  A surrogate for reaching this energy is the surface circular orbit speed of 7.913 km/s.  To this must be added losses-to-cover.  These losses I presumed as 5.6% of surface circular each,  for the gravity and drag losses.  That gets us to just about 8.8 km/s to just barely reach orbit,  to include a circularization burn. 

Once in orbit,  there needs to be a dV budget for rendezvous with “something” in order to be useful,  plus a deorbit burn budget.  I used 0.6 (arbitrary but ballpark) and 0.1 km/s (fairly precise) for these,  respectively.  Excluding any sort of landing burns,  the mission mass ratio-effective dV is thus just about 9.5 km/s.  That’s close enough for a realistic bounding calculation.

The other point to be made is the variation of propulsion Isp with increasing altitude during the ascent.  This is sketched in a sort-of-sketch plot in the lower right of the figure.  Behavior is quite different for engines with conventional bells versus engines with free-expansion nozzle designs.  The point is this:  you need an ascent-averaged Isp to use in the rocket equation SSTO model. 

However, the figure makes clear that regardless of what inert mass fraction you might presume to be credible,  only nuclear propulsion could supply enough Isp to make a reusable lifting body SSTO feasible at 15-20% inert.  Chemical propulsion is probably only feasible for expendable SSTO designs,  with the rather low stage inert mass fraction that is typical of such things. 

Alternatively,  you might increase the payload fraction a little bit at 5% inert,  in an expendable LOX-LH2 SSTO,  possibly to as high as 5-6%.  That would be an equal payload fraction to an expendable TSTO.  But,  expendable is just not inexpensive to use,  because you are losing all the hardware!  The TSTO would be cheaper use even if only its first stage were reusable,  and that has already been demonstrated feasible at low inert fractions by SpaceX with its Falcons!

Details (read on only if you are interested in the details)

Isp is not solely a property of the propellant combination,  but characteristic velocity c* is,  contrary to prevailing opinions.  This and the mixture ratio are weak functions of the downstream chamber pressure Pc that feeds the nozzle (whatever it is).  Isp would be proportional to c* if all else were equal,  but it is not!  The thrust coefficient that is achievable with the expansion is just as important to Isp as is c*. 

Once you determine an expansion thrust coefficient,  then Isp actually is proportional to c*.  See Figure 2 below for typical c* data (blue) versus propellant combinations.  At any design point Pc,  c* = (c* at 1000 psia)*(Pc/1000 psia)^m.  Thrust coefficient is C_F = Fth/Pc At = (Ae/At)*(Pe/Pc)[1 + γ η_KE Me²] – (Pa/Pc)*(Ae/At),  where the expansion details set your values of Me,  Ae/At,  and Pe/Pc,  plus your nozzle kinetic energy efficiency η_KE.

The basic message here is that once the expansion and thrust coefficient are defined (along with the dumped bleed fraction BF),  you can rough-estimate Isp for a different propellant combination as being proportional to c*.  That is because Isp = C_F c* (1 – BF)/g_c C_D.  In a decent design,  C_D ~ .99.

Designing fixed-bell engines can be to a fixed area expansion ratio Ae/At,  or to a fixed expanded pressure ratio Pe/Pc (both depend explicitly on Me).  The ambient atmospheric pressure Pa also affects the value of C_F,  as indicated above.  A traditional sea level-optimized design sets its expansion such that Pe = Pa at sea level,  figured at a suitable design value of Pc.  What I call a “compromise bell” does that same thing,  just at an altitude above sea level,  such as 10,000 feet,  20,000 feet,  or even 30,000 feet.  The “compromise bells” get higher values of Ae/At (and higher resulting vacuum Isp performance but lower near sea level) at the cost of part-throttle flow separation at sea level while testing in “open-air nozzle” mode. 

Figure 2 – Models for c* and r Versus Some Propellant Combinations (1969 P&W Handbook)

There are multiple kinds of free-expansion nozzles.  These always reach Pe = Pa at any altitude,  thus varying the effective Ae/At.  This has to be measured at the last point of physical contact with the expansion spike or surface. The shape of that expanded Ae varies,  as does the divergence angle of the outer streamline with respect to the axial thrust direction. 

The nozzle kinetic energy efficiency η_KE is the integrated average of all the cosine factors-to-axial of all the streamlines in the exiting stream.  This is fixed for a fixed bell,  and highly-variable with altitude (becoming more extreme at very high altitudes) for a free expansion nozzle of any kind.  The effect is quite serious in a simple coaxial spike design with a sonic-only gas generator,  as shown in Figure 3 below.  Such are really-lousy vacuum engines.

I explored multiple design variations of free-expansion nozzles,  until settling on an aerospike design fed with multiple gas generators.  The streamline divergence was extreme until I added some fixed-bell supersonic expansion,  before further expanding that supersonic exit plume against the aerospike.  That was the key to improved performance flying out into vacuum,  because there was reduced potential for Prandtl-Meyer expansion effects.  See Figure 4 below.  These are actually “good” vacuum engines,  but they still fall slightly short of fixed-bell vacuum Isp.

These partial-supersonic free-expansion estimates are somewhat over-estimates,  since they do not account for the oblique shocks incurred turning a supersonic stream.  But this design approach gets an Isp trend vs altitude that ascent-averages pretty much the same as that of a good compromise bell.  I do not show a distinct advantage of either nozzle type,  either way,  for average performance during ascent!  However,  the “trap” is not properly optimizing the free-expansion design.  It “falls off the cliff” easier than the fixed bell,  if you fail to optimize it “just so”.

I sized most of these nozzles at modest Pc values,  using LOX-RP1,  at a nominal c* = 5900 sec.  To approximately convert that trend to LOX-LH2,  just multiply the Isp values by 7950/5900 = 1.347.  For LOX-LCH4,  multiply by 6120/5900 = 1.037.  It is actually better to just do the ballistics.

For the compromise bell at 100,000 feet in Figure 4,  325 s max Isp on LOX-RP1 becomes about 438 sec with LOX-LH2.  For the partial-supersonic aerospike,  350 sec max Isp on LOX-RP1 becomes about 472 sec with LOX-LH2.  Those numbers might get you to 10% inert (not very credible for reusability) in the Figure 1 plot of inert fraction vs ascent-average Isp.  They do not get you even close to the more-credible 15-20% inert range!  That requires 530+ sec of Isp,  which is nuclear.

Figure 3 – Free Expansion Nozzle Performance Falls Quickly With Sonic-Only Gas Generators

Figure 4 – No Advantage Either Way With Bells Vs Partial-Supersonic Aerospike Free Expansion

References:

All of these listed below are earlier articles posted on this site that relate to this topic in some way.  Use the blog archive on the left side of this page as a fast navigation tool to find them quickly.  All you need is the article title and its posting date.  Click on the year,  then on the month,  then on the title if more than one article was posted that month.  The 2-4-23 article (bold italic) has a lot of the earlier work used here.  The other 11-12-18 article (also bold italic) was the first to explore free expansion. 

3-3-24                 Launch to Low Earth Orbit:  1 or 2 Stages?

3-4-24                 Launch to Low Earth Orbit: Fixed Geometry Options

6-20-23               TSTO Launch Fundamentals

2-4-23                 Rocket Nozzle Types (bells and aerospikes)

10-1-22               Rocket Engine Calculations

2-9-21                 Rocket Vehicle Performance Spreadsheet

2-16-20               Solid Rocket Analysis (solid ballistics and more)

11-12-18             How Propulsion Nozzles Work