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.

On Pressure Vessels

One can build an unpressurized liquid tank to any desired shape,  unless it is so large that the depth of liquid inside exerts a significant pressure all by itself.  The same is not true of pressurized vessels,  even small ones,  beyond a “single handful of psi” gage pressure inside it.  There are very good reasons these tanks are only made in spherical shapes,  or as right circular cylinders with hemispherical or spherical-segment ends. 

This document presumes the reader knows what “gage pressure” means,  and what mechanical stresses and strains are.

Figure 1 shows a cylindrical pressure vessel holding gage pressure P inside.  This could be a tank,  or even a pipe or a tube.  The “hoop stress” is given by the formula shown top right,  which is sometimes referred to as Barlow’s pipe stress formula,  which is a measure of the stress trying to  split the cylinder open longitudinally.  Technically,  you use the cylinder inside diameter for this formula (a requirement working with pipe and tubing),  but as long as the material thickness is small compared to the diameter,  something true of rocket cases and propellant tanks,  there is little-or-no perceptible difference between inside,  outside,  and average diameters. 

Bottom left,  the cross-section of the cylinder,  or any section through a sphere,  is shown.  For a cylinder,  the “axial stress” is that which resists the pressure trying to part one end of the vessel from the other.  For a sphere,  it’s just the “membrane stress” that resists splitting the sphere apart,  no matter the section orientation.  That stress works out to be just half the hoop stress,  as shown.

Figure 1 – Pressure Vessel Stresses In the Vessel Material

For a cylindrical pressure vessel fitted with hemispherical ends,  Figure 2 takes this notion a bit further,  showing as it does the stress distributions on a tiny patch of material,  located on either side of the joint.  The axial stresses match,  while the hoop direction stresses differ by a factor of 2.  There are (at least theoretically) no stresses perpendicular to the material itself (the radial direction from the cylinder axis,  or from the hemisphere’s center). 

This hoop stress mismatch at the joint corresponds to a strain mismatch in the circumferential direction,  in turn corresponding to a radial displacement mismatch between the cylinder material,  and the end (or “head”) material.  The cylinder swells radially under pressure twice as much as the end or “head” swells radially,  as measured at the joint.  This distorts both the cylinder and the head locally at the joint,  as the materials bend locally,  in order to try to stay joined.  This induces large bending stresses locally,  which add in certain ways to the hoop and axial (or membrane) stresses already described.  That makes the joint quite vulnerable to local overstress failure. 

The ”fix” for this is to locally thicken the cylinder and head materials at the joint.  In effect,  the extra material “sops up” the extra imposed stresses.  For boilers,  there are very specific guidelines for how much local thickening is needed,  and how far “local” extends away from the joint.  Those rules are the ASME boiler code,  which is legally mandatory everywhere in the country,  for designing and building boilers.  Every provision represents a life lost learning that lesson.  This is serious business!

Figure 2 – There Are Distortions With Extra Stresses At the Joints

There are choices allowed for how to implement those localized thickenings at the joints.  Those are depicted in Figure 3,  and also apply to solid rocket motor case designs.  You can increase the thickness toward the inside,  or toward the outside,  or even some of both,  just as long as enough  extra material is supplied.  Finite-element stress-strain analysis can refine this further.

The same figure also shows a variation on the spheroidal end,  where only a sphere segment is used as the end membrane.  This requires a connection ring that resists radial swelling at about the same rate as the end resists radial swelling at its attachment joint.  That way,  no thickening of the membrane is required,  the ring supplies that for the membrane.  You still need a local thickening of the cylinder material at its attachment to the ring,  because of the radial swelling mismatch.

This spherical segment and ring approach lets one enclose more volume within a given length,  without making the assembly any heavier than a full hemispherical end.  This is how most solid rocket motor case closures are designed.  It is a well-proven solution.  The flatter the membrane,  the heavier the ring gets,  though.  It’s a trade-off.

Figure 3 – Full Hemispherical Ends Vs. Spherical Segment Ends With Rings

There are many possible reasons for wanting to use other shapes for one or both cylindrical pressure vessel end heads.  As long as membrane stress is insignificant (meaning very low pressure indeed!),  you can do that!  But as soon as the pressure (and the stresses it induces) become significant,  those other shapes rapidly become infeasible.  This is shown in Figure 4,  where the two spheroidal options are depicted,  along with elliptical and conical ends.  A flat end fails even worse than the elliptical,  from similar stresses that are just higher,  plus a sharp corner effect that locally greatly magnifies the local stresses even further,  right at the corner.

There is one positive benefit to a conical end,  if an axially-directed load must be carried from the cone tip into the cylinder.  This is an efficient load path for such a load.  But it is still a lousy pressure vessel choice!  It will require a lot of internal stiffeners to keep it from trying to “go round”.  Those are going to add significant weight,  there is no way around that problem!  You must trade off the axial load path advantage against the big weight gain incurred to make a conical shape a pressure vessel.

Figure 4 – Which End Shapes Work and Which Do Not,  and Why

It is a common belief that an elliptical shape is as good as the ring and spherical segment.  This is not true from a pressure vessel design standpoint,  as the figure shows.  The elliptical head will try to “go round”,  inducing severe bending stresses.  It is volumetrically efficient,  which is why many unpressurized railroad tank cars use elliptical heads on cylindrical bodies.  But these tank cars are not pressurized!  Their evident abundance is deceptive regarding the pressure vessel issues.

That same effect is why you want to use circular cylinders as your basic tank body for a pressure vessel,  not some elliptical (or other) cross section shape.  Those other shapes will try to “go round” upon pressurization,  leading to enormous bending stresses and very rapid failure.  If you must put pressurized storage within some oddly-shaped volume,  you must fill it with multiple small circular cylinders!  The non-circular cross section is not,  and will never be,  a successful pressure vessel design!  This is why air mattress floats are made the way that they are,  for example:  multiple cylinders,  connected at the ends so as to fill together all at once.

Addendum:  Exact Analysis

The exact analysis for right circular cylinders and spheres is given in Figure 5,  along with the geometries that allow these formulations to be made from very simple measurements.  

Figure 5 – Exact Formulations

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Air Launch to Low Earth Orbit

There is a conundrum associated with launching to low Earth orbit from an airplane.  The illustration tries to sum up the various parts of it.  That is not to say that it cannot be done,  because it already has.  But,  it may,  or may not,  be an attractive way to do the mission.

The first part of this conundrum is the low speed of the launch aircraft (which for the Pegasus system is a wide-body subsonic airliner).  That forces the dropped rocket vehicle to be at least two-stage,  despite the advantage of the low stratospheric launch altitude.  As it says in the illustration,  speed at drop is the biggest influence on the rocket vehicle design,  and altitude the least,  although both are beneficial.  Mach 0.85 at 45,000 feet is but 822 feet/sec (0.25 km/s).  The drag loss of the rocket vehicle is (at least theoretically) less,  because it starts in thinner air up high.

The second part of this conundrum is not so obvious:  the level path angle of the carrier airplane at the drop point.   A low-loss non-lifting ballistic trajectory begun at stratospheric altitude would need a path angle at ignition on the order of 45 degrees,  maybe even a little more.  So,  either the carrier airplane,  or the rocket vehicle,  has to pull up rather sharply,  to reach that path angle from level flight.  One or the other must do this!

The usual airplane flying high in the stratosphere is at or near its “service ceiling”,  where there is barely enough wing lift being produced at an efficient angle of attack,  to hold up the weight,  and essentially all the thrust the airbreathing engines can make is just overcoming drag at the flight speed!  The airplane can neither accelerate path-wise,  nor can it climb!  That is the definition of “service ceiling”,  and for most planes,  it falls in the 45,000-55,000 foot altitude range,  at high subsonic speeds.  There have been exceptions:  the U-2 variants and the SR-71 variants could fly higher,  being very specialized designs.

Left unaddressed in the airplane,  the service ceiling problem puts the sharp pull-up task squarely upon the rocket vehicle to be dropped.  There are only two choices:  put wings on the rocket vehicle,  or fly it at very large angles of attack,  so that the cross-path vector component of its thrust is effectively a large lift force. 

Pegasus used large wings,  on the first stage of a two-stage rocket vehicle.  Those add both weight and drag,  especially drag-due-to-lift at the large lift coefficient needed to pull up sharply.  That pretty-well eats up the advantage gained by launching the rocket at elevated altitude in the thin air.  The wings are bigger than you would want,  precisely because of that thin air!  And that problem is why there have just not been that many Pegasus launches.

Leaving the wings off of the rocket vehicle forces you to pitch it up to very large angles of attack,  in the 45-75-degree range,  to get enough of a cross-path vector component of the rocket thrust,  to serve as the necessary lift force for a sharp pull-up maneuver.  That reduces the path-wise vector component of thrust,  while at the same time greatly increasing vehicle drag.  So,  you accelerate slowly( if at all) in rocket thrust during the pull-up maneuver,  using up a great deal of rocket propellant that adds nothing to your speed.  That also eats up any advantage of launching in the thin air,  way up high!

The only other feasible alternative is to add another large source of thrust to the launch airplane,  so that it can execute the pull-up maneuver into a zoom climb,  without stalling and falling out of the sky,  out-of-control.  Generally speaking,  you would add a source of thrust immune to the service ceiling effect,  and that is rocket thrust!  Your launch airplane would have to be modified for mixed (parallel-burn) rocket and gas turbine propulsion,  somewhat similar to the NF-104 and some of the early high-speed X-planes. 

So far,  no air-launch carrier plane has had this design approach,  but it certainly would be possible!  And it would take care of the high path angle requirement that is second only to speed at launch in importance,  while keeping the wings on the airplane where they belong,  and not on the rocket vehicle!

That leaves speed at launch,  the most important variable affecting the rocket vehicle design.  There are (or have been) very few supersonic aircraft designs that are also large enough to serve as a drop aircraft for a rocket vehicle of any significant size.  Those would include the B-58 Hustler (long-retired,  and none are left),  the SR-71 (also retired,  but very expensive to operate indeed),  and the B-1B bomber (currently in service as a military strategic bomber).  

The modifications to include rocket propulsion to the SR-71 likely would not fit within its very-critical shape.  The M-21 variant that launched the D-21 drone was limited in payload size,  to the size of that drone (not very large).  A rocket might be added in the tail cone of a B-1B,  but its payload would be limited to that which would fit in its bomb bay.  That B-1B option would reach a low supersonic launch speed at the high path angle needed,  with a rather-dangerous zoom climb and recovery after drop.

That brings up the danger of supersonic store separation.  There is a very good reason that most military aircraft,  even those capable of supersonic flight,  are limited to high-subsonic weapon release speeds.  That is because the inherent wobbles of a released store will include pitch-up,  thus developing lift.  At high enough speeds,  that lift generated by the wobbling store will exceed its weight,  and it can easily fly up and collide with the drop aircraft,  before the store’s drag can pull it behind. 

It cost a destroyed airplane and the life of one of the two crew,  to learn this lesson with the M-21 trying to launch a D-21 drone (without a booster) at just about Mach 3.   That is why the drone was re-fitted with a big booster,  and launched subsonically from B-52’s instead.  It’s not that supersonic store separation cannot be done (because that booster separated at Mach 3 from the D-21).  But successful supersonic store separation is very difficult to achieve,  and the risks of doing it are inherently very high.

So how fast a drop speed can be obtained?  That depends upon the gas turbine engines powering the launch aircraft.  Those are seriously limited by the high air temperatures associated with capturing supersonic air.  Most are limited to about Mach 2.5.  There are a very few that went faster:  those powering the XB-70 at Mach 3,  those powering the SR-71 variants at Mach 3.2,  and the 500 hour short-life,  replace-don’t-overhaul engines in the Mig-25 at Mach 3.5.  So,  to have a wide range of possible engines available for new designs,  it looks like Mach 2.5 at drop is “about it” with gas turbine.  Maybe Mach 3.

So,  the answer would seem to be a mixed-propulsion airplane with gas turbine propulsion,  augmented by parallel-burn rocket propulsion,  added to enable the zoom-climb by a sharp pull-up maneuver.  This would be at high altitude near 45,000 feet,  for the drop of the rocket vehicle.  To do this successfully,  the very difficult supersonic store separation problem must be very carefully addressed!  Both aircraft and crews are at serious risk.

Mach 2.5 at that altitude would be 2419 feet/second (0.737 km/s),  less than 10% of low circular orbit speed,  so one is still looking at a two-stage rocket vehicle to reach orbit.  Deliverable payload would be limited in size by the size of the drop aircraft,  since that in turn limits the size of the rocket vehicle it can drop.

In a word,  this has already been done with subsonic carrier aircraft,  although it has proven no more attractive than vertical rocket launch,  at best.  The supersonic release has yet to be tried,  and will prove both difficult and dangerous,  although the improvement in attractiveness may be worth that effort and risk.  No one yet knows. 

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Search code 01082025;  

search keywords:  aerothermo,  airplanes,  launch,  space program  

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Update 8-2-2025:  Please do not misunderstand,  air launch to LEO is possible and in fact has been done more than once!  It's just not easy,  because many of the problems associated with it are hard.  They are hard enough that the attractiveness of this approach is still in question,  relative to the tried-and-true vertical rocket launch. 

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Update 8-4-2025:  For an air launch-to-orbit carrier aircraft,  the gas turbine speed limitation could be gotten around by instead using ramjet propulsion,  which for a true high speed design might reach speeds between Mach 3 and Mach 4 in the stratosphere,  limited mainly by atmospheric drag of something inherently not a “clean” missile shape. 

One would still need the rocket component of a mixed-propulsion parallel-burn scheme to achieve the necessary climb angle at launch of the rocket payload,  and one would still need to solve the dangerous supersonic store separation problem.  But this would get the highest possible speed at launch,  at the right launch angle,  and at an altitude high enough to be beneficial.

The downside is that ramjet has no static thrust!  You will need some sort of booster to reach ramjet takeover speed,  and the necessary high-speed ramjet design is going to have a takeover speed in the Mach 1.8 to 2.5 range.   Given that rocket is needed to reach the high climb angle at launch,  that same rocket is likely the propulsion needed to reach takeover speed. 

Speeds will be limited by the percentage of frontal blockage area occupied by each of the two propulsion systems.  The airbreather is fundamentally lower in frontal thrust density than is the rocket,  so it needs to occupy the larger fraction of the total frontal blockage area. 

Being a lower percentage of vehicle frontal blockage area than the ~100% of a “clean” missile design,  the max possible speed capability of a ramjet (near Mach 6) cannot be reached with this kind of a vehicle.  But the ramjet weighs far less than any possible turbojet propulsion!  That makes a smallish rocket system feasible for getting off the ground with wings,  and reaching Mach 1.8 to 2.5 takeover speed at relatively low altitude. 

From there,  you climb in ramjet to high altitude at speeds near Mach 2.5,  and pull over level to accelerate to top speed in the thin air.  Fire up the rocket to climb steeply for the supersonic store separation,  then shut down the rocket and throttle-back the ramjet to execute a zoom climb and descent back into air dense enough to support controlled flight.  Cruise back in ramjet,  then glide to a landing with the rocket in reserve for go-around capability.

The real trade-off here,  yet to be evaluated,  is whether to integrate the two propulsion systems into some sort of combined-cycle rocket-ramjet,  or leave them as separate systems to be operated entirely separately.  Combined-cycle usually seriously compromises the performance of both components,  while parallel-burn does not,  instead running into the fraction-of-frontal area problem. 

And there is also the problem of there being “no such thing as cooling air” above about Mach 3 to 3.5 in the stratosphere.  Vehicle designs flying faster than that will need one-shot ablatives for their ramjet combustor and nozzle heat protection.  Which means you must swap-out the entire combustor and nozzle unit after every flight!  Given that eventuality,  you could do a solid propellant integral booster in the combustor and nozzle unit,  like a great big JATO motor,  for the initial takeoff.  That reduces the volume (and cross-sectional area) of the on-board propellants for the liquid rockets.  

None of these issues have been resolved for an air launch-to-orbit application. 

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Starship Explosion

Update 7-5-2025:  search code 23062025.

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From AIAA’s “Daily Launch” email newsletter for Monday,  6-23-2025.  This ship was intended for Flight Test 10.  It blew up before they ever ignited the engines.  Quote: 

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SpaceX traces Starship test-stand explosion to failure of pressurized nitrogen tank

By Mike Wall published 3 days ago  (on Space.com)

"Initial analysis indicates the potential failure of a pressurized tank known as a COPV."

 

SpaceX thinks it knows why its newest Starship spacecraft went boom this week.

The 171-foot-tall (52-meter-tall) vehicle exploded on a test stand at SpaceX's Starbase site late Wednesday night (June 18) as the company was preparing to ignite its six Raptor engines in a "static fire" trial.

 

A day later, SpaceX narrowed in on a likely cause.

 

"Initial analysis indicates the potential failure of a pressurized tank known as a COPV, or composite overwrapped pressure vessel, containing gaseous nitrogen in Starship's nosecone area, but the full data review is ongoing," the company wrote in an update on Thursday (June 19).

 

"There is no commonality between the COPVs used on Starship and SpaceX's Falcon rockets," the company added. So, launches of the workhorse Falcon 9, which has already flown 75 times in 2025, should not be affected.

 

The Starship explosion did not cause any reported injuries; all SpaceX personnel at Starbase are safe, according to the update. People living around the site, which is near the border city of Brownsville, shouldn't be worried about contamination from the incident, SpaceX said.

"Previous independent tests conducted on materials inside Starship, including toxicity analyses, confirm they pose no chemical, biological, or toxicological risks," the company wrote. "SpaceX is coordinating with local, state, and federal agencies, as appropriate, on matters concerning environmental and safety impacts."

That said, the explosion did damage the area around the test stand, which is at Starbase's Massey site (not the orbital launch mount area, from which Starship lifts off).

"The explosion ignited several fires at the test site which remains clear of personnel and will be assessed once it has been determined to be safe to approach," SpaceX wrote in the update. "Individuals should not attempt to approach the area while safing operations continue."

Wednesday night's explosion occurred during preparations for Starship's 10th flight test, which SpaceX had hoped to launch by the end of the month. (Static fires are common prelaunch tests, performed to ensure that engines are ready to fly.) That timeline will now shift to the right, though it's not clear at the moment by how much.

The incident was the latest in a series of setbacks for Starship upper stages. SpaceX lost the vehicle — also known as Ship — on the last three Starship flight tests, which launched in January, March and May of this year.

Starship's first stage, called Super Heavy, has a better track record of late. For example, on Flight 7 and Flight 8, the huge booster successfully returned to Starbase, where it was caught by the launch tower's "chopstick" arms as planned.

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My take:  if the description “in the nosecone” for the location of the COPV is correct,  then it is located very close to the oxygen header tank (as the version 1 with 1200 metric tons propellant capacity was laid out),  which is also in the nose of the vehicle,  ahead of the “cargo bay” area.  Such a COPV explosion would easily rupture that oxygen header tank.  Compressed gases drive great explosive violence (with shrapnel) when such vessels burst.  See Figure 1.

Figure 1 – Inboard Profile of Starship Version 1

There would seem to be an oxygen header tank transfer piping line down the windward “belly” of the cargo bay section,  based on descriptions I have read.  In the explosion slow-motion video,  the cargo bay splits open through its heat shield,  right where that transfer line supposedly is,  with gush of something white (not fire) bursting through,  followed immediately by an explosion engulfing about the top half of the vehicle,  and a second or so later by a second explosion seemingly centered lower down.    

The main propellant tanks below the cargo bay would be the main methane tank forward,  with the methane header tank located inside,  at the base of that tank,  and finally the main oxygen tank,  just ahead of the engine bay.  The upgraded version 2 has a bigger propellant capacity,  but should be laid out similarly.  

I would hazard the guess that the COPV explosion and bursting oxygen header tank somehow put a large force on the transfer line,  which split open the belly at the cargo bay,  allowing liquid (and vapor) oxygen out through that split,  as well as releasing a few tons of liquid oxygen to fall down on top of the main methane tank. 

My guess is that spilled header oxygen and vented methane vapors are much of the first explosion.  Bear in mind that the impact of a few tons of liquid oxygen on the top of the main methane tank would rupture it as well,  adding some fuel to that first explosion pulse.  That first explosion pulse would massively rupture the main methane tank,  and also likely the main oxygen tank below it.  That’s the second pulse of the explosion,  which was larger and longer,  reflecting the larger mass of reactants. 

All of that scenario is just an educated guess on my part. 

As for the nitrogen tank,  said to be a “COPV”,  or “composite overwrapped pressure vessel”,  maybe that is not the right choice this early in the flight test program.  Such a design is a metal shell that is simply too thin to hold the pressure,  overwrapped by a yarn or fabric-reinforced composite material,  to bring it up to strength at a lighter weight. 

Here’s the problem:  no composite material has a large plastic (post-yield) strain capability.  If the COPV over-pressures for any reason whatsoever,  failure will be sudden,  without any warning!  Maybe a heavier all-metal nitrogen tank,  one with much more plastic strain capability,  would be a better choice until the other bugs all get worked out.  At least you could see it stretch before it explodes.  You do not want to fly even experimentally,  with too many possible failure modes!  See Figure 2.  

Figure 2 – Stress-Strain Curves for Low and High Plastic Strain Capability

Lots of things look good “on paper”,  but there are a lot of other things to worry about,  many of which cannot be put on that paper.  This is where the “older hands”,  with many years of school-of-hard-knocks experiences,  can be effectively very much wiser than youngsters fresh out of school.  SpaceX has no “old hands” on its staff:  they hire no one over about age 40 or 45.  There’s no gray heads visible anywhere in that organization.

PS:  note in Figure 1 the "typical" inert mass of the Starship Version 1 upper stage,  as 120 metric tons.  I do not have a figure for the Version 2 inert mass,  but it simply cannot be very much different from Version 1!  They are all built the same way.  The hearsay bandied about on the internet,  about 80 tons,  or even less,  is simply BS!