Getting To Low Earth Orbit and Back

There are two pictures and a lot of notations in the figure.  Near center is a plot of altitude versus velocity,  similar in format to a standard flight envelope presentation,  except that the values extend to low circular Earth orbit.  Top right is a diagram of Earth with low circular orbit at 300 km shown,  plus a transfer ellipse that grazes the surface at its perigee,  and grazes low circular at its apogee. 

Typical non-lifting vertical launch pretty much reaches the vicinity of the transfer orbit at ~90 km altitudes at low drag loss,  and accelerates onto it exo-atmospherically at no drag, by about 200 km altitude,  give or take.  From there,  the vehicle loses a little speed coasting to apogee at circular orbit altitude,  where a small burn speeds it back up to circularize.  

Typical entries start with a small deorbit burn onto the transfer orbit,  where significant entry forces begin at the “entry interface” altitude of 140 km.  From there,  peak heating precedes the peak deceleration,  which finally ends at the end-of-hypersonics point near Mach 3 speed and 35-40 km altitude,  as indicated on the altitude-speed plot.  This was figured for an Apollo capsule.  Be aware that the “circularize” and “deorbit” points need not be co-located around the circular orbit. 

Also shown is a winged lifting spaceplane ascent trajectory to orbit.  This trajectory is limited above by too little lift with a practical wing area size,  and by heating too intense to endure below.  It is essentially entry flown in reverse!  The point reached on this trajectory by the fastest of the X-15 rocket plane flights is also shown on the altitude-speed plot.   If your vehicle takes off horizontally,  it will inherently have to use this entry-in-reverse ascent trajectory,  incurring devastatingly-huge drag losses.

Vertically-launched non-lifting vehicles endure only very modest ascent heating,  with windblast forces being just as important for payload protection.  They must only endure high heating during entry,  where orbital-class speeds still exist at much lower altitudes.  In contrast,  the winged lifting ascent vehicle must endure entry-class heating (and drag) on that ascent,  and on the descent.   Its heat shielding must endure two,  not just one,  entry-level heating episode per flight,  which means it will “wear out” twice as fast!

This is why I strongly recommend that winged spaceplane designs be vertically launched on non-lifting trajectories!  

Why Vote For He Who Will Hurt You?

I found this on LinkedIn.  It’s too good not to re-post here.  While it is pointless trying to change the minds of Trump-cult believers,  perhaps there are non-Trump-cult Republicans,  and independents like me,  who might be willing to listen.

I would only add one thing that was left out of the image.  This is a man so hungry for power (and trying to stay out of jail) that he would incite an insurrection in an attempt to overthrow his own government,  in order to become a dictator-for-life.  He would do this by using martial law “to quell the violence” as the excuse.  He will do that again,  or whatever else he thinks is needed,  to become that dictator-for-life (and stay out of jail),  if given the chance!

Please do NOT vote for this person!  Do please vote,  just NOT for him!  We had enough chaos the first time around!

 

Drugged Spiders?

My wife found this somewhere on Facebook.  It was too funny not to repost here. 

While intellectually I know it is likely that different species will respond to drugs in different ways,  it was still startling to see caffeine apparently disrupt web-spinning,  while speed and marijuana apparently do not! 

If this is real research being done somewhere,  it is a contender for the Ig Nobel!

Starship/Superheavy Flight Test 5, 13 October 2024

Note:  a version of this article appeared as a board-of-contributors column in the Waco "Tribune-Herald" for Tuesday,  15 October,  2024.

--------------------------------------

I found out after the fact that the test took place this morning.  I watched the SpaceX videos to find out what happened.  While not perfect,  the test was a resounding success! 

The launch was normal with all 33 Raptors working in the Superheavy booster stage.  Hot staging was successful,  and they seemed to keep control of the propellant ullage problem by running 3 booster engines all during the staging event and the flip-around.  The Starship upper stage pulled away on all 6 of its Raptor engines successfully.  Staging took place at approximately 67 km altitude and 5200 km/hr (1.44 km/s) speed (which is actually a bit less speed than I expected to see).

The booster successfully flew back to the South Texas “Starbase” facility,  and successfully made the landing burns (initially 13 engines,  finally 3 engines) and the tower catch,  which was utterly amazing to see!   I did see the methane plume from one vent burning with air along one side of the stage near its base.  That vented material continued to burn for some time after the landing.  (They might consider adding a water spray on the tower to put such fires out.)

The Starship upper stage successfully made the same kind of almost-an-orbit suborbital trajectory,  to come down in the ocean on the other side of the world.  There is no need for a deorbit burn on this trajectory:  entry is automatic.  The video was astonishingly good.  I saw no visible plasma effects at the nominal entry interface altitude 140 km.  Speed was somewhere around 27,000 km/hr (7.5 km/s) at this point,  although I did not recover the speed data on the screen.

I saw a visible plasma glow under the tail and portside aft flap,  starting at about 102 km altitude and at a speed of about 26,727 km/hr (7.43 km/s).  The announcer said the flaps were in control of vehicle attitude at about 85 km altitude and 26,720 km/hr (7.42 km/s).  I started to see the speed readout begin dropping at a noticeable rate (indicating significant deceleration beginning) at about 75 km altitude and 26,350 km/hr (7.32 km/s).  The vehicle is generating lift at about 60 degrees angle of attack,  which shallows the descent angle and makes the entry process longer in time. 

When the announcer said peak heating occurred,  the ship was at about 70 km altitude,  and about 25,500 km/hr (7.08 km/s) speed.  This always occurs before the “max dynamic pressure” (or max deceleration gees) point.  I never heard the announcer say where max dynamic pressure occurred.  But I saw one of the flaps develop a hinge line burn-through!  I’m not sure which one,  there were 4 views of 4 flaps,  the other 3 were unlabeled as to which they were.  That was at roughly 45 km altitude and 9500 km/hr (2.64 km/s),  probably substantially after the max dynamic pressure point.  There is still very significant heating going on,  just not the maximum amount. 

According to the announcer,  the ship was down to about Mach 2,  which I read off the screen as about 25 km altitude and 1400 km/hr (0.39 km/s) speed.  He indicated the ship was in the subsonic “belly-flop”,  for which I read the screen as 3 km altitude and 400 km/hr (0.11 km/s) speed. 

The ship fired up its 3 sea level Raptors successfully,  and flipped tail first rather quickly,  at very low altitude (apparently per plan),  and touched down on the ocean in the proper attitude (nose high).  It hit the target zone,  and a camera on one of the target zone buoys recorded a huge steam cloud obscuring everything,  then a fiery explosion.  Apparently the ship broke up and exploded when it tipped over onto the water.  There was burning cylindrical wreckage visible,  sticking up out of the water at about a 45 degree angle.  The last recorded speed,  which I took to be speed at touchdown,  was 8 km/hr  (2.2 m/s).

All in all,  this was an astonishingly successful test flight.  Kudos to SpaceX,  they did good,  really good!  And,  they are doing things no one has done before,  and accomplishing them faster than anyone has a right to expect.  Well done!

The heat shielding at the flap hinge lines obviously still needs some improvement (see figure).  I do believe it is time to try landing and recovering the Starship on land,  and it is time to try doing Raptor restart burns in space.  Solve those issues,  and they are ready to attempt propellant transfer tests “for real”.

Update 10-15-2024:  Here is the simplest notion I could come up with,  for a “fix” to stop the hinge line burn-through problems.   It’s not a reusable part,  you would install these on the 4 flaps before every launch.  But these are not large items;  they would not “break the bank” on launch costs.  My materials selections are just a start point.  But the final product might be very similar to my sketch.

Elliptic Capture

This concept applies to either arrivals or departures from a planetary body.  Arrival and departure would take place at the periapsis of an extended elliptic orbit about the planet,  where the orbit speed is closest to any required arrival or departure spacecraft speed,  all measured with respect to the planet.  The min and max radii of the ellipse determine both its shape and its speed distribution.

Figure 1 shows everything you need to compute from min and max distance any of the elliptic orbit parameters,  given a mass and radius for the central body,  and a value for the universal gravitation constant.  Figure 2 shows a numerical example computed for a nominal 4-day extended ellipse about the Earth,  with a perigee altitude of 300 km,  to match “typical” low circular Earth orbit conditions.

From low circular orbit,  the ideal dV required to get onto the co-planar ellipse from circular orbit would be the ellipse perigee speed minus low circular orbit speed.  Any interplanetary trajectory has a “near-the-Earth” speed requirement that is always beyond escape speed,  sometimes significantly.  To get onto that interplanetary trajectory from the extended ellipse,  the ideal dV is trajectory speed minus perigee speed,  a smaller number.  To get onto that same trajectory directly from low circular orbit,  the ideal dV would be near-Earth trajectory speed minus circular orbit speed,  which is a much larger number!  Same applies to arrivals,  just reversed in direction.

To get a perigee speed that is close to escape at perigee altitude,  the ellipse has to be quite extended.  Its apogee distance will generally be well outside the geosynchronous distance,  and it will usually fall outside the outer Van Allen radiation belt as well.  But such an extended ellipse will transit the two major Van Allen belts twice each passage,  both outbound,  and both inbound.  (Note that the geosynchronous distance actually does fall within the outer Van Allen belt.)

Figure 1 – All the “How-To” for Elliptic Capture Orbits Anywhere

Figure 2 – An Elliptic Capture Orbit of 4 Day Period About the Earth

These dV values are ideal in the sense that they are astronomically-derived values.  Depending upon whether the propulsion is “impulsive” or not,  these may need to factored-up higher for the mass ratio-effective values necessary to use the rocket equation for sizing stages or vehicles.  This allows only for gravitational losses,  since there are no drag losses outside of the atmosphere.  If the propulsion is “impulsive”,  a factor f = 1 may be used.  For electric propulsion as we currently know it,  use a factor f between 1.5 and 2.  I prefer the more conservative 2,  some others recommend only 1.5.

The value I use for the universal gravitation constant is G = 6.6732 x 10^-11 N-m²/kg².   Values for planetary body masses M and radii are given in the table just below.  The orbit equations work in N, kg,  m,  and sec units of measure.  However,  it is customary to show distances in km not m,  and speeds in km/s not m/s.  The values for G and planet M “go together”,  in the sense that it is their combined effect that determines orbit characteristics.  Do not mix values from different sources!

Figuring interplanetary speed requirements is out-of-scope here.  Suffice it to say that differences V_far between body and spacecraft with respect to the sun must be corrected for 3-body attraction as distances close,  which is then the V_near needed at departure.  V_near² = V_far² + V_esc²,  escape figured at periapsis,  not the surface.  The magnitudes are accurate,  but there is no direction information!

More Detail

Note that elliptic capture (or departure) takes place at the extended elliptic orbit’s periapsis,  where speed is the highest and closest to arrival and departure speed requirements.  It does not take place at the ellipse’s apoapsis,  where speeds with respect to the planet are lowest,  and very far indeed from interplanetary arrival or departure speeds (as measured with respect to the planet).

Note also that for the extended ellipse,  the periapsis speed is very close to the escape speed at that altitude.  That means there is a large dV required to get onto the ellipse from circular orbit,  very nearly the difference between escape speed and circular orbit speed (usually near 30% of escape).  This also applies going back to circular from the ellipse,  just reversed in direction.  To do this,  the burns must take place at periapsis,  not apoapsis.  That is part of the fundamentals of making changes to orbits: a burn changing the speed at one end,  changes the distance at the other end.

Now,  there is one additional nuance of using an extended elliptical capture orbit,  and that would be to put its periapsis altitude down in the atmosphere,  somewhere between the entry interface altitude (140 km for Earth) and the surface (0 km altitude).  The idea would be to use an aerobraking periapsis pass to decelerate from interplanetary arrival speed (near the planet) to the calculated periapsis speed for the ellipse penetrating the atmosphere.  Instead of departing hyperbolically,  the vehicle then follows an extended ellipse out of the atmosphere,  without making any burn at all.  See Figure 3. 

If nothing else is done,  the vehicle will inevitably enter the atmosphere,  upon returning to periapsis passage,  which is still within the atmosphere.  If instead,  the orbit needs to be modified to move that periapsis above the atmosphere,  that is done with an apoapsis burn (of rather modest magnitude).  But the move to a low circular orbit will still require a periapsis burn of rather large magnitude (roughly 30% of escape),  no matter what.  See Figure 4.

This one aerobraking pass into an extended ellipse that gets modified into a stable orbit with a small apoapsis burn,  is an attractive capture method,  excepting the difficulty of reaching this orbit from either the surface or low orbit,  because the dV to reach it terminates very nearly in escape speed,  not just low orbit speed!  That difference is always inherently on the order of 30% of escape at the periapsis altitude.  If the vehicle so capturing must also contain the propellant to change to a low orbit,  this attractiveness entirely evaporates!  It is actually “cheaper” by the value of the stabilizing apoapsis burn,  to decelerate directly into low orbit from the interplanetary trajectory,  always outside the atmosphere (and thereby eliminating the need for a heat shield).

One way around this dilemma is as follows:  after the initial hard aerobraking pass (which always requires a deep penetration,  or you will NOT get the deceleration!!!),  you adjust periapsis with a small apoapsis burn,  such that the periapsis altitude just barely falls below the entry interface altitude.  That way,  over repeated passes,  the drag deceleration at periapsis reduces apoapsis altitude on each pass,  ultimately toward the desired final circular orbit value.  Then one small burn at that final desired apoapsis altitude pulls your periapsis up out of the atmosphere,  for a stable low circular orbit.  See Figure 5.

Implicit in this process is the requirement to decelerate hard on the first aerobrake pass,  deep in the atmosphere.  This is imposed by a required deceleration dV on the order of a nontrivial fraction of escape,  or else you will not capture at all!  The subsequent apoapsis-lowering decelerations can be much smaller,  not nearly so deep in the atmosphere.  Gentler is more such orbital passes,  over a longer period of time,  though.  It’s a tradeoff.

The hard braking required on that first pass deep in the atmosphere implies that the vehicle must have a very significant heat shield!  You do not get such deceleration amounts unless you go deep in the atmosphere:  there is no “shallow-skimming” that gets you any significant deceleration!  And if you get deceleration,  you WILL get heating!  The peak heating pulse precedes the peak deceleration pulse in all entries. 

Further,  if the vehicle uses multiple shallow passes to adjust apoapsis after the initial deep deceleration pass,  its heat shield must remain serviceable for multiple entries,  each a little less demanding than the preceding one.  The initial one occurs at near-escape speed at periapsis.  As each subsequent shallower pass decreases the apoapsis,  periapsis speed decreases toward circular orbit speed,  which is still quite the demanding entry in terms of heating (at Earth). 

Finally,  in Earth’s atmosphere,  which has repeatable and predictable density vs altitude at entry altitudes,  this multi-pass aerobraking elliptic capture process has much merit. 

But on Mars,  where the high-altitude densities vary through factors of plus-or-minus two (or more),  rather unpredictably,  this aerobraking deceleration process has far less merit!  As you leave the atmosphere on the first pass,  if your speed is still too high,  you will have to burn to decelerate,  lest you not capture at all,  becoming quite literally “lost in space”!  (Mars entry interface is 135 km, Earth’s is 140 km.)

If you have to be prepared to do that,  you might as well avoid the uncertainty and risk,  and just decelerate with a burn directly into some orbit,  all outside the atmosphere.   And not need any heat shield!

Figure 3 – Aerobraking Elliptic Capture With No Burn,  Resulting in a Second-Pass Entry

Figure 4 – Aerobraking Elliptic Capture,  With A Burn To Raise Periapsis Out Of the Atmosphere

Figure 5 – Aerobraking Elliptic Capture With A Burn To Allow Subsequent Braking Passes

Yet More Detail

The aerobraking capture notion (into an extended capture ellipse) is conceptually illustrated in Figure 6.  The simple 2-D Cartesian entry spreadsheet model that I have,  cannot do this analysis,  although a typical entry plot is shown in that figure for a simple direct entry off Hohmann at Mars,  with a big heavy object,  which did use the simple spreadsheet analysis.  The point of including it was to show that peak heating is seen before significant deceleration gees are seen!  And that will always be true,  in any entry! 

In addition,  at the figure’s bottom,  a couple of sketches are provided to conceptually show what an actual aerobraking capture event must entail.  All the dV to capture into the extended ellipse must be obtained in the first pass,  by definition!  Or else it is not capture at all!

The entering object has a significant dV requirement in order to capture,  which must be produced by aerobraking drag,  by definition in this scenario.  To accomplish that requires that the entering object experience significant deceleration gees,  although perhaps not actually the peak possible value in a straight entry.  To obtain those significant levels of deceleration in only the 1 pass,  it must inherently dive rather deep into the atmosphere,  to somewhere near (or even deeper than) where peak convective heating will occur.  (If at Earth or Venus where speeds exceed 10 km/s at entry interface,  there will also be even-larger amounts of plasma radiation heating.) 

If the required deceleration dV is not achieved in the first pass,  the object is literally lost in space,  likely exiting at a speed still above escape,  unless it can quickly burn to make up the difference.   Bear in mind that the extended-ellipse periapsis speed is only very slightly below escape speed at that entry interface altitude,  while the hyperbolic approach speed off Hohmann transfer will be significantly above escape speed at that same altitude.  For faster transfers,  the approach speeds are,  in point of fact,  well above escape,  easily by as much as 50-60% of escape!

Figure 6 – Summary of the Aerobraking Capture Process Into an Extended Ellipse

Conclusions

1.      (1) Elliptic capture makes sense at Earth where high-altitude densities are reliably predictable.  The “right” process is that in Figure 5,  where one deep pass captures into the ellipse,  with an apogee-raising of the perigee to a higher altitude,  but still in the atmosphere,  where the apogee altitude can be reduced by drag deceleration at perigee in multiple circuits. 

2.     (2)  Elliptic capture does not make sense at Mars,  where high-altitude densities can vary up or down by a factor of 2 or more,  erratically and unpredictably,  in terms of current known science.  If you must be prepared to burn to make up an unpredictably-deficient first pass deceleration,  you might as well just burn directly into orbit,  all outside the atmosphere,  entirely eliminating the need for any heat shield.   The dV is about the same either way.  

3.      (3)  There is no “shallow skimming” of the upper atmosphere to decelerate in one pass into elliptic capture,  without the need for a full entry-capability heat shield.  You either decelerate significantly for capture (and suffer heating) during the initial pass,  or you escape into “lost in space” status,  if you cannot burn to make up the deceleration deficit.  The very significant heating occurs before you ever see any significant deceleration gees,  and you must see significant deceleration gees,  in order to capture at all,  per Figure 6.

4.     (4)  Further:  to use repeated shallow passes to reduce elliptic capture apogee per Figure 5,  your full entry-capability heat shield must be capable of surviving repeated heating episodes,  although only the first pass is the worst.  Materials embrittled upon cooling after the first (deep) pass will fall apart under the influence of pressure forces and re-heating effects,  even during shallow subsequent passes.

Other Final Comments

1.      (1)  Entries off the interplanetary trajectories at Mars (from Earth) take place somewhere in the 5 to 8 km/s range,  depending upon the speed of the interplanetary trajectory.  Only convective heating is significant in this speed range,  which varies roughly proportional to speed at entry interface cubed.  The plasma sheath is more-or-less transparent to infrared,  so that refractory heat shields that re-radiate to cool are feasible,  just as they are at Earth from low Earth orbit,  at about 8 km/s at entry interface (this is not true of entry from extended ellipses about the Earth).

2.    (2)   Entries at Earth and Venus off of interplanetary trajectories take place at speeds in the 12-17 km/s range,  for which plasma radiation heating dominates over convective by far,  varying as some very high exponent equal to or exceeding 6,  of speed at entry interface.  Under these conditions,  the plasma sheath is not at all transparent to infrared re-radiation,  so that only ablatives are feasible,  according to all known technologies.

3.     (3)   The new heat shield concepts based around carbon fabrics on spars (resembling umbrellas),  or around carbon or other materials extended as inflatables,  are NOT reusable!  Per the designs and the testing,  they are only good for one heat exposure!  There is shrinkage,  cracking,  and embrittlement,  in all of those materials,  after only one heating exposure.  Re-heating on a subsequent exposure means a re-exposure to the wind forces,  whereupon the embrittled or shrinkage-cracked materials will fall apart!

 

What Went Wrong?

The Starliner debacle is now past,  with the crew still on the International Space Station,  but the capsule safely landed in New Mexico.  With all the problems that cropped up on this test flight,  NASA thought it best not to risk the crew’s lives riding home in Starliner,  despite their both being former Navy test pilots and former NASA astronauts. 

One must ask the question:  what went wrong?  As near as I can tell,  there were some very serious plumbing problems.  There are two systems involved.  One is the helium gas pressurant that drives the propellants to the attitude control thrusters and maneuvering engines,  and the other is in the delivery lines for the nitrogen tetroxide oxidant those thrusters and engines use.

Those devices use nitrogen tetroxide oxidizer,  and one of the hydrazine fuels,  as propellants that ignite all by themselves upon contact.  Being helium pressure-driven instead of turbo-pumped,  they are extremely-simple rocket designs!  This technology dates back over 6 decades,  to the earliest manned space flights in the very early 1960’s.  From about 1965 forward,  this technology has been very dependable and reliable. 

So,  what went wrong with Starliner?  And why?

One problem was multiple helium leaks.  There was one known before launch,  and more appeared during the flight to the space station,  including one that was rather large.  Closing valves at the supply tanks stops the leaks,  but one must worry about how long the helium supply will last,  when you open those valves to use the thruster systems,  and the leaks resume!   

The other problem was thrusters erratically underperforming.  The trouble was traced to reductions in nitrogen tetroxide flow because of some sort of seals swelling when hot,   and acting to reduce the oxidizer flow.  Thrusters way-underperform when that happens. 

These thruster propellants are very toxic,  and they react quite readily and rather extremely with the materials one uses in the plumbing design.  In particular,  nitrogen tetroxide has a history of this,  even more so than hydrazine!  This is not a new problem,  based on the history,  back to the early 1960’s. 

It was no longer a problem with Apollo or the Space Shuttle.  It is apparently not a current problem with crew Dragon.  But it has once again become a problem with Starliner!  Why?

There are two suspects for this:  one is corporate greed over-riding any ethics,  and the other is an interruption of the passing-on of prior knowledge and history to the current generation of engineering designers.  Both are at work here!

The corporate greed problem traces directly to the anonymity of working in a very large organization.  People in large organizations are effectively separated from face-to-face contact with the people their actions might hurt.  So they will do evils,  as part of the organization,  that they would never do one-on-one,  with members of the public.  That frees the corporate bigwigs to act in any manner that maximizes money made,  so long as they do not get caught violating any laws.

The problem of an interruption in the passing-on of prior knowledge is also related to money!  There is an old saying that I like to quote:   “Rocket science is not really science.  It is only about 40% science,  that being the stuff written down.  It is about 50% art,  that being the stuff never written down,  because no one would pay to have it written down.  And,  it is about 10% blind dumb luck”. 

I would only add that’s in production work;  in development,  the art and luck percentages are even higher!  That engineering art is not taught in schools.  It is passed-on one-on-one,  on-the-job,  from the old hand to the newbie.  Except,  it is only passed-on,  if the company hasn’t gotten rid of all the old hands as too-expensive!

Boeing was once renowned for its high-quality aircraft.  Bombers included the B-17,  the B-29,  the B-47,  and the B-52.  Transports included the B-377/C-97 that saw long service as the KC-97 tanker,  the B-707/720,  the B-727,  the early B-737 models,  the B-747,  the B-757,  the B-767,  and the B-777. 

However,  in recent years,  Boeing transports have become more problematical,  and they don’t make bombers anymore (the last being the B-1B actually built by North American Rockell,  whom Boeing bought).

Most folks are aware of the serious troubles with the late-model B-737MAX series.  They might not be aware of troubles with the B-777-9 certification,  the serious quality issues plaguing B-787 Dreamliner assembly,  and the KC-46 tanker (based on the B-767) that could not effectively serve as a tanker for 2 years after entering service.  We will see how the earlier 737 models work out as the new Navy P-8 patrol plane (update 9-20-2024). 

These troubles (and the SLS rocket that is too expensive to use,  plus the failure-prone Starliner) emerged after Boeing absorbed its last transport competitor,  McDonnell-Douglas,  and a merged  corporate management regime took over.   The new regime bragged about converting the company from a quality product producer,  to one that maximized shareholder value.  They also moved headquarters away from where engineering was done,  to Chicago (update 9-20-2024),  finally to Virginia,  to better lobby Congress.

All this shows in the stock price history plotted over time.  Growth in stock price mostly surged after the new regime took over,  punctuated primarily  by the Covid-induced downturn that affected everybody.  Clearly,  money matters,  but lives do not,  not anymore!  And that simply cannot be tolerated in a manufacturer of aircraft,  or spacecraft!

There are some regulators who need to account for letting this happen!

Rocket Equation-Based Launch Vehicle Analyses

The rocket equation (as used in hand calculations) is shown in Figure 1.  It inherently goes with a vehicle weight statement,  also shown,  in both mass and mass-fraction form.  Whenever more than one weight statement applies,  you may only do one rocket equation calculation for each weight statement.  Velocity increments may be summed only if they occur within one weight statement!

You have a garbage-in, garbage-out (GIGO) problem with rocket equation answers,  if that weight statement is not realistic!  The propellant mass fraction and vehicle mass ratio are utterly intrinsic to each other,  as the illustrated derivation confirms.  The calculated delta-vee (dV) capability requirement must be sufficient to cover the required mission plus any gravity and drag losses.

Figure 1 – Basic Rocket Equation Items

The basics for Earth launch eastward to low (about 300 km) circular orbit at low inclination are shown in Figure 2 below.  Launch is vertical,  upward against gravity,  but rapidly bends over in a non-lifting thrusted gravity turn.  Zeroing lift minimizes the drag.  The trajectory leaves the sensible atmosphere at a still-modest supersonic speed,  with only a slight upward angle “a” above the local horizontal.  If a two-stage-to-orbit (TSTO) vehicle,  this region is where staging usually occurs. 

For high-inclination orbits,  the dV should be increased by the eastward speed of the Earth’s rotation at the launch site.  For westward launch,   add two of those rotation speeds.  Launch from other bodies is similar,  except that the numbers are different,  including those for the empirical fractions from which gravity and drag losses are figured.  Ratio those Earth values,  by the ratios of the body’s surface gravity and “air” density to Earth standard values,  for a first approximation.

In addition to determining dV capability from the rocket equation,  one must also size the thrust of the engines in each stage.  This is very important,  as the thrust required is determined from the stage ignition mass,  allowing for the number of engines the stage must have.  Engine mass affects the inert mass and inert mass fraction in the weight statement!  Design iteration will be involved!

At launch,  you want a vehicle thrust/weight ratio at or above 1.5,  in order not to burn so much propellant only just above the launch pad,  where speed is still slow.  Otherwise,  you need to use a bigger gravity loss fraction to figure the gravity loss you must cover with your dV capability.  Effectively,  you want the net vertical acceleration above gravity to be half a standard gee or more,  to use your propellant more efficiently.  This is empirical,  but it has long-proven to be necessary.

Similarly,  if your vehicle is two stages,  there is a similar empirical acceleration requirement at stage 2 ignition,  in order to use your propellant more efficiently.  You want half a standard gee net above the vector component of gravity along your flight path.  The simple equation is in the figure.

Figure 2 – Launch Mission Basic Items

The first figure showed how to obtain the effective exhaust velocity Vex from the published rocket engine specific impulse Isp.  Vex is what Isp would be,  if it were defined in terms of dimensionally-consistent units.  It is not:  lbs of thrust divided by lbm/sec of flow rate,  and dividing-out lb with lbm to obtain seconds,  is not dimensionally consistent!  Nor is kg-f (or metric tons-force) of thrust divided by kg/sec (or metric tons/sec) of flow rate,  and dividing-out kg-f with kg (or metric ton-force with metric ton)  to obtain seconds!  However,  that tradition is too old and well-established to change.  Just deal with it!  In typical metric units,  g_c = 9.80667 m/s².  In US customary,  it is g_c = 32.174 ft/sec².

Clearly one needs a good figure for Isp,  in order to have a realistic value of Vex.  The published textbook data versus propellant combinations is crudely in the ballpark,  but not good enough for actual design!  It is figured for fixed chamber pressures that may not be the ones you want.  It is figured for fixed expansion ratios that are unlikely to be the ones you want.  It is figured for 100% nozzle kinetic energy efficiency which no real nozzle has.  It is figured for zero turbopump drive bleed gas dumped overboard,  which might not be the turbopump drive cycle in your engine.  All these things affect Isp quite strongly,  except nozzle efficiency,  which is usually near 98-99%.

The best Isp estimate comes from doing the engine ballistics yourself,  per Figure 3 below.  The first thing to go look up is the data representing your desired propellant combination.  That would be the chamber characteristic velocity c*,  the mixture ratio r,  and the combusted gas ratio of specific heats γ.  You need a delivered c*,  not a thermochemical theoretical value.  The c* value and its efficiency are weak power functions of chamber pressure,  at around 95-98% efficiency.

The second thing to quantify is the nozzle entrance chamber pressure Pc (and pressure turndown ratio P-TDR) and dumped bleed fraction BF corresponding to the engine and its turbopump drive cycle that you desire to model.  You need to define max,  min,  and an intermediate Pc,  values.

The third thing is to quantify the characteristics of your nozzle: the throat discharge efficiency coefficient C_D,  the two bell half-angles a1 and a2 (if a conical nozzle,  then they are the same,  at the cone half-angle),  the geometric throat area At and the geometric exit area Ae,  plus the ambient atmospheric pressure Pa.  C_D is effective throat flow area divided by geometric throat area.  It is usually in the vicinity of 98-99.5%,  if the profile into,  through,  and beyond the throat is smooth and well-radiused.  The profile radius to throat diameter ratio should be about 1,  with a short throat flat.

The fourth thing is to do the compressible flow nozzle analysis,  which is usually best done iteratively in a spreadsheet,  the details of which are not given here.  That analysis determines the average expansion bell half angle “a”,  the nozzle kinetic energy efficiency η_KE,  and the expanded Mach number Me and pressure ratio Pe/Pc,  using the expansion area ratio Ae/At and the specific heat ratio γ.  From those the thrust coefficient C_F is determined as shown. This includes the vacuum thrust coefficient C_Fvac,  and the backpressure correction term,  which is zero,  if out in vacuum.

The fifth (and final) thing is the estimation overall engine performance values.  Thrust down in the atmosphere is computed from chamber pressure Pc,  nozzle throat area At,  and the thrust coefficient C_F as shown.  The relevant specific impulse for the rocket equation must use the propellant flow rate actually drawn from the tanks w_tot.  Isp can be computed either way,  as shown:  from thrust and flow rate,  or from C_F and c* plus a couple of other factors.  For an estimate of vacuum performance,  use instead C_Fvac as your thrust coefficient,  and figure the values with it.  In that way,  sea level and vacuum performance are very easily and quickly estimated. 

Not shown in the figure is something easy to generate,  if the nozzle expansion is analyzed with a spreadsheet.  By means of a standard table of atmospheric pressure values,  versus a list of altitudes from sea level to space,  it is quite easy to obtain thrust and Isp performance versus altitude,  and even to plot it.  Calculate the constant vacuum thrust and subtract from it the varying backpressure correction term.  Doing that kind of thing reliably may require estimating the values of backpressure that cause bell flow separation!  The entirely-empirical equation for that is not in a figure,  instead it is given here:

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

That being said,  optimizing a nozzle is a separate topic in and of itself,  not shown here.  Suffice it to say that a sea level-optimized nozzle would have Pe = Pa at the nozzle design value of Pc.  There is no such thing as a “vacuum-optimized” nozzle!  There is only the max exit area ratio that results in engines that will geometrically fit within the base volume of the stage,  allowing for gimballing.  

How to optimize Pc and nozzle expansion for best ascent-averaged Isp is a separate topic,  not illustrated here.   By sacrificing some sea level thrust and Isp,  vacuum thrust and Isp can be improved over the vacuum performance of a standard sea level design.  In effect,  the nozzle is “designed” at a higher altitude than sea level,  and operates over-expanded at sea level,  but very definitely not so over-expanded that it risks separation at sea level, for useful chamber pressures.

The next separate topic not shown is exactly how to estimate ascent-averaged Isp.  Some simply arithmetically-average sea level and vacuum Isp,  but that is really a lower bound,  not a proper and representative average.  The “right” value would have to come from a trajectory simulation,  which defeats the purpose of simple “by-hand” estimates!  The procedure recommended here would be to compute performance versus altitude in a table in the spreadsheet-based nozzle analysis,  and simply average all the values in the table.   It’s not “right”,  but it really is fairly close.

Figure 3 – The Ballistics of Liquid Rocket Engine Performance

The vehicle inert mass buildup is absolutely crucial to getting a reliable weight statement,  so that the rocket equation result is also reliable.  This is complicated by the fact that mass ratio is related to mass fractions as well as masses,  but you have to start somewhere.  It is further complicated by the fact that you need the propellant mass in order to estimate the inert mass of the tanks that hold it,  and you need the ignition mass to estimate the stage thrust,  from which you estimate the inert mass of the engines that produce it.  There are also interstage rings and skirts to estimate,  as well as fixed equipment items like guidance and control hardware,  and the electrical power source that runs it.    All of this is indicated in Figure 4.

Figure 4 – Dealing With Vehicle Inert Mass Items

Usually,  the analyst is interested in delivering a certain payload mass to orbit.  Guess a payload mass fraction,  and a stage inert mass fraction,  and compute the propellant fraction from them as propellant fraction = 1 – payload fraction – inert fraction.  Convert the propellant fraction to mass ratio MR as shown in the first figure above. 

See if the resulting dV covers the mission requirements;  if not,  adjust payload fraction (and perhaps also the Isp) until it does.  Then use those fractions and that payload to run the actual masses in the weight statement,  and use those to estimate the inert mass buildup,  and the resulting inert fraction.  Keep iterating until the inert fraction as used by the rocket equation agrees with the inert fraction from directly estimating inert masses of tanks,  engines,  equipment,  and the various interstages and skirts.  The payload mass and its mass fraction give you the ignition mass.  That and the fractions for propellant and inerts give you those masses.  Always do the check sums to see if you did the arithmetic right.

This weight statement convergence process is very iterative,  and the best way to do it is with a spreadsheet.    See Figure 5.  You have to do two such convergence analyses for a two-stage vehicle.  The payload mass for the first stage is the fully loaded second stage.  Do the second stage first,  because of that linkage,  then do the first stage using that second stage ignition mass result.

Figure 5 – Recommended Procedure to Converge the Stage Weight Statement

There is one final aspect for sizing-out a design concept,  short of actual detailed design.  You need to estimate tank volumes and other geometry,  to estimate a length and a diameter for the vehicle at launch.  As indicated in the second figure above,  this overall geometry impacts the drag loss very significantly.  This is therefore a very important step,  and a point of further design iteration.  Again,  this might best be done in a spreadsheet.

Per Figure 6 below,  split your propellant mass Wp into oxidizer and fuel masses,  then use their specific gravities to turn those masses into volumes.  As a function of diameter,  estimate the tank overall lengths for the stage,  and add them.  Select a number of engines to produce the thrust.  Add in a good guess for the summed length of (1) an interstage with the G&C and power,  (2) a between-tanks interstage,  and (3) another good guess for the engine overall length.  That last can be modeled as twice the expansion bell length,  to cover both the turbopump drive cycle equipment and chamber,  and the gimballing engine mount.  Then compute stage L/D,  and select from the list the diameter that gives the right stage L/D.  

For a single-stage-to-orbit (SSTO) vehicle,  this would be about stage L/D ~ 6 without the payload,  producing a still-larger,  and definitely lower-drag,  L/D with the payload atop the stage. 

For a TSTO,  do the same procedure on the first stage,  using a desired L/D in the 3-4 range.  Then make the second stage the same diameter and find its L/D.  Sum the L/D’s for the two stages together:  you will again want about 6 for the two stages together,  without the payload. 

Figure 6 – Running the Geometry from the Weight Statement

You will have already needed to set a number of engines for each stage,  and to re-scale their dimensions from the ballistics analysis thrust level to the as-sized thrust level.  Dimensions scale proportional to the square root of thrust. 

Once the vehicle is sized,  check to make sure your engines will actually fit the bases of the stages.  You may need to revise vacuum engine expansion ratio to change the exit diameter and bell length.  Or perhaps change the number of engines (and rescale them).  Or both.

The result of doing all of this correctly is the roughed-out sizing of a credible launch vehicle concept,  and good ballpark estimates of its performance and payload capability.  Such is now ready for you to invest in actual detailed design and analysis,  especially if it is the best concept you have evaluated,  among several competing concepts. 

Doing this “up-front” stuff by-hand (assisted by spreadsheets) is the way to reduce the cost of evaluating multiple concepts inexpensively and quickly enough,  to permit raising the odds of success,  by using true brainstorming to generate multiple concepts.  See Figure 7,  which applies to all sorts of engineering development efforts,  not just sizing launch vehicles.  

Figure 7 – General Development Process That Includes Brainstorming Cost-Effectively

Where to Obtain the Spreadsheet Tools

There are three general-application spreadsheet tools that support doing this kind of analysis.  There is a fourth spreadsheet tool that is not general,  but is useful for making oversimplified bounding calculations,  for Earth launch SSTO and TSTO concepts.  See Figure 8.

Figure 8 – Available Spreadsheet Tools and Supporting Materials

The first general tool is a 2-body elliptic orbits calculator that can be used to determine the theoretical velocity requirements of space missions.  Its results must be tempered with some empirical estimates for losses,  in order to determine the mass ratio-effective delta-vee (dV) numbers.  Those loss estimates and that orbit analysis are not covered here.

Another general tool is a rocket engine performance calculator in which the propellant combination is modeled with a specific heat ratio γ,  a chamber c* velocity as a function of chamber pressure,  and an oxidizer/fuel mass flow ratio “r” that is also a function of chamber pressure.  The nozzle is modeled with half-angles for its kinetic energy efficiency,  and a throat area massflow discharge coefficient (or area efficiency).  The nozzle expansion conditions and thrust coefficient are sized from an appropriate chamber pressure level,  and either a design expanded pressure,  or a design expansion area ratio.  The engine cycle is modeled by the throttleable range of chamber pressures,  and the dumped-bleed fraction of turbopump drive gases.

The third general tool is a reentry dynamics and stagnation convective heating calculator that is based on the simple model used by H. Julian Allen for warhead entry back in the 1950’s.  To it,  I have added a stagnation plasma radiation heating estimate.  The atmosphere is modeled by a “scale height” type of simple exponential model for density versus altitude,  and an altitude at which “entry interface” begins at orbital-class speeds.  The entry conditions are speed and path angle at entry interface.  The entering object is modeled by its hypersonic ballistic coefficient and (only for heating) the effective value of its “nose tip” radius. 

The oversimplified rocket vehicle sizing tool for bounding calculations presumes that there is only one weight statement involved,  so that all the dV values may be summed into a simple calculation for each stage.  It does include the sizing of stage thrust values,  which impacts inert masses,  and it does estimate tankage volumes,  which also impact inert masses.  This is done in a way that allows the user to force clean,  low-drag shapes,  of adequate but not excessive slenderness ratio.  But it is also only a bounding calculation,  a good start-point for more investigative design.  The numbers are indicative,  but not really trustworthy.

These tools are available as free downloads by way of the Mars Society’s “New Mars” forums,  as indicated in Figure 9 below.  Those same postings also offer a whole series of course materials as free downloads,  that teach the user how to do these various things,  in greater detail. 

For the four spreadsheet tools discussed here,  the links from the New Mars forums postings are reproduced here:

Item                                                                     name (link below)

2-body orbits spreadsheet                        orbit basics.xlsx https://www.dropbox.com/scl/fi/cxtpsx2ne7ltg77syavxt/orbit-basics-spreadsheet.xlsx?rlkey=w6g4p5et7yacnxrwls5inmkmv&dl=0

 

 

Rocket performance spreadsheet          liquid rockets.xlsx

https://www.dropbox.com/scl/fi/mcmgvky9p1xukj6m6nmqo/20240330-liquid-rockets.xlsx?rlkey=zypo0wegrzzhtudesw1fiqvjk&dl=0

 

 

Rocket user manual document               user man.pdf

https://www.dropbox.com/scl/fi/z01cwbeqesk8fnhy83o0n/20240330-User-manual-liquid-rockets.pdf?rlkey=xuoyzafq4og2tyiev3hnhpi33&dl=0

 

 

Results reporting image                             engine sizing report.png

https://www.dropbox.com/scl/fi/9scws7nh485qs9xaqfwi3/20240330-engine-sizing-report.png?rlkey=pynvmgdkohwb0gixlvrnwb7fi&dl=0

 

 

Entry spreadsheet                                        entry estimates.xlsx

https://www.dropbox.com/scl/fi/9nqdv47z0md8hb7zofa0x/20240427-Entry-Estimates.xlsx?rlkey=0y8df5x9x5var2jg7vp21s5ic&st=8xtf6avf&dl=0

 

Entry user manual document                   user manual revised entry.pdf

https://www.dropbox.com/scl/fi/dw1o8endbjz5txhkm3fir/20240427-User-Manual-Revised-Entry.pdf?rlkey=ueqqmnaodwmtrab8napm1tbr8&st=j2g0c208&dl=0

Oversimplified rocket vehicle sizing tool            launch sizing.xlsx

https://www.dropbox.com/scl/fi/ufyre58dtxp4ly76tcrsq/20240525launch-sizing.xlsx?rlkey=w7cjw0ricawafisvbnv5v8rb2&st=ynb99zms&dl=0 

 

User Manual part 1 of 3                                             user manual launch sizing.pdf

https://www.dropbox.com/scl/fi/tfquke9u7y7bfbljg1z8a/20240525User-Manual-Launch-Sizing.pdf?rlkey=tdd5gr6yr9hzbyxvjl69simpx&st=7uq0t8y5&dl=0

 

User manual part 2 of 3                                            addendum to user manual.pdf

https://www.dropbox.com/scl/fi/kion5sgdclqluj4vbt7rt/20240525Addendum-to-User-Manual.pdf?rlkey=4j05tgxilqsn9goxop5c8uizr&st=z410gpvt&dl=0

 

User manual part 3 of 3                                            addendum2.pdf

https://www.dropbox.com/scl/fi/nxefg4say6icdufjomkg2/20240525Addendum-2.pdf?rlkey=2zw5vnrmnmfrzwe0zrw7xh897&st=suznj4yk&dl=0

Figure 9 – Where to Find Links to the Tools on the New Mars Forums

How to Do the Initial Bounding Analysis for SSTO

One can get a crude approximation to get started,  by doing a rather simple bounding calculation with the rocket equation.  The vehicle sizing / bounding calculation spreadsheet tool can be used for this.  The numbers are not trustworthy for design,  but they are indicative.   Figure 10 shows the sort of inputs that you need.  LOX is liquid oxygen.  LH2 is liquid hydrogen.  LCH4 is liquid methane.  RP1 is rocket-grade kerosene. 

Figure 10 – Data Sources for the SSTO Bounding Analysis

The rocket equation-in-reverse is used to determine the propellant mass fraction,  and from that the “allowance” that you have for the sum of inerts and payload.  Results are given in Figure 11. 

Where the allowance curve is above the inerts-only band,  there is the possibility of carrying payload.  The bands for inerts-only fractions in the plot at lower right are based on miscellaneous modern stages for the expendables,  Falcon cores for the reusable first stages that enter at low speeds,  and good educated guesses for fully reusable items capable of orbital entry and some sort of landing. 

Note that reusable first stage cores do not apply for SSTO bounding purposes,  as they are not orbital entry-capable.  They tend to hit atmosphere at only about Mach 3,  whereas full orbital entry hits atmosphere at about Mach 25.  The entry heating is vastly different!

The figure shows a spreadsheet image and 3 plots made from it,  annotated appropriately.  Required mass ratio MR is shown versus propulsion specific impulse Isp,  which is what most people look at,  but is the least informative.  There is a plot of both propellant mass fraction Wp/Wig and the “allowance” for the sum of inert and payload fractions,  but the scale is inconvenient.  The most informative plot is only “allowance” versus Isp in the lower right,  where bands of inert fraction Winert/Wig at zero payload fraction Wpay/Wig can be spotted upon it,  as well as bands of possible Isp versus propellant combination.  That is the most useful for bounding calculation purposes.

Figure 11 – Results for Initial SSTO Bounding Calculation

What we actually see from the lower-right plot is that an expendable SSTO can be propelled with any of the 3 propellant combinations investigated.  However the payload fraction potential with LOX-RP1 is essentially nil,  and with LOX-LCH4 it is very small (on the order of only 1-2%).  With LOX-LH2,  expendable SSTO payload fraction potential is significant (on the order of 4-7%). 

It is very unlikely that the inert fraction of a reusable SSTO powered by LOX-LH2 could ever be as low as 10%,  but if it were,  the payload potential might be on the order of 1%,  and that only with the highest-possible engine performance. 

It is far more likely that the inert fraction of a reusable LOX-LH2 SSTO would fall in the range of 15-20% (or even more,  depending upon how the landing is to be done,  and whether it is a lifting body or has wings),  which falls above the required allowance curve!  That very clearly precludes the possibility of successful reusable SSTO design even with LOX-LH2 propulsion, and even at zero payload fraction!  Such a design would need something significantly better than chemical rocket propulsion as we know it today (required Isp in the 500-1000 sec range,  or more).

Details:  More About Doing Engine Ballistics

The liquid rockets.xlsx spreadsheet does liquid rocket engine chamber ballistics (from propellant and cycle information) and expansion nozzle compressible flow analysis (to include both sizing the expansion,  and calculating resulting performance).  Results from both topics combine to produce the engine specific impulse values. 

The propellant information includes models for c* and r versus chamber pressure Pc,  where that is at the entrance to the nozzle,  downstream all of the engine cycle features and components.  The pressure feeding-in to those features from the pumps can be,  and most often is,  substantially higher!  Chamber ballistics is primarily determining the throat area and nozzle propellant flow rate to meet a thrust requirement at a certain Pc.  The throat area discharge coefficient C_D is a real-world efficiency factor that is part of the nozzle flow determination. 

There is no need to model all the details of the engine cycle that drives the turbopumps.  This usually involves hot gas taps,  preburners,  and/or stages of combustion at off-design mixture ratios.  All we need to know to estimate engine specific impulse Isp is the fraction of massflow dumped overboard (BF) relative to the massflow drawn from tankage.  The sum of nozzle massflow and dumped massflow equals the massflow drawn from tankage.

The nozzle is a converging-diverging channel for which the diverging portion usually has one conical half-angle near the throat,  and another smaller one at the exit.  These two angles are used to estimate rather closely the kinetic energy efficiency η_KE of the nozzle,  a real world efficiency factor that helps determine thrust coefficient accurately.  It represents the exit area average of the cosine component factors of all the exiting streamline velocity vectors.  The effects of fluid friction are essentially zero compared to this streamline divergence effect,  which is exactly why propulsion nozzle expansions can be modeled quite well with isentropic compressible flow models.

The thrust coefficient provides a very convenient way to directly relate thrust to chamber pressure,  and to hardware size (in the form of geometric throat area).  It is composed of a vacuum thrust coefficient,  less a backpressure correction term.  The vacuum thrust coefficient depends only upon the expansion geometry Ae/At and kinetic energy efficiency η_KE.  The backpressure correction term depends upon the expansion geometry Ae/At,  the chamber pressure Pc,  and the ambient atmospheric pressure Pa.  It is zero,  out in vacuum where Pa = 0 by definition.

Vehicle performance depends upon the thrust,  and vehicle mass depends upon the propellant drawn from the tankage.  So for the rocket equation model to apply,  the engine specific impulse must be based upon the massflow drawn from tankage,  not just the nozzle massflow,  which is smaller if BF is not zero.  In such cases,  specific impulse is higher if computed with the nozzle-only massflow,  something often reported for marketing-hype purposes!  But using it leads to incorrect answers in rocket equation vehicle performance estimates!

This picture is represented conceptually in Figure 12.

Figure 12 – How Liquid Engine Ballistic and Performance Estimates Are Done

The solid rocket has quite different engine ballistics from the liquid,  so the liquid engines.xlsx spreadsheet is entirely inappropriate for calculating their estimates!  Only the nozzle sizing and performance calculations are identical,  which deceives many newcomers to this field.  To analyze vehicles powered by solid rockets,  you need a reliable specific impulse value,  but how you get it is entirely different from the way you get it for a liquid. 

The solid has exactly the same nozzle massflow equation for a given chamber pressure,  which in steady operation must exactly match the massflow coming off the burning surface of the propellant grain (a term for the propellant charge).  That burning surface can vary during the burn,  sometimes drastically.  But the main quandary is that the propellant burn rate is a power function of chamber pressure (for no erosive burning)!  A lesser complicating factor is that the chamber c* velocity used in the nozzle massflow equation is also a weak power function of chamber pressure. 

For no erosive burning,  there is a balance of the nozzle and propellant grain massflows,  with chamber pressure appearing in 3 places,  distributed on both sides of the steady balance equation.  (This gets to be transcendental with erosive burning,  making the direct solution for Pc impossible.) 

Solving this balance for the chamber pressure reveals an exponential sensitivity of equilibrium pressure to variations in any of the factors in the equation.  The higher the burn rate exponent,  the more sensitive this balance is.  Also,  the motor blows up anytime the sum of burn rate exponent and c* exponent equals or exceeds 1!  See Figure 13.

Figure 13 – The Ballistic Balance in a Solid Rocket Motor

The basic message here is that you need a solid rocket analysis code or spreadsheet to properly determine the sizing and performance of a solid motor.  An introduction to what that really looks like is given in the “exrocketman” article “Solid Rocket Analysis”,  posted 16 February 2020.  (That site is http://exrocketman.blogspot.com.)   One can quickly navigate to the article using the archive tool left side of page.   Click on the year,  then the month,  then the title if need be.  In this case,  there is no need to click on the title,  the article was the last thing posted that month, top-of-list. 

This is a very large topic and article!  Note that with solids,  there is no bleed fraction BF and there is no pressure turndown ratio P-TDR;  plus,  the oxidizer/fuel mixture ratio r is meaningless.

The hybrid rocket has quite different engine ballistics from both the liquid and the solid,  so the liquid engines.xlsx spreadsheet is entirely inappropriate for calculating their estimates!  The regression rate for an unoxidized fuel looks like the erosive burn rate correlations for solid propellants.  For an under-oxidized fuel,  it has both the exponential pressure dependence term, and the erosive burning term,  for a two-term regression correlation in 2 different variables. 

Only the nozzle sizing and performance calculations are identical,  which again deceives many newcomers to this field.  To analyze vehicles powered by hybrid rockets,  you need a reliable specific impulse value,  but how you get it is entirely different,  and is not discussed here at all.  Apparently,  only numerical simulations really work at all.

Details:  Analyzing and Optimizing Conventional Fixed-Bell Nozzles

Nozzle sizing and performance for rocket engines is best done with the compressible flow model,  at no gain in entropy (“isentropic flow”),  albeit with empirical corrections for the effective throat area in terms of massflow (discharge coefficient C_D),  and with empirical corrections for streamline divergence effects at the exit area (nozzle kinetic energy efficiency η_KE).  That is how one obtains a reliable figure for the actually-achieved estimated value of thrust. 

For sizing,  there is (1) the expansion area ratio Ae/At and its effects upon expanded Mach number Me and pressure ratio (Pe/Pc or Pc/Pe),  and (2) the effects of hardware size (embodied as throat geometric area At) upon design thrust F at some design chamber pressure Pc.

Nozzle sizing can be done in two fundamentally-different ways,  as indicated in Figure 14.  One is to calculate to a design expanded pressure in the exit plane for a given design chamber pressure,  the other is to calculate to a design expansion area ratio.  There are in turn two different ways to do the design expanded pressure:  either perfect expansion at an elevated altitude,  or incipient flow separation at sea level.  Both generate roughly the same sorts of designs. 

Figure 14 – Calculating Sizing and Performance of Nozzles

It is possible to determine the expanded Mach number Me from the design value of the expanded pressure ratio,  in a closed-form solution,  using Pc/Pe = [1 + (γ – 1) Me²/2]^{γ/(γ – 1)}.  The easiest way to do this is to convert the pressure ratio to a temperature ratio Tc/Te = (Pc/Pe)^{(γ – 1)/γ} = [1 + (γ – 1) Me²/2].  Then solve the temperature ratio equation for the Mach number: Me = [2*(Tc/Te – 1)/(γ – 1)]^0.5 .  The expansion area ratio can then be calculated with that Mach number from the compressible streamtube area ratio equation:  Ae/At = (1/Me)[2*(Tc/Te)/(γ + 1)]^{0.5(γ + 1)/(γ – 1)}.   These are the standard isentropic relations.

It is not possible to determine the expanded Mach number directly from the area ratio as a closed form solution.  The compressible streamtube area ratio equation is transcendental in Mach number,  so that solution is inherently iterative.  That is where software or a spreadsheet can be used with great labor savings.  Once that Mach number is known,  it is used in the pressure ratio equation to determine the exit expanded pressure.

“Sea level-optimized” designs are generally perfectly expanded (Pe = Pa) at sea level,  for the design value of the chamber pressure.  That might (or might not) be the max value the engine is capable of,  but it will be close,  something like 80+% of max.  These designs then have the lowest area expansion ratios Ae/At,  and the least vacuum thrust (where the exhaust is inherently underexpanded,  to that least expansion ratio). 

There is no such thing as a “vacuum-optimized” nozzle design!  Such a thing would have an infinite expansion ratio (and dimensions) in order to reach a zero expanded pressure from any finite chamber pressure!  In the real world,  there is some finite space at the rear of the stage,  into which some appropriate number of “vacuum engines” must actually fit.  That fit behind the stage is what limits the area expansion ratio.

The limitations are both the stage diameter and the length of the space into which these engines must go,  and how that fit is determined also depends on the number of engines selected, allowing for gimballing some,  but perhaps not all,  of the engines.  Ultimately,  this is driven by the thrust required at stage ignition,  the stage diameter to meet fineness ratio requirements,  and the length available between that stage’s aft tank bulkhead and the top of the lower stage.  

My point:  vehicle design requirements determine allowable vacuum engine expansion ratio,  not the engine expansion sizing alone.  “Vacuum” engines have larger expansion ratios,  and better thrust in vacuum,  at the same flow rate and chamber pressure,  than their sea level counterparts.  But in general,  they cannot be fired in the open air at sea level,  due to flow separation in the bell that is induced by too much ambient atmospheric backpressure!  Such separation leads to destruction the bell in a matter of seconds,  due to localized overheating at the location where the shockdown occurs inside the bell.

These sea level and vacuum designs are the endpoints of a possible spectrum of designs that could serve as “ascent engines” between sea level and vacuum (outside the sensible atmosphere).  Such ascent engines could be operated at sea level near (or at) full chamber pressure,  with near-incipient separation,  resulting in an expansion ratio intermediate between the optimized sea level and vacuum designs.  If operated at sea level at low chamber pressure,  separation ensues!  Sea level thrust is less than that of a true sea level design,  because the exhaust is overexpanded,  and suffers a very large backpressure correction term on thrust.  But performance going out into vacuum is much better than a sea level design,  while not quite as good as a “real” vacuum design.

See Figure 15 for typical results,  calculated for engines all sized to the same chamber pressure and massflow rate (different bells fitted to the same power head and throat area).  The baseline powerhead was the sea level engine.  Thrust level for sizing was set such that the same throat diameter and total flow rate was obtained for all 3 engine designs.  The 10 metric ton-force thrust class was simply chosen arbitrarily.  Rescale it at need.  Dimensions vary with square root of thrust.  Flow rates vary with thrust.  Isp does not rescale.

Figure 15 – Comparison of Sea Level,  Ascent,  and Vacuum Designs 

Bear in mind that there are multiple detailed choices for the various parameters,  so that the final answers obtained are not always the same as these initial rough estimates,  although they are always rather close to what is depicted here.  I sized the sea level and ascent designs at max chamber pressure for this study. 

80% max Pc might have been a better choice,  especially for a higher-technology design at higher max Pc.  It is easier to ignite and quickly ramp up to something like 80% power,  than it is to ignite and quickly ramp up all the way to full power.  Full power starts have been a “no-no” for many decades now,  starting with the engines in Von Braun’s V-2 rockets. 

Regardless,  the trend is clear regarding good ascent engines for first stages and SSTO designs:  you sacrifice a little thrust and Isp at sea level for much better thrust and Isp out in vacuum,  compared to a real sea level design.  The vacuum performance of the ascent engine design is still less than the performance of the “real” vacuum design,  but it is a lot closer than the sea level design ever could be.  The ascent-averaged Isp of the ascent design is rather close to its vacuum level,  and not all that far from the Isp of the typical fit-limited “real” vacuum design.

Case Study:  All-Expendable SSTO

The overall bounding study indicated that only the LOX-LH2 expendable SSTO looks to be competitive with an all-expendable TSTO.  Accordingly,  I came up with a modest-technology LOX-LH2 ascent engine sizing to support such a design traceably,  using the “r noz alt mod” worksheet in the “liquid rockets.xlsx” spreadsheet.  See Figure 16.  I then used the “launch sizing.xlsx” spreadsheet’s “SSTO exp” and “tank sizing” worksheets to size a vehicle of the correct fineness ratio,  complete with a total thrust specification at liftoff.  See Figure 17.  

Figure 16 – Engine Sizing Report for LOX-LH2 Ascent Engines of Modest Technology

Figure 17 – Vehicle Sizing For an Expendable LOX-LH2 SSTO

Case Study:  All-Expendable TSTO

The overall bounding study indicated that the upper stage of an all-expendable TSTO should be powered with modest-technology vacuum LOX-LH2 engines,  while the lower stage could be powered by either modest-technology LOX-LCH4 or LOX-RP1 ascent engines.  I chose RP1 to get the benefit of the higher fuel density.   

Accordingly,  I came up with a modest-technology LOX-RP1 ascent engine sizing to support such a design traceably,  using the “r noz alt mod” worksheet in the “liquid rockets.xlsx” spreadsheet.  See Figure 18.  I used the same tool to size a modest technology LOX-LH2 vacuum engine.  See Figure 19.  I then used the “launch sizing.xlsx” spreadsheet’s “TSTO exp” and “tank sizing” worksheets to size a vehicle of the correct fineness ratio,  complete with a total thrust specification at liftoff,  and at second stage ignition.  See Figure 20.  

Figure 18 – Engine Sizing Report for Modest-Technology LOX-RP-1 Ascent Engine

Figure 19 – Engine Sizing Report for Modest-Technology LOX-LH2 Vacuum Engine

Figure 20 – Vehicle Sizing Report for an Expendable LOX-RP1/LOX-LH2 TSTO

Case Study:  Feasibility of Reusable SSTO?

I used the “SSTO reU” worksheet in the “launch sizing.xlsx” spreadsheet to investigate the possibility of a reusable SSTO,  as a lifting body entry craft in order to minimize its likely inert mass fraction.  Lifting body craft will have landing speeds in the 300 mph class.  True winged craft will have landing speeds in the 200 mph class,  but also higher inert mass fractions,  to support wings extended in near-broadside hypersonic flow without them being ripped off.   The worksheet has a means to crudely estimate what those inert mass fractions might be.  Strictly speaking,  it is limited to the high engine thrust/weight ratios of chemical engines,  especially in smaller vehicle sizes. 

The first configuration estimated used the same modest-technology LOX-LH2 ascent engine sizing that was used for the expendable SSTO.  Those results show clear infeasibility,  as the leftover payload fraction is negative,  as indicated in Figure 21. 

The second configuration in Figure 22 is the same lifting body,  just with the ascent-averaged Isp raised (by about 27 sec) to represent a higher-technology LOX-LH2 engine.  It still shows as infeasible,  and by an amount comparable to the modest-technology option.  A little more Isp (maybe 10 s) might be available,  but that clearly will not help. 

The third configuration simply raises the ascent-averaged Isp to a value more-or-less representative of what was achieved by the old NERVA nuclear thermal rocket engine technology,  circa 1974.  This shows to be feasible,  and at an attractive payload mass fraction,  although this performance is likely over-estimated,  since the low thrust/weight ratio of the solid core nuclear engine is not correctly modeled in the inert mass buildup options!  See Figure 23 for the (rather unreliable) numbers.  They really only indicate some level of feasibility,  but not really by how much!

Bear in mind that the NERVA exhaust was radioactive,  so this kind of nuclear technology is not something you would really want to use in a reusable SSTO!  However,  the Isp modeling this,  is in a class not approachable with any available chemical rocket technologies known today!  And THAT is the real message here!

Figure 21 – Bounding Calculation for Modest-Technology LOX-LH2 Reusable SSTO: Infeasible

Figure 22 – Bounding Calculation for High-Technology LOX-LH2 Reusable SSTO Still Infeasible

Figure 23 – Bounding Calculation for NERVA-Based Nuclear Reusable SSTO:  Feasible

About the “Orbits+” Courses:

This document details the how-to that corresponds to certain lessons in the “orbits+” course set,  that is available for free download via the Mars Society’s “New Mars” forums.  Those lessons on vehicle sizing,  engine performance estimation,  and launch (specifically from Earth) correspond to what is discussed here.  This document presumes the reader is already familiar enough with orbital mechanics to know how to estimate the velocity requirements,  that are factored-up to cover losses.  If not,  the first few lessons in the course set,  cover that topic.

By-hand analysis techniques still have a useful role to play in all sorts of engineering disciplines,  not just space launch as is covered here.  The models are simple,  they have few inputs,  and they do not require that a real design with proper drawings has already begun.  The results are less precise,  but they are good enough to tell a good idea from a bad one.  See Figure 24.

What that makes possible is two-fold:  (1) you can quickly and cheaply screen a lot of brainstormed ideas up-front,  enabling you to commit the significant resources of starting a real design,  to only the best one or two ideas,  and (2) if you can obtain approximate by-hand estimates,  you can better recognize garbage-in/garbage-out problems with the computer-generated results of the full-blown design process.  Those two possibilities can save you a ton of grief and money,  over the long run. 

Figure 24 – Where By-Hand Estimates Fit Into the “Big Picture”

Where most people get into this launch stuff is by asking the question “how big a rocket do I need to get to my desired destination?”  The answer is usually obtainable with the classical rocket equation,  but only if you know how to use it properly!  It needs the correct inputs for engine performance and velocity requirements,  and the correct weight statement inputs to represent the various stages or vehicles.  These inputs in part are theoretical (orbital mechanics),  and in part are quite empirical (loss factors,  inert mass buildups,  etc.).   The different topics (among those needed) can lead the student to entirely-different destinations in terms of skills,  as shown in Figure 25. 

Figure 25 – Where Most People Start:  Asking Questions About Rockets

I laid out a series of course lessons aimed at bringing the novice up-to-speed quickly and efficiently,  based on no math more difficult than high school algebra.  Classical orbital mechanics requires multiple presentation lessons to get the basics of the physics and how things interact across to the student,  before the student is ready to attempt solving problems with it.  This is indicated in Figure 26 as lessons 1-3 for the presentations,  and lesson 3B as the corresponding problem-solving session. 

There is a spreadsheet tool that makes running the 2-body orbit numbers far easier in the 3B problem session.  The problems start with solved demo problems to show the student exactly how to do this,  then proceed to similar assigned problems for the student to work.  The solutions to the assigned problems are provided,  in order for the student to self-assess how well he/she did.

Launch is a more empirical topic,  complete with ways to estimate what are called gravity and drag losses.  Those losses have to add to the velocity requirements that size the vehicle. This is covered in lesson 4,  with 4B as the problem-solving session.

The problem of interplanetary transfers is covered for min-energy Hohmann transfers in lessons 5 and 5B,  and for faster transfers in lessons 5.5 and 5.5B.  There are both theoretical orbital mechanics and some empirical real-world effects that are parts of these lessons (such as mid-course correction budgets). The lessons suffixed with B’s are the problem sessions.

Lessons 6 and 6B cover the basics of the entry,  descent,  and landing problem,  to include some very empirical real-world items necessary for doing any of this.  The more detailed ability of estimating entry dynamics and heating is covered in lessons 7 and 7B,  supported by a spreadsheet adaptation of the old by-hand estimates made in the mid-1950’s for warhead entry,  updated.

Lessons 8 and 8B cover the basics of doing custom spreadsheets to link multiple rocket equation burn analyses and weight statements together,  in order to determine estimated vehicle performance (the topic of this article.  There is an oversimplified spreadsheet tool available for bounding calculations,  but the student will,  by this time,  need to do his own custom spreadsheets,  with multiple rocket equation analyses linked together and done in correct order.

Lessons 9 and 9B cover how to create reliable estimates of liquid rocket engine performance from few inputs.  The engine “cycle” that drives the turbopumps is not addressed,  except that its dumped gas bleed fraction is needed to get a reliable specific impulse figure.  Scope is limited to fixed expansion bells,  and some appropriate empirical factors are included.  

Figure 26 – What Are The “Orbits+” Course Offerings

All the course topic areas are covered in the same way,  as indicated in Figure 27.  There is one or more “present the basics” lessons that resemble lectures,  followed by one (or more) problem-solving sessions.  These include working a demo for the students,  followed by assigning similar problems to the students.  Solutions to the student problems are provided for self-check on how well the student did. Each lesson has a course text document,  and an associated slide set for teaching it to others,  plus supporting spreadsheets and user manuals as needed. As indicated in Figure 28,  the course materials can be obtained via links posted on the Mars Society’s New Mars forums.  Direct links to the supporting spreadsheets are located just ahead of Figure 9 above.

Figure 27 – How the Courses Are Structured

Figure 28 – Where Most of the Course Materials Can Be Downloaded For Free

Final Comments

I first did this kind of rocket equation thing for a living,  as a graduate student while taking a summer off,  to work at what was then LTV Aerospace’s Marshall Street Facility,  working in the “Scout” launcher group.  I sized 1-off configurations and advanced launcher designs,  plus I helped with orbital analysis for customer satellite launches.  I did exactly these kinds of hand calculations in order to set up design models of what I wanted to “fly” with LTV’s “NEMAR” launch trajectory code.

This sort of by-hand analysis to set up real design efforts was exactly what I did working for Rocketdyne/Hercules in McGregor,  Texas,  once I left graduate school.  I did this for both rockets and ramjets,  particularly the booster rockets for the ramjets.  Once again,  this sort of thing sets up the actual design efforts,  the exploratory testing,  and the planning of what to pursue next.

I did some of this same by-hand work at what was then Tracor Aerospace in Austin,  Texas,  between two tours at the McGregor rocket plant. I did this kind of thing for both propulsive decoy items,  and for towed decoy items (a different subject not involving the rocket equation).  Again,  it helped screen candidate ideas in a cost-effective fashion,  it helped set up real design efforts,  it helped determine exploratory testing,  and it helped guide the planning. 

I (and all the colleagues around me) started out doing this sort of thing pencil-and-paper,  running numbers with a slide rule,  most of the time.  Only occasionally,  when there was no other way,  we would run a software code model on the computer,  which in those days was an enormous power-consumptive mainframe,  which you communicated with by batch runs of punch cards in a steel tray,  and waited hours before your results ever came back. 

We quickly converted to scientific calculators,  but it was still literally pencil-and-paper work.  Desktop computers were still years in the future,  and spreadsheet software years after that.  But since those times,  I have maintained my abilities to do by-hand engineering design analysis,  assisted now by both calculators and modern spreadsheet software.  The spreadsheets offer a fast way to iterate solutions,  and to plot your data. 

To see what the old style tools once were,  see Figure 29.  Some of you may not know what a slide rule is!  It did what today’s calculators do,  for about 300 years before there was any such thing as a calculator.

Other than that,  I would invite interested persons to visit the New Mars forums site for more than just the free downloads of the stuff I have put together on this rocket vehicle sizing subject.  Poke around,  and see all the various things forums participants have discussed.  Such might be of considerable interest to you.  Consider also becoming a member and participating.  To do so,  contact newmarsmember@gmail.com.  

Figure 29 – Old Style Engineering Tools,  Circa 1960’s and 1970’s