Bounding Calculations for SSTO Concepts

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

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

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

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

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

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

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

Figure 1 – Overall Results for SSTO Bounding Calculations

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

References:

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

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

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

6-20-23               TSTO Launch Fundamentals

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

10-1-22               Rocket Engine Calculations

2-9-21                 Rocket Vehicle Performance Spreadsheet

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

11-12-18             How Propulsion Nozzles Work