Monday, December 17, 2018

The "Train Wreck" Nears in DC


I saw for George H. W. Bush's state funeral more crowd turn-out and "to-do" than I have seen for any other state funeral,  since the JFK assassination.  It would appear that many regular Americans,  regardless of their politics,  really do know somewhere in their hearts what kind of a person makes a fine President.  


Mr. Bush's kindness and humility,  his knowledge of how to govern,  and his prioritization of good-of-the-country over party (or personal) advantage,  seem quite well known.  The contrast between Bush 41 and Trump 45 could not be more stark!  


Those three contrasting points are three very good reasons not to have elected Mr. Trump to the Presidency in the first place.  However,  egged-on by Russian propaganda spread mostly by social media,  a big chunk of the US electorate became disenchanted with both parties,  and voted for him instead. 

They voted massively in the primaries so that Trump clinched the Republican nomination.  His supporters hoped to send him to Washington to disrupt the “establishment”.  Enough of these disenchanted Trump supporters voted for him in rural America,  to send him to the White House.  And they got their wish:  he has disrupted everything he comes in contact with. 

They have also begun to pay the price for his trade wars and tariffs.  So,  there has to be a limit to how far their support for him,  which is still rabid,  will go.  The rest of us are appalled by the encouragement he provides to racist groups in their demonstrations-gone-violent in the streets (evocative of early 1930’s Germany),  and the obviously racist nature of his southern border policies. 

“Truth” for Mr. Trump quite demonstrably has nothing to do with the factualness of anything,  it only has to do with what benefits his image among his supporters.  It has only taken not-quite-two-years for his own words and statements to reveal publicly that he is an egregiously-chronic liar,  and that essentially 100% of what he claims is false.

Now,  the various ongoing federal and state investigations have revealed certain patterns.  More is to come,  as all of these are not done,  yet.  There seem to be some serious campaign finance violations related to the hush money paid to his mistresses.  There may be obstruction of justice associated with his firing of James Comey,  and more obstruction of justice related to the way he has since repeatedly threatened the Mueller investigation.  We’ll soon see,  in that case.

There may,  or may not,  have been collusion between his campaign and the Russians,  since Mueller has yet to report.  But things do not look good on that score for Mr. Trump,  as it would appear that (among many things) the infamous Steele Dossier has slowly been proven true,  point by point.  If there was provable collusion,  that is indeed a very,  very serious crime.  So is a US President being vulnerable to Putin’s control,  by extortion. 

In my opinion,  no one of these is grounds,  all by itself,  for impeachment.  Taken together,  they are indeed very good grounds.  Plus,  there is also one other thing no one talks about anymore:  treason of the “aid and comfort to the enemy” type.  Mr. Trump has very publicly damaged our relations with our allies,  and thus weakened our alliances,  very particularly NATO. 

One former high security official indeed termed his behavior at Helsinki “treasonous”,  and lost his clearance for it,  simple retaliation by Mr. Trump.  No one since has used that word in public.  It is my opinion they should have,  and should still.  Loudly.

Damaging or weakening the NATO alliance is something the Russians (under one or the other government) have tried and failed to accomplish for about 7 decades.  Mr. Trump did it for them in about a year and a half,  while cozying up to Russia’s Putin,  among other evil dictators. 

There is no doubt that Putin’s Russia is a foe (synonym for “enemy”).  Millions (if not billions) have witnessed on television Mr. Trump’s words and deeds that weakened the NATO alliance.  This seems to meet the Constitutional definition of treason (as “aid and comfort to the enemy”) and far exceeds its witness requirements (only two).

Treason is,  all by itself,  a very serious impeachable offense;  that no one can deny.  Even if Mr. Trump has not quite actually committed treason,  he has skirted close enough to it,  for that close approach to be an impeachable offense.  So,  I strongly recommend we add treason,  or at least near-treason,  to the impeachment grounds list.

Returning to Mueller’s investigation,  it is my unsupported belief that his probe is very nearly completed (and his report likely mostly written,  at least in draft form),  as evidenced by the recent flurry of indictments and convictions of Trump underlings.  This probe has been quite professional,  as evidenced by the utter lack of leaks from Mueller’s team,  unlike most activities in Washington,  DC. 

A professional effort like that will produce a whole lot (that we have yet to see) that is provable to courtroom standards,  and a lot more still to consider,  that is perhaps not so provable at the courtroom standard,  but still relevant to an impeachment process.  All of this will be in Mueller’s report to Congress.  Nothing short of this would be of professional quality.

Now,  both the House and Senate are currently controlled by the Republican Party,  in turn demonstrably dominated by its extreme right wing of rabid Trump supporters.  Mr. Mueller would be foolish to present his report to that Congress,  as the Republicans in control would just bury it (most likely by security classification) to protect Mr. Trump “at all costs”.  And so,  the public would never learn its real contents. 

Later in January,  control of the House will switch to the Democrats.  I believe Mr. Mueller is waiting until after that happens,  so that his report actually cannot be buried and kept from the public.  (That is what I would do,  if it were my investigation,  so as to make sure the public learns the real truth.)  That report-to-Congress event and subsequent public revelations is when the storm will break!

It is a majority of the House that passes Articles of Impeachment,  more or less equivalent to a grand jury indictment in the courts.  The case is tried in the Senate,  with the Chief Justice of the Supreme Court presiding,  not the Vice President,  who usually presides over Senate activities.  It takes a two-thirds majority of Senators to convict. 

After the Mueller report storm breaks,  Senate conviction will be far more likely,  due to the inevitable public outcry.  This is quite distinct from most Senate actions now,  as controlled by Mitch McConnell (who demonstrably values party advantage above the good of the Republic,  but that’s another subject for some other time and place). 

It has been recently reported that Mr. Trump is privately concerned about being impeached,  despite his public statements to the contrary.  I just gave you multiple very good reasons for his concern. 

Once the Mueller storm breaks,  somewhere around the end of January or early February (if I am right),  I urge you to contact your Representatives,  and especially your Senators,  about following-through on this urgent matter (my own have been pointedly silent in reply so far,  and I will not forget that silence).  Our country has suffered more than enough damage at the hands of Mr. Trump.


Monday, November 12, 2018

How Propulsion Nozzles Work

This article applies to anything that thrusts from a subsonic chamber through (at least) a sonic throat.  It is intended to give readers a means to compute accurate and realistic thrusts.  This plus a knowledge of chamber characteristic velocity c* is sufficient to do very elementary rocket ballistics.

Update 11-16-18:  see stuff added at the very end,  below,  past the original figures.  

Most (but not all) nozzles that have a sonic throat also have a supersonic expansion bell.  Scramjet is excluded as being without a sonic throat:  the feed to the nozzle inlet is already supersonic,  and there is no contraction in flow area to a throat. 

Rockets of any type are typically high pressure ratio PR chamber-to-exit,  and high area ratio AR exit-to-throat.  These can be ablatively cooled,  or actively liquid-cooled.

Gas turbine engine nozzles are typically low pressure ratio PR chamber-to-exit,  and low area ratio AR exit-to-throat.  These are usually air-cooled,  and variable geometry:  anything from convergent-only to a mild supersonic expansion bell.  Lower turbine inlet temperatures require lean mixtures and cooler flames,  making air cooling possible,  as long as the air itself isn’t too hot.  That high speed air heat effect limits the flight speeds achievable with gas turbine engines.

Ramjet engine nozzles are typically low pressure ratio PR chamber-to-exit,  and low area ratio AR exit-to-throat.  Modern missile designs are usually ablative.  Some of the oldest designs were air-cooled,  similar to gas turbines,  but this approach is severely limiting in a modern ramjet design,  which can run far richer,  and at far higher flight speeds,  where the air itself is far hotter.   

Fundamentals

Conservation of mass:  the same massflow exists throughout the nozzle (any air cooling bleed effects or other injections or leaks are ignored,  if any exist at all).

Conservation of momentum:  a control volume drawn about the rocket engine is pierced by the exit stream exactly at its exit area,  and the momentum of the propellant feeds are either inconsequential,  or perpendicular to the thrust axis,  or they come from tanks inside the control volume.  This could be any combination of those situations,  or even all three.  Balancing stream momentum and the pressure forces against a restraining force,  leads to evaluating the thrust.

Conservation of energy:  the drop (from chamber to exit) in enthalpy,  as measured by the drop in static temperature,  equals the increase in kinetic energy of the stream,  with essentially zero kinetic energy inside the chamber.  There is an accompanying drop in static pressure,  in an amount defined by the ideal gas assumptions and the corresponding equation of state.  See Figure 1.  All figures are at the end.

We use enthalpy “h” instead of internal energy “u”,  because it includes the effects of pressure change upon energy content,  and internal energy does not.  Enthalpy difference Δh is essentially the temperature difference ΔT,  multiplied by the specific heat at constant pressure cp.  (Internal energy change uses the specific heat at constant volume cv.)

Book-keeping:  this is done the easiest way in Mach number-pressure-temperature variables,  instead of the primitive variables,  as long as the ideal gas assumption applies.  That last means we may use as our equation of state P = ρ R T,  and we may use as the change in enthalpy Δh = cp ΔT. 

In this book-keeping scheme,  we make good use of total (or stagnation) pressures Pt and temperatures Tt.  Assuming no appreciable friction losses,  flow is “isentropic”,  meaning total pressure and total temperature are constant through the nozzle,  a very good assumption in almost every conceivable case. 

The ratio of specific heats γ = cp/cv becomes a very useful value to relate totals to statics.  At a location where the Mach number is M,  the total/static temperature ratio TR = 1 + 0.5*(γ – 1) M2,   and the total/static pressure ratio is PR = TRexp,  where exp = γ / (γ – 1).    

The streamtube area model is more complicated than the simple mass conservation-derived relation in incompressible flow,  and is based off of sonic conditions at the throat area At.  If you know the Mach number M at another station where the area is A,  you can find that area ratio AR = A/At as easily as the total/static ratios TR and PR.  If you know instead the area ratio A/At,  finding the Mach number M is inherently a transcendental (iterative) solution: 

                A/At = (1 / M) [TR / const1]const2
where TR is defined as above,  const1 = 0.5 (γ + 1),  and const2 = 0.5 (γ + 1) / (γ – 1)

Heat transfer:  this is driven not by static temperature but by recovery temperature!   We must do this because the supersonic flow in the nozzle is both highly compressible,  and highly dissipative.  At any given Mach number M,  recovery temperature Tr is very nearly the same as total temperature Tt.  How it varies depends upon laminar versus turbulent flow,  and the gas property Prandtl number Pr:

                Tr = T + r (Tt – T)  where the recovery factor r = Pr0.5 laminar,  Pr0.33 turbulent

Only the heat transfer film coefficient h responds significantly to the varying Mach number,  pressure,  and temperature down the nozzle profile.  It does this in a very empirical way.  Multiple models exist for this,  not covered here.  The local heat flux at any station is of the form:

                Q/A = h (Tr – Ts) where Ts is the material surface temperature

For heat transfer purposes,  failing real data,  you can estimate Prandtl number Pr = 4 γ /(9γ – 5).

Conventional Nozzle Thrust Coefficient CF

Your ideal gas model of the gas flowing through the nozzle comprises its constant specific heat ratio γ,  and its constant molecular weight MW.  These can come from thermochemistry calculations,  and need to reflect the high temperatures involved. 

The thrust F of an idealized nozzle evaluated at its exit plane is the momentum of the exiting gas,  plus the exit area Ae times the difference in pressure between the exiting stream static pressure Pe and the ambient backpressure of the surroundings Pb.  Ideally,  all the streamlines are parallel to the thrust axis.  

In the real world,  they are not.   This streamlines-off-angle effect is modeled with the nozzle kinetic energy efficiency factor ηke.  It applies to the momentum term in thrust,  but not the pressure term,  as long as the exit plane is perpendicular to the axis.

                F = ηke m Ve + (Pe – Pb) Ae where m = mass flow rate and Ve = exit velocity

To convert this to compressible flow variables,  we make use of the m = ρe Ae Ve massflow definition,  the ideal gas equation of state Pe = ρe R Te with R = Runiv/MW,  and the exit plane speed of sound for an ideal gas ce = (γ gc R Te)0.5.  The variable gc is the “gravity constant” to make the equation consistent with inconsistent mass and force units.  If those units are consistent,  gc will be 1.

                F = ηke (Pe / R Te) Ve2 Ae + (Pe – Pb)Ae using massflow,  then equation of state
                F = γ ηke Pe Ae Me2 + (Pe – Pb)Ae  using speed of sound

Note that the first term in the equation just above is the momentum term,  and the second term is the static pressure difference term.  Distribute the Ae so that there are 3 separate terms,  and regroup. 

                F = Pe Ae [1 + γ ηke Me2] – Pb Ae  recombining terms such that Pe Ae factors out

Here,  inside the bracket,  the 1 now corresponds to the exit static pressure term with Pe Ae factored out,  and the γ ηke Me2 corresponds to the momentum term with Pe Ae factored out.  The backpressure effect is still a separate force term,  with the recombined bracket-containing force term really just being thrust into vacuum.

Now we introduce the definition of thrust coefficient CF = F / Pc At with the understanding that the Pc is the total (stagnation) pressure feeding the nozzle.  If the contraction from chamber to throat area is large enough,  there is no measurable difference between total and static pressure at the nozzle entrance.

                CF = F / Pc At = (Pe Ae / Pc At)[1 + γ ηke Me2] – (Pb Ae / Pc At)
                CF = (Pe/Pc)(Ae/At)[1 + γ ηke Me2] – (Pb / Pc)(Ae / At)    regrouping P’s and A’s together
                CF = (AR / PR) [1 + γ ηke Me2] – AR / PRop     (the thrust coefficient equation)
    with PRop = Pc/Pb using the actual design Pc and Pb
                with PR = Pc/Pe = (1 + 0.5 (γ – 1) Me2)exp = TRexp   where exp = γ / (γ – 1)
                and AR = Ae/At = (1/Me)[ TR/const1]const2
    with const1 = 0.5 (γ + 1) and const2 = 0.5 (γ + 1) / (γ – 1)

This last formulation is particularly convenient when one wants a certain exit Mach number Me,  because AR = Ae/At and PR = Pc/Pe are easily calculated from Me using the ideal gas γ.  Otherwise,  if conditions at a certain AR are desired,  one iteratively tries Me values until the desired AR obtains,  then computes PR.  Essentially,  Me and Pc/Pe are “locked in” by the AR value regardless of the value of Pc,  although they are not most conveniently figured in that order. 

The “operating pressure ratio” PRop = Pc/Pb depends directly upon your design choices for Pc and Pb.  One had to choose a Pc to do the thermochemistry,  and Pb is set by the altitude,  or else 0 if vacuum.

Once γ,  Me,  PR,  AR,  and PRop are all known,  evaluating CF is easy,  per the above equation.  If you have used a value of c* to size a throat At elsewhere in your fundamental ballistics,  then the nozzle thrust is easily obtained as F = CF Pc At.  From ballistics,  choked nozzle massflow w = Pc CD At gc / c*,   see Figure 2 below.  CD is the nozzle throat’s discharge coefficient (or efficiency).

If all the hot gas generated in the engine workings upstream of the nozzle entrance goes through the nozzle,  then Isp = CF c* / gc.  If not,  you must ratio down your calculated Isp,  F,  and At by 1 + f,  where f is the fraction of generated hot gas massflow that does not go through the nozzle.

Example Problem:  Conventional Nozzle,  Sea Level and 20 kft Designs

I automated these calculations into a spreadsheet,  and verified the numbers with hand calculations.  An image of the spreadsheet for the sea level design is given in Figure 3.  In the spreadsheet,  items highlighted yellow are the user inputs,  and items highlighted blue are the principal outputs from the sizing calculations.  These are used to generate the performance table versus altitude,  which is not highlighted. 

For this example,  I assumed Pc = 1800 psia,  and a conical nozzle of 15 degree half angle.  I used specific heat ratio γ = 1.20,  and a c* = 5900 ft/sec so that specific impulse would be near 300 sec,  similar to LOX-RP1.  I used At = 1.0 square inch,  with a nozzle CD = 0.99 to size flow rate.  The resulting design is a nominal 3000 lb thrust design,  completely immune to backpressure-induced separation,  since it is never over-expanded.  How the nozzle kinetic energy efficiency is calculated from half angle is discussed below.

Keeping all the data the same except for the design backpressure,  I ran the spreadsheet again for perfect expansion at 20 kft instead of sea level.  The effect is to increase the expansion ratio for a higher momentum term,  and then accept the negative pressure difference term reducing thrust below 20 kft altitude.  The gas generating chamber and throat are exactly the same.  An image of the 20 kft design spreadsheet is given as Figure 4.  The spreadsheet includes a separation backpressure estimate (see that discussion just below),  which shows the risk starts at pressures about 9 psi larger than sea level air pressure.  So,  this design is also very likely immune to backpressure-induced separation risks.

Flow Separation Risks

These can only be estimated empirically.  There are many correlations.  My preferred one uses the inverse of PR = Pc/Pe.  Psep is the estimated backpressure,  at and above which nozzle flow separation is to be expected.  It is empirical,  and it is a rough estimate.  The designer should allow significant margin. 

                Psep / Pc = (1.5 Pe/Pc)0.8333

For the 20 kft design example just above,  Pc/Pe = 266.3,  so that Pe/Pc = 0.003755.  Thus Psep/Pc = 0.013355,  and for Pc = 1800 psia,  the expected Psep = 24.04 psia,  quite a margin above sea level pressure.  We can conclude that there is no risk of separation in the example nozzle,  all the way down to sea level,  where Pb is only 14.7 psia.  The risky backpressure is even higher at about 45 psia for the sea level design.

KE-Efficiency Correlations

Most conventional nozzles are axisymmetric.  Those streamlines near the axis are aligned along that axis,  so that the cosine factor for off-angle alignment is cos(0o) = 1.00.  Those near the nozzle wall are aligned at the angle of that wall off the axis.  For a conical nozzle,  this is the half angle of the cone.  The cosine factor for off-angle alignment is cos(a) where “a” is the half angle of the cone.  See Figure 1 again.

Thus,  there is a distribution of local off-axis alignments for the streamlines across the exit plane.  While the “correct” way to determine the effective cosine factor for the distribution would be to integrate them for an average,  there is an easier model that is just as good.  Simply compute the arithmetic average of the centerline cosine factor value (1.00) and the wall cosine factor value cos(a),  and call that the nozzle kinetic efficiency factor:

                ηke = 0.5 [1 + cos(a)]  where “a” is the effective average half-angle of the nozzle wall

For a conical nozzle,  “a” is the cone’s geometric half-angle.  At 15 degrees,  ηke = 0.983.  For a curved bell,  there is a local “a” near the throat,  and a smaller local “a” at the exit lip.  One simply averages the two local a’s,  and uses that average as “a” in the kinetic energy efficiency formula.  For most practical curved bell designs,  that average “a” won’t be far from 15 degrees.  See again Figure 1.

Free-Expansion Designs By “Last Point of Contact = Perpendicular Exit Plane Model”

There are multiple techniques and geometries by which a nozzle can be made self-compensating for perfect expansion at any altitude backpressure.  They all share two features:  (1) a free streamtube surface unconfined by a physical shell before the “exit plane”,  and (2) a point of last contact with physical structure that is wetted by the propulsion stream that locates the exit plane.   We want the components of the actual distribution of exhaust velocities,  that are aligned with the engine axis. 

Most,  if not all,  these free-expansion designs can be analyzed for expected performance using the very same ideal gas compressible flow techniques used for conventional nozzles.  It is just that the order in which things need to be done is revised.  Note that the very same gas-generating chamber and throat area serves as the feed to the free-expansion “nozzle” at all values of Pb. 

Conceptually,  we are interested in an effective planar exit area located at the “point of last contact” (just as the exit lip is the “last point of contact” with conventional bell nozzles), and oriented perpendicular to the engine axis.  This is shown in Figure 5. 

Unlike conventional nozzles,  these are always perfectly expanded,  so that Pe = Pb,  as long as Pb is not exactly zero!  Once a Pb is known,  then PR = Pc/Pe = Pc/Pb is known.  One solves the PR equation for Me at this value of PR,  which is not a transcendental iteration,  just a simple direct solution:

                Me =  { 2/(γ – 1) [PR(γ-1)/γ – 1]}0.5

With Me now known,  find the area ratio from the streamtube relation,  and use it with the throat area to find the effective value of the exit area Ae:

                TR = 1+ 0.5 (γ – 1) Me2
                const1 = 0.5 (γ + 1)
                const2 = 0.5 (γ + 1) / (γ – 1)
                AR = (1 / Me) [TR / const1]const2
                Ae = AR At

Referring again to Figure 5,  there is obviously a distribution of streamline directions at the exit plane,  which is different for each backpressure.  Each geometry is different,  but the idea is to find the largest half-angle off of axial and use it as “a”.  This goes into the correlation for kinetic energy efficiency.  That correlation is generally for “a” < 30 degrees,  so we are misusing this here!  But,  it is the best I have at this time to offer.  Any such “a”-dependent model,  even if flawed,  is better than no model at all!

                ηke = 0.5 [1 + cos(a)]

Instead of a thrust coefficient,  we estimate thrust directly from the calculated exit plane conditions,  remembering that Pe = Pb,  and from that thrust,  the thrust coefficient (to use with c* for Isp):

                F = ηke γ Pe Ae Me2
                CF = F / Pc At

One should note that neither Ae nor ηke are constants here,  as Pb changes.  At high backpressures (low altitudes),  “a” is small,  ηke is high,  and Ae and Me are lower.  At low backpressures (high altitudes),  “a” is quite large,  ηke is lower,  and Ae and Me are high.  Exactly how “a” varies is quite geometry-dependent. 

If Pb = 0 (vacuum of space),  PR = infinite,  leading to infinite Me and Ae.  There can be no planar exit plane,  and Prandtl-Meyer expansion says “a” > 90 degrees by a small amount.  There is no point trying to use this compressible flow analysis technique on a free-expansion nozzle in vacuum,  quite unlike a conventional nozzle!  (Which means this free-expansion design approach is inappropriate in vacuum!)

However,  for an axisymmetric center-spike design (aerospike nozzle),  one could estimate a = tan-1[(Re-Rt)/Lspike].  For this,  Re = (Ae/pi)0.5,  and Rt = approximately (At/pi)0.5.  Lspike is the distance from throat plane to exit plane.  Longer is lower effective “a”,  but higher weight,  and a tougher cooling design.

I made another worksheet in the spreadsheet for axisymmetric aerospike nozzles,  embodying the above calculation techniques,  and I verified it with hand calculations.  It lays out differently,  since the sequence is different,  and more items vary with altitude.  The same grouping of design point data vs altitude performance is maintained,  and the same color-coding for highlighted items.   However,  the volume of data is larger,  requiring two figures (vs one) to display herein.

Example Axisymmetric Aerospike Problem

The fairest way to compare this type of nozzle design with any conventional nozzle design is to size both with the same Pc,  At,  and γ.  If thrust is the issue,  and it usually is for launch vehicles,  then the preferred performance variable to examine is thrust. 

For the example problem,  we use Pc = 1800 psia,  At = 1.0 in2,  and γ = 1.20,  same as the conventional nozzle examples earlier.  The same c* and nozzle throat CD are used.  In effect,  this engine shares the very same gas generator as the two conventional examples.  The same altitude backpressures are also used,  so that this design can be compared directly to the earlier examples,  except that vacuum performance cannot be included.

The spreadsheet results are given in Figures 6 and 7 below.  The two figures together provide the image of the spreadsheet.  I have repeated the altitude data in Figure 7 for convenience. 

Comparisons Among the Example Nozzle Designs

How these designs compare,  especially as regards altitude performance,  does not “jump off the page” from tabular data.  That takes plots,  something this spreadsheet software offers.  I used the same altitudes and air pressure data for all 3 examples.  Copying selected data from each worksheet into yet another worksheet provides a way to directly plot performance from all 3 nozzles on the same page.  I did this for thrust,  specific impulse,  thrust coefficient,  and nozzle kinetic energy efficiency. 

Bear in mind that all three share the same gas generator at Pc = 1800 psia,  At = 1 square inch,  γ = 1.20,  chamber c* = 5900 ft/sec,  and nozzle throat discharge coefficient CD = 0.99.  All three are roughly the same 3000 lb thrust at their design points,  within a percentage point or three. 

The thrust comparison is given in Figure 8 below.  The conventional sea level design has slightly better thrust at sea level ( by about 82 lb out of a nominal 3000 lb) than the conventional 20 kft design.  This reflects the effects of the negative pressure difference term at sea level,  for the slightly-overexpanded 20 kft design. 

The 20 kft design has about a 107 lb thrust advantage,  above 100 kft,  over the sea level design.  This reflects the larger expansion ratio of the 20 kft design,  and the fact that the exit momentum term dominates by far over the pressure difference term,  in thrust.

The axisymmetric aerospike design is “right in there” with the other two,  up to about 50 kft or 60 kft altitude.  Then its performance drops dramatically with increasing altitude,  something the free expansion is supposed to compensate!  It is a little better than the conventional sea level design at sea level,  and it remains superior all the way up to about 55 kft.  It is equivalent or very slightly better to the conventional 20 kft design at sea level,  and remains essentially equivalent to about 20 kft.  Its downturn in thrust performance is quite dramatic,  and starts at about 40 kft or 50 kft. 

It should not surprise anyone that the specific impulse trends in Figure 9 tell the same tale as the thrust in Figure 8,  since all three share the same gas generator with the same propellant massflow.  Nor should it surprise anyone that the thrust coefficient trends in Figure 10 also tell exactly the same tale,  since all 3 designs share the same gas generator operating at the same chamber pressure. 

The reason for the dropoff in aerospike performance,  versus the conventional designs,  traces directly to the trends of nozzle kinetic energy efficiency,  something that in turn depends upon the effective average half-angle of the propulsion stream bondary.  This is really nothing but the cosine factors of streamlines that are aligned off-axis.  Kinetic energy efficiency trends are given in Figure 11.

Remember,  for the conventional designs,  half-angle is locked-in by the physical bell,  right up to the exit plane.  Downstream of the exit lip,  gas expands laterally into the vacuum,  but this happens downstream of the “last point of contact”,  where thrust is actually calculated.  This is implied by how we draw the control volume about the engine and nozzle,  something shown in the lower right corner of Figure 1,  touching at that last point of contact.

For the axisymmetric aerospike free-expansion design,  the last point of contact is the tip of the spike.  The free expansion surface of the plume is inside the control volume,   as is the bell of the conventional nozzle.  At high altitudes where the air pressure is low,  the plume boundary must expand quite far laterally,  between the throat,  and the “exit plane” at the last point of contact.  This is precisely how large AR and Me are achieved,  in order to match Pe = Pb.  Since the length of the free-expansion zone is fixed,  the boundary half-angle must be quite large at high AR.  That reduces kinetic energy efficiency. 

The two conventional designs share a constant kinetic energy efficiency of 98.3%,  as shown.  The aerospike starts out slightly better at 99.1% (due to the choice of Lspike used),  but drops below conventional at about 20 kft,  and falls ever more rapidly to only about 77.7% at 100 kft.  This traces directly to the effective half-angle of the plume boundary between the throat,  and the exit plane at last point of contact. 

That is why I included a plot of the axisymmetric aerospike half-angle vs altitude as Figure 12.  Looking at this,  please remember that half-angle is constant-with-altitude at 15 degrees for the two conventional designs.  At 100 kft,  cos(56.335o) = 0.5543.  Averaging that with 1 inherently produces ηke = 77.7%.

Conclusion

I don’t see any significant advantage to the free-expansion nozzle approach.  The small performance improvement is restricted to the lower atmosphere,  and this design approach is entirely inappropriate for use in vacuum!  The complications with cooling the spike outweigh any tangible performance benefits,  which are low (unless you cheat by not accounting for the streamline divergence effects).  

Update 11-16-18:  This conclusion is correct as far as it goes,  but it is also incomplete.  See the update just below the original 12 figures.  I explored the free-expansion nozzle approach a lot further,  in two different forms.  It can be made to work better at high altitudes,  but NOT in vacuum,  and not really significantly better than a conventional nozzle sized at the highest altitude consistent with not flow-separating at sea level.


 Figure 1 – Nozzle Fundamentals



 Figure 2 – Modeling Nozzles with Compressible Flow



 Figure 3 – Spreadsheet Image for 15 Degree Conical Nozzle As Sea Level Design



 Figure 4 – Spreadsheet Image for 15 Degree Conical Nozzle As 20 Kft Design



Figure 5 – Analogous Procedure for Free-Expansion Designs



 Figure 6 – Example Axisymmetric Aerospike Nozzle Results,  Part A




 Figure 7 – Example Axisymmetric Aerospike Nozzle Results,  Part B



Figure 8 – Thrust Comparison Among the 3 Designs vs Altitude



Figure 9 – Specific Impulse Comparison Among the 3 Designs vs Altitude




Figure 10 – Thrust Coefficient Comparison Among the 3 Designs vs Altitude



Figure 11 – Nozzle Kinetic Energy Efficiency Comparison Among the 3 Designs vs Altitude




Figure 12 – Trend of Effective Half-Angle “a” for Axisymmetric Aerospike Design



Update 11-16-18:  
                                                                         
When I did the original article,  I made the axisymmetric aerospike design at the same spike length as the nozzle bell length for the 20 kft conventional nozzle design.  This made its performance comparable to the 20 kft conventional design up to around 40 kft altitude,  then above that,  the ever-increasing streamline divergence angles “killed” its nozzle kinetic energy efficiency,  reducing its performance below conventional,  at the higher altitudes.  

This design approach essentially determined 20 kft as the altitude at which the effective average boundary half-angle of the circular cross-section free-expansion streamtube was 15 degrees,  same as the 15 degree conical conventional nozzles.  The fixed input here is the aerospike expansion surface’s length.  The area ratio (and expanded diameter) are determined by expanding to local pressure.  The difference between exit and throat diameters (ignoring the spike),  divided by the spike length,  is the tangent of the streamtube half-angle,  presuming a conical shape. 

I did not originally check separation pressures for the conventional nozzles,  but added that later.  As it turns out,  for this study’s 1800 psia chamber pressure,  you do not want to attempt conventional designs above about 30 kft.  This is because the margin between expected separation backpressure and sea level pressure gets to be too small to trust.

To the three designs in the original article,  I added a 30 kft conventional design,  and a revised axisymmetric aerospike with a much longer spike that has a 15 degree half angle at 100 kft altitude.  These five designs (are depicted in Figure 13 (all figures at the end of this update).

There is a different idea about the free-expansion aerospike geometry that limits expanded half-angle better:  I call it the “twin aerospike”.  Instead of being axisymmetric with the spike immersed along the plume centerline,  in the twin aerospike,  the nozzle bell is conceptually cut away,  from throat to exit,  top and bottom,  leaving two symmetrically-placed spikes along each side,  at the physical bell half angle.  This is shown in Figure 14.  

I have never before seen a proposal like this;  therefore, it is my idea.  Please give me credit for it,  if you pursue it. 

This alternate approach leads to an elliptical plume cross section,  with one diameter fixed by the position of the twin spikes.  The other diameter is smaller at low altitudes,  and larger at high altitudes.  The effective average boundary half angle is the arithmetic average of the fixed spike angle,  and the variable angle produced by plume diameter,  throat diameter,  and spike length.  The plume cross section is circular at the design point,  where the free expansion half angle matches the physical spike half-angle. 


I ran this twin spike design approach at a spike half-angle same as the conventional bell (15 degrees),  and two spike lengths,  one set by a design altitude the same as the 30 kft conventional nozzle,  the other 100 kft,  same as the added axisymmetric aerospike design.  That makes a total of 7 designs to compare,  as given in the following table:


It is very important to understand what happens to effective average boundary half-angle for these various designs,  in order to understand what happens to nozzle kinetic energy efficiency.  That in turn governs the thrust and impulse performances that can be achieved. 

Accordingly,  the first comparison plot (Figure 15) is effective average boundary half-angle vs altitude for the 7 designs.  Its cosine averaged with 1 is the nozzle kinetic energy efficiency.  That item is plotted vs altitude in the second comparison plot (Figure 16) for all 7 designs.   The other performance measures are thrust,  specific impulse,  and thrust coefficient,  as given in Figures 17, 18,  and 19

Looking at Figure 15,  note that all three conventional designs share exactly the same 15 degree half-angle.  They fall right on top of each other,  so only the 30k design is visible in the plot,  being the last group plotted.

Both of the lower-altitude free-expansion designs show similar upward trends of half-angle to about 70-80 kft,  with only the twin aerospike 30k design trend “bending over” with increasing altitude.  This is the beneficial effect of the elliptical cross-section shape,  with one longitudinal section fixed at 15 degrees,  and averaged with the other that has a variable boundary half angle,  to meet exit area.

The two high-altitude free-expansion designs have the same design altitudes of 100 kft,  and very similar trends of effective average boundary half-angle with increasing altitude.  The differences attribute to the averaging of effective boundary half-angle that takes place in the twin aerospike,  but not in the axisymmetric aerospike. 

I did not have available atmosphere data higher than 200 kft;  indeed such is questionable,  as we are heading into something more like free-molecule flow,  than continuum flow,  at such altitudes.  Had there been atmospheric pressure data available between 200 kft and 300 kft,  the twin 100k design would have “flattened out” the way the twin 30k design did.  Neither axisymmetric design can do that.

What those half-angle data represent is the effective average boundary values whose cosines average with 1 in the nozzle kinetic energy efficiency correlation equation.  Those data for the 7 designs are given in Figure 16.  Note that all 3 conventional designs share a constant half angle of 15 degrees,  and thus a constant nozzle efficiency of 98.3%.  There is not much room above that for the free-expansion designs to do “better” at low altitudes,  although they certainly do. 

However,  above about 30-40 kft,  the two low-altitude free-expansion designs (“axi a 20k” and “twin 30k”) have efficiencies that drop below conventional.  These drop to around 80% efficiency,  although at two different altitudes.  That difference is an artifact of the two design altitudes,  and (more importantly) how the plume cross section forces the averaging (or not) of the effective half angle.  The “twin 30k” design sort-of “bottoms out” around 80% efficient,  while the “axi a 20k” design does not.  We could “eyeball-extrapolate” the “twin 30k” design to an efficiency near 80% at 300 kft (effectively out in space);  the “axi a 20k” design’s efficiency evidently continues to drop.

Designing the free expansion nozzles at higher altitude (getting far longer spikes) is quite evidently a better deal from a fluid mechanics standpoint.  The nozzle efficiencies in Figure 16 show both those designs with similar trends to 200 kft altitude.  An “eyeball guess” says the twin 100k design might “bottom out” somewhere near 70% efficiency at 300 kft,  while the “axi a 100k” design’s efficiency will probably just continue to fall precipitously. 

Free expansion nozzle efficiencies are equal to,  or very slightly better than,  conventional,  all the way up to the design altitude;  and backpressure-induced separation is by definition no risk at all,  all the way down to sea level.  Then,  at altitudes above the design point,  the nozzle efficiencies must inherently fall off,  according to one or the other behavior,  depending upon whether an axisymmetric aerospike or a twin aerospike.  There is an altitude above which free-expansion efficiency is less than conventional.

These half-angle and nozzle efficiency behaviors combine with the fluid mechanics of expansion,  to produce the thrust vs altitude curves of Figure 17These show quite clearly that we do not want the lower-altitude free-expansion designs,  since by about 40 kft,  the conventional nozzles outperform them,  just as I said in the original article.  The 30k twin aerospike design does better than the 20k axisymmetric aerospike design,  mainly because of the half-angle averaging,  but neither maintains equal or better performance than conventional,  past 50 kft “for sure”. 

The two high-altitude free-expansion designs do indeed show equal or slightly superior performance to the conventional designs,  but only up to about 170-180 kft altitudes.  They “peak” at their design altitudes of 100 kft.  The “axi a 100k” design’s performance will fall precipitously past 200 kft,  while the “twin 100k” design’s performance may (or may not) “bottom out” well below conventional performance levels,  at 300 kft. 

This outcome also shows in the delivered specific impulse and thrust coefficient trends of Figures 18 and 19.  That is because “all else really is equal”,  in particular chamber pressure,  throat area,  c* velocity,  and thus nozzle mass flow rate. 

Could we “push” the free-expansion advantage to higher altitudes still?  Probably,  by making the effective average boundary half angle = 15 degrees at higher altitudes than 100 kft.  The cost,  as indicated in Figures 13 and 14,  is very long spikes indeed!  There is no fluid mechanical optimum here! 

But,  increasingly-long spikes are increasingly infeasible from both the constructional,  and the thermo-structural,  viewpoints.  It is that trade-off,  not fluid mechanics,  which determines whether either of the free-expansion design approaches is “better” than a conventional nozzle design.  And,  as shown in Figures 17-19,  the performance advantage of the free-expansion design over a 30 kft conventional design is never very large at all!  Just as the original article conclusion indicated.

Updated Conclusions

The conclusions in the original article are correct,  but a bit incomplete.  Free-expansion designs can NEVER outperform conventional designs in vacuum,  just as originally concluded.  The trends are just wrong to support such a conclusion.   

Designed at sufficiently high altitude,  the performance of free-expansion designs can be made to equal,  or to slightly exceed,  conventional designs up to some fairly-useful altitude in the 200-300 kft range.  But,  there will always be a performance penalty to pay in full vacuum!

This performance gain at some altitudes with free-expansion designs comes at a cost:  an ever-longer expansion spike (or spikes).  These can easily be infeasible for constructional,  or thermo-structural reasons.  Fluid mechanics does not limit this!

Of the two free-expansion designs considered here,  the twin aerospike approach offers somewhat better performance potential than the axisymmetric aerospike.  This is because of the way the fundamentally-elliptic cross-section of the twin aerospike forces the average boundary half-angle to behave,  relative to the fundamentally-circular cross-section of the axisymmetric aerospike. 

Two-dimensional linear aerospike geometries were not included in this study.  However,  their performance characteristics should fall within the bounds of the axisymmetric aerospike and the twin aerospike.  There will be no “breakthroughs” with the 2-D linear form,  relative to the other two. 

Conventional nozzle technology is well-established and has been flying for over a century.  Free-expansion nozzle designs (of any kind) have never been flown up to this time.  They thus cannot be considered a well-established technology,  the failed X-33 program notwithstanding. 

So,  for the time being,  my personal recommendation is just continue with conventional nozzles,  designed to (at most) about 30 kft perfect expansion,  as long as the somewhat lower sea level thrust is tolerable,  relative to a sea level-expanded design.  If not,  reduce the design altitude,  and accept the penalty at high altitudes.  Simple.  Effective.  Well-proven.  “KISS”,  which means “Keep It Simple,  Stupid”.  


 Figure 13 – Pertinent Dimensions and Conditions for Conventional and Axisymmetric Aerospike Designs


 Figure 14 – Pertinent Dimensions and Conditions for Twin Aerospike Designs


Figure 15 -- Comparison of Effective Half-Angle vs Altitude for the 7 Designs

 Figure 16 -- Comparison of Nozzle Kinetic Energy Efficiency vs Altitude for the 7 Designs

 Figure 17 – Comparison of Thrust vs Altitude for the 7 Designs


 Figure 18 -- Comparison of Specific Impulse vs Altitude for the 7 Designs



Figure 19 -- Comparison of Thrust Coefficient vs Altitude for the 7 Designs