Rocket Nozzle Types

I have corresponded with multiple friends recently over the merits of using an aerospike engine versus a conventional bell nozzle engine for flying from Earth’s surface into low Earth orbit.  I conducted a design analysis using such nozzles,  without pushing the state-of-the-art right to the edge,  and I found even the sea level conventional bell to be much superior in vacuum. 

What I uncovered during my design sizing analyses was:  (1) not only was unconfined streamline divergence a very serious problem for aerospike designs,  as I have maintained for some years now,  but also (2) there is a strong effect favoring lower chamber pressures!  Unless one sharply reduces design chamber pressures,  the streamline divergence problem degenerates into complete infeasibility at very low altitudes indeed.  That chamber pressure reduction has a big negative effect upon the thrust and specific impulse that one can achieve,  including the effects of somewhat lower chamber c* velocity.

I did in fact confirm that my “roughly 60,000 foot altitude” point,  beyond which aerospike performance falls while conventional bell performance does not,   is indeed correct!  While still a rather fuzzy boundary,  my analyses do show poor aerospike performance not far above that critical altitude.

Here Is Where I Started

I had done some nozzle evaluations and published an earlier article on this topic (ref. 1).  The free-expansion designs I evaluated for that article were a twin-spike single-throat approach,  not an annular or linear aerospike,  but the behavior and physics are quite similar.  That is the genesis of Figs. 1 and 2.

Figure 1 – Basic Flow Physics of Conventional Bell Nozzles

Figure 2 – Basic Flow Physics of Aerospike Nozzles and Other Free-Expansion Designs

The fundamental lessons as I initially understood them are:

#1. Conventional bells have inherently-limited streamline divergence effects,  with a fixed (locked-in) momentum term in thrust;  plus an exit pressure term in thrust that differences expanded pressure and ambient atmospheric pressure acting on the fixed exit area.  If the ambient pressure is too high for the expanded pressure,  bell flow separates and “kills” the momentum and pressure terms. 

#2. Free-expansion designs,  including the aerospikes,  have ever-increasing streamline divergence as ambient atmospheric pressure drops,  while the expansion Mach number increases.  This leads to an ever-increasing potential momentum term (and there is no pressure term) in thrust.  However,  the streamline divergence angles quickly lead to low cosine-components of the streamtube momentum vectors in the axial thrust direction.  At higher altitudes,  this divergence inefficiency effect completely overwhelms the larger momentum effect,  with the result that performance actually falls with altitude. 

This Is What I Did

For the conventional bell cases,  I used “typical” chamber c* = 5900 ft/sec for LOX-RP-1 at 1000 psia Pc,  from ref. 2,  a modest modern max Pc = 3000 psia,  a modest pressure turndown ratio (TDR) of 3,  and a massflow bleed fraction of 5% to drive the turbopumps.  I did not change the c* for the min Pc value,  as it is a small effect over that small a Pc range.  The other variables were much more important.   

I used the sea level chamber and throat design as the basic common gas generator,  by forcing the design thrust levels for two higher-altitude designs (60,000 feet and 30,000 feet) to produce the same throat area At and flow rate values as the sea level design.  That produces a very fair comparison,  conceptually just substituting one bell design for another onto the same chamber,  throat,  and pump assembly,  while also operating at the same chamber pressures and propellant flow rates. 

I used a simple empirical equation to estimate separation pressure ratios from the bell’s average half angle.  It works very well for conical nozzles,  and runs slightly conservative with curved bells:

Psep/Pc = (1.5 * Pe/Pc)^0.8333 

For the aerospike nozzle,  I started with that same Pc = 3000 psia gas generator,  and sized low in the stratosphere,  but the streamline divergence effects were infeasibly extreme at 28 x 2 degrees.  The numbers simply made little sense.  I did notice a definite improvement at the min Pc over max Pc! 

Therefore,  I revised the gas generator design to a max Pc = 300 psia,  c* = 5700 ft/sec (reflecting the drastically-lowered pressure range),  TDR = 3 as before,  and the same 5% bleed fraction.  I sized for 100,000 lb thrust at 10,000 feet,  using an aerospike that started at 56 degrees to axial,  to zero the fan angle at design.  That gave me numbers that actually made sense,  and looked very realistic. 

I selected that design point thrust so that the sea level thrust was comparable to the sea level bell design at 100,000 lb.  That gave me flow rates somewhat higher than the fixed-bell designs,  primarily because of the lower c* associated with the order-of-magnitude-lower range of chamber pressures. 

I used the Prandtl-Myer flow model of supersonic expansion around a corner to estimate the flow divergence angles at the edge of the plume,  based on the expected expansion Mach number as determined by the Pc/Pa ratio.  The equations for Prandtl-Meyer expansion come from Ref. 3. 

The orientation of the gas generator throat axis is the same as the slope of the aerospike at its forward end,  so that the attaching stream starts out parallel to the adjacent surface.  If it were to impinge more directly,  that would induce a strong shock wave train on the aerospike surface,  as it turns toward the axis direction,  with corresponding large pressure losses,  disrupting the expansion.

I finally picked a 56 deg x 2 deg shape for the aerospike,  with the ring of thrusters 56 deg off axis at its start.  The Prandtl-Meyer angle gets that 56 deg subtracted off,  because of thruster orientation,  to determine the actual lateral divergence fan angle of the plume relative to the thrust axis.  Below design altitude,  you get negative fan angle data,  because the plume geometrically contracts due to the changing shape of the aerospike.  The end of the aerospike is a small angle whose cosine is always near 1.  Nozzle kinetic energy efficiency is just the average of the cosines of the inner and outer angles. 

At design,  I used straight axial and the aft aerospike angle to calculate the effective nozzle kinetic energy efficiency.  Both below and above design altitudes where the plume edge is off axial,  I used the average of the fan angle cosine and the 2 deg aerospike cosine,  for my effective nozzle kinetic energy efficiency.  This efficiency is just a cosine component correction to the plume momentum. 

I needed a nonzero ambient pressure at 300 kft altitude,  instead of just using zero representing vacuum.  That zero works fine for conventional bells,  but is inappropriate for estimating free-expansion designs.  It drives the expanded area and Mach numbers to infinity.  Accordingly,  I looked up a “standard atmosphere” model in ref. 4 that extended all the way up to 300 kft geometric altitude.  It’s not an exact match to the standard atmosphere table from ref. 2 that I used,  but it’s still “in-the-ballpark”,  and gave me realistic numbers.  The ref. 2 data only extended up to 200 kft.

Here Are The Results I Found

I selected relevant data for comparison of the conventional bell designs,  and arranged those as 3 plots versus altitude on a single figure for each design.  I created the same plots for the aerospike,  but needed a second figure to display the variable expansion data and the streamline divergence data. 

Figure 3 shows the baseline sea level conventional bell,  Figure 4 a bell sized at 60,000 feet as if it were a “vacuum” design,  and Figure 5 a bell sized at 30,000 feet,  representing a “compromise vacuum” design that could actually be static-fired at sea level without separating.  Figures 6 and 7 show the results for the aerospike design,  with 6 showing the same content in the same format as the conventional bells. 

The sea level conventional bell in Figure 3 does not separate at full power or min power,  at any altitude.  The exit pressure term on thrust shows significant effect on thrust coefficient,  thrust,  and specific impulse up to around 60,000 feet,  and almost no effect above that.  The change from sea level to vacuum thrust and specific impulse (Isp) is quite modest,  as would be expected from the very limited expansion available for the fixed momentum term of thrust. 

The same gas generator fitted with a “vacuum” bell (the 60,000 foot design in Figure 4) shows a potential for very significant thrust increase with increasing altitude (due to the pressure term acting on a larger exit area,  along with a larger momentum term).  This obtains up to about 60,000 feet.  There is very little effect from there to vacuum.  The problem is that much of this potential is unrealizable for launch,  due to flow separation in the bell near sea level,  even at max Pc,  as noted in the figure. 

The “compromise vacuum” design sized at 30,000 feet in Figure 5 shows behavior intermediate between the other two extremes.  It has an intermediate momentum term and an intermediate exit area.  The thrust increase with altitude due to the pressure term is realizable at full power,  but not at min power due to flow separation,  as shown in the figure. 

The results for the aerospike design are given in Figures 6 and 7.  More detail is required to understand the expansion and flow divergence phenomena,  which is why Figure 7 is included.  As a reminder,  the data in Figure 6 are the same content and format as that presented for the bell nozzles. 

The thrust coefficient,  thrust,  and specific impulse data increase to peak values in the stratosphere,  then decrease from there into vacuum!  The peak is near 50-60,000 feet at full Pc,  and nearer 80-90,000 feet at min Pc,  so there is a strong pressure effect favoring lower chamber pressures!  That is why I had to reduce the Pc range to 300-to-100 psia,  from 3000-to-1000 psia for the conventional bells.

The main difference between the aerospike and the conventional bell designs is the effective nozzle kinetic energy efficiency data that is shown in the same plots with thrust coefficient versus altitude.  The conventional bells all have constant kinetic energy efficiency,  at a rather high value,  all the way out into vacuum.  This reflects the confined plume following bell angles,  right to the exit point (last point of contact).  The exiting plumes suddenly spread wide,  out in vacuum,  but since that occurs downstream of the last point of contact,  it does not affect the exit flow condition results.

The aerospike plume is unconfined laterally,  and spreads very wide as altitude increases.  This shows up as an initially-high effective nozzle kinetic energy efficiency,  that starts decreasing about 50-60,000 feet.  It falls to drastically-low values as altitude increases into space!  The details in Figure 7 show why:  the fan-out angles get very large,  and simply overwhelm the increasing expansion,  quite rapidly.  

Figure 3 – Analysis Results For Sea Level Bell With Common Gas Generator Design

Figure 4 – Analysis Results For High-Altitude Bell With Common Gas Generator Design

Figure 5 – Analysis Results For Modest-Altitude Bell With Common Gas Generator Design

Figure 6 – Analysis Results For Low-Altitude-Sized Aerospike With Reduced-Pressure Gas Generator

Figure 7 – Analysis Results Details For Aerospike

The only thing that I found that I did not really expect initially,  was just how sensitive the aerospike is to the aggravation of plume spreading effects at higher chamber pressures!  I could not get a feasible design until I reduced the chamber pressures from thousands to only hundreds of psia.  That’s a factor-10 reduction required!  Otherwise,  what I thought going into this analysis turned out to be true.  The list of revised lessons follow (#3 Is the new one,  no change to #1 or #2):

#1. Conventional bells have inherently-limited streamline divergence effects,  with a fixed (locked-in) momentum term in thrust;  plus an exit pressure term in thrust that differences expanded pressure and ambient atmospheric pressure acting on the fixed exit area.  If the ambient pressure is too high for the expanded pressure,  bell flow separates and “kills” the momentum and pressure terms. 

#2. Free-expansion designs,  including the aerospikes,  have ever-increasing streamline divergence as ambient atmospheric pressure drops,  while the expansion Mach number increases.  This leads to an ever-increasing potential momentum term (and there is no pressure term) in thrust.  However,  the streamline divergence angles quickly lead to low cosine-components of the streamtube momentum vectors in the axial thrust direction.  At higher altitudes,  the divergence inefficiency effect completely overwhelms the larger momentum effect,  with the result that performance falls with altitude. 

#3. Aerospikes,  and presumably all the free-expansion designs,  benefit strongly from reducing gas generator chamber pressures by around an order of magnitude below modern rocket practice.  This acts to somewhat-limit the adverse plume spread laterally,  at higher altitudes approaching vacuum.  You want that fan angle to be zero at your design point,  which sets your forward spike and thruster angle.

Final Remarks

Bear in mind that I already know how to optimize the designs of conventional bell nozzles.  I used to do that for a living,  long ago.  I do not yet know how to optimize the design of free-expansion nozzle configurations,  including specifically the aerospike nozzles examined here. 

The aerospike configuration I came up with “worked”,  but can hardly be said to be optimal!  I had to compromise it severely by lowering chamber pressure by a factor of 10 to match up plume boundary expansion effects at the design point,  with the requirement that the plume boundary fan angle be zero.  That also forced a very large initial spike angle and mounting angle for the gas generator chambers adjacent to it.  And it lowers chamber c*,  and thus specific impulse.

Therefore,  do not put much credence in the lower specific impulse I got near design at low altitudes,  lower than with any of the conventional bell nozzles.  That is very likely an artifact of my not knowing how to optimize the design of aerospike engines. 

Put your credence into the strongly-decreasing performance trends at higher altitudes as you fly out into vacuum!  That is real,  and even an optimized design will show a similar trend!  It is inherent that the plume boundary will spread straight out to the side as you fly an aerospike into vacuum,  and it is also inherent that this phenomenon will affect the thrust level that can be achieved. 

The plume inherently spreads laterally precisely because of the physics embodied in Prandtl-Meyer expansion around a corner.   It does not matter if that model needs modification to tailor it to this application or not,  it will still show the same basic plume-spreading trend. 

Because this plume spreading takes place upstream of the last point of stream contact with the engine hardware,  it inevitably must affect the thrust!  It is nothing more than velocity vector component effects at off-axis angles.  That’s just the physics of compressible flow.  No one can argue otherwise.  But it takes place while the expansion is still occurring,  which in turn is what creates the thrust,  which must be measured at that last point of contact. 

As a result,  the effective nozzle kinetic energy efficiency,  the achieved thrust for the flow rate,  and the delivered specific impulse,  will inherently show downward trends as one flies out into space.  Whether I got the exact right numbers is irrelevant.  That downward trend,  and the physics underlying it,  are real!

Aerospike nozzles show excellent fluid mechanical performance from the surface up to the stratosphere,  probably better than with bell nozzles,  if they can be correctly optimized.  But,  the free-expansion nozzles will always show severe performance degradation as you fly from the stratosphere out into vacuum!  It is inherent,  and it is unavoidable.  It is a big effect!

The most important take-away:  aerospike nozzles are simply NOT good vacuum nozzles,  despite what is often claimed.  They inherently cannot be. 

The better application for aerospikes is between the surface and the stratosphere.  That is where the ambient atmospheric pressures are high enough to limit the plume lateral expansion,  which greatly improves the effective nozzle kinetic energy efficiency.   I rather suspect that is true of any free-expansion design approach that lets the plume boundary adjust prior to last point of contact with engine structure. 

The only thing I can think of to investigate further is to add some nozzle expansion past the sonic throat of the gas generator chambers,  in an effort to limit the Prandtl-Meyer fan expansion effect to lesser values,  at least initially.  This might also allow an increase in chamber pressure,  that being a lesser effect per Figure 7 above.  About the largest expansion to add would be a sea level expansion.  This does raise the risk of compression shocks on the spike,  as the Mach number at impingement is higher. 

Aside

As an aside,  the aerospike nozzle is in fairly wide use in some aircraft turbine engines that lack afterburners.   Those would be the ones with a conical spike sticking out past the “turkey-feather” exit.  These work from sea level to the lower stratosphere.  Stream pressures approaching the nozzle are much lower than they are in typical modern rockets.  All of that is favorable to aerospike behavior.

When the turkey feathers form a convergent nozzle,  and the internal stream pressures are high enough to more-than-just-barely-choke that exit,  this rig functions very well as an aerospike nozzle facilitating a supersonic plume expansion to the last point of contact:  the tip of the exit spike.  That increases engine overall thrust and performance by increasing the nozzle thrust term in the airbreathing thrust equation.

My Qualifications to Say These Things

My original college and graduate school education was in high-speed compressible aerodynamics and thermodynamics/heat transfer,  much of it oriented toward propulsion.  I spent 20 years in aerospace defense work doing compressible flow mechanics,  including specifically the operation of all kinds of nozzles for rockets,  ramjets,  and other propulsion items,  some rather unconventional because of throat area modulation devices.

References

#1. G. W. Johnson,  “How Propulsion Nozzles Work”,  posted on “exrocketman” 12 November 2018.

#2. Pratt and Whitney “Aeronautical Vest-Pocket Handbook”,  12^th edition,  21^st printing,  December 1969;  from “Theoretical Rocket Engine Propellant Summary” page 92,  for LOX-RP1 at 1000 psia;  and from “U.S. Standard Atmosphere – 1962” pages 4 – 9 for pressure ratio versus altitude. 

#3. Ames Research Staff,  National Advisory Committee For Aeronautics (NACA) Report 1135 “Equations,  Tables,  and Charts For Compressible Flow”,  1953;  specifically “Prandtl-Meyer Expansion”, page 14.

#4. Chemical Rubber Company (CRC) “Handbook of Chemistry and Physics”,  53^rd edition 1972-1973,  published by CRC Press;  section F page F-171,  metric or English abbreviated tables of the US Extension to the ICAO Standard Atmosphere,  for the pressure ratio at 300 kft geometric altitude.

Related Articles

There are lists of articles on this site,  organized by the topic areas they pertain to.  These are given in an article titled “Lists of Some Articles By Topic Area”,  dated 21 October 2021.  Related topic areas might include “ramjet”,  “aerothermo”,  and “rocket performance”.  I have added this article to those lists.

The fastest way to access any given article on this site is to jot down the dates and titles you want,  and use the fast navigation tool on the left side of this page.  Click on the year,  the month,  then the title if need be (if there was more than one posted that month). 

When looking at any given article,  it is possible to see all the figures enlarged,  by clicking on any one of them.  You click on any of the small images bottom of page,  to see any one of the figures enlarged.  There is an X-out feature to page top right,  that takes you right back to the article.  You may enlarge and X-out multiple times,  as desired.  

There is also a list of keywords at the end of each article.  If you click on a keyword,  you will see only those articles bearing that same keyword.  Top of the list is a “show all posts” option.

Appendix

Here are images of the spreadsheet worksheets I used to generate the plots given above,  except that I did not include the worksheet-generated plots in these images.  The name of the spreadsheet file is “nozzles.xls”.  Figure 8 is the worksheet for the sea level bell design.  Figure 9 is the 60,000 foot bell as “vacuum engine”,  and Figure 10 the 30,000 foot bell as the “compromise vacuum engine.  Figure 11 is the worksheet used for the aerospike.  It has to be laid out differently,  as the expansion is not geometrically fixed.   Figure 12 shows exactly how nozzle efficiencies were computed,  and what assumptions were made to do the analyses,  for both the bells and the aerospikes.

Figure 8 – Sea Level Bell Worksheet

Figure 9 – 60,000 Foot Bell Worksheet

Figure 10 – 30,000 Foot Bell Worksheet

Figure 11 – Aerospike Worksheet

Figure 12 --  Nozzle Efficiency Calculations and Assumptions Made

Update 6 Feb 2023:          

Doing exactly what was suggested above,  I designed a revised aerospike nozzle that uses some bell-confined supersonic expansion out of the gas generator,  before doing the free expansion on the spike from there to ambient.  This actually did reduce the expanded plume fan angles at high altitude,  enough to raise the design altitude,  and to re-raise the gas generator chamber pressure from 300 psia back to 3000 psia.   These changes were beneficial enough to restore much of the compromised off-design performance seen with the sonic-only gas generator design. 

Results for thrust coefficient and nozzle efficiency,  thrust,  and specific impulse are given in Figure 13,  which one should compare to the sonic-only gas generator design in Figure 6 above.  Results for expanded Mach number,  expanded area ratio,  and the plume fan angles are given in Figure 14,  which one should compare to the sonic-only gas generator values in Figure 7 above.  Formats are identical.

The best of the three fixed-bell designs was the one designed for perfect expansion at a modest altitude,  which gets a lot of improved vacuum performance,  while still being testable at sea level in the open air,  without flow separation in the bell.  Those performance numbers are given in Figure 5 above. 

In Figure 15,  data are plotted for direct comparison of the best fixed bell design,  and the best aerospike design,  for both at max Pc = 3000 psia.  In Figure 16,  the same comparison plots are given with both operating at min Pc = 1000 psia.  Bear in mind these are LOX/RP-1 designs that I arbitrarily roughed out.  The “best bell” was sized to perfect expansion at 30,000 feet so that it could be fired in the open air at sea level,  at full Pc.  At min Pc,  it must be at or above almost 30,000 feet in order not to separate.  It was sized with Fth = 106,230 lb.  The best aerospike was the revised spike with the supersonic-bell gas generators,  sized for an axial plume boundary at 60,000 feet,  and a nominal thrust Fth = 100,000 lb.

There is still performance degradation with the revised aerospike below fixed-bell levels,  while flying out into vacuum,  but it is not nearly as degraded as with the earlier sonic-only gas generator aerospike design described above.  This improvement in performance was afforded by the limited supersonic expansion bell on the gas generator,  which limits how adversely-lateral the plume angle can spread at lowest backpressures.  The aerospike itself is a little less extreme in its initial angle,  as well.

That this revised aerospike is a near-optimal design is confirmed by its specific impulse performance very slightly exceeding the best fixed bell,  from sea level to about 200,000 feet.  For ascents,  this aerospike might be competitive in terms of performance,  since the specific impulse advantage in the stratosphere offsets the specific impulse deficit out in vacuum,  as long as it is not used for too much impulse delivery out in vacuum.  For routine use out in vacuum,  the fixed bell is still better.

The original conclusion above that aerospikes are not good vacuum nozzles really is confirmed in this update.  However,  the vacuum shortfall definitely can be made more modest than was originally indicated in the sonic-only gas generator version.  One does that by fitting a sea level bell upon the gas generator,  allowing a reduction in initial spike surface angle off of axial.  The mechanism of the fan angle reduction is the direction-confining action of the modest bell,  limiting the further Prandtl-Meyer expansion angle from there to ambient,  and which then also allows higher chamber pressure.

The higher chamber pressure raises both c* and thrust coefficient,  which in turn acts to raise specific impulse.  The more-limited plume fan-out angle in vacuum raises the effective nozzle kinetic energy efficiency,  which acts to raise thrust and specific impulse.  Of the two effects,  the fan angle dominates.

Figure 13 – Basic Results for Aerospike with Supersonic-Expanding Gas Generator

Figure 14 – Detail Results for Aerospike with Supersonic-Expanding Gas Generator

Figure 15 – Comparison of Best Bell and Best Aerospike at Max Pc

Figure 16 --  Comparison of Best Bell and Best Aerospike at Min Pc

As before,  the better measures of relative performance lie in the nondimensional factors like thrust coefficient.  The differences in sized thrust show up in the sized throat areas (a sum of multiple gas generators in the aerospike case),  which directly affect thrust and flow rate in the same way.  These areas are close but not equal,  in the comparison given here.  That’s why one should put less credence in the calculated thrust trends as a direct measure of comparison.  However,  one can put full credence in the specific impulse trends,  because the throat area effects on thrust and massflow rate divide out.

The basic message here is that by sizing the throat areas correctly,  the thrust shortfall evident even in the stratosphere in Figures 15 and 16 can be eliminated!  This will not really change the thrust coefficient and specific impulse trends!  Those show the revised aerospike (with combined supersonic bell and free-expansion spike) can equal or exceed the performance of the “best” fixed bell up to the outer stratosphere (around 200,000 feet),  but will inevitably fall short of fixed-bell performance from the outer stratosphere on out into vacuum! 

Those statements are made for a “modest vacuum bell” design,  sized to operate over-expanded at sea level up to its design perfect expansion altitude of 30,000 feet.  From there it operates under-expanded all the way to vacuum,  at the highest expanded-momentum term available.  That design selection is limited by being able to test fire in the open air without flow separation at sea level and full power.  It cannot be test-fired at sea level like that,  at min throttle.

The ”full vacuum bell” can equal aerospike performance in the stratosphere,  as indicated in Figure 17,  but cannot be operated at sea level.  However,  note that out in vacuum,  even a sea level bell outperforms the aerospike!  Aerospikes are quite simply not good vacuum engines!

Figure 17 – 3-Way Comparison In Terms of Specific Impulse