Primer on Ramjets

Update 23 March 2024:  For the readers of this and other similar articles about ramjet propulsion,  be aware that GW’s ramjet book is finally available as a self-published item.  Its title is “A Practical Guide to Ramjet Propulsion”.  Right now,  contact GW at gwj5886@gmail.com to buy your copy. 

He will,  upon receipt of payment by surface mail or Western Union (or similar),  manually email the book to you as pdf files.  This will take place as 9 emails,  each with 3 files attached,  for a total of 27 files (1 for the up-front stuff,  1 each for 22 chapters,  and 1 each for 4 appendices).  The base price is $100,  to which $6.25 of Texas sales tax must be added,  for an invoice total of $106.25. 

This procedure will get replaced with a secure automated web site,  that can take credit cards,  and automatically send the book as files.  However,  that option is not yet available.  Watch this space for the announcement when it is.  

GW is working on a second edition.  No projections yet for when that will become available.

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Update 6-19-18:  This article seems to be drawing a lot of readership lately.  I hope readers are finding it useful.  The booster issues discussed in the comments and replies to this article are thoroughly covered in my book.  

That book,  submitted to AIAA for publication,  is "still out for review" every time I have inquired about its status,  for the last several months.  You might contact AIAA and inquire about it yourself,  if you are interested in obtaining a copy.  The title as submitted is "A Practical Guide to Ramjet Propulsion".  

They might move more quickly if they thought there was real demand.
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Update 11-28-17:  The last draft materials (that 4th appendix and a revised table of contents) have been submitted to AIAA.  I haven't yet heard back on a publishing decision.  
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Update 9-19-17:  Application materials and most of the book went to AIAA today.  Their timelines say weeks for approval,  months for actual publication.  
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Update 5-17-17:  My ramjet book is nearly completed.  I have drafted 22 of 22 chapters,  and 3 of 4 appendices.  I am in process of applying to publish this work as a book in AIAA's "education series".  

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Many folks have looked at my Ramjet Cycles Analysis posting on this site,  dated 12-21-2012.   There is a whole lot more to engineering ramjet propulsion than just cycle analysis.  The following is but a primer on this very large topic.  It is not comprehensive,  nor does it have the real "how-to".  But is does introduce the real operative concepts.  For the details and "how-to",  you'll have to wait for my book.  

To see the figures enlarged,  click on one,  and you can see them all.  The "escape key" gets you out of the enlarged figures and back to the normal view.  

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There are two different speed ranges for conventional subsonic combustion ramjets,  with two different sets of appropriate design features.  There’s no point trying to use either one outside its appropriate speed range. 

Low Speed Ramjets

The low speed range extends from very high subsonic to about Mach 2,  no more than about 2.5,  and such designs have a simple pitot/normal shock inlet,  and a convergent-only nozzle that is not always choked.  That nozzle in a well-designed system begins to choke in the vicinity of Mach 1 to 1.1,  which limits combustor flow speeds to values compatible with successful flameholding (no more than about Mach 0.42).  These have been off-the-shelf ready-to-apply liquid-fueled ramjet technology since WW2.  Example:  Red-Head / Roadrunner,  Gorgon-IV.  Rather volatile fuels like gasoline or the wide-cut fuel JP-4 are required.

High-Speed Ramjets

The high-speed designs extend from just under Mach 2 to speeds above Mach 4,  to maybe Mach 6.  These have supersonic inlets with external compression features that don't work at all below about Mach 1.7,  give or take ~ 0.2.  The nozzle is an always-choked convergent-divergent design with a very modest expansion ratio (near 1.5-1.7).  In a properly-sized system,  combustor flow speeds are flameholding-limited to the same values as in a low-speed design,  but pressures are higher because the incoming inlet pressures and air massflow are higher,  so performance potential is higher than the low speed designs.  These have been off-the-shelf,  ready-to-apply liquid-fueled ramjet technology since the 1950's.  Examples:  Bomarc,  Talos,  SA-4,  SA-6,  Bloodhound,  Navajo.

Superior flame stabilization techniques,  solid gas generator-fed fuel options,  and better ways to add the needed booster rockets got added in the 1960's,  and have been off-the-shelf ready-to-apply technologies since about 1970.  Examples:  SA-6,  ALVRJ,  ASALM-PTV,  Kh-31,  Kh-41,  Kh-61,  and Meteor.  The prime innovation was the integral booster.  There are also ejectable booster nozzles,  nozzle-less boosters,  and inlet port cover design approaches,  all associated with those integral boosters.  The other booster designs are stage-off (Talos) and carry-along (Bomarc).  Slightly-less volatile kerosene and kerosene-like fuels may be used in these designs,  and solid gas generator-fed fuels are feasible.  There are severe geometry restrictions with those solid fuels that require flameholding,  less with the hypergolic solids. 

Characteristics of Ramjets

Neither type (low- or high-speed) operates at constant fuel flow rate,  unless you only fly at only one speed and altitude.  Inlet captured airflow varies very strongly with speed of flight and with ambient air pressure,  which at high altitudes is very low indeed.  The variations are not linear,  reflecting both atmospheric variations and actual inlet hardware characteristics,  as well as engine operating parameters like fuel flow. 

What you want is operation at a constant fuel/air ratio,  thus your fuel flow varies exactly as wildly as your airflow capture does.  A thrust-intense accelerator mixture might be 10% rich.  Higher fuel economy performance is available leaned-back in steady cruise at around 10-15% lean.  This leaned value varies a lot from design to design.  The rich value is an almost universal “knee-in-the-curve” item.

For both kinds of design,  frontal thrust density (thrust per unit cross section area) varies more-or-less in proportion to operating combustor pressure,  which at high altitudes is very low indeed.  Weight does not 

reduce with reducing air pressure,  so there quickly comes a point where you cannot generate enough thrust to actually fly level,  much less climb or accelerate.  Very few designs can successfully operate much above about 80-100,000 feet.  There have been a very few exceptions,  but nothing above 125,000 feet.

“Scramjets”

Supersonic combustion ramjets (“scramjets”) operate from about Mach 4 on up,  albeit at lower fuel economy performance than high speed ramjets have in their speed range.  They also have the same low frontal thrust density limitations as the conventional ramjet.  They must also variably-meter fuel at constant mixture ratio with a highly-variable captured airflow,  but beyond that,  there is no resemblance to conventional ramjet.  Try to run one below Mach 4,  and it explodes,  according to the test data.

Scramjets share only a few inlet features with high speed conventional ramjets.  The design analysis techniques,  and the basic hardware components for the combustor and nozzle,  are entirely different from what is used for conventional ramjet. Scramjet cycle analysis is very effort-intensive and usually based on computer fluid dynamic-type analysis. Conditions are so far outside what these codes were developed for,  your analyses must be validated by test before you can trust them.  That has been rather unreliable.  Scramjets require a long constant-Mach isolator duct,  while ramjets require a terminal shock in a divergent subsonic diffuser.  These are fundamentally-incompatible geometry requirements.  Scramjets just explode without the isolator duct.  Ramjets cannot function without the subsonic diffuser.  Simple as that. 

None of that scramjet “sophisticated non-ideal-gas” analysis is needed for conventional ramjet,  coming as it does from the pencil/paper/slide rule days,  and operating in regimes where the ideal gas models still apply.  While ramjet can be done with scramjet analysis tools,  it is hardly worth the effort and cost to do so.  It is simply far more effective and efficient to “do it the old way”. There is no way around that little fact of life.

The recent test flight vehicles X-43A and X-51A did demonstrate successful,  but very experimental scramjet burns.  But neither design actually accelerated at all as an airbreather.  Scramjet is NOT an off-the-shelf ready-to-apply technology.  It just barely works at all,  in a few highly-experimental and hugely expensive flight tests.  X-43A flew twice out of 3 attempts,  netting two 3-second burns at constant speeds of Mach 7 and 10,  with hydrogen fuel.  X-51A flew twice out of four attempts,  netting two 3-minute burns at constant speeds of Mach 5,  with hydrocarbon fuel.  

These scramjets can indeed be developed,  yes,  but I wouldn't hold my breath waiting for a ready-to-apply technology!  I have personally watched this endeavor for over half a century,  without seeing anything but highly experimental,  only-partially successful results,  and only in the last 12 years.  Nothing before that.

            Acceleration in Airbreather Mode

(Conventional,  subsonic-combustion) ramjet is something I did for a living for ~20 years in defense work.  Most of these were designed for max speeds of Mach 3 to Mach 4.  Although,  one test vehicle (ASALM-PTV) reached an unintended Mach 6 on a short transient.  It accelerated airbreather-only from Mach 2.5 takeover to Mach 6,  in a matter of several seconds!  Vehicle thrust minus drag,  divided by vehicle weight,  was a fair fraction of one full gee!  Like I said above,  ramjet works,  is off-the-shelf,  and is very definitely effective and ready-to-apply. 

            Combined-Cycle Engines

Combined cycle engine ideas are like scramjet,  just experimental toys.  Nothing is off-the-shelf ready-to-apply.  I've also been watching that effort for nearly half a century.  The closest thing to reality is the ejector ramjet,  but you'll actually get better overall performance if you just physically separate the rocket from the ramjet,  which then offers some unique advantages for parallel burn.  Each can then assist the other anywhere needed,  all along the trajectory,  and this happens at the best (uncompromised) performance from each.  

There are fundamentally two kinds of combined-cycle engines:  the turbine-based and rocket-based combined-cycle engines.  It would be hard enough to do either of these as turbine- or rocket- combined with ramjet,  but the trend in recent decades has been to try to blend them with scramjet.  This has turned out to be utter nonsense so far,  not just because combined-cycle makes fatally-severe compromises to individual component performance,  but also because scramjet is just fundamentally unready to apply,  even in its pure forms.

The rocket-based combined cycle with conventional subsonic-combustion ramjet is also known as the ejector ramjet.  This one hides a rocket engine within the ramjet engine,  in an effort to provide meaningful static thrust for takeoff.  It actually can work,  but the performance levels are abysmally low.  It’s just not worth the losses.

The rocket’s thrust performance is greatly reduced by the jet drag against the ramjet structures around it.  Any thrust-augmenting airflow induced through the engine by the rocket jet is drastically reduced for all inlets with external compression features,  simply because there are no attached-shock solutions for the external compression features.  That’s just physics,  there is no way around it.  That effect inherently “kills” ingested air flow.  And the flameholding flow pattern of the ramjet component is inherently and fatally disturbed by the presence of the rocket jet,  with which it is coaxial. 

The concept of separate rocket and ramjet engines capable of “parallel burn” is just the better deal,  and by far!  The individual components yield full performance levels,  and can be operated simultaneously as well as sequentially,  which is far greater flexibility.  Parallel burn with separate devices is also how a self-boosting ramjet airplane becomes a feasible thing to attempt.  You need to retain enough rocket propellant to provide a safe and practical landing capability:  “go around” or “divert” on rocket propulsion.  This is probably best done with small liquid propellant rocket engines to provide boost,  landing,  and any parallel-burn mission capabilities.  These can use the same fuel as the ramjet,  so that only the oxidizer need be added. 

Ramjet Heat Protection

Up to now,  ramjet has been applied to one-shot missile designs.  Combustor heat protection is best done as DC-93-104 silicone ablative,  but retained in place once charred-through,  by kinked stainless steel ribbons spot-welded to the case ID.  This retained char becomes brittle and falls apart after the burn,  upon cool-down,  so it simply cannot be reused!  The kinked-ribbon retention idea has been implemented by very few outfits!

However,  by means of acid-etched Teflon film and appropriate primers,  conventional rocket propellant can be cast in place on DC-93-104 as a fully-case-bonded integral booster rocket.  Without the Teflon separator,  this does not work,  as the silicone chemistry of the DC-93-104 is fundamentally incompatible with the hydrocarbon chemistry of CTPB,  HTPB,  and PBAN propellant binder systems.  (I leave out monopropellant explosive GAP binder as simply too hazardous to work with,  like raw NG,  but it would also need the Teflon barrier film.)  The most practical propellant grain design for ramjet-like L/D’s and high boost pressures is the keyhole slot. 

High-density “super-ceramics” are just not a feasible option for a combustor packaged inside an airframe,  precisely because they have high thermal conductivity (inherently because of their high density).  The heat flow through such a wall is catastrophically large,  destroying anything near the incandescent combustor shell.  The only “option” would be regenerative cooling,  but unlike rocket,  the fuel flow is far smaller and so unable to accept the large quantity of heat.  This same consideration applies to nozzle and inlet structures. 

There are a couple of low-density ceramic solutions,  but these are very experimental,  and preclude the use of an integral booster packaged in the combustor.  For surface temperatures under 3200 F,  an all-alumino-silicate solution is available as a ceramic composite made of pipe insulation paste and aircraft fire curtain cloth.  This has been done once successfully,  in a miniature combustor whose flight speed did not exceed Mach 0.9. 

For surface temperatures exceeding 3200 F,  some kind of reduced-density zirconia composite is required.  The material choices are fewer,  and far more expensive.  Thermal conductivity is very much higher,  so the required thickness is far greater.  As near as I can tell,  this was actually attempted once,  but has never been done successfully.  There are also very serious concerns about usable lifetime,  on the part of the zirconia materials maker.

Neither ceramic solution is compatible with the integral booster concept.  All such materials,  being so porous,  are inherently very fragile.  Booster rocket pressures would utterly destroy them in microseconds.  For either ceramic solution,  the booster rocket must be located outside the combustor,  which means either a stage-off booster or else a parallel-burn rocket of some kind.  There is simply no way around that. 

Ramjet missiles usually feature exposed martensitic stainless-steel structures,  good to about 1000-maybe-1200-F material temperatures.  Plain carbon steel and titanium are only good to about 700-800 F material temperatures.  That 1200 F limit with stuff like D6ac is why no operational ramjet missiles have ever exceeded about Mach 4 cruise velocity.  To fly faster requires some way to limit material temperatures.  Air temperatures in the stratosphere are around 3000 F at Mach 6.  And that’s just skins.  Leading edges are more demanding. 

The only other ramjet heat protection scheme is the perforated cooling air sleeve,  something common in designs from the 1950’s and 1960’s.  It is still seen in jet engine afterburner ducts.  Overall mixture must be lean to have the excess air needed for the cooling sleeve,  and this air must be cool enough to actually serve effectively as a coolant for an item washed by flame on the other side.  Such schemes have never served at speeds over Mach 3 because of the air temperatures.  A perforated sleeve like this is as incompatible with an integral booster as are the low-density ceramics. 

Ramjet Flame Stability

For all the non-hypergolic fuels,  there must be properly sized and located flameholding recirculation zones,  or else the igniter must fire massively throughout the flight.  There is no way around that,  it is just physics and chemistry.  The hypergolic fuels are vapor magnesium and TEB or TEB/TEA blends.  These are low energy,  low stoichiometry,  low-Isp (very high TSFC) fuels.  They are also the only feasible flight igniter materials. 

No flameholding ramjet (gasoline,  wide-cut,  kerosene,  or solid gas generator-fed) was ever successful with a ramjet throat area / combustor area ratio exceeding 65%,  because the balance of inlet and duct sizes becomes impossible at feasible flow speeds above that limit.  Few were ever successful with that ratio below about 55-60%,  because frontal thrust densities fell too low to be useful.  It makes sense to size at 65%,  and set inlet/throat area ratio so as not to spill,  throughout the flight envelope,  and duct area so as not to be unignitable or suffer flashback from high or low duct velocities. 

Hypergolic (magnesium vapor) gas generator-fed systems can use throat area ratios up to ~90%,  and need no flameholder recirculation zones at all.  All that is important is fine-scale mixing for combustion efficiency.  But hypergolic systems are the exception,  not the rule.  The SA-6 was one of these.  There have been no others,  with the possible exception of the ramjet variant of the AA-12 “Adder”.

There are two practical kinds of flameholder:  baffles and “sudden-dump” combustors.  The actually-implemented form of the baffle is the V-gutter,  used in the 1950’s and 1960’s systems,  and still in use today in jet engine afterburners.  These are very sensitive to the speed in the duct approaching the V-gutter.  They do require inlet air temperatures low enough to be an effective coolant,  because these structures are bathed in flame on the downstream side.  They have never been successfully used as bare metal items above about Mach 3 or so,  because of the high inlet air temperatures.  Their area blockage is usually no more than about 10-15% of the inlet duct area. These effects depend upon absolute size:  smaller is less stable.

The sudden dump combustor,  whether center or side entry,  has only insulated combustor surfaces facing the combustor flame.  These can be used to at least Mach 6,  which is about the practical limit with subsonic-combustion ramjet anyway.  They are relatively insensitive to approaching duct velocity,  and thus more stable for a wider range of conditions.  It is low pressure and low temperature in the oncoming air that have the greatest negative impact on flame stability in dump combustors. The sudden expansion area ratio must fall in the feasible range for this to work.  These effects also depend upon absolute size:  smaller is less stable.

In ASALM-PTV,  the fuel spraybar assembly was configured to shed a wake resembling that of a V-gutter,  but it was insulated on the downstream side,  and located right in the dump plane,  in order to support the inlet port cover during boost.  The purpose of that spraybar wake turbulence was additional mixing energy to offset the effects of a very short combustor,  not to enhance flameholding. 

The other practical baffle flameholder (besides the V-gutter) is the “colander” burner,  whether inverted or not.  This was last successfully used in a ramjet in Talos in the 1950’s,  and that proved to be fraught with problems in testing.  It finds wide application in gas turbine engine work,  where flowing conditions are just not quite so extreme.  It does not find application in afterburner ducts,  which are actually quite similar to ramjets. 

            Ramjet Engine Sizing

Ramjet operation and performance is almost completely determined by what the inlet can do.  If the pressure demanded of the inlet is too high,  the shock system is expelled out in front of the cowl lip,  leading to spillage of decelerated air.  That spillage incurs drag without the benefit of producing any thrust. 

Things that raise the pressure demanded of the inlet are (1) too large an inlet relative to the other proportions,  (2) too rich a mixture,  and (3) too small a ramjet throat relative to the other proportions.  Inlet,  throat,  and mixture strength must be carefully determined to meet requirements and also maximize performance while doing so.  This is not a trivial exercise;  it requires both appropriate knowledge and real experience. 

The subsonic inlet duct must also be sized in correct proportion,  so that it has an appropriate flow speed,  both burning,  and in cold flow for ignition.  Flameholding is a part of this (upper speed limits),  as is the risk of flashback (lower speed limits),  which can destroy inlet structures.  So also is the flame stabilizer pressure loss a function of chosen duct speed.  Most dump-stabilized systems do well if the ratio of duct area to combustor area falls between 45% and 50%.

The only pressure-rise item in a ramjet is the inlet recovery.  The combustor entry and flameholder are pressure losses,  as is the combustion zone itself.  This is quite unlike the gas turbine,  in which the pressure rise effect of the compressor dominates,  and by far.  This compressor pressure rise is always much larger than the pressure drop across the turbine that drives the compressor,  in turn the dominant pressure reduction effect.  

Ramjet thrust and performance maximize for maximum captured air massflow,  with maximum feasible pressure recovery.  You cannot truly maximize both simultaneously,  and massflow is the more important item in ramjet.  Therefore,  one operates with a critical-to-slightly-supercritical inlet for max air ingestion at good pressure recovery.  This is quite distinct from gas turbine,  which requires spillage to match inlet and engine massflows:  an always-subcritical inlet.  The inlets of ramjets and gas turbines look the same (in the same speed ranges) because they utilize the same components,  but these are actually used quite differently. 

            Ramjet Ground Testing

Testing ramjets on the ground is best done in a direct-connect test facility,  as long as the inlet performance is already well-defined.  Most of these use “vitiated” (combustion-heated) air with make-up oxygen to achieve the needed temperatures. This is appropriate for the liquid hydrocarbon fuels,  but not the metallized gas generator fuels.  The active metals utilize the excess carbon dioxide and water vapor in vitiated air as additional oxygen,  thus getting bad test results.  Similar problems increasingly cloud results even with the liquid fuels at inlet air temperatures beyond about 2500 F,  due to dissociation and ionization chemistry effects.

One overcomes this by means of a pebble bed-type “clean air” heater instead of combustion vitiation.  Achievable temperatures are somewhat lower with this design approach,  but the delivered air really is air.  This is crucial with the metallic hypergolic fuels,  like magnesium.  Even aluminum and boron are questionable. 

Analysis of ground test data is not done by comparison to predictions from a cycle code.  Instead,  many of the same mathematical models are incorporated into a test data analysis program that computes independent estimates of combustion performance from both combustor pressure and test article thrust,  as completely-separate sources of data.  When these agree,  you know you did everything right.  Of the two,  getting reliable thrust data is far more difficult,  because of the difficulty of calibrating all the possible facility tare forces.  There is no such thing as a tare pressure,  though.  Trust pressure-derived performance in preference to thrust-derived performance,  always.

            Subscale Test Scaling

Scaling down to subscale test is more complicated than generally thought.  What you want is the same pressure and speed distributions inside the subscale representation of your engine.  If no other considerations were important,  then all you need do is match the air and fuel flow rates per unit cross section area,  the inlet total temperature,  and the geometric proportions of the engine.  But,  other considerations do matter.  Once you scale down too far,  the residence time distribution will be fatally wrong compared to full scale!  This is because chemistry rates do not scale with size.

There is a minimum size below which you have to distort the engine geometry in order to maintain feasible residence time distributions.  Only certain kinds of distortions are effective.  The details of this are entirely different for each geometry class (baffle versus coaxial dump versus side entry dump). There is no one general procedure to use!  The criteria are entirely empirical and unique in each geometry class.  Both overall residence time and flameholder recirculation zone residence time are critical items to address.  These are computed quite differently,  and in the case of flameholder residence time,  methods differ by geometry class.

            Running Studies Requires Modeling Inlets

A lot of the folks who want to do this,  do not have real wind tunnel data on real inlets available to them.  As long as the studies are “ballpark” explorations,  and not real system predictions,  there is a way to adequately estimate inlet performance for “new designs” based on past historical data.  This estimating technique is based on the “shock-on-lip design Mach number” of the inlet,  which applies to high-speed inlet designs only.   

I have a curve composited from old data that is fairly universal,  if used as the pressure recovery PRCR for Mach numbers below shock-on-lip.  In that regime,  they’re basically all just about the same.  A factor taken from a second curve applies to the PRCR value at shock-on-lip,  to create the individualized PRCR curve at Mach numbers above shock-on-lip.  That factor varies with flight-minus-design difference in Mach numbers. 

I have two other curves that model streamtube area ratio ARCR trends with that same difference of flight minus design Mach.  One is for round inlet cross-section shapes,  the other is for two-dimensional inlet cross-section shapes.  The basis area is A_C,  defined as the swept-out area:  this is the cross-section of the cowl entry channel plus the frontal blockage of the external compression surfaces,  at zero angle of attack. 

These techniques work fairly well for near-zero angle of attack with side-mounted inlets and chin inlets,  and pretty much up to 15 or 20 degree angle of attack with nose inlets.  It is easy to use this technique in a trade study to help define what the “best” inlet shock-on-lip design speed is,  for any given problem. 

The missing piece is additive drag coefficient,  which does not usually apply to nose inlets at all,  may or may not apply to a chin inlet,  but is quite important for side-mounted inlets.  This usually represents the ”pre-entry” drag on the entering streamtube,  where it is in contact with vehicle surfaces and influenced by the vehicle bow and forebody shock and expansion field.  It does not include the spillage drag when operating subcritically.  The reference area for this is also A_C,  as defined above.  This coefficient,  the A_C,  and the freestream dynamic pressure multiply together for the additive drag force.

Often,  the effects of boundary layer diverter drag,  and the spillage drag of capture enhancing bleeds located near the cowl lip,  are included lumped-in with the pre-entry drag into the additive drag data.  I have a set of real wind tunnel additive drag data that includes pre-entry,  diverter,  and capture-enhancing bleed drags for a real design actually tested. There is a knee in this curve at the design shock-on-lip speed.  For trade study purposes,  I just shift this curve left or right to put that knee at the shock-on-lip speed in my study problem. 

Subcritical spillage drag coefficient is easy to estimate as twice the subcritical spillage margin.  The area basis is A_C,  and those with the free stream dynamic pressure gets you to a drag force for the subcritically-spilled air. 

Many systems use air bled from the subsonic diffuser aft of the terminal shock,  to power pneumatically-operated machinery.  This bleed reduces the air actually fed to combustor from that captured by the inlet,  by an amount called the “bleed fraction”.  As long as your ram drag (inlet air momentum) is based on all the air captured,  you have already accounted for the drag of scooping up your machinery bleed air. 

Pitot/normal shock inlets for low-speed designs are far easier to estimate.  These are almost invariably nose inlets without any capture-enhancing bleeds,  so the additive drag is zero. 

Pick a “high” number like 98% to represent the subsonic diffuser PRCR.  From Mach one on up,  multiply that 98% factor by the total pressure ratio across a normal shock at each Mach number.  That product is your supersonic PRCR,  rather closely. 

Your pitot/normal shock ARCR is just another pretty-constant “high” factor like 98%,  across the board from subsonic to supersonic. Base your A_C on the dividing-streamline “highlight” defined by the inlet lip radius. 

            Thrust-Drag Accounting

There are two systems:  (1) net jet,  and (2) installed.  As regards ramjet propulsion, net jet thrust is nozzle thrust minus the ram drag of the captured airflow.  The additive and subcritical spillage drags must be added to the airframe drag.  Installed thrust is nozzle thrust minus ram drag,  minus additive drag,  and minus subcritical spillage drag.  The airframe drag is unchanged.  Do not mix definitions!  Be consistent!  Net jet is popular among propulsion specialists,  while the vehicle aerodynamicists and trajectory dynamics folks prefer installed. 

            Engine Flight Envelope

The standard presentation is altitude on the vertical axis,  and speed on the horizontal axis.  For supersonic-capable systems of all kinds,  the preferred form for speed is Mach number.  You can create one such plot for each day-type model (such as a “standard day”) that you choose to use. 

The minimum ramjet operating speed on this plot may or may not be constant with altitude.  Generally it is not the absolute minimum operating speed for thrust equal to drag,  but something higher set by adequate vehicle acceleration capability.  The maximum ramjet operating speed is very likely to be determined by thermal protection risks.  Something like 1200 F inlet air total temperature is usually a good representation of this. 

The “ceiling” of this operating envelope is fairly likely to be either a scooped air massflow contour,  or a flight dynamic pressure contour.  Flameholder stability and fuel turndown ratio limitations may clip off corners or zones from this basic envelope,  as your study proceeds.  The process of drawing it is iterative.

A variation on this uses the scaled-down inlet A_C to generate contours of constant airflow on a flight envelope that corresponds to subscale test hardware.  These and the constant inlet temperature contours get plotted as an easy way to relate ground test conditions (airflow and temperature) to simulated flight conditions (Mach number and altitude) for that particular design.  Facility limits usually exist in the form of air flow rate and temperature limits for direct comparison.  Open-air nozzle choke limits can usually be determined in terms of a critical airflow value.  Where you can simulate in test is thus a sub-envelope,  generally. 

            Selecting Propulsion

Gas turbine is available basically in two forms:  (1) high-bypass ratio “fanjets” that offer high economy,  but only at subsonic speeds,  and (2) low bypass ratio “turbojets” that offer supersonic flight at the cost of substantially-lower fuel economy.  Both types can be thrust-augmented with afterburners,  at the cost of very low economy. In practical terms,  gas turbine is limited by excessive inlet air temperatures to maximum feasible speeds near Mach 3.3-3.5 in the stratosphere. 

High-speed range ramjets are useful from near Mach 2,  to at least Mach 4,  and perhaps to Mach 6.  They are far simpler and lighter than gas turbines,  but resemble afterburning low-bypass gas turbines in terms of fuel economy.  Simple,  lightweight,  inexpensive,  and more fuel-economical than solid rocket propulsion,  that is why ramjets are often employed as missile propulsion. 

Ramjets are not as simple to design,  or to incorporate into a compact missile,  as solid rockets,  so they are not generally selected for the shorter range tactical missiles.  For the longer stand-off range tactical missions,  ramjets are well worth the trouble to incorporate,  especially the modern integral-booster forms.  Ramjets will cover the range in smaller packages,  arriving at higher speeds for better maneuverability,  and will do so in shorter flight times that greatly enhance the survivability of the launch aircraft,  ship,  or site. 

For tactical missile work,  I’d recommend selecting solid rocket for ranges under ten miles,  ramjet for over ten miles,  generally speaking.  For aircraft designs,  select ramjet if you need to fly faster than Mach 3.3-3.5.  Otherwise,  use an afterburning low-bypass gas turbine if supersonic,  fanjet if subsonic.

Once there really are ready-to-apply scramjet and combined-cycle engine technologies,  you will have more options to choose from.  That time is not yet.

            Concluding Comments

The details of all items just discussed were obviously not included.  There are many more details and issues associated with ramjet-propelled vehicles,  all quite critical to success.  The sum of all that is far larger than can be put into a few paragraphs here.

My ramjet book is still in work,  but it is more than half-written now.  It addresses all these issues,  and much more besides,  in a very hands-on / how-to manner.  In it,  I tried to include not only the science,  the applications,  real examples,  and the history,  but also a lot of the engineering art of ramjet propulsion that I do happen to know.  That art is the part not written down,  but passed-on in the workplace directly from the seasoned hand to the newbie. 

I hope to find the proper publishing outlet,  get this book finished,  and get it published and available,  during 2017. 

As regards engineering art,  I am fond of saying that “rocket science ain’t science,  it’s only about 40% science.  It’s about 50% art,  and about 10% blind dumb luck”.  I would also add two extra points to that statement: 

(1) It applies to production work.  In development work,  the art and luck factors are even higher. 

(2) It applies to just about all of engineering,  not just rocketry or ramjetting. 

G. W. Johnson,  PE,  PhD