Ramjet Cycle Analyses

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 9-19-17:  Application materials and most of the book went today to AIAA for approval for publication.  Timeline is weeks for approval,  months for actual publication.
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Update 5-17-17:  Ramjet book is nearly done. 22 of 22 chapters drafted,  3 of 4 appendices drafted.  I am in process of applying to publish this as part of AIAA's "education series" books.  
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Update 12-10-12:  I have put up a posting titled "Primer on Ramjets",  dated 12-10-12.  Cycle analysis as discussed here is but a small portion of actual ramjet propulsion engineering.  Take a look at the primer to see where cycle analysis (as one design analysis tool) fits into a much bigger picture. My ramjet book is still in work,  hopefully to be published in 2017.  I am well over half-done writing it now.
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Update 6-5-2016:  this is one of the most popular articles on the entire website. I hope readers have found it useful.  Just be aware that there is a whole lot more to engineering a ramjet propulsion system than just cycle analysis.  But,  understanding the cycle analysis is the first step toward understanding how these things really work.  
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I have noticed than many people have located this site by using "ramjet performance",  "ramjet efficiency",  or similar keywords. Ramjet engine performance is just not characterizable by a single value. Accordingly, I have posted here a basic cycle analysis document describing what is important, and trying to convey to the reader just how variable these parameters are. How an effective cycle analysis is actually best implemented is one of the subjects in a book on ramjet engineering that I am writing.  The subscript and superscript formatting in the source document does not translate well into html for this posting.  For that I apologize. 

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basic cycle analysis document for ramjet engines:
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updates 1-17-13 in red below

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addendum 1-19-13 in purple at end
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part 2 on compressible flow analysis added at end 5-4-13
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update 5-17-14 in blue below
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update 5-30-14 in green below
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Update 6-6-14 in orange below
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Update 7-11-14 in black below
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Update 9-5-14 in black below
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Update 9-20-14 in black below
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Update 12-12-15 here:  I now have 5 computer codes for design analysis and testing purposes.  For the high-speed range of ramjet designs (near M2 to M6 with supersonic inlets and always-choked nozzles) I have a sizing code RJHISZ and a point-performance code RJHIPF.  For the low-speed rang of ramjet designs (high subsonic to near M2 with pitot inlets and sometimes unchoked nozzles) I have a sizing code RJLOSZ and a point-performance code RJLOPF.  The fifth code models test articles complete with an inlet choke block assembly in ground test,  specifically direct-connect ground test.  I have invented a unique and effective simplified model of fuel-air thermochemistry to support calculations with these codes at about 2% accuracy in computed values.  These are written in an obsolete language that allows writing "spaghetti code",  because the engine balance pretty much requires that you program that way.
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Most of the modern cycle analyses available for ramjet, in most of the modern textbooks and their associated software, are one-step modifications of the ideal ramjet cycle analysis, which is not-at-all realistic, as these texts have freely admitted for decades. For ramjets, that one step is essentially nothing but a combustion efficiency and a bunch of total pressure recovery ratios for the inlet, the flameholder, the combustion zone, and (if you are lucky) the exit nozzle. See Fig. 1.


Figure 1 – Zones and Station Numbers in a Generic Ramjet Engine

Mex^2 = (2/(γ-1))[(1 + 0.5*(γ-1)*Mo^2)*(rd*rc*rn*Po/pex)^((γ-1)/γ) – 1] upper limit, (γ is sp. heat ratio)

rd = pt2/pto this is the inlet total pressure recovery ratio, usually between 0.9 and 0.1

rc = pt4/pt2 combustor /flameholder combined total pressure recovery, near 0.8-0.9

rn = pt6/pt4 this is the total pressure recovery ratio across the exit nozzle, above 0.95

Vex = Mex [ γRTt4/(1 + 0.5*(γ-1)*Mex^2) ]^0.5

f/a = [Tt4/Tto - 1]/[ηb*LHV/cp*Tto – Tt4/Tto] this is fuel/air ratio by mass, expressed in terms of desired total temperature, not usually the way we really do it

ηb = this is effectively the temperature-rise combustion efficiency, not stream thrust efficiency

Fnj = wa[(1 + f/a)Vex - Vo] net jet thrust, not including any propulsive drags

net jet Isp = Fnj/wf = 3600/TSFC where fuel flow wf = wa*f/a, and wa is the inlet air flow

inlet air flow is always less than, or at most equal to, that swept out by the cowl lip in a ramjet, utterly unlike gas turbine, where air flow is set by engine speed, size, and inlet air density

My old engineering school propulsion textbook (ref. 1) has this very same ratio-modified cycle analysis in it. So do more modern references like the very famous ref. 2, and also ref. 3, which has some cycle analysis software associated with it. The main implication of this approach is that this combustion efficiency and these pressure ratios can be input as “typical” constant values for a reasonably accurate result. The truth is, for ramjet they simply cannot be modeled as constants,  although for gas turbine, to some extent, they can. This is because the compressor pressure ratio dominates the analysis for all gas turbine engines. There is no such compressor in a ramjet. See Fig. 2 to compare engines.


Figure 2 – A Comparison of Generic Ramjet and Turbojet Engines

Ramjet is quite different. Too much depends upon the detailed internal flow aerodynamics of all of these components, since there is no other compression mechanism than the inlet, and its compression “gain” must overcome the losses in all the other components. That internal flow state-dependence also means none of these ratios are remotely constant at “typical” values, excepting perhaps the nozzle.

Scope here is limited to subsonic-combustion ramjet only, so that conventional adiabatic compressible flow analysis, and well-known component loss models, may be applied. Supersonic combustion ramjet (scramjet) is quite different: it is not adiabatic at all, and the component loss models are not very well-known. That topic is not covered here at all.

Further, what is modeled with this typical textbook cycle analysis is “net jet thrust” only. This “puts the onus” on the user to include additive (pre-entry) drag, bleed drag, spillage drag, and diverter drag in the vehicle drag buildup. These are all propulsion-related drag items with which few vehicle aerodynamicists are familiar, and they are very, very significant in magnitude. Neglecting them is a very major mistake. This topic is also well-covered in my low-speed cycle analysis document, something not posted here.

Depending upon the inlet configuration, sometimes additive (pre-entry) drag does not apply (nose pitot inlet). But when it does, it is very significant, and it is very flight speed dependent, as well. So also is the basic maximum pressure recovery of the inlet, and in many configurations, the stream tube area ratio, free stream to inlet capture. All of these inlet recovery-related items are major influences upon performance. Neglecting any of them is a major analysis mistake. They are very strong empirical functions of flight Mach number and attitude angles.

Ramjet flameholders can be baffles, perforated cans, or sudden dumps, or combinations of these approaches. (Even in magnesium-fueled ramjets like the Russian SA-6, while flameholding itself is not an issue, the total pressure losses associated with air entry geometry are.) All of these air entry geometries share a total pressure loss dependence upon the square of the approaching internal Mach number (which means the detailed internal flow state at that place must be analyzed). This is a very significant effect, and it is very dependent upon both flight conditions and throttle setting, and sometimes in an indirect fashion.

Combustors have a total pressure loss across the flame zone that depends dramatically upon the ratio of the combusted total temperature to the inlet total temperature, and to the fuel-to-air ratio by mass. It also depends less dramatically upon the change in gas properties across the flame zone, and for gas generator-fed ramjets, the effluent injection momentum’s axial component relative to that of the air. These are all definitely not constant items, at all conditions. What this means is that the internal flow states must be analyzed in detail at the inlet entry,  the forward end of the combustor,  and the aft end of the combustor (which in turn interacts strongly with the flow states in the nozzle). 

Nozzles (only if choked) come the closest to modeling as single-constant parameters, until you introduce variable geometry, although backpressure-induced flow separation will upset that picture quite substantially. An unseparated fixed-geometry nozzle has a kinetic energy efficiency that is quite effectively modeled as a constant value. For conical expansion bells between 0 and 20 degrees half-angle, this kinetic energy efficiency depends almost solely upon that half angle:

ηnoz = 0.5*[1 + cos(half-angle)]

For curved-profile bell nozzles and the use of the above equation, the effective half-angle is the arithmetic average of the post-throat and the exit half-angles.

Since the most common half-angle (or effective half-angle) is near 15 degrees, most folks use a nozzle kinetic energy efficiency of 98%. This should be applied to only the momentum term of vacuum thrust, for the particular chamber conditions. It does not apply to the exit expanded static pressure term, nor to the backpressure term. At ramjet pressures (nearer 50-100 psia at low altitudes), both of these terms are significant relative to the momentum term, not so very much with higher-pressure rockets (nearer 2000 psia). At high altitudes, ramjet combustor chamber pressures can be below 10 psia.

Convergent-only nozzles of the low-speed designs are often unchoked.  The performance characteristics of these can only be determined by a detailed internal flow analysis. 

Variable-Geometry Ramjet Nozzles

Here follows some engineering art regarding variable ramjet nozzle geometry. It derives from contract efforts related to the fixed-nozzle ASALM-PTV ramjet vehicle that flew in tests about 1980. One of those efforts undertook to determine the “how-to” of reversible two-position variable-geometry nozzle technology for the ASALM ramjet engine. See Fig. 3. This was a high speed range device.


Figure 3 – Two-Position Paddle Concept for Variable-Geometry Nozzles in Ramjets

The basic idea was that high thrust, wide-open nozzle conditions were necessary for climb-to-altitude, and for high-speed terminal dive. During cruise flight at throttled-back conditions, a smaller ramjet nozzle (1) reduced flow losses from the terminal shock position in the divergent duct of the inlet, and also (2) reduced the losses of the sudden dump flameholder, by lowering its approach Mach number significantly. These two effects raised cycle specific impulse and increased range, at the cost of extra weight and complexity. (Ultimately, this was not selected for inclusion in the seven ASALM flight tests.)

I got to work on these variable-geometry nozzle efforts directly. We found in model tests, and confirmed in full-size 20-inch diameter direct-connect ramjet tests, that a rotating paddle-shaped throat blockage could modulate the ramjet throat area by around 4:1, at typical kinetic energy efficiencies of 95-96%, versus the typical 98% for a clean, fixed-geometry nozzle.

The paddle structure had a stainless-steel core, covered over with “heat shield” pieces of silica phenolic ablative. It had two positions, streamline and broadside, just 90 degrees apart from a hydraulic actuator standpoint. Both positions fell into the 95-96% kinetic energy efficiency range in full scale test, even after the phenolic was “all chewed up” from a 200+ second cruise burn.

For the record, this particular variable geometry approach is entirely unsuitable for the changes needed between rocket boost and ramjet sustain. That area ratio is far larger than 4:1, plus the booster pressures are far nearer 2000 psi than they are to the 50-100 psi of ramjet pressures low altitudes, even lower pressures at high altitudes. The materials simply cannot handle boost loads like that.

Low speed range devices with a convergent-only nozzle could use “turkey feathers” exactly like those on an afterburner-equipped turbojet.

Ramjet Design Analysis and Testing Analysis

As you might by-now guess, the classic textbook cycle analysis of a ramjet is too undiscriminating to use for test purposes, just as it is inadequate for actual performance prediction of real systems. The only way to discriminate what is going on inside the collection of components that is a ramjet during a test, is a collection of accurate models for all of those components. There is no way around this dilemma.

There are two design options here, related primarily to the speed regime for which the ramjet system is designed. One is the “low speed regime”, ranging from high subsonic to low supersonic speeds (approximately Mach 0.7 to no more than about Mach 2.5, usually less). These designs will feature pitot (normal shock) inlets and convergent-only nozzles that are not always choked. See my low-speed cycle analysis document for the details of that case. This is best done with the most primitive variables and equations, as is done in that document.

The other design option is the high-speed design regime, ranging from low supersonic to low hypersonic speeds (approximately Mach 1.5 to about Mach 5 to 6). These designs will feature inlets with external compression devices (cone or ramp shock-generating surfaces ahead of the cowl lip), and convergent-divergent nozzles that are always choked. For this case, empirical correlations for combined flameholder and burner pressure recoveries become feasible, precisely because the nozzle is always choked (a part of the definition of the correlating parameter). The rest is similar to the low-speed case. See my high-speed cycle analysis document for the details of how things are really calculated.  (This document has not been posted,  but it is to be a part of my book.) 

See Fig. 4 to compare typical engine component selections for the two basic speed ranges.


Figure 4 – Comparison of Component Selections for Low and High Speed Ramjet Designs

For testing purposes, variations of either the low-speed cycle analysis or the high-speed cycle analysis can be used, as appropriate. Performance needs to be computed independently from both combustor static pressures and properly-calibrated thrust stand forces, and then compared. Until these agree, you can be quite sure that you did not do something correctly, somewhere, usually in the thrust stand design and calibration. Most folks do choked sonic-only throat tests, for greater analysis simplicity. See also ref. 4 for ramburner data reporting standards that are generally accepted. (My station numbering matches that standard for ramjet, a lot of these textbook cycle analyses do not.)

References:

1. Hill, Phillip G., and Peterson, Carl R., “The Mechanics and Thermodynamics of Propulsion”, Addison-Wesley Publishing, 1965 (second printing 1967). Often referred to as the “Hill and Peterson” text.

2. Oates, Gordon C., “Aerothermodynamics of Gas Turbine and Rocket Propulsion (revised and enlarged)”, AIAA Education series, published by AIAA (American Institute of Aeronautics and Astronautics, 1988 (second printing). Applies a general cycle analysis formulated for gas turbine, to ramjet, almost as an afterthought.

3. Mattingly, Jack D., Heiser, William H., and Daley, Daniel H., “Aircraft Engine Design”, AIAA Education series, published by AIAA (American Institute of Aeronautics and Astronautics, 1987 (third printing); comes with associated ONX (on-design) and OFFX (off-design) software. Uses the equations in Oates (ref. 2) for the book presentation, and in the software.

4. Chemical Propulsion Information Agency (CPIA) Publication 276 “Recommended Ramburner Test Reporting Standards”, March 1976. Defines the recovery and efficiency parameters, and the station numbering.

Addendum 1-19-13:

Stringing models of components together allows one to compute the internal flow states through the engine.  These flow states are actually what the most important individual component pressure ratios depend upon:  the air entry pressure recovery,  the combustion zone pressure ratio,  and the exit nozzle performance.  Inlet pressure recovery is best modeled from empirical wind tunnel test results,  especially for high-speed designs that have inlets with the external compression features. Only the very simplest nose pitot inlet designs can be "reliably" estimated from first principles,  and even those are better modeled from real test data.

A typical inlet on a high-speed design will have a maximum captured streamtube area ratio (to inlet cowl area) that is usually a very strong function of flight Mach number,  and a weaker function of vehicle attitude angles.  Actual ingestion can be less,  if there is subcritical spillage.  The associated propulsive drag items vary with the same variables.  Streamtube area recovery sets the air massflow ingested,  and freestream conditions the stagnation temperature. 

That same inlet will have a maximum stagnation pressure recovery ratio (to freestream) that is also typically a very,  very sensitive function of flight Mach number,  and a weaker function of vehicle attitude angles.  Actual recovery can be less,  even far less,  if the inlet has a large supercritical pressure margin (meaning the terminal shock wave is located deep within the final diffuser,  in order to match low delivered static pressures).  

The inlet (or inlets) can enter the combustor in a variety of geometries.  Most modern designs use a sudden dump geometry that also provides flameholding in the associated separated-wake zones surrounding the entering air streams.  Sizing the dump area ratio is a complex tradeoff,  since large expansions lead to larger flameholders,  but also to much higher pressure losses.  

For a selected geometry,  the Mach number of the internal flow approaching the dump depends upon the downstream static pressure that is set by engine operation,  while upstream operating conditions set the stagnation temperature and pressure,  and the massflow.  Low static pressures associate with high Mach numbers approaching the dump,  in turn associated with high stagnation pressure losses that vary as the square of approach Mach number. 

The combustor flame zone represents a balance of mass continuity,  momentum conservation,  and energy conservation (accounting for fuel energy release plus significant change in gas properties).  In a low-speed design,  the exit nozzle throat is often unchoked,  making combusted Mach number yet another variable,  governed more by nozzle conditions.  A primitive variable balance is the best analysis approach with these designs.  In high-speed designs,  the nozzle throat is choked,  thus locking in a fixed combusted Mach number at any given fuel/air ratio (gas properties).  In that case,  a correlation of combustor stagnation pressure ratio with operating variables is possible.  This short-cuts the difficult primitive-variable balance process,  but only if you already have empirical test data for that specific geometry. 

The nozzle for low-speed designs is convergent-only,  and often unchoked.  Flowing static pressure in the exit plane must equal ambient atmospheric,  up to the choking point.  That makes the nozzle entrance Mach number and total massflow passed,  functions of both combusted properties and exit plane conditions,  right up to that choke point,  where they finally lock in at the combusted condition.  In always-choked supersonic designs,  the nozzle is convergent-divergent,  but of very modest exit bell expansion ratio (under 2).  Combusted conditions and massflow for a given pressure are always locked in,  but exit bell performance is subject to atmospheric backpressure-induced flow separation,  a very complex and difficult topic not covered here.  

Now to the engine balance.  You should always start at "critical inlet" (both streamtube area ratio and stagnation pressure recovery ratio at maximum).  The matchpoint is the combusted conditions existing at the combustor exit (which is also the nozzle entrance).  

The inlet sets the airflow.  Fuel/air ratio sets the fuel flow.  Fuel/air ratio and air stagnation temperature set the combusted gas properties.  Properties and flow rates and delivered inlet conditions set the combusted stagnation pressure.  From those same properties,  and the throat size,  the massflow that the nozzle can pass can be determined.  If ingestion exceeds nozzle capability,  the inlet must go subcritical and spill air,  thus reducing total massflow.  If ingestion is less than nozzle capability,  the inlet must go supercritical and reduced delivered stagnation pressure,  which reduces nozzle massflow capability.  How the fuel/air ratio varies with this depends upon how the fuel control scheme actually operates,  another item not covered here.  

This calculation is a very laborious iterative process if done manually.  And I have not yet even mentioned considering the massflow coming off any ablative materials lining the combustor,  or nozzle entrance.  In some designs,  this ablative massflow can be comparable to the fuel flow itself,  so neglecting it can be a really,  really serious error.  It doesn't really figure into the combusted gas properties that depend upon fuel/air ratio and inlet air stagnation temperature,  but it does figure into the massflow that the exit nozzle throat must pass. 

So NOW you know why the typical textbook simple cycle analysis models (that really do work acceptably well with turbine engines) are so inappropriate for ramjets.  That's why you'll probably want my book,  when I get finished writing it.  Happy calculating!

GW
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PART 2 --  On Compressible Flow Analysis -- 5-4-13
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Effective ramjet cycle analysis demands that the flow state be analyzed throughout the engine.  This is because component total (stagnation) pressure losses in a component depend very strongly upon the flow state entering that component.  That dependence makes the constant pressure loss inputs of the typical “classic” textbook cycle analysis very unrealistic indeed.  This situation obtains because ramjets have no mechanical compressor,  whose pressure rise in gas turbines dominates all else in the analysis.  

Flow speeds inside the burner and nozzle entrance contraction of a propulsive ramjet are typically from a third of a Mach number to sonic.  In some designs,  there are zones in the inlet with supersonic flow and shock waves.  It is not possible to model such conditions in any way realistically,  with ordinary incompressible flow models.  Even in the subsonic zones,  speeds are usually high enough to warrant compressible flow models as the far-better analysis.  

Classic compressible flow analysis works in local Mach number,  local flow area ratioed to the sonic area,  mass flow rate,  total temperature,  and total pressure.  The static variables are computed from these.  All of this depends upon the ideal gas model (simple equation of state,  and constant properties).  The component to be analyzed is a control volume,  and is termed “adiabatic” if there is no heat transfer across the lateral boundaries.  (That concept excludes the sensible heat contained in the entering and exiting streams.)  The usual models exclude transient effects,  being steady flow analyses.    

There are increasing difficulties of analysis as more real-world effects are included in the model.  These variations include isentropic stream tube analysis (adiabatic,  no heat release in the stream,  no total pressure loss),  simple friction- or flow separation-induced total pressure loss  (adiabatic,  no heat release),  and combustion heat release inside an otherwise adiabatic component.  See Figure 1.  

 

                Figure 1 – Basic Compressible Flow Analysis

The fundamental analysis assumptions that one must make are circled to the upper left in Figure 1.  The realistic analysis options are in the bottom half.  None of this could be successfully (or even realistically) used in the analysis of a supersonic-combustion ramjet (scramjet),  because neither the adiabatic wall or the ideal gas assumptions apply under those conditions.  So,  scramjets are out-of-scope here.

Handling classic streamtube analysis (isentropic flow) is the easiest,  because Mach number and static properties vary with area ratio,  but stagnation temperature and pressure are constant through the component.  Calculating area ratio from Mach number is straightforward,  but the reverse (the most common case) requires a transcendental numerical solution.  See Figure 2 and Reference 1.

                Figure 2 – Isentropic Streamtube Model

For this model the equations are:

                T_t/T = 1 + 0.5*(γ – 1)*M²     where γ = specific heat ratio for the ideal gas

                P_t/P = (T_t/T)^{γ/(γ – 1)}

                A/A* = (1/M)*[(T_t/T)/(0.5*(γ + 1))]^{0.5*(γ+ 1)/(γ – 1)}

and the static pressure and temperature are ratioed from the total values with T_t/T and P_t/P.  Speed of sound can be computed from the static temperature and gas properties.  It and the Mach number give flow velocity.  The static pressure and temperature,  and the ideal gas properties,  give the density from the ideal-gas equation of state.  

There are only a very few items in a practical ramjet to which the simple streamtube analysis would apply.  One is the flow contraction from the combustor into the nozzle throat,  most properly done with combusted-gas properties.  Sometimes there are entering airflow streams,  prior to any physical capture or any external shock waves,  and prior to wetting any external surface,  that model this way.  All other components have at least a friction- or separation-induced total pressure loss.  Some (like at least the internal portion of the inlet) usually have shock losses,  too.

To analyze a component with a total pressure loss but no heat release,  it is useful to re-express the basic mass conservation equation in terms of these variables,  and isolate all the Mach number items into a function on one side.  See Figure 3.

                Figure 3 – Compressible Flow with Total Pressure Loss Only

That basic mass flow definition at any one station is:

                w = ρ*V*A          where ρ is the ideal gas density

This rearranges to what I like to call the f₅ function,  although that name is arbitrary:

                [w/(P_t*A)]*[(R*T_t)/(γ*g_c)]^0.5 = M*(T_t/T)^{0.5*γ/(γ – 1)}  =  f₅

If you already know the flow state entering the component,  you simply compute f₅ from the Mach number.  If not,  compute it from the massflow and total properties.  At constant massflow,  it ratios across the “lossy” component as shown in the figure,  which includes any area change as well.  

                Sta.b f₅ = Sta.a f₅*(A_a/A_b)*(P_tb/P_ta)

From the f₅ function at the outlet (station b),  the Mach number can be transcendentally determined.  From it the total/static ratios are computed,  and all the static properties determined.

This kind of component analysis applies to inlets downstream of terminal shock waves,  through all the subsonic inlet ducting components,  and including the air entry into the combustor.  It is quite easily used to model baffle-type flameholders in constant area ducts,  or sudden-dump flame stabilization schemes.  Even side-entry inlet elbows are amenable to analysis with this model.  

The total pressure loss is typically understood as proportional to the approaching dynamic pressure in incompressible flow.  In compressible flow this takes a different functional form,  right up to Mach 1 on the approach,  yet the proportionality constant is the same as the one used in incompressible analysis.  That allows the correlations for a variety of geometries,  already characterized for incompressible conditions,  to be used in the compressible models  required for ramjet work (see also Figure 4).

                incompressible     P_tb – P_ta  = K*q_a   where q_a = 0.5*ρ*V_a²  and K is the empirical loss coefficient

                compressible        P_tb/P_ta  =  exp(-0.5*N_D*γ*M_a²)   where exp is the base-e exponential and N_D is the “dissipation number”.  Plus:

                N_D = K

                Figure 4 – Pressure Loss Factors for Compressible Flow

The combustor itself is a device with (effectively) both mass addition (the injection of fuel) and heat release by combustion in the flow,  although the lateral boundaries are typically quite adiabatic (being heat protected in some way).  For that kind of analysis,  there are two options:  (1) the total pressure ratio can be correlated for a choked exit nozzle,  and (2) the total pressure ratio has to be computed from first principles (required for unchoked nozzles).  In passing,  one should note that the massflow generated from any ablative heat protection should be included as another injected massflow item.  

The second case (“first principles”) depends on another arbitrarily-named flow function f₆,  derived from the ratio of the mass conservation equation and the momentum equation (in the form of a stream thrust).  It also depends upon the combusted stagnation temperature and gas properties.    The burner duct area is assumed to be constant.  

                f₆  =  M*[γ*(T_t/T)]^0.5/(1 + γ*M²)

                sta.b f₆  =  sta.a f₆*(1 + f/a)*[(R_b*T_tb)/(R_a*T_ta)]^0.5

One computes the f₆ function at station a,  from its Mach number.  Then one ratios that with the fuel/air and total temperature factors to produce the f₆ function at station b.  The f₆ definition is then transcendentally solved for the Mach number at station b.  

There is a relation between Mach numbers and static pressures across a heat release device.  It is essentially just the momentum equation.  The station-a static pressure,  and both Mach numbers,  are now known,  so the static pressure at station b can thus be found.  This can be ratioed-up to find station b total pressure.  The subscripts on specific heat ratio are there to emphasize that the uncombusted gas properties at station a are different from the combusted gas properties at station b.   This applies to the f₆ definitions,  too.

                P_a*(1 + γ_a*M_a²)  =  P_b*(1 + γ_b*M_b²)

This process involving the “primitive” relations is depicted in Figure 5.  It is applicable to either case,  but is most useful for the lower-speed designs that feature unchoked nozzles.  The momentum equation that is part of the f₆ function,  and which is used explicitly in the P_b determination,  would be different if there were significant momentum in the injected fuel stream,  such as in the solid propellant gas generator-fed ramjet (ducted rocket).  That change is beyond scope here.

                Figure 5 – “Primitive” Burner Model

It is possible to correlate the burner total pressure recovery ratio P_tb/P_ta in terms of parameters that depend upon thermochemically-calculable variables:  the combusted temperature and gas properties.  Those correlation parameters typically depend upon a choked ramjet throat in their very definition,  so this approach can be used for that class of designs only.  However,  this does make the burner component analysis easier,  if such a correlation from test data is available.  

For that (choked,  correlated) case,  the burner exit (station b) Mach number is “locked-in” at a single value by the contraction ratio to the choked throat,  as in classic streamtube analysis.  That sets the T_t/T and P_t/P ratios.  If the P_tb/P_ta ratio is known from a correlation,  then P_tb itself is directly calculable.  Thus the static variables can be determined,  as is usual.  Unlike typical loss data,  the total pressure ratio correlation must be determined from actual test data in the actual geometry.  But,  once available,  this is a much easier way to run performance predictions with the cycle analysis.  See Figure 6.

                Figure 6 – Correlated Burner Model

Inlets are a bit unique.  These could be analyzed with the basic pressure-loss model at constant stagnation temperature,  but there are no correlations actually kept for real inlets in terms of a loss coefficient K_T or N_D.  Subsonic inlets are often analyzed in terms of a static pressure rise versus a convenient dynamic pressure somewhere in the component.  This definition is not very useful (or even very repeatable) above approaching-stream speeds of Mach 1.  

None of this helps to determine the size of the actually-captured airstream,  either.  That depends primarily upon downstream flow demand or flow resistance (see the discussion in Reference 2 for an alternative incompressible formulation leading to the same conclusion).  

The supersonic inlet performance definition is not particularly discriminating at subsonic approach speeds (but it is still useful).  It works very well for all supersonic approach speeds,  and is also very repeatable at those conditions.  The effects of inlet-intrinsic performance are separated from the effects of downstream flow demand-or-resistance quite easily by means of spillage margin or supercritical pressure margin parameters.  The inlet-intrinsic effects correlate quite easily with approach Mach number and attitude angles,  and there are now a lot of wind tunnel data available for a variety of geometries.  

In its basic form,  there are two correlation parameters of interest,  the maximum-available streamtube area capture ratio (A∞/A_C)_CR or ARCR,  and the maximum recoverable total (stagnation) pressure ratio (P_t2/Pt∞)_CR or PRCR.  Both of these are correlated from wind tunnel tests versus the primary variable freestream Mach number M∞.  Secondary variables (plotting parameters) would be the pitch,  yaw,  and roll attitude angles.  The actual streamtube area recovery ratio AR is related to ARCR by the spillage margin SM.  The actual total pressure recovery PR is related to PRCR by the supercritical pressure margin PM.  One or the other margin must be zero for the other to be non-zero.  “Critical inlet” is when both margins are zero.  See Figure 7.

                AR  =  A∞/A_C      = ARCR*(1-SM)                                where SM is a decimal,  not a percentage

                PR  =  P_t2/P_t∞     =  PRCR*(1-PM)                               where PM is a decimal,  not a percentage

                Figure 7 – Supersonic Inlet Model Is Also Usable At Subsonic Speeds

What is so very peculiar about this,  is the AR-ARCR-SM model for ramjets.  Ramjets can only “sweep out” their air,  they cannot “pump” it in.  Only the cowl area A_C can be “swept out”,  which is why ARCR is limited to values of no more than 1 for ramjet application.  This is quite different from the behavior of the very same inlet components with turbine engines,  where the ingested massflow is set by the engine demand (essentially by the rotation speed).  At very subsonic approach speeds,  turbine inlet AR can easily far exceed unity.  (The PR-PRCR-PM model is the same for both ramjet and turbine applications.)  

Turbine installations must always have the latitude to spill excess air that the engine cannot demand.  Therefore,  turbine installations always operate at PM=0 and SM > 0.  The reverse is true of properly-sized ramjets:  SM = 0 while PM > 0,  but PM must still be minimized to maximize performance.  This affects the size and placement of external compression features in supersonic designs:  ramjet needs shock-on-lip or shock-inside-lip to ingest air at maximum streamtube recovery,  while turbine always needs shock-outside-lip to freely spill while operating at maximum pressure recovery.  

You cannot guess even semi-realistic values for the ARCR and PRCR trends-with-Mach for anything other than a nose-mounted pitot (normal shock) inlet at anything other than very low attitude angles (under 10 degrees).  A flow-aligned nose pitot inlet that is well-designed will show PRCR in the 0.98 to 0.995 range subsonically,  and very close to (but a little below) the theoretical normal shock total pressure ratio supersonically.  Its ARCR will be very close to 1.00 both subsonically and supersonically,  for which the A_C is the so-called “highlight” area.  

These pitot inlets are not very useful (due to high shock losses) beyond about Mach 2 flight speeds.  However,  they cover very well the range from high subsonic to that maximum (“low speed range designs”).  Podded-engine thrust is generally greater than pod drag even at low subsonic speeds,  but engine specific impulse is generally less than solid rocket,  from about half a Mach number-on-down.  

“Supersonic” inlets have some sort of external shock-wave compression features.  These are usually either conical center bodies,  or two-dimensional ramp surfaces,  that extend forward of the cowl lip.  There is a shockwave-detachment “lower speed limit” with all such designs,  usually in the neighborhood of Mach 1.4 to 1.6,  as can be seen in the ramp and cone shock figures of Reference 1.  

From this limit downward,  PRCR may approach unity,  but ARCR trends rapidly to near-zero.  (For some designs,  this same behavior starts at much higher speeds,  where the ramp or cone shock falls ahead of the cowl lip,  called “shock-on-lip speed”.)  Inlets of this external-compression type are generally useful from about Mach 1.6,  to as high as Mach 6,  that being pretty close to the very fuzzy “limit” for getting useful thrust out of a subsonic-combustion ramjet.  These are “high speed range designs”.  

That fuzzy “limit” depends more upon the engine installation drag than it does upon anything intrinsic to achievable engine thrust production.  The thrust-achievable diverges more and more from what these flow models can predict,  above around Mach 5 or so.  That effect is due mainly to ionization of high-temperature gases in the combustor.  This is,  in turn,  due to high inlet total temperature effects upon the thermochemistry.  Recombination of ionized species cannot contribute to nozzle thrust production.

Nozzles present quite a dichotomy for design analysis purposes.  There are two types of interest to subsonic-combustion ramjet designs.  Low speed-range designs feature pitot inlets and convergent-only nozzles that are not always choked (under Mach 1 at minimum area).  High speed range designs feature external compression features and convergent-divergent nozzles that are always choked.  It is imperative that any convergent-divergent nozzle must be choked,  otherwise the divergent bell actually reduces thrust,  and quite substantially.  See Figure 8,  which matches the station numbering of Reference 3.

The choked convergent-divergent nozzle is the generally-easier case to analyze,  excepting for flow separation effects in an over-expanded divergent bell,  which is beyond scope here.  Rather than use a primitive isentropic model of the contraction coupled with a primitive minimal-pressure loss model for the bell,  the methods developed for rocket nozzle analysis have become standard,  just at far lower pressure ratios and much smaller divergent bell expansion ratios in the case of ramjet application.  

Essentially,  this rocket nozzle approach is really the same thing as the coupled primitive models,  except that a nozzle kinetic energy efficiency parameter is used instead of a bell friction loss parameter.  The contraction is analyzed as isentropic,  with the approach Mach number “locked in” by the contraction ratio.  The nozzle kinetic energy efficiency applies to the gas flow momentum term in the nozzle thrust (momentum conservation) equation,  not to any of the (pressure)*(area) terms.  It is commonly correlated from the exit cone half angle (or effective half angle if a curved bell) as:

                η_ke  =  0.5*(1 + cos(α))                   where α  =  half angle                                                                                   

                F_noz  =  η_ke*w_e*V_e  +  P_e*A_e  -  P_amb*A_e     for unseparated flow,  “e” denotes exit, “amb” ambient

It can be shown (not here) that this primitive formulation for nozzle thrust can be transformed to the far-more-convenient compressible-flow variables,  using something called the nozzle thrust coefficient, which depends upon pressure ratio,  kinetic energy efficiency,  and expansion ratio:

                F_noz  =  C_F*P_o*C_D* A_t                        where “o” is chamber stagnation,  “t” is throat

                C_F  =  (A_e/A_t)[(P_e/P_o)(1 + η_ke*γ*M_e²) – (P_amb/P_o)]

for which C_D is the nozzle massflow discharge efficiency,  and A_e/A_t,  M_e,  and P_e/P_o are related as follows:

                P_e/P_o = (1 + 0.5*(γ-1)*M_e²)^(-γ)/(γ-1)

                A_e/A_t = (1/M_e)[(1 + 0.5*(γ-1)*M_e²)/(0.5*(γ+1))]^{0.5*(γ+1)/(γ-1)}

                Figure 8 – The Two Types of Nozzles

For the case of estimating the performance of a given (and properly-sized) geometry,  the expansion ratio A_e/A_t,  gas specific heat ratio γ,  chamber pressure P_o,  effective throat area C_D*A_t,  and ambient backpressure P_amb will be known inputs.  The exit bell geometry has a conical half angle α (or its effective value),  from which η_ke can be quickly calculated.  The ratio P_amb/P_o is also easily calculated.  

The isentropic streamtube relation transcendentally yields a supersonic M_e for input A_e/A_t and γ.  The isentropic static/total pressure relation yields P_e/P_o for input M_e and γ.  For known A_e/A_t,  P_e/P_o,  γ,  η_ke,  and M_e,  the C_F equation yields a C_F value.  From it and the values of P_o and C_D*A_t,  F_noz is easily calculated.  

Nozzle thrust,  inlet ram drag,  and (depending upon the thrust-drag accounting system) other propulsive drag items must all be combined to compute the ramjet engine thrust.  That topic is not addressed here.  

The areas A_e and A_t in the thrust equation and expansion ratio really should be effective values corrected from physical geometry by the effects of boundary layer displacement thickness through the engine and nozzle.  However,  those correction factors are typically numbers very nearly unity,  and therefore essentially the same size,  so that they very nearly divide-out of the expansion ratio,  but not the throat massflow equation.  The same is true of the isentropic contraction toward the throat.  

Thus,  ratioed areas may be computed directly from geometric values without any appreciable error in the analysis.  That is most definitely not true for the mass conservation relation,  as it depends directly upon the effective size of the throat area C_D*A_t:

                w_t  = ρ_t*V_t*C_D*A_t             where C_D is the discharge efficiency of the throat,  typically near 0.98

At constant massflow for a choked throat,  the throat velocity is always sonic,  simplifying the equations considerably.  Gas properties and total temperature can be lumped into a variable called c* (the “characteristic velocity”),  so that the choked-throat mass flow relation becomes proportional to P_o:

                w_t = w_o = w_e = P_o*C_D*A_t*g_c/c*    where g_c is the gravitational constant for units conversion

                c*  =  {[(0.5*(γ+1))^{(γ+1)/(γ-1)}]*g_c*R*T_tot/γ}^0.5

For purposes of balancing the ramjet engine,  both the massflow and the total pressure approaching the nozzle will be known at the combustor exit (usually “station 4” in a ramjet,  see also Reference 3),  being calculated from inlet ingestion and fuel injection control,  and all the component efficiencies or loss factors.  At that same total pressure,  the nozzle has a massflow capacity,  calculable as just above.  

If the nozzle massflow capacity is less than the engine massflow,  the spillage margin must increase to reduce ingested air massflow,  and so the balance is iterative.  On the other hand,  if the nozzle massflow capacity is greater than the engine massflow,  the inlet supercritical margin must increase,  so as to reduce engine total pressure (used as nozzle inlet total pressure),  thus reducing the nozzle massflow capacity to match.  This is also an iterative balance.  

The choked convergent-only nozzle works exactly the same way,  except that A_e/A_t,  M_e,  and η_ke are all unity,  for some simplification.  The engine balance also works exactly the same way,  but only as long as the nozzle is choked.  The sonic static pressure P_crit at the throat (which needs to be at least equal to ambient in convergent-only geometries),  ratioed to approach total,  is given by:

                P_crit/P_o = [0.5*(γ+1)]^(-γ)/(γ-1)

The unchoked convergent-only nozzle presents an additional level of complexity,  since the nozzle massflow capacity is no longer “locked in” proportional to P_o by a fixed throat Mach number,  at any given value of approach total pressure.  For an unchoked convergent-only nozzle,  P_amb = P_thr always.  Therefore,  the total and ambient pressures may be used to compute subsonic throat Mach number:

                P_o/P_amb = (1  +  0.5*(γ-1)*M_thr²)^γ/(γ-1)

Once the throat Mach number M_thr is known,  the throat static temperature T can be ratioed from the total temperature T_tot,  and used with the gas properties to compute throat speed of sound,  and then throat velocity:

                T_tot/T = 1  + 0.5*(γ-1)*M_thr²

                T  =  T_tot/(T_tot/T)

                c  =  [γ*g_c*R*T]^0.5                   where g_c is the gravitational constant for units consistency

                V_thr  = Mthr * c

The throat density comes from throat static properties and the ideal gas law:

                ρ_thr  =  P_amb/RT

and the throat massflow capacity (at the oncoming total pressure) comes from the primitive variables:

                w_thr  =  ρ_thr*V_thr*C_D*A_t

Once again, if the throat massflow capacity is less than the oncoming engine massflow,  then air must be spilled to match massflows,  in an iterative balance.  For properly-sized components,  this is what will always occur in low speed designs,  especially at subsonic flight speeds.  On the other hand,  if the throat massflow capacity is greater than the oncoming engine massflow,  then oncoming total pressure must be reduced (requiring an increase in pressure margin).  

For designs operating at subsonic flight speeds,  there is no shock wave mechanism by which to raise pressure margin,  there is only “diffuser stall” (flow separation in the divergent portion of the inlet diffuser).  That flow separation leads to very severe flameholding,  combustion stability,  and heat protection problems (none of which are amenable to one-dimensional flow analysis of any type),  so this case must be avoided at all costs by properly sizing the components.  

A summary of the flow functions used in this cycle analysis is given in Figure 9.  Besides those,  the definition of dynamic pressure is:

                q  =  0.5*ρ*V²  =  0.5*γ*P*M²       where both ρ and P refer to static (thermodynamic) values

                Figure 9 – Summary of Compressible Flow Functions

References:

1. National Advisory Council for Aeronautics (NACA),  Report 1135 Equations,  Tables,  and Charts for Compressible Flow”,  Ames Research Staff,  1953.

2. Sighard F. Hoerner,  “Fluid Dynamic Drag”,  published by the author 1965,  and later by his widow;  specifically Chapter IX “Internal-Flow Systems”.    

3. Chemical Propulsion Information Agency (CPIA) Publication 276 “Recommended Ramburner Test Reporting Standards”,  March 1976.

 
Update 5-17-14:

I have programmed a sizing code and a performance code for the low-speed range ramjets,  using an obsolete programming language with which I am nevertheless familiar.  The performance code includes also routines for flame stability estimates,  and for combustor-cooling heat transfer estimates.  

These codes are working and verified,  so if someone needs a design for an engine flying between high subsonic and about Mach 2 or 2.5,  please contact me.  I can help you.  (But I don't do it for free.)  

Plans call for the corresponding high-speed range codes to be operational in the next several days.  That would be for engines flying between about Mach 1.5 and Mach 6.  Will post a notice here when that is done. 

Update 5-30-14:

I have a working high-speed range sizing code.  It currently has 9 different inlet models to choose from,  some nose-mounted,  some side-mounted.   Just barely got started on the high-speed range performance code.

Update 6-6-14:

I made my first execution run of the high-speed performance code today.  It needs some investigation to verify proper operation,  but those first run results look very,  very good.  

I also found in some old reference materials in my data library a way to estimate fairly closely the recovery curves of an inlet with an arbitrary design (shock-on-lip) Mach number,  something I had forgotten that I had done.  Since this was almost 4 decades ago,  I guess I can be forgiven for having forgotten.  

I also worked out a way to rescale some real additive drag data to other design Mach numbers.  That got incorporated into the high-speed sizing and performance codes,  where it is applicable,  which run on a selection of different inlet models pre-built into the codes.  

All of these inlet performance estimation techniques will soon become an inlet system estimation code that is a companion to the suite of low-speed range and high-speed range sizing and performance codes.  In that upcoming inlet code (watch this space for further updates),  I will include the best estimates that I have for the loss factors associated with all the various flameholding and combustor entry design options.  

The now-operational sizing and performance codes mean that I can size liquid-fueled ramjet engines for any application that anyone might have.  These would be basic cycle analysis results,  adjusted to provide some crude guidance for flame stability issues,  and with a combustor heat transfer estimate as an option.  Those last two items are things no one else's cycle analysis has ever offered.  

That means I can now estimate the overall engine geometry and performance parameters fairly quickly,  for pretty much any application of ramjet propulsion.  The art of getting fuel vaporization,  real flame stability,  and real turbulent mixing and efficiency are not addressed by this augmented cycle analysis capability.  And they really are arts,  not so much science.  I do know those arts,  and I can do that work,  but I don't do it "for free".  

Please contact me if this kind of ramjet engine design work is something you need done.  But,  be aware that this topic is one covered very extensively by the US "ITAR" rules.  I cannot do it for most foreign inquiries.  

Update 7-11-14:

I have updated something I call the "TWP correlation" to a verifiably far-more-accurate form,  and modified the high-speed range ramjet codes for sizing and performance accordingly.  This package of modifications is a way to input the equivalent of a 600+ data-item table-look-up file for thermochemical calculations of combusted-gas temperature and properties,  but with only 11 numbers input to the ramjet analysis code,  and yet still not sacrifice any accuracy.  These are 1%-class design analysis tools!  (I still need to modify the low-speed range codes to this new configuration.)

There is not,  and has never ever been,  anything like this available in industry or the literature.  Watch for my ramjet book.  This way of handling thermochemical data will be in it.  I am still working on that book.  It'll be the "Hoerner Drag Bible" of ramjet propulsion.

There are four models pre-loaded in my codes this way for JP-5/Jet-A kerosene,  JP-4/Jet-B wide-cut distillate,  JP-7 high-temperature/low volatility kerosene,  and RJ-5/Shelldyne-H high-density synthetic.  Given thermochemical data to correlate,  I can add more.  It takes about half a day to accurately correlate the data for a fuel in this way.  Ballpark-ish approximations are possible far more rapidly,  given nothing more that stoichiometric air/fuel ratio,  lower heating value,  and a specific gravity.

Update 9-5-14:

I have worked out an elegantly-simply way to analyze the massflow capability of unchoked convergent-only ramjet nozzles for the low speed-range designs.  This is fully-compatible with the new updated-TWP method of handling thermochemical data with very few inputs.  I have looked carefully at all the calculation steps,  and verified that there are no unexpected sources of data error.

I will input this modification to the low-speed codes when I revise them for the updated TWP method.  That has not been done yet (watch this space for future updates).  The high-speed codes are already fully modified for the updated-TWP method,  and for them,  unchoked-nozzle analysis is not (and never has been) an issue.

Update 9-20-14:

I have found in my old inventory (and re-activated) an old computer code I wrote decades ago,  that analyzes the production and decay of turbulence in ramjet combustors.  Such turbulence decaying to the viscous dissipation scale is what actually produces the mixing necessary to get good combustion efficiency in pre-mixed fuel-and-air flames,  typical of liquid-fueled ramjet systems.

I have three old empirical "rules-of-thumb" from liquid ramjet work long ago,  that give very crude indications of the combustor length that might be required to get good mixing and combustion. When I plug estimates like this into my old turbulent mixing code,  I can determine what might be an appropriate pressure loss across the the flame stabilizing element.

It is that pressure loss that determines the turbulence kinetic energy available to do the mixing.  More pressure loss is better for mixing,  but worse for engine cycle efficiency.  This is a design trade-off that one MUST address.  This is not a cycle code feature,  it is a separate analysis!  Even so,  these things must be done together,  as the engine design process proceeds.

For solid gas generator-fed systems (the so-called "ducted rocket",  not really the same as the so-called "air-augmented rocket"),  the required mixing length is somewhat longer,  but otherwise the combustor design process is just about the same.  That kind of ramjet is also complicated by the heterogeneous-phase composition of the fuel effluent to be burned in the airbreathing combustor. And THAT is also dependent upon the formulation of the fuel propellant in the gas generator,  completely unlike the typical liquid system.

With low-metal "hydrocarbon" fuels for reduced or minimum smoke applications,  only certain inlet entry integrations will actually work.  I cut my teeth in ramjet work finding out exactly what works and does not work,  decades ago.  With high-magnesium metallized fuels,  that design answer is very different,  and I did that,  too. With metallized formulations involving boron,  the answer is yet still different from the other two cases,  and I also did a lot of work there.  I actually have extensive experience with all three types,  obtained decades ago.  

If you need help in these areas (flame stabilization and combustor mixing design,  or solid gas generator-fed ramjets of any of the three kinds),  please contact me. But,  be aware that all of this falls under the US "ITAR" rules.  I CANNOT do this work for foreign clients.