**Application materials and most of the book went today to AIAA for approval for publication. Timeline is weeks for approval, months for actual publication.**

__Update 9-19-17__:----------------------------------------------------------------

**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.**

__Update 5-17-17__:----------------------------------------------------------------

**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.**

__Update 12-10-12__:-----------------------------------------------------------------

**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.**

__Update 6-5-2016:__------------------------------------------------------------------

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.

*The subscript and superscript formatting in the source document does not translate well into html for this posting. For that I apologize.*

**How an effective cycle analysis is actually best implemented is one of the subjects in a book on ramjet engineering that I am writing.**------------------------------------------------

basic cycle analysis document for ramjet engines:

------------------------------------------------

updates 1-17-13 in red below

------------------------------------------------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

*, or*

**less than***equal to, that swept out by the cowl lip in a ramjet,*

**at most***, where air flow is set by engine speed, size, and inlet air density*

**utterly unlike gas turbine**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.

*, 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.*

**The truth is, for ramjet they simply cannot be modeled as constants**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

*combustor static pressures*

**both***properly-calibrated thrust stand forces,*

**and***. 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*

**and then compared***, 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.)*

**choked sonic-only throat tests****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)*M^{2 }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 f

_{5}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)}= f_{5}
If you already know the flow state entering the
component, you simply compute f

_{5}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 f

_{5}= Sta.a f_{5}*(A_{a}/A_{b})*(P_{tb}/P_{ta})
From the f

_{5}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_{a}^{2}and K is the empirical loss coefficient
compressible P

_{tb}/P_{ta}= exp(-0.5*N_{D}*γ*M_{a}^{2}) 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 f

_{6}, 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.
f

_{6}= M*[γ*(T_{t}/T)]^{0.5}/(1 + γ*M^{2})
sta.b f

_{6}= sta.a f_{6}*(1 + f/a)*[(R_{b}*T_{tb})/(R_{a}*T_{ta})]^{0.5}
One computes the f

_{6}function at station a, from its Mach number. Then one ratios that with the fuel/air and total temperature factors to produce the f_{6}function at station b. The f_{6}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 f

_{6}definitions, too.
P

_{a}*(1 + γ_{a}*M_{a}^{2}) = P_{b}*(1 + γ_{b}*M_{b}^{2})
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 f

_{6}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,**. But,***the total pressure ratio correlation must be determined from actual test data in the actual geometry*__once available__, this is a much easier way to run performance predictions with the cycle analysis. See Figure 6.
Figure
6 – Correlated Burner Model

**. 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**

*Inlets are a bit unique*_{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**for ramjet application. This is***no more than 1*__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

**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***cannot guess*__even semi-realistic__values__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

**is the generally-easier case to analyze, excepting for flow separation effects in an over-expanded divergent bell,***choked convergent-divergent nozzle***. 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.***which is beyond scope here*
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_{e}^{2}) – (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_{e}^{2})^{(-γ)/(γ-1)}
A

_{e}/A_{t}= (1/M_{e})[(1 + 0.5*(γ-1)*M_{e}^{2})/(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

**works exactly the same way, except that A***choked convergent-only nozzle*_{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

**presents an additional level of complexity, since the***unchoked convergent-only nozzle*__nozzle massflow capacity is no longer “locked in” proportional to P__, at any given value of approach total pressure. For an_{o}by a fixed throat Mach number__unchoked convergent-only nozzle__, P_{amb}= P_{thr}**. Therefore, the total and ambient pressures may be used to compute subsonic throat Mach number:***always*
P

_{o}/P_{amb}= (1 + 0.5*(γ-1)*M_{thr}^{2})^{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_{thr}^{2}
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*V

^{2 }= 0.5*γ*P*M^{2 }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.

**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.**

*The now-operational sizing and performance codes mean that I can size liquid-fueled ramjet engines for any application that anyone might have.*__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.

**I do know those arts, and I can do that work, but I don't do it "for free".**

*And they really are arts, not so much science.*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.

Gary;

ReplyDeleteYou wrote:

"See my low-speed cycle analysis document for the details of that case"

I've skimmed the blog but can't locate an entry on that. Can I get a pointer? :)

Randy

Randy:

ReplyDeleteSorry, the low-speed cycle analysis document isn't posted anywhere. At least not yet. That one will likely be broken into two documents, one on cycle analysis, the other on hardware design implementations. Both are intended to be chapters in my book.

GW

Ahhh,

ReplyDeleteMight want to point that out then :)

Randy

Randy:

ReplyDeleteI did. See the updates and changes.

GW