**In a liquid rocket equipped with a side-inserted
pintle valve** (Figure
1) for its choked throat, the
propellant flow rate is (at least conceptually) independent of the operating
chamber pressure. The steady-flow
chamber pressure is “set” by flow rate,
throat area, and chamber characteristic
velocity c*, which is a weak power
function of chamber pressure. This situation
models with only the choked-nozzle massflow equation:

w, lbm/s
= (Pc, psia)(C_{D})(At, sq.in)(gc = 32.174 ft-lbm/lb-s^{2})/(c*,
ft/s)

where
c*, ft/s = k (Pc, psia)^{m } with exponent “m” a small number on the order
of 0.01 or so

This variation of flow rate with chamber pressure is very nearly linear, since m is such a small number. Therefore the rate of change of pressure with pintle insertion dPc/dX is a modest number, and not all that nonlinear in its behavior (pressure inversely proportional to throat area at constant flow rate), especially since for the pintle valve, throat area itself varies linearly with insertion over much of the range of insertion.

Figure 1 – Side Inserted Pintle Valve

**In a solid propellant device, such as a fuel-rich gas generator, the internal ballistics with a side-inserted
pintle valve are vastly different and much more complicated.** There must be a

__steady-flow balance__of the massflow through the nozzle and the massflow generated by the burning solid propellant grain. The chamber pressure will rise just high enough to do that, for any given throat area and chamber c* (unless the propellant burning rate exponent is too high, in which case the motor explodes at ignition). Here are the modeling equations:

w_{noz},
lbm/s = (Pc, psia)(C_{D})(At, sq.in)(gc = 32.174 ft-lbm/lb-s^{2})/(c*,
ft/s)

where
c*, ft/s = k (Pc, psia)^{m } = c*_{1000} (Pc/1000 psia)^{m}

with exponent “m” a small number on
the order of 0.01 to 0.05

w_{grain},
lbm/s = (ρ,
lbm/cu.in) (η_{exp}) (S, sq.in) (r, in/s)

where r,
in/s = r_{factor} r_{77F}
and r_{77F} = a Pc^{n} = r_{ref} (Pc/P_{ref})^{n}

with the fairly large burn rate
exponent 0.2 < n < 0.7 rather
typical

and S is
a function of how far into the propellant we have burned, often a very strong function

The item named r_{factor} is the ratio of burn rate
at some other temperature to the burn rate at room temperature, taken to be 77F = 25C. This is most straightforwardly-modeled using
the propellant burn rate temperature sensitivity factor “σ_{p}”. That equation is:

r_{factor}
= exp[σ_{p}(T
– 77F)] which is < 1 for colder than
77F, and > 1 for hotter than 77F

where σ_{p} is usually
between 0.1%/F and 0.2%/F, which is
0.001/F to 0.002/F

and the
notation “exp” means the base-e exponential function

__A throttleable solid propellant device will inherently
operate over a very broad range of chamber pressures Pc__. Whether fuel-rich or not, it is quite rare for the 77F burn rate to
correlate as a single slope n __on a log-log plot of burn rate vs pressure__. There is usually a breakpoint pressure P_{ref}
above which the correlating slope is n_{high}, and below which the correlating slope is n_{low}. The most straightforward way to write down
this behavior is

r_{77F},
in/s = (r_{ref}, in/s) (Pc / P_{ref}, psia)^{n}

where n
= n_{high} for Pc > P_{ref},
and n = n_{low} for Pc < P_{ref}

and r_{ref},
in/s is the burn rate seen experimentally at P_{ref} and 77F.

For this situation,
one can determine the appropriate values for the power law curve fit
constant “a” (in r_{77F} = a Pc^{n}) quite easily from the r_{ref}, the P_{ref}, and the appropriate exponent n:

a_{high}
= (r_{ref}, in/s)/P_{ref}^n_{high} for Pc > P_{ref}

a_{low} = (r_{ref}, in/s)/P_{ref}^n_{low} for Pc < P_{ref}

Chamber c* velocity as delivered in tests typically __does
not show a slope breakpoint__ on a log-log plot of c* vs Pc. Usually,
average c* is determined from lab motor tests at average motor pressure
Pc, for a variety of average pressures
from multiple tests, and these are
plotted on a log-log plot. The slope of
the data trend on that log-log plot is the value of the exponent m. This is quite similar to determining slope(s) n
on a log-log burn rate vs pressure plot.
The same math model power function applies.

With end burning grain designs, there can be a c*-knockdown factor early in
the burn, something generally not seen
at all in internal-burning grain designs,
because of the larger initial motor free volume. That issue was ignored for this study, in the name of simplicity. It only lasts for a few seconds.

The usual reference pressure for c* (and burn rate) is 1000
psia in the US. If the slope break of
burn rate is at a different value, use
that for P_{ref}. But for
c*, the usual quotation is a c* velocity
at 1000 psia, and the slope constant
m. This goes in either form of the c*
model:

c*, ft/s
= k (Pc, psia)^{m} = (c*_{1000}, ft/s)(Pc/1000 psia)^{m}

so that
k = (c*_{1000}, ft/s)/(1000 psia)^{m}

where c*_{1000},
ft/s is the test value of c*, ft/s, at
1000 psia

Values for c*_{1000} are usually in the vicinity of
2000-2500 ft/sec at 1000 psia for fuel-rich propellants, and nearer 4000-4500 ft/s at 1000 psia for
fully oxidized propellants.

The expulsion efficiency η_{exp}
= W_{expelled}/W_{prop}, where W_{expelled} is the change in
motor weight (in a specified configuration) from before firing to after
firing. W_{prop} is the loaded
propellant weight. Both are usually
listed as “lbm”, but being a ratio, the units do not matter, as long as they are consistent. This is obviously an experimental value. It is size-dependent: higher in larger sizes, up to around 6 or 7 inches diameter, above which size usually matters little, if at all.
In that size, expulsion efficiency
is usually 0.98 to 1.00, for a
production-ready propellant formulation.
It can slightly exceed 1.00, if
there is significant mass lost from an ablative insulator inside the motor
case.

The solid propellant density ρ
is best measured in units of lbm/cu.in for the US customary unit choices used
in this article. If you divide that
value by the density of room temperature water (0.03611 lbm/cu.in), you get the specific gravity of the
propellant. From there, you can convert to any units you desire to use.

The burning surface S, sq.in, is a strong (sometimes extremely strong) function
of the distance burned into the propellant charge, known as “web”, typically measured in inches in the US. Theoretically, for a flat-faced end-burner, S is constant across all the values of
web, where the final web number is the
physical length of the grain.

In the real world, __this is almost never true__, there being bondline burn rate augmentation, due to particle packing effects (finer
oxidizer is usually higher burn rate) at the bondline all along the outer
periphery of the grain assembly. This
causes a surface coning effect, leading
to somewhat higher S late in the burn.
The end of the burn starts at a web value less than the grain length, because of this, and this phenomenon also produces a “tailoff
sliver” instead of a sharp burnout event.

For an internal burner, the “full-length” web is some fraction of the
radius of the grain assembly, not its
length, and the surface S is a very
strong function of web burned. This can
be sort-of “rainbow neutral” for appropriately-proportioned cylindrical
segments and “keyhole slots”, or it can
be generally two-level (initially-high,
finally-low) for finned-tube (or slotted-tube) grain designs. That subject is immense, and far beyond scope here. See also Refs. 1 and 2.

What you need is a table of surface
S vs web, in the spreadsheet, for the grain design of interest. The lowest value, and the largest value, are of interest sizing the pressures, flow rates,
and throat area requirements for a gas generator grain design using a
side-inserted pintle valve as the nozzle.

That design situation is because it
is quite common that there is a required max flow rate out of the gas generator
at ignition, which must be obtained
fully-cold-soaked, but within the
pressure limits for the motor case design.
That pressure limit factored up slightly is MEOP for “max expected
operating pressure”, which structurally
designs the case. The required pressure
is highest when the propellant grain is cold-soaked, because of the effects of cold r_{factor}
< 1 on burn rates, while the required
flow rate is usually a fixed value.

There is a pressure factor P_{factor}
applied to MEOP for the ballistic calculations,
to reduce the initial operating pressure from MEOP. This is mostly to compensate for larger S
values a bit later in the burn. P_{factor}
= 1.1 to 1.2 is pretty common for fairly neutral-burning grain designs. It can be a lot higher.

**Sizing the Generator and Valve**

The usual sizing requirements
derive from the specific application for the solid propellant gas generator
device. If fuel-rich, it could be the fuel supply for a gas
generator-fed ramjet system. There would
be a ** larger flow rate** required for low-altitude ramjet ignition
in the dense air, and a much

**required to fly at high altitudes in the thin air. Both flight velocity ratio and density ratio are of interest.**

*lower flow rate*One needs to achieve that ** ignition
max massflow even with the propellant soaked out cold**, and at an acceptable high generator chamber
pressure. The size of the burning
surface S and the achievable burn rate r are very integral to this. The grain design and achievable burn rates
must be compatible with that max flow requirement, at that high-end value for Pc.

One figures the burn rate and c*
from the design pressure Pc at ignition.
These combine with the burning surface S and density and expulsion
efficiency to set the flow rate, __using
the grain massflow generation equation__.
One must adjust burn rate and/or surface S to achieve that flow rate
which is required. __Then the necessary
throat area gets sized by the nozzle massflow equation__.

It will be the __minimum__
throat area needed of the valve! That
seems intuitively wrong, if one knows
nothing about solid propellant internal ballistics. Why that is true depends upon the simultaneous
balancing of both massflow equations,
needed to determine performance vs valve throat area, as described further below.

The ** minimum massflow has to
be obtainable with the grain soaked out hot,
for which the pressure must be far lower** to reduce the higher
burn rate associated with being hot, as well as just producing a low flow rate. This needs to occur at some perhaps-larger
surface S late in the burn. It also has
to occur at a pressure Pc high enough for the fluid mechanics of using the
generator effluent stream to be practical.
That is particularly important for fuel injection into a gas
generator-fed ramjet.

Typically, the desired massflow turndown ratio TDR is
specified, or the minimum flow rate
specified directly. For the gas
generator-fed ramjet application, it is
rarely feasible to use a generator pressure less than about 50 psia, and still
choke both the valve and the fuel injector to which it is coupled. One __uses the grain massflow equation__
at final surface S to determine a burn rate r,
and the low value of pressure Pc required to reach it while
hot-soaked. Then one __uses the nozzle
massflow equation__ to determine the necessary (large) throat area. That is the maximum throat area the valve
needs to be capable of supplying.

The ratio of max throat area to min
throat area (from these sizings) is the minimum area turndown ratio (TDR)
required of the valve, which sets the
ratio of pintle diameter to passage diameter for a side-inserted pintle valve, __whose tip radius equals the passage radius__.
From there, one decides whether to use all the pintle
travel, or just the portion with linear
area variation. Then one sets the
passage size to get the necessary max and min areas.

I set up a spreadsheet “GG
throttle” that does all of this automatically.
It has two worksheets, “geometry”
and “ballistics”. One sizes the valve turndown
in “ballistics”, with the max and min
flowrate calculation blocks. Then one
runs “geometry” to set the pintle/passage diameter ratio for the TDR
required, and the passage size to get
the max and min areas required. These
results are copied from “geometry” and pasted back into “ballistics”, for the performance vs insertion calculation
block.

The worksheet “geometry” is
actually a duplicate of the same worksheet in the throttled-throat liquid
rocket spreadsheet “tthr valve nozzle”.
That one is for liquid rockets with the vastly-different ballistics.

**Doing GG Performance
Calculations vs Valve Insertion**

At any given moment, whether throttled or not, the pressure in a solid propellant device
reflects a steady-flow balance between the massflow generated from the
propellant grain, and the massflow going
through the nozzle. __These
massflows must be exactly equal, or
there is no equilibrium.__

w_{noz}
= Pc C_{D} At g_{c} / c*
with c* = k Pc^{m}

w_{grain}
= ρ
η_{exp}
S r_{factor} r with r = a Pc^{n}
and rfactor = exp[σ_{p} (T – 77F)]

w_{noz}
= w_{grain} is required for equilibrium, so ….

Pc
C_{D} At g_{c}/c* = ρ η_{exp} S r_{factor}
r

Now, for any given burn, r_{factor} is a constant. For a brief interval about any given time
point during the burn, S is essentially
constant, although from time point to time
point, S does change as the grain burns
back. The same is true of throat area
At: even with throttling, the At is essentially constant for a brief
interval about any given time point during the burn.

All we need do is then
substitute-in the Pc-dependent models for r and c*:

Pc
C_{D} At g_{c} / k Pc^{m} = ρ η_{exp} S r_{factor}
a Pc^{n}

It is easy enough to gather all the
Pc factors in one place, and combine
them using the rules of exponents, so
that we can solve for Pc:

Pc^{1-n-m}
C_{D} At g_{c} / k = ρ η_{exp} S r_{factor}
a

Pc^{1-n-m}
= (ρ
η_{exp}
S r_{factor} a k) / (C_{D} At g_{c})

**Pc = [(****ρ ****η _{exp} S r_{factor} a k) / (C_{D}
At g_{c})]^{1/(1-n-m)}
(equilibrium equation)**

This **very nonlinear**
result is the expression for chamber pressure equilibrium. A typical value of n in a throttling system
might be 0.7. A typical value of m might
be 0.05. So the exponent of the argument
in the equilibrium expression would be 1/(1 - .7 - .05) = 1/(1 - .75) = 1/.25 =
4.

**At n = .3 and m = .05, it is still 1.54. For n = 0.95 and m = 0.05, it is infinity!**

*That large exponent explains very neatly why solid propellant devices are so sensitive to changes in throat area and burning surface.*The literature generally says “n
<1 is required”. ** Actually, as the equilibrium equation shows, it is n + m < 1 that is required.** Otherwise,
the denominator of the exponent goes to 0, and the exponent goes to infinity.

__Which is a mathematical way of saying the motor will explode immediately upon ignition__.

In the performance vs insertion
block in the spreadsheet, one must be
sure to use the correct value of “a” in the argument, depending upon what the Pc result turns out
to be. That ensures that the correct
value of burn rate gets used. This is
important if there is a slope break in the burn rate vs Pc log-log plot.

Once the equilibrium pressure at
any given At has been determined (for appropriate values of S and r_{factor}), then one computes the flow rate. Either the w_{grain} or w_{noz}
equations could be used, but the nozzle flow
can be calibrated experimentally, while
the grain S vs web variation cannot. ** So
I recommend that you use the w_{noz} equation.** That

__is__what I put into the spreadsheet.

**Testing Gas Generators**

Most early experimental gas
generator tests are with fixed throats.
If you have the throat geometry and an estimate of its discharge
coefficient, you have the data you need
to size a throat for any given grain geometry and burn rate. Even with side-inserted pintles as
throats, you just size the insertion that
gets the fixed throat area you desire,
and fire the unit that way.

The problem occurs once you put the
inserted pintle under some sort of active control during the burn. The least risky is pre-determined commanded
pintle positions, without any control of
pressure or flow rate. As long as you
estimate the various throat areas correctly,
you can pre-program small pintle movements to produce those areas, and observe the GG response. That is in fact how development efforts
began. The most risky thing to do is
commanding an arbitrary pressure or an arbitrary flow rate. See below for an explanation of why that is
so.

**Example Case**

Size the side-inserted pintle valve
for a fuel-rich end-burning solid propellant gas generator that is to be the
fuel supply for a gas generator-fed ramjet propulsion system. Max required flow rate is 1.4 lbm/s at sea
level takeover. The altitude-compensating
fuel flow rate turndown ratio required is 12:1.
Max gas generator case pressure is MEOP = 2200 psig. The min acceptable gas generator pressure is
about 50 psia. End-burning propellant
grain diameter is 6.5 inches. Evaluate
77 F performance vs insertion at min and max burn surface values, -65 F performance at min S, and +145 F performance at max S.

Figure 2 shows a partial image from the
“ballistics” worksheet where inputs are entered and the max and min flow rate
sizing calculations are made. Yellow
highlighted items are the user inputs. Significant
outputs are highlighted blue and green.
For this process, the area
turndown ratio is the necessary result,
needed for running “geometry” in the next step.

Figure 3 is an image of the “geometry”
worksheet, for which the only inputs
(yellow highlighted) are passage diameter and pintle diameter. 100 mm is an arbitrary but convenient input
for passage diameter, so that the pintle
diameter input has the same digits as the Dpin/Dpass ratio being modeled. Significant outputs are blue
highlighted. The Dpin/Dpass ratio and
the normalized (nondimensional) area vs insertion model are the outputs needed
for this case study. The tabular model
results needed for “ballistics” include the columns in the tabular model for x
= X/Dpass, y = At/Acirc, “condition”,
and the area turndown ratio At/minAt.

Figure 2 – Part of the “Ballistics”
Worksheet Showing Valve Sizing at Max and Min Flow Rates

Figure 3 – Image of “Geometry”
Worksheet

For this case study, I chose to use only the insertion range for
linear variation of At with insertion X.
One copies the normalized model from “geometry” into the correct
location in “ballistics”, and the
“ballistics” worksheet creates the correct absolute-units version. These inputs to “ballistics” are depicted in Figures 4 and 5. The results are depicted in Figure 6. Note that you have to select the correct
value of n, depending upon what P turns
out to be, relative to Pref. Then depending upon whether P is above or
below Pref, you must input the correct
values of “a” and “k” into the appropriate cells for P in each “P, psia”
column.

Figure 4 – Image of the “Geometry”
Inputs Copied to “Ballistics” Worksheet

Figure 5 – Plot of the Normalized
At vs X Model Computed by “Geometry”

Figure 6 – Image of the Performance
vs Insertion Portion of the “Ballistics” Worksheet

There are 4 models computed in the performance vs insertion section of “ballistics”. The first two are both done with 77 F burn rates, one at min S, the other at max S. There is a cold-soaked model done at min S, and a hot-soaked model done at max S. These last two bound the problem in terms of variation. The significant results are Pc, flow rate w, and the derivative of Pc with X, as a sort of sensitivity of the system to changes in throttle position, similar to the gain factor in a control system. These results are plotted in Figures 7, 8, and 9.

Figure 7 – Predicted Chamber
Pressure Pc vs Insertion for 4 Cases

Figure 8 – Predicted Flow Rate
Delivery vs Insertion for 4 Cases

Figure 9 – Predicted Sensitivity of Pc With X, vs X, for 4 Cases

The first impression looking at Figure 7 is that max
insertion (to min throat area) will over-pressure the gas generator (to
destruction) for every condition except the cold size point. That cold size point turns out very close to
the sizing calculations at just over 1.4 lbm/s at just over 2000 psia, as the flow data in Figure 8 and the tabular data in Figure 6 indicate.

For warmer propellant, or higher surface S, your throttle control must simply (and
reliably) stay well away from max insertion (where the tip of the pintle
contacts the far passage wall). Judging by the Figure 7 plots,
max insertion is about 0.43 inches for hot operation at max S, 0.44 inches for 77 F operation at max S, and about 0.45 inches for 77 F at min S
operation. The cold sizepoint is “on the
far wall” at 0.51 inches insertion.

You cannot see the hot min flow
point at max S in Figure 7, at the scale of the plot. But you can see it in the tabular data of Figure 6. That works out very close to the sizing
calculations, at 51 psia Pc (versus a
requirement of at least 50 psia), and a
flow rate that is almost “dead nuts on” for the required 12:1 massflow
turndown. None of the other 3 cases show sufficient Pc or w to be feasible at
the min insertion point, so your control
system will have to reliably stay more inserted than this value.

**Extremely Nonlinear Controls Are
Required**

** One thing that should be
apparent from these plotted results is the extreme nonlinearity of equilibrium
operation versus pintle insertion, just
like the math equations indicate**.
How quickly this pressure equilibrium shifts with pintle position is
indicated by the derivative of the Pc vs X curve with respect to insertion X. Those dPc/dX data are plotted in Figure 9 vs X, for the 4 cases. The variation is not only extreme, but extremely nonlinear. Judging by the tabular data in Figure 6, this derivative varies extremely nonlinearly
between about 500 psi/inch and over 29,000 psi/inch.

*The extreme nonlinearity of
the behavior and its sensitivity to position should explain why a linear or
linearized control never worked with this type of throttle decades ago.*__Those motors always exploded__! I am no controls expert, but I did the ballistics (like these) that
supported the development of a very nonlinear control system with an adaptive
gain factor. __Those motors actually
worked, and precision control was
achieved, along with repeatable reliability__! It was __not__ easy! There were many development failures before
the true nature of the adaptive gain was determined.

This type of throttle valve and
associated nonlinear controls was done between about 1979 and 1994 at Hercules
in McGregor , Texas, for a possible ramjet propulsion upgrade to
the AIM-120 AMRAAM missile. See Figure 10 for a
conceptual sketch. Both gas generators
were end-burners, but the
strand-augmented version was called the “strand-augmented end burner”
(SAEB). The SAEB let us divorce the
required effluent fuel properties from the required burn rate ballistics, and let the fully-oxidized strand propellant limit
motor effective burn rate temperature sensitivity with its lower σ_{p}.

This throttling system was
successfully used with a variety of different fuel-rich propellants in full
scale static (gas generator-only) and direct-connect (ramjet) tests. The missile prime was Hughes Aircraft, and a Hughes employee developed the nonlinear
adaptive-gain controls as a subcontractor to us. My roles in this were (1) generator internal
ballistics, (2) developing fuel-rich
propellants, (3) making the throttle
work with a fuel injection nozzle (patented),
(4) planning and executing the ramjet tests, (5) evaluating and improving the stability and
efficiency of the airbreathing flame in the ramjet combustor, and (6) evaluating predicted weapon
performance with a trajectory code.

One should note that, while the ramjet AMRAAM was never flown by
USAF, essentially the same system is now
operational as the gas generator-fed ramjet “Meteor” in Europe. To the best of my knowledge, the throttle valve in “Meteor” is not a
side-inserted pintle. I do not know the
nature of its control system, but it has
to be very nonlinear, __because the very
same solid propellant ballistics apply__.

Figure 10 – Conceptual Sketch of
Gas Generator-Fed Ramjet Technology for AMRAAM

**Spreadsheet Availability Note:**

The current location of the
spreadsheet “GG throttle.xlsx” is on my laptop,
in the folder “engineering files”,
subfolder “GG throttle valve”. It
merely represents a fast way to do what I once did pencil-and-paper, with nothing more than a scientific pocket
calculator.

**Integration of Choked Throttle
Valve With the Necessary Fuel Injection Geometry**

The actual fuel injection geometry
into a gas generator-fed ramjet has much to do with ramjet performance and
ramjet flameholding, as evidenced in Refs. 3 and 4. There is also the issue of the compressible
fluid mechanics getting from the throttle valve throat, to the actual fuel injection ports into the
combustor. That is an exceedingly
difficult compressible internal flow problem.

The best flameholding geometries that
are known for solid gas generator-fed ramjets which are not hypergolic
magnesium-fueled, are the “dual
adjacent” and “5-ported” injection schemes,
both used in two-inlet dump combustor geometries that are
asymmetrical, at inlets 90 degrees
apart. As for why that is true, see again Refs. 3 and 4.
Be aware that I played a key role in determining that, as well.

The same asymmetric twin inlet
scheme can be successfully used with a single centerline gas generator port on the
combustor centerline (whether choked or unchoked), but only at a performance decrement of around
5% relative to the other two options. At
least, it is only a 5% decrement! Which makes it a usable and very convenient screening
test scheme, despite the small loss in
performance.

It is the single flow passage from
gas generator to combustor that best integrates with a gas generator throttle
valve, avoiding all the
sudden-acceleration and shock-down problems of bifurcated geometries. This should be obvious to the casual
observer, especially one who has ever
dabbled in compressible flow calculations with shock waves, plus a mix of supersonic and subsonic flow
zones.

That is why the selected fuel injection
geometry with the ramjet AMRAAM was for using the 5-ported injector, off-centerline, as a single flow passage from gas generator
to the combustor injection ports.
However, the pressure drop
downstream of the throttle valve pintle would most often lead to supersonic
flow followed by shock-down to an all-subsonic flow, in turn located inside the 5-ported injector. It had to be subsonic upstream of the lateral
injection ports.

The problem was that after bleeding
off some flow through a lateral port,
the remaining injector core flow would reaccelerate supersonic inside
the fuel injector. Where injector
internal flow was supersonic, it could
not effectively “make the turn” to flow out of the lateral ports on the
injector, completely unlike the
fixed-flow form originally developed by CSD (Chemical Systems Division, United Technologies Corporation). In that form,
the injection ports literally are the gas generator throat.

Flow rates delivered to various
regions in the flameholder were then very definitely NOT proportional to the
injector port areas, which they were in the
fixed-flow designs without a throttle valve.
That made the distribution of fuel to the flameholder
unpredictable. Such an error can be (and
often is) fatal to flameholding. At the
very least, ramjet engine performance
suffers.

The required adaptation was
something to enforce all-subsonic flow within the fuel injector, even with a throttle valve pintle
upstream. That requires two different
things: (1) duct area reductions down
the injector, as mass is bled off at
each port location, and (2) a total port
area just barely small enough to enforce subsonic flow throughout the overall
passage downstream of the pintle,
excepting that small region just downstream of the throttle pintle
itself.

That type of design, if done correctly, isolates the port bleed phenomena from the pintle
shockwave phenomena. The stepped
internal diameter is what prevents reacceleration to internal supersonic flow. ** However, the ports cannot be too small, as that would unchoke the throttle valve
upstream.** The correct solution
to this problem resulted in my throttled fuel injector patent, cited here as Ref. 5.

Clearly, the compressible fluid mechanics of the
injector downstream of the valve influence the behavior of its injected streams
into the combustor. The behavior, location,
and proportioning of those streams of fuel have a major, critical influence upon the flameholding and
performance in the ramjet combustor.
This all has to interact correctly to get combustion at all, and must be “tuned up” to get good
performance out of that combustor. And
it has to be adapted slightly for each different fuel propellant. That process is more fully addressed in the
flameholding article, Ref. 3, and it was a big part of the work described
for the ramjet AMRAAM engine in Ref. 4. See Figure 11 for an
illustration.

Figure 11 – The Interrelationship
Between Injection and Flameholding Fluid Dynamics

**There Is Yet Another GG
“Throttle” Approach**

This article covers only the choked-throat
throttle valve for the gas generator-fed ramjet (also known as the “ducted
rocket”). Not covered here, but implied by the ballistics, is the fixed-throat gas generator that has a
fixed flow rate delivery history. That
history can be tailored by the burning surface vs web that is designed into the
gas generator propellant grain design.

There is also the unchoked
generator throat (which has no valve).
This can be a constant fuel/air ratio “throttle”, if the propellant burn rate exponent is high
enough (essentially 1). Achievable burn
rates usually restrict this choice to internal-burning grain designs, as the generator pressure is quite low at
essentially the ramjet combusted total pressure.

Such an unchoked gas generator-fed
ramjet was actually test-flown by the French under the name “Rustique”, and I extensively ground-tested it in ramjet
tests. It works very well indeed, if you have the high-exponent propellants
(which the French did not have, but we
did). That unchoked-throat “throttle
control” topic will be covered elsewhere,
at a future date.

**References:**

#1. G. W. Johnson, “Solid Rocket Analysis”, 16 February 2020, published here on “exrocketman”.

#2. W. T. Brooks, “Solid
Propellant Grain Design and Internal Ballistics”, NASA SP-8076 (monograph
on solid ballistics), March 1972.

#3. G. W. Johnson, “Ramjet Flameholding”, 3 March 2020,
published here on “exrocketman”.

#4. G. W. Johnson, “The Ramjet I Worked On The Most”, 2 August 2021, published on “exrocketman”.

#5. G. W. Johnson, “Fuel Injector for Ducted Rocket Ramjet Motor”, US patent 4,416,112, 1981 (assigned to employer).