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)(CD)(At, sq.in)(gc = 32.174 ft-lbm/lb-s2)/(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:
wnoz, lbm/s = (Pc, psia)(CD)(At, sq.in)(gc = 32.174 ft-lbm/lb-s2)/(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
wgrain, lbm/s = (ρ, lbm/cu.in) (ηexp) (S, sq.in) (r, in/s)
where r, in/s = rfactor r77F and r77F = a Pcn = rref (Pc/Pref)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 rfactor 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 “σp”. That equation is:
rfactor = exp[σp(T – 77F)] which is < 1 for colder than 77F, and > 1 for hotter than 77F
where σp 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 Pref above which the correlating slope is nhigh, and below which the correlating slope is nlow. The most straightforward way to write down this behavior is
r77F, in/s = (rref, in/s) (Pc / Pref, psia)n
where n = nhigh for Pc > Pref, and n = nlow for Pc < Pref
and rref, in/s is the burn rate seen experimentally at Pref and 77F.
For this situation, one can determine the appropriate values for the power law curve fit constant “a” (in r77F = a Pcn) quite easily from the rref, the Pref, and the appropriate exponent n:
ahigh = (rref, in/s)/Pref^nhigh for Pc > Pref
alow = (rref, in/s)/Pref^nlow for Pc < Pref
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 Pref. 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 ηexp = Wexpelled/Wprop, where Wexpelled is the change in motor weight (in a specified configuration) from before firing to after firing. Wprop 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 rfactor < 1 on burn rates, while the required flow rate is usually a fixed value.
There is a pressure factor Pfactor 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. Pfactor = 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 lower flow rate required to fly at high altitudes in the thin air. Both flight velocity ratio and density ratio are of interest.
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.
wnoz = Pc CD At gc / c* with c* = k Pcm
wgrain = ρ ηexp S rfactor r with r = a Pcn and rfactor = exp[σp (T – 77F)]
wnoz = wgrain is required for equilibrium, so ….
Pc CD At gc/c* = ρ ηexp S rfactor r
Now, for any given burn, rfactor 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 CD At gc / k Pcm = ρ ηexp S rfactor a Pcn
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:
Pc1-n-m CD At gc / k = ρ ηexp S rfactor a
Pc1-n-m = (ρ ηexp S rfactor a k) / (CD At gc)
Pc = [(ρ ηexp S rfactor a k) / (CD At gc)]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. That large exponent explains very neatly why solid propellant devices are so sensitive to changes in throat area and burning surface. At n = .3 and m = .05, it is still 1.54. For n = 0.95 and m = 0.05, it is infinity!
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 rfactor), then one computes the flow rate. Either the wgrain or wnoz 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 wnoz 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.
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 σp.
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.
#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).