Friday, October 1, 2021

Use of the Choked Pintle Valve for a Solid Propellant Gas Generator Throttle

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, = 32.174 ft-lbm/lb-s2)/(c*, ft/s)

               where c*, ft/s = k (Pc, psia) 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, = 32.174 ft-lbm/lb-s2)/(c*, ft/s)

               where c*, ft/s = k (Pc, psia) = 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/ (ηexp) (S, (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/ 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/,  you get the specific gravity of the propellant.  From there,  you can convert to any units you desire to use. 

The burning surface S,,  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. 

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 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).

Thursday, September 16, 2021

Cassandra Speaks (Yet Again!)

 For those who do not know,  the “Cassandra” in the title was a character from Greek mythology who had the gift of prophecy.  No one believed her,  mostly because they did not like what she prophesied,  and so did not want to believe.   I know how she felt about that.


I foresee what no one wants to see,  so no one sees it coming :  there will NEVER be a final end to the SARS-CoV-2 / covid-19 pandemic! 

It is time to start the next “Operation Warp Speed” and rapidly develop upgraded vaccines that counter these newer,  more contagious,  more deadly variants,  that we can already see coming.  That’s the trend:  toward ever more contagious and ever more deadly forms of this SARS-CoV-2 virus. 

There is no longer any defense for being younger:  the Delta variant is circulating wildly among our children in schools,  and it is killing some of them.  And some of their teachers. 

That is just the nature of coronaviruses:  they mutate fast,  and they vary a lot in their threat characteristics from strain to strain.  The rapid mutation rate is the real enemy here;  it is also partly why we have never developed a cure for the common cold,  which is also mostly caused by several different coronaviruses. 

The other part is that they change so fast that any acquired immunity doesn’t last very long,  which is why you can catch cold after cold on a time scale of only months.   We are already seeing that short immunity effect with the covid-19 booster shot issue. 


It being the case that this virus will be with us forever,  the best weapon we have is new vaccine after new vaccine,  from now on!  Forever!  Which means we start now,  and we do everything we can to speed the process without undermining its integrity.  But history shows these work well,  only when just about everybody gets vaccinated. 

The second best weapon we have is the ancient one of masking / distancing / handwashing,  augmented in modern times by an understanding of microbes and disease.  We will be using those ancient weapons from now on,  whenever an outbreak of a new variant not covered by current vaccines occurs.  This tool also only works well when just about everybody does it.  You can only stop when everybody is vaccinated.

The third best tool we have is shutting everything down and quarantining at home;  and we surely don’t want to have to do that again!  Not unless we are utterly forced to do so!   You avoid it by using the other two tools as effectively as you can.

There are no other tools for fighting such plagues.


As with other lethal or intolerably-dangerous viruses and other microbes that we cannot yet eradicate,  we have to make the vaccines for this plague mandatory,  and allow only a few exceptions,  very few indeed!  Examples of things we already treat in this mandatory way would be whooping cough,  polio,  etc. 

When outbreaks occur with a new variant that resists our current vaccines,  the old masking / distancing / handwashing tools also have to be mandatory until the new vaccine is fully-administered.  It’s not about freedom-to-choose,  it’s about the public safety of all of us:  something long established in our society,  and its many predecessors over the centuries.

Your freedom to choose not to mask,  or not to vaccinate,  ENDS BEFORE you infect others!  Just like your freedom to swing your fist ENDS BEFORE you strike others.  And just like your freedom to yell “fire” when there is no fire inside a crowded theater,  ENDS BEFORE you injure or kill somebody in the panic that ensues.  Anyone who tells you different is lying to you.

The sooner everybody comes to realize the real truth about this “freedom to choose” issue,  the better off we will all be.


Bottom line:  the sooner we implement the continuing upgrades to these vaccines as an ongoing program,  and the sooner we make the resulting vaccines mandatory,  the freer we will all actually be. 

It really is that simple. 


And so also you can figure out when you are being lied-to,  for political or monetization purposes.  In particular a lot of social media sources (and some others,  particularly politicians) profit in some way from feeding you more of what you want to hear,  without regard to truth,  and not feeding you any of the truth that you need to hear. 

Which is something you should never,  ever tolerate!


A final thought: 

Denying schools the ability to control infection rates in their facilities with the vaccination and masking tools,  is essentially criminal child endangerment.  It is essentially criminal assault when businesses are prevented from using these tools.  Both child endangerment and criminal assault are very serious crimes,  but this is happening in a lot of states,  and for no better reason than a faulty political ideology based on misinformation.   There is no real excuse for such dangerous behavior.

I suggest that the local district attorneys in the places where these schools and businesses are located,  should file criminal charges against the politicians in power who are trying to stop the use of these tools against the virus. 

Specifically,  the officials who sign the offending orders,  and the officials who attempt to enforce them,  should be arrested on criminal charges.  Not civil lawsuits,  criminal charges!  There is no office-holder immunity against that. 

And it needs to go through the justice system process,  all the way through trial.  Guilt is pretty clear,  just ask the medical professionals.  We need to make some examples out of those who behave so dangerously.  Then that dangerous nonsense would stop.


Update 9-19-2021:  a shorter and substantially-edited version of this article appeared as a board-of-contributors item in the Sunday 9-19-2021 Waco Tribune-Herald newspaper.  

Wednesday, September 1, 2021

Making Stiff Blends At The Gas Pump

I have experimented with ethanol conversions and stiff ethanol blends in cars,  tractors,  and lawn equipment,  since 2006.  I did this with quite a variety of equipment.  The two full E-85 conversions were a 1944 Farmall-H farm tractor,  and a 1973 VW (air-cooled) Beetle. 

The vehicles I used for stiff blend research included that same 1973 VW Beetle,  a 1960 VW Beetle,  a 1995 Ford F-150 pickup,  a 1998 Nissan Sentra sedan,  and a 2005 Ford Focus sedan.  These were not modified in any way from stock for the blend tests or for routine use. 

The ’73 Beetle was restored to its E-85 conversion,  while the 1960 Beetle was left stock,  when these two vehicles were re-mothballed.  The rest are still in service.

I also have been using E-30 to E-35 blends in two lawnmowers and a wood chipper,  all 4-stroke,  all completely unmodified.  If there were a problem with this,  then after 15 years,  I would know about it.  There is not. 

The automotive manufacturers,  and the 4-stroke lawn and garden equipment manufacturers,  went to ethanol-compatible materials many years ago,  not long after the advent of unleaded gasoline.  The high aromatic content in the unleaded gasolines was just almost as harsh to the older materials as the ethanol.  (Methanol is far worse yet.)

The 2-stroke manufacturers,  and the boat motor people (and the old-time aircraft manufacturers),  did not update to ethanol-tolerant materials.  They are the source of all the tales about how “bad” ethanol is for engines.  And for them (and only them) it is true,  precisely because they never updated their materials selections!  These materials actually go “bad” rather rapidly on ethanol-free gasoline,  just because of the high aromatic content.  The ethanol in an E-8-to-E-10 ULR just causes these already-vulnerable materials to fail only a little faster.

The net result of all this is that I can recommend stiff blends up to E-35 maximum,  as drop-in fuels in any spark-ignition automobile,  no modifications required.  You will not be able to tell the difference in power,  or in fuel economy,  between these blends and straight gasoline.  That is because the natural scatter in your mileage data from tank to tank is larger than any effects of the blends

If you do this in a 4-stroke lawnmower,  do it with a not-so-stiff blend nearer only E-20.  If you do an E-35,  the over-simplified carburetor may need its screws reset. It won’t idle right.  The idea was “drop-in” fuel.  I’ve used E-35 in them,  but not-so-stiff is more of a drop-in fuel.

In the cars,  you are essentially trading increases in energy conversion efficiency for the decreases due to lower heating value (this was also the conclusion of my PhD dissertation,  which was based on earlier work I did with ethanol in piston engine airplanes). 

You will see cleaner spark plugs and a potential increase in your oil change interval,  because your oil will not darken as quickly.  Those of you who do your own repairs and take the cylinder heads off,  will find combustion chambers and piston faces that are cleaner of carbon deposits,  than any you ever saw that were operated on gasoline.

Really old cars,  from before the switch to unleaded fuels,  might be vulnerable if the factory-stock carburetor and fuel pump parts are still installed.  However,  these were replaced in most such vehicles long ago,  due to simple age,  and normal wear-and-tear.  The replacement parts were made from ethanol-compatible materials,  starting many years ago.

How To Do This

To do this reliably,  you need to “calibrate” the marks on your fuel tank quantity gauge.  When you pull into the filling station,  you need to be able to read the gauge and know how many gallons it will take to fill you up.  I leave that determination to you;  everybody has a different vehicle,  and needs to do this for themselves. (See Update 9-5-21 below,  at end of article.)

 Once you know how many gallons you are going to put in,  all you need to do is figure out how many of those gallons need to be E-85,  and how many need to be unleaded regular gasoline (ULR).  That is quite easy to figure,  as explained below.

Be aware that the maximum ethanol (“max eth”) content of E-85 is 85%,  and that of ULR (which is a nominal E-10 material) is maximum 10%.  The minimum ethanol (“min eth”) content of the E-85 product in wintertime is actually about 70%.  Most of the time when I tested ULR,  I found its ethanol content to be less than maximum,  at about 8%.

There is a volume ratio R of ULR to E-85 that is associated with the blend strength you want.  This is depicted in Figure 1.  Use the “max eth” curve data for this,  to limit just how strong your blend will be.  R = 2 will get you a blend strength between E-35 (the “max eth” curve value) and E-29 (the “min eth” curve value).  R = 6 will get you a blend strength between E-21 and E-17.  R = 4 will get you a blend between E-25 and E-20. 

Figure 1 --  Blend Number vs Volume Ratio R = ULR to E-85

Once you know your desired value of R,  divide your intended gallons-to-fill by R+1,  to get the gallons of E-85 to load.  Your fill-up gallons minus your E-85 gallons is your ULR gallons to load.  It really is that simple!  Most of your cell phones have a calculator,  with which you can make these calculations,  standing right there at the fill-up pump. 

Don’t exceed a “max eth” blend strength of E-35!  That works even in winter,  although you may have to start the engine twice (or even 3 times),  before it sustains.  That’s the maximum practical blend strength for cold starting.  (It is still short of where mileage and power suddenly drop,  which is just about E-42.)  You can use a lower blend strength,  but do not go higher!

To aid your selection of an R value,  the max eth and min eth data,  from which the curves in the figure were plotted,  are included here as Table 1.  Use the max eth data and your max desired blend strength to set R.  Then use the min eth data at your R to find your minimum blend strength,  which is effectively a winter value with typical winter product formulations.

Table 1 – Data From Which The Curves Were Plotted

Related Articles (all are located on this site)

There is a navigation tool on the left side of this page.  Use the list below (jot them down on scratch paper).   Click on the year,  then on the month,  then on the title. 

12-15-09….Red Letter Day:  Ethanol VW Experiment Complete…[ethanol ’73 VW]

11-12-10….Stiff Blend Effects in Gasoline Cars…[’73 VW,  ’60 VW,  F-150,  Nissan]

11-17-10….Nissan Mileage Results on Blends…[Nissan blend experiments]

2-12-11…...“How-To” For Ethanol and Blend Vehicles…[converting the ’73 VW,  and more]      

5-5-11…….Ethanol Does Not Hurt Engines…[general]

6-12-11…..Another Red-Letter Day…[re-mothballing the VW’s]

5-4-12……Energy Storage: Batteries vs Unpressurized Liquid Fuels…[related data]

8-9-12……Biofuels in General and Ethanol in Particular…[related data]

Update 9-5-21:  How to Calibrate Your Analog Fuel Quantity Gauge

You need two things:  a fuel mileage log,  and a set procedure during fill-up to fill the tank to the same mark every time.  There is no way to properly calibrate the gauge without these two things,  so just make up your mind to do it!  You can only do this with an analog gauge that uses a needle with a scale of marks.

Fill it to the same mark (such as 3+ click-offs of the pump;  pick a number and live by it) every single time!  That takes it somewhere slightly above the full mark on the gauge.  Log the gallons filled and the odometer reading every single time.  EVERY SINGLE TIME! 

Between fill-ups,  watch your gauge reading as you roll up miles traveled.  Record the odometer reading as the needle locates at every mark,  down to about the ¼ (or possibly 1/8),  tank marks on the gauge.  There is no other way to do this,  so just make up your mind to do it!  Take care and watch the needle closely over long time baselines,  as turning accelerations,  throttle accelerations,  and braking decelerations will cause it to misread temporarily. 

Do this for at least 10,  and preferably 20-or-more,  tanks of fuel.  That way you get statistical reliability.  Then reduce your data.  Here is how: 

The mileage difference between two fill-up odometer readings is the miles traveled on the gallons loaded at the current fill-up.  Divide miles traveled by gallons loaded,  for the average mpg on that tank.  Do this to about 4 significant figures (example:  20.57 mpg).  Use the corresponding mpg value for each tank in the calculations that follow,  not some overall average. 

The mileage difference between the fill-up odometer reading,  and the odometer reading at the full mark,  is the miles traveled on the overfill quantity.  That miles traveled divided by the average mpg for that tank is the estimated gallons of overfill above the full mark.

The mileage difference between the full and ¾ tank marks is the miles traveled on the top quarter of a tank.  Those miles divided by the average mpg for that tank is an estimate of the gallons in that top quarter of the tank between those marks.  (You will find that a “quarter of a tank” isn’t really 25% of a tankful.  Surprise,  surprise!)

The mileage difference between the 3/4 and 1/2 tank marks is the miles traveled on the second quarter of a tank.  Those miles divided by the average mpg for that tank is an estimate of the gallons in that second quarter of the tank between those marks.

The mileage difference between the 1/2 and 1/4 tank marks is the miles traveled on the third quarter of a tank.  Those miles divided by the average mpg for that tank is an estimate of the gallons in that third quarter of the tank between those marks.

If you have the data (if not,  skip),  the mileage difference between the 1/4 and 1/8 tank marks is the miles traveled on that part of the tank.  Those miles divided by the average mpg for that tank is an estimate of the gallons in that portion of the tank between those marks.

If you do this data reduction at every fill-up,  it is not so onerous.  But either way,  it must be done!  Once you have this data for 10 to 20+ fill-ups,  then average the estimated gallons for overfill-to-full,  for full-to-3/4,  for ¾ to ½,  for ½ to ¼,  and (if you have the data) for ¼ to 1/8 tank.  Those will be the best estimates you can make. 

Now your gauge is “calibrated”.  If you add these things up,  then you know statistically how many gallons it will take to fill,  from each mark,  starting at zero for the overfill condition.

One fill-up simply won’t do it!  You need the statistical reliability from many fill-ups before you can trust this data!  10 is the utter minimum,  and it is not very good.  You really need 20+.  Trust me,  I’ve done this many times with many vehicles.  Done right,  it really works. 

Wednesday, August 18, 2021

Propellant Ullage Problem and Solutions

For liquid rockets employing free-surface tanks sitting on the launch pad or in thrusted flight,  the propellants in the tanks are pushed by gravity or the vehicle acceleration into the same position covering the drains into the engine pumps.  But between burns,  the vehicle is essentially in free-fall.  The propellants,  because they no longer fill the entire volume,  get pulled by surface tension into multiple spherical globules floating around inside the tanks. 

This situation essentially removes the propellants from the inlets to the engine pumps.  The engines cannot be restarted without correcting this situation.  Nor could a ship-to-ship refueling operation be conducted,  as the drain pipe inlets are dry.  No pump of any kind can pull propellant out of tanks if the pump inlet has only vapor in it.

Initially,  rocket stages needing to ignite in free fall were the upper stages of multi-stage launch vehicles.  The first solution to this problem was adding small solid propellant rocket cartridges called “ullage motors” to the stage.  This is depicted in Figure 1.  The solid propellant cartridges were entirely unaffected by either gravity or its total absence. 

When the ullage motors fired,  their total thrust accelerated the vehicle by an amount equal to total thrust divided by vehicle mass.  After a period of time,  the globules settled into pools of liquid in the tank bottoms,  as depicted in the figure.  This has a long history of success with both storable liquids and with cryogenics,  over fairly short time intervals.  Attitude thrusters can also be used for this.

Figure 1 – The Ullage Problem and the Ullage Motor Solution

A “time constant” for this process would be the time for a globule to fall from one end of the tank to the other,  under the small acceleration induced by the ullage thrust. A figure-of-merit for the settling time into a pool with no voids in it,  would be roughly three times the time constant.  That plus the time it takes to get the engine fully ignited and stabilized,  is the min burn time required of the ullage motors.

The pressure at the main engine pump inlet (for each line) is the pressure inside the tank,  plus an increment that is the depth shown  in the figure times the liquid density times the vehicle acceleration.  That pressure divided by Earth gravity-times-the-density must equal or exceed the “net positive suction head” specified for the engine (or refill) propellant pump.  The pressure in the tank is that of the vapor,  plus that of any injected pressurant gas.

The first ullage motors were solid propellant devices,  but that is not the only way to provide ullage thrust.  Liquid propellant attitude thrusters can be used for this purpose,  if designed to operate in free-fall.  These are usually bladdered-tank systems such as those in Figures 2 and 3 (discussed just below),  and which are almost universally pressure-fed instead of pumped. 

Bladdered Tank Approaches

Bladdered tank designs contain the liquid inside a bladder,  in turn inside the tank.  Pressurant gas injected between the tank and bladder squeezes it to the liquid,  preventing the formation of void space in which free-floating globules can form.  The difference in pressure between the pressurant gas and engine pump inlet drives the expulsion of liquid from the bladdered tank.  This has a long history of success with near-room-temperature storable liquid propellants,  but none with cryogenics. 

Figure 2 – The Bladdered Tank Solution,  Done As Axial Eversion

This can be done in pretty much any geometry,  but such a bladder as it crushes under pressure will crumple and wrinkle,  which significantly lowers the liquid expulsion efficiency of the design.  Expulsion efficiency is defined here as volume expelled / volume loaded.  A way to achieve high expulsion efficiency is to “evert” the bladder,  so that one half of it collapses inside the other half.  If the symmetry of this eversion can be preserved,  the expulsion efficiency can theoretically be very nearly perfect.  There are two eversion geometries for cylindrical tanks:  axial (Figure 2),  and lateral (Figure 3). 

Figure 3 – The Bladdered Tank Solution,  Done As Lateral Eversion

The axial eversion path offers propellant expulsion from a tank end with a centered connection,  but offers a long eversion path,  increasing the probability of asymmetric eversion with wrinkles.  That leads to less expulsion from the tank than desired.  The shorter lateral eversion path offers higher symmetric eversion probability,  for higher expulsions nearer those desired.  However,  it requires side feed and expulsion connections.

Both geometries require the bladder be bonded to the tank on one side or one end.  Both feature a very sharp bend with a very short radius of curvature indeed,  at the eversion point.  The trick for reusability is to ensure the strain at the eversion point is elastic,  otherwise,  the bladder will become the wrong shape,  and will no longer fit the tank correctly.  That guarantees wrinkles and lower expulsion than design.  It also increases the likelihood of bladder failure.  Note also that the eversion point moves!

Key here is very large elastic strain values for the bladder material.  Generally speaking,  with most elastomers,  these strain capabilities are quite large at ordinary temperatures,  but quite low at cryogenic temperatures.  That is why this bladdered expulsion propellant system has historically been used with more-or-less room temperature storable propellants,  but not with cryogens!  If the right material with the right properties can be found,  it would then work with cryogens. 

Corrosiveness of the propellant is also an issue to consider.  This is especially important with the nitric acid systems,  and to some extent with the hydrazines.  Reliability of the design,  especially if it is to be reusable,  is extremely important.  This is especially true for very toxic propellants such as NTO and the hydrazines.  Leaks simply cannot be tolerated.

In any event,  the pressure of the expulsion gas that is fed in must exceed whatever propellant pressure exists within the bladder.  If not otherwise controlled,  the rate of propellant expulsion is proportional to the square root of the difference between the feed gas pressure and the tank pressure inside the bladder.  It is also proportional to the propellant outlet area. 

The feed gas volume flow rate at pressure,  must be as large as the volume rate of flow of expelled propellant,  while still providing that expulsion pressure difference.  This is generally a rather significant pressure difference,  in order to achieve useful expulsion rates.

The bladder material must be able deform easily,  so that it does not resist this pressure difference,  instead just resting against the liquid while the gas moves them both. It therefore cannot be stiff,  despite whatever thickness it must have,  to survive.   We are talking about materials with high tensile strength,  very low Young’s modulus,  and truly enormous elastic strain capability,  at all the cryogenic temperatures it will see in service. 

The radius of curvature at the eversion point,  right on the inside of that eversion bend,  is essentially zero.  That puts the outer side of the bend into considerable tensile stress and strain.  If the material is going to split,  that is where it will happen.  The thicker the bladder has to be,  the worse this eversion point bend-splitting risk is.

Piston-Driven Displacement

About the only remaining alternative approach for positive expulsion would be a gas-driven piston,  essentially a syringe.  This is depicted in Figure 4.  As far as I know,  this approach has never been used in a flying system,  other than as an engine start primer. 

There are multiple constraints on this type of design for a rocket vehicle.  One tank dome has to be inverted,  in order for there to be a place to locate the piston skirt,  when the tank is full,  such that max liquid volume is obtained.  The piston must have such a skirt,  in order to remain properly aligned,  and not jam. 

At the other end of the piston travel,  the piston face must match the contour of that dome,  so that maximum liquid volume may be expelled. 

The piston skirt and side must be of nontrivial thickness in order to house the groove (or grooves) for the O-ring seals.  Remembering that the square edges of the grooves are stress concentrators,  there must be enough “meat” to sustain the loads on the piston,  repeatedly for reusability. 

The depth of the groove plus the wall clearance has to be such that the O-ring seal(s) is compressed radially enough to seal.  That compression distance is small if the O-ring is hard,  but the compression force is high.  The compression force is low if the O-ring is soft,  but the O-ring may not be compressed enough to effectively seal,  if this is taken too far. 

High compression force is high friction force to move the piston.  In particular,  the static friction is typically much higher than sliding friction,  leading to slip-and-jerk behavior characteristics,  which are quite undesirable. 

The width of the O-ring groove has to be wide enough not to compress the O-ring,  so that pressurant gas can fill one side of the groove evenly,  and force the O-ring against the other side,  thus effecting the seal.  This works correctly if there is only one O-ring.

If there are multiple O-ring seals,  pressurant gas cannot reach those seals further away from the pressurant gas side.  If such seals are installed,  their grooves must be narrower,  so that radial compression forces deform the O-ring enough to seal against both sides of the groove.  This requires high radial pressures,  obtaining large friction values,  and causing severe slip-and-jerk behavior.  Net result:  multiple-seal redundancy is not always a good idea!

Figure 4 – Positive Displacement With A Piston

Venting Boiloff Vapors With Cryogens

All the issues and observations made so far apply to near-room-temperature storable liquid propellants.  The boiloff behavior of cryogenic propellant materials introduces yet another very complicated issue to deal with:  adequate venting of boiloff vapors.

In the free-surface tank where ullage thrust is used,  the tank venting system is located on the forward dome.  The tank may be filled fully at launch,  or very nearly so,  but when in free-fall,   there will be considerably more physical tank volume than there is liquid volume inside it.  Vapors add to the atmosphere not occupied by liquid,  which must be vented periodically,  if tank pressure is not to rise rapidly. The forward dome location is the logical place to install such venting equipment. 

When venting,  ullage thrust must be applied to resettle the globules into a pool of liquid,  with a separate vapor atmosphere.  Otherwise,  liquid as well as vapor will be vented. 

We may conclude that free-surface tanks with ullage thrust are inherently compatible with cryogenic propellants.  History bears that conclusion out.  Periodic venting will also require ullage thrust,  in addition to engine ignitions and propellant transfer operations.

Using cryogenics in the bladdered tank approach is going to require a flexible vent line between the bladder and a location on the tank shell.  The equipment to control tank venting can be mounted at the shell location.  But,  there are two very serious problems:  (1) the vent line is quite long,  essentially full tank length,  if axial eversion is used,  and (2) how does one ensure that only vapor enters the vent line,  when vapor can form essentially anywhere within the bladder?

Problem 1 can be reduced in severity by using lateral eversion.  The vent line is much shorter,  but must still be flexible,  and it will affect the everting bladder geometry,  requiring an inconvenient pocket in the tank shell to hold it,  when the bladder is filled.  There is no practical way to put the vent line back into the pocket during tank refill,  without opening the tank at the pocket location,  and physically flaking the flexible vent line in place. 

Problem 2 has no known solution,  yet,  other than the application of ullage thrust.  But if you add ullage thrust,  you might as well just build a simple free-surface tank! 

The flexible vent line is as severe a cryogenic elastic strain capability problem,  as is the bladder itself.  These are technologies requiring development and demonstration.  They are not ready to apply!

We may therefore conclude that neither bladdered tank approach is compatible in any practical way with a boiloff vapor vent,  which is absolutely required if cryogenic propellants are to be used.  There are materials technologies that must be developed and demonstrated to enable this design approach.

The piston displacement approach will require a venting installation on the piston itself,  as there is no other feasible place to put it.  This may prevent the piston from recessing fully into the forward dome when the tank is full,  thus lowering the volume of liquid propellant that could otherwise be loaded into a tank of a given volume. 

It also requires a long flexible vent line from the piston to the forward dome,  where the venting controls can be mounted.  This flexible line will also act to hold the piston off the forward dome.  There is no way to flake this vent line into position between the piston and forward dome during refill,  without opening an access port in that forward dome.

This idea also suffers the cryogenic elastic strain capability problem for the vent line,  which the piston was supposed to eliminate by eliminating the bladder.  And,  it still suffers the same problem with how to ensure only venting vapor,  when liquid is inherently adjacent to the piston,  and the vapor can form throughout the liquid volume.  As with the bladder tank,  you could apply ullage thrust,  but then it would be easier just to build a free-surface tank. 

The flexible line technology at cryogenic temperatures requires development and demonstration.  It is not ready to apply!

We may therefore conclude that the displacement piston approach is also incompatible with a boiloff vapor vent,  which is required if cryogenic propellants are to be used.  There are materials technologies that must be developed and demonstrated to enable this design approach.

Bottom Line:

The most practical solution to the ullage problem when using cryogenic propellants,  is the free-surface tank with ullage thrust provided.  This is also the historically-proven solution,  and all the technologies to enable it are ready to apply.  It is compatible with boiloff vapor venting.  Ullage thrust must be supplied for every free-fall engine ignition,  every tank refill in free fall,  and every boiloff vapor venting event.

The bladdered tank and piston displacement approaches are not compatible with boiloff vapor venting installations,  which are required,  and they run severe cryogenic elastic strain capability risks for the various structures that are required to be flexible at cryogenic temperatures.  Such materials technologies are not yet ready to apply.


Update 2 October 2021

There is another option for providing an ullage solution in a free-surface tank with cryogenic propellants.  That is to spin the tank to provide “artificial gravity”,  so that a free surface forms again.  The dynamics of spinning objects are stable only about those axes with maximum and minimum mass moments of inertia.  For objects that are cylindrical,  that would be spinning end-over-end (like a baton) at maximum moment of inertia,  or spinning about the long axis like a rifle bullet,  for minimum inertia. 

The two geometries are quite different in their effects. If you spin end-over-end,  the result is as illustrated in Figure 5.  The moving tank walls “intercept” the floating globules,  enforcing their acceleration into the spinning motion.  The result is the formation of a new free surface inside the tank,  reflecting the direction of the centrifugal force of the spinning motion.   It is likely there will be one tank on each side of the center of gravity,  so that propellants are slung to opposite ends of the tanks,  as illustrated.  The drain and vent roles will be reversed for at least one set of tank plumbing connections.

Because the mass moment of inertia is maximum for this spin direction,  a maximum torque-time product is required to spin-up the vehicle to any given rotation speed.  However far the spin-up thruster is from the center of gravity is the moment arm for the torque that thruster provides.  The torque-time product divided by the moment arm length is the thrust time product (total impulse) required of the thruster to spin-up the vehicle for ullage.  A similar total impulse is required to de-spin.  Note that engine restarts cannot be done successfully while the vehicle is spinning,  although propellant transfers might be. 

Figure 5 – What Happens With End-Over-End Spin

The other option is rifle-bullet spin,  as illustrated in Figure 6.  This one slings the propellants radially outward against the periphery,  as shown,  to form a cylindrical free surface inside each tank.  The total impulse required to spin-up and de-spin is less,  because this is the minimum mass moment of inertia.  The pre-existing vent connections can still serve that role in both tanks,  as long as they are near the center of the tank dome.  The drain connections will require a second set of drains along the tank peripheries,  where the liquid pools are located,  in addition to the aft dome center connections,  used when under thrust.  It would best preserve spin stability to install the periphery drain connections in a symmetrical manner.  Otherwise cross products of inertia become large instead of zero.

There is one other problem to solve,  associated specifically with this spin direction.  There are no surfaces that “intercept” the globules when you spin-up the vehicle!  In effect,  you spin-up the hardware,  but not the propellant.  Only the random motions of the globules bring them into contact with the moving tank wall.  Friction forces collect some of the spatter from those collisions onto the wall.  In this way,  eventually the propellant finally gets “spun up” and affixed to the tank wall.  But this random and inefficient process takes a very long time indeed!

The solution is a set of radially-oriented perforated baffles,  similar to anti-slosh baffles.  When you spin-up the hardware,  these baffles “intercept” the floating globules,  forcing their immediate spin-up,   and thus their immediately getting slung outward against the tank wall.  This is shown in Figure 7

This solution costs some extra inert mass in the form of the extra peripheral drain connections,  and the radial perforated baffles.  But it does reduce the mass of spin and de-spin thruster propellant that needs to be budgeted.  Given a fast switch from peripheral to aft dome drains,  this rifle-bullet spin geometry might possibly serve for engine reignitions as well as for propellant transfers.  For transfers,  the very convenient nose-to-nose or tail-to-tail docking geometry could serve.   See Figure 8

Figure 6 – What Happens With Rifle-Bullet Spin

Figure 7 – Spinning-Up the Propellant As Well As the Hardware

Figure 8 – Using Rifle-Bullet Spin For Propellant Transfers Between Two Docked Vehicles

Updated Bottom Line:

The rifle-bullet spin technique could easily supplant the application of ullage thrust for propellant transfers between docked vehicles.  It might not serve as well for engine reignitions,  where simple ullage thrust is already well-proven and easily had.   While this spin technique needs demonstration,  there is nothing here to suggest that it wouldn’t be a successful and short effort.

The obvious application here is the tanker problem for refilling SpaceX “Starships” on-orbit.  Such is not required for low orbit operations,  but it is required for high-orbit operations,  and for outside-of-orbit operations.  That last includes trips to the moon and to Mars (or anywhere else outside of Earth orbit).  This spin technique may help make possible the otherwise-very attractive tanker scenarios I have already explored in multiple other articles on this site.

A not-so-obvious (but still important) application would be for an on-orbit propellant depot facility.  Such a depot would have to serve a variety of vehicles,  and so would have to store a variety of propellant combinations.  Some of those are going to be cryogenics.  The bladdered tank solutions in Figures 2 and 3 apply to the storable materials,  but the cryogenic materials s could be stored in tanks that spin like rifle bullets.  That is an idea worth exploring further,  perhaps in a future article.

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