Thursday, October 21, 2021

Lists of Some Articles By Topic Area

I was once an all-around ramjet design,  development,  and test engineer,  among many other things,  including rocket work.  This was mostly at a plant in McGregor,  Texas,  once known as Rocketdyne or Hercules.  Part of that reservation is where SpaceX tests rockets now. 

I did just about everything there was to do,  for this ramjet work.  There are very few indeed with knowledge and experience this comprehensive,  I was definitely not a narrow specialist!  But my knowledge and abilities,  in each of all these different specialty disciplines,  was actually quite substantial and deep!  

My design analyses usually took the form of custom hand-calculations,  not just sitting there blindly running other people’s computer codes.  (Although,  I did use computer codes,  and even wrote some myself.)  I have informally published several articles on my blog site that describe how some of this ramjet work was done.

Ramjet & Closely-Related Articles (there are others,  but these are the best):

11-2-21                The “Warm Brick” Ramjet Device (nonpropulsive application to an infrared decoy)                                 [also the 11-2-21 update to this catalogue list]

10-1-21                Use of the Choked Pintle Valve for a Solid Propellant Gas Generator Throttle

8-2-21                  The Ramjet I Worked On the Most

7-1-21                  Another Ramjet I Worked On

11-9-20                Fundamentals of Inlets

3-3-20                  Ramjet Flameholding

2-16-20                Solid Rocket Analysis (applies to ramjet for boosters)

2-4-20                  One of Several Ramjets That I Worked On

1-2-20                  On High-Speed Aerodynamics and Heat Transfer

11-12-18              How Propulsion Nozzles Work

7-4-17                  Heat Protection Is the Key to Hypersonic Flight

6-12-17                Shock Impingement Heating Is Very Dangerous

12-10-16              Primer on Ramjets

12-21-12              Ramjet Cycle Analyses

These are located on http://exrocketman.blogspot.com,  along with many others on a wide variety of subjects. 

There is a navigation tool on the left of that page.  For the article you want,  you only need its publication date and its title.  Use the navigation tool:  click on the year,  then the month.  Then click on the title if you need to.  The data you need are in these lists.

If you click on one of the figures,  you can see all of them enlarged.  You see nothing but the figures,  though.  There is an “X-out” from this view,  upper right of screen.

At the end of any given article,  there is also a list of search keywords assigned to it.  If you click on “ramjet”,  you will only see the articles bearing that keyword.  The same is true of the other keywords.

Here follows a photo of one of the ramjets I worked on:  ASALM-PTV.  It is hanging under the wing of an A-7 Corsair-II,  an aircraft my father designed.  I always considered this photo a sort of “family portrait”. 

 

ASALM-PTV Ramjet Vehicle Underwing of A-7 Corsair-II

Some of those ramjet articles overlap with the next list.  That next list is of aerothermodynamics and heat transfer-related articles.  Some of these relate to high-speed atmospheric flight,  and others to atmospheric entry from space.  Those two scenarios are quite different,  in that atmospheric flight is a steady-state equilibrium problem,  while atmospheric entry is mostly a transient heat-sinking problem.  The search keyword for these is “aerothermo”.  Clearly,  I was adept at multiple specialties.

Aerothermodynamics & Heat Transfer Articles:

4-1-20                  Entry Heating Estimates

1-2-20                  On High-Speed Aerodynamics and Heat Transfer

1-9-19                  Subsonic Inlet Duct Investigation

1-6-19                  A Look At Nosetips (Or Leading Edges)

1-2-19                  Thermal Protection Trends For High-Speed Atmospheric Flight

11-12-18              How Propulsion Nozzles Work

7-4-17                  Heat Protection Is the Key to Hypersonic Flight

6-12-17                Shock Impingement Heating Is Very Dangerous

11-17-15              Why Air Is Hot When You Fly Fast

8-4-13                  Entry Issues

3-18-13                Low-Density Non-Ablative Ceramic Heat Shields

1-21-13                BOE Entry Analysis of Apollo Returning From the Moon

1-21-13                BOE Entry Model User’s Guide

8-19-12                Ballute Drag Data

8-19-12                Blunt Capsule Drag Data

7-14-12                “Back Of the Envelope” Entry Model

I was also a rocket propulsion engineer,  mostly in solid composite propellants.  However,  from the chamber outlet through the nozzle,  the ballistics of all rockets are the same,  including liquid propellant rockets.  If you can allow for any gas bled off and dumped overboard for turbopump operation,  then the very same ballistics apply,  right down to the chamber pressure vs flow rate calculation. 

Further,  the estimation of vehicle performance from the simple rocket equation can be made quite accurate,  if you know how to apply “jigger factors” in the appropriate places for gravity and drag losses,  and if you know what values of these “jigger factors” to apply.  I have been very successful at doing this kind of work. The following list shows that,  and mostly shares the “launch” and “space program” keywords.

Rocket Ballistics and Rocket Vehicle Performance articles:

3-15-21                Reverse Engineering Estimates: Starship Lunar Landings

3-9-21                  Reverse-Engineering Starship/Superheavy 2021

3-5-21             Fundamentals of Elliptic Orbits (delta-vee requirements)

2-9-21                  Rocket Vehicle Performance Spreadsheet (rocket vehicle performance)

7-13-20                Non-Direct to the Moon with 2020 Starship

7-5-20                  How the Spreadsheet Works (Starship to Mars)

7-5-20                  2020 Starship/Superheavy Estimates for the Moon

7-3-20                  Cis-Lunar Orbits and Requirements

6-21-20                2020 Starship/Superheavy Estimates for Mars

5-25-20                2020 Reverse Engineering Estimates for Starship/Superheavy

2-16-20                Solid Rocket Analysis (solid ballistics & more)

11-21-19              Interplanetary Trajectories and Requirements

10-22-19              Reverse-Engineering the 2019 Version of The Spacex “Starship” / “Super Heavy” Design

9-26-19                Reverse-Engineered “Raptor” Engine Performance (liquid ballistics)

9-16-19                Spacex “Starship” as a Ferry for Colonization Ships

9-9-19                  Colonization Ship Study

11-12-18              How Propulsion Nozzles Work (rocket, ramjet, & turbine; plain & free-expansion)

9-11-18                Velocity Requirements for Mars

8-23-18                Back-of-the-Envelope Rocket Propulsion Analysis (rocket vehicle performance)

4-17-18                Reverse Engineering the 2017 Version of the Spacex BFR

10-23-17              Reverse-Engineering the ITS/Second Stage Of the Spacex BFR/ITS System

3-18-17                Bounding Analysis for Lunar Lander Designs (rocket vehicle performance)

3-6-17                  Reverse-Engineered “Dragon” Data (rocket vehicle performance)

8-31-13                Reusable Chemical Mars Landing Boats Are Feasible (rocket vehicle performance)

In 2009,  I attended an asteroid defense conference in Granada,  Spain,  as a poster paper presenter.  I have since written some articles about asteroid defense.  Unfortunately,  the asteroid defense capability picture hasn’t changed much since my 2009 attendance at that conference.  Again,  the latest are the best and most up-to-date.   Be aware that “NEO” (Near Earth Object) includes comets as well as asteroids as threats.  Comets may be the more difficult to defend against,  because of the surprise nature of the detection and orbits.  These articles all share the “asteroid defense” keyword.

Asteroid Defense Articles:

8-30-20                Asteroid Threats  (current status assessment:  not good)

6-3-20                  On the Manned Spacex Launch

7-14-19                Just Mooning Around

12-13-13              Mars Mission Study 2013

4-21-09                On Asteroid Defense and a Good Reason for Having National Space Programs

I have also applied my wide-ranging knowledge to the problems of atmospheres to breathe while in space,  and the kinds of spacesuits that might best serve our needs.  Again,  the latest is the best and most up-to-date.  But I have been looking into these issues for some time,  as indicated by the dates on these articles.  These all share the “spacesuit” keyword.

Space Suits and Atmospheres Articles:

3-16-18                Suit and Habitat Atmospheres 2018

11-23-17              A Better Version of the MCP Spacesuit?

2-15-16                Suits and Atmospheres for Space

1-15-16                Astronaut Facing Drowning Points Out Need for Better Space Suit

11-17-14              Space Suit and Habitat Atmospheres

2-11-14                On-Orbit Repair and Assembly Facility

12-13-13              Mars Mission Study 2013

1-21-11                Fundamental Design Criteria for Alternative Space Suit Approaches

One of my favorites is the MCP (mechanical counter pressure) version of the spacesuit.  This was pioneered by Dr. Webb in the 1960’s as a possible suit for the Apollo missions to the moon.  It is not a full pressure suit at all,  but essentially a tight garment that simply squeezes the body.  It is porous,  so that you sweat right through it to cool,  just like ordinary street clothing.  But this design was tested quite successfully in 1968 for 30 minutes in a vacuum chamber,  at way above the equivalent “vacuum deathpoint” altitude.  Photo follows:


Webb’s MCP Space Suit:  Helmet,  Backpack,  and Supple Garment Total 85 Lbs

Besides vacuum death and microgravity disease,  there is also a radiation hazard to worry about in space.  But,  it is not quite what you think:  there are two completely different hazards to worry about.  On Earth,  we have two kinds of protection:  the atmosphere,  and the magnetic field.  In low Earth orbit,  we have only the magnetic field.  Outside the magnetic field,  going to the moon or anywhere else,  there is no protection.  Yet these things can be quantified,  and some of it shielded fairly effectively.  What got me started on this topic were the dangers posed by the nuclear disaster in Fukushima,  Japan.  Keyword “radiation”.

Radiation Hazard Articles:

10-5-18                Space Radiation Risks:  GCR vs SFE

4-11-15                Radiation Risks for Mars Trip

5-2-12                  Space Travel Radiation Risks

3-24-11                Radiation and Humans

3-17-11                Follow-Up On the Japan Nuclear Crisis

3-15-11                On the Nuclear Crisis In Japan

On a lighter note,  I have long been interested in pulsejet engines,  especially valveless pulsejets.  While teaching math at TSTC,  Waco,  I became involved with mentoring a student who was also interested in pulsejets.  I and a colleague assisted this student in making his own valveless pulsejet engines,  which attention and involvement also turned this student into an “A” student in math!  Keyword “pulsejet”.

That student built a small engine that eventually pushed an old golf cart around,  and then a much bigger engine which we together fired up out here on my farm homestead.  Photos of the two engines follow:

Smaller Student-Built Valveless Pulsejet Engine (Later Pushed a Golf Cart)


Larger Student-Built Valveless Pulsejet Engine

Pulsejet Articles:

5-20-12                Recommended Broad Design Guidelines For Valveless Pulsejet Combustors

4-30-12                Big Student Pulsejet an Even Larger Hit at TSTC

3-6-12                  Student Pulsejet a Hit at EAA Meeting

11-12-11              Student Pulsejet Project

I have been interested in ethanol fuels since my early days in college.  When I went to work for what is now Minnesota State University,  after my 20-year career in aerospace defense work ended,  I got more serious about it.  My next job was at Baylor University in Waco,  Texas,  and it dealt directly in alternative fuels for aircraft.  The scope of that included ethanol (and an ether) as piston-engine fuels,  and biodiesel-jet fuel blends as turbine fuels,  plus STC work with the FAA,  and also experimental engineering research work,  as well as classroom teaching. 

Not too long after leaving Baylor,  I began my own experimental engineering research at home,  using E-85 ethanol fuel,  and stiff ethanol blends,  in a variety of vehicles.  Those would include straight E-85 ethanol fuel in an old farm tractor and in an old-time air-cooled VW beetle,  plus stiff ethanol blends in a variety of completely-unmodified cars and 4-stroke lawn and garden equipment.  I basically recommend up to E-35 blend strength,  as a “drop-in” fuel,  for just about any 4-stroke piston engine. 

The keywords are “ethanol” and “old cars” for most of these articles.  Once again,  the latest is the best and most up-to-date.

Articles About Ethanol and Ethanol Blends in Vehicles:

9-1-21                  Making Stiff Blends At the Gas Pump

11-3-13                Aviation Alternative Fuel Compatibility Issues

11-2-13                An Update on Ethanol Fuel Use

8-9-12                  Biofuels in General and Ethanol in Particular

5-4-12                  Energy Storage: Batteries vs Unpressurized Liquid Fuels

6-12-11                Another Red-Letter Day

5-5-11                  Ethanol Does Not Hurt Engines

2-12-11                “How-To” For Ethanol and Blend Vehicles

11-17-10              Nissan Mileage Results on Blends

11-12-10              Stiff Blend Effects in Gasoline Cars

12-15-09              Red Letter Day:  Ethanol VW Experiment Complete

7-1-09                  Another Antique Comes Out of Storage

I have returned part-time out of retirement to help a friend with his auto repair business.  I was once ASE-certified as a condition of employment while teaching at Minnesota State in its Automotive and Manufacturing Engineering Technology department.  Before that,  I did most of my own automotive maintenance and repair work.  Accordingly,  I have posted some articles about basic car care,  plus one funny.  These all share the “old cars” and “fun stuff” keywords. 

Automotive Care Articles:

12-3-20                Blinker Fluid (the “funny”,  and it is a sight gag)

8-20-20                Underhood Check

7-25-20                Taking Care of Car Batteries

When I returned to the rocket plant in McGregor for my second employment there,  the family and I acquired an old farm outside McGregor as our home.  We have been there ever since.  This place was largely covered in shin- to knee-high prickly pear cactus,  so thick there were few trails through it.  After grubbing it out of the house’s back yard with hand tools,  I decided there had to be a better way to do this cactus eradication. 

I tried a variety of mechanical drags behind my old farm tractor for some 15 years without success.  The results were always the same:  it looked better for a while,  but returned worse than ever before,  within months.  My neighbor was trying shredding at 1 inch off the ground.  Eventually that worked,  but required the neighbor to be out there shredding,  every single day,  the same ground over and over,  for 6 (or more) years.  The neighbor also tried spraying herbicides on one patch of ground,  which took 3 years to show results,  but then totally reinfested within another 2 years.

I then tried to build a “scooper-upper” out of scrap steel.  The idea was to bust the aboveground cactus loose from its roots,  and catch it on a tarp towed behind the “scoop-upper”,  for disposal in a burn pit.  It completely failed to work,  because when the tool hit the cactus and busted it loose from its roots,  it fell forward in front of the tool,  instead of backward onto the deck.  The tool then just ran over the top of the cactus debris.  I gave up in disgust when this failure-to-scoop happened.

I went back up a few months later to salvage the steel,  and saw something totally unexpected:  the cactus was dead and gone wherever the tool had been towed!  Grass was growing in the cow pasture where the cactus had been.  It did not take very long to understand that the aboveground cactus foliage had been crushed and damaged passing underneath the heavy tool,  such that the pads dried out and died,  before they could put down new roots from the thorn sites in contact with soil.  They had completely composted away over those months.

I “played” with this tool to get it just “right”,  and started killing acres of prickly pear quite effectively,  and with very little time and effort involved.  In fact,  I still have this very same experimental prototype,  and it still works today.  This prototype led to me filing a patent on the cactus tool in 2002. 

I revised the design to something more producible from real steel stocks,  and built two production prototypes that worked just as well as the original experimental prototype,  but were easier to build.  Then,  with the patent in hand as of 2004,  I began building and selling these tools to the public.  My first customer wouldn’t wait for a real production tool,  and insisted on buying one of the two production prototypes.  I still have the other one.  I still use it,  and it now serves as an experimental test bed for new features,  too.

As time went by,  it quickly became apparent that other folks had rockier land,  or land with tree stumps.  I changed the design twice,  to add a heavier stabilizing snout,  plus a “barge front” wedging surface to get over small rock outcrops.  This was quite successful,  and is embodied in the tools still built and sold today.

A close friend wanted to do cactus-killing for hire,  and bought a “one-off” design from me.  I also helped him build and modify a few more tools,  until the “commercial version” was defined:  a really tough snout,  a big “barge front”,  and retractable wheels to facilitate stepping over obstacles,  plus easier loading up ramps onto trailers. 

When that friend retired,  I revised his “commercial” design into something that used a common core tool chassis with my “homeowner grade” plain tool.  This common core chassis had the big barge front,  and used either a tough snout for the “plain tool”,  or a longer tough snout for the “hydraulic tool”,  that was also fitted with retractable wheels operated hydraulically.  I sell both versions to this very day.  Both are towed on a chain bridle behind a farm tractor’s drawbar.

I am working on a third version that could be an alternative implement affixed to the hydraulic boom of a skid-steer loader.  It uses an already-available “universal” adapter plate to accomplish this,  as a quick-change item.  There is nothing to report here yet about that project,  but the “plain” and “hydraulic” tools are well-described in a series of articles on “exrocketman” under the keyword “cactus-killing”. 

These two versions are shown in the photo,  with the plain tool in the foreground,  and the hydraulic tool in the background.

Foreground:  Plain Tool;  Background:  Hydraulic (Wheeled) Tool

Articles Related to Cactus Eradication:

2-9-17                  Time Lapse Proof It Works

7-30-15                New Cactus Tool Website

1-8-15                  Kactus Kicker Development

1-8-14                  Kactus Kicker:  Recent Progress

10-12-13              Construction of the Plain Cactus Tool

5-19-13                Loading Steel Safely (Cactus Tool)

12-19-12              Using the Cactus Tool or Tools

11-1-12                About the Kactus Kicker

12-28-11              Latest Production Version of the Kactus Kicker

 









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, sq.in)(gc = 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, sq.in)(gc = 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/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. 

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.

References:

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

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

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

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

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