Sunday, February 16, 2020

Solid Rocket Analysis

I promised in another article to do an article on solid rockets.  Here it is.  For a bit of context,  I explore up front where solids fit among all rockets.  But the bulk of the article is just about solid rockets.

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Update 2-22-20:  Three forgotten items have been inserted where needed below.  These are the purpose and effects of propellant metal contentcombustion instability,  and how to get propellant characteristics from lab motor tests,  particularly empirical c*.  All three are labeled as “Update 2-22-20/topic title”. 

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Most of what applies to feasibility calculations for liquid and solid rocket vehicles (and hybrids) is given in the first two items in the following list of articles on this site (highlighted).  The nozzle article applies to all propulsion items,  not just rockets,  and details thrust,  and thrust coefficient,  calculations.  The second applies to any sort of rocket,  and details how to do mass ratio/delta-vee calculations. 

That second article is the compendium of what I currently use to size rocket systems and estimate their performance.  The other items in the list are earlier iterations of this same process for various applications,  for which I had not yet fully generated my list of appropriate assumptions. 

Here is the list:

11-12-18  How Propulsion Nozzles Work  (applies to rockets,  ramjets,  gas turbines,  or anything else with a steady-flow propulsive jet)

8-23-18   Back of the Envelope Rocket Propulsion Analysis (how to obtain mass-ratio-effective delta-vee from ideal delta-vee,  what assumptions to make,  and how to size the rocket vehicles to such delta-vees)

11-26-15  Bounding Analysis: Single Stage to Orbit Spaceplane,  Vertical Launch

8-16-14   The Realities of Air Launch to Low Earth Orbit

10-2-13   Budget Moon Missions

9-24-13   Single Stage Launch Trade Studies

8-31-13   Reusable Chemical Mars Landing Boats Are Feasible

12-14-11  Reusability in Launch Rockets

I'm not “big” into hybrids,  but I am an authentic expert in solids,  as well as ramjets.  At a more superficial level,  I also know about liquid rockets:  not the detailed cycles driving the pumps,  but their overall effects (mainly modeled as what fraction of generated hot gas massflow actually goes through the propulsion nozzle).  This is covered in the propulsion analysis article,  second item in the list.

Chamber Massflow Balance (Liquids and Solids)

Any rocket chamber whatsoever has a massflow input,  a massflow output,  and a transient mass storage term.  Using w to represent flow rate,  that relationship is

winpt = wout + wstor where wstor = (Vfree/RT) dP/dt

This is a very good approximation (since temperature and molecular weight are but weak and extremely-weak functions of pressure,  respectively).  Vfree is the free volume of the chamber,  which may (or may not) vary with time,  and certainly does vary (quite drastically) in a solid.

For the steady state balance:  dP/dt = 0,  so winpt = wout

The nozzle article (first item on the list above) indicates the nozzle output flow to be wout = Pc CD At gc / c*,  for a choked nozzle throat.  In most textbooks on the subject,  both At and c* are assumed to be constants,  and CD is utterly ignored as being essentially 1.  Variations in the geometric area At during the burn,  due to erosion or slag deposition,  are completely ignored.

In the real world,  those are all bad assumptions!  Those constants are all really significant variables.  CD usually doesn't vary very much,  but it is rarely actually 1,  just usually close.  At varies from the start to the finish of the burn,  as nozzle slagging and/or erosion effects change the effective throat diameter. 

This effective throat diameter usually varies as something close to the square root of burn time,  if erosion dominates,  so that a linear variation of At from its initial to its final value is quite the realistic model.  If slagging dominates,  all bets are off,  as there is usually a slag accumulation that erratically sloughs off suddenly during the burn.

The characteristic velocity c* is the square root of actual chamber temperature,  and a weaker function of the hot gas properties:  molecular weight and specific heat ratio.  Empirically,  c* is just a power function of the chamber pressure c* = c*ref (Pc/Pcref)m = K Pcm,  where m is a small number on the order of 0.1,  or even a little smaller.  You get it directly from test firing data,  not theoretical calculations! 

For liquids and for solid internal burners (with significant internal free volume even at ignition),  this empirical power function is an adequate model of c* variation.  But,  for solid end burners,  the initial free volume is quite small,  and until it becomes significant,  c* is reduced further than just the pressure dependence would indicate!  A knockdown factor varying with time or free volume is a good empirical model for that effect.   Past the knockdown-modeling point,  it just takes on the value of 1.

Liquid Rockets

For liquid rockets,  winpt is a value determined by the operator of the engine.  Varying propellant flow rate is exactly how a liquid rocket engine is throttled. Lower flow rate is lower chamber pressure by the nozzle equation,  and lower thrust (by the methods given in the nozzle article).  Flow rate is linearly proportional to chamber pressure in the textbooks,  not quite linearly proportional in the real world. 

Where liquids differ so sharply from solids is in the meaning of “flow rate”.  Most liquid engines tap off hot gas from the chamber to run the propellant pump assemblies (usually turbopumps).  What exactly is done with the tapped-off massflow varies from cycle to cycle. 

But for any given cycle,  there is a tapped-off percentage of generated hot gas that does not go through the propulsion nozzle,  even if that percentage is near or at zero.  You have to reduce the propellant flow from tankage by the tapped-off amount to accurately model what goes through the nozzle. 

The rocket propulsion article (item 2 in the list above) covers how to do that.  Doing it correctly affects your effective specific impulse and thrust values,  as well as your engine throat and exit sizing.

Solid Rockets

Solids are quite different,  as the input massflow is generated by the burning of the solid propellant within the chamber (called a "motor case" in solids).  There is no tapped-off hot gas massflow for anything.  This gas generation process is a distinctly pressure-dependent process, since the propellant burn rate behavior is usually modeled as a power function of the chamber pressure:

winpt = ρ ηexp S r 

where ρ is the solid propellant density,  ηexp is the experimental expulsion efficiency (weight expelled/weight of propellant),  S is the (instantaneous) burning surface,  and r the burn rate.

The usual burn rate model is:

r = fT rref (Pc/Pcref)n,  where n is the burn rate exponent,  also expressible as r = fT a Pn

Quite often,  different values of n apply in different ranges of pressure.  The rref at Pcref version of the burn rate model is useful for this situation.  Otherwise a = rref/Pcrefn.  Values of n typically fall in the 0.2 to 0.7 range.  Pc is the chamber total or stagnation pressure,  usually indistinguishable from the static chamber pressure at most practical nozzle contraction ratios.

Burn rate is also a strong function of the soaked-out temperature of the solid propellant.  The factor fT models that effect,  scaling the reference burn rate up and down with soak temperature.  However, this is a very nonlinear-with-temperature effect:

fT = EXP[σP(T - Tref)],  where “EXP” represents the base “e” exponential function 

For this,  σP models the burn rate sensitivity to temperature,  usually a number in the range of 0.002 per degree F,  and usually a bit larger if very fuel-rich in formulation.  In US units,  Tref is usually taken to be 77 F.  For metric degree-C temperatures,  use σP values 1.8 times larger,  and Tref = 25 C.  Values of σP are often expressed on a percentage basis,  such as 0.2%/F for 0.002/F.

For the steady-state case of chamber mass balance,  that puts power functions of chamber pressure on both sides of the mass balance equation,  as well as a linear Pc dependence on one side:

ρ ηexp S fT a Pcn = Pc CD At gc / K Pcm

Solving this equation for Pc gets a very informative equation for the equilibrium motor burning pressure:

Pc = [(ρ ηexp S K fT a)/(CD At gc)]{1/(1 - n - m)}

Note that the exponent value is rather large at 2.0,  for n = 0.4 and m = 0.1.  This explains the exponentially-sensitive behavior of motor pressure to small changes in At,  variations of r and c* with pressure,  and especially the variation in burning surface S,  which is usually the very largest effect. 

Note that the equilibrium motor pressure equation "blows up" if the sum n + m ever equals 1.  That is the motor instability point usually quoted in the textbooks as a "max stable burn rate exponent is less than 1".  If you allow for variation of c* with motor pressure (which those texts do not),  then it isn't just n,  but n + m,  that cannot reach 1 in a stable choked motor.

Consequences of violating this stability limit are quite severe:  usually a motor explosion within at most a very few milliseconds of ignition.

Burning Surface Variation in Solids

This is a consequence of the as-cast geometry and the law that solid propellant burnback is always perpendicular to the local surface.  Flats stay flat.  Concavities increase in radius of curvature by the current distance burned.  Convexities transform to sharp cusps as the radius of curvature decreases to zero by the distance burned.  This is known as Piobert’s Law,  originally formulated in the 19th century for gunpowder rockets,  but since shown to apply to all solids.   

But,  this also gets complicated by some very real-world effects.  Those include (1) bondline burn rate augmentation and (2) erosive burning.  Bondline burn rate augmentation is unavoidable in end burners,  but is usually of no consequence in internal burners.  Erosive burning is generally something to be avoided in all solids.  It occurs in internal burners,  but generally cannot occur in end burners.  Erosive burning is usually modeled as an extra term in the burn rate model:

r = a Pcn + C (w/A)s = a Pcn + C (ρ V)s,  with exponent s being a number not far from 0.3

In true hybrids with unoxidized fuel grains,  the regression rate is the second term just above,  with “a” in the first term zeroed.  This regression rate acts only on surfaces actually scrubbed by the flow of hot gas (which is thus a violation of Piobert’s Law in terms of how the surface regresses,  because not all surfaces experience such scrubbing).

End Burners

The simplest concept is the flat-faced end-burning propellant grain geometry.  Ideally,  this burns only on the one face,  not down the sides of the grain or the forward face,  which are inhibited by being bonded to the insulated motor case wall. For this idealized case,  burning surface area S is a constant.  The instantaneous distance burned through the propellant is called the instantaneous "web".  The total distance burned through the propellant is its total web,  which for this ideal case is just the physical length of the propellant grain.

Real-world end-burners do NOT follow the ideal case,  because of bondline burn rate augmentation.  The burn rate at the bondline is higher than that of the bulk propellant.  This is because the local packing of the propellant solids particles (mostly oxidizer) against the wall favors more fines in the local distribution of particle sizes,  and empirically,  more oxidizer fines favors higher burn rates.

As a result,  the burn rate right along the bondline leads that of the bulk propellant,  leading to a coned burning surface,  and a conical "sliver" instead of a sudden burnout.  Surface versus web is "progressive" (increasing) until the equilibrium cone is established. This is sketched in Figure A.


 Figure A – Ideal and Real-World End-Burner Behavior

Either way,  the integral of the surface vs (instantaneous) web trace must be the original as-cast volume of the propellant grain.  And,  the integral over time of the (bulk) burn rate must equal the (total) web burned. 

Simple Tube Segment Grains

One of the simpler grain designs is the internal-burning tube most famously used in the Shuttle SRB segments.  This makes a wonderful lab motor geometry,  because both the motor hardware and the propellant cast tooling are so very simple.  Grains are cast into hard sleeves for “cartridge load” into usually-insulated lab motor cases,  or they can be case-bonded to the motor case segments,  as they were in the Shuttle SRB. They burn on the bore and end surfaces,  but not the outer cylindrical surface.

For best results,  the bore diameter is about 25% of the grain outer diameter (to control propellant stresses hot and cold),  and the finished grain length is chosen for maximum "neutrality" of the surface-web trace.  For the bore and both ends burning,  that's a length about 162% of the outer propellant diameter.  See Figure B for the typical results of those design selections in a nominal 6-inch lab motor. 

 Figure B – The Simple Segmented-Tube Grain

For a nominal 6-inch diameter lab motor grain,  that's a length of about 9.5 inches and the bore just about 1.50 inches ID.  The cast sleeve is 6.00 inches OD and 0.075 inches thick,  for a finished outer propellant diameter of 5.85 inches.  Bore dia/outer dia = 0.256,  and L/outer dia = 1.624.  The web to be burned is half the propellant diameter difference = 2.175 inches,  so the web fraction WF = web/outer propellant radius = 0.744.  For a reference area equal to outer propellant dia x propellant length = 55.575 sq.in,  the average surface to reference area ratio is 1.9735.  The cross-sectional propellant loading is grain end area/circle area = 0.934,  which is quite large. 

As the web burns,  the bore diameter increases by two instantaneous web distances.  The grain length decreases by two instantaneous web distances.  You figure the current burning perimeter of the bore,  and multiply it by the current length,  to get the current bore area.  You use the current bore diameter and the (constant) outer diameter to figure the end area burning surface.  There are two of these. 

The sum of those three varying areas is the total burning surface S as a function of web burned.  When the web equals half the original diameter difference,  burnout occurs.  It is rather sharp (see Figure B again). This surface-vs-web trace shape is called "rainbow-neutral" from the "bowed" shape of the curve.  It is gently progressive to the max surface at mid-burn,  then gently regressive to the end of burn.

The max-to-min surface ratio for this design is about 120%.  All else being equal,  that’s a factor of 1.44 max to min equilibrium Pc,  for n = 0.4 and m = 0.1.  It's a much higher pressure ratio yet,  if either n or m are significantly higher.  For example,  n = 0.7 and m = 0.1:  equilibrium exponent is 5.00,  and for a surface ratio of 1.2,  the equilibrium pressure ratio is 2.49,  instead of the 1.44 at the lower n.

Having the correct length to outer grain diameter ratio L/D is crucial to getting rainbow neutrality,  in which the initial and final surfaces are the same.  This simple tube design is rather sensitive to incorrect values of L/D.  Max to min surface ratios get significantly larger when the L/D is wrong for rainbow neutrality. 

With the grain L/D for rainbow neutrality being near 1.6,  and most missile motor designs being around L/D = 5 to 10,  there is a mismatch problem requiring the use of multiple segments.  Unless your motor case is segmented (as with the Shuttle SRB),  or you can cartridge-load multiple grains in their own cast sleeves,  this is otherwise a very difficult design to actually build,  in practical real-world proportions.  

Having the correct bore diameter for a given outer diameter is also crucial to a practical design.  This is measured by the "web fraction",  which is the ratio of web to be burned divided by the outer grain radius (half the outer diameter).

Propellant shrinks faster than motor case materials shrink,  upon soaking cold.  This puts the bore surface into tension,  which can crack,  exposing extra surface,  and potentially leading to a motor explosion.  It can also stress the bondline.  That can rupture,  also exposing extra surface,  and leading to the same catastrophic outcome.

The "trick" is not having too high a web fraction.  The "correct" value is dependent upon propellant physical and structural properties,  as well as geometry.  But,  as a rule-of-thumb for circular segment grains in lab motors,  web fraction is 1 – bore dia/outer grain dia,  which should not exceed about 75%,  based on experience. That’s why the bore diameter fraction is what it is (about 25%). 

Keyhole Slot Grain Design

This one has an inhibited outer cylindrical surface,  and a centered circular bore,  like the circular segment grain design.  But,  along one side only,  there is also a slot in the propellant from the bore to the case wall,  and from one end to the other.  This design burns on both ends,  and on the bore and slot surfaces.  See Figure C.  The slot width ought the fall in the range of 33-67% of the bore diameter.  

 Figure C – The Keyhole-Slot Grain Design

The instantaneous web adds to the bore radius and subtracts from the length twice,  just like the segment grain design.  But,  the slot width widens by two instantaneous webs, which reduces the burning perimeter of the bore.  However,  the increase in bore radius also reduces the slot height,  and thus the burning perimeter contributions,  of the slot sides.  Basically,  one has to keep track of the “corner” where the moving vertical slot side intersects with the expanding bore circle,  in order to compute the total burning perimeter accurately. 

This design can be configured for exact “rainbow neutrality” at a grain L/D nearer 2.3 than 1.6.  Its max/min surface ratio is lower than the segment grain,  and it is substantially less sensitive to being “off” in grain L/D than the segment grain.  In point of fact,  it is only a little progressive,  if used in L/D ratios up to 4 or even 5.  That plus its large web fraction (high cross-sectional loading of propellant) makes it very attractive for integral boosters in flameholding ramjet combustors (usually L/D = 3 to 4).

The effect of the slot reduces bore tensile stresses and bondline stresses somewhat,  when soaked cold.  That means the bore/outer diameter ratio can be a bit smaller than the segment grain,  at about 20%,  leading to slightly higher web fraction (about 80%) and propellant cross-sectional loading.

For a specific example,  consider a keyhole slot with outer propellant diameter 17.68 inches (case-bonded in a 20-inch OD insulated case),  propellant length 74 inches,  bore diameter 4.0 inches,  and slot width 2.0 inches.  Web is 6.840 inches.  Bore/outer diameter ratio is 0.226.  Grain L/D is 4.186,  which is quite far from the rainbow-neutral proportion.  Slot width/bore dia is 0.500.  Web fraction WF is 0.774. 

The reference area DxL is 1308.32 sq.in.  Min S is the initial S = 1818.50 sq.in.  Average S = 2135.77 sq.in.  Max S is 2261.76 sq.in.  Final S = 2187.35 sq.in.  The final/initial S ratio is 1.203,  net progressive.  The max/min S ratio is 1.244,  while the maxS/avgS ratio is only 1.059,  compared to 1.068 for the segment grain.  The avg S/ref area ratio is 1.632. Cross sectional loading is 0.804.  A smaller bore diameter and narrower slot width might increase this further.

Other Grain Designs

There are two basic internal-burner situations to consider:  the more-or-less neutral internal burner,  and the two-level (“boost-sustain”) grain design.  The former are covered by properly-proportioned full-length “dendrite” or “wagon-wheel” grain designs,  and the latter by the partially-slotted tube designs.  See Figure D.  Mis-proportioning dendrite or wagon wheel designs can make them boost-sustain as well. 

 Figure D – Two Other Classes of Internal Burners

The best source of grain design ideas and proper ballistic analysis practice is the NASA monograph on the subject.  This is Brooks,  W. T.,  “Solid Propellant Grain Design and Internal Ballistics”,  NASA SP-8076 (monograph on solid ballistics),  March 1972.  W. T. “Ted” Brooks was a friend and colleague at Rocketdyne / Hercules - McGregor,  and he personally taught me the interior ballistics of solid rockets,  when I was a young engineer,  new out of college. The hardest part of the whole process is keeping track of the details of a changing propellant grain geometry,  which is usually fundamentally three-dimensional in nature.  Such math is just never easy.

               Dendrite or Wagon-Wheel Designs

The dendrite or wagon-wheel designs use multiple thin “branches” of propellant cross section,  separated by empty spaces into which the hot gases can go.  They have the same basic cross-section shape throughout the grain length.  This has the effect of providing a very large burning perimeter,  for a very large average burning surface,  but at very low values of total web,  and thus very low web fraction.  The cast tooling for this can be quite complicated. 

By the way,  those spaces into which the hot gas goes have to be large enough to limit the flow speeds in those spaces to avoid erosive burning and pressure-drop-induced grain geometry distortions.  The usual rule-of-thumb about that is “bore area never less than twice the throat area,  and usually quite a bit larger than that,  more like 5-10 times the throat area”.

Cold soakout has very little impact on grain stress with these designs,  but they are very vulnerable to fracture if subjected to high acceleration gees or high mechanical shock,  since the propellant “branches” are relatively unsupported.  Loss of a chunk not only opens up extra burning surfaces,  it also presents a nozzle-plugging hazard as the lost chunk tries (and fails) to go through the nozzle throat.  The motor explosion risk of this should be quite obvious to the casual observer.   

The structural weakness of these designs shows up in another risk:  vibration during carriage on underwing pylons.  The unsupported branches flex during vibration,  creating localized internal material friction heating at the flex points.  If that heating is too large,  the propellant can depolymerize and liquify locally,  leading again to grain failure and loss of propellant chunks.

               Slotted-Tube (Finocyl) Designs

The partially-slotted tube designs are long tubular grains with a circular bore,  which feature two or more longitudinal slots in the aft portion of the grain assembly.  Another descriptive term is “fin-cylinder”,  or “finocyl”.  The slotted aft portion has high surface at relatively low total web,  while the forward bore-only portion has the lower average surface at higher total web.

There are two distinct levels of burning surface as the web burns,  initially higher,  then lower for the remainder,  usually by around a factor of 2 to 3.  The lower final bore-only surface is almost always somewhat progressive.

The slot geometry is relatively simple,  so that there are few risks associated with unsupported branches of propellant,  the way that there are such risks in dendrite and wagon-wheel designs.  These slotted tubes are better for withstanding shock and vibration.  Cold soak bore cracking is the usual limiting factor for the forward bore diameter.  Cast tooling is fairly simple. 

NEVER EVER use slots in the forward portion of the grain,  with the plain tubular bore located aft!  Doing this puts essentially the entire grain massflow through the smallest possible flow channel area.  That leads (at least) to the highest-possible pressure drop along the bore,  and quite probably to pressure drop-induced grain distortion that further reduces bore diameter.  That is an unstable positive feedback that generally guarantees a motor explosion.

Most tactical missile motors are now slotted-tube designs in typical motor L/D’s near 5 to 10,  because of the simplicity,  the robustness,  and the rather common mission need for initial short higher thrust,  followed by a longer-but-lower sustaining thrust.

Solid Propellant Types

There are fundamentally two types:  composite and double-base.  Composite uses solid ingredients dispersed evenly in a polymer binder.  These solids include the oxidizer,  any metal,  and other minor additives.  The binder requires both a chemical cure-hardening agent,  and oven heat,  to cure properly (the analogue to vulcanization of rubber). 

The double-base propellants are made of pelletized nitrocellulose (plus minor solid additives) flooded with liquid nitroglycerin.  The nitroglycerin reacts physically with the nitrocellulose to form a single plastic-like material.  It bleeds out again,  once the material is past its maximum service age,  much like the way nitroglycerin bleeds out of over-age dynamite sticks.  That situation is quite dangerous.

The double base propellants can have oxidizer and/or metal powders and any minor solid additives dry-mulled into the pelletized nitrocellulose,  before nitroglycerin is added.  These are called “composite-modified double-base” (CMDB) propellants.  It’s still the same reaction to a single plastic-like material,  however.  The added solids are just distributed within it the same way the solids are evenly distributed within a composite. 

Composites usually use a rubber-like polymer for the binder system.  The most common are CTPB (carboxy-terminated polybutadiene),  HTPB (hydroxy-terminated polybutadiene),  and PBAN (polybutadiene acrylonitrile),  with GAP (glycidyl azide polymer) an alternative that has liquid explosive characteristics.  Cure agents are usually isocyanates,  and the oven cure temperatures are usually near 250 F or more,  well above any hot service temperature the motor is ever likely to encounter. 

There have been many composite oxidizer materials,  but the two receiving most use are ammonium nitrate (AN) and ammonium perchlorate (AP).  Both are monopropellant explosives capable of mass detonation at one or another level of sensitivity (AP is the more hazardous for mass detonation,  and the most sensitive in any of the other safety tests).  

Other solids have included metal powders (primarily aluminum),  plastic resins,  and other high-yield monopropellant explosives like HMX and RDX.  Minor ingredients include opacifying carbon black,  and iron oxide as a burn rate “catalyst” for higher burn rates.

Processing this stuff is always done with remotely-operated equipment,  because of the fire and explosion,  even mass detonation,  hazards.  This is NOT stuff anyone would ever want to “cook up” on their stove top!  

Usually,  for simple gravity-driven sleeve-cast capability,  total solids content in the composite propellant must be well under 75%.  Otherwise,  the mix is simply too viscous for such casting.  Casting into vacuum instead of air reduces bubbles and voids.   The analogue here is the use of a water-rich “wet” mix of concrete,  to allow its gravity-sleeve casting in construction,  at low viscosity.  That behavior is similar to low-solids composite propellant. Mixing is also done under vacuum to avoid entraining air as bubbles.

Higher solids propellants (up to 87 or 88% solids) can be made by forced extrusion (“pressure”) casting with subsequent “pressure-packing”.  The casting vessel has a piston driven by air pressure that forces the thick mix directly into the motor/cast tooling assembly.  This has its analogue in casting a thick “dry” concrete mix,  which must be cast direct from the mixer barrel,  and hand-packed into the forms.  This casting operation MUST be done under high vacuum in the motor case to reduce bubbles and voids!  Once the motor unit is full,  it is exposed to the full casting pressure for a time,  which forces-closed any remaining bubbles or voids.  Only then is the loaded unit “cooked” to its full cure in the oven.

The composite-modified double-base propellants use similar oxidizers and other solid ingredients,  as are used in the composite propellants.  AN was long used,  along with aluminum if smoke was not an issue.  More recently,  AP oxidizer has been used in composite-modified double-base.  Carbon black is a minor opacifying agent,  whether in plain double-base,  or composite-modified double base.  Iron oxide can be used as a burn rate catalyst,  in the oxidizer-bearing CMDB formulations.

Whether nitrocellulose,  or nitrocellulose plus oxidizer and other solids,  the mixed powders are loaded dry into the motor,  with its cast tooling in place.  Nitroglycerin is flooded in from the bottom,  until all the dry powders are wetted.  The reaction between nitrocellulose and nitroglycerin is allowed occur to completion.  This produces a plastic-like material in which all the other solids are suspended. 

Typically,  the double-base propellants have lower specific impulse than the composite propellants,  because of the higher oxidizer content in the composite propellant (nitrocellulose and nitroglycerin contain oxygen,  but overall they are fuel-rich).  Adding oxidizer to make a composite-modified double-base,  makes up some of that difference in specific impulse.  Double base burn rates are grossly similar to burn rates in AP-oxidized composites. 

The AN composite formulations have almost an order of magnitude lower burn rates than the AP composite formulations,  so there is a wide range of burn rates available in composite propellants.  AN composites also produce significantly-lower specific impulse than AP composites.  Double base specific impulse generally falls in the same range as AN-oxidized composites.  Specific impulse of composite-modified double base looks more like that of AP-oxidized composites,  especially if the CMDB oxidizer is AP.  CMDB with AN falls a bit short of that.

In terms of the various hazards to resist,  the composites are more benign than double base in the fragment impact and sympathetic detonation tests,  and the double bases more benign in the fuel fire cook-off tests,  particularly what is called “slow cookoff”.  Nearly all the composites classify as “class 1.3 explosives”.  Some of the double base propellants classify as “class 1.1 explosives”,  and the rest as “class 1.3 explosives”.  Class 1.1 is a greater mass detonation hazard than class 1.3.

Representative numbers for burn rate,  specific impulse,  density,  and hazard classifications can be found in the “AIAA Aerospace Design Engineers Guide”.  Mine is a third edition from 1993.  The section is 10 “Spacecraft Design”,  subsection “propulsion systems”.  Burn rates are plotted on page 10-37.  There are two tables of properties and performance on pages 10-38 and 10-39.  Page numbers may vary in other editions.  See Figures E1,  E2,  and E3,  copied from my copy of the reference. 

Note that the burn rates shown plotted in Figure E1 were plotted on a log-log graph.  The reason for this is very simple:  power function models produce straight lines on log-log plots.  So,  data that correctly models as a power function will correlate very closely as straight lines on such a plot.  Pearson’s r-squared will be a very high number,  well above 0.98 usually,  if one does the statistics for curve-fitting. 

In the “old days”,  we just made and presented the burn rate data plots,  and measured the slopes on the plots for n.  The tight fit was obvious from the plot,  so the statistics were unnecessary.  It is also easy to see on such plots slope breaks that correspond to different values of n in different regions of Pc.

At high solids loading,  I have seen rates almost as high as the top dashed line labeled high burn rate composite in Figure E1.  They look a lot like the JPN-type DB curves in the figure,  which are are the ones near 0.6 ips at 1000 with steep slope n.  The curves near 0.3 ips at 1000 psia with low slopes n are listed as AP composite.  They look typical of low solids-loading AP composites.

I know little about the “XLDB composite” except that “XLDB” stands for “cross-linked double base”.  How that is also a “composite” makes no sense to me.

The composite ammonium nitrate curves look pretty typical to me.  It is also my understanding that the multiple-slope curve labeled “Plateau DB” is pretty typical of a lot of double base formulations.  Many of those DB formulations have two slopes with a near-zero n range in the middle between them,  which is what that curve illustrates.

Figure E2 shows tabular data for an AN composite and two AP composite formulations.  The higher binder listing of 18% corresponds to what is called in the industry a lower solids loading, in this case 82%.  The tradeoff is less AP for more aluminum,  and vice versa.  The other listing for 12% binder is the high solids loading of 88%.  There were very few manufacturers who could process material this thick.  Performance is higher if the solids loading is feasibly processed.  The same tradeoff of AP versus aluminum applies.

Be aware that data comparison tables like this may list Isp for perfect expansion from 1000 psia to 14.7 psia,  but those data are not corrected for (1) realistic nozzle kinetic energy efficiency,  and (2) the c* values are theoretical not empirical.  Of the two error effects,  the theoretical c* value is likely the larger error.  And this c* error is motor size-dependent:  SRB-sized c* and Isp will be higher than tactical-size c* and Isp,  yet both will be less than the theoretical c*.

Figure E3 lists typical tabular data ranges for AN and AP composites,  DB propellants,  and CMDB propellants.  Three of the top 4 in the list are DB or CMDB materials,  and require extrusion processing.  The fourth is a not-very-common composite that also can require extrusion processing.  All the rest in the list are composite propellants,  which require more-or-less “ordinary” cast processing. 

The PBAN material in the second group,  and the CTPB and HTPB materials in the third group are the most common propellants in use today.  The last one in the list is a very old AN composite.  The more modern AN composites have Isp performance rather close to the DB at the top of the list. 

 Figure E1 – Plots of Burn Rate Trends of Various Propellants from AIAA Handbook

 Figure E2 – Typical Composite Propellant Data from AIAA Handbook


Figure E3 – Typical Data on Composite,  DB,  and CMDB Propellants from AIAA Handbook

There are two kinds of rocket plume smoke:  particulates and secondary condensation.  Metal oxide particles and sometimes soot particles comprise the particulate smoke.  The presence of chlorine atoms as HCl molecules in the exhaust plume,  combines with atmospheric water vapor,  to produce what is called secondary condensation smoke.

Any propellant containing metallization will produce particulate smoke.  This metal is usually aluminum,  producing aluminum oxide particulates in the plume,  a dense white smoke.  Metal content almost never exceeds 20% by mass in the propellant. 

Any propellant containing AP as its oxidizer will produce chlorine atoms in its exhaust plume,  thus causing white secondary condensation smoke because of the humidity in the atmosphere.  Denser smoke correlates with higher humidity. 

Other particulates might come from the trace additives,  or from carbon soot resulting from fuel-rich combustion.  Soot produces a dark gray or black smoke. 

The term “reduced smoke” refers to nonmetallized propellants that may or may not contain AP oxidizer.  There is at most only a very little bit of particulate smoke,  and perhaps also significant secondary condensation smoke,  if AP is present and humidity is high. 

The terms “smokeless” or “min smoke” refer to nonmetallized propellants that also do not have AP in the oxidizer,  so that there is no secondary condensation smoke or any significant particulate smoke. 

There are both US government,  and NATO,  standards for reduced and min smoke propellants.  These are defined in terms of things that can be measured in testing.

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Update 2-22-20/Metallization: 

The metal added to most solid propellants is aluminum powder,  to at most 20% of the propellant by mass.  This is usually done as AP replacement in AP-bearing composites,  or as nitrocellulose replacement in double base propellants.  In fuel-rich propellants intended for airbreathing combustors,  the metals can include magnesium,  aluminum,  or even boron or related boron-bearing compounds.

In the usual solid rocket propellants,  aluminum addition has the effect of increasing chamber temperature and empirical c* substantially.  That second item is reduced a bit below just the temperature effect,  by also increasing effluent molecular weight.  The overall effect is somewhat higher specific impulse,  all else being equal,  at the cost of substantially-increased particulate smoke. 

Metallization also greatly reduces the susceptibility to combustion instability.  The metal oxide particles are a cloud within the motor free volume that inertially resist the oscillating gas movement of such instability.  The basic coupling is by particle drag,  a dissipative effect.  In the fuel-rich airbreathing propellants,  the same effect also traces to the solid soot particles as well as any metal oxide particles.

Update 2-22-20/Combustion Instability:

I only wish to give a cursory discussion of this topic here.  The actual topic is huge,  and the subject of enormous amounts of ongoing research and development. 

Combustion instability can occur whenever a frequency within the inherent noise spectrum created by combustion,  matches the frequency of some acoustic mode available within the combustion cavity.   This matchup happens all the time,  so there is a required second enabling item:  that being a feedback from the energy released by combustion,  to the energy contained within the susceptible vibration mode.  This is a phenomenon that can happen in any combustion system,  but solid propellant rockets are particularly notorious for it.  In solids,  this is traceable to burn rate and exposed burning surface.

Combustion noise is a sort of white noise spectrum from around a hundred Hertz to at most around 10,000 Hertz (10 KHz). It shows up as pressure oscillations of several to a few tens of psi,  superposed upon the basic motor pressure signal of several hundred to a few thousand psi.   These correspond to a ratio of oscillation amplitude to basic signal level,  of something on the order of 1%.  (Seeing something over 10% is likely indicative of instability.)

Explanations for the source of combustion noise vary,  but my favorite is vortices in the necessarily turbulent combustion environment.  That turbulence is required for mixing and the related combustion completeness,  almost regardless of what system we are talking about.  Some (perhaps many) of these vortices are enriched in fuel or oxidant species over their surroundings.  As they spin away from contact with an adjacent surface,  the surrounding gases are drawn into the vortex,  and that vortex may then ignite and explode.  It is as good a model as any.

Acoustic modes as explained in physics books focus upon organ pipes and similar tubular devices.  These are all longitudinal modes,  meaning the oscillation is end-to-end.  In rocket motors,  there are also various radial and circumferential modes to consider,  which means there are quite a lot of possible vibration modes,  with quite a lot of possible excitation frequencies.  Plus,  the shape of the cavity (and its selection of possible modes) varies quite drastically as the propellant surface burns back. 

Usually only the fundamental (lowest-frequency) mode and the first couple of harmonics (multiples of 2 higher in frequency) of each mode type (longitudinal,  radial,  and circumferential) are of technological interest,  but that is an empirical observation,  not a hard-and-fast rule!

Enabling items include both localized burn rate enhancement and localized high values of burning surface exposed in the susceptible regions.  Burn rate is normally a power function of pressure,  but past a certain threshold value of scrubbing action,  it is also a power function of mass flux (or density x velocity).  Mass flux w/A and density-velocity ρV are equivalent measures of this scrubbing action.

At pressure nodes for the oscillation,  the amplitude of the pressure variation (high to low) is largest,  and so is the basic burn rate variation,  based on the exponential dependence upon pressure r = a Pcn.  Similarly,  at pressure antinodes the velocity variation amplitude is highest,  for enhanced erosive burning effects Δr = C (w/A)s = C (ρ V)s.  Both of these act in phase,  so the feedback effect is enhanced by either occurring.

There are places within the motor cavity where flow velocities are inherently highest.  These are where the maximum massflow must pass through the minimum flow channel areas exclusive of the nozzle.  That is where the erosive burning effect has its highest potential.  This effect does not occur if the scrubbing action is below the minimum threshold value.  The direct pressure effect on burn rate always occurs.

Either way,  these enhanced burn rate effects upon motor pressure maximize when there is maximum affected burning surface area.  That can be the maximum exposed burning surface at a pressure node,  or it can be maximum exposed burning surface at (and just upstream of) a minimum flow channel area location.

Recognizing instability depends mostly upon the frequency response of your data acquisition and processing equipment,  which may or may not be the same equipment.  A very high frequency response is required to discern properly high-frequency pressure variations (the noise “hash” superposed upon the basic pressure level signal).

One way to achieve this is to record the pressure trace signal at just about a 1 megahertz response level,  as an analog recording.  This can be played back through any desired equipment,  depending upon what it is that you wish to see,  and to do,  with the data.  Most digital processing equipment which is affordable cannot provide this level of frequency response,  so you need a way to plot the analog data directly from that high frequency-response analog recording,  in order to see combustion instability effects directly.  

Combustion instability usually shows up as a sudden increase in average pressure level simultaneous with a substantial increase in the ratio of oscillation amplitude to average pressure level.  This sudden increase in average pressure level may (or may not) lead to a motor explosion.  But the point here is recognizing the simultaneous increases in average level and oscillation amplitude ratio. If you are lucky,  both effects (pressure rise and increased “hash” amplitude percentage) are apparent before the motor explodes.

You cannot see both of these effects in the usual all-digital system plot of test pressure!  You might (or might not) see the sudden increase in average pressure,  but resolving the high-frequency “hash” is just beyond most of the affordable all-digital equipment.  Most of the time,  the higher-amplitude “hash” occurs at a definite dominant frequency that can only be measured from the analog plot.  This is a crucial piece of data when instability is suspected,  but it is not sufficient to determine a solution!


Empirical means to combat instability in solids includes changes to the cloud of solids particles in the cavity,  changes in the cavity geometry,  and changes in the distribution of burning surface geometry within that cavity.  The first one is affected by metal content in the propellant,  up to about a maximum practical 20% aluminum,  in solid rocket propellants that must generate c* and specific impulse.  The other two require major changes in the selected propellant grain design.  Not much else is known to have any beneficial effects. 

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Roughing-Out the Design of a Solid Motor to An Impulse Requirement

The usual design requirement is essentially a thrust-time trace required of the motor,  whose integral is the total impulse required of the motor:  Favg tb = Itot.  This thrust and total impulse must be produced at a specific altitude-of-flight condition,  usually characterized by some ambient atmospheric backpressure Pa.  There is usually also some specification for how smoky the exhaust plume can be (which sets propellant type and metallization levels). 

There are also usually motor outer diameter and overall length limits driven by the mission and missile design.  Most motor designs must meet some sort of hot and cold soak criteria,  and also some sort of survivability criteria against things like mechanical shock and vibration.   For military motor hardware,  some version of Mil Std 210 is usually required for these things.  Depending upon which version,  hot soak temperatures can be 145 F to 165 F.  Cold soak temperature is usually -65 F. 

One starts with a “typical” max expected operating pressure (MEOP) for the motor design,  and its outer motor case diameter from the motor requirements.  This is strongly size dependent:  120-inch diameter motors generally operate under 1000 psia,  while nominal 6 inch motors operate at or above 2000 psia.  This is because basic material strength does not change with size,  but the applied loads do. 

The structures and thermal people use this MEOP plus a margin-of-safety factor (from 1.0 to 1.1,  usually) to determine the thickness and material of the motor case.  They have to do this at the expected hot operating conditions for that material,  which reduce its strength.  So,  too,  does fabrication method affect this result:  such as strength in welds versus strength in the parent material.  Usually,  it is hoop stress in the case that governs,  which is best modeled by this form of Barlow’s equation:

Pdesign ID = 2 σhoop tcase,  where ID = OD – 2 tcase and Pdesign = factor x MEOP

The hot case temperature depends upon both internal heating and external aeroheating effects,  versus re-radiation to the environment.   Detailing that topic is out of scope here.  But the main result for internal ballistics purposes is the finished propellant outer diameter,  which is the motor case outer diameter less twice the sum of case and insulation thicknesses.

The motor internal ballistics people use the same MEOP value,  the hot soak temperature,  and a “typical” max to average burn surface ratio of perhaps 1.5 or thereabouts,  to figure a “typical” average equilibrium chamber pressure Pc at 77 F. 

This average 77 F chamber pressure Pc and the ambient pressure at the design point Pa,  plus the hot gas properties for the selected propellant (specific heat ratio near 1.20 is nearly always “well inside the ballpark”),  set the nozzle expansion ratio Ae/At and thrust coefficient CF.  Assuming perfect expansion at the design point Pe = Pa:

Me = {[(Pc/Pe)(ϒ-1)/ϒ – 1](2/(ϒ – 1))}0.5  where expanded pressure Pe = ambient pressure Pa
Ae/At = (1/Me){[1 + 0.5(ϒ – 1)Me2]/[0.5(ϒ + 1)]}(ϒ + 1)/(2(ϒ – 1))
ηKE = 0.5(1 + cos a) where a = average exit cone half-angle = usually about 15 degrees
Estimate CD (usually 0.98 to 0.995 for a smooth nozzle profile)
CF  =  [Pe/Pc (1 + γ Me2 ηKE) – Pa/Pc] Ae/At

Now,  the same internal ballistics people rough-out the basic motor characteristics in terms of propellants being considered:

Estimate chamber c* = K Pcm  from empirical test data for the propellant being considered
Estimate Isp = CF c*/ gc (which assumes 100% of generated massflow goes through the nozzle)
Size At = req’d Favg / Pc CF (by definition of CF)
Size Ae = At (Ae/At)  (by definition of expansion ratio Ae/At)
Size average flowrate w = Pc CD At gc / c*  (assuming steady-state operation)
Estimate Wp = Itot/Isp  or as Wp = w tb where Favg tb = Itot (and check the other for consistency)
Using a lab propellant density ρ, estimate the propellant volume Vp = Wp/ρ
Using empirical test data for expulsion efficiency ηexp,  determine the r S product from the flow rate w:    r S = w /ρ ηexp

From here,  the same internal ballistics people determine the characteristics that the grain design must actually have.  The finished grain overall circular cross section area is Acirc = π D2/4,  where D is the finished propellant outer diameter (and R = D/2 its radius).  The finished grain length L is some appropriate fraction of the overall motor length,  usually something like 90-95%. 

The propellant grain design as-cast end area Aend = Vp/Lgrain.  Aend/Acirc is the propellant cross sectional loading required of the grain design.

Any given grain design has a max distance through the propellant that is to be burned,  including “sliver”,  if any.  That distance is the total web to be burned,  of that grain design.  That value of total web divided by the outer grain radius R is the web fraction (WF) of that grain design.

Propellant volume divided by that same total web is the average burning surface of that grain design.  That average burning surface ratioed to a convenient reference area,  is a relative measure of the size of the average burning surface to the overall motor design size constraints.  A convenient reference area is the flat rectangle of grain outer diameter D times grain length L.  Thus,  the relative average surface ratio is average burn surface divided by grain D times grain L:  avg. S/(D L). 

The average 77 F burn rate required of the propellant is the grain design web divided by the burn time tb of the required thrust-time trace:  ravg at 77 F Pc = grain design web/tb. 

For the propellant under consideration,  there are max and min feasible values of burn rate available at the average 77 F motor pressure for this problem.  That required burn rate at 77 F needs to fall within the min to max feasible 77 F burn rate range for the propellant under consideration.  The burn rate exponent n can be used to correct these values at average 77 F Pc to “standard” burn rate at 1000 psia and 77 F values,  for easier general comparison:  77 F r1000 = 77 F r-at-Pc (1000 psia/Pc)n.    

The internal ballisticians then look through their repertoire of grain designs for cross-sectional loadings,  web fractions,  and relative average surface ratios,  that match these requirements.  They must also consider whether the required burn rate falls in the feasible range for the propellant under consideration. 

Where all four values match up with requirements and constraints,  is a combination of grain design approach and propellant identity,  that is feasible.  There is often more than one such combination that is feasible.

Detailing the Design

From there,  further design iteration is required to set all the details for each feasible candidate combination,  primarily the effect of actual max/average surface ratio for the candidate grain design upon the appropriate value of 77 F Pc.  That change affects everything done so far.  This process is thus inherently iterative,  and so for multiple iterations,  on each feasible candidate. 

The final design is then chosen from among those few feasible combinations,  considering all the other motor design requirements. That finalized selected ballistic design is both a grain design,  and a propellant selection and specification.

Miscellaneous

Many grain designs are possible.  Those explored herein are but a few,  each with many possible variations.  Some representative data follows:

Figure F – Normalized Ballistics Parameters for Design Selection

For the end-burner in the table,  the bondline augmentation burn rate ratio is assumed to be 1.3.  That corresponds to a final coned surface 39.7 degrees off flat.  For the average surface,  I used the arithmetic average of initial flat and final coned surfaces.  This is a sort of worst-case level of augmentation.  It is usually only about factor 1.15 or 1.2 on bulk rate.

For the keyhole slot in the table,  I used a large grain L/D.  This gave about the same max/average surface ratio as the rainbow-neutral segment grain in the table,  but at a much more usable L/D proportion for the IRR booster application.  The truly rainbow-neutral keyhole slot has an L/D closer to 2.3,  but the keyhole slot is in general a lot less sensitive to L/D than the segment grain design is.

The heading Aend/Acirc is the cross-sectional loading afforded by the grain design.  The heading tot. web/R is the web fraction afforded by the grain design (necessarily a function of L/D for an end-burner).  The heading avgS/DL is the relative ratio of average surface to the DL reference area afforded by the grain design (necessarily a function of L/D for an end burner).  The heading Smax/Smin is the max-to-min surface ratio afforded by the grain design.  The heading Smax/Savg is the ratio of max S to average S afforded by the grain design.   The last heading is grain L/D,  necessarily a function for the end-burner.

Developing State-of-the-Art Propellants

This is NOT something amateurs can do.  It takes expensive equipment and facilities to do this work,  as remote operations for safety,  and in explosion-resistant revetment cells.  

You need mixers at the pint scale,  the 1-gallon scale,  and the 5-gallon scale in your propellant development lab.  You’ll need mixers at the 25-gallon scale and the 300-gallon scale in your production area.  Everything about your facility should be capable of handling liquid explosives.  And,  if you handle GAP or nitroglycerin,  you will have fairly-frequent explosions and equipment losses!

You’ll need a strand bomb with an inert atmosphere for testing burn rates of small strands.  These burn from break wire to break wire inserted through the strand,  after being ignited at one end,  with all sides inhibited.  Such strands can be cast in a pan and cut,  from a pint,  or even a half-pint,  mix.

You’ll need 2-inch burn rate motor hardware and a safe place to fire them,  knowing that motor explosions will be frequent,  until you characterize what amounts to the propellant c* values for selecting nozzle sizes.  2-inch burn rate motors are nominally 2.00 inch case ID,  with a 1.00 inch bore ID,  and a grain sleeve length just about 4.24 inches long. 

These are rainbow-neutral segment grains cast in hard sleeves,  and cartridge-loaded into the test case.  Nozzle assemblies are ejected in the event of an over-pressurization (explosion) event.  In that way,  hardware is not damaged,  and can be reused indefinitely,  regardless of the explosions.  Many 2-inch burn rate motors and a strand pan sample can be cast from a 1-gallon mix.  The same plus a couple of 4 or 6 inch lab motors can be cast from a 5-gallon mix.

You will need both 4.00 inch and 6.00-inch diameter lab motor hardware,  and a safe place to test them,  knowing that explosions will occur.  These need to be heavyweight cases and closures held together by neckdown bolts that break before the cases can burst.  In that way hardware is not damaged by the inevitable motor explosions,  and thus can be reused indefinitely.  These lab motors need to be able to handle either internal-burning or end-burning test grains of propellant,  cast into appropriate sleeves or boots.  Nozzles are best made as drilled orifices through half-inch thick monolithic graphite discs,  backed up by washers or the nozzle housing steel shell itself.

4 inch motors can indicate the size scale-up effects on the propellant burn rate curve,  and an early (and not very reliable) indication of empirical c*.  End-burning grains will help quantize the bondline burn rate augmentation,  once a bondline system has been selected,  should the application be an end-burner.

6-inch motors can also confirm burn rates as scaled-up,  and provide a much better indication of the empirical c* to expect in the flight design.  Usually,  the burn rates and c* correlate very well between 6-inch lab motors and actual tactical-size flight motors.

Everywhere a motor of any type is to be fired,  you will need a safe control room,  protected from blast  noise,  shrapnel,  and fumes.  You will need some sort of safe-arm protection for crews working on the test stand.  You need a fire department with trucks to put out grass fires,  because your motor explosions will cause them.  You will need digital data acquisition equipment that can also be used to analyze your motor ballistics with appropriate software that you provide.  This is custom programming work.  And for the 4 and 6-inch motor tests,  you will need video,  with an option for very high-speed video,  of such tests.  Pressure is more important than thrust data for lab motor tests,  but both need collection redundantly.

You will need weather-tight revetments in which to store ingredients,  particularly the explosive ones.  You DO NOT store ingredients in a mix cell,  and you DO NOT put but one mixer in a mix cell!  There will be fires and explosions,  and you do not want such an event to “take out” more than one mixer or cell at a time.

You will need a way to remotely transport explosive ingredients from storage to a mix cell,  particularly nitroglycerin.  You DO NOT want humans pushing carts loaded with this stuff.

You will need 2 or 3 propellant formulation chemists and perhaps half a dozen technicians in your propellant laboratory.  You will need a couple of test engineers and perhaps half a dozen technicians in your test department.  You will need a full-capability machine shop to make motor cases and parts (that’s near a hundred people). 

You will need a full-blown manufacturing department that includes mix-and-cast people,  fabrication-and-assembly people,  cure oven operators,  and quality control people.  That’s another hundred or more.

You will need engineers in your engineering department to cover every specialty.  That’s a couple of dozen high-talent people. And you will maintenance and security operations.

This is NOT an operation for the faint-of-heart or the low-of-budget. 

Propellant development for a single application is usually a $250,000+ operation over more-than-a-year,  using $millions in facilities and equipment,  for composites.  Double base is more expensive yet.  Anything more complicated than a very simple basic design application magnifies these expenses by factors up to 10.

“Them’s just the ugly little facts of life!  So,  get used to it.”

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Update 2-22-20/Lab Motor Test Results Characterize Propellant Ballistically:

What one wants out of a lab motor is a sudden surface burnout and a very fast pressure trace tailoff.  This is available with internal-burning segment grains,  and not generally available with end burners (because of the coning induced by bondline rate augmentation).  That is part of why most empirical correlations from lab motor tests use the internal-burning segment grain designs.  The remainder of “why” is the simplicity of cast tooling and the associated lab motor hardware design. 

The data collected during such a lab motor test comprises thrust and pressure,  and in the case of 2-inch burn rate motors,  usually only pressure.   Plus,  there is the weight actually expelled from the motor versus the installed propellant weight,  and the before-and-after measurements of the nozzle throat diameter. 

The weight measurements Wfinal and Winitial must be in the same hardware configuration before-and-after,  and the as-cast propellant charge weight Wp and its web must be measured and recorded,  as well.  In the case of Wp,  this quite often the as-cast grain-in-boot weight minus the empty boot weight.

Conceptually,  the nozzle massflow equation can be easily integrated with time under certain convenient assumptions,  so that the pressure trace integral can be related to expelled weight and c*:

∫w dt  =   ∫[Pc CD At gc / c*] dt   where w = nozzle flowrate and gc is the gravity constant for the units

Wexp = [∫Pc dt] avgCD avgAt gc / avgc*   where Wexp is final motor weight minus initial motor weight

What is assumed here is that the variations in CD,  At,  and c* are trivial during the test,  and so these variables can be replaced with their average values,  treated as constants.  The integral of the pressure trace is quite often directly available in digital data acquisition systems. 

The initial and final throat areas At are computed from the initial and final throat diameter measurements,  then averaged.  The average value of CD is assumed based upon the nozzle profile shape.  For a smooth approach profile,  this is usually in the neighborhood of 0.98 in small sizes,  and up to 0.99 in larger sizes.  Drilled orifice nozzles in graphite discs will have CD close to 0.80,  however.  Flow calibration is required to set this value. 

But in any case,  the integrated nozzle massflow can then be solved for the effective average c* during the test,  and the pressure traces and motor data used very effectively to define propellant characteristics:

avg c* = [∫Pc dt] avgCD avgAt gc / Wexp   where Wexp = Wfinal – Winitial and avgAt = (Atintial + Atfinal)/2
avg Pc = [∫Pc dt] / tb  where tb is time from ignition to the aft tangent at web burnout
ηexp = Wexp/Wp
avg r = web/tb

The term “aft tangent” refers to an empirical (and very “hands-on” manual) means to determine exactly where on the motor Pc-time trace web burnout point actually occurs.  There is a sudden drop in pressure during tailoff that is distinct from motor behavior just prior.  One “fairs-in” on a paper copy of the trace the tangents just-prior and just-after this change in behavior.  These tangent lines cross,  creating an angle between them,  which can be bisected.  Where the bisector touches the motor pressure trace,  is the point whose time coordinate is used for tb.

One accumulates across multiple firings of the same size lab motor,  tables of avg r vs avg Pc,   avg c* vs avg Pc,  and the list-averaged value of ηexp.  The r-Pc and c*-Pc tables can be plotted on log-log paper (or just curve fitted) to determine n and m,  respectively.  Burn rate firings must be made at three fully-soaked and measured temperatures to generate 3 curves:  one each for 77 F,  -65 F,  and something hot near 145-165 F. The change in burn rate hot-to-cold is used to define σP:

define fT = average value of rhot/rcold at any one Pc
then σP = [LN (avg fT)]/(Thot – Tcold)  where LN is the base “e” logarithm function

That is how one uses lab motor tests to empirically define 77 F r = a PcnσPηexp,  and c* = K Pcm for any given propellant. 

Be aware that burn rates from strands are the least representative of real motor burn rate,  while 2-inch burn rate motor tests are far closer.  Usually,  burn rates in 2-inch,  4-inch,  and 6-inch motors will correlate well with each other,  and with real motors.  Strand rates will be “in the ballpark”,  but usually a little different from the rest.

The correlated values of c* will differ significantly among the motor sizes.  They will be significantly-lower in 2-inch burn rate motors,  and somewhat low in 4-inch motors.  The correlation is usually fairly close between 6-inch motors and real motors.


Values of expulsion efficiency ηexp may be erratic in 2-inch motors because of the difficulty in routinely making accurate measurements for things that small.  4-inch motor expulsion is “in the ballpark” for real motor expulsion,  and 6-inch motor expulsion is usually pretty close to real motor expulsion.

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Final Remarks

There are obviously many more layers of detail about each and every one of the topics discussed above.  Should there be interest from any readers,  I can add some of those extra details in future updates.  But in any event,  I would like some reader feedback about this article. Please feel free to comment.