--------

**Update 2-22-20:**Three forgotten items have been inserted where needed below. These are the purpose and effects of propellant

__metal content__,

__combustion 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”.

--------

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

w

_{inpt}= w_{out}+ w_{stor}where w_{stor}= (V_{free}/RT) dP/dt
This is a very good approximation (since temperature and
molecular weight are but weak and extremely-weak functions of pressure, respectively). V

_{free}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 w_{inpt}= w_{out}
The nozzle article (first item on the list above) indicates
the nozzle output flow to be w

_{out }= Pc C_{D}A_{t}g_{c}/ c*, for a choked nozzle throat. In most textbooks on the subject, both A_{t}and c* are assumed to be constants, and C_{D}is utterly ignored as being essentially 1. Variations in the geometric area A_{t}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__. C

_{D}usually doesn't vary very much, but it is rarely actually 1, just usually close. A

_{t}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 A

_{t}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/Pc_{ref})^{m}= K Pc^{m}, 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**

**Varying propellant flow rate is**

*For liquid rockets, w*_{inpt}is a value determined by the operator of the engine.__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

**(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:***input massflow is generated by the burning of the solid propellant within the chamber*
w

_{inpt}= ρ η_{exp}S r
where ρ is the solid propellant density, η

_{exp}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 = f

_{T}r_{ref}(Pc/Pc_{ref})^{n}, where n is the burn rate exponent, also expressible as r = f_{T}a P^{n}
Quite often,
different values of n apply in different ranges of pressure. The r

_{ref}at Pc_{ref}version of the burn rate model is useful for this situation. Otherwise a = r_{ref}/Pc_{ref}^{n}. 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.**The factor f**

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

f

_{T}= EXP[σ_{P}(T - T_{ref})], where “EXP” represents the base “e” exponential function
For this, σ

_{P}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, T_{ref}is usually taken to be 77 F. For metric degree-C temperatures, use σ_{P}values 1.8 times larger, and T_{ref}= 25 C. Values of σ_{P}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 η

_{exp}S f_{T}a Pc^{n}= Pc C_{D}A_{t}g_{c}/ K Pc^{m}
Solving this equation for Pc gets a very informative
equation for the equilibrium motor burning pressure:

Pc = [(ρ η

_{exp}S K f_{T}a)/(C_{D}A_{t}g_{c})]^{{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 A

_{t}, variations of r and c* with pressure,__and especially__the variation in burning surface S, which is usually the very largest effect.**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 (**

*Note that the equilibrium motor pressure equation "blows up" if the sum n + m ever equals 1.*__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 19

^{th}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 Pc

^{n}+ C (w/A)^{s}= a Pc^{n}+ 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.

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.

**Max to min surface ratios get significantly larger when the L/D is wrong for rainbow neutrality.***This simple tube design is rather sensitive to incorrect values of L/D.*
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 "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%).**

*The "trick" is not having too high a web fraction.***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.

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.

The best source of grain design ideas and proper ballistic analysis
practice is the NASA monograph on the subject.
This is

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

*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.

**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.**

*NEVER EVER use slots in the*__forward__portion of the grain, with the plain tubular bore located aft!
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 I

_{sp}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 I_{sp}will be higher than tactical-size c* and I_{sp}, 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 I

_{sp}performance rather close to the DB at the top of the list.
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.

-------

-------

**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.*is a sort of*

__Combustion noise____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.

*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.*

__Acoustic modes__
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!

*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.*

__Enabling items__
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 Pc

^{n}. 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.

*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).*

__Recognizing instability__**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**

*One way to achieve this is to record the pressure trace signal at just about a 1 megahertz response level, as an analog recording.***, in order to see combustion instability effects directly.**

*you need a way to plot the analog data directly from that high frequency-response analog recording*
Combustion instability usually shows up as a

**. 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.***sudden increase in average pressure level*__simultaneous with__a substantial increase in the ratio of oscillation amplitude to average pressure level
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!*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*

__Empirical means to combat instability____major changes__in the selected propellant grain design. Not much else is known to have any beneficial effects.

------

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

**This thrust and total impulse must be produced at a specific altitude-of-flight condition, usually characterized by some ambient atmospheric backpressure P**

*The usual design requirement is essentially a thrust-time trace required of the motor, whose integral is the total impulse required of the motor: F*_{avg}t_{b}= I_{tot}._{a}. 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:

P

_{design}ID = 2 σ_{hoop}t_{case}, where ID = OD – 2 t_{case}and P_{design}= 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 P

_{a}, 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 A_{e}/A_{t}and thrust coefficient C_{F}. Assuming perfect expansion at the design point P_{e}= P_{a}:
M

_{e}= {[(Pc/P_{e})^{(}^{ϒ-1)/}^{ϒ}– 1](2/(ϒ – 1))}^{0.5}where expanded pressure P_{e}= ambient pressure P_{a}
A

_{e}/A_{t}= (1/M_{e}){[1 + 0.5(ϒ – 1)M_{e}^{2}]/[0.5(ϒ + 1)]}^{(}^{ϒ + 1)/(2(}^{ϒ – 1))}
η

_{KE}= 0.5(1 + cos a) where a = average exit cone half-angle = usually about 15 degrees
Estimate C

_{D}(usually 0.98 to 0.995 for a smooth nozzle profile)
C

_{F}= [P_{e}/Pc (1 + γ M_{e}^{2}η_{KE}) – P_{a}/Pc] A_{e}/A_{t}
Now, the same internal
ballistics people rough-out the basic motor characteristics in terms of
propellants being considered:

Estimate chamber c* = K Pc

^{m}from empirical test data for the propellant being considered
Estimate I

_{sp}= C_{F}c*/ g_{c}(which assumes 100% of generated massflow goes through the nozzle)
Size A

_{t}= req’d F_{avg}/ Pc C_{F}(by definition of C_{F})
Size A

_{e}= A_{t}(A_{e}/A_{t}) (by definition of expansion ratio A_{e}/A_{t})
Size average flowrate w = Pc C

_{D}A_{t}g_{c}/ c* (assuming steady-state operation)
Estimate W

_{p}= I_{tot}/I_{sp}or as W_{p}= w t_{b}where F_{avg}t_{b}= I_{tot}(and check the other for consistency)
Using a lab propellant density ρ, estimate the propellant volume V

_{p}= W_{p}/ρ
Using empirical test data for expulsion efficiency η

_{exp}, determine the r S product from the flow rate w: r S = w /ρ η_{exp}
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 A

_{circ}= π D^{2}/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 A

_{end}= V_{p}/L_{grain}.*A*_{end}/A_{circ}is the**required of the grain design.***propellant cross sectional loading*
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

**of that grain design.***total web divided by the outer grain radius R is the web fraction (WF)*
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 t

_{b}of the required thrust-time trace:**.***r*_{avg}at 77 F Pc = grain design web/t_{b}
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 r*_{1000}= 77 F r-at-Pc (1000 psia/Pc)^{n}
The
internal ballisticians then look through their repertoire of grain designs for

**,***cross-sectional loadings***, and***web fractions***, that match these requirements. They must also consider whether the***relative average surface ratios***for the propellant under consideration.***required burn rate falls in the feasible range*
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 A

_{end}/A_{circ}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**

**It takes expensive equipment and facilities to do this work, as remote operations for safety, and in explosion-resistant revetment cells.**

*This is NOT something amateurs can do.*
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.”

------

**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 W

_{final}and W_{initial}__must be in the same hardware configuration__before-and-after, and the as-cast propellant charge weight W_{p}and its web must be measured and recorded, as well. In the case of W_{p}, 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 C

_{D}A_{t}g_{c}/ c*] dt where w = nozzle flowrate and g_{c}is the gravity constant for the units
W

_{exp}= [∫Pc dt] avgC_{D}avgA_{t}g_{c}/ avgc* where W_{exp}is final motor weight minus initial motor weight
What is assumed here is that the variations in C

_{D}, A_{t}, 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 average value of C**

*The initial and final throat areas A*_{t}are computed from the initial and final throat diameter measurements, then averaged._{D}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 C

_{D}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] avgC

_{D}avgA_{t}g_{c}/ W_{exp}where W_{exp}= W_{final}– W_{initial}and avgA_{t}= (A_{tintial}+ A_{tfinal})/2
avg Pc = [∫Pc dt] / t

_{b}where t_{b}is time from ignition to the__aft tangent__at web burnout
η

_{exp}= W_{exp}/W_{p}
avg r = web/t

_{b}
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 t*_{b}.
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 η

_{exp}. 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 σ_{P}:
define f

_{T}= average value of r_{hot}/r_{cold}at any one Pc
then σ

_{P}= [LN (avg f_{T})]/(T_{hot}– T_{cold}) where LN is the base “e” logarithm function

*That is how one uses lab motor tests to empirically define 77 F r = a Pc*^{n},

*σ*_{P},

*η*_{exp}, and c* = K Pc^{m}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 η

_{exp}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.
------

**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.