**Update 4-8-2024:**
Should any readers want to learn how to do what I do (estimating
performance of launch rockets or other space vehicles), be aware that I have created a series of
short courses in how to go about these analyses, complete with effective tools for actually
carrying it out. These course materials are
available for free from a drop box that can be accessed from the Mars Society’s
“New Mars” forums, located at http://newmars.com/forums/, in the “Acheron labs” section, “interplanetary transportation” topic, and conversation thread titled “orbital
mechanics class traditional”. You may
have scroll down past all the “sticky notes”.

The first posting in that thread has a list of the classes
available, and these go far beyond just the
two-body elementary orbital mechanics of ellipses. There are the empirical corrections for
losses to be covered, approaches to use
for estimating entry descent and landing on bodies with atmospheres, and spreadsheet-based tools for estimating
the performance of rocket engines and rocket vehicles. The same thread has links to all the materials
in the drop box.

The New Mars forums would also welcome your
participation. Send an email to newmarsmember@gmail.com to find out
how to join up.

A lot of the same information from those short courses is
available scattered among the postings here.
There is a sort of “technical catalog” article that I try to main
current. It is titled “Lists of Some
Articles by Topic Area”, posted 21
October 2021. There are categories for
ramjet and closely-related,
aerothermodynamics and heat transfer,
rocket ballistics and rocket vehicle performance articles (__of
specific interest here__), asteroid
defense articles, space suits and
atmospheres articles, radiation hazard
articles, pulsejet articles, articles about ethanol and ethanol blends in
vehicles, automotive care articles, articles related to cactus eradication, and articles related to towed decoys. All of these are things that I really
did.

To access quickly any article on this site, use the blog archive tool on the left. All you need is the posting date and the
title. Click on the year, then click on the month, then click on the title if need be (such as if
multiple articles were posted that month).
Visit the catalog article and just jot down those you want to go see.

Within any article,
you can see the figures enlarged, by the expedient of just clicking on a
figure. You can scroll through all the
figures at greatest resolution in an article that way, although the figure numbers and titles are
lacking. There is an “X-out” top right
that takes you right back to the article itself.

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**Three forgotten items have been inserted where needed below. These are the purpose and effects of propellant**

__Update 2-22-20__:__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”.

**These solid ballistics apply directly to the choked variable-throat area throttle valve technology developed for ramjet AMRAAM. Exactly how that works, and what it looks like, is documented in “Use of the Choked Pintle Valve for a Solid Propellant Gas Generator Throttle”, dated 10-1-21, and published on this same site.**

__Update 10-1-2__1:**Chamber Massflow Balance (Liquids and Solids)**

_{inpt}= w

_{out}+ w

_{stor}where w

_{stor}= (V

_{free}/RT) dP/dt

_{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.

__steady state balance__: dP/dt = 0, so w

_{inpt}= w

_{out}

_{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.

_{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.

_{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!**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.

*You have to reduce the propellant flow from tankage by the tapped-off amount to accurately model what goes through the nozzle.***Solid Rockets**

**(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*_{inpt}= ρ η

_{exp}S r

_{exp}is the experimental expulsion efficiency (weight expelled/weight of propellant), S is the (instantaneous) burning surface, and r the burn rate.

_{T}r

_{ref}(Pc/Pc

_{ref})

^{n}, where n is the burn rate exponent, also expressible as r = f

_{T}a P

^{n}

_{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:

_{T}= EXP[σ

_{P}(T - T

_{ref})], where “EXP” represents the base “e” exponential function

_{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.

_{exp}S f

_{T}a Pc

^{n}= Pc C

_{D}A

_{t}g

_{c}/ K Pc

^{m}

_{exp}S K f

_{T}a)/(C

_{D}A

_{t}g

_{c})]

^{{1/(1 - n - m)}}

_{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__at most__a very few milliseconds of ignition.

**Burning Surface Variation in Solids**

^{th}century for gunpowder rockets, but since shown to apply to all solids.

^{n}+ C (w/A)

^{s}= a Pc

^{n}+ C (ρ V)

^{s}, with exponent s being a number not far from 0.3

**End Burners**

*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.

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

**Simple Tube Segment Grains**

**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.***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**

__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.

**Other Grain Designs**

**. 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*

*Slotted-Tube (Finocyl) Designs***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!**Solid Propellant Types**

_{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*.

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

-------

**Update 2-22-20/Metallization:**

**Update 2-22-20/Combustion Instability:**

__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.)

*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__*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__^{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.

*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***. 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__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).

__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.

_{design}ID = 2 σ

_{hoop}t

_{case}, where ID = OD – 2 t

_{case}and P

_{design}= factor x MEOP

__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.

_{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}:

_{e}= {[(Pc/P

_{e})

^{(}

^{ϒ-1)/}

^{ϒ}– 1](2/(ϒ – 1))}

^{0.5}where expanded pressure P

_{e}= ambient pressure P

_{a}

_{e}/A

_{t}= (1/M

_{e}){[1 + 0.5(ϒ – 1)M

_{e}

^{2}]/[0.5(ϒ + 1)]}

^{(}

^{ϒ + 1)/(2(}

^{ϒ – 1))}

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

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

_{F}= [P

_{e}/Pc (1 + γ M

_{e}

^{2}η

_{KE}) – P

_{a}/Pc] A

_{e}/A

_{t}

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

_{sp}= C

_{F}c*/ g

_{c}(which assumes 100% of generated massflow goes through the nozzle)

_{t}= req’d F

_{avg}/ Pc C

_{F}(by definition of C

_{F})

_{e}= A

_{t}(A

_{e}/A

_{t}) (by definition of expansion ratio A

_{e}/A

_{t})

_{D}A

_{t}g

_{c}/ c* (assuming steady-state operation)

_{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)

_{p}= W

_{p}/ρ

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

_{exp}

_{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%.

_{end}= V

_{p}/L

_{grain}.

*A*_{end}/A_{circ}is the**required of the grain design.**

*propellant cross sectional loading***of that grain design.**

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

*relative average surface ratio is average burn surface divided by grain D times grain L: avg. S/(D L)*_{b}of the required thrust-time trace:

**.**

*r*_{avg}at 77 F Pc = grain design web/t_{b}__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}**,**

*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***Detailing the Design**

__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.

*That finalized selected ballistic design is both a grain design, and a propellant selection and specification.***Miscellaneous**

_{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.*__will__have fairly-frequent explosions and equipment losses!

__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.

**.**

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

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

*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.*_{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.

_{D}A

_{t}g

_{c}/ c*] dt where w = nozzle flowrate and g

_{c}is the gravity constant for the units

_{exp}= [∫Pc dt] avgC

_{D}avgA

_{t}g

_{c}/ avgc* where W

_{exp}is final motor weight minus initial motor weight

_{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.

_{D}avgA

_{t}g

_{c}/ W

_{exp}where W

_{exp}= W

_{final}– W

_{initial}and avgA

_{t}= (A

_{tintial}+ A

_{tfinal})/2

_{b}where t

_{b}is time from ignition to the

__aft tangent__at web burnout

_{exp}= W

_{exp}/W

_{p}

_{b}

__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}._{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}:

_{T}= average value of r

_{hot}/r

_{cold}at any one Pc

_{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._{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**