Erosive Burning Design Criteria

8/3/2019 Erosive Burning Design Criteria

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by Charles E. Rogers

Introduction

Tis article is part of a continuing series inHigh Power Rocketry Magazine on the solidrocket motor. In this series solid propellantselection and characterization, internal ballisticsand grain design, erosive burning design criteria,and solid rocket motor performance analysis andprediction will be covered in extensive detail.Previous installments of th is series were publishedin the following issues of High Power RocketryMagazine; Performance Analysis of the Id eal RocketMo tor (in retrospect Part "0" of the se ries), in theJanuary 1997 issue; Parts 1 and 2, Solid PropellantSelection and Characterization, in the February (A)2001 and February (B) 2001 issues; Part 3, SolidPropellant Grain Design a n d Int ern al Ballistics, inthe O ctoberINovember 2002 issue; and Parts 4 and

5, Depa rtures Jom Idea l Performance for ConicalNozzles a nd Bell Nozzles, Str aigh t-Cut Throats an dRounded Throats, in the October 2004 andNovember 2004 issue s.In this installment of the series erosive burningdesign criteria for high power and experimental1amateur solid rocket motors will be presented.Design criteria for motors tha t will essentially burnnon-erosive, and design criteria for maximum rec-ommended erosive burning will be presented.Combined core Mach numberlcore mass flux erosiveburning design criteria will be presented which willallow the effective design of non-erosive an d erosivesolid rocket motors whethe r the m otor propellant issensitive to velocity-based or mass flux-based ero-sive burning. Of particu lar intere st to high pow erand experimentallamateur rocketeers is that theseerosive burning design criteria will allow theLength-to-Diameter ratio (LID) of high power andexperimentallamateur solid rocket motors to be

increased, or will allow an increase in the am oupropellant installed in a motor for a given mLID, increasing the motor installed performanminimum diameter rockets andlor increasingmotor total impulse. Finally, a constant core flux core design will be presented tha t once a glevel of erosivity or non-erosivity for the motobeen set, will allow a further increase in the mLID or motor installed propellan t, an d hence an greater increase in m otor performance.The present author would like to give spthanks to Derek Deville from EnvironmAerosciences Corporation (EAC) for providingmass flux erosive burning data from propecharacterization tests performed as part ofdevelopment of the solid rocket motor for the Cproject, and Gary Rosenfield of AeroTech, IncRCS R ocket Motor Com ponents, Inc., for provmotor drawings from the RCS Resource LiCompact Disk (CD), modified versions of which used in several of the figures in this article.

Erosive BurningWhen high velocity or high mas s flow rate hofrom upstream combustion passes over a dstream burning surface in a solid rocket molocal increase in propellant burning rate resThis local increase in propellant burning racalled erosive burning. The propellant burning(r,) in a particular location in a solid rocket mconsists of the non-erosive burning rate (r,), anaddition to the burning rate due to erosive bur

(re).

Where:rb = propellant burning rate, mlsec (inlsec)

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Copyright 2005 by Charles E. Rogers. All rights reserved. Published with permission.


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re = addition to propellant burning rate due toerosive burning, m /sec (inlsec)ro = non-erosive propellant burning rate,m/sec (inlsec)The non-erosive propellant burning rate is a func-tion of chamber pressure. The most widely usedanalytical expression for the non-erosive propellantburning rate a s a function of chamber pressure is deSaint Robert's law (Eq. (2)).

Where:a = coefficient of pressure, also called burningrate constantn = burning rate pressure exponentpc= chamber pressure , Pa (lb/in2)The coefficient of pressure (also called the burn-

ing rate constant) and the pressure exponent for thepropellant are part of what is known as the propel-lant internal ballistic characteristics, which areempirically meas ured characteristics of the propel-lant, and w hich vary from propellant to propellant.There are two basic types of erosive burning;velocity-based erosive burning a nd m ass flux-basederosive burning. Some propellants are more sensi-tive to the effect of the velocity of the hot gas flow-ing over the burning surface of the propellant(velocity-based erosive burning), some propellantsare more sensitive to the effect of the mass flux ofthe hot gas over the burning surface (mass flux-based erosive burning). Mass flux is the mass flowrate per squ are inch of port area (core cross-section -al area) flowing over the propellant burning surface.Mass flux is calculated by taking the propellantmass flow rate from the burning surfaces upstreamof the location for which the mass flux is beingdetermined, divided by the port ar ea a t the locationfor which the m ass flux is being determined.

Where:A, = port area (core cross-sectional are a),m2 (in2)G = mass flux, kglsec-m2 (lblsec-in2)

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f i rb propellant mass flow rate upstream of mflux location, kglsec (lblsec)r = core radius (circular port), m (in)As will be seen for velocity-based erosive burthe speed of the gas flow over the propellant bing surface will be characterized based on Mnumber rather than velocity, as the Mach numcan be directly tied to the motor core geometry

thus is a more intuitive and more direct paramto track for velocity-based erosive burning. Whethe erosive burning of a propellant is velodependent or mass flux-dependent can onlydetermined from erosive burning characterizatests.The approach tha t will be presented in this aris the use of combined Mach number-based mass flux-based erosive burning design criteriassis t in the design of solid rocket motors that heither essentially no erosive burning, or a reaable amount of erosive burning for maximum formance with good design margins for safe option of the motor. Since combined Mach numbased (replacing velocity) and mass flux-basedsive burning design criteria are used, these decriteria can be used for either velocity-sensitivmass flux-sensitive erosive burning propellantfor propellants where the sensitivity of the prolant to velocity-based and mass flux-based eroburning has not yet been determined.

Erosive Burning Motor Design IssuesFigures 1 and 2 illustrate som e of the solid ro

motor design issues t ha t arise from erosive burnFrom Figure 1 one of the primary high powerexperimentallamateur rocket design techniquemaximize performance for minimum diameter rets is to place the maximum motor total impbehind the smallest possible rocket frontal producing the minimum aerodynamic drag forrocket. The installed total impulse is maximizethe rocket for a given frontal area by lengthethe motor; i.e., by increasing the motor LengthDiameter ratio (LID). Thus a primary questionthe high power or experimentallamateur solid et mo tor designer is how high of an LID ca nmotor be designed to; how much can the mlength be increased for a given mo tor diameterAs will be seen later in this article, velocity-berosive burning is a function of the port areacore cross-sectional area) divided by the th roat (the port-to-throat area ratio). The lower the porthroat area ratio, the higher the velocity-based


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Erosive Burning Motor Design Issues - 1High Length-to-Diameter (UD) Motors lncrease Rocket Flight Performance.Maximizes Total Impulse within a Given Frontal Area,Minimizes Aerodynamic Drag in Minimum Diameter Rockets.

Keeping he Core Diameter the Same, How Much Can Motor Length Be IncreasedMotor Propellant Grain Length is Increased. . For a Given Motor Diameter?

For Velocity-Based Erosive Burning:1) lncreased Propellant Grain Lengthlncreases Propellant Surface Area.2) For Same K,,, ncreased Propellant SurfaceArea Requires ncrease in Throat Area (Ath).3) lncreased Throat Area ApproachesPort Area (Ap, the Core Cross-Sectional Area).Port-to-Throat Area Ratio (APIAth)Decreases,Core Mach Number Increases, lncreasedVelocity-Based Erosive Burning.

Figure I. Erosive Burningsive burning, From Figure 1 , when the m otor lengthis increased for a given motor diameter (the motorLID is increased), the motor grain length increaseswhich increases the motor propellant burning sur-face area. If the motor K, (the propellant burningsur face are a divided by the thro at are a, A,/A,) iskept constant to maintain the same m otor chamberpressure, a s the propellant burning surface area isincreased the motor throat area will have toincrease to maintain the same motor K,. If themotor core diameter is held cons tant, as the motorthroat are a is increased with the sam e port area, theport-to-throat area ratio will be decreased. Thisreduced port-to-thr oat area ratio will increase veloc-ity-based erosive burning, which will set a limit tohow much the m otor lerigth (a nd motor LID) can beincreased.

As will be seen later in this article, mass flux-based erosive burning is a function of the core massflux G (Eq. ( 3 ) ) , he m ass flow rate in the core divid-ed by the port area (the core cross-sectional area).The higher the core mass flux, the higher the massflux-based erosive burning. From Figure 1 , when themotor length is increased for a given motor diame-

For Mass Flux-Based Erosive Burning:I)ncreased Propellant Grain Length lncreasesPropellant Surface Area.2) lncreased Propellant Surface Area lncreasesMass Flow Rate Down Core.3) With Same Core Diameter, Port Area (Core CroSectional Area) Remains the Same.lncreased Core Mass Flow Rate through SameCore Cross-Sectional Area Results inlncreased Core Mass Flux, lncreased MassFlux-Based Erosive Burning.Motor Design Issues - I.ter, the increased motor grain length will incthe propellant burning surface are a, increasinmas s flow rate in th e core. If the motor core diter is held constan t, when this increased m assrate passes through the same port area (the core cross-sectional area ), the m ass flux in th ewill increase, increasing mass flux-based erburning. This increased mass flux-based erburning will set a limit to how much the mlength can be increased.From Figure 2 another solid rocket motor dtechnique to increase the perform ance of high pand experimentallamateur rockets is to reduccore diameter for a given motor length an d diamGiven a fixed motor case length and diametfixed motor outside envelope and internal voluif the core diameter is decreased there will be tional installed propellant in the motor, thuincreased total im pulse for the motor. Note th atfor a high LID motor like that shown in Figuonce the motor length h as been set, decreasinmotor core diameter increases the a mo unt of prlant installed in the motor, increasing the mtotal impu lse. Thus for any motor (sh ort or long

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rosive Burning Motor Design lssues - 2For Any Motor Length (Short or Long), Reducing Core Diameter IncreasesPropellant Loading, More Propellant Loaded within Fixed Motor Volume.Maximizes Volumetric Loading (Total Impulse Installed within a Fixed Volume).lncreased Rocket Flight Performance rom Higher Total Impulse Installed withinVolume or Length Available for Motor.

How Much CanMotor Core DiamBeReduced?

For Velocity-Based Erosive Burning:1) Reduced Core Diameter ReducesPort Area (Ap, the Core Cross-Sectional Area).Port Area Begins to Approach Fixed Throat Area (A,).Port-to-Throat Area Ratio (Ap Ath) Decreases,Core Mach Number Increases.2) lncreased Core Mach Number,lncreased Velocity-Based Erosive Burning.

Figure 2. Erosive BurningLID or high LID), once the motor le ngth an d diam e-ter and thus the motor volume have been fixed,decreasing the core diameter increases the motortotal impulse.As noted previously, for velocity-based erosiveburning the lower the port-to-throat area ratio, thehigher the velocity-based erosive burning. If themotor throat area is held constant, for essentiallythe same K, and th us essentially the same chamberpressure, as th e core diameter is decreased the port-to-throat area ratio will decrease, increasing veloci-ty-based erosive burning. This increased velocity-based erosive burning will set a limit to how muchthe core diameter can be decreased.From Figure 2, for mass flux-based erosive burn-ing as the core diameter is reduced, there is adecrease in the mass flow rate in the core becausethe propellant burning surface area in the core(ignoring the propellant burning surfaces on thesides of the individual grains in Figure 2) isdecreased. The propellant burning surface area inthe core is directly proportional to the core radius r,(the core propellant burning surface area is equal tothe circumferen ce of the core [2nrc] times t he len gthof the core), as the core diameter decreases the corepropellant burning surface area decreases. But theport area (the core cross-sectional area, nr,2)decreases at a faster rate, as the port area is direct-ly proportional to the core radius squared. The neteffect of a decrease in core diameter is an increasein core mass flux, increasing mass flux-based ero-sive burning. This increased mass flux-based ero-sive burning will set a limit to how much th e motor

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For Mass Flux-Based Erosive Burning:1) Reduced Core Diameter Reduces PropellantSurface Area, Reducing Core Mass Flow Rabut Port Area (Core Cross-Sectional Area)is Reduced at a Greater Rate.Result is an Increase in Core Mass Flux.2) lncreased Core Mass Flux, lncreasedMass Flux-Based Erosive Burning.

Motor Design Issues - 2.core diameter can be decreased.How much the motor LID can be increased given motor diameter, and how much th e core deter can be decreased for a given motor lengthvolume, are important design issues facing power and experimental/amateur solid rocket mdesigners. After the basic motor design andoverall grain geometry design are completedmotor designer faces a fundamental issue, small or large does the motor core need to bethis article design criteria will be presentedvelocity-based and mass flux-based erosive burnincluding combined core Mach number/core flux design criteria which will allow solid romotor designers to configure motor cores for eessentially non-erosive burning, or to achieve mmum recommended, yet conservative erosivitthe m otor core design.

Velocity-Based Erosive BurningData show ing the typical augmentation of prlant burning rate with combustion gas velocitcomposite propellants sensitive to velocity-berosive burning is presented in Figure 3 (reprod

from Ref. 1, original reference Ref. 2). The comtion ga s velocity (u,) is the local'velocity of thgas flow down the core of the solid rocket mwhich varies from zero velocity at the head enthe core to a m aximum velocity at the aft end ocore. The vertical scale is the ratio of the propeburning rate including erosive burning (r , , se


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Combust ion gas ve loc i t y , u f t / sec9Figure 3. Typical Effect of Combustion Gas Velocity on Burning Rate Augmentation - eloBased Erosive Burning.(1)) divided by the non-erosive burning rate (r,).rb/ro = 1.0 indicates non-erosive burning.Note from Figure 3 tha t a t low velocity the propel-lant burning rate actually dips below the assumednon-erosive value. This opens up the interestingpossibility that experimentallamateur rocketeersmay be performing propellant characterization testswith the core gas flow velocities too low, and whileassuming that they are measuring the non-erosivepropellant burning rate, are actually measuring alower propellant burning rate from the dip in thecurve in Figure 3. The present author , for the variouspropellant characterization tests he h as participatedin, has never seen this phenomenon of the dip inpropellant burning rate shown in Figure 3.Experimentallamateur rocketeers when performingpropellant characterization tests should add addi-tional tests to verify tha t this possible dip in propel-lan t burning rate show n in Figure 3 is not occurringfor the propellant being tested. Techniques forchecking for a possible dip in propellant burningrate a t low core gas flow velocities when performingpropellant characterization tests will be presentedlater in th is article.

The threshold velocity (u,,) is the combustion gasvelocity where velocity-based erosive burning

ation in threshold velocity between two diffepropellants (Propellants A and B ), again emphing th at velocity-based erosive burning is propedependent, and can vary widely from propellanpropellant.Using analysis methods th at will be subsequepresented, the experimentalJamateur rocketeercalculate the local flow velocity at the aft (noend of the core of the solid rocket motor, andcompare the flow velocity to the threshold veloas a function of chamber pressure in Figure 3will be seen the Mach number at the aft end ocore, which is the region which has the highest of erosive burning, is a function of the ratio oport area to the thr oat area. Using the Mach numat the aft end of the core a s an erosive burning and design criteria is much more useful than uflow velocity, and the present author recommthat the Mach number at the aft end of the corused as the primary design criteria for velobased erosive burning.

Core Mach Number Design Criteria forVelocity-Based Erosive Burning

Propellant B varies from 1700 ftlsec to 2050 ft/secas a function of chamber pressure. AS shown in burning begins of 1700 ft/sec to 2050 fFigure 3 for Propellant B, the higher the motor ~ h ~ w nn Figure 3 at first appears quite high. chamber pressure the lower the propellant threshold ~ h ~ k e dlow a t the r ~ ~ ~ l ehroat (Mach 1 at thevelocity. Figure 3 also s hows the possible wide vari- zle throat), the flow in the solid rocket motor 20 High Power Roc


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will be subsonic. It sho uld be remembered that thetemperature of the gas flow in the motor core, dueto the combustion of the solid propellant, is quitehigh , 2000 -650 0 OR (1540-60 40 OF). Thus the speedof sound for the gas flow in the motor core will bequite high.The speed of sound is a function of the ratio ofspecific heats , a ga s constant, and tem perature. Thegas constant (R) is the universal gas constant (R')divided by the molecular mass m.In SI Units:

Where:a = speed of sound, mlseca= molecular ma ss, kglkg-molR = gas constant per unit weight, ~1kg-OKR' = universal ga s c ons tan t, 83 14 .3 Jlkg mol-OK7- = abso lute tem pera ture, OKy = ratio of specific he ats , dimensionlessIf English Units are used the molecular mass isreplaced with molecular weight. To calculate thespeed of sound for the gas flow in the motor core,the ratio of specific heats and the ga s con stant (theuniversal gas constant divided by the molecularmass) for the combustion products making up the

gas flow in the core are used in Eq. (4). The temper-ature used in Eq, (4) is the adiabatic equilibriumflame temperature (T,) , the combustion tempera-ture. When several values are available from ther-mochemical equilibrium theoretical specific impulseprograms for adiabatic equilibrium flame tempera-ture, ratio of specifics heats , an d the ga s properties(typically for the chamber, throat, a nd nozzle exit),the chamber values should be used in Eq. (4) for thespeed of sound calculations for the gas flow in themotor core.Table 1 presents speed of sou nd da ta for represen-tative composite solid propellants, and representa-tive high power and experimental/amateur solidpropellants for the gas flow in the core of a solidrocket motor. The propellant adiabatic equilibriumflame temperature an d gas properties used in Eq. (4)for the speed of sound are included in Table 1. Twoof the propellants are representative composite solidpropellants taken from Sutton, Rocket PropulsionJanuary 2005

Elements (Ref. 3) , the re st of the propellant dafrom Appendix A of Parts 1 and 2 of TheSolid RoMotor series (Ref. 4).Note from the data in Table 1 that the speesound for the gas flow in the m otor core is primly a function of the propellant adiabatic equilibflame temperature, and that the speed of soundthe gas flow in the motor core is quite high, 23500 ftlsec.Based on the Propellant B da ta in Figure 3, ua representative chamber pressure of 500 psirepresentative threshold velocity (ufv) or the oof velocity-based erosive burning is 1950 ftUsing a 20% increase in propellant burning (rb/rO= 1.2) as a reasonable upper bound for mmum recommended erosivity for velocity-basedsive burning, based on the dat a plotted in Figufor Propellant B at a chamber pressure of 500 the com bustion ga s velocity (u,) for maximumommended erosivity for velocity-based erosive bing is 2500 ftlsec. These two velocity limits caconverted to Mach number limits by dividing vity by the speed of sound.

Mach number for flow in motor corbased on combustion gas velocity,dimensionlessthreshold Mach number for flow inmotor core based on threshold velofor onse t of velocity-based erosiveburning, dimensionless

u, = combustion gas velocity, mlsec (ftlsec)ufV threshold velocity for onset of velocity-berosive burning, mlsec (ftlsec)Note that the threshold velocity for velocity-berosive burning and the velocity for the maxirecommended erosivity can occur anywhere in

motor core, but they will be reached first in the est velocity location in the core which is at th(nozzle) end of the core. Thus the Mach numbinterest is the Mach number at the aft end ocore, where the Mach number and the combugas velocity will be highest and w here there wthe highest level of velocity-based erosive bur


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Tc(OK)(Note 3)(Note 4)

3542.8

Y(Note 4)

__I1.13

(Ma)&(u&=1950Wsec)(Note 5)-.56

(Ma)(ug =2500Wse(Note-0.72

Propellant AeroTech/ISP TcPropellant (OR)I.D. Number (Note 3)(Notes 1,2) (Note 4)

a(Wsec)(Note 4)-489.3eroTechASPHigh SolidsLoadingHigh Specific Impulse

Organic PolymerBinderIAPlAL18% Binder66-78% AP4-20% AL(Sutton [Ref. 31)Polymer Binder1APIAL12% Binder68434%AF'4-20% AL(Sutton [Ref. 31)AeroTechBlue ThunderAeroTechWhite LightningAeroTechBlack JackNotes:(1) All AeroTechIISP propellants use Hydroxy-Terminated Polybutadiene (HTPB) binder and Amm oniumPerchlorate (AP) oxidizer.(2) First two num bers of Propellant I.D. Number are solids loading. Exam ple; 884 3 - 88% solids load in(12%HTPB binder).(3)Adiabatic equilibrium flame tempera ture.(4 ) p, = 1000 psia, equilibrium flow.(5) Core Mach num ber for threshold velocity (u,, = 1950 ftlsec) for onset of velocity-based erosive burn(6) Core Mach number for combustion gas velocity for maximum recommended erosivity (u, = 2500 ftfor velocity-based erosive burn ingNA = Not ApplicableAL = aluminumAP = ammonium perchlorateHTPB = hydroxyl-terminated polybutadiene

Table I.Adiabatic Equilibrium Flame Temperature, Ratio of Specific Heats, Gas PropertSpeed of Sound, and Velocity-Based Erosive Burning Threshold and Maximum RecommenErosivity Mach Number Limits for Representative Composite Solid Propellants and High Poand ExperimentalIAmateur Solid Propellants.22 High Power Rock


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Table 2. Variation of Adiabatic Equilibrium Flame Temperature, Ratio of Specific Heats,Gas Properties, and Speed of Sound with Chamber Pressure for Combustion Gas Flowfor a Representative Composite Solid Propellant.and which fortuitously as it turns out is the M achnumber in the core that is the easiest to calculate.Because the core Mach number at the aft end ofthe core is used almost exclusively over the Machnumbers elsewhere in the core, the term "core Machnumber a t the af t end of the core" is often truncatedto just "core Mach num ber," in particular whendescribing a motor design. It is im portant to remem-ber that when a motor design is stated to have adesign "core Mach numbe r," this "core Mach num ber"is the Mach number at the aft end of the core atmotor ignition.Core Mach number limits based on the thresholdvelocity for velocity-based erosive burning and thevelocity for maximum recommended erosivity areincluded for the example propellants in Table 1.Note that for the AeroTechIISP propellants inTable 1 the adiabatic equilibrium flame tempera-ture, ratio of specific heats and g as properties werecalculated based on p, = 1000 psia, while the repre-sentative thresh old velocity for the on set of velocity-based erosive burning and the maximum recom-mended erosivity gas flow velocity were based on p,= 500 psia. (Both chamber pressures based on theavailable d ata.) The chamber pressure for the repre-sentative composite solid propellant data fromSutto n, Rocket Propulsion Elements, wa s no t speci-fied. The p, = 1000 psia adiab atic equilibrium flametemperatur e data , ratio of specific heats and ga sproperties data can still be used for the speed ofsound calculations and the (Ma),, and (Ma), calcula-tions in Table 1 because there is little variation inthe adiabatic equilibrium flame temperature, theratio of specific heats and the ga s properties withchamber pressure. Table 2 shows the variation inthe adiabatic equilibrium flame temperature, theratio of specific heats , the gas properties, an d thespeed of sound for the combustion gas flow as afunction chamber pressure for a representative m et-alized HTPBIAP composite propellant. Note that

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decreasing the chamber pressure from 1000 ps500 psia reduces the speed of sound only 0.4%Based on the d ata presented in Table 1 , the pent author recommends the following core Mnumber design criteria for velocity-based eroburning for high power and experimentallamasolid rocket motors:Non-Erosive:

Core Mach Number I 0.50Maximum Recommended Erosivity:

Core Mach Number = 0.70From Table 1 it can be seen that many of thepellants have a speed of sound for the combugas flow in the motor core of 3400-3500 ftwhich when combined with the threshold velfor the onset of velocity-based erosive burnin1950 ftlsec, results in a core Mach number of 00.57. For design margin on avoiding the onsvelocity-based erosive burning the present ausets a non-erosive core Mach number limit threduced approximately 10% o a core Mach nuof 0.50. From Table 1 the 2500 ftlsec combugas velocity for maximum recommended erosfor velocity-based erosive burning, combined the speed of sound of 3400-3500 ftlsec for manthe propellants in Table 1 , results in a core Mnumber of 0.72-0.73, rounded dow n by the prauthor to 0.70. Note from Table 1 that the hig

adiabatic equilibrium flame tempe rature propelset the lowest core Mach number limits. Folower adiabatic equilibrium flame temperaturepellants Table 1 shows that the core Mach nulimits are higher, thus using core Mach numberits based on the highest propellant adiabatic librium flame temperatures sets conservative


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Mach number limits for all propellants.Could the maximum recommended erosivity coreMach number be increased, for a more aggressive'design in terms of velocity-based erosive burning?As will be shown in the next section, with a coreMach number of 0.70 the port are a ( the core cross-sectional area) is only 10% larger than the thr oatarea. If the port area is reduced to the throat area,the mo tor will fire nozzleless; i.e. a t ignition Mach 1will be reached at the end of the core rather than atthe throat. Thus with a core Mach number of 0.70the port area is nearly a s small as the throat area,and th us little can be gained from fur ther reductionsin the port area, and wha t gains could be made (10%or less) would come at th e cost of much more risk interms of much higher velocity-base d erosive burnin gat ignition of the motor.

Core Mach Number as a Function ofPort-to-Throat Area Ratio

The Mach number at the aft (nozzle) end of thecore can be determined from the port area (the corecross-sectional area) divided by the throa t area (theport-to-throat area ratio) using Eq. (7) from Section6.5.2 of Space Propukion AnaZysis and Design (Ref.5)

Where:A, = port area (core cross-sectional ar ea) ,m2 (in2)A, = nozzle throat ar ea (thro at cross-sectionalarea ), m2 (in2)A, /Afh= port-to-throat are a ratio, dimensionlessMa = core Mach number a t aft end of core,dimensionlessr = core radius (circular port), m (in)rfh = nozzle throat radius, m (in)Eq. (7) gives the core Mach number a t the a ft endof the core, which a s discussed previously is known

a s th e "core Mach number." The ratio APIA fh sport-to-throat area ratio, and is a key parametethe motor, as it determines the core Mach numbUsing Eq. (7) if the desired core Mach numbknown, and the port-to-throat area ratio (A, needs to be determined, the core Mach numbersimply be entered into Eq. (7 ) to determine t he pto-throat area ratio. The more common situatiotha t the port-to-throat are a ratio is known, an dthe core Mach number t hat needs to be determiSimple iterative methods can be used to solve(7) for the core Mach number given the porthroat area ratio, with the simplest method bsimply iterating the core Mach number from using an increment of 0.01 until the port-to-tharea ratio is matched.Note that since Eq. (7) is for the Mach numbthe aft end of the core for the combustion gas down the c ore, th at if several ratios of specific hare available, the chamber value should be uversus u sing the throat or nozzle exit values.Note that the derivation of Eq. (7) from Reassume s a constant port area, i.e., a cons tant csectional area,of the core. A more complex mefor tapered cores is presented in Section 6.5.2 of5. All cores that have erosive burning at igniwill have a tapered core later in the burn, and in this article core designs will be presented feature an increase in port area for the grains aaft end of the motor, but the present auth or recmends th at Eq. (7) be used for all solid romotors for determining the core Mach number, the caveat that the equation is exact for conscross-sectional area cores and non-erosive moand approximate for tapered or staggered coreshighly erosive motors.Eq. (7) is somewhat complex, and needs tsolved iteratively, although the equation canbuilt into mo tor simula tion program s. For core Mnumber-based erosive burning design criteriacore Mach number is only needed at the aft enthe core and only at ignition, thus h and c alculatcan be performed using Eq. (7) using the mgrain geometry at ignition. Eq. (7) can be soiteratively using hand calculations, but to simthe process the present auth or has created FiguFigure 4 is a plot of the core Mach number a s a ftion of APIAfh ased on Eq. (7) assuming a ratspecific heats (y) equal to 1.2 (a good representavalue for solid rocket motors). Experimental/teur rocketeers can us e Figure 4 for rapid determtion of core Mach num ber for a given A, /Amotor ignition.Using Eq. (7), and assuming y = 1.2, the pously recomm ended core Mach number design cria can be converted to the following recommeport-to-throat area ratios (APIAfh)or velocity-b

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; y 4 . 2a versus _.

AthFigure 4. Nir, as a function of Ap /Ath , y = 1.2.

erosive burning for high power and experimental1 core). The slope of the core Mach number with amateur solid rocket motors: to-throat area ratio (APIA,) shown in Figubecomes very steep for port-to-throat area rNon-Erosive: approaching 1.0, reaching very high core Mnumbers for port-to-throat area ratios less Core Mach Num ber 2 0.50 1.10. Thus as the port area approaches the tharea and the port-to-throat area ratio approaFor y = 1.2; APIA, 2 1.36 1.0, the velocity-based erosive burning will becvery severe, not making it worth taking advanMaximum Recommended E rosivity: of a further possible 10% eduction in port areaCore Mach Num ber = 0.70

Mass Flux-Based Erosive BurningFor y = 1.2; APIA, = 1.10Note that for the maximum recommended erosiv-ity core Mach number of 0.70 the port area is only10% arger than the throat area. Little therefore canbe gained from further reductions in the port area.If the port area is reduced to the throat area themotor will fire nozzleless (Mach 1 at the end of the

Figure 5 (reproduced from Ref. 1, original rence Ref. 6) show s the typical augm entation ofpellant burning rate with mass flux for a comppropellant sensitive to mass flux-based eroburning. The mass flux shown on the horizscale is the mas s flux over a particular section opropellant. For downstream (toward s the noJanuary 2005


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Symbol0A0*AXe

0 1 3 5 7Mass f lux, &/A, Ib/sec-in. 2

Figure 5. Typical Ef fect of Mass Flux on Burning Rate Augmentation- Mass Flux-Based EroBurning.

end) sections of the propellant grain the mass fluxover the section arises from the m ass flow from non-erosive burning, and the additional mass flow fromany erosive burning that may be occurring furtherupstream in the core. As in Figure 3 the verticalscale is the ratio of the propellant burning rateincluding erosive burn ing (r,) divided by the non-erosive burning rate (r,). r, lr, = 1.0 indicates non-erosive burning.One of the item s of interes t in Figure 5 is the m assflux for the o nset of m ass flux-based erosive burn-ing (th e thre sho ld ma ss fl ux, G,,). As can be seen inFigure 5, the mass flux for the onset of mass flux-based erosive burning is a function of cham ber pres-sure; the higher the chamber pressure the higher themass flux required for the o nset of erosive burning.Table 3 presents summary data for the thresholdmass flux for the onset of mass flux-based erosiveburning based on the data for the representativecomposite propellant plotted in Figure 5.Note from Table 3 that the threshold m ass flux forthe onset of mass flux-based erosive burning is thesame for both p, = 1350 psia and p, = 1500 psia,for which the present author has no explanationbeyond possible scatter in the experimental data.The present author recommends that the thresholdmass flux value of 2.15 lblsec-in2 be used for p, =1400 psia, an approximate average of 1350 psia and1500 psia.The increase with chamber pressure of the thresh -old mass flux (G,,) for the onset of mass flux-basederosive burning is quite appar ent from the dat a plot-ted in Figure 5 and the summary data in Table 3.Higher chamber pressures can allow the motordesigner to design to a higher core mass flux and

still avoid mass flux-based erosive burning. apparent from the data plotted in Figure 5 is tcore mass flux 5 1.0 lblsec-in2 is no n-erosive fchamber pressures for the particular propellantted in Figure 5. The present author proposes tcore mass flux 5 1 O lblsec-in2 will likely be nonPC GtvChamber Pressure Mass Flux for On(psi$ ofMass Flux-BasErosive Burning(lb/sec-in2)

1500 1 2.15 (Note 1)

Notes:(1) Present author recommends that thesedata points be replaced with a mass flux of lblsec-in2 a t p, = 1400 psia.Table 3. Threshold Mass Flux for the OnMass Flux-Based Erosive Burning as a Funof Chamber Pressure.

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sive for all solid propellants at all chamber pres-sures.Based on the da ta presented in Figure 5 and Table3, the present author recommends the followingcore mass flux design criteria for mass flux-basederosive burning for high power and experimentallamateur solid rocket motors:

Non-Erosive:p, = 400-600 psia;

Core Mass Flux I 1.0 Ibtsec-in2p, = 800 p sia; Core Mass F lux 2 1.75 Iblsec-in2p, = 1400 psia; Core Mass Flux 2 2.0 Ibtsec-in2

Maximum Recommended Erosivity:p, = 400 p sia; Core Mass Flux = 2.0 Ibtsec-in2p, = 600 psia; Core Mass Flux = 2.5 Iblsec-in2p, 2 800 p sia; Core Mass Flux = 3.0 Iblsec-in2Core Mass Flux limits for MaximumRecommended Erosivity should not be exceededunless Erosive Burning Characterization Testsare performed for propellant.

To arrive a t the above core mass flux design crite-ria generally the core mass flux values from Figure5 an d Table 3 have been rounded to the nearest 0.25Iblsec-in2. The present author has also been some-what conservative in setting the above design crite-ria due to the steep slopes and u ncertaintylscatter inthe data at some chamber pressures for the datapresented in Figure 5. Generally the maximum rec-ommended erosivity core mass flux values are setbased on a 20% increase in propellant bur ning rate(r , /rO= 1.2), again set conservatively due to thesteep slopes of the propellant erosive burning ratewith mass flux at some chamber pressures. (Thep, = 400 psia d ata being a n excellent example; if thecurve is shifted to the left by 0.5 Ibtsec-in2 due tovariation s between propellants, there can be a majormisprediction in the erosive burning ra te.) Note th atwhile the p, = 1750 psia data is clearly non-erosiveat a core mass flux of approximately 3.0 Iblsec-in2and would have a r, lro = 1.2 at 5.0 Ib/sec-in2, thedata above p, = 800 psia (p, = 1350-1500 psia) isinconclusive, and in terms of maximum recom-mended erosivity the present author recommendstha t a core mass flu x of 3.0 Iblsec-in2not be exceed-ed even at high chamber pressures unless empiricalmass flux-based erosive burning data is availablefor the propellant.

January 2005

For the core mass flux design criteria that been presented, the present author recommthat the maximum recommended erosivity mass flux limits not be exceeded unless eroburning characterization tests are performedobtain empirical data for mass flux-based eroburning for the particular propellant, to determthe actual threshold mass flux for the onset of mflux-based erosive burning, and to determine a sonable maximum mass flux limit for mass fbased erosive burning.

CSXT 0 8 Propellant Mass Flux-BasedErosive Burning Characterization Data

As part of the motor development program untaken by the Environmental Aerosciences Corption (EAC) for the Civilian Space exploration T(CSXT) experimentallamateur rocket, the first professional/non-governmental rocket launche dspace which reached an altitude of 379,900 feetmiles), a series of propellant characterization were performed by Derek Deville of EAC, with sultation by the present author, to characterizepropellant for the CSXT solid rocket moPropellant characterization tests for chamber psure versus K, and burn rate versus chamber psure were performed. Additionally mass flux-berosive burning characterization t ests were also formed, the results from which are presentedFigure 6. Figure 6 presents the burn rate vechamber pressure for the CSXT D 8 propelincluding the basic non-erosive propellant brate, and the variation of propellant burn rate wcore mass flux. Note that the mass flux values ed in Figure 6 are core mass flux. Additionally that using a technique recommended in a later tion of this article, the mass flux values listeFigure 6 and referred to below are the initial mass flux at m otor ignition at the a ft (nozzle) enthe core, calculated based on the mass flow down the core being equal to the propellant frate, with the propellant flow rate calculated baon the propellant non-erosive burning rate attest chamber pressure (core mass flux at ignibased on non-erosive burn rate).Due to the tight development schedule forCSXT solid rocket motor, there was time to perfonly a limited number of propellant erosive burcharacterization tests. Thus for the burn rate vechamber pressure and core mass flux data forCSXT D 8 propellant presented in Figure 6, bothcore mass flux and the chamber pressure were ied during the tests, versus the more rigorous tnique of performing multiple mass flux tests atsame chamber pressure. The basic non-erosive b


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CSXT D8 PROPELLANT BURN RATE VERSUS CHAMBER PR ESSUREWlTH VARIATION WlTH CORE MASS FLUX(MASS FLUX-BASED EROSIVE BURNING)

CHAMBER PRESSURE (psia)Figure 6. CSXT D8 Propellant Burn Rate Versus Chamber Pressure with Variation with CoreFlux.rate versus chamber pressure for the D8 propellantis shown as a straight line on the log-log plot inFigure 6. The p, = 409 psia da ta point had a n initialcore mass flux of 0.86 lblsec-in2 (less than 1.0lblsec-in2, clearly non-erosive), the p, = 933.4 psiadata point had an initial core mass flux of 1.69lblsec-in=.For the p, = 933.4 psia data point thecore mass flux initially looks high for a non-erosiveburn rate test, but using the previously presentednon-eros ive core mass flux design criteria with a p,= 933.4 psia the threshold core mass flux for thebeginning of m ass flux-based erosive burning wouldbe expected to be between 1.75 lblsec-in2 and 2.0lblsec-in2, making the p, = 933.4 psia data pointalso non-erosive.Note th at the data plotted in Figure 6 is the aver-age burn rate and the average chamber pressurefrom each of the propellant characterization testmotors fired. As the propellant characterization testmotors were designed to produce a nearly flatthrust-time curve (a nearly constant thrust andchamber pressure), for the non-erosive tests there islittle error from using the average burn rate andaverage chamber pressure versus any instantane ousvalues for the burn rate and chamber pressure withtime during each test. Once the threshold mass fluxfor mass flux-based erosive burning is exceed, theactua l erosive burn rate a t ignition of the propellantcharacterization test motor will be higher than theaverage burn rate value shown in Figure 6. The pro-

pellant characterization test motors were deswith a shor t burn time to minimize this effect, is important to note that while the threshold flux for mass flux-based erosive burning Figure 6 is accurate, once mass flux-based erburning begins the actual erosive burn rate attion of the motor will be higher than the vbased on average burn rate th at are plotted in F6. Summarizing, it is very difficult and reqcareful set-up of the propellant characterizatioto accurately measure the instantaneously higtial burn rate when threshold values are excand erosive burning is present.Note that the characteristic shape of the incin burn rate with mass flux once the threshold flux is exceeded as was seen in Figure 5 is swha t camouflaged in Figure 6, because a s the cber pressure increases the threshold mass flumass flux-based erosive burning also increaseFigure 5 and Table 3. But eventually for theplotted in Figure 6 the increase in the core masovertakes the increase in the threshold masswith chamber pressure to create the characteincreased burn rate with mass flux shape whicbe clearly seen in Figure 6. Note tha t the thremass flux at a given chamber pressure is fexceeded at a m ass flux of 1.8 lblsec-in2 at a1297 psia, values nicely bracketed by the chapressure/threshold mass flux design criteria foonset of mass flux-based erosive burning pres

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Table 4. Core Mach Numbers for Test Motors Fired for CSXT D8 Propellant Characterization T

previously (p, = 800 psia, threshold mass flux 1.75Ib/sec-in2 ; p, = 1400 psia, threshold mass flux 2.0lblsec-in2) based on d ata from Figure 5 and Table 3.The core Mach number (core Mach number a t theaft end of the core) at ignition for each of the testmotors fired for the CSXT D8 propellant characteri-zation tests are presented in Table 4. The two non-erosive test motors and the four mass flux-basederosive burning characterization test motors corre-spond with the burn rate with chamber pressure an dmass flux data plotted in Figure 6.Note tha t the core Mach numbers for the non-ero-sive test motors were clearly non-erosive (Ma


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mended DesignErosive Internal Ballistics SimulationPrograms Combined with ErosiveBurning Design Criteria

Non-Erosive Intern al Ballistics Simu lationPrograms Combined with Erosive Burning DesignCriteria

The detailed m odeling of erosive burning in com-puter simulations of solid rocket motors is quitecomplex, and even with th e creation of detailed ero-sive burning models extensive calibration of themodels is required. Co mparisons of predicted thru st-time curves with actual thrust-time curves over aseries of erosive burning tests are required to fullycalibrate the models. On the other hand there aremany non-erosive internal ballistics solid rocketmotor computer simulations available to the highpower and experimentallamateur rocket communi-ties. The first internal ballistics solid rocket motorsimulation program released to the high power andexperimentallamateur rocket communities was theENGMOD program developed by the present authorand Fred Brennion and released in 1988. Manyinternal ballistics solid rocket mo tor simulation pro-grams are now available to high power and experi-mentallamateur rocketeers, and many experimen-tallam ateu r rocketeers us e MicrosoftTM ExcelTMspreads heets to perform these calculations.In internal ballistics solid rocket motor sim ulationprograms the m otor grain geometry inputs a re usedto calculate the prop ellant burning su rface area. Thepropellant burning surface area is then divided bythe throat area to determine the motor Kn.Usingstored or inpu tted propellant internal ballistic char-acteristics (chamber pressure versus Kn, burn rateversus chamber pressure), the chamber pressure isdetermined given the Kn With the chamber pressureand atmospheric pressure and using a thrust coeffi-cient model the thrust is determined. The chamberpressure is then used with the propellant internalballistic characteristics to determine the propellantburning rate. With the propellant burning rateapplied over a given time step the propellant burn-ing surfaces are receded, the new pro pellant burningsurface area is determined, and the computationalprocess begins again until the propellant burningsurfaces reach the case (or propellant liner) innerwall. Note that in these non-erosive internal ballis-tics motor simulation programs erosive burning isnot included, all propellant burning surfaces havethe same non-erosive propellant burning rate.The design method recommended by the presentauthor is to combine non-erosive internal ballistics

solid rocket motor simulation prog rams with eroburning design criteria. This method takes adtage of the large body of non-erosive internal b atics solid rocket motor simulation programs avable to high power and experimentallamateur reteers, by taking the application of these progrto a new level through the application of eroburning design criteria to the design of the morun on the simulation programs.Using the recommended combined core Mnumberlcore mass flux non-erosive design criwhich will be presented, the experimentallamarocketeer can design solid rocket motors w hichhave little or no erosive burning. Th ese solid romotor designs can then be run on existing non-sive internal ballistics simulation programsaccurate predictions of the m otor thrust-time cuCombined core Mach num berlcore mass flux decriteria will also be presented for the maximamount of erosive burning recommended bypresent author. Even for these maximum recmended erosivity motor designs, non-erosive innal ballistics solid rocket motor simulation grams can be used for the overall design ofmotor and will produce reasonable predictionsthe overall total impulse of the motor, althothere will be sub stan tial mispredictions in the inthrust and the tail-off of the motor due to eroburning. Even for maximum recommen ded erosmotor designs, combining erosive burning decriteria with non-erosive internal ballistics srocket motor simulation programs can result ieffective solution to the erosive burn ing solid romotor design problem.

Erosive Burning D esign Criteria Based on CoreMach Number and Core Mass Flux at Aft End o

The highest combustion gas velocity andhighes t mass flux in the core of a solid rocket mwill be reached a t the af t (nozzle) end of the comotor ignition. Thus the threshold values for vity (u,,) and mass flux (G,,) for velocity-basedsive burning and mass flux-based erosive burwill be reached first at the aft end of the core. Ithreshold values for velocity or core mass fluxerosive burning are exceeded prior to the aft enthe core, the highest velocity-based or mass based erosive burn ing will occur at the aft end ocore. As noted in a p revious section, core Mach nber is a much more useful parameter than velowith the core Mach number also reaching its mmum value a t the aft end of the core at motor tion. Thus the present author proposes that the core Mach number or core mass flux of interes

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These Propellant Burning SurfacesDo Not Contribute to Core Mass Flux Inhibited Propellant SurfacesDo Not Contribute to Core Mass Flux

Core Mass Flux Based On Mass Flow RateFrom Upstream Propellant Burning SurfacesCore Mass Fluxat Aft End of Core.Commonly Shortened to:"Core Mass Flux"

For Erosive Burning Design CriteriaCore Mass Flux Based On Core Mass Flow RBased On Non-Erosive Propellant Burn RateMost Important Core Mass Flux Condition isthe Highest Core Mass Flux, Which Occursat Aft End of Core at Motor Ignition.

Figure 7. Propellant Burning Surfaces Contributing to Core Mass Flux a t Aft End of Core. Core Flux Based on Non-Erosive Burn Rate.erosive burning design criteria are the core Machnumber and core mass flux at the very end (the aftend) of the core at motor ignition.The core Mach number at the aft end of the coreis determined u sing Eq. (7). The core Mach n umberdesign criteria, based on the core Mach number atthe aft end of the core, were presented previously.

Core M ass Flux a t Aft End of Core Based o n Non-Erosive Burn RateFigure 7 shows the mass flows from the variouspropellant burning s urfaces which comb ine to formthe total mass flow rate for the core mass flux at theaft end of the core. Any inhibited propellant s urfac esdo not contribute to the core mass flux. Note thatthe aft-facing propellant surfaces located aft (dow n-stream) of the end of the core do not contribute tothe ma ss flow rate in the core, and hence do not con-tribute to the core mass flux.An important technique now introduced by thepresent author is to calculate the core mass flux atthe aft end of the core (the "core mass flux" for themotor) for motor ignition only, and to bas e the core

mass flux on the propellant flow rate based on thenon-erosive burn rate. The highest core mass flux,and th us the highest m ass flux-based erosive burn-ing will be at motor ignition. Using the mass flowrate down the core based on the non-erosive burnrate greatly simplifies the calculation of the coremass flux, since including the effect that upstreamerosive burning increases the mass flux for down-January 2005

stream propellant surfaces greatly complicatescalculation of the core mass flux. If the core mflux limit for a given chamber pressure for nonsive burning is not exceeded, then the fact thacore mass flux calculations were done assuminon-erosive propellant burning rate will haveffect since there will be no erosive burning anyUsing the recommended core mas s flux design cria and through experience the experimentallateur rocketeer can calibrate wh at level of core mflux based on non-erosive propellant burning produces acceptable levels of erosive burning.From conservation of mass, the mass flow through the aft end of the core will be equ al tomass flow rate of the propellant being consufrom the propellant burning s urfaces. Includingthe propellant burning surfaces upstream of thend of the core, and a s recommended by the preauthor using the non-erosive propellant burrate (r,), the core mass flux equation, Eq. (3),

Eq. (3): G = n i t b / APand the core mass flow rate now become;

Where:A', = propellant burning surface areaups tream of aft end of core, m2 (in2)


17/23

(Propellant burning surface a reasdown stream of aft end of core shown inFigure 7 are not included.)pp = propellant density, kg/m3 (lb/in3)Since the propellant burning surface area down-stream of the aft end of the core tha t is not includ-ed in A', (sh ow n in Figure 7) is a sm all part of thetotal propellant burn ing su rface area (A,), for quickapproximate calculations the propellant mass flow

rate based on the total propellant burning surfacearea A, can be used in place of the propellan t mas sflow rate based o n A', in Eqs. (8) and (9 ),which alsoallows the core mass flux calculations to be tieddirectly to the m otor Kn .With n?, = n?', , Eqs. (3), 8) and (9) now become;

Where:A, = propellant burning surface area, m2 (in2)(All propellant b urning surfaces.)A, = nozzle throat area (throat cross-sectionalarea), m2 (in2)Kn = propellant burning surface area divided bythro at area (A, /Ath),dimensionlessn i b = propellant m ass flow rate, kg/sec (lblsec)

(Total propellant m ass flow rate from allpropellant burning surfaces.)

Combined Core Mach N umberICore M ass FluxErosive Burning Design CriteriaThe present author recommends that combinedcore Mach numberlcore mass flux erosive burningdesign criteria be used for high power and experi-mentallamateur solid rocket motors. As shown inFigure 8 the com bined core Mach num berlcore massflux erosive burning design criteria should beapplied a s follows:

(1) The solid rocket motor should have a constantcore diameter from the head end to the a ft end of thecore.(2) The initial sizing of the core port-to-thro at arearatio (A, /Afh) s based on the non-erosive or maxi-

mum recommended erosivity core Mach numand port-to-throat area ratio values presented pously an d repeated in Figure 8. The core Mach ber limits are based on the core Mach numb er aaft end of the core (Eq. (7)) a t motor ignition.(3)The initial (at motor ignition) core mass flthe aft end of the core is determined based onnon-erosive propellant burning rate (Eqs. (8)(g) ), with the propellant burnin g surfa ces aft oend of the core not included (see Figure 7).approximate calculations the entire propellant bing surface area can be used to calculate the mass flux (Eqs. (10)-(1 2)).(4) A design point core mass flux for the motestablished based on recommended non-erosivmaximum recommended erosivity core mass values presented previously and repeated in F8.(5) The initial core mass flux at the aft end ocore is checked again st the desired design pointmass flux. If the design point core mass fluexceeded, the motor core is opened up (the mcore diameter is increased) until the initial mass flux is reduced to the design point core flux value.

The primary advantage of using combined Mach numb erlcore mas s flux erosive burn ing decriteria for either non-erosive or maximum remended erosivity motor designs is that irregarof whether the motor propellant is susceptibvelocity-based or mass flux-based erosive burprovisions for both have been included a s bothMach number limits and core mass flux limits been considered in the sizing of the motor coreConstant Core Mass Flux Core Design

An improved solid rocket motor core designposed by the p resent auth or, that for a given leverosivity maximizes the length and thus LID motor, or maximizes the amount of propeinstalled in a motor for a given length, is thesta nt core m ass flu x core design. Once a level osivity is established for the o verall motor desig nconstant core mass flux core design maximizeam ou nt of installed propellant to maximize theformance of the motor.As shown in Figure 9, when using a constanmass flux core design, once a design point ma ss flux is achieved based o n the level of erosthe motor is being designed to, the core is opened up (the core diameter is increased) to m

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sign Criteriaiameter Cores

Step 1:Initia l Core Diameter Sized Based On Core Mach Number and Core Mass FluxCore Mach Number Conditions are at Motor IgnitionNon-Erosive; Ma= 0.50

y = 1.2; APIAth= 1.36Max Erosive; Ma= 0.70

Core Mass Flux Values Based onNon-Erosive Propellant Burn Rate

Maintain Constant Core DiameterFrom Head End to Aft End of Core

Step 2:Check Core Mass Flux at Aft End of CoreIf Core Mass Flux at Aft End of Core Higher Than Design Point Core Mass Flux,Increase Core Diameter to Reduce Core Mass Flux to Design Point Value.Design Point Core Mass Flux (Recommended Values)Non-Erosive; p, = 400-600 psia

p, = 800 psiap, = 1400 psiaMax Erosive; p, = 400 psia

p, = 600 psiap, 2 800 psia

Figure 8. Combined Core Mach NumberICore

Core Mass Flux 1 1.0 Iblsec-in2Core Mass Flux1 .75 Iblsec-in2Core Mass Flux 1 .0 Iblsec-in2Core Mass Flux = 2.0 Iblsec-in2Core Mass Flux = 2.5 Iblsec-in2Core Mass Flux = 3.0 Iblsec-in2Vlass Flux Erosive Burning Design Criteria for M

tain the same core mass flux down the rest of thecore. As more propellant burning surface area isadded down the core, the core has to be proportion-ately opened up to hold the core mass flux constant.As shown in Figure 9 the consta nt core mass fluxcore design is implemented as follows:(1) Initially a constant core diameter is used fromthe head end to the aft end of the core.(2) The initial sizing of the core port-to-throat arearatio (A, /A,) is based on the non-erosive or maxi-mum recommended erosivity core Mach numbersand port-to-throat area ratio values presented previ-ously and repeated in Figure 9.(3) A design point core mass flux for the motor isestablished based on recommended non-erosive ormaximum recommended erosivity core mass flux

January 2005

values presented previously and repeated in Fi9.(4) Based on the non-erosive propellant burrate, the point down the core where the design pcore mass flux is achieved is identified. For point down the core where the core mass flux isculated, only the upstream propellant burningfaces are included in the calculated core mass f(5)Once the design point core mass flux is achiethe core diameter is increased to increase the area (the core cross-sectional area) to maintaconstant core mass flux at the design point mass flux value.(6)All core mass flu x values, and th e opening uthe core to maintain a constant core mass fluxbased on the ignition core mass flux and the incore geometry a t ignition.


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Constant Core Mass FluxCore

(7) For the BATES grains used in the generic motordesign presented in Figure 9 , the last gra in intersec-tion upstream of the point where the design pointcore mass flux is achieved sets the point where alldownstream grains will have their core diametersincreased. Once the design point core mass flux isachieved, the core mass flux halfway down thelength of each grain determines the required corediameter for each grain to maintain the constantcore mass flux.

For the BATES grain generic motor design pre-sented in Figure 9 the core mass flux versus motorlength will "stair-step" in discrete steps as each s ub-sequent grain core diameter increases based on thecore mass flux halfway down the length of eachgrain. Note that the constant core mass flux coredesign can also be used for monolithic grains, wherethe port area can be gradually increased once thedesign point core mass flux is achieved, versus thestep-wise increase in core diameter for the BATESgrain s in Figure 9.

Effect of Core Complexity on Ap = xrc2Erosive BurningX = Q2 1 (47cAp)= (2xrC)2 (47c2r,2)= 1 OAn important n ote for the erosive burning materi-al presented previously in this article is that the

34 High Power Ro

Core Mach Number and Core Mass FluxDesign Point Conditions are at Motor IgnitionCore Mass Flux Values Based onNon-Erosive Propellant Burn Rate

Provides Maximum Motor LengMinimum Motor Core DiameterMaximum Propellant Loading fa Given Level (Design Point) ofErosive Burning

Core Diameter Increased Past This Pointto Maintain Constant I

Design Point Core Mass Flux Achieved1 Initial Core Diameter Based OnDesign Point Core Mach NumbDesign Point Core Mass Flux (Recommended Values) Non-Erosive; M, = 0.50Non-Erosive; p, = 400-600 psia Core Mass Flux 51.0 Iblsec-in2 y = 1.2; APIAth= 1.36p, = 800 psia Core Mass Flux 5 1.75 Iblsec-in2p, = 1400 psia Core Mass Flux 5 2.0 Iblsec-in2 Max Erosive; Ma= 0.70y = 1.2; APIAth= 1.10Max Erosive; p, = 400 psia Core Mass Flux = 2.0 Iblsec-in2p, = 600 psia Core Mass Flux = 2.5 Iblsec-in2p, 1 00 psia Core Mass Flux = 3.0 Iblsec-in2

Figure 9. Constant Core Mass Flux Core Design.

velocity-based and m ass flux-based erosive budata presented in Figures 3 and 5, the core number and core mass flux erosive burning dcriteria developed from the data in Figures 3 and Tables 1 and 3, and the example mass based erosive burning data for the CSXT proppresented in Figure 6, are all for solid rocket mwith cylindrical cores (circular ports ). Eq. (13) Ref. 1 , original referen ce Ref. 7) provides a remeasure of configuration complexity for the coa solid rocket motor.

Where:Q = burning perimeter of grain, m (in)X = configuration factor, dimensionless

For a grain with a circular port;


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Where:r = core radius (circular port), m (in)Thus a circular port has a configuration factor

(X) of 1 O. For a non-circular port the co nfigurationfactor will be over 1.0. Generally grain geom etrieswith more complex cores (a higher configurationfactor) will be more susceptible to erosive burning,and will have more severe erosive burning ifthreshold Mach number or threshold core massflux values a re exceeded. Because of th is core com-plexity effect, when designing motors with non-cir-cular ports the erosive burning characterizationtest s for the motor propellant sh ould be performedusing test motors with the same core shape andthe same configuration factor. If erosive burningcharacterization data for a propellant is based ontest motors using cylindrical cores, and the propel-lant is then used in a motor using a grain designwith a more complex core with a much higher con-figuration factor, significant errors in the predictederosive burning may result.

KnLID-M

January 2005

SummaryA solid rocket motor design method for hpower and experimentallamateur rocket mowas presented where non-erosive internal batics solid rocket motor simulation programscombined with erosive burning design criteriathe effective consideration of erosive burndesign issues in the design of high power experimentallamateur solid rocket motors. Decriteria for non-erosive and maximum recommed erosivity for velocity-based and mass flux-baerosive burning were presented. The use of cMach number as a design criteria for velocbased erosive burning, the use of the core Mnumber and core mass flux at the aft end ofcore as design criteria, and the basing of the mass flux design criteria on the core mass frate based on the non-erosive propellant burnrate were presented. Combined core Mach nberlcore mass flux erosive burning design critwere presented, which allow the effective designon-erosive and erosive solid rocket mowhether the motor propellant is sensitivevelocity-based or ma ss flux-based erosive burnFinally, a constant core mass flux core design presented that once a given level of erosivitynon-erosivity has been set, maximizes the amoof propellant in a solid rocket motor and hemaximizes the motor performance.

Glossarycoefficient of pressure, a lso called burning rate constan tspeed of soun d, mlsecpropellant burn ing surfa ce area , m2 (in2)propellant burnin g surfa ce area up stream of aft end of core, m2 (in2)port area (core cross-sectional a rea ), ma (in2)nozzle throat area (throat cross-sectional area ), m2 (in2)port-to-throat area ratio, dimensionlessmass flux, kglsec-m2 (lblsec-in2)threshold mass flux for the onset of mass flux-based erosive burning,kglsec-m2 (lblsec-in2)propellant burning su rface area divided by throat ar ea (Ab A,), dimensionlessLength-to-Diameter ratio, dimensionlessmolecular mass, kglkg-mol


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core Mach number a t aft end of core, dimensionlessMach number for flow in mo tor core based on combu stion ga s velocity, dimensionlthreshold Mach num ber for flow in motor core based on threshold velocity for onsevelocity-based erosive burning, dimensionlesspropellant mass flow rate, kg/sec (lblsec)propellant mass flow rate upstream of mass flux location, kg/sec (lblsec)burning rate pressure exponentchamber pre ssure, Pa (lb/in2)burning perimeter of grain, m (in)ga s c ons tant per unit weight, J/kg-OKuniversal gas constant, 8314.3 J/kg mol-OKpropellant burning rate, mlsec (inlsec)core radius (circular port), m (in)addition to propellant burning rate due to erosive burning, mlsec (inlsec)nozzle throat radius, m (in)non-erosive propellant burning rate, m/sec (in/sec)absolute temperature, O Kadiabatic equilibrium flame temperature (combustion temperature), O K (OR)combustion gas velocity, m/sec (ft/sec)threshold velocity for onset of velocity-based erosive burning, m/sec (ftlsec)configuration factor, dimensionlessratio of specific heats, dimensionlesspropellant density, kg/m3 (lb/in3)

References

1.Solid Rocket Motor Pet;formance Analys is an d 3. Sutto n, George I?, and Biblarz, Oscar, RockePrediction, NASA Space Vehicle Design Criteria Propulsion Elements, 7th Edition, Joh n Wile(Chemical Propuls ion) , NASA SP-8039, May 1971. Sons, Inc., 2001.

2. Zucrow, M, J., Osborn, J. R., and Murphy, J. M., An 4. "Solid Propellant Selection an d CharacterizaExperimentalInvestigation ofth e Erosive Burning excerpts from NASA SP-8064 with additionaCharacteristics o a Non -Homogeneous Solid mate rial, forward by Rogers, Charles E., HigPropella nt, AIAA Preprint 64- 107, AIAA Solid Power Rock etry Magazine, Vol. 15 , No. 6,Propellant Rocket Conference (Palo Alto, CA), February (A) 2001, p ages 42-64, and Vol. 1January 29-3 1, 1964. No. 7, February (B) 2001, pages 45-72.36 High Power Ro


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5. Humble, Ronald MT, Henry, Gary N., and Larson,Wiley J. , Space Propulsion Ana lysis and Design,The McGraw-Hill Companies, Inc., 1995.6. Kreidler, J. W;, Erosive Burning: New ExperimentalTechniques and Methods ofAnalysis, AIAAPreprint 64-155, AIAA Solid Propellant RocketConference (Palo Alto, CA), January 29-31, 1964.7. Fenech, E. J., and Billheimer, J. S., Use oflnterna lGrain ConJguration to Predict Erosive BurningConstant,Paper 61-8, Western States Section, TheCombustion institute (Pacific Palisades, CA), April

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Erosive Burning Design Criteria

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