7 Combustion of Energetic Materials - Princeton University...• A major appli ilication of solid...

7

Combustion of Energetic gMaterials

Richard A. YetterPenn State Universityy

Princeton‐Combustion Institute 2018 Summer School on Combustion

June 25‐29, 2018Princeton NJPrinceton, NJ

1

Additional Reading and ReferencesAdditional Reading and References• Bulusu, S., ed., Chemistry and Physics of Energetic Materials, Kluwer

Academic Publishers Boston 1990Academic Publishers, Boston, 1990.• Clark, J.D., Ignition! An Informal History of Liquid Rocket Propellants,

Rutgers University press, New Brunswick, NJ, 1972.• Davenas, A., “Development of Modern Solid Propellants,” Journal of , , p p , f

Propulsion and Power, 19 (6), 1108‐1128, 2003.• Dreizin, E.L., “Metal‐based Reactive Nanomaterials,” Progress in Energy

and Combustion Science, 35, 141‐167, 2009.• Edwards T “Liquid Fuels and Propellants for Aerospace Propulsion 1903• Edwards, T., Liquid Fuels and Propellants for Aerospace Propulsion 1903‐

2003,” Journal of Propulsion and Power, 19 (6), 1089, 2003. • Glassman, I., Yetter, R.A., and Glumac, N., Combustion, 5th Edition,

Academic Press, Waltham, MA, 2015• Jackson, T.L., “Modeling of Heterogeneous Propellant Combustion: A

Survey,” AIAA Journal 50, 5, May 2012, 993• Kubota, N., Propellants and Explosives: Thermochemical Aspects of

Combustion 2nd Edition Wiley‐VCH Verlag GmbH & Co KGaA WeinheimCombustion, 2 Edition, Wiley VCH Verlag GmbH & Co. KGaA, Weinheim, 2007.

2

Additional Reading and ReferencesAdditional Reading and References

• Kuo, K.K. and Summerfield, eds., Fundamentals of Solid Rocket P ll t C b ti A i I tit t f A ti dPropellant Combustion, American Institute of Aeronautics and Astronautics, Reston, VA, 1984.

• Kuo, K.K., and Acharya, R., Applications of Turbulent and Multiphase Combustion John Wiley and Sons Hoboken NJ 2012Multiphase Combustion, John Wiley and Sons, Hoboken NJ, 2012.

• Sundaram, D., Yang, V., and Yetter, R.A., “Metal‐based Nanoenergetic Materials: Synthesis, Properties, and Applications,” Progress in Energy and Combustion Science 61, 293‐365, 2017.

• Sutton, G.P. and Biblarz, O., Rocket Propulsion Elements: An Introduction to the Engineering of Rockets, 9th Edition, John Wiley and Sons, New York, 2017.Y V B ill T B d R W Z d S lid P ll Ch i• Yang, V., Brill, T.B., and Ren, W.‐Z., eds., Solid Propellant Chemistry, Combustion, and Interior Ballistics, American Institute of Aeronautics and Astronautics, vol. 185, Reston, VA, 2000.

• Yetter R A Risha G A and Son S F “Metal Particle CombustionYetter, R.A., Risha, G.A., and Son, S.F., Metal Particle Combustion and Nanotechnology,” Proceedings of the Combustion Institute 32, 1819‐1838, 2009.

3

A diAppendixes

4

Constant Volume Temporal Reacting System

( = 1, ..., K)jj j

K

dYMW j

dtυω= T: temperature

Yk: mass fraction of jth speciesMWk: molecular weight of jth species

1

K

v j j jj

dTC e MWdt

υ ω=

= − ∑k

t: timeυ: specific volumeV: volume of systemm: mass of systemV / mυ =m: mass of systemek: internal energy of jth speciesCv: constant volume heat capacity

: molar rate of production of jth speciesjω' "M M , 1,...,n n

ji j ji j i mν ν↔ =∑ ∑M: specieskf, kb: forward and backward rate constants for the ith reactionv: stoichiometric coefficients for the ith

1 1ji j ji j

j j= =∑ ∑

( ) ( ), ,1 1

M Mji jin n

i f i j b i jj j

q k k′ ′′

= −∏ ∏ν ν

v: stoichiometric coefficients for the ireaction and jth species of the reactants or products

1 1j j= =

ji ji ji i ji iq q′′ ′⎡ ⎤= − =⎣ ⎦ω ν ν νm

∑1

j ji ii

q=

=∑ω ν

5

Model for RDX Combustion by LiauModel for RDX Combustion by Liauand Yang

6

Model for RDX Combustion by Liau and Yang ‐ The Gas‐Phase• The gas‐phase continuity equationThe gas phase continuity equation

• The gas phase species conservation equations are:

( ) 0ut xρ ρ∂ ∂+ =

∂ ∂• The gas‐phase species conservation equations are:

( ) ( ) ( )1 2ii i i

YY u V i , ,...N

t xρ

ρ ω∂ ∂

+ + = =⎡ ⎤⎣ ⎦∂ ∂• The gas‐phase energy conservation equation

( ) ( ) N

k k ke ue T uY V h p

tρ ρ

λ ρ∂ ∂ ⎛ ⎞∂ ∂ ∂

+ = − −⎜ ⎟∂ ∂ ∂ ∂ ∂⎝ ⎠∑

1k k k

kt x x x x=⎜ ⎟∂ ∂ ∂ ∂ ∂⎝ ⎠

∑• The gas‐phase enthalpy of the kth species

( )T⎡ ⎤

⎢ ⎥∫ ( ) 0

ref

k p ,k s ,k ref f ,kT

h C dT h T h⎢ ⎥= + + Δ⎢ ⎥⎣ ⎦∫

• The diffusion velocity is evaluated as1; k k k

k kk j jk

j k

D X YV DX x X / D

≠

∂ −= − =

∂ ∑7

• The equation of state for a multicomponent system

Model for RDX Combustion by Liau and Yang ‐ The Gas‐Phase

• The equation of state for a multicomponent system

h l h l d1

Nk

ukk

Yp R TMW

ρ=

= ∑• The general chemical reaction is represented as

' "

1 1

fj

bj

N Nk

i i i iki i

M Mν ν= =

⎯⎯→←⎯⎯∑ ∑• The reaction rate constant for the jth forward and backward

reactions are expressed in Arrhenius form

1 1i i= =

E⎛ ⎞

• The rate of change of molar species i by the jth reaction is

jabj j

u

Ek A T exp

R T⎛ ⎞

= −⎜ ⎟⎜ ⎟⎝ ⎠

• The total rate of change of species i is( ) ( ) ( ), ,

1 1

ij ij

ij i i

n n

M ij ij f j M b j Mi i

C v v k C k Cν ν′ ′′

= =

⎡ ⎤′′ ′= − −⎢ ⎥⎣ ⎦∏ ∏

1

R

ij

N

i i Mj

MW Cω=

= ∑8

Model for RDX Combustion by Liau and Yang‐ The Foam Layer Region

• A two phase fluid dynamic model using a spatial averaging technique was employed to formulate the physicochemical processes in the foam layer regionfoam layer region

• These processes include thermal decomposition, evaporation, bubble formation, gas‐phase reactions in bubbles, and interfacial transport of mass and energy between gas and condensed phasesmass and energy between gas and condensed phases

• The formulation is based on the control volume approach for conservation equations with the control volumes for gas bubbles and condensed phases complementary to each othercondensed phases complementary to each other

• The Dupuit‐Forchheimer assumption is used that allows the fractional‐volume void fraction (or porosity) to be extended to a fractional area void fraction definitionvoid fraction definition

where φ is the void fraction, Ag is the fractional cross‐sectional area consisting of gas bubbles in foam layer and A is the cross sectional

gA Aφ=

consisting of gas bubbles in foam layer, and A is the cross‐sectional area of the propellant

9


• The gas‐phase mass conservation equation

( )gg g c gu

φρφρ ω →

∂ ∂+ =

• The gas‐phase species conservation equations are:

( )g g c gut x

φρ ω →+∂ ∂

( ) ( ) ( )1 2g giYY V i N

φρφ

∂ ∂ ⎡ ⎤+ +

• The gas‐phase energy conservation equation

( ) ( ) ( )1 2g gg gi g gi gi gY u V i , ,...N

t xφρ ω⎡ ⎤+ + = =⎣ ⎦∂ ∂

( ) ( ) gNe u e Tφρ φρ ⎛ ⎞∂ ∂ ∂∂( ) ( )

( )1

gg g g g g g

g g gk gk gkk

g

e u e TY V h

t x x xu

p q A h T T

φρ φρφλ φ ρ

φ ω

=

⎛ ⎞∂ ∂ ∂∂+ = −⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠

∂+ +

∑

( )c g c g s c c gp q A h T Tx

φ ω → →− + + −∂

where qc→g is the heat evolved during pyrolysis of condensed‐phase to gas‐phase, is the rate of conversion of condensed phase c gω →g p pmaterial to gas‐phase products, As is the interface area between bubbles and liquid per unit volume, and hc is the heat transfer coefficient

c g→

10


• The condensed‐phase mass conservation equation

( ) ( )( )11c u

φ ρφ ρ ω

∂ − ∂+

• The condensed‐phase species conservation equations

( ) ( )( )1 c c c gut x

φ ρ ω →+ − = −∂ ∂

( )( )1 Yφ∂ ∂

• The condensed‐phase energy conservation equation

( )( ) ( ) ( ) ( )1

1 1 2c cic ci c ci ci c

YY u V i , ,...N

t xφ ρ

φ ρ ω∂ − ∂

+ − + = =⎡ ⎤⎣ ⎦∂ ∂

p gy q

( )( ) ( )( ) ( ) ( )1

1 11 1

cNc c c c c c

c c ck ck ckk

e u e T Y V ht x x xφ ρ φ ρ

φ λ φ ρ=

∂ − ∂ − ⎛ ⎞∂∂+ = − − −⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠

∑( ) ( )1 c

c g c g s c c gup q A h T Tx

φ ω → →

⎝ ⎠∂

− − − − −∂

11

Model for RDX Combustion by Liau and Yang‐Reactions in The Foam Layer Region

• RDX(l) → 3CH2O + 3N2O

( ) ( ) ( )131 1 6 10 36 0k k / s exp kcal / mol / R Tω φ ρ= − = × −⎡ ⎤⎣ ⎦• RDX(l) → 3HCN + 1.5NO2 + 1.5NO + 1.5H2O

( ) ( ) ( )1 1 11 1 6 10 36 0c uk , k / s exp . kcal / mol / R Tω φ ρ= = × ⎡ ⎤⎣ ⎦

( ) ( ) ( )162 2 21 1 16 10 45 0c uk , k / s exp . kcal / mol / R Tω φ ρ= − = × −⎡ ⎤⎣ ⎦

• A secondary reaction was also considered

NO2 + CH2O → NO + CO + H2O

( ) ( ) ( )2 2 2 6 0 5 0c uk , k / s exp . kcal / mol /ω φ ρ ⎡ ⎤⎣ ⎦

( ) ( )

2 2

2 2

3 3

3 2 77802 13 73

g CH O g NO

CH O NO

Y Yk ,

MW MWk / l T k l / l / R T

ρ ρω φ=

⎡ ⎤• In addition to thermal decomposition, the thermodynamic

phase transition from liquid to vapor RDX is considered

( ) ( )3 2 773 802 13 73.

uk cm / mol s T exp . kcal / mol / R T− = × −⎡ ⎤⎣ ⎦

p q p

RDX(c) ↔ RDX(g)12

Model for RDX Combustion by Liau and Yang‐Evaporation and Condensation Considerations of RDX

• The condensation flux can be characterized in terms of the rate at which vapor molecules collide and stick to the interface

p

0814

ucondensation

R T pm sn MW s XMW RTπ

+⎛ ⎞⎛ ⎞′′ ′′= = ⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠• If thermodynamic phase equilibrium is achieved evaporation

81 v equ pR T pMW⎛ ⎞ ⎛ ⎞⎛ ⎞′′ ′′ ′′ ⎜ ⎟ ⎜ ⎟⎜ ⎟

• If thermodynamic phase equilibrium is achieved, evaporation proceeds at the same rate as condensation

4v ,equ

evap condensationppm m sn MW s

MW RT pπ⎛ ⎞⎛ ⎞′′ ′′ ′′= = = ⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠⎝ ⎠

• The vapor pressure can be approximated by the ClausiusClape ron eq ation

( )0v ,eq vap up p exp H / R T⎡ ⎤= −Δ⎣ ⎦

Clapeyron equation

• At nonequilibrium conditions the net evaporation rate is• At nonequilibrium conditions, the net evaporation rate is

net evap condensationm m m′′ ′′ ′′= −13

Model for RDX Combustion by Liau and Yang‐Evaporation and Condensation Considerations of RDX

• Thus, the specific mass conversion rate due to evaporation is

p

c g s netA mω ′′=

• The specific surface area is a function of the void fraction and number density of bubbles

( )1 3 2 336 1 2/ /s bA n /π φ φ= <

( ) ( )1 3 2/336 1 1 2/s bA n /π φ φ= − ≥

where nb is the number density of bubbles determined bempirically

14

Model for RDX Combustion by Liau and Yang‐Boundary Conditions

• With application of conservation laws to the propellant surface, the matching conditions at the gas‐foam layer interface are:

• Mass Flux:( ) ( )001 c c g gu u uφ ρ φρ ρ +−⎡ ⎤− + =⎣ ⎦

• Species Flux:

E Fl

( ) ( ) ( ) ( )( )001

i i i ic c c c g g g g i iu V Y u V Y u V Yφ ρ φρ ρ +−⎡ ⎤− + + + = +⎣ ⎦

• Energy Flux:

( ) ( ) ( ) ( )1 10 0

1 1gc

i i i i i i

NNgc

c c c c c c g g g g g gi i

N

dTdT u V Y h u V Y hdx dx

φ λ φ ρ φλ φρ− −= =

⎡ ⎤⎡ ⎤− − − + + − +⎢ ⎥⎢ ⎥

⎢ ⎥⎣ ⎦ ⎣ ⎦⎡ ⎤

∑ ∑

( )1 0

N

i i ii

dT u V Y hdx

λ ρ+=

⎡ ⎤= − +⎢ ⎥⎣ ⎦

∑

• Phase transition from liquid to vapor RDX at the interface:• Phase transition from liquid to vapor RDX at the interface:

( ) 01 c c netu mφ ρ − ′′− =⎡ ⎤⎣ ⎦

15

Model for RDX Combustion by Liau and Yang‐Boundary Conditions

• In the foam layer, it is assumed Tc = Tg

• The far field conditions for the gas phase require that the gradients of the flow properties be zero as x→∞

YT ∂∂ ∂ ∂

• The conditions at the cold boundary for the condensed phase

0iYu Tx x x xρ ∂∂ ∂ ∂= = = =

∂ ∂ ∂ ∂

• The conditions at the cold boundary for the condensed phase (x → ‐∞) are

0c iT T xφ= = → ∞and atc i φ

16

Application of Energetic Materials toApplication of Energetic Materials to Rocket Propulsion

17

Solid Propellants Used in RocketsA j li i f lid ll i i k• A major application of solid propellants is in rockets

• Burning solid propellants produces high‐temperature combustion products, which can be expanded through acombustion products, which can be expanded through a converging‐diverging nozzle to generate thrust for rocket propulsion

ll h f l d ll h• Generally, there are two types of solid propellant grains that are used in rockets– Side burning type:g yp

– End burning type:Side Burning Type of Solid Propellant Rocket Motor

End Burning Type of Solid Propellant Rocket MotorFrom Kubota, 2007 18

Thrust of a Solid Rocket Motor• Thrust is a result of pressure force distribution over interior

and exterior surfaces of the motor

• It can be expressed in an equation as follow:( )p e e e ambF PdA m V A P P= = + −∫ (36)

• The thrust is also expressed in terms of a dimensionless thrust• The thrust is also expressed in terms of a dimensionless thrust coefficient CF, the nozzle throat area At, and the average chamber pressure Pc

F t cF C AP= (37)

19

Thrust of a Solid Rocket Motor (cont.)Nozzle exit has a divergence angle α which implies that notNozzle exit has a divergence angle αd, which implies that not all the jet momentum ( ) is in the axial direction. Therefore, a λ parameter is introduced to account for loss of axial momentum in the thrust calculations

p em V

axial momentum in the thrust calculations.This parameter λ can be evaluated from:

1 cos dαλ+

=(38)

With this correction, the thrust of a rocket motor can be evaluated from

2

(39)From the isentropic flow relationships the nozzle exit flow

( )p e e e ambF m V A p pλ= + −

From the isentropic flow relationships, the nozzle exit flow velocity is determined from:

(40)

1

2 pγγγ−⎡ ⎤

⎛ ⎞⎢ ⎥ (40)2 11

ee c

c

pV RTp

γγγ

⎛ ⎞⎢ ⎥= − ⎜ ⎟⎢ ⎥− ⎝ ⎠⎢ ⎥⎣ ⎦ 20

Thrust of a Solid Rocket Motor (cont.)The mass generation rate of a non‐metallized solid propellant must be equal to the mass flow rate through a choked nozzle under steady‐state operation. Then, using the choked flow equation, can be related to th h b P d t t T gmthe chamber pressure Pc and temperature Tc as:

( )

g

( ) c tp g d

p Am m mRT

γ= = = Γ(41)

where Γ is defined as:cRT

( )( )

12 12γγ+−⎛ ⎞

Γ(42)

Using Eqs. (38), (40) and (41) into Eq. (30), we have the following i f h h

( )( )2

1γ γ

γ⎛ ⎞

Γ = ⎜ ⎟+⎝ ⎠

expression for the thrust:

( )1

2 12 1 e e e ambp A p pF A

γγγλ

+−

⎧ ⎫⎡ ⎤⎪ ⎪⎛ ⎞ ⎛ ⎞⎢ ⎥Γ +⎨ ⎬⎜ ⎟ ⎜ ⎟ (43)1

1e e e amb

c tc t c c

p p pF p Ap A p p

γλγ

⎢ ⎥= Γ − + −⎨ ⎬⎜ ⎟ ⎜ ⎟⎢ ⎥− ⎝ ⎠ ⎝ ⎠⎪ ⎪⎢ ⎥⎣ ⎦⎩ ⎭ 21

Thrust of a Solid Rocket Motor (cont.)A i f E (37) i h E (41) i ld h f ll iA comparison of Eq. (37) with Eq. (41) yields the following expression for the thrust coefficient CF:

( )1

2 12 p A p p A p pγγγ+−

⎧ ⎫⎡ ⎤⎪ ⎪⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎢ ⎥

(44)

The mass balance in the rocket motor can be written as:

( )2 1

02 1

1e e e amb e e amb

F Fc t c c t c c

p A p p A p pC Cp A p p A p p

γγλ λγ

⎪ ⎪⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎢ ⎥= Γ − + − = + −⎨ ⎬⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥− ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎪ ⎪⎢ ⎥⎣ ⎦⎩ ⎭

(45)( )g CVCV

g d

d Vdm m mdt dt

ρ= = −

The rate of mass generation by solid‐propellant burning for a non‐metallized propellant is:

(46)m m A rρ= = (46)

The rate of mass discharge through the nozzle is:

(47)

g p p b bm m A rρ= =

d D t cm C A p= ( )d D t cp

22

Total Impulse of a Solid Rocket MotorThe mass flow factor CD in Eq. (47) is:

(48)( )112 Dimensions ofMW timeC C

γγγ

γ

+−Γ ⎛ ⎞ ⎡ ⎤

≡ ⎜ ⎟ ⎢ ⎥

Note that the parameter CD should not be confused with the di i l di h ffi i t C

Dimensions of 1D D

u cc

C CR T lengthRT

γγ

≡ = ⎜ ⎟ ⎢ ⎥+⎝ ⎠ ⎣ ⎦

dimensionless discharge coefficient CdFor steady‐state burning conditions, the mass balance equation Eq. (45) can be written as: By utilizing Eq. (37), we have:

g dm m=have:

(49)D

p b b D t cF

CA r C A p FC

ρ = =

Total impulse is the thrust force integrated over burning time:

(50)U it (N )t

I Fdt∫ ( )

0

Units: (N-s)tI Fdt= ∫23

Specific Impulse of a Solid Rocket MotorSpecific Impulse of a Solid Rocket Motor

The specific impulse is defined as the total impulse per unit p p p pweight of propellant burnt:

0 U it ( )

t

t t

FdtI II

∫(51)

0

00

0

Units: (s)t tsp t

p pg

Ig m W

g m dt≡ = =

∫

where g0 (=9.8066 m/s2 or 32.175 ft/s2) is the gravitational

acceleration at sea levelacceleration at sea level

24

Other Performance Parameters• If we consider constant thrust level for the majority of motor• If we consider constant thrust level for the majority of motor

operation time as shown in the following figureru

st t0: Initiation timetA: Start of thrust rise due to igniter

•

• ••

sure

or T

hr A gtB: Start of the propellant burningtC: Time when pressure or thrust is equal to half of the steady‐state value

tD: End of chamber volume filling periodt End of the propellant burning

• •• •

•

Pres

Tit t

tB

t t t t t

tE

tE: End of the propellant burningtF: Point of maximum rate of change of curvature during tail‐off period

tG: Fixed percentage of pavg or pmaxtH: End of motor thrust

• In the above case of static firing of the solid rocket motor, the average thrust can be defined as:

Timet0 tA tC tD tFtG tH

Si il l h i d fi d

1 E

C

t

avgE C t

F Fdtt t

=− ∫ (52)

• Similarly, the average pressure is defined as:1 E

C

t

avgE C t

p pdtt t

=− ∫

(53)

25

Other Performance Parameters of Solid Rocket MotorsIn case of constant thrust operation the specific impulse is:In case of constant thrust operation, the specific impulse is:

(54)0

spp

FIm g

≅

From Eq. (49), the thrust can be written as:

(55)F

pCF mC

= (55)

Substituting Eq. (55) in Eq. (54), we have:

pDC

FCI =(56)

Substituting CD in the above equation, we have:0

spD

IC g

=

(57)

11

or

2 1

c cFsp sp

T TCI IMW MW

g

γγ

γ

+−

= ∝

⎛ ⎞⎜ ⎟ (57)

Propellants with high Tf and low MW products are desirable

0 1 ug

Rγ

γ⎜ ⎟+⎝ ⎠

26

Variation of Thrust Coefficient with Area Expansion RatioExpansion Ratio

From Kuo and Acharya, 2012

• For maximum CF, it is desirable to have combustion products expanded to the ambient pressure.

• At optimum expansion , Pe = Pamb 27

Variation of Thrust Coefficient with Area Expansion Ratio at Several Specified Pressure Ratiosp

pc/pa=25

pc/pa=50pc/pa=100

pc pa

pc/pa=10

• The maximum on each curve represents the optimum expansion p p pcondition

• Note that during a flight of a rocket, optimum expansion is achieved only at one altitude From Kuo and Acharya, 2012 28

Dependency of Isp on Tc/MW

Tcsp

TIMW

∝

• Under optimum expansionUnder optimum expansion, i.e., Pe = Pamb

• The effect of Pc/Pe can also b b d f hi lbe observed from this plot

• It can be seen that Tc/MW has a much stronger effect on Ispsp

Note MW = Mw

From Kuo and Acharya, 2012b 29

Variation of Flame Temperature and Specific Impulse with NG Concentration of a Double‐Base Propellantp

From Kuo and Acharya, 2012 30

Variation of Isp and Tf with AP or RDX Weight Percentage of CMDB Propellantsg p

• CMBD = composite modified double base propellant


p p p• At very high RDX wt %, the material properties are poor; hence they cannot be used as true propellants

• Solids loadings above ~ 88 wt % not practical 31

Density Ispk f d b h• For some rockets, performance is measured by the

“density‐Isp”, which is defined as the product of propellant density and specific impulse; i.e.,DIsp ≡ ρIsp = ρp × Isp (58)

• In order to accommodate a large weight of propellant in a given vehicle tank space a dense propellant isin a given vehicle tank space, a dense propellant is preferred. This permits smaller vehicle size and weight, which also results in lower aerodynamic drag.Th ll t d it h i t t ff t• The average propellant density has an important effect on the maximum flight velocity and range of any rocket‐powered vehicle

• The average propellant density can be increased by adding heavy materials such as aluminum powders into the propellant mixturep p

32

Characteristic Velocity C*

A characteristic velocity C* is defined as a measure of energy available y gyto generate thrust after combustion of the propellant. It is defined as:

Et

c tp A dt∫(59)

If the chamber pressure is constant for major part of the rocket *

0t*

pC

m=

operation, then C* can be written as:

1u* c t cRp A TCMW C

= = =Γ (60)

C* is a fundamental performance parameter, similar to the Isp. Both of

p Dm MW CΓ

pthese parameters depend on by a linear relationship.Typical values for C* range from 800 to 1,800 m/s. Higher values correspond to more energetic propellants, which can produce greater thrust and impulse Using the definition of C* thrust can be expressed

cT / MW

thrust and impulse. Using the definition of C , thrust can be expressed as:

(61)*

p FF m C C=33

Characteristic Velocity, C*, for a non‐Metallized Composite PropellantMetallized Composite Propellant

1540

1500

1520

s

1460

1480

C*,

m/s

HTPB/AP

1420

1440binder: HTPB C

10H

15.4O

0.07

oxidizer: APp

c = 70 bar

78 80 82 84 86 88 90 92AP weight percentage

• C* is a function of propellant formulation as shown above• The solids loading should be approximately 88% for maximum performance


Characteristic Velocity, C* , for a Metallized Composite PropellantMetallized Composite Propellant

1580

1590

1570

1580

s

1550

1560

C*,

m/s

HTPB/Al/AP 88% solids

154012% binder: HTPB C

10H

15.4O

0.07

88% solids: AP + Al powderp

c = 70 bar

1530 0 5 10 15 20Al weight percentage

• Note the upper limit for Al powder is near 18% by weight for maximum performance


Effect of Pressure Exponent on Stability of a Solid Rocket MotorStability of a Solid Rocket Motor

• Mass discharge rate from rocket motor is proportional to pressure i e

ng p bm A apρ= d D tm C A p= to pressure, i.e.,

• The mass generation rate from propellant is

l

d cm P∝

( ) <1gm n

g p b pρ

d i d b proportional to Pn, i.e.,

• If n > 1 any pressure

ng cm P∝

( )g

Stableg dm m=

pc determined by

dm

gmor

• If n > 1, any pressure fluctuation in the motor will lead to either overpressure or a dramatic decrease in chamber pressure resulting

( )>1gm n

dm

chamber pressure resulting in extinction of the solid propellant combustion processTh lid ll f

Pressure, p

Unstable

• Thus, solid propellants for rocket motors should have n< 1

After Kubota, 2001 36

Pressure Sensitivity of Burning RateThe parameter Kn is defined as the ratio of burning surface area of the solid propellant to the nozzle throat area of a rocket motor; i.e.,

(62)n b tK A / A≡The pressure sensitivity πk of the rocket motor combustor is defined as:

(63)

1k

i

pp T

π⎡ ⎤∂

≡ ⎢ ⎥∂⎣ ⎦ (63)The pressure in the solid‐propellant rocket motor combustion chamber can be expressed:

(64)

ni Kp T∂⎣ ⎦

( )( )f k fp p exp T Tπ= − (64)Therefore, the pressure sensitivity of a rocket motor can be written as:

( )( )ref k i i ,refp p exp T Tπ=

( )refln p / pπ =

(65)The relationship between σp and πk is given by the following equation with pressure exponent n treated as a constant:

( )ki i ,refT T

π =−

with pressure exponent n treated as a constant:(66)( )1p knσ π= −

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Thrust Coefficient, CF – Efficiency, ηThe thrust coefficient for a period of operation is defined as:The thrust coefficient for a period of operation is defined as:

(67)

( )0

Et

tF ex

F t dt

C =∫

(67)

Recall that the theoretical thrust coefficient C based upon Eq (44)

( ) ( )0

EF ,ex t

t ct

C

A t p t dt∫Recall that the theoretical thrust coefficient CF,th based upon Eq. (44) can be written as:

0e,avge amb

F th FpA pC C

Aλ

⎛ ⎞= + −⎜ ⎟⎜ ⎟

(68)In the theoretical calculations of CF,th, no erosion of the nozzle throat is assumed For experimental test conditions the nozzle throat size

0F ,th Ft c ,avg c,avgA p p⎜ ⎟⎜ ⎟⎝ ⎠

is assumed. For experimental test conditions, the nozzle throat size could increase due to thermo‐chemical erosion. The experimental value of CF, can be evaluated from Eq. (67). The average thrust‐

ffi i t ffi i th t ffi i i th d fi dcoefficient efficiency or thrust efficiency is then defined as:

(69)F ,exp erimental F ,ex

F ,theoretical F ,th

C CC C

η ≡ =38

Properties and Performance Parameters of Different Oxidizers at p = 70 atmp

Oxidizer ChemicalFormula

State ρ(g/cm3)

ΔHf(298)(cal/mol)

Tc(K)

MW(g/mol)

Isp(s)

NC(12 6%N) C H O NO ) S 1 66 160 2 2586 24 7 230NC(12.6%N) C6H7.55O5(NO2)2.45 S 1.66 ‐160.2 2586 24.7 230NC(14.14%) C6H7.0006N2.9994O10.9987 S 1.66 ‐155.99 3025 26.8 243NG C3H5O3(NO2)3 L 1.60 ‐9.75 3289 28.9 244TMETN C5H9O3(NO2)3 L 1.47 ‐97.8 2898 23.1 2535 9 3( 2)3TEGDN C6H12O4(NO2)2 L 1.33 ‐181.6 1376 19.0 183DEGDN C4H8O3(NO2)2 L 1.39 ‐103.5 2513 21.8 241ADN H4N4O4 S 1.82‐1.84 ‐36.01 2051 24.8 202AN NH4NO3 S 1.73 ‐87.37 1247 22.9 161AP NH4ClO4 S 1.95 ‐70.73 1406 27.9 157HNF CH5H5O6 S 1.87‐1.93 ‐17.22 3082 26.4 254HNIW(CL‐20) C6H6N12O12 S 2 04 99 35 3591 27 4 273HNIW(CL 20) C6H6N12O12 S 2.04 99.35 3591 27.4 273NP NO2ClO4 S 2.22 8.88 597 36.4 85RDX C3H6N3(NO2)3 S 1.82 14.69 3286 24.3 266HMX C4H8N4(NO2)4 S 1.90 17.92 3278 24.3 266TAGN CH9H7O3 S 1.59 ‐11.5 2050 18.6 231HNIW: HexanitrohexaazaisowurtzitaneTAGN: Triaminoguanidinim Nitrate

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Variation of MW and Tf of Several HTPB‐based Solid Propellants with Oxidizer Concentrationp

P 7 MPPc = 7 MPa


Variation of Isp of Several HTPB‐based Solid Propellants with Oxidizer Concentrationp

P = 7 MPaPc 7 MPaPe = 0.1 MPa


Nitronium perchorate (NP) shows the highest Isp, but it is not a common oxidizer due to its low temperature decomposition initiating at 50oC.

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