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Modeling Combustion Instability in Aluminized Hybrid Propellants

Areth Biancotti1 and Dario Pastrone2

Politecnico di Torino, 10129 Torino, Italy

An analytical formulation is developed to model the radiation due to aluminum/alumina particles in order to investigate the effects of the radiative heat feedback on low frequency combustion instability in hybrid rockets using aluminized propellants. A quasi-steady aluminum combustion model is considered, with a delay that accounts for the adjustment of radiating particle clouds to the changes in the regression rate. A linearized treatment shows that low frequency combustion instability can be driven by the coupling of the thermal transient in the solid fuel grain and the radiation heat feedback related to the distributed combustion of aluminum particles.

Nomenclature a = two-flux absorption coefficient B = two-flux backscatter fraction Cd = drag coefficient dflame = flame distance from surface Di = particle diameter Ea = activation energy fvi = volume fraction fmAl = mass fraction of aluminum in the solid grain Gcl, Gfb,Gop= transfer function (closed, feedback, open) Gox = oxidizer mass flux I+ , I-, Ib = total radiant intensity (forward, backward, blackbody) k = emission coefficient krad = imaginary part of the complex refraction index L = optical thickness m& = mass flow rate nrad = real part of the complex refraction index p = pressure Q = heat flux r& , R = dimensional and nondimensional regression rate s = two-flux scattering coefficient or spectral variable T, T* = temperature, T/1000 U = velocity yAl = aluminum flame thickness α = absorptivity ε = emissivity µ = dynamic viscosity ρ = density or reflectivity τu = dynamic response time χ = parameter, see Eq. (5)

1 Graduate Student, Corso Duca degli Abruzzi, 24 Torino. 2 Associate Professor, Dipartimento di Energetica, Corso Duca degli Abruzzi, 24 Torino. AIAA Senior Member.

43rd AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit 8 - 11 July 2007, Cincinnati, OH

AIAA 2007-5356

Copyright © 2007 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

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Subscripts 0 = initial 1 = perturbation Al = aluminum ave = averaged conv = convection f = vaporized fuel flame = oxidizer/volatile fuel flame ghtpb = pyrolyzed HTPB at surface rad = radiation shtpb = solid HTPB st = steady state surface = solid fuel grain surface tot = total yAl = burning aluminum region Superscript _ = normalized value ave = averaged

I. Introduction adiative effects in non-metalized hybrid rocket combustion are generally neglected due to the fact that previous studies1 showed that radiation heat flux is small when compared to the amount of total heat transfer to the fuel

grain surface. Indeed, radiations of hot gaseous products are spectrally selective and the emission bands often overlap, causing negligible radiation. This approximation cannot be done when metalized fuel grains are used, since the presence of a solid metallic phase causes potentially significant levels of radiant heat flux. Aluminized fuel grains can be used in hybrid rockets to improve performance, due to the high reaction heat and to the high density of aluminum, and, hopefully, to enhance the grain regression rate. The presence of aluminum affects the combustion process and the grain properties: the performance and the dynamics of the combustor are consequently modified. As far as the combustion process is concerned, aluminum behaves very differently from other more volatile ingredients of the solid grain. Due to its high boiling point, aluminum emerges from the solid grain surface. Depending on the surface conditions, it could accumulate and coalesce before leaving the surface. Metallic particles ignite when they reach the flame zone in the boundary layer, where temperatures are sufficiently high and the presence of oxidizer allows the combustion to occur. As a result of aluminum combustion, condensed-phase oxides which are carried away by the gaseous flow and ejected through the nozzle are produced. Condensed particles dampen the pressure oscillation by removing heat at the propellant surface and by scattering the acoustic waves that might arise. On the other hand phenomena such as the distributed combustion of aluminum and the radiation heat flux coming from condensed particles, are suspected to drive combustion instabilities. This paper focuses on the last of these mechanisms: the radiative effects, due to hot solid/liquid particles of aluminum/alumina, on the low-frequency combustion dynamics in aluminized hybrid rockets are investigated. The approach here used owes mainly to the models developed by Karabeyoglu et alii.2,3 To perform a similar linearized treatment, an analytical formulation is developed to evaluate the radiation heat feedback. The results of the analytical approach have been compared with numerical results obtained using a modified version of the simple, yet realistic, model proposed by Brewster and Parry.4 The radiation dynamics has been modeled assuming a quasi-steady response of the aluminum combustion and a delay, with respect of the regression rate oscillations, equal to the time needed for the aluminum to reach the flame zone in the boundary layer. Finally, the coupling of the radiation heat feedback dynamics and the thermal lag in the solid propellant grain is analyzed.

II. Combustion – Radiation analytical model The following section outlines the analytical model that has been developed to evaluate the radiation heat

feedback dynamics due to aluminum/alumina particles. Figure 1 shows a schematic of the combustion process. The origin of the x coordinate is at the grain leading edge and the origin of the y coordinate is at the surface of the propellant, which is stationary and considered to move upward with a speed equal to the regression rate r& . The volatile vaporized fuel is convected upward to the flame zone, where it reacts with the oxidizer which is transported

R

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towards the flame by diffusion and flow turbulence. The flame position dflame at a given x coordinate is evaluated according to Ref. 5. The aluminum particles, considered to be mono-disperse, are accelerated by the gas phase and ignite as they reach the flame zone. Then at least three regions can be considered. Under the flame gaseous products due to fuel pyrolysis and non-burning aluminum particles are present. In the following a gaseous oxygen (GOX) and hydroxyl terminated polybutadiene (HTPB) propellant combination is considered, and 1,3 butadiene gas, the most abundant pyrolysis product of HTPB, is assumed to constitute the gas phase under the flame. A second region exists where the distributed burning of aluminum droplets occurs and aluminum oxide particles are generated. Finally, in the third region, alumina particles are the only condensed phase. A one-dimensional model (y direction) is developed to evaluate aluminum velocity at the flame position. A constant averaged gas phase velocity Ug,ave is considered. The value of Ug,ave is obtained considering an average of two limiting cases: an adiabatic flow with drag due to condensed phase, and a frictionless constant-area flow with stagnation temperature change due to flame heating. We assume that drag, evaluated using the Stoke’s law, is the only force acting on the aluminum droplets. Applying Newton’s second law to the droplet gives

dt

dUDUUD Al

AlAlAlavegAl !"

µ" 6

)( 3 30,,0, =# (1)

Equation 1 can be integrated from grain surface to flame position, assuming a constant value of dynamic viscosity and neglecting the aluminum velocity at the grain surface, to yield

!!"

#$$%

&''=

u

flame

surfacegavegflameAl

tUUU

(exp ,,,

where µ

!"

18

20, AlAl

u

D

= (2)

Remembering that UAl = dy/dt, the time tflame needed for aluminum particles to reach the flame (y = dflame) can be obtained. Integration of Eq. (2) yields

aveg

u

flame

usurfacegflame

flameU

tUd

t,

, exp !!"

#$$%

&''

=(

(

(3)

which easily converges assuming an initial trial value of tflame. Since the main contribution to the radiation flux is given by the flame region, tflame can be used to represent the radiation time-delay with respect to regression rate fluctuations. Once that UAl,flame is obtained, the thickness of the distributed aluminum combustion yAl can be approximated by multiplying this aluminum speed by the droplet burning time. The aluminum burning rate has been modeled using the following relation,6

( ) 5.1

1

5.1

0, tDDAlAl

!"= (4) where the burning rate constant β depends on the mole fractions of the oxidizers, including O2, H2O and CO2; in the considered case β is assumed to be 1.0924e-3 mm1.5/ms. As far as radiative transfer is concerned, the heat flux contribute of the aluminum and alumina particles in each of the aforementioned regions is evaluated. The emissivity of a cloud of particles7 is

Figure 1: Schematic representation of the physical situation (blue= notignited aluminum, red = burning aluminum, green = alumina)

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)/ 5.1exp(1 !"# ! Lfv$$= (5) where fv is the volume fraction of the considered specie, L is the optical thickness of the cloud, and λ is the wavelength. For the sake of simplicity the asymptotic form of the Mie equations is considered. In this limit χ is a function of the complex refraction index of the particle (nrad – i krad)

( ) 222

242

24

radradradrad

radrad

knkn

kn

++!

="

# (6)

For Al particles the complex refraction index is assumed to be (1.5 – i 1.5), whereas for Al2O3 the following analytical relations8 are used:

( )[ ]473.0*0202.01

4.321

281.5

01225.0

058.1

00376.0

024.11

5.0

222

2 !+"#

$%&

'()

*+,

-

!+

!+

!+= Tn

rad

.... (7)

where T*=T/1000

( ) ( )[ ]95.2*847.1exp17.006.0002.0 2!++= Tk

rad"" (8)

For the thermal wavelength interval (0.2 to 6 µm), we obtained mean integral values of nrad e krad respectively equal to 2.67 and 0.001, in agreement with other studies found in literature.7-8 The radiative heat flux of each species in the three main regions is then evaluated assuming that particle clouds behave as a grey body at an average temperature. Since it is not possible to analytically integrate Eq. (5), in order to preserve an analytical formulation, the total emissivity is evaluated assuming that ε = ε λ (Λ) where Λ is a proper wavelength, giving

)/ 5.1exp(1 !""= Lfv #$ (9) The total radiative heat flux is computed summing the different cloud contributions, neglecting all interactions.

III. Combustion – Radiation numerical model In order to cross-check the analytical model, the radiation heat feedback is also evaluated using a numerical

approach, based on the one-dimensional model proposed by Brewster and Parry.4 Their model has been simplified and modified in order to keep into account the peculiar hybrid combustion process. Two sets of equations are used. The first set describes the flowfield, and is used in the region from the grain surface up to the flame due to GOX/HTPB combustion, evaluating the aluminum velocity at y = yflame. The second set of equations is used to model the radiative transfer. These two sets of equations can be decoupled, assuming that radiation has negligible influence on energy transport within the gas phase. The flowfield is first solved, and the results thus obtained are used in the radiative two-flux model to solve the forward and backward radiation intensity in order to assess the radiant flux incident on the propellant surface. A brief description of the model is given in the following. Further details can be found in Ref. 4.

A. Combustion - Flow Model In the region under the flame, a two-phase non-reacting flow is considered. Retaining the assumptions presented

in the previous section, the condensed phase consists of aluminum particles which are assumed to be spherical and mono-disperse, whereas the gas phase consists of 1,3 butadiene. Aluminum particles and gas phase are supposed be in thermodynamic equilibrium. The energy equation is substituted by a proper temperature profile. The drag on the aluminum particles is modeled according to Stoke’s law, assuming a constant gas viscosity. The momentum equations are used both for aluminum particles and gas

( )

( )!!"

!!#

$

%%%%=

%%=

dx

dpDCUUUUfv

dx

dUfvU

DCUUUUdx

dUU

AldAlgAlgAlg

g

ggg

AldAlgAlggAl

AlAl

0,

0,

43

43

&&

&& (10)

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Figure 2: Aluminum and gas velocity under the flame.

Figure 3: Velocity of aluminum particles at the flame: numerical and analytical model comparison.

together with the perfect gas equation

T

dTd

p

dp+=

!

! (11)

and the species conservation equations

0)()(==

dx

Ufvd

dx

Ufvd gggAlAlAl!! (12)

Through the continuity equation it is also possible to obtain the evolution of the aluminum volumetric fraction in the direction y, normal to the fuel surface. Proper initial conditions are given, including pressure, initial velocities and volume fractions. The initial aluminum velocity is the fuel grain regression rate, whereas its volumetric fraction is the same as in the solid grain. Figure 2 shows an example of the computed velocity profile, up to the flame region, for a given aluminum mass fraction and aluminum particle diameter: the heating due to the flame accelerates the gas phase, which in turn, drags the aluminum particles. Figure 3 compares the numerical and analytical evaluation of the aluminum velocity at the flame as a function of the aluminum particle diameter. Above the flame, Eqs. (10) and (11) are no longer integrated, assuming that the burning aluminum particles have a constant velocity along with the produced aluminum oxide. Finally, in the region where the aluminum combustion occurs, Eqs. (12) are no longer true and the volumetric fractions are to be evaluated considering the aluminum consumption rate

al

AlAlAlAlAl

D

fv

dy

Ufd !""667.1

)(#= (13)

B. Radiation Model The radiant flux is essentially given by burning aluminum and alumina particles, which are both considered spherical and mono-disperse. The governing equations of the two-flux radiation model,4 are integrated to evaluate the forward (I+ ) and backward (I-) radiant intensity

( ) ( )

( ) ( )!!

"

!!

#

$

%&'

()*+++++++=

++

%&'

()*+++++++=

+

gT

bI

OAlk

sT

bI

AlksIIsa

dy

dI

gT

bI

OAlk

sT

bI

AlksIIsa

dy

dI

32

32 (14)

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Figure 5: Comparison between numerical and analytical evaluation of the radiative heat flux at surface

Figure 4: Example of radiative heat flux results using the two flux model

provided proper boundary conditions are given. A first boundary condition, (I+)y=0 = 0 is posed, assuming that the propellant surface behaves as a blackbody at zero degrees, with zero radiative emissions and reflection. This condition can be changed later adding the actual emitting characteristics of the solid fuel. The backward radiation intensity is an unknown of primary interest thus a boundary problem arises. The boundary condition suggested by Brewster and Parry,4 dI+/dy (x! ) = 0, is substituted by (I+) = (I-) at a given y which corresponds to the engine symmetry axis . The blackbody intensities Ib are evaluated at the gas temperature for non-burning aluminum and aluminum oxide, whereas, following Ref. 4, Ts = 2320 K for burning aluminum. The absorption, emission and scattering effects are accounted for, using the relative coefficients,

32323232

32

323232

32

3232

33

3

3

OAlOAlOAlOAlAlAlAlAl

OAl

OAlOAlOAl

Al

AlAlAl

iOAl

OAlOAl

Al

AlAl

Dåfvk and Dåfvk

D

Bñfv

D

Bñfvs

D

áfv

D

áfva

==

!!"

#$$%

&+=

!!"

#$$%

&+=

where the values of the parameters, derived from Ref. 4, are given in Table 1.

Figure 4 shows an example of the forward and backward radiative heat flux, Q = π Ι, as a function of the distance from the grain surface, whereas a comparison of the numerical vs. the analytical evaluation model of the radiative heat feedback at grain surface Qrad is shown in Fig. 5, as a function of the initial aluminum particle diameter: a qualitative agreement (order of magnitude and trend) is obtained, whereas a quantitative agreement is reached in the 30-45 microns range.

Table 1: Properties of Al and Al2O3 particles

α ρ ε B Al 0.1 0.9 1.0 burning

0.1 non burning 0.50

Al2O3 0.45 0.55 0.45 0.30

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IV. Effect on combustion instability Karabeouglou and Altman2 developed a quasi-steady combustion model, coupled with the thermal lag system with boundary layer delays to account for the adjustment of the boundary layer to the changes in the freestream condition and the blowing from the surface. Later3 the combustion chamber gasdynamics was also taken into account. A comprehensive dynamic model is out of the scope of this paper which aims at analyzing the coupling of radiative dynamics due to condensed particles and the thermal-lag. A thermal-lag model detailed description can be found in Ref. 2, whereas the radiation dynamics and coupling with thermal lag are presented here: first a perturbation solution is searched to evaluate the radiative heat dynamics due to regression rate fluctuation (open loop), and then the regression rate feedback is considered (closed loop), coupling the radiation dynamics with the thermal lag model. Marxman10 has analyzed the coupling between radiative and convective heat, giving the following relation

!!"

#

$$%

&+''(

)**+

,-=

conv

rad

conv

radconvtot

Q

Q

Q

QQQ exp (15)

which considers how the addition of radiative heat transfer increases the regression rate which in turn inhibits the convective heat transfer. Then, within the limit of a quasi-steady solution, the total heat flux perturbation can be linked to the radiative perturbation through the following normalized and linearized equation

[ ] )()exp(1)( 11111 tQBBAtQ radtot !!= (16)

where

stconv

rad

sttot

conv

Q

QB

Q

QA !

!"

#$$%

&=!

!"

#$$%

&=

11 and (17)

The subscript 1 and st, indicate, respectively, the first-order perturbation term and the steady state solution. The radiative perturbation is found using the governing equations seen in the previous sections. These equations have been normalized, linearized, and finally perturbed around the steady-state reference point. Since the Stefan Boltzmann grey body emission equation has been considered, using an average temperature for each particle species, the radiative heat flux perturbation is equal to the emissivity perturbation of each particle cloud, which can be obtained through the normalization and linearization of Eq. (9). For example, the emissivity of aluminum in the distributed combustion region, leads to the following expression

( ) )exp()()(1

)( 11111 CtytfvCt AlyAlAl

styAlAlyAlAl !+=

"" (18)

where

[ ]st

AlAlyAlAl yfvC != /5.11

" (19)

To evaluate the perturbation of the aluminum volume fraction, a space-averaged expression is derived

( )!

!"d

D

UD

y

fvfv

Aly

Al

flameAlAl

Al

flameAl

yAlAl #$

=

0

30,

2

,5.1

0, (20)

which yields

!!!

"

#

$$$

%

&

!!

"

#

$$

%

&+'=

2

,

2

,

22 1

flameAl

Al

flameAl

Al

flameAlyAlAlU

yC

U

yBfvAfv (21)

where

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styAlAl

flameAl

fv

fvA

!!

"

#

$$

%

&=

2

st

flameAl

Al

Al U

y

DB

!!

"

#

$$

%

&=

,5.10,

2

' 2

,30,

2

23

stflameAl

Al

Al U

y

DC

!!

"

#

$$

%

&=

' (22)

finally

)()()()( 1111 tUtfvtytfv AlflameAlAlyAlAl !+= (23)

According to the present model, the perturbation of the distributed combustion region thickness, yAl, results to be the perturbation of the velocity of particles at the flame, which, in turn, can be obtained through the linearization of Eq. (2), leading to

( ) )()1()( 154321

,,

154

flamerKK

st

stflameAl

stAl ttReKKrKKK

U

rtU st !+!=

&&

& (24)

where

Al

ghtpb

shtpb

Al fmfmK +!="

")1(1

12

!=ghtpb

shtpbK

"

" !!

"

#

$$

%

&'=

uaveg

flame

U

dK

(,

3 exp avegUKK ,24 /=

!!"

#$$%

&'=

u

flametK

(exp5

The time needed for aluminum particles to reach the flame, is assumed to represent the time-delay for the adjustment of the particle clouds to the changes in the regression rate. Using the same procedure for the other regions and using the Laplace transform, it has been possible to obtain the final transfer function of the open loop Gop that links the total heat flux at the surface (output) with the regression rate (input).

)exp()(

1

)(1 stBAsR

stotQ

G flameop !!== (25)

where A and B are two coefficients resulting from equations (16 - 24), whose values are ruled by the regression rate, the aluminum mass fraction fmAl in the fuel, and the diameter of the particles. Reference 2 gives the transfer function between the regression rate perturbation and the applied heat flux perturbation. This transfer function, here called Gfb, depends on the activation energy and the latent heat of fuel gasification, and represents the feedback of the present model. The scheme of the considered closed loop is shown in Fig. 6: the relative transfer function has been obtained through equilibrium at node A

fbop

optottot

GG

G

sR

sQG

!==1)(

)(

1

1 (26)

The presence of a branch point in the

thermal lag transfer function may significantly alter the nature of the response. Those points are characterized by the fact that around any contour enclosing their branch point the complex function will be multi-value. The problem is that common types of functions with branch points will not be meromorphic and thus the Cauchy integral theorem, and consequently criteria such as the Nyquist one, cannot be used over the

Figure 6: Scheme of the coupling between radiative heat feedback and the thermal lag

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Figure 7: Transfer Function of the closed loop system (Activation energy = 60 kcal/mole)

entire domain even though the poles are isolated. Hopefully, it can be demonstrated that the sufficient condition for the system to be unstable is satisfied if the system has at least one of its poles in the positive right half plane (although the reverse statement is not true in general).

Figure 7 shows that the closed loop system presents poles in the positive real part over the frequency domain, determining instability of the system, whose frequency is about 10 Hz in the considered case. Physically, the complete system is unstable because of the coupling between the delay of the metallic particles in reaching the flame from the surface and the thermal lag of the grain, which requires a certain delay to fit the boundary thermal conditions.

V. Conclusion The development of an analytical formulation for the evaluation of radiation heat feedback in aluminized hybrid

propellant is put forward. The radiation dynamics is represented with a time-delay related to the time that aluminum particles need to reach the flame zone into the boundary layer. Based on this formulation, the linearized treatment of the coupling of the grain surface incident heat flux coming from aluminum/alumina particles with the thermal lag in the grain has led to the conclusion that the presence of aluminum can play a role in the generation of low-frequency combustion instabilities in hybrid rockets.

VI. Aknowledgments Support for this work from Ministero dell’Università e della Ricerca (MIUR) is gratefully acknowledged .

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Journal of Propulsion and Power, Vol.16, No.1, 2000, pp.125-137.. 2 Karabeyoglu M.A., and Altman D., “Dynamic Modeling of Hybrid Rocket Combustion,” Journal of Propulsion and Power,

Vol.15 No.4, 1997, pp.562-571. 3 Karabeyoglu M.A., De Zilwa S., Cantwell B., and Zilliac G., “Modeling of Hybrid Rocket Low Frequency Instabilities,”

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Thermophysics and Heat Transfer, Vol.2, No.2, 1988, pp.123-130. 5 Marxman, G.A., Wooldridge, C.E., and Muzzy, R.J., “Fundamentals of Hybrid Boundary-Layer Combustion”, Progress in

Aeronautics and Astronautics, Vol. 15, 1964, pp. 485-522. 6 Brooks K.P., and Beckstead M.W., “Dynamics of Aluminum Combustion,” Journal of Propulsion and Power, Vol.11,

No.4, 1995, pp.770-781. 7 Sarofim A.F., “Flame emissivities: Alternative Fuels,” Alternative hydrocarbon fuels: Combustion and Chemical Kinetics,

Progress in Astronautics and Aeronautics, AIAA, Vol.62, 1967, pp.206-207. 8 Duval R., Soufiani A., and Taine J., “Coupled Radiation and Turbulent Multiphase Flow in an Aluminized Solid Propellant

Rocket Engine,” Journal of Quantitative Spectroscopy & Radiative Transfer, Vol.84, 2004, pp.513-526. 9 Rieger T.J., “On the emissivity of Alumina/Aluminum composite particles,” Journal of Spacecraft and Rockets, Vol.16

No.6, 1979, pp.438-439. 10Marxman G.A., and Gilbert M., “Turbulent Boundary Layer Combustion in the Hybrid Rocket,” 1963, Ninth National

Symposium on Combustion, pp.371-380. 11 Marxman G.A., “Combustion in the turbulent boundary layer on a vaporizing surface,” Tenth international Symposium on

Combustion, 1965, The Combustion Institute, pp.1337-1349.