Theoretical study on premixed flames of nano aluminum
particles and water mixture
Jun-Su Shin1,a, Hong-Gye Sung1,b
1
Department of Aerospace & Mechanical Engineering, Korea Aerospace University
100 Hanggongdae gil. Hwajeon-dong, Deogyang-gu, Goyang-city, Gyeonggi-do 412-791, Korea
a
[email protected], [email protected]
Keywords: aluminum particle combustion, nano particle combustion, hydro-reaction, premixed
flame
Abstract. A theoretical model is proposed to investigate premixed combustion characteristics of
nano aluminum particles - water mixture. The effects of particle size, initial pressure, and temperature
were considered as well. Computational domain is divided into 3 regions; preheat zone 1, preheat
zone 2, and reaction zone. No reaction occurs in either of the preheat zones. Reaction zone, consisting
of nano aluminum particles–steam mixture and the combustion products, is the region where reaction
and heat-release occurs. Energy conservation is considered separately at each zones. The flame speed
and temperature distribution are derived by solving the energy equation in each regime and matching
the temperature and heat flux at the interfacial boundaries. Combustion time correlation of nano
aluminum particle is also considered to imply complex aluminum combustion kinetics. Flame speed
normalized by maximum flame speed is calculated as a function of pressure, initial particle diameter,
and equivalence ratio and compared with experimental data, and represent conversion factor of flame
speed in the function of polynomial.
Introduction
Metal particles, especially aluminum particle have been attractive in the rocket society since
1960’s as it provides significantly high energy density during combustion. Due to its favourable
characteristics of high volumetric heat release associated with combustion, aluminum particle has
been used as additives in solid rockets to increase Isp and most of the previous researches are
dedicated to study on the combustion behavior and characteristics of aluminum particle in the solid
rocket. [1] Metal particles such as aluminium, boron, and magnesium have much higher heat of
reaction compared to conventional hydrocarbon fuels. [2] Furthermore, metals can react with liquid
water, which is one of the most attractive characteristics of the metal fuels. Another characteristics of
metal particle combustion especially of aluminum and magnesium is that when they react with
oxygen or water it inherently produces alumina(Al2O3) and magnesia(MgO), which are
environmentally not harmful products respectively. It is quite different from conventional
hydrocarbon fuels used in our daily lives which generate carbon dioxide and monoxide.
Price et al. investigated aluminum particles as an additive to increase the thrust and Isp of the solid
rockets. [3] Analytic study of aluminum particle combustion including radiation effect was conducted
by Bidabadi et al. [4,5] Perturbation technique were used to obtain exact solution of sets of ordinary
differential equations. Following and improving his theoretical formulations Ying et al. [6] have
conducted analytical research on the combustion of aluminum particle laden flow considering heat
interaction between particle phase and gas phase. Recently Sundaram et al. [7] conducted similar
study on nano-sized particles. There is a similar study [8] with magnesium particle as well. Risha et al.
[9,10] implemented experimental studies on nano aluminum combustion with water. They found
linear and mass burning rates of quasi-homogeneous mixtures of nano aluminum(38~130nm) and
water as a function of pressure, mixture composition, particle size. In our previous studies,
combustion characteristics of bimodal aluminium particle and air mixture [11], radiation effects on
combustion of nano aluminium and water mixture[12] were investigated from theoretical point of
1
view. In this study, non-dimensional flame speed of nano aluminium and water mixture is obtained
by solving sets of ordinary differential equations in the function of particle size, pressure, equivalence
ratio, and flame temperature is calculated by using CEC(Chemical Equilibrium Code). Conversion
factor in the function of equivalence ratio is proposed. Dimensional flame speed can be obtained by
multiplying non-dimensional flame speed and conversion factor.
Theoretical formulation
Theoretical model is considered to simulate aluminum particles - water mixture as premixed
combustion. The model allows for investigation into the effect of particle size, equivalence ratio,
chemical kinetics, and oxidizer (water or air) on the combustion characteristics such as flame
propagation speed and flame profile of aluminum and oxidizer mixture.
Fig. 2. Computational domain
Fig. 1. Physical domain
Assumptions for the theoretical equations are steady and adiabatic. Other assumptions are
following; the combustion wave propagates uniformly, allowing for the equations to be written in the
moving coordinate system, where the initial mixture moves with constant velocity (flame propagating
velocity) relative to the stationary flame front. Energy conservation equation is used in a way of
one-equation model. Assumptions for one-equation model are; 1) Isotropic and homogeneous porous
medium (aluminum particles are uniformly distributed with oxidizer) 2) one dimensional 3)
Collisions and interactions between burning particles are neglected 4) Viscous dissipation and
pressure work are negligible 5) The surface porosity is equal to the porosity 6) Radiation heat transfer
between fluid and solid phases are neglected for simplicity. Applying local thermal equilibrium that
the temperature difference between the solid phase (aluminum particle) and the gas phase (oxidizer)
is negligible, system temperature can be expressed as representative temperature T.
Effective thermal conductivity which combines the thermal conductivity of solid and fluid
appropriately as one single model is one of the most important factors on the governing equation.
Hamilton-Cross(H-C) model is used for effective thermal conductivity. Zhang et al. had reported a
theoretical relation based on the H-C model to compute the effective thermal conductivity of the
nano-fluid mixture. [13]
  p  ( n  1) f  ( n  1)v ( f   p ) 
eff   f 
  p  ( n  1) f  v ( f   p ) 
  p  2 f  2v ( f   p ) 
 f 
  p  2 f  v ( f   p ) 
(1)
n 3
Here,  f ,  p and  v are the thermal conductivity of the fluid and particles, volume fraction of
particle, respectively. When the shape of the particles are spherical, the theoretical result shows that n
is equal to 3 and independent of both the particle size and the ratio of the conductivities. [13]
In order to simulate flame profile and flame speed of aluminum reaction in water, flame structure
can be divided into 3 regions; preheat zone 1 (water is in liquid phase), preheat zone 2 (water is in
2
Preheat zone 1
dT
 mix S L c p , mix
dx
d 2T
 mix 2
dx
Preheat zone 2
(2)
 mix S L c p , mix
 mix
Tign
dT
dx
d 2T  H 2 O S L h fg

dx 2
t
 0.087 

 exp 

 log  2r0  106   7.28 


Reaction zone
 mix S L c p , mix
(3)
c 
Bu  Q
c
 C3
d 1.8
C1T0 0.2 X eff P 0.1
(7)
(5)
(6)
Q   mix  c p , mix  (Tad  Tu )
wF  Q  
(4)
dT
d 2T
 mix 2  wF  Q
dx
dx
dC
C

3 c
dx
(9)  c 
d 0.3
C2e  Eb / RT X eff P 0.48
(8)
(10)
vapor phase), and reaction zone.
Preheat zone 1 consists of aluminum particles and liquid state water. Its temperature range lies
from initial temperature to boiling point of water. Assuming steady state, governing equation for this
region is as follows, Temperature of aluminum and water mixture is governed by convection and

c
diffusion. In Eqn.2, mix is density of mixture, S L is laminar flame speed, p,mix is specific heat of
mixture.
Preheat zone 2 consists of aluminum particles and gas state water (vapor). Its temperature range
lies from boiling point of water to ignition temperature of aluminum particle. Ignition temperature of
nano aluminum particle which depends on the size of the particle proposed by Sundaram et al. [7] In
Eqn.3,  H O is density of water, h fg is vaporization enthalpy of water. In Eqn.4, r0 is initial particle
2
radius in μm. This region includes the effect of phase change, vaporization enthalpy of water, and
ambient pressure. Thickness of preheat zone 2(t) is not known a priori, and it can be obtained by
solving heat flux matching equation at the interface of Zone 1 and 2. In this study, Bisection method is
implemented to obtain thickness of preheat zone 2(t).
In the reaction zone particles react with water vapor and heat is released by combustion.
Governing equation for this region is as follows. Temperature dependencies of thermodynamic
properties are considered in terms of library so as not to increase the complexity of the calculation
process. Heat release by combustion is applied as a source term in Eqn.5 in the form of wF  Q where,
wF is reaction rate [kg/s] and Q is heat release per unit mass and calculated in terms of density,
specific heat and adiabatic temperature, Tad, and Tu is temperature of preheat zone 1. Adiabatic
temperature is determined by CEC. Bu is initial mass per unit volume [kg/m3] in unburned region
respectively. Eqn.6 is source term in Eqn.5. Eqn.9[1] is combustion time(  c ) for large size particles
(>10μm) and Eqn.10[7] is for smaller size particles(<10μm). Constants in Eqn. 9, 10, C1=0.00735,
C2=8.72105. R is gas constant, Eb=73.6 kJ/mol, and Xeff is the effective mole fraction of oxidizer, d is
particle diameter in μm. In order to close the formulation, a particle mass consumption rate is solved,
which is given by Eqn.8. C is defined as ratio of the instantaneous particle diameter to initial particle
diameter.
Eqn. 5, 8 are reduced to a system of first ordinary differential equations and are solved using
Rosenbrock method. The system of equations described above poses an Eigenvalue problem with the
flame speed treated as the Eigenvalue. Newton’s method has been employed to determine the roots.
Results
The theoretical formulation proposed has been validated with strand burner (experimental) result
conducted by Risha et al. [9,10] They studied combustion characteristics of nano aluminium and
water mixture experimentally. Effects of various particle sizes of 38, 80, 130nm, equivalence ratio in
lean burn condition, and pressure in the range from 0.12 to 15 MPa were investigated.
Figure 4 represents normalized flame speed in the function of particle diameter at a pressure of
3.65 MPa and stoichiometric condition. Data of experimental and theoretical results are normalized
by the flame speed of particle size of 38 nm.. As the particle size increases 38nm to 130nm, the flame
speed decreases. This approximately corresponds to a d-1.0 law for the flame speed. Theoretical results
have similar trends compared with experiment results.
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Fig. 3 Schematic diagram of windowed pressure vessel (left) and result (right)[9,10]
Fig. 4. Variation of non-dimensional flame speed
with particle size at  =1.0 and p=3.65 MPa
Fig. 5. Variation of non-dimensional flame
speed with pressure at  =1.0
Fig. 6. Variation of non-dimensional flame
speed with equivalence ratio at p=3.65MPa
and d p =38nm
Fig. 7. Conversion factor for experimental
results at 38nm, 3.65MPa
The variation of flame speed with pressure and particle diameter is shown in figure 5 at
stoichiometric condition and particle size of 38nm. The experimental measurements indicate that the
pressure exponent in the burning rate law is about 0.5. The pressure exponents predicted by the model
are 0.44 of the 38nm, 0.27 of the 80nm and 0.31 of the 130m. The pressure constants are varied with
pressure 0.46 to 2.46. Thus, the model rather over-predicts the dependence of burning rates on
pressure. However, both the prediction and experimental results reveal that combustion mechanism
of aluminum and water is dominated by collision of water molecules on the particle surface.
Figure 6 shows the burning rate with equivalence ratio at a pressure of 3.65 MPa for 38nm particle
size. Present results are normalized by the values of equivalence ratio of 1.25. It shows that the
difference between two cases increase as decreasing with equivalence ratio.
Flame speed of experimental results is rather high compared with theoretical result because of
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adiabatic temperature obtained from CEC, which is expected larger than actual flame temperature.
Conversion factor   0.023  4.818  8.344 2  3.891 3
(11)
Conversion factor of experimental result at a condition of 38nm, 3.65MPa is shown in figure 7 and
Eqn.11. The conversion factor has the lowest value at stoichiometric condition because of more solid
particles in the unit volume at high equivalence ratio. So it can be more influenced by agglomeration
effect which was neglected in this study.
Conclusion
The analytical model to investigate combustion characteristics of nano aluminum particles in
liquid water was proposed and validated flame speed in terms of pressure, particle size, and
equivalence ratio. Present result shows marginal agreement in tendency of experimental data. The
major investigations of the present work are as follow;
1. The predicted burning rates roughly follow a d-1.0 law for nano aluminum in the size range
38-130nm. Flame speed and conversion factor are obtained and in reasonably good agreement
with experimentally measured values.
2. Pressure exponent and constant varies with the particle size. Range of pressure exponent is
0.27-0.44 and pressure constant is 0.46-2.46.
3. Different from aluminum particle combustion in air which adiabatic flame temperature barely
changes with the equivalence ratio, adiabatic flame temperature of aluminum combustion in
water changes drastically. The more accurate calculation of flame propagation speed of
aluminum-water reaction with various equivalence ratio requires experimental flame
temperature data.
Acknowledgements : This work is sponsored by fundamental research project of the National
Research Foundation of Korea(NRF) under contract no. 20090070395.
References and Notes
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