PERGAMON
Progress in Energy and Combustion Science 27 (2001) 1–74
www.elsevier.com/locate/pecs
Fundamental aspects of the heterogeneous flame in the
self-propagating high-temperature synthesis (SHS) process
A. Makino*
Department of Mechanical Engineering, Faculty of Engineering, Shizuoka University, Hamamatsu 432-8561, Japan
Received 30 April 1999; received in revised form 7 February 2000; accepted 7 February 2000
Abstract
Recent progress on understanding fundamental mechanisms governing the Self-propagating High-temperature Synthesis
(SHS) process, which is characterized by the flame propagation through a matrix of compacted reactive particles and is
recognized to hold the practical significance in producing novel solid materials, is reviewed. Here the focus is not only on
the theoretical description of the heterogeneous nature in the combustion wave, which has not been captured by the conventional premixed-flame theory for a homogeneous medium, but also on the extensive comparisons between the predicted and
experimental results in the literature. Topics included are the statistical counting procedure used for deriving governing
equations of the heterogeneous theory, flame propagation in the adiabatic condition, flame propagation and extinction under
heat loss conditions, effects of bimodal particle dispersion on the combustion behavior, those of external heating by electric
current, the transition boundary from steady to pulsating combustion, and the initiation of the combustion wave by use of the
external heating source. The importance of heterogeneity in the combustion wave, that is the particle size of the nonmetal or the
higher melting-point metal, has been emphasized for fundamental understanding of such combustion behavior as flame
propagation, extinction, and initiation. Potentially promising research topics are also suggested. 䉷 2000 Elsevier Science
Ltd. All rights reserved.
Keywords: Combustion synthesis; Self-propagating high-temperature synthesis; Flame propagation; Burning velocity; Range of flammability;
Extinction; Ignition; Ignition delay time
Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2. The SHS process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1. General characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2. Scope of the present survey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3. Phenomenological description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3. Formulation of the heterogeneous theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1. Surface regression rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2. Statistical counting procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.1. Change of distribution function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.2. Conservation equations in general form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.2.1. Overall continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.2.2. Species conservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.2.3. Momentum conservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.2.4. Energy conservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
* Tel.: ⫹ 81-53-478-1050.
E-mail address: [email protected] (A. Makino).
0360-1285/00/$ - see front matter 䉷 2000 Elsevier Science Ltd. All rights reserved.
PII: S0360-128 5(00)00004-6
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3.3. Heterogeneous theory for the SHS process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.1. Size distribution function in quasi-one-dimensional form . . . . . . . . . . . . . . . . . . . . . . .
3.3.2. Conservation equations for steady, one-dimensional form . . . . . . . . . . . . . . . . . . . . . . .
3.3.2.1. Overall continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.2.2. Species conservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.2.3. Momentum conservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.2.4. Energy conservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.3. Further simplification of the governing equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Flame propagation in the adiabatic condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1. Dominant parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2. Burning velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3. Range of flammability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4. Theoretical and experimental results for Ti–C system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5. Applicability of the heterogeneous theory for other systems . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5.1. Available experimental data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5.2. Effects of mixture ratio, m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5.3. Effects of degree of dilution, k . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5.4. Effects of initial temperature, T0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5.5. Effects of particle radius, R0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.6. Approximate expression for the burning velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.6.1. Derivation of the approximate expression for m ⱕ 1 . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.6.2. General behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.6.3. Comparisons with numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.6.4. Approximate expression for m ⱖ 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.7. Some remarks on the flame propagation in the adiabatic condition . . . . . . . . . . . . . . . . . . . . . .
Flame propagation in the nonadiabatic condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1. Governing equations and boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2. Characteristics in the nonadiabatic condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3. Turning-point determined by the thermal theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4. Experimental comparisons for Ti–C system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4.1. Range of flammabilty and extinction limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4.2. Burning velocity under heat loss condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.5. Experimental comparisons for other systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.5.1. Boride synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.5.2. Synthesis of intermetallic compounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.6. Approximate expression of the heat loss parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.6.1. Asymptotic expansion analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.6.2. Correction term for the heat loss parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.6.3. General behavior of the heat loss parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.6.4. Applicability of the analytical expression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.7. Some remarks on the flame propagation under heat loss condition . . . . . . . . . . . . . . . . . . . . . . .
Other aspects of the nonadiabatic flame propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.1. Bimodal particle dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.1.1. Equivalent particle radius . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.1.2. Experimental comparisons for spherical carbon particles . . . . . . . . . . . . . . . . . . . . . . . .
6.1.3. Experimental comparisons for diamond particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2. Representative length of the cross-sectional area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3. Correspondence between the heterogeneous theory and the homogeneous theory . . . . . . . . . . . .
6.4. Field activated SHS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.4.1. Heat-input parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.4.2. Temperature profiles outside the combustion wave . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.4.3. Effective range of electric field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.4.4. Experimental comparisons for Si–C system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.5. Some remarks on the several, other, important factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Boundary between steady and pulsating combustion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.1. Criterion for the appearance of cracks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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7.2. Linear stability analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2.1. Basic solution for planar propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2.2. Asymptotic solution and linear stability boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2.3. Experimental comparisons for the boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2.4. Boundary at the adiabatic condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.3. Transverse effects on the range for steady combustion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.4. Some remarks on the boundary of steady combustion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8. Initiation of the combustion wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.1. Flame initiation induced by igniter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.1.1. Inert stage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.1.2. Transition stage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.1.3. Limiting solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.1.4. Experimental comparisons for the ignition delay time . . . . . . . . . . . . . . . . . . . . . . . . . .
8.1.5. Expression for the ignition energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.1.6. Results for the ignition energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.2. Flame initiation by use of radiative heat flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.2.1. Model definition and inert stage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.2.2. Transition stage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.2.3. Results and experimental comparisons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.2.3.1. Situation without heat loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.2.3.2. Situation with heat loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.3. Some remarks on the flame initiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9. Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.1. Summary of the present survey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.2. Area for further research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1. Introduction
There has been increasing interest in the combustion
synthesis of new materials which would play important
roles in future technologies in various fields. Many of the
materials available today, such as ceramics (inorganic
compounds) and intermetallic compounds, fall into a category of chemical compounds, so that we can expect that
combustion would make a great contribution in producing
new materials and/or enhancing productivity of these materials, because of the existence of strong chemical reactions
accompanied by the combustion. The combustion synthesis,
defined as “materials synthesis” by combustion, is therefore
an interdisciplinary research subject in the fields of combustion and material science, for producing ceramics and/or
intermetallic compounds in lumps, thin films of oxides
and/or diamond, ultra-fine powder of inorganic compounds,
etc. as combustion products.
Research for the combustion synthesis has been very
active in the field of material science. On the other hand,
research from the viewpoint of combustion has not been so
active, in spite of the important fact that synthesis of materials by combustion can only be accomplished by virtue of
combustion. Then, it remains unsolved whether the knowledge of combustion hitherto obtained is applicable to the
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combustion synthesis in exploring and/or explaining its
combustion phenomena. This may be attributed to the fact
that the main concern of the combustion has been on producing heat, for the sake of heating and/or heat-supply to heat
engines. As for the combustion synthesis, its main concern
is not the heat, but the materials produced as combustion
products.
However, it should be recognized that combustion synthesis is one of the research subjects of combustion science
and technology because of the existence of combustion
phenomena, although objects to be produced by combustion
might be very different from each other. Furthermore, since
it is a new research field in combustion, combustion synthesis should be studied systematically, from the viewpoint
of combustion science and technology. This trend appeared
in the late 1980s and has been reflected by the establishment
of a session on Materials Synthesis in the International
Symposium on Combustion, since 1990.
Generally speaking, methods of combustion synthesis can
be divided into two main classes: one is burner flames which
produce thin films of oxides and/or diamond, and ultra-fine
powder of inorganic compounds; the other method is to
produce ceramics (inorganic compounds) and/or intermetallic compounds in lumps, by use of a kind of powder metallurgy called the Self-Propagating High-Temperature
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Nomenclature
A
a
a0
B
B~
b
C
c
D
Da
E
F
f
fst
G
H
h
hrad
I
Jn
K
k
k~
L
Le
M
m
n
p
P
Q
q0
R
R_
r
S
s
s~
T
T0
Ta
Td
t
U
u
v
W
X
x
Reduced surface Damko¨hler number ‰ˆ Da exp…⫺Ta =T†Š
Parameter …a3 ˆ f ⫺ 1† in Section 4; length of the rectangular cross section in Section 6; thermometric
conductivity in Section 8.1; radius of heating area by radiative heat flux in Section 8.2
Leading term of the normalized mass burning rate …ˆ m=ma † in Sections 5 and 7
Frequency factor for the surface reaction in Section 3; constant in Section 7
Frequency factor for the “reaction” term in the conventional homogeneous theory
Parameter …b3 ˆ f ⫹ 2g ⫹ 1† in Section 4; width of the rectangular cross section in Section 6
Constant; maximum temperature in Section 7
Specific heat
Mass diffusivity in Sections 3–7; constant in Section 8.
Surface Damko¨hler number; …ˆ BR=D†
Electric field in Section 6.4; energy for establishing combustion wave in Section 8
Thermal runaway criterion in Section 8.2
Distribution function
Stoichiometric mass ratio
Size distribution function
Heat-input parameter in Section 6.4
Enthalpy
Radiative heat-transfer coefficient
Integrated quantity
Bessel function of the first kind of order n in Section 8.2.2
Constant in Section 7; derivative of temperature in nondimensional form in Section 8.1.2
Inverse of the arithmetic mean of melting point and adiabatic combustion temperature in nondimensional form
Over-all transverse wave-number in Section 7.3; nondimensional parameter related to normalized mass burning
rate in Section 8.1
Volumetric heat loss
Lewis number
Melting parameter
Mass burning rate for deflagration
Number of N particles
Static pressure in Section 3; perimeter of rectangular cross section in Section 6.3
Size ratio defined in Eq. (143) in Section 6.1.1
Heat flux
Heat of combustion per unit mass of N species
Average radius of N particles
Rate of change of particle size R …ˆ dR=dt†
Radius of compacted specimen
Cross-sectional area of compacted specimen
Variable which represents a change of particle size in Section 3; stretched variable of time in reactive/diffusive
region in Section 8.2
Stretched variable of time in transient/diffusive region in Section 8.2
Temperature
Standard temperature
Activation temperature for the reaction
Activation temperature for the condensed phase mass diffusivity
Stretched variable of temperature in Section 5.6.1; time in Sections 7 and 8.
Diffusion velocity of species
Heterogeneous burning velocity
Velocity of a particle
Molecular weight
Normalized heat loss, defined as X ˆ exp…⫺2bC=a 02 † ˆ exp…⫺C1 † in Sections 5 and 7; stretched variable of
location in reactive/diffusive region in Section 8.2
Physical spatial coordinate
A. Makino / Progress in Energy and Combustion Science 27 (2001) 1–74
Y
y
Z
Mass fraction
Stretched variable of mass ratio of fluid in Section 5.6.1
Mass ratio of fluid
Vectors and tensors
E
F
f
P
q
T
U
u
v
Unit tensor
Force vector
Body force vector
Total pressure tensor
Heat flux vector
Viscous stress tensor
Diffusion velocity vector
Veclocity vector for fluid
Veclocity vector for a particle
Greek symbols
a
b
G
g
D
d( )
d
e
z
h
u
k
L0
L
l
l u, l d
m
n
j
1⫺j
r
s
sc
s ST
s
t
t xx
f
w
w~
x
C
c
V
v
Normalized temperature rise
Spalding transfer number in Section 3; Zeldovich number in Sections 4–8
Mass ratio of small particles to large particles in Section 6
Mass ratio of fluid to solid
Damko¨hler number for flame initiation in Section 8
Delta function
Thickness of combustion wave
Emissivity
Normalized mass fraction
Variable which represents a change of particle size in Section 3; inner variable of length in Section 5.6.1
Normalized temperature
Degree of dilution defined as the initial mass fraction of the diluent
Mass burning rate eigenvalue
Heat penetration coefficient in Section 8
Thermal conductivity
Exponents which represent temperature distribution in Section 6.4 and Section 7
Mixture ratio defined as the initial molar ratio of nonmetal, N, to metal, M, divided by the corresponding
stoichiometric molar ratio
Stoichiometric coefficient
Normalized mass ratio of fluid
Normalized mass ratio of solid
Density
Nondimensional coordinate
Electric conductivity in Section 6.4
Stefan–Boltzmann constant
Stretched coordinate in the downstream in Section 5.6.1.
Nondimensional time
xx component of the viscous stress tensor in Section 3.
Inverse of the mixture ratio m in Section 4.6 and 5.6; heat loss in Section 8.2.
Function which represents dependence of the “reaction” term on reactants
Nondimensional parameter related to heat loss in Section 8.1
Surface regression rate
Heat-loss parameter
Variable for particle size distribution in Section 3; stretched variable for temperature in Section 8
Constant
Mass rate of production by homogeneous chemical reaction in Section 3; stretched variable of location in
transient/diffusive region in Section 8.2
5
6
A. Makino / Progress in Energy and Combustion Science 27 (2001) 1–74
Subscripts
a
Adiabatic condition
c
Initiation of the combustion wave
cr
State of extinction
d
Downstream
E
Equivalent
e
Edge of boundary
f
Fluid
I
Inert heating
ig
Igniter
in
Material flow to the fluid from particles
L
Large particles
M
Metal or lower melting-point metal
m
Melting point
max
Maximum
N
Non metal or higher melting-point metal
P
Combustion product
p
Perturbed term
r
Representative value in Section 6.4; radial direction in Section 8.2
rel
Relative
S
Small particles
s
Solid or surface
sb
Neutral stability boundary
sp
Specimen, that is, medium to be ignited
TMD
Theoretical maximum density
t
Differentiation with respect to time
u
Upstream
z
Differentiation with respect to location in the axial direction
j
With respect to mass fraction of fluid Z
u
With respect to temperature
t
Ignition delay
0
Initial or unburned state
∞
Final or burned state in the adiabatic condition
Superscripts
t
Nondimensional
—
Average over all velocities in Section 3.
Synthesis (SHS) process. In the present review, emphasis is
put on the latter, because of its various advantages, to be
examined.
2. The SHS process
2.1. General characteristics
Among various types of combustion synthesis, wide
attention has been given to the SHS process, proposed by
Merzhanov and co-workers [1–3] in Russia in the late
1960s, because not only bulk processing of materials, but
also formation of some elemental combinations not
previously synthesized, can be performed [4–9] by initiating and passing a combustion wave through a matrix of
compacted reactive particles (cf. Figs. 1 and 2). From the viewpoint of combustion science and/or technology, as shown in
Fig. 2 photographically, the SHS process falls into a category
of the flame propagation, because a reaction initiated at one
end of a compacted medium self-propagates through an
unburned medium, in the form of a combustion wave.
The velocity of the combustion wave, depending on
systems, varies from 1 to 250 mm/s. The high temperature
needed for synthesis (perhaps more than 2000 K for inorganic compounds and more than 1000 K for intermetallic
compounds) can be supplied by the self-sustained exothermic chemical reactions. The potential advantages of this
process are as follows [4–9]: rapid synthesis; self-heating;
energy savings; self-purification due to enhanced impurity
outgassing; near-net-shape fabrication, etc. In addition,
more than 500 kinds of materials, including carbides,
A. Makino / Progress in Energy and Combustion Science 27 (2001) 1–74
Fig. 1. Schematic drawing of combustion synthesis by the SHS
process [5].
7
borides, silicides, nitrides, sulfides, hydrides, intermetallics,
and complex composites, are reported [10] to be synthesized
by applying the SHS process not only for solid–solid
systems but also for solid–gas and/or solid–liquid systems.
Table 1 shows only a few of examples of inorganic
compounds synthesized by the SHS process. Compounds
synthesized are being considered for use as electronic materials, materials resistant to wear, corrosion, and heat.
Furthermore, it can even be applied to the synthesis of
shape-memory alloys, hydrogen-storage alloys, and hightemperature superconductors [7–9]. Thus, it is now well
recognized that the SHS process can be of practical significance in producing novel solid materials.
However, if restricted to synthesizing homogeneous
materials in composition and/or texture, as is the case
for conventional materials, the SHS process would not
be attractive to researchers in the field of combustion. It
was in the late 1980s when various attempts began to
discover possibilities for improving the specific nature
of the conventional, homogeneous materials for the
purpose of producing advanced materials with useful,
multifunctional characteristics generated by heterogeneity in composition and/or texture. Among these
advanced materials, Functionally Graded Materials
(FGMs), which are composed of different material
components such as ceramics and metals with continuous profiles in composition, structure, texture, material
strength, and thermophysical properties, have attracted
special interest as advanced heat-shielding/structural
materials in future space applications. At that time,
Fig. 2. A sequence of the advance of a combustion wave in the SHS process for Ti–C system; stoichiometric mixture without dilution; diameter
is 18 mm and length is about 50 mm; the initial temperature T0 is 300 K and radius R0 of the carbon particle is 5 mm; the maximum temperature
is about 3200 K and the burning velocity u is 17 mm/s. (a) Just after the ignition; (b) 1 s; (c) 2 s; and (d) 3 s.
TiB2 –Al2O3, etc.
MoB–Al2O3,
NiS, Ni3S2,
CoS, CoS2
MnS
WC–Al2O3,
MoS2, WS2,
MoSe2, WSe2
SiC–Al2O3, SiC–MgO,
TiC–ZrO2,
CsH2
TiC–TiB2,
Hydride
Compound
TiC–SiC, TiC–Al2O3,
PrH2, NdH2
TiH2, ZrH2
B4C–Al2O3,
TaS2,
NbSe2, TaSe2
TiS2, ZrS2,
PbS, TiSe2
ZnS, CdS,
ZnSe, CdTe
Cu2S,
CuInSe2
Chalcogenide
CeS
FeSi2, CoSi2
MnSi
CrSi2, MoSi2,
Mo3Si, WSi2
VSi2, V3Si,
NbSi2, TaSi2
TiSi, TiSi2,
ZrSi, ZrSi2
Mg2Si, CaSi2
Cu2Si
Silicide
YSi2, LaSi2,
CeSi2, DySi2
CrN
VN, NbN,
TaN, Ta2N
Si3N4, TiN,
ZrN, HfN
Mg3N2, SrN
BaN
CuN
Nitride
BN, AlN,
NdN
CaC2
Carbide
B4C, Al4C3
SiC, TiC,
ZrC, HfC
VC, Nb2C,
NbC, TaC
Cr3C2, MoC,
Mo2C, WC
Mn7C3
FeB, NiB
MnB, MnB2
CrB2, MoB,
Mo2B5, WB
VB, NbB,
NbB2, TaB
TiB, TiB2,
ZrB2, HfB2
AlB2, LaB6,
CeB2
MgB2, MgB4,
MgB6, CaB6
Boride
VII
VI
V
IV
III
II
I
Group in periodic table
Complex
Table 1
Examples of inorganic compounds synthesized by the SHS process
Fe4N, Fe8N
A. Makino / Progress in Energy and Combustion Science 27 (2001) 1–74
VIII
8
production of FGMs by use of the SHS process
(especially for producing ceramics in FGMs with gradual
profiles in composition) was considered because of its rapidity in production. Although it has proven effective, it was
also recognized that in order to control the manufacturing
process, dependence of flame propagation speed on various
dominant parameters is indispensable. In addition, range of
flammability with respect to various parameters is urgently
required because the combustion wave cannot be maintained outside the range of flammability. Of course, dependence on various parameters is also required in preventing
flame extinction. From an academic point of view, these
areas are within the parameters combustion researches
have pursued. That is, researchers in the field of combustion
are strongly encouraged to clarify these problems through
knowledge of combustion.
In addition to its practical utility, the SHS process
commands fundamental interest because of its interesting
and diverse phenomena, such as steady planar propagation,
pulsation, spinning, and repeated combustion, which can be
observed during the flame propagation in the solid compacted
medium [1–3,11,12]. In pulsating combustion which is also
called self-oscillation combustion, the combustion wave
travels in a planar, but pulsating manner, which frequently
results in materials with a laminated structure (cf. Fig. 3). In
spinning combustion the combustion wave is nonplanar, and
one or more hot spots are observed to move along a spiral path
over the surface of the sample specimen, inside of which reactants experience only heating, without reaction. In repeated
combustion, after the passage of the combustion wave, another
combustion wave is initiated and propagates through the
burned medium, yielding the final combustion products from
the intermediates. Fundamental understanding of flame propagation, extinction, and/or initiation in the SHS process is not
only of interest for its own sake, but also of great use for
potential improvement in the manufacturing process.
Various experimental works have been conducted, and
many of the accomplishment have been summarized in
several good review papers [5–9]. In order to elucidate
effects of various system parameters on the flame behavior,
theoretical studies have also been conducted keeping up
with experiments. The basic theory, originally derived by
Novozhilov [13] for the combustion of solid propellant, and
extended by Merzhanov [14] for the SHS process, has been
applied to examining stability [15] and unsteady behaviors,
such as pulsating [16] and spinning [17] combustion. Effects
of competing reactions [18], particle size distribution [19],
heat loss [20,21], Arrhenius mass diffusion [22], etc. have
also been studied. Subsequent accomplishments by theoretical works are well summarized in recent review papers
[23,24]. Note here, however, that theoretical description in
those works has primarily been based on the premise that
synthesis is accomplished through the passage of a premixed
flame in a homogeneous medium in which reactants are well
mixed at the molecular level before arrival of the flame front
of the combustion wave.
A. Makino / Progress in Energy and Combustion Science 27 (2001) 1–74
9
Fig. 3. Laminated structure in the combustion products; Ti–C system for stoichiometric mixture without dilution; the initial temperature T0 is
300 K. (a) Radius of the carbon particle R0 is 0.5 mm and the relative density r rel is 0.56. (b) R0 ˆ 0:5 mm and rrelˆ 0:63: (c) R0 ˆ 0:5 mm and
rrel ˆ 0:72: (d) R0 ˆ 2:5 mm and rrel ˆ 0:47: (e) R0 ˆ 2:5 mm and rrel ˆ 0:64:
2.2. Scope of the present survey
Although much work has been conducted both experimentally and theoretically, a general lack of consistency
has been noted [25] in the combustion data and no firm
understanding of the combustion mechanism has been
achieved. This may be attributed to the fact that comparisons between experimental and theoretical results have been
rare, regardless of their importance for comprehensive
understanding of various combustion characteristics.
Through agreement between theoretical and experimental
results over wide range of parametric quantities for various
combinations of metal and nonmetal (or another metal of a
higher melting point) systems, appropriateness of theories
should therefore be examined.
When such comparisons are conducted with conventional
models in the homogeneous premixed-flame theory, there
exists an inherent deficiency that cannot account for size
effect of particles, while experiments have firmly established that the flame propagation speed increases with
decreasing particle size in a sensitive manner [26–28].
This inherent deficiency comes from insufficient treatment,
based on the homogeneous premixed-flame theory. Even if
fine powder is used, for example 0.1 mm in particle
diameter, each particle contains more than 10 8 atoms. This
situation is far from that to which the homogeneous
premixed-flame theory can be applied. That is, we cannot
regard the SHS process as a homogeneous phenomenon.
Therefore, we are urgently required to re-examine the
appropriateness of describing the SHS process as the
premixed flame propagation in the homogeneous medium.
In another mathematical model for the SHS process, it is
proposed to assume the compacted medium as laminae of
reactants [29,30]. The combustion situation for the usual
SHS process, however, is quite different from this model
because the combustion wave propagates through the
medium of compacted reactive particles. Again we are
required to construct an appropriate model which can properly represent the combustion situation in the SHS process,
by taking its heterogeneity into account in an essential
manner.
The present paper, restricted to theoretical accomplishment, aims to complement the extensive reviews by Frankhouser et al. [5], Munir and Anselmi-Tamburini [7],
Merzhanov [8] and Varma and Lebrat [9] regarding the
experimental accomplishment, and those by Merzhanov
[4], Merzhanov and Khaikin [23] and Margolis [24] regarding the theoretical accomplishment. Indeed, because of the
broad coverage of some of these reviews, a narrow range of
topics will be discussed here, focused on the heterogeneity
in the SHS process, to which little attention has been paid,
although it is indispensable for realistic descriptions of
combustion behavior. The present survey also emphasizes
understanding of the fundamental mechanisms governing the flame propagation, extinction, and/or initiation in
the SHS process, by being restricted to the solid–solid
systems. It is also suggested that the reader refers to
Ref. [31] for topics about the SHS process for the
gas–solid systems, which are not covered in the present
review.
In the next section, a phenomenological description of the
flame propagation in the SHS process is presented. In
Section 3, the statistical counting procedure is reviewed
and its usefulness is assessed. In Section 4, theoretical
approaches are primarily aimed at studying the SHS flame
propagation in the adiabatic condition. Following this,
several important effects on the SHS flame propagation
(such as the heat loss, size distribution of particles, and
external heating by electric current) are discussed in relation
to flame extinction. In Section 7, the transition boundary
between steady and pulsating combustion, which is indispensable in bulk processing, is discussed. Up to this point,
10
A. Makino / Progress in Energy and Combustion Science 27 (2001) 1–74
Fig. 4. Schematic flame structure in the heterogeneous flame (crystallization not described) [32].
discussion is confined to the flame propagation and extinction. In the next section, initiation of combustion wave is
discussed for two different ignition methods: igniter and
radiative heat flux.
2.3. Phenomenological description
In order to recognize differences between the heterogeneous flame in the SHS process and the homogeneous
premixed flame, let us first make a detailed phenomenological consideration for the SHS flame propagation. It can be
considered that the SHS process typically involves reaction
between particles of a metal and a nonmetal (or another
metal of a higher melting point). Generally speaking, before
the arrival of the combustion wave, the effects of the reaction cannot be remarkable due to the low temperature and
small area of contact between particles. However, as this
particle matrix is heated by the approaching combustion
wave, the lower melting point metal melts first, resulting
in slurry consisting of the higher melting-point nonmetal
particles suspended in the molten metal. In addition, due
to an increase in the area of contact, as well as that in
temperature, the reaction between nonmetal particles and
molten metal is greatly enhanced. Then, it is anticipated
that the subsequent reaction between them will take place
mainly over the surface of the nonmetal particles; this situation is schematically shown in Fig. 4 [32]. Provided that the
nonmetal particles do not dissolve in the molten liquid, and
that they are not too small, compared to the thickness of the
combustion wave, it is easily seen that the flame propagation
through the compacted medium is basically heterogeneous
in nature, involving a premixed-mode of propagation for the
bulk flame supported by the nonpremixed reaction of the
dispersed nonmetal particles in the liquid metal. Therefore,
it is reasonable to anticipate that the resulting flame structure and response should be quite different from those
obtained by assuming a premixed-mode of reaction at the
molecular level. That is, the present phenomenological
consideration strongly suggests that a satisfactory formulation must take into account the nonpremixed mode of particle reaction.
Here, it may be of great importance to recognize that the
flame propagation in the SHS process is actually quite similar to that in fuel spray combustion, if we identify the fuel
droplets as the nonmetal (or higher melting point metal)
particles and the oxygen in air as the molten metal, restricting ourselves to the flame zone in the combustion wave. It is
also appropriate to recognize that across the combustion
wave, dozens of nonmetal particles [33] can exist in the
“reaction” zone in which the temperature is higher than
the melting point of metal. This can easily be confirmed
by comparing a representative particle size, for instance
10 mm, to the approximate “flame thickness” d of about
0.3 mm, evaluated from a relation d ⬇ l=…rcu† derived by
the phenomenological analysis for gaseous premixed flames
[34], with representative values of thermometric conductivity l /(r c) of 3 × 10⫺6 m2 =s and the burning velocity u of
0.01 m/s for Ti–C system. This recognition further suggests
that a statistical counting procedure for spray combustion,
explained in Ref. [35], is indispensable and of great use in
describing the combustion behavior of the heterogeneous
flame in the SHS process. The degree of similarity is achievable if we restrict ourselves to fundamental principles without probing too far into specific details of the physical
processes.
Another important factor that can influence the SHS flame
propagation is the mass diffusion in liquid phase, which is
closely related to the nonpremixed reaction of nonmetal
particles in the combustion wave. Since the mass diffusivity
in the SHS process is anticipated to increase markedly over
a relatively thin, high temperature combustion wave, use of
the temperature-dependent liquid-phase mass-diffusivity in
Arrhenius fashion [22] is appropriate and indispensable.
As for the surface reaction on each nonmetal particle,
related to the nonpremixed reaction in the combustion
wave, it is recognized to be quite similar to the combustion
A. Makino / Progress in Energy and Combustion Science 27 (2001) 1–74
Ta;s
;
Ts
of a single solid particle, having been examined in the field
of solid combustion, if the oxygen in air is identified as the
molten metal. This recognition further facilitates inclusion
of the finite-rate reaction at the particle surface when formulation is conducted.
Finally, in closing this section, it may be informative
to describe a simplified model [32] which has been
adopted for investigating combustion behavior of the
heterogeneous flame in the SHS process. As shown in
Fig. 4, the combustion situation modeled is the planar,
heterogeneous flame in an infinite domain of a
compacted medium, originally consisting of a mixture
of particles of nonmetal N, metal M, and an inert which
can be the product P of the reaction according to
nM M ⫹ nN N ! nP P; where n i is the stoichiometric coefficient. For simplicity, it is assumed that there is no
reaction until the mixture has been heated to the melting point Tm of the metal, at which all the metal particles melt instantly. The reaction that follows is assumed to
take place at the particle surface at a finite rate and proceed
until all of the nonmetal (or higher melting-point metal) particles, or metal species, are consumed.
A ˆ Das exp ⫺
3. Formulation of the heterogeneous theory
r~ ˆ
r
vR
Y
n W
; vr ˆ r ; Y~ M ˆ M ; fst ˆ M M ;
R
De
fst
nN WN
3.1. Surface regression rate
r~ ˆ
rM
D
; D~ ˆ
:
De
rM;e
Before describing the statistical counting procedure used
in spray combustion, let us first confirm the description of a
single solid particle undergoing combustion. Since it is
unnecessary to consider the velocity difference between
particles and molten metal in the SHS combustion wave,
only combustion behavior in a quiescent fluid will be examined. For a single, spherical, nonpermiable, solid particle of
radius R in a quiescent fluid, the rate of change of its mass
caused by the surface reaction nM M ⫹ nN N ! nP P is
expressed as
_ ˆ ⫺ d rN 4 pR3 ˆ ⫺rN 4pR2 R;
_
M
…1†
dt
3
which yields
⫺RR_ ˆ
rM D
x~ ;
rN
x~ ⬅
_
M
:
4prM DR
…2†
Here, x~ is the surface regression rate, which coincides with
the definition of the nondimensional combustion rate,
commonly used in the conventional solid combustion.
Then, we have
x
R_ ˆ ⫺ ;
R
xˆ
rM D
x~ :
rN
…3†
The specific form of x~ can be obtained as [32,36]
x~ ˆ A
Y~ M;e ⫺ b
;
1⫹b
11
b ˆ exp…x~ † ⫺ 1;
…5†
by solving the quasi-steady liquid-phase conservation equation in spherical coordinates
!
~
~
~ d r~2 dY M ⫹ x~ dY M ˆ 0;
⫺…r~ D†
…6†
dr~
dr~
d~r
under the isothermal condition, with the boundary condition
at the particle surface …~r ˆ 1†
!
~
~ s dY M ⫹x~ Y~ M;s ˆ ⫺x~ ;
⫺…r~ D†
x~ ⬅ r~ vr r~2 ˆ AY~ M;s ;
dr~ s
…7†
and that at the outer edge …~r ! ∞† of the boundary layer
around the particle
Y~ M ˆ Y~ M;e :
…8†
In the above, A is the reduced surface Damko¨hler number,
Das the surface Damko¨hler number ( ˆ B·R/D), b the Spalding transfer number, and other variables and parameters are
defined as
where r is the radial coordinate, vr the radial velocity, Y~ M the
stoichiometrically weighted mass fraction, fst the stoichiometric mass ratio, W the molecular weight, D the mass
diffusivity, and the subscript s and e designate, respectively,
the particle surface and the outer edge of the boundary layer.
An introduction of the isothermal condition, which
enables us to put r~ ˆ 1 and D~ ˆ 1; can be justified when
the thickness of the bulk flame is much larger than the
particle size because we can anticipate that both a nonmetal
solid particle and molten metal around the particle are
heated equally as the combustion wave arrives. The use of
the conventional constant property assumption can also be
justified if attention is confined to a restricted region around
the single particle. Note that changes in thermophysical
properties in the bulk during the flame propagation can be
incorporated.
The analytical expression in Eq. (4), however, is an implicit expression with respect to the surface regression rate,
which is difficult to have a clear image of its dependence on
other parameters. Therefore, an attempt has been made to
obtain an explicit, approximate expression [32,36] as
A ~
Y M;e ;
…9†
x~ ⬅ ln…1 ⫹ b† ⬇ ln 1 ⫹
1⫹A
by use of the approximate relation
…4†
b
ˆ 1 ⫺ exp…⫺x~ † ⬇ x~
1⫹b
…10†
12
A. Makino / Progress in Energy and Combustion Science 27 (2001) 1–74
for small x~ ; as is the case for most solid combustion. Note
that the expression in Eq. (9) for the surface regression rate
has the same form as that commonly used in droplet
combustion. The error between Eqs. (4) and (9) is within
2% for Ti–C system. It may be informative to note that Y~ M;e
in Eq. (9) can change as the combustion wave propagates,
being governed by the species conservation equations, to be
mentioned in the next section.
3.2. Statistical counting procedure
3.2.1. Change of distribution function
In spray combustion [35] a statistical description is given
by the distribution function
f …R; x; v; t† dR dx dv
which is the probable number of particles in the radius range
dR about R, located in the spatial range dx about x with
velocities in the range dv about v at time t. Here, dx and
dv are the three-dimensional elements of physical space and
velocity space, respectively. The variables R, x, and v must
appear in the distribution function because conditions are
not known well enough to permit specification of the exact
size, position, or velocity of each particle.
An equation governing the time rate of change of the
distribution function f, which is called the spray equation
in spray combustion, is given as
2f
2 _
ˆ⫺
…Rf † ⫺ 7 x ·…vf † ⫺ 7 v ·…Ff †;
2t
2R
…11†
when there are no particle formation/destruction by
processes nor collisions with other particles. Here, R_
( ˆ dR/dt) is the rate of change of the particle size R at (R,
x, v, t) and F …ˆ dv=dt† the force per unit mass on this
particle. The terms in the RHS respectively represent the
changes in f resulting from the change of particle size, the
motion of particles into and out of the spatial element dx by
virtue of their velocity v, and the acceleration of particles in
the velocity element dv caused by the force F. Note that both
R_ and F are allowed to depend on R, x, v, t, and the local
properties of the fluid.
3.2.2. Conservation equations in general form
In analyzing behavior of the combustion wave, hydrodynamic equations for the fluid are necessary. If we restrict
ourselves to the situation in which the statistical fluctuations
in the fluid, induced by the random motion of individual
particles, may be neglected, these governing equations for
the local average properties in the fluid are equivalent to the
ordinary equations of fluid dynamics, with suitably added
source terms accounting for the average effect of the
dispersed particles.
3.2.2.1. Overall continuity By adding the mass of material
per unit volume per unit time from the particles to the fluid,
the overall continuity equation for the fluid is given as
ZZ
2rf
_ dR dv;
rN 4pR 2 Rf
⫹ 7 x ·…rf u† ˆ ⫺
…12†
2t
where r f is the fluid density defined as the mass of fluid per
unit volume of physical space. Note that the fluid and liquid
densities are related by the expression
ZZ 4
pR3 f dR dv :
rf ˆ rM 1 ⫺
3
…13†
3.2.2.2. Species conservation When chemical reactions
occur at the particle surface and species M is
consumed at the surface reaction according to
n MM ⫹ n NN ! n PP, the species conservation equation
for component M is given as
2
…r Y † ⫹ 7 x ·‰rf …u ⫹ UM †YM Š
2t f M
ZZ
n W
_ dR dv;
ˆv⫹ M M
rN 4pR 2 Rf
nN WN
…14†
where YM is the mass fraction of species M in the fluid,
UM the diffusion velocity, and v the mass rate of
production of species M by homogeneous chemical
reactions in the fluid. Note that for steady flow, by
use of Fick’s law, UM is expressed as
UM YM ˆ ⫺D…7 x YM †:
…15†
3.2.2.3. Momentum conservation
conservation equation is given as
rf
The
momentum
N
X
2u
⫹ rf u·7 x u ˆ ⫺7 x ·P ⫹ rf
Yk f k
2t
kˆ1
⫺
⫺
ZZ
ZZ
4
rN pR 3 Ff dR dv
3
_ ⫺ u†f dR dv;
rN 4pR2 R…v
…16†
in which the third term in the RHS represents the average
force per unit volume exerted on the particles by the
surrounding fluid, and the last term accounts for the
momentum carried to the fluid by the material from the
particles. In the above, P is the total pressure tensor
( ˆ pE ⫹ T), related to the hydrostatic pressure p and the
viscous stress tensor T, where E is the unit tensor. In
addition, fk is the external body force per unit mass acting
on species k in the fluid.
3.2.2.4. Energy conservation The energy conservation
A. Makino / Progress in Energy and Combustion Science 27 (2001) 1–74
equation is given as
"
!#
"
!#
2
u2
u2
r h ⫹
⫹ 7 x · rf u hf ⫹
2t f f
2
2
and the bar denotes an average over all velocities, that is
1 Z_
1 Z
Rf dv;
v ˆ
vf dv:
…20†
R_ ˆ
G
G
Eq. (18) can be further simplified as
N
X
2p
⫹ rf
Yk …u ⫹ Uk †·f k
ˆ ⫺7 x ·q ⫺ 7 x ·…T·u† ⫹
2t
kˆ1
2 _
1 2
ˆ 0;
…RG† ⫹
…SvG†
2R
S 2x
ZZ
4
rN pR 3 …F·v†f dR dv
3
!
ZZ
v2
2_
⫺
rN 4pR R hin ⫹
f dR dv;
2
⫺
…17†
where hf is the total enthalpy per unit mass of the fluid, q the
heat flux vector, and hin the total enthalpy per unit mass of
material flowing from the vicinity of a particle to the fluid.
In the above, the last two terms accounts for the work done
on the fluid by particle and the energy added to the fluid by
the material from the surface.
3.3. Heterogeneous theory for the SHS process
In the SHS process, it is usual to consider a situation in
which a reaction, initiated at one end of a matrix of
compacted reactive particles, moves through the adjacent
unburned matrix in the form of a self-sustained combustion
wave. In this situation, even a quasi-one-dimensional treatment has practical importance, and its simplicity is favorable for clarifying dominant parameters that influence the
flame propagation.
3.3.1. Size distribution function in quasi-one-dimensional
form
The velocity dependence of the distribution function f,
which is not of primary interest, can be eliminated from
the governing equations by integrating Eq. (11) over all
velocity space. Since f ! 0 as 兩v兩 ! ∞ for all physically
reasonable flows, the integral of the last term in Eq. (11)
is zero. As for the unsteady change of the distribution function f, it is of considerable significance in spray combustion,
in relation to such phenomena as diesel-engine combustion
or combustion instability in liquid-propellant rocket motors.
However, in the SHS process it may be excluded from our
consideration, because a matrix of compacted reactive particles through which a combustion wave propagates is
expected to have a certain distribution of particles when it
is produced by compaction, as far as the distribution of
particle sizes is concerned. Then, we have
2 _
…RG† ⫹ 7 x ·…vG† ˆ 0;
2R
13
…18†
where the number of particles per unit volume per unit range
of radius is
Z
G ˆ f dv;
…19†
…21†
when the local cross-sectional area S(x) only depends on the
variable x. Here, v is the x component of v ; and the quantities
_ v;
and G are averages over the cross section, independent
R;
of the spatial coordinates normal to x.
By using a relation that R_ ˆ ⫺x=R in Eq. (3), Eq. (21)
becomes
⫺x
2c
2c
⫹ vR
ˆ 0;
2R
2x
…22†
where
c⬅
SvG
:
R
…23†
Upon transformation to the new independent variables
Zx x Zx x s 2 ˆ R2 ⫺ 2
dx;
dx;
h2 ˆ R2 ⫹ 2
v
v
⫺∞
⫺∞
…24†
Eq. (22) becomes 2c=2s ˆ 0; which means that c only
depends on h . Letting the subscript 0 identify conditions
at the unburned state, we have c ˆ c 0(h ). Since h ˆ R0
as x ! ⫺ ∞, we see from Eq. (23) that the expression
S 0 v0
R
Gˆ
G …h†
…25†
Sv
h 0
determines the size distribution G(R, x) at any position x in
terms of the distribution G0(h ) at the unburned state …x !
⫺∞†:
3.3.2. Conservation equations for steady, one-dimensional
form
When we restrict ourselves to steady, one-dimensional,
constant-area flows in which all particles travel at the same
velocity, the distribution function f is expressed as
f …R; x; v† ˆ G…R; x†d…v ⫺ v†;
…26†
where x is the flow direction, v the velocity of all particles,
and d (·) the delta function.
3.3.2.1. Overall continuity The integral over v in Eq. (12)
yields
Z∞
d
_ dR;
…rf u† ˆ ⫺
…27†
rN 4pR 2 RG
dx
0
where u is the x component of u. The RHS in Eq. (27) also
appears in
Z∞
d
_ dR;
ˆ
…r v†
rN 4pR2 RG
…28†
dx s
0
14
A. Makino / Progress in Energy and Combustion Science 27 (2001) 1–74
which is derived by multiplying Eq. (11) by r N(4/3)pR 3 and
integrating over v and R. Note that r s is the mass of solid per
unit spatial volume defined as
Z∞ 4
pR3 G dR:
rs ˆ rN
0 3
rf u hf ⫹
u2
2
!
⫹ rs v hN ⫹
v2
2
!
⫹ qx ⫹ txx u ˆ constant:
…37†
…29†
Eqs. (27) and (28) show that the overall continuity equation
is written in the form
rf u ⫹ rs v ˆ m ˆ constant;
which yields
…30†
where m is the total mass flow rate (i.e. mass burning rate)
per unit area.
Note that in deriving Eq. (36), use has been made of the
relation
Z∞
_ in GdR ˆ d …rs vh
N †;
rN 4pR2 Rh
…38†
dx
0
obtained by equating
Z∞
_ N G dR
N† ˆ
7 x ·…rs vh
rN 4pR2 Rh
0
3.3.2.2. Species conservation When there exists no
homogeneous chemical reactions in the fluid, the species
conservation Eq. (14) is simplified as
d
nM WM d
‰rf …u ⫹ UM †YM Š ˆ
…r v†;
…31†
dx
nN WN dx s
which yields
rf D
dYM
⫺ …rf uYM ⫺ rs v fst † ˆ constant;
dx
…32†
3.3.2.3. Momentum conservation The momentum
conservation Eq. (16), by neglecting influences of body
forces fk, becomes as
…33†
which yields
rf u2 ⫹ rs v2 ⫹ p ⫹ txx ˆ constant;
…34†
where t xx is the xx component of the viscous stress tensor T.
Note that in deriving Eq. (33), use has been made of the
relation
Z∞
0
4
dv
rN pR3 FG dR ˆ rs v ;
3
dx
…35†
which is the momentum equation of the dispersed particles,
derived by multiplying Eq. (11) by rN …4=3†pR3 v and
integrating over v and R, as well as the relations in Eqs.
(27) and (28).
3.3.2.4. Energy conservation When radiative heat transfer is
neglected, the energy conservation Eq. (17) becomes as
"
!
!#
d
u2
v2
⫹ rs v hN ⫹
rf u hf ⫹
dx
2
2
dq
d
ˆ⫺ x ⫺
…t u†;
dx
dx xx
and
Z∞
0
where UM is the x component of UM and can be expressed by
Fick’s law.
d
dp
dt
…r u2 ⫹ rs v2 † ˆ ⫺
⫺ xx ;
dx f
dx
dx
⫹
…36†
ZZ
4
2 hN
rN pR3 R_
⫹ v·7 x hN ⫹ F·7 v hN f dR dv;
3
2R
…39†
_ in G dR ˆ
rN 4pR2 Rh
⫹
ZZ
Z∞
0
_ N G dR
rN 4pR2 Rh
4
2h
rN pR3 R_ N ⫹ v·7 x hN ⫹ F·7 v hN f dR dv:
2R
3
…40†
Here, Eq (39) is obtained by multiplying Eq. (11) by
rN …4=3†pR3 hN and integrating over v and R, while Eq.
(41) is obtained by multiplying
d
4
…r pR3 hN †
dt N 3
_ N ⫹ rN 4 pR3 R_ 2hN ⫹ v·7 x hN ⫹ F·7 v hN ;
ˆ rN 4pR2 Rh
3
2R
…41†
_ in ˆ
rN 4pR2 Rh
by f and integrating over v and R. Note that Eq. (41) gives a
relation between hin and hN.
3.3.3. Further simplification of the governing equations
In the SHS process, all particles in the combustion wave
are reasonably assumed to travel at the same velocity as the
fluid; that is, v ˆ u: Then, the overall mass conservation in
Eq. (30) becomes
rt u ˆ m ˆ constant;
…42†
where rt ⬅ rf ⫹ rs is the total density. As for the mass
consumption of solid N particles, by introducing the mass
fraction of fluid as
rf
Zˆ
;
…43†
rf ⫹ rs
Eq. (28) reduces to
d…1 ⫺ Z†
1 Z∞
_ dR:
ˆ⫺
r 4pR2 RG
dx
m 0 N
…44†
Substituting Eqs. (3) and (25) into Eq. (44) and by
A. Makino / Progress in Energy and Combustion Science 27 (2001) 1–74
integrating, we have
2!
d…1 ⫺ Z†
4p Z∞
u0
R
ˆ⫺
G0 …h† dR:
rN
x
dx
m
u
h
0
species conservation equations, such as
…45†
When the particles are monodispersed with number density
n0, that is G0 …R† ˆ n0 d…R ⫺ R0 †; Eq. (45) reduces to
d…1 ⫺ Z†
4prN n0 R0
u0
1 ⫺ Z 1=3
ˆ⫺
x
;
…46†
m
u
dx
1 ⫺ Z0
where the following relation has been made use of
!
Zx x 1=2
Z∞ R2
dx
G0 …h† dR ˆ n0 R20 ⫺ 2
v
h
0
⫺∞
!
Zx x 3=2
1⫺Z
dx ;
R30 ˆ R20 ⫺ 2
1 ⫺ Z0
v
⫺∞
…48†
…50†
when u2 =2 p hf and/or hN. The enthalpy of the fluid is given
as
hf ˆ
N
X
Yk hk ; hk ˆ h0k ⫹ c…T ⫺ T 0 †; k ˆ M; P
…51†
kˆ1
and that of the particles is
hN ˆ h0N ⫹ cN …T ⫺ T 0 †:
{YP ⫺ …1 ⫹ fst †}Z ⫹
as well as the relation between the heat of combustion and
enthalpies as
⫺fst hM ⫹ …1 ⫹ fst †hp ⫺ hN
ˆ
nP WP hP ⫺ nM WM hM ⫺ nN WN hN
nN WN
…56†
…l=c† dT~
ˆ …T~ ⫺ T~ 0 † ⫺ …Z ⫺ Z0 †;
…57†
m dx
where T~ is the nondimensional temperature ‰ˆ cT=q0 Š and
q 0 the heat of combustion per unit mass of N species. Here, it
is assumed that enthalpy of phase change is negligible
because it is usually much smaller than the heat of combustion, that thermal conductivity across the combustion wave
is constant, and that the specific heats are the same, that is
c ˆ cN :
Note that in the nonadiabatic condition, it is necessary to
take account of heat loss L in the lateral direction, so that the
energy conservation equation is expressed as
"
#
d …l=c† dT~
L
⫺ …T~ ⫺ T~ 0 † ⫹ …Z ⫺ Z0 † ˆ
:
…58†
dx
m dx
mq0
In closing this section, it should be mentioned that these
governing equations can simply be derived if it is a priori
assumed that N particles are the same size in counting for
the heat and mass balance around them [38,39]. Regardless,
here statistical counting procedure has been used, to show
the mathematical description in a general form, which is
applicable to systems in which particle size distribution is
not monodispersed.
4. Flame propagation in the adiabatic condition
Mathematical description [32,37] of the flame propagation
in the SHS process has been made by use of Eq. (46) for Nconsumption, Eq. (49) for M-conservation, and Eq. (57) 1 for
energy conservation, as well as the relation of constant mass
1
N
X
dT
⫹
hk rf Yk Uk ;
dx
kˆ1
rf YP UP
ˆ {YP;0 ⫺ …1 ⫹ fst †}Z0 ; …55†
m
…52†
As for the x component of heat flux vector, it is expressed as
qx ˆ ⫺l
…54†
Eq. (50) reduces to
where Y~ M is the stoichiometrically weighted mass fraction
‰ˆ YM =fst Š and the constant of integration is evaluated at the
unburned state …x ! ⫺∞†: Note that the use of the Arrhenius mass-diffusivity defined as D ˆ D0 exp…⫺Td =T† is
indispensable for the appropriate description of the SHS
process.
The momentum conservation equation has already been
expressed in Eq. (34), which states that constant pressure is a
good assumption when the velocity is low and the velocity
gradient is small.
The energy conservation equation (37) in the adiabatic
condition becomes
qx
ˆ Z0 hf;0 ⫹ …1 ⫺ Z0 †hN;0 ;
m
rf YM UM
ˆ …YM;0 ⫹ fst †Z0 ;
m
ˆ ⫺q0 ⫹ …c ⫺ cN †…T ⫺ T 0 †;
derived by the integration of Eq. (29).
The species conservation equation (32) for M species is
given as
rf D dY~ M
…49†
ˆ …Y~ M ⫹ 1†Z ⫺ …Y~ M;0 ⫹ 1†Z0 ;
m
dx
Zhf ⫹ …1 ⫺ Z†hN ⫹
…YM ⫹ fst †Z ⫹
…47†
and
rt u
r t;0 u0
15
…53†
when there is no radiation heat transfer. Then, by the use of
It should be noted that the energy conservation equation first
reported [32] was incorrect, due to inappropriate treatment in
accounting for a term representing the sensible heat of N particles.
This was pointed out and corrected in the subsequent paper. [37]
Figs. to be presented in this review have already been corrected by
use of the energy conservation Eq. (57) although quantitative trends
are the same.
16
A. Makino / Progress in Energy and Combustion Science 27 (2001) 1–74
burning rate expressed in Eq. (42). The combustion situation
modeled, as shown in Fig. 4, is that of the steady, adiabatic,
one-dimensional, planar, heterogeneous flame propagation in
an infinite domain of a compacted medium, originally consisting of a mixture of particles of nonmetal N, metal M, and an
inert which can be the product P of the reaction according to
vM M ⫹ vN N !vp P: Then the flame propagation can be examined by solving these governing equations, subject to boundary
conditions at the unburned and burned states.
4.1. Dominant parameters
Before describing numerical results, let us first reconfirm
dominant parameters, because it is important in analyses to use
parameters which can fairly represent system conditions for
the initial and final states of the SHS process, as well as their
clear definitions from the physical point of view. Furthermore,
in order to make comparisons between theoretical and experimental results, definite relations between parameters used in
analyses and those in experiments are strongly required.
At the burned state, we have
T~ ∞ ˆ Z∞ ⫺ Z0 ⫹ T~ 0 ;
…59†
Z
Y~ M;∞ ˆ …Y~ M;0 ⫹ 1† 0 ⫺ 1;
Z∞
…60†
which are obtained, respectively, from the energy conservation equation (57) and the species conservation equation
(49) at a completely reacted state. We see that both T~ ∞
and Y~ M∞ are expressed in terms of not only their initial
values, but also the mass fractions of fluid, Z, at the burned
and unburned states. As for Z∞, by counting for the mass
fraction of the the remaining N particles after the completion
of the reaction, we have
8
…m ⱕ 1†
>
<1
Z∞ ˆ
;
…61†
1 ⫺ Z0
>
…m ⱖ 1†
: Z0 ⫹
m
where m is the mixture ratio defined as the initial molar ratio
of nonmetal, N, to metal, M, divided by the corresponding
stoichiometric molar ratio. Note that it is common in experiments to specify the nature of the initial (unburned)
compact by this mixture ratio m and the degree of dilution
k , defined as the initial mass fraction of the diluent. These
experimental parameters are related to the initial mass
fraction of N particles, 1 ⫺ Z0, used in analyses, as [37]
1 ⫺ Z0 ˆ
m…1 ⫺ k†
:
m ⫹ fst
4
rN pR30 n0
1
3
mˆ
4 3
…nN WN †=…vM WM †
r YM;0 1 ⫺ pR0 n0
3
ˆ
…64†
4
r YP;0 1 ⫺ pR30 n0
3
kˆ ˆ …1 ⫺ YM;0 †Z0 ; …65†
4 3
4
r 1 ⫺ pR0 n0 ⫹ rN pR30 n0
3
3
where r is the fluid density that consists of metal M and inert
P. It is seen that by specifying m and k , both the mass
fraction of N particles, 1 ⫺ Z0 ; and the mass fraction YM,0
of M species in fluid at the unburned state can be specified.
It may be noted that the present consideration of dilution,
through which the mass fraction of molten M is reduced due
to addition of combustion product, implicitly assumes that
the product is soluble in the liquid metal, and that the dissolution process occurs much faster than the reaction of N
particles. The first requirement depends on the specific
metal/nonmetal (or metal/metal) system, and can be
assessed from its phase diagram, while the second requirement is favored for fast rates of interfacial dissolution and
dispersion through mass diffusion, especially for small
product particles. For example, in the Ti–C system, TiC
melts in the range 1918–3340 K, depending on composition. Since Ti melts at about 1950 K, and the mixture
temperature increases due to reaction, TiC in the combustion wave is expected to exist in the fluid phase.
As another dominant parameter, we can point out the
initial temperature T0. The effect of T0, which determines
T∞ through Eq. (59), may be significant because T∞ can
exponentially influence the reaction rate and mass diffusivity in the condensed phase. It should be noted that T0 is the
temperature of the unburned medium, just before the arrival
of the flame front of the combustion wave.
4.2. Burning velocity
Because the spatial coordinate x appears in none of these
equations, except as d/dx, the problem can further be
reduced to having Z instead of x as the independent coordinate, for adiabatic flame propagation. Dividing Eqs. (49)
and (57) by Eq. (46), and defining the nondimentional
variables as
…62†
This relation can be derived from the following expressions
for their definitions [32]:
4
r 1 ⫺ pR30 n0
3
Z0 ˆ ;
…63†
4 3
4
r 1 ⫺ pR0 n0 ⫹ rN pR30 n0
3
3
fst 1 ⫺ Z0
;
YM;0 Z0
uˆ
T~ ⫺ T~ 0
;
T~ ∞ ⫺ T~ 0
zˆ
Y~ M ⫺ Y~ M;0
;
Y~ M;∞ ⫺ Y~ M;0
jˆ
Z ⫺ Z0
;
Z∞ ⫺ Z0
…66†
we have two first-order differential equations as [32]
!
L …r =r †
du
T~ d …u ⫺ j†
ˆ 0 t;0 1=3t exp
;
…67†
dj
…1 ⫺ j† x~
T~ …1 ⫺ Z 0 †
A. Makino / Progress in Energy and Combustion Science 27 (2001) 1–74
dz
Le 0 L0 …rt;0 =rt † 2
2T~ d
ˆ
exp
1=3
dj
{Z0 ⫹ …1 ⫺ Z0 †j}…1 ⫺ j† x~
T~
Z∞
Z0z
z⫺
;
j⫹
Z∞ ⫺ Z0
1 ⫺ Z0
!
…68†
when the mixture ratio is smaller than or equal to unity.
When the mixture ratio is larger than unity, we have [40]
!
L 0 …rt;0 =rt †
du
…u ⫺ j†
T~ d
;
ˆ 1=3 exp
dj
Z∞ ⫺ Z0
T~ …Z ∞ ⫺ Z0 †
1⫺
j
x~
1 ⫺ Z0
…69†
dz
ˆ
dj
Le 0 L0 …rt;0 =rt † 2
1=3
Z ∞ ⫺ Z0
{Z0 ⫹ …Z∞ ⫺ Z0 †j} 1 ⫺
j
x~
1 ⫺ Z0
!
2T~ d
Z∞
Z0z
:
exp
z⫺
j⫹
Z∞ ⫺ Z0
Z∞ ⫺ Z0
T~
…70†
In the above, L 0 is the mass burning rate eigenvalue defined
as
L0 ˆ
…Z∞ ⫺ Z0 † 2 m2a
;
4p…rM D0 †…l=c†n0 R0
…71†
Le0 the Lewis number as
Le0 ˆ
…l=c†
;
…rt D†0
…72†
and x~ the surface regression rate as expressed in Eq. (9). In
Eq. (71), ma is the total mass burning rate in the adiabatic
condition. The boundary conditions are
j ˆ 0;
u ˆ um;
j ˆ 1;
u ˆ 1;
z ˆ 0;
z ˆ 1:
…73†
…74†
Note that the cold boundary difficulty [34] can be eliminated
by assuming that the reaction is initiated at the melting point
Tm of the metal [32].
Therefore, it is seen that the problem is reduced to solving
the mass burning rate ma from the two first-order equations
of Eq. (67) and (68) for under-stoichiometric mixture ratios,
or Eqs. (69) and (70) for over-stoichiometric mixture ratios,
satisfying the four boundary conditions in Eqs. (73) and
(74). For spray combustion, the surface regression rate x~
is independent of the droplet size and L 0 becomes the eigenvalue. Since n 0 R03 is fixed for a given stoichiometry, L 0
implies that ma ⬇ R⫺1
0 : For the present problem, the surface
Damko¨hler number A, and the surface regression rate x~ ;
depend on the particle radius R because of the existence
of the surface reaction; the particle radius varies as R ˆ
R0 …1 ⫺ j† 1=3 : Then the dependence of ma on R0 is more
complex. In the limit of small R0, x~ ⬃ A ⬃ R 0 ; that is L 0/
R0 becomes the eigenvalue such that ma ⬃ R⫺1=2
: In the
0
17
diffusion-controlled limit, x~ becomes independent of R0
and we have ma ⬃ R⫺1
0 :
Once the specific value of L 0 is obtained by a numerical
calculation, the burning velocity is then obtained as [40]
p s
1 ⫺ Z0
rM =rN
;
…75†
…u0 ·R0 † ˆ
…D0 3L0 Le0 †
Z∞ ⫺ Z0
1 ⫺ Z0
which is simply expressed as [32,37]
s
p rM =rN
…u0 ·R0 † ˆ D0 3L0 Le0
;
1 ⫺ Z0
…76†
when the mixture ratio is under-stoichiometric. We see that
the burning velocity is inversely proportional to the particle
size. The parameter u0·R0 is sometimes called the “SHS rateconstant”. Note that in the experiments the burning velocity
u0 should be determined as the velocity at which the
dazzling combustion wave moves normal to its surface
through the adjacent unburned compacted medium.
As for the effect of the relative density r rel, which is also
one of the important system parameters in experiments and
is defined as the ratio of the apparent density of a compacted
medium to the theoretical maximum density r TMD, its influence on the burning velocity is anticipated to appear through
the thermophysical properties in the Lewis number Le0 ‰ˆ
…l=c†=…rt D†0 Š which appears in Eqs. (75) or (76). Since the
total density r t is expressed as rt ˆ rrel ·rTMD and the
specific heat c is proportional to r rel, while the thermal
conductivity is a weak function of r rel, the burning velocity
is considered to be inversely proportional to the relative
density r rel, as the first approximation for the usual range
of r rel in the SHS process.
4.3. Range of flammability
The regime in which a solution of the present problem can
exist is anticipated to coincide with the range of flammability. If we consider the general restriction that the temperature of the completely reacted state should be higher than the
melting point …T∞ ⱖ Tm † [32,37], we have
fst
! ⱕ m ⱕ 1;
1⫺k
⫺1
T~ m ⫺ T~ 0
f
0 ⱕ k ⱕ 1 ⫺ 1 ⫹ st …T~ m ⫺ T~ 0 †;
m
…77†
…78†
for the under-stoichiometric mixture ratios;
1ⱕmⱕ
1⫺k
⫺ fst ;
T~ m ⫺ T~ 0
0 ⱕ k ⱕ 1 ⫺ …m ⫹ fst †…T~ m ⫺ T~ 0 †;
for the over-stoichiometric mixture ratios.
…79†
…80†
18
A. Makino / Progress in Energy and Combustion Science 27 (2001) 1–74
Fig. 5. Representative flame structure for the temperature u , the carbon concentration …1 ⫺ j†; and the reaction rate dj /ds , along the
nondimensional coordinate s , for the stoichiometric mixture …m ˆ 1:0† without dilution …k ˆ 0† when the initial temperature T 0 ˆ 450 K;
the Lewis number Le0 ˆ 100; the initial Damko¨hler number Da0 ˆ 10 9 ; and the adiabatic mass burning rate eigenvalue L0 ˆ 6:109 × 10⫺5 :
Dashed curves are the results in the adiabatic condition [32] and solid curves are those in the nonadiabatic condition [64].
4.4. Theoretical and experimental results for Ti–C system
Eqs. (67) and (68) have been numerical integrated to
determine the mass burning rate eigenvalue for Ti–C system
[32]. Values of the physicochemical parameters are: q ˆ
15 MJ=kg; c ˆ 1 kJ=…kg:K†; rM ˆ 4:50 × 103 kg=m3 ; rN ˆ
2:25 × 103 kg=m3 ; WN ˆ 47:9 × 10⫺3 kg=mol; WM ˆ 12:0 ×
10⫺3 kg=mol; Td ˆ 1973 K; Ta ˆ 3 × 104 K: For the mass
diffusivity D ˆ D0 exp…⫺1:66 × 104 =T† m2 =s is used,
following Hardt and Phung [29]. Since apparent thermometric conductivity l /(r tc) is of the order of 10 ⫺5 m 2/s at
3000 K [41], the Lewis number Le0 is estimated to be 50 or
more. Total densities before and after combustion are
assumed to be equal. Note that the thermophysical properties used here implicitly account for effects of compact
density, gases in void spaces, gas evolution, etc.
Fig. 5 shows a representative flame structure with profiles
of temperature u , carbon mass fraction, 1 ⫺ j; and the particle consumption rate dj /ds , along the nondimensional
distance s ˆ ‰m a …1 ⫺ Z0 †=…l=c†Šx: The dashed curves represent the results in the adiabatic condition, while solid
curves are those of the nonadiabatic condition, to be
discussed in Section 5. Position at which u ˆ u m is set at
s ˆ 0: With the monotonic increase in the temperature u
from the melting point u m to the adiabatic combustion
temperature …u ˆ 1†; the mass fraction of carbon 1 ⫺ j
rapidly decreases from 1 to 0. The particle consumption
rate dj /ds exhibits the initial increase due to carbon depletion and consequent decrease due to reactants depletion, as
R ! 0: Its profile spans approximately the same range over
which the temperature increases. Compared to the typical
premixed flame structure, characterized by a narrow, highly
peaked reaction rate profile located close to the maximum
temperature, the present result demonstrates the diffusive
nature of the heterogeneous flame structure supported by
particle reaction, so that the region of the particle consumption in the combustion wave can be called the “consumption
zone.” For the prescribed conditions with R 0 ˆ 10 mm; the
flame thickness and burning velocity are predicted to be
about 1 mm and 10 mm/s, respectively.
Fig. 6 shows mass burning rate eigenvalue L 0 as a function of the initial surface Damko¨hler number Da0 ‰ˆ
B·R0 =D0 Š for an initial temperature T~ 0 ˆ 0:03 …T0 ˆ
450 K†; with the Lewis number Le0 and the mixture ratio
m taken as parameters. With increasing Da0, L 0 initially
increases and attains a constant value for Da0 ⬎ 10 5.
This implies that for higher values of Da0, the reaction,
and hence the mass burning rate, becomes diffusion
limited, herein A=…1 ⫹ A† ! 1 in x~ which appears in
Eqs. (67) and (68). Since Da0 is closely related to B·R0,
its value is also shown in the abscissa. Fig. 6 further
shows that due to the very large value of Le0, L 0 is insensitive to Le0. It is, however, significantly increased with
increasing m because of the increased amount of
volumetric heat generation.
Since L 0 is basically independent of Le0, in order to
determine appropriate values of B and Le0, numerical results
are compared with the experimental results [33] for various
radii (from 0.5 to 12.5 mm) of specimen and it is confirmed
[32] that close correlation can be obtained for Le0 ˆ 100
and B ⱖ 10 3 m=s when the relative density r rel is about 0.6.
Note that the value of B ⱖ 103 m=s implies that the combustion is diffusion-controlled. Fig. 7 shows the effect of the
initial radius R0 of carbon particle on the burning velocity u0
of the stoichiometric mixture …m ˆ 1† for various B. The
comparison between the predicted and experimental results
[26–28,33,42] demonstrates the importance of heterogeneity in the flame propagation process, and shows the fact
A. Makino / Progress in Energy and Combustion Science 27 (2001) 1–74
Fig. 6. Mass burning rate eigenvalue L 0 for Ti–C system as a
function of the initial surface Damko¨hler number Da0 for the
nonmetal particle, with the Lewis number Le0 and the mixture
ratio m taken as parameters [32]; the initial temperature T~ 0 ˆ 0:03
(T0 ˆ 450 K).
Fig. 7. Burning velocity u0 for Ti–C system as a function of the
initial radius R0 of carbon particle, with the frequency factor B taken
as a parameter [32]; the Lewis number Le0 ˆ 100; the initial
temperature T~ 0 ˆ 0:03 (T0 ˆ 450 K), and the mixture ratio m ˆ
1:0: Data points are experimental [26–28,33,42]; solid curves are
calculated from theory.
19
that u0 is inversely proportional to R0 as shown in Eqs. (75)
or (76). The theory breaks down as R0 ! 0; and ma ! ∞;
which corresponds to the theoretical fact that the flame
propagation mode should become homogeneous and the
combustion be reaction-controlled (cf. Fig. 6). Experimental
observation [33], however, shows that there occurs a transition from the steady, one-dimensional flame propagation
to the so-called pulsating combustion [1–3] when the particle radius is less than about 2.5 mm.
Fig. 8 shows u0 as a function of the mixture ratio m , with
R0 taken as a parameter. Comparison between the calculations and experimental results [33] is again satisfactory. The
lower limit of flammability given by Eq. (77) is 0.448, which
is close to the lowest m of about 0.47 obtained experimentally [33]. Calculations have also been performed with a
constant diffusivity of D ˆ 10⫺9 m2 =s; which is representative of its value in the consumption zone. Results for the
constant diffusivity are more “curved” than the somewhat
linear variations shown in both the experimental data and
calculated results with Arrhenius mass-diffusivity. This
therefore demonstrates the need [22] to realistically include
the temperature-dependent mass diffusivity, without which
u0 becomes large because of the overestimated mass diffusivity at relatively low temperatures.
As for the dependence of u0 on m in its wide range, Fig. 9
shows the SHS rate-constant u0,a·R0 as a function of 2m=…1 ⫹
m† for T0 ˆ 450 K; with the degree of dilution k taken as a
Fig. 8. Burning velocity u0 for Ti–C system as a function of the
mixture ratio m , with the initial particle radius R0 and the frequency
factor B taken as parameters [32]; the Lewis number Le0 ˆ 100 and
the initial temperature T~ 0 ˆ 0:03 (T0 ˆ 450 K). Data points are
experimental [33]; solid curves are calculated with Arrhenius diffusivity; dashed curves are calculated with representative diffusivity.
20
A. Makino / Progress in Energy and Combustion Science 27 (2001) 1–74
Fig. 9. Burning velocity multiplied by particle radius (SHS rate-constant) u0·R0 for Ti–C system as a function of 2m=…1 ⫹ m†; with the degree of
dilution k taken as a parameter [40]; the initial temperature T0 ˆ 450 K: Data points are experimental [33,43] for different particle radii of
R0 ˆ 5 mm and 12.5 mm.
parameter. Introduction of 2m=…1 ⫹ m†; which is a harmonic
mean of m and 1=m; enables us to discuss the effect of m ,
which can change from zero to infinity, in a finite domain.
With increasing 2m=…1 ⫹ m†; the SHS rate-constant u0,a·R0
first increases, reaches the maximum, and then decreases.
The increase in u0,a·R0 is attributed to an increase in the
volumetric heat generation, as well as an increase in the
heat generation rate in the combustion wave, related to the
total surface area of N particles. The peak of u0,a·R0 occurs
on the rich side …m ⬎ 1† of N particles, because of the
increase in the total surface area of N particles, in spite of
the decrease in volumetric heat generation. Further increase
in 2m=…1 ⫹ m† results in a decrease in u0,a·R0, because of the
reduced volumetric heat generation. It is also observed that
u0,a·R0 decrease with increasing k , and that dilution moves
the position of the maximum u0,a·R0 towards the stoichiometric mixture. Experimental data [33,43] for two different
particle sizes are also shown in Fig. 9 and fair agreement is
observed as far as the general trend and the approximate
magnitude are concerned. It is pointed out, however, that
discrepancies near the maximum u0,a·R0 may be attributed to
the assumption of constant thermophysical properties
throughout the combustion wave. Overestimation of the
range of flammability by Eqs. (77) and (79) is attributed to
the heat loss to be discussed in Section 5.
Fig. 10 shows u0 as a function of the initial temperature
T0, for m ˆ 0:6 and 1.0. The trend of increasing u0 with
increasing T0 is as expected, as the preheating effect. The
comparison with experimental data [33,44] is again satisfactory.
Dilution by the final product (TiC), to control the burning
velocity and/or combustion temperature has also been investigated; in the field of material processing, this corresponds
to controlling grain size in materials through control of the
maximum temperature. Fig. 11 shows u0 as a function of k
for m ˆ 1:0; with T0 taken as a parameter. It is seen that u0
gradually decreases until k reaches the dilution limit given
by Eq. (78), due to reduced volumetric heat generation.
Experimental results shown in Fig. 11 are those by
Bloshenko et al. [44] at T0 ˆ 973 K and those by Makino
et al. [33] at T0 ˆ 450 K; the experimental results
[26,45,46] with carbon particles of 0.1 mm in diameter are
excluded because steady propagation cannot be expected.
Fig. 12 shows u0 as a function of the relative density r rel
for m ˆ 1:0; with 2R0 taken as a parameter; u0 gradually
decreases when 2R0 is larger than 10 mm and the steady
Fig. 10. Burning velocity u0 for Ti–C system as a function of the
initial temperature T0, with the mixture ratio m and the frequency
factor B taken as parameters [32]; the Lewis number Le0 ˆ 100 and
the initial radius R0 ˆ 10 mm: Data points are experimental [33,44];
solid curves are calculated from theory.
A. Makino / Progress in Energy and Combustion Science 27 (2001) 1–74
21
densities it is caused by reduced temperature in the combustion wave due to enhanced heat transfer ahead of the
combustion wave.
4.5. Applicability of the heterogeneous theory for other
systems
Agreement with a single system is not adequate to establish
the validity of the heterogeneous theory, especially since this
theory represents a fundamentally different description of the
SHS process. To this aim, the heterogeneous theory has been
applied to compare with experimental results obtained from
the syntheses of several borides and intermetallic compounds
for which reliable experimental data are available. It is attributed that the viability of the theory will be demonstrated
through extensive, quantitative comparisons in the range of
flammability with respect to effects of mixture ratio, degree
of dilution, initial temperature, and/or particle size on the burning velocity.
Fig. 11. Burning velocity u0 for Ti–C system as a function of the
degree of dilution k , with the initial temperature T0 and the
frequency factor B taken as parameters [32]; the Lewis number
Le0 ˆ 100 and the mixture ratio m ˆ 1:0: Data points are experimental [33,44]; solid curves are calculated from theory.
propagation is maintained. When the particle size is small,
pulsating combustion occurs and the trend is quite different
from that of the steady propagation. It is reported [26] that
the reduction of the burning velocity is caused by the
reduced contact area between the molten metal and solid
particles for low relative densities, as well as the reduced
volumetric heat generation, whereas for high relative
4.5.1. Available experimental data
Borides and intermetallic compounds have been chosen
[47] because of the practical interest for advanced applications. For example, the boride ceramics of titanium diboride
(TiB2), zirconium diboride (ZrB2), and hafnium diboride
(HfB2), due to their refractory nature and hardness, can be
used as heating elements for high-temperature electricresistance furnace, as components and reinforcements for
furnaces in metal industry, and as cutting tools and abrasives. Further, TiB2 is considered an important component
in the fabrication of Functionally Graded Materials (FGMs),
for use as heat-shielding/structural materials in future space
applications.
Alloys based on the intermetallic compound of nickel
aluminide (NiAl) possess an attractive combination of
Fig. 12. Burning velocity u0 for Ti–C system as a function of the relative density r rel, with particle diameter 2R0 taken as parameters; the Lewis
number Le0 ˆ 100 and the mixture ratio m ˆ 1:0: Data points are experimental [26,33]; open symbol designates the steady propagation and
half-open symbol the pulsating combustion.
A. Makino / Progress in Energy and Combustion Science 27 (2001) 1–74
3:00 × 104
3:83 × 104
3:73 × 104
4:79 × 104
1:68 × 104
4:28 × 104
5:61 × 104
Azatyan et al. [46]
Borovinskaya et al. [51]
Borovinskaya et al. [51]
Borovinskaya et al. [51]
Naiborodenko and Itin [58]
obtained with data from Itin et al. [53]
obtained with data from Itin et al. [53]
mechanical and physical properties, such as high strength,
low density, and high thermal conductivity, that make them
ideal candidates for a number of high-temperature applications in aviation, automobile, and nuclear power engineering.
The intermetallic compounds such as titanium cobaltide
(TiCo) and titanium nickelide (TiNi) are of interest because
of their shape-memory properties. It is even suggested [5]
that formation of these compounds by the SHS process has
distinct advantages over other conventional methods in
terms of stronger shape-restoring force and enhanced homogeneity.
The values of physicochemical parameters for these
systems are listed in Table 2. Some of the transport parameters listed in Table 3 are those in the solid phase of
the metal near the melting point, because of very limited
published data for the mass diffusivity. For the activation
energies, it has been found that they have nearly the same
values as those obtained from the Arrhenius plot of (u/T) vs.
(1/T) at high temperatures in which mass diffusivity can be
expected to influence the burning velocity, where u is the
burning velocity and T the maximum temperature. The thermometric conductivity l=…rt c† are those at the melting point
of the lower melting-point metal [48]. Representative values
of the Lewis number Le0 are also listed.
In order to apply the heterogeneous theory [32], which
allows for the presence of finite-rate surface reactions,
kinetic parameters for the individual surface reactions are
also required. However, because of the high temperature in
the consumption zone in the SHS process, for instance,
higher than about 2000 K for boride synthesis and 1000 K
for the intermetallic compounds, the reaction is likely to be
fully activated, such that the combustion behavior is
expected to proceed in the diffusion-controlled regime.
Satisfactory agreement in the subsequent comparisons
substantiates this assumption.
a
b
M-species represents metal or lower melting-point metal.
N-species represents nonmetal or higher melting-point metal.
1973
1973
2125
2500
934
1767
1728
12:0 × 10⫺3
10:8 × 10⫺3
10:8 × 10⫺3
10:8 × 10⫺3
58:7 × 10⫺3
47:9 × 10⫺3
47:9 × 10⫺3
47:9 × 10⫺3
47:9 × 10⫺3
91:2 × 10⫺3
178:5 × 10⫺3
27:0 × 10⫺3
58:9 × 10⫺3
58:7 × 10⫺3
2:25 × 103
2:35 × 103
2:35 × 103
2:35 × 103
8:85 × 103
4:50 × 103
4:50 × 103
15.0
12.9
14.9
16.6
2.0
2.6
2.9
Ti ⫹ C ! TiC
Ti ⫹ 2B ! TiB2
Zr ⫹ 2B ! ZrB2
Hf ⫹ 2B ! HfB2
Al ⫹ Ni ! NiAl
Co ⫹ Ti ! TiCo
Ni ⫹ Ti ! TiNi
1.00
0.99
0.46
0.45
1.18
0.55
0.52
4:50 × 103
4:50 × 103
6:53 × 103
13:09 × 103
2:69 × 103
8:90 × 103
8:85 × 103
Mol. wt. of
N-species
WN (kg/mol)
Mol. wt. of
M-species
WM (kg/mol)
Density of
N-species
r N (kg/m 3) b
Density of
M-species
r M (kg/m 3) a
Specific heat
c (kJ/kg ·K)
Heat of
combustion
q 0 (MJ/kg)
Reaction
nM M ⫹ nN N ! nP P
Table 2
List of physicochemical parameters [47]
Melting point
of M-species
Tm (K)
Activation
temperature of
reaction Ta (K)
Remarks with respect to Ta
22
4.5.2. Effects of mixture ratio, m
Fig. 13(a) shows the burning velocity u0 for Ti–B system
as a function of stoichiometry, with initial temperature T0
taken as a parameter. It has been found that flame propagation becomes diffusion-controlled when the frequency factor
B exceeds 10 4 m/s. The particle radius is set to R0 ˆ 0:5 mm
in accordance with experimental data; some of the experiments [45,49,50] were conducted with 1 mm boron particle
in diameter, others [51,52] with 0.1 mm boron particle. The
comparison between predicted and experimental results is
considered to be satisfactory, hence demonstrating the
importance of stoichiometry in the flame propagation
process. The trend of decreasing u0 with decreasing m is
as expected, because of reduced volumetric heat generation,
and the lower limits of flammability as given by Eq. (77)
fairly agree with those of the experiments. As for the
preheating effect, it is found to be relatively small although
u0 increases with increasing T0 at a certain stoichiometry.
Further, since there is no discernible difference in the
A. Makino / Progress in Energy and Combustion Science 27 (2001) 1–74
burning velocities for different sizes of the boron particles, it
is pointed out that there can be agglomeration for small
particles in the experiments.
Fig. 13(a)–(d) shows similar comparisons for Zr–B,
[3,45,50–52] Hf–B, [51,52] and Co–Ti [53,54] systems.
The values of R0 used are indicated in the respective figures.
The comparison between predicted and experimental
results, including the limits of flammability, can be considered to be satisfactory.
Hardt and Phung [29]
Hardt and Phung [29]
Hardt and Phung [29]
Samsonov and Vinitskii [62]
Naiborodenko and Itin [58]
Graham [63]
Graham [63]
Touloukian et al. [41] and Touloukian et al. [48]; values at the melting point of the M-species.
Touloukian et al. [41] and Touloukian et al. [69].
a
1:0 × 10⫺5
1:6 × 10⫺5
1:8 × 10⫺5
1:8 × 10⫺5
7:0 × 10⫺5
8:0 × 10⫺6
1:6 × 10⫺5
1:66 × 104
1:54 × 104
1:74 × 104
2:40 × 104
9:01 × 103
1:87 × 104
1:92 × 104
2:0 × 10⫺7
8:9 × 10⫺9
1:3 × 10⫺8
5:0 × 10⫺7
4:8 × 10⫺6
1:1 × 10⫺5
2:9 × 10⫺5
Ti–C
Ti–B
Zr–B
Hf–B
Al–Ni
Co–Ti
Ni–Ti
Activation
temperature Td (K)
Pre-exponential
factor D0 (m 2/s)
4.5.3. Effects of degree of dilution, k
Fig. 14(a) shows the burning velocity u0 for Ti–B system
as a function of k for m ˆ 1:0; with T0 taken as a parameter.
The experimental results are from Refs. [45,51,52,55,56]. It
is seen that the burning velocity gradually decreases until k
reaches the dilution limit given by Eq. (78), due to reduced
volumetric heat generation, as well as the reduced mass
diffusivity caused by the reduced temperature in the
consumption zone. Fig. 14(b)–(e) shows similar comparisons for Zr–B [3,45,50–52], Hf–B [50–52], Al–Ni [57],
and Co–Ti [53] systems. Fair agreement is demonstrated
as far as the trend and approximate magnitude are
concerned.
b
0.4
0.6
0.7
0.7
0.8
0.8
0.8
100
1000
500
4
10
10
4
Remarks with respect to D0
and Td
Emissivity, e b
Representative Lewis
number Le0
Thermometric conductivity
l /(r t c) (m 2/s) a
Diffusivity
System
Table 3
List of transport parameters [47,66]
23
4.5.4. Effects of initial temperature, T0
Initial temperature T0 is also one of the controllable parameters in experimentation, and its effects on the burning
velocity have been examined extensively. Fig. 15(a) shows
u0 for Al–Ni system as a function of T0, for stoichiometric
mixture …m ˆ 1:0†: Values of B in the diffusion-controlled
regime is 10 5 m/s and more. The particle radius is set to
R0 ˆ 10 mm in accordance with the experimental data;
Hardt and Holsinger [45] used the Ni particle of 6 mm in
diameter, Naiborodenko and Itin [58,59] used 20 mm Ni
particle, and Maslov et al. [57] used particle less than
45 mm in diameter; the relative density was about 0.4. The
trend of increasing u0 with increasing T0 is due to the
preheating effect, as expected. Fair agreement is demonstrated when T0 is lower than about 600 K. The deviation
at high temperatures can be attributed to the experimental
determination of T0; that is, the initial temperature T0, which
should be that before arrival of the combustion wave, could
be higher than the ambient temperature, which might be the
reported T0, because of the heat associated with the ignition
source. Fig. 15(b) shows similar comparison for Co–Ti
system. The particle radius is chosen to be R0 ˆ 30 mm:
Experimental results are from Ref. [53]. The trend is the
same as that of Al–Ni system.
4.5.5. Effects of particle radius, R0
Particle radius R0 is the important parameter which specifies the heterogeneity in the combustion wave, such that the
burning velocity can be controlled by varying the particle
size. Fig. 16(a) shows the effect of particle radius R0 on u0
for Co–Ti system when m ˆ 1:0 and T0 ˆ 573 K: The
comparison between predicted and experimental [53] results
24
A. Makino / Progress in Energy and Combustion Science 27 (2001) 1–74
Fig. 13. Burning velocity u0 as a function of the mixture ratio m , with the initial temperature T0 taken as parameters [47]; solid curves are
calculated from theory. (a) Ti–B system for Le0 ˆ 103 and R0 ˆ 0:5 mm; (W) data from Hardt and Holsinger [45]; (A) from Kirdyashkin et al.
[49]; ( ) from Zenin et al. [50]; (K) from Borovinskaya et al. [51]; (S) from Novikov et al. [52]. (b) Zr–B system for Le0 ˆ 500 and R0 ˆ
0:5 mm; (A) data from Merzhanov and Borovinskaya [3]; (W) from Hardt and Holsinger [45]; ( ) from Zenin et al. [50]; (K) from Borovinskaya
et al. [51]; (S) from Novikov et al. [52], (c) Hf–B system for Le0 ˆ 4 and R0 ˆ 0:5 mm; (K) data from Borovinskaya et al. [51]; (S) from
Novikov et al. [52]. (d) Co–Ti system for Le0 ˆ 10; R0 ˆ 30 mm and 50 mm; ( ) data from Itin et al. [53]; (A, W) from Itin et al. [54].
Fig. 14. Burning velocity u0 as as a function of the degree of dilution k , with the initial temperature T0 taken as a parameter [47]; stoichiometric
mixture …m ˆ 1:0†; solid curves are calculated from theory. (a) Ti–B system for Le0 ˆ 103 and R0 ˆ 0:5 mm; (W) data from Hardt and
Holsinger [45]; (K) from Borovinskaya et al. [51]; (S) from Novikov et al. [52]; ( ) from Shcherbakov and Pityulin [55]; (L) Maksimov
et al. [56]. (b) Zr–B system for Le0 ˆ 500 and R0 ˆ 0:5 mm; (A) data from Merzhanov and Borovinskaya [3]; (W) from Hardt and Holsinger
[45]; ( ) from Zenin et al. [50]; (K) from Borovinskaya et al. [51]; (S) from Novikov et al. [52]. (c) Hf–B system for Le0 ˆ 4 and R0 ˆ
0:5 mm; ( ) from Zenin et al. [50]; (K) from Borovinskaya et al. [51]; (S) from Novikov et al. [52]. (d) Al–Ni system for Le0 ˆ 10 and R0 ˆ
10 mm; (A, S) data from Maslov et al. [57] for T0 ˆ 298 K and 740 K. (e) Co–Ti system for Le0 ˆ 10 and R0 ˆ 30 mm; (W) data from Itin et al.
[53] for T0 ˆ 873 K:
A. Makino / Progress in Energy and Combustion Science 27 (2001) 1–74
25
26
A. Makino / Progress in Energy and Combustion Science 27 (2001) 1–74
Fig. 15. Burning velocity u0 as a function of the initial temperature
T0 [47]; stoichiometric mixture …m ˆ 1:0†; solid curves are calculated from theory. (a) Al–Ni system for Le0 ˆ 10 and R0 ˆ 10 mm;
(W) data from Hardt and Holsinger [45]; (A) from Maslov et al.
[57]; ( ) from Naiborodenko and Itin. [58,59]. (b) Co–Ti system
for Le0 ˆ 10 and R0 ˆ 30 mm; (W) data from Itin et al. [53] for k ˆ
0:0; ( ) from Itin et al. [53] for k ˆ 0:2:
can be considered to be satisfactory, hence demonstrating
the essentially heterogeneous nature of the flame propagation process. Specifically, u0 is inversely proportional to R0
when combustion is diffusion-controlled (say, B ⱖ
107 m=s†: Fig. 16(b) shows similar comparison for Ni–Ti
system [60,61] when m ˆ 1:0 and T0 ˆ 750 K: The inverse
dependence of u0 on R0 is again observed.
4.6. Approximate expression for the burning velocity
Mass burning rate eigenvalue L 0, the square root of
which is the nondimensional burning velocity, has been
obtained numerically [32,47] and hence presented
Fig. 16. Burning velocity u0 as a function of the initial particle
radius R0 [47]; stoichiometric mixture …m ˆ 1:0†; solid curves are
calculated from theory. (a) Co–Ti system for Le0 ˆ 10 and T0 ˆ
573 K; data from Itin et al. [53]; open symbol designates the steady
propagation and solid symbol the unsteady combustion. (b) Ni–Ti
system for Le0 ˆ 4 and T0 ˆ 750 K; (K) data from Bratchikov et al.
[60]; (W) from Itin et al. [61].
parametrically, by solving the two first-order equations of
(67) and (68) under the four boundary conditions in Eqs.
(73) and (74). An approximate analytical expression for the
burning velocity is also obtainable, which can explicitly
display the functional dependence of the burning velocity
on the various system parameters.
4.6.1. Derivation of the approximate expression for m ⱕ 1
To this aim, we should first note that for the usual SHS
process the diffusion-controlled reaction prevails, as
mentioned in the previous sections, and the Lewis number
is very large, say, Le ⬎ 104 in the consumption zone. By
A. Makino / Progress in Energy and Combustion Science 27 (2001) 1–74
assuming diffusion-controlled limit and infinite Lewis
number, we have
x~ ⬇ ln…1 ⫹ Y~ M †
z⬇
…81†
j
Z 0 ⫹ …1 ⫺ Z0 †j
…82†
respectively. The former is derived from Eq. (9) by setting
the reduced Damko¨hler number A to be infinity. The latter is
derived by setting the last part of the RHS in Eq. (68) to be
zero, because of the excessively slow rate of diffusion as
compared to thermometric conductivity and reaction rate.
Numerical results obtained by using Eqs. (81) and (82),
instead of Eqs. (9) and (68), are even found to be practically
indistinguishable.
By introducing these relations, Eq. (67) is expressed in
integral form as
!
Z1
T~ d
exp ⫺
du
T~
um
ˆ …1 ⫹ g†L0
rt;0 Z1
rt
0
…t ⫺ j†
dj
f⫺j
…1 ⫺ j† 1=3 ln 1 ⫹
j⫹g
(83)
where
1
fˆ ;
m
Z0
…mk ⫹ f †
gˆ
ˆ
1 ⫺ Z0
m…1 ⫺ k†
…84†
27
and the complementary error function
2 Z∞
exp…⫺t2 † dt
erfc…z† ˆ p
p z
…89†
In integrating the RHS of Eq. (83), an attempt [37] has been
made to express u with 1 ⫺ j and …1 ⫺ j 1=3 †; and it is
reported that by comparing the numerical results, the relation
…1 ⫺ u† ˆ
1 ⫺ um
…1 ⫺ j†
2
…90†
fairly represents the temperature profile at high temperatures
(cf. Fig. 17), which is needed for the fair evaluation of the
Arrhenius nature for the liquid-phase mass-diffusion, to
determine L 0. Note that the proportionality constant is the
arithmetic mean of the constants determined at j ˆ 0 and 1.
When Eq. (90) is substituted into Eq. (83) and the first two
terms of the following mathematical formula is used
ln…1 ⫹ z†
(
ˆ2
z
z3
z5
⫹
⫹
⫹…
3
2⫹z
3…2 ⫹ z†
5…2 ⫹ z†5
…91†
)
z2n⫺1
…
⫹
⫹
;
…2n ⫺ 1†…2 ⫹ z†2n⫺1
The integral Iu in the LHS of Eq. (83) is evaluated by
performing an asymptotic expansion for large values of
the Zeldovich number
bˆ
T~ d …T~ ∞ ⫺ T~ 0 †
T~ ∞2
…85†
and retaining up to the second term in the expansion. That is,
by using the relation
o
T~ d
T~ n
ˆ d 1 ⫹ a…1 ⫺ u† ⫹ a2 …1 ⫺ u†2 ⫹ …
T~
T~ ∞
"
#
b 3
1 2
⫹ a2 …1 ⫺ u† ⫹
⬇
;
a 4
2a
we have
p
p
3b
p exp ⫺
Iu ⬇
4a
2b a=b
1
1 ⫹ 2a…1 ⫺ um †
p
erfc p ⫺ erfc
2 a=b
2 a=b
…86†
…87†
where
aˆ
T~ ∞ ⫺ T~ 0
;
T~ ∞
…88†
Fig. 17. Temperature profile in the reaction zone, with the mixture
ratio m taken as a parameter [37]. Solid curves are numerical results
and dashed curves the assumed profiles to obtain the approximate
analytical expressions for the mass burning rate eigenvalue L 0.
28
A. Makino / Progress in Energy and Combustion Science 27 (2001) 1–74
p
Fig. 18. Nondimensional burning velocity L0 for Ti–C system [37]; (a) as a function of the mixture ratio m ; and (b) as a function of the degree
of dilution k . The initial temperature T0 is taken as a parameter. Solid curves are numerical results and dashed curves are those with analytical
expression.
initial temperature T0, the mixture ratio m , and the degree of
dilution k are operating parameters when the experimental
conditions are specified.
Keeping only the leading terms in Iu and Ij in Eqs. (87)
and (92), respectively, the functional dependence of L 0 on
the system parameters can be displayed as
the integral Ij in the RHS of Eq. (83) is expressed as
"
!
1 ⫹ um
1
…f ⫺ 1†
Ij ⬇
…f ⫹ g† 2 ⫺
⫹
5…f ⫹ g†
2a
2
(
!
p )
p ⫺1 2 ⫺ a
…1 ⫹ a† 3
3
p
ln
p
⫺
⫺ 2 3 tan
3
f
3a
…f ⫹ 2g ⫹ 1†
⫹
6b
(
! p )#
p ⫺1 2 ⫹ b
…b ⫺ 1†3
3
p
× ln
p ;
⫺
⫹ 2 3 tan
3
f ⫹ 2g
3b
…92†
where a3 ˆ f ⫺ 1; and b3 ˆ f ⫹ 2g ⫹ 1:
Then, from Eqs. (87) and (92), we have
r
Iu
I
L0 t;0 ˆ
ˆ …1 ⫺ Z 0 † u ;
rt
…1 ⫹ g†Ij
Ij
…93†
which is found to be surprisingly close to that obtained by a
numerical calculation. This good agreement further demonstrates the importance of the describing processes well at the
hot boundary, rather than the cold boundary (cf. Fig. 17).
Since Eq. (93) is relatively simple, its use could facilitate
analysis and interpretation.
4.6.2. General behavior
An explicit form of L 0 provides useful insight into the
effects of the reduced parameters (a , b , u m, f , and g ),
which depend on the system parameters (fst, Td, Tm, T0, m ,
and k ) used in the experiments. Among these parameters,
the stoichiometric mass ratio fst, the activation temperature
Td in the mass diffusivity, and the melting point Tm of the
lower melting-point metal are physicochemical parameters
when the two components of the mixture are specified. The
L0 ⬃
m 2 …1 ⫺ k†2 exp…⫺b=a†
…m ⫹ fst †‰…1 ⫹ fst † ⫺ k…1 ⫺ m†Šb…1 ⫹ um †
…94†
As far as m and k are concerned, L0 ⬃ m2 =…m ⫹ fst † for
2
k ˆ 0; and L0 ⬃ …1 ⫺ k†p
for m ˆ 1: That is, the nondimen
sional burning velocity L0 decreases with decreasing m
and 1 ⫺ k: Its quantitative dependence on m and 1 ⫺ k;
however, is more complicated because volumetric heat
generation, which is a function of m and 1 ⫺ k; also influences a and b through T∞. The effect of T0, which determines T∞, may also be significant because T∞ can
exponentially influence L 0 through a and b . Note that b
also depends on the choice of the mixture components.
When we compare L 0, Ti–C for Ti–C system with L 0, Ti–B
for Ti–B system, (L 0, Ti–B/L 0, Ti–C) 1/2 is about 4 for m ˆ
1; k ˆ 0; and T0 ˆ 450 K; because of the smaller value of
b for Ti–B system.
4.6.3. Comparisons with numerical results
Applicability of the analytical expression has been examined by comparing it with the numerical results [32,47] for
TiC, TiB2, ZrB2, HfB2, and TiCo systems in Sections 4.4 and
5.5, and a fairpagreement
has been reported in Ref. [37]. Fig.

18(a) shows L0 for Ti–C system as a function of m , with
T0 taken as a parameter. Solid and dashed curves are respectively numerical and analytical results.pExcept
near stoichio
metry …m ˆ 1†; both results show that L0 decreases until m
reaches the limit given by Eq. (77), and that they mostly
A. Makino / Progress in Energy and Combustion Science 27 (2001) 1–74
29
p
Fig. 19. L0 for Ti–B system. Notation is the same as that in Fig. 18.
agree well with each other. Although the trend in the analytical expression near m ˆ 1 is opposite to that of the numerical result, the quantitative comparison can be considered to
be fair (⬍7% error). This discrepancy is attributed to the
overestimated temperature profile given by Eq. (90), which
causes an overestimation of Ij , and hence an underestimation of L 0. Fig. 18(b) shows the similar comparison with
respect to k . The analytical expression again agrees well
with the numerical results, to be within 7%. Figs. 19 and
20 show similar comparisons for Ti–B and Co–Ti
systems. The extent of agreement is similar to that for
Ti–C system.
4.6.4. Approximate expression for m ⱖ 1
When the mixture ratio is over-stoichiometric, the
normalized concentration z of M-species is
z⬇
Z∞j
;
Z 0 ⫹ …Z∞ ⫺ Z0 †j
…95†
then, we have
r
I
fI u
L 0 t;0 ˆ …Z∞ ⫺ Z0 † u ˆ
;
rt
Ij
…1 ⫹ g†Ij
…96†
where the relation has been made use of in Eq. (61). The
integral Iu is the same as that in Eq. (87), while the integral
Ij is expressed as [40]
!
!
1 ⫹ um
f⫹g
Ij ⬇
2
f2
"
…3 ⫺ 5f† ⫺ 3…1 ⫺ f†a2
…f ⫹ g†
× 2…1 ⫺ a2 † ⫹
⫹
10…f ⫹ g†
3b
(
ln
2…f ⫹ g†
f ⫹ 2g
b⫺1
a⫹b
!3
p
⫺2a ⫹ b
p
⫺ 2 3 tan⫺1
3b
p
2⫹b
⫹2 3 tan⫺1 p
3b
!)#
p
Fig. 20. L0 for Co–Ti system. Notation is the same as that in Fig. 18.
:
!
(97)
30
A. Makino / Progress in Energy and Combustion Science 27 (2001) 1–74
p
Fig. 21. L0 for Ti–C system as a function of m when T0 ˆ 450 K; with k taken as a parameter. Solid curves are numerical results and dashed
curves are those with analytical expressions.
p
Dependence of L0 for Ti–C system on m in its wide range
is shown in Fig. 21, with k taken as a parameter. The extent
of agreement in m ⱖ 1 is similar to that in m ⱕ 1:
4.7. Some remarks on the flame propagation in the adiabatic
condition
Calculated results for several ceramics and intermetallic
compounds, such as Ti–C, Ti–B, Zr–B, Hf–B, Al–Ni, Co–
Ti, Ni–Ti systems, using consistent sets of physicochemical
parameters, respectively, show satisfactory agreement with
experimental results in the literature, if we use the heterogeneous theory in the adiabatic condition. An important
relation, that the burning velocity is inversely proportional
to the nonmetal particle size, has been discovered. In addition, these comparisons suggest that the nonmetal (or
higher melting-point metal) particles in these systems
react in the diffusion-controlled limit, indicating that finite
rate chemical reaction is not the controlling factor in the
propagation of the bulk flame. Furthermore, it has been
suggested that concentration of the molten metal is mainly
affected through consumption, because of the excessively
slow rate of diffusion compared to the thermometric
conductivity and reaction rate, due to excessively large
Lewis number in the combustion wave. These identifications further enable us to obtain an approximate expression
for burning velocity, which can show clear dependence of
the burning velocity on the dominant parameters.
5. Flame propagation in the nonadiabatic condition
The heterogeneous theory [32] described in Section 4 is
an adiabatic one, and hence is incapable of describing the
experimentally observed phenomena of flame extinction due
to heat loss from the high-temperature combustion process.
From a practical point of view, extinction is to be properly
described and hence controlled in the SHS process. Toward
this aim, it is required to extend the adiabatic theory to allow
for the volumetric heat loss, to properly describe flame
extinction and the various parameters that characterize it.
It will be emphasized here that not the reaction, but the
temperature sensitive Arrhenius liquid-phase mass-diffusion
is responsible for the flame extinction, because in the SHS
process, combustion is basically diffusion controlled.
5.1. Governing equations and boundary conditions
A one-dimensional model with heat loss has been used to
discover the dominant parameters which determine the limit
of steady flame propagation. Although heat loss perpendicular to the propagation direction might lead to the curvature of the combustion wave, a choice of this model
simplifies the formulation and hence meets the present
purpose. Therefore, regardless of the existence of heat
loss, the governing equations (except for the energy conservation equation) have the same form as those in the adiabatic condition, that is Eq. (46) for N-consumption and Eq.
(49) for M-conservation [64]. The energy conservation
equation is now not Eq. (57), but Eq. (58).
Defining the nondimensional variable and parameter
…Z ∞ ⫺ Z0 †ma
…l=c†L
…98†
sˆ
F…u† ˆ 2 0
x;
…l=c†
m a q …T~ ∞ ⫺ T~ 0 †;
in addition to those in Eqs. (66), (71) and (72), we have the
governing equations as
"
#
~ m Le 0 …rt;0 =rt † exp…T~ d =T†
dz
ˆ
ma
ds
Z0 ⫹ …Z∞ ⫺ Z0 †j
Z∞
Z0z
z⫺
;
…99†
j⫹
Z∞ ⫺ Z0
Z∞ ⫺ Z0
d2 u
m d
u⫺j
F…u†
ˆ
⫺
;
m a ds Z ∞ ⫺ Z0
ds 2
…Z ∞ ⫺ Z0 † 2
…100†
A. Makino / Progress in Energy and Combustion Science 27 (2001) 1–74
dj
ˆ
ds
"
x~
~
…m=m a †L0 …rt;0 =rt † exp…T~ d =T†
31
#
1=3
Z∞ ⫺ Z0
1⫺
j
:
1 ⫺ Z0
…101†
Eqs. (99)–(101), together with the mass burning rate m, are
to be solved subject to the boundary conditions
sˆ0:
s!∞:
u ˆ u m ; z ˆ 0; j ˆ 0;
…102†
du
F…u†
dz
dj
;
ˆ⫺
ˆ 0;
ˆ 0:
…m=m a †…Z∞ ⫺ Z0 † ds
ds
ds
…103†
In the same manner as that in the adiabatic flame propagation, the cold boundary difficulty is eliminated by specifying that the reaction is initiated at the melting point of M.
The downstream boundary condition for u , based on
convection-loss balance, requires a short domain of integration, and is preferred over the statement of T ˆ T 0 …u ˆ 0†;
which would require an excessively large domain of integration. If we further assume the heat loss to be Newtonian,
it is expressed as F…u† ˆ C·u; where C is called the heat
loss parameter.
5.2. Characteristics in the nonadiabatic condition
Values of the physicochemical parameters employed are
those for Ti–C system and the Lewis number Le0 is set to be
100 [32] as described in Section 4.4. Representative flame
structure for the stoichiometric mixture …m ˆ 1† without
dilution …k ˆ 0† has already been shown in Fig. 5 by solid
curves, when the nonadiabatic situation is close to the
extinction state. In contrast to the adiabatic case in which
temperature increases monotonically to the adiabatic flame
temperature, the temperature gradually decreases to the
ambient value after the maximum flame temperature.
Decrease in the temperature u reduces the particle consumption rate dj /ds and hence the normalized burning-rate m/ma.
The reactant consumption zone is then broadened due to
heat loss.
By obtaining profiles of the four terms constituting Eq.
(100), which, respectively, represent diffusion d 2u /ds 2,
convection ⫺‰…m=ma †=…Z∞ ⫺ Z0 †Š…du=ds†; consumption
‰…m=ma †=…Z∞ ⫺ Z0 †Š…dj=ds†; and heat loss ⫺F…u†=…Z ∞ ⫺
Z0 †2 ; a further examination has been made to identify the
flame structure for the nonadiabatic case in Fig. 5. As shown
in Fig. 22, the combustion wave consists of a relatively thin
zone of diffusion and consumption/convection, which is
followed by a much thicker zone of convection and heat
loss. The existence of an extended zone of downstream
heat loss is similar to the nonadiabatic homogeneous
gaseous flame.
Fig. 23 shows the variation of the normalized mass burning rate m/ma,st as a function of the normalized heat loss
C ⴱ ‰ˆ …m a =ma;st †2 ·CŠ with m and k taken as parameters.
Fig. 22. The representative profiles of the diffusion, consumption,
convection, and loss terms in the energy conservation equation,
along the nondimensional distance [64]. Conditions are the same
as those in the nonadiabatic situation in Fig. 5; m=ma ˆ 0:6 and C ˆ
3:347 × 10 ⫺2 :
Here, ma,st is the mass burning rate for the stoichiometric
mixture in the adiabatic condition. Results exhibit the characteristic extinction turning point behavior, with the upper
branch being the stable solution and the turning point designating the state of extinction. With decreasing m or increasing k , the mass burning rate is gradually reduced, until
extinction occurs. Since not only m/ma,st but also (ma/
ma,st) 2·C strongly depends on the mass burning rate m, the
parameters m/ma and C are usually used to obtain the
extinction condition; the values of (m/ma)cr corresponding
to flame extinction ranges from 0.57 to 0.70, depending on
m and k .
Here it may be informative to note the significant implication of Eq. (81). Since extinction is frequently associated
with the nonlinear, highly temperature-sensitive nature of
the Arrhenius kinetics, the disappearance of its influence in
the diffusion-controlled limit implies that extinction as
shown in Fig. 23 is actually caused by an alternate
factor, which also has a nonlinear influence on flame
propagation. This factor is the liquid mass-diffusivity
D ‰ˆ D 0 exp…⫺Td =T†Š which varies in a sensitive, nonlinear,
Arrhenius manner with temperature. Thus in the SHS
process, even in the diffusion-controlled limit, combustion
behavior, as well as the particle regression rate, is still an
Arrhenius function of temperature, with the large activation
temperature for diffusion, Td, exerting the dominant
temperature sensitivity.
Fig. 24(a) and (b), respectively, show the effect of m and
k on the critical heat loss parameter Ccr ; with the initial
temperature T0 taken as a parameter. The change in Ccr is
gradual when either m or k is away from the flammability
limit, while Ccr decreases rapidly as this limit is approached.
It is also seen that Ccr increases with increasing T0 for a given
m or k . This suggests that even if a heat loss exists, flame
32
A. Makino / Progress in Energy and Combustion Science 27 (2001) 1–74
Fig. 23. Normalized mass burning rate m/ma,st as a function of the normalized heat loss C ⴱ ˆ …m a =ma;st †2 ·C; with the mixture ratio m and the
degree of dilution k taken as parameters when the initial temperature T 0 ˆ 450 K [64]. The values of numerically obtained (m/ma,st)cr
corresponding to the turning point are indicated by the horizontal solid lines; those of analytical ones are indicated by the dashed lines.
propagation can be sustained when T0 is high. Note that
there exists a maximum in C cr at high T0.
5.3. Turning-point determined by the thermal theory
The heterogeneous SHS flame propagation has some
special features, as shown in Eqs. (81) and (82), which
can be contrasted with the homogeneous gaseous flame
propagation. Nevertheless, an explicit relation for the state
of extinction can be derived [64] in the same manner as that
for the homogeneous premixed flame [65].
In the thin diffusion consumption/convection zone,
contribution of the heat loss term is very small, such that
Eq. (100) becomes
du
m
u⫺j
:
…104†
⬇
ma
ds
Z ∞ ⫺ Z0
Eliminating s from Eq. (104) with Eq. (101) and conducting
integration, we have
!
rt;0
T~ d
exp‰⫺b…1 ⫺ u†Š
m 2
exp
ˆ
L0
ma
b
rt
T~ ∞
×
Zj
0
u⫺j
1=3 dj;
Z∞ ⫺ Z0
…Z ∞ ⫺ Z0 † 1 ⫺
j
x~
1 ⫺ Z0
…105†
where the relation has been used in Eq. (86), retaining up to
the first term. When je ⬇ 1 at the downstream edge of the
consumption zone, by taking account of the adiabatic condition, that is u ˆ j ˆ 1 and m ˆ m a ; we obtain the following relation to the lowest order:
m 2
ˆ exp‰⫺b…j e ⫺ ue †Š:
…106†
ma
Fig. 24. Critical heat loss parameter C cr [64]: (a) as a function of the mixture ratio m when the degree of dilution k ˆ 0; and (b) as a function of
k when m ˆ 1:0: The initial temperature T0 is taken as a parameter.
A. Makino / Progress in Energy and Combustion Science 27 (2001) 1–74
33
Fig. 25. Comparisons between numerical and analytical solutions for the critical value of the mass burning rate ratio (m/ma)cr [64]; (a) as a
function of m ; and (b) as a function of k . The initial temperature T0 is taken as a parameter.
On the other hand, if we integrate and evaluate Eq. (100) at
the downstream edge of the consumption zone, we have
…V=b†
…j e ⫺ ue † ˆ
;
…m=m a † 2
where (V /b ) is expressed as
Zu e
V
f…u†
du ⫹ f…ue †;
ˆ
b
u
0
…107†
…108†
by the use of the upstream relation …du=ds† ˆ …m=m a † ‰u=…Z∞ ⫺ Z0 †Š and the downstream boundary condition
(103).
Combining Eqs. (106) and (108), we obtain the relation
m 2
m 2
ln
ˆ ⫺V;
…109†
ma
ma
from which we have …m=m a †cr ˆ e ⫺1=2 ˆ 0:6065 at the turning point, corresponding to the state of extinction. Fig. 23
shows that this analytical extinction state fairly represents
the numerical value, as the first approximation.
Fig. 25(a) and (b) further show (m/ma)cr as functions of m
and k , with T0 taken as a parameter. Although the numerical
and analytical solutions for (m/ma)cr differ by 15%, this
difference is reduced to less than 4% in determining C cr.
A more precise treatment will be given in Section 5.6.
5.4. Experimental comparisons for Ti–C system
There exist several sources of experimental data which
can be compared with the calculated results, as in
Refs.[64,66], by keeping as many parameters as fixed
possible. For the heat loss, only thermal radiation is considered, because of the high temperature in the consumption
zone, say, higher than 2000 K. Then the lateral heat loss for
a cylindrical compact with radius r can be expressed as
4h rad
Lˆ
…110†
…T ⫺ T0 †;
2r
where
hrad ˆ es SB …T 2 ⫹ T02 †…T ⫹ T0 †
…111†
is a radiative heat-transfer coefficient, e the emissivity, and
s SB the Stefan–Boltzmann constant. As for the heat loss
parameter C , it is expressed as
# 2!
"
l
4es SB …T 2 ⫹ T02 †…T ⫹ T0 †
R0
Cˆ
:
…112†
rtc
2r
…rt c†…u0;a R0 †2
Note here that C depends not only on the physicochemical parameters, but also on the particle radius R0 and the
compact diameter 2r. It is also inferred that R 02 =…2r†; which
is sometimes called the “Heat Loss Index”, can be a useful
parameter in correlating size effects related to the heat loss,
from a practical point of view. This parameter R20 =…2r† can be
understood as follows, from a physical point of view; the
increase in R0 decreases the total surface area of N particles
in the combustion wave, thereby reducing the volumetric
heat generation rate; the decrease in r increases the
surface-volume ratio of the compact, thereby enhancing
heat loss rate.
5.4.1. Range of flammabilty and extinction limit
Predicted results are compared with experimental data for
Ti–C system [64]. In addition to the values of physicochemical parameters, described in Section 4.1, use has been made
of l=…rt c† ˆ 10⫺5 m2 =s [41], e ˆ 0:4 [41], and s SB ˆ
5:67 × 10 ⫺11 kJ=…m2 :s:K4 †: The compact radius is set to be
r ˆ 9 mm and its relative density 0.6, in accordance with the
experiments [33,43]. As for the heat loss parameter C , it is
evaluated at the maximum temperature of the combustion
wave.
Fig. 26(a) shows the range of flammability when T 0 ˆ
450 K; the abscissa is the mixture ratio m and the ordinate
the particle radius R0. Extinction is anticipated to occur
when C in Eq. (112) reaches the critical heat loss parameter
C cr in Fig. 24(a). We see that while the range of flammability spans from 0.448 to 1 when R 0 ! 0; it contracts with
34
A. Makino / Progress in Energy and Combustion Science 27 (2001) 1–74
Fig. 26. Nonadiabatic combustion behavior for Ti–C system [64];
solid curves are calculated from theory; data points are experimental
[33,43]. (a) Range of flammability with respect to the mixture ratio
m and the particle radius R0; (W) designates the steady propagation,
(Z) the pulsating combustion, (K) the flame extinction during the
propagation, and ( × ) the non-ignition. (b) SHS rate-constant u0·R0
as a function of m , with R0 taken as a parameter.
increasing R0. It also shows that the combustion wave
cannot propagate through a compact when R0 is larger
than 20.9 mm. Experimental results [33,43] are also shown
in Fig. 26(a). Outside the range of flammability, even if the
flame propagation might be initiated by a powerful ignition
source, it loses its self-sustained nature and is extinguished
in the course of the propagation. Fair agreement is demonstrated for particles smaller than 7.5 mm, while discrepancy
becomes large for R0 ˆ 12:5 mm: This discrepancy is attributed to the effect of smaller particles, because of the existence of particle-size distribution. It is also pointed out that
the extinction limit obtained here corresponds to that for the
steady flame propagation. When the process occurs in an
unsteady mode, due to disappearance of stability of the
steady propagation, the limit of self-sustained combustion
could be modified accordingly. Fig. 26(b) shows the SHS
rate-constant u0·R0 in the adiabatic condition as a function of
m . The agreement is again fair, as far as the trend and
approximate magnitude are concerned, suggesting that
u0·R0 is one of the important parameters in correlating
experimental results.
The range of flammability with respect to the degree of
Fig. 27. Nonadiabatic combustion behavior for Ti–C system [64].
(a) Range of flammability with respect to the degree of dilution k
and the particle radius R0; notation is the same as that in Fig. 26(a).
(b) u0·R0 as a function of k , with R0 taken as a parameter.
dilution k is shown in Fig. 27(a), and Fig. 27(b) shows u0·R0
as a function of k . The same comments as those for Fig.
26(a) and (b) can be made.
Fig. 28(a) shows the range of flammability for various m
and k , with R0 taken as a parameter [66]. The range of
flammability becomes narrow with increasing R0. Experimental results [33,43] in Fig. 28(a) are those of R0 ˆ
5 mm: Fig. 28(b) shows a similar comparison [40] for the
dependence of the range of flammability on m in its wide
range when R0 ˆ 12:5 mm: In the range from m ˆ 1 to 2, the
range of flammability with respect to k is wider than that for
the stoichiometric mixture. This is attributed to the increase
in u0,a·R0 (cf. Fig. 9), which reduces C , and hence retards the
flame extinction. Discrepancy between the predicted and
experimental results for the under-stoichiometric mixtures
…m ⬍ 1† is attributed to the effect of smaller particles
because of the existence of particle size distribution. Discrepancy for the over-stoichiometric mixtures …m ⬎ 1†
might be attributed to an initiation of a radial crack, which
inevitably appears in pulsating combustion [1–3] resulting
in a laminated structure with separable layers.
Munir and Lai [67] measured the boundaries of the SHS
for Ti–C system at several ambient (or initial) temperatures.
Their test specimen (19 mm in diameter and 0.65 in relative
density) is composed of tens of layers (5 mm in thickness)
A. Makino / Progress in Energy and Combustion Science 27 (2001) 1–74
35
with increasing initial temperature T0, as expected, due to
preheating. This suggests that even if there is heat loss,
flame propagation can be sustained when T0 is high. It is
also seen that for a given T0, the range of flammability
narrows with increasing R0.
Shkiro and Borovinskaya [26] reported experimental data
for Ti–C system with varying degree of dilution k and
compact diameter 2r. The relative density is 0.56. Fig.
30(a) shows the range of flammability as functions of k
and 2r, with T0 taken as a parameter. The carbon particle
size and the initial temperature are estimated to be R 0 ˆ
2:5 mm and T0 ˆ 300–450 K; respectively, by examining
the dependence of the reported burning velocity on k and
T0. It is seen that with increasing k , the compact diameter in
which the combustion wave can propagate increases first
gradually and then rapidly. Further, an increase in T0
expands the range of flammability. Since it has been found
[66] that their experimental data correspond well to the
calculated result for T0 ˆ 300 K when k is less than about
0.2, while the comparison is better for T0 ˆ 450 K near the
limit of flammability, by examining the dependence of u0 on
k and T0, further confirmation has been made by investigating
the effect of the compact diameter 2r on the burning velocity
u0. Fig. 30(b) shows that u0 decreases with decreasing 2r
Fig. 28. Range of flammability for Ti–C system as functions of m
and k , with R0 taken as a parameter: (a) for R0 ˆ 5 mm [66]; (b) for
R0 ˆ 12:5 mm [40]. Data points are experimental [33,43] and notation is the same as that in Fig. 26(a).
with a gradual increase in the degree of dilution. The average
diameter of the carbon particles is reported to be 5 mm. Fig. 29
shows the limit of dilution with the radius of carbon particles,
R0, taken as a parameter. The range of flammability expands
Fig. 29. Range of flammability for Ti–C system as functions of k
and T0, with R0 taken as a parameter [66]. Data points are experimental [67]; (W) designates the steady propagation, (L) the
unsteady propagation, and ( × ) the flame extinction.
Fig. 30. Nonadiabatic combustion behavior for Ti–C system [66].
(a) Range of flammability as functions of k and the compact
diameter 2r, with T0 taken as a parameter, when R0 ˆ 2:5 mm;
data points are experimental [26] and notation is the same as that
in Fig. 26(a). (b) Burning velocity u0 as a function of 2r, with k
taken as a parameter; T0 ˆ 450 K:
36
A. Makino / Progress in Energy and Combustion Science 27 (2001) 1–74
because the heat loss rate is enhanced due to an increase in
the surface-to-volume ratio. Fair agreement is demonstrated
between the experimental and predicted results.
5.4.2. Burning velocity under heat loss condition
Here, let us investigate the effect of the heat loss on the
burning velocity. As pointed out in Section 5.4, not only the
physicochemical properties, but also sizes of N particles and
test specimen influence C through Eq. (112), hence the
burning velocity. Fig. 31(a) shows the SHS rate-constant
u0·R0 as a function of 2r=R 02 ; which is the inverse of the
heat loss index, for T0 ˆ 450 K and k ˆ 0; with m taken
as a parameter [40]. With decreasing 2r=R20 ; u0 ·R0 decreases
because of the increase in the heat loss rate, and reaches the
extinction limit shown by a dashed curve. We see that
effects of the compact diameter 2r and the particle radius
R0 on the burning velocity u0 can generally be discussed by
the use of parameters 2r=R20 and u0·R0. Experimental results
[33,43] for various R0 are also shown in Fig. 31(a). Fair
agreement is shown, as far as the general trend and approximate magnitude are concerned. Fig. 31(b) shows a similar
comparison for the stoichiometric mixture …m ˆ 1†; with k
taken as a parameter. Again, usefulness of the SHS rate-
Fig. 31. SHS rate-constant u0·R0 for Ti–C system as a function of 2r/
R02 for T0 ˆ 450 K [40]: (a) with m taken as a parameter; and (b)
with k taken as a parameter. Data points are experimental [33,43]
for various sizes of N particles.
constant u0·R0 and the heat loss index R20 =…2r† is demonstrated, as the correlating parameters for experimental
results.
5.5. Experimental comparisons for other systems
The nonadiabatic theory can also be applied to other
experimental results obtained from the synthesis of several
borides and intermetallic compounds for which reliable
experimental data are available. Through extensive comparisons [64,66] for the range of flammability with respect to
effects of dominant parameters (m , k , T0, R0, and 2r) on the
burning velocity u0, it has been demonstrated that the theory
is indeed viable, thereby allowing the synthesis of the
diverse set of experimental results within a coherent framework of description. In the same manner as that for the Ti–C
system, only thermal radiative heat loss is considered,
because of the high temperature in the combustion wave,
say, higher than about 1000 K.
5.5.1. Boride synthesis
Munir and Sata [68] measured the boundary of SHS in the
boride synthesis of Ti, Zr, and Hf at several ambient (or
initial) temperatures. The test specimen was prepared in
the same manner as that explained for Fig. 29, with amorphous boron of about 1 mm diameter and crystalline boron
particles whose diameter is less than 42 mm. Fig. 32(a)
shows the limit of dilution for Ti–B system [64] obtained
Fig. 32. Range of flammability for Ti–B system; (a) as functions of
the degree of dilution k and the initial temperature T0, with the
initial particle radius R0 taken as a parameter [64]; and (b) as functions of k and R0, with T0 taken as a parameter [66]. Data points are
experimental [68] and notation is the same as that in Fig. 29.
A. Makino / Progress in Energy and Combustion Science 27 (2001) 1–74
37
parameter. The same trend as that in Fig. 27(a) can be
observed and the range of flammability expands with
increasing T0. Experimental data with crystalline boron
particles have been plotted for R0 ˆ 5 mm because sieved
powders of ⫺325 mesh (⬍44 mm) size are estimated to have
about 10 mm in diameter. The deviation at large particles
could be caused by the difference of crystallization and/or
the existence of small particles, as well as the experimental
determination of T0, as inferred in Section 4.5.
Figs. 33 and 34 show similar comparisons for Zr–B
system [45,50,68] and Hf–B system [68]. Extent of agreement is similar to that for the Ti–B system.
Fig. 33. Range of flammability for Zr–B system [66]; (a) as functions of k and T0, with R0 taken as a parameter; data points are
experimental [68] and notation is the same as that in Fig. 29; (b)
as functions of k and R0, with T0 taken as a parameter; datum for
k ˆ 0 is from Hardt and Holsinger [45]; data for k ˆ 0:3 and 0.5 are
from Zenin et al. [50].
by the use of physicochemical parameters in Table 2 and the
transport parameters in Table 3. The same trend as that in
Fig. 29 is observed. Data points are experimental with boron
particles of about 1 mm diameter. It is pointed out that the
limit of flammability observed by Munir and Sata [68] could
have been affected by other effects not considered in the
theory such as agglomeration of the particles, as inferred
in Section 4.5.2.
Fig. 32(b) shows the range of flammability for Ti–B
system [66] as functions of k and R0, with T0 taken as a
Fig. 34. Range of flammability for Hf–B system as functions of k
and T0, with R0 taken as a parameter [64]. Data points are experimental [68] and notation is the same as that in Fig. 29.
5.5.2. Synthesis of intermetallic compounds
Naiborodenko and Itin [58] measured the burning velocities for stoichiometric Al–Ni system by varying the
diameter of compact whose relative density is 0.4. The
diameter of the Ni powder was reported to be of the order
of 20 mm. Fig. 35 shows that u0 decreases with decreasing
compact diameter because of the enhanced heat loss rate;
the boundary of the steady propagation is shown by a dashed
curve [66]. The agreement is fair for R0 ˆ 15 mm:
Itin et al. [53] measured the burning velocities for Co–Ti
system by varying the compact diameter and the particle
size of Ti. The compact for the stoichiometric mixture had
a relative density of 0.6; diameters of Ti particles were from
100 to 160 mm; initial temperature was 573 K. Fig. 36(a)
and (b), respectively, show the range of flammability and the
burning velocity [66]. We see that the range of flammability
expands with increasing compact diameter and that the
burning velocities agree well for the 130 mm particles,
which is representative of the average particle diameters.
Fig. 35. Burning velocity u0 in the nonadiabatic condition for Al–Ni
system as a function of the compact diameter 2r, with the initial
particle radius R0 taken as a parameter [66] the mixture is stoichiometric and the initial temperature T0 ˆ 300 K; data are from
Naiborodenko and Itin [58].
38
A. Makino / Progress in Energy and Combustion Science 27 (2001) 1–74
Fig. 36. Nonadiabatic combustion behavior for Co–Ti system with
stoichiometric mixture at T0 ˆ 573 K [66]. (a) Range of flammability as functions of 2r and R0; data are from Itin et al. [53] and
notation is the same as that in Fig. 29. (b) Burning velocity u0 as
a function of 2r, with R0 taken as a parameter; data points are
experimental [53].
Bratchikov et al. [60] measured the burning velocities for
stoichiometric Ni–Ti system with a relative density of 0.6
by varying the compact diameter and the particle size of Ti.
The initial temperature was 573 K. Fig. 37(a) shows the
range of flammability as functions of 2r and R0 [66]. Particle
size of Ti is estimated from their related paper [70] which
states that Ti particles were from 100 to 160 mm in diameter.
Fig. 37(b) shows that the burning velocities are correlated
fairly well with the predictions for R0 ˆ 100 mm:
Fig. 37. Nonadiabatic combustion behavior for Ni–Ti system with
stoichiometric mixture at T0 ˆ 573 K [66]. (a) Range of flammability as functions of 2r and R0; data are from Bratchikov et al. [60];
datum for R0 ˆ 25 mm is from Itin et al. [71]; notation is the same
as that in Fig. 29. (b) Burning velocity u0 as a function of 2r, with R0
taken as a parameter; data points are experimental [60].
for the temperature, as the outer solutions, in the upstream
and downstream regions where the consumption term dj /ds
is exponentially small:
C
C
a0
u ˆ 1 ⫹ 1 ⫹ 22 ⫹ … exp
s …s ⬍ 0†;
b
…Z∞ ⫺ Z0 †
b
…113†
C
C
1
bC
u ˆ 1 ⫹ 1 ⫹ 22 ⫹ … exp ⫺
s
…Z∞ ⫺ Z0 † a 0
b
b
5.6. Approximate expression of the heat loss parameter
…s ⬎ 0†:
Extinction of the combustion wave in the SHS process
has been obtained numerically and hence presented parametrically. An approximate analytical expression for the
extinction criterion [72] which explicitly displays its functional dependence on the various system parameters, is also
obtainable by referring to an analytical criterion for the
“premixed” SHS flame propagation obtained by Kaper et
al. [73] by the use of matched asymptotic expansion method.
The analysis and results presented herein is an extension of
those described in Section 5.3.
Here, b is the Zeldovich number defined in Eq. (85),
5.6.1. Asymptotic expansion analysis
Following Ref. [73], we have the following expressions
s ˆ
(114)
s
m
a
a
;
ˆ a0 ⫹ 1 ⫹ 22 ⫹ …;
b ma
b
b
…115†
and C1 ; C2 ; … are constants.
As for the inner solution, with the inner variables being
uˆ1⫹
sˆ
h
;
b
t1
t
y
y
⫹ 22 ⫹ …; j ˆ y 0 ⫹ 1 ⫹ 22 ⫹ …;
b
b
b
b
…116†
we obtain the following equations for the first-order inner
A. Makino / Progress in Energy and Combustion Science 27 (2001) 1–74
structure
have
d 2t1
a0
dy0
⫹
ˆ 0;
…Z∞ ⫺ Z0 † dh
dh2
exp ⫺
…117†
"
#
r
dy
3b
b 1
at 2
⫺ 1
bL0 t;0 a0 0 ˆ exp ⫺
⫺
a 2
rt
dh
4a
b
3
2
Z∞ ⫺ Z0
1=3
f
⫺
y
6
Z∞ ⫺ Z0
1 ⫺ Z0 0 7
7;
× 1⫺
y0
ln 6
1
⫹
5
4
Z ⫺ Z0
1 ⫺ Z0
y0
g⫹ ∞
1 ⫺ Z0
…118†
2bC
a 02
39
!
ˆ a20 k…1 ⫺ M† ⫹ M;
…125†
which has nearly the same form as that presented by Kaper
et al. [73] except for the factor k. The parameter M identified
in Ref. [74] is called the melting parameter because it corresponds to the energy to heat the unburned mixture from T0 to
Tm. Consequently, 1 ⫺ M is the normalized driving force for
the adiabatic flame propagation. The turning point
corresponding to the flame extinction, then, appears at
dC=da 02 ˆ 0; which yields
Xcr …1 ⫹ ln Xcr † ˆ M;
…126†
where a is defined in Eq. (88) and f ˆ 1=m:
Matching the outer and inner solutions in the limit of h !
^∞; we have
as well as the following relawhere ln Xcr ˆ
tions for the critical parameters at extinction:
a0
h ! ⫺∞ : t1 ! C1 ⫹
h; y0 ! 0;
…Z∞ ⫺ Z0 †
…a0 †2cr ˆ
h ! ∞ : t1 ! C1 ; y0 ! 1;
…119†
…120†
where the constant C1 is given by the second-order inner
problem as
C1 ˆ ⫺
2bC
:
a 02
…121†
By using the matching conditions, Eq. (117) can be integrated once to yield
dt1
a0
ˆ
…1 ⫺ y0 †:
dh
…Z∞ ⫺ Z0 †
…122†
Eliminating h from Eq. (122) with Eq. (118) and conducting
integration, we have
"
#
…Z∞ ⫺ Z0 † ZC1
3b
b 1
a t1 2
dt1
⫺
exp ⫺
⫺
4a
b
a 2
bL0 …rt;0 =rt †a20 ⫺ ∞
ˆ
Z1
0
…1 ⫺ y0 †
2
3 dy0 :
Z∞ ⫺ Z0
1=3
f
⫺
y0
7
6
Z∞ ⫺ Z0
1 ⫺ Z0 7
ln 6
1⫹
y0
1⫺
5
4
Z∞ ⫺ Z0
1 ⫺ Z0
g⫹
y0
1 ⫺ Z0
…123†
If we note here that L 0 is expressed in Eq. (93), that Iu in Eq.
(87), and that Ij in Eq. (92) for m ⬉ 1 and Ij in Eq. (97) for
m ⱖ 1; then Eq. (123) becomes
"
!#
2a 02
b
1⫺
bIu exp
1 ⫹ um
a
ˆ
Z0
C1
at
dt1 ⬇ 1 ⫺ exp…C1 †:
exp t1 1 ⫺ 1
b
…124†
Putting k ˆ 2=…1 ⫹ um † and 1 ⫺ M ˆ bIu exp…b=a†; we
⫺2bC cr =…a0 †cr2 ;
⫺Xcr ln Xcr
X …ln Xcr † 2
; C cr ˆ cr
:
k…1 ⫺ M†
2bk…1 ⫺ M†
…127†
Note that hte maximun flame temperature is given by
umax ˆ 1 ⫹ …ln Xcr =b†:
Knowing the critical heat loss parameter C cr and the
maximum flame temperature u max, the critical size of N
particles and the critical compact diameter for the extinction
are determined through
!
C cr …rt c†…u0;a R0 † 2
R 02
; …128†
ˆ
2 ⫹ T 2 †…T
2r cr ‰l=…rt c†Š4es SB …Tmax
max ⫹ T0 †
0
derived from Eq. (112) when only the thermal radiative heat
loss from the cylindrical surface of a compact with radius r
is considered [64,66]. It is informative to recall that R20 is
closely related to the heat generation rate in the consumption zone and 2r the heat loss rate. In addition, the SHS rateconstant in the adiabatic condition, which is the product of
the burning velocity u0,a and the initial radius R0 of N
particles, is expressed as
!
1 ⫺ Z0
Iu
2
2 rM
…u0;a ·R0 † ˆ 3Le0 D0
;
…129†
rN
Z∞ ⫺ Z0
Ij
by combining Eqs. (75) and (96).
5.6.2. Correction term for the heat loss parameter
Predictions based on Eq. (126) have been found to deviate
substantially from the numerical results shown in Section
5.4. The discrepancies are at least 50% for the heat loss
parameter C cr, and are even more for the critical particle
size and/or compact diameter. Since this discrepancy is
expected to be caused by ignoring the O(b ⫺1) term in
obtaining the RHS of Eq. (118), which would have modified
the RHS of Eqs. (125) and (126) by an amount ⫺O‰…1 ⫺
M†=bŠ; we are therefore required to add the term (1 ⫺ M)/b
to Eq. (126) as
1⫺M
X cr …1 ⫹ ln Xcr † ˆ M ⫺
:
…130†
b
40
A. Makino / Progress in Energy and Combustion Science 27 (2001) 1–74
have been used. If we further use …1 ⫹ um †=2 ⬃ 1; a ⬃ 1; and
T~ d …T~ ∞ ⫺ T~ 0 †
T~ d
T~ d
⬃
ˆ
2
~
~
~
Z∞ ⫺ Z0
T∞ ⫺ T0
T∞
1 ⫺ Z0
m ⫹ fst ~
T ;
ˆ
Z∞ ⫺ Z0
m…1 ⫺ k† d
bˆ
…135†
we have the following relations with respect to m and k :
!
(
)
1
m
e⫺2
m
1⫺
…m ⱕ 1†;
C cr ⬃
m ⫹ fst
2eT~ d m ⫹ fst
T~ d
…136†
Fig. 38. Critical value of the normalized heat loss, Xcr ˆ
exp‰⫺2bC cr =…a0 †cr2 Š; as a function of the melting parameter M,
with the measure of activation energy relative to thermal energy,
b , taken as a parameter [72]. Solid curves are numerical results and
dashed curves are results by the explicit approximate analytical
expression.
With this modification, the discrepancies can be reduced
to less than 20%. Furthermore, Xcr is now explicitly
expressed in terms of M as
Xcr ˆ
p !
1=2
1 ⫹ 2e
1
1⫺M
M2;
1 ⫹ 2e M ⫺
⫹ 1⫺
e
e
b
…131†
which yields a discrepancy of less than 1% (cf. Fig. 38), in
the range of practical values of M.
5.6.3. General behavior of the heat loss parameter
An explicit form of C cr provides useful insight into the
effects of the reduced parameters (a , b , and u m), which
depend on the system parameters (fst, Td, Tm, T0, m , and
k ), in the same manner as that for the mass burning rate
eigenvalue L 0 in Section 4.6.2. Keeping only the leading
terms, the functional dependence of C cr on the system parameters can clearly be displayed as
C cr ⬃
1 ⫹ um
4eb
1⫺
2a…e ⫺ 1† ⫺ e
:
b
…132†
Here the following relations
…1 ⫺ M† ˆ bIu exp
b
a
⬃1⫺
2a
b
…133†
and
Xcr ⬃
1
e…2a ⫺ 1†
1⫹
b
e
…134†
C cr ⬃
1
2eT~ d
1
m ⫹ fst
(
1⫺
e⫺2
T~ d
!
1
m ⫹ fst
)
…m ⱖ 1†
…137†
when there is no dilution …k ˆ 0†; when the mixture is
stoichiometric …m ˆ 1†; we have
!
(
)
1
1⫺k
e⫺2
1⫺k
: …138†
1⫺
C cr ⬃
1 ⫹ fst
2eT~ d 1 ⫹ fst
T~ d
That is, C cr increases with increasing m when m ⱕ 1; while it
decreases with increasing k or m when m ⱖ 1; although with
progressively diminishing rate. The effect of T0, which determines T∞, also appears because T∞ influences C cr through a
and b .
5.6.4. Applicability of the analytical expression
Applicability of the analytical expression has been examined by comparing it with the numerical results for the
syntheses of TiC, TiB2, ZrB2, HfB2, and AlNi, because of
fair agreement between numerical and experimental results,
as shown in Sections 5.4 and 5.5. Fig. 39(a) shows the
critical value of the heat loss parameter C cr for Ti–C system
as a function of m , with k ˆ 0 and T0 taken as a parameter.
Solid and dashed curves are respectively numerical and
analytical results. Fig. 39(b) shows C cr as a function of k ,
with m ˆ 1 and T0 taken as a parameter. The analytical
result mostly agrees with the numerical result, within 17%
error.
Fig. 40(a) shows the critical value of the heat loss index
‰R 02 =…2r†Šcr as a function of m , with k ˆ 0 and T0 taken as a
parameter. The agreements between the numerical and
analytical results are generally good, except near the stoichiometric state at which the analytical result shows a
reverse trend, resulting in about 18% under-prediction at
m ˆ 1: The discrepancy here is attributed to the overestimation in Ij in obtaining the approximate expression of L 0, as
mentioned in Section 4.6. Fig. 40(b) shows ‰R 02 =…2r†Šcr as a
function of k , with m ˆ 0:95 and T0 taken as a parameter.
When m is 0.95 or less, agreement is good, with less than
14% error.
Figs. 41 and 42 show similar comparisons for Hf–B and
Al–Ni systems, as examples, when the degree of dilution is
A. Makino / Progress in Energy and Combustion Science 27 (2001) 1–74
41
Fig. 39. Critical heat-loss parameter, C cr, for Ti–C system [72]; (a) as a function of the mixture ratio m , with the initial temperature T0 taken as a
parameter when the degree of dilution k ˆ 0; and (b) as a function of k , with T0 taken as a parameter when m ˆ 1: Solid curves are numerical
results and dashed curves are those with the analytical expression.
varied. The extent of agreement, including Ti–B and Zr–B
systems, is similar to that for Ti–C system.
5.7. Some remarks on the flame propagation under heat loss
condition
A nonadiabatic, heterogeneous theory for the SHS process
with heat loss has been formulated and the range of flammability can be determined from the characteristic extinction turning-point. A particularly significant identification is that the
process responsible for its temperature sensitivity and ultimate
extinguishment is the Arrhenius liquid-phase mass-diffusion.
Fair agreement between predicted and experimental results
for the extinction limits, in trend and in approximate magnitude, suggests that this theory has captured the essential
features of the heterogeneity in the SHS process. A useful
parameter called the “heat loss index”, which is defined as
the square of the nonmetal particle size divided by the
compact diameter, has also been identified. Furthermore,
under the appropriate and realistic assumptions of diffusion-limited reaction and infinite Lewis number, an approximate criterion for the extinction has been derived, which can
yield useful insight into manipulating the flame extinction.
6. Other aspects of the nonadiabatic flame propagation
In this section, several other important effects, such as the
bimodal particle distribution, representative length of the
cross-section of a compacted medium, on the SHS flame
Fig. 40. Critical heat-loss index [R02/(2r)]cr for Ti–C system [72]; (a) as a function of m , with T0 taken as a parameter when k ˆ 0; and (b) as a
function of k , with T0 taken as a parameter when m ˆ 0:95: Notation is the same as that in Fig. 39.
42
A. Makino / Progress in Energy and Combustion Science 27 (2001) 1–74
Fig. 41. Critical values of parameters for stoichiometric Hf–B system as a function of k , with T0 taken as a parameter: (a) C cr; (b) [R02/(2r)]cr.
Notation is the same as that in Fig. 39.
propagation and/or extinction are described. Also described
here is the correspondence between the heterogeneous
theory and the homogeneous premixed-flame theory, from
the analytical point of view. Effects of the electric field on
the flame propagation and/or extinction are also described in
this section.
6.1. Bimodal particle dispersion
Analyses of the heterogeneous flame, hitherto
described, have assumed a monodisperse distribution
of the nonmetal particles. However, there exist practical
interests in having a bimodal particle distribution. For
example, such a distribution would allow smaller particles to fill in part of the inter-particle spacing between
large particles, hence allowing for a denser, and
possibly more desirable, packing. Since not only the
increase in the burning velocity but also the extension
of the flammability limit is anticipated by reducing the
particle size, for a fixed amount of the dispersed phase,
it is considered to be desirable to have small particle
size for the propagation of the combustion wave.
However, preparation of the small particles can be
quite energy intensive. Therefore, preparation of the
bimodal mixture, with only a fraction of the N particles
having small sizes, can be less energy intensive without
substantially affecting the burning intensity. Further,
since SHS processes have been observed to possess a
variety of flame instability phenomena, which are invariably dependent on the burning intensity and hence the
particle size, such instabilities could be modulated for
bimodal mixtures. In view of the above considerations,
Fig. 42. Critical values of parameters for stoichiometric Al–Ni system as a function of k , with T0 taken as a parameter: (a) C cr; (b) [R02/(2r)]cr.
Notation is the same as that in Fig. 39.
A. Makino / Progress in Energy and Combustion Science 27 (2001) 1–74
formulation has been extended to bimodal systems
under nonadiabatic situations [75,76].
6.1.1. Equivalent particle radius
The problem of interest is basically the same as that
described in Section 5.1. The bimodal system for N particles
is considered to consist of small particles of density r N,S,
initial radius RS,0, and number density nS,0, and large particles with the corresponding parameters r N,L, RL,0, and nL,0.
In spite of the existence of the bimodal particle distribution,
the governing equations and boundary conditions are identical to those for the monomodal case, if we introduce the
equivalent particle radius and number density as (n0R0)E and
express the conservation equation for the N-Consumption
formally as [75,76]
d…1 ⫺ Z†
4prM D
u
1 ⫺ Z 1=3
ˆ⫺
…n0 R0 †E 0 x
:
u
dx
m
1 ⫺ Z0
…139†
Thus, our primary concern is to find out the specific form of
(n0R0)E in Eq. (139), as a function of the bimodal parameters. The description for the bimodal system has first
been made for the system with the same kind of N particles
[75] and then extended for the system with different kinds of
N particles [76].
The parameter (n0R0)E can be obtained by using the size
distribution function
rN G0 …R† ˆ rN;S nS;0 d…R ⫺ RS;0 † ⫹ rN;L nL;0 d…R ⫺ RL;0 †;
…140†
where d (·) is the delta function. By substituting it into the
general N-consumption equation in Eq. (45), we have
"
!
!2 #
rN;L
RL;0
…n0 R0 †E ˆ 1 ⫹
PG
rN;S
RS;0
1⫹G
1 ⫹ P 3G
1=3
…n L;0 RL;0 †
…141†
where
Gˆ
3
r N;S nS;0 RS;0
rN;L nL;0 R3L;0
…142†
is the mass ratio, P ˆ …hS =RS;0 †=…hL =RL;0 † the size ratio, and
hi =Ri;0 …i ˆ S; L† the normalized particle size defined as
!2
hi
2 Zx
x
dx …i ˆ S; L†:
…143†
ˆ1⫺ 2
Ri;0
Ri;0 ⫺ ∞ u
If we further take account of the relation for the mass fraction of N particles as
4
3
rt;0 …1 ⫺ Z0 † ˆ
†
…144†
prN;L …1 ⫹ G†…n L;0 RL;0
3
the mass burning rate eigenvalue becomes
"
#
…Z∞ ⫺ Z0 †2 m2a
L0 ˆ
4p…rM D0 †…l=c†…n0 R0 †E
#
"
Z∞ ⫺ Z0 2 …1 ⫺ Z0 †u20;a …RL;0 =F†2
ˆ
1 ⫺ Z0
3Le0 D20 …rM =rN;L †
where RL,0/F is the equivalent particle radius and
!
!2
rN;L
RL;0
1⫹
PG
rN;S
RS;0
2
F ˆ
:
…1 ⫹ G† 2=3 …1 ⫹ P3 G† 1=3
43
…145†
…146†
We see that u0,a·(RL,0/F) is independent of the bimodality,
and that Eq. (143) suggests P ˆ O…R S;0 =RL;0 †: Note that
G=…1 ⫹ G† is the mass fraction of small particles and can
be called the degree of small particle content. As for the
heat loss parameter C , it is given as
#" 2
#
"
…RL;0 =F†2
l
4es SB …T 2 ⫹ T02 †…T ⫹ T0 †
Cˆ
:
rtc
2r
…rt c†…u20;a …RL;0 =F†2
…147†
It depends not only on the physicochemical parameters but
also on the equivalent particle radius RL,0/F and the compact
diameter 2r.
6.1.2. Experimental comparisons for spherical carbon
particles
There exist several sources that can be compared with the
numerical results. Makino et al. [43,77,78] using various
radii of spherical carbon particles, measured the range of
flammability and the burning velocity of the SHS process
for Ti–C system. The representative Lewis number of Le0 ˆ
25; which corresponds to l=…rt c† ˆ 5 × 10⫺6 m2 =s and has
been determined through correlation similar to that in
Section 4.4 with experimental results [43,77,78] (0.6 relative density) of monomodal system, has been used in numerical calculations [75] because of the fair correlation for
various radii of spherical carbon particles and various
compact diameters, as shown in Figs. 43 and 44.
Fig. 43(a) shows the range of flammability for the monomodal system with respect to the mixture ratio m and the
heat loss index R20 =…2r†; with T0 taken as a parameter [79].
Experimental results [43,77,78] for various particle radii
(1.5, 2.5, 5, 10, and 15 mm) and various compact diameter
(5, 7, 10, and 18 mm) at T0 ˆ 300 K show that not only the
range of flammability but also the range of pulsating
combustion extends with decreasing R20 =…2r†: Fig. 43(b)
shows the similar comparison with respect to the degree
of dilution k and R20 =…2r†: Fair agreement is demonstrated
between the predicted and experimental results.
The choice of Le0 ˆ 25 for the spherical carbon particles
requires further comments because Le0 ˆ 100 has been used
in comparisons in Section 4.4 and 5.4. As a remarkable
difference, compared to the irregular particles in Refs.
44
A. Makino / Progress in Energy and Combustion Science 27 (2001) 1–74
Fig. 43. Range of flammability for Ti–C system with monomodal particles; (a) with respect to the mixture ratio m and the heat-loss index R02/
(2r), with the initial temperature T0 taken as a parameter [79]; and (b) with respect to the degree of dilution k and the heat-loss index R02/(2r).
Data points are experimental at T0 ˆ 300 K [43,77,78]: (W) designates the steady-propagation; (Z) the pulsating combustion; (S) the quasisteady, multipoint combustion; (K) the extinction during the flame propagation, and ( × ) the non-ignition.
[33,43], the spherical particles are reported to have smooth
surfaces and very narrow size distribution [77]. This
suggests that the value of Le0 ˆ 100 for the irregular particles implicitly accounts for the effects of surface roughness
and the relatively wide size distribution because these
effects, which appear in (n0R0)E in Eq. (139), are grouped
into l /c when only the particle size is known.
Fig. 44(a) shows the SHS rate-constant u0·R0 for the
monomodal system as a function of the inverse of the heat
loss index R20 =…2r† at T0 ˆ 300 K; with m taken as a parameter [79]. The trend is the same as that in Fig. 31(a). Data
points are experimental [43,77,78] and the agreement is
again considered to be satisfactory. Fig. 44(b) shows a similar comparison, with k taken as a parameter.
Having confirmed the appropriateness of the choice of
several physicochemical parameters, comparisons for the
bimodal system have been conducted, with the size ratio
P taken to be P ˆ RS;0 =RL;0 : Fig. 45(a) shows the range
of flammability for the bimodal system with respect to the
degree of small particle content G=…1 ⫹ G † and the degree of
dilution k , for T 0 ˆ 300 K; m ˆ 1; RS;0 ˆ 1:5 mm; RL;0 ˆ
10 mm; and 2r ˆ 18 mm: It is seen that the range of flammability, which spans from k ˆ 0 to 0.178 when G=…1 ⫹ G † ˆ
0; gradually extends with increasing G /(1 ⫹ G ) and finally
spans from k ˆ 0 to 0.400 when G=…1 ⫹ G † ˆ 1: Extension
of the range of flammability is attributed to the decrease in
the equivalent particle radius RL,0/F. The limit of dilution
…k ˆ 0:446† without heat loss suggests that even at G=…1 ⫹
G † ˆ 1: the heat loss affects the range of flammability. It is
seen that as far as the trend and approximate magnitude are
concerned, we can fairly predict the experimentally
obtained extinction limit [43,77].
Fig. 45(b) shows u0,a in the adiabatic condition as a
function of G=…1 ⫹ G † for T 0 ˆ 300 K and m ˆ 1; with k
taken as a parameter. Owing to the decrease in the equivalent particle radius, u0,a increases first gradually and then
rapidly with increasing G=…1 ⫹ G †: At a certain G=…1 ⫹ G †;
u0,a decreases with increasing k because of the reduced
amount of chemical heat generation. The dashed curve is
the extinction limit under heat loss conditions. A fair degree
of agreement is shown.
The appropriateness of the equivalent particle radius can
further be shown in Fig. 46. It is, however, important to note
that since the present formulation is based on the statistical
analysis, a few tens of large particles should exist at least
across the consumption zone, even for the bimodal system.
Violation of this restriction results in the breakdown of the
present formulation, such that discussion with the equivalent
particle radius becomes less useful.
6.1.3. Experimental comparisons for diamond particles
There exists another source for the bimodal system.
Makino et al. [77] measured the range of flammability and
the burning velocity of the SHS process for Ti–diamond
system, as a basic research to fabricate the diamond
embedded TiC/Ti composites by the SHS process, to
achieve advanced cutting tools and/or abrasives. In this
fabrication, the SHS flame must propagate through a
medium of compacted particles of titanium, carbon (or
graphite), and diamond, without participation of the
diamond particles in the SHS process.
By comparing the predictions [76] with the experimental
results [43,77] it has been determined to use the physicochemical parameters, described in Sections 4.4 and 5.4.1,
except for r N ˆ 3:51 × 10 3 kg=m3 : A Lewis number of
Le0 ˆ 100 is used because of the fair correlation with the
A. Makino / Progress in Energy and Combustion Science 27 (2001) 1–74
Fig. 44. SHS rate-constant u0·R0 for Ti–C system with monomodal
particles, as a function of the inverse of the heat loss index R02/(2r) at
T0 ˆ 300 K [79]; (a) with m taken as a parameter and (b) with k
taken as a parameter. Data points are experimental [43,77,78]; open
symbol designates the steady propagation and half-open symbol the
pulsating combustion.
experimental results [43,77] (0.6 relative density) for the
various radii of diamond particles, as shown in Figs. 47
and 48. The compact diameter 2r is 18 mm in accordance
with the experiments.
Fig. 47(a) shows the range of flammability for monomodal compacts with respect to m and R0, for T0 ˆ
300 K and k ˆ 0: Experimental results [43,77] for the
steady propagation, quasi-steady multipoint combustion,
extinction, and non-ignition show that the range of
flammability extends with the decreasing R0. Fig.
47(b) shows u0,a·R0 as a function of m , with R0 taken
as a parameter. Compared to the Ti–C system, the burning velocity for Ti–diamond system is reduced because
of the increase in the nonmetal density r N, as shown in
Eq. (76), which suggests that u0,a is inversely proportional to (r N) 1/2. The increase in r N, thus, reduces the
total surface area of particles in the combustion wave,
and hence, suppresses the volumetric heat-generation
rate. Fig. 48 shows a similar comparison with respect
45
Fig. 45. Combustion behavior for Ti–C system with bimodal particles [75]; (a) Range of flammability with respect to the degree of
small particle content G=…1 ⫹ G † and the degree of dilution k , for
T 0 ˆ 300 K; m ˆ 1; RS;0 ˆ 1:5 mm; and RL;0 ˆ 10 mm; data points
are experimental [43,77] and notation is the same as that in Fig. 43.
(b) Burning velocity u0,a in the adiabatic condition as a function of
G=…1 ⫹ G †; with k taken as a parameter; notation is the same as that
in Fig. 44.
to k . Data points are experimental [43] and the same
comments as those for Fig. 47 can be made.
Having confirmed the fair agreement for the monomodal
system, it was intended to make experimental comparisons
for the bimodal system. Fig. 49(a) shows the range of
flammability with respect to G=…1 ⫹ G † and k , and Fig.
49(b) shows u0,a as a function of G=…1 ⫹ G †; for the stroichiometric mixture …m ˆ 1† at T 0 ˆ 300 K with RS;0 ˆ 3 mm;
Fig. 46. SHS rate-constant u0,a·(RL,0/F) in the adiabatic condition as
a function of the degree of dilution, with T0 taken as a parameter.
Data points are experimental [43,77] and notation is the same as that
in Fig. 44.
46
A. Makino / Progress in Energy and Combustion Science 27 (2001) 1–74
Fig. 47. Combustion behavior for Ti–diamond system with monomodal particles [76]. (a) Range of flammability with respect to m
and R0, for T0 ˆ 300 K without dilution …k ˆ 0†; data points are
experimental [43,77] and notation is the same as that in Fig. 43.
(b) u0,a·R0 as a function of m , with R0 taken as a parameter.
RL;0 ˆ 15 mm; and rN;S ˆ rN;L : Extent of agreement
between the predicted [76] and the experimental [77] results
is similar to that of Fig. 45.
The appropriateness of the equivalent particle radius
RL,0/F can be further shown by comparing the experimental results of R0 ˆ 8 mm for the monomodal system
in Fig. 48(a) with the predicted results for the bimodal
system of G=…1 ⫹ G † ˆ 0:463; which corresponds to
R L;0 =F ˆ 8 mm; in Fig. 49(a). As mentioned in Section
6.1.2, this discussion with the equivalent particle radius
becomes less useful when a number of large particles
across the combustion wave becomes too small to apply
the statistical counting method.
In order to further study a fundamental aspect of the
practical fabrication of the diamond embedded TiC/Ti
composite, the bimodal system with large diamond and
small carbon particles has been used. Here, the diamond
particles are large enough to prevent the SHS flame propagation, even for the stoichiometric mixture. Fig. 50(a) shows
the range of flammability with respect to G=…1 ⫹ G † and k ,
and Fig. 50(b) shows u0,a as a function of G=…1 ⫹ G †; for the
stoichiometric mixture …m ˆ 1† at T 0 ˆ 300 K with RS;0 ˆ
2:5 mm; RL;0 ˆ 15 mm; rN;S ˆ 2:25 × 10 3 kg=m3 ; and
rN;L ˆ 3:51 × 103 kg=m3 : The same trends as those shown
in Fig. 45(a) and (b) are obtained, respectively, for the
Fig. 48. Combustion behavior for Ti–diamond system with monomodal particles [76]. (a) Range of flammability with respect to k
and R0, for stoichiometric mixture …m ˆ 1† at T0 ˆ 300 K; data
points are experimental [43,77] and notation is the same as that in
Fig. 43. (b) u0,a·R0 as a function of k , with R0 taken as a parameter.
present bimodal system with different densities of N particles. Data points are experimental [77] and agreement is
fair, as far as the trend is concerned.
6.2. Representative length of the cross-sectional area
Thus the far, shape of a compacted medium has been
considered to be cylindrical in evaluating the radiative
heat loss from the side surface of the compacted medium.
Then, the representative length of the cross-sectional area
has been the compact diameter, as has appeared in the heat
loss parameter C in Eq. (112). Here the choice of a representative length for the noncircular cross-sectional area is
examined, with a rectangular compact taken as an example.
When the radiative heat loss was introduced into the
formulation in Section 5.1, the heat loss L per unit time
and per unit volume was considered. When a sample specimen has a cross-sectional area S and perimeter p, the heat
loss per unit time is expressed as LSd , where d is the thickness of a volume considered along the x-coordinate, say, the
thickness of the combustion wave or the thickness of the
consumption zone. As for the heat loss per unit time transferred from the surface to the ambience by radiation, it is
A. Makino / Progress in Energy and Combustion Science 27 (2001) 1–74
47
Fig. 49. Combustion behavior for Ti–diamond system with bimodal particles [76]. (a) Range of flammability with respect to
G=…1 ⫹ G † and k , for m ˆ 1 at T 0 ˆ 300 K with RS;0 ˆ 3 mm;
RL;0 ˆ 15 mm; and rN;S ˆ rN;L ; data points are experimental [77]
and notation is the same as that in Fig. 43. (b) Burning velocity u0,a
in the adiabatic condition as a function of G=…1 ⫹ G †; with k taken
as a parameter; notation is the same as that in Fig. 44.
Fig. 50. Combustion behavior for Ti–C–diamond system [76]. (a)
Range of flammability with respect to G=…1 ⫹ G † and k , for m ˆ 1 at
T 0 ˆ 300 K with RS;0 ˆ 2:5 mm; RL;0 ˆ 15 mm; rN;S ˆ 2:25 × 10 3
kg/m 3, and rN;L ˆ 3:51 × 103 kg/m 3; data points are experimental
[77] and notation is the same as that in Fig. 43. (b) Burning velocity
u0,a in the adiabatic condition as a function of G=…1 ⫹ G †; with k
taken as a parameter; notation is the same as that in Fig. 44.
expressed as es ST …T 4 ⫺ T04 †pd: Then, we have
Fair agreement is again obtained, as far as the trend and
approximate magnitude are concerned.
es ST …T 4 ⫺ T04 †
;
S=p
by which the heat loss parameter is expressed as
Lˆ
l
Cˆ
rtc
"
4es SB …T 2 ⫹ T02 †…T ⫹ T0 †
…rt c†…u0;a R0 †2
#
!
R20
:
4S=p
…148†
…149†
When the cross-sectional area is circular with radius
r, we have S ˆ pr2 and p ˆ 2pr; which yields 4S=p ˆ
2r: When the cross-sectional area is rectangular with a
in length and b in width, we have S ˆ ab and p ˆ 2…a ⫹
b†; which yields 4S=p ˆ 2ab=…a ⫹ b†: It may be informative to note that this is a harmonic mean of a and b,
and is called the equivalent diameter in the field of the
heat-transfer engineering.
Fig. 51(a) and (b) shows the range of flammability for Ti–
C system as a function of m and k , respectively. Values of
the physicochemical parameters are as those in Sections 4.4
and 5.4.1 Data points are experimental [80] at T0 ˆ 450 K
with various a and b, as well as the various radii of (irregular) carbon particles. Appropriateness of the choice of the
representative length is demonstrated. Fig. 52(a) shows u0 as
a function of m , with k taken as a parameter, and Fig. 52(b)
shows u0 as a function of k , with m taken as a parameter.
6.3. Correspondence between the heterogeneous theory and
the homogeneous theory
Much of the theoretical analyses on the flame propagation
in SHS, as reviewed in Ref. [24], are based on the classical
treatment of the premixed flame propagation, characterized
by a large activation energy Arrhenius reaction rate and
large Lewis number heat and mass diffusion. Although
these analyses have successfully captured the temperaturesensitive nature of the flame propagation, and consequently
have been able to describe such empirically observed
phenomena as abrupt flame extinction due to heat loss,
and instability of the combustion wave in the form of pulsating and spinning propagation, as explained in Refs. [4–9],
there are several fundamentally unsatisfactory aspects as
mentioned in the previous sections. While it is pleasing
that the heterogeneous theory has been successful in
explaining the experimental results, especially the temperature sensitivity, one must nevertheless confront the somewhat curious result that the characterization of an Arrhenius
diffusion process can be similar to that of an Arrhenius
reaction process. To resolve this interesting correspondence,
we shall demonstrate [81] that the governing equations for
the heterogeneous theory can indeed be cast in a form which
48
A. Makino / Progress in Energy and Combustion Science 27 (2001) 1–74
Fig. 51. Range of flammability for Ti–C system, with T0 taken as a parameter; (a) with respect to m and R20 =‰2ab=…a ⫹ b†Š; and (b) with respect to
k and R20 =‰2ab=…a ⫹ b†Š: Data points are experimental [80] and notation is the same as that in Fig. 43.
can be considered to be functionally identical to that
describing the large activation-energy premixed-flame
propagation in a condensed medium.
To this aim, the N-consumption equation (46) must first
be rewritten as
d…1 ⫺ Z†
T
ˆ ⫺rt B~ w exp ⫺ d ;
…150†
rt u
dx
T
where
B~ ˆ
wˆ
!
!
3
rM
rt
…1 ⫺ Z0 †D0
;
rN
rt;0
R20
1⫺Z
1 ⫺ Z0
Y~ M ˆ
1=3
ln 1 ⫹
A ~
YM ;
1⫹A
1⫺m
…1 ⫺ Z0 † ⫹ …1 ⫺ Z†
m
:
1 ⫺ …1 ⫺ Z†
…151†
…152†
…153†
In deriving Eq. (153), the following relation has been used
Z
…1 ⫹ Y~ M † ˆ …1 ⫹ Y~ M;0 † 0 ;
…154†
Z
derived from Eq. (49) under the assumption of the negligible
diffusion effect. The particle consumption term in the RHS
of Eq. (150) therefore corresponds to the “reaction” term in
the homogeneous premixed-flame theory, with B~ being the
“frequency” factor, w the “reactant concentration”, and Td
the “activation temperature”. As for the energy conservation
equation, from Eq. (58), we have
d
l d
d
T
…cT† ⫺ rt u …cT† ⫹ q0 rt B~ w exp ⫺ d
dx
dx
c dx
T
ˆ L:
(155)
The boundary conditions are as follows:
x ˆ 0 : T ˆ Tm ; 1 ⫺ Z ˆ 1 ⫺ Z0
x!∞:
d
L
d…1 ⫺ Z†
…cT† ˆ ⫺
ˆ 0:
;
dx
…rt u†
dx
…156†
…157†
We see that a set of governing equations and boundary
conditions is exactly the same as that for the conventional
homogeneous theory (cf. Ref. [4]). Although differences are
minor from a mathematical point of view, this identification
is valuable for the physical interpretation of the combustion
behavior. First, it should be noted that w in the “reaction”
term is a function of the mass fraction of solid N particles,
1 ⫺ Z: In addition, if we note that the SHS proceeds under
diffusion-controlled situation, as is the case for usual SHS,
we have A=…1 ⫹ A† ! 1; which suggests that the mass diffusivity in the Arrhenius fashion exerts nonlinear influence on
flame propagation and/or extinction. That is, the activation
temperature which can influence the nonlinear, highly
temperature-sensitive nature of the SHS flame propagation
is not the activation temperature Ta for the reaction, but Td
for the liquid-phase mass-diffusivity. As for the effect of
particle size, it has been incorporated into the frequency
~ tacitly.
factor B;
Since it has been confirmed that the set of the governing
equations and boundary conditions in the heterogeneous
theory is identical to that in the conventional homogeneous
theory, results of theoretical analyses hitherto obtained by
the homogeneous premixed-flame theory, as reviewed in
Refs. [23,24], can be anticipated to be qualitatively valid
if we take note of the differences involved.
6.4. Field activated SHS
For low exothermic systems, in order to sustain flame
A. Makino / Progress in Energy and Combustion Science 27 (2001) 1–74
49
described in Section 5.1, except for the energy conservation
equation, which is now not Eq. (58) but
d
dT
⫺ mc…T ⫺ T0 † ⫹ mq0 …Z ⫺ Z0 † ˆ L ⫺ P …158†
l
dx
dx
when it is expressed in dimensional form [89] where P is the
energy supplied by the electric field. When we consider the
situation that only N particles participate in the external
heating due to electric field, the energy supply rate P is
expressed as
P ˆ …1 ⫺ Z†
scE 2
;
2
…159†
where E is the electric field (V/m) and s c the electric
conductivity (W ⫺1/m). It may be informative to note that
the electric field induced by the alternative current must
p be
evaluated as the effective value with a factor of 1= 2: In
addition, it is anticipated that when the temperature is lower
than the melting point of M species, there is insufficient
energy supply by electric current, because of the contact
resistance among particles. When the combustion product,
such as SiC has been produced, its electric resistance is too
high to allow for electric current.
By introducing a nondimensional parameter
Hˆ
Fig. 52. SHS rate-constant u0·R0 for Ti–C system as a function of
the inverse of the heat loss index R20 =‰2ab=…a ⫹ b†Š at T0 ˆ 450 K;
(a) with m taken as a parameter; and (b) with k taken as a parameter.
Data points are experimental [80] and notation is the same as that in
Fig. 44.
propagation, it is usual to heat a compacted medium externally by preheating and/or by applying an electric field.
Since preheating in a long period of time could induce
production of undesired compounds which might inhibit
initiation of the combustion wave, it is reported to be preferable [82,83] for low exothermic systems to apply the electric
field, which can also exert a great influence on the flame
propagation [84,87] especially when the electric conductivity in the combustion wave is much higher than those in the
unburned and burned states. Although there exist several
analytical [88] and/or numerical [84,86] works on this
subject based on the homogeneous premixed-flame theory,
heterogeneity involved has not been explored. In this
section, focusing on the effect of the electric field on the
burning velocity, let us examine it by use of the heterogeneous theory.
6.4.1. Heat-input parameter
The problem of interest is basically the same as that
…l=c†s c E 2
;
m2a q0
…160†
in addition to other variables and parameters in Sections 4.2
and 5.1, Eq. (158) becomes
!
d du
m=m a
m=ma
⫺
u⫹
j
ds ds
Z∞ ⫺ Z0
T~ ∞ ⫺ T~ 0
ˆ
C·u
H…1 ⫺ j†
;
⫺
…Z ∞ ⫺ Z0 † 2
…Z ∞ ⫺ Z0 †…T~ ∞ ⫺ T~ 0 †
…161†
where H can be called the heat-input parameter. The other
governing equations are the same as those in Eqs. (99) and
(101). The boundary conditions are also the same as those in
Eqs. (102) and (103).
6.4.2. Temperature profiles outside the combustion wave
As mentioned in Section 5.2, in the limit of large Zeldovich number, melting, diffusion, and consumption/convection are confined to a thin layer in the neighborhood of the
combustion wave. Outside this layer, the diffusion and
consumption/convection terms are exponentially small. If
we consider the situation that the combustion wave locates
at s ˆ s f and that the electric field exerts influence only in
the region 0 ⬍ s ⬍ s f ; temperature profile outside the
combustion wave is expressed as
s ⱕ 0 : u ˆ A u exp…lu s†; j ˆ 0;
…162†
50
A. Makino / Progress in Energy and Combustion Science 27 (2001) 1–74
0 ⬍ s ⬍ s f : u ˆ Ah exp…lu s† ⫹ B h exp…ld s†
(170) as
H…Z ∞ ⫺ Z0 †
⫹
; j ˆ 0;
C…T~ ∞ ⫺ T~ 0 †
(163)
s ⱖ s f : u ˆ C exp…ld s†; j ˆ 1;
…164†
where
s)
m=ma
4C
;
lu ˆ
1⫹ 1⫹
2…Z∞ ⫺ Z0 †
…m=m a † 2
(
…165†
s)
(
m=ma
4C
:
1⫺ 1⫹
ld ˆ
2…Z∞ ⫺ Z0 †
…m=m a † 2
Relations among constants Au, Ah, Bh, and C are determined by use of the continuities of temperature and heat flux
at s ˆ 0; respectively, the continuity of temperature at the
combustion wave …s ˆ s f †; and the jump condition of du /
ds in the combustion wave as
du
du
m=ma
⫺
ˆ
;
…166†
ds s f⫺
ds s f⫹ T~ ∞ ⫺ T~ 0
which is obtained by integrating Eq. (161) and evaluating at
s ˆ s f : Then, we have
Au ˆ
1
m=ma
exp…⫺lu s f †
lu ⫺ ld T~ ∞ ⫺ T~ 0
⫺
Ah ˆ
ld
H…Z∞ ⫺ Z0 †
‰1 ⫺ exp…⫺lu s f †Š;
lu ⫺ ld C…T~ ∞ ⫺ T~ 0 †
1
m=ma
ld
H…Z∞ ⫺ Z0 †
⫹
lu ⫺ ld T~ ∞ ⫺ T~ 0
lu ⫺ ld C…T~ ∞ ⫺ T~ 0 †
exp…⫺lu s f †;
Bh ˆ ⫺
Cˆ
lu
H…Z∞ ⫺ Z0 †
lu ⫺ ld C…T~ ∞ ⫺ T~ 0 †
…167†
…168†
…169†
1
m=ma
exp…⫺ld s f †
lu ⫺ ld T~ ∞ ⫺ T~ 0
⫺
lu
H…Z∞ ⫺ Z0 †
‰1 ⫺ exp…⫺ld s f †Š:
lu ⫺ ld C…T~ ∞ ⫺ T~ 0 †
…T~ ∞ ⫺ T~ 0 † ˆ …Z∞ ⫺ Z0 † 1 ⫹
sf
ˆ ln
Z∞ ⫺ Z0
"
Hs f
;
Z∞ ⫺ Z0
#
…1 ⫺ H†…Z∞ ⫺ Z0 †
;
…T~ m ⫺ T~ 0 † ⫺ H…Z∞ ⫺ Z0 †
…172†
…173†
in the limit of C ! 0: We see that the adiabatic flame
temperature increases with increasing strength of the electric field. It may be informative to note that the relation in
Eq. (59), which gives a relation between the temperature and
the mass fraction of fluid cannot be used any more when heat
is supplied externally.
6.4.3. Effective range of electric field
In order for the electric field to be effective for increasing
combustion temperature, the flame location must be positive, which requires that the logarithmic term in Eq. (173)
must be positive. From this requirement, the range in which
the heat-input parameter H can exist is determined as
0ⱕH⬍
T~ m ⫺ T~ 0
⬍ 1:
Z∞ ⫺ Z0
…174†
In the same manner, when there exists heat loss, we have
from Eq. (171) the following relation between C and H.
(
)
1 …m=m a † 2
H…Z∞ ⫺ Z0 †
⫺
⫺
2
4
T~ m ⫺ T~ 0
v
(
)
u
u …m=ma †2 …m=ma †2
H…Z
⫺
Z
†
∞
0
⫺t
⫹
8
8
T~ m ⫺ T~ 0
)
…m=m a † 2
H…Z∞ ⫺ Z0 †
⫺
4
T~ m ⫺ T~ 0
v
(
)
u
u …m=ma †2 …m=ma †2
H…Z
⫺
Z
†
∞
0
⫹t
: …175†
⫹
8
8
T~ m ⫺ T~ 0
⬍C⬍⫺
1
2
(
Note that when C has the value of the limit, length between
the location of melting and that of the combustion wave
becomes infinite.
…170†
If we further note that the temperature at s ˆ 0 is the
melting point, we can determine the flame location s f as
3
m
ld
⫹
H…Z
⫺
Z
†
∞
0
7
6
1
ma
C
7;
sf ˆ
ln 6
5
4
l
lu
d
~
~
…T m ⫺ T 0 †…lu ⫺ ld † ⫹
H…Z ∞ ⫺ Z0 †
C
…171†
2
which is obtained from Eq. (167) by setting Au ˆ um : The
flame temperature in the adiabatic condition, which is indispensable for nondimensionalization, is obtained from Eq.
6.4.4. Experimental comparisons for Si–C system
Values of physicochemical parameters employed are
those for Si–C system: q 0 ˆ 6:23 MJ=kg; c ˆ 1 kJ=…kg=K†;
r M ˆ 2:34 × 103 kg=m3 ; rN ˆ 2:25 × 103 kg=m3 ; WM ˆ
28:1 × 10⫺3 kg=mol; WN ˆ 12:0 × 10⫺3 kg=mol; Tm ˆ
1681 K:
For
the
mass
diffusivity,
D ˆ 3:3 ×
10⫺5 exp…⫺3:39 × 104 =T† m2 =s is used [90]. As for the
representative Lewis number, Le0 ˆ 1; because experimental results [85] have been correlated for this value.
Fig. 53 shows the SHS rate-constant u0·R0 for stoichiometric Si–C system, as a function of the square root of the
heat-input parameter H. A solid curve is u0,a·R0 in the adiabatic condition and a dashed curve is that at the extinction
A. Makino / Progress in Energy and Combustion Science 27 (2001) 1–74
51
In this context, the heterogeneous theory is indeed viable in
examining the SHS process. Correspondence between the
heterogeneous theory and the homogeneous premixed-flame
theory has further been examined. In spite of the minor
differences, from the viewpoint of the mathematics, the
important identification related to the physical interpretation, pointed out in the previous sections, has been reconfirmed.
7. Boundary between steady and pulsating combustion
Fig. 53. SHS rate-constant u0·R0 for stoichiometric Si–C system, as
a function of the square root of the heat-input parameter H [89]; a
solid curve is that in the adiabatic condition and a dashed curve is
that at the limit of flame propagation. Data points are experimental
[85]; (S) designates data for rrel ˆ 0:52 and (W) for rrel ˆ 0:64:
limit. A range surrounded by these curves corresponds to
that for flame propagation under electric field. With increasing H, up to the limit determined by Eq. (174), u0·R0
increases first gradually and then rapidly. In the abscissa,
for convenience, the strength of electric field is shown,
obtained by use of the electric conductivity of carbon, s c ˆ
1:21 × 10 4 ⫹ 6:25…Tr ⫺ 450† (W ⫺1/m), based on the data in
the literature, [91] with the representative temperature Tr
taken as the arithmetic mean of the melting point of Si
and the maximum combustion temperature. The initial
carbon radius R0 of 2 mm is evaluated from the reported
burning velocity [85] that was sustained in the lowest electric field of about 0.8 kV/m. Data points are experimental
[85] and fair degree of agreement is demonstrated, as far as
the trend and approximate magnitude are concerned.
It is also reported that the characteristic extinction turning-point behavior described in Section 5.2 can exist when
the heat-input parameter H is less than about 0.2, while it
ceases to exist and the flame extinction occurs at the limit in
Eq. (175) when H ⬎ 0:2: It may be informative to note that
since the heat-input parameter H depends on the particle
size, only the heterogeneous theory could captured the
heterogeneous feature of the field activated SHS.
6.5. Some remarks on the several, other, important factors
Related to the SHS flame propagation and/or extinction,
other important factors, such as the bimodal particle distribution, the representative length of the cross-section, and
the field activation are examined. It has been shown that
these effects on the flame propagation and/or extinction
are also closely related to the particle size of the nonmetal.
So far, flame propagation and/or extinction in steady
combustion has been examined. However, SHS flame
propagation can be either steady or nonsteady, depending
on the system parameters. Since nonhomogeneity in
combustion products induced by nonsteady propagation is
often responsible for the weaknesses in the solid materials
synthesized, understanding of the mechanisms that trigger
instabilities is expected to give appropriate directions for
controlling the SHS process.
The nonsteady flame propagation can assume two characteristic modes [3,7,9] namely pulsating and spinning
combustion. In pulsating combustion, the combustion
wave travels in a planar but pulsating manner as a result
of the one-dimensional instability of the planar combustion
wave to longitudinal perturbations. In spinning combustion,
the combustion wave is nonplanar, and one or more hot
spots are observed to move along a spiral path over the
surface of the sample specimen as a consequence of the
instability of the one-dimensional oscillatory to transverse
perturbations. It has been observed that spinning combustion occurs in low exothermic systems with excess amounts
of either one of the reactants, or the diluent, or the large
particles, at low preheat temperatures. As for pulsating
combustion, it has been observed in energy intensive
systems with stoichiometric mixtures of small particles at
elevated preheat temperatures.
Since rapid synthesis is preferred from a practical point of
view, and it can be accomplished by using small particles
and/or high preheat temperatures, pulsating combustion
frequently occurs and results in materials with a laminated
structure [3] (cf. Fig. 3), which cannot be used as a bulk. As
such, the study and control of transition from steady
combustion to pulsating combustion are of greater practical
relevance than those of the transition from steady combustion to spinning combustion.
In order to identify and gain an understanding on the
boundary governing the transition from steady to pulsating
combustion, we are required to extend the steady heterogeneous theory of SHS flame propagation developed in Refs.
[32,64] to allow for a nonsteady effect, as has been done in
Ref. [92]. In the following, first by considering the material
strength and flame instability, description for the boundary
between steady and pulsating combustion is made in terms
of the system parameters, which consist not only of the
52
A. Makino / Progress in Energy and Combustion Science 27 (2001) 1–74
physicochemical parameters, but also of the operating parameters, such as the mixture ratio, degree of dilution, particle size, and initial temperature. From this identification,
operating parameters at which steady flame propagation is
maintained, even at the adiabatic condition, are obtained, to
elucidate the necessary condition for pulsating combustion,
in order to identify the useful parameter in suppressing
pulsating combustion. Finally, transverse effects related to
heat loss from the side surface of the compact are examined
to assess the range of steady combustion.
7.1. Criterion for the appearance of cracks
Pulsating combustion is characterized by the periodic
movement of the combustion wave, with the laminated
structure of the combustion product being caused by the
presence of cracks in the radial direction. Although it has
been pointed out [51] that both thermal and mechanical
effects induce pulsating combustion, the main concern has
been directed to the thermal effect, and various numerical
and analytical investigations have been conducted, as
reviewed by Margolis [24]. However, the appearance of
radial cracks in the combustion products implies that
mechanical effects caused by pressure increase in pores
[93,94] and/or thermal expansion, can exert strong influence
on the combustion behavior. When there exists temperature
distribution along the axial direction, there appears not only
a normal stress corresponding to macroscopic pressure
distribution along the axial direction, due to desorbed gas
in pores, but also a shear stress corresponding to a difference
of thermal expansions in the radial direction. When strains
caused by these stresses are kept below a certain value Ksb,
determined by the material fracture toughness, as well as the
coefficient of thermal expansion, radius of the curvature of
the pores, compact diameter, and so on, we can anticipate
that radial cracks can be suppressed and steady combustion
can be maintained. Since both of these stresses are related to
a difference of the temperature gradients along the axial
direction, and the difference is the maximum at the propagating combustion wave, the criterion that steady combustion is maintained can be expressed as [95]
2T
2T
ⱕ Ksb :
⫺
…176†
2x 0⫺
2x 0⫹
In evaluating the temperature gradients, an attempt [92]
has been made to apply and extend the linear stability analysis within the framework of the heterogeneous theory, by
linearizing the problem about the basic solution and seeking
harmonic solutions through the perturbation quantities u p(s ,
t ) and ap(t ) as
u ˆ u 0 …s† ⫹ u p …s; t†; …m=m a † ˆ a0 ⫹ ap …t†;
…177†
as has been made by Margolis [24,74] and Kaper et al. [73]
based on the homogeneous premixed-flame theory. Note
that the perturbation quantities are O(b ⫺1), where b is the
Zeldovich number.
7.2. Linear stability analysis
7.2.1. Basic solution for planar propagation
Prior to the linear stability analysis, let us first confirm the
governing equations and boundary conditions. By introducing the nondimensional time defined as
tˆ
…Z∞ ⫺ Z0 † 2 m2a t
;
…l=c†rt
…178†
in addition to the other variables and parameters introduced
in Sections 4.2 and 5.1, the energy conservation equation
and the N-concentration equation are expressed as
2
…m=m a † 2
22u
F…u†
…j ⫺ u† ⫹
…j ⫺ u† ⫹
ˆ
;
2t
…Z∞ ⫺ Z0 † 2s
…Z ∞ ⫺ Z0 † 2
2s 2
…179†
2j
…m=m a † 2
Z0
⫹
j⫹
…Z∞ ⫺ Z0 † 2s
2t
Z∞ ⫺ Z0
1=3
x~
Z∞ ⫺ Z0
ˆ
1⫺
j
:
~
1 ⫺ Z0
…Z∞ ⫺ Z0 †L0 …rt 0=rt † exp…T~ d =T†
…180†
The boundary conditions are as follows:
s ! ⫺∞ :
u ˆ 0; j ˆ 0;
…181†
sˆ0:
u ˆ u max ;
…182†
s!∞:
2u
F…u†
2j
ˆ⫺
ˆ 0:
;
…m=m a †…Z∞ ⫺ Z0 † 2s
2s
…183†
The mass burning rate m…ˆ r t u†; subject to heat loss L,
would emerge from the solution as an eigenvalue, with ma
and L 0 being, respectively, the mass burning rate and the
mass burning rate eigenvalue in the steady, adiabatic limit.
The SHS rate-constant u0·R0 is obtained from this eigenvalue.
In conducting the linear stability analysis, we are first
required to obtain the basic solution u 0(s ) and a0. Since
the perturbation quantities are O(b ⫺1), the basic solution
is exactly the same as that for the asymptotic “outer” solution in the steady state, as presented in Eqs. (113) and (114),
in the limit of large Zeldovich number b . In Ref. [73], the
basic solution is expressed in a slightly different form as
u ˆ C exp…l u s†; j ˆ 0; …s ⬍ 0†;
…184†
u ˆ C exp…l d s†; j ˆ 1; …s ⬎ 0†;
…185†
where
1
a0
1⫹
2 Z∞ ⫺ Z0
1
a0
1⫺
ld ˆ
2 Z∞ ⫺ Z0
lu ˆ
1
;
C
1
;
C
…186†
and C is the maximum temperature in the normalized form,
A. Makino / Progress in Energy and Combustion Science 27 (2001) 1–74
expressed as
s
4C
C 1 ⫹ 2 ˆ 1;
a0
…187†
which is obtained from the jump in du /ds at s ˆ 0: Note
that by conducting a similar treatment in Eq. (125), we have
the relation
m 2 exp ‰b…C ⫺ 1†Š ⫺ M
…188†
ˆ
ma
k…1 ⫺ M†:
Having obtained the basic solution, we can now evaluate
the difference of the temperature gradients in the criterion in
Eq. (176). Then, we have
a0 …Z∞ ⫺ Z0 † ⫹ O…b⫺1 † ⱕ ‰a0 …Z∞ ⫺ Z0 †Šsb ⬅
l=c
Ksb
ma …q0 =c†
…189†
for steady combustion. We see that the appearance of cracks
in pulsating combustion can be primarily controlled by
a0 …Z∞ ⫺ Z0 † because the perturbed O(b ⫺1) term is much
smaller than the first term. Note here that Z∞ ⫺ Z0 is related
to the heat of combustion, which depends on the mixture
ratio m , degree of dilution k , and initial temperature T0, and
that a0 is related to the heat loss which depends on the
particle size R0 and the compact diameter 2r. When Z∞ ⫺
Z0 is kept constant, we can anticipate that steady combustion
occurs in the range a0 ⬍ …a0 †sb ; which is equivalent to C ⬎
Csb ; if we recall the relationship between C and a0. When
the heat loss condition is kept constant, the steady combustion range is Z ∞ ⫺ Z0 ⬍ …Z∞ ⫺ Z0 †sb ; which yields m ⬍ msb
for under-stoichiometric mixture …m ⬍ 1† and m ⬎ msb for
an over-stoichiometric mixture …m ⬎ 1† at a certain k ; for
constant m , k ⬎ ksb :
Eq. (189) further suggests that when Z∞ ⫺ Z0 ⬍ …Z∞ ⫺
Z0 †sb ; steady combustion can be maintained, even if the
perturbed O(b ⫺1) term is positive; likewise, when Z∞ ⫺
Z0 ⬎ …Z∞ ⫺ Z0 †sb ; steady combustion cannot be maintained
even if the perturbed term is negative. In this context, the
condition for no amplification, nor attenuation of the
perturbed term is anticipated to provide information about
the transition boundary, and then we are required to obtain
the specific form of the perturbed term in Eq. (189).
7.2.2. Asymptotic solution and linear stability boundary
In obtaining asymptotic solution, we should note that the
first order inner solution is identical to that for the adiabatic
combustion when the Zeldovich number is large, because of
a higher order effect of heat loss, while the outer problem is
influenced by both heat loss and unsteadiness. The outer
problem is then given by [92]
2u
…a0 †
2u
2 2u
C·u
⫹
ˆ
⫺
;
2
…Z∞ ⫺ Z0 † 2s
2t
2s
…Z ∞ ⫺ Z0 † 2
…190†
u ˆ 0 as s ! ^∞:
…191†
53
In examining the linear stability of the steady planar SHS
flame, it is the standard to linearize the nonlinear problem
(Eqs. (179)–(183)) regarding the basic solution (Eqs. (184),
(185), (187) and (188)) and to seek harmonic solutions.
Substituting the perturbation quantities in Eq. (177) into
Eq. (190), we obtain
2u p
2up
ap
a0
du0
⫹
⫹
2t
…Z∞ ⫺ Z0 † 2s
…Z∞ ⫺ Z0 † ds
ˆ
2 2up
C· u p
⫺
;
2
2s
…Z∞ ⫺ Z0 † 2
…192†
up ˆ 0 as s ! ^∞:
…193†
Here, a relation between the perturbed quantities is given as
[73]
(
)
B
b
M
ˆ
1⫹ 2
a0
2
a0 k…1 ⫺ M†
a p ˆ Bup …0; t†;
…194†
where use has been made of Eq. (188). Note that, since 1 ⫺
M is the normalized driving force for the flame propagation,
as mentioned in Section 5.6.1, B/a0 is a measure of the
energy generated in the combustion wave compared to the
driving force for the nonadiabatic flame propagation, that is
a measure of the driving force for instability.
A normal mode solution for u p(s , t ) is then found to be a
scalar multiple of
"
1
iv
u p …s; t† ˆ exp …ivt† ⫺
(
× exp…lu s† ⫹ 1 ⫹
1
iv
!
a0
Z∞ ⫺ Z0
!
a0
Z∞ ⫺ Z0
!
B
l C
a0 u
!
)
B
lu C
a0
#
× exp …pu s† ; …s ⬍ 0†;
"
(195)
1
iv
a0
Z∞ ⫺ Z0
1
× exp …ld s† ⫹ 1 ⫹
iv
#
a0
Z∞ ⫺ Z0
u p …s; t† ˆ exp…ivt† ⫺
(
× exp…pd s† ; …s ⬍ 0†;
where
!
!
!
B
l C
a0 d
!
)
B
lC
a0 d
(196)
54
A. Makino / Progress in Energy and Combustion Science 27 (2001) 1–74
8
s
9
<
=
1
a0
1
Z∞ ⫺ Z0 2
pu ˆ
⫹4
…iv† ;
1⫹
2
:
;
2 Z∞ ⫺ Z0
a0
C
…197†
8
9
s
<
=
1
a0
1
Z∞ ⫺ Z0 2
pd ˆ
⫹4
…iv† :
1⫺
2
;
a0
2 Z∞ ⫺ Z0 :
C
…198†
Having obtained the perturbation quantities u p(s ,t ) and
ap(t ), we can now evaluate the perturbed term in Eq. (176)
as
"
!2
!
1
a0
B
a 0 …Z∞ ⫺ Z0 † exp…ivt† ⫺
2…iv† Z∞ ⫺ Z0
a0
(
1
⫹ 1⫹
2…iv†
a0
Z∞ ⫺ Z0
!2
! )
B
C
a0
…199†
v
!2
#
u
u 1
Z ⫺ Z0
t 2 ⫹ 4 ∞
…iv† :
a0
C
Since the jump condition with respect to 2u=2s at s ˆ 0
gives iv as [73]
! 2" (
!2
! )
1
a0
1
B
B
C
iv ˆ
⫺
⫹4
⫺
8 Z∞ ⫺ Z0
a0
a0
C2

!v
!2
! #
u
1
B u
1
B
B
t
^
⫺
⫹
C ;
⫺8
C
a0
C
a0
a0
…200†
we see that the perturbed term attenuates in the regime
0
s1
p !
6
B
1 A
1
0ⱕ
;
1
ⱕ
⬍
;
⬍ 2C@1 ⫹ 1 ⫹
a0
C
2
4C4
Fig. 54. Location of the neutral stability boundary [92].
where Xsb is determined by solving the neutral stability
condition
b
X
2 X⫺M
8
9
s
2=
<
2
1
2
ˆ s 1 ⫹ 1 ⫹
1 ⫺ ln X
; …206†
:
;
4
b
2
1 ⫺ ln X
b
with X ⬅ exp…⫺2bC=a 02 † ˆ a20 k…1 ⫺ M† ⫹ M: In obtaining
Eq. (206), use has also been made of Eqs. (187) and (194).
Fig. 55 shows the neutral stability boundary in the b –M
plane, with X being taken as a parameter.
Once the critical values of the heat-loss parameter C sb,
and the flame temperature u sb, are obtained, the critical size
of N particles and compact diameter for steady combustion
…201†
0ⱕ
B
a0
⬍
1
;
C…1 ⫺ C2 †
!
p
1
6
⬍
;
2
C
…202†
because of the negative value of the real part of iv , as shown
in Fig. 54. Restricting ourselves to the neutral stability
boundary on which there is neither attenuation nor amplification of the burning velocity, we have a set of critical
values as
…a0 †2sb ˆ
Xsb ⫺ M
;
k…1 ⫺ M†
…203†
C sb ˆ ⫺
…a0 †sb2
ln Xsb ;
2b
…204†
usb ˆ 1 ⫹
ln Xsb
b
…205†
Fig. 55. Neutral stability boundary in the b –M plane [92].
A. Makino / Progress in Energy and Combustion Science 27 (2001) 1–74
Fig. 56. Ranges of flammability and instability for Ti–C system
with l=…rt c† taken as a parameter [92]; the initial temperature T0 ˆ
300 K and the compact diameter 2r ˆ 18 mm; (a) as functions of
the mixture ratio m and the radius R0 of N particles, with l=…rt c†
taken as a parameter; and (b) as functions of the degree of dilution k
and R0. In the region below the dashed curve, pulsating combustion
proceeds. In the region between the solid (or dotted) and dashed
curves, stable and steady propagation occurs. Data points are
experimental [43,77] and notation is the same as that in Fig. 43.
are determined through
!
C sb …rt c†…u0;a R0 † 2
R 02
ˆ
2r sb ‰l=…rt c†Š4es SB …Tsb2 ⫹ T02 †…Tsb ⫹ T†
…207†
Note that the SHS rate-constant u0,a·R0 in the adiabatic
condition is given by Eq. (129).
7.2.3. Experimental comparisons for the boundary
In order to demonstrate the appropriateness of the transition boundary determined above, a comparison has been
made with Ti–C system taken as an example. Values of
the physicochemical parameters are the same as those
described in Sections 4.4 and 5.5, while the Lewis number
Le0 ˆ 25 in accordance with experiments with spherical
carbon particles.
Fig. 56(a) shows the ranges of flammability and
instability with respect to the mixture ratio m and the radius
R0 of N particles for an initial temperature T0 of 300 K. In
accordance with the experiments [77] the compact diameter
55
2r is 18 mm. The solid and dotted curves are respectively
the numerically [64] and analytically [72] obtained
extinction limits. The dashed curves are the neutral
stability boundaries, with l /(r tc) taken as a parameter.
Results show that, except near stoichiometry …m ˆ 1†;
the range of instability with respect to R0 is reduced
with decreasing mixture ratio. When m ⬍ 0:673;
instability is suppressed, even for R0 ! 0; because of
the decrease in the volumetric heat-generation with
decreasing m . We also see that the instability can be
suppressed with increasing l=…rt c†: This is attributed to
the fact that heat transferred from the consumption zone
increases with increasing l=…rt c†: Likewise, a reduction
of T0 is also expected to suppress instability, although
the range of flammability is reduced. The slight increase
in the range of instability with decreasing m near stoichiometry is attributed to the overestimation in Ij in Eq.
(92), as noted in Section 4.6. Experimental results [77]
also show that the range of pulsating combustion
extends with decreasing R0. The decrease in R0
increases the total surface area for surface reactions in
the combustion wave, and hence, makes the mixture
more energy intensive. Since instability is invariably
dependent on the burning intensity, mixtures with
small particles have wider ranges for pulsating combustion. Experimental results suggest that the limit of
instability predicted by the present analytical result is
fair.
Fig. 56(b) shows the similar comparison with respect
to the degree of dilution k . We see that the range of
instability with respect to R0 is narrowed with increasing k and that instability is suppressed even at R 0 ! 0
when k ⬎ 0:280: Again, fair agreement is demonstrated
with the experimental results [43,77] in the trend and
the approximate magnitude.
Fig. 56(a) and (b) show that the predicted limit of instability with l=…rt c† ˆ 5 × 10 ⫺6 m2 =s fairly represents the
experimental results, while the limit of extinction has been
obtained with l=…rt c† ˆ 2:5 × 10⫺6 m2 =s: This difference
may be caused by the different temperatures at which
these two phenomena occur; that is, the temperature at
which the pulsating combustion ceases to occurs is about
200 K higher than that for flame extinction.
Fig. 57 shows the ranges of flammability and instability
with respect to m and k , with R0 ˆ 1:5 mm at T0 ˆ 300 K:
The solid curve is the limit of flammability [64] the dashed
curve the boundary for R0 ˆ 1:5 mm obtained by numerically solving Eq. (206) with l=…rt c† ˆ 5 × 10⫺6 m2 =s; and
the dotted curve the boundary for the adiabatic condition to
be mentioned in the next section. It is seen that the pulsating
combustion can be suppressed with decreasing m and/or
increasing k because of the decrease in the volumetric
heat-generation. Fair agreement is again demonstrated
between the predicted and experimental [77] results for
spherical carbon particles (3 mm in diameter). Since the
condition for these experimental results is considered to
56
A. Makino / Progress in Energy and Combustion Science 27 (2001) 1–74
Fig. 57. Ranges of flammability and instability for Ti–C system as functions of m and k , for T0 ˆ 300 K and 2r ˆ 18 mm [95]. The solid curve
is the limit of flammability [64], the dashed curve the numerically obtained boundary [92] for R0 ˆ 1:5 mm; and the dotted curve the predicted
result [95] in the adiabatic condition. Data points are experimental [77] for R0 ˆ 1:5 mm and notation is the same as that in Fig. 43.
be close to the adiabatic condition, as shown in Fig. 56(a)
and (b), heat-loss effect on this boundary is relatively small.
7.2.4. Boundary at the adiabatic condition
In the SHS process, reduction of particle size, which
reduces heat-loss effect, facilitates the occurrence of pulsating combustion when the system parameters, such as the
physicochemical and operating parameters in Sections
4.6.2 or 5.6.3, are given. However, even if we use small
particles, there is a possibility that steady combustion is
maintained because appearance of pulsating combustion
also depends on the operating parameters, such as the
mixture ratio m , degree of dilution k , and initial temperature
T0, through the heat of combustion when the physicochemical parameters for a specific system are given. Such that,
operating parameters at which steady combustion proceeds
even at the adiabatic condition (i.e. X ˆ 1) has been
obtained in order to elucidate the necessary condition for
pulsating combustion, under which the only way effective in
suppressing the appearance of pulsating combustion is
increasing size of N particles. In order for Xsb to be unity,
the critical values of b and M must satisfy
p
b=2
ˆ 2 ⫹ 5;
…208†
1 ⫺ M sb
which has already been identified in the field of stability
analysis. If we recall the expression 1 ⫺ M ˆ
bIu exp…b=a† in Section 5.6.1 and Iu in Eq. (87), we can
obtain …T∞ ⫺ T0 †sb because a and b are defined by use of
T∞ ⫺ T0 : Note that when the normalized melting-point u m is
smaller than about 0.7, the following equation fairly represents the relation between a and b .
p
p
b=a
5 ⫹ 2 p exp…y2 † erfc…y†
; y⬅
p
a⬇
:
…209†
y
2
2
Since T∞ ⫺ T0 is related to the system parameters as [40]
8
m…1 ⫺ k†
>
>
…m ⱕ 1†
>
< m ⫹ fst
T~ ∞ ⫺ T~ 0 ˆ Z∞ ⫺ Z0 ˆ
;
…210†
>
…1 ⫺ k†
>
>
…m ⱖ 1†
:
m ⫹ fst
when there is no external heating, we can obtain the critical
values for these operating parameters.
Fig. 58(a) shows the boundary between steady combustion and pulsating combustion in the adiabatic condition, as
functions of the mixture ratio m and the initial temperature
T0, when there is no dilution …k ˆ 0†: The solid curve is the
limit of flammability numerically obtained [64] and the
~ 0 is the boundary obtained with Eq.
dashed curve for kˆ
(208) [95]; the chain curve for k~ ˆ k~lower is the boundary
when the transverse effect is the largest, which will be examined in the next section. We see that the pulsating combustion regime expands with increasing T0, and that the steady
combustion regime contracts. Experimental results
[33,43,77] for small carbon particles, with diameters less
than 3 mm, are also shown because these data are considered
to be influenced very little by the heat loss (cf. Fig. 57). Fair
agreement is demonstrated as far as the general trend is
concerned. Fig. 58(b) shows the transition boundary as function of k and T0, when the system is stoichiometric. The
extent of agreement is similar to that in Fig. 58(a).
7.3. Transverse effects on the range for steady combustion
The transition from steady combustion to pulsating
combustion is in general accompanied by the appearance
of nonplanar spatial perturbations. Through various analyses
for examining the transition boundary by use of the multidimensional media with cylindrical [74,96,97] or rectangular [98–100] shapes, it is reported that this kind of
A. Makino / Progress in Energy and Combustion Science 27 (2001) 1–74
57
Fig. 58. Ranges of flammability and instability for Ti–C system [95]; (a) as functions of m and T0; (b) as functions of k and T0. The solid curve is
the limit of flammability [64] numerically obtained, while the dashed curve for k~ ˆ 0 and the chain curve for k~ ˆ k~lower are the numerically
obtained boundaries [95] between steady and pulsating combustion; k~ ˆ 0 means that there is no transverse effect, while k~ ˆ k~lower means the
transverse effect is the largest. Data points are experimental [33,43,77] when R0 is less than 1.5 mm; notation is the same as that in Fig. 43.
perturbation can be properly described by introducing overall transverse wave-number k~ into the linear stability analysis. When this effect is taken into account, the boundary
under heat-loss conditions is given by the following relation:
[73,74]
3
1
b
X
C⫺
C
2 X⫺M
2
1
1
b
X
⫹ 2 ⫹ 4k~2
⫹ ⫺4C C ⫺
C
2 X⫺M
C
1
1
1
b
X
⫹ 4C C ⫺
⫺
⫹ 4k~2
2
C
C
2 X⫺M
C
2
1
⫺
⫹ 4k~2 ˆ 0:
(211)
C2
given by solving
b
X
4
ˆ s ;
2 X⫺M
2
1 ⫺ ln X
b
…213†
~ which is specified as k~lower : For the
appears at a certain k;
stoichiometric Ti–C system, we have b=2 ˆ 2:29 and M ˆ
0:240 at T0 ˆ 300 K: In this system, Xsb is 0.5792 when
~ is given, the
When a set of parameters (b , a , M, and k)
critical value of Xsb can be determined. As reported in
Refs. [73,74] and as shown in Fig. 59, with an increasing
k~ for fixed Xsb, b /2 first decreases, reaches a minimum, and
then increases. The minimum value of b /2 exists at k~ determined by solving
4k~2 ˆ 2 ⫺
1
:
C2
…212†
We see in Fig. 59 that b /2 decreases with decreasing Xsb
when k~ is fixed.
It may be common in practical situations that the system
parameters such as b , a , and M are given by specifying
combinations of metal and nonmetal species, initial
temperature, and so on. Thus, it is of interest to examine
~ with b /2 taken as a
changes in Xsb due to an increase in k;
~
constant. With increasing k for fixed b /2, Xsb determined
from Eqs. (211) and (212) first increases, reaches a maximum, and then decreases. The maximum of Xsb, which is
Fig. 59. Neutral stability boundary for stoichiometric Ti–C system,
~ the half of the Zeldovich number
as functions of the wave-number k;
b , and the normalized heat loss Xsb at the critical condition. The
initial temperature T0 is 300 K and there is no dilution.
58
A. Makino / Progress in Energy and Combustion Science 27 (2001) 1–74
there are no effects of the transverse wave-number …k~ ˆ 0†;
~ k~lower ˆ 0:4489; as shown in Fig.
while Xsb ˆ 0:6412 for kˆ
59. This result suggests that the steady combustion regime is
the broadest when k~ ˆ k~lower :
The boundary between the steady and pulsating combustion determined with k~ ˆ k~lower without heat loss …X ˆ 1† is
also shown in Fig. 58(a) and (b). Although k~ depends on the
compact diameter and the transverse mode of the combustion wave, because it is influenced by the boundary conditions in the radial direction such as the heat-loss
condition on the side surface of the compact, we see that
the difference between the boundary for k~ ˆ 0 and that for
k~ ˆ k~lower is relatively small.
Fig. 60(a) shows the ranges of flammability and instability for Ti–C system at T0 ˆ 300 K as functions of m and
R20 =…2r† [95]. The solid curve is the limit of flammability
numerically obtained [64] the dotted curve is that analytically obtained [72]; the dashed curve is the boundary
between the steady and pulsating combustion for k~ ˆ 0;
and the chain curve is that for k~ ˆ k~lower ; by use of Eqs.
(206) and (211), respectively. Data points are experimental
[43,77,78] for various particle radii (1.5, 5, 10, 15 mm) and
compact diameters (3, 5, 7, 10, 18 mm). We see that the
range of instability is extended with decreasing R20 =…2r†
because the effect of heat loss is reduced due to the increase
in the volume-to-surface ratio. Fair agreement between
the predicted and the experimental results suggests that
the heat loss index R02/(2r) is useful in correlating
experimental results. Fig. 60(b) shows the similar plot
for k and R20 =…2r†:
Finally, it may be informative to note that bifurcation
[74,96,97,99] of pulsating and spinning combustion, and
chaotic combustion [98,100] can occur at a certain con~ as shown in
dition, because of the trend between b and k;
Fig. 59. Topics about bifurcation and the chaotic combustion, which are related to the nonlinear stability, are not
covered here; readers are recommended to refer Ref. [24].
7.4. Some remarks on the boundary of steady combustion
Use of small particles enhances the SHS flame propagation speed and necessarily induces pulsating combustion,
such that the identification obtained here offers the limit
of the use of small particles when other system parameters
are kept unchanged. As for the effects of transverse wave
number, which are related to heat loss on the side surface of
the compacted medium on the location of the transition
boundary, it has been found to be relatively small. Further,
appropriateness of the transition boundary, which has been
demonstrated by comparing the predictions with experimental results in the literature, suggests that the heterogeneous
theory has captured the essential, heterogeneous feature of
the SHS flame propagation and that the results are of practical importance in controlling the combustion behavior to
be steady.
Fig. 60. Ranges of flammability and instability for Ti–C system for
T0 ˆ 300 K [95]; (a) as functions of m and R20 =…2r†; and (b) as
functions of k and R20 =…2r†: Data points are experimental
[43,77,78] for various particle radii and various compact diameter;
notation is the same as that in Fig. 43. The solid curve is the limit of
flammability numerically obtained [64], the dotted curve is that
analytically obtained [72], and the dashed curve for k~ ˆ 0 and the
chain curve for k~ ˆ k~lower are the numerically obtained boundaries
[95].
8. Initiation of the combustion wave
Besides flame propagation, initiation of the combustion
wave (which is indispensable for the SHS process) has also
been investigated extensively because the energy that must
be supplied externally in the SHS process is basically that
for initiating the combustion wave. Although much of the
theoretical accomplishment about the initiation of the
combustion wave has been reviewed by Barzykin et al.
[101] and Barzykin [102] all are based on the homogeneous
premixed-flame theory, and hence the heterogeneity has not
A. Makino / Progress in Energy and Combustion Science 27 (2001) 1–74
59
over a short period when the activation energy (or temperature) is large. Let tc denote an ignition delay time, at which
temperature rises rapidly. During most of time up to the
ignition, t ⬍ tc ; the specimen experiences only inert heating
because heat-generation term is exponentially small due to
large activation energy, and reactant consumption can be
neglected. Then, Eq. (214) becomes as
Fig. 61. Model used for analyzing initiation of the SHS combustion
wave in the specimen, induced by another SHS flame that has
passed through the igniter [103].
aj
2T~
2T~
1
4hrad ~
⫹
ˆ
…T ⫺ T~ 0 †;
2t
…rt †j c·2r
2x2
…j ˆ ig; sp†;
…216†
been clarified. The heterogeneous theory, which has
succeeded in describing dependence of the flame-propagation process on the particle size, is also expected to be
applied for examining ignition phenomena. First, we shall
consider the initiation of the combustion wave induced by
an igniter [103] in order to examine heterogeneity that has
not been examined in a series of analyses [104–110]. After
that, initiation of the combustion wave by use of the radiative heat flux will be examined.
where
T~ ˆ …cT†=q0 ;
a ˆ l=…rt c†:
The boundary conditions at x ˆ 0 are as follows:
~ ig ˆ …T†
~ sp ;
…T†
l
8.1. Flame initiation induced by igniter
One-dimensional, planar, heterogeneous flame initiation
has been considered in a semi-infinite medium, which is
heated by an SHS flame passed through the other semi-infinite medium, called an igniter; cf. Fig. 61. It is supposed that
the region x ⬎ 0 is the igniter with thermophysical properties of l ig, (r t)ig, and cig, and that the region x ⬍ 0 is the
medium to be ignited, called a specimen, with l sp, (r t)sp, and
csp. The contact surface is set to be x ˆ 0 and the moment
that the SHS flame reaches the surface is t ˆ 0: The igniter
and the specimen are compacted media respectively,
consisting of particles of nonmetal (or higher meltingpoint metal) N, lower melting-point metal M, and an inert
I that can be the product P of the reaction between N and M
according to nM M ⫹ nN N ! nP P: Monodisperse size distribution is considered for N particles, with an initial radius R0
and number density n0.
Governing equations for these media, from Eqs. (179) and
(180), are expressed in a dimensional form as follows.
Energy:
rt
2
22 T
2Z
…cT† ˆ l 2 ⫹ q0 rt
⫺ L:
2t
2t
2x
N-consumption:
rt
2Z
rt
ˆ …4prN n0 R0 †
x
2t
rt;0
!
1⫺Z
1 ⫺ Z0
:
…215†
8.1.1. Inert stage
Ignition processes can usually be characterized by a
gradual increase in temperature followed by a rapid increase
2T~
2x
!
ˆ l
ig
2T~
2x
!
:
…218†
sp
As the initial conditions, we have
"
x ⬍ 0 : …T~ ⫺ T~ 0 † ˆ …T~ max ⫺ T~ 0 † exp
#
…m†ig
x ; Z ˆ Z0 ;
…l=c†sp
…219†
x ⬎ 0 : …T~ ⫺ T~ 0 †
"
1
ˆ …T~ max ⫺ T~ 0 † exp ⫺
…m†ig
#
4hrad
x ; Z ˆ Z∞ ;
c·2r
…220†
which are obtained for the steady SHS flame propagation
(e.g. Ref. [73]), under an assumption of large activation
energy for the rate-limiting process, by noting that melting,
diffusion, and consumption/convection are confined to a thin
layer in the combustion wave. Putting
sˆ
…Z ∞ ⫺ Z0 †sp …ma †sp x
…Z∞ ⫺ Z0 †sp2 …ma †2sp t
; tˆ
;
…l=c†sp
…l=c†sp …rt †sp
uˆ
⫺ T~ 0
T~ ⫺ T~ 0
T~
; us ˆ max
;
…T~ ∞ ⫺ T~ 0 †sp
…T~ ∞ ⫺ T~ 0 †sp
…214†
1=3
…217†
…m†ig =…ma †sp
C
k~ ˆ
; w~ ˆ
;
~ ∞ ⫺ Z0 †sp2
…Z∞ ⫺ Z0 †sp
k…Z
Cˆ
a sp …4hrad †
…rt c†sp …u0 ·R0 †sp2
!
R20
l
; L ˆ p
2r
a
60
A. Makino / Progress in Energy and Combustion Science 27 (2001) 1–74
and using the Laplace transformation, we have
!
us
s2
~
uI ˆ
⫺ w~ k t
exp ⫺
‰1 ⫹ …Lsp =Lig †Š
4t
(
×
L sp
1
1⫹
2
Lig
"
!
diffusive region near the contact surface, stretched variables
c ˆ b…u ⫺ u I †; X ˆ ⫺bKz …s ⫺ s c †;
are introduced, where b is the Zeldovich number defined as
! 2#
p
s
p ⫹ k~ t
2 t
!
!
p
L sp
s
1
~
p ⫹ k t ⫺
1⫺
2
2 t
Lig
× erfc
"
bˆ
exp
p
s
⫺ p ⫹ k~ t
2 t
!
p
s
~
× erfc ⫺ p ⫹ k t
2 t
⫹ exp
d2 c
D
ˆ ⫺ t ec⫺X ;
2
dX 2
!
2
x~ c
Dt ˆ
;
bKz2 …Z∞ ⫺ Z0 †L0 …rt;0 =rt † exp…T~ d =T~ c †
! 2#
× erfc
which is subject to
dc
…c† Xˆ0 ˆ 0;
ˆ 0:
dX X!∞
s! 2 #
a ig
s
⫺ p ⫹ w~
t
2 t
asp
s!)
a ig
s
⫺ p ⫹ w~
t :
2 t
asp
(221)
Here, L ig and L sp are the heat penetration coefficients. In the
following, temperature rise in the inert, heating stage is
designated with the subscript I. Note that temperature
response for a finite size of the igniter can also be obtained
in the same manner, as shown by Strunia et al. [106]
8.1.2. Transition stage
In analyzing the transition from inert heating to vigorous
reaction, temperature deviation must be taken into account
because the heat generation term becomes important as
temperature rises. Since departure from the inert heating is
anticipated to occur at time and position such that uI ⫺ …uI †c
is close to zero, we have the lowest order solution as
uI ˆ …uI †c ⫹ Kt …t ⫺ tc † ⫹ Kz …s ⫺ s c †;
…222†
by expanding Eq. (221), where Kt ˆ …uI =ut†c ; Kz ˆ
…2uI =2s† c : Note that (·)c designates the condition corresponding to the rapid increase in temperature, which occurs
at t ˆ tc and s ˆ s c ; …uI †c is the maximum temperature in
the inert heating. Since temperature at a certain position in
the specimen first increases and then decreases, the most
plausible condition for the ignition is that for 2uI =2t ˆ 0;
at which the position has its maximum temperature.
The inner problem for the asymptotics is then as
2u
2 2u
x~
:
ˆ
⫹
~
2t
2s z2
…Z∞ ⫺ Z0 †L0 …rt0 =rt † exp…T~ d =T†
T~ d …Z∞ ⫺ Z0 †
:
T~ c2
…225†
Retaining only nonvanishing terms on the exponential, we
have
× exp
"
…224†
…223†
Here, reactant consumption and heat loss are ignored, as is
the case for usual ignition phenomena. Identification of Eq.
(223) infers that the present problem parallels the earlier
development [104–110]. In analyzing the reactive–
Eq. (226) can be solved analytically and we have
"
(
!2 )#
1
⫺1 ⫹ C e ⫺X
;
c ˆ X ⫹ ln
1⫺
Dt
1 ⫹ C e⫺X
p
1 ⫺ 1 ⫺ Dt
p
Cˆ
1 ⫹ 1 ⫺ Dt
…226†
…227†
…228†
…229†
(e.g. Ref. [111]). Since there is no solution for Dt ⬎ 1; the
critical condition is as follows:
!
2
x~ c
Dt ⬅
ˆ 1; …230†
bKz2 …Z∞ ⫺ Z0 †L0 …rt;0 =rt † exp…T~ d =T~ c †
where
2uI
2s
Kz ˆ
(
!
ˆ
c
us
exp…⫺w~ k~tc †
1 ⫹ …Lsp =Lig †
×
p
k~ exp…k~ 2 tc † erfc…k~ tc †
⫺ w~
!
s
s!)
aig
aig
aig
:
exp w~ 2
tc erfc w~
t
asp
asp
asp c
(231)
Recalling definitions of parameters, we see that the ignition
delay time can be expressed as functions of not only physicochemical parameters, but also operating parameters characteristic to the SHS process, such as the mixture ratio,
degree of dilution, initial temperature, and particle radius,
through the eigenvalue L 0 and Z∞ ⫺ Z0 :
It may be informative to note that the criterion used by
Averson et al. [104] and/or Strunia et al. [106] was Dt ˆ 2
which was obtained by Enig [112] by the use of the so-called
integral method by equating a transient–reactive relation at
the flame initiation and a transient–diffusive relation in the
inert heating. Such that, compared to the criterion in
A. Makino / Progress in Energy and Combustion Science 27 (2001) 1–74
61
We see that the ignition delay time is proportional to the
square of the particle size, which has not been captured in
previous theories.
Fig. 62. Ignition delay time as a function of the mixture ratio m ig in
the igniter, with the emissivity e taken as a parameter [103]. Data
points are experimental [113].
Eq. (230) by the asymptotics, the ignition delay time is
overestimated by a factor of 2 when k~ ! ∞ and w~ ˆ 0:
8.1.3. Limiting solution
It may be informative to consider a limiting situation for
k~ ! ∞ and w~ ˆ 0;
…232†
which corresponds to the flame initiation caused by a spontaneous contact with a hot substance in adiabatic condition.
The critical condition yields explicit dependence of the ignition delay time on other parameters as
tc ˆ
R20 exp…T~ d =T~ c †
T~ d …T~ max ⫺ T~ 0 †2
:
2
2
~
2p‰1 ⫹ …Lsp =Lig †Š T c 3D0 …1 ⫺ Z0 †sp …rM =rN †x~ c
…233†
8.1.4. Experimental comparisons for the ignition delay time
In numerical calculations, the situation considered is that
both the igniter and the specimen consist of mixtures of
titanium and carbon particles, although their mixture ratios
and degrees of dilution are different. In accordance with
experiments [113] the initial temperature before arrival of
the SHS flame is set to be 450 K, the compact diameter is
18 mm, and particle radius of carbon particles in the igniter
is 5 mm. The Lewis number (Le0)ig for the igniter is set to be
100 [32] while that for the specimen, (Le0)sp, is 25 [74]
because different kinds of carbon powder are used in an
experiment; that in the igniter has an irregular shape,
while that in the specimen has a spherical shape. Ratio of
the heat penetration coefficients L sp/L ig is set to be 0.5,
accordingly.
Fig. 62 shows the ignition delay time tc as a function of
the mixture ratio m ig in the igniter, with the emissivity e
taken as a parameter, which is equivalent to varying heat
loss rate. The particle radius (R0)sp in the specimen is 10 mm.
With decreasing m ig, tc increases first gradually, then
rapidly, and reaches the ignition limit, due to the decrease
in heat supplied from the igniter. We see that the decrease in
the heat loss makes tc shorter, because the extent of heat
used for initiating the combustion wave becomes large. A
further decrease in the heat loss brings about a situation at
which the ignition limit locates beyond the extinction limit
for the combustion wave in the igniter, because of the heat
loss during the flame propagation [64]. Data points are
experimental for …R0 †sp ˆ 10 mm [113]. We see that the
ignition delay times are fairly correlated with e ˆ 1:0
when m ig is higher than about 0.8, while they are correlated
with e ˆ 0 as m ig decreases. This is attributed to the
Fig. 63. Ignition delay time tc for e ˆ 1, with (R0)sp in the specimen taken as a parameter [103]; (a) as a function of the mixture ratio m sp; and (b)
as a function of the degree of dilution k sp. Data points are experimental [113].
62
A. Makino / Progress in Energy and Combustion Science 27 (2001) 1–74
2
Fig. 64. Ignition delay time tc as a function of (R0)sp
, with e taken as
a parameter [103].
maximum temperature of the SHS flame that has passed
through the igniter because influence of heat loss can be
strongly suppressed as the flame temperature decreases by
reducing m ig in the igniter. Note that tc becomes short as
…R0 †sp in the specimen is reduced, because a decrease in the
particle size increases the total surface area which participates in the reaction.
Fig. 63(a) shows tc for e ˆ 1:0 as a function of m sp in the
specimen with …R0 †sp taken as a parameter. The mixture ratio
m ig in the igniter is 0.8. With decreasing m sp, tc for …R0 †sp ˆ
10 mm increases first gradually, then rapidly, and reaches
the ignition limit because of a decrease in the heat generated
during the flame initiation. As …R0 †sp is reduced, tc becomes
short. In addition, with decreasing m sp, tc gradually increases
and before reaching the ignition limit, encounters the flame
extinction limit [64] corresponding to …R0 †sp ; beyond which
establishment of the combustion wave is impossible, even
though initiated. Data point are experimental [113] and a
fair degree of agreement is demonstrated. Fig. 63(b) shows
the similar comparison with respect to k sp. It is also reported
that the same trends can be observed as those in Fig. 63(a)
and (b) by reducing the heat loss, instead of the particle size.
As for the effects of particle size on the ignition phenomena, little attention has been paid in the previous theoretical
works, as was reviewed [101,102]. However, particle size is
expected to exert great influences in the flame initiation no
less than in the flame propagation and/or extinction, because
it is closely related to the total surface area which participates in reaction [32,64]. Fig. 64 shows tc as a function of
…R0 †2sp ; with e taken as a parameter when mig ˆ 0:8 and
msp ˆ 1: As shown in Eq. (233), tc is proportional to
~ say,
…R0 †2sp when e ˆ 0; because of the large value of k;
larger than about 10. With increasing heat loss, tc becomes
large, because the extent of heat used for initiating combustion wave becomes small. At a certain heat loss, with
increasing …R0 †2sp ; tc increases first gradually, then
rapidly, and reaches the ignition limit. The increase in
tc is attributed to the decrease in the total surface area
for the reaction, caused by the increase in …R0 †sp : Note
that the flame extinction limit with respect to the particle radius is 20.9 mm when the compact diameter 2r is
18 mm [64].
Fig. 65(a) shows the ignitable range as functions of m sp
and R20 =…2r†: When there is no heat loss …e ˆ 0†; the ignitable
range coincides with the flammable range because ignition
is limited only by the flame extinction after the establishment of the combustion wave. The ignitable range is
contracted as m sp is reduced. In addition, it narrows drastically as the heat loss increases, because of the appearance of
the ignition limit. Experimental results [113] in Fig. 65(a)
also represent a fair agreement. Fig. 65(b) shows the similar
Fig. 65. Ignitable range with heat loss taken as a parameter [103]; (a) as functions of the mixture ratio m sp and the heat loss index ‰R20 =…2r†Šsp ; and
(b) as a functions of the degree of dilution k sp and the heat loss index ‰R20 =…2r†Šsp : Data points are experimental [113]. (W) designates the flame
initiation and ( × ) the non-ignition.
A. Makino / Progress in Energy and Combustion Science 27 (2001) 1–74
comparison with respect to k sp and again demonstrates a fair
degree of agreement.
8.1.5. Expression for the ignition energy
Concerning the ignition delay time tc, fair agreement has
been demonstrated between the predicted and experimental
results. Now, let us evaluate the ignition energy [114] for
establishing combustion wave in the specimen, by integrating heat flux through the contact surface [104,106,108].
Note here that the ignition energy consists of two parts:
one is the energy supplied after arrival of the SHS flame,
which has passed through the igniter, at the contact surface;
and the other is supplied during the flame propagation in the
igniter before arrival of the SHS flame at the contact surface.
By the use of the temperature profile in the inert stage and
the ignition delay time, the energy Ec, supplied after the
arrival of the SHS flame, is obtained as
E~ c ˆ
us
…1 ⫹ Lsp =Lig †
"
1
~ tc } erfc…k~ptc †
exp{…k~2 ⫺ w~ k†
k~ ⫺ w~
s
p
q #
aig =asp
k~
⫺1 ⫹
{1 ⫺ erfc… w~ k~tc †} ⫺
w~
w~ …aig =asp † ⫺ k~
"
(
! )
s!
aig
2 aig
~
× exp
w~
⫺ w~ k tc erfc w~
t ⫺1
asp
asp c
s!
r!
cig
4w~
4w~
⫺
1⫹ 1⫹
1⫹ 1⫹
csp
k~
k~ig
s
;
CA ˆ ⫺
cig
4w~
2
1⫹
csp
k~ig
…237†
s!
r!
cig
4w~
4w~
1⫹ 1⫹
1⫺ 1⫹
⫺
csp
k~
k~ig
s
CB ˆ
;
cig
4w~
2
1⫹
csp
k~ig
…l=c†sp ~
k~ig ˆ
k:
…l=c†ig
…238†
In deriving Eq. (236), a temporal variation of temperature in
the specimen is first obtained as
0
q 1
~ ⫹ k~ 2 ⫹ 4w~ k~
k
B
C
…239†
s ⱕ 0 : u ˆ B u exp @
sA;
2
B u ˆ 2us
(
Dˆ
s!
cig
4w~ 1
1⫹
;
csp
k~ig D
q
k~ig ⫹ k~ig2 ⫹ 4w~ k~ig
!
× exp
…235†
s!)
cig
4w~
⫺
exp
1⫹ 1⫹
csp
k~ig
It may be informative to note that the expression of Ec
obtained by Strunia et al. [106] is slightly different from
that of Eq. (234) because their expression was derived
under an assumption that there is no temperature distribution in the compacted medium, although they have taken the
temperature distribution into account in obtaining the ignition delay time.
As for the energy E0, supplied before the arrival of the
SHS flame, although it has not been taken into account in the
literature [106] is expressed as
s!
1 …rt †ig
4w~ us
E~ 0 ˆ
1⫹ 1⫹
2 …rt †ig
k~ CB
"
!
(
s!⫺1 )#
n
∞
X
CA
2
4w~
1 ⫹ …2n ⫹ 1† 1 ⫹
;
⫺
×
CB
k~ig
k~ig
nˆ0
…236†
…240†
s!
s!)
cig
4w~
4w~
⫺
1⫺ 1⫹
1⫹ 1⫹
~k
csp
k~ig
…234†
where
Ztc 2t …ma †sp
E~ c ⬅
Ec ˆ ⫺
du:
0
2s sˆ0
…l=c†sp q …rt †sp
0
where
where
s
q #!
w~ aig
⫹
{1 ⫺ erfc… w~ k~tc †} ;
k~ asp
63
2
(
sf ⫺
s!
4w~
1⫹ 1⫹
k~
q
k~ig ⫺ k~ig2 ⫹ 4w~ k~ig
2
!
sf ;
…241†
when the propagating SHS flame in the igniter locates at
s ˆ s f …⬎0†: After that by assuming constant speed of the
flame propagation, the following relation has been used
s f ˆ ⫺k~
…rt †sp
t
…rt †ig
…242†
which is equivalent to xf ˆ ⫺…u†ig t:
8.1.6. Results for the ignition energy
In numerical calculations, the situation considered is the
same as that in Section 8.1.4. Fig. 66 shows the ignition
energy E as a function of m ig in the igniter, with e taken
as a parameter. It is seen that with decreasing m ig, the ignition energy E increases first gradually and then rapidly until
the ignition limit, and that the contribution of E0 supplied
before arrival of the SHS flame is relatively large. In addition, in spite of the decrease in the heat of combustion in
the igniter, the energy E0 increases with decreasing m ig. This
64
A. Makino / Progress in Energy and Combustion Science 27 (2001) 1–74
trend is similar to that for the ignition delay time tc in Fig.
63(a). Fig. 67(b) shows the similar results with respect to
k sp. The same trends as those in Fig. 67(a) and (b) can be
observed by reducing heat loss, instead of reducing particle
size.
Fig. 68(a) shows E as a function of …R0 †sp2 ; with e taken as
a parameter when mig ˆ 0:8 and msp ˆ 1: The general trend
is similar to that for tc in Fig. 64. Fig. 68(b) shows the similar
results for E at mig ˆ 0:65: The same trend as that in Fig.
68(a) is observed although the ignition energy is increased
and the ignitable range is contracted. This is attributed to the
decrease in the maximum temperature in the SHS flame that
has passed through the igniter, due to the decrease in m ig in
the igniter.
8.2. Flame initiation by use of radiative heat flux
Fig. 66. Ignition energy E as a function of m ig in the igniter, with e
taken as a parameter [114]. The specimen is stoichiometric with
…R0 †sp ˆ 10 mm: Energy E0 is that supplied before the arrival of
the SHS flame in the igniter.
is attributed to the gradual heating of the specimen due to
slow burning velocity of the SHS flame in the igniter. As for
the energy supplied after the arrival of the SHS flame, it
increases first gradually and then rapidly, with decreasing
m ig, due to a decrease in the maximum temperature of the
SHS flame in the igniter. It is also seen that at a certain m ig, a
decrease in the heat loss makes the ignition energy smaller,
because extent of heat used for initiating combustion wave
becomes large. A further decrease in the heat loss brings
about a situation at which the ignition limit locates beyond
the extinction limit.
Fig. 67(a) shows E for e ˆ 1:0 as a function of m sp in the
specimen, with …R0 †sp taken as a parameter. The general
Combustion wave in the SHS process can also be initiated
by use of radiative heat flux, instead of using igniters.
Although numerical results based on the heterogeneous
theory are reported [115] it is considered that an extension
of the heterogeneous theory for the ignition phenomena
should be conducted by including analytical results related
to the ignition of reactive condensed materials, such as
explosives and solid propellants [116–120]. For this aim,
it is considered the situation that the front surface of a
compacted medium experiences partial, stepwise heating
by the radiative heat flux [121–123]. In analyzing the
inert stage, use has been made of the formulation presented
by Niioka and Hasegawa [124]. In the analysis for the transition stage, related to the initiation of the combustion wave,
results of the asymptotics [116] have been introduced into
the heterogeneous theory. Comparisons between predicted
and experimental results are also conducted.
Fig. 67. Ignition energy E under heat loss condition …e ˆ 1†; with (R0)sp in the specimen taken as a parameter [114]; (a) as a function of m sp in the
specimen; (b) as a function of k sp in the specimen. The mixture ratio m ig in the igniter is 0.8.
A. Makino / Progress in Energy and Combustion Science 27 (2001) 1–74
65
Fig. 69. Model used for analyzing the SHS flame initiation by use of
radiative heat flux [124].
with large activation energies, the compacted medium
experiences only inert heat conduction during most of the
time that t ⬍ tc because the heat generation term, related to
reactant consumption, is exponentially small due to the large
activation energy. Then, the equation of heat conduction in
the axisymmetric coordinate becomes as
2u
22 u
22 u
1 2u
ˆ
⫹
⫹
⫺ fu;
2t
s r 2s r
2s z2
2s r2
…243†
which is subject to the initial and boundary conditions as
t ˆ 0; u ˆ 0;
…I:C:†
…B:C:†
Fig. 68. Ignition energy E as a function of …R0 †2sp ; with e taken as a
parameter [114]; (a) the mixture ratio m ig in the igniter is 0.8; (b)
mig ˆ 0:65:
~
s z ˆ 0; 兩s r 兩 ⱕ a;
2u
~
ˆ ⫺Q;
2s z
…245†
~
s z ˆ 0; 兩s r 兩 ⬎ a;
2u
ˆ 0;
2s z
…246†
s z ! ∞; s r ! ∞; u ˆ 0;
…247†
where
si ˆ
8.2.1. Model definition and inert stage
The problem of interest is the radiative heating of the
front surface of an axisymmetric semi-infinite compacted
medium (Fig. 69), originally consisting of a mixture of N
particles, metal M, and an inert I that can be the product P of
the reaction nM M ⫹ nN N ! nP P. The front surface of the
compacted medium is subjected to a partial, stepwise heating with a constant heat flux Q for time t ⬎ 0; the diameter
of heating area is 2a. Monodisperse size distribution is
considered for N particles, with an initial radius R0 and
number density of n0.
As described in Section 8.1.1, in the ignition processes
…244†
a~ ˆ
…Z∞ ⫺ Z0 †ma xi
Q
; …i ˆ z; r†; Q~ ˆ
;
…l=c†
…Z∞ ⫺ Z0 † 2 q0 ma
…Z∞ ⫺ Z0 †ma a
;
…l=c†
C
;
…Z ∞ ⫺ Z0 † 2
# 2!
"
l
4es SB …T 2 ⫹ T02 †…T ⫹ T0 †
R0
Cˆ
:
rtc
2r
…rt c†…u0;a R0 †2
fˆ
Noted that the inert problem described has a known solution
66
A. Makino / Progress in Energy and Combustion Science 27 (2001) 1–74
and can be written as [123,125]
(
q !
Q~ a~ Z∞
~ exp ⫺ s z …l 2 ⫹ f†
uI ˆ
J …ls r †J1 …la†
2 0 0
q!
q!
sz
2
p ⫺ e t…l ⫹ f† ⫺ exp s z …l 2 ⫹ f†
× erfc
2 t
q!)
sz
dl
p ⫹ …l 2 ⫹ f† p ;
× erfc
(248)
2 t
l2 ⫹ f
variables are to be introduced.
where Jn is the Bessel function of the first kind of order n.
Note that temperature rise in the inert, heating stage is designated with the subscript I.
where b is the Zeldovich number defined in Eq. (225). By
retaining only the first nonvanishing terms on the exponential, the energy conservation Eq. (223) yields
8.2.2. Transient stage
In the same manner as that described in Section 8.1.2, the
transition stage can be analyzed. The only difference is the
position at which departure from the inert heating occurs;
that is, s z ˆ s r ˆ 0: Therefore, Eq. (248) is expanded as
uI ˆ …uI †c ⫹Kt …t ⫺ tc † ⫺ Kz s z;c ;
…249†
where
(
p
Q~
…uI †c ˆ p 2 ⫺ 2erfc … ftc †
2 f
p
⫺ exp…⫺a~ f† erfc
p
a~
p ⫺ ftc
2 tc
!
p
p
a~
⫹ exp…a~ f† erfc p ⫹ ftc ;
2 tc
Kt ˆ
2uI
2t
!)
(250)
…251†
⫺Kz ˆ
!
~
ˆ ⫺Q:
…253†
s ˆ bKt …t ⫺ tc †
(
!
!)
1
x~
T~ d
p 2
⫹ ln
;
exp ⫺
bKz …Z∞ ⫺ Z0 †L0 …rt;0 =rt †
T~ c
…254†
X ˆ bKz s z;c ;
d 2c
1
ˆ ⫺ p exp…c ⫺ X ⫹ s†;
b
dX 2
which is subject to
dc
ˆ c…∞; s† ˆ c…X; ⫺∞† ˆ 0:
dX Xˆ0
…255†
…256†
…257†
The condition as s ! ⫺∞ is obtained from matching to the
inert stage.
By using c ⫺ X as the dependent variable, Eq. (256) is
integrated to produce
s
dc
2
ˆ 1 ⫺ 1 ⫹ p {exp…c 0 ⫹ s† ⫺ exp…c ⫺ X ⫹ s†};
dX
b
…258†
(
!)
Q~
a~ 2
exp…⫺ftc †;
ˆ p 1 ⫺ exp ⫺
ptc
4tc
c
2uI
2s z
c ˆ b…u ⫺ u I †;
…252†
c
Terms relevant to s r can be ignored because temperature
distribution in the radial direction is flat around s r ˆ 0:
Here, (·)c designates the condition corresponding to t ˆ tc
and s z ˆ s r ˆ 0:
Since the independent variables in the transition stage are
just t and s z, the use of a set of one-dimensional, transient
governing equations in the heterogeneous theory [92,95] can
be justified. If we further ignore the heat loss effect, as is the
case for usual inner problems related to ignition, the energy
conservation equation coincides with Eq. (223).
This identification infers that the present problem closely
parallels the earlier development [116] for the ignition of
solid propellants. In analyzing the reactive-diffusive region
near the surface of the compacted medium, new stretched
where c 0 ˆ c…0; s†: This solution must match to that of the
transient-diffusive region which has the spatial variable
p
p
bKt
v ˆ bK t s z ˆ
X:
…259†
bKz
Application of matching and introduction of the time variable
Kz
s~ ˆ s ⫹ ln p
…260†
Kt
results in the transient–diffusive problem as
2c
2 2c
2c
ˆ
;
ˆ ⫺exp c…0; s~† ⫹ s~ ;
2
2~s
2
v
2v
vˆ0
…261†
c…∞; s~† ˆ c…v; ⫺∞† ˆ 0;
which has already been solved by Lin˜a´n and Williams
ÿ [116].
It is reported that the thermal runaway, that is c 0; s~ ! ∞;
occurs at a finite time s~ ˆ ⫺0:431: Since it is defined that
t ˆ t c at ignition, Eqs. (254) and (260) yield an implicit
formula for the ignition delay time as
!
!
1
x~
T~
p
exp ⫺ d ˆ e ⫺0:431
Kz bKt …Z∞ ⫺ Z0 †L0 …rt0 =rt †
T~ c
ˆ 0:650:
(262)
A. Makino / Progress in Energy and Combustion Science 27 (2001) 1–74
67
We see that the ignition delay time can be expressed as
functions of not only Td, Q, and a, as pointed out by Niioka
and Hasegawa [124] but also the operating parameters characteristic to the SHS process, such as the mixture ratio m ,
the degree of dilution k , the initial temperature T0, and the
initial particle radius R0. This can be seen from the definition
of t , which is expressed as
!
rt;0
rM
t
…263†
…1 ⫺ Z0 †
t ˆ 3D0 L0
rt
rN
R20
by using Eq. (71), such that tc ⬃ tc R20 for a given stoichiometry and/or a degree of dilution, once t c is obtained by
solving Eq. (262). Note also that …1 ⫺ Z0 † ˆ m…1 ⫺ k†=…m ⫹
fst †; shown in Eq. (62).
The dependence of the nondimensional parameters on the
operating parameters, in the same manner as that in Eq.
(263), is represented as follows. For Q~ and a~ we have
Q~ ˆ
…QR0 †
q0 r t;0 D0 …Z∞ ⫺ Z0 †{3Le0 L0 …1 ⫺ Z0 †…rM =rN †}1=2
⬃ …QR0 †;
a~ ˆ
1=2 3L0
r
a
a
⬃
;
…1 ⫺ Z0 † M
R0
R0
Le0
rN
…264†
…265†
which suggest that for a given stoichiometry and/or a degree
of dilution, both Q~ and a~ depend on the particle radius R0
even when the heat flux Q and the radius a of the heating
area are fixed, as are the cases for usual experimental conditions. The energy supplied per unit area in the heating
surface, required for initiating the combustion wave, is
given by
1=2 1
3L0
r
Qtc
Q~ tc ˆ 0
…1 ⫺ Z0 † M
rN
R0
q rt …Z∞ ⫺ Z0 † Le0
Qtc
⬃
;
R0
…266†
while the total energy supplied is simply
!
Qtc a
2
~
…Qtc †…pa~ † ⬃
:
R30
…267†
It is clear that the particle radius R0 is an essential parameter
affecting initiation of the combustion wave.
8.2.3. Results and experimental comparisons
In calculations, use has been made of physicochemical
parameters for Ti–C system [32,64] as described in Sections
4.4 and 5.4.1; a Lewis number of Le0 ˆ 25 is used [75] in
accordance with an experiment. Let us first examine the
situation without heat loss, and then with heat loss.
8.2.3.1. Situation without heat loss Fig. 70(a) shows the
ignition delay time tc as a function of the mixture ratio m ,
Fig. 70. Ignition delay time tc for Ti–C system with R0 ˆ 5 mm;
with the radiative heat flux Q taken as a parameter [121]; (a) as a
function of the mixture ratio m ; (b) as a function of the degree of
dilution k . Data points are experimental [126]; open symbol designates the ignition and flame establishment, while solid symbol
designates the ignition but extinguishment.
with the radiative heat flux Q taken as a parameter. The
initial radius R0 of N particles in the compacted medium is
set to be 5 mm. We see that the ignition delay time increases
with decreasing m , due to a decrease in heat generated
during the flame initiation. A further decrease in m makes
the combustion wave impossible to propagate, for the same
reason as that in Fig. 63(a). As for the effect of Q, we see that
tc increases with decreasing Q, because its decrease prolongs
the heating period required for the front surface to be heated
to the ignition temperature. Data points are experimental
[126] for different heat fluxes (3.3 and 2.57 MW/m 2) and
a fair degree of agreement is demonstrated. Here, it may be
informative to note that in general, tc for the radiative
heating is much longer than that for the igniter. This may
be attributed to the fact that heat supply by the SHS flame
with high temperature can be made much more rapidly than
by the radiative heat flux. Fig. 70(b) shows a similar
comparison with respect to the degree of dilution k . The
same comment can be made as that in Fig. 70(a).
Fig. 71 shows tc for m ˆ 1 as a function of R0, with Q
taken as a parameter. We see that tc increases with increasing R0, in the same reason as that in Fig. 64. It is important to
emphasize that such a significant parametric dependence
cannot be described by the homogeneous premixed-flame
theory. The decrease in Q causes tc longer, because of the
prolonged heating period of time.
Fig. 72 shows tc for m ˆ 1 with R0 ˆ 5 mm; as a function
of the radius a of the heating area, with Q taken as a parameter. With decreasing a, tc becomes longer, first gradually
and then rapidly, due to a decrease in the heating area at the
front surface.
Fig. 73(a) and (b) shows tc as functions of m and k ,
68
A. Makino / Progress in Energy and Combustion Science 27 (2001) 1–74
Fig. 71. Ignition delay time tc for Ti–C system of the stoichiometric
mixture …m ˆ 1†; as a function of R0, with Q taken as a parameter
[121]. Data points are experimental [126].
respectively, with Q and a taken as parameters. Because of
the strong effect in decreasing the heating area, as shown in
Fig. 72, tc becomes longer. In addition, this effect becomes
remarkable as m decreases or k increases, because the
increase in m or decrease in k reduces the heat generated
during the flame initiation.
Fig. 74 shows tc for m ˆ 1 as a function of R0, with Q and
a taken as parameters. It is further confirmed that the effect
of the heating area on tc is strong, as well as the effect of R0.
8.2.3.2. Situation with heat loss Now, let us examine effects
Fig. 73. Ignition delay time tc for Ti–C system with R0 ˆ 5 mm; as a
function of k , with Q and a taken as parameters [122]; (a) as a
function of m ; and (b) as a function of k . Data points are experimental [126].
Fig. 72. Ignition delay time tc for Ti–C system of m ˆ 1 with R0 ˆ
5 mm; as a function of the radius a of the heating area, with Q taken
as a parameter [121]. Data points are experimental [126].
of heat loss on the ignition delay time. Since the ignition
delay time t c is determined by solving the criterion in Eq.
(262), dependence of the LHS of Eq. (262) on the heat loss
f is to be examined, with t c taken as a parameter. As shown
in Fig. 75, with increasing f , the LHS of the criterion,
designated by F, first decreases slightly and then increases
A. Makino / Progress in Energy and Combustion Science 27 (2001) 1–74
69
which initiation of the combustion wave cannot be realized.
The state of the turning point can be determined by dF=df ˆ
0; which yields
duc
ˆ⫺
df
t=2
⬃ 0:
Z∞ ⫺ Z0
b⫹
T~ c
…268†
It is informative to note that as a first approximation,
p
a~
ft ˆ p
2 t
Fig. 74. Ignition delay time tc for Ti–C system of m ˆ 1 with R0 ˆ
5 mm; as a function of R0, with Q and a taken as parameters [122].
Data points are experimental [126].
rapidly. When tc ˆ 2:4317; F for f ˆ 0 is 0.650, which
corresponds to the value for the thermal runaway in the
adiabatic limit. Another solution for tc ˆ 2:4317 exists
when f is about 0.093, at which heat loss is quite large
and it is considered to be unrealistic. It is further seen that
for F ˆ 0:650; there exist two solutions for 2:4317 ⱕ tc ⬍
2:7252; one solution at tc ˆ 2:7252; and there is no solution
for tc ⬎ 2:7252: Thus, there exists an ignition limit, beyond
…269†
can be a solution for determining the state of the turning
point.
Fig. 76 shows the ignition delay time tc as functions of the
mixture ratio m and degree of dilution k , with the radiative
heat flux Q and the heat loss taken as parameters. The conditions are the same as those in Fig. 70. Note that the emissivity e represents the extent of the heat loss. It is seen that,
as the emissivity e increases, which results in an increase in
the heat loss, the ignitable range becomes narrow although
the ignition delay time itself is not influenced very much by
the heat loss.
Fig. 77 shows tc for the stoichiometric mixture as a function of the radius R0, with the radiative heat flux Q and the
emissivity e taken as parameters. The results clearly demonstrate that the system ignitability can be profoundly affected
by the particle size in the presence of heat loss. Since heat
loss is invariably present, the existence of the loss induced
ignition limit is particularly significant in assessing the
global performance of a synthesis system.
8.3. Some remarks on the flame initiation
In the SHS process, the energy required is basically for
initiating SHS flame, such that effects of various parameters
Fig. 75. Behavior of the thermal runaway criterion on the heat loss [123], for the stoichiometric mixture of Ti–C system with R0 ˆ 5 mm; heated
by the radiative heat flux of 3.3 MW/m 2, with 10 mm in heating diameter.
70
A. Makino / Progress in Energy and Combustion Science 27 (2001) 1–74
Fig. 76. Ignition delay time tc for Ti–C system, with the radiative
heat flux Q and the emissivity e taken as parameters [123]; (a) as a
function of the mixture ratio m ; (b) as a function of the degree of
dilution k . Conditions are the same as those in Fig. 70.
on the flame initiation are indispensable for practical utility.
Investigation based on the heterogeneous theory suggests
that the ignitable range is limited by the heat loss during
flame initiation when the particle size is large and the
compact diameter is small; while it is the heat loss during
flame propagation that confines the ignitable range when the
particle size is small and the compact diameter is large. This
identification provides useful insight in controlling ignition
phenomena in the SHS process by varying particle size,
which is expected to make a contribution in applying the
SHS process for near-net-shape fabrications by use of the
SHS process.
9. Concluding remarks
In the following the current status of SHS research on
flame propagation, extinction, and initiation by use of the
heterogeneous theory is summarized, and areas which
require further research are suggested.
9.1. Summary of the present survey
From a phenomenological consideration of flame propagation in the SHS process, it may appear in the combustion
wave as a slurry-like configuration, which consists of the
higher melting-point nonmetal particles and the molten
metal. This consideration leads to the important
identification about the SHS process that, although the
flame propagation in the bulk appears like that of the
premixed flame, this bulk flame propagation is eventually
supported by the nonpremixed reaction of the dispersed
nonmetal particles in the molten metal. This nonpremixed
mode has not been incorporated in formulations in the
previous theoretical works based on the theory for the
Fig. 77. Ignition delay time tc for Ti–C system of the stoichiometric
mixture …m ˆ 1†; as a function of the particles radius R0, with Q and
e taken as parameters [123].
premixed flame in the homogeneous medium. So that
heterogeneous features of the combustion wave have been
unresolved in previous theoretical works.
Recognizing the limitation of a homogeneous premixedflame theory in describing the heterogeneous nature related
to the SHS process (especially on the influence of particle
size), the existing theory on flame propagation in fuel sprays
has been extended to describe such processes. Specific
modifications included are finite rate of reaction at the particle surface and Arrhenius mass diffusion in the liquid
phase of molten metal, so that the particle consumption
rate controlled by both diffusion and reaction can be incorporated into the formulation, and prediction and characterization of such important bulk flame propagation as the
flame propagation speed, the range of flammability, and
the flame extinction limit, have successfully been made in
relation to the heterogeneity in the combustion wave.
Calculated results for several representative ceramics and
intermetallic compounds, such as Ti–C, Ti–B, Zr–B, Hf–B,
Al–Ni, Co–Ti, Ni–Ti systems, using consistent sets of
physicochemical parameters, respectively, show satisfactory agreement with experimental results in the literature.
An important relation that the burning velocity is inversely
proportional to the nonmetal particle size has been
confirmed. These comparisons further suggest that the
nonmetal (or higher melting-point metal) particles in these
systems might react in the diffusion-controlled limit, indicating that finite rate chemical reaction is not the controlling
factor in the propagation of bulk flame. Under such situations, the Arrhenius liquid-phase mass-diffusivity exerts the
dominant temperature sensitivity on the flame propagation
in the SHS process. In addition, because of the excessively
slow rate of diffusion compared to the thermometric
A. Makino / Progress in Energy and Combustion Science 27 (2001) 1–74
conductivity and reaction rate (suggested by the excessively
large Lewis number in the combustion wave), it has been
clarified that concentration of the molten metal is mainly
affected through consumption.
A nonadiabatic, heterogeneous theory for the SHS
process with volumetric heat loss, has further been formulated and shown that the characteristic extinction turning
point exists in the same manner as that in the premixed
flame. It has also been found that the range of flammability
under heat loss conditions can well be described, in terms of
its dependence on stoichiometry, dilution, initial temperature, particle size, and compact diameter. A particularly
significant identification is that the process responsible for
its temperature sensitivity and ultimate extinguishment is
the Arrhenius liquid-phase mass-diffusion. The fair agreement between predicted and experimental results for the
extinction limits in trend and in approximate magnitude
suggests that this theory has captured the essential features
of the nonadiabatic heterogeneous flame propagation in the
SHS process. A useful parameter called the heat loss index,
which is defined as the square of the nonmetal particle size
divided by the compact diameter, has also been found.
Based on these identifications, not only an approximate
expression for burning velocity in the adiabatic condition,
but also for extinction criterion in the nonadiabatic condition, has been derived under the appropriate and realistic
assumptions of diffusion-limited reaction and infinite Lewis
number. Effects of the dominant parameters on the burning
velocity and/or extinction criterion have been clarified,
yielding useful insight into manipulating the SHS flame
propagation and extinction. Since a fair degree of agreement
with numerical results has been demonstrated by examining
applicability of these expressions, these expressions are
expected to have general, qualitative and quantitative utility,
in each case.
Bimodal particle dispersion for the nonmetal particles has
also been taken into account in the nonadiabatic heterogeneous theory. It has resulted that the mathematical problem
can be rendered identical to the monomodal case by introducing an equivalent particle radius expressed as a function
of the bimodal parameters. This identification has further
provided useful insight into manipulating the flame propagation and/or extinction by making the system bimodal,
especially in recognizing that because of the increase in
the total surface area of particles in the combustion wave,
the reduced equivalent particle size in the bimodal system
increases the burning velocity and broadens the range of
flammability. Furthermore, flame instability, usually caused
by the high energy-release rate in the combustion wave
when the preparation is made with small particles, can
even be suppressed by increasing the equivalent particle
size in the bimodal system.
The heterogeneous theory which has succeeded in capturing the essential feature of the heterogeneity in the SHS
process has further been examined to elucidate differences
from the homogeneous premixed-flame theory. Although
71
differences are minor from the mathematical point of
view, important identification which has already been
pointed out for the physical interpretation has been reconfirmed. So that qualitative validity of theoretical results
based on the homogeneous premixed-flame theory has
been shown, if the differences involved are noted.
The field activated SHS, which is effective especially for
low exothermic systems to sustain flame propagation, can
also be examined by use of the heterogeneous theory, by
introducing an additional term due to external heating by
electric current. It has been shown that the external heating
enhances the burning velocity and extends the range of
flammability, as expected. Significant identifications
obtained are that extent of these effect is strongly influenced
by the particle size, as well as the physicochemical parameters, and that the range of flammability is still restricted
even under external heating conditions, neither of which
have been captured by the homogeneous premixed-flame
theory. In this context, the heterogeneous theory is indeed
viable in examining the field activated SHS.
The study for the transition boundary from steady to
pulsating combustion by use of the neutral stability analysis,
has also been conducted and it has been shown that there
exists a limit in the use of small particles, which is an
important suggestion for near-net-shape fabrication by use
of the SHS process. As for the effects of transverse wave
number, which are related to heat loss on the side surface, on
the location of the boundary, it has been found to be relatively small from a practical point of view.
In addition to the flame propagation and/or extinction,
initiation of the combustion wave has been examined
because energy required is basically that for initiating
SHS flame and effects of various parameters on the flame
initiation are indispensable for practical utility. It has been
found that the ignitable range is limited by heat loss during
flame initiation when the particle size is large and/or
compact diameter is small; while it is heat loss during
flame propagation that confines the ignitable range when
the particle size is small and/or compact diameter is large.
This identification provides useful insight to controlling
ignition phenomena in the SHS process by varying particle
size, which is expected to make various contributions in
practical applications.
9.2. Area for further research
In view of the recent interest to produce Functionally
Graded Materials (FGMs) by use of the SHS process, a
candidate for research is the examination of flame
propagation in a medium with distributed reactant concentrations, not only in the longitudinal direction but also in the
lateral directions. In this situation, contour of the combustion wave is no longer planar and the combustion wave
spreads two-dimensionally, because of the distributed heat
generation rate and/or two-dimensional temperature distribution, caused by the distributed reactant concentrations.
72
A. Makino / Progress in Energy and Combustion Science 27 (2001) 1–74
Since flame strength is not constant along the curved
combustion wave, the concept of flame stretch might be of
great use in fundamental understanding of this flame propagation and its related extinction.
Another candidate for research related to producing
FGMs is that of examining flame propagation for multicomponent system. Since both diffusion-controlled combustion
for some components and reaction-controlled combustion
for other components can proceed simultaneously in the
combustion wave, recognition of dominant parameters for
each component and/or each competing reaction might be
indispensable for conducting theoretical works and fundamental understanding. This identification would further
facilitate exploration of such combustion mechanism as
bulk flame propagation, which proceeds prior to the other
bulk flame propagation, observed for multicomponent
system.
In order to gain fundamental understanding for the
SHS flame propagation, it might be of great importance
to incorporate results of microscopic observations into
formulations of theoretical works, because information
indispensable for constructing the heterogeneous theory
is that of microscopic observations. In this context,
further accumulation of experimental results through
microscopic observations is strongly advised, in order
to refine the heterogeneous theory.
In addition to microscopic observations, experiments in
the gravity free environment might be of great value in
obtaining further fundamental understanding, although this
experiments cannot be easily conducted on earth. For the
SHS process, natural convection would exert little influence
in removing heat from the combustion wave, compared to
that of thermal radiation, but density difference between the
nonmetal particles and the molten metal in the combustion
wave (by yielding local change of stoichiometry, thickness
of the combustion wave, etc.), could exert influences,
because of the existence of gravity.
Although not specifically covered in the present
review, the field activated SHS might also prove a
fertile area of research because this is indispensable
for low exothermic systems to initiate and/or sustain
combustion wave. In addition to the effects of external
heating, if influences of electric and/or magnetic fields
on profiles of reactant concentrations could be utilized
in producing advanced materials, useful and multifunctional characteristics would be generated by inducing
heterogeneity in composition and/or texture.
Finally, it may be informative to point out the similarity
between spray combustion and the SHS process, as has been
mentioned in the present review. This similarity also
suggests that the SHS process might be used for the basic
research of spray combustion, especially for identifying
effects of dominant parameters on the flame propagation,
extinction, and/or initiation, because influence of thermal
expansion which usually conceal those of the other
parameters is relatively weak in the SHS process.
Acknowledgements
It is a pleasure to acknowledge the continuing interest and
support that the author has received for this research on
flame propagation, extinction, and/or initiation of the
combustion wave in the SHS process, through a series of
the Grant-in-aid for Scientific Research of the Ministry of
Education, Science, Sports, and Culture, the Iwatani Naoji
Foundation’s Research Grant, the Research Grant of Tokai
Foundation for Promoting Industrial Technologies, and the
Mechanical Industry Development and Assistance Foundation’s Research Grant, Japan.
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Could a new proposed particle help to detect dark matter?

D