Acta mater. 49 (2001) 1095–1103
www.elsevier.com/locate/actamat
IGNITION PHENOMENA AND CONTROLLED FIRING OF
REACTION-BONDED ALUMINUM OXIDE
M. J. WATSON‡, M. P. HARMER†‡, H. M. CHAN‡ and H. S. CARAM
Materials Research Center, Lehigh University Bethlehem PA, 18015 USA
( Received 13 March 2000; received in revised form 6 October 2000; accepted 6 October 2000 )
Abstract—The reaction-bonded aluminum oxide (RBAO) process utilizes the oxidation of attrition-milled
Al/Al2O3/ZrO2 powder compacts, that are heat treated in air, to make alumina-based ceramics. A simultaneous
mass and energy balance has been used to model the propagation of the ignition front that has been observed
during reaction-bonding. The model is used to determine conditions under which ignition can be avoided.
 2001 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved.
Keywords: Firing; Computer simulation; Kinetics; Thermally activated processes; High temperature;
Reaction-bonded aluminum oxide
1. INTRODUCTION
The reaction-bonded aluminum oxide (RBAO) process, developed by the Advanced Ceramics Group of
the
Technical
University
Hamburg–Harburg
(Hamburg, Germany), has been shown to have many
advantages over conventional ceramic processing [1–
4]. The RBAO process starts with intensely milled
aluminum/Al2O3 compacts, that are heat treated in air,
to produce Al2O3-based ceramics.
Several models have been used to describe the
reaction behavior during firing. A continuum model,
using macroscopic simultaneous mass and energy balances, has been used to describe how the temperature
and concentration of oxygen and aluminum vary with
time along the width of an infinite flat slab [5]. This
model has been used to design heating cycles that
avoid shrinking core reaction behavior, and has
resulted in the successful firing of larger samples. A
kinetic equation was proposed based on a parabolic
rate law and the particle size [4], and the mechanism
for reaction-bonding was explained in terms of a critical particle size: particles below the critical particle
size react completely below the melting point (solid–
gas reaction) whereas particles above the critical particle size are oxidized via a liquid–gas reaction.
During reaction-bonding, aluminum oxidizes
according to the reaction:
† To whom all correspondence should be addressed. Tel.:
⫹1-610-758-4227; fax: ⫹1-610-758-4244.
E-mail address: [email protected] (M.P. Harmer)
‡ Member, American Ceramic Society.
4Al ⫹ 3O2→2Al2O3.
It had been assumed before that the temperature and
concentration profiles would follow the symmetry of
the sample. Our observations show, however, that
ignition starts at one point of the sample and propagates as a front as a consequence of the highly exothermic nature of aluminum oxidation.
A similar phenomenon is observed in self-propagating high-temperature synthesis (SHS), a process
that relies on a reaction front propagation to form products. Ignition front propagation is used to produce
several materials through several combustion synthesis reactions [6, 7] to produce borides (TiB2), carbides (TiC, SiC, WC), nitrides (TiN, ZrN, Si3N4), silicides (MoSi2) and intermetallics (FeAl). The
reactions are usually solid–solid and are characterized
by high activation energy and heat of reaction. The
diffusion of a gaseous reactant into the sample is not
considered for a solid–solid reaction. This means that
the composition in the plane perpendicular to the
direction of the reaction front is equal over the entire
plane. In other words, in a cylindrical geometry there
is no significant radial variation in composition if the
ignition front is propagating in the axial direction.
Unlike the SHS process, an ignition wave front is
undesirable for the RBAO process because both thermal and chemical stresses are developed. The thermal
stresses are transitory and are caused by the large
temperature difference between the hot reaction zone
and the cooler unreacted zone that occurs over a short
distance. The chemical stresses are caused by the
large volumetric expansion (28%) associated with the
1359-6454/01/$20.00  2001 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved.
PII: S 1 3 5 9 - 6 4 5 4 ( 0 0 ) 0 0 3 4 3 - 8
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WATSON et al.: IGNITION PHENOMENA AND CONTROLLED FIRING OF RBAO
oxidation of aluminum. The chemical stresses are not
transitory, but remain in the wake of the ignition
wave front. They are caused by the steep composition
gradients between a completely oxidized shell, and a
partially reacted core. While sample breakage has not
been observed during wave front propagation, it is
usually observed 1–5 min after ignition. This is probably caused by a coefficient of thermal expansion
mismatch that develops between the reacted shell and
partially oxidized core thereby creating stresses as the
sample cools.
The steep composition gradients are shown in Fig.
1, which is a light optical micrograph of the crosssection of a rod after an ignition front has passed.
There are three distinct regions: an incompletely oxidized core, a fully reacted shell, and a transition region
between the core and shell. The completely reacted
shell consists of Al2O3 and ZrO2, and the depth of
the reacted shell is restricted by the radial diffusion
of oxygen into the rod. The incompletely oxidized
core consists of ZrO2, Al2O3, Al3Zr and any remaining unreacted aluminum [8]. The composition and
morphology of the transition zone is unknown, but is
probably caused by the partial oxidation of Al3Zr.
Oxygen must filter through the porous structure of
the RBAO sample to oxidize the aluminum. We propose that ignition is caused by the exothermic oxidation of aluminum on the surface of the sample. This
creates enough heat for the ZrO2 within the core of
the sample to react with the aluminum according to
the reaction:
13Al ⫹ 3ZrO2 → 2Al2O3 ⫹ 3Al3Zr.
When this occurs, an ignition wave propagates across
the sample.
In this paper the velocity and temperature of the
ignition front are measured to characterize the kinetic
parameters, namely the activation energy and preexponential factor. Simultaneous mass and energy
balances are used to model the propagation of the
ignition front. The experimental results are compared
with the continuum model, to ensure a reasonable
match-up between the numerical calculations and the
experimental observations. The model is then used to
determine under what conditions ignition can be avoided. Specifically, a furnace heating cycle which avoids ignition and cracking is calculated and tested
numerically and experimentally.
2. EXPERIMENTAL
2.1. Powder preparation
Green RBAO powder is prepared by co-attrition
milling of aluminum, Al2O3 and ZrO2 in mineral spirits with 2 and 3 mm ZrO2 balls. The resulting slurry
is separated from the milling media, vacuum filtered,
rinsed with acetone and allowed to dry. The dry filter
cake is then ground and sieved. A more detailed
description of the powder preparation is provided
elsewhere [9].
Three powders initially containing 20 vol% ZrO2
and 30, 45 and 55 vol% aluminum (labeled MS30,
MS45 and MS55, respectively) were used to characterize the ignition behavior. The remainder was
Al2O3. The powder composition changes slightly during milling due to the partial oxidation of aluminum
and wear from the ZrO2 milling balls. The compositions after milling, calculated from thermogravimetric (TG) measurements, are shown in Table 1.
2.2. Ignition measurements
Rods measuring 3.6 mm in diameter and approximately 50–60 mm long were prepared by cold isostatically pressing the powder to 290 MPa. Three
thermocouples were attached to the surface of the rod
so that the junction of the thermocouple remained in
contact with the surface of the sample, as shown in
Fig. 2. The rod and the thermocouples were placed
in a tube furnace and heated at 5°C/min, while the
output from each thermocouple was recorded using a
chart recorder.
3. MODEL DEVELOPMENT
The following mathematical model describes a
reaction front propagating along a long, thin rod of
RBAO. Specifically, the model describes changes in
temperature and aluminum concentration with time
along the axial direction of the rod. Figure 3 shows
Table 1. Post-milling powder composition
Composition (volume %)
Fig. 1. Cross-section of a RBAO rod after an ignition front has
passed.
Concentration
(mol/m3)
Powder
Al2O3
ZrO2
Al
Al
MS30
MS45
MS55
60.48
49.32
37.41
19.48
19.32
19.42
20.04
31.36
43.16
12,050
18,850
25,950
WATSON et al.: IGNITION PHENOMENA AND CONTROLLED FIRING OF RBAO
R ⫽ koexp
Fig. 2. Schematic of a thermocouple connected to a sample.
冉 冊
⫺Ea
C ,
RgT Al
1097
(3)
and is expressed on a per mole aluminum basis.
CAl is the concentration of aluminum, T is the
absolute sample temperature, Rg is the gas constant, k0 is the pre-exponential factor, and Ea is the
activation energy. The expression for the rate equation is empirical and does not attempt to describe
any complex microstructural reaction mechanisms.
4. The heat of reaction is the volume weighted average of the two reactions given above.
5. Other than the change in reaction products, there
is no aluminum concentration or temperature variation in the radial-direction.
6. Material properties such as specific heat, density
and conductivity remain constant.
3.1. Simultaneous mass and energy balance
The aluminum reacts to form Al2O3 and Al3Zr giving:
∂CAl
⫽ ⫺R
∂t
(4)
where t represents time.
The energy balance is given by:
Fig. 3. Schematic of a reacting RBAO rod.
an ignition front propagating along a RBAO sample
of length L and radius R. In the wake of the ignition
front, gaseous oxygen from the air has reacted with
the aluminum near the surface of the rod. In the core
of the sample, aluminum is reacting with ZrO2, forming Al2O3 and Al3Zr.
To maintain the one-dimensionality of the model,
the following assumptions were made:
1. The reaction of aluminum with gaseous oxygen
only occurs in a thin outer volume, Vd, of the rod.
Within Vd there is no resistance to oxygen diffusion, and beyond this the resistance is infinite.
In this way an oxygen mass balance does not need
to be considered in the material balance.
2. The reaction of aluminum with ZrO2 only occurs
within the inner core of the rod and aluminum is
the limiting reactant.
3. As a first approximation, an overall rate is used
for both of the reactions:
4Al ⫹ 3O2 → 2Al2O3
(1)
13Al ⫹ 3ZrO2 → 2Al2O3 ⫹ 3Al3Zr
(2)
The rate, R, at which aluminum reacts is
described by:
∂T
2
∂2T
rcp ⫽ l 2 ⫹ (⫺⌬He)R⫺ [h(T⫺T⬁)
∂t
∂z
R
⫹ s⑀(T4⫺T4⬁)]
(5)
where r is the density, cp is the specific heat, l is the
thermal conductivity, z is the axial coordinate, and
(⫺⌬He) is the effective heat of reaction. The heat loss
terms consist of convective and radiative components
and are expressed in terms of the furnace temperature, T⬁, the convective heat transfer coefficient, h,
the Stefan–Boltzman coefficient, s, the emissivity, ⑀,
and the radius of the rod, R. The effective heat of
reaction, (⫺⌬He), is given by
(⫺⌬He) ⫽
Vd
V⫺Vd
(⫺⌬H1) ⫹
(⫺⌬H2)
V
V
(6)
where (⫺⌬H1) is the heat of reaction (per mole of
aluminum) for reaction (1) and (⫺⌬H2) is the heat of
reaction (per mole of aluminum) for reaction (2).
3.2. Initial and boundary conditions
Many possible variations in experimental conditions are expressed in terms of the initial and
boundary conditions. Specifically, changes in the
initial concentration of aluminum, the temperature
program of the furnace and temperature variations
across the length of the sample are described in the
initial and boundary conditions.
The initial concentration of aluminum is calculated
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WATSON et al.: IGNITION PHENOMENA AND CONTROLLED FIRING OF RBAO
from TG analysis and it is assumed to be constant for
the entire length of the rod:
CAl(0,z) ⫽ CAl,0.
(7)
The initial temperature along the length of the rod is
also assumed to be constant:
T(0,z) ⫽ T0.
(8)
The boundary conditions are obtained by balancing
the heat flux via convection and radiation into the
ends of the rod (z ⫽ 0,L) with heat flux conducted
away from the end:
∂T
l (t,0) ⫽ h(T(t,0)⫺T⬁(t,0)) ⫹ s⑀(T4(t,0)⫺T4⬁(t,0)),
∂z
l
∂T
(t,L) ⫽ ⫺h(T(t,L)⫺T⬁(t,L))⫺s⑀(T4(t,L) (9)
∂z
⫺T4⬁(t,L)).
Different furnace heating cycles and spatial variation
of the furnace temperature may be investigated by
expressing T⬁ ⫽ T⬁(t,z).
3.3. Model parameters
The model parameters are listed in Table 2. Where
possible the parameters were measured quantities or
literature tabulated values. The length and radius of
the rod were measured directly with Vernier calipers.
The density was measured using Archimedes’
method, with water as the immersion medium. The
specific heat and thermal diffusivity, a ⫽ (l/rcp),
were measured using the laser flash method
(Thermaflash 2200, Holometrix, Bedford, MA), and
the thermal conductivity was calculated based on
these measurements. The initial aluminum concentration was calculated from TG analysis (STA 409C,
Netzsch Instruments, Paoli, PA). The heat transfer
Table 2. Ignition model parameters
Property
Symbol
Value
Unit
Length
Radius
Density
Specific heat
Conductivity
Heat of reaction
(1) [17, 18]
Heat of reaction
(2) [19]
Volume ratio
Pre-exponential
factor
Activation energy
Gas constant
Stefan Boltzmann
coefficient
Emissivity
Heat transfer
coefficient
L
R
r
cp
l
(⫺⌬H1)
0.06
0.0018
2460
1000
1.14
835,000
m
m
kg/m3
J/(kgK)
W/(m K)
J/(mol Al)
(⫺⌬H2)
43,770
J/(mol Al)
[(Vd)/V]
k0
0.05
274e6
m3/m3
s⫺1
Ea
Rg
s
170,000
8.314
5.67⫻10⫺8
J/mol
J/(mol K)
W/(m2K4)
⑀
h
0.05
14
W/(m2K)
coefficient was estimated from correlations [10]. The
volume ratio of reacted to unreacted sample was
determined from the cross section of the rods after an
ignition front had passed, similar to the cross-section
shown in Fig. 1, where the ratio of the white area to
the total area gives Vd/V. The pre-exponential factor,
k0, and the activation energy, Ea, were left as adjustable parameters, and the values listed in Table 2
showed good agreement with the experimental
results.
4. EXPERIMENTAL RESULTS
4.1. Ignition measurements
Figure 4 shows an example of the temperature
measurements as a function of time. In this example
there is a temperature difference, at the beginning of
the process, of about 80°C from the first to the third
thermocouple which are 44 mm apart. The temperature difference is caused by a steep temperature gradient along the length of the tube furnace where the
samples were fired. The ignition starts ahead of the
first thermocouple and propagates along the rod.
When the ignition front reaches the first thermocouple, the temperature departure is recorded. After
the ignition front has passed, that section of the rod
cools over the next 50 s. Approximately 10 and 20 s
after the ignition front has passed the first thermocouple, the ignition front reaches the second and third
thermocouples, respectively. The velocity is the distance between the thermocouples divided by the time
between the peaks. Interestingly, in this experiment
the maximum, or combustion, temperature decreases
as the ignition front moves to the cool end of the
furnace. This is caused by the lower initial temperature and the increased heat loss to the furnace as the
ignition front propagates to the cool end of the furnace. A good match between the model (equations
(4)–(9), discussed in Section 3) and the experimental
data is achieved when the appropriate model parameters and boundary conditions are chosen.
Simplified solutions to equations (4) and (5) exist
if the ignition wave is assumed to propagate at a con-
Fig. 4. Temperature measurements from chart recorder.
WATSON et al.: IGNITION PHENOMENA AND CONTROLLED FIRING OF RBAO
1099
stant velocity. The solution gives the velocity of the
ignition front in terms of the reaction parameters [11]:
v2 ⫽
冉 冊
RgT2c
⫺Ea
2l
k exp
(⫺⌬He)CAl,0 Ea 0
RgTc
(10)
where v is the velocity of propagation, and Tc is the
combustion (maximum) temperature. This assumes
that the activation energy is high, the width of the
reaction zone is infinitely thin, and there is no reactant
consumption either side of the reaction zone.
Equation (10) gives a convenient set of axes on
which to represent ignition measurements. Measurements of the propagation velocity, v, and combustion
temperature, Tc, of samples with varying aluminum
content, CAl,0, are represented on a plot of
ln(v2CAl,0/T2c) against 1/Tc. This is shown in Fig. 5 for
MS30, MS45, and MS55, and the numerical results
shown for comparison. The numerical results will be
discussed in detail in Section 5.
5. NUMERICAL RESULTS
To evaluate the accuracy of the model, equations
(4)–(9) are solved numerically and compared with the
ignition measurements above. Once the accuracy of
the model is confirmed, it can be used to predict when
ignition will occur under a variety of experimental
conditions (furnace temperature gradient, heating
rate, initial aluminum content).
Fig. 6. Distribution of sample temperature during ignition.
tions are solved using the radau5 stiff integrator
[13]. The integrator uses an implicit Runge–Kutta
method of order 5 with step size control. Finite differences [14] are used to calculate the first and secondorder spatial derivative in the axial direction. After
every five integrator time steps, the spatial grid is
adapted so that the nodes are grouped closer together
at the ignition front. The spatial adaptation algorithm
is described elsewhere [15] and utilizes equidistribution principles based on the magnitude of the second
spatial derivative.
5.2. Ignition velocity
The spatial co-ordinate, z, is divided into 101 spatial points, and discrete, finite difference approximations are used to calculate the spatial derivatives.
In this way a partial differential equation is approximated by 101 ordinary differential equations. This is
known as the numerical method of lines [12] and is
used to solve the simultaneous mass and energy balances (equations (4) and (5)) with the appropriate
initial and boundary conditions (equations (7)–(9)).
The resulting set of 202 ordinary differential equa-
To initiate wave front propagation, a furnace temperature difference of 10°C was assumed to exist
from one side of the sample to the other. This is small
compared to the temperature gradient measured in
Fig. 4, however, numerical results indicate that the
ignition front velocity does not depend strongly on
the temperature gradient. The value of T⬁ was raised
from 225°C to 725°C at 5°C/min to simulate the
experimental furnace heating rate. The model-predicted temperature and aluminum concentration distribution were recorded every 1.5 s.
Figures 6 and 7 show the temperature and alumi-
Fig. 5. Ignition results for (䉫) MS30, (䊊) MS45, and (왕)
MS55. Solid symbols are numerical results.
Fig. 7. Distribution of aluminum concentration during ignition.
5.1. Method of solution
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WATSON et al.: IGNITION PHENOMENA AND CONTROLLED FIRING OF RBAO
num concentration, respectively, as a function of distance along the rod over a series of discrete time
intervals. The first two time intervals are 130 s and
3000 s, respectively, showing a progression from the
initial condition as the rod is heated. The time interval
for the remainder of the distributions is 1.5 s. The
initial aluminum concentration was 18850 mol/m3,
corresponding to the initial concentration of the MS45
powder. The line at 225°C corresponds to the initial
temperature with the 10°C difference across the sample. The sample temperature rises with the furnace
temperature to approximately 500°C. Soon after this
(130 s) an ignition front is seen propagating from left
to right along the rod. The velocity is the distance
between the peaks divided by the time interval
between each temperature distribution (1.5 s).
Figure 7 shows the how the concentration of aluminum varies as a function of time and distance along
the rod during ignition. There is, as expected, a strong
correspondence between the consumption of aluminum and the temperature at a given point along the
rod.
Comparisons with the temperature measurements
from the chart recorder are made by interpolating the
temperature distributions onto an evenly spaced axial
grid. For discrete axial positions corresponding to the
ignition experiment (z ⫽ 9, 30, and 53 mm), the temperature–time history is plotted and compared with
the experimental results. The numerical results are
shown as dotted lines in Fig. 4, and a good match
between the experimental and numerical results can
be achieved.
The velocities of the ignition fronts were calculated
from the numerical results for the three initial aluminum concentrations given in Table 1 corresponding
to the powders MS30, MS45 and MS55. The combustion temperatures were also recorded and the data are
shown in Fig. 5 as solid symbols. The simulations
match closely with the experimental measurements.
The combustion temperature predicted from the
model is higher than the combustion temperature that
was measured for the MS55 sample. This is probably
due to an underestimate of the radiative heat transfer
term, from the surface of the rod to the surrounding
furnace, in the boundary conditions and energy balance.
When the rate of heat generation exceeds the rate of
heat loss, thermal explosion occurs. The theory
ignores heat transfer via conduction along the sample,
and instantaneous changes in the reaction rate due to
the consumption of aluminum. The heat conduction
term,
l
∂2T
,
∂z2
and radiative heat loss term,
2s⑀ 4 4
(T ⫺T⬁),
R
are neglected in this analysis because they are only
significant when the ignition front has already begun
to propagate and this analysis is used to determine
the onset of ignition. Figure 8 shows how the heat
generation and heat loss vary as a function of sample
temperature for various furnace temperature conditions.
When T⬁ ⫽ Tf1, the sample will slowly heat and
react at the intersection of the heat loss and generation
curves corresponding to T ⫽ TA. This is a stable reaction regime, because the rate of heat loss is greater
than the rate of heat generation for small increases in
temperature. The second intersection point, T ⫽ TB,
is an unstable regime, because a slight decrease in
temperature will force the temperature to fall to TA,
whereas a slight increase in the temperature will
cause the sample to ignite. If the furnace temperature
is increased to T⬁ ⫽ Tf3, the reacting sample will
ignite because the rate of heat generation is always
greater than the rate of heat loss. The highest furnace
temperature for which a steady reaction regime exists
is the thermal explosion limit, corresponding to
T⬁ ⫽ Tf2 in Fig. 8, when the heat generation curve is
tangent to the heat loss curve [16]. The sample tem-
5.3. Predicting ignition
To determine under what conditions the onset of
ignition will occur, Semenov’s theory of thermal
explosion [16] is used. The basic idea is to compare
the rate of heat generation,
冉
冉 冊 冊
(⫺⌬H)k0 exp
⫺Ea
C ,
RgT Al
with the rate of heat loss,
冉
冊
2h
(T⫺T⬁) .
R
Fig. 8. Thermal explosion analysis: heat loss curve for (– – –
) Tf1 ⫽ 419°C, (– · –) Tf2 ⫽ 439°C, (· · ·) Tf3 ⫽ 459°C, and
(———) heat generation curve.
WATSON et al.: IGNITION PHENOMENA AND CONTROLLED FIRING OF RBAO
1101
perature at the point of tangency in Fig. 8 is 466°C
and this is the ignition temperature.
For a given concentration of aluminum, the thermal
explosion limit is found from the tangency condition:
冉 冊
冉 冊 册 冋
册
(⫺⌬H)k0 exp
冋
⫺Ea
2h
C ⫽ (T⫺T⬁)
RgT Al
R
(11)
∂
⫺Ea
∂ 2h
(T (12)
(⫺⌬H)k0 exp
C ⫽
∂T
RgT Al
∂T R
⫺T⬁) .
Table 3 shows the solutions to the tangency condition
for the three powders. There is reasonable agreement
between the thermal explosion theory predictions and
the experimental data when the kinetic constants
listed in Table 2 are used to estimate the ignition temperature.
Equations (11) and (12) represent the tangency
condition for any aluminum concentration, as well as
a number of different experimental conditions. For
example, the effect of increasing the sample radius,
R, or decreasing the heat transfer coefficient, h, is to
lower the ignition temperature for a given concentration of aluminum. Equations (11) and (12) also
state that the furnace temperature can be increased
as aluminum is reacted. If the change in aluminum
concentration with time is incorporated into the
analysis, it is possible to predict furnace heating
cycles that will avoid ignition, as long as the temperature–concentration trajectory remains below the
ignition limit. This is discussed in the following section.
5.4. Furnace temperature program
A furnace temperature program is calculated using
the thermal explosion analysis of Section 5.3 and the
kinetic constants of Table 2. The consumption of
aluminum is incorporated into the calculation by simultaneously solving the equations for the aluminum
concentration and thermal explosion limit (equations
(4), (11) and (12)). The resulting furnace heating program is on the thermal explosion limit. Figure 9
shows the furnace temperature program on the thermal explosion limit for the three initial aluminum
concentrations of MS30, MS45 and MS55.
Experiments reveal that the sample will ignite
Table 3. Tangency condition for ignition
Sample
Aluminum
concentration
(mol/m3)
Calculated
temperature
(°C)
Measured
temperature
(°C)
MS30
MS45
MS55
12,050
18,850
25,950
479
466
457
500±10
510±40
510±20
Fig. 9. Furnace program on the thermal explosion limit for (–
– –) MS30, (– · –) MS45, and (———) MS55.
when the furnace temperature programs of Fig. 9 are
used. We believe that this is because the temperature
programs are calculated based on the thermal
explosion limit and there is no room for minor furnace temperature fluctuations, temperature gradients
or mismatch between experiment and prediction. For
this reason, a safety factor, fⱖ1, is incorporated into
the tangency conditions of equations (11) and (12),
which gives a more conservative estimate of the
ignition temperature:
冉 冊
冉 冊 册 冋
f(⫺⌬H)k0 exp
冋
⫺Ea
2h
C ⫽ (T⫺T⬁)
RgT Al
R
(13)
册
∂
⫺Ea
∂ 2h
(T⫺T⬁) .
f(⫺⌬H)k0 exp
C ⫽
∂T
RgT Al
∂T R
(14)
Figure 10 shows the calculated and measured sample temperatures as a function of time for a safety
factor of f ⫽ 2 and f ⫽ 4. The sample was MS45.
Note that the furnace temperature programs that are
used are an approximation to those calculated using
equation (13) because the furnace was programmed
with linear segments. Also, in both experiments the
furnace was heated at 5°C/min to the initial temperature. The sample still ignites when f ⫽ 2 although the
severity of the temperature departure (苲150°C) is
diminished considerably compared with that of Fig.
4 (苲600°C). No ignition occurs when f ⫽ 4 and the
sample did not crack or break.
Figure 11 shows the model predicted sample temperatures as a function of time for a safety factor of
f ⫽ 2 and f ⫽ 4, based on the experimental furnace
temperature programs that were used to generate Fig.
10. The temperature program calculated using equation (13) are included for comparison. The model
1102
WATSON et al.: IGNITION PHENOMENA AND CONTROLLED FIRING OF RBAO
Fig. 10. Furnace temperature programs with and without
ignition for MS45. Safety factor of f ⫽ 2 (· · ·) calculated and
(———) measured, and f ⫽ 4 (– · –) calculated and (– – –)
measured.
mine under what conditions ignition will occur, a
mathematical model was developed. The numerical
solutions to the model equations showed both qualitative and quantitative agreement with the experimental measurements of ignition velocity and combustion temperature. The numerical results indicated
that only a small temperature difference of 10°C
across the sample is required for an ignition front to
propagate across the sample when the furnace temperature is increased at a rate of 5°C/min. Experimental measurements show that furnace temperature
gradients well in excess of 10°C can occur.
The conditions for ignition were estimated from
thermal explosion theory. A furnace heating cycle
that avoids ignition was estimated when a safety factor is incorporated in the thermal explosion theory.
The safety factor gives a more conservative estimate
of the furnace heating cycle so as to overcome temperature differences within the furnace. Experimental
results and model prediction show that ignition was
avoided when a safety factor of 4 was used to calculate the furnace temperature program, while ignition
still occurred to a lesser degree when a safety factor
of 2 was used.
Acknowledgements—Financial support provided by the United
States Office of Naval Research under Grant No. N0014-961-0426.
REFERENCES
Fig. 11. Model predicted ignition results for MS45. Safety factor of f ⫽ 2 (· · ·) and f ⫽ 4 (– · –) calculated from equation
(13) and safety factor of f ⫽ 2 (———) and f ⫽ 4 (– – –) calculated from equations (4) and (5).
predicted sample temperature was taken at the center
of the sample, corresponding to z ⫽ 0.03 m.
The full model, equations (4)–(9) (Fig. 11), agrees
reasonably well with the experimental measurements
of Fig. 10 when it is assumed there is a 10°C temperature difference across the sample. The model-predicted temperature spike for f ⫽ 2 occurs earlier than
the experimental data. However, the magnitude of the
temperature spike for f ⫽ 2 and the lack of any
ignition for f ⫽ 4 agrees with the experimental
results.
6. CONCLUSIONS
Ignition can occur during firing of RBAO samples,
and ignition causes cracking and breakage. To deter-
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