Ignition phenomena and controlled firing of reaction-bonded aluminum oxide

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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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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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

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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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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

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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-

1. Claussen, N., Janssen, R. and Holz, D., J. Ceram. Soc. Japan, 1995, 103(8), 1. 2. Claussen, N., Wu, S. and Holz, D., J. Eur. Ceram. Soc., 1994, 14, 97. 3. Holz, D., Wu, S., Scheppokat, S. and Claussen, N., J. Am. Ceram. Soc., 1994, 77(10), 2509. 4. Wu, S., Holz, D. and Claussen, N., J. Am. Ceram. Soc., 1993, 76(4), 970, (April). 5. Gaus, S. P., Harmer, M. P., Chan, H. M. and Caram, H. S., J. Am. Ceram. Soc., 1999, 82(4), 897 (April). 6. Munir, Z. A., Ceram. Bull., 1988, 67(2), 342. 7. Yi, H. C. and Moore, J. J., J. Mater. Sci., 1990, 25, 1159. 8. Sheedy, P. M., Caram, H. S., Chan, H. M. and Harmer, M. P., J. Am. Ceram. Soc., 2001, accepted for publication. 9. Watson, M. J., Chan, H. M., Harmer, M. P. and Caram, H. S., J. Am. Ceram. Soc., 1998, 81(8), 2053. 10. Janna, W. S., Engineering Heat Transfer, SI edn. Van Nostrand Reinhold International, London, 1988. 11. Varma, A. and Mobidelli, M., Mathematical Methods in Chemical Engineering. Oxford University Press, New York, 1997. 12. Schiesser, W. E., The Numerical Method of Lines: Integration of Partial Differential Equations. Academic Press, San Diego, 1991. 13. Hairer, E. and Wanner, G., in Solving Ordinary Differential Equations II. Stiff and Differential-Algebraic Problems, Springer Series in Computation Mathematics, Vol. 14. Springer-Verlag, Berlin, 1991. 14. Fornberg, B., Math. Comput., 1988, 51(184), 699. 15. Saucez, P., Vande Wouwer, A. and Schiesser, W. E., J. Comput. Phys., 1996, 128, 274. 16. Zel’dovich, Y. B., Barenblatt, G. I., Librovich, V. B. and Makhviladze, G. M., The Mathematical Theory of Combustion and Explosion, English edn. Consultants Bureau, New York, 1985.

WATSON et al.: IGNITION PHENOMENA AND CONTROLLED FIRING OF RBAO 17. Perry, R. H. and Green, D., Perry’s Chemical Engineers’ Handbook, 6th edn. McGraw-Hill Book Company, New York, 1984. 18. Weast, R. C., Astle, M. J. and Beyer, W. H., eds., CRC

1103

Handbook of Chemistry and Physics, 67th edn. CRC Press, Boca Raton, FL, 1986–1989. 19. Kematick, R. J. and Franzen, H. F., J. Solid-State Chem., 1984, 54, 226.