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The numerical simulation of the porosity effect on the unstable propagation during micropyretic synthesis
Scripta Materialia 50 (2004) 999–1002 www.actamat-journals.com
The numerical simulation of the porosity effect on the unstable propagation during micropyretic synthesis H.P. Li
*
Jin-Wen Institute of Technology, Office of Research and Development, 99 An-Chung Road, Hsintien Taipei County 231, Taiwan Received 24 November 2003; received in revised form 22 December 2003; accepted 1 January 2004
Abstract The present study uses a numerical simulation to investigate the porosity effect on the unstable Ti + 2B micropyretic synthesis. The results show that the porosity affects the increment rate of pre-heat zone thickness, further influencing the average propagation velocity and the propagation fashion of TiB2 unstable combustion front. 2004 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Micropyretic synthesis; Self-propagating high-temperature synthesis (SHS); Borides; Simulation
1. Introduction Many exothermic non-catalytic solid–solid or solid– gas reactions, after being ignited locally, can release enough heat so as to sustain the self-propagating combustion front throughout the specimen without additional energy [1–14]. Since the 1970’s, this kind of exothermic reaction has been used in the process of synthesizing refractory compounds in the former Soviet Union [13]. This novel technique, so-called micropyretic synthesis/combustion synthesis, has been intensively studied for process implication [1–14]. This technique employs exothermic reaction processing, which circumvents difficulties associated with conventional methods of time and energy-intensive sintering processing. The advantages of micropyretic synthesis also include the rapid net shape processing and clean products. When compared with conventional powder metallurgy operations, micropyretic synthesis not only offers shorter processing time but also excludes the requirement for high-temperature sintering. In the solid–solid micropyretic reaction, it has been reported that the compact porosity has a significant effect on the micropyretic reaction [15–20]. Rice [16] has indicated that the effect of compact porosity on ignition and propagation is attributed to a balance between have en-
*
Tel.: +886-932383482; fax: +886-223813621. E-mail address: [email protected] (H.P. Li).
ough particle contact to aid reaction but not too much to lead to excessive thermal loss from the reaction zone. The study of Deevi [20] has also shown that with a decrease in the compact porosity, the combustion propagation velocity increases up to an optimum value and then decreases. This phenomenon is attributed to the thermal conductivity change. With a decrease in the compact porosity, the interfacial contacts increases, thus, increasing the efficiency of heat transfer to the pre-heat zone. This accelerates the heat transfer to the next region and helps the combustion temperature and the propagation velocity to increase. However, as the compact porosity is continuously decreased, the contact areas increase significantly, resulting in an increase in the effective thermal conductivity of the compact. The increased rate of heat transfer from the combustion front (reaction zone) to the reactants ahead of the combustion front (pre-heat zone) decreases the temperature which is necessary in the reaction region to maintain a steady state propagation of the combustion front. This decreases the average propagation velocity and the combustion temperature, and creates instability in the combustion front. When the conductivity is increased to a higher level, it may cause extinction of the combustion front. On account of the higher activation energy of Ti + 2B micropyretic reaction, the propagation of a combustion front occurs in an unstable oscillatory fashion. It has been found that the change of porosity not only affects the temperature, but also changes the propagation fashion during TiB2 unstable propagation [15–20]. This
1359-6462/$ - see front matter 2004 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.scriptamat.2004.01.001
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H.P. Li / Scripta Materialia 50 (2004) 999–1002
Nomenclature Cp E K0 Q P R T T0
heat capacity of product (general form), kJ/ kg/K activation energy, kJ/kg pre-exponential constant, (s1 for zero order reaction) heat of reaction, kJ/kg porosity, % gas constant, kJ/kg/K temperature, K initial temperature, K
study uses a numerical simulation to systematically investigate the porosity effect on the average propagation velocity and the oscillatory frequency of TiB2 unstable propagation. The variations in the thermal profile and the pre-heat zone thickness caused by the porosity change have also been investigated.
volume fraction,% dimensional coordinate, m diameter of the specimen, m surface heat transfer coefficient, J/m2 /K/s time, s density, kg/m3 thermal conductivity (general form), kJ/m/K/ s g fraction reacted UðT ; gÞ reaction rate, 1/s
V z d h t q j
density at node j: X ðqs V ð1 P Þ=ð1 þ ðP =2ÞÞÞ q¼
ð3Þ
s
thermal conductivity at node j: X j¼ ðjs V ð1 P Þ=ð1 þ ðP =2ÞÞÞ
ð4Þ
s
2. Numerical calculation procedure During the passage of the combustion front in the reaction, the energy equation for transient heat conduction, including the source term, containing heat release due to the exothermic micropyretic reaction is given as [10–13]: o j oT oT 4hðT T0 Þ oz qCp þ qQUðT ; gÞ ð1Þ ¼ ot d oz Each symbol in the equation is explained in the nomenclature section. The reaction rate, UðT ; gÞ, in Eq. (1) is given as: og E UðT ; gÞ ¼ ¼ K0 ð1 gÞ exp ð2Þ ot RT In this study, a numerical calculation for Eq. (1) is carried out with the assumption of the first order kinetics. In the Eq. (1), the energy required for heating the synthesized product from the initial temperature to the adiabatic combustion temperature is shown on the left-hand side. The terms on the right-hand side are the conduction heat transfer term, the surface heat loss parameter, and the heat release due to the exothermic combustion reaction, respectively. The surface heat loss is assumed radically Newtonian and is taken to be zero in this study. In the computational simulation, a one-dimensional sample of 1 cm long is divided into 1201 nodes (regions) to calculate the local temperature. The porosity effects of the reactants and product that influence the density (q) and thermal conductivity (j) profiles are considered in the numerical calculation:
where s denotes the components involved in the reaction, including Ti and B; and P is the porosity of a specific material. The effect of melting of reactants and product is also included in the calculation procedure. During the numerical calculation, the various thermophysical/chemical parameters, such as thermal conductivity, density and heat capacity of the reactants and product, are assumed to be independent of temperature, but they are different in each state. The average values of these parameters vary as the reaction proceeds, depending upon the degree of reaction. The parameter values used in the computational calculation and detailed numerical procedure have been introduced in the previous studies [1–3,5–7]. The calculated results generated in this numerical simulation have a good agreement with the experimental observations [2,3,5–7]. In this study, the combustion temperature is defined as the highest reaction temperature during micropyretic synthesis, and the propagation velocity is the velocity of the combustion front propagation. The reaction zone is defined as 0:01 < g < 1 and the pre-heat zone starts from the end of reaction zone until the position where the temperature is decreased to the original substrate temperature. In addition, the higher pre-exponential factor (K0 ) value, 1.5 · 1010 1/s, is used to be capably of illustrating the variation of the propagation velocity for the TiB2 unstable micropyretic reaction. 3. Results and discussion Fig. 1 shows the temperature profiles of combustion fronts at different time points along the specimens with 5%, 15%, and 25% porosities, respectively. The com-
H.P. Li / Scripta Materialia 50 (2004) 999–1002
bustion reaction is ignited at an initial position of 0 cm and the combustion front propagates from left to right. The numbers 20 and 40 shown in the figure, respectively, denote the 20th time step (0.04 s) and the 40th time step (0.08 s) of the temperature profiles after ignition. Since the activation energy for Ti + 2B micropyretic reaction is relatively higher than other micropyretic reactions, the combustion front of TiB2 system has been found to propagate in a rather unstable manner [1,8,9]. In such an unstable propagation, the temperature and the propagation velocity of the combustion front are periodically changed with the distance. Therefore, it can be found from Fig. 1 that the TiB2 combustion front oscillates periodically in the succession of rapid and slow movements. Fig. 1 also shows that the average propagation velocity and the oscillatory frequency of temperature variation initially increase (Fig. 1(b)) and then decrease (Fig. 1(c)) with an increase in the compact porosity. The variations in the pre-heat zone thickness, enthalpy, and combustion temperature with the reaction times for the specimens with 15% compact porosity are shown in Fig. 2. It is noted from Fig. 2 that at the beginning of each oscillatory cycle, the reaction is ignited and the combustion temperature (i.e., the highest temperature of the reaction for a given time step) as well as the enthalpy are sharply enhanced, resulting in a higher cooling rate. A previous numerical study [5] has shown that the higher combustion temperature and cooling rate result in more heat loss, thus, the pre-heat zone thickness is noted to decrease sharply at the initial stage of each oscillatory cycle (Fig. 2(a)). As the reaction proceeds, the combustion temperature and the enthalpy are noted to decrease due to heat loss (Fig. 2(b) and (c)). A corresponding decrease in the cooling rate is expected, which reduces the rate of heat loss and the pre-heat zone thickness is found to decrease gradually. When the combustion temperature is decreased to a low level, the propagation of combustion front almost stops. The lower
Fig. 2. The variations of the pre-heat zone thickness, enthalpy, and combustion temperature (Tc ) with the reaction time for the specimen with 15% porosity.
reaction temperature and rate, which correspond to less heat loss, lead to an accumulation of the excess enthalpy in the pre-heat zone and the pre-heat zone thickness starts to increase. When the pre-heat zone thickness is increased to a critical value, the excess enthalpy in that zone is sufficient to ignite the new reaction. Thus, the combustion front starts to propagate again. The new oscillatory cycle starts and the combustion temperature reaches to a previous high value, as shown in Fig. 2(c). The combustion temperature and the average propagation velocity are noted to increase and decrease alternately, causing oscillatory propagation. Several studies have indicated that the porosity affects the thermal conductivity of compact, further influencing the excess enthalpy and the changes in the thickness of pre-heat zone [15,16,20]. Thus, it is noted from Fig. 3 that the increment rate of pre-heat zone thickness is initially increased from 4.83 to 4.97 cm/s and then decreased to 4.50 cm/s with an increase in the porosity. The higher increment rate aids to increase the pre-heat zone thickness up to a critical value which can ignite the new micropyretic reaction. The new oscillatory cycle is
Increment Rate of Zone Thickness, cm/s
Fig. 1. Time variations of the combustion front temperature along the TiB2 specimen. The interval time between two consecutive time steps (profiles) is 0.002 s. The numbers 20 and 40 respectively denote the 20th time step (0.04 s) and the 40th time step (0.08 s) after ignition. The porosities in (a)–(c) are 5%, 15%, and 25%, respectively.
1001
5.4
5.0
4.6
4.2
3.8 0
5
10
15
20
25
30
35
Porosity, %
Fig. 3. Plots of the increment rate of the pre-heat zone thickness in each oscillatory cycle as a function of porosity.
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H.P. Li / Scripta Materialia 50 (2004) 999–1002
Fig. 4. Plots of (a) oscillatory frequency of temperature variation and (b) average propagation velocity of combustion fronts as functions of porosity.
formed rapidly and the oscillatory frequency of unstable propagation is increased correspondingly. Normally, an increase in the oscillatory frequency of the unstable propagation is equivalent to an increase in the reaction rate [1,9], further enhancing the average propagation velocity of unstable propagation. Therefore, it is expected that the average propagation velocity and the oscillatory frequency of unstable propagation, like increment rate of pre-heat zone thickness, are also nonlinear functions of porosity, as shown in Fig. 4. The maximum oscillatory frequency and the maximum average propagation velocity of unstable propagation are found at the 10–15% porosity level where the maximum increment rate of pre-heat zone thickness is occurred. It explains why the average propagation velocity and the oscillatory frequency of temperature variations are initially increased and then decreased with an increase in the porosity, as shown in Fig. 1. Such a change in the calculated propagation velocity caused by the porosity effects agrees very well with the experimental observations [15,17–19].
4. Summary and conclusions The porosity effect on the unstable propagation during TiB2 micropyretic reaction has been investigated using a numerical simulation. The numerical results show that the average propagation velocity and oscillatory frequency of the combustion front initially in-
crease and then decrease upon increasing the porosity. Such changes in the velocity and frequency are attributed to the variations in the increment rate of the pre-heat zone thickness. The maximum oscillatory frequency and the maximum average propagation velocity of unstable propagation are found at the 10–15% porosity level where the maximum increment rate of preheat zone thickness is occurred. It is inferred that the propagation fashion of the unstable micropyretic synthesis is influenced by the increment rate of pre-heat zone thickness. This study has also shown that the porosity directly affects the increment rate of the preheat zone thickness. Therefore, the stability of the micropyretic reaction can be enhanced by changing the porosity of the compact.
Acknowledgements The support from the National Center for HighPerformance Computing in Taiwan (account number: y48hpl00) is acknowledged.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]
Li HP. J Mater Res 2002;17:2515. Li HP. Acta Mater 2003;51:3213. Li HP. Metall Trans A 2003;34A(9):1969. Munir ZA, Anselmi-Tamburini U. Mater Sci Rep 1989;3:277. Li HP. Mater Chem Phys 2003;80(3):758. Li HP. J Mater Sci 2004;39(2):707. Li HP. J Mater Res 1995;10:1379. Li HP, Bhaduri SB, Sekhar JA. Metall Trans A 1992;23A:251. Li HP, Sekhar JA. J Mater Res 1993;8:2515. Lakshmikantha MG, Bhattacharys A, Sekhar JA. Metall Trans A 1992;23A:23. Lakshmikantha MG, Sekhar JA. J Mater Sci 1993;28:6403. Lakshmikantha MG, Sekhar JA. J Am Ceram Soc 1994;77:202. Merzhanov AG, Khaikin BI. Prog Energy Combust Sci 1988;14:1. Munir ZA. Am Ceram Bull 1988;67:342. Rice RW, Richardson GY, Kunetz JM, Schroeter T, McDonough WJ. Adv Ceram Mater 1987;2:222. Rice RW. J Mater Sci 1991;26:6533. Munir ZA, Holt JB. J Mater Sci 1987;22:710. Naiborodenko YS, Itin VI. Combust Explos Shock Wave 1975;11(3):293. Nekrasov EA, Maksimov YM, Ziatdinov MK, Shteinberg AS. Combust Explos Shock Wave 1978;14:575. Deevi SC. J Mater Sci 1991;26:2662.
The numerical simulation of the porosity effect on the unstable propagation during micropyretic synthesis H.P. Li
*
Jin-Wen Institute of Technology, Office of Research and Development, 99 An-Chung Road, Hsintien Taipei County 231, Taiwan Received 24 November 2003; received in revised form 22 December 2003; accepted 1 January 2004
Abstract The present study uses a numerical simulation to investigate the porosity effect on the unstable Ti + 2B micropyretic synthesis. The results show that the porosity affects the increment rate of pre-heat zone thickness, further influencing the average propagation velocity and the propagation fashion of TiB2 unstable combustion front. 2004 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Micropyretic synthesis; Self-propagating high-temperature synthesis (SHS); Borides; Simulation
1. Introduction Many exothermic non-catalytic solid–solid or solid– gas reactions, after being ignited locally, can release enough heat so as to sustain the self-propagating combustion front throughout the specimen without additional energy [1–14]. Since the 1970’s, this kind of exothermic reaction has been used in the process of synthesizing refractory compounds in the former Soviet Union [13]. This novel technique, so-called micropyretic synthesis/combustion synthesis, has been intensively studied for process implication [1–14]. This technique employs exothermic reaction processing, which circumvents difficulties associated with conventional methods of time and energy-intensive sintering processing. The advantages of micropyretic synthesis also include the rapid net shape processing and clean products. When compared with conventional powder metallurgy operations, micropyretic synthesis not only offers shorter processing time but also excludes the requirement for high-temperature sintering. In the solid–solid micropyretic reaction, it has been reported that the compact porosity has a significant effect on the micropyretic reaction [15–20]. Rice [16] has indicated that the effect of compact porosity on ignition and propagation is attributed to a balance between have en-
*
Tel.: +886-932383482; fax: +886-223813621. E-mail address: [email protected] (H.P. Li).
ough particle contact to aid reaction but not too much to lead to excessive thermal loss from the reaction zone. The study of Deevi [20] has also shown that with a decrease in the compact porosity, the combustion propagation velocity increases up to an optimum value and then decreases. This phenomenon is attributed to the thermal conductivity change. With a decrease in the compact porosity, the interfacial contacts increases, thus, increasing the efficiency of heat transfer to the pre-heat zone. This accelerates the heat transfer to the next region and helps the combustion temperature and the propagation velocity to increase. However, as the compact porosity is continuously decreased, the contact areas increase significantly, resulting in an increase in the effective thermal conductivity of the compact. The increased rate of heat transfer from the combustion front (reaction zone) to the reactants ahead of the combustion front (pre-heat zone) decreases the temperature which is necessary in the reaction region to maintain a steady state propagation of the combustion front. This decreases the average propagation velocity and the combustion temperature, and creates instability in the combustion front. When the conductivity is increased to a higher level, it may cause extinction of the combustion front. On account of the higher activation energy of Ti + 2B micropyretic reaction, the propagation of a combustion front occurs in an unstable oscillatory fashion. It has been found that the change of porosity not only affects the temperature, but also changes the propagation fashion during TiB2 unstable propagation [15–20]. This
1359-6462/$ - see front matter 2004 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.scriptamat.2004.01.001
1000
H.P. Li / Scripta Materialia 50 (2004) 999–1002
Nomenclature Cp E K0 Q P R T T0
heat capacity of product (general form), kJ/ kg/K activation energy, kJ/kg pre-exponential constant, (s1 for zero order reaction) heat of reaction, kJ/kg porosity, % gas constant, kJ/kg/K temperature, K initial temperature, K
study uses a numerical simulation to systematically investigate the porosity effect on the average propagation velocity and the oscillatory frequency of TiB2 unstable propagation. The variations in the thermal profile and the pre-heat zone thickness caused by the porosity change have also been investigated.
volume fraction,% dimensional coordinate, m diameter of the specimen, m surface heat transfer coefficient, J/m2 /K/s time, s density, kg/m3 thermal conductivity (general form), kJ/m/K/ s g fraction reacted UðT ; gÞ reaction rate, 1/s
V z d h t q j
density at node j: X ðqs V ð1 P Þ=ð1 þ ðP =2ÞÞÞ q¼
ð3Þ
s
thermal conductivity at node j: X j¼ ðjs V ð1 P Þ=ð1 þ ðP =2ÞÞÞ
ð4Þ
s
2. Numerical calculation procedure During the passage of the combustion front in the reaction, the energy equation for transient heat conduction, including the source term, containing heat release due to the exothermic micropyretic reaction is given as [10–13]: o j oT oT 4hðT T0 Þ oz qCp þ qQUðT ; gÞ ð1Þ ¼ ot d oz Each symbol in the equation is explained in the nomenclature section. The reaction rate, UðT ; gÞ, in Eq. (1) is given as: og E UðT ; gÞ ¼ ¼ K0 ð1 gÞ exp ð2Þ ot RT In this study, a numerical calculation for Eq. (1) is carried out with the assumption of the first order kinetics. In the Eq. (1), the energy required for heating the synthesized product from the initial temperature to the adiabatic combustion temperature is shown on the left-hand side. The terms on the right-hand side are the conduction heat transfer term, the surface heat loss parameter, and the heat release due to the exothermic combustion reaction, respectively. The surface heat loss is assumed radically Newtonian and is taken to be zero in this study. In the computational simulation, a one-dimensional sample of 1 cm long is divided into 1201 nodes (regions) to calculate the local temperature. The porosity effects of the reactants and product that influence the density (q) and thermal conductivity (j) profiles are considered in the numerical calculation:
where s denotes the components involved in the reaction, including Ti and B; and P is the porosity of a specific material. The effect of melting of reactants and product is also included in the calculation procedure. During the numerical calculation, the various thermophysical/chemical parameters, such as thermal conductivity, density and heat capacity of the reactants and product, are assumed to be independent of temperature, but they are different in each state. The average values of these parameters vary as the reaction proceeds, depending upon the degree of reaction. The parameter values used in the computational calculation and detailed numerical procedure have been introduced in the previous studies [1–3,5–7]. The calculated results generated in this numerical simulation have a good agreement with the experimental observations [2,3,5–7]. In this study, the combustion temperature is defined as the highest reaction temperature during micropyretic synthesis, and the propagation velocity is the velocity of the combustion front propagation. The reaction zone is defined as 0:01 < g < 1 and the pre-heat zone starts from the end of reaction zone until the position where the temperature is decreased to the original substrate temperature. In addition, the higher pre-exponential factor (K0 ) value, 1.5 · 1010 1/s, is used to be capably of illustrating the variation of the propagation velocity for the TiB2 unstable micropyretic reaction. 3. Results and discussion Fig. 1 shows the temperature profiles of combustion fronts at different time points along the specimens with 5%, 15%, and 25% porosities, respectively. The com-
H.P. Li / Scripta Materialia 50 (2004) 999–1002
bustion reaction is ignited at an initial position of 0 cm and the combustion front propagates from left to right. The numbers 20 and 40 shown in the figure, respectively, denote the 20th time step (0.04 s) and the 40th time step (0.08 s) of the temperature profiles after ignition. Since the activation energy for Ti + 2B micropyretic reaction is relatively higher than other micropyretic reactions, the combustion front of TiB2 system has been found to propagate in a rather unstable manner [1,8,9]. In such an unstable propagation, the temperature and the propagation velocity of the combustion front are periodically changed with the distance. Therefore, it can be found from Fig. 1 that the TiB2 combustion front oscillates periodically in the succession of rapid and slow movements. Fig. 1 also shows that the average propagation velocity and the oscillatory frequency of temperature variation initially increase (Fig. 1(b)) and then decrease (Fig. 1(c)) with an increase in the compact porosity. The variations in the pre-heat zone thickness, enthalpy, and combustion temperature with the reaction times for the specimens with 15% compact porosity are shown in Fig. 2. It is noted from Fig. 2 that at the beginning of each oscillatory cycle, the reaction is ignited and the combustion temperature (i.e., the highest temperature of the reaction for a given time step) as well as the enthalpy are sharply enhanced, resulting in a higher cooling rate. A previous numerical study [5] has shown that the higher combustion temperature and cooling rate result in more heat loss, thus, the pre-heat zone thickness is noted to decrease sharply at the initial stage of each oscillatory cycle (Fig. 2(a)). As the reaction proceeds, the combustion temperature and the enthalpy are noted to decrease due to heat loss (Fig. 2(b) and (c)). A corresponding decrease in the cooling rate is expected, which reduces the rate of heat loss and the pre-heat zone thickness is found to decrease gradually. When the combustion temperature is decreased to a low level, the propagation of combustion front almost stops. The lower
Fig. 2. The variations of the pre-heat zone thickness, enthalpy, and combustion temperature (Tc ) with the reaction time for the specimen with 15% porosity.
reaction temperature and rate, which correspond to less heat loss, lead to an accumulation of the excess enthalpy in the pre-heat zone and the pre-heat zone thickness starts to increase. When the pre-heat zone thickness is increased to a critical value, the excess enthalpy in that zone is sufficient to ignite the new reaction. Thus, the combustion front starts to propagate again. The new oscillatory cycle starts and the combustion temperature reaches to a previous high value, as shown in Fig. 2(c). The combustion temperature and the average propagation velocity are noted to increase and decrease alternately, causing oscillatory propagation. Several studies have indicated that the porosity affects the thermal conductivity of compact, further influencing the excess enthalpy and the changes in the thickness of pre-heat zone [15,16,20]. Thus, it is noted from Fig. 3 that the increment rate of pre-heat zone thickness is initially increased from 4.83 to 4.97 cm/s and then decreased to 4.50 cm/s with an increase in the porosity. The higher increment rate aids to increase the pre-heat zone thickness up to a critical value which can ignite the new micropyretic reaction. The new oscillatory cycle is
Increment Rate of Zone Thickness, cm/s
Fig. 1. Time variations of the combustion front temperature along the TiB2 specimen. The interval time between two consecutive time steps (profiles) is 0.002 s. The numbers 20 and 40 respectively denote the 20th time step (0.04 s) and the 40th time step (0.08 s) after ignition. The porosities in (a)–(c) are 5%, 15%, and 25%, respectively.
1001
5.4
5.0
4.6
4.2
3.8 0
5
10
15
20
25
30
35
Porosity, %
Fig. 3. Plots of the increment rate of the pre-heat zone thickness in each oscillatory cycle as a function of porosity.
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H.P. Li / Scripta Materialia 50 (2004) 999–1002
Fig. 4. Plots of (a) oscillatory frequency of temperature variation and (b) average propagation velocity of combustion fronts as functions of porosity.
formed rapidly and the oscillatory frequency of unstable propagation is increased correspondingly. Normally, an increase in the oscillatory frequency of the unstable propagation is equivalent to an increase in the reaction rate [1,9], further enhancing the average propagation velocity of unstable propagation. Therefore, it is expected that the average propagation velocity and the oscillatory frequency of unstable propagation, like increment rate of pre-heat zone thickness, are also nonlinear functions of porosity, as shown in Fig. 4. The maximum oscillatory frequency and the maximum average propagation velocity of unstable propagation are found at the 10–15% porosity level where the maximum increment rate of pre-heat zone thickness is occurred. It explains why the average propagation velocity and the oscillatory frequency of temperature variations are initially increased and then decreased with an increase in the porosity, as shown in Fig. 1. Such a change in the calculated propagation velocity caused by the porosity effects agrees very well with the experimental observations [15,17–19].
4. Summary and conclusions The porosity effect on the unstable propagation during TiB2 micropyretic reaction has been investigated using a numerical simulation. The numerical results show that the average propagation velocity and oscillatory frequency of the combustion front initially in-
crease and then decrease upon increasing the porosity. Such changes in the velocity and frequency are attributed to the variations in the increment rate of the pre-heat zone thickness. The maximum oscillatory frequency and the maximum average propagation velocity of unstable propagation are found at the 10–15% porosity level where the maximum increment rate of preheat zone thickness is occurred. It is inferred that the propagation fashion of the unstable micropyretic synthesis is influenced by the increment rate of pre-heat zone thickness. This study has also shown that the porosity directly affects the increment rate of the preheat zone thickness. Therefore, the stability of the micropyretic reaction can be enhanced by changing the porosity of the compact.
Acknowledgements The support from the National Center for HighPerformance Computing in Taiwan (account number: y48hpl00) is acknowledged.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]
Li HP. J Mater Res 2002;17:2515. Li HP. Acta Mater 2003;51:3213. Li HP. Metall Trans A 2003;34A(9):1969. Munir ZA, Anselmi-Tamburini U. Mater Sci Rep 1989;3:277. Li HP. Mater Chem Phys 2003;80(3):758. Li HP. J Mater Sci 2004;39(2):707. Li HP. J Mater Res 1995;10:1379. Li HP, Bhaduri SB, Sekhar JA. Metall Trans A 1992;23A:251. Li HP, Sekhar JA. J Mater Res 1993;8:2515. Lakshmikantha MG, Bhattacharys A, Sekhar JA. Metall Trans A 1992;23A:23. Lakshmikantha MG, Sekhar JA. J Mater Sci 1993;28:6403. Lakshmikantha MG, Sekhar JA. J Am Ceram Soc 1994;77:202. Merzhanov AG, Khaikin BI. Prog Energy Combust Sci 1988;14:1. Munir ZA. Am Ceram Bull 1988;67:342. Rice RW, Richardson GY, Kunetz JM, Schroeter T, McDonough WJ. Adv Ceram Mater 1987;2:222. Rice RW. J Mater Sci 1991;26:6533. Munir ZA, Holt JB. J Mater Sci 1987;22:710. Naiborodenko YS, Itin VI. Combust Explos Shock Wave 1975;11(3):293. Nekrasov EA, Maksimov YM, Ziatdinov MK, Shteinberg AS. Combust Explos Shock Wave 1978;14:575. Deevi SC. J Mater Sci 1991;26:2662.