Ind. Eng. Chem. Res. 2008, 47, 4379–4386
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Temperature Profile Analysis of the Citrate-Nitrate Combustion System Marjan Marinsˇek,† Jana Kemperl,† Blazˇ Likozar,*,‡ and Jadran Macˇek† UniVersity of Ljubljana, Faculty of Chemistry and Chemical Technology, AsˇkercˇeVa cesta 5, 1000 Ljubljana, SloVenia
The citrate-nitrate combustion reaction was used for mixed NiO-YSZ (yttria-stabilized zirconia) preparation. This system was chosen because of its nonviolent nature and its potential use in nanosized YSZ-based cermets and in some electrocatalytic applications such as fuel cells. The kinetic and thermodynamic parameters of the combustion system were studied using wave velocity analysis, the Boddington method, and the Freeman-Carroll method. The calculated activation energies of the combustion system using these three kinetic analysis approaches were 65.4, 61.4, and 60.6 kJ/mol, respectively. The obtained thermodynamic and kinetic parameters of the citrate-nitrate system were also used for a computer simulation of the combustion temperature profiles and the apparent conversion profiles. Introduction Combustion synthesis or self-propagating high-temperature synthesis (SHS) provides an attractive practical alternative to the conventional methods of producing advanced materials, such as ceramics, composites, and intermetallic compounds.1–3 The combustion process involves decomposition of a reduction/ oxidation reaction system, which then proceeds as a selfsustaining front throughout the reactant gel mixture. The reaction conditions and the large amount of heat evolved during the reaction enable the direct production of a large number of singleor multicomponent powders that are crystalline and homogeneous and have a narrow particle size distribution.4–8 From the variety of possible reduction/oxidation reaction systems, a combustion mixture based on the citrate-nitrate combination can potentially be applied for the production of large quantities of mixed oxides owing to its relatively nonviolent combustion.9 Theoretical physical models that explain, simulate, and predict SHS reaction phenomena are based on energy and mass balances.10 The principal reasons for which kinetic analysis should be carried out are both to describe the process itself in terms of the reactions that control the synthesis and to establish the rate constant(s) and the rate law in general of prevalent reaction(s) in order to successfully scale up the process. The latter is of vital importance, especially because of the characteristics and the promising fields of application of the products synthesized by such processes. The reaction kinetics is ratelimiting to the overall process owing to the swiftness of the combustion process. To obtain the kinetic parameters of an SHS reaction, two approaches are primarily utilized, namely, wave velocity measurements and temperature profile determinations. In regard to the former, the first mathematical model for an SHS reaction was proposed by Booth.11 His one-dimensional model of a burning cylindrical rod neglected the heat losses due to radiation, the effect of the diffusion of certain material originating species, and the heat loses from the side of the rod. This model was subsequently further developed for various SHS reaction systems by many researchers,12–19 and in some cases, special mathematical techniques were employed to solve the model. The temperature dependence of the wave velocity and kinetics, where the temperature dependence is ordinarily * To whom correspondence should be addressed. Tel.: +386-1-2419-540. Fax: +386-1-24-19-541. E-mail:
[email protected]. † Chair of Inorganic Chemical Technology and Materials. ‡ Chair of Polymer Engineering, Organic Chemical Technology and Materials.
described using the Arrhenius equation whereas the conversion dependence is ordinarily described using a power law, are usually used to calculate the activation energy of the process.20 The heat of reaction (J/g) normally changes with inert diluting agent, and thus, the wave velocity equation is written as20
( )
CP R u2 -E exp (1 - λ) ) f(η)K0R 2 Q E RT TC 0 C
(1)
where u is the propagation wave velocity (cm/s), f(η) is a function that depends on the kinetics of the reaction, R is the thermal diffusivity of the product (cm2/s), CP is the heat capacity of the product [J/(g K)], Q0 is the heat of reaction for an undiluted system (J/g), λ is the weight fraction of diluting agent, E is the activation energy (kJ/mol), TC is the maximum combustion temperature (K), R is the universal gas constant [J/(mol K)], and K0 is an arbitrary constant (s-1). As the reaction system is diluted with an inert substance, parameters u and TC are altered, and the activation energy can be calculated from the slope of a plot of ln(u/TC) versus 1/TC. In the case when the reaction rate is used to calculate the kinetic term, the mathematical analysis of the measured temperature profiles is utilized. These profiles can be considered as temporal or spatial, as the combustion wave is being considered and the coordinate (x) and time (t) are interrelated through the wave velocity (u). Mathematically, there are two different approaches for the temperature profile analysis: one was provided by Boddington et al.21 and the other by Zenin et al.22 The so-called Boddington method is particularly valuable because preliminary thermal conductivity determinations are not necessary. Likewise, measurements of other material properties such as density and heat capacity are also not needed. For a reliable comparison, the latter two properties have to be determined independently; yet for the analysis itself, this is not mandatory. This fact is useful as, upon application of the analysis, process average material properties are obtained, whereas upon independent estimation of both reactant and product properties, the dilemma of their utilization remains unresolved. Moreover, the Boddington method is not based on a specific kinetic model but predicts the same kinetic behavior of the system (diluted or undiluted) at a certain value of reactantto-product conversion. The Boddington method uses the following heat-balance equation κ
∂T ∂2T + Φ ′ (T,η) - h(T - T0) ) 0 - CPF ∂t ∂x2
10.1021/ie800296m CCC: $40.75 2008 American Chemical Society Published on Web 06/10/2008
(2)
4380 Ind. Eng. Chem. Res., Vol. 47, No. 13, 2008
where CP is the heat capacity of the product [J/(g K)], F is its density (g/cm3), κ is the thermal conductivity [W/(m K)], Φ(T,η) is the rate of heat generation (W/cm3), h is the axial heat-transfer coefficient [W/(cm3 K)], T is the temperature (K), T0 is the initial pellet temperature (K), η represents the conversion at some unspecified height (or at any specified height at the specific time) in the pellet, t is the time (s), and x is the coordinate along which the wave is propagating (cm) (the radiant heat losses are assumed to be negligible). By performing the Boddington analysis of the measured temperature profiles, the estimation of the reaction rate, ∂η/∂t, and the conversion, η, versus t is eventually possible. The activation energy is then calculated from the slope of a plot of ∂η/∂t) versus the reciprocal of the absolute temperature at a fixed value of η. A good example of the utilization of the Boddington analysis for a combustion system was presented by Dunmead et al.27,28 When data in terms of the conversion, η, versus T or t are accessible, several approaches for kinetic analysis are available. A particularly interesting approach to kinetic parameter determination,primarilybecauseofitssimplicity,istheFreeman-Carroll method.29 This method is based on the temperature dependence of the reaction rate, described using the Arrhenius equation, and on the conversion dependence of the reaction rate, described by the power law, and therefore assumes the form dη E ) K′0 exp (1 - η)n dt RT
(
)
(3)
where K0′ is a constant (s-1) and n is an exponential factor in a function of the kinetic order. In the present work, a citrate (fuel)-nitrate (oxidant) combustion synthesis method was employed to produce NiO-(Y2O3)0.1(ZrO2)0.9 oxide mixture. This system was chosen due to the fact that fine cermets based on yttria-stabilized zirconia (YSZ) have potential uses in some electrocatalytic applications such as anode material in solid oxide fuel cells, especially if they are reduced to Ni-YSZ.23 In such Ni-YSZ cermets, particle size and phase distribution are essential. The ideal microstructure is composed of small randomly mixed and well-connected Ni and YSZ particles with a highly developed triple-phase boundary (boundary between the electronic conductor Ni, the ionic conductor YSZ, and the adjacent fuel). In this view, combustion-derived oxide mixtures have a number of clear advantages over those obtained by other preparation routes. Specifically, citrate-nitrate combustion for the preparation of mixed NiO-YSZ enables nanosized mixtures.24 Furthermore, the combustion method is very simple, and as such, it is also appropriate for the preparation of larger quantities of mixed oxides. Because of some advantages regarding material homogeneity and particle size distribution that can be ensured by choosing a combustion synthesis route, this article also addresses the kinetic analysis of the burning citrate-nitrate system. The kinetic parameters of the citrate-nitrate combustion system were calculated using different approaches, i.e., the Boddington method and the Freeman-Carroll method, and subsequently validated in a computer simulation of the combustion. Experimental Section NiO-YSZ composite powders were prepared by a modified combustion synthesis. The combustion system was based on the citrate-nitrate reduction/oxidation reaction. The starting substances for reactive gel preparation were ZrO(NO3)2 · 6H2O, Y(NO3)3 · 6H2O, Ni(NO3)2 · 6H2O, citric acid, and nitric acid (analytical reagent grade). ZrO(NO3)2 · 6H2O, Y(NO3)3 · 6H2O, Ni(NO3)2 · 6H2O, and citric acid were dissolved separately with
minimum additions of water in amounts that provided the desired Y2O3 content in YSZ (90 mol % ZrO2 and 10 mol % Y2O3) and the final NiO content in NiO-YSZ (56 vol % of NiO in the final composite). HNO3 (aqueous, 65%) was then mixed with the zirconyl nitrate aqueous solution to ensure an initial citrate/nitrate molar ratio of 0.18. All precursors were mixed together to prepare a solution that was kept over a water bath at 60 °C under vacuum (5 mbar) until it transformed into a bright green xerogel. More detailed citrate-nitrate reactive gel preparation and its analysis are described elsewhere.25 In order to extract the kinetic parameters from wave velocity measurements, the reaction system was diluted with different additions of R-Al2O3 (from 0 to 20 wt % Al2O3 in the reactive mixture). Al2O3 was mixed into the system before the reactive mixture was dried. In this way, a uniform distribution of Al2O3 in the xerogel was ensured. If the additions of Al2O3 exceeded 20 wt % the combustion system did not react in a self-sustaining mode. The dried and diluted gels were gently milled and homogenized in an agate mortar and subsequently uniaxially pressed (17 MPa) into pellets (diameter, 16 mm; height, ∼30 mm). The pellets were ignited at the top to start a self-sustaining combustion reaction producing NiO-YSZ oxide mixtures. Temperature profiles of burning tablets were measured using an optical pyrometer (Ircon, IPE 140, based on sample brightness). The measuring range of this pyrometer is from 50 to 1200 °C, and it has a very quick response time (1.5 ms). The accuracy of the optically measured temperatures was (2.5 °C below 400 °C and (0.4% of the measured value (in °C) above 400 °C. Because the measured systems were all ceramics whose exact emissivities were unknown, the emissivity was set to 0.85 and was kept constant for all measurements. This value is an average of the cited emissivities of the products NiO, YSZ, and Al2O3 in the measured temperature range. Temperature profiles of the measured systems were taken from a distance of 10 cm, resulting in a space resolution of the temperature measurements (size of the measured spot on the sample surface) of 0.3 mm. The experimentally measured temperature profiles were smoothed in the zone of rapid temperature change prior to the Boddington kinetic analysis. A smoothing procedure is essential owing to the fact that the measured values can oscillate around the average value, which is reflected in difficult temperature profile analysis. Slight temperature oscillations were ascribed to surface roughness and the fact that the synthesized material exhibits substantial porosity. Mathematica 5.0.0.0 was employed for the simulation of the temperature profile of the citrate-nitrate combustion system. Results and Discussion The general reaction between nitrate and citrate can be represented as25,26 aC6H8O7 + bNO3- f cC + dCO + eCO2 + f H2O + gOH- + hNO (4) where stoichiometric factors a-h depend on the conditions during the overall reaction. With respect to the investigated citrate-nitrate ratio, the stoichiometric factors a-h were 1, 4, 1, 1, 4, 2, 4, and 4, respectively. However, a possible reaction of any residual fuel with oxygen from air cannot be excluded. In that case, even though the residual product after combustion is highly porous (Figure 1), the overall reaction might be influenced by the mass transfer of oxygen as well. Nevertheless, considering the relatively rapid principal reaction (eq 4), even though gas-phase oxygen probably participates in unwanted
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Figure 1. Course of the citrate-nitrate combustion reaction (λ ) 0.15 wt %) and microstructure details of the synthesized product. Table 1. Reaction Parameters of the Citrate-Nitrate SHS Systems λ (wt %) TC (K) µ × 103 (cm/s) (∂T/∂t)max (K/s) (∂T/∂x)max (K/cm) 0 5 10 15 20
Figure 2. Measured temperature profiles of citrate-nitrate SHS systems with different additions of diluent.
parallel reactions, the latter are limited in extent. This presumption can be justified with several additional facts. First, the reactions with gas-phase oxygen can occur solely on the outer interface of the reacting material, whereas the bulk-phase reactions mainly correspond to the mechanism embodied in eq 4. Furthermore, even on the outer surface interface, the gasphase oxygen reactions are limited by the mass transfer of oxygen from the ambient air bulk phase to the reactive surface, thus rendering even the surface reactions mainly those that can be represented by eq 4. Finally, the extent of the reactions with gas-phase oxygen is diminished in practice, as evolving gases (right-hand side of eq 4) from the combusting material sweep aside the nearby air, thus preventing the oxygen from reaching the reacting material. An example of a burning system and the measured temperature profiles of the chosen citrate-nitrate combustion mixtures with different additions of Al2O3 diluting agent (λ) are shown in Figures 1 and 2, respectively. The temperatures in Figure 2 were measured at a certain height of the pellet and are not averages over the entire pellet; nevertheless, because combustion wave propagation is assumed, the coordinate-time superposition
1426 1381 1349 1318 1289
47.8 40.3 37.1 32.1 26.4
1164 1061 994 884 802
24351 26328 26792 27539 30378
can be applied to the temperature at each position on the pellet surface and at the specific time. Altering the amount of diluting agent influences many parameters pertinent to combustion reactions, such as wave velocity and maximum combustion temperature, as summarized in Table 1. According to the results in Figure 2 and Table 1, greater additions of diluting agent lower both the peak combustion temperature and the rate of propagation of the combustion reaction. However, the general shape of temperature profiles did not change with the variation in the addition of diluting agent. An interesting feature is the relatively rapid product cooling. The samples cooled from their peak temperature to 50 °C in approximately 80 s. Such fast cooling was ascribed to the very porous structure formed after the combustion (Figure 1). The determination of parameters TC and u is essential for the wave velocity analysis. Because the generated heat per unit mass of catalyst, though not the reaction enthalpy, is strongly affected by the amount of added diluting agent, eq 1 was used for the calculation of the activation energy, and both obtained parameters were plotted in terms of ln[u(1 - λ)0.5/TC] versus 1/TC (Figure 3). From the best-fit linear slope of the experimental combustion parameters, the apparent combustion activation energy was calculated to be 65.1 kJ/mol. To obtain the relevant kinetic information from temperature profiles, a variety of calculations were performed on the measured experimental data. First, the temperature profiles in the zone of rapid temperature change were smoothed using the fast Fourier transform algorithm for each measured temperature over a range of 20 adjacent measured temperatures. From the smoothed version of the temperature profiles, the first and second derivatives, ∂T/∂t and ∂2T/∂t2, of each smoothed profile were calculated. The peak value in the plot of ∂T/∂t versus t corresponds to the maximum heating rate achieved during the
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Figure 3. Wave velocity analysis for the combustion reaction of citrate and nickel, zirconium, and yttrium nitrates. Table 2. Experimentally Determined Parameters of the Boddington Analysis for Citrate-Nitrate Combustion λ (wt %) 0 5 10 15 20
tr (s)
td (s)
tx (s)
t* (s)
τad(K) R × 102 (cm2/s)
5.0075 5.2383 5.2438 5.3419 5.4348
17.794 18.382 18.416 18.692 18.762
12.786 13.144 13.172 13.350 13.327
6.9686 7.3259 7.3313 7.4795 7.6511
1533 1471 1438 1394 1362
1.5922 1.1898 1.0091 0.7707 0.5333
reaction. If this value is divided by the measured wave velocity value (u), the maximum thermal gradient, (∂T/∂x)max, in terms of the propagating coordinate can be calculated (Table 1). The highest value of (∂T/∂x)max was calculated for the maximally diluted sample. Such a high value is mainly the consequence of it having the lowest reaction propagation rate. Subsequently, a complete Boddington analysis of the obtained temperature profiles was performed, including the calculation of the parameters tr and td (the rise time of a general adiabatic wavefront and the remote decay time, respectively) which are necessary for determining two other characteristic parameters, namely, tx (thermal relaxation time) and t* (inverse of the difference between the inverse decay and rise times), using the relationships tx ) td - tr and 1/t* ) 1/tr - 1/td. The parameters tx and t* incorporate thermodynamic properties and can be further expressed as t* ) R/u2 and tx ) FCP/h, where R represents the effective thermal diffusivity (Table 2). The remote rise zone interval was determined by the dynamic scanning calorimetry-evolved gas analysis (DSC-EGA) technique as described elsewhere.25 These measurements showed that the citrate-nitrate reaction in the presence of Ni, Zr, and Y ions started at 180 °C, which was also the upper limit value of the remote heat-affected zone. The temperature rise of the combustion system without any heat losses, i.e., τad (Table 2), was calculated by integration of the temperature profile as described in the literature.27 If the conditions of the combustion system are adiabatic, the value of τad should be close to the value of TC. However, if the data from Tables 1 and 2 are compared, τad and TC can be seen to differ substantially. One reason for the difference is that the conditions during the reaction are only partially adiabatic. During the citrate-nitrate combustion, some volatile products are also formed,26 carrying off a certain amount of heat. This fact implies that eq 2 should be changed in such a way as to describe the heat balance of the citrate-nitrate combustion. Furthermore, in certain cases (intensively forced convective regimes), the formation of volatile products affects both the numerical analysis and the extrapolative model predictability of the system. Nonetheless, eq 2 can be used in the kinetic
Figure 4. Temperature profile and time dependence of conversion for the undiluted sample.
analysis, as it is still one of the best and most widely used combustion models, which can definitely be applied under natural convection-determining boundary conditions, which was the case during our citrate-nitrate combustion experiments. Therefore, the proposed Boddington heat-balance equation can still be used for the kinetic analysis and the estimation of the η versus T (or t) relationship. The next step in the kinetic analysis of the system was the formation of a power function G as described in the literature.28 This function was calculated in accordance with the following rearranged Boddington heat-balance equation G ) tx-1τ + ˙τ - t*τ¨
(5)
where τ stands for (T - T0) and τ˙ and τ¨ represent its first and second derivatives, respectively, with respect to time. Because G is also equivalent to τad(∂η/∂t), it was possible to calculate the rate of the reaction by dividing the values of G by the adiabatic temperature rise. Finally, by numerically integrating the experimentally determined reaction rate, the conversion, η, was calculated (Figure 4). The calculated η versus t data show that the citrate-nitrate combustion is a relatively rapid process. In an undiluted sample, the elapsed time interval for conversion to increase from 10% to 90% was approximately 0.8 s. In the case of the maximally diluted sample (λ ) 20 wt %), the time interval for the completion of the reaction from 10% to 90% conversion was extended to 1.1 s. The obtained kinetic parameters were used to calculate the activation energy for the citrate-nitrate combustion process.
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Figure 5. Temperature dependence of the reaction rate ln(∂η/∂t) for different η values in the citrate-nitrate combustion system.
According to Boddington and co-workers,21 because of the interdependence of the T-η relationship and the additions of inert diluting agent, the activation energy can be calculated by plotting the experimentally determined values of ∂η/∂t versus the reciprocal of the absolute temperature at a fixed value of η. The magnitude of the slope in such a constructed diagram should equal -E/R. The temperature dependence of the rate of reaction at different η values is shown in Figure 5. It is evident from these data that the slopes of the data sets for the middle range of conversion (from η ) 0.2 to η ) 0.8) agree fairly well with one another. On the other hand, the slope at the value for η ) 0.1 and the slope at the highest η values (η > 0.8) exhibited somewhat different values than the middle-range conversion data sets. The reason for such behavior was pointed out by Dunmed et al.28 They suggested that a low reaction rate (i.e., the reaction rate at very low or very high η values) should result in a higher error in determining the time at which the specific value of η occurs. Additionally, the different magnitudes of constructed slopes imply that, at low or high η values, a different solidstate mechanism could be involved than at the middle η values.31 Taking into account only the slopes of the middle-range conversion data sets, the activation energy for the citrate-nitrate combustion system was calculated as 61.4 kJ/mol. The third technique used for the kinetic analysis of the chosen citrate-nitrate system was the Freeman-Carroll method.29 When the predetermined ∂η/∂t versus T data were used, this method enabled the activation energy to be determined, as well as the parameters n and K0′ in eq 3. First, the parameter n was calculated from the slope of a plot of ∆[ln(dη/dt)]/∆(1/T) versus ∆[ln(1 - η)]/∆(1/T). Afterward, the activation energy of the system and the parameter K0′ were calculated from a plot of ln[(dη/dt)/(1 -η)n] versus the reciprocal of the absolute temperature (Vachuska-Voboril method30). The magnitude of the slope in such a diagram is equal to -E/R, and the intercept on the ordinate gives the value of ln K0′ (Figure 6). Parameters n, K0′, and E were determined for the undiluted system, as well as for all systems with different additions of diluting agent. The average n, K0′, and E values were calculated to be 0.940, 578 s-1, and 60.6 kJ/mol, respectively. The results of the activation energy calculations obtained from the wave velocity measurements (65.4 kJ/mol), the temperature profile analysis (61.4 kJ/mol), and the Freeman-Carroll kinetic analysis (60.6 kJ/mol) are in good agreement, implying that any of the three described techniques can be used for the determination of kinetic parameters. Nonetheless, because all of these methods are based on similar assumptions concerning the kinetic
Figure 6. Determination of n, K0′, and E according to the Freeman-Carroll method, demonstrated on the data for the sample with λ ) 10 wt %: (a) n, (b) K0′ and E.
law, the above experimentally determined values of the kinetic parameters (n, K0, and E) should be related to those extracted by other methods in order to confirm their validity. The equal values of the three estimated activation energies imply only that the apparent activation energy can be estimated with any of these methods. Moreover, they do not imply that all of the kinetic parameters are correct. Nevertheless, if a certain parameter is independently determined by three different procedures to have highly similar values, the extracted parameters can be considered as the best suitable values for a given system upon application of a given model. As stated in the Boddington analysis model, the processes that govern the rate of conversion are assumed to be the chemical reaction(s) and the diffusion process. Thus, in using this model, all of the reactions are treated simultaneously by means of the lumped generalized function f(η), which, all things considered, describes the time dependence of the conversion equivalent, η, sufficiently. The determined value of the exponent n also implies that there might be different processes that determine the rate of conversion. Either more than one reaction is occurring, each subject to its own kinetic law, or the diffusion process is the rate-limiting factor. The latter possibility would explain the relatively low value of the determined activation energies. Experimentally determined kinetic and thermodynamic values were also used for a simulation of the combustion wave (Figure 7). The variation of the temperature profile and the time dependence of the conversion were investigated using eqs 2 and 3, which were rewritten as
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κ ∂2T Q ∂T -E h ) (T - T0) + K′ exp (1 - η)n ∂t CPF ∂x2 CP 0 RT CPF
( )
(6) The parameters used to resolve eqs 3 and 6 as an illustration for λ ) 0 were CP ) Q/τad ) 1.434 J/(g K), κ ) RCPF ) 1.638 W/(m K), h ) CPF/tx ) 0.080 W/(cm3 K), E ) 63.0 kJ/mol, K0′ ) 598 s-1, and n ) 0.940 (appearing in eqs 13, as well). The heat generated during the reaction, Q ) 2199 J/g (λ ) 0 wt %), was determined by the DSC technique, and the product (oxide mixture) density of F ) 0.7170 g/cm3 was determined as tap the density. The obtained values of CP were higher than those reported in the literature.32 For instance, CP for neat zirconia should be in the range between 0.455 J/(g K) (standard conditions) and 0.647 J/(g K) (at TC). Nevertheless, the apparent parameters depend on other contributing factors as well, such as the influence of the evolving gases inflating the solid-state matrix. The gases occupying the porous matrix, however, have less of an impact on the thermal conductivity κ, because the conductive heat transfer is strongly influenced solely by the solid-state thermal conductivity. Hence, the obtained κ results correspond with the literature values, the latter being between 1.5 and 2.1 W/(m K).33 The value of the heat-transfer coefficient, h, cannot be compared to the literature values is as straightforward a manner as the other parameters, as the majority of the most intense convective heat transfer takes place in the combustion zone, the relative width of which is essential for the estimation of h. If, on the other hand, the combustion zone width is eliminated from the equation by division, that is, if the literature values for h [W/(cm2 K)]34 are divided by the determined values [W/(cm3 K)], the quotient represents exactly one-quarter of the pellet diameter, V/A. With such an estimation, the pellet diameter is quite similar to the measured value. The activation energy of the reaction (generalized in eq 4) was obtained using various approaches and was estimated to be within the range of 60-66 kJ/mol, which is of the same order of magnitude as the value established for the combustion synthesis of γ-lithium aluminate by Kim et al. (54 kJ/mol).35 The activation energy estimated for the present system is significantly lower than that of diffusion (several hundreds of kilojoules per mole) in a normal solidstate reaction, but somewhat higher than those for the combustion reaction of fuel (20-30 kJ/mol)36 and the evaporation of water (about 40 kJ/mol).37 The activation energy, 60-66 kJ/ mol, is thought to be that of product formation (eq 4). The simulated curve of the undiluted system presented in Figure 7 predicts the zone of the remote wavefront where reactant preheating takes place, the zone of rapid temperature increase, and the decay zone where the reaction products cool. Generally, the shapes of all of the simulated curves imply that the Boddington heat-balance equation can, in principle, be successfully used for point temperature-change predictions. However, when the simulated curve is compared to the measured experimental data, some discrepancies, especially in the region of the initial temperature rise zone, can be observed. Specifically, the two curves are in very good agreement only in the zone of the cooling of the reaction products. During the early stages, the simulated curve predicts a broader remote wavefront zone and a more intense temperature rise in this zone. Consequently, in the rapid temperature rise zone, the steepness of the simulated curve is lower than that of the measured temperature profile. This inconsistency during the early stages of the temperature rise of the combustion system can be ascribed to two general causes: (i) Equation 6 does not take into account any heat loss
Figure 7. Experimentally determined (O) and simulated (s) temperature profiles, along with the calculated ∂(t) curves obtained from the experimental data (O) and simulated ∂(t) for the sample with λ ) 0 wt %.
from the system due to volatile products formation and, as such, does not completely represent the citrate-nitrate combustion system. (ii) Some escaping gases might also cause difficulties during the temperature measurement itself, resulting in less accurate temperature profile determinations. Analogously to the simulated temperature profile, a plot of the time dependence of conversion was created as well (Figure 7). The η(t) dependence was described using eq 3. The simulated curves generally predict a steady reactant-to-product conversion. When conversion starts to increase steadily in the zone of the rapid temperature rise, the simulation predicts a very fast combustion reaction. The time interval needed for conversion of reactants to products from 10% to 90% is approximately 24 ms. The reaction time obtained from the simulation (24 ms) is probably an order of magnitude smaller than the experimental values (0.8 s) because of the nature of the algorithm used to simulate the temperature profiles by application of eq 6. The steeper rise in the simulated conversion corresponds to the solution of the stiff system of ordinary differential equations (eqs 3 and 6), which was obtained by simultaneously solving the temperature profile equation (eq 6) and the conversion profile equation (eq 3) using the nonstiff Adams method and a stiff Gear backward differentiation formula method for the former (a variable coordinate step with a corresponding variable time step) and a forward differentiation formula method for the latter (a coordinate step of 2 × 10-6 m with a corresponding time step of 4 × 10-3 s). Thus, the parametric sensitivity of conversion on activation energy is expressed even more strongly, resulting in a sharper increase. The model was accordingly applied to all formulations containing different amounts of diluting agent. The agreement between experimentally measured data and model-predicted results was satisfactory. For all formulations, the simulation predicted a steeper increase before the maximum temperature and, consequently, more gradual decrease thereafter. Regardless of the satisfactory agreement, the results of the simulation might even be improved upon through the implementation of a parameter-fitting procedure. Nevertheless, the aim was to perform the simulation using the independently determined parameters in order to demonstrate the extrapolative predictability of the model, whereas for the accurate interpolative purposes, some of the parameters ought to be subjected to a fitting algorithm. As we independently and simultaneously fitted the parameters of eqs 3 and 6, taking the independently determined parameter values as the initial fitting approximations
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using the Levenberg-Marquardt fitting algorithm, the results were as follows: The obtained solution appeared to be less affected by the material properties of the system (including density, specific heat, and so on), but on the other hand, it was relatively sensitive to variations in the kinetic parameters, especially the activation energy, E. An excellent fit was obtained even for slight deviations (1-5%) of E from its initially approximated value, whereas other parameters converged to unrealistic values only to render a much poorer agreement overall. Thus, the model could generally be employed even for other combustion synthesis systems, regardless of whether the parameters (i.e., material and kinetic properties) are estimated independently or result from simultaneous model fitting. An arbitrary choice of the mode of model implementation rests upon the desired manner in which the model is intended to be used and its corresponding accuracy. There are numerous different systems to which this methodology could be extended, either for special ceramics preparation (e.g., perovskites or other complex oxides) or alongside different citrate-nitrate reactions or chemically different reduction/oxidation reaction systems (e.g., glycine-nitrate, urea-nitrate, and other systems). Owing to the advantages of the use of the Boddington method (described in the Introduction), the methodology could be extended in a straightforward manner in a broader sense. Neither reactant nor product material properties need to be established. Moreover, should combustion synthesis of any of the mentioned systems be considered, especially other reduction/oxidation reaction systems, the parameters determined for our system could readily be applied even for other more or less similar systems, at least as initial approximations for the subsequently established system-specific characteristics. Conclusions Analysis and simulation were used for the citrate-nitrate combustion system that can be used to produce materials employed in applications such as oxygen sensors and solid oxide fuel cells because of its high oxygen ion conductivity. Because of the high demands regarding desired material properties in these fields of application, predictive and descriptive modeling of material preparation is of vital importance. The citrate-metal nitrate combustion system was employed for the synthesis of a NiO-(Y2O3)0.1(ZrO2)0.9 oxide mixture. To perform kinetic analysis, temperature profiles of several reaction mixtures with different additions of Al2O3 diluent were measured. The maximum combustion temperature and wave propagation velocity decreased with increasing Al2O3 addition. The maximum diluent addition that still ensured the SHS reaction mode was found to be 20 wt %. The change in peak combustion temperature and the propagation of the combustion zone with the amount of diluent were used for wave velocity analysis, and the resulting activation energy was calculated to be 65.4 kJ/mol. A complete Boddington analysis was performed on the prepared combustion systems, resulting in the determination of some thermodynamic parameters (CP, κ, and h) and the estimation of the activation energy as 61.4 kJ/mol. The results for the time/distance dependence of the conversion obtained by the temperature profile analysis were also used for the determination of kinetic parameters according to the Freeman-Carrol method. The activation energy, kinetic constant K0′, and exponent factor n calculated by the latter method were found to be 63.0 kJ/mol, 598 s-1, and 0.940, respectively. The kinetic and thermodynamic parameters determined using different analytical approaches were also used for a computer simulation of the combustion temperature profile and the dependence of
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ReceiVed for reView February 20, 2008 ReVised manuscript receiVed April 16, 2008 Accepted May 2, 2008 IE800296M
