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Determination of aerosol kinetics of thermal ZnO dissociation by thermogravimetry
Chemical Engineering Science 62 (2007) 5952 – 5962 www.elsevier.com/locate/ces
Determination of aerosol kinetics of thermal ZnO dissociation by thermogravimetry Christopher Perkins, Paul Lichty, Alan W. Weimer ∗ Department of Chemical and Biological Engineering, University of Colorado at Boulder, Boulder, CO, USA Received 23 October 2006; received in revised form 15 June 2007; accepted 27 June 2007 Available online 10 July 2007
Abstract The thermal decomposition of ZnO is the high temperature solar step in a two-step water splitting process for sustainable H2 production. To optimize aerosol solar reactor design, it is desired to understand the forward kinetics of this reaction in an aerosol configuration. Non-isothermal thermogravimetric (TG) experiments were conducted to determine the applicability of TG kinetic data to aerosol reactor environments. It was found that the differentiating heat and mass transfer factors—initial loaded mass, particle size, and heating rate—had no statistically significant effect on the activation energy or pre-exponential factor. This allowed TG data to be applied to the aerosol case. Isothermal TG experiments were subsequently performed to determine the kinetic rate parameters. Using the model expression d = k ∗ eEa /R(1/T −1/T0 ) (1 − )2/3 , dt with T0 = 1895 K, Ea was found to be 353 ± 25.9 kJ/mol, and k ∗ was found to vary inversely with diffusion distance in the TG crucible. The specific rate constant for a diffusion distance of 1 in was k0 = 3.15 × 106 ± 5.54 × 105 s−1 . Both of these results are in agreement with L’vov theory, and a simple electrostatic dissociation mechanism was proposed. 䉷 2007 Elsevier Ltd. All rights reserved. Keywords: Solar thermal; Water-splitting; Zinc oxide; Hydrogen; Renewable energy
1. Introduction and background As oil supplies drop and sea levels rise, it is clear that a replacement energy source must be found. Hydrogen has often been cited as this fuel of the future (Turner, 2005; Winter, 2005). When used, it forms only water as a waste product, and it can be used in high efficiency fuel cells. However, the vast majority of hydrogen (95%) in use today comes from steam methane reforming of natural gas, an inherently non-renewable process, (US Department of Energy, 2006). For hydrogen to truly be a fossil fuel replacement, some renewable, sustainable method for generating it must be developed. One such method is the solar thermochemical Zn/ZnO cycle for splitting water (Palumbo et al., 1998; Steinfeld et al., 1998; ∗ Corresponding author. Tel.: +1 303 492 3759; fax: +1 303 492 4341.
E-mail address: [email protected] (A.W. Weimer). 0009-2509/$ - see front matter 䉷 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.ces.2007.06.039
Steinfeld, 2002; Perkins and Weimer, 2004). In the first step of the cycle, ZnO is dissociated at high temperature in a solar thermal reactor: solar thermal energy
ZnO −−−−−−−−−−−→ Zn + 21 O2 , H = 456 kJ/mol, 1700.2000 ◦ C.
(1)
In a second, exothermic step, the Zn products from the first reaction are combined with water to yield hydrogen and zinc oxide: Zn + H2 O → H2 + ZnO, H = −104 kJ/mol, 350.400 ◦ C.
(2)
The ZnO is fed back into the first step of the cycle; the net process consumes no zinc. The net reaction is the splitting of water into hydrogen and oxygen, allowing hydrogen to be derived from only water and sunlight.
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Knowledge of the kinetics of the ZnO dissociation reaction is critical. The efficiency of high temperature solar reactors depends heavily on the view factor for re-radiation. As much of the cost of such a system is in the tracking solar reflectors, minimizing the number of these collectors is of paramount importance to achieving a profitable plant (Hammache and Bilgen, 1988). Knowledge of the kinetics of the reaction will allow the reactor size to be minimized, and thus optimal efficiency and minimal cost of the final hydrogen product may be obtained. A number of the sources in the literature report problems with recombination of zinc metal and product oxygen (Palumbo et al., 1998; Steinfeld, 2002). This recombination is parasitic of yield, and should be minimized. However, the reaction processes leading to recombination cannot be accurately understood before those comprising the dissociation are also understood. This gives yet another reason for studying the dissociation kinetics. An aerosol transport tube reactor, where small particles are entrained in a gas flow and carrier downward through a reaction zone, could be an ideal geometry for solar chemistry on the industrial scale. Such reactors have been implemented industrially by the Dow Chemical Company to produce tungsten carbide, by various researchers to produce other ceramics (e.g. NbB2 , TiB2 ), as well as by Dahl et al. to dissociate methane in a solar-thermal configuration. (Matovich, 1976; Maeda et al., 1994; Saito et al., 1997, Dahl et al., 2001; Dahl et al., 2002; Dahl et al., 2004a–d). Dispersion of small reactant particles in an aerosol allows for rapid heat and mass transfer, giving kinetics that are mainly chemically controlled. An indirectly heated tubular geometry allows for isolation of the chemical reaction environment from that of the solar receiver. This makes a windowless receiver possible, which increases receiver efficiency and decreases cost. Contamination of the reaction products is eliminated, and their collection becomes much simpler than in systems where the reaction takes place in a windowed receiver zone. Finally, as particle feed to aerosol reactors is relatively easy to start, stop, or scale, this system could be integrated better with intermittent solar heat than proposed batch reactors. Unfortunately, studying thermal decomposition kinetics at ultra high temperatures in aerosol flow is extremely challenging. In situ gas concentration measurement at temperatures exceeding 1700 ◦ C is essentially impossible, and flow is often in the laminar regime. These two factors make residence time (and thus reaction time) very difficult to define; as a result, uncertainty in kinetic measurements will be large. Likewise, recombination of oxygen and zinc during product cooling could give forward reaction rates that were skewed to low values. Instead, other techniques need to be used to explore ZnO decomposition kinetics, and the results applied to the aerosol case. There have been a few previous studies of ZnO decomposition kinetics reported in the literature. Grunze and Hirschwald (1974) studied the kinetics by thermogravimetric analysis (TGA) at relatively low temperatures (450–950 ◦ C) and absolute pressures 10−2 –5 torr. The kinetic theory developed by (L’vov, 1997) for dissociating solids suggests separate mechanisms for vacuum and inert gas environments. From a practical perspective, solar ZnO dissociation would likely be operated
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at atmospheric pressure to avoid the expense and complication of vacuum systems. Likewise, inert gases will be needed to effectively sweep product gases out of the reactor system and to shift reaction thermodynamics to lower operating temperatures. In light of this, it makes sense to expand the kinetic work of Grunze and Hirschwald to inert gas, atmospheric pressure regimes, as discussed by L’vov. Moller and Palumbo (2001) studied the rate of surface retraction of solid blocks of ZnO illuminated by high flux solar radiation. While the heating rates and temperatures examined were in the range of typical solar operation (> 102 K/s, 1950–2400 K), the use of solid blocks of ZnO could lead to different rate controlling mechanisms than with high surface area particles. Specifically, the lower mass transfer resistance of diffusion at the surface of the particles could lead to faster kinetics for dispersed particles than observed in the illuminated plate configuration used by Moller and Palumbo. The kinetic expressions determined in that work would likely not be applicable to decomposition of ZnO particles in an aerosol. Weidenkaff et al. (2000) performed ZnO dissociation in a thermogravimetric analyzer. They found rates of reaction to increase as initial mass of sample decreased, suggesting solid diffusion controls, but did not explore the lower limits of initial mass to determine if the reaction rates asymptotically approached a regime where intrinsic chemical controls dominated the solid diffusion effects. Likewise, as this work was not focused on the aerosol configuration, limiting factors differentiating TGA from an aerosol case were not explored. No single order of reaction was selected, and the physical applicability of the general rate expression was not discussed in detail. A range of values was given for model parameters, but no uncertainty reported on these values. Their work was expanded upon in this study to more fully determine the ZnO decomposition kinetics. The study reported below was conducted to more fully understand the effects of factors differentiating conditions in thermogravimetry from those in an aerosol reactor on reaction kinetics. With a firm grasp of how these factors affected (or did not affect at all) the observed reaction kinetics, isothermal TGA experiments were conducted to select a kinetic rate expression and determine the parameters for that expression. Finally, these parameters were compared with a mechanistic theory for dissociation of ZnO. 2. Experimental Thermogravimetric experiments were conducted in a Theta Gravitronic II thermogravimetric analyzer (TGA), shown in Fig. 1. The analyzer consisted of a graphite element electrical resistance furnace capable of high heating rates (> 50 ◦ C/min) and temperatures (2000 ◦ C). Dissociating samples were isolated from the heating element by an Al2 O3 protection tube. This protection tube was continuously purged with 200 SCCM argon gas, flowed downward across the crucible. Al2 O3 crucibles (McDanel Technical Ceramics #ACN-3758, 0.75 in diameter, 1 in height) were used to contain the dissociating ZnO sample, and these crucibles were suspended inside the protection tube by a molybdenum oxide hangdown wire. Al2 O3 is
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Fig. 1. Schematic of thermogravimetric analyzer.
thermodynamically predicted to react with ZnO to form ZnAl2 O4 at the temperatures of interest, as shown by simulation with the Facility for the Analysis of Chemical Thermodyamics (FACT) computer software. Due to this, it was important to determine if this was having a major effect on the kinetic results. As ZnAl2 O4 is solid and stable at temperatures up to the maximum value explored (1750 ◦ C), any formation of this product would manifest itself in final mass losses of less than 100%. As will be seen below, this was not seen for the results in the current work, indicating that the reaction between ZnO and the sintered 99.8% Al2 O3 was much slower than the decomposition of ZnO and that this reaction did not affect kinetic results. This hangdown wire was connected to a Cahn D-101 microbalance, which recorded the mass of the sample throughout the experiment. For ZnO dissociation, products are all in the gas phase, so fractional mass loss was equal to fractional conversion. Experiments were performed in two distinct phases. In the first phase, non-isothermal experiments were conducted over a range of possible rate limiting conditions to determine which of these had statistically significant effects on the reaction kinetics. Because non-isothermal thermogravimetric experiments are run at a large number of temperature points, the specific rate constant k can be determined from a single experiment. This made comparison of limiting factors very efficient. The
factors selected were those that differentiated TGA conditions from aerosol conditions. In the second experimental phase, isothermal experiments were performed to more accurately determine the parameters in the kinetic rate law and to quantify the effects of any statistically significant factors identified in the first phase. Performing decomposition in isothermal conditions eliminates the heating rate uncertainty in non-isothermal experiments, as well as allowing for a fuller spectrum of fractional conversion at any given temperature. This was expected to give better estimates of experimental uncertainty and repeatability, as well as reducing the size of the confidence bands around the rate parameters. 2.1. Non-isothermal experiments The environment inside the TGA does not exactly match that inside an aerosol reactor. Therefore, when using TGA to study aerosol rates, the effects of these differences must be considered. The most important factor is the difference in mass and heat transfer rates between the two configurations. Inside an aerosol reactor, the fine dispersion of small particles reduces the resistance of mass transfer of oxygen and zinc vapor away from the reacting particles. Likewise, radiation heat transfer to particles with Biot numbers much less than unity should result in heating rates on the order of 105 K/s. (Dahl et al., 2002)
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Inside a TGA, particles confined within a crucible naturally sit in a small pile. If diffusion effects within the solid pile are important, then the experiment could give slower “apparent” kinetics than would be realized in the aerosol case. Likewise, physical limitations of the TGA furnace prevent it from realizing the ultra-fast heating rates observed in the aerosol condition. To investigate whether these differences between the TGA and aerosol case have a major effect on the calculated kinetic model parameters, a central composite design, as described by Montgomery, was performed to determine the significance of the differentiating factors. (Montgomery, 2001) Three factors were chosen: initial loaded mass of particles, heating rate of the sample, and the particle size of ZnO used. As the initial mass of particles increases, the resistance to mass transfer of product gas should increase as well. If this resistance is a controlling factor in the kinetics, it could result in the observation of slower reaction rates. Indeed, this effect was reported by Weidenkaff et al. (2000), but should be influenced by both the mass of particles used and the TGA geometry. Diffusion controls should become less important as initial mass is decreased, and the rates should approach some asymptotic value. Initial masses between 62.5 mg and 232.5 mg were explored in this study. Particle size can also affect mass transfer control of reaction kinetics. As particle size is decreased, the specific surface area of the particles increases. This will decrease diffusion resistance to the gas phase and could result in faster reaction rates. Two particle sizes were explored in the experiment, 50 nm (Aldrich #544906) and 1m (Sigma-Aldrich #255750). Finally, there exists an inherent difference between the ultra-fast particle heating rates observed in an aerosol reactor and the heating rates existing in the TGA. Although this factor was not expected to control the kinetics, it was included for completeness. Heating rates between 10 and 30 ◦ C/ min were examined in the experiment. The experiments in the central composite design were performed non-isothermally. The sample was heated at a constant rate from room temperature to 1750 ◦ C, where the temperature was held constant for 20 min. Because rate data are collected over a wide range of temperatures, all necessary information to fit a kinetic rate model was contained in each experiment. The non-isothermal method then provided a very efficient path to detect the effects of the aforementioned factors. A drawback to the method was increased uncertainty, as random error in the heating rate and the lack of temperature replicates in each experiment lead to rather large confidence intervals. Therefore, the method was well suited to exploration of factor effects, but determination of rate parameters required an experimental method less prone to experimental uncertainty. 2.2. Isothermal experiments To determine the kinetic rate parameters without the uncertainty associated with non-isothermal experiments, isothermal kinetic experiments were performed. 150 mg of ZnO powder (50 nm, Aldrich #544906) were loaded into Al2 O3 crucibles of the same diameter used in the non-isothermal experiments. For the isothermal experiments, two heights of crucible were used,
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1 in (“full”) and 0.5 in (“half”). The “half” crucibles were fabricated by grinding off the top 0.5 in of the “full” crucibles. Gas diffusion resistance in the crucible could then be tested as a controlling kinetic factor, as it was not examined in the nonisothermal experimental matrix. The TGA furnace was heated to the temperature of interest at 30 ◦ C/ min; this heating rate was not expected, from experience with the non-isothermal experiments, to produce significant conversion before the investigated temperature was achieved. The temperature was held constant until complete reaction was observed. Temperatures between 1440 and 1750 ◦ C were studied. 3. Results and discussion 3.1. Non-isothermal experiments Experiments were performed at the points shown in Table 1, and TG curves generated for each experiment. One such curve is shown in Fig. 2. As the temperature was increased, the mass in the crucible began to decrease, corresponding to decomposition of the ZnO powder. For this experiment, it can be seen that this decomposition begins near 1400 ◦ C. It is important to note that the sample does not fully convert in the non-isothermal portion of the experiment. Conversions at the end of this segment ranged between 0.25 and 0.6, depending on the heating rate of the sample. For each experiment, the reaction went to completion during the hold phase. Before analyzing the TG curves, an appropriate rate model must be chosen. The rate model most often chosen in the literature for solids decomposition is (Galwey and Brown, 1999) d = k0 e−Ea /RT f (), dt
(3)
where is the fractional conversion. The temperature dependent factor is the familiar Arrhenius form, with pre-exponential factor k0 and activation energy Ea . The factor (f ) allows for rate controls similar to concentration effects in gas-based reactions. It most commonly has an “order of reaction” form: f () = (1 − )n ,
(4)
where n is the “order of reaction”. (Galwey and Brown, 1999) The choice of n depends on the physical assumptions of the reaction model. For this case, n was chosen to be 23 , corresponding to a constant rate of radial retraction of spherical particles. This is appropriate for reactions where dissociation occurs only at the particle surface and the rate of dissociation is controlled by surface temperature and chemical properties, as in a shrinking core model. (Levenspiel, 1999) As the literature has suggested that metal oxide dissociations occur preferentially at surfaces (due to the high stabilization energy for ions at the center of the lattice), this is a reasonable choice of model. (Maciejewski, 1992; Galwey and Brown, 1999) This choice of model will not affect conclusions made from the non-isothermal factor study, but will need to be justified when reporting model parameters from the isothermal experiments. In order to avoid the highly eccentric joint confidence region associated with the Arrhenius form at high temperature, and
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Table 1 Experimental results for non-isothermal ZnO decomposition Initial mass (mg)
Particle size (nm)
Heating rate (◦ C/ min)
Ea , linear fit (kJ/mol)
k ∗ , linear fit (s−1 × 10−4 )
Ea , nl regression (kJ/mol)
k ∗ , nl regression (s−1 × 10−4 )
100 200 100 238.5 62 150 200 150 150 200 200 150
1000 1000 50 1000 50 50 50 50 50 1000 50 1000
14 14 14 20 20 20 14 20 10 26 26 20
366 425 350 358 334 321 390 390 302 342 405 334
3.73 1.34 9.24 3.40 2.55 2.37 2.47 2.79 5.18 5.37 4.60 5.35
347 367 397 359 340 363 422 351 303 342 405 334
3.39 1.62 7.90 3.57 2.80 2.28 2.40 3.10 5.18 5.37 4.60 5.35
Fig. 2. TG curve for a non-isothermal experiment (14 ◦ C/ min heating rate, 100 mg initial mass, 1 m particle size).
the concomitant “compensation effect” which seems to relate the model parameters linearly, a transformation was performed on the model equation: d = k ∗ e−Ea /R(1/T −1/T0 ) f (), dt
(5)
k ∗ = k0 e−Ea /RT 0 ,
(6)
where T0 is some central temperature. For these experiments, it was chosen to be 1597 ◦ C, approximately midway between the onset of reaction and the maximum temperature. Such a transformation has been shown to condition the joint confidence interval, de-correlating estimates for the model parameters. (Himmelblau, 1970). The non-isothermal TG curves were analyzed by two methods. The first follows from a rearrangement of the model
equation: Ea 1 1 (d/dT ) , = ln k ∗ − − ln f () R T T0
(7)
where is the constant rate of temperature increase used in the heating program. A plot of the left hand term in Eq. (7) against 1/T − 1/T0 should be linear, with slope −Ea /R and intercept ln(k ∗ ). Derivatives of fractional conversion with respect to temperature were calculated using a central difference formula (Gerald and Wheatley, 1994). For most of the non-isothermal experiments, less than half of the conversion was realized during the non-isothermal phase. The above method might give misleading model parameter fits. To model the entire dissociation curve, non-linear regression was used, allowing kinetic parameters to be fit for a superposition of the non-isothermal and isothermal decomposition
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Table 2 p -values from ANOVA significance test for limiting TGA factors
Fig. 3. Arrhenius plot for a non-isothermal experiment (14 ◦ C/ min heating rate, 100 mg initial mass, 1 m particle size).
regimes. The model equation was solved numerically using the Euler method, and the sum of squares of the residuals from the experimental data was minimized by varying the model parameters. Obviously, this method could not be used to test the effect of heating rate (an inherently non-isothermal factor), but was used to investigate the significance of the other two factors. Values of the activation energy and pre-exponential factor calculated from both methods are shown in Table 1. One plot for the linearization is shown in Fig. 3, demonstrating the linearity of the data and the applicability of this method. The significance of the heating rate was difficult to determine, as at high heating rates (> 25 ◦ C/min) the conversion reached during the non-isothermal phase of the experiments was very small (∼0.15). As a result, the high heating rate samples had relatively few points from which to determine rate parameters, and linear regression on the data points yielded large confidence intervals. To analyze the effect of this factor, a 22 factorial design was examined. Particle size was the other factor, with the two values mentioned above. Initial mass was constant at 200 mg, and the two levels of the heating rate were 14 and 26 ◦ C. An analysis of variance (ANOVA) method was used to analyze the responses, as described by (Montgomery, 2001). The application of the ANOVA method was completed using MINITAB software. (Minitab, 2003) A p-value for the ANOVA analysis of 0.613 showed the factor was not significant on Ea , to 95% confidence. Likewise, it is seen to be unimportant with respect to k ∗ , with a p-value of 0.190. Although the evidence for the significance tests is sparse, the low level of conversion during the non-isothermal phase of the experiments would tend to diminish the importance of heating rate as a factor. It is entirely unimportant for isothermal determinations of model parameters, as there is no effective heating rate. From the evidence above, it is concluded that it is safe to ignore this parameter when examining the other two factors. The significance of the particle size and initial mass factors on the model parameters was examined by constructing a
Rate parameters
Initial mass p-value
Particle size p-value
Ea , linear fit k ∗ , linear fit Ea , non-linear regression k ∗ , non-linear regression
0.384 0.184 0.627 0.290
0.966 0.504 0.107 0.611
22 factorial design with replicates and center points from the original central composite design. The predictions for activation energy and pre-exponential factor determined from both data analysis methods were used as experimental responses in an ANOVA significance test. The ANOVA test was completed using MINITAB software (Minitab, 2003). The p-values for ANOVA analysis of the fractional factorial design are shown in Table 2. As none of the p-values are smaller than 0.05, neither particle size nor initial loaded mass are expected to be significant factors to 95% confidence for estimation of Ea . The same conclusion can be made for estimation of k ∗ . The residuals from the ANOVA model predictions were normally distributed, verifying the central assumption of nonstructured, normally distributed error inherent to the ANOVA approach and analysis of the factorial design. As the initial loaded mass had no effect on the model parameter estimation, it could be concluded for this range of initial masses that mass diffusion through the solid particle pile is not controlling the rate of dissociation. Likewise, in the size range of particles employed, there was no controlling effect on the kinetics. Therefore, by use of particles and initial masses in the ranges explored above, kinetic models obtained from TGA data can be applied to aerosol dissociation without modifications for these physical effects. 3.2. Isothermal experiments In order to experimentally verify the form of f () chosen above, a reduced time analysis of the isothermal data was performed. (Galwey and Brown, 1999) A transformation of the time variable can be made such that t − t0.25 tred = , (8) t0.8 − t0.25 where t0.25 and t0.8 are the times at which 25% and 80% of the sample has dissociated, respectively. Integrating the differential equation model (Eq. (3)): g() = kt, d g() = . o f ( )
(9) (10)
When calculating tred , the temperature dependent rate constant k is eliminated. Therefore, for decompositions of solids with consistent mechanisms for dissociation, plots of conversion against reduced time should be identical, regardless of the temperature at which the decomposition occurred. Comparison of these plots for experiments at different temperatures against a
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Fig. 4. Reduced time conversion response for full-size crucible.
Fig. 5. Reduced time conversion response for half-size crucible.
master curve for the model employed gives an indication of the appropriateness of the model. Figs. 4 and 5 show plots of fractional conversion against reduced time for the isothermal experiments completed in the fullsized and half-sized crucibles, respectively. The figures show very close agreement between the curves across experiments, with deviations at the center point of less than 3% conversion. In addition, there was no ordering of the curves with respect to temperature. This verifies that the mechanism of dissociation does not change with temperature, within the resolution of the data. As can also be seen, the data matches the master curve for
n = 23 much better than for the most commonly chosen order of reaction, n = 1, and there is little difference between the fits for n = 21 and 23 . Statistically, this is borne out by average deviations, in terms of fractional conversion, of 0.00299 and 0.00340 for the n = 21 and n = 23 models in the full crucible, respectively, and average deviations of 0.00232 and 0.00323 for the n = 21 and n = 23 models in the half crucible, respectively. As the uncertainty in the conversion at any point was calculated to be 0.0057 from the aggregate of model data, the differences in deviation are smaller
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Fig. 6. Linear fit for determination of specific rate constant k from an isothermal experiment (temperature of 1550 ◦ C).
Fig. 7. Arrhenius plot for the determination of the kinetic rate constants.
than the uncertainty in the measurements, and the models are indistinguishable. The n = 23 model was chosen for the physical reasons listed above: that the particles are assumed spherical and are expected to react only on the surface. To determine the model parameters from the isothermal TGA data, the integrated form of the rate law is used (Eq. (9)). If the choice of rate law is appropriate, then the graph of g() against time will yield a straight line with slope k. Fig. 6 shows such a plot for one of the isothermal experiments. Deviation of the intercept from zero is due to the fact the time is not corrected to the initiation time of the reaction. A linear least-squares re-
gression yielded the rate constant k at the experimental temperature, and computation of the confidence intervals for the regression yielded the uncertainty for each of these measurements. Rearranging the form of the rate constant: Ea ln k = ln k − R ∗
1 1 − T T0
.
(11)
The rate parameters can easily be determined from the linear plot of the natural logarithm of the rate constant against the corrected reciprocal temperature. Fig. 7 shows this plot for both
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the “full” and “half” crucible experiments. The values of Ea and k ∗ calculated for these experiments were: Ea,full = 353 ± 25.9 kJ/mol, k ∗ = 3.58 × 10−4 ± 6.30 × 10−5 s−1 , Ea,half = 346 ± 42.0 kJ/mol, k ∗ = 5.86 × 10−4 ± 9.90 × 10−5 s−1 . Uncertainty for each parameter corresponds to the 95% confidence interval for the linear regression. These activation energies are in line with those found in previous literature. Weidenkaff et al. stated that the activation energy ranged between 312 and 376 kJ/mol, and Hirschwald found a similar range for activation energy (Hirschwald and Stolze, 1972; Weidenkaff et al. 2000). Möller and Palumbo found an activation energy of 328 kJ/mol, in line with what was found in this study (Moller and Palumbo, 2001). Based on the L’vov theory described below, it makes sense that activation energy is configuration independent; the confidence bands given in the current work place the activation energy in a range consistent with previous literature. As for pre-exponential factor, this has been shown by the dependence on crucible diffusion distance to be strongly configurationally dependent. Previous literature can be difficult to compare, as pre-exponential factors were not always expressed in equivalent terms. For example, Möller and Palumbo express this factor in terms of a mass flux (g m−2 s−1 ), whereas the current model requires it to be expressed in reciprocal seconds (Moller and Palumbo, 2001). Weidenkaff et al., did not report values of the pre-exponential factor, so comparisons cannot be made (Weidenkaff et al., 2000). However, based on the strong configurational dependence of this parameter, meaningful comparisons would be difficult to make. The values of Ea for the full and half crucible are, within uncertainty, identical. However, the value of k ∗ is, from a statistical viewpoint, significantly larger for the “half” crucible than for the “full” crucible, as the confidence intervals for the two measurements do not intersect. This suggests that gas diffusion out of the crucible is important with respect to the kinetic rate. Further, as it only affects k ∗ , the activation energy calculated from the TGA experiment is gas diffusion independent and reflects the intrinsic kinetic limitations of the ZnO dissociation. 3.3. Mechanistic model for reaction kinetics L’vov, et al. suggested a model based on equilibrium thermodynamics to predict the rates of metal oxide dissociations, specifically for those dissociations with all products in the gas phase. ZnO falls into this category, and the development sketched below closely follows that shown in L’vov’s paper (L’vov, 1997). The model assumes that only lattice members on the surface of the solid are available for dissociation, and that the rate of this dissociation is dependent only upon the surface temperature and the chemical properties of the solid. This assumption makes sense, as ions within the lattice will have to
overcome a larger energetic barrier to escape, due to electrostatic interactions with surrounding ions, than those ions on the surface subjected to electric field contributions from only one side. At reaction equilibrium, the maximum rate of dissociation is limited to the rate of condensation, which can be calculated from statistical mechanics. This L’vov based model is based on the Hertz–Knudsen– Langmuir model for a condensing substance in a inert foreign gas environment. It predicts the Arrhenius parameters from an equilibrium basis as Ea =
HT◦ , (a + b) a/(a+b)
k0 =
NA DM
(12) b/(a+b)
DO qzRT
a (a a bb )1/a+b
◦
eST /R(a+b) , (13)
where D is the binary diffusion coefficient of the oxygen (O) or metal (M) in the foreign gas, z is the diffusion distance to the portion of the foreign gas where the concentration of the substance drops to zero, q is the number of metal atoms per unit surface area of solid, NA is Avogadro’s number, a and b are the stoichiometric coefficients of metal and oxygen in the starting oxide, respectively, R is the ideal gas constant, T is the temperature of decomposition, HT◦ is the standard enthalpy of reaction at temperature T , and ST◦ is the standard entropy of reaction at temperature T . The temperature dependence in the denominator is mostly offset by temperature dependence of the binary diffusion coefficients, given by the Chapman–Enskog theory (Bird et al., 1960). This makes the pre-exponential factor independent of temperature over small temperature ranges (500 K). For the dissociation of ZnO, the rate limiting step could come from two different reactions: ZnO −→ Zn + O, ZnO −→ Zn + 21 O2 ,
HT◦ = 712 kJ/mol, HT◦ = 470.6 kJ/mol.
(14) (15)
For the first of these reactions, a + b = 2, and the activation energy would be predicted by the L’vov formulation as 356 kJ/mol. For the second, a+b=1.5, and the activation energy would predicted to be 314 kJ/mol. The isothermal experiments give a value in close agreement with the first equation, and the second activation energy is outside the confidence bounds for the parameter determined experimentally for the “full” crucible experiments. This suggests that the rate limiting step is the dissociation of Zn and O atoms (or ion pairs) from the surface, and that formation of diatomic oxygen occurs in the gas phase away from the surface. A purely electrostatic mechanistic interpretation of the L’vov approach can be described as follows. For a completely ionic lattice, the energy required to remove an atom is approximately equal to the electrostatic stabilization energy achieved when that ion to enter the lattice and the associated ionization energies required to form the ion. This can be calculated from a Born–Haber cycle for metal oxide formation. For equally charged ions, under this mechanistic picture, this will be equal to the stoichiometric contribution to the enthalpy of reaction
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for that ion (Huheey et al., 1993). If we model the system as a set of point charges, the energy of the system increases monotonically as we remove an ion to infinity from the surface. Because the energy increases monotonically, no “transition state” of higher energy exists, and the activation energy should be equal to the enthalpy of reaction for the rate limiting step, in this case the removal of one cation or anion. For solids with some covalent boding, predictions from this mechanism should deviate from what is expected. The bonds become more directional, and the simplistic point charge picture no longer applies. The covalent picture would be expected to show some kind of transition state as these bonds are stretched to the intermediate point between bond breakage and formation. To reach this point, an activation energy greater than the enthalpy of reaction would be required. Indeed, the data of L’vov shows an experimental deviation to higher activation energies for HgS, a solid with large amounts of covalent bonding (L’vov, 1997). The authors originally adjusted the stoichiometric ratio to give a better fit to the data, but for physical reasons the above interpretation makes more sense. Zn has an electronegativity (Pauling scale) difference with O of 1.79, placing it in the ionic bonding regime (Huheey et al., 1993). It has some covalent bonding, evidenced by its behavior as a high bandgap semiconductor, but its large electronegativity difference justifies treating it as ionic when examining dissociation. The above mechanistic interpretation is admittedly simplistic, and more experimental work over a wide range of metal oxides decompositions should be performed to verify its claims. This work would include measurement of any intermediates at the surface, comparison of compounds that are strongly in the ionic bonding regime with those whose bonding is more strongly covalent, and computational chemical calculations of the energies for possible reaction pathways at the surfaces for different metal oxides. The L’vov method prediction for k0 depends on the diffusion distance to a concentration sink. In the TGA, inert gas is always flowing across the top of the crucible, carrying reaction products with it. Therefore, the concentration of reaction products should go to zero there, and the distance z will be equal to the crucible height. Using the L’vov predicted activation energy, the “full” crucible measured value is k0 = 3.15 × 106 ± 5.54 × 105 s−1 and the “half” crucible measured value is k0 = 5.15 × 106 ± 8.71 × 105 s−1 . As can be seen, the “full” values are about one half as great as the “half” values, as predicted by the L’vov theory. In an aerosol reactor, it is expected that the values of z will be much smaller, due to the smaller diffusion film existing around the small, dispersed particles. As such, the preexponential factor (and consequently, the reaction rates) should be much larger in an aerosol reactor. In order to determine these values of z, careful experimentation in an aerosol reactor should be performed. However, the mechanism of dissociation will not change, and the equations determined by the above work would still be applicable to this geometry; only the diffusion distance need still be determined, and the more difficult task of finding an appropriate kinetic expression is already completed.
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4. Conclusions Non-isothermal TGA experiments to determine the effect of heating rate on the measured kinetic rate constants showed that this factor was not significant. This conclusion is admittedly blurred somewhat by the small conversions reached during the non-isothermal phase of the experiments for many high heating rate trials, but the overall statistics show little significance of the factor. Similar experiments to test the effects of particle size and initial loaded mass showed no significance for these factors on the measured model parameters. The kinetic rates were concluded to be independent of the physical diffusion phenomena associated with these factors (i.e. diffusion of gas through the solid phase and particle interstices), and the TGA results were seen as applicable to an aerosol situation. This is in contrast with the rate controls for diffusion through the gas phase, which were seen to be significant in the isothermal TGA experiments. Isothermal TGA experiments were conducted to measure the kinetic model parameters. Reduced time analysis was applied, and a consistent reaction mechanism was seen across temperatures for both “full” and “half” crucible sizes. An order of reaction model with 23 was seen to match the data well. A linear Arrhenius plot on the “full” crucible yielded rate parameters of 356 ± 25.9 kJ/mol for the activation energy and 3.58×10−4 ±6.30×10−5 s−1 for the pre-exponential factor k ∗ . The values of activation energy closely matched L’vov kinetic theory, including the predicted doubling of the pre-exponential factor for reduced diffusion distance. The close fit to the L’vov predictions suggests that the mechanism of reaction is a reversible thermal perturbation of zinc and oxygen atoms from the solid surface, followed by formation of diatomic oxygen. Acknowledgements The authors would like to acknowledge the National Science Foundation for its support through the Graduate Research Fellowship program, as well as the US Department of Energy for its support under Grant # DE-FG36-03GO13062. References Bird, R.B., Stewart, W.E., Lightfoot, E.N., 1960. Transport Phenomena. Wiley, New York. Dahl, J.K., Tamburini, J., et al., 2001. Solar-thermal processing of methane to produce hydrogen and syngas. Energy & Fuels 15 (5), 1227–1232. Dahl, J.K., Barocas, V.H., et al., 2002. Intrinsic kinetics for rapid decomposition of methane in an aerosol flow reactor. International Journal of Hydrogen Energy 27 (4), 377–386. Dahl, J.K., Buechler, K.J., et al., 2004a. Rapid solar-thermal dissociation of natural gas in an aerosol flow reactor. Energy 29 (5–6), 715–725. Dahl, J.K., Buechler, K.J., et al., 2004b. Solar-thermal dissociation of methane in a fluid-wall aerosol flow reactor. International Journal of Hydrogen Energy 29 (7), 725–736. Dahl, J.K., Weimer, A.W., et al., 2004c. Sensitivity analysis of the rapid decomposition of methane in an aerosol flow reactor. International Journal of Hydrogen Energy 29 (1), 57–65. Dahl, J.K., Weimer, A.W., et al., 2004d. Dry reforming of methane using a solar-thermal aerosol flow reactor. Industrial & Engineering Chemistry Research 43 (18), 5489–5495.
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Facility for the Analysis of Chemical Thermodynamics (FACT), Centre de Recherche en Calcul Thermochimique, Ecole Polytechnique Montreal. Galwey, A.K., Brown, M.E., 1999. Thermal Decomposition of Ionic Solids. Amsterdam, Elsevier Science B.V. Gerald, C.F., Wheatley, P.O., 1994. Applied Numerical Analysis. Reading, MA, Addison-Wesley. Grunze, M., Hirschwald, W., 1974. Vacuum microbalance investigations on heterogeneous surface-reaction mechanisms. Journal of Vacuum Science & Technology 11 (1), 424–428. Hammache, A., Bilgen, E., 1988. Evaluation of Thermal Efficiency and Cost of High-Temperature Solar Heat from Central Receiver Systems to Use in Hydrogen Producing Thermochemical Processes. International Journal of Hydrogen Energy 13 (9), 539–546. Himmelblau, D.M., 1970. Process Analysis by Statistical Methods. Hirschwald, W., Stolze, F., 1972. Kinetics of thermal dissociation of zinc oxide. Zeitschrift fur Physikalische Chemie-Frankfurt 77 (1–6), 21–42. Huheey, J.E., Keiter, E.A., Keiter, R.L., 1993. Inorganic Chemistry: Principles of Structure and Reactivity. HarperCollins College Publishers, New York. Levenspiel, O., 1999. Chemical Reaction Engineering. Wiley, Hoboken, NJ. L’vov, B., 1997. Interpretation of atomized mechanisms in electrothermal atomic absorption spectrometry by analysis of the absolute rates of the processes. Spectrochemica Acta Part B 52, 1–23. Maciejewski, M., 1992. Somewhere between fiction and reality—the usefulness of kinetic data of solid-state reactions. Journal of Thermal Analysis 38 (1–2), 51–70. Maeda, H., Yoshikawa, T., et al., 1994. Synthesis of ultrafine NbB2 powder by rapid carbothermal reduction in a vertical tubular reactor. Journal of alloys and compounds 215 (1–2), 127–134. Matovich, E., 1976. High temperature chemical reaction processes utilizing fluid-wall reactors. U. S. P. a. T. Office. Thagard Technology Company, USA.
Minitab, I., 2003. MINITAB, Minitab, Inc. Moller, S., Palumbo, R., 2001. Solar thermal decomposition kinetics of ZnO in the temperature range 1950–2400 K. Chemical Engineering Science 56 (15), 4505–4515. Montgomery, D.C., 2001. Design and Analysis of Experiments. Wiley, New York. Palumbo, R., Lede, J., et al., 1998. The production of Zn from ZnO in a high-temperature solar decomposition quench process—I. The scientific framework for the process. Chemical Engineering Science 53 (14), 2503–2517. Perkins, C., Weimer, A., 2004. Likely near-term solar-thermal water splitting technologies. International Journal of Hydrogen Energy 29 (15), 1587–1599. Saito, T., Fukuda, T., et al., 1997. Synthesis of ultrafine titanium diboride particles by rapid carbothermal reduction in a particulate transport reactor. Journal of Materials Science 32 (15), 3933–3938. Steinfeld, A., 2002. Solar hydrogen production via a two-step water-splitting thermochemical cycle based on Zn/ZnO redox reactions. International Journal of Hydrogen Energy 27 (6), 611–619. Steinfeld, A., Kuhn, P., et al., 1998. Solar-processed metals as clean energy carriers and water-splitters. International Journal of Hydrogen Energy 23 (9), 767–774. Turner, J., 2005. The sustainable hydrogen economy. Geotimes 50 (8), 7. US Department of Energy, E. E. a. R. E., 2006. Natural gas reforming, US Department of Energy. 2006. Weidenkaff, A., Reller, A., et al., 2000. Thermogravimetric analysis of the ZnO/Zn water splitting cycle. Thermochimica Acta 359 (1), 69–75. Winter, C., 2005. Into the hydrogen energy economy—milestones. International Journal of Hydrogen Energy 30 (7), 681–685.
Determination of aerosol kinetics of thermal ZnO dissociation by thermogravimetry Christopher Perkins, Paul Lichty, Alan W. Weimer ∗ Department of Chemical and Biological Engineering, University of Colorado at Boulder, Boulder, CO, USA Received 23 October 2006; received in revised form 15 June 2007; accepted 27 June 2007 Available online 10 July 2007
Abstract The thermal decomposition of ZnO is the high temperature solar step in a two-step water splitting process for sustainable H2 production. To optimize aerosol solar reactor design, it is desired to understand the forward kinetics of this reaction in an aerosol configuration. Non-isothermal thermogravimetric (TG) experiments were conducted to determine the applicability of TG kinetic data to aerosol reactor environments. It was found that the differentiating heat and mass transfer factors—initial loaded mass, particle size, and heating rate—had no statistically significant effect on the activation energy or pre-exponential factor. This allowed TG data to be applied to the aerosol case. Isothermal TG experiments were subsequently performed to determine the kinetic rate parameters. Using the model expression d = k ∗ eEa /R(1/T −1/T0 ) (1 − )2/3 , dt with T0 = 1895 K, Ea was found to be 353 ± 25.9 kJ/mol, and k ∗ was found to vary inversely with diffusion distance in the TG crucible. The specific rate constant for a diffusion distance of 1 in was k0 = 3.15 × 106 ± 5.54 × 105 s−1 . Both of these results are in agreement with L’vov theory, and a simple electrostatic dissociation mechanism was proposed. 䉷 2007 Elsevier Ltd. All rights reserved. Keywords: Solar thermal; Water-splitting; Zinc oxide; Hydrogen; Renewable energy
1. Introduction and background As oil supplies drop and sea levels rise, it is clear that a replacement energy source must be found. Hydrogen has often been cited as this fuel of the future (Turner, 2005; Winter, 2005). When used, it forms only water as a waste product, and it can be used in high efficiency fuel cells. However, the vast majority of hydrogen (95%) in use today comes from steam methane reforming of natural gas, an inherently non-renewable process, (US Department of Energy, 2006). For hydrogen to truly be a fossil fuel replacement, some renewable, sustainable method for generating it must be developed. One such method is the solar thermochemical Zn/ZnO cycle for splitting water (Palumbo et al., 1998; Steinfeld et al., 1998; ∗ Corresponding author. Tel.: +1 303 492 3759; fax: +1 303 492 4341.
E-mail address: [email protected] (A.W. Weimer). 0009-2509/$ - see front matter 䉷 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.ces.2007.06.039
Steinfeld, 2002; Perkins and Weimer, 2004). In the first step of the cycle, ZnO is dissociated at high temperature in a solar thermal reactor: solar thermal energy
ZnO −−−−−−−−−−−→ Zn + 21 O2 , H = 456 kJ/mol, 1700.2000 ◦ C.
(1)
In a second, exothermic step, the Zn products from the first reaction are combined with water to yield hydrogen and zinc oxide: Zn + H2 O → H2 + ZnO, H = −104 kJ/mol, 350.400 ◦ C.
(2)
The ZnO is fed back into the first step of the cycle; the net process consumes no zinc. The net reaction is the splitting of water into hydrogen and oxygen, allowing hydrogen to be derived from only water and sunlight.
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Knowledge of the kinetics of the ZnO dissociation reaction is critical. The efficiency of high temperature solar reactors depends heavily on the view factor for re-radiation. As much of the cost of such a system is in the tracking solar reflectors, minimizing the number of these collectors is of paramount importance to achieving a profitable plant (Hammache and Bilgen, 1988). Knowledge of the kinetics of the reaction will allow the reactor size to be minimized, and thus optimal efficiency and minimal cost of the final hydrogen product may be obtained. A number of the sources in the literature report problems with recombination of zinc metal and product oxygen (Palumbo et al., 1998; Steinfeld, 2002). This recombination is parasitic of yield, and should be minimized. However, the reaction processes leading to recombination cannot be accurately understood before those comprising the dissociation are also understood. This gives yet another reason for studying the dissociation kinetics. An aerosol transport tube reactor, where small particles are entrained in a gas flow and carrier downward through a reaction zone, could be an ideal geometry for solar chemistry on the industrial scale. Such reactors have been implemented industrially by the Dow Chemical Company to produce tungsten carbide, by various researchers to produce other ceramics (e.g. NbB2 , TiB2 ), as well as by Dahl et al. to dissociate methane in a solar-thermal configuration. (Matovich, 1976; Maeda et al., 1994; Saito et al., 1997, Dahl et al., 2001; Dahl et al., 2002; Dahl et al., 2004a–d). Dispersion of small reactant particles in an aerosol allows for rapid heat and mass transfer, giving kinetics that are mainly chemically controlled. An indirectly heated tubular geometry allows for isolation of the chemical reaction environment from that of the solar receiver. This makes a windowless receiver possible, which increases receiver efficiency and decreases cost. Contamination of the reaction products is eliminated, and their collection becomes much simpler than in systems where the reaction takes place in a windowed receiver zone. Finally, as particle feed to aerosol reactors is relatively easy to start, stop, or scale, this system could be integrated better with intermittent solar heat than proposed batch reactors. Unfortunately, studying thermal decomposition kinetics at ultra high temperatures in aerosol flow is extremely challenging. In situ gas concentration measurement at temperatures exceeding 1700 ◦ C is essentially impossible, and flow is often in the laminar regime. These two factors make residence time (and thus reaction time) very difficult to define; as a result, uncertainty in kinetic measurements will be large. Likewise, recombination of oxygen and zinc during product cooling could give forward reaction rates that were skewed to low values. Instead, other techniques need to be used to explore ZnO decomposition kinetics, and the results applied to the aerosol case. There have been a few previous studies of ZnO decomposition kinetics reported in the literature. Grunze and Hirschwald (1974) studied the kinetics by thermogravimetric analysis (TGA) at relatively low temperatures (450–950 ◦ C) and absolute pressures 10−2 –5 torr. The kinetic theory developed by (L’vov, 1997) for dissociating solids suggests separate mechanisms for vacuum and inert gas environments. From a practical perspective, solar ZnO dissociation would likely be operated
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at atmospheric pressure to avoid the expense and complication of vacuum systems. Likewise, inert gases will be needed to effectively sweep product gases out of the reactor system and to shift reaction thermodynamics to lower operating temperatures. In light of this, it makes sense to expand the kinetic work of Grunze and Hirschwald to inert gas, atmospheric pressure regimes, as discussed by L’vov. Moller and Palumbo (2001) studied the rate of surface retraction of solid blocks of ZnO illuminated by high flux solar radiation. While the heating rates and temperatures examined were in the range of typical solar operation (> 102 K/s, 1950–2400 K), the use of solid blocks of ZnO could lead to different rate controlling mechanisms than with high surface area particles. Specifically, the lower mass transfer resistance of diffusion at the surface of the particles could lead to faster kinetics for dispersed particles than observed in the illuminated plate configuration used by Moller and Palumbo. The kinetic expressions determined in that work would likely not be applicable to decomposition of ZnO particles in an aerosol. Weidenkaff et al. (2000) performed ZnO dissociation in a thermogravimetric analyzer. They found rates of reaction to increase as initial mass of sample decreased, suggesting solid diffusion controls, but did not explore the lower limits of initial mass to determine if the reaction rates asymptotically approached a regime where intrinsic chemical controls dominated the solid diffusion effects. Likewise, as this work was not focused on the aerosol configuration, limiting factors differentiating TGA from an aerosol case were not explored. No single order of reaction was selected, and the physical applicability of the general rate expression was not discussed in detail. A range of values was given for model parameters, but no uncertainty reported on these values. Their work was expanded upon in this study to more fully determine the ZnO decomposition kinetics. The study reported below was conducted to more fully understand the effects of factors differentiating conditions in thermogravimetry from those in an aerosol reactor on reaction kinetics. With a firm grasp of how these factors affected (or did not affect at all) the observed reaction kinetics, isothermal TGA experiments were conducted to select a kinetic rate expression and determine the parameters for that expression. Finally, these parameters were compared with a mechanistic theory for dissociation of ZnO. 2. Experimental Thermogravimetric experiments were conducted in a Theta Gravitronic II thermogravimetric analyzer (TGA), shown in Fig. 1. The analyzer consisted of a graphite element electrical resistance furnace capable of high heating rates (> 50 ◦ C/min) and temperatures (2000 ◦ C). Dissociating samples were isolated from the heating element by an Al2 O3 protection tube. This protection tube was continuously purged with 200 SCCM argon gas, flowed downward across the crucible. Al2 O3 crucibles (McDanel Technical Ceramics #ACN-3758, 0.75 in diameter, 1 in height) were used to contain the dissociating ZnO sample, and these crucibles were suspended inside the protection tube by a molybdenum oxide hangdown wire. Al2 O3 is
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Fig. 1. Schematic of thermogravimetric analyzer.
thermodynamically predicted to react with ZnO to form ZnAl2 O4 at the temperatures of interest, as shown by simulation with the Facility for the Analysis of Chemical Thermodyamics (FACT) computer software. Due to this, it was important to determine if this was having a major effect on the kinetic results. As ZnAl2 O4 is solid and stable at temperatures up to the maximum value explored (1750 ◦ C), any formation of this product would manifest itself in final mass losses of less than 100%. As will be seen below, this was not seen for the results in the current work, indicating that the reaction between ZnO and the sintered 99.8% Al2 O3 was much slower than the decomposition of ZnO and that this reaction did not affect kinetic results. This hangdown wire was connected to a Cahn D-101 microbalance, which recorded the mass of the sample throughout the experiment. For ZnO dissociation, products are all in the gas phase, so fractional mass loss was equal to fractional conversion. Experiments were performed in two distinct phases. In the first phase, non-isothermal experiments were conducted over a range of possible rate limiting conditions to determine which of these had statistically significant effects on the reaction kinetics. Because non-isothermal thermogravimetric experiments are run at a large number of temperature points, the specific rate constant k can be determined from a single experiment. This made comparison of limiting factors very efficient. The
factors selected were those that differentiated TGA conditions from aerosol conditions. In the second experimental phase, isothermal experiments were performed to more accurately determine the parameters in the kinetic rate law and to quantify the effects of any statistically significant factors identified in the first phase. Performing decomposition in isothermal conditions eliminates the heating rate uncertainty in non-isothermal experiments, as well as allowing for a fuller spectrum of fractional conversion at any given temperature. This was expected to give better estimates of experimental uncertainty and repeatability, as well as reducing the size of the confidence bands around the rate parameters. 2.1. Non-isothermal experiments The environment inside the TGA does not exactly match that inside an aerosol reactor. Therefore, when using TGA to study aerosol rates, the effects of these differences must be considered. The most important factor is the difference in mass and heat transfer rates between the two configurations. Inside an aerosol reactor, the fine dispersion of small particles reduces the resistance of mass transfer of oxygen and zinc vapor away from the reacting particles. Likewise, radiation heat transfer to particles with Biot numbers much less than unity should result in heating rates on the order of 105 K/s. (Dahl et al., 2002)
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Inside a TGA, particles confined within a crucible naturally sit in a small pile. If diffusion effects within the solid pile are important, then the experiment could give slower “apparent” kinetics than would be realized in the aerosol case. Likewise, physical limitations of the TGA furnace prevent it from realizing the ultra-fast heating rates observed in the aerosol condition. To investigate whether these differences between the TGA and aerosol case have a major effect on the calculated kinetic model parameters, a central composite design, as described by Montgomery, was performed to determine the significance of the differentiating factors. (Montgomery, 2001) Three factors were chosen: initial loaded mass of particles, heating rate of the sample, and the particle size of ZnO used. As the initial mass of particles increases, the resistance to mass transfer of product gas should increase as well. If this resistance is a controlling factor in the kinetics, it could result in the observation of slower reaction rates. Indeed, this effect was reported by Weidenkaff et al. (2000), but should be influenced by both the mass of particles used and the TGA geometry. Diffusion controls should become less important as initial mass is decreased, and the rates should approach some asymptotic value. Initial masses between 62.5 mg and 232.5 mg were explored in this study. Particle size can also affect mass transfer control of reaction kinetics. As particle size is decreased, the specific surface area of the particles increases. This will decrease diffusion resistance to the gas phase and could result in faster reaction rates. Two particle sizes were explored in the experiment, 50 nm (Aldrich #544906) and 1m (Sigma-Aldrich #255750). Finally, there exists an inherent difference between the ultra-fast particle heating rates observed in an aerosol reactor and the heating rates existing in the TGA. Although this factor was not expected to control the kinetics, it was included for completeness. Heating rates between 10 and 30 ◦ C/ min were examined in the experiment. The experiments in the central composite design were performed non-isothermally. The sample was heated at a constant rate from room temperature to 1750 ◦ C, where the temperature was held constant for 20 min. Because rate data are collected over a wide range of temperatures, all necessary information to fit a kinetic rate model was contained in each experiment. The non-isothermal method then provided a very efficient path to detect the effects of the aforementioned factors. A drawback to the method was increased uncertainty, as random error in the heating rate and the lack of temperature replicates in each experiment lead to rather large confidence intervals. Therefore, the method was well suited to exploration of factor effects, but determination of rate parameters required an experimental method less prone to experimental uncertainty. 2.2. Isothermal experiments To determine the kinetic rate parameters without the uncertainty associated with non-isothermal experiments, isothermal kinetic experiments were performed. 150 mg of ZnO powder (50 nm, Aldrich #544906) were loaded into Al2 O3 crucibles of the same diameter used in the non-isothermal experiments. For the isothermal experiments, two heights of crucible were used,
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1 in (“full”) and 0.5 in (“half”). The “half” crucibles were fabricated by grinding off the top 0.5 in of the “full” crucibles. Gas diffusion resistance in the crucible could then be tested as a controlling kinetic factor, as it was not examined in the nonisothermal experimental matrix. The TGA furnace was heated to the temperature of interest at 30 ◦ C/ min; this heating rate was not expected, from experience with the non-isothermal experiments, to produce significant conversion before the investigated temperature was achieved. The temperature was held constant until complete reaction was observed. Temperatures between 1440 and 1750 ◦ C were studied. 3. Results and discussion 3.1. Non-isothermal experiments Experiments were performed at the points shown in Table 1, and TG curves generated for each experiment. One such curve is shown in Fig. 2. As the temperature was increased, the mass in the crucible began to decrease, corresponding to decomposition of the ZnO powder. For this experiment, it can be seen that this decomposition begins near 1400 ◦ C. It is important to note that the sample does not fully convert in the non-isothermal portion of the experiment. Conversions at the end of this segment ranged between 0.25 and 0.6, depending on the heating rate of the sample. For each experiment, the reaction went to completion during the hold phase. Before analyzing the TG curves, an appropriate rate model must be chosen. The rate model most often chosen in the literature for solids decomposition is (Galwey and Brown, 1999) d = k0 e−Ea /RT f (), dt
(3)
where is the fractional conversion. The temperature dependent factor is the familiar Arrhenius form, with pre-exponential factor k0 and activation energy Ea . The factor (f ) allows for rate controls similar to concentration effects in gas-based reactions. It most commonly has an “order of reaction” form: f () = (1 − )n ,
(4)
where n is the “order of reaction”. (Galwey and Brown, 1999) The choice of n depends on the physical assumptions of the reaction model. For this case, n was chosen to be 23 , corresponding to a constant rate of radial retraction of spherical particles. This is appropriate for reactions where dissociation occurs only at the particle surface and the rate of dissociation is controlled by surface temperature and chemical properties, as in a shrinking core model. (Levenspiel, 1999) As the literature has suggested that metal oxide dissociations occur preferentially at surfaces (due to the high stabilization energy for ions at the center of the lattice), this is a reasonable choice of model. (Maciejewski, 1992; Galwey and Brown, 1999) This choice of model will not affect conclusions made from the non-isothermal factor study, but will need to be justified when reporting model parameters from the isothermal experiments. In order to avoid the highly eccentric joint confidence region associated with the Arrhenius form at high temperature, and
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Table 1 Experimental results for non-isothermal ZnO decomposition Initial mass (mg)
Particle size (nm)
Heating rate (◦ C/ min)
Ea , linear fit (kJ/mol)
k ∗ , linear fit (s−1 × 10−4 )
Ea , nl regression (kJ/mol)
k ∗ , nl regression (s−1 × 10−4 )
100 200 100 238.5 62 150 200 150 150 200 200 150
1000 1000 50 1000 50 50 50 50 50 1000 50 1000
14 14 14 20 20 20 14 20 10 26 26 20
366 425 350 358 334 321 390 390 302 342 405 334
3.73 1.34 9.24 3.40 2.55 2.37 2.47 2.79 5.18 5.37 4.60 5.35
347 367 397 359 340 363 422 351 303 342 405 334
3.39 1.62 7.90 3.57 2.80 2.28 2.40 3.10 5.18 5.37 4.60 5.35
Fig. 2. TG curve for a non-isothermal experiment (14 ◦ C/ min heating rate, 100 mg initial mass, 1 m particle size).
the concomitant “compensation effect” which seems to relate the model parameters linearly, a transformation was performed on the model equation: d = k ∗ e−Ea /R(1/T −1/T0 ) f (), dt
(5)
k ∗ = k0 e−Ea /RT 0 ,
(6)
where T0 is some central temperature. For these experiments, it was chosen to be 1597 ◦ C, approximately midway between the onset of reaction and the maximum temperature. Such a transformation has been shown to condition the joint confidence interval, de-correlating estimates for the model parameters. (Himmelblau, 1970). The non-isothermal TG curves were analyzed by two methods. The first follows from a rearrangement of the model
equation: Ea 1 1 (d/dT ) , = ln k ∗ − − ln f () R T T0
(7)
where is the constant rate of temperature increase used in the heating program. A plot of the left hand term in Eq. (7) against 1/T − 1/T0 should be linear, with slope −Ea /R and intercept ln(k ∗ ). Derivatives of fractional conversion with respect to temperature were calculated using a central difference formula (Gerald and Wheatley, 1994). For most of the non-isothermal experiments, less than half of the conversion was realized during the non-isothermal phase. The above method might give misleading model parameter fits. To model the entire dissociation curve, non-linear regression was used, allowing kinetic parameters to be fit for a superposition of the non-isothermal and isothermal decomposition
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Table 2 p -values from ANOVA significance test for limiting TGA factors
Fig. 3. Arrhenius plot for a non-isothermal experiment (14 ◦ C/ min heating rate, 100 mg initial mass, 1 m particle size).
regimes. The model equation was solved numerically using the Euler method, and the sum of squares of the residuals from the experimental data was minimized by varying the model parameters. Obviously, this method could not be used to test the effect of heating rate (an inherently non-isothermal factor), but was used to investigate the significance of the other two factors. Values of the activation energy and pre-exponential factor calculated from both methods are shown in Table 1. One plot for the linearization is shown in Fig. 3, demonstrating the linearity of the data and the applicability of this method. The significance of the heating rate was difficult to determine, as at high heating rates (> 25 ◦ C/min) the conversion reached during the non-isothermal phase of the experiments was very small (∼0.15). As a result, the high heating rate samples had relatively few points from which to determine rate parameters, and linear regression on the data points yielded large confidence intervals. To analyze the effect of this factor, a 22 factorial design was examined. Particle size was the other factor, with the two values mentioned above. Initial mass was constant at 200 mg, and the two levels of the heating rate were 14 and 26 ◦ C. An analysis of variance (ANOVA) method was used to analyze the responses, as described by (Montgomery, 2001). The application of the ANOVA method was completed using MINITAB software. (Minitab, 2003) A p-value for the ANOVA analysis of 0.613 showed the factor was not significant on Ea , to 95% confidence. Likewise, it is seen to be unimportant with respect to k ∗ , with a p-value of 0.190. Although the evidence for the significance tests is sparse, the low level of conversion during the non-isothermal phase of the experiments would tend to diminish the importance of heating rate as a factor. It is entirely unimportant for isothermal determinations of model parameters, as there is no effective heating rate. From the evidence above, it is concluded that it is safe to ignore this parameter when examining the other two factors. The significance of the particle size and initial mass factors on the model parameters was examined by constructing a
Rate parameters
Initial mass p-value
Particle size p-value
Ea , linear fit k ∗ , linear fit Ea , non-linear regression k ∗ , non-linear regression
0.384 0.184 0.627 0.290
0.966 0.504 0.107 0.611
22 factorial design with replicates and center points from the original central composite design. The predictions for activation energy and pre-exponential factor determined from both data analysis methods were used as experimental responses in an ANOVA significance test. The ANOVA test was completed using MINITAB software (Minitab, 2003). The p-values for ANOVA analysis of the fractional factorial design are shown in Table 2. As none of the p-values are smaller than 0.05, neither particle size nor initial loaded mass are expected to be significant factors to 95% confidence for estimation of Ea . The same conclusion can be made for estimation of k ∗ . The residuals from the ANOVA model predictions were normally distributed, verifying the central assumption of nonstructured, normally distributed error inherent to the ANOVA approach and analysis of the factorial design. As the initial loaded mass had no effect on the model parameter estimation, it could be concluded for this range of initial masses that mass diffusion through the solid particle pile is not controlling the rate of dissociation. Likewise, in the size range of particles employed, there was no controlling effect on the kinetics. Therefore, by use of particles and initial masses in the ranges explored above, kinetic models obtained from TGA data can be applied to aerosol dissociation without modifications for these physical effects. 3.2. Isothermal experiments In order to experimentally verify the form of f () chosen above, a reduced time analysis of the isothermal data was performed. (Galwey and Brown, 1999) A transformation of the time variable can be made such that t − t0.25 tred = , (8) t0.8 − t0.25 where t0.25 and t0.8 are the times at which 25% and 80% of the sample has dissociated, respectively. Integrating the differential equation model (Eq. (3)): g() = kt, d g() = . o f ( )
(9) (10)
When calculating tred , the temperature dependent rate constant k is eliminated. Therefore, for decompositions of solids with consistent mechanisms for dissociation, plots of conversion against reduced time should be identical, regardless of the temperature at which the decomposition occurred. Comparison of these plots for experiments at different temperatures against a
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Fig. 4. Reduced time conversion response for full-size crucible.
Fig. 5. Reduced time conversion response for half-size crucible.
master curve for the model employed gives an indication of the appropriateness of the model. Figs. 4 and 5 show plots of fractional conversion against reduced time for the isothermal experiments completed in the fullsized and half-sized crucibles, respectively. The figures show very close agreement between the curves across experiments, with deviations at the center point of less than 3% conversion. In addition, there was no ordering of the curves with respect to temperature. This verifies that the mechanism of dissociation does not change with temperature, within the resolution of the data. As can also be seen, the data matches the master curve for
n = 23 much better than for the most commonly chosen order of reaction, n = 1, and there is little difference between the fits for n = 21 and 23 . Statistically, this is borne out by average deviations, in terms of fractional conversion, of 0.00299 and 0.00340 for the n = 21 and n = 23 models in the full crucible, respectively, and average deviations of 0.00232 and 0.00323 for the n = 21 and n = 23 models in the half crucible, respectively. As the uncertainty in the conversion at any point was calculated to be 0.0057 from the aggregate of model data, the differences in deviation are smaller
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Fig. 6. Linear fit for determination of specific rate constant k from an isothermal experiment (temperature of 1550 ◦ C).
Fig. 7. Arrhenius plot for the determination of the kinetic rate constants.
than the uncertainty in the measurements, and the models are indistinguishable. The n = 23 model was chosen for the physical reasons listed above: that the particles are assumed spherical and are expected to react only on the surface. To determine the model parameters from the isothermal TGA data, the integrated form of the rate law is used (Eq. (9)). If the choice of rate law is appropriate, then the graph of g() against time will yield a straight line with slope k. Fig. 6 shows such a plot for one of the isothermal experiments. Deviation of the intercept from zero is due to the fact the time is not corrected to the initiation time of the reaction. A linear least-squares re-
gression yielded the rate constant k at the experimental temperature, and computation of the confidence intervals for the regression yielded the uncertainty for each of these measurements. Rearranging the form of the rate constant: Ea ln k = ln k − R ∗
1 1 − T T0
.
(11)
The rate parameters can easily be determined from the linear plot of the natural logarithm of the rate constant against the corrected reciprocal temperature. Fig. 7 shows this plot for both
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the “full” and “half” crucible experiments. The values of Ea and k ∗ calculated for these experiments were: Ea,full = 353 ± 25.9 kJ/mol, k ∗ = 3.58 × 10−4 ± 6.30 × 10−5 s−1 , Ea,half = 346 ± 42.0 kJ/mol, k ∗ = 5.86 × 10−4 ± 9.90 × 10−5 s−1 . Uncertainty for each parameter corresponds to the 95% confidence interval for the linear regression. These activation energies are in line with those found in previous literature. Weidenkaff et al. stated that the activation energy ranged between 312 and 376 kJ/mol, and Hirschwald found a similar range for activation energy (Hirschwald and Stolze, 1972; Weidenkaff et al. 2000). Möller and Palumbo found an activation energy of 328 kJ/mol, in line with what was found in this study (Moller and Palumbo, 2001). Based on the L’vov theory described below, it makes sense that activation energy is configuration independent; the confidence bands given in the current work place the activation energy in a range consistent with previous literature. As for pre-exponential factor, this has been shown by the dependence on crucible diffusion distance to be strongly configurationally dependent. Previous literature can be difficult to compare, as pre-exponential factors were not always expressed in equivalent terms. For example, Möller and Palumbo express this factor in terms of a mass flux (g m−2 s−1 ), whereas the current model requires it to be expressed in reciprocal seconds (Moller and Palumbo, 2001). Weidenkaff et al., did not report values of the pre-exponential factor, so comparisons cannot be made (Weidenkaff et al., 2000). However, based on the strong configurational dependence of this parameter, meaningful comparisons would be difficult to make. The values of Ea for the full and half crucible are, within uncertainty, identical. However, the value of k ∗ is, from a statistical viewpoint, significantly larger for the “half” crucible than for the “full” crucible, as the confidence intervals for the two measurements do not intersect. This suggests that gas diffusion out of the crucible is important with respect to the kinetic rate. Further, as it only affects k ∗ , the activation energy calculated from the TGA experiment is gas diffusion independent and reflects the intrinsic kinetic limitations of the ZnO dissociation. 3.3. Mechanistic model for reaction kinetics L’vov, et al. suggested a model based on equilibrium thermodynamics to predict the rates of metal oxide dissociations, specifically for those dissociations with all products in the gas phase. ZnO falls into this category, and the development sketched below closely follows that shown in L’vov’s paper (L’vov, 1997). The model assumes that only lattice members on the surface of the solid are available for dissociation, and that the rate of this dissociation is dependent only upon the surface temperature and the chemical properties of the solid. This assumption makes sense, as ions within the lattice will have to
overcome a larger energetic barrier to escape, due to electrostatic interactions with surrounding ions, than those ions on the surface subjected to electric field contributions from only one side. At reaction equilibrium, the maximum rate of dissociation is limited to the rate of condensation, which can be calculated from statistical mechanics. This L’vov based model is based on the Hertz–Knudsen– Langmuir model for a condensing substance in a inert foreign gas environment. It predicts the Arrhenius parameters from an equilibrium basis as Ea =
HT◦ , (a + b) a/(a+b)
k0 =
NA DM
(12) b/(a+b)
DO qzRT
a (a a bb )1/a+b
◦
eST /R(a+b) , (13)
where D is the binary diffusion coefficient of the oxygen (O) or metal (M) in the foreign gas, z is the diffusion distance to the portion of the foreign gas where the concentration of the substance drops to zero, q is the number of metal atoms per unit surface area of solid, NA is Avogadro’s number, a and b are the stoichiometric coefficients of metal and oxygen in the starting oxide, respectively, R is the ideal gas constant, T is the temperature of decomposition, HT◦ is the standard enthalpy of reaction at temperature T , and ST◦ is the standard entropy of reaction at temperature T . The temperature dependence in the denominator is mostly offset by temperature dependence of the binary diffusion coefficients, given by the Chapman–Enskog theory (Bird et al., 1960). This makes the pre-exponential factor independent of temperature over small temperature ranges (500 K). For the dissociation of ZnO, the rate limiting step could come from two different reactions: ZnO −→ Zn + O, ZnO −→ Zn + 21 O2 ,
HT◦ = 712 kJ/mol, HT◦ = 470.6 kJ/mol.
(14) (15)
For the first of these reactions, a + b = 2, and the activation energy would be predicted by the L’vov formulation as 356 kJ/mol. For the second, a+b=1.5, and the activation energy would predicted to be 314 kJ/mol. The isothermal experiments give a value in close agreement with the first equation, and the second activation energy is outside the confidence bounds for the parameter determined experimentally for the “full” crucible experiments. This suggests that the rate limiting step is the dissociation of Zn and O atoms (or ion pairs) from the surface, and that formation of diatomic oxygen occurs in the gas phase away from the surface. A purely electrostatic mechanistic interpretation of the L’vov approach can be described as follows. For a completely ionic lattice, the energy required to remove an atom is approximately equal to the electrostatic stabilization energy achieved when that ion to enter the lattice and the associated ionization energies required to form the ion. This can be calculated from a Born–Haber cycle for metal oxide formation. For equally charged ions, under this mechanistic picture, this will be equal to the stoichiometric contribution to the enthalpy of reaction
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for that ion (Huheey et al., 1993). If we model the system as a set of point charges, the energy of the system increases monotonically as we remove an ion to infinity from the surface. Because the energy increases monotonically, no “transition state” of higher energy exists, and the activation energy should be equal to the enthalpy of reaction for the rate limiting step, in this case the removal of one cation or anion. For solids with some covalent boding, predictions from this mechanism should deviate from what is expected. The bonds become more directional, and the simplistic point charge picture no longer applies. The covalent picture would be expected to show some kind of transition state as these bonds are stretched to the intermediate point between bond breakage and formation. To reach this point, an activation energy greater than the enthalpy of reaction would be required. Indeed, the data of L’vov shows an experimental deviation to higher activation energies for HgS, a solid with large amounts of covalent bonding (L’vov, 1997). The authors originally adjusted the stoichiometric ratio to give a better fit to the data, but for physical reasons the above interpretation makes more sense. Zn has an electronegativity (Pauling scale) difference with O of 1.79, placing it in the ionic bonding regime (Huheey et al., 1993). It has some covalent bonding, evidenced by its behavior as a high bandgap semiconductor, but its large electronegativity difference justifies treating it as ionic when examining dissociation. The above mechanistic interpretation is admittedly simplistic, and more experimental work over a wide range of metal oxides decompositions should be performed to verify its claims. This work would include measurement of any intermediates at the surface, comparison of compounds that are strongly in the ionic bonding regime with those whose bonding is more strongly covalent, and computational chemical calculations of the energies for possible reaction pathways at the surfaces for different metal oxides. The L’vov method prediction for k0 depends on the diffusion distance to a concentration sink. In the TGA, inert gas is always flowing across the top of the crucible, carrying reaction products with it. Therefore, the concentration of reaction products should go to zero there, and the distance z will be equal to the crucible height. Using the L’vov predicted activation energy, the “full” crucible measured value is k0 = 3.15 × 106 ± 5.54 × 105 s−1 and the “half” crucible measured value is k0 = 5.15 × 106 ± 8.71 × 105 s−1 . As can be seen, the “full” values are about one half as great as the “half” values, as predicted by the L’vov theory. In an aerosol reactor, it is expected that the values of z will be much smaller, due to the smaller diffusion film existing around the small, dispersed particles. As such, the preexponential factor (and consequently, the reaction rates) should be much larger in an aerosol reactor. In order to determine these values of z, careful experimentation in an aerosol reactor should be performed. However, the mechanism of dissociation will not change, and the equations determined by the above work would still be applicable to this geometry; only the diffusion distance need still be determined, and the more difficult task of finding an appropriate kinetic expression is already completed.
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4. Conclusions Non-isothermal TGA experiments to determine the effect of heating rate on the measured kinetic rate constants showed that this factor was not significant. This conclusion is admittedly blurred somewhat by the small conversions reached during the non-isothermal phase of the experiments for many high heating rate trials, but the overall statistics show little significance of the factor. Similar experiments to test the effects of particle size and initial loaded mass showed no significance for these factors on the measured model parameters. The kinetic rates were concluded to be independent of the physical diffusion phenomena associated with these factors (i.e. diffusion of gas through the solid phase and particle interstices), and the TGA results were seen as applicable to an aerosol situation. This is in contrast with the rate controls for diffusion through the gas phase, which were seen to be significant in the isothermal TGA experiments. Isothermal TGA experiments were conducted to measure the kinetic model parameters. Reduced time analysis was applied, and a consistent reaction mechanism was seen across temperatures for both “full” and “half” crucible sizes. An order of reaction model with 23 was seen to match the data well. A linear Arrhenius plot on the “full” crucible yielded rate parameters of 356 ± 25.9 kJ/mol for the activation energy and 3.58×10−4 ±6.30×10−5 s−1 for the pre-exponential factor k ∗ . The values of activation energy closely matched L’vov kinetic theory, including the predicted doubling of the pre-exponential factor for reduced diffusion distance. The close fit to the L’vov predictions suggests that the mechanism of reaction is a reversible thermal perturbation of zinc and oxygen atoms from the solid surface, followed by formation of diatomic oxygen. Acknowledgements The authors would like to acknowledge the National Science Foundation for its support through the Graduate Research Fellowship program, as well as the US Department of Energy for its support under Grant # DE-FG36-03GO13062. References Bird, R.B., Stewart, W.E., Lightfoot, E.N., 1960. Transport Phenomena. Wiley, New York. Dahl, J.K., Tamburini, J., et al., 2001. Solar-thermal processing of methane to produce hydrogen and syngas. Energy & Fuels 15 (5), 1227–1232. Dahl, J.K., Barocas, V.H., et al., 2002. Intrinsic kinetics for rapid decomposition of methane in an aerosol flow reactor. International Journal of Hydrogen Energy 27 (4), 377–386. Dahl, J.K., Buechler, K.J., et al., 2004a. Rapid solar-thermal dissociation of natural gas in an aerosol flow reactor. Energy 29 (5–6), 715–725. Dahl, J.K., Buechler, K.J., et al., 2004b. Solar-thermal dissociation of methane in a fluid-wall aerosol flow reactor. International Journal of Hydrogen Energy 29 (7), 725–736. Dahl, J.K., Weimer, A.W., et al., 2004c. Sensitivity analysis of the rapid decomposition of methane in an aerosol flow reactor. International Journal of Hydrogen Energy 29 (1), 57–65. Dahl, J.K., Weimer, A.W., et al., 2004d. Dry reforming of methane using a solar-thermal aerosol flow reactor. Industrial & Engineering Chemistry Research 43 (18), 5489–5495.
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