9778
J. Phys. Chem. B 2001, 105, 9778-9784
Kinetic Analysis of Temperature-Programmed Reduction: Behavior of a CrOx/Al2O3 Catalyst Jaana M. Kanervo* and A. Outi I. Krause Department of Chemical Technology, Helsinki UniVersity of Technology, P.O. Box 6100, FIN-02015 HUT, Finland ReceiVed: April 16, 2001; In Final Form: July 9, 2001
The kinetic modeling of hydrogen reduction was studied with the use of experimental data obtained by temperature-programmed method. The reduction of a CrOx/γ-Al2O3 catalyst prepared by atomic layer deposition (ALD) was used as the test reaction. Reduction kinetic models reported in the literature were evaluated by model estimation techniques. It was found that an experimental data set with three different heating rates was needed to discriminate between the models. The Avrami-Erofeyev model, which assumes the reaction to proceed via nucleation and nuclei growth, was found to be promising, and it was investigated in detail. Problems often arise in the application of this model to temperature-programmed reduction kinetics: (1) the physical meaning of the model and the parameters are often poorly defined, (2) the temperature dependencies of the rates are not correctly included, and (3) various limiting cases of the model are customarily used without consideration of the underlying assumptions. These problems were considered in this work, and revisions were made in the model. The reduction kinetics of the catalyst was best described with the model assuming instantaneous nucleation and two-dimensional nuclei growth.
Introduction Thermoanalytical techniques are among the most important and widely used methods for the characterization of solid materials. Temperature-programmed reduction (TPR) is a convenient technique for characterizing supported metal oxide catalysts. Generally, TPR is used to provide information on the influence of support materials, preparation and pretreatment procedures, and metal additives on catalyst reducibility. The directly observable quantities include the total consumption of reducing agent and the temperatures of the reduction rate maxima. Qualitatively the character of the TPR pattern contains information on the nature of the reduction process. Robertson et al.,1 in 1975, first described the presently used TPR technique. The technique is intrinsically quantitative and also produces kinetic information on the processes involved in reduction. The TPR technique with its kinetic aspects has been described and reviewed thoroughly.2-6 In 1982 Hurst et al.2 reviewed the thermodynamics, kinetics, and mechanisms of reduction with illustrative examples taken from the TPR of many supported and unsupported oxides. Numerous attempts have been made to extract quantitative kinetic information from TPR data, but the kinetic modeling of TPR patterns has received relatively little attention. Chromium oxide supported on Al2O3 or SiO2 is an important catalytic material. The dehydrogenation of light alkanes (C3C5), a selective method for the industrial production of light alkenes, is commercially carried out with either supported chromium or platinum catalysts. Chromium oxide catalysts have been intensively studied over the years, as recently reviewed by Weckhuysen and Schoonheydt.7 The identification of active sites and the influence of preparation and pretreatment on chromium oxide species and on the catalytic activity and * Corresponding author. Fax: +358 9 451 2622. E-mail: kanervo@ polte.hut.fi.
selectivity still continue to be of considerable interest. After their preparation, CrOx/Al2O3 catalysts contain chromium in oxidation states Cr6+ and Cr3+ (determined by XPS).8 Cr3+ is considered to be the catalytically active species in dehydrogenation,8 and the catalysts are therefore reduced before use. Reducibility studies furnish information helpful to understanding the nature of the active sites on chromia catalysts. This study was carried out to increase understanding of the reduction of a supported oxide catalyst by applying kinetic modeling of H2TPR data. Data on dynamic hydrogen consumption was studied by traditional methods and model estimation techniques. Modeling Kinetic Information from H2-TPR. Two in principle different techniques are described in the literature for determining kinetic parameters from TPR experiments. One requires TPR data collected at different heating rates and utilizes only one point from each TPR curve, and the other is based on computersimulated nonlinear regression and exploits the whole experimental curve or curves. The techniques that extract quantitative information from TPR patterns without complete model fitting were originally developed for other thermoanalytical techniques such as differential thermal analysis (DTA) and temperature-programmed desorption (TPD). In these methods a simple rate expression is established and algebraically manipulated in order to estimate the apparent activation energy by linear regression. Kissinger,9 in 1957, presented a technique to obtain the apparent activation energy from DTA data for nth order decomposition reactions. The Kissinger peak analysis was later introduced to TPR by Wimmers et al.,10 who extended its applicability, under certain assumptions, beyond a specific reduction mechanism. The activation energy estimate (E) is obtained from the shift of the rate maximum temperature (Tm) against heating rate (β). If the
10.1021/jp0114079 CCC: $20.00 © 2001 American Chemical Society Published on Web 09/19/2001
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J. Phys. Chem. B, Vol. 105, No. 40, 2001 9779
plot of ln(Tm2/β) versus 1/(Tm) results in a straight line, the slope of plot is E/R, where R is the gas constant. The conversion at rate maximum is assumed to be independent of the heating rate. The Kissinger technique has been used both independently and as a complementary means for mechanistic modeling. Gentry et al.11 determined the activation energy from TPR experiments analogously to the procedure for TPD data presented by Cvetanovic and Amenomiya.12 Despite the different approaches, the equations applied by Gentry et al.11 and Wimmers et al.10 are very similar. Another direct method relying on multiple heating rates was applied for TPR studies by Tarfaoui.13 In this technique, ln(dR/ dt) is plotted against 1/T at selected conversions, and the slope gives -E/R. The rate of conversion, dR/dt, is obtained from the experiments. This method of constant conversion, originally presented by Friedman14 for polymer thermal decomposition, is more informative than the Kissinger peak analysis since it provides apparent activation energy estimates at different degrees of reduction. The second type of methods to obtain kinetic parameters involves nonlinear regression. In these complete model-fitting techniques, a dynamic model for the solid-phase conversion is postulated and its relation to the consumption of the reducing agent is established. This requires a reactor material balance with chemical kinetic expressions. Typically, a pseudohomogeneous differential reactor with uniform well-controlled temperature is assumed. The model is simulated and its solution is compared with the experimental data. The kinetic parameters are adjusted and the model solving is repeated until optimal agreement is reached. In practice, the model equations are solved by numerical computer codes and new parameter values are generated by an optimization algorithm. Kinetic Models for Reduction Reactions. The reduction of a solid by hydrogen is a multistep process. Not only the intrinsic kinetics of reduction but also adsorption, desorption, diffusion, and topochemical aspects may affect the observed rate of the reduction. In contrast to a homogeneous reaction, a topochemical reaction is a reaction that is localized at the interface between a solid substrate and a solid product. Different mechanisms may be operative, depending on the nature of reducible solid. Reduction kinetics of bulk reducible compounds, reducible species in solid solutions, and supported species are usually described by different models. Supported oxidessthe special concern of this worksmay be present as three-dimensional islands, in which case they may be reduced in a manner similar to that of unsupported oxides, and the support functions only as dispersing agent. Alternatively, metal oxide species may be evenly distributed across the support surface and interactions between the support and the reducible oxide may be significant for the reduction kinetics. The reduction of these species may proceed via reduction of individual metal ions or clusters.2 If oxide species are bound to the carrier in many different ways, the picture becomes further complicated. The kinetic modeling of TPR is commonly attempted by expressing the overall reduction rate as the product of a temperature-dependent rate coefficient k, a function of the degree of reduction of the solid material f(R), and a function of the gas-phase composition f2:10,15
dR ) k(T) f(R) f2(CH2,CH2O) dt
(1)
The rate coefficient k(T) is expressed using the Arrhenius equation k(T) ) A exp(-E/(RT)). Function f2 is often ignored in models when the experiments are carried out with one feed
TABLE 1: Functions f(r) for Typical Reduction Kinetic Models10 random nucleation contracting area contracting volume 2-dimensional nuclei growth (2D Avrami-Erofeyev) 3-dimensional nuclei growth (3D Avrami-Erofeyev)
1-R (1 - R)1/2 (1 - R)2/3 2(1 - R)(-ln(1 - R))1/2 3(1 - R)(-ln(1 - R))2/3
composition only and differential conditions prevail in the reactor. Different mechanistic assumptions are tested by altering f(R) in the model estimation. The kinetic models for TPR are either of empirical nature or they utilize mechanisms of gassolid reactions that are older than the TPR technique.16,17,28,29 A collection of a few reduction rate laws f(R) that have been tested for the reduction of supported metal oxide is presented in Table 1. Diffusion models are not included because in our test system the reducible species are located on the surface of the carrier oxide as a thin chromia layer. The model expressed by (1) is well suited for parameter estimation when the rate process occurs as a single well-defined process, with one clear rate-determining step. The effects of particle size as well as other material heterogeneities are neglected. In the case of a multipeak TPR pattern, more complex modelling strategies than (1) are called for. Multistep models with intermediate products13,18 or models with multiple independently reducible species6,19 may then be required. It is also possible that the dominating overall reduction mechanism changes with temperature or degree of reduction. In theory it is possible to construct models by combining contributions of more than one type such as the nucleation and growth law and the phase boundary limited reaction rate law.20 Nucleation and Nuclei Growth (N/NG) Mechanism and Kinetics. Gas-solid reactions proceed by nuclei formation (nucleation) and subsequent nuclei growth. Nucleation is a dynamic process that initiates the phase change. The conversion process then continues with the growth of the activated nuclei. The relative rates of these two processes and the density of potential nucleus-forming sites (germ nuclei) determine the macroscopic conversion-time behavior. The micromechanisms of nucleation are not fully understood, but in the case of crystalline metal oxides, nucleation is believed to take place in localized imperfections, the number of which is very small relative to the total number of atoms in the oxide specimen. The number of germ nuclei can be controlled by pretreatment. The nuclei growth rate has been observed to be constant at a certain temperature and for a particular composition of the gas phase. The energetics of the overall reduction process is primarily dependent upon the rate-determining process.17 N/NG characteristics appear in the literature dealing with reduction kinetics in Avrami-Erofeyev (A-E) models. These A-E models have been proposed for reduction kinetics for both bulk oxides Fe2O3,10 VOx,21,22 CuO,13 and MnOx13 and supported oxides VOx/TiO222 and VOx/Al2O3.13 Avrami23,25,26 has presented mathematical descriptions for both nucleation and nuclei growth and formulated equations for conversion-time relationships. Characteristic of Avrami’s approach is that the nucleation and unlimited growth are described with specific rate laws, and the possible overlap of the growing nuclei and the ingestion of the germ nuclei before activation are taken into account by expressing the relation between the extended (unlimited) and true conversion.23 Erofeyev24 derived analogous models for the solid conversion from completely different considerations.
9780 J. Phys. Chem. B, Vol. 105, No. 40, 2001
Kanervo and Krause
Despite the conceptual difference, Erofeyev’s probabilistic approach inherently includes the same assumptions as Avrami’s treatment. The N/NG models of Avrami23,25,26 were originally used to describe phase transformations of steel. Since then they have found numerous applications in crystallization, precipitation, decomposition of various solid materials, thin film growth, and polymerization. For certain bulk reducible metal oxides, micromechanistic information is available on the intrinsic rates of reduction, including microscopic evidence for the N/NG mechanisms.27 However, to the best of our knowledge, no direct proofs have been published of whether supported metal oxide catalysts are reduced by nuclei growth mechanism. The usual way to apply the Avrami-Erofeyev model with two- or three-dimensional nuclei growth is to introduce the following mass action functions f(R) into the model (1):15
dR ) k(t)[2(1 - R)(-ln(1 - R))1/2] S dt
∫0tk(t) dt)2]
(2a)
∫0t k(t) dt)3]
(2b)
R(t) ) 1 - exp[-( dR ) k(t)[3(1 - R)(-ln(1 - R))2/3] S dt R(t) ) 1 - exp[-(
The rate coefficient is now intentionally expressed as a function of time, the condition that arises under temperature programming. How these expressions (2a,b) are obtained from first principles is not self-evident. The derivation of a general dynamic equation for interface advance reactions begins with a definition of the total volume of the reacted material:28,29
∫
dN Vtot(t) ) 0 V(t, y) dy dt t)y t
( )
(3)
This equation is a convolution integral that is composed of a law of nucleation (dN/dt)t)y and a law of a nuclei growth V(t,y), where N denotes the number of active (growing) nuclei and V(t,y) gives the volume at time t for a nucleus that became activated at time y. In principle, (3) describes a class of models with different nucleation and growth processes. If the radius of the nucleus r(t,y) increases with rate kl, the nucleus volume becomes28,29
V(t,y) ) Kg[r(t,y)]m The radius of the nucleus is r(t,y) ) ∫ytkl(t*) dt*. For time-invariant rate kl the volume becomes V(t,y) ) Kg(kl(t - y))m. The growth takes place m-dimensionally in isotropic material. Kg is a shape factor; for example Kg is π for disks and (4/3)π for spheres. Incorporating the first order nucleation with rate coefficient k2 and N0 potential nucleus-forming sites, we obtain
dN dN ) k2(N0 - N) S ) k2N0e-k2y dt dt t)y
( )
(4)
and with m-dimensional nuclei growth, (3) becomes
Vtot(t) )
∫0tKg(∫ytkl(t*) dt*)mk2N0e-k y dy 2
(5)
and for the isothermal case
Vtot(t) )
∫0tKg(kl(t - y))mk2N0e-k y dy ) t KgN0klmk2∫0 (t - y)me-k y dy 2
2
which integrates to
[
]
m KgN0klmm! (k2t)i m+1 -k2t i Vtot(t) ) (-1) e - (-1) i! i)0 k2m
∑
(6)
(7)
Both sides of the equation are then divided by V0, the initial volume of the converting substance, and the left-hand side is marked by Rex:
[
]
m KgN0klmm! (k2t)i m+1 -k2t i (-1) e - (-1) Rex(t) ) i! i)0 V0k2m
∑
(8)
So far, the nuclei have been allowed to grow in an unlimited way according to their growth law. Eventually, however, the growing nuclei reach the boundaries of the converting material and each other’s boundaries and they may also ingest germ nuclei. To account for these phenomena, the extended rate is to be projected to the true rate by the relation proposed by Avrami:28
dR ) dRex (1 - R) S Rex ) -ln(1 - R) where R represents the actual degree of conversion. Application of (9) to (8) gives
[
(9)
]
m KgN0klmm! (k2t)i m+1 -k2t i (-1) e - (-1) -ln(1 - R(t)) ) i! i)0 V0k2m (10)
∑
which is solved for R. Equation 10 gives the conversion-time behavior when nuclei are formed as a first order process with rate constant k2 and the radii of formed nuclei grow at the constant rate of kl. According to Avrami, the presented treatment assumes that the distribution of germ nuclei is locally random. Different limiting cases can be derived from (10). Commonly, the right-hand side of (10) is approximated by the term with the highest order of t, for the limiting case of large k2t. Then, solving for R, the expression becomes28
R(t) ) 1 - e-Kt
m
(11)
The parameter K is the rate of the nuclei growth kl grouped together with constant (KgN0/V0)1/m, and m ()1, 2, 3) is the dimensionality of the nuclei growth. Equation 11 is valid if the rate of nucleation k2 is high. In addition, (11) describes the conversion-time behavior for large values of t, that is, the later stages of conversion when ingestion and nuclei overlap play a significant role. The accordance of (2a,b) with (11) is directly observed in the isothermal case when k in (2a,b) is the constant K in (11). Avrami’s rigorous treatment was not restricted to isothermal conditions. It was actually originally derived for isokinetic case. Avrami introduced a new scaling of time k2 dt ) dτ, where τ is the characteristic time of nucleation. If all the variables in
TPR of CrOx/Al2O3 Catalyst
J. Phys. Chem. B, Vol. 105, No. 40, 2001 9781
(5) are expressed in the new τ system and (9) is applied, then (9) becomes
-ln(1 - R(τ)) )
(
N0Kg V0
k
∫0τ ∫zτk2l du
)
m
e-z dz
(12)
If there existed an isokinetic range such that the activation energies of the nucleation and the growth would coincide, the ratio kl/k2 could be denoted by constant L, and the previous equation would simplify to
-ln(1 - R(τ)) )
N0KgLm V0
∫
τ
(τ - z)me-z dz 0
rate of reduction that is responsible for the nuclei growth becomes
( )
rate ) kl ) Ag′ exp -
dT ) β, T(0) ) T0 dt
[
]
i
m KgN0L m! (τ) (-1)m+1 e-τ - (-1)i V0 i! i)0 (14)
∑
Analogously to the isothermal case, limiting cases for the large and small τ values can now be derived. If the last term is again taken for a sufficient approximation of the right-hand side of (14) and the equation is solved for R, we get
R(τ) ) 1 - e-Aτ
m
p (2πkT)1/2
(
(16)
Consequently, under a temperature program a constant feed composition results in time-varying gas-phase mass action on the rate of reduction. This is now included, such that the intrinsic
(
A
E
∫yt(βτ + gT )1/2 exp - R(βτ +g T ) 0
0
) )
m
dτ
(19)
The nucleation dynamics is still modeled as a first order decay of germ nuclei. However, the time-varying nucleation rate coefficient k2 ) k2(t) is taken into account, and consequently, the rate of nucleation corrected by the Arrhenius temperature dependence becomes
()
dN dN ) k2(t)(N0 - N) S dt dt k2(y)N0 exp(-
(15)
where A ) L(KgN0/V0)1/m. If conversion in ordinary time t is desired, the relation k2(t) dt ) dτ is used. The form (15) is also in accordance with expression (2a,b) under the isokinetic conditions if the rate coefficient k(t) in (2a,b) is given the meaning A1/mk2(t). As can be seen, the commonly applied forms (2a,b) are obtained from the isothermal or the more general isokinetic conversion-time behavior by truncating the exact solution. The isothermal equation for conversion is clearly inadequate to model TPR: since the rates of chemical reactions increase with temperature, they are time-dependent (k ) k(t)) during the temperature programming. Even more general treatment than the isokinetic case may be desirable when dealing with the transience of the TPR technique. Furthermore, the limiting cases of the model may be inadequate when the task is to describe the whole range of conversions. The temperature dependencies of the involved rate processes are now considered as independent of each other. The concentration of the reducing agent is likely to affect the intrinsic rate of reduction. As a common practice, the concentration is either ignored or included as a constant within the preexponential factor. According to the kinetic theory of gases, the average velocity of molecules is related to temperature, and the immensity of the flux is proportional to the pressure divided by the square root of temperature:
F)
(18)
The volume of a nucleus in time-dependent form now becomes
(13)
This equation is integrated analogously to (6) to give the conversion R as a function of τ:
-ln(1 - R(τ)) )
(17)
Assuming that temperature changes at a constant rate, then
V(t,y) ) Kg
m
( )
Eg Ag Eg F ) 1/2 exp RT RT T
)
(
t)y
E
∫0yk2(t*) dt*) ) N0AN exp - R(βy +N T )
(∫
exp -
y
0
(
AN exp -
) )
0
EN R(βt* + T0)
)
×
dt* (20)
and the total N/NG equation becomes
Vtot(t) )
(
A
(
E
))
∫0tKg ∫yt (βτ + gT )1/2 exp - R(βτ +g T ) dτ 0
() 0
dN dt
m
×
dy (21) t)y
By grouping all the constants together and incorporating the extended reduction degree, we obtain
-ln(1 - R(t)) ) A°
(
∫0t ∫yt(βτ +1T )1/2 ×
(
0
) )( )
E dτ exp R(βτ + T0)
m
dN dt
dy (22) t)y
As can be seen, the refinements needed to account for the time dependence of the rates are now included in the model. Presumably an analytical solution is not attainable, but the parameter estimation with numerical integration of (22) can be performed with appropriate software for nonlinear regression. Experimental Section Catalysts. The catalyst was prepared by atomic layer deposition (ALD), a technique for preparing highly dispersed supported catalysts. The catalyst preparation is described in detail elsewhere.30 The catalyst sample contained 7.5 wt % chromium, evidently corresponding closely to the monolayer concentration.29 After preparation, CrOx/Al2O3 catalysts contain chromium in oxidation states Cr6+ and Cr3+.30 According to UV-vis
9782 J. Phys. Chem. B, Vol. 105, No. 40, 2001 spectrophotometry the amount of Cr6+ species was 2.9 wt %. The XRD analysis showed no crystalline chromia phases on the catalyst. H2-TPR Experiments. The H2-TPR measurements were performed with the Altamira Instruments AMI-100 catalyst characterization system. The catalyst samples (30 mg) were flushed with argon and heated from 30 to 115 °C at a rate of 11 °C/min, and the samples were held at 115 °C for 60 min. After that the samples were heated from 115 to 50 °C at a rate of 11 °C/min under a flow of 5.0% O2/Ar and kept for 30 min. The samples were cooled to 30 °C in 5.0% O2/Ar flow. TPR was performed at heating rates of 6, 11, and 17 °C/min up to 590 °C under a flow of 11.2% H2/Ar (50 cm3/min). The consumption of hydrogen was monitored by a thermal conductivity detector (TCD) and recorded at a signal rate of 6 points/ min. The hydrogen consumption was quantified by a pulse calibration. The temperature was measured adjacent to the catalyst bed and followed a strictly linear trend. In the H2-TPR modeling, the rate of reduction was unequivocally related to the observed rate of H2 consumption. Separate mass spectrometer analyses during the TPR showed that water was retained on the catalyst, and water production was not an appropriate measure for the progress of reduction. The retaining of water has also been reported earlier.31 The selection of experimental conditions was in agreement with the criterion developed by Malet and Caballero, P ) βS0/(FC0) , 20 K.6 The system was not however in agreement with the characteristic number given by Monti and Baiker,5 K ) S0/(FC0), which should remain between 55 and 140 s, and our value was below the lower limit. These Monti-Baiker limits suggest that hydrogen consumption at rate maximum should be between 1/10 and 2/3 of the available hydrogen to ensure the detection sensitivity and to prevent hydrogen exhaustion. The instantaneous maximum conversion of hydrogen was less than 5.5% in our system, and the detector sensitivity was sufficient. In addition, the differential reactor assumption was valid in our system. The diffusion rate of the gas in the catalyst pores was considered in terms of the Weisz-Prater criterion.32 The value of the criterion (,1) indicated that the observed reaction rate was free of the intraparticle mass transfer resistance of the reactant. Parameter Estimation. The parameters were estimated by nonlinear regression. The integral equation (20) was numerically solved by a trapezoidal method, the conventional equations of form (1) were solved by an adaptive Simpson quadrature, and the object function minimization was carried out by the NelderMead search method. The criterion of optimization was the sum of squared residuals (SSR) between the measured hydrogen consumption data and the corresponding model output. In the multiresponse fitting, a combined criterion was formed by adding up the SSRs. All the computations were performed in the MATLAB6 (MathWorks Inc.) environment. The temperature mean centering was done for the rate parameters k(T) ) kref exp[(E/R)(1/Tref - 1/T)] in order to enhance the parameter identifiability. Results and Discussion Thermograms and Their Conventional Investigation. The H2-TPR thermograms (β ) 6, 11, 17 °C/min) measured for the CrOx/Al2O3 catalyst are shown in Figure 1. The experimental reduction data clearly exhibited a single-peak behavior. Comparison of the total hydrogen consumption and the analyzed amount of reducible chromia strongly supports the presumed reduction stoichiometry from Cr6+ to Cr3+. From this it was
Kanervo and Krause
Figure 1. Measured H2-TPR data for CrOx/Al2O3, heating rates 6, 11, 17 K/min.
Figure 2. (a) Application of Friedman method of constant conversion. (b) Activation energy as a function of R.
concluded that only one clearly dominant reduction step was taking place and the reducible material did not contain serious heterogeneities. The rate maxima occurred at conversions Rp ) 0.61, 0.64, and 0.66 and at temperatures Tp ) 298, 319, and 331 °C for heating rates 6, 11, and 17 °C/min, respectively. The results were highly reproducible with temperatures of rate maxima remaining within (2 °C. The apparent activation energy (E) was extracted using the Kissinger method,11 which is based on the shift of the temperature (Tm) of the rate maximum as a function of heating rate (β). Plotting ln(Tm2/β) versus 1/(Tm) gave a straight line (correlation coefficient > 0.99). By the assumptions of the Kissinger analysis the slope of this plot equals E/R, yielding the activation energy of 87 kJ/mol. Another technique utilizing different heating rates to extract activation energy, the Friedman’s14 method of constant conversion, was applied by plotting ln(dR/dt) against 1/T at fixed conversions of 0.2-0.8. Also this method produced straight lines (correlation coefficients > 0.9) (Figure 2a). The Friedman analysis gave activation energy values of about 90 kJ/mol for the reduction degrees below 0.6, and then higher (100-115 kJ/mol) values for the higher reduction degrees (Figure 2b). The results of the Kissinger and Friedman analyses (for R < 0.6) agreed well, evidently because the two techniques assume a similar general form (1) for the reduction
TPR of CrOx/Al2O3 Catalyst rate, and the rate maxima occurred below conversion 0.7. The results of the Friedman analysis show, however, that the reduction process is not satisfactorily explained by any expression of type (1) with the same energetics for all reduction degrees. The increase in the activation energy as a function of reduction degree may imply a shift in the relative rates of the elementary reactions. Alternatively the increase in activation energy might be explained by the slight material heterogeneity of chromia: a small part of the species may be less reducible. Yet another possible explanation could be that, at the later stages of reduction, water accumulates on the catalyst. It has been observed that water production is delayed in the reduction of CrOx/Al2O3 and some water is retained in the system.31 The effect of water is negligible at the beginning of the reduction but becomes more pronounced at the end. Accumulated water may retard the reduction reaction by kinetic effect. Thermodynamics favors the reduction of Cr6+ to Cr3+ even in watercontaining systems. If the hydrogen reduction of the corresponding bulk chromium oxide occurs under conditions similar to those of our test system, the water-to-hydrogen pressure ratio in the gas phase would not constitute a thermodynamic limitation for the reduction. Kinetic Model Estimation. Kinetic modeling of the TPR patterns was performed with nonlinear regression analysis. Various typical reduction models (Table 1) such as the random nucleation law, the phase boundary controlled reaction law (contracting area or volume), and the two- and three-dimensional nuclei growth laws (also known as Avrami-Erofeyev models) were tested with model type (1). When the parameter estimation was carried out on the basis on a single TPR, all the abovementioned models described the observations well and equally, as measured by the value of the object function. The estimates of activation energy varied in a wide range from one model to another. Evidently, a single TPR pattern is not sufficient for model discrimination. When the model was simultaneously fitted to an experimental data set with three different heating rates, only the nuclei growth models (2a,b) were able to satisfactorily describe the experimental TPR data. The two-dimensional model (2a) was better than the three-dimensional model (2b). In addition, an empirical noninteger dimensionality for the nuclei growth was tested. With the 2.4-dimensional nuclei growth model, an estimate of apparent activation energy of 86 kJ/mol was obtained. This value is close to the value obtained by applying the Kissinger peak method. The fit was good (rootmean-square error, rms ) 0.006 µmol/s) and the 2.4D model predicted the rate maxima most accurately. Due concern over the models (2a,b), expressed above, led us to test more general nuclei growth models of the form (22). Four parameters, preexponential constants and the activation energies for both the first order nucleation process and the nuclei growth, were estimated on the basis of the three measured TPR patterns. Close fits (rms ) 0.006 umol/s) were found for twodimensional nuclei growth. The activation energy of the nuclei growth for model (22) with m ) 2 was 89 ( 1 kJ/mol. The parameter estimates of nuclei growth were well identified, but this was not the case for the parameter estimates of nucleation. In model fitting the nucleation parameters obtained a multiplicity of values so that the nucleation became instantaneous and virtually complete before the nuclei growth started. Thus the first order decay law of germ nuclei degenerated to an impulse function and the overall rate of conversion was limited solely by the rate of nuclei growth. The nucleation thus lost its character as a dynamic process in the time scale of the growth in this study.
J. Phys. Chem. B, Vol. 105, No. 40, 2001 9783
Figure 3. Measured data (O) and description given by the N/NG model (s).
Let us consider the model (22) with sudden and complete nucleation
-ln(1 - R(t)) )
Kg V0
(
A
∫0t ∫yt(βx + gT )1/2 ×
(
exp )
(∫
A gK gN 0 V0
) )
0
Eg
R(βx + T0)
1 × (βx + T0)1/2
t
0
(
exp ) A°
(∫
m
dx N0δ(t) dy
Eg R(βx + T0)
1 × (βx + T0)1/2
) )
m
dx
t
0
(
exp -
Eg R(βx + T0)
) )
m
dx
(23)
where δ(t) is the Dirac delta function. Equation 23 is mathematically analogous to (2a,b), which explains the nearly equal performance of the models (2a) and (23) in this case study. The only slight difference arises from the appearance of (17) in (23). The parameter estimates for (23) obtained the optimal values A° ) (1.4 ( 0.2) × 108 and Eg ) 100 ( 2 kJ/mol. The measured data and the solution of the model with the optimal parameters are presented in Figure 3. These reduction kinetic investigations support the N/NG reduction mechanism for the supported chromia catalyst. The model of two-dimensional nuclei growth, alongside the concept of extended reduction degree, gives a good description of the experimental results. Furthermore, the two-dimensional model outperformed the three-dimensional model. The two-dimensional growth of converting centers is highly acceptable for the supported well-dispersed catalyst with Cr6+ on the surface. The chromia exists on alumina in different amounts as monomeric, dimeric, and polymeric species, as a function of loading.33 We suggest that these species do not behave as independent entities under reduction, but the reduction advances as growing centers. The model that represents the homogeneous (random) reduction of species, namely the first order rate law (random nucleation) was clearly inadequate to describe the experimental data. The slight deviation between the N/NG model and the measured data could be due to the simple reduction rate law that is
9784 J. Phys. Chem. B, Vol. 105, No. 40, 2001 incorporated in the nuclei growth law. This rate is modeled to be proportional to the rate coefficient and the gas phase hydrogen concentration, but the real reaction rate is probably more complex. Nucleation/Nuclei Growth Models. As has been seen, the N/NG concept in fact comprises a class of models with different limiting cases. The inherent assumptions need to be known when the models are applied. The typically applied forms (2a,b) originate as approximations from the isokinetic case or from the general form (22) with the assumption of instantaneous nucleation. When reduction kinetic modeling based on dynamic hydrogen consumption data is undertaken, the model needs to cover the whole range of conversionssa challenging proposition. In the topochemical gas-solid reaction, the rate is strongly a function of the sample history and the limiting forms of the model may lead to misinterpretations. A convenient methodology would be to start the parameter estimation with a sufficiently general model and then to conduct model reduction if unnecessary dynamics appears in it. The equations (2a,b) may be used for fast characterization of ideal TPR patterns. The same goes for the Kissinger and Friedman analyses, which are the fastest ways to extract the apparent activation energy from the TPR data. The numerical integration of general forms of the N/NG model should not constitute an obstacle because the integrands arising in applications are typically smooth and regular functions. If we consider general applicability of nuclei growth models to kinetics of reduction, the three-dimensional growth model is likely feasible only for reduction of certain bulk oxides. The applicability of two-dimensional nuclei growth model is probably confined to reduction of supported oxide systems, where the oxide material has a tendency to form large monolayer clusters. The amount of the oxide on the support might also be meaningful. Conclusions This case study of kinetic modeling of reduction of supported oxide (CrOx/Al2O3) was carried out with H2-TPR data. Typical kinetic models were tested and the activation energies of reduction were calculated by known techniques. It was observed that an experimental data set with three different heating rates was needed to distinguish between the kinetic models. The wellknown Avrami-Erofeyev model was found to describe the experimental data best and the foundations of the model were then investigated in detail. A revised form of the nucleation and nuclei growth model was set up by omitting the isokinetic assumption. The temperature dependencies of the nucleation and the nuclei growth were considered independently of each other. The numerical solution of the model was implemented in MATLAB6 and parameter estimation was carried out. The test system was best described in terms of the instantaneous nucleation and two-dimensional nuclei growth. An activation energy of 100 kJ/mol was obtained with this model. We suggest that this nuclei growth reduction model may suitably describe the reduction of supported monolayer catalysts.
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