680
Ind. Eng. Chem. Res. 2002, 41, 680-690
Kinetics of the Total Oxidation of Methane over a La0.9Ce0.1CoO3 Perovskite Catalyst Ro´ bert Auer and Fernand C. Thyrion* Chemical Engineering Institute, Universite´ Catholique de Louvain, 1, Voie Minckelers, B-1348 Louvain la Neuve, Belgium
The kinetic study of the total oxidation of methane over a La0.9Ce0.1CoO3 catalyst has been achieved. A discrimination among the five multistep rate-controlling models, preselected in a preceding study (Stud. Surf. Sci. Cat. 2001, 133, 599-604), was carried out on the basis of experimental data from an integral fixed-bed reactor using a sequential experimental design. According to the selected model, the overall rate is controlled by the rate of three elementary steps, i.e., (a) active surface oxygen formation, (b) the reaction between surface oxygen species and gas-phase methane, and (c) the desorption of reaction products. The kinetic constants of the model were determined using weighted regression. The experimental conversions could be predicted with an average error of lower than 5%. Both reaction products inhibit the reaction, although H2O causes much stronger inhibition. When the partial pressure of water in the feed is significantly higher than the one produced in the reaction, the desorption becomes an equilibrium process and the overall rate is determined only by the two elementary steps a and b. 1. Introduction
Table 1. Kinetic Models Reported in the Literature: Effect of the Reactants
Catalytic combustion is an important environmental technique that has the potential to reduce NOx emission.1 Perovskite-type oxides, represented by the general formula ABO3, are very attractive candidates for the total oxidation; many recent research papers deal with their preparation, deposition on a support, and thermal and poisoning resistance, using methane combustion as the probe reaction.2-7 Although a few reaction kinetic models for the total oxidation of methane over different perovskite catalysts have been proposed in the literature,8-14,28-33 a comprehensive kinetic study has not yet been presented. The mechanism of methane combustion over Pd catalyst is supposed to proceed through mild oxidation in a parallel, consecutive reaction scheme.15 Nevertheless, by far our knowledge, mechanistic studies for identifying intermediary species formed over perovskite catalysts has not yet been published. The absence of CO or other partial oxidation products formation over a wide range of investigated perovskites was reported.6-19 A first discrimination among 23 rival kinetic models has been published previously,24 where account was taken of the influence of reactant molecule partial pressures. The present paper is devoted to the final discrimination among models retained from the preceding study by means of sequential experimental design (SED). Then the selected model is completed with terms related to the inhibition caused by the product molecules. 1.1. Kinetic Models Proposed in the Literature. In Table 1 rate equations which take into account the influence of the reactants are presented. Two EleyRideal (ER) models, either with dissociative (ED) or nondissociative (EN) adsorption of oxygen, were proposed. Arai et al.10 put forward a parallel mechanism, * To whom correspondence should be addressed. E-mail:
[email protected]. Fax: +32 (10) 47.23.21.
code
catalyst
rate equation
perovskite8,9 CuO/Al2O328
r)
EN
NiO/Al2O329
r)
LAT
perovskite10,11,12
r)
ED
LD
Pt/Al2O310
S1
perovskite13 Pt/Al2O330
MVKn
perovskite14
r)
r)
r)
ksrPCH4xKO2PO2 1 + xKO2PO2 ksrKO2PCH4PO2 1 + K O 2 PO 2 ksrPCH4xKO2PO2 1 + xKO2PO2
+ kLPCH4
ksrKCH4PCH4xKO2PO2 (1 + KCH4PCH4 + xKO2PO2)2 k1k2PO2PCH4 k1PO2 + νk2PCH4 k1k2PO2nPCH4 k1PO2n + νk2PCH4
which states that CH4 can react with oxygen either from the surface or from the lattice; the corresponding rate equation in this case is indicated by LAT. A LangmuirHinshelwood-Hougan-Watson (LHHW) model with dissociative adsorption of oxygen (LD) and two Marsvan Krevelen mechanisms where the reaction is first order (S1) and nth order (MVKn) were also proposed. A few rate equations in which the influence of the reaction products was taken into account were also reported (Table 2). Besides a power law and a semiempirical equation, an ER model assuming dissociative oxygen adsorption and occupation of active sites by the product molecules was proposed. The inhibition was
10.1021/ie0104924 CCC: $22.00 © 2002 American Chemical Society Published on Web 01/18/2002
Ind. Eng. Chem. Res., Vol. 41, No. 4, 2002 681 Table 2. Kinetic Models Reported in the Literature: Effect of the Reaction Products catalyst
rate equation
PdO/Al2O331
rCH4 ) k(PCH4)1(PO2)0.1(PH2O)-0.8
PdO/Al2O332
rH2O )
CuO/Al2O333 rCH4 )
kPCH4 1 + KH2OPH2O k(KO2PO2)0.5PCH4 1 + (0.5KO2PO2)0.5 + KCO2PCO2 + KH2OPH2O
attributed only to H2O over palladium catalysts,31,32 whereas either CO2 or H2O inhibited the reaction over the metal oxide catalyst.33 1.2. Kinetic Models Considered in This Work. The total oxidation is supposed to undergo the classical sequence of elementary steps adapted to the case of methane (eqs 1-4), i.e.,
(0) An eventual weak adsorption of methane on the catalyst surface k0
CH4 + [ ] 79 98 [CH4] k 0b
(1)
(1) A nondissociative adsorption of oxygen on active vacant sites, [ ], followed by its fast transformation to active oxygen species [O]* k1
k′1 fast
98 [O2] 98 [O]* O2 + [ ] 79 k 1b
(2)
Because there is no experimental evidence about the nature of [O]*, which can be [O2-], [O22-], [O-], or [O2-] according to Golodets,25 the stoichiometry of step 1′ is not specified.
(2) The reaction between CH4 and the active oxygen forming an intermediary [I] that is quickly converted into reaction products [P], H2O, and CO2 +[O]* k2′ fast
k2
CH4 + [O]* 98 [I] 98 [P]
(3)
anism of oxygen adsorption, which could be dissociative or nondissociative, and to the stoichiometry of the ratedetermining step. An overview of the models was presented elsewhere,24 while the corresponding rate equations were derived following the approach of the rate-controlling step of Yang and Hougan.26 (II) Six multistep rate-controlling (MSRC) models developed for catalytic oxidation by Golodets25 denoted by S1-S6 have also been considered (Table 3). The corresponding rate equations are derived by assuming that the elementary steps 1-3 (eqs 2-4) are quasiirreversible; viz., the forward rates of the elementary steps are substantially higher than the backward rates. For the development of the rate equation of the model S1 given in the table, which is formally equivalent to the one based on the oxidoreduction mechanism proposed by Mars and van Krevelen,27 the following assumptions are made: (a) Methane reacts from the gas phase. (b) The adsorption of oxygen is nondissociative. (c) The elementary steps are of first order. (d) The transformation of the adsorbed oxygen into active surface species (step 1′ in eq 2) is fast. (e) The desorption of the reaction products is fast. As a consequence, the catalyst surface is supposed to be partially covered by the active oxygen species [O]* while the coverage by the other molecules is negligible. Thus, the site balance is given by
θ + θO ) 1
where θO indicates the surface coverage by [O]*, whereas θ denotes the fraction of active vacant sites. The rates of oxygen adsorption and methane reaction are defined by eqs 6 and 7.
-
-
k3
98 P + [ ] [P] 79 k 3b
(4)
The forward rate constants of the elementary steps are marked by k0 to k3, while the ones for backward rates are identified by the supplementary index b. Various kinetic models could be derived based on the aforementioned sequence of elementary steps. They are classified according to the number of rate-limiting steps, i.e., (I) single-step and (II) multistep rate control. (I) Models of a single rate controlling step fall into two categories: (I.1) ER models in which methane is assumed to react from the gas phase and (I.2) LHHWtype mechanisms involving methane that is weakly adsorbed on the catalyst surface (eq 1) prior to the reaction with surface oxygen species. These models were further divided into groups according to the ratedetermining step, i.e., the adsorption of reactants, the surface reaction, or the desorption of products. Subcases within the groups were defined according to the mech-
dPO2 dt
dPCH4 dt
) k1PO2θ
(6)
) k2PCH4θO
(7)
The overall rate expressed as a function of the gasphase partial pressure of CH4 and O2 is written as
r)-
(3) Product desorption
(5)
dPCH4 dt
)-
1 dPO2 ν dt
(8)
where ν ) 2 is the stoichiometric coefficient. The substitution of eqs 6 and 7 into eq 8 yields
1 r ) k2PCH4θO ) k1PO2θ ν
(9)
Equation 9 is an algebraic equation with two unknown variables (θ and s). The exact solution requires the use of the site balance (eq 5). After having resolved the equation system, one obtains the rate as
r)
k1k2PO2PCH4 k1PO2 + νk2PCH4
(10)
The other MSRC rate equations (S2-S6) were derived on the basis of other assumptions given below. The changes in the aforementioned elementary steps, the rate-controlling steps, and the site balance, which lead to the rate equations, are summarized in Table 3.
Ind. Eng. Chem. Res., Vol. 41, No. 4, 2002
νk2PCH4
k 1 PO 2
+
k1PO2 + νk2KCH4PCH4
r) r ) k2KCH4PCH4θO ) 1/νk1PO2θ (0) CH4 + ( ) S (CH4) dual site S6
θ+θO+θCH4)1
[
k1k2PO2KCH4PCH4
x(
2νk2PCH4
)
2
k 1 PO 2
k1PO2 1 r ) k1PO2 1 + + ν 2νk2PCH4 r ) k2PCH4θO2 ) 1/νk1PO2θ (2) CH4 + 2[O] f [I] S5
θ + θO ) 1
k2PCH4
2k1PO2
r) r ) k2PCH4θO ) 1/νk1PO2θ2 (1) O2 + 2[ ] f 2[O]* S4
θ + θO ) 1
) k1 P k′1 O2 1-
( k1PO2 + νk2PCH4
k1k2PO2PCH4
r) r ) k2PCH4θO ) 1/νk1PO2θ ) 1/νk′1θθO (1′) [O2] + [ ] f 2[O] slow S3
θ + θ O + θO 2 ) 1
νPk1k2 PO2PCH4 k3 k1PO2 + νk2PCH4 +
k1k2PO2PCH4
r) θ + θ O + θP ) 1 r ) k2PCH4θO ) 1/νk1PO2θ ) νP/νk3θP (3) [P] f P + [ ] slow S2
k1k2PO2PCH4
θ + θO ) 1 r ) k2PCH4θO ) 1/νk1PO2θ S1
change in the elementary steps code
Table 3. Description of the MSRC Models
rate-controlling steps
surface coverage site balance
r)
k1PO2 + νk2PCH4
rate equation
]
(k1PO2 + νk2PCH4 - xνk2PCH4(νk2PCH4 + 4k1PO2))
682
(S2) The product desorption step 3, is not fast; as a consequence, a substantial part of the surface is occupied by the product(s). (S3) The transformation of [O2] into [O]* (step 1′) is not fast, so the two types of oxygens coexist on the surface. (S4) Dissociative adsorption of oxygen occurs in step 1. (S5) The reaction step 2 takes place between CH4 and two molecules of active [O]* species. (S6) Methane adsorbs on the catalyst surface (step 0) via equilibrium by a dual-site mechanism. 1.3. Preliminary Model Discrimination Study.24 Preceding works20-23 revealed that, among a series of perovskite-type oxides described by the general formula La1-xCexCoyMn1-yO3, La0.9Ce0.1CoO3 exhibited a fairly good activity for methane combustion and resistance to SO2 poisoning. The first discrimination among 23 plausible kinetic models was based on results obtained from two experimental studies, i.e., kinetic data analysis of integral reactor data and an initial rate study carried out in a Micro-Berty reactor. (1) In the fixed-bed reactor experiments, three parameters, i.e., the inlet molar fraction of methane and oxygen, yCH4in (0.39-3.0% by volume), yO2in, and the space time, W/FCH4in (0.044-0.35 g‚s/µmol), were varied according to a central composite design at five temperature levels (360, 390, 420, 460, and 500 °C), while the total pressure was kept constant at 1 bar. After the kinetic constants of the rival models had been estimated, the model adequacy was verified by a lack of fit (F) test and the 95% confidence interval of the parameters was estimated. Models exhibiting a lack of fit or whose parameter(s) were nonsignificantly different from zero or negative were rejected. Only three MSRC models, S1-S3, as well as a LHHW model, which assumes nondissociative oxygen adsorption and surface reaction control, passed these statistical verifications. However, the LHHW model had to be rejected as well because ln(KO2) and ln(KCH4) decreased against 1/T, which is opposite to the van’t Hoff law. On the other hand, the rate constants of the models S1-S3 showed a good agreement with the Arrhenius law. (2) In the initial rate study, three variables, the space time (0.14-1.26 g‚s/µmol), the total pressure (5-13.3 bar), and the temperature (320-360 °C), were modified systematically, while the inlet composition, yCH4 ) 1.0% by volume, and yO2 ) 1.16 % by volume as well as the agitation speed of the Berty reactor (3000 rpm) were kept constant. According to the method,26 the forms of theoretical and experimental initial rate curves vs total pressure were compared. The results suggested that, among the models retained from the fixed-bed reactor analysis, S2 and S3 are more probable than S1. 1.4. SED. The application of SED for model discrimination and precise parameter estimation in the field of kinetic modeling has been reviewed in several papers.34-36 Although, the essential features for discrimination will be recalled below. The iterative procedure of SED for model discrimination (Figure 1) consists of the following sequence: (1) After a number of experiments, (2) the parameters of the rival models have to be updated, then (3) the model probabilities can be estimated, and finally (4) the operating variables of the next experiment are selected by a divergence criterion. Steps 1-4 must be repeated
Ind. Eng. Chem. Res., Vol. 41, No. 4, 2002 683
where m is the number of models, pi is the number of parameters in model i, and si2 is the unbiased estimate of the error variance, obtained as n
(Xi - X0i )2 ∑ i)1
si2 )
(16)
n - pi
The probability density38 of model i after n experiments (pdi,n) is obtained according to the Bayes theorem (eq 17). The calculation of the relative a posteriori probability (eq 18) of the ith model after n experiments, denoted by πi,n (%), necessitates the input of the a priori value, πi,n-1.
pdi,n ) Figure 1. Algorithm for model discrimination using SED.
until the ending condition is fulfilled or, in the worst case, no further improvement can be attained. In what follows, the elements of the design will be described. Parameter estimation is achieved by integral kinetic data treatment. To obtain the predicted methane conversion, X0, the plug-flow differential equation (eq 11) is numerically solved for the boundary condition (eq 12). The parameters are estimated by nonlinear regres-
dX0 )r d(W/FCH4)
(11)
X00 (W/FCH4 ) 0) ) 0
(12)
sion (see section 1.5). The obtained residual sum of squares associated with model i (RSQi) are passed to the probability calculation. Probability calculation requires another input, viz., the estimated pure error mean square, se2, which is calculated by eq 13 using repeated experimental conversion, where nrep is the number of repeated experi-
se2 )
nrep - 1
(13)
ments and X h rep is the average repeated conversion. The experimental conversion (%) is defined as
X ) 100(PCH4in - PCH4out)/PCH4in
(14)
If repeated data were not available, se2 would be replaced by the pooled estimate of variance and lack of fit,37 obtained as m
js2 )
(n - pi)si2 ∑ i)1 m
n - pi ∑ i)1
x2 × 3.14 × [se2 + si2] πi,n )
(15)
exp
{
}
(X - Xi(n))2 2[se2 + si2]
πi,n-1pdi,n
(17)
(18)
m
πj,n-1pdj,n ∑ j)1 Initially, all probabilities are set to be equal, viz.,
πi,0 ) 100/m
(19)
where m is the number of rival models. Ending Condition. If the relative probability of one model exceeds an arbitrarily selected value of 99%, the discrimination procedure could be stopped. Otherwise, πi,n values are sent to the divergence calculation. Divergence Calculation. A grid is defined over the feasible operating variable space. The next, (n + 1)th, experiment is selected in the grid point, g, where the Box-Hill38 divergence criterion (eq 20) is maximal. m-1
D(g) )
m
∑ ∑ πi,nπj,n i)1 j)i+1
(
(X0i (g) - X0j (g))2
nrep
∑ Xh - Xh rep h)1
1
{
(sp,i2(g) - sp,j2(g))2
1 2
+
(se2 + sp,i2(g))(se2 + sp,j2(g)) 1
+ 2
se + sp,i (g)
2
se + sp,j2(g)
)}
(20)
The criterion takes into account the differences in prediction, the probabilities, and the estimated variance corresponding to the rival models. The estimation of predicted variance (eq 21) for the nonlinear model i in the grid point g, sp,i2(g), needs the numerical estimation of the Jacobian matrix (Ji), which will be described below.
sp,i2(g) ) vi(g)T(JiTJi)-1vi(g) se2
(21)
where the vi(g) vector is the gth line of the matrix. The Box-Hill criterion was applied in an experimental study,39 while the use of different criteria was reported in other papers. The so-called T criterion37,40 (eq 22) takes into consideration the differences in prediction and the variances but not the model probabilities. Other criteria,41,42 i.e., eqs 23 and 24, measure only the divergences in predicted values which, besides
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Ind. Eng. Chem. Res., Vol. 41, No. 4, 2002
conversion, might be space time or rate depending on the mechanism of kinetic analysis. m-1
m
∑ ∑ (Xi(g) - Xj(g)) i)1 j)i+1
T(g) )
(m - 1)(mse + m-1
∑ ∑ i)1 j)i+1
D(g) )
m-1
D(g) )
m
(
W
(22)
Fiin
si (g)) ∑ i)1
(g) -
2
W Fjin
(g)
)
m
(24)
n
(25)
experimental points. When the residuals (eq 26) obtained from unweighted regression (eq 25) are not equally distributed with one operating variable, e.g., temperature, a weighted regression procedure as illustrated below should be applied.
resi ) Xi - X ˆi
(26)
Note that in what follows, the errors are assumed to be uncorrelated. The mean square of the residuals, sr2, at a given temperature is
2
sr )
∑ i)1
2
n(T) - 1
(27)
where n(T) is the number of experimental points at T and res is the average residual. The mean square (sr2) vs T has to be fitted e.g., by a polynomial, and then reestimated as a function of T. The weight related to the temperature is obtained as
wi )
1
xsr2(T)
Then, the weighted sum of squares to be minimized becomes
(31)
(32)
where b is a vector of estimated parameters of length p, which indicates the number of parameters; s2 represents the error variance (eq 16). J in eq 32 is the Jacobian matrix of size n × p, where p is the number of parameters. The columns of the matrix are obtained as the partial derivatives of X with respect to parameter bk, i.e.,
J bk )
∂X ˆ ∂bk
(33)
To estimate Jbk, the following differential equation has to be solved:
d(∂X/∂bk) d(W/FCH4)
)
∂rate ∂bk
(34)
where the right-hand side of eq 34 might be analytically determined, e.g., by symbolic calculus. With boundary conditions such as those in eq 35, the predicted value is obtained at the experimental space-time value by eq 36.
∂X (W/F)0) ) 0 ∂bk
(35)
[( ) ( ) ] W W ) FCH4 FCH4in
(36)
exp
The variances of parameters, var, are situated on the diagonal elements of the V(b) matrix, whereas the covariances between parameters, cov, are found in the nondiagonal elements. The correlation coefficients, which can take up values in the range 0-1, between bi and bj parameters are given as
cov(bi,bj)
|
xvar(bi) var(bj)
(37)
The confidence interval, ci, of parameter bj was estimated while keeping other estimates constant in their optimal value:
ci(bj) ) bj ( var(bj) t(n-p;1-R/2)
n
wi(Xi - X ˆ i)2 f min ∑ i)1
)]
V(b) ) (JTJ)-1s2
cor(bi,bj) ) | (28)
Ea 1 1 R T Tavg
Correlation Coefficient and Confidence Interval of Parameters.47 The variance-covariance matrix of the parameters for models that are nonlinear in parameters is approximated as
∂X ∂X ˆ ) ∂bk ∂bk
n(T)
(resi - res)
[ (
(23)
Discrimination by probabilities was employed in several studies,37,39,41 while another methodology,35 in which models could be rejected by F or Bartlett’s χ2 lack of fit tests, was used elsewhere.40,42 The model selection procedure, described above, is suitable for single reactions, i.e., the combustion of methane to form H2O and CO2, whereas SED for simultaneous reactions was treated by other authors.43,44 1.5. Kinetic Data Treatment. Weighted Regression.45 Nonlinear regression is required for estimating the parameters wherein the residual sum of squares (eq 25) has to be minimized, where n is the number of
∑ i)1
(30)
Reparametrization of the Kinetic Parameters. To get less correlated parameter estimates, the use of reparametrized temperature scale46 is recommended, i.e.,
k ) k0ref exp -
2
∑ ∑ |ratei(g) - ratej(g)| i)1 j)i+1
(Xi - X ˆ i)2 f min
resiw ) xwi(Xi - X ˆ i)
2
m
2
Then weighted residuals are obtained as
(38)
(29) where the quantity of t(n-p;1-R/2) is the tabulated
Ind. Eng. Chem. Res., Vol. 41, No. 4, 2002 685
Student distribution value with n - p degrees of freedom and R risk selected as 0.05. Calculation Methods. The nonlinear least-squares minimization of eqs 25 and 29 was achieved by a largescale algorithm based on the interior reflective Newton method. The differential eqs (11) and (34) were resolved with an algorithm that applies an explicit RungeKutta(4,5) formula, the so-called Dormand-Prince pair. All computations have been achieved by MATLAB software. 2. Experimental Section Catalyst Preparation and Characterization. The catalyst was prepared by the citrate method. The amorphous precursor was calcined at 700 °C for 5 h in air. The obtained powder was pressed, crushed, and sieved; a fraction of 40-100 µm was used. The resulting pellets had a 10.2 m2/g Brunauer-Emmett-Teller (BET) specific surface area. The catalyst showed a welldefined rhombohedral perovskite structure in the X-ray diffraction pattern. The catalyst was pretreated prior to the experiments under a flow of 100 cm3/min air at 600 °C for 2 h. Neither structural modification nor loss of surface area was observed after the catalytic tests.20 A “W”-shaped quartz reactor of internal diameter 10 mm, with a built-in porous frit no. 1 to support the catalyst load, was used. It was fed in a downflow direction. A thermocouple was placed at the bottom of the catalyst bed, whereas another one allowed the measurement of the gas-phase temperature 5-10 mm above the bed. The isothermal operation was ensured by a cylindrical ceramic oven (750 W) equipped with a regulated upflow of air, which allowed maintenance of a maximum difference of 1 °C between the two thermocouples. Furthermore, the catalyst was diluted by carborundum. An Omron E5CK-T digital controller regulated the temperature of the catalyst bed. Gases N2 (N28), O2 (N25), CH4 (N30), CO2 (N25), and dried air (N25) supplied by Air Liquide were used without any further purification. The flow rate of the gases was controlled by mass flow controllers. Water vapor was introduced by passing a flow of nitrogen via a humidifier that was maintained under isothermal conditions. The concentration of methane was measured by a flame ionization detector gas chromatograph using a Chromosorb 102 (2 m, 80/100 mesh) packed column. A Chromjet integrator from Spectraphysics made the analysis and controlled the automatic injections. 3. Results and Discussion Preliminary experiments and calculations ensured that under the selected operating conditions intrinsic kinetic data could be obtained. A calculation spreadsheet developed by EUROKIN48 allowed the verification of plug-flow operation and the absence of heat- and mass-transfer limitations. The absence of the limitations has been verified also by experimental tests recommended elsewhere.34 The presence of CO in the effluents was never detected (Neotronics CO101 detector). The catalyst showed a very stable operation because neither loss of activity nor structural modification of the catalyst was observed after 100 h on stream. The estimated kinetic parameters obtained from the preceding study24 were further tested. All models containing adsorption equilibrium constants, i.e., the LHHW and ER models as well as a MSRC model (S6), could be
Figure 2. Position of the experimental points on the grid at 390 °C. P1-P5: preliminary experiments; 1-7, sequentially selected experiments. Table 4. Experimental Conditions CH4 flow oxygen source total flowb temperature levels catalyst weight carborundum weight catalyst + diluent L a
0.769 cm3/min (STP)a O2/dried air 19.2-192 cm3/min (STP) 360, 390, 420, 460, and 500 °C 0.1 g 0.04 g 40-100 µm
STP ) standard temperature and pressure. b Balance: N2.
Table 5. Lack-of-Fit (F) Test of the MSRC Models Fcalc ) si2/se2 T (°C)
se2
Fcrit
S1 (p ) 2)
S2 (p ) 3)
S3 (p ) 3)
S4 (p ) 2)
S5 (p ) 2)
360 390 420 460 500
0.22 0.18 0.66 0.46 1.08
8.7 8.8 8.7 8.9 9
0.81 7.00 5.25 38.2 62.8
0.19 0.88 0.93 0.75 0.82
0.74 5.25 3.62 16.4 20.9
0.64 5.78 4.39 33.5 60.2
0.74 6.44 4.97 35.1 59.6
rejected because the values of ln(K) for oxygen or methane did not increase or in some cases even decreased against 1/T, in violation with the van’t Hoff law. On the other hand, the kinetic constants of the models S1-S5 obeyed the Arrhenius law. Therefore, even though some lack of fit for models S4 and S5 was revealed at the two highest investigated temperatures, these models are also taken into account for the final discrimination. The first goal of the present study is to select the most appropriate model among those of S1S5 at five different temperatures. 3.1. Model Selection by SED. The space time was fixed at 0.191 g‚s/µmol by maintaining the weight of the catalyst, diluted by carborundum and the flow of methane. An experimental grid of yCH4in and yO2in (Figure 2) was built, wherein the values were varied at six levels. The other experimental conditions are summarized in Table 4. After five preliminary experiments, marked by P1-P5 in Figure 2, were carried out, the iterative procedure (section 1.5) was followed. The pure error mean square (eq 14), required for probability and divergence calculations, was estimated for the different temperatures by means of repeated conversion values obtained from the previous study24 (Table 5). The location of the next experiment was chosen by the Box-Hill divergence criterion (eq 20) because faster convergence in discrimination was expected because of weighing by model probabilities; however, the other
686
Ind. Eng. Chem. Res., Vol. 41, No. 4, 2002
Figure 3. (a) Progress of model probabilities at 390 °C. (b) Evolution of the probability of model S2 at different temperatures.
Figure 4. (a) Arrhenius plot for model S1. (b) Arrhenius plot for model S2.
criteria (eqs 22-24) were also evaluated. In the first two iterations, the same grid point was selected by all criteria, but then different locations were suggested, probably because of the contribution of probabilities in eq 20. The progress of model probabilities with a number of experiments at 390 °C is shown in Figure 3a. Initially, all probabilities were set to 20%. The value obtained after the five preliminary experiments is indicated at experiment no. 1; sequential experiments are numbered onward from 2. A competition between models S2 and S3, which have been found to be most probable according to the initial rate study,24 can be observed, while the probability of the other models declines rapidly. Finally, the model S2 emerges as the best. Figure 3b shows that model S2 is appropriate at all temperatures. A relatively small number of sequentially selected values led to discrimination; this number decreases with temperature. Further statistical and thermodynamical tests justified the adequacy of model S2. The F test35 revealed a very strong lack of fitting for all models except S2 at 460 and 500 °C (Table 5). Indeed, when the calculated ratio (Fcalc) si2/se2 exceeds the critical F value (Fcrit), the model i could be rejected with 95% certainty. The estimated experimental variance (Table 5) for model S2 is substantially lower than the ones obtained for other models at all temperatures. Furthermore, the temperature dependence of the rate constant k1 for models S1, S4, and S5 was not conformed to the Arrhenius law, as is shown in Figure 4a as an example for S1, while for S2 (Figure 4b) and S3, a correct behavior was observed. Note that a reparametrized temperature scale was applied. 3.2. Estimation of the Kinetic Parameters of Model S2. To estimate the preexponential factors, k0i, and activation energies, Eai (where i ) 1-3), of the elementary steps of the selected model, data available
Table 6. Kinetic Parameters and Confidence Intervals for Model S2a value ci (95%)
r)
k0ref1
k0ref2
k0ref3
E a1
Ea2
E a3
347.4 (121,7
71.97 (11.26
2.316 (0.343
82.6 (32.4
131 (15.2
44.7 (5.9
k1k2PO2PCH4
; νPk1k2 PO2PCH4 k3 Ea 1 1 k ) k0ref exp ; R T Tavg
k1PO2 + νk2PCH4 +
[ (
)]
Tavg ) 699.1 K
[k01, k02] ) µmol/(s‚gcat‚bar), [k03] ) µmol/(s‚gcat), [Ea1-3] ) kJ/mol.
from the SED experiments were used. The conversion was averaged over repeated experiments. Experimental data obtained at 360 °C were omitted because poorer precision in the gas chromatographic analysis was obtained than at higher temperatures because of the low methane conversion level (1.0-2.9%). Using these data would have increased the uncertainty of the estimation. Weighted regression (eqs 27-30) was applied because ˆ i) increased with temperature (see the residuals (Xi - X ref 49 for more details). The calculated reparametrized kinetic constants (eq 31) along with their 95% confidence interval are tabulated in Table 6. Very good agreement between experimental and predicted conversion was obtained as indicated by the parity plot in Figure 5. The error bars in the figure, i.e., the 95% confidence interval of the prediction (cip) and the experimental error interval (ciexp), were estimated by eqs 39 and 40, respectively.
(
cip ) t n-p,1-
)x
R 2
sp2 n-p
(39)
where sp,i2 is calculated by eq 21 and t is the tabulated
Ind. Eng. Chem. Res., Vol. 41, No. 4, 2002 687
Figure 5. Parity plot for model S2. W/FCH4in ) 0.19 g‚s/µmol, and T ) 390-500 °C. Error bars: horizontal, 95% confidence interval of the experimental error; vertical, 95% confidence interval of the predicted conversion.
Student distribution value with n - p degrees of freedom and an R risk selected as 0.05.
)x
se2 nrep - 1
R ciexp ) t nrep-1,12
(
(40)
where se2 is estimated by eq 14 and nrep is the number of repeated experiments. The average percent error, PE (eq 41), amounted to 5%, while the average value of the correlation coefficients among the estimated parameters amounted to 0.35. Execution of SED experiments for precise parameter estimation34,35 would perhaps lead to the obtainment of less correlated parameters of smaller confidence intervals.
PE )
100 n
n
∑ e)1
|Xe - X ˆ e| Xe
(41)
To conclude, the kinetics is driven by a MSRC mechanism (model S2, Table 1), where besides the oxygen adsorption and the reaction steps, the desorption of the products controls the overall reaction rate. Hence, the influence of the product molecules on the kinetics will be investigated. 3.3. Effect of H2O and CO2 on the Reaction Kinetics. Experiments with two different oxygen concentrations, either in deficit or in excess with respect to the stoichiometry, were made, keeping the space time and methane concentration constant (W/FCH4in ) 0.398 gcat‚s/µmol and yCH4in ) 1.02% by volume). The inlet concentration of water and carbon dioxide was varied at three levels. In one experiment, both compounds were fed together at their highest concentration as well. A 2 times higher water concentration than that of CO2 was applied in order to match the ratio in their formation by the total oxidation of methane. Figure 6 shows that CO2 inhibits the methane combustion less strongly than H2O does; the conversion in the presence of CO2 is very close to the lower limit of the error bars (eq 40) of the experimental conversion obtained in the absence of product molecules in the feed stream. However, the inhibition by the two molecules is additive. To compare the inhibition by H2O and CO2 under the different inlet oxygen concentrations and different temperatures, the relative conversions, i.e., X/X0 were calculated. Methane conversion values obtained under
Figure 6. Effect of the reaction products on methane conversion. yCH4in ) 1.02% by volume, yO2in ) 0.84% by volume, and W/FCH4in ) 0.398 g‚s/µmol. Error bar: 95% confidence interval of the experimental error. Table 7. Methane Conversion (X0, %) in the Absence of CO2 and H2O at the Feed (W/FCH4in ) 0.398 gcat‚s/µmol and yCH4in ) 1.02 % by Volume) yO2in
390 °C
420 °C
460 °C
500 °C
0.84 8.02
6.32 10.2
12.8 18.7
28.5 42.6
37.2 68.1
the two oxygen feed concentrations in the absence of H2O and CO2 (X0) are presented in Table 7. This quantity is plotted against the inlet concentration of CO2 and H2O under a deficit (a and c) and under a large excess of oxygen (b and d) in Figure 7. The comparison of parts a and b of Figure 7 as well as parts c and d of Figure 7 reveals that modification in the oxygen concentration affects only slightly the inhibition. The somewhat stronger inhibition when using an excess of oxygen might be explained by the fact that more H2O and CO2 are formed because of the higher conversion. Moreover, the inhibition caused by H2O diminishes with the temperature, while the tendency is not clear for CO2. Kinetic Modeling. In the development of model S2, the desorption rate was assumed to be substantially higher than that of the readsorption of the products. However, the inhibition by the product molecules, as shown above, suggests that the readsorption is not negligible. The occupation of the surface by the two products appears to be independent because the inhibition was found to be additive (Figure 6). These pieces of information lead to the assumption that the inhibition arises because of the competitive adsorption between O2 and H2O as well as CO2 for the active vacant sites. Two elementary steps, the adsorption of oxygen and the reaction, control the rate (eq 9), while the desorption of H2O and CO2 is supposed to attain the equilibrium. The site balance is modified as
θ + θO + θW + θC ) 1
(42)
where θW and θC are the surface coverage by water and carbon dioxide, and they are expressed as
θW ) KWPWθ
(43)
θC ) KCPCθ
(44)
where KW and KC are the adsorption equilibrium constants for H2O and CO2, respectively.
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Figure 7. Relative inhibition due to the reaction products. yCH4in ) 1.02% by volume, and W/FCH4in ) 0.398 g‚s/µmol. For X0 values, see Table 7. (a, c) yO2in ) 0.84% by volume. (b, d) yO2in ) 8.02% by volume. Solid lines are fitted by eq 45.
The following rate equation can be derived by resolving the system of eqs 9 and 42-44, viz.,
r)
k1k2PO2PCH4 k1PO2 + νk2PCH4(1 + KWPW + KCPC)
(45)
Estimation of the Kinetic Constants. The adsorption constant of CO2 could not be determined because of the too small variation of the methane conversion with yCO2in (cf. Figures 6 and 7). A rough estimation of KW was carried out by means of an unweighted nonlinear regression over the data obtained from the SED study, and the experiments were carried out in the presence of water. The logarithm of the adsorption constant (KW) increases with 1/T, giving a good straight-line fitting (Figure 8) in agreement with the van’t Hoff law. This confirms the assumption that H2O desorption reaches the equilibrium under the examined reaction conditions. The predicted relative conversion (X/X0), indicated by solid lines in Figure 7, gives a satisfactory agreement with the experimental values. The adsorption enthalpy and entropy were estimated from the slope and intercept of the graph. However, a more detailed study would be useful because rather few experimental data on the inhibition are available. Precise parameter estimation would have required that the experimental variables, i.e., yCH4in, yO2in, and yH2Oin, were varied simultaneously. Moreover, the role of CO2 on the inhibition should be clarified.
Figure 8. Arrhenius plot of the estimated equilibrium constant of water adsorption [Kw] (bar-1).
In this investigation two limiting cases have been considered, i.e., a feed without water and carbon dioxide and a feed containing them at a significantly higher concentration than that produced by the methane oxidation. In the first case, desorption of the reaction products is significantly slower than their readsorption. In the second case, however, the high partial pressure of H2O resulted in the adsorption-desorption equilibrium being reached and eq 45 being suitable for predicting the rate. Although, a feed containing H2O and CO2 at a relatively low concentration level would allow investigation of the elementary steps separately. 4. Conclusions (i) The kinetics of the total oxidation of methane over a La0.9Ce0.1CoO3 catalyst reflects very accurately a
Ind. Eng. Chem. Res., Vol. 41, No. 4, 2002 689
steady-state oxidation mechanism. Three elementary steps with a first-order rate control the overall reaction rate, i.e., (a) active surface oxygen formation, (b) reaction between the surface oxygen species and methane from the gas phase to form an intermediary that is transformed to H2O and CO2 by fast reaction, and (c) product desorption. (ii) The corresponding kinetic model (S2) was selected by a discrimination from five SSO models, which were retained from 23 plausible models in a preceding study,1 using SED. This selection was confirmed by further statistical and thermodynamical tests as well as by the initial rate study. The model was able to predict the experimental conversion within a 5% average error. (iii) Methane conversion is inhibited by CO2 and H2O. H2O causes far stronger inhibition than CO2 does. The inhibition is independent from the inlet concentration of oxygen. (iv) When the partial pressure of water in the feed is significantly higher than the one produced by the reaction, the desorption becomes an equilibrium process. Only the two elementary steps then determine the overall rate, i.e., the adsorption of oxygen, which competes for vacant sites with H2O and CO2, and the reaction between active surface oxygen and methane from the gas phase. Acknowledgment The authors are indebted to J. Kirchnerova and B. Delmon for their valuable suggestions. We thank M. Alifanti for the catalyst preparation and characterization. This work was supported by a European contract (ENV4-CT97-0599). Nomenclature b ) parameter ci ) confidence interval CV ) confidence volume D ) divergence criterion for model discrimination Dis ) discrimination function (JDC) DP ) divergence criterion for parameter estimation Ea ) activation energy, kJ/mol est ) parameter estimation function (JDC) g ) gridpoint I ) intermediate J ) jacobian matrix k ) kinetic constant, µmol/(s‚gcat‚bar) k0 ) preexponential factor, µmol/(s‚gcat‚bar) ki ) set of kinetic parameters for model i m ) number of models n ) number of experiments ng ) number of gridpoints nv ) number of experimental variables npr ) number of preliminary experiments nrep ) number of repeated experiments p ) number of parameters P ) partial pressure, bar pd ) probability density function r ) rate, µmol/s‚gcat R ) resistances of the elementary steps or general gas law constant (8.31), kJ/mol‚K RSQ ) residual sum of squares s2 ) error variance sj2 ) pooled estimate of variance se2 ) pure error mean square sp2 ) predicted variance in divergence calculation sr2 ) mean square of the residuals t ) tabulated Student test value
T ) T divergence criterion (eq 22) T ) temperature, K v ) vector of partial derivatives of X ˆ with parameters W/F ) space time, gcat‚s/µmol of CH4 X ) experimental conversion, 100(PCH4in - PCH4out)/PCH4in, % X ˆ ) predicted conversion, % y ) volumetric fraction, %v Greek Letters R ) risk level for the confidence interval ν ) stoichiometric coefficient for O2 νp ) stoichiometric coefficient for products (H2O, 2; CO2, 1) π ) relative probability, % σ2 ) pure error variance θ ) ideal surface coverage Subscripts and Superscripts 0-3 ) index of elementary steps avg ) average b ) backward CH4 ) methane des ) desorption e ) index of experiment number est ) estimated i ) index of model in ) inlet k ) index of parameters O ) active surface oxygen O2 ) oxygen P ) product Abbreviations LHHW ) Langmuir-Hinshelwood-Hougan-Watson mechanism MSRC ) multistep rate control SED ) sequential experimental design
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Received for review June 4, 2001 Revised manuscript received October 17, 2001 Accepted October 24, 2001 IE0104924
