Ind. Eng. Chem. Res. 2008, 47, 5879–5891
5879
Single-Event Microkinetic Model for Fischer-Tropsch Synthesis on Iron-Based Catalysts Gisela Lozano-Blanco,† Joris W. Thybaut,*,† Karine Surla,‡ Pierre Galtier,‡ and Guy B. Marin† Laboratory for Chemical Technology, Ghent UniVersity, Krijgslaan 281, S5 B-9000 Ghent, Belgium, and Institut Franc¸ais du Pe´trole, B.P. 3, 69390 Vernaison, France
A single-event microkinetic (SEMK) model was developed for Fischer-Tropsch synthesis and applied to experimental data obtained on an iron-based catalyst in a temperature range from 523 to 623 K, total pressures from 0.6 to 2.1 MPa, and H2/CO inlet ratios from 2 to 6 mol/mol. The use of the single-event concept allowed a significant reduction of the number of adjustable parameters. The single-event pre-exponential factors were calculated based on statistical thermodynamics. The reaction enthalpies, as well as initial guesses for the activation energies, were obtained through the unity bond index-quadratic exponential potential (UBI-QEP) method. Ten activation energies of the kinetically relevant reaction families and four atomic chemisorption enthalpies remained as adjustable parameters, the latter corresponding to so-called catalyst descriptors. The SEMK model describes well the product distribution over a wide range of operating conditions with physically sound kinetic parameters. The reductive elimination toward n-alkanes and the β-hydride elimination involved in the formation of 1-alkenes with activation energies amounting to 117.8 and 96.3 kJ/mol are the two most kinetically significant steps determining the product distribution. In particular, the symmetry effects specifically accounted for by the single-event concept appeared critical in the interpretation of the deviations from the Anderson-Schulz-Flory distribution at low carbon numbers. Because of its fundamental character, the SEMK model developed here can be easily applied to describe Fischer-Tropsch synthesis over other catalysts. 1. Introduction Fischer-Tropsch synthesis is the conversion of synthesis gas, which consists of carbon monoxide and hydrogen, into “clean” transportation fuels and chemicals. Synthesis gas can be obtained from any carbonaceous material, preferentially containing hydrogen.1 Coal and natural gas are the most common sources to produce synthesis gas. Advantages of the Fischer-Tropsch hydrocarbons compared to crude oil derivatives are, e.g., the absence of sulfur, nitrogen, and the low aromatic content.2 The increasing prices and depleting reserves of crude oil, together with the necessity of monetizing “stranded” natural gas resources have renewed the interest in this technology in the last decades.1,3 Iron-based catalysts are preferred in commercial operations for Fischer-Tropsch synthesis when using coal-derived COrich syngas.1 The more elaborated kinetic models reported in literature for iron-based catalysts consist of fundamental models where some of the elementary steps are assumed as kinetically significant steps while the rest of elementary reactions are assumed as quasi-equilibrated.4–8 Also for cobalt catalysts, fundamental or detailed models have been proposed.9–11 In this work, none of the elementary reactions is assumed quasiequilibrated and the actual rate of every elementary step is calculated.12 The number of adjustable kinetic parameters is reduced by applying the single-event concept, which filters out symmetry effects from the kinetic coefficient.13 The so-called single-event kinetic coefficient remains and is constant for each reaction family. Hence, single-event microkinetic (SEMK) models are particularly useful to model reaction networks including homologous series of hydrocarbons such as FischerTropsch synthesis.12 * To whom correspondence should be addressed. Tel.: +32 92644519. Fax: +32 92644999. E-mail:
[email protected]. † Ghent University. ‡ Institut Franc¸ais du Pe´trole.
It is known that iron-based catalysts are difficult to model because phase changes may occur with varying operating conditions.14 The latter is known as extrinsic relaxation15 and corresponds to variations in the chemical or phase composition of the surface by the effect of reaction conditions. Then, the number of active sites may change for example at different hydrogen to carbon monoxide inlet ratios. In the experimental data used in this work, however, the catalyst, consisting of iron carbide and magnetite,16 exhibited a quasi-stable behavior and these phenomena could be neglected. In a previous publication,12 the construction of an SEMK model for Fischer-Tropsch synthesis was presented. This work presents the procedures to calculate and estimate the kinetic coefficients. In addition, the SEMK model simulations are discussed and compared to iron-based experimental results. 2. Procedures 2.1. Operating Conditions, Catalyst, and Products. Experiments were performed in a tubular fixed-bed reactor using a commercial precipitated iron catalyst at steady-state under isothermal and nondeactivating conditions.16 The operating conditions studied vary in the following ranges: T ) 523-623 K, ptot ) 0.6-2.1 MPa, H2/CO inlet ratio ) 2-6 mol/mol, 0 ) 9.2-82.0 kg s/mol. The total number of investigated W/FCO conditions was 90 where the conversion of carbon monoxide ranged from 12 to 100%. The detailed experimental program, description of the experimental setup, and characterization of the catalyst have been published elsewhere.14,16,17 The tubular reactor (0.27 m in length and 0.021 m inside diameter) was loaded with 5 g of catalyst. The mean diameter of the catalyst pellets was 0.4 mm. For the dilution of the catalyst bed 50 g of steatite of aluminum oxide of the same diameter size was used. The catalyst used was a commercial, precipitated, promoted iron catalyst containing 60.3 wt % Fe2O3, 11.1 wt % SiO2, 3.1 wt % CuO, 2.0 wt % K2O, and 0.1 wt % Na2O. As
10.1021/ie071587u CCC: $40.75 2008 American Chemical Society Published on Web 07/02/2008
5880 Ind. Eng. Chem. Res., Vol. 47, No. 16, 2008
mentioned in the introduction, the catalyst pretreatment is important for this type of catalysts which composition and activity may change with the reaction conditions. In this case, Fe2O3 was reduced with H2 into Fe3O4. Iron carbides and magnetite (Fe3O4) are in equilibrium with the gas phase. There is no deactivation by free carbon. According to Mo¨ssbauer spectrum results, the transformation in the iron phase occurs mostly in the pretreatment stage and the kinetic analysis is performed on a quasi-stable catalyst consisting of iron carbide and magnetite. Approximately 75% of the surface of the catalyst is present as carbide phase with Fe2.23C as overall composition after 200 h of reaction, the remaining 25% being magnetite.14 Magnetite is a cubic mineral with an inverse spinel structure.18 In a closed packing of oxygen atoms, ferric (Fe3+) and ferrous (Fe2+) ions are situated in octahedral sites and ferric ions in tetrahedral sites.19 In these materials a fast electron hopping occurs between the two cations in the octahedral sites. The dominant lattice faces of magnetite are (111) and (110), however, none of them have neutral charge and the surface tends to reconstruct. As a result, the actual surface structure of magnetite, and of spinel metal oxides in general, is hard to elucidate.18 Data on carbon monoxide adsorption indicated that 120 mmol CO/kg of catalyst or 285 mmol CO/kg of iron was chemisorbed. This corresponds to 3% exposed iron atoms. The specific iron surface area amounts to 12 m2/g of catalyst or 28 m2/g of iron, which is similar to other reported values.14 Assuming that one carbon monoxide molecule is chemisorbed on two iron atoms, the moles of iron atoms on the surface amount to 240 mmol Fe/kg catalyst.14 The intrinsic character of the observed reaction kinetics was verified by applying the criteria of Weisz and Prater and Mears20 at 623 K and 2.1 MPa, assuming that the pores were filled with hydrocarbon wax. This hydrocarbon wax was analyzed and revealed to consist of n-alkanes and 1-alkenes up to carbon number 100.1,14,16 In addition, experiments with different catalyst pellet diameters were performed at 623 K, 2.1 MPa, hydrogen to carbon monoxide inlet ratio 2 mol/mol, and space time of 8 kg s/mol. The results indicated that no diffusional limitations occurred for the catalyst pellet diameter used in the kinetic study.16 For the lower temperature studied, i.e., 523 K, product condensation inside the reactor was excluded applying the criteria of Caldwell and van Vuuren.21 The most abundant carbon-containing products were nalkanes and 1-alkenes up to carbon number 10 and carbon dioxide. Initial rate studies evidenced that carbon dioxide was probably formed on different sites than the hydrocarbons, i.e., on magnetite, in agreement with other experimental observations reported in literature.7 Branched hydrocarbons and alcohols were observed only in minor quantities and the corresponding yields were added to that of the n-alkanes. Also, the conversion to 1-alkenes and iso-alkenes was grouped and reported as the conversion toward 1-alkenes.22 2.2. Conversion and Yields. The conversion of carbon monoxide is defined as follows: XCO )
0 FCO - FCO 0 FCO
FH0 2 - FH2 FH0 2 FH0 2
0 FCO
Yi )
Fi
2.3. Regression Analysis. The estimation of the kinetic parameters is performed by minimizing the objective function SSQ(β) with respect to the model parameter vector β(bl, l ) 1,..., npar):23 nresp nresp
SSQ(β) )
nob
∑ ∑ ∑ (Y
ij - Yij)(Yik - Yik)
σjk
j)1 k)1
ˆ
ˆ
β
98 minimum
i)1
(4) where Yij and Yik are respectively the jth and kth experimental response values in the ith observation, Yˆij and Yˆik are the calculated jth and kth response values for observation i, and σjk is the element for the jth and kth responses of the inverse of the variance-covariance matrix of the experimental errors. In the present work, the responses are the molar conversions or molar yields of the gas-phase components. The elements σjk of the inverse error covariance matrix are calculated from replicate experiments, or when no replicate experiments are available, they can be estimated from the observed and calculated response values. These elements are obtained from an iterative procedure in which new estimates are calculated using the response values obtained with the previous estimates for σjk. Initially, a diagonal matrix is used in which the elements correspond to the reciprocal of the squared mean value of the responses. Multiresponse Rosenbrock and Levenberg-Marquardt algorithms24,25 were combined to minimize the objective function. An in-house developed code for the Rosenbrock algorithm is used to find an adequate direction leading to a possible global optimum. The latter is reached by the ODRPACK package,26 which is an advanced implementation of the Levenberg-Marquardt algorithm and is available in the Netlib library.27 On the basis of mass balance verifications 79 observations out of 90 are used in the regression of the SEMK model on an iron-based catalyst. Depending on the operating conditions, n-alkanes and 1-alkenes up to carbon number 8, 9, or 10 are reported. A reaction network up to 10 carbon atoms is used and for the molecules not observed in the particular experiment, the corresponding elements σjk of the inverse error covariance matrix are set to zero. 2.4. Reactor Model. The experimental data were obtained in a fixed bed reactor operating in the integral regime.20,28 In order to calculate the conversions or molar yields of the gasphase components identified as responses in the objective function (eq 4) a pseudohomogeneous one-dimensional model, i.e. assuming ideal plug-flow, was applied. Accordingly, for a given experiment, the continuity equation of a reference component j over an infinitesimal catalyst mass element can be written as
0 d(W ⁄ FCO )
(1)
(2)
(3)
0 FCO
dYj
The molar yield of hydrogen with respect to CO is calculated as follows: XH2 )
The conversion into the products or molar yields are defined as follows:
) Rj
(5)
with 0 )0 Yj(0) ) Y0j when W/FCO
(6)
as an initial condition. nresp continuity equations are numerically integrated up to the 0 corresponding space time (W/FCO ) for each observation.
Ind. Eng. Chem. Res., Vol. 47, No. 16, 2008 5881
The pseudo-steady-state approximation is applied to all the surface species resulting in a set of nonlinear algebraic equations, where: Rj ) 0
(7)
Rj being the net production rate of the surface species j in the observation. These nonlinear algebraic equations are solved simultaneously with the continuity equations for the gas-phase components (eq 5). Net rates of formation are function of temperature, total pressure, partial pressure of every component, and concentration of the intermediate species on the surface. In the objective function defined in eq 4, the calculated values Yˆ follow from the solution of the set of ordinary differential equations (eq 5) and nonlinear algebraic equations (eq 7). This set of equations is solved simultaneously with the numerical subroutine DASPK29–31 from Netlib software library.27 DASPK uses variable-stepsize backward differentiation formulas applying either direct linear system methods or a preconditioned Krylov iterative method.31 In the present work, the direct method was applied. When using DASPK, the integration must start with a consistent set of initial conditions. The initial concentration of the gas-phase components is known, vide eq 6. However, the initial concentration of the surface species is unknown. The subroutine DASPK provides optional strategies for solving the initialization problem when some of the variables are unknown in the initial point. In this case, the unknown variables are the variables associated with the algebraic equations. As the latter are nonlinear, reasonable initial guesses must be provided as input to the solver in order to reach convergence. Hence, the numerical subroutine DNSQE, also available at Netlib library,27 is used to solve the set of nonlinear algebraic equations by implementation of a hybrid Powell method.32 This subroutine provides reasonable initial values for the variables associated with the algebraic equations required by DASPK to converge efficiently for different kinetic parameter values. For the reaction network up to 10 carbon atoms used in the regression, the total number of equations to solve is 52, from which 23 are ordinary differential equations and 29 are algebraic equations. 3. Single-Event Microkinetic Model 3.1. Reaction Network. The elucidation of the mechanism is the first step to construct a SEMK model for metal catalysis.12 The iron carbide phase of the catalyst contains the active sites for the formation of hydrocarbons.7,33 As mentioned earlier, carbon dioxide is mostly formed by the water-gas shift reaction which takes place on the iron oxide phase of the catalyst, more specifically on magnetite (Fe3O4).7,34–36 The selection of a reliable mechanism for the hydrocarbonforming reactions was already discussed elsewhere.12 The socalled “carbene” mechanism is the most popular mechanism for Fischer-Tropsch synthesis. However, Gaube and Klein37 recently claimed that both “carbene” and CO insertion mechanisms take place. The “carbene” mechanism is attributed to the products with lower carbon number as it is used in this work for hydrocarbons up to carbon number 10, while the CO insertion mechanism is attributed to the products with higher carbon numbers. Fischer-Tropsch synthesis follows a polymerization type kinetics, with initiation, propagation, and termination elementary reactions. The corresponding products can be described by the so-called Anderson-Schulz-Flory (ASF) distribution.38 The latter is characterized by a chain growth
probability R which can be expressed as the ratio between the rate of propagation and the sum of rates of propagation and termination: Rn )
Rprop,n Rprop,n + Rterm,n
(8)
However, the ASF product distribution cannot account for all experimentally observed features. In particular, the overall product carbon number distribution on a molar basis is high for C1, has a minimum at C2, has a maximum again at C3 or C4, and then decreases monotonically with increasing the carbon number.39 The SEMK model can describe these experimentally observed deviations by accounting for the symmetry effects in the kinetic coefficient (vide section 3.2) as will be shown in section 4.2. Also at higher carbon numbers, i.e. from 8 to 10 onward, deviations are observed from the ASF distribution. Physisorption, solubility effects, and mass transport limitations have been reported as possible explanations for the decreasing alkene to alkane ratios with the carbon number.40 The description of this phenomenon is, however, beyond the scope of this study. The so-called chain initiation steps, i.e., the formation of the building-blocks, corresponds to the consecutive hydrogenation of carbide species generated from the chemisorption and dissociation of carbon monoxide by dissociatively chemisorbed hydrogen: H2 + 2M / 2MH CO + 2M / MMCO MMCO + 3M / MMMC + MMO MMMC + MH / MMMCH + M MMMCH + MH / MMCH2 + 2M MMCH2 + MH / MCH3 + 2M
(9) (10) (11) (12) (13) (14)
The formation of water occurs through consecutive hydrogenation of oxygenated species on the surface: MMO + MH / MOH + 2M MOH + MH / H2O + 2M
(15) (16)
The “carbene” mechanism consisting of the insertion of methylene species into growing metal-alkyl species is applied to describe the hydrocarbon chain formation, e.g.:
Alkanes are formed through reductive elimination, e.g.:
While alkenes are produced through β-hydride elimination followed by desorption, e.g.:
5882 Ind. Eng. Chem. Res., Vol. 47, No. 16, 2008
The water-gas shift reaction is an exothermic (∆H0R ) -40.6 kJ/mol) and reversible reaction where carbon monoxide and water produce carbon dioxide and hydrogen:34 CO + H2O / CO2 + H2
(21)
Two reaction mechanisms are generally accepted for the water-gas shift on metal oxides, i.e., the so-called regenerative mechanism and the formate mechanism.34 The regenerative mechanism implies the direct oxidation of CO with O giving CO2 and occurs through the consecutive oxidation and reduction of the surface, i.e., H2O oxidizes the surface giving H2 and CO reduces the surface producing CO2. The formate mechanism involves the formation of a formate as intermediate species. The electron hopping between the ferrous and ferric ions in the octahedral sites of the lattice facilitates the regenerative mechanism. However, the presence of other metal oxides in the catalyst, e.g., Al2O3, TiO2, or SiO2 supports, stabilize the ferrous ions by forming mixed oxides.41 The regenerative mechanism cannot occur if iron cations do not undergo changes in oxidation state41 and, hence, the formate mechanism is the most presumable mechanism for supported magnetite.4,7,41 For the catalyst used in this study, silicon was added as promoter.14 The formate mechanism according to Rethwisch and Dumesic41 consists of six elementary reactions where two different types of sites are identified, i.e., the metal atom and the oxygen anion: _ +O _ /M _ - OH + O _ -H H2O + M
(22)
CO + M _ /M _ - CO
(23)
M _ - CO + O _ -H/O _ - CHO - M _
(24)
O _CHO - M _ +M _ - OH + O _ /O _ - COOH - M _ +O _ -H+M _ (25) O _ - COOH - M _ / CO2 + O _ -H+M _
(26)
_ 2O _ - H / H2 + 2O
(27)
The underlined atoms denote atoms belonging to the metal oxide lattice. 3.2. Rate Equations. In a microkinetic model the rate of every elementary step is explicitly accounted for. By using single-event microkinetics, elementary steps involving compounds belonging to homologous series can be grouped into reaction families if the chemical transformation occurs on analogous reactive moieties. As a result, the number of kinetic parameters is, in principle, equivalent to the number of reaction families. The rate coefficient k in terms of the single-event rate coefficient k˜ is derived from the transition state theory:13 k)
( ) (
˜0,* σgl,r kBT ∆S ∆H0,* exp exp σgl,* h R RT σgl,r k) k˜ σgl,*
)
(28)
The law of mass action is applied to calculate the rates of the elementary steps in the reaction network. The net formation rate of every species is obtained by summation of the rates of the elementary steps in which these species are involved. The probability of finding adjacent occupied or unoccupied atoms is included in the rate equations.42 The number of nearest neighbor atoms z surrounding a particular atom on the carbide and oxide phase of the catalyst is assumed equal to four. This number is divided by two for not counting the neighbor atoms twice due to their indistinguishability. The configuration of the surface of magnetite under Fischer-Tropsch conditions is unknown. However, the surface of the catalyst used in this study is occupied mainly by Fe2+ ions14 in line with other studies.41 Hence, the assumption is that oxygen and metal atoms alternate on the surface lattice forming an electronically neutral surface. The net formation rates for carbon monoxide and hydrogen are given by RCO )
RH2 )
σgl,r z σgl,r ˜ k˜des,COCMtotθMMCO k C p θ 2+ σgl,* 2 σgl,* chem,CO Mtot CO M σgl,r σgl,r k˜2 C θ k˜ C p θ (30) σgl,* K ˜ M_ tot M_ -CO σgl,* 2 M_ tot CO M_ 2 z σgl,r ˜ z σgl,r ˜ k C θ 2k C p θ 2+ 2 σgl,* des,H2 Mtot MH 2 σgl,* chem,H2 Mtot H2 M ˜ z σgl,r ˜ z σgl,r k6 k6CO_ totθO_ -H2 p C θ 2 (31) 2 σgl,* 2 σgl,* K ˜ H2 O_ tot O_ 6
where CMtot is the concentration of iron atoms in the carbide phase, equal to 180 mmol Fe/kg catalyst, CMtot and COtot are the concentrations of iron and oxygen atoms in the oxide phase, which are both equal to 60 mmol/kg catalyst. The kinetic and equilibrium coefficients related to the elementary steps involved in the water-gas shift, eqs 22–27, have been numbered according to the order displayed. The net formation rates for an alkane l, an alkene k, carbon dioxide, and water are obtained from: nM-alkyls
Ralkane,l )
∑ i)1
z σgl,r ˜ k C θ θ 2 σgl,* re,M-alkyls Mtot M-alkyl,ifl M-H z σgl,r ˜ k C p θ 2 (32) 2 σgl,* oa,alkanes Mtot alkane,l M
Ralkene,k )
RCO2 )
σgl,r k˜ C θ σgl,* des,M-alkenes Mtot M-alkene,jfk σgl,r k˜ C p θ σgl,* chem Mtot alkene,k M
(33)
˜ σgl,r z σgl,r k5 k˜5CM_ totθO_ -COOH-M_ C p θ θ σgl,* 2 σgl,* K˜ M_ tot CO2 O_ -H M_ 5
(34)
(29)
Hence, according to the single-event concept, differences in rate coefficients corresponding to the same reaction family are described by the ratio between the global symmetry number of the reactant σgl,r and transition state σgl,*. The calculation of the global symmetry numbers is given elsewhere.12 The catalyst is considered symmetrical and uniform. This means that perpendicular symmetry axes in the adsorbate are preserved during the chemisorption process while parallel symmetry axes are lost.
RH2O )
z σgl,r ˜ C θ k θ 2 σgl,* re,MOH Mtot MOH MH
˜ z σgl,r ˜ z σgl,r k1 koa,H2OCMtotpH2OθM2 + C θ θ 2 σgl,* 2 σgl,* K˜ M_ tot O_ -H M_ -OH 1
z σgl,r ˜ kC p θ θ (35) 2 σgl,* 1 M_ tot H2O O_ M_ The balance of iron atoms in the iron carbide phase is required to calculate the concentration of free iron atoms in this phase:
Ind. Eng. Chem. Res., Vol. 47, No. 16, 2008 5883 nM-alkyls
θMtot ) θM +
∑
nM-alkenes
θM-alkyl,i +
i)1
∑
θM-alkene,j + θMH +
j)1
2θMMCO + 3θMMMC + 2θMMO + 3θMMMCH + 2θMMCH2 + θMOH (36) Mass balances are also required for the two types of active sites in the iron oxide phase: θM_ tot ) θM_ + θM_ -OH + θM_ -CO + θO_ -CHO-M_ + θO_ -COOH-M_ (37) θO_ tot ) θO_ + θO_ -H + θO_ -CHO-M_ + θO_ -COOH-M_
(38)
Affinities are calculated for every elementary reaction in order to assess whether or not the elementary steps involved are equilibrated. The affinity of an elementary reaction is the Gibbs free energy with the minus sign.42 The temperature dependence of the single-event rate coefficients is expressed through the Arrhenius relationship. As a result, two types of kinetic parameters arise: single-event preexponential factors and activation energies. Single-event preexponential factors are calculated with equations derived from statistical thermodynamics, as explained in the following section. The activation energies are initially calculated by a phenomenological method43 and further estimated by regression to experimental data. On the other hand, the thermodynamic consistency of the individual elementary reactions and the overall Fischer-Tropsch reaction can be easily satisfied by systematically using the principle of microscopic reversibility. According to this principle, single-event pre-exponential factors and activation energies of the reverse step are calculated as a function of single-event standard entropies and standard enthalpies of elementary reactions.12 The temperature dependence of the thermodynamic and kinetic parameters in heterogeneous catalysis is calculated using the heat capacities of the surface species. Since these heat capacities are not known, the heat capacity for a given surface species is assumed equal to the heat capacity of the corresponding gas-phase species. Standard heat capacities for alkanes, alkenes, and alkyl radicals are calculated with group contribution methods.44–47 Heat capacities for small molecules and radicals, e.g., carbon monoxide, methylidyne, methylene,..., etc., are taken from literature sources.48,49 3.3. Single-Event Pre-exponential Factors. The preexponential term in the single-event kinetic coefficient is for an elementary step j given by:
( )
˜ 0,* kBT ∆S j exp (39) h R The single-event standard activation entropy for an elementary step j, ∆S˜j0,*, between nreact reactant(s) and the corresponding transition state is calculated as follows: ˜ for ) A j
nreact 0 ˜ 0,* ) ˜ S TS ∆S j
∑ υ ˜S
0 i surf,i
(40)
i)1
0 Single-event entropies of surface species S˜surf and transition 0 ˜ states STS are calculated from single-event entropies of the 0 associated molecules in the gas phase S˜gas and the single-event standard entropy change corresponding to the chemisorption step ∆S˜0chem. For instance, the single-event entropy of a surface 0 species i, S˜surf,i , follows from
0 0 ˜ ˜0 + ∆S )˜ S gas,i S surf,i chem,i
(41)
The same is formally valid for transition state species. Single-event pre-exponential factors for the reverse elementary steps are calculated according to the principle of microscopic reversibility: ˜ for A j (42) ˜0 ∆S r,j exp R The single-event standard reaction entropy of the elementary 0 reaction j, ∆S˜r,j , is calculated as follows: ˜ rev ) A j
( )
nprod
˜0 ) ∆S r,j
nreact
∑ υ ˜S - ∑ υ ˜S 0 i i
i)1
0 k k
(43)
k)1
Standard entropies of the associated molecules of the chemi0 sorbed species in gas phase S˜gas are obtained directly from empirical sources48,49 or are calculated by group additivity methods.44–47 Single-event entropies for small molecules and radicals (CO, · CH, · CH2, · OH,..., etc.) are taken from open databases.48,49 However, single-event entropies for alkyl radicals must be calculated. In this case, a group additivity method is applied.45 The entropies calculated for alkyl radicals do not contain the correction for the symmetry contribution, i.e., single-event entropies are directly obtained. Single-event entropies of alkanes and alkenes are also calculated with the same group additivity method.44,46,47 In general, the entropy change associated to a chemisorption step ∆S˜0chem is dominated by the loss of translational entropy 0 Strans .50,51 This is more evident for small molecules, where the entropy changes related to rotational and vibrational degrees of freedom can be neglected compared to changes with respect to translation.50 For larger molecules, rotational and vibrational motions may become more important52 and the corresponding entropy contribution can be comparable to that corresponding to one degree of translational freedom. High surface coverages reduce the configurational entropy of the chemisorbed layer and the motion of the surface species.50 Under Fischer-Tropsch conditions, high surface coverages have been reported10 and, hence, the loss of two or three degrees of translational motion upon chemisorption is a reasonable approximation. In general, upon chemisorption, the loss of two translational degrees of freedom has been assumed. For chemisorbed carbon monoxide, however, no translational and rotational degrees of freedom are expected53 and, hence, the loss of the three degrees of translational freedom from the gas-phase molecule is assumed. Oxygen and carbon are less mobile adatoms on the surface than hydrogen adatoms. As a consequence, a loss of three degrees of translational freedom is adopted for oxygen and carbon while for hydrogen only the loss of two degrees of translational freedom is assumed. For the transition state species involved in carbon monoxide dissociation and β-hydride elimination, the loss of entropy corresponding to the translational motion of the corresponding gas phase species has been applied. In the case of carbon monoxide dissociation this means that the activation entropy is equal to zero since the same entropy is assumed for reactant and transition state. Indeed, substantial C-O and C-H bond stretching occurs during carbon monoxide dissociation52 and β-hydride elimination,54 which denotes tight transition states. The same assumption has been applied to the reductive elimination of the oxygen adatom. For the other reaction families, a loss of two degrees of translational freedom upon
5884 Ind. Eng. Chem. Res., Vol. 47, No. 16, 2008
transition state formation has been adopted. In general, these assumptions have also been made to satisfy the pre-exponential factor ranges reported in literature.51,53 The translational entropy of a gas-phase molecule is calculated with the Sackur-Tetrode equation:55
( (
0 Strans ) R ln
RT 2π(MW ⁄ NA)kBT p0NA h2
)) 3⁄2
5 + R 2
(44)
where p0 is the standard pressure, NA is the Avogadro constant, and MW is the molecular mass. One or two degrees of translational freedom are approximated as 1/3 or 2/3 of the full translational entropy value. Of course, single-event pre-exponential factors and singleevent standard reaction entropies are calculated for all the considered elementary reactions at the reaction temperature. 3.4. Activation Energies and Reaction Enthalpies. Activation energies for the reverse steps are calculated by applying the principle of microscopic reversibility:
equation
weak
A
QAB )
weak
A-B
QAB )
a)
j
j
k
k
(46)
(47)
Q0A + DAB n
ab(a + b) + DAB(a - b)2 ab + DAB(a + b)
(48)
(49)
(Q0A + Q0B)2
Q0B2(Q0B + 2Q0A)
(50)
(Q0A + Q0B)2
weak
A
Q A2 )
(9/2)Q0A2 (3Q0A + 8DA2)
(51)
strong
A
QAB )
QA2 QA + DAB
(52)
intermediate
A
QAB )
k)1
Although the chemisorption step is exothermic, chemisorption enthalpies Q are taken positive following Shustorovich formulas. Chemisorption enthalpies are calculated as a function of the atomic chemisorption enthalpies and gas-phase bond energies based on the unity bond index-quadratic exponential potential (UBI-QEP) method developed by Shustorovich.43 The use of ab initio methods for obtaining the desired data has been discarded because of the computational cost, especially for the larger species involved. Depending on the electronic configuration of the adsorbate, the UBI-QEP method is derived differently in order to account for weak, strong or intermediate chemisorption enthalpies, which are calculated at zero coverage.43 The corresponding equations are given in Table 1. Atomic chemisorption enthalpies could be considered as adjustable parameters since the actual values for a specific catalyst are normally not available. For the hydrocarbon-forming reactions, atomic chemisorption enthalpies on the iron carbide phase are adopted as adjustable parameters in the model. The latter are catalyst descriptor parameters since they specifically account for the interaction between the reacting species and the catalyst surface. For the water-gas shift reaction on the iron oxide phase, literature values for the chemisorption enthalpies of the gas-phase species involved are used to calculate the chemisorption enthalpies of the surface species. This allows us to decrease the number of adjustable parameters. For closed-shell molecules or molecular radicals with strongly delocalized unpaired electrons, relatively low chemisorption enthalpies (∼40-140 kJ/mol) are expected. In the Fischer-Tropsch reaction network, species such as carbon monoxide, molecular hydrogen, water, and alkanes belong to this category. Molecular radicals with localized unpaired electrons such as hydroxyl, methylidyne, and methylene present an atomic like bonding and display in general strong chemisorption enthalpies (∼140-500 kJ/mol). Monovalent radicals, such as alkyl species, present intermediate bonding where the chemisorption enthalpy can be
Q0A2
Q0A2(Q0A + 2Q0B)
nprod
∑υQ -∑υ Q j)1
atom(s) bound to the surface
(45)
Standard enthalpies of surface reactions ∆H0r,surf are calculated through linear combinations of chemisorption enthalpies Q and reaction enthalpies of analogous molecules in the gas phase 0 ∆Hr,gas : nreact
chemisorption enthalpy type
b)
for 0 Erev a ) Ea - ∆Hr
0 0 ∆Hr,surf ) ∆Hr,gas +
Table 1. Equations Used to Calculate the Chemisorption Enthalpies of Surface Species Involved in the Fischer-Tropsch Network43
(
Q0A2 QA2 1 + 2 Q0A QA + DAB + DAB n
)
(53)
obtained from the average of the weak and the strong bond enthalpy calculations. Standard enthalpies of formation of hydrocarbons and radicals are calculated with a group additivity method.44–47 No difference between sp2 and sp3 radicals was considered.45 Standard enthalpies of formation of small molecules and radicals in gas phase are taken from literature databases.48,49 4. Results and Discussion 4.1. Molecular Chemisorption Enthalpies. Chemisorption enthalpies of all surface species involved in the hydrocarbonforming reactions up to four carbon atoms are reported in Table 2. These values were calculated using the estimated atomic chemisorption enthalpies obtained by regression to experimental data given in section 4.2. The coordination type of the adsorbate is indicated as ηnµm where the superscript n on η indicates the number of atoms of the molecule which are bound to the surface and the subscript m on µ represents the number of metal atoms involved in the bonding. Numerous studies can be found in the literature dealing with the chemisorption of carbon monoxide on transition metal surfaces. The chemisorption enthalpy on the iron catalyst calculated with the UBI-QEP method is 116.1 kJ/mol. In general this value is consistent with reported values in literature. However, chemisorption enthalpies of carbon monoxide are rather sensitive to surface structure, coverage, and calculation method and, hence, reported values can differ substantially. Experimentally obtained chemisorption enthalpies at low cover-
Ind. Eng. Chem. Res., Vol. 47, No. 16, 2008 5885 Table 2. Standard Gas-Phase Formation Enthalpies and Chemisorption Enthalpies of the Surface Species Involved in Fischer-Tropsch Synthesis on the Iron Carbide Phase up to Four Carbon Atoms (QFe-C ) 639.5 kJ/mol, QFe-H ) 249.2 kJ/mol, QFe-O ) 578.8 kJ/ mol) species CO H2 CH CH2 OH alkyls CnH2n+1
alkenes CnH2n alkanes CnH2n+2
a
n)1 n)2 n)3 n)4 n)2 n)3 n)4 n)1 n)2 n)3 n)4
0
coord type
∆fHgas (kJ/mol)
Q (kJ/mol)
η1µ2 η1µ1 η1µ3 η1µ2 η1µ1 η1µ1 η1µ1 η1µ1 η1µ1 η2µ1 η2µ1 η2µ1 η1µ1 η1µ1 η1µ1 η1µ1
-110.5 0.0 594.1 386.4 39.0 146.9 118.8 97.9 77.0 52.5 20.2 0.1 -74.6 -83.7 -104.6 -125.5
116.1a 106.0a 468.2b 314.8b 332.7b 164.8c 168.9c 169.6c 169.6c 62.1d 59.1e 59.2e 65.8a 67.9a 68.1a 68.1a
Equation 47. b Equation 52. c Equation 53. d Equation 51. e Equation
48.
age (θ ) 0.12) equal 119.6 kJ/mol on Fe(110) surfaces56 and from 7557 to 104.6 kJ/mol58 on Fe(100) surfaces. Applying different computational methods, on Fe(110) surfaces, chemisorption enthalpies equal to 163.9 and 150.4 kJ/mol on a 2-fold bridge at 0.25 and 0.5 ML, respectively,59 were calculated, while, on Fe(100) structures, values ranging from 103.3 to 137.6 kJ/mol on a 2-fold site and low coverage (θ ) 0.25)60 and 172.3, 162.6, and 128.9 kJ/mol at 0.25, 0.5, and 1 ML, respectively, on the bridge site61 were obtained. The UBI-QEP chemisorption enthalpy for methylidyne is 468.2 kJ/mol. This value is lower than the chemisorption enthalpies found in the literature, e.g., 661.9 kJ/mol (θ ) 0.25) on Fe(100)62 and 569 kJ/mol on Fe(110).63 Likewise, a lower chemisorption enthalpy for methylene is predicted with the UBI-QEP method, i.e., 314.8 kJ/mol, compared to literature reported values. Chemisorption enthalpies equal to 416.7 kJ/mol (θ ) 0.25) on a hollow site on Fe(100)62 and 405 kJ/mol on Fe(110)63 were calculated. The UBI-QEP chemisorption enthalpy for metal methyl is equal to 164.8 kJ/mol. This value is slightly lower than reported values from literature. For instance, 182.4 kJ/mol (θ ) 0.25) was calculated on Fe(100)62 and 251 kJ/mol on Fe(110).63 UBI-QEP chemisorption enthalpies for the rest of metal alkyls are around 169 kJ/mol. These values are independent of the carbon number and in line with reported chemisorption enthalpies. For instance, experimentally determined chemisorption enthalpies of ethyl on H-presaturated Fe(100) surfaces was equal to 158.99 ( 16.74 kJ/mol.64 UBI-QEP chemisorption enthalpies for alkenes are nearly constant with the carbon number, ranging from 59 to 62 kJ/ mol. In the literature, chemisorption enthalpy of ethylene on Fe(100) coadsorbed with hydrogen was measured as 33.47 kJ/ mol;64 the low value obtained was attributed to the competitive adsorption with the hydrogen atom. UBI-QEP chemisorption enthalpies for alkanes on iron catalysts range from 66 to 68 kJ/mol. The lower value corresponds to methane and from carbon number three onward the calculated chemisorption values are constant, since also virtually constant gas-phase bond energies (used in eqs 47–53) are calculated with the group additivity method. The calculated values are in line with other UBI-QEP values found in literature.10,65
Table 3. Standard Chemisorption Enthalpies for Reactants and Products of Water-Gas Shift and Chemisorption Enthalpies of the Surface Species Involved in the Chemisorption Step on Magnetite molecule CO H 2O CO2 H2
0 ∆Hchem (kJ/mol)
species
Q (kJ/mol)
-46.066 -65.067 -60.066 27.668
M-CO M-OH O-COOH-M O-H
46.0 359.6 222.7 204.2
Chemisorption enthalpies are expected to decrease with chain length because of higher repulsive interactions with the metal electronic structure. This becomes unimportant for surface reactions where the enthalpy change between reactant(s) and product(s) maintains virtually constant. Standard chemisorption enthalpies for carbon monoxide, molecular hydrogen, carbon dioxide, and water on magnetite are given in Table 3. From these values, chemisorption enthalpies of the surface species involved in the specific chemisorption step can be easily calculated by applying a Born-Haber cycle. The chemisorption enthalpy of the remaining surface species (formate) is assumed equal to the chemisorption enthalpy of bicarbonate. For carbon dioxide adsorbed as bidentate (eq 26), a desorption enthalpy from 60 to 120 kJ/ mol was measured.66 In the present work, the lower value was adopted. 4.2. Regression Results. The UBI-QEP method can also be used to calculate activation energies for surface elementary steps by assuming the energy level of a particular transition state.43 In the present work, activation energies for the forward elementary steps were calculated by the UBI-QEP method and used as initial guesses of the regression. A nonactivated dissociative chemisorption of hydrogen was calculated by the UBI-QEP method. Hydrogen chemisorption is usually characterized by high sticking coefficients.50 Hence, during the parameter estimation this step was assumed as nonactivated. In general, molecular chemisorption within the UBI-QEP framework is assumed as a nonactivated step. This assumption has been applied in this work to the chemisorption of carbon monoxide and alkenes. Carbon monoxide chemisorbs molecularly on all the transition metals at low temperatures52 while high sticking coefficients have been found for alkenes, e.g., for ethylene on Fe(100) surfaces,64 and, hence, low or no activation barriers are expected. The UBI-QEP activation energies give a qualitative view of the energetics of the reaction pathway but for quantitative purposes, further adjustment by regression with experimental data is required. Storsaeter et al.10 have recently proposed a microkinetic model with different reaction pathways for the production of Fischer-Tropsch products with one and two carbon atoms on cobalt catalysts. The “carbene” mechanism was rejected because of the high UBI-QEP activation energy predicted for carbon monoxide dissociation and first C-H bond formation. These values are overpredicted by UBI-QEP calculations, e.g., a tight transition state corresponds in general to low activation energies for dissociation, which is not accounted for by the UBI-QEP method. Only the calculated conversion of carbon monoxide and hydrogen were compared to the observed conversions at 483 K, 0.185 MPa, and a hydrogen to carbon monoxide inlet ratio of 10. Also in their case, some UBI-QEP calculated activation energies required further adjustment with experimental data. Regarding the water-gas shift reaction, an activated dissociation step is reported for hydrogen chemisorption,68 vide Table 3. Hydrogen is weakly chemisorbed on magnetite, and an activation barrier is possible. However, since no further
5886 Ind. Eng. Chem. Res., Vol. 47, No. 16, 2008 Table 4. Single-Event Forward and Reverse Pre-exponential Factors (Calculated at 623 K) and Estimated Forward Activation Energies and Atomic Chemisorption Enthalpies with the 95% Confidence Interval and Reverse Activation Energies for an Iron-Based Catalyst, by Regression 0 of Experimental Data at T ) 523-623 K, ptot ) 0.6-2.1 MPa, H2/CO ) 2-6 mol/mol, and W/FCO ) 9.2-82.0 kg s/mol ave A˜for (1/s or 1/MPa s) A˜rev or A˜rev (1/s or 1/MPa s) Ea,for/Q (kJ/mol) Ea,rev (kJ/mol)
reaction family/elementary reaction
chain initiation 3.1 × 109 2.2 × 108 1.3 × 1013 8.8 × 1014 5.9 × 1011 2.3 × 1011
(1) H2 + 2M / 2MH (2) CO + 2M / MMCO (3) MMCO + 3M / MMMC + MMO (4) MMMC + MH / MMMCH + M (5) MMMCH + MH / MMCH2 + 2M (6) MMCH2 + MH / MCH3 + 2M
6.4 × 1011 1.0 × 1016 1.0 × 1014 1.3 × 1013 1.3 × 1013 1.3 × 1013
0 0 56.81 ( 0.53 77.66 ( 0.70 11.94 ( 0.10 61.88 ( 0.50
58.34 116.08 78.51 1.71 39.34 124.80
1.2 × 1013
44.79 ( 0.43
152.73 139.71 138.94
1.4 × 108 8.9 × 107
117.75 ( 0.67
144.54 125.53 124.75
2.4 × 109
96.27 ( 0.50
8.2 × 10
62.09 59.08
83.31 91.55 0.00
chain growth (“carbene” mechanism) (7) MCnH2n+1 + MMCH2 / MCn+1H2n+3 + 2M
(n ) 1) (n ) 2) (n ) 3-9)
1.4 × 1010 8.8 × 109 7.4-8.8 × 109 chain termination alkanes formation
(8) MCnH2n+1 + MH / CnH2n+2 + 2M
(n ) 1) (n ) 2) (n ) 3-10)
3.3 × 1010 2.4 × 1010 1.9-2.1 × 1010 alkenes formation
(9) MCnH2n+1 + M / MCnH2n + MH (10) MCnH2n / CnH2n + M
(n ) 2) (n ) 3-10) (n ) 2) (n ) 3-10)
1.7 × 1010 1.4-0.8 × 1010 1.3 × 1013 1.3 × 1013
7
carbon dioxide formation (11) O-CHO-M + M-OH + O / O-COOH-M + O-H + M 1.7 × 1014
2.8 × 1011
138.95 ( 1.15
102.42
2.2 × 1010 3.4 × 108
103.80 ( 0.96 86.22 ( 0.62
40.78 8.23
water formation (12) MMO + MH / MOH + 2M (13) MOH + MH / H2O + 2M
1.3 × 1012 2.4 × 1011
atomic chemisorption enthalpies Fe2.23C-C Fe2.23C-H Fe2.23C-O Fe3O4-H
experimental evidence has been found regarding the existence of an activation energy for this step, the chemisorption enthalpy of hydrogen on magnetite is assumed as adjustable parameter in the microkinetic model. For water, carbon monoxide and carbondioxidechemisorption,anonactivatedstepwasassumed.66,67 In particular, sticking coefficients close to 0.1 were measured for carbon dioxide chemisorption.66 According to the UBI-QEP predictions, the bicarbonate formation from formate (eq 25) and the carbon dioxide desorption from bicarbonate (eq 26) are the most activated forward steps, in agreement with other studies.4,7,41 By fixing the chemisorption enthalpy of carbon dioxide at its lowest value (60 kJ/mol), the elementary carbon dioxide chemisorption reaction (eq 26) is found to be quasi-equilibrated as verified by the affinity value.42 Hence, the only activation energy to be adjusted for the water-gas shift reaction is the activation energy for the formation of bicarbonate (eq 25). The total number of adjustable parameters in the SEMK model for Fischer-Tropsch synthesis amounts to 14, i.e., 10 activation energies and 4 atomic chemisorption enthalpies. For the hydrocarbon-forming reactions, the activation energies for the dissociation of carbon monoxide (eq 11), reductive elimination of metal carbide (eq 12), metal methylidyne (eq 13), metal methylene (eq 14), metal oxide (eq 15) and metal hydroxyl (eq 16), methylene insertion (eq 17), reductive elimination toward alkanes (eq 18), and β-hydride elimination (eq 19) need to be adjusted. Also, the chemisorption enthalpies of carbon, hydrogen, and oxygen must be adjusted by comparison with experimental data.
639.47 ( 2.07 249.18 ( 0.61 578.77 ( 0.86 222.89 ( 1.51
Following the procedure discussed in section 2.3 for parameter estimation, convergence was achieved in the regression results after only one iteration on the variance-covariance matrix, i.e., subsequent variance-covariance matrices recalculated with the new estimated parameters only lead to marginal changes in the parameter estimates. The parameter estimates and their corresponding individual 95% confidence interval are given in Table 4 together with the calculated single-event pre-exponential factors (at 623 K) for all the forward and reverse elementary steps leading to n-alkanes and 1-alkenes up to 10 carbon atoms. In general, reasonable pre-exponential factors have been calculated, within the range of values reported in literature for the same type of elementary step.51,53 For instance, preexponential factors for carbon monoxide dissociation on transition metals are usually smaller than pre-exponential factors for carbonmonoxidedesorptionwitharatiolyingwithin10-1-10-4.52,69 In our case, the ratio between the dissociation and desorption pre-exponential factors is 10-3. Tight transition states are expected for carbon monoxide dissociation due to the stabilization of carbon and oxygen atoms with the metal surface as C-O bond stretches in the transition state.52 The pre-exponential factor for the association of carbon and oxygen is 1 order of magnitude higher (1014 1/s) than the pre-exponential factor for the dissociation (∼1013 1/s), which is also in agreement with the literature.52 In general, for the hydrogenation of chemisorbed species, the observed pre-exponential factors are around 1011 1/s,69 in correspondence with the pre-exponential factors obtained for
Ind. Eng. Chem. Res., Vol. 47, No. 16, 2008 5887
Figure 1. Experimental and calculated product yields of n-alkanes (]) and 1-alkenes (×) containing up to 10 carbon atoms at T ) 523-623 K, ptot ) 0 0.6-2.1 MPa, H2/CO ) 2-6 mol/mol, and W/FCO ) 9.2-82.0 kg s/mol. Product yields are calculated with the set of parameters given in Table 4 by integration of eq 5 and simultaneously solving eq 7 with the corresponding net production rates eqs 32 and 33.
the hydrogenation of methylidyne, methylene, and oxide. The pre-exponential factor for the hydrogenation of metal carbide (∼1014) is higher compared to the other carbonaceous species, i.e., methylidyne and methylene, because of the low mobility assumed for the metal carbide, as discussed in section 3.3. In principle, single-event pre-exponential factors are calculated for every forward and reverse step. Single-event preexponential factors for molecules with more than two carbon atoms are calculated with entropies derived from group additivity values, as explained in section 3.3. The variation of these values with the molecular mass of the particular molecule is similar and, hence, the corresponding activation and reaction entropies for the elementary steps within a particular reaction family are practically constant. This can be observed in Table 4 for methylene insertion (step 7), reductive elimination toward alkanes (step 8), β-hydride elimination (step 9), and metal alkene desorption (step 10). Consequently, an average pre-exponential factor for all the forward and reverse elementary steps within the individual reaction families is assumed, in correspondence with the single-event concept. The only exception is applied to the reductive elimination toward methane. Activation energies for the reverse elementary steps, calculated with the reaction enthalpy at 623 K with eq (45) are also included in Table 4. Individual reaction enthalpies for the different carbon numbers are used, so different reverse activation energies are applied. The parity diagrams for n-alkanes and 1-alkenes, at 623, 573, 553 and 523 K, are given in Figure 1. The fit is reasonably good taking into account the wide range of experimental conditions described. Examples of experimental and calculated conversions and molar yields for the nonhydrocarbon molecules are given in Figure 2 at different temperatures. n-Alkanes and 1-alkenes product distributions also at different temperature conditions are shown in Figure 3. The F value for the significance of the regression is 278, exceeding the tabulated value of 2.79. As shown in Table 4, narrow confidence intervals are obtained implying good parameter significance for a 95% probability level. All the calculated t values are considerably higher than the critical value of 1.961. The higher t values are obtained for the atomic chemisorption enthalpies and for the activation energies of the reductive elimination toward alkanes and the β-hydride elimina-
Figure 2. Calculated and experimental conversions of carbon monoxide (2), hydrogen (]), carbon dioxide ([), water (0), and methane (O) with respect to the space time at (a) T ) 623 K, ptot ) 2.1 MPa, H2/CO ) 3; (b) T ) 573 K, ptot ) 1.1 MPa, H2/CO ) 3; (c) T ) 553 K, ptot ) 2.1 MPa, H2/CO ) 3, and (d) T ) 523 K, ptot ) 2.1 MPa, H2/CO ) 3. Lines are calculated with the set of parameters given in Table 4 by integration of eq 5 and simultaneously solving eq 7 with the corresponding net production rates eqs 30–35.
Figure 3. Experimental (open symbols) and calculated (closed symbols) product distributions for n-alkanes and 1-alkenes up to nine carbon atoms 0 at (a) T ) 623 K, ptot ) 2.1 MPa, H2/CO ) 3, W/FCO ) 24 kg s/mol; (b) 0 T ) 573 K, ptot ) 1.1 MPa, H2/CO ) 3, W/FCO ) 49 kg s/mol; (c) T ) 0 553 K, ptot ) 2.1 MPa, H2/CO ) 3, W/FCO ) 36 kg s/mol; and (d) T ) 0 523 K, ptot ) 2.1 MPa, H2/CO ) 3, W/FCO ) 48 kg s/mol. Closed symbols are calculated with the set of parameters given in Table 4 by integration of eq 5 and simultaneously solving eq 7 with the corresponding net production rates eqs 32 and 33.
tion. The smallest t value in this case was 203, obtained for the methylene insertion step. Strong binary correlations were not
5888 Ind. Eng. Chem. Res., Vol. 47, No. 16, 2008
found for the parameter estimates, with a maximum binary correlation of 0.77. As observed in Figure 3, carbon number independent parameters are sufficient to adequately describe the molar yields of n-alkanes and 1-alkenes. The deviation observed from the ASF product distribution for methane and ethylene, i.e., higher methane molar yield and lower ethylene molar yield than expected from the theoretical distribution, is adequately described by the model as a consequence of the different global symmetry numbers found for the C1 and C2 species. As introduced in section 3.2, the ratio between global symmetry numbers of reactant(s) and transition state determines the selectivity within a reaction family, vide eq 29. The higher selectivity toward methane is a consequence of the symmetry loss of the corresponding transition state in the reductive elimination step, as shown in eq 54, while for the rest of n-alkanes, the global symmetry number in the transition state is unaltered, vide eq 55. The 3-fold symmetry of the methyl group is hindered by the newly C-H bond formed in the transition state for the metal methyl species (vide eq 54) while the terminal methyl group is free to rotate for the longer metal alkyl species (vide eq 55). As a result, the rate coefficient of the reductive elimination toward methane is three times higher than the rate coefficient of the reductive elimination toward the rest of n-alkanes. Similarly, the ratio of the global symmetry numbers for 1-alkenes and the transition states prior to chemisorption is higher for ethylene, i.e., 2, than for the rest of alkenes, i.e., 1. This is because ethylene loses the 2-fold symmetry parallel to the surface when chemisorbing while the other alkenes maintain the 3-fold symmetry of the corresponding terminal methyl group. Hence, ethylene is more reactive and reincorporates faster to the chain growth than the remaining 1-alkenes.
selectivities toward alkenes for the operating conditions studied22 by the same author. The activation energy for the water-gas shift reaction obtained through regression amounts to 139.0 kJ/mol, which is higher than the values typically found in the literature, i.e., from 27.7 to 125 kJ/mol.5 This high activation energy for the water-gas shift implies that this reaction is rather slow and that thermodynamic equilibrium may not be reached under all the operating conditions studied. The presence of potassium in the catalyst (vide section 2.1)14 may slow down the water-gas shift reaction.70 According to affinity calculations, the kinetically significant elementary steps are the dissociation of carbon monoxide (eq 11), the partial hydrogenation of the CHx (x ) 0-2) species (eqs 12–14), methylene insertion (eq 17), reductive elimination toward alkanes (eq 18), β-hydride elimination (eq 19), and the formation of bicarbonate from formate for the water-gas shift reaction (eq 25). The other elementary reactions, i.e., hydrogen chemisorption/desorption (eq 9), carbon monoxide chemisorption/desorption (eq 10), metal alkene desorption/chemisorption (eq 20), reductive elimination/oxidative addition of metal oxide and metal hydroxyl (eqs 15 and 16) are fast reactions close to equilibrium. Smaller absolute affinity values are obtained at higher temperatures, as expected. The relative concentration of surface species on the carbide phase is shown in Figure 4 with respect to the space time and temperature. Over the studied iron catalyst, the oxygenated species are the most abundant species on the surface. This is due to the quasi-equilibrium of the elementary reactions involved in the formation of water (eqs 15 and 16). The lower concentration of metal alkyls and metal methylene species at higher temperatures explain why shorter hydrocarbons are observed at higher temperatures. Estimated atomic chemisorption enthalpies are within ranges of reasonable values, as observed in Table 5 where the estimated chemisorption enthalpies and other reported values are shown. Differences with literature values can be due to the effect of promoters and to the high surface coverage found on the surface. 4.4. Prediction of Experimental Trends. The estimated activation energies increase according to the following: Ea,mi (45 kJ/mol)
