detailed kinetics of fischer-tropsch synthesis on an industrial

Detailed Kinetics of Fischer-Tropsch Synthesis on an IndustrialFe-Mn Catalyst

Jun Yang,†,‡ Ying Liu,†,‡ Jie Chang,† Yi-Ning Wang,† Liang Bai,† Yuan-Yuan Xu,†Hong-Wei Xiang,† Yong-Wang Li,*,† and Bing Zhong†

Group of Catalytic Kinetics & Theoretical Modeling, State Key Laboratory of Coal Conversion,Institute of Coal Chemistry, Chinese Academy of Sciences, Taiyuan 030001, People’s Republic of China, andDepartment of Chemistry, Jinan University, Guangzhou 510632, People’s Republic of China

The detailed kinetics of the Fischer-Tropsch synthesis over an industrial Fe-Mn catalyst wasstudied in a continuous integral fixed-bed reactor under the conditions relevant to industrialoperations [temperature, 540-600 K; pressure, 1.0-3.0 MPa; H2/CO feed ratio, 1.0-3.0; spacevelocity, (1.6-4.2) × 10-3 Nm3 kg of catalyst-1 s-1]. Reaction rate equations were derived on thebasis of the Langmuir-Hinshelwood-Hougen-Watson type models for the Fischer-Tropschreactions and the water-gas-shift reaction. Kinetic model candidates were evaluated by the globaloptimization of kinetic parameters, which were realized by first minimization of multiresponseobjective functions with a genetic algorithm approach and second optimization with theconventional Levenberg-Marquardt method. It was found that an alkylidene mechanism basedmodel could produce a good fit of the experimental data. This model shows that the desorptionof the products and the insertion of methylene into the metal-alkylidene bond are the rate-determining steps. The activation energy for olefins formation is 97.37 kJ mol-1 and smallerthan that for the paraffin formation (111.48 kJ mol-1). In this model, the readsorption andsecondary reactions of olefins are taken into account, and deviations of hydrocarbon distributionfrom the conventional ASF distribution can therefore be quantitatively described. However, thedeeper information for the olefin-to-paraffin ratio has not intrinsically been described in thepresent stage, leaving for the further improvements in models to consider the transportation-enhanced readsorption and secondary reaction of olefins more practically in the reactor modelingstage.

1. Introduction

Fischer-Tropsch synthesis (FTS) is an industriallyimportant process for the conversion of syngas (H2/CO)derived from carbon sources such as coal, peat, biomass,and natural gas into hydrocarbons and oxygenates.Today, it continuously attracts interest as an option forthe production of clean transportation fuels and chemi-cal feedstocks.1-3 The FTS product is composed of acomplex multicomponent mixture of linear and branchedhydrocarbons and oxygenated products, the majority ofwhich are linear hydrocarbons.3,4

The FTS kinetics has extensively been studied, andmany attempts have been made for the rate equationsdescribing the FTS reactions.4-13 In most cases, thehydrocarbon products were lumped according to thecarbon number of hydrocarbon molecules with an idealAnderson-Schulz-Flory (ASF) distribution. Althougha few of the available kinetic models were developedbased on the detailed mechanism of the Langmuir-Hinshelwood-Hougen-Watson (LHHW) type,4,10-13 onlytwo of them simultaneously considered both the syn-gas conversion rates and the hydrocarbon formationrates.11-13 We have pointed out the deficiencies existingin the conventional lumped models and those tailingsyngas conversion rates with a carbon number distribu-tion formula when a comprehensive simulation isrequired.13

The quality of a detailed kinetic model is closelyrelated to the understanding of the mechanism in theFTS catalytic reaction system, in which a polymeriza-tion process has been recognized to be dominant;however, sufficient details on the FTS mechanisticaspect are not yet fully understood.10,14,15 In 1946,Herington first treated the molar distribution of hydro-carbons from FTS in terms of a polymerization mech-anism.15,16 The same formulation was rediscovered byAnderson et al. in 1951 and named the ASF distribu-tion.7,16 In the ASF model, the formation of hydrocarbonchains was assumed as a stepwise polymerization pro-cedure and the chain growth probability was assumedto be independent of the carbon number. However,significant deviations from the ideal ASF distributionhave been observed in many studies.17-20 Pichler et al.21

for the first time reported the deviations of experimentalresults from the ASF distribution. The usual deviationsof the distribution of the linear hydrocarbons are arelatively higher selectivity to methane, a relativelylower selectivity to ethane, and an increase in thechain growth probability with increasing molecular sizein comparison to the ideal ASF distribution. Severaldifferent explanations about the cause of these devia-tions have been proposed.22-24 Some authors25,26 inter-preted the deviations from the standard ASF distribu-tion by the superposition of two ASF distributions. Theysuspected the existence of two sorts of sites for the chaingrowth on the catalyst surface and, therefore, proposedthat each site might individually yield the ideal ASF

† Chinese Academy of Sciences.‡ Jinan University.

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distribution with different chain growth probabilities.However, this explanation cannot interpret the increaseof the paraffin/olefin ratio with the chain length. On thebasis of the experiments with cofed olefins, it was notedthat the readsorption and secondary reaction of olefinshad a great influence on the products distribution ofFTS.27 Some researchers28-30 proposed a more plausibleexplanation for these deviations and suggested that theoccurrence of secondary reactions of the olefins causedthe deviations from the ASF distribution. Iglesia et al.30

developed a model describing the olefin readsorptioneffect enhanced by intraparticle and interparticle trans-port processes. They suggested that the diffusion limita-tion within liquid-filled pores slowed the removal of1-olefins, which caused an increase of their residencetime within the catalyst pores. In their model, the COhydrogenation model and the olefin readsorption modelwere treated separately. In 1992, Zimmerman andBukur31 developed a model for both the formation oflinear hydrocarbons and the water-gas-shift (WGS)reaction over iron catalysts. In their model, the read-sorption and secondary reaction of olefins were takeninto account and the increase of the paraffin/olefin ratiowith the carbon number can be predicted. However, theresults displayed significant deviations between modelpredicted and experimental mole fractions, especiallyfor methane and ethene. Kuipers et al.32 proposed achain-length-dependent olefin readsorption mechanismto explain the fact that the paraffin/olefin ratio increaseswith the carbon number. On the basis of the olefincofeeding kinetic experiments over cobalt catalyst in acontinuously operated and well-mixed slurry reactor,Schulz and Claeys33 developed a kinetic model for theFischer-Tropsch reaction system.

It is well accepted that the FTS reaction is a surfacepolymerization reaction, but its detailed mechanism isstill not fully understood. Recent mechanistic studiesshow new evidence in favor of a mechanism start withCO dissociation. It is generally agreed that the FTSproceeds via the dissociation of CO, further formingcarbide on the surface in sequence of the hydrogenationof this carbide.10,34,35 Despite considerable researchefforts, uncertainties still remain about the mechanismof chain growth in the FTS reaction. Brady and Pettit36

proposed an alkyl mechanism and suggested that thechain growth is initiated by the insertion of methyleneinto the adsorbed alkyl. Martinez et al.37 proposed achain growth mechanism, in which the chain growth isinitiated by an R-vinyl and the polymerization processproceeds through the reaction between methylene andan alkenyl. Joyner38 developed an alkylidene mecha-nism and assumed the chain growth process propagatedvia the successive insertion of methylene into themetal-alkylidene bond. With this model, one can easilyexplain the formation of R-olefins as the primaryproducts in FTS. These proposed models have provideda basis for a workable kinetic model derived fromexperimental data. However, models are still far frommeeting the demands for both better mechanism un-derstanding and applications in engineering scale-up,leaving a great deal of work under the expectation fora better treatment in considering both the fundamentalmechanism and self-consistency in rate expressions.10,13

Lox and Froment11,12 studied the FTS kinetics overan iron catalyst. On the basis of the LHHW scheme,they developed several sets of “elementary steps”, fromwhich detailed kinetic models were derived by assuming

the carbide mechanism for the FTS and the formateintermediate scheme for the WGS reaction. The ratesof CO or H2 consumption and product formation wereunified in these models. Experimental data were re-gressed with large-scale nonlinear optimization ap-proaches for determining the “best” kinetic parameters,and statistical analysis was applied to validate thefeasibility of both models and parameters in them. Thefinal model could describe the distribution of linearparaffins and olefins of FTS obeying the ideal ASFdistribution at the level of surface reaction. However,the deviations, observed in many experiments, from theideal ASF distribution were totally neglected in theirmodel. Very recently, we have proposed a systematicapproach for considering more complicated olefin read-sorption phenomena in kinetic modeling on the basisof ideas similar to those of Froment.13 Although pri-mary, namely, a limited number of mechanism sets,from which only about 10 kinetic models were scanned,were considered, our optimal model for an industrialiron catalyst showed a better description for the non-ideal ASF distribution than that from Lox and Froment.However, the models based on the detailed mechanismscan for FTS processes need to first describe the mostintrinsic factors involved, namely, those defined at thesurface mechanism level, and the solubility/diffusivity-enhanced paraffin/olefin ratio dependence on the carbonnumber should be described through modification of themodels with intrinsic significances. This modificationneeds to be carefully arranged at both the surfacemechanism and the reactor simulation levels becausethe transportation-enhanced phenomena may not fun-damentally be reflected solely either in an “intrinsic”kinetic model or in reactor models using the intrinsickinetics correctly. This is because reactor models canfundamentally account for transportation phenomena,and at the same time long-chain olefins staying in thecatalyst pores filled with wax are difficult to removeinstantly as “defined” even in the case of kineticexperimental conditions. In fact, this is a dilemma forthe kinetics modeling of FTS. The difficulties arise,making kinetic modeling impossible, if one wants tofundamentally take the transportation phenomena intoaccount for the modeling of a kinetic reactor. On thebasis of this understanding, we first make an attemptto grasp the most significant intrinsic information atthe mechanism kinetic level, and later efforts will bedevoted to solving the modification problem for consid-ering the transportation-enhanced phenomena with thecombination of kinetics and reactor simulations.

The goal of this paper is, therefore, to establish andtest detailed mechanistic kinetic models for the Fis-cher-Tropsch system over an industrial Mn-Fe cata-lyst while considering many more possibilities in mech-anism combinations than before. The new mechanisticmodel, in which the olefin readsorption and secondaryreactions are taken into account, combines all of the FTSreactions with WGS reaction in a self-consistent way,leaving the transportation-enhanced olefin readsorptionfactor for further work in the combination of kineticsand reactor simulations.

2. Experimental Section

The experimental setup used in this study is shownin Figure 1. A micro-fixed-bed reactor (a 1.0 m longstainless steel tube of 0.012 m in internal diameter with

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a temperature-controlled molten salt bath to achieve auniform temperature profile in the catalyst bed) wasused for all kinetic experiments. The catalyst used inthe kinetic study is an industrial manganese-promotediron catalyst prepared by Institute of Coal Chemistry(ICC), Chinese Academy of Sciences. The preparationmethod has been patented, and some open informationcan be found in the literature.39,41 The fresh catalystwas crushed and sieved to particles with diameters of0.25-0.36 mm (40-60 ASTM mesh), which has beenproved to be a compromising particle size safe forneglectable intraparticle transfer limitations and prom-isingly easy operations to the reactor during theexperiment.10-13 A 3.58 g (3 cm3) catalyst was used forthe kinetic experiments. The catalyst was diluted by 24cm3 inert silica sand [(catalyst:sand ) 1:8 (v/v)] withthe same mesh size range. Before the reaction, thecatalyst was reduced in situ with syngas (H2/CO ) 2.0)in the reactor.

In a typical reduction procedure, operation conditionswere adjusted to 0.25-0.30 MPa and 1000 h-1 at 513K, and then the temperature was raised to 548 K atthe rate of 1 K h-1 and retained at that temperaturefor the whole reduction stage. After reduction for 36 h,the temperature was lowered to 513 K and then theoperation conditions were adjusted to the desired valuefor the FTS. For intrinsic kinetic studies, a high spacevelocity is needed to exclude the external diffusionlimitation. In our work, the space velocity of (1.6-4.2)× 10-3 Nm3 kg of catalyst-1 s-1 is much higher thanthose reported by others,5,13 ensuring the kinetic experi-ments with the neglected external diffusion effect.

The syngas was prepared by blending of pure CO andH2 (>99.99% purity). The CO and H2 passed through aseries of an oxygen-removal trap, a heated silica gelmolecular sieve trap, and an activated charcoal trap toremove tiny amounts of oxygen, water, and otherimpurities. The flow rates of CO and H2 were controlledby two Brooks 5850E mass flow controllers. The outletof the reactor is connected with a hot trap (420 K) andthen an ice trap (273 K) at the system pressure. Afterthese product collectors, the pressure was releasedthrough a backpressure regulator. The flow rate of thetail gas was monitored by a wet test gas meter. Gas

samples for the gas chromatograph were collected afterremoval of the water.

For kinetic studies, a stabilization period of more than1000 h was used to ensure that the stable catalyticphases were established. This could avoid the sharpchange in crystalline structures observed in previousexperimental studies by other researchers.40,41-45 Ki-netic samples were cumulatively collected during atypical period of 10-16 h. For each operation condition,it took at least 10 h to ensure the steady-state behaviorof the catalyst after a change of the reaction conditions.

The products of the synthesis were separated intothree portions: gas phase, liquid phase (from the icetrap), and wax phase (from the hot trap). Both thepurified syngas and the tail gas were analyzed on a gaschromatograph. H2, O2, N2, CH4, and CO were separatedon a 13× molecular sieve packed column (1.5 m × 3 mmi.d., Ar carrier flow) and detected with a thermalconductivity detector (TCD). C1-C5 hydrocarbons in thegas phase were analyzed on a C22

0/C-22 (170-250 µm)packed column (7.2 m × 3.2 mm i.d.) flame ionizationdetector (FID) with N2 as the carrier. CO2 was measuredon a silica gel packed column (1 m × 3 mm i.d, H2carrier) with TCD and quantified by an external stan-dard method. The oil product from the cold trap wasseparated on a 60 m × 0.25 mm (i.d.) OV-101 capillarycolumn (N2 carrier, FID) with the temperature pro-grammed from 333 K (maintained for 20 min) to 563 Kat the rate of 3 K min-1. The wax product from the hottrap was first dissolved in CS2 (0.5-1.0 mass %) andthen analyzed on a OV-101 capillary column, FID, andN2 carrier with the temperature programmed (1 Kmin-1) from 343 to 563 K. The water phase wasseparated by using a BD-wax (GW, US) capillarycolumn (N2 carrier, FID). All of the data in steady-statereactions showed promising material balances withcarbon, hydrogen, and oxygen material balances be-tween 95 and 104%.

The experiences from our intensive experimentalinvestigations of the catalytic performances of variousFTS catalysts showed that better material balances aredifficult to reach in practice. This is partially due to thefact that small fluctuations during a sampling timebrought about certain experimental errors for the finalresults and partially due to the fact that the complexityof the FTS products caused difficulties in analyses. Inaddition, detailed kinetic modeling requires the infor-mation of the compositions of the reactants and almostall FTS products (hydrocarbons relevant) in each stream.It is estimated that about 5% error in the total materialbalances could produce very large errors (sometimesseveral times) for the contents of components with verysmall mole fractions in a stream. This error amplifica-tion phenomenon in FTS makes it even more difficultto achieve high modeling accuracy in detailed kinetics.

3. Kinetic Models

3.1. Active Site Assumption. The overall FTSreactions can be simplified as the combination of FTSreactions and the WGS reaction.11,12

Figure 1. Experimental setup: (1) gas cylinders; (2) pressuremeters; (3) pressure regulators; (4) pressure regulators before thereactor; (5) mass flow controllers; (6) the reactor; (7) wet flowmeter;(8) needle valves; (9) backpressure regulator; (10) purificationcolumns; (11) wax condenser; (12) oil/water condenser; (13) waxtrap (hot trap); (14) cold trap; (15) high-temperature ball valves;(16) ball valves; (17) salt bath.

paraffin formation: nCO + (2n + 1)H2 )CnH2n+2 + nH2O (1)

olefin formation: nCO + 2nH2 ) CnH2n + nH2O (2)

WGS reaction: CO + H2O ) CO2 + H2 (3)

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Active sites for the above reactions in FTS are stillnot clear especially in the cases of iron catalysts. Thisis because of the fact that the iron-based catalystsstarting from their oxide precursors have experimen-tally proved to have complex phase transfer during thereduction as well as synthesis operation.40,41-45 Thecontroversy in the active sites (or phases) in the FTSsystem appeals the need for much attention to be paidto the understanding of the FTS mechanism at a deeplevel; for example, the work for Co or Ru catalysts, forwhich metal phases are believed to be active phases forFTS, has been considered at even a molecular level withmany interesting results achieved, casting an encourag-ing light on the detailed mechanism understanding.46

For iron catalysts, however, we expect many morechallenges in this direction.41-49 Here, we will not gointo these controversies and instead will use experi-mental data to fit many possible mechanism-derivedkinetic models, which are anyhow expected to reflectthe most important points of FTS catalysis and chem-istry. Beyond the complexities in the FTS catalysis withiron catalysts, it is generally accepted that reactions ineqs 1-3 can be assumed to take place on two kinds ofactive sites. The active sites for the hydrocarbon forma-tion and the WGS reaction are iron carbides andmagnetite (Fe3O4), respectively.4,11-13,47-49

3.2. Kinetic Models for the Formation of LinearHydrocarbons. 3.2.1. Elementary Reactions of theFTS. The FTS is a complex network of parallel andseries reactions involving different extents and deter-mining altogether the overall catalyst performance. Thewhole synthesis reaction can be simplified as thecombination of FTS reactions and the WGS reaction.The FTS reactions considered here consist of surfacesteps in five categories:50 (1) adsorption of reactants (H2and CO); (2) chain initiation; (3) chain propagation; (4)chain termination and desorption of products; (5) read-sorption and secondary reaction of olefins.

The CO adsorbed on the catalyst surface either in themolecule state or in the dissociated state18,51-53

where s1 is an empty catalytic site, on which hydro-carbon can be formed. The dissociative adsorption ofhydrogen takes place on two adjacent free activesites.51-53

For the further formation of hydrogenated carbonspecies, mechanism studies for FTS often assumed theformation of CH2 and CH3 species,4,10-13,55 while recentcharacterization and theoretical studies in energeticsdebate for the increased possible formation of CHspecies on several transition-metal surfaces.46,57 For ironcatalysts, the situation considered in theoretical models(or the model catalysts and operation conditions incharacterization) describing the catalyst surfaces isdefinitely too far from a real working catalyst, whichcontinuously changes in both the bulk phase andsurfaces due to the carbonization of the iron and thehydrogenation of carbon species on the surface duringreduction and reactions.40-45,47,52 The carbides on thecatalyst surface may exchange carbon and hydrogensources with reactants and surface intermediates during

Fischer-Tropsch catalysis cycling. The precise defini-tion of the catalysis cycle during FTS on iron catalystsis, therefore, impossible with the current understandingof the FTS catalysis. Nevertheless, some conventionalideas/treatments may serve for the development ofkinetic models. We here take advantage of the conven-tional idea that the building block “CH2” is formed bythe reaction of a surface carbon with dissociatedhydrogen4,10-13,50,54,56 to build the kinetic models. It isthus assumed that surface carbon species undergoes areaction with surface hydrogen:

or with molecular hydrogen according to an Eley-Rideal(ER) mechanism:

Another possible pathway of the formation of “CH2”starts with molecularly adsorbed carbon monoxide andsuccessive hydrogen-assisted dissociation,4,18,55

or assisted by molecular hydrogen via the ER mecha-nism:

There is still a controversy about the mechanism ofchain growth in the FTS. The alkyl mechanism proposesthat the reaction is initiated by the formation of amethyl species and that chain growth takes place bythe successive insertion of methylene into the metal-alkyl bond:36

The alkylidene mechanism proposes that the forma-tion of the adsorbed ethylidene initiates the chainformation and that chain growth is facilitated by meth-ylene insertion into the metal-alkylidene bond:38

Another mechanism assumes that chain growth isinitiated by the adsorption of CO on the active sitesalready containing a hydrocarbon intermediate and thenby a sequence of hydrogenation:56,58

CO + s1 ) COs1 (4)

COs1 + s1 ) Cs1 + Os1 (5)

H2 + 2s1 ) 2Hs1 (6)

Cs1 + Hs1 ) CHs1 (7)

CHs1 + Hs1 ) CH2s1 (8)

Cs1 + H2 ) CH2s1 (9)

COs1 + Hs1 ) HCOs1 + s1 (10)

HCOs1 + Hs1 ) Cs1 + H2Os1 (11)

Cs1 + Hs1 ) CHs1 (12)

CHs1 + Hs1 ) CH2s1 (13)

COs1 + H2 ) H2COs1 (14)

H2COs1 + H2 ) CH2s1 + H2O (15)

CH2s1 + CnH2n+1s1 ) Cn+1H2n+3s1 + s1 (16)

CH2s1 + CnH2ns1 ) Cn+1H2n+2s1 + s1 (17)

CO + CnH2n+1s1 ) CnH2n+1s1CO (18)

CnH2n+1s1CO + H2 ) CnH2n+1s1C + H2O (19)

CnH2n+1s1C + H2 ) CnH2n+1s1CH2 (20)


Termination of the chain growth can occur viaseveral routes. For the desorption of an olefin, asingle-site mechanism was assumed that R-olefins (1-alkenes) may be formed by a â-hydride eliminationreaction

or by the desorption of adsorbed alkylidene

For the desorption of a paraffin, both a single-sitereaction with molecular hydrogen

and a dual-site reaction with an adsorbed hydrogenatom

were considered.It is generally agreed that the R-olefins desorbed from

the catalyst surface can readsorb on the active sites andtake place as secondary reactions, which have signifi-cant influences on the product distribution of theFTS.29,59

On the basis of the above five categories of elementaryreactions, we can define 10 possible mechanisms forFTS, as shown in Table 1.

3.2.2. Rate Expressions of the HydrocarbonFormation. On the basis of the detailed elementaryreaction steps of FTS listed in Table 1, the formation

Table 1. Elementary Reaction Sets for FTS

FT I 1 CO + s1 ) COs1 FT VI 1 CO + s1 ) COs12 COs1 + H2 ) H2COs1 2 COs1 + s1 ) Cs1 + Os13 H2COs1 + H2 ) CH2s1 + H2O 3 Cs1 + H2 ) CH2s14 H2 + 2s1 ) 2Hs1 4 Os1 + H2 ) H2O + s15(n) CH2s1 + Hs1 ) CH3s1 + s1 5 H2 + 2s1 ) 2Hs1

CH2s1 + CH3s1 ) CH3CH2s1 + s1 6(n) CH2s1 + Hs1 ) CH3s1 + s1CH2s1 + CnH2n+1s1 ) CnH2n+1CH2s1 + s1 CH2s1 + CH3s1 ) CH3 CH2s1 + s1

6(n) CnH2n+1s1 + Hs1 ) CnH2n+2 + 2s1 CH2s1 + CnH2n+1s1 ) CnH2n+1CH2s1 + s17(n) CnH2n+1s1 ) CnH2n + Hs1 7(n) CnH2n+1s1 + H2 ) CnH2n+2 + Hs1

FT II 1 CO + s1 ) COs1 8(n) CnH2n+1s1 ) CnH2n + Hs12 COs1 + H2 ) H2COs1 FT VII 0 H2 + 2s1 ) 2Hs13 H2COs1 + H2 ) CH2s1 + H2O 1(n) CO + Hs1 ) Hs1CO4 H2 + 2s1 ) 2Hs1 CO + CH3s1 ) CH3s1CO5(n) CH2s1 + Hs1 ) CH3s1 + s1 CO + CnH2n+1s1 ) CnH2n+1s1CO

CH2s1 + CH3s1 ) CH3CH2s1 + s1 2(n) Hs1CO + Hs1 ) Hs1C + HOs1CH2s1 + CnH2n+1s1 ) CnH2n+1CH2s1 + s1 CH3s1CO + Hs1 ) CH3s1C + HOs1

6 (n) CnH2n+1s1 + H2 ) CnH2n+2 + Hs1 CnH2n+1s1CO + Hs1 ) CnH2n+1s1C + HOs17 (n) CnH2n+1s1 ) CnH2n + Hs1 3(n) CnH2n+1s1C + Hs1 ) CnH2n+1s1CH + s1

FT III 1 CO + s1 ) COs1 4(n) CnH2n+1s1CH + Hs1 ) CnH2n+1s1CH2 + s12 COs1 + H2 ) H2COs1 5(n) CnH2n+1s1CH2 ) CnH2n+1CH2s13 H2COs1 + H2 ) CH2s1 + H2O 6 HOs1 + Hs1 ) H2O + 2s14 H2 + 2s1 ) 2Hs1 7(n) CnH2n+1s1 + Hs1 ) CnH2n+2 + 2s15(n) CH2s1 + CH2s1 ) CH2CH2s1 + s1 8(n) CnH2n+1s1 ) CnH2n + Hs1

CH2s1 + CnH2ns1 ) CnH2nCH2s1 + s1 FT VIII 1 CO + s1 ) COs16(n) CnH2ns1 + Hs1 ) CnH2n+1s1 + s1 2 COs1 + s1 ) Cs1 + Os17(n) CH3s1 + Hs1 ) CH4 + 2s1 3 Cs1 + Hs1 ) CHs1 + s1

CnH2n+1s1 + Hs1 ) CnH2n+2 + 2s1 4 CHs1 + Hs1 ) CH2s1 + s18(n) CnH2ns1 ) CnH2n + s1 5(n) CH2s1 + Hs1 ) CH3s1 + s1

FT IV 0 H2 + 2s1 ) 2Hs1 CH2s1 + CH3s1 ) CH3CH2s1 + s11 CO + s1 ) COs1 CH2s1 + CnH2n+1s1 ) CnH2n+1CH2s1 + s12 COs1 + Hs1 ) HCO s1 + s1 6 H2 + 2s1 ) 2Hs13 HCOs1 + Hs1 ) Cs1 + H2Os1 7 Os1 + Hs1 ) HOs1 + s14 H2Os1 ) H2O + s1 8 HOs1 + Hs1 ) H2O + 2s15 Cs1 + Hs1 ) CHs1 + s1 9(n) CnH2n+1s1 + Hs1 ) CnH2n+2 + 2s16 CHs1 + Hs1 ) CH2s1 + s1 10(n) CnH2n+1s1 ) CnH2n + Hs17(n) CH2s1 + Hs1 ) CH3s1 + s1 FT IX 1 CO + s1 ) COs1

CH2s1 + CH3s1 ) CH3CH2s1 + s1 2 COs1 + s1 ) Cs1 + Os1CH2s1 + CnH2n+1s1 ) CnH2n+1CH2s1 + s1 3 Cs1 + Hs1 ) CHs1 + s1

8(n) CnH2n+1s1 + Hs1 ) CnH2n+2 + 2s1 4 CHs1 + Hs1 ) CH2s1 + s19(n) CnH2n+1s1 ) CnH2n + Hs1 5(n) CH2s1 + CH2s1 ) C2H4s1 + s1

FT V 0 H2 + 2s1 ) 2Hs1 CH2s1 + CnH2ns1 ) CnH2nCH2s1 + s11(n) CO + Hs1 ) Hs1CO 6 H2 + 2s1 ) 2Hs1

CO + CH3s1 ) CH3s1CO 7 Os1 + Hs1 ) HOs1 + s1CO + CnH2n+1s1 ) CnH2n+1s1CO 8 HOs1 + Hs1 ) H2O + 2s1

2(n) Hs1CO + H2 ) Hs1C + H2O 9(n) CnH2ns1 + Hs1 ) CnH2n+1s1 + s1CH3s1CO + H2 ) CH3s1C + H2O 10(n) CnH2n+1s1 + Hs1 ) CnH2n+2 + 2s1CnH2n+1s1CO + H2 ) CnH2n+1s1C + H2O 11(n) CnH2ns1 ) CnH2n + s1

3(n) Hs1C + H2 ) Hs1CH2 FT X 1 CO + s1 ) COs1CH3s1C + H2 ) CH3s1CH2 2 COs1 + s1 ) Cs1 + Os1CnH2n+1s1C + H2 ) CnH2n+1s1CH2 3 Cs1 + H2 ) CH2s1

4(n) CnH2n+1s1CH2 ) CnH2n+1CH2s1 4(n) CH2s1 + CH2s1 ) CH2CH2s1 + s15(n) CnH2n+1s1 + H2 ) CnH2n+2 + Hs1 CH2s1 + CnH2ns1 ) CnH2nCH2s1 + s16(n) CnH2n+1s1 ) CnH2n + Hs1 5 H2 + 2s1 ) 2Hs1

6 Os1 + H2 ) H2O + s17(n) CnH2ns1 + Hs1 ) CnH2n+1s1 + s18(n) CnH2n+1s1 + H2 ) CnH2n+2 + Hs19(n) CnH2ns1 ) CnH2n + s1

CnH2n+1s1 + Hs1 ) CnH2n+2 + 2s1 (24)

CnH2n+1s1 ) CnH2n + Hs1 (21)

CnH2ns1 ) CnH2n + s1 (22)

CnH2n+1s1 + H2 ) CnH2n+2 + Hs1 (23)


rates of the linear paraffins and olefins were derived.For each model, the possible rate-determining steps(RDSs) were identified, while all other steps wereassumed to be at quasi-equilibrium.

Before we derive the rate expression of hydrocarbonformation, first, it is assumed that the steady-stateconditions are reached for both the surface compositionof catalyst and the concentration of all of the intermedi-ates involved. Second, it is assumed that the rateconstant of the elementary steps for the formation ofhydrocarbons is independent of the carbon number ofthe intermediates involved in the elementary reactionsexcept for methane. Third, there are two types ofuniformly distributed active sites respectively for FTSand WGS reactions on the catalyst surface.4

For the derivation of the rate expressions, FT IIIin Table 1 will be demonstrated. It is assumedthat the RDSs are steps 5, 7, and 8 (model FT III).The remaining steps can be considered to be rapidand at equilibrium. The rates of formation of paraffinsand olefins with n carbon atoms can thus be writtenas

The pseudo-steady-state conditions are applied to theconcentration of surface intermediates [CnH2ns1]:

After rearrangement, eq 28 yields

where k5[CH2s1] is related to the chain growth ofhydrocarbon and k8

+(1 - k8-PCnH2n[s1]/k8

+[CnH2ns1]) isthe rate of olefin formation, which is the net effect ofdesorption and readsorption of olefins with n carbon

atoms. Here we introduce a readsorption factor ân,which is defined as follows:

The chain growth probability for the carbon chainwith n carbon atoms is

The concentration of surface intermediates can beexpressed as a function of the partial pressure of CO,H2, and H2O by applying the pseudo-equilibrium rela-tion

Substitution of eqs 32 and 33 in eq 31 yields

Equation 28 can be rearranged as

Equation 35 can be rewritten in the following form:13

where Xn, RA, Yn, and B are defined as follows:

RCH4) k7M[CH3s1][Hs1] )

k7MK6[CH2s1][Hs1]2/[s1] (25)

RCnH2n+2) k7[CnH2n+1s1][Hs1] )

k7K6[CnH2ns1][Hs1]2/[s1] (n g 2) (26)

RCnH2n) k8

+[CnH2ns1] - k8-PCnH2n

[s1] (n g 2) (27)

-d[CnH2nS1]

dt) k5[CnH2ns1][CH2s1] -

k5[Cn-1H2n-2s1][CH2s1] + k7[CnH2n+1s1][Hs1] +

k8+[CnH2nS1] - k8

-PCnH2n[s1] ) k5[CnH2ns1][CH2s1] -

k5[Cn-1H2n-2s1][CH2s1] + k7K6[CnH2ns1][Hs1]2/[s1] +

k8+[CnH2ns1] - k8

-PCnH2n[s1] ) 0 (n g 2) (28)

[CnH2ns1]

[Cn-1H2n-2s1])

k5[CH2s1]

k5[CH2s1] + k7K6[Hs1]2/[s1] + k8

+(1 -k8

-PCnH2n[s1]

k8+[CnH2ns1]

))

k5[CH2s1]

k5[CH2s1] + k7K6[Hs1]2/[s1] + k8

+(1 - ân)(n g 2) (29)

ân )k8

-

k8+

PCnH2n[s1]

[CnH2ns1](n g 2) (30)

Rn )k5[CH2s1]

k5[CH2s1] + k7K6[Hs1]2/[s1] + k8

+(1 - ân)(n g 2) (31)

[CH2s1] ) K1K2K3

PH2

2PCO

PH2O[s1] ) K′3

PH2

2PCO

PH2O[s1] (32)

[Hs1] ) xK4PH2[s1] (33)

Rn )[CnH2ns1]

[Cn-1H2n-2s1])

k5K′3PH2

2PCO

PH2O[s1]

k5K′3PH2

2PCO

PH2O[s1] + k7K6K4PH2

[s1] + k8+(1 - ân)

(n g 2) (34)

[CnH2ns1] )k5[CH2s1][Cn-1H2n-2s1]

k5[CH2s1] + k7K6[Hs1]2/[s1] + k8

+ +

k8-PCnH2n

[s1]

k5[CH2s1] + k7K6[Hs1]2/[s1] + k8

+ (n g 2) (35)

Xn ) RAXn-1 + BYn (36)

Xn ) [CnH2ns1] (n g 2) (37)

RA )k5[CH2s1]

k5[CH2s1] + k7K6K4PH2[s1] + k8

+ )

k5K′3PH2

2PCO

PH2O[s1]

k5K′3PH2

2PCO


[s1] + k8+

(38)


From eq 36, we get

where X1 and Yn-i+2 are defined as follows:

Thus, the concentration of [CnH2ns1] can be rewrittenas

Substituting eqs 32 and 47 into eq 30 yields

According to the definition of chain growth probabilityRn in eq 34, we have

Thus, the expression of the concentration of [CnH2n+1s1]is

The concentrations of the surface intermediates [CH3s1],[COs1], and [H2COs1] can be derived on the basis of theelementary steps 1-4 and 7 in FT III.

Normalization of the concentration of the sites on thecatalyst surface leads to

Combining eqs 32, 33, and 49-53 with eq 54 gives

The concentration of free active site [s1] can thus beexpressed as follows:

[CnH2n+1s1] ) K6[CnH2ns1][Hs1]/[s1] )

K6K40.5K′3

PH2

2.5PCO

PH2O

[s1] ∏i)2

n

Ri (n g 2) (50)

[CH3s1] ) K6K1K2K3K40.5

PH2

2.5PCO

PH2O[s1] )

K6K′3K40.5

PH2

2.5PCO

PH2O[s1] (51)

[COs1] ) K1PCO[s1] (52)

[H2COs1] )K2PH2

[COs1] ) K1K2PCOPH2[s1] ) K′2PCOPH2

[s1] (53)

1 ) [s1] + [Hs1] + [COs1] + [CH2s1] + [H2COs1] +

[CH3s1] + ∑i)2

n

[CnH2ns1] + ∑i)2

n

[CnH2n+1s1] (54)

1 ) [s1] + xK4PH2+ K1PCO[s1] +

K′3PH2

2PCO

PH2O

[s1] + K1K2PCOPH2[s1] +

K6K40.5K′3

PH2

2.5PCO

PH2O

[s1] + K′3PH2

2PCO

PH2O∑i)2

n

∏j)2

i

(Rj)[s1] +

K6K40.5K′3

PH2

2.5PCO

PH2O∑i)2

n

∏j)2

i

(Rj)[s1] (55)

[s1] ) 1/[1 + xK4PH2+ K1PCO +

K′3PH2

2PCO

PH2O

+ K1K2PCOPH2+

K6K40.5K′3

PH2

2.5PCO

PH2O

+ K′3PH2

2PCO

PH2O∑i)2

n

∏j)2

i

(Rj) +

K6K40.5K′3

PH2

2.5PCO

PH2O∑i)2

n

∏j)2

i

(Rj)] (56)

B )k8

-

k5[CH2s1] + k7K6K4PH2[s1] + k8

+ )

k8-

k5K′3PH2

2PCO


[s1] + k8+

(39)

Yn ) PCnH2n[s1] (n g 2) (40)

X2 ) RAX1 + BY2 (41)

X3 ) RA2X1 + RABY2 + BY3 (42)

X4 ) RA3X1 + RA

2BY2 + RABY3 + BY4 (43)

............

Xn ) RAn-1X1 + B∑

i)2

n

RAi-2Y(n-i+2) (n g 2) (44)

X1 ) [CH2s1] ) K ′3PH2

2PCO/PH2O[s1] (45)

Yn-i+2 ) PC(n-i+2)H2(n-i+2)[s1] (n g 2) (46)

[CnH2ns1] )

RAn-1[CH2s1] + B[s1]∑

i)2

n

RAi-2PC(n-i+2)H2(n-i+2)

)

RAn-1K′3PH2

2PCO/PH2O[s1] + B[s1]∑i)2

n

RAi-2PC(n-i+2)H2(n-i+2)

(47)

ân ) (k8-/k8

+){PCnH2n/[RA

n-1K′3PCOPH2

2/PH2O +

k8-

k5K′3PCOPH2

2/PH2O[s1] + k7K6K4PH2[s1] + k8

+×

∑i)2

n

(RAi-2PC(n-i+2)H2(n-i+2)

)]} (n g 2) (48)

[CnH2ns1] ) [CH2s1] ∏i)2

n

Ri )

K′3PH2

2PCO

PH2O

[s1]∏i)2

n

Ri (n g 2) (49)


Substituting eq 56 into eqs 49 and 50 yields

Substituting eqs 57 and 58 into eqs 25-27, we have

3.3. Kinetic Models for the WGS Reaction. Thekinetics of the WGS reaction has been intensivelystudied by many researchers, and several mechanismswere proposed.60-64 It is generally accepted that theWGS reaction over supported iron catalysts proceeds viaa mechanism of formate species due to a limited changeof oxidation states of the iron cations.4,60-64 Rethwischand Dumesic61 suggested that the WGS reaction pro-ceeds on active sites different from those for FTS, andfor supported iron catalysts, the magnetite is the mostactive phase for the WGS. On the basis of the formateintermediate mechanism, the elementary steps of theWGS, and corresponding expressions of rates, equilib-rium constants are summarized in Table 2.

If we assumed that the slowest step in the WGS isstep IV (WGS 3), the rate of CO2 formation can bewritten as follows:

From the elementary step listed in Table 3, we have

On the basis of eqs 63-66, the concentrations of thesurface intermediates [CO-s2], [OH-s2], and [COOH-s2] can be derived as follows:

If Kp is used to represent the equilibrium constant ofthe WGS reaction, it can be expressed by the equilib-rium partial pressures of CO, CO2, H2, and H2O.

[CnH2ns1] )

K′3PH2

2PCO

PH2O∏i)2

n

Ri/[1 + xK4PH2+ K1PCO +

K′3PH2

2PCO

PH2O

+ K1K2PCOPH2+ K6K4

0.5K′3PH2

2.5PCO

PH2O

+

K′3PH2

2PCO

PH2O∑i)2

n

∏j)2

i

(Rj) + K6K40.5K′3

PH2

2.5PCO

PH2O∑i)2

n

∏j)2

i

(Rj)](n g 2) (57)

[CnH2n+1s1] )

K6K40.5K′3

PH2

2.5PCO

PH2O∏i)2

n

Ri/[1 + xK4PH2+ K1PCO +

K′3PH2

2PCO

PH2O

+ K1K2PCOPH2+ K6K4

0.5K′3PH2

2.5PCO

PH2O

+

K′3PH2

2PCO

PH2O∑i)2

n

∏j)2

i

(Rj) + K6K40.5K′3

PH2

2.5PCO

PΗ2Ã∑i)2

n

∏j)2

i

(Rj)](n g 2) (58)

RCH4)

k7MK4K6K′3PH2

3PCO

PH2O

/[1 + xK4PH2+ K1PCO +

K′3PH2

2PCO

PH2O

+ K1K2PCOPH2+ K6K4

0.5K′3PH2

2.5PCO

PH2O

+

K′3PH2

2PCO

PH2O

(1 + K6xK4PH2)∑i)2

n

∏j)2

i

(Rj)]2

(59)

RCnH2n+2)

k7K4K6K′3PH2

3PCO

PH2O∏j)2

n

Rj /[1 + xK4PH2+ K1PCO +

K′3PH2

2PCO

PH2O

+ K1K2PCOPH2+ K6K4

0.5K′3PH2

2.5PCO

PH2O

+

K′3PH2

2PCO

PH2O

(1 + K6xK4PH2)∑i)2

n

∏j)2

i

(Rj)]2

(60)

RCnH2n) k8

+(1 - ân)K′3PH2

2PCO/PH2O∏j)2

n

Rj /

[1 + xK4PH2+ K1PCO + K′3

PH2

2PCO

PH2O

+

K1K2PCOPH2+ K6K4

0.5K′3PH2

2.5PCO

PH2O

+

K′3PH2

2PCO

PH2O

(1 + K6xK4PH2)∑i)2

n

∏j)2

i

(Rj)] (61)

RCO2) r4 ) kWGS4[COOH-s2] -

k-WGS4PCO2[H-s2] (62)

[CO-s2] ) KWGS1PCO[s2] (63)

KWGS2PH2O[s2]2 ) [OH-s2][H-s2] (64)

KWGS3[OH-s2][CO-s2] ) [COOH-s2][s2] (65)

[H-s2] ) xKWGS5PH2[s2] (66)

[CO-s2] ) KWGS1PCO[s2] (67)

[OH-s2] ) KWGS2KWGS5-0.5PH2OPH2

-0.5[s2] (68)

[COOH-s2] )

KWGS1KWGS2KWGS3KWGS5-0.5PH2OPCOPH2

-0.5[s2] (69)

Kp )PCO2

/ PH2

/

PCO/ PH2O

/(70)


According to eqs 63, 64, and 66, the partial pressuresof CO, H2O, and H2 can be written as follows:

If we assumed that step IV reached the equilibriumstate, the expression of partial pressure of CO2 at theequilibrium state (PCO2

/ ) can be obtained:

Substituting eq 71-74 into eq 70 gives

When eq 65 is substituted into eq 75, Kp can beexpressed as

Equation 76 can rewritten as follows:

Substituting eq 77 into eq 69 yields

As a result of the assumption that the RDS is stepIV, it can be considered that the concentration of theadsorbed species [COOH-s2] is much larger than thoseof the other adsorbed species.

The concentration of the empty active site, [s2], can beobtained by substituting eq 78 into eq 79

Thus, the expressions of [H-s2] and [COOH-s2] can beobtained by substituting eq 80 into eqs 66 and 78.

Finally, the rate expression of CO2 formation can beexpressed as

where Kv ) KpKWGS4KWGS50.5 and kv ) kWGS4Kv.

By different assumptions of the RDS in WGS, otherrate expressions of CO2 of the WGS reaction can bederived, and the results are tabulated in Table 3. Theequilibrium constant Kp of the WGS reaction can becalculated by the following relation:11-13

A total of 39 kinetic models of FTS can be obtained bythe combination of 13 FTS models with 3 WGS models.

4. Results and Discussion

4.1. Experimental Results. Iron-based Fischer-Tropsch catalysts, once exposed in the synthesis gasunder typical reduction or initial reaction conditions,are normally reduced by H2 and CO, transforming fromtheir oxide phases to metallic and carbide phases.40,41-45

During this transformation stage, the phases of cata-lysts greatly change with time on stream as well aschanges in operation conditions, leading to irreproduc-ible experimental data. To minimize (it can never becompletely eliminated in FTS over iron catalysts) theeffect of a “sharp” phase change on the quality of kineticdata, the catalyst used for kinetic tests was reduced andthen stabilized at a fixed operation condition for about1000 h, aiming at the complete establishment of thecatalyst phase for FTS. After this stabilization stage,operation conditions were switched according to kinetic

Table 2. Elementary Steps and Corresponding Expressions of Rates and Equilibrium Constants for the WGS Reaction

step elementary reaction expressions of rates and equilibrium constants

I CO + s2 ) COs2 KWGS1 ) [COs2]/PCO[s2]II H2O + 2s2 ) OHs2 + Hs2 KWGS2 ) [OHs2][Hs2]/PH2O[s2]2

III COs2 + OHs2 ) COOHs2 + s2 KWGS3 ) [COOHs2][s2]/[COs2][OHs2]IV COOHs2 ) CO2 + Hs2 r4 ) kWGS4[COOHs2] - k-WGS4PCO2[Hs2]

KWGS4 ) kWGS4/k-WGS4

V 2Hs2 ) H2 + 2s2 1/KWGS5 ) PH2[s2]2/[Hs2]2

Table 3. Rate Expressions for the WGS Reaction

model RDS rate expression

WGS1 step I RCO2 ) kv(PCO - PCO2PH2PH2O-1/Kp)/(1 + KvPCOPH2/PH2O)

WGS2 step III RCO2 ) kv(PCOPH2O/PH20.5 - PCO2PH2

0.5/Kp)/(1 + KvPH2O/PH20.5)2

WGS3 step IV RCO2 ) kv(PCOPH2O/PH20.5 - PCO2PH2

0.5/Kp)/(1 + KvPCOPH2O/PH20.5)

[H-s2] )xKWGS5PH2

1 + KpKWGS4KWGS50.5PCOPH2O/PH2

0.5(81)

[COOH-s2] )

KpKWGS4KWGS50.5PCOPH2O/PH2

0.5

1 + KpKWGS4KWGS50.5PCOPH2O/PH2

0.5(82)

RCO2)

kv(PCOPH2O/PH2

0.5 - PCO2PH2

0.5/Kp)

1 + KvPCOPH2O/PH2

0.5(83)

Kp ) 5078.0045T

- 5.8972089 + 13.958689 ×10-4T - 27.592844 × 10-8T 2 (84)

PCO/ ) [CO-s2]/(KWGS1[s2]) (71)

PH2O/ ) [OH-s2][H-s2]/(KWGS2[s2]

2) (72)

PH2

/ ) [H-s2]2/(KWGS5[s2]

2) (73)

PCO2

/ ) [COOH-s2]*/(KWGS4[H-s2]*) (74)

Kp )KWGS1KWGS2[COOH-s2]*[s2]*

KWGS4KWGS5[CO-s2]*[OH-s2]*(75)

Kp )KWGS1KWGS2KWGS3

KWGS4KWGS5(76)

KWGS3 )KWGS4KWGS5KP

KWGS1KWGS2(77)

[COOH-s2] )

KpKWGS4KWGS50.5PCOPH2OPH2

-0.5[s2] (78)

[COOH-s2] + [s2] ) 1 (79)

[s2] ) 11 + KpKWGS4KWGS5

0.5PCOPH2O/PH2

0.5(80)


sampling arrangements. The reaction results during thestabilization stage are summarized in Table 4 andFigure 2. From these results, it can be found that thecatalyst reached a stable stage after about 500 h onstream under the conditions used. The selectivity to CO2was maintained at the level of 40 ( 2%, that to CH4 at16 ( 1%, that to C5

+ at 50 ( 2%. For the activity, COconversion reached to a level of 95 ( 1%. This confirmedthat kinetic data sampling may be based on a ratherstable stage of the catalyst in the following kineticexperimental stage.

Kinetic sampling conditions were arranged accordingto orthogonal arrangement of sampling points, enablingefficient and optimal distribution of experimental points.13

The reaction condition variables that need to be con-sidered are temperature, pressure, and H2/CO ratio. Foreach variable, four values were planned, bringing about16 experimental points (corresponding to an orthogonal

table of L16(43)). The experimental results obtained inthis investigation for different reaction temperatures,total reaction pressures, space velocities, and H2/COratios are given in Table 5. Five more points were addedto consider more reaction conditions, among which onepoint (no. 16 in Table 5) was the reference point (no. 12in Table 5). The order of kinetic experiments was fromno. 1 to no. 21 listed in Table 5. From the results listed,it can be found that the CO and H2 conversions at no.16 well reproduced those at no. 12, indicating that thecatalytic phase reached a stable state.

4.2. Estimation of the Kinetic Parameters. 4.2.1.Reactor Model of the Fixed Bed. The reactor modelfor describing the kinetic experimental conditions isassumed to be a plug-flow homogeneous state. At thisstage, the transportation in catalyst pores and thesolubilities of different hydrocarbons are not consideredin order to avoid the unsolvable difficulties in the

Figure 2. Catalyst stability with time on stream (H2/CO ) 2.0; P ) 23 bar; GHSV ) 5.1 × 10-4 Nm3 kg of catalyst-1 s-1).

Table 4. Catalyst Performance with Time on Stream during the Stability Test

time on stream (h)

34 83 180 273 404 516 641 845 960

temperature (K) 543 543 556 556 556 556 556 556 556pressure (MPa) 2.30 2.30 2.30 2.30 2.30 2.30 2.30 2.30 2.30GHSV (10-4 Nm3 kg of catalyst-1 s-1) 5.1 5.1 5.1 5.1 5.1 5.1 5.1 5.1 5.1CO conversion % 94.8 94.6 96.1 96.9 95.9 96.4 96.0 96.1 96.0H2 conversion % 43.7 40.1 44.3 47.6 41.7 44.6 47.4 45.8 44.8H2/CO (in feed gas) 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0H2/CO (in tail gas) 21.55 22.13 28.33 34.35 28.61 30.43 25.95 27.25 28.21selectivity CO2 (%) 36.09 40.88 38.71 39.13 36.73 39.27 40.37 38.70 38.80selectivity (wt %)

C1 7.51 9.39 13.35 14.44 15.62 16.80 16.76 16.84 16.90C2

0 3.00 4.26 6.17 7.04 7.41 8.20 8.20 8.24 8.30C2

) 5.31 5.18 5.35 4.81 4.87 4.91 4.85 4.39 4.41C3

0 1.76 2.18 2.77 2.94 3.32 3.55 3.63 3.59 3.63C3

) 7.63 8.15 8.82 8.91 8.61 9.08 9.28 8.47 8.50C4

0 1.51 1.83 2.07 2.11 2.31 2.51 2.51 2.49 2.48C4

) 6.04 6.19 6.58 6.46 6.68 6.63 7.05 6.37 6.40C2-4

)/tol‚HC (wt %) 18.99 19.51 20.75 20.18 20.16 20.62 21.18 19.23 19.21C2-4

)/C2-40 2.89 2.19 1.70 1.48 1.38 1.28 1.30 1.18 1.17

C5+ 67.24 62.82 54.89 53.29 51.19 48.33 47.72 49.61 49.40



integration of the reactor model embedded in a param-eter optimization procedure. This simplification maylose the transportation-enhanced readsorption phenom-ena, and further modification to the kinetic modelsobtained is needed. This will be discussed elsewhere.

From the group of equations (1)-(3), the total numberof the components, Nc, and of the reactions, NR, are 2N+ 3 and 2N, respectively, where N is the maximumcarbon number of hydrocarbons considered. The matrixof the stoichiometric coefficients between the productsand reactions according to the FTS and the WGSreaction is listed in Table 6.11-13

The model of the fixed-bed reactor used in our kineticstudy can be described as follows:11-13

with the initial condition

where W is the weight of the catalyst used, mi and mi,0are the mole flow rates of component i along the reactoraxis and at the inlet of the reactor, NR is the number ofreactions involved, Nc is the total number of compo-nents, Rij is the stoichiometric coefficient for the ithcomponent in reaction j, and Rj is the rate of reaction j

(RCnH2n+2, RCnH2n, RCO2, ...). The partial pressure of theith component can be calculated by using the followingformula:

where PT is the total pressure in the reactor.Numerical integration of the continuity equation (85)

is performed by using Gear’s method.65

4.2.2. Optimization Method. In the estimation ofparameters of the kinetic model, the Levenberg-Mar-quardt (LM) algorithm still plays an important role.However, in most cases, the objective function basedupon the nonlinear and experimental data frequentlycontains more than one minimum.66 As a generalalgorithm, LM-type continuum methods often fail inlocating global minima. In our paper, to avoid gettingtrapped in local minima, the parameters of the variousrival models in this paper were estimated in a first stepusing the genetic algorithm (GA) approach that wasdeveloped in this group67 and then using the LMalgorithm to make refined optimization, after whichthe statistical tests and the physiochemical constraintsare used to evaluate the significance of models andparameters. The GA algorithm encoded in this groupusing a real-number code chromosome representation to

Table 5. Operation Conditions and Resulting Product Quantities

no.T

(°C)P

(MPa)H2/COratio

Fin(mL/min)

reactiontime (h)

CO convn(%)

H2 convn(%)

Vexit(mL/min)

oil(g)

wax(g)

water(g)

1 283.0 1.98 2.05 342.0 10.60 40.9 28.2 308.9 2.46 2.14 7.202 283.2 2.51 2.62 348.6 10.67 57.0 28.9 292.5 3.62 2.83 8.043 283.3 3.05 3.08 435.6 7.01 57.6 26.8 375.8 2.16 1.85 8.404 283.1 1.50 1.03 343.6 12.86 21.8 21.0 341.6 1.92 3.36 5.105 283.2 3.05 3.06 349.3 11.33 70.1 28.9 287.8 3.52 3.24 14.306 297.0 2.02 1.03 449.7 6.02 35.8 27.6 404.3 2.28 3.06 4.507 297.2 2.51 3.13 462.9 10.83 69.0 25.8 408.7 4.21 3.41 15.528 297.1 3.01 2.58 456.1 7.51 74.9 36.6 346.7 3.89 2.64 10.029 297.2 3.02 2.59 606.9 10.22 59.2 33.7 482.6 5.83 4.27 15.77

10 297.1 1.50 2.07 454.7 10.51 35.2 26.8 411.5 2.71 2.02 8.2011 297.0 2.05 2.05 342.0 6.02 66.1 34.5 270.2 2.46 1.62 4.6012 312.2 2.02 3.05 684.6 7.03 63.3 27.7 575.0 3.02 0.78 12.7013 312.3 3.02 2.04 686.9 13.25 70.9 38.0 492.5 12.56 4.55 27.5914 312.1 2.50 1.02 678.6 11.04 42.6 31.1 599.4 9.18 7.49 12.6015 312.0 1.50 2.55 686.9 9.01 38.9 17.1 672.4 3.23 0.58 11.5016 312.2 2.02 3.05 684.8 7.02 63.8 27.1 569.2 3.53 0.68 13.2317 328.5 2.50 2.04 892.6 5.51 67.4 35.0 713.7 5.75 1.96 14.0018 328.4 2.02 2.55 894.1 7.01 59.3 25.5 788.7 4.60 1.20 15.4019 328.3 1.51 3.05 896.1 8.05 45.0 17.3 866.4 2.82 0.83 13.3020 328.4 3.02 1.02 908.4 8.04 63.9 47.2 665.2 13.33 6.72 15.1021 328.1 2.50 2.55 1378.9 4.50 66.1 32.4 1123.5 5.25 1.20 16.50

Table 6. Stoichiometric Coefficients of the Matrix for FTS Reactions

reaction pathcomponent

reactants CO + H2O CO + 3H2 2CO + 4H2 2CO + 5H2 nO + 2nH2 nO + (2n + 1)H2

products CO2 + H2 CH4 + H2O C2H4 + 2H2O C2H6 + 2H2O CnH2n + nH2O CnH2n+2 + nH2OCO -1 -1 -2 -2 -n -nH2 1 -3 -4 -5 -2n -(2n + 1)CO2 1 0 0 0 0 0H2O -1 1 2 2 n nCH4 0 1 0 0 0 0C2H4 0 0 1 0 0 0C2H6 0 0 0 1 0 0...CnH2n 0 0 0 0 1 0CnH2n+2 0 0 0 0 0 1

dmi

dW) ∑

j)1

NR

RijRj (i ) 1, 2, ..., Nc; j ) 1, 2, ..., NR) (85)

W ) 0; mi ) mi,0 (86)

Pi )mi

∑i)1

Nc

mi

PT (i ) 1, 2, 3, ..., Nc) (87)


ensure a wide searching space for optimal kineticparameters. The GA procedure consists mainly of fourparts: randomly producing an initial population, ran-domly selecting two individuals from the populationspace to crossover (using certain operator), producing thenext generation of population, nonuniform mutation, andrepeating these steps to scan the whole searching space.

For estimation of the models, 20 model responsesin the regression procedure were the outlet concen-trations of 15 paraffins and olefins (such as CH4, C2H4,C2H6, C3H6, C3H8, C4H8, C4H10, etc.) selected as themost significant Fischer-Tropsch products, along withthe concentrations of CO2, the conversions of CO andH2, and the overall concentrations of C5

+ and C11+

hydrocarbons. The objective function is defined asfollows:

where mi,exp and mi,cal are the experimental and calcu-lated values of conversions of reactants or the selectivi-ties of products, respectively, and Nexp is the totalnumber of experimental runs. Because of the com-plexity of the models, a multiresponse objective functionshould be introduced, which is expressed in the follow-ing form:

where Wi represents the weighting factor of the ithobjective function (it expresses the relative importanceof the ith response). Those responses with the mostaccurate measurement and/or with special significancein the regression are provided with greater weights):Nresp is the number of responses in the system, mij,exp isthe experimental value of the ith response for the jthkinetic experiment, and mij,cal is the calculated value ofthe ith response for the jth kinetic experiment.

The adjustable model parameters for the severalkinetic models were calculated by minimizing FTol,objwith the GA and LM methods.

The accuracy of the fitted model relative to theexperimental data was obtained from the MARR (meanabsolute relative residuals) function:

The relative residual (RR) between experimental andcalculated values of responses will be used to show thedeviations between the model and experiment

The dependence of the reaction rate parameters ontemperature can be described by the Arrhenius law.

4.2.3. Parameter Estimation. For scanning themodels by parameter optimization, several basic physi-cal criteria are applied: the rate constants and equi-librium constants should be positive, and the activationenergies for the paraffins and olefin formation and forthe WGS reaction should be in the range of valuesreported by other researchers. The optimization proce-dure by the GA and LM methods has been set to findthe minimum, which provided (1) a reasonable fit to theexperimental data, (2) physically meaningful values ofthe model parameters, and (3) acceptable values ofstatistical parameters, such as F values for the modelsand t values for the parameters.

A total of 25 kinetic models are rejected because ofunreasonable values of the parameters. The discrimina-tion of the remaining 14 rival models is performed bycomparing first their MARR and second statistical testson models (F test) and parameters (t test). The resultshows that only two kinetic models’ MARR are withinthe 20%. The two models are FT III WGS3 (MARR18.6%) and FT VI WGS3 (MARR 19.2%).

4.3. Results and Discussion. The parameters ofthe remained best model are listed in Table 7, andall of them are statistically significant. The estimatedactivation energy for chain growth, E5, is 75.52 kJmol-1, indicating that the methylene insertion intothe metal-alkylidene bond has a moderate height ofenergy barrier that it needs to overcome. The activationenergy for methane formation, E7M, is 97.39 kJ mol-1,which is in good agreement with that reported byVannice58 (89 kJ mol-1) and smaller than that for otherparaffin formation, E7, with a value of 111.48 kJ mol-1,which can explain the higher selectivity of methane thanthose of other paraffins. The activation energy of olefinformation is 97.37 kJ mol-1, much smaller than that ofparaffins, which can interpret the general fact that theselectivity is much higher to alkenes on the Fe-Mncatalysts than on other iron-based catalysts.67 This isdifferent from Dictor and Bell69 (80-90 kJ mol-1 forparaffins and 100-110 kJ mol-1 for olefins) as well asLox and Froment11,12 (94.5 kJ mol-1 for paraffins and

Table 7. Values of the Parameters for the Mechanism FT III WGS3a

parameter value unit parameter value unit

k5,0 7.88 × 103 mol g-1 s-1 bar-1 Ev 58.43 kJ mol-1

E5 75.52 kJ mol-1 k-8 2.77 × 10-5 mol g-1 s-1 bar-1

k7M,0 2.01 × 106 mol g-1 s-1 bar-1 Kv 2.76 × 10-2 bar-0.5

E7M 97.39 kJ mol-1 K1 2.59 bar-1

k7,0 1.10 × 106 mol g-1 s-1 bar-1 K2 1.67 × 10-3 bar-1

E7 111.48 kJ mol-1 K3 8.34 × 10-2

k8,0 8.79 × 103 mol g-1 s-1 K4 1.21 bar-1

E8 97.37 kJ mol-1 K6 0.10kv,0 3.42 mol g-1 s-1 bar-1.5

a All of the energetic values are estimated to be in the 95% confidence level, and frequencies may be lowered to 90%.

fi,obj ) (mi,exp - mi,cal

mi,exp)2

(i ) 1, 2, ..., Nexp) (88)

FTol,obj ) ∑i)1

Nresp

∑j)1

Nexp

Wi(mij,exp - mij,cal

mij,exp)2

(i ) 1, 2, ..., Nresp; j ) 1, 2, ..., Nexp) (89)

MARR ) ∑j)1

Nresp

∑i)1

Nexp(mi,exp - mi,cal

mi,exp) 1

NrespNexp

× 100 (90)

RR )mi,exp - mi,cal

mi,exp× 100 (91)

ki(T) ) ki,0 exp(-Ei/RT) (92)


132.3 kJ mol-1 for olefins) over a Fe-Cu-K catalyst. Itis evident that all of the estimated activation energiesof the hydrocarbon formation are mostly within thosereported in the literature (all between 70 and 105 kJmol-1).11,12,17,69 The activation energy of 58.43 kJ mol-1

for the WGS reaction in this work is comparable withthe values reported in the literature.6,8,11,12,70

A comparison between the experimental and calcu-lated values of conversion of CO and selectivities of CO2are presented in Figures 3 and 4. The figures show thatthe RRs between the model and experiment are mostlywithin 20%.

Figures 5-10 show the comparison of experimentaland calculated product distributions. Figure 5, 7, and 9are predicted by model FT III WGS3 and Figure 6, 8,and 10 by Lox and Froment’s ASF model. The ASFmodel appears to give a strong deviation for the selec-tivity to hydrocarbons, lower to methane and higher toother hydrocarbons. As shown in these figures, theselectivities to olefins predicted with the ASF type modelare lower than those to paraffins, in contrast with theexperimental results. The modeled product distributionsusing FT III WGS3 are in good agreement with theexperimental selectivities, and the deviation for meth-ane is described fairly accurately.

It should be noted that the current kinetic model hasnot considered the effect of diffusivity and solubility ofolefins with different chain sizes on the paraffin/olefinratio on the basis of the understanding of different

orientations of kinetic and reactor models in consideringthe intrinsic mechanism information and the transpor-tation effects. The kinetic model developed is significantonly for the cases excluding the diffusivity and solubilityfactors, which are believed to be significant in enhancingthe olefin readsorption and secondary reaction and,therefore, in changing the olefin and paraffin selectivi-ties. However, kinetic experiments can never excludethe diffusivity and solubility (mobility) especially of

Figure 3. Comparison of calculated and experimental CO conver-sions (FT III WGS3).

Figure 4. RRs of calculated and experimental CO2 selectivity (FTIII WGS3).

Figure 5. Comparison of the calculated and experimental productdistribution (FT III WGS3 reaction conditions: T ) 556 K, P )2.51 MPa, H2/CO ) 2.62, GHSV ) 1.6 × 10-3 Nm3 kg of catalyst-1

s-1).

Figure 6. Comparison of the calculated and experimental productdistribution (Lox and Froment’s model reaction conditions: T )556 K, P ) 2.51 MPa, H2/CO ) 2.62, GHSV ) 1.6 × 10-3 Nm3 kgof catalyst-1 s-1).

Figure 7. Comparison of the calculated and experimental productdistribution (FT III WGS3 reaction conditions: T ) 585 K, P )3.02 MPa, H2/CO ) 2.04, GHSV ) 3.2 × 10-3 Nm3 kg of catalyst-1

s-1).







heavy olefins. Figure 11 shows the dependence of theolefin-to-paraffin ratio on the carbon number. It can beseen that the kinetic model without considering thetransportation effect (diffusivities and solubilities) hasnot predicted the correct dependence compared withexperimental results. This again proved the understand-ing in FTS that pure olefin readsorption and secondaryreaction have little consequence to the observed olefin/

paraffin ratio dependence on the carbon number if thetransportation enhancement is not considered.19 Wehave recently tested the kinetic models developed in thisgroup in a reactor simulation task, in which all of thetransportation effects are comprehensively consideredthanks to the nonintrinsic factor excluded kineticmodels, and the primary results recovered the experi-mentally observed olefin/paraffin ratio dependence cor-rectly. This later work is too extensive to present hereand will be discussed elsewhere.

5. Conclusions

Kinetic experiments of the Fischer-Tropsch reactionover an industrial Fe-Mn ultrafine particle catalyst arecarried out for a wide range of industrially relevantconditions. Different reaction equation combinationswere evaluated by global parameter optimization, whichinvolved the minimization of the multiresponse objectivefunction by a GA approach and a LM method. It wasfound that a kinetic model for the FTS based on thealkylidene mechanism gives the best fit of the experi-mental data. The best model shows that two elementarysteps, the insertion of methylene into the metal-alkylidene bond and the desorption of hydrocarbonproducts, are intrinsically slower than the others in FTSand the RDS of the WGS reaction is the desorption ofCO2 via formate intermediate species. The activationenergy for alkene formation is 97.37 kJ mol-1 and ismuch smaller than that for paraffins formation, 111.48kJ mol-1, which can interpret the higher alkene selec-tivity on Fe-Mn catalysts than on other iron-basedcatalysts.

Acknowledgment

Financial support from the Chinese Academy ofSciences (Project No. KGCX1-SW-02), Committee ofScience and Technology of China via 863 plan (ProjectNo. 2001AA523010), Shanxi Natural Science Founda-tion (20031032), and the National Natural SciencesFoundation of China (Project No. 29673054) is gratefullyacknowledged.

Figure 8. Comparison of the calculated and experimental productdistribution (Lox and Froment’s model reaction conditions: T )585 K, P ) 3.02 MPa, H2/CO ) 2.04, GHSV ) 3.2 × 10-3 Nm3 kgof catalyst-1 s-1).

Figure 9. Comparison of the calculated and experimental productdistribution (FT III WGS3 reaction conditions: T ) 601 K,P ) 1.51 MPa, H2/CO ) 3.05, GHSV ) 4.2 × 10-3 Nm3 kg ofcatalyst-1 s-1).

Figure 10. Comparison of the calculated and experimentalproduct distribution (Lox and Froment’s model reaction condi-tions: T ) 601 K, P ) 1.51 MPa, H2/CO ) 3.05, GHSV )4.2 × 10-3 Nm3 kg of catalyst-1 s-1).

Figure 11. Comparison of the calculated and experimental olefin/paraffin ratios for different carbon numbers (FT III WGS3 reactionconditions: T ) 585 K, P ) 3.02 MPa, H2/CO ) 2.04, GHSV )3.2 × 10-3 Nm3 kg of catalyst-1 s-1).






Appendix: Rate Expressions of the Different Models of FTS

Model FT I: RDS, steps 5-7.

Model FT II: RDS, steps 5-7.

RCH4) k6MK4PH2

R1/(1 + xK4PH2+ K1PCO + K′3PH2

2PCO/PH2O + K1K2PH2PCO + xK4PH2 ∑

i)1

n

(∏j)1

i

Rj))2 (A1)

RCnH2n+2) k6K4PH2∏

j)1

n

Rj /(1 + xK4PH2+ K1PCO + K′3PH2

2PCO/PH2O + K1K2PH2PCO + xK4PH2 ∑

i)1

n

(∏j)1

i

Rj))2 (n g 2)

(A2)

RCnH2n) k7

+(1 - ân)xK4PH2∏j)1

n

Rj / [1 + xK4PH2+ K1PCO + K′3PH2

2PCO/PH2O + K1K2PH2PCO +

xK4PH2 ∑i)1

n

(∏j)1

i

Rj)] (n g 2) (A3)

ân ) (k7-/k7

+){PCnH2n/[R1RA

n-1 +k7

-

k5K′3PCOPH2

2

PH2O

[s1] + k6xK4PH2[s1] + k7

+

∑i)2

n


)]} (n g 2) (A4)

Rn )k5K′3PH2

2PCO/PH2O

k5K′3PH2

2PCO/PH2O + k6xK4PH2+ k7

+(1 - ân)/[s1](n g 2) (A5)

R1 )k5K′3PH2

1.5PCO/PH2O

k5K′3PH2

1.5PCO/PH2O + k6xK4

(A6)

RA )

k5K′3PH2

2PCO

PH2O[s1]

k5K′3PH2

2PCO

PH2O[s1] + k6xK4PH2

[s1] + k7+

(A7)

RCH4) k6MK4

0.5PH2

1.5R1/[1 + xK4PH2+ K1PCO + K′3

PH2

2PCO

PH2O

+ K1K2PH2PCO + xK4PH2 ∑

i)1

n

(∏j)1

i

Rj)] (A8)

RCnH2n+2)

k6K40.5PH2

1.5∏j)1

n

Rj /[1 + xK4PH2+ K1PCO + K′3

PH2

2PCO

PH2O


i)1

n

(∏j)1

i

Rj)] (n g 2) (A9)

RCnH2n)

k7+(1 - ân)xK4PH2 ∏

j)1

n

Rj /[1 + xK4PH2+ K1PCO + K′3

PH2

2PCO

PH2O


i)1

n

(∏j)1

i

Rj)] (n g 2)

(A10)

Rn )

k5K′3PH2

2PCO

PH2O[s1]

k5K′3PH2

2PCO

PH2O[s1] + k6PH2

+ k7+(1 - ân)

(n g 2) (A11)


Model FT III: RDS, steps 5, 7, and 8.

R1 )

k5K′3PH2

2PCO

PH2O[s1]

k5K′3PH2

2PCO

PH2O[s1] + k6

(A12)

RA )

k5K′3PH2

2PCO

PH2O[s1]

k5K′3PH2

2PCO

PH2O[s1] + k6PH2

+ k7+

(A13)

ân ) (k7-/k7

+){PCnH2n/[R1RA

n-1 +k7

-

k5K′3PCOPH2

2

PH2O

[s1] + k6PH2+ k7

+

∑i)2

n


)]} (n g 2) (A14)

RCH4) k7MK4K6K′3

PH2

3PCO

PH2O

/[1 + xK4PH2+ K1PCO + K′3

PH2

2PCO

PH2O

+ K1K2PCOPH2+ K6K4

0.5K′3PH2

2.5PCO

PH2O

+

K′3PH2

2PCO

PH2O

(1 + K6xK4PH2)∑i)2

n

∏j)2

i

(Rj)]2

(A15)

RCnH2n+2) k7K4K6K′3

PH2

3PCO

PH2O∏j)2

n

Rj /[1 + xK4PH2+ K1PCO + K′3

PH2

2PCO

PH2O

+ K1K2PCOPH2+ K6K4

0.5K′3PH2

2.5PCO

PH2O

+

K′3PH2

2PCO

PH2O

(1 + K6xK4PH2)∑i)2

n

∏j)2

i

(Rj)]2

(n g 2) (A16)

RCnH2n) [k8

+(1 - ân)K′3PH2

2PCO/PH2O∏j)2

n

Rj]/[1 + xK4PH2+ K1PCO + K′3

PH2

2PCO

PH2O

+ K1K2PCOPH2+

K6K40.5K′3

PH2

2.5PCO

PH2O

+ K′3PH2

2PCO

PH2O

(1 + K6xK4PH2)∑i)2

n

∏j)2

i

(Rj)] (n g 2) (A17)

Rn )

k5K′3PH2

2PCO

PH2O

k5K′3PH2

2PCO

PH2O+ k7K6K4PH2

+ k8+(1 - ân)/[s1]

(n g 2) (A18)

ân ) (k8-/k8

+){PCnH2n/[RA

n-1K′3PCOPH2

2/PH2O +k8

-

k5K′3PCOPH2

2/PH2O[s1] + k7K6K4PH2[s1] + k8

+×

∑i)2

n


)]} (n g 2) (A19)

RA )k5K′3PH2

2PCO/PH2O[s1]

k5K′3PH2

2PCO/PH2O[s1] + k7K6K4PH2[s1] + k8

+ (A20)


Model FT IV: RDS, steps 7-9.

Model FT V(I): RDS, steps 1, 5, and 6.

RCH4) k8MK0PH2

R1/(1 + xK0PH2+ K1PCO(1 + K2xK0PH2

) + PH2O/K4 +

K4K3K2K1K0PCOPH2/PH2O(1 + K5xK0PH2

+ K6K5K0PH2) + xK0PH2 ∑

i)1

n

(∏j)1

i

Rj))2 (A21)

RCnH2n+2) k8K0PH2 ∏

j)1

n

Rj /(1 + xK0PH2+ K1PCO(1 + K2xK0PH2

) + PH2O/K4 +


+ K6K5K0PH2) + xK0PH2 ∑

i)1

n

(∏j)1

i

Rj))2 (n g 2) (A22)

RCnH2n) k9

+(1 - ân)xK0PH2 ∏j)1

n

Rj /[1 + xK0PH2+ K1PCO(1 + K2xK0PH2

) + PH2O/K4 +


+ K6K5K0PH2) + xK0PH2 ∑

i)1

n

(∏j)1

i

Rj)] (n g 2) (A23)

Rn )k7K′3PH2

2PCO/PH2O

k7K′3PH2


+(1 - ân)/[s1](n g 2) (A24)

R1 )k7K′3PH2

2PCO/PH2O

k7K′3PH2

2PCO/PH2O + k8xK1PH2

(n ) 1) (A25)

ân ) (k9-/k9

+){PCnH2n/[R1RA

n-1 +k9

-

k7K′3PH2

2PCO/PH2O[s1] + k8xK0PH2[s1] + k9

+∑i)2

n


)]} (n g 2)

(A26)

RA )k7K′3PH2

2PCO/PH2O[s1]

k7K′3PH2


+(A27)

RCH4) k5MPH2xK0PH2

R1/[1 + xK0PH2+ (1 +

1

K2K3K4

PH2O

PH2

2+

1

K3K4

1

PH2

+1

K4)xK0PH2 ∑i)1

n

(∏j)1

i

Rj)] (A28)

RCnH2n+2) k5PH2xK0PH2 ∏

j)1

n

Rj /[1 + xK0PH2+ (1 +

1

K2K3K4

PH2O

PH2

2+

1

K3K4

1

PH2

+1

K4)xK0PH2 ∑i)1

n

(∏j)1

i

Rj)] (n g 2) (A29)

RCnH2n) k6

+(1 - ân)xK4PH2∏j)1

n

Rj /[1 + xK0PH2+ (1 +

1

K2K3K4

PH2O

PH2

2+

1

K3K4

1

PH2

+1

K4)xK0PH2 ∑i)1

n

(∏j)1

i

Rj)] (n g 2)

(A30)

Rn )k1PCO

k1PCO + k5PH2+ k6

+(1 - ân)(n g 2) (A31)

R1 )k1PCO

k1PCO + k5PH2

(n ) 1) (A32)

RA )k1PCO

k1PCO + k5PH2+ k6

+ (A33)


Model FT V(II): RDS, steps 2, 5, and 6.

Model FT V(III): RDS, steps 3, 5, and 6.

ân ) (k6-/k6

+){PCnH2n/[R1RA

n-1 +k6

-

k1PCO + k5PH2+ k6

+∑i)2

n


)]} (n g 2) (A34)

RCH4) k5MPH2xK0PH2

R1 /[1 + xK0PH2+ (1 +

1

K3K4PH2

+1

K4)xK0PH2 ∑i)1

n

(∏j)1

i

Rj) +

K1PCOxK0PH2(1 + ∑

i)1

n

(∏j)1

i

Rj))] (A35)


j)1

n

Rj /[1 + xK0PH2+ (1 +

1

K3K4PH2

+1

K4)xK0PH2 ∑i)1

n

(∏j)1

i

Rj) +

K1PCOxK0PH2(1 + ∑

i)1

n

(∏j)1

i

Rj))] (n g 2) (A36)

RCnH2n) k6

+(1 - ân)xK0PH2 ∏j)1

n

Rj /[1 + xK0PH2+ (1 +

1

K3K4PH2

+1

K4)xK0PH2 ∑i)1

n

(∏j)1

i

Rj) +

K1PCOxK0PH2(1 + ∑

i)1

n

(∏j)1

i

Rj))] (n g 2) (A37)

Rn )k2PCOPH2

k2PCOPH2+ k5PH2

+ k6+(1 - ân)

(n g 2) (A38)

R1 )k2PCO

k2PCO + k5(n ) 1) (A39)

RA )k2PCOPH2

k2PCOPH2+ k5PH2

+ k6+ (A40)

ân ) (k6-/k6

+){PCnH2n/[R1RA

n-1 +k6

-

k2PCOPH2+ k5PH2

+ k6+

∑i)2

n


)]} (n g 2) (A41)

RCH4) k5MPH2xK0PH2

R1/[1 + xK0PH2+ (1 +

1

K4)xK0PH2 ∑

i)1

n

(∏j)1

i

Rj) +

xK0PH2(K1PCO +K1K2PCOPH2

PH2O)(1 + ∑

i)1

n

(∏j)1

i

Rj))] (A42)


j)1

n

Rj /[1 + xK0PH2+ (1 +

1

K4)xK0PH2 ∑

i)1

n

(∏j)1

i

Rj) +


PH2O)(1 + ∑

i)1

n

(∏j)1

i

Rj))] (n g 2) (A43)

RCnH2n) k6

+(1 - ân)xK0PH2 ∏j)1

n

Rj /[1 + xK0PH2+ (1 +

1

K4)xK0PH2 ∑

i)1

n

(∏j)1

i

Rj) +


PH2O)(1 + ∑

i)1

n

(∏j)1

i

Rj))] (n g 2) (A44)


Model FT VI: RDS, steps 6-8.

Rn )k3PCOPH2

2/PH2O

k3PCOPH2

2/PH2O + k5PH2+ k6

+(1 - ân)(n g 2) (A45)

R1 )k3PCOPH2

2/PH2O

k3PCOPH2

2/PH2O + k5M

(n ) 1) (A46)

RA )k3PCOPH2

2/PH2O

k3PCOPH2

2/PH2O + k5PH2+ k6

+ (A47)

ân ) (k6-/k6

+){PCnH2n/[R1RA

n-1 +k6

-

k3PCOPH2

2/PH2O + k5PH2+ k6

+∑i)2

n


)]} (n g 2) (A48)

RCH4) k7MK5

0.5PH2

1.5R1/[1 + xK5PH2+ K1PCO + K4

-1PH2O

PH2

+ K1K2K4

PH2PCO

PH2O

+ K′3PH2

2PCO

PH2O

+

xK5PH2 ∑i)1

n

(∏j)1

i

Rj)] (A49)

RCnH2n+2) k7K5

0.5PH2

1.5 ∏j)1

n

Rj /[1 + xK5PH2+ K1PCO + K4

-1PH2O

PH2

+ K1K2K4

PH2PCO

PH2O

+ K′3PH2

2PCO

PH2O

+

xK5PH2 ∑i)1

n

(∏j)1

i

Rj)] (n g 2) (A50)

RCnH2n) k8

+(1 - ân)xK5PH2 ∏j)1

n


-1PH2O

PH2

+ K1K2K4

PH2PCO

PH2O

+ K′3PH2

2PCO

PH2O

+

xK5PH2 ∑i)1

n

(∏j)1

i

Rj)] (n g 2) (A51)

Rn )

k6K′3PH2

2PCO

PH2O[s1]

k6K′3PH2

2PCO

PH2O[s1] + k7PH2

+ k8+(1 - ân)

(n g 2) (A52)

R1 )

k6K′3PH2

PCO

PH2O[s1]

k6K′3PH2

PCO

PH2O[s1] + k7

(n ) 1) (A53)

RA )k6K′3PH2

2PCO/PH2O[s1]

k6K′3PH2

2PCO/PH2O[s1] + k7PH2+ k8

+ (A54)

ân ) (k8-/k8

+){PCnH2n/[R1RA

n-1 +k8

-

k6K′3PCOPH2

2/PH2O[s1] + k7PH2+ k8

+∑i)2

n


)]} (n g 2) (A55)


Model FT VII(I): RDS, steps 1, 7, and 8.

Model FT VII(II): RDS, steps 2, 7, and 8.

RCH4) k7MK0PH2

R1/(1 + xK0PH2+

PH2O

K6xK0PH2

+

( PH2O

K2K3K4K5K02PH2

2+

1

K3K4K5K0PH2

+1

K4K5xK0PH2

+1

K5

+ 1)xK0PH2 ∑i)1

n

(∏j)1

i

Rj))2

(A56)


j)1

n

Rj /(1 + xK0PH2+

PH2O

K6xK0PH2

+

( PH2O

K2K3K4K5K02PH2

2+

1

K3K4K5K0PH2

+1

K4K5xK0PH2

+1

K5

+ 1)xK0PH2 ∑i)1

n

(∏j)1

i

Rj))2

(n g 2) (A57)

RCnH2n) k8

+(1 - ân)xK0PH2∏j)1

n

Rj /[1 + xK0PH2+

PH2O

K6xK0PH2

+

( PH2O

K2K3K4K5K02PH2

2+

1

K3K4K5K0PH2

+1

K4K5xK0PH2

+1

K5

+ 1)xK0PH2 ∑i)1

n

(∏j)1

i

Rj)] (n g 2) (A58)

Rn )k1PCO

k1PCO + k7xK0PH2[s1] + k8

+(1 - ân)(n g 2) (A59)

R1 )k1PCO

k1PCO + k7xK0PH2[s1]

(A60)

RA )k1PCO


+(A61)

ân ) (k8-/k8

+){PCnH2n/[R1RA

n-1 +k8

-


+∑i)2

n


)]} (n g 2) (A62)

RCH4) k7MK0PH2

R1/(1 + xK0PH2+

PH2O

K6xK0PH2

+

(K1PCO +1

K3K4K5K0PH2

+1

K4K5xK0PH2

+1

K5

+ 1)xK0PH2 ∑i)1

n

(∏j)1

i

Rj))2

(A63)


j)1

i

Rj /(1 + xK0PH2+

PH2O

K6xK0PH2

+

(K1PCO +1

K3K4K5K0PH2

+1

K4K5xK0PH2

+1

K5

+ 1)xK0PH2 ∑i)1

n

(∏j)1

i

Rj))2

(n g 2) (A64)


Model FT VIII: RDS, steps 5, 9, and 10.

RCnH2n) k8

+(1 - ân)xK0PH2 ∏j)1

i

Rj /[1 + xK0PH2+

PH2O

K6xK0PH2

+

(K1PCO +1

K3K4K5K0PH2

+1

K4K5xK0PH2

+1

K5

+ 1)xK0PH2 ∑i)1

n

(∏j)1

i

Rj)] (n g 2) (A65)

Rn )k2K1xK0PH2

PCO[s1]

k2K1xK0PH2PCO[s1] + k7xK0PH2

[s1] + k8+(1 - ân)

(n g 2) (A66)

R1 )k2K1xK0PH2

PCO

k2K1xK0PH2PCO + k7xK0PH2

(n ) 1) (A67)

RA )k2K1xK0PH2

PCO[s1]


[s1] + k8+

(A68)

ân ) (k8-/k8

+){PCnH2n/[R1RA

n-1 +k8

-


[s1] + k8+

∑i)2

n


)]} (n g 2)

(A69)

RCH4) k9MK6PH2

R1/(1 + xK6PH2+ K1PCO + K2

1PH2

PCO

PH2O

+1

K6K7K8

PH2O

PH2

+PH2O

K8xK6PH2

+ K′3PH2

1.5PCO

PH2O

+

K′4PH2

2PCO

PH2O

+ xK6PH2 ∑i)1

n

(∏j)1

i

Rj))2

(A70)

RCnH2n+2) k9K6PH2∏

j)1

i

Rj /(1 + xK6PH2+ K1PCO + K2

1PH2

PCO

PH2O

+1

K6K7K8

PH2O

PH2

+PH2O

K8xK6PH2

+ K′3PH2

1.5PCO

PH2O

+

K′4PH2

2PCO

PH2O

+ xK6PH2 ∑i)1

n

(∏j)1

i

Rj))2

(n g 2) (A71)

RCnH2n) k10

+(1 - ân)xK6PH2∏j)1

i


1PH2

PCO

PH2O

+1

K6K7K8

PH2O

PH2

+PH2O

K8xK6PH2

+

K′3PH2

1.5PCO

PH2O

+ K′4PH2

2PCO

PH2O

+ xK6PH2 ∑i)1

n

(∏j)1

i

Rj)] (n g 2) (A72)

Rn )k5K′4PH2

2PCO/PH2O

k5K′4PH2


+(1 - ân)/[s1](n g 2) (A73)

R1 )k5K′4PH2

1.5PCO/PH2O

k5K′4PH2

1.5PCO/PH2O + k9xK6

(n ) 1) (A74)


Model FT IX: RDS, steps 5, 10, and 11.

where K′4 ) K1K2K3K4K62K7K8.

Model FT X: RDS, steps 4, 8, and 9.

RA )k5K′4PH2

2PCO/PH2O[s1]

k5K′4PH2


+(A75)

ân ) (k10-/k10

+){PCnH2n/[R1RA

n-1 +k10

-

k5K′4PH2


+∑i)2

n


)]} (n g 2)

(A76)

RCH4) k10K9K6

0.5K′4PH2

2.5PCO

PH2O

/(1 + xK6PH2+ K1PCO + K2

1PH2

PCO

PH2O

+ K22

PH2O

PH2

+PH2O

K8xK6PH2

+ K′3PH2

1.5PCO

PH2O

+

K′4PH2

2PCO

PH2O

+ (1 + K9xK6PH2)K′4

PH2

2PCO

PH2O∑i)2

n

∏j)2

i

Rj)2

(A77)

RCnH2n+2) k10K9K6K′4PH2

3PCO/PH2O∏i)2

n

Ri /(1 + xK6PH2+ K1PCO + K2

1PH2

PCO

PH2O

+ K22

PH2O

PH2

+PH2O

K8xK6PH2

+

K′3PH2

1.5PCO

PH2O

+ K′4PH2

2PCO

PH2O

+ (1 + K9xK6PH2)K′4

PH2

2PCO

PH2O∑i)2

n

∏j)2

i

Rj)2

(n g 2) (A78)

RCnH2n) k11

+(1 - ân)K′4PH2

2PCO/PH2O∏i)2

n

Ri /[1 + xK6PH2+ K1PCO + K2

1PH2

PCO

PH2O

+ K22

PH2O

PH2

+PH2O

K8xK6PH2

+

K′3PH2

1.5PCO

PH2O

+ K′4PH2

2PCO

PH2O

+ (1 + K9xK6PH2)K′4

PH2

2PCO

PH2O∑i)2

n

∏j)2

i

Rj] (n g 2) (A79)

ân )

(k11-/k11

+){PCnH2n/[K′4PH2

2PCO/PH2ORAn-1 +

k11-

k5K′4PH2

2PCO/PH2O[s1] + k10K9K6PH2[s1] + k11

+∑i)2

n

RAi-2PC(n-i+2)H2(n-i+2)]}

(n g 2) (A80)

Rn )k5K′4PH2

2PCO/PH2O

k5K′4PH2

2PCO/PH2O + k10K9K6PH2+ k11

+(1 - ân)/[s1](n g 2) (A81)

RA )k5K′4PH2

2PCO/PH2O[s1]

k5K′4PH2

2PCO/PH2O[s1] + k10K9K6PH2[s1] + k11

+ (A82)

RCH4) k8MK7K5

0.5K′3PH2

2.5PCO/PH2O/[1 + xK5PH2+ K1PCO + K2

1PH2

PCO

PH2O

+ K22

PH2O

PH2

+ K7K′3K50.5

PH2

2.5PCO

PH2O

+

K′3PH2

2PCO

PH2O

+ (1 + K7xK5PH2)K′3PH2

2PCO/PH2O ∑i)2

n

∏j)2

i

Rj] (A83)


Nomenclature

E5 ) activation energy for chain growth (kJ mol-1)E7M ) activation energy for methane formation (kJ mol-1)E7 ) activation energy for paraffin formation (kJ mol-1)E8 ) activation energy for olefin formation (kJ mol-1)Ev ) activation energy for the WGS reaction (kJ mol-1)fi,obj ) objective function for the ith responseFTot,obj ) multiresponse objective functionkWGS4 ) rate constant of the forward reaction in step IV

(mol g-1 s-1)k-WGS4 ) rate constant of the reversible reaction in step

IV (mol g-1 s-1)k5 ) rate constant of chain growth (mol g-1 s-1 bar-1)k5,0 ) preexponential factor of chain growth (mol g-1 s-1

bar-1)k7M ) rate constant of methane formation (mol g-1 s-1

bar-1)k7M,0 ) preexponential factor of methane formation (n g

2) (mol g-1 s-1 bar-1)k7 ) rate constant of paraffin formation (mol g-1 s-1 bar-1)k7,0 ) preexponential factor of paraffin formation (n g 2)

(mol g-1 s-1 bar-1)k8 ) rate constant of olefin formation (mol g-1 s-1)k8,0 ) preexponential factor of olefin formation (n g 2) (mol

g-1 s-1)kv ) rate constant of CO2 formation (mol g-1 s-1 bar-1.5)kv,0 ) preexponential factor of CO2 formation (mol g-1 s-1

bar-1.5)k-8 ) rate constant of olefin readsorption reaction (mol g-1

s-1 bar-1)

K1 ) equilibrium constant of the elementary reaction 1 forFTS (bar-1)

K2 ) equilibrium constant of the elementary reaction 2 forFTS (bar-1)

K3 ) equilibrium constant the elementary reaction 3 forFTS

K4 ) equilibrium constant the elementary reaction 4 forFTS (bar-1)

K6 ) equilibrium constant the elementary reaction 6 forFTS

Kp ) equilibrium constant of the WGS reactionKv ) group of constants of the WGS reaction (bar-0.5)KWGS1 ) equilibrium constant of CO adsorption elementary

stepKWGS2 ) equilibrium constant of H2O adsorption elemen-

tary stepKWGS3 ) equilibrium constant of surface reaction elemen-

tary stepKWGS4 ) equilibrium constant of CO2 desorption elementary

stepKWGS5 ) equilibrium constant of H2 desorption elementary

reactionmi ) mole flow rate of component i (mol s-1)MARR ) mean absolute relative residualsN ) maximum carbon number of the hydrocarbons in-

volvedNc ) total number of components involvedNR ) total number of reactions involvedNexp ) total number of experimentsNresp ) total number of responses for parameter estimiza-

tion

RCnH2n+2) k8K7K5

0.5K′3PH2

3.5PCO/PH2O∏i)2

n

Ri/[1 + xK5PH2+ K1PCO + K2

1PH2

PCO

PH2O

+ K22

PH2O

PH2

+

K7K′3K50.5

PH2

2.5PCO

PH2O

+ K′3PH2

2PCO

PH2O

+ (1 + K7xK5PH2)K′3PH2

2PCO/PH2O ∑i)2

n

∏j)2

i

Rj] (n g 2) (A84)

RCnH2n) k9

+(1 - ân)K′3PH2

2PCO/PH2O∏i)2

n

Ri/[1 + xK5PH2+ K1PCO + K2

1PH2

PCO

PH2O

+ K22

PH2O

PH2

+

K7K′3K50.5

PH2

2.5PCO

PH2O

+ K′3PH2

2PCO

PH2O

+ (1 + K7xK5PH2)K′3PH2

2PCO/PH2O ∑i)2

n

∏j)2

i

Rj] (n g 2) (A85)

Rn )k4K′3PH2

2PCO/PH2O

k4K′3PH2

2PCO/PH2O + k8K7K5PH2+ k9

+(1 - ân)/[s1](n g 2) (A86)

RA )

k4K′3PH2

2PCO

PH2O[s1]

k4K′3PH2

2PCO


[s1] + k9+

(A87)

ân ) (k9-/k9

+) ×

{PCnH2n/[K′3

PH2

2PCO

PH2O

RAn-1[s1] +

k9-

k4K′3PH2

2PCO

PH2O

[s1] + k8K7K5PH2[s1] + k9

+

[s1] ∑i)2

n

RAi-2PC(n-i+2)H2(n-i+2)]} (n g 2)

(A88)


Pi ) partial pressure of component i (bar)PT ) total pressure of the reaction system (bar)R ) gas constant (J mol-1 K-1)Rj ) overall reaction rate of reaction path j (mol g-1 s-1)Ri ) rate of formation of component i (mol g-1 s-1)RR ) relative residual between the experimental and

calculated values of the responses1 ) active site for hydrocarbon formations2 ) active site for the WGS reactionT ) temperature of the reaction system (K)Wi ) weight of the ith responseW ) weight of the catalyst used (g)

Greek Symbols

R1 ) chain growth factor for carbon number 1Rn ) chain growth factor for carbon number n (n g 2)RA ) chain growth factor without olefin readsorptionRi,j ) stoichiometric coefficient for the ith component in the

jth reactionân ) readsorption factor of 1-olefin with carbon number n

(n g 2)

Superscripts and Subscripts

cal ) calculated valueexp ) experimental valuei ) index indicating reactionsj ) index indicating componentsM ) methanen ) number of carbon atoms* ) equilibrium state

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Received for review February 14, 2003Revised manuscript received June 12, 2003

Accepted June 13, 2003

IE030135O


detailed kinetics of fischer-tropsch synthesis on an industrial

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