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Notes on the issues of equilibrium in the Fischer-Tropsch synthesis.

INTRODUCTION

The conversion of CO and [H.sub.2] to hydrocarbons has been studied extensively since it was first reported by Fischer and Tropsch (1926). The process has been operated commercially both in Germany and South Africa, and the standard catalysts are either iron or cobalt based with current interest in ruthenium (Schultz, 1999). The molecular weight distribution (MWD) has become of more interest in recent years as demonstrated Glebov and Kliger (1994).

The mathematical derivation of the FT molecular weight distribution is similar to that for polycondensation and polymerization processes. The probability of chain growth, [alpha], is given by the ratio

[alpha] = [r.sub.p]/[r.sub.p] + [r.sub.t] (1)

where [r.sub.p] is the rate of propagation and [r.sub.t] is the rate of termination of the chain. This gives the equation for molecular weight distribution

ln [x.sub.i] = i ln [alpha] + ln (1 - [alpha]/[alpha]) (2)

The focus on a kinetic approach rather than a thermodynamic approach can be traced to the early work of Friedel and Anderson (1950), Storch et al. (1951), and Anderson (1956). These calculations demonstrated that global equilibrium is not achieved within the FT process. Each product was considered as a different stoichiometric reaction derived from CO and [H.sub.2] and the free energy change for each reaction was considered. This work mimicked the approach taken in the much earlier work of Smith (1927). More recently, the equilibrium product as a function of [H.sub.2]/CO ratio was studied by Tillmetz (1975). All studies demonstrate that methane is the predominant hydrocarbon if global equilibrium were to be achieved.

Despite this, an equilibrium explanation for the FT synthesis has been hinted at by various workers. Stenger and Askonas (1986) used a free energy minimization technique for a family of chemical products. The base chemical equation was

CO + U[H.sub.2] [right arrow] [[infinity].summation over (i=2)] [a.sub.i][C.sub.i][H.sub.2i] + [[infinity].summation over (i=2)] [b.sub.i][C.sub.i][H.sub.2i+2]+ [[infinity].summation over (i=2)] [c.sub.i][C.sub.i][H.sub.2i+1]OH +d[H.sub.2]O + eC[O.sub.2] (3)

where U is the usage ratio, and [a.sub.i], [b.sub.i], [c.sub.i], d, and e are undetermined stoichiometric coefficients. The system free energy was minimized subject to the mass balance constraints; the ratio of CO/[H.sub.2] was dictated by the parameter U. The calculated product distributions matched those from experimental work in that the concentrations of n-alkanes, n-alkenes, and n-alcohols were found to give a straight line on a log-linear plot. In essence, this approach implies a stoichiometric restriction to an equilibrium system (Norval et al., 1992).

Norval and Phillips (1990) demonstrated that an equation relating equilibrium concentration and carbon number could be derived from thermodynamics. The equation has the same form as that of the Anderson-Schultz-Flory (ASF) product distribution, albeit with a two parameter model--the two parameters being the chemical potentials of C and H in the system. This form of the product distribution holds true for n-alkane, n-alkene, and n-alcohol families.

Bell (1995) investigated the thermodynamics of the Fischer-Tropsch process including the state of the metal catalysts. The equilibria between reduced metal, metal carbides and metal oxides are related to the gas composition. Thermodynamics dictates that iron and cobalt can convert CO and [H.sub.2] to hydrocarbons, whereas Ni can only convert CO and [H.sub.2] to methane, consistent with experimental observations. In addition, thermodynamics dictates that as the C[O.sub.2] content in the gas phase increases, the metal becomes increasingly oxidized. There is a C[O.sub.2] concentration at which point the gas phase becomes oxidizing relative to the metal. Thermodynamic calculations correctly predict that iron requires operating temperatures roughly 50-100[degrees]C greater than for cobalt catalysts. Liaw and Davis (2000) arrive at a similar equilibrium conclusion, albeit with a different derivation.

There continues to be much work on the Fischer-Tropsch synthesis with a key goal being to investigate the impact of changes in feed composition on the reaction rate and product distributions. This continued search for a kinetics explanation ignores the issue that the Fischer-Tropsch process has many features of an equilibrium controlled system. This article has two objectives: to demonstrate that the one-parameter ASF distribution can be developed from a thermodynamic basis, and to further expand on the implications of equilibrium in this process.

DERIVATION

The Fischer-Tropsch process is usually defined by chemical equations such as

nCO + (2n + 1)[H.sub.2] [right arrow] Cn[H.sub.2n+2] + n[H.sub.2]O (4a)

nCO + 2n[H.sub.2] [right arrow] Cn[H.sub.2n] + n[H.sub.2]O (4b)

nCO + 2n[H.sub.2] [right arrow] Cn[H.sub.2n+2]O + (n - 1)[H.sub.2]O (4c)

Norval and Phillips (1990), demonstrated that an equilibrium approach gave the equations:

ln [x.sub.ip] = i([[PSI].sub.Cp] + 2[[PSI].sub.Hp] - [B'.sub.p]) - [A'.sub.p] - ln P + 2[[PSI].sub.Hp] (5a)

ln [x.sub.io] = i([[PSI].sub.Co] + 2[[PSI].sub.Ho] - [B'.sub.o]) - [A'.sub.o] - ln P (5b)

ln [x.sub.ia] = i([[PSI].sub.Ca] + 2[[PSI].sub.Ha] + [[PSI].sub.oa] - [B'.sub.a]) - [A'.sub.a] - ln P + 2[[PSI].sub.Ha] (5c)

where [[PSI].sub.Cp], [[PSI].sub.Hp], [[PSI].sub.Co], [[PSI].sub.Ho] are the Lagrange multipliers for carbon and hydrogen in the paraffin, alkene and alcohol series, respectively. The constants [A'.sub.p], [B'.sub.p], [A'.sub.o], [B'.sub.o], [A'.sub.a], and [B'.sub.a] are the values of the constants for the free energies of formations of the alkane, alkene, and alcohol homologous series (Alberty, 1983; Alberty and Gehrig, 1984; Alberty et al., 1987).

There is no requirement that the components of a system be the chemical elements; the sole requirement is that the components be linearly independent and satisfy the material balance requirements of the formula matrix (Smith and Missen, 1982). Consequently, it is possible to select the system components to be C[H.sub.2], [H.sub.2] and O; these are termed "pseudo-elements". On this basis, the formula vector for CO becomes (1, -1, 1), and that for propane becomes (3, 1, 0). With this definition, the formula vectors for the chemical species are positive or negative integers.

The Fischer-Tropsch process operates at elevated pressures; the ideal gas model for chemical potential is not expected to be valid. A general approach to the non-ideality is

[[mu].sub.*.sub.i] = [[mu].sub.o.sub.i] + f (P) (6)

where [[mu].sub.*.sub.i] is the chemical potential of species i at temperature T and pressure P, [[mu].sub.o.sub.i] is the chemical potential of species i at temperature T, and f(P) is an appropriate equation of state.

Equations (5a)-(5c) are readily converted into the following set of equations

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7a)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7b)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7c)

The thermodynamic derivation gives a log-linear relationship between mole fraction and carbon number. The slope of the line is the term ([PSI]C[H.sub.2] - B'), the Lagrange multiplier for C[H.sub.2] less the constant B'--the slope of the chemical potential with carbon number, which is a function of T only. This means that the value of [alpha] is directly related to the chemical potential of the C[H.sub.2] monomer in the Fischer-Tropsch products.

DISCUSSION

Element Abundance Representation

The standard Fischer-Tropsch feed is described in terms of the [H.sub.2]/CO ratio. The work of Stenger and Satterfield (1985a,b) is typical. The solvent used for a slurry reactor system was varied, and the rate of reaction was found to vary by a factor of roughly 20 due to the insolubility of the higher hydrocarbon products in the lower activity solvent. Regardless of the system activity, the slopes of the ASF plot were identical. The solvent activities correlated with the alkene to alkane ratios, with more alkenes observed from the more active catalysts. The systems all had the same [H.sub.2]/CO ratio, but the higher catalyst activity gives a lower [H.sub.2] partial pressure in the reactor due to the greater CO conversion. There was an inverse relationship between alkene to alkane ratio and [H.sub.2] partial pressure. This work is one example of many that demonstrate the importance of knowing the actual chemical composition in the reactor. Describing the elemental abundances as C[H.sub.2]/[H.sub.2]/O is in keeping with these works; the feed is described as potential product with a [H.sub.2]/O ratio, which is independent of conversion.

Equilibrium Effect in Oligomerization

Henrici-Olive and Olive (1976) made the link between the product distribution of the Fischer-Tropsch synthesis and that of Schultz-Flory oligomerizations. Anderson (1978) noted that such a chain growth mechanism had been proposed by various authors. It is worth noting that the polymerizations in the 1930s, were performed at lower pressures than in current industrial practice (Flory, 1936). The product molecular weights would be lower, and the chain lengths of a 1930s polymer would have been closer to that of modern Fischer-Tropsch oils.

The average molecular weights of modern linear condensation polymers are significantly greater due both to changes in the catalyst system and the operating conditions; it is not unusual to see average molecular weights of the order of 10 000-20 000. The mathematics behind the molecular weight distributions are still based on that of Flory (cf Ch. 6, Rodriguez, 1982). Regardless, the polymer shows an ASF type decrease in mole fraction above the average carbon number, with an exponential growth leading up to the average carbon number.

The implication is that equilibrium is a factor in the molecular weight distributions of Schultz-Flory oligomers and polymers. Norval et al. (1989) demonstrated that oligomerization systems should be treated as pseudo-one-element systems--Equations (7a)-(7c) therein is readily converted to a log-linear pattern. The alkane and alcohol homologous series do not lend themselves to the simple oligomerization derivation due to the stoichiometry. Regardless, the pseudo-element approach used herein ties the three Fischer-Tropsch product families together with the olefin oligomerizations and the alkene to gasoline process (Tabak, 1981).

Mims et al. (1990) performed studies of the Fisher-Tropsch system over ruthenium catalysts into which isotopically labelled olefins were injected into the system. The results demonstrate that injected 1-hexene and 1-octene become involved both in polymerization and depolymerization reactions. They concluded that the polymerization rates are greater than the rates of chain initiation. A log-linear distribution of the hydrocarbon products was observed, consistent with equilibration within the hydrocarbon pool. The isotopic distributions within specific products were not random, demonstrating that rates were fast enough to achieve chemical equilibrium but not sufficiently fast for total equilibration within the carbon pool. This study is consistent with the earlier results of Yamasaki et al. (1981) and Kobori et al. (1981) that demonstrated that catalyst surface has a small coverage of growing polymer chains surrounded by a large pool of CO species. These kinetics measurements are consistent with chemical equilibrium being a reasonable explanation for the product distribution.

Iron-Based Catalysts

Iron catalysts are a mixture of magnetite and iron carbide. The results of Lee et al. (1990) demonstrated that the reaction rate, product distribution and catalyst change as a function of temperature, pressure and [H.sub.2]/C[O.sub.2] ratio. Increased pressures lead to increased fractions of [Fe.sub.3][O.sub.4] and decreased fractions of [Fe.sub.5][C.sub.2]. This agrees with the results of Riedel et al. (2003) who were able to correlate the induction time for Fischer-Tropsch to begin with the creation [Fe.sub.3][O.sub.4] and [Fe.sub.5][C.sub.2] phases for both CO and C[O.sub.2] hydrogenation. Wu et al. (2005) demonstrated that catalysts with high initial fractions of [Fe.sub.3][O.sub.4] resulted in low [Fe.sub.5][C.sub.2] fractions even after significant time on stream with synthesis gas.

It is useful to note that thermodynamic studies of the system C-H-O over iron have been performed for a variety of non-catalytic purposes. Schechter and Wise (1979) produced phase diagram (triangular graphs) showing the regions of stable [Fe.sub.3][O.sub.4] and [Fe.sub.5][C.sub.2] for a variety of temperatures and pressures. Typical Fischer-Tropsch feeds fall in the region where [Fe.sub.3][O.sub.4] and [Fe.sub.5][C.sub.2] coexist as the thermodynamically stable phases. The results agree with those of Taylor (1981) who used thermodynamics to demonstrate that "breakaway" oxidation of mild steel by C[O.sub.2] can only occur when the gas is both oxidizing and carburizing.

It is plainly evident that the stable iron phases formed in the Fischer-Tropsch process are those that would be in equilibrium with the gas composition. [H.sub.2] is reducing, [H.sub.2]O is oxidizing, just as CO is reducing relative to C[O.sub.2]. Not surprisingly, iron catalysts suffer from product inhibition due to the product water (Schultz, 1999)--the gas phase becomes more oxidizing. Satterfield et al. (1986) compared the effect of added [H.sub.2]O on both the product distribution and the catalyst through use of Mossbauer spectroscopy. The key observations were that while added water increased the alkene to alkane ratio, increased the yield of alcohols, decreased the yield of methane and the rate of reaction, the slope of the ASF plot was not impacted by the addition of [H.sub.2]O. In other words, product equilibration is rapid relative to monomer formation.

Xu et al. (1997) studied the influence of C[O.sub.2] content in the synthesis gas on both the rate and the distribution of products. At high C[O.sub.2]/CO ratios, the water gas shift reaction dominates over the Fischer-Tropsch reaction. When the C[O.sub.2]/CO ratio is low, C[O.sub.2] contributes to chain initiation (after water-gas shift), but it is not associated with chain propagation. Riedel and Schaub (2003) reported varying effects of C[O.sub.2]/CO ratio, depending on the promoters used in the catalyst preparation. All of these workers anticipate a kinetic effect of C[O.sub.2], rather than an equilibrium effect due to the gas phase becoming more oxidizing.

Jalama et al. (2007) studied the effect of ethanol addition to a synthesis gas feed ([H.sub.2]/CO ratio of 2). The ethanol ([C.sub.2][H.sub.6]O) is more reducing than the synthesis gas. As such, it is not surprising that ethanol addition to that feed impacted the catalyst reactivity, as it changes the nature of gas phase, and thereby, the catalyst surface itself.

In all cases, changing the feed composition changes the catalyst reactivity, but not the ASF distribution. The hydrocarbon product achieves an equilibrium distribution, and a second equilibrium occurs between the gas phase and the catalyst.

Cobalt-Based Catalysts

Bell (1995) considered the chemical potential of cobalt metal and promoted cobalt oxides, and suggested that the promoters make it more difficult for reduction of cobalt oxide to cobalt metal. The active catalyst would be a promoted oxide of cobalt. Bremaud et al. (2005) report that catalyst deactivation is more likely to occur as the [H.sub.2]/CO ratio increases. Zhang et al. (2002) report that Co/Si[O.sub.2] deactivation is slower at higher [H.sub.2]O/[H.sub.2] ratios. Van de Loosdrecht et al. (2007) conclude that "the large discrepancies in the literature on the oxidation behaviour of cobalt are likely due to the lack of direct characterization of the cobalt oxidation state," and that "the oxidation of cobalt can be prevented by the correct combination of the reactor partial pressures of hydrogen and water." More recently, Svoboda et al. (2008) demonstrated that cobalt spinel (CoAl2[O.sub.4]) is thermodynamically favoured under typical Fischer-Tropsch conditions, supported Bell's premise. All of these observations demonstrate that equilibration with the gas phase is key for cobalt catalysts.

Claeys and van Steen (2002) report that the addition of [H.sub.2]O to a synthesis gas feed results in improved reaction rates as well as reducing the methane selectivity on a Ru/Si[O.sub.2] catalyst, similar to the observations noted above.

Carbon Deposition

All catalysts suffer from carbon deposition to various extents. Tevebaugh and Cairns (1965) calculated the phase boundaries for which graphite will form as a function of the CHO composition of a gas phase. For standard FTS temperatures, all mixtures of CO and [H.sub.2] fall within the stable graphite area. Thus, thermodynamics clearly explains coke deposition during the Fischer-Tropsch synthesis.

Despite this, Fisher-Tropsch catalysts are incredibly resistant to coke formation. From reaction 4b, it is evident that as the extent of CO conversion increases, the gas phase becomes increasingly oxidizing. There are many molecules of [H.sub.2]O produced for each molecule of hydrocarbon produced. This shifts the gas to a more oxidizing gas phase and reduces the thermodynamic tendency to form graphite.

Yates and Satterfield (1989) demonstrated that product inhibition is a reaction that must be considered. The question is whether it is product inhibition or a change in catalyst surface due to the change in gas composition that is the root cause.

Thermodynamic Sensitivity Analysis of [alpha]

When a system tends to equilibrium, sensitivity analysis can be used to estimate the impact of changes in temperature, pressure or compositions on the system (Smith and Missen, 1982, p. 192). This predicts the changes in system behaviour based solely on thermodynamic data.

The value of [alpha] increases when the [H.sub.2]/CO ratio or reactor temperature decrease (Espinoza et al. 1999). The value of B' increases with increasing temperature (the free energy of formation increases with temperature)--thus a decrease in temperature must increase the slope of Equations (5a)-(5c) and Equations (7a)-(7c). The change in [alpha] is consistent with that predicted by thermodynamics.

Sensitivity with respect to changes in pressure is a more difficult case to demonstrate. The alkene homologous series has the set of species [(C[H.sub.2]).sub.n], with n > 2, and is a pseudo-one-element system (Norval et al., 1989). The equations that need be solved are

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8a)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8b)

These equations assume that the species is an ideal gas, for simplicity. When solved, they give the general solution

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

Recognize that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. There will be a value of j, the number of C[H.sub.2] units in the oligomer, at which the sign of the partial derivative changes from negative to positive. This means that higher pressures will produce more product with a higher molecular weight; in other words, the value of [alpha] increases with pressure. This impact of changes in pressure is observed in both the Fischer-Tropsch synthesis as well as oligomerization and polymerization systems.

The impact of changes in feed composition can be demonstrated a priori. A hypothetical FT systemwas generated for an alkane system ([C.sub.1]-[C.sub.10]), with CO, [H.sub.2], C[O.sub.2], and [H.sub.2]O, the ASF parameter was 0.7, and 20% conversion of the CO was assumed. The product composition was determined, and the set of first order derivatives with respect to feed composition ([partial derivative][PSI]/[partial derivative]b and [partial derivative]ln [n.sub.t]/[partial derivative]b) were then calculated. These were used to calculate the values of the first order sensitivity coefficients ([partial derivative][n.sub.i]/[partial derivative]b), which are presented in Table 1.

The sensitivity coefficients show that when the molar abundance of C[H.sub.2] is increased, the moles of all products increase, as do CO and [H.sub.2], while C[O.sub.2] and [H.sub.2]O decrease (a positive partial derivative means a molar increase, and a negative partial derivative means a molar decrease). If Reaction (4a) was considered to be at equilibrium, it is apparent that an increase in C[H.sub.2] (column [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] in Table 1) means an increase in product, which shifts the reaction to the left, increasing CO, [H.sub.2] and decreasing [H.sub.2]O. The reduction in C[O.sub.2] is the subsequent impact on the water gas shift equilibrium.

Increasing the molar abundance of hydrogen (column [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] in Table 1) increase the moles of all species save for CO and C[O.sub.2]. Increasing the molar abundance of O (column [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] in Table 1) decreases the moles of all product species, and increases the molar abundances of CO, [H.sub.2], C[O.sub.2] and [H.sub.2]O. Again, these trends match experimental observations for systems in which the water gas shift equilibrium occurs.

Depending on the catalyst and operating conditions, the product liquid can contain alkanes, alkenes, and/or alcohols. The value of the ASF parameter [alpha] for each homologous series is basically the same (e.g., Figure 6 in Glebov and Kliger, 1994). This suggests that the chemical potential of C[H.sub.2] (the Lagrange multiplier) is basically constant, regardless of the product mix.

There is discussion in the literature of the "double-alpha" model which addresses the problem that in a number of cases, the product liquid tends to have more higher hydrocarbons than would be expected from the simple ASF model. The intersection of the two alphas tends to be at a carbon number of 6 (Glebov and Kliger, 1994; Patzlaff et al., 1999). As the number of available isomers increases, the free energy of the isomeric mix decreases. The value of the constant B' is smaller for isomeric mixes, relative to the n-isomers. For a given value of the Lagrange multiplier [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], it is readily apparent that value of the term ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]) must increase when the value of B' decreases. The consequence is that the mole fraction will slightly increase at the higher carbon numbers when any isomerization occurs. The observation that equilibrium theory can explain the double alpha phenomena is consistent with the work of Zhan and Davis (2002) in which diffusion control was shown not to be a valid explanation for double alpha.

Limitations on the Equilibrium Approach

There are two difficulties with extending this analysis to greater levels of detail. The lack of a validated equation of state prevents exact determine of the partial derivatives of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] with respect to T and P.

Most importantly, the system is not at global equilibrium. Global equilibrium means that all species in the system have achieved equilibrium. Patterson et al. (2003) shows that C[H.sub.4] is the sole product formed over molybdenum and tungsten carbides. Global equilibration is a more consistent explanation of methanation catalysis, consistent with the calculations reported by Smith (1927), Anderson (1956), and Tillmetz (1975).

A partial equilibrium approach means that some of the species in the system have achieved equilibrium, while others have not (Norval et al., 1992). The Fischer-Tropsch log-linear distribution demonstrates that equilibrium is achieved within the homologous series, but global equilibrium is not achieved. Thus, the product homologous series are one (or more) partial equilibrium subsystem.

The water gas shift reaction is a second partial equilibrium subsystem depending on the catalyst. The water gas shift reaction does not occur over cobalt catalysts, but is quite rapid over iron catalysts (Riedel et al., 1999). The cobalt catalyst sees some rate inhibition due to competitive C[O.sub.2] adsorption at high C[O.sub.2] contents in the feed gas. The yield of hydrocarbons was observed to be substantially lower with the iron catalyst when compared with the cobalt catalyst for high C[O.sub.2] contents in the feed gas. Despite this, the hydrocarbon distribution was not impacted by the C[O.sub.2]/CO ratio.

Xu et al. (1997) had similar observations regarding the difference between cobalt and iron catalysts. The study included [sup.14]C labelled C[O.sub.2] additions with the result that [sup.14]C appeared in the Fischer-Tropsch product--generally evenly distributed. It was noted that the rate of the water-gas shift reaction was fast relative to the Fischer-Tropsch reaction. Not surprisingly, the work of Shi et al. (2002) with [D.sub.2]O in the feed, shows an even distribution of D in the hydrocarbon products. These distributions are consistent with equilibration.

There is ample evidence that the water-gas shift reaction approaches equilibrium over iron catalysts. This equilibrium does not interact with the equilibrium within the homologous series. These are multiple independent partial equilibria subsystems.

The catalyst changes with changes in operating conditions (vide supra). The catalyst itself is in equilibrium with the gas/liquid phase that surrounds it. It remains active, but the activity changes depending on temperature, pressure and CHO ratio. It is reasonable to conclude that catalyst is in equilibrium with the Fischer-Tropsch reactants and products.

The one unresolved issue relates to why alcohol products are formed in greater amounts at elevated pressures. Higher system pressures will change the balance of phases on the catalyst surface (Lee et al., 1990). Equilibrium theory is able to determine the stable phases of a catalyst in equilibrium with a gas of a CHO composition, and it is able to determine the equilibrium composition of a reacting gas mixture. It is not yet able to allow estimation of stable catalyst phases while also performing reacting gas calculations. The Fischer-Tropsch process exceeds the limits of current thermodynamic abilities.

Implications for Reactor Modelling

Reactor models are developed to allow for improved estimation of the system performance and for reactor design. Ammonia synthesis and S[O.sub.2] oxidation are classical equilibrium controlled systems. The reactor model has a fluid mechanics portion, a reaction rate expression, usually containing a product inhibition term. The reaction proceeds to equilibrium.

If the Fischer-Tropsch process were considered to be a similar equilibrium-limited process, the model would have a simple rate expression for hydrocarbon yield, with the hydrocarbons then equilibrating. The reaction rate would be a function of gas composition, which, in effect, accounts for the change in reaction rate due equilibration of the catalyst with the changing gas composition. Such a reactor model would have few adjustable parameters.

In contrast, current FT reactor models are substantially more complex (Keyser et al., 2000; Anfiray et al., 2007) and use many adjustable "rate" parameters, rather than a single "equilibrium" parameter. Modern computers enable one to estimate the values of kinetic parameters for complex kinetic schemes. In particular, given the complex hydrodynamics of a slurry reactor, and the problems of estimating gas diffusion rates, a equilibrium approach will provide an equal model, with fewer adjustable parameters.

CONCLUSIONS

This work demonstrates that a 1 parameter log-linear relation ship between product composition and carbon number can be derived from a thermodynamic basis. The implication is that the hydrocarbon product of the Fischer-Tropsch synthesis is an equilibrium mixture of hydrocarbons. The equilibrium model correctly predicts a variety of experimental observations.

The Fischer-Tropsch process is best considered as multiple subsystems each working to achieve a partial equilibrium. The hydrocarbon homologous series are the first partial equilibrium. Over iron catalysts, the water-gas shift reaction is rapid, giving a second partial equilibrium. The third partial equilibrium analysis involves the gas composition, whether oxidizing or reducing, and the catalyst surface.

Bell (1995) concluded that the Fischer-Tropsch system is an example of "thermodynamically controlled catalysis." Schultz (2003) noted that a "dynamic thermodynamically controlled equilibrium of surface segregation is attained," and that "these sites cooperate in the Fischer-Tropsch mechanism as for chain growth and product desorption". The unsteady-state vapour-liquid equilibrium model of Raje and Davis (1996) uses vapour-liquid equilibrium to account for negative deviations in slurry reactor systems. Equilibrium theory explains far more observations in Fischer-Tropsch chemistry than has previously been thought. The Fischer-Tropsch system is unique, and thermodynamic calculations should be included as part of future kinetic data interpretation. The implications of equilibrium need to be considered prior to ascribing the observations to the more traditional kinetics effects, and in particular before building a highly complex mechanistically based reactor model.
NOMENCLATURE

A constant in free energy of formation data of Alberty

A' dimensionless Alberty constant

[a.sub.ki] subscript to the kth element in the molecular formula
 of species i

B slope in Free energy of formation data of Alberty

B' dimensionless Alberty slope

[b.sub.i] abundance of element i (moles)

f(P) equation of state

[G.sub.i] Gibbs energy of species i

g(P) modified equation of state (Equations 7a-7c)

P pressure (kPa)

R ideal gas constant (8.314 J/mol K)

r rate of reaction

T Temperature (K)

U usage ratio (Equation 1)

[x.sub.i] mole fraction of species i

Greek Letters

[alpha] probability of chain growth

[[mu].sub.i] chemical potential of species i

[[PSI].sub.i] Lagrange multiplier for element i

Superscripts

o standard or reference state (function of T only)

* standard or reference state (function of T and P)

Subscripts

a alcohols

f formation

i, j index

o alkene

p propagation step

p alkanes

t termination step


Manuscript received 2 April 2007; revised manuscript received 10 September 2008; accepted for publication 26 June 2008

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([dagger]) This paper is submitted in memory of Ronald W. Missen and Diran Basmadjian, who both passed away in the winter of 2007, and were tremendous role models.

Graeme W. Norval * Department of Chemical Engineering and Applied Chemistry, University of Toronto, 200 College St., Toronto, Ontario, Canada M5S 3E5

* Author to whom correspondence may be addressed. E-mail address: Graeme.norval@utoronto.ca
Table 1. Stoichiometry, composition and first order derivatives for an
alkane system ([alpha]=0.7); feed [H.sub.2]=2, CO=1, C[O.sub.2]=0.01

Species Stoichiometry Stoichiometry
 [CH.sub.2] [H.sub.2]

CO 1 -1
[H.sub.2] 0 1
C[0.sub.2] 1 -1
[H.sub.2]0 0 1
[CH.sub.4] 1 1
[C.sub.2][H.sub.6] 2 1
[C.sub.3][H.sub.8] 3 1
[C.sub.4][H.sub.1O] 4 1
[C.sub.5][H.sub.12] 5 1
[C.sub.6][H.sub.14] 6 1
[C.sub.7][H.sub.16] 7 1
[C.sub.8][H.sub.18] 8 1
[C.sub.9][H.sub.20] 9 1
[C.sub.l0][H.sub.22] 10 1

Species Stoichiometry O Moles

CO 1 8.00E-01
[H.sub.2] 0 1.50E+00
C[0.sub.2] 2 1.00E-02
[H.sub.2]0 1 2.00E-01
[CH.sub.4] 0 5.10E-02
[C.sub.2][H.sub.6] 0 2.53E-02
[C.sub.3][H.sub.8] 0 1.26E-02
[C.sub.4][H.sub.1O] 0 6.24E-03
[C.sub.5][H.sub.12] 0 3.10E-03
[C.sub.6][H.sub.14] 0 1.54E-03
[C.sub.7][H.sub.16] 0 7.64E-04
[C.sub.8][H.sub.18] 0 3.80E-04
[C.sub.9][H.sub.20] 0 1.88E-04
[C.sub.l0][H.sub.22] 0 9.36E-05

Species [MATHEMATICAL [MATHEMATICAL
 EXPRESSION NOT EXPRESSION NOT
 REPRODUCIBLE REPRODUCIBLE
 IN ASCII] IN ASCII]

CO 2.45E-01 -9.80E-02
[H.sub.2] 1.96E-01 7.50E-01
C[0.sub.2] -9.56E-03 -1.21E-03
[H.sub.2]0 -2.26E-01 1.00E-01
[CH.sub.4] 7.10E-02 2.54E-02
[C.sub.2][H.sub.6] 6.72E-02 1.26E-02
[C.sub.3][H.sub.8] 4.93E-02 6.23E-03
[C.sub.4][H.sub.1O] 3.23E-02 3.08E-03
[C.sub.5][H.sub.12] 2.00E-02 1.53E-03
[C.sub.6][H.sub.14] 1.19E-02 7.56E-04
[C.sub.7][H.sub.16] 6.86E-03 3.74E-04
[C.sub.8][H.sub.18] 3.88E-03 1.85E-04
[C.sub.9][H.sub.20] 2.17E-03 9.17E-05
[C.sub.l0][H.sub.22] 1.19E-03 4.54E-05

Species [partial deri-
 vative][n.sub.i]/
 [partial deri-
 vative][b.sub.O]

CO 6.36E-01
[H.sub.2] 5.47E-01
C[0.sub.2] 2.04E-02
[H.sub.2]0 3.23E-01
[CH.sub.4] -4.50E-02
[C.sub.2][H.sub.6] -5.40E-02
[C.sub.3][H.sub.8] -4.25E-02
[C.sub.4][H.sub.1O] -2.89E-02
[C.sub.5][H.sub.12] -1.82E-02
[C.sub.6][H.sub.14] -1.10E-02
[C.sub.7][H.sub.16] -6.40E-03
[C.sub.8][H.sub.18] -3.65E-03
[C.sub.9][H.sub.20] -2.05E-03
[C.sub.l0][H.sub.22] -1.13E-03
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Date:Dec 1, 2008
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