Modeling of transport phenomena in fixed-bed reactors for the Fischer-Tropsch reaction: a brief literature review

2.1 Proposed mechanisms for the FTS reaction

Four mechanisms have been postulated to explain hydrocarbon formation during the FTS reaction; nonetheless, great uncertainty remains regarding exactly how the reaction occurs on the catalyst. The mechanisms are based on the proposed monomer for the reaction (e.g. van der Laan and Beenackers 1999, Storsæter et al. 2006, Visconti et al. 2010, van Santen et al. 2013, and references therein). These proposals include the following: (a) a “carbide” mechanism, where CO is dissociated and forms a metal carbide, and posterior hydrogenation results in CH_x species, which is proposed as the monomer; (b) an “enol” mechanism, where the adsorbed CO reacts with hydrogen to form oxymethylene (CHOH), which acts as an initiator of chain growth; (c) a “CO insertion” mechanism, where CO is inserted into a metal-methyl or metal-methylene carbon bond, and further hydrogenation creates terminated hydrocarbons; and (d) an “alkenyl” mechanism, where a surface vinyl species reacts with a surface methylene to form an allyl species.

van der Laan and Beenackers (1999) summarized two mechanisms for the formation of linear hydrocarbons as supported by experiments. FTS is a polymerization reaction and is believed to occur under the following steps: (i) reactant adsorption, (ii) chain initiation, (iii) chain growth, (iv) chain termination, (v) product desorption, and (vi) readsorption and further reactions. Different adsorbed species have been proposed to be responsible for chain initiation and growth. Figure 3 depicts the species that have been proposed so far. The reaction forms a complex mixture that mainly comprises linear and branched hydrocarbons and oxygenates (R-CH₃ and R-CH=CH₂; R-CH₂-OH and H₂O, where R is a proton or alkyl).

Figure 3:

Proposed and observed chemisorbed species during FTS reaction.

Adapted from van der Laan and Beenackers (1999).

Once the monomer is formed on the active site, the polymerization reaction ensues, resulting in the formation of long chains of hydrocarbons. The distribution of the products of the polymerization reaction can be calculated with the so-called Anderson-Schulz-Flory (ASF) equation [Equation (6)], which ideally estimates the product distribution when the value of α (chain growth probability) is known.

As can be expected from Equation (6), the value of α is a function of the catalyst type. This parameter represents the fraction of hydrocarbon chains with n-carbon atoms that grow by reacting with the monomer. The FTS reaction commonly occurs on catalysts that are based on Ru, Fe, or Co as the active metal because these three elements have shown the greatest activity for this reaction. Ru-based catalysts have the highest activity toward the formation of hydrocarbons with high molecular weight; however, their high cost and low availability inhibit their use at an industrial scale (Dry 2002). However, a report regarding hybrid catalysts that use Ru as a promoter has shown interesting properties, such as suitable activity and selectivity for the production of liquid hydrocarbons based on commercial demands (Kibby et al. 2013). Thus, the value of α is large for a highly active catalyst. For example, researchers have reported the values of α for Ru (α=0.85–0.95), Co (α=0.70–0.80), and Fe (α=0.50–0.70), which agree with the well-known activities of each metal for FTS, and have concluded that the chain growth probability decreases with increasing reactor temperature and at higher H₂/CO ratios (van der Laan and Beenackers 1999).

On the contrary, deviations in the product distribution as predicted by the ASF equation have been found. Among the main observed deviations is an overproduction of methane compared to the ASF-predicted fraction, whereas the amounts of C₂ and C₃ are usually smaller than the ASF-predicted fractions (van der Laan and Beenackers 1999, Puskas and Hurlbut 2003). The main causes of this behavior have been attributed to the assumption of a constant polymerization factor in the ASF equation; products can be readsorbed under real conditions, so several values for the polymerization parameter can coexist during the FTS reaction.

Furthermore, some modified ASF equations in the literature minimize the deviations in the predicted fractions of reaction products. For example, two α-ASF distribution [Equation (7)] fundamentally assumes the existence of two superimposed polymerization processes during the reaction. This equation has been useful to describe experimental data for the product distribution.

A more recent work (Förtsch et al. 2015) proposed a relationship that enhances the prediction of hydrocarbon fraction production. Although these authors’ proposed equation [Equation (8)] does not provide more insight into the understanding of the FTS mechanism, the equation can be useful to describe long-term or transient experimental data.

where n̅ is defined as the average carbon number. This last equation may be useful to describe the behavior of the FTS reaction under non-steady-state conditions, where the selectivity of the reaction can be increased, as will be discussed in Section 3.

2.2 Kinetic and reaction rates for the FTS reaction

As mentioned above, the product distribution of the FTS reaction highly depends on the catalyst (Ru, Fe, or Co as the active metal) that is used in the reaction. Fe-based catalysts are used in industries for their low cost compared to the other metals and their high selectivity to olefins. Additionally, these catalysts’ high activity to the WGS reaction makes this element ideal when the synthesis gas has a low H₂ concentration (van der Laan and Beenackers 1999, Dry 2002). However, Fe-based catalysts are sensitive to the presence of H₂O; this compound inhibits the FTS reaction because of its adsorption onto active sites. On the contrary, Co-based catalysts have been reported to have the highest yield and selectivity toward linear paraffin hydrocarbons compared to Fe-based catalysts, exhibit low selectivity to the WGS reaction, and are not inhibited by water that forms as a by-product of the FTS reaction (van der Laan and Beenackers 1999).

According to the literature, most of the published kinetic rate equations are Langmuir-Hinshelwood-type equations. Comparing these equations is difficult because the reaction conditions (catalyst type, reactor feed composition, temperature, pressure, etc.) and the reactor type are usually quite different. Table 1 shows some reaction rates for Fe-based catalysts. The reaction rates of Langmuir-Hinshelwood-type equations involve the partial pressure of water in the denominator because this compound [which forms during the FTS reaction; Equations (1)–(3)] inhibits the reaction on this type of catalyst. In some of the kinetic equations, the partial pressure of water is related to the H₂ partial pressure term, which is written as PH2O/PH2 and allows an analysis of the kinetic effects of H₂O on the FTS reaction using the equilibrium of the WGS reaction. In contrast, the most predominant term in the Langmuir-Hinshelwood-type equations for Co-based catalysts is carbon monoxide (CO; Table 2) because the H₂/CO ratios are higher than those for Fe-based catalysts. These differences in the reaction rate equations in Tables 1 and 2 can be explained by the different proposed reaction mechanisms for the FTS reaction and the variety of studied catalysts. A summary of kinetic parameters for Fe-based catalysts can be consulted in the works of Atwood and Bennett (1979) and Huff and Satterfield (1984). For example, it has been found that the activation energy for the FTS reaction is in the range of 37–103 kJ/mol for Fe-based catalysts and in the range of 37–100 kJ/mol for Co-based catalysts (e.g. Atwood and Bennett 1979, Yates and Satterfield 1991, Jess and Kern 2009, Chabot et al. 2015). On the contrary, the heat of the adsorption/desorption process (equilibrium constants) has been reported with a value of -100 kJ/mol (Atwood and Bennett 1979) and -8 kJ/mol (Huff and Satterfield 1984) for Fe-based catalysts and a value of -68.5 kJ/mol (Jess and Kern 2009) and 20 kJ/mol (Chabot et al. 2015) for Co-based catalysts.

Nevertheless, the variety of reaction rate equations allows inferring that these equations have decisive and different effects on predictions of the FTS reaction’s behavior. Moreover, special care should be taken when the reaction rate equation is determined experimentally for the FTS reaction because the catalyst can change the activity and selectivity as a function of time. Usually, the stabilization of the catalytic surface takes some time because of the formation of different chemisorbed species on the active sites and the occurrence of different transport phenomena, which can significantly affect the reaction. These effects can be considered reversible, but some changes at the catalytic sites are irreversible and negatively affect the activity of the catalyst because the activity sites are modified. Thus, the kinetic data from the literature were reported after a “prudent” amount of time had elapsed since the beginning of the reaction. For example, some works (e.g. Pour et al. 2014) reported that a “steady state” in the activity measurements was obtained after 4 h of operation. In other works (e.g. Morales et al. 2006, den Breejen et al. 2011), dozens of hours were required for the catalyst to reach “stationary” activity. In still other works (e.g. Bezemer et al. 2006), catalyst deactivation resulted in changes in the activity measurements; therefore, the “true” activity (in “steady state”) of the catalyst could only be obtained after several days of operation.

Broadly speaking, the deactivation of the catalyst during the FTS reaction can occur through different mechanisms, including poisoning, sintering, the reoxidation of active sites, and coke formation. Usually, the poisoning of catalysts occurs because of the presence of sulfur compounds in the reactor feed. The sintering of catalysts can occur during overheating in the reactor. Meanwhile, the reoxidation of active sites along the catalyst mainly occurs because of the presence of H₂O, with Fe-based catalysts being more affected by this type of deactivation than Co-based catalysts. Coke production is strongly affected by the reaction temperature, and its accumulation on catalysts can cause the catalyst to break up (Espinoza et al. 1999). For example, it is reported that there are three causes of deactivation for the Fe-based catalysts in fixed-bed reactors; at the front of the reactor, sulfur poisoning and coke deposition are the two observed factors of the catalyst deactivation, whereas the occurrence of hydrothermal sintering and/or oxidation of the active sites predominates at the end of the reactor; this last factor is promoted by the increasing amount of H₂O along the reactor (Dry 1990, Duvenhage and Coville 2006).

On the contrary, studies that discuss the transport phenomena that occur during the FTS reaction coupled with the kinetics of catalyst deactivation are scarce. Representative examples of these works include those of Hallac et al. (2015), who studied the loss in activity by fouling (coke) for an Fe-based catalyst in a TFBR, and Sadeqzadeh et al. (2012), who studied the deactivation of a Co-based catalyst by sintering in a TFBR. Thus, using the correct reaction rate equation is very important to obtain useful data from simulations.

Catalyst deactivation is an important issue in the design of chemical reactors and must be studied in depth because this process is multifactorial (e.g. catalyst composition, operating conditions, reactor type, time on stream, and feed composition), which could lead to misinterpretations of the kinetic data and erroneous reactor designs. Hence, the kinetic data should be carefully analyzed to obtain a true representation of the reaction through the reaction rate equation.

4.1 Microscale (catalytic particles or pellets)

Studies on the catalytic particle (or pellet) scale have garnered great interest because several physical and chemical phenomena occur during the FTS reaction. In general, the mechanism by which synthesized gas molecules react on the catalytic surface of the particle is as follows: (i) the transport of reactive gas molecules from the bulk gas phase to the external catalyst surface, (ii) the adsorption of reactant molecules onto the active sites, (iii) the reaction of the adsorbed molecules on the catalytic surface, (iv) the desorption of products from the active sites, and (v) the transport of products to the bulk gas phase. Points (i) and (v) represent external mass transport and are generally not considered in modeling because they are not an industrially limiting stage due to the high gas hourly space velocity (GHSV) that is employed in industrial reactors (Shen et al. 1996a). Points (ii) to (iv) are normally involved in the reaction kinetics, the complexity of which is reflected in the variety of equations that are found in the literature, some of which are shown in Section 2.

Before reaction, the molecules should travel through the pores (usually with an intricate morphology); once the molecules reach the catalytic surface, the reaction proceeds by forming hydrocarbon chains of different sizes. Depending on the reaction conditions, the products can condense (capillary condensation) and/or produce liquid waxes that fill the pores of the pellet (Post et al. 1989, Shen et al. 1996a,b, Wang et al. 2001, Jung et al. 2010), as shown in Figure 4. Subsequently, new reactants in the gas phase must first solubilize in the liquid wax. Because of the existence of a concentration gradient, these reactants are then transported by diffusion through the liquid wax (e.g. n-C₂₈H₅₈; Wang et al. 2001) from the outside of the pores to the catalytic surface, where they react to produce hydrocarbon chains.

Figure 4:

Illustration of a catalyst particle during the FTS reaction showing the regions where are frequently considered the transport phenomena balances.

Furthermore, the diffusivity coefficient is a property that measures the transport ability of a solute within a solvent or dissolution. When the solute and the diffusion media are in the same phase, the diffusivity is known as molecular. Froment et al. (2011) reported that the flux of the ith component in a mixture of N components can be driven by its own concentration gradient and those of all other species in the mixture; thus, the so-called Maxwell-Stefan equation [Equation (9)] can be obtained for ideal gases.

However, the diffusivity coefficient can be estimated by assuming a mean binary diffusivity of species i through a mixture (D_im); however, individual diffusivity coefficients are needed because of the difficulty in determining the flux from Equation (9). If the gas mixture in the system is considered ideal and the liquid wax that forms during the FTS reaction (where the ith compound must diffuse) is assumed to be a stagnant medium (the flux ratios are zero for k=2, 3, …), the mean binary diffusivity can be determined using Equation (10).

Equation (10) is useful to estimate the mean diffusivity of a diluted compound through a liquid mixture. It is not easy to estimate the molecular diffusivity of CO or H₂ in the operation conditions of the Fischer-Tropsch reaction (mixture of hydrocarbons at high temperature and pressure). Furthermore, the reported values in the literature are scarce and usually are for very specific cases. For that reason, Makrodimitri et al. (2011) made a study using molecular dynamics simulations, where they performed the theoretical determination of the Maxwell-Stefan diffusion (D^MS) coefficients of H₂ and CO in two pure alkanes (n-C₁₂ and n-C₂₈). This coefficient is a kinetic factor, and according to the Fick’s law, it is related to the Fick diffusion coefficient (D^Fick) with the relation DABFick=QDABMS, where Q is a thermodynamic factor. They showed that the D^MS coefficient has a strong dependence on the molar gas composition (CO or H₂) and the type of liquid hydrocarbon considered (see Table 3) as well as the temperature and pressure. It can be seen from Table 3 that Makrodimitri et al.’s predicted values are close to those obtained experimentally, showing that the use of theoretical calculations can be a useful tool for the estimation of diffusivity coefficient in the conditions in where the FTS reaction occurs. Moreover, they found that, in diluted mixtures (molar gas fraction<0.1), the Q factor is close or equal to unity; therefore, the D^MS and D^Fick diffusion coefficients are equal. This result is in agreement with the broad consensus in the literature to consider a diluted H₂ (or CO) solution in the estimation of the diffusivity coefficient. In practical circumstances during the FTS reaction, it is expected that low concentrations of CO and H₂ exist in the liquid wax, as the solubility of these gases decreases with the temperature and with the number of carbon atoms (molecular weight) of the liquid hydrocarbon mixture (Torres et al. 2013, Trinh et al. 2016).

Nevertheless, the Maxwell-Stefan model does not adequately describe systems in which molecules collide frequently (gas phase) with the walls of the porous media. In such cases, the molecular and Knudsen diffusivities have an important effect on the mass transport; consequently, the dusty gas model [DGM; Equation (11)] could be suitable to describe these systems (Froment et al. 2011).

The last term in Equation (11) corresponds to viscous flow, which is normally only considered when (Bopt/μD′K,i)>10–20 and corresponds to micrometer-size pores. The DGM could be applied to the FTS reaction because the pores of the catalyst particles were not completely filled with the liquid wax that was produced during the reaction. Although this model considers the possible interactions and collisions of molecules with the walls of the porous media (and therefore could accurately predict the gas transport inside the pores), it is not frequently used because of its inherent complexity for multicomponent gas system modeling.

A simplified equation for the description of molecular diffusion can be obtained from Equation (11) by neglecting the viscous flow (such as for nanometer-size pores), considering that reactant A diffuses through the liquid wax B (assuming a binary mixture), and assuming an equimolar counterdiffusion (N_B=-N_A), which results in Equation (12).

Equation (12) is the sum of the resistances for the transport of gas molecules according to the molecular and Knudsen diffusivities and is known as the Bosanquet equation [Table 4, Equation (4.6)].

The literature does not frequently contain analyses of the prediction capability of different diffusive flux models. Among the few examples, Nanduri and Mills (2015) compared the prediction accuracy of five diffusion models for the FTS reaction with porous particles in an Fe-based catalyst: (i) the diffusion flux equation by Wang et al. (2001) [Table 4, Equation (4.3)], which was also used by Wu et al. (2013); (ii) the Wilke model [Equation (10)], which was also used by Mamonov et al. (2013) and Chabot et al. (2015); (iii) the Wilke-Bosanquet model [Equation (12); see Table 4, Equation (4.6)], which was also used by Hallac et al. (2015); (iv) the Maxwell-Stefan model [Equation (9)]; and (v) the DGM [Equation (11)]. Only the DGM and Wilke-Bosanquet model include both the molecular and Knudsen diffusivities.

The above authors performed numerical simulations with a 1D model to evaluate the ability of the five models to predict the concentration profiles for key compounds inside particles (spherical geometry) during the FTS reaction, assuming isothermal operation (493 K), a pressure system of 2.5 MPa, a molar feed ratio H₂/CO=2, and a mean diameter pore of 25 nm. The vapor-liquid equilibrium (VLE) was estimated using the Soave-Redlich-Kwong (SRK) equation. All the models predicted a minimum concentration of CO and H₂ at the center of the particle, whereas the H₂O profile always presented a maximum at the center. However, the concentration profiles of CO₂ were a function of the diffusion flux model that was used. For instance, the CO₂ concentration profile exhibited a maximum at the center of the particle for the Wilke, Wilke-Bosanquet, and Maxwell-Stefan models, whereas the CO₂ concentration increased to a maximum and then continuously decreased toward the center of the particle when using the Wang model and DGM. The latter CO₂ concentration profile can be explained by the occurrence of the reverse WGS reaction [Equation (5)], where the CO₂ that is produced is consumed to produce CO, which is used for the polymerization (FTS) reaction. Thus, this work demonstrated that only the DGM and Wang model predicted the WGS occurrence.

Moreover, the authors evaluated the prediction capability of the intraparticle liquid-vapor (L/V) ratios of the five models. Their results showed that the Wang model predicted the highest L/V ratio among the models. This result and the fact that the CO profile from the same model rapidly approached zero inside the particle suggest that important diffusional effects occurred because of the liquid wax that formed, whereas the Wilke-Bosanquet model and DGM exhibited similar trends for the CO concentration. However, the diffusional limitations of the last two models were attributed to Knudsen diffusion inside the particle pores.

The results of Nanduri and Mills (2015) showed the importance of understanding the coupled effect of the Knudsen and molecular diffusivities on a catalytic porous particle and considering the phase of the FTS reaction products because different kinetics can occur during each phase. Thus, from the point of view of mass transport, the pores of the catalytic particles are not completely filled with liquid wax for low degrees of polymerization in the FTS reaction (as with Fe-based catalysts). Thus, the Knudsen diffusivity contributes to the transport of gas compounds through the catalyst’s pores, and this transport is followed by the molecular diffusion of the compounds in the liquid wax. On the contrary, the heavy compounds that form under high degrees of polymerization (as with Co-based catalysts) are found in a liquid state; therefore, the main resistance to the mass transport will be from molecular diffusion.

Thus, the system could be divided into three regions to analyze the mass transport during the FTS reaction on a microscale level: (I) the gas mixture, which has a boundary at the gas-liquid wax interface; (II) the liquid wax between two boundaries, one that is located at the gas mixture-liquid wax interface and the other at the liquid wax-catalytic surface interface; and (III) the liquid wax at the interface with the external catalytic surface of the particle, where the FTS reaction occurs. These three regions are depicted in Figure 4. Region I is characterized by a convective process of reactants from the gas bulk toward the gas-liquid wax interface, where the solubilization of gas molecules occurs in the liquid wax. In region II, the molecules are transported by molecular diffusion in a liquid phase to the catalytic surface if the pores of the particle are totally filled with liquid wax because of differences in the concentrations between the two boundaries. Conversely, if the pores are not totally filled with liquid wax, both molecular and Knudsen diffusion are important for the transport of reactive molecules to the catalyst’s reactive surface. Finally, molecular diffusion through the liquid wax inside the pores and reactions on the catalytic surface occur in region III. The thickness of both the gas film (mass transfer external resistance) and the liquid wax are usually neglected during the analysis of the mass transport process because these values are very small; thus, only the diffusion effects inside the pores are usually considered.

On the contrary, several proposals from the literature evaluate the diffusion coefficients during the FTS reaction. Erkey et al. (1990) showed that the diffusivity of the synthesis gas in the liquid wax that forms during the FTS reaction can be assessed as the diffusion of synthesis gas through a pure hydrocarbon (e.g. n-alkane), with an average number of carbon atoms that is distinctive of the liquid wax. Some studies considered icosane (C₂₀H₄₂) as the representative liquid wax media (Hallac et al. 2015, and references therein). This hydrocarbon compound is typically obtained in FTS when the reaction is performed with Fe-based catalysts. Once the reactants diffuse through the liquid wax and react on the catalytic surface, the gaseous compounds that form are first solubilized in the liquid wax and then travel out of the pores by diffusion. Basically, the reactants diffuse toward the catalytic surface and the products counterdiffuse to the bulk gas phase of the reaction media. This counterdiffusion has not been studied in the literature (at least to our knowledge) possibly because the field flow within nanometer-size pores is quite complex.

Because the reaction system is heterogeneous (fluid phases plus a solid), the estimation of the diffusivity coefficient must consider the properties of the solid, which is known as the effective diffusivity. Several proposals have been presented in the literature, some of which are listed in Table 4.

The path that molecules traverse inside the catalyst pellet increases with the porosity. The trajectory that the molecules follow in their diffusion through the pores is not straight but instead follows a sinuous path. A measure of this sinuosity is the tortuosity, and the estimation of the effective diffusivity must include the difficulty of the transit within the pores of the compounds. For example, the effective diffusivity can be approximated by multiplying the molecular diffusivity by the relationship between the particle porosity (0<ε_p<1) and the tortuosity (τ>1; Shen et al. 1996a,b, Jung et al. 2010). This relationship implies that the effective diffusivity will be lower when the intricacy of the path increases because the ratio ε_p/τ will decrease as the tortuosity increases (see Table 4).

The FTS reaction is nonisothermal, so some authors have considered the effect of temperature on the diffusivity coefficient (e.g. Wang et al. 2001, Wu et al. 2013) using an Arrhenius-type model, which means that the process is thermally activated. Another method to estimate the effective diffusivity coefficient considers the composition (number of carbon atoms) of the liquid wax that forms during the FTS reaction. Two such models have been proposed: one by Iglesia et al. (1993), which predicts a strong dependence of the diffusivity with the number of carbon atoms [exponential type, Table 4, Equation (4.9)], and another that correlates the diffusivity with the size of the hydrocarbons using a power law equation [Table 4, Equation (4.10); Zhan and Davis 2002], which is less dependent on the composition of the liquid wax than the exponential model. Additionally, an analysis of these two proposed models shows that the true dependence of the diffusivity coefficient with the number of carbon atoms is less than the predicted value that is obtained with Equation (4.9) (Kuipers et al. 1995).

Furthermore, Hallac et al. (2015) calculated the effective diffusivity using a relationship between the molecular diffusivity (D_AB) and the Knudsen diffusivity (D_K) with the approximation of Bosanquet [Equation (12)]. These authors found that the Knudsen diffusivity was typically two orders of magnitude larger than the molecular diffusivity for the conditions of their study, so the contribution of D_K to the effective diffusivity was negligible. This result agrees with the findings of Brunner et al. (2012), who also found that D_K (3×10^-6 m/s²) was two orders of magnitude larger than D_AB (2×10^-8 m/s²). This relationship coincides with models that only consider the molecular diffusivity with the physical argument that reactants must diffuse through pores that are completely filled with liquid wax; thus, the low solubility of the reactants in the liquid wax slows the mass transport process (Zhan and Davis 2002). Therefore, determining the solubility of reactants in the liquid wax (see Figure 4, region II) is essential to construct a phenomenological model that can describe the diffusional effects on the catalytic reaction.

Knowledge of the physical state (gas or liquid) in which the compounds are found inside the reaction medium is fundamental to obtain the best phenomenological description of the reaction system. Depending on the reaction conditions, some of the hydrocarbons that are produced could condensate, whereas the water and incondensable gasses (e.g. H₂, CO, and CO₂) that form always remain in a gaseous phase. The physical properties of the phases must be estimated to determine the diffusivity and heat transfer coefficients. Therefore, a descriptive model is desirable to evaluate the VLE in a multicomponent mixture because obtaining experimental data directly from the FTS equipment during operation is difficult and expensive. The determination of properties such as the solubility of the reactant compounds presents at least two challenges (Wang et al. 1999): (i) defining the liquid wax is not an easy task because this material is a complex mixture of hydrocarbons and its composition (and therefore its physical characteristics) is a function of the reaction conditions. To overcome this difficulty, the hydrocarbon mixture that comprises the liquid wax must be frequently lumped into one pseudo-component to simplify the analysis (Leibovici 1993, Lox and Froment 1993). (ii) Because of the substantial differences in the molecular properties of light compounds (gas phase) and the compounds that comprise the liquid wax, conventional thermodynamic descriptions such as the cubic equation of state are difficult to apply because their scope is not designed for systems where the gas-liquid mixtures have such asymmetrical properties.

Recently, Haarlemmer and Bensabath (2016) proved that estimating the flash parameters with the SRK and Peng-Robinson (PR) equations was possible. However, these equations have high computational costs; instead, these authors used Antoine’s law for their VLE calculations. Wang et al. (1999) developed a generalized correlation using a cubic equation of state to estimate the VLE (with an emphasis on the solubility) in the FTS reaction. These authors suggested that their correlation could be used for systems with a wide range of solutes, including CO, H₂, CO₂, CH₄, C₂H₄, and C₂H₆, and heavy wax solvents in the C₂₀–C₆₁ range. These authors found that their equation provided good correlations for 406 experimental data points of 29 binary systems that were reported in the literature. The fit of the experimental data resulted in an absolute deviation of <6%, which was mentioned to be sufficient to meet the engineering requirements for the Fischer-Tropsch process design (Wang et al. 1999). Additionally, Wang et al. (2001) considered the thermodynamic equilibrium of the FTS reaction compounds that are present in both gas and liquid wax phases in the pores of the catalyst particle. The compounds’ concentrations were estimated using the modified SRK equation of state (MSRK EOS); this equation demonstrated a suitable representation of the diffusional effects for their study on the FTS reaction with data from an industrial Fe-Cu-K catalyst.

On the contrary, Zhan and Davis (2002) analyzed the diffusional limitations during the FTS with data from an Fe-based catalyst and determined the effective diffusivity coefficient by considering an exponential dependence with the amount of carbon atoms [Table 4, Equation (4.9)]. Evaluating this equation for a hydrocarbon with n+1 carbon atoms resulted in Equation (13):

Equation (13) shows that the diffusivity of heavier hydrocarbons is always lower and can be approximated using a simple relationship between diffusivities: D_n+1/D_n≈0.74. This result agrees with the hydrocarbon chain growth probability that is predicted by the ASF equation because the formation of heavier compounds decreases when the diffusion of the compounds inside the pores decreases. Moreover, these authors concluded that the diffusion of heavy products in the liquid wax is not necessarily more difficult compared to that of light hydrocarbons because the solubility of the former compounds should also be considered. Thus, only a solubilized compound can diffuse inside and outside the catalyst’s pores, and the effect of the solubility prevails over that of all the other processes under the most common operating conditions for the FTS reaction. Consequently, if the solubility of heavier compounds is high enough, their concentration will be homogeneous in the liquid wax, and their diffusional limitations become less important than those of light hydrocarbons (Zhan and Davis 2002).

Because the liquid wax film on the catalyst is considered very thin (see Figure 4, region II), the diffusion of the components through the liquid is neglected. Thus, the presence of diffusional limitations inside the particle (see Figure 4, region III) is commonly determined using the dimensionless Thiele modulus (Φ), which was originally defined for a homogeneous first-order reaction (Thiele 1939) and is expressed in Equation (14):

where L_p is a characteristic length of the reaction system, k_A is the kinetic rate constant for a homogeneous first-order reaction, and D_e,A is the effective diffusivity. Aris (1957) noted that the effectiveness factor for different catalyst geometries can be approximated by a single function of the Thiele modulus if the length parameter L_p in Equation (14) is defined as the ratio between the volume of the pellet (V_p) and its external surface area (S_p). Then, the Thiele modulus is redefined as Equation (15):

The simplest model from the literature that was designed to study the importance of mass transport restrictions during the FTS reaction analyzes a single catalytic particle or pellet, which allows the study of the different phenomena that occur inside the pores. Thus, a mathematical model for the diffusion of molecules inside pores and the occurrence of reactions on the catalytic surface (see Figure 4, region III) can be expressed in Equations (16) to (18) by considering isothermal and steady-state operating conditions:

where q is a geometric factor that is used to adjust Equation (16) to rectangular, cylindrical, or spherical coordinates; ξ is the spatial coordinate; ℓ_c is the characteristic distance to the external catalyst surface; c_A is the mass concentration; c_As is the mass concentration at the catalytic surface; and R^A is the reaction rate. Equation (16) indicates that the mass transport of species A by diffusion is determined by the rate at which this species can be transformed on the catalytic surface. B.C. 1 in Equation (17) establishes that the concentration of species A in the center of the particle is finite. B.C. 2 [Equation (18)] indicates that the concentration of species A on the catalyst’s surface is determined by the affinity of species A and the catalytic sites.

Furthermore, the effectiveness factor (η) should be evaluated to quantify the mass transfer resistance inside the particle (Figure 4, region III). This factor is defined according to Equation (19) as the ratio between the actual or observed reaction rate (R^A) and the rate when mass transfer resistance is absent (R^AO), which is calculated over the average particle volume (V).

If η tends close to unity in Equation (19), the mass transfer resistance inside the particle is negligible; if the value tends to zero, high resistance to mass transfer is present inside the catalyst particle (intraparticle mass transfer resistance). Additionally, if the Thiele modulus (Φ) in Equation (14) tends to zero, the intraparticle mass transfer resistance is negligible, and the process is governed by the kinetics of the chemical reaction; otherwise (Φ much >0), the process is governed by intraparticle mass transfer.

Table 5 demonstrates the influence of the catalyst particle’s geometry in the solution of Equations (16) to (18) on the concentration profile and the effectiveness factor, where r_p is the radius of the cylinder or sphere according to the coordinate system; Φ is the Thiele modulus as defined in Equation (14); I₀ and I₁ are the modified Bessel functions of zero and first orders, respectively; and η_a is an approximation of the effectiveness factors. The equations were solved by assuming isothermal and isobaric operating conditions and a homogeneous first-order kinetic rate.

Gonzo and Gottifredi (1983) and Wu et al. (2013) reported that the magnitude of internal mass transport limitations in a spherical catalytic particle can be determined using the mathematical inequality |1-η|≤0.05; thus, one can estimate the effectiveness factor in the absence of internal mass transfer resistances. When the value of Φ is >2, the value of η is approximately 0.4, which corresponds to severely restricted internal mass transport. Additionally, if Φ>2, the value of the η factor can be estimated by the above criterion or the approximation of Equation (5.5) in Table 5, indicating severe diffusional resistances. Moreover, the estimated value for the diffusivity is larger for ideal catalytic pores compared to that for a real porous particle because the tortuous nature of the latter pores must be considered.

An example of the effect of intraparticle resistance on the FTS reaction is the work by Post et al. (1989), who conducted an experimental study with both Fe- and Co-based catalysts in a fixed-bed microreactor system. In their study, the catalyst particle size varied between 0.2 and 2.6 mm (with a pore diameter between 10 and 220 nm), and the diffusional limitations were determined using the Thiele modulus [Equation (14)]. The main considerations for their model were the following: spherical particles, H₂ as a representative reaction molecule, and a homogeneous pseudo-first-order reaction. Their results demonstrated that the reaction rate was severely limited by the intraparticle diffusion of H₂ when the catalyst particle’s diameter was larger than 1 mm. Their analysis suggests that the effective reaction rate decreases significantly when the catalyst pores are filled with high-molecular-weight hydrocarbons that form during the FTS reaction, which agrees with the idea that molecular diffusivity is the main resistance to mass transfer when the pore is completely filled with liquid wax, as discussed above. To simplify the analysis of the FTS reaction, some researchers have considered that all the reaction products are present in a single phase, either vapor or liquid, during the development of their mathematical models (Shen et al. 1996a,b). Although this proposal is useful to analyze problems that consider that reaction products are present in a single phase, this approach is an oversimplification of the problem because it ignores the effect of the VLE in the FTS reaction. Thus, Shen et al. (1996a) analyzed the temperature dependence of η for the FTS reaction using data from an Fe-Cu-K precipitated catalyst. These authors studied the physical state of the reaction mixture in two different situations: one in which the compounds were present only in a gas phase and another in which the compounds were present in gas and liquid phases. Their results showed that the η values in the former case were always higher than those when considering the combination of the two phases, which reinforces the importance of considering the VLE in modeling the system.

On the contrary, Jung et al. (2010) solved the model of Equations (16) to (18) for a single cylindrical pore, only assuming diffusion along the axial coordinate with the parameters ℓ_c=L and q=0 using a heterogeneous kinetic rate equation [e.g. Table 1; Equation (1.5)], and compared this solution to that from a pseudo-homogeneous first-order kinetic rate. These authors observed that the concentration profile from the pseudo-homogeneous first-order kinetic rate was always higher compared to that from the heterogeneous kinetic equation. This result agrees with those from Shen et al. (1996a) in terms of the error that is introduced by considering only a gas phase as the reaction media and not considering a reaction in a combined gas/liquid phase.

Nevertheless, the two above studies are an approximation because having a simple first-order kinetic reaction or a single heterogeneous kinetic rate in the FTS reaction is not realistic. In fact, both the kinetic rate of the main reaction (FTS reaction) and the kinetic rates of methane formation and the WGS reaction should be considered in the model, mainly when Fe-based catalysts are used to perform the FTS reaction. Shen et al. (1996a) used experimental information from an industrial Fe-Cu-K catalyst to study a combination of the FTS and WGS chemical reactions; their proposal suggested that a “single” reaction occurs during the FTS process. In this case, these authors considered the heat and mass transport resistances in the catalyst particle. The FTS and WGS reactions [Equations (1) and (5)] were summed, which can only be done if the ratio of the reaction rates is known or can be estimated. The resulting chemical equation is expressed in Equation (20):

where χ is the WGS reaction rate (R^WGS) and the FTS reaction rate (R^FT) ratio (χ=R^WGS/R^FT). The parameter χ retrieves the limit cases for both reactions. The above authors proposed the use of representative families of compounds to facilitate the analysis; thus, C₁–C₆ hydrocarbons were chosen as representatives of the gas phase and C₁₆₊ hydrocarbons as representatives of the liquid phase. The authors’ simulations determined that significant mass transfer limitations occurred when a hydrocarbon with high molecular weight was chosen as the representative compound, which decreased the effectiveness factor and thus indicated strong diffusional limitations. For example, an η value of 0.82 was obtained for C₂H₆, whereas the effectiveness factor dropped to 0.77 when the C₇H₁₆ compound was chosen to represent the reaction medium. The authors extrapolated the behavior of η in the reactor and concluded that this factor had an average value of approximately 0.4 along the catalyst bed under optimal conditions during reactor operation because the PH2O/PCO ratio [a term in their WGS kinetic rate equation; e.g. Table 1, Equations (1.6) and (1.7)] increased along the catalyst bed, decreasing the reaction rate as H₂O formed and inhibited the reaction. Nevertheless, the disadvantage of this proposal is the lumping of reaction products, which does not allow the different reaction products to be discriminated and therefore prevents the reaction selectivity from being studied.

Jung et al. (2010) simulated a cylindrical pore (average length of 675 mm) that was filled with liquid wax to assess the mass transfer restrictions from the limited dissolution of H₂ and CO during FTS. These authors used a kinetic rate equation obtained with an Fe-based commercial catalyst [Table 1, Equation (1.5)]. This study proposed the existence of concentration gradients along the axial direction inside the pore without considering the concentration gradients along radial coordinates because of the insignificant size of the pore diameter (nanometer size) compared to the pore length. The authors found that only 13% of the pore was effectively used for the FTS reaction and attributed this result to the existence of severe diffusional limitations.

This result agrees with the findings of Wang et al. (2001), who simulated the FTS reaction and recorded experimental data from a spherical pellet of an industrial Fe-Cu-K catalyst. These authors introduced the rate equations for the formation of methane, alkanes (C_nH_2n+2, n≥2), and alkenes (C_nH_2n, n≥2) and the rate equation for CO₂ reactions. Comprehensive kinetic rate equations were used to solve the model [see Table 6, Equations (6.1)–(6.6)] into their modeling. Their proposed model is expressed in Equations (21) to (26).

Mass transport balance

Energy transport balance

Equation (21) is the mass transport equation for the species i and considers the diffusion into the particle and the rate of generation/consumption. B.C. 3 [Equation (22)] establishes that the concentration in the center of the pellet is finite. The above authors considered that the thickness of the liquid wax film that covers the catalyst pellet is very small [(l_film/d_p)<<1], so the external mass transfer limitation can be neglected. Consequently, k_m in Equation (23) is very large because (c_s,i-c)≈0, which agrees with reports that external mass resistances under industrial operation can be neglected (Shen et al. 1996a). Thus, rather than consider B.C. 4, the above authors considered that the solubilization of gaseous reactants in the liquid wax is the prevalent effect in the absence of external mass transfer resistances. With this idea, the authors proposed the thermodynamic equilibrium expression yi=(φiL/φiV)xi to estimate the compounds’ composition (c_s,i) in the liquid wax. The transport of energy [Equation (24)] in the pellet considers that the entirety of the reaction’s heat is transferred by conduction in the radial coordinates of the particle. B.C. 5 [Equation (25)] establishes that the temperature in the center of the pellet is finite. According to B.C. 6 [Equation (26)], the heat conduction from the particle dissipates at the solid-liquid wax interface by convection with the fluid around the catalyst particle. Thus, Wang et al. (2001) solved the model by considering a millimeter-size particle. These authors found that the effectiveness factor for CO (η_co) ranged from 0.14 to 0.28 for industrial-size pellets with diameters of approximately 2–4 mm, indicating severe diffusional limitations (Figure 5).

Figure 5:

Variation of effectiveness factor with pellet size.

Reprinted with permission from Wang et al. (2001, p. 4324). © Copyright 2001 American Chemical Society.

The authors also found that a catalytic reaction only occurred along 20% of the external section of the pellet (r/r_p=0.8–1); thus, the interior of the particle was practically inert during the reaction, which they called “inert core thickness”. Figure 6 demonstrates this result: selectivity toward C₅₊ hydrocarbon compounds was only favored along the particle’s surface. Thus, the authors referred to this behavior of the catalyst particle as an eggshell-type catalyst, which appears to be a very attractive solution to lessen the intraparticle transport restrictions during the FTS reaction. With the existence of an inert core, the catalyst pellet (or eggshell) would avoid the depletion of CO inside the particle, thus maintaining the molar H₂/CO ratio along the inner surface of the pellet according to differences in the obtained ratio from a uniform pellet. Therefore, an eggshell-type pellet could suppress the chain termination reaction in the deep catalytic area of the pellet, leading to the formation of heavier hydrocarbons (see Figure 6).

Figure 6:

Effect of inert core radius on C₅₊ selectivity.

Reprinted with permission from Wang et al. (2001, p. 4324). © Copyright 2001 American Chemical Society.

As would be expected from the above results, the authors also found a negative impact on the selectivity because of strong diffusional limitations. Light paraffinic hydrocarbons formed easily in the inner section of the catalyst pellet, whereas heavier hydrocarbons preferentially formed in the outer region because of the existence of severe intraparticle diffusion limitations. This result is reflected in the selectivity toward the desired (Figure 7A) and undesired (Figure 7B) products as the pellet size increased. Nevertheless, the authors mentioned that the size of the pellet must range from 2 to 4 mm under practical conditions to maintain a low pressure drop and effective heat removal. This last point suggests that a strong compromise must be made to achieve the best conditions between the catalyst’s performance and the pressure drop in the reactor.

Figure 7:

Selectivity variations of C₂₊ and C₅₊ products with pellet radius (A), and selectivity variations of CO₂ and CH₄ with pellet radius (B).

Reprinted with permission from Wang et al. (2001, p. 4324). © Copyright 2001 American Chemical Society.

In terms of the energy balance in the catalyst pellet, the above authors’ simulations determined that the temperature difference between the bulk gas and the outer surface of the catalyst pellet was 2 K, whereas the temperature difference between the outer surface of the pellet and its center was <0.02 K because of the excellent thermal conductivity of the catalyst (Fe-Cu-K). This last point indicates that the thermal resistance inside the catalyst pellet can be neglected and only the external thermal resistance must be considered when analyzing this reaction (Carberry 2001). This concept is relevant because the selectivity toward products is very sensitive to changes in the reaction temperature. An example is illustrated in Figure 8 through two cases, namely, for an Fe-based catalyst (Figure 8A) and for a Co-based catalyst (Figure 8B), where the selectivity of the reaction was estimated as a function of the reactor temperature. Although the selectivity of the reported compounds exhibited minor changes, the selectivity to olefins decreased significantly with only a 20-degree difference in the temperature of the reactor, which reflects the high sensitivity of the FTS reaction to the reaction temperature (van der Laan and Beenackers 1999).

Figure 8:

Effect of the reactor temperature on the selectivity to C₂–C₄, C₅–C₉, CH₄, and olefin/paraffin ratio for Fe-based catalyst (A and B, respectively; filled symbols: H₂/CO=2; empty symbols: H₂/CO=0.2) and selectivity to C₅₊, C₂₅₊, CH₄, and olefin/paraffin ratio for Co-based catalyst (C and D, respectively; H₂/CO=2.1).

Data taken and adapted from Feyzi and Jafari (2012) for Fe-based catalyst and from Visconti et al. (2010) for Co-based.

On the contrary, Wu et al. (2013) simulated the FTS reaction using data from a micrometer-size Fe-based catalyst. Their work developed a comprehensive model for the FTS reaction that incorporated a mechanistic kinetic model and the probability of chain growth using a modified ASF distribution with two α values. As an approximation, the consumption of CO was estimated by the sum of both formation rate equations, namely, the equation for hydrocarbons [Equation (1)] and the reverse WGS reaction [Equation (5)]. Their multicomponent diffusion-reaction model analyzed the relationship between diffusion and the reaction and the influence of the reaction’s temperature and pressure and the size of the catalyst particle (micrometers) on the FTS reaction. Their mathematical model basically followed the form of Equations (21) to (26); however, these authors expressed their model in terms of the mole fraction and defined a dimensionless variable for the radial coordinate as r*=r/r_p, in which not all of the volume of the particle is used during the reaction (Wang et al. 2001). Additionally, these authors considered that the thermal resistance between the external surface of the catalyst particle and the surrounding film of liquid wax could be neglected in Equation (26) (h_f→∞). Additionally, the products were lumped by compound families because the authors considered including the nondominant products of the reaction unnecessary; this consideration simplified the problem and allowed the reaction system to be expressed in a useful form. Thus, the authors took the following compounds as the main representative products: CH₄, C₃H₈, C₁₀H₂₂, C₂₀H₄₂, CO₂, and H₂O. These representative compounds were used to propose a system of independent reactions that could characterize the FTS reaction system when an Fe-based catalyst was used [Equations (27)–(31)].

The results of Wu et al. (2013) for the concentration gradients for CO, H₂, and chosen products showed that the FTS reaction mainly occurred in the outer shell of the catalyst particle, which agrees with the results by Wang et al. (2001). In their simulations, Wu et al. (2013) found that the temperature difference between the center of the catalyst particle and its surface was equal to 0.012 K because of the small particle diameter in their study (d_p=115–120 μm). Likewise, the authors analyzed the effect of the temperature on the FTS reaction by changing the temperature of the bulk gas phase (503, 513, and 523 K) and observed that increasing the temperature decreased the production of long-chain hydrocarbons, whereas the selectivity toward methane increased. Their explanation for this trend was that the increase in the bulk gas-phase temperature promoted the WGS reaction, which in turn increased its participation in the consumption of the carbon source (CO) during the formation of heavier hydrocarbons. When the pressure in the reaction system varied (1.5, 2.1, and 3 MPa), no significant effect on the concentration profiles was observed for the reactants or products; the authors mentioned that this result mainly occurred because the pressure only exerts small effects on the diffusivities of the reactants in the liquid wax that is produced. Furthermore, the concentration profiles in the catalyst particles were calculated, which demonstrated that the CH₄ and C₁₀H₂₂ concentrations were higher than the C₃H₈ concentration. The authors mentioned that this result occurred because of the high concentration of H₂ that was available for the reaction. However, the behavior that was reported for the concentrations contrasted the expected behavior for these compounds (particularly the C₃H₈ compound), even when high deviations in the ASF distribution were present (e.g. Puskas and Hurlbut 2003). This result may have been a consequence of the representative compounds that were chosen because the range between the chosen hydrocarbons was very broad. Clearly, adequately estimating the selectivity using lumping methods is very complex.

4.2 Mesoscale: novel approach (microreactors)

Technological advances have led to the development of reaction systems that consist of microchannels to minimize the heat and mass transport resistances in the catalysts. The results obtained to date show that a smaller catalyst particle size implies less intraparticle resistance to mass transfer; therefore, in this condition, the intrinsic reaction rate would be obtained.

Thus, the development of heterogeneous reactors that are able to minimize the internal mass transfer in the catalyst presents an alternative to conventional slurry-phase reactor (SPR) and fixed-bed reactor. For example, Derevich et al. (2013) studied the hydrodynamic behavior during the FTS reaction in a vertical microchannel reactor with cylindrical geometry. In their study, a Co-based catalyst deposited in microparticles (50–300 µm) on the interior surface of the microchannels was used. Similarly to small catalyst particles, the hydrodynamic resistance could negatively affect the reactor operation when the microchannel mean diameter decreased below 0.4 mm. It is also necessary to consider the hydrodynamic effects on the smooth- and rough-walled microchannels for this type of reactor because these effects can be entirely different in both surfaces, and significant errors can be obtained if the ideal smooth-walled model is used for the pressure drop calculation instead of a more realistic model (rough-walled model).

Becker et al. (2014) conducted a simulation study to determine the form of the minimized diffusional restrictions in a microchannel reactor with a rectangular geometry. For this study, they assumed the Co-based catalyst adhered to the inner surface of the microchannels. The intention in this type of reactor is that the diffusional resistance of the reactants and products inside the catalyst layer is the only resistance to mass transport; thus, the porous network is the main variable to be optimized to achieve the maximum productivity for the desired hydrocarbons (C₅₊). In this regard, their work focused on the design of the catalyst coating rather than optimizing the process conditions. Thus, to study the reduction of diffusional limitations, they introduced the concept of “transport pores” (TP). Figure 9 shows that this type of reactor can be modeled on different scales. The different scales and their corresponding characteristic parameters are as follows:

1. W_M, L_M, H_M.

2. W_m, L_m, H_m.

3. w_m, h_m.

4. l_cat, l_TP.

Figure 9:

Representation for the different scale-levels for transport phenomena analysis in a microstructured reactor.

(A) Micro-structured reactor, (B) Monolith, (C) One-channel, (D) Catalyst-coated and the gas flow and (E) Gas-wax-catalyst.

The magnitude order of these scales is depicted in Figure 9. Their mathematical model was established on the mesoscale of the reaction system. The transport balance was made on the structured catalyst layer (Figure 9E) instead of the macroscale (the whole reactor; Figure 9A). Becker et al. (2014) considered isothermal operation conditions and neglected the convective mass transport within the channels to simplify the model as much as possible. The concentration of reactants was assumed constant in the gas-liquid interface, meaning that the mass resistance due to the liquid wax film can be neglected (as is the case when the liquid wax film is very thin; otherwise, the external mass transport should be included in the model). They studied the mass transport for the reactive species (CO and H₂) and the effect of diffusion on the reaction rate and selectivity in addition to the performance of the catalyst layer with and without TP.

In their model, it is supposed that the synthesis gas (CO and H₂) flows inside the microchannels (convective mass flow) and that transversal diffusion occurs toward the catalyst layer (with a micrometer thickness), where the FTS reaction occurs. After the FTS reaction occurs, the compounds that form the liquid wax remain on the catalyst surface, and the reactant gases must first diffuse into the liquid wax before they reach the inner section of the TP to react at the catalytic surface.

Then, to develop the differential equations on the structured catalyst layer, the volume average process (Whitaker 1999, 2006) is applied to the local mass conservation equations for both the liquid wax and the solid catalytic phases. Thus, the volume fractions are defined as ε_i=V_i/V, where the volume V is defined as V=V_TP+V_cat and the subindex i represents either cat (catalyst) or TP (transport pore). The mass balance in the inert TP filled with the liquid wax resulted in Equations (32) to (34):

where D_e,TP is the effective diffusivity of the ith compound in the TP, calculated with the relation D_e,TP=D_i(ε_TP/τ_TP); n_cat·J_i,cat is the diffusive mass flux in the normal direction to the catalyst surface; n_cat is the unit normal vector, which in this case corresponds to -δ_x according to Figure 8; and D_e,cat is the effective diffusivity of the ith compound on the catalyst. B.C. 7 indicates the absence of mass transfer limitations between the gas and liquid phases, and finally, B.C. 8 indicates whether diffusion of the compounds stops at the end of the catalyst layer or in the monolith wall. Additionally, the mass balance inside the catalyst resulted in Equation (35):

The authors did not provide the boundary conditions that they used for solving the differential equation; thus, we think that the following boundary conditions could apply [Equations (36) and (37)]:

In this case, the unit normal vector for the TP is opposite in direction to the catalyst n_TP=-n_cat. The mass balance in the catalyst represented in Equation (35) indicates that the mass transport by diffusion and by the concentration gradient at the liquid-solid interface equals the reaction rate occurring on the catalyst. B.C. 9 in Equation (36) indicates that the mass transport of species i by diffusion is determined by the rate (supposing that is the effective rate on the TP volume) at which this species can be transformed on the catalytic surface considering that the TP is sufficiently small (d_TP≥l_TP), and B.C. 10 indicates whether diffusion of the compounds stops at the end of the catalyst layer or in the monolith wall.

The models described in Equations (32) to (37) and in Equations (21) to (26) are different because they are estimated at different scales in the reaction system. Equations (32) to (37) are at the scale of the catalyst layer (micrometer), whereas Equations (21) to (26) are defined at the catalyst porous particle scale. Furthermore, in Equations (32) to (34), the interfacial mass transfer part that connects the mass balance in the catalyst with the mass balance in the TP comes from the effective mass balance in the system rather than a boundary condition (see Figure 9).

From these results, it can be observed that this configuration of the reaction system enhanced the diffusion of the reaction compounds because the compounds can diffuse easily inside of the catalyst layer (τ→1) due to the regular geometry of the microchannel. The results suggest that the productivity can be high if the fraction of TPs in the reactor is optimized because this arrangement can provide a large catalyst surface area in contact with the reactants, resulting in an enhanced reaction rate (Becker et al. 2014, and references therein).

Although the TP (microchannels) improves the CO and H₂ mass transport toward the catalytically active centers, these microchannels presents a practical disadvantage because the intrinsic catalytic activity is only achieved at small catalyst thicknesses (~100 mm). For larger catalyst thicknesses, the selectivity for desired fraction (e.g. C₅₊) decreases considerably. These adverse effects become more evident with the use of highly active catalysts because the polymerization reaction occurs rapidly, leading to light compounds, which is in agreement with the effect observed with the size of the catalyst pellets discussed above (see Figure 7). Therefore, Becker et al. (2014) suggested that a combination of highly active catalyst with a layer of 100 mm would lead to the best performance of the microstructured reactors. However, the main disadvantage with this type of reactor is the amount of catalyst that can be deposited in the walls of the channels because the amount will always be less than the amount of catalyst used in conventional reactors, which essentially limits the reactor productivity.

Once the behavior of the FTS reaction on the microscale (particle or pellet) and/or mesoscale is known, the next target is designing the reactor at the macroscale to achieve the best performance in productivity and selectivity. Then, simulations performed on the reactor scale can be used to determine the appropriate values of the parameters to enhance the catalyst lifetime, conversion, and selectivity, reducing the amount required for the expensive experimentation.

4.3 Macroscale: modeling of FTS reactors

4.3.2 Modeling of the FTS reaction on a TFBR

4.3.2.1 1D TFBR models

There are several studies that model the FTS reaction in the TFBR. Most of these studies use different approaches to explain the phenomena that occur during the chemical reaction and perform the modeling to achieve a description of the operational variables of the reactor. In this regard, the works mentioned herein studied the behavior of the reactor (operating temperature, conversion, yield, and selectivity) while varying the diameter of the catalyst particle (d_p), the internal diameter of the reactor tube (D_T), the height or length of the reactor (L_T), the synthesis gas molar feed ratio (H₂/CO), the coolant temperature (usually boiling water: T_cool), the mass flow rate of the reactor feed (F), and the recycle ratio (R).

Atwood and Bennett (1979) published a study on FTS reaction modeling at a time when computation capability was difficult to obtain and great effort was necessary to obtain the numerical results. They conducted an experimental study to measure the distribution of products and the reaction rates for the FTS reaction using both commercial Fe- and Co-based catalysts. Their simulations using a heterogeneous kinetic model considered a plug-flow velocity profile; however, the longitudinal mass and heat dispersions, as well as the radial temperature gradients inside the reactor tube, were not considered to simplify the analysis of their 1D model. Instead, the mass and energy balances were developed in the entire reactor (macroscale balance). The internal mass transfer resistance for the catalyst pellet was evaluated through the effectiveness factor, and the intraparticle temperature gradient was neglected because the heat generation function (β=rate of heat generated by chemical reaction/rate of heat transported by thermal conduction) was considered small. In essence, β is a measure of the nonisothermal effects and was calculated with the approximation β≈(T_center-T_sup)/T_sup=ΔT_max/T_sup (Weisz and Hicks 1962); thus, when β→0, an isothermal condition in the pellet or particle catalyst is reached. From their analysis, the interfacial resistance (gas-particle) was considered through the material and energy balance results in Equations (38) and (39).

(38)kmaν(cAs-cAb)=-R^i

(39)hfaν(Tsup-Tb)(-ΔHi)R^i

These equations are driving linear force-type relationships that simplify the analysis by assuming that all the catalyst particles are subjected to the same concentration and temperature throughout the external surface. In the simulations, they used a H₂/CO=2 and a reactor operating pressure of 2 MPa to study the behavior of the FTS reaction at different temperatures of synthesis gas (523, 553, and 583 K). Temperatures above 583 K created an unstable region in the reactor. They analyzed the effect of the particle diameter (d_p: 2, 4, and 6 mm), flow rate (Re_p: 25, 50, 75, and 100), tube diameter (D_T: 20, 40, and 60 mm), and tube length (L_T: 1.65–10 m) on the effective heat exchange area of the reactor (the sum of the area of all tubes in the reactor) and found that this area was minor with an increase in L_T because the heat exchange was increased with the enlarged surface of the tubes and when the flow (F) in each tube of the reactor was decreased, which indicated a smaller amount of reagents reacting on the reactor, resulting in less heat that has to be removed. Finally, the use of small d_p led to a decrease in the production of heavy hydrocarbons, resulting in less heat production; therefore, less heat exchange area was required.

Wang et al. (2003b) proposed a pseudo-homogeneous reactor model comprising mass and energy balances in the reactor [Equations (40)–(43)]. They called this a “heterogeneous model” despite the main supposition that the reaction system has pseudo-homogeneous behavior. An important consideration here is that the analysis of the heat and mass transfer in the reactor are obtained only by the balances in the axial coordinate of the reactor, neglecting the effects of the radial coordinate. This method resulted in a 1D model that can be solved “easily”:

Mass balance in the reactor

(40)ddz(usci)=24dp3ρp(1-εB)∫0rp(∑j=1NRαijR^j) r2dr (i=1,NPG)

(41)B.C. 11: z=0, ci=ci0

Energy balance in the reactor

(42)ρgCp,musdTbdz=24dp3(1-εB)∫0rp[∑j=1NR(-ΔHj)R^j] r2dr+4UDT(Tcool-Tb)

(43)B.C. 12: z=0, Tb=To

On the contrary, in the FTS reaction, the concentration gradients and temperature profiles inside the catalyst particles have important implications on the selectivity for the reaction products, which should be determined. The corresponding mass and heat transfer balances in the catalyst pellets are expressed in Equations (44) to (49).

Mass balance in the catalyst particle or pellet

(44)Deff,i1r2ddr(r2dcs,idr)=-ρp∑j=1NRαijR^j

(45)B.C. 13: r=0, dcs,idr=0

(46)B.C. 14:  r=rp, -Dedcs,idr=km(cs,i-ci)

Energy balance in the catalyst particle or pellet

(47)keff1r2ddr(r2dTsdr)=-ρp∑j=1NR(-ΔHj)R^j

(48)B.C. 15: r=0, dTsdr=0

(49)B.C. 16: r=rp, -keffdTsdr=hf(Ts-Tg)

Wang et al. (2003b) assumed that the external mass transfer limitations can be considered absent because the film of heavy hydrocarbons (liquid wax) around the catalyst particle is very thin. Therefore, the mass transfer coefficient in Equation (46) can be supposed to be large, and then the boundary condition results in c_s,i=c_i. This concentration could be explained by assuming that there is a gas-liquid equilibrium at the interface between the catalytic particle and the gaseous phase (this is a reasonable supposition for the TFBR; Froment et al. 2011), taking into account the solubilization of the gas-phase compounds into the liquid wax [yi=(φiL/φiV)xi]. Thus, the value of c_i can be obtained from the thermodynamic equilibrium. This consideration is only acceptable in a strongly exothermic reaction, if the operation conditions allow the supposition that there is not a sufficient temperature gradient in the catalyst pellet (e.g. small catalyst pellet size and/or high thermal conductivity in the pellet), leaving the main resistance to the heat transfer in the region between the liquid wax and the external surface of the catalyst particle (Carberry 1961, 2001, Weisz and Hicks 1962).

To obtain the solution of the model, momentum balance is also required. Almost all 1D models reviewed here used the momentum balance in the reactor given by Equations (50) and (51):

(50)dPdz=-fp(G2/ρg)dp1-εε3

(51)B.C. 17: z=0, P=P0

where f_p=f_p(Re_p) is a function of the Reynolds number based on the catalyst particle (Re_p) defined in Equation (52):

(52)Rep=dpμG1-ε

where G=ρu_s is the mass flux (or mass flow per unit area) in the axial direction and u_s is the superficial gas velocity. Most studies also used the Ergun correlation (Ergun 1952) expressed in Equation (53), but Wang et al. (2003b) used the correlation of Hicks (1970) expressed in Equation (54). There is a substantial difference in the behavior of both equations when the Reynolds number increases. In the case of the Equation (53), a continuous drop until an asymptotic value is observed, whereas Equation (54) always predicts an increase, which is the best physical description of the flow during the reactor operation (Hicks 1970).

(53)fp=1.75+150Rep

(54)fp=6.8(Rep1-ε)0.8

Essentially, all revised 1D transport phenomena models for the TFBR considered a plug-flow regime, which simplifies the analysis of the problem. However, Mamonov et al. (2013) simulated the FTS process with a Co-based catalyst [see Table 6, Equations (6.12)–(6.16)] for a laboratory-scale reactor without consideration of the plug-flow regime because they considered this assumption to be an oversimplification of the flow pattern. Instead, they used an expression for the linear gas velocity that depends on the temperature, pressure, and the conversion of CO, taking into account that these parameters vary throughout the reactor length. Equivalently, Moutsoglou and Sunkara (2011) conducted a simulation study in a multitube packed-bed reactor with an Fe-based catalyst, considering the changes in the mass flow rate of the gas through the packed tube as a function of the adsorption of CO and H₂ on the catalytic surface and the incorporation into the fluid stream of the produced hydrocarbons (olefins and paraffins), some in the gas phase (C_n, n<20) and others in the liquid phase (C_n, n≥20). These changes in the molar volume of the mixture promote important changes in the flow pattern inside the reactor that have to be taken into account to achieve a proper description of the reactor behavior. For example, Mamonov et al. (2013) made use of the ratio between the molar flux of the ith component (Φ_i) and the linear velocity (u) to estimate the concentration of the compounds (C_i=Φ_i/u) along the reactor length. The concentration in this case has a strong relationship with the temperature, pressure, and CO conversion; therefore, its variation along the reactor will not be linear or a simple function of the reaction parameters.

Mazzone and Fernandes (2006) conducted a study to describe the polymerization of the CO and the distribution of products during the FTS reaction when the contribution of the WGS reaction is significant (which is the real situation during the FTS reaction on Fe-based catalysts). They supposed that alkyl and alkenyl mechanisms acted together during the FTS reaction. Their model assumes that the heat generated during the reaction is completely removed by the cooling water in the reactor jacket, allowing for consideration of isothermal operation in the reactor (543 K). The external mass transfer resistance between the catalyst and the gas phase, as well as the resistance to mass transfer within the particle, was considered negligible due to the very small diameter of particle used (d_p=70 µm) in their simulations. To simplify the study, the concentration and temperature gradients within the tube in the radial coordinate were not considered, and it was assumed that the produced hydrocarbons were in equilibrium in the gas and liquid phases at the reactor outlet. The reactor pressure (1–4 MPa), the gas velocity (1–10 m/s), and the H₂/CO feed ratio (0.5–2) were varied to study the influence of these parameters on the conversion, production, and distribution (selectivity) of the produced hydrocarbons. The total system pressure and the H₂/CO ratio were the main factors that affected the reactor performance and should be adjusted depending on the desired hydrocarbon compound fraction. For example, the production of gasoline was favored by high pressures and high gas velocities, producing 4.8 wt% compared to only 1.9 wt% of diesel. Additionally, the selection of an adequate catalyst is critical to achieve the best performance in the reactor.

Moutsoglou and Sunkara (2011) conducted a study with an Fe-based catalyst using the rate equations derived from the alkyl and alkenyl reaction mechanisms (similar to Mazzone and Fernandes 2006) to predict the formation of paraffin and olefin compounds. Their model considered isothermal conditions and the kinetic effects derived from the hydrocarbon desorption and their integration into the bulk gas phase. They conducted simulations to evaluate the effects of varying the H₂/CO ratio, pressure at the reactor inlet, and reactor length parameters on the selectivity and product distribution. The rate of carbon conversion increased with an increase in the inlet pressure, in agreement with the results of Wang et al. (2003b), and the same occurred with an increase of the reactor length and the H₂/CO ratio. The selectivity to paraffins increased with an increase in the H₂/CO ratio and length of the reactor (L_T), whereas the selectivity to olefins increased with an increase in the inlet pressure of the reactor.

The control of the reaction temperature is not only performed using a cooling jacket in the TFBR; recirculating a fraction of the reactor effluent facilitates temperature control in the reactor. There is an increase in the temperature difference between the reactor core and the outer surface of the reactor when D_T (diameter of the reactor tube) increases. The control of the reactor temperature is a key factor in the FTS reaction because the temperature determines the productivity and selectivity of the process. Thus, there is a necessity to develop mathematical models that can help in the understanding of the reactor behavior during the FTS reactions to economize and improve the design and scale-up of the reactors.

For example, Wang et al. (2003b) conducted a study in which CO, H₂, CO₂, H₂O, N₂, and hydrocarbons (n-paraffins and n-olefins) of up to 50 carbon atoms were selected as key compounds in the development of a mathematical model to evaluate the effects of the main process parameters on the FTS reaction using a recycling operation. They assumed that the particle bed does not offer resistance to heat transport through the reactor tube wall; in other words, they stated that the effective radial thermal conductivity in the particle bed tends to infinity (λ_rad→∞), and the resistance to heat conduction due to the reactor wall (λ_W→∞) was neglected. An excess of heat could increase the reactor temperature, causing the occurrence of a hotspot (Wang et al. 2003b). They observed that, when D_T increases, the hotspot moves forward in the axial coordinate of the reactor (Figure 10A), severely affecting the selectivity to C₅₊ products, which finally leads to a large production of unwanted products (CH₄ and CO₂). This result is in agreement with the result reported by Atwood and Bennett (1979), who suggested that it is undesirable to use tubes with a D_T >40 mm to perform the FTS reaction.

Figure 10:

Effect of process parameters on the axial temperature profile in the fixed-bed reactor. Base conditions: dt=D_T=32 mm, L=7.0 m, GHSV=500 h^-1, R_cyc=R=3.0, T_W=523 K, P_in=2.5 MPa, fresh synthesis gas composition (%); CO: 30.59; H₂: 57.75; CO₂: 7.0; N₂: 4.08; CH₄: 0.58.

Reprinted with permission from Wang et al. (2003b). © Copyright 2003 Elsevier.

Change of temperature along the reactor as function of: (A) Catalyst particle diameter, (B) Recycle ratio, (C) Cooling temperature, and (D) Reaction pressure.

When the recirculating effect was studied by Wang et al. (2003b), it was observed that, without recycling (R) to the reactor, the temperature near the reactor inlet increases rapidly, leading to a hotspot formation or thermal instability of the reactor (Figure 10B). The recirculation of a fraction of unreacted synthesis gas (diluted with light hydrocarbons) to the reactor inlet causes an increase in the linear velocity of the synthesis gas inside the reactor, which reduces the concentration of reactants and the reaction rate, preventing reactor overheating. As can be seen in Figure 10B, the temperature of the hotspot decreased and its location in the reactor moved backward as the value of R was increased, showing that the recycle operation is important to maintaining the thermal stability of the TFBR because recycling serves to boost the heat transfer parameters in the reactor. The last allows a greater control over the selectivity towards the desired products (e.g. C₅₊) and consequently prevents the formation of unwanted products (i.e. CH₄ and CO₂; Wang et al. 2003b).

The temperature in the cooling jacket (T_cool) also has a significant effect on the operation temperature and reactor performance. The heat load generated by the reaction must be absorbed by the refrigerant fluid in the cooling jacket. The rate of heat removal depends mainly on T_cool; therefore, when this temperature is lower than the temperature of the feed gas, there are no hotspots at the inlet reactor (see Figure 10C, where T_W=T_cool). If T_cool increases (or consequently the reactor temperature increases), then the synthesis gas conversion is favored, but this has a negative effect on the selectivity to the desired products (Wang et al. 2003b). Nevertheless, under these operation conditions, the removal of heat by the cooling jacket does not have the same rate as that generated by the chemical reaction; hence, a hotspot could be formed. This hotspot could move toward the beginning of the catalyst bed with an increase in the reactor temperature (Wang et al. 2003b, Mamonov et al. 2013), affecting the selectivity towards the desired products (C₅₊). Therefore, it is necessary to properly select the operation conditions for each reactor to ensure efficient heat removal from the catalytic zone so that good conversion of the synthesis gas can be achieved in addition to maintaining an adequate selectivity and avoiding the appearance of hotspots. Additionally, increasing the reactor pressure increases the concentration of the gas reactants on the liquid wax due to the VLE that surrounds the catalyst particles, resulting in a greater availability of the reactants on the catalytic surface. This high reactant concentration increases the reaction rate; therefore, an additional generation of reaction heat is observed. Thus, if the pressure in the reactor increases, the temperature of the hotspots also increases, and their position in the reactor could move towards the reactor inlet (Wang et al. 2003b; see Figure 10D). However, an increase in pressure commonly improves the conversion of CO and the overall performance for producing C₅₊ products.

Since the discovery of the FTS reaction, the reaction rate has been extensively studied, and the number of publications that propose a reaction rate equation is large. However, there are few publications that report the quantitative value of the parameters for the rate equations for commercial catalysts (Jess and Kern 2009). For example, Lee and Chung (2012) found that, in most previous investigations of the FTS reaction, the reaction rate equations were developed with the use of lumped reaction models, with evident disadvantages due to their inability to predict the exact amount of heat released by the FTS reaction and therefore the real consumption of H₂ and CO. They mentioned the differences that appear when only the reactant conversion is considered in the determination of the reaction heat released; for instance, if the same consumption amount of CO (10 moles) is considered, the formation of n-decane and methane results in a heat release of 1560 and 2060 kJ (at STP conditions), respectively. This difference demonstrates the need to obtain reaction rate equations that truly represent the FTS reaction and allow that the temperature of the reactor to be satisfactorily controlled if the heat released by the chemical reaction is accurately determined.

4.3.2.2 2D TFBR models

The 1D models proposed so far in the literature for the FTS reaction (e.g. Wang et al. 2003b, Mamonov et al. 2013) are useful because they provide understanding of the transport phenomena and the kinetic effects that occur as a result of the combination of reaction parameters, such as temperature, pressure, and particle size. However, on the macrolevel, these 1D models do not provide an accurate way to quantify the energy transport because the radial coordinate is not taken into account. Therefore, it is necessary to develop and study 2D models that consider the transfer of energy in both the axial and radial coordinates of the reactor to properly evaluate (as quantitatively as possible) the mass and energy transport.

Since the pioneering work of Bub et al. (1980), who discussed a 2D model for an FTS reactor, there have been few works reporting the 2D analysis of the reactor. Computer processing capacity has increased substantially in recent years, as well as the development of commercial simulators that can solve systems of partial differential equations, allowing for the solution of the complex equations that result from the momentum, mass, and energy balances. Thus, several studies for the FTS reaction that comprise 2D models have been published recently (e.g. Everson et al. 1996, Jess et al. 1999, Wu et al. 2010, Jess and Kern 2012a,b, Lee and Chung 2012, Aligolzadeh et al. 2015, Chabot et al. 2015). Examples of the 2D studies include the papers of Jess and Kern (2009, 2012a,b), who found that the prediction of the ignition temperature in the reactor by a 1D model can have an appreciable difference compared to the prediction using a 2D model. As an example, the first case (1D) resulted in a temperature of 528 K, whereas the second case (2D) resulted in a temperature of 520 K, which is a clear indication that the mathematical models must be solved considering as many physical aspects as possible regarding the FTS reaction and reactor. Recently, Ermolaev et al. (2015) published a work where they made use of a comprehensive 2D heterogeneous model using a Co-based catalyst [see Table 6, Equations (6.7)–(6.11)]. They supposed in their model that the pores of the catalyst particles were filled with liquid wax and water due to capillary condensation and validated their model against experimental data from a laboratory equipment and pilot-scale plant. Figure 11 shows a comparison of the temperature results along the reactor length for the predicted data and the experimental measurements (gas flow from left to right). Because there is a high conversion at the entrance of the reactor (due to the high amount of reactants in the fluid stream in contact with the catalyst), the temperature in that zone of the reactor increases suddenly, accompanied by the formation of a hotspot, such as is shown in Figure 11 (see also Figure 10). It can be observed from the theoretical results of Ermolaev et al. (2015) that the proposed model underestimates the temperature in the reactor zone where the increasing temperature occurs (thermocouples index 1–6). The authors indicated that the good prediction of the model after the seventh thermocouple index is due to the small thermal difference between the cooling jacket and the catalytic bed (~3°C), which provides efficient heat removal. Conversely, at the entrance of the reactor, error was introduced in the data fed to the model due to the estimation of the cooling water temperature (the average value of the temperature at the inlet and the outlet of the jacket). Despite this error, the proposed model has good prediction ability for the temperature in the reactor, providing valuable information and a good demonstration (via an experimental validation) that a 2D model is necessary for the simulation of the thermal behavior in a TFBR.

Figure 11:

Temperature profile inside the single-tube reactor (2.1 m long; black, solid) and the water jacket temperature profile (black, dotted).

The solid line (line red in the electronic version) is temperature profile along the reactor tube estimated by the theoretical model.

Reprinted with permission from Ermolaev et al. (2015). © Copyright 2015 Elsevier.

A review of the emerging 2D modeling papers for the FTS reaction involves great effort in the analysis of the proposals to highlight the key differences and advantages in each particular case because the diversity of operating conditions, catalysts, and reactor types makes a straightforward comparison difficult. However, this effort should be conducted to gain better comprehension of the phenomena occurring during the FTS reaction in the pellet and during reactor operation.