Thermodynamic Modeling of Low-Temperature Fischer–Tropsch Synthesis: A Gibbs Free Energy Minimization Study for Hydrocarbon Production

Abstract

Fischer–Tropsch synthesis (FTS) facilitates the conversion of syngas, derived from feedstocks such as biomass, coal, and natural gas, into valuable hydrocarbons (HCs). This investigation employed optimization methods, specifically Gibbs energy minimization, to perform a thermodynamic characterization of the low-temperature Fischer–Tropsch (LTFT) reaction for HC generation. The CONOPT3 solver within GAMS 23.2.1 software was utilized for solving the developed model. To represent the complex FTS product spectrum, twenty-three compounds, encompassing C₂–C₂₀ aliphatic hydrocarbons, were considered using a stoichiometric framework. The study explored the impact of operational parameters, including temperature (350–550 K), pressure (5–30 bar), and H₂/CO molar feed ratio (1.0–2.0/0.5–1.0), on hydrocarbon synthesis. Evaluation of the outcomes focused on HC yield and product characteristics. A significant sensitivity of the reaction to operating parameters was observed. Notably, lower temperatures, elevated pressures, and a H₂/CO ratio of 2.0/1.0 were identified as optimal for fostering the formation of longer-chain HCs. The developed model demonstrated robustness and efficiency, with rapid computation times across all simulations.

1. Introduction

Based on 2022 data reported by the IEA [1], fossil energy resources (coal, oil, and natural gas) account for approximately 80.9% of global energy consumption. The remainder is derived mainly from biomass and waste (8.8%), nuclear energy (4.7%), and hydropower (2.5%). Multiple factors underscore the need for a more extensive use of renewable energy sources. These include the depletion of fossil fuel reserves, the growing global demand for fuels, and the alarming levels of greenhouse gases resulting from increased carbon dioxide (CO₂) emissions. These challenges have motivated research on renewable energies over the last two decades [2,3].

Among several emerging strategies to optimize or replace the use of fossil fuels, economic and environmental factors have renewed the interest in developing and improving gas-to-liquid (GTL) technologies, particularly the Fischer–Tropsch synthesis (FTS) process. Recent advancements have focused on improving efficiency and sustainability in converting methane to liquid hydrocarbons (HCs), with innovations in catalyst design and operational conditions [4,5]. Such efforts address the growing demand for cleaner fuels and the imperative to reduce carbon emissions associated with conventional fossil fuels [5].

In this process, synthesis gas, a mixture primarily consisting of hydrogen (H₂) and carbon monoxide (CO), which can be derived from various renewable sources such as biomass, is converted into synthetic fuels, including methanol, ethanol, gasoline, and diesel [6,7]. This process can be simplified as shown in Figure 1. Recent studies emphasize that biomass gasification remains a promising route to produce clean bio-syngas suitable for the FTS process, yielding renewable liquid fuels that meet stringent environmental standards [5,8].

The synthetic HCs produced are cleaner, with lower sulfur, nitrogen, aromatic, and heavy metal contents compared to petroleum-derived HCs. Additionally, they are reported to be easier and more cost-effective to transport. Due to these characteristics, synthetic fuels produced via the FTS process are considered green fuels [9,10].

The composition and yield of the products are governed by various mechanisms and different kinetic factors. The catalysts employed, the type of reactor, and the operating conditions, such as temperature, pressure, and the composition of syngas in the process feed, exert a significant influence on the composition and characteristics of the resulting HC products, as emphasized by Farias et al. [11]. Accurate modeling of these systems is essential, including everything from equilibrium thermodynamic analyses, such as the one proposed in this work, to complex reactor models that consider transport and pressure drop phenomena, as demonstrated by Jess and Kern [12].

There are essentially two operational modes for the FTS reaction: low-temperature Fischer–Tropsch (LTFT) and high-temperature Fischer–Tropsch (HTFT). The main characteristics of these processes are summarized in Figure 2.

In the high-temperature Fischer–Tropsch (HTFT) process, synthesis gas is typically reacted in fluidized bed reactors in the presence of iron-based catalysts, yielding hydrocarbons primarily in the C₁–C₁₅ range. This process is primarily utilized for the production of liquid fuels and high-value chemical compounds, such as α-olefins. Oxygenated compounds are obtained in the liquid stream and are subsequently separated and purified to produce alcohols, acetic acid, and ketones [13,14].

In contrast, the low-temperature Fischer–Tropsch (LTFT) process, which is commonly a catalytic process, usually employs iron or cobalt-based catalysts (or both) for the synthesis of long-chain HCs, waxes, and paraffin. High-quality, sulfur-free diesel is produced from this process [15]. Most of the technology developed for the FTS process is based on LTFT technology. Due to their high stability, high reagent conversions, and elevated selectivity for HCs, cobalt catalysts are the most widely used in the LTFT process [16].

Within the LTFT process, the reactor operates under conditions ranging from 20 to 40 bar, with temperatures around 340 °C (613.15 K), utilizing iron-based catalysts. Steynberg et al. [17] emphasize that the catalyst-to-gas ratio nearly doubles when comparing Sasol Advanced Synthol (SAS) reactors to conventional circulating fluidized bed (CFB) reactors. Furthermore, the SAS reactor allows for better energy recovery within the reactor and lower production costs. A simplified design of the reactor used in SAS technology is presented in Figure 3.

Table 1. Selectivity of the SAS reactor.

Product	(%)
Methane	7
Olefins C2–C4	24
Paraffins C2–C4	6
Gasoline	36
Middle distillates	12
Waxes	9
Water-soluble oxygenates	6

shows the selectivity of the products obtained with the SAS reactor on a carbon basis. An evaluation of the products indicates that it is possible to simultaneously produce chemicals and fuels. A more detailed view of the oxygenated products obtained is presented in

Table 2. Main oxygenated products.

Main Non-Acidic Chemicals	% Mass
Acetaldehyde	3
Acetone	10
Ethanol	55
i-Propanol	3
n-Propanol	13
i-Butanol	3
n-Butanol	4
Acids	% Mass
Acetic acid	70
Propionic acid	16
Butyric acid	9
Valeric acid and higher	5

.

The low-temperature Fischer–Tropsch (LTFT) process typically aims to produce high-molecular-weight hydrocarbons in fixed-bed multitubular reactors. Several commercial processes utilize this technology. Sasol operates a plant that employs LTFT technology for wax production, operating at 230 °C (503.15 K) and 27 bars using iron-based catalysts. Shell also has a plant that uses LTFT technology; however, instead of iron catalysts, the plant installed in Malaysia in 1993 uses cobalt catalysts [18,19].

The active plants of Sasol and Shell utilize natural gas (used in Shell’s plants) and coal (used in Sasol’s plants) as feedstocks to produce synthesis gas. The integration of biomass gasification processes with Fischer–Tropsch synthesis (FTS) to produce liquid fuels has not yet been implemented at an industrial level. The stage that presents the greatest challenges in integrating these two processes lies in the cleaning and purification of the produced bio-syngas.

This occurs because the catalysts used are highly sensitive to the presence of compounds that can contaminate them, even when these compounds are present in small quantities. Examples of such compounds include sulfur and nitrogen compounds that can be formed during the gasification process from the sulfur present in biomass, as explained by Borg et al. [20]. There are commercially developed processes for the purification and removal of impurities from synthesis gas; however, these technologies were designed for other processes and operational conditions, making their application not entirely viable for use in bio-syngas production.

Within this context, studying the reaction conditions and the influence of composition on the behavior of the products emerges as a necessity for a better understanding of the more favorable characteristics of this type of reaction, especially considering the LTFT process. This information is valuable in the development of new processes and technologies to promote the LTFT FTS process, as well as in the development of more selective catalysts to better promote the economic viability of existing processes. In contrast to kinetic models, which demand comprehensive information on elementary reaction steps, catalyst surface properties, and rate constants, Gibbs energy minimization models rely solely on mass and energy conservation and the thermodynamic stability of species. This makes it highly suitable for preliminary process evaluation, feasibility studies, and parametric sensitivity analyses, where rapid estimation of system behavior under varying conditions is required, such as for the LTFT FTS system.

In this work, optimization techniques were applied to minimize Gibbs energy to thermodynamically characterize the Fischer–Tropsch synthesis (FTS) reaction aimed at producing HC from the LTFT FTS process. The software GAMS 23.2.1 and the CONOPT3 solver were used to solve the proposed problems, employing the non-negativity constraint of the number of moles and the stoichiometric balance. Twenty-three selected compounds were used to represent the possible products of the reaction, including linear aliphatic hydrocarbons in the range of C₂–C₂₀.

The main advance in this study in relation to previous works is a greater representative capacity of the proposed predictive Gibbs energy minimization thermodynamic model. The model developed here is based on a representation of the equilibrium composed of a solid phase (s) to predict the possible formation of solid carbon in the reactor, two liquid phases (l₁ and l₂) intended to predict the formation of HC and water, and a vapor phase (v) to represent all possible combinations of products and phases. The non-idealities of the vapor phase were represented by the virial equation, and the liquid phases were both represented as ideal and immiscible. This model combination enables good predictive capacity combined with low computational times and represents an evolution of the model proposed by Freitas and Guirardello [21].

2. Materials and Methods

2.1. Gibbs Energy Minimization (minG) Model

The Gibbs energy minimization methodology was applied to conduct a thermodynamic analysis of hydrocarbon synthesis from syngas. This approach is based on the principle that, at equilibrium, the total Gibbs energy of the system reaches its minimum value. The total Gibbs energy can be calculated using Equation (1).

$$G=\sum _{i=1}^N\sum _{k=1}^F{n}_i^k{\mu }_i^k$$

(1)

In this context, i represents the selected compounds, and k represents the phases of the system. In this study, the phases considered for formation were a solid phase (represented solely by coke (C_(s))), two liquid phases (l₁ and l₂), and a gas phase (V). Here, G denotes the total Gibbs energy of the system, $n_i^k$ denotes the number of moles of component i in phase k, and ${\mu }_i^k$ represents the chemical potential of component i in phase k. The chemical potential can be calculated using Equation (2):

$${\mu }_i^k=G_i^{°}+RTln\left({\displaystyle \frac{f_i^k}{f_i^{°}}}\right)$$

(2)

In Equation (2), R represents the universal gas constant, T is the temperature of the system, and $f_i^k$ denotes the fugacity of component i in phase k, with the superscript (⁰) indicating the thermodynamic standard state. By substituting Equation (2) into Equation (1) and defining the phases, Equation (3) can be derived.

$$G=\sum _{i=1}^{NC}{n}_i^g\left({\mu }_i^0+RT\left(lnP+ln{y}_i+ln{\varphi }_i\right)\right)+\sum _{i=1}^{NC}{n}_i^l\left({\mu }_i^0+RT\left(lnP+ln{x}_i+ln{\gamma }_i\right)\right)+\sum _{i=1}^{NC}{n}_{C_{\left(s\right)}}^s{\mu }_{C_{\left(s\right)}}^0$$

(3)

The variable ${\mathit{y}}_{\mathit{i}}$ denotes the composition of component i in the gas phase, while x_i represents its composition in the liquid phase. The analysis assumes ideality in the liquid phases, with activity coefficients (γ_i) set to 1. Additionally, the model considers complete immiscibility between two liquid phases (represented as l₁ and l₂), representing the liquid–liquid equilibrium commonly observed between water and hydrocarbons in real FTS systems. This approach simplifies the equilibrium calculations by avoiding the need for a predictive model to address the complex interactions in mixed-phase systems during liquid phase formation.

Non-ideality in the gas phase was represented by the fugacity coefficient (${\varphi }_i$), calculated using the virial equation truncated at the second virial coefficient. This equation is based on the correlation proposed by Pitzer [22] and modified by Tsonopoulos [23], as shown in Equation (4).

$${ln\widehat{\varphi }}_i=\left[2\sum _j^m{y}_j{B}_{ij}-B\right]{\displaystyle \frac{P}{RT}}$$

(4)

This combination of thermodynamic formulation and state equations has previously been used and a good predictive ability reported in the results presented by Freitas and Guirardello [24] and Dias et al. [25] under supercritical conditions, indicating the good capability of virial EoS to represent reactive systems at moderate and high pressures while maintaining the mathematical robustness of the thermodynamic model. By considering the ideality of the liquid phases, the total immiscibility model for both formed liquid phases, and the virial equation to account for non-idealities in the vapor phase, Equation (3) can be reformulated as a minimization problem, as presented in Equation (5):

$$\begin{gathered} minG=\sum _{i=1}^{NC}{n}_i^g\left({\mu }_i^0+RT\left(lnP+ln{y}_i+ln{\varphi }_i\right)\right) \\ +\sum _{i=1}^{NC}{n}_i^{l1}\left({\mu }_i^0+RT\left(lnP+ln{x}_i\right)\right)+\sum _{i=1}^{NC}{n}_i^{l2}\left({\mu }_i^0+RT\left(lnP+ln{x}_i\right)\right)+\sum _{i=1}^{NC}{n}_{C_{\left(s\right)}}^s{\mu }_{C_{\left(s\right)}}^0 \end{gathered}$$

(5)

Following the previous equation, liquid phase 1 (l₁) supports the formation of aliphatic hydrocarbons, while liquid phase 2 (l₂) facilitates water formation within the reaction system. The thermodynamic relationships needed to determine the chemical potentials of the components are presented below by Equations (6) and (7):

$${{\displaystyle \frac{\partial }{\partial T}}\left({\displaystyle \frac{{\mu }_i^{°}}{RT}}\right)}_P=-{\displaystyle \frac{H_i}{R{T}^2}}$$

(6)

$${\left({\displaystyle \frac{\partial H_i}{\partial T}}\right)}_P={Cp}_i$$

(7)

where H_i represents the enthalpy and Cp_i represents the heat capacity of compound i. The equation used for Cp is presented by Equation (8).

$${Cp}_i={Cpa}_i+{Cpb}_{i}T+{Cpc}_i{T}^2+{Cpd}_i{T}^3$$

(8)

Equation (5) can be minimized to determine the equilibrium composition of the system in question; however, some constraints need to be imposed on the system. The first constraint that needs to be added is the non-negativity constraint of the number of moles of each of the components in each of the phases, represented by Equation (9).

$$n_i^k\ge 0$$

(9)

Another restriction is necessary for the model and refers to the mass balance that needs to be imposed on the system. The balance considered in this work and its characteristics are presented below.

2.2. Thermodynamic Approach—Mass Balance of the Reactive System

The literature presents two principal methodologies for ensuring mass balance in reactive minG methodologies: the stoichiometric and the non-stoichiometric formulations. The stoichiometric formulation, which was employed in this study, specifically accounts for the chemical reactions that can transpire during the optimization process. This approach is mathematically described by Equation (10).

$$\sum _{k=1}^{NP}{n}_{ik}=n_i^0+\sum _{j=1}^{NR}{\vartheta }_{ij}{\xi }_j$$

(10)

where NP is the number of phases and NR is the number of reactions involved. ${\mathit{\vartheta }}_{\mathit{i}\mathit{j}}$ is the stoichiometric coefficient of component i in reaction j, and ${\mathit{\xi }}_{\mathit{j}}$ is the degree of advance of the reaction.

The general equation used to represent the formation of hydrocarbons by the FTS process is presented in Equation (11). In addition to this reaction, the possibility of the occurrence of the water gas shift reaction (WGS) was considered (Equation (12)):

$$nCO+\left(2n+1\right)H_2\leftrightarrow C_n{H}_{2n+2}+n{H}_{2}O. (1\le n<20)$$

(11)

$$CO+H_{2}O\leftrightarrow {CO}_2+H_2$$

(12)

In addition, the formation of the 23 compounds listed in

Table 3. Compounds considered in the simulations and their symbols.

Compound	Symbol	Compound	Symbol
Carbon monoxide	CO	n-Decane	C10
Carbon dioxide	CO2	n-Undecane	C11
Water	H2O	n-Dodecane	C12
Hydrogen	H2	n-Tridecane	C13
Ethane	C2	n-Tetradecane	C14
Propane	C3	n-Pentadecane	C15
n-Butane	C4	n-Hexadecane	C16
n-Pentane	C5	n-Heptadecane	C17
n-Hexane	C6	n-Octadecane	C18
n-Heptane	C7	n-Nonadecane	C19
n-Octane	C8	n-Eicosane	C20
n-Nonane	C9	-	-

was considered, including unbranched hydrocarbons in the range between C₂–C₂₀. Table 3 also presents the nomenclature used for each of the compounds throughout this work. These compounds were selected based on some experimental studies in the literature that emphasize that the main hydrocarbons produced comprised paraffins in the range between C₂–C₂₀ [26,27].

The saturation pressure was determined using Antoine’s equation, which is

$$ln{P}_i^{sat}=a_i-{\displaystyle \frac{b_i}{c_i+T}}$$

(13)

All thermodynamic data used in the calculations, including Gibbs energy and enthalpy values in the reference state, as well as parameters for the Antoine equation and the equation used to calculate heat capacity, were obtained from Reid et al. [28].

2.3. Resolution of the Model and Conditions for Conducting the Thermodynamic Analysis

The problems were defined as nonlinear programming optimization challenges, with solutions obtained via GAMS 23.2.1 software utilizing its CONOPT3 solver. During the optimization, the actual formation of any given phase was decided by the solver’s internal generalized reduced gradient (GRG) methodology.

The consideration of coke formation was included in the simulations, represented as pure solid carbon (C_(s)). This type of formulation has been applied to a wide range of systems, with good results when compared with experimental data obtained in the literature [29,30,31].

The reaction conditions analyzed here were temperatures between 350 and 550 K, pressures between 5 and 30 bar, and a molar composition of H₂/CO in the feed of 1.0–2.0/0.5–1.0. The formation of CH₄ was inhibited during the simulations to represent the behavior of the catalyst in the system; this behavior is reported by some experimental works in the literature [32].

The production of olefins was not considered, and the data regarding the presence of these compounds were suppressed from the thermodynamic model because some studies in the literature emphasize that these compounds are formed in concentrations significantly lower than those of paraffins, especially when processes at temperatures below 220 °C (LTFT process) are used [33,34].

This research introduces an enhanced thermodynamic model that builds upon the Gibbs energy minimization framework for FTS process evaluation presented by Freitas and Guirardello [21]. Key advancements in the current model include the incorporation of vapor phase non-ideality, achieved by calculating fugacity coefficients using the virial equation. Furthermore, it addresses liquid phase non-ideality by enabling phase separation through the inclusion of two immiscible liquid phases (l₁ and l₂). Significantly, these improvements were integrated without compromising the model’s robustness while enhancing the thermodynamic model’s predictive capabilities for FTS systems. Incorporating two immiscible liquid phases (l₁ and l₂) into the thermodynamic model of FTS at low temperatures significantly enhances its predictive capability by providing a more accurate description of phase behavior and product distribution while demanding low computational times. With rapid convergence, the Gibbs energy minimization model could be embedded as a soft sensor or model predictive control (MPC) module, providing near-instantaneous predictions of phase behavior, product distribution, and thermodynamic limits under varying operating conditions. This allows operators to adjust parameters proactively to maintain optimal product yields and minimize the formation of undesired byproducts.

Reaction performance parameters were determined based on compositional data of products and reactants. The conversion of the analyzed reagents, in this case CO and H₂, was calculated using Equation (14), and the yield of H₂ in HC was calculated using Equation (15).

$$X_{Conv. i}={\displaystyle \frac{n_{in i}-n_{out i}}{n_{in i}}}.100\%$$

(14)

$$Yield. H_2 in HC=100-{\displaystyle \frac{n H_2{O}_{out}}{n H_{2, in}-{n H}_{2, out}}}.100\%$$

(15)

3. Results and Discussion

In the following sections, we evaluate the effects of modifications in temperature, pressure, and the composition of the feed reagents on the composition of the HC products. Low computational times were observed in all cases (less than 5 s), highlighting the robustness of the proposed thermodynamic model.

3.1. Model Validation

The thermodynamic minG model proposed in the present study was validated using data from Rafiq et al. [35], who investigated the Fischer–Tropsch synthesis (FTS) reaction using a syngas mixture composed of 33% H₂, 17% CO, and 50% N₂. The reaction conditions were a wall temperature of 473 K, a pressure of 20 bar, and a gas hourly space velocity (GHSV) ranging from 37 to 180 NmL.gcat⁻¹·h⁻¹. The FT reactor was a shell-and-tube system, with high-pressure boiling water circulating through the shell to maintain temperature control. A spherical, unpromoted cobalt catalyst was employed to facilitate the FTS reaction.

In addition to experimental data, the authors also presented results from a two-dimensional pseudo-homogeneous kinetic model, which incorporated both transport and reaction rate equations. Both the experimental results and those obtained from the 2D model were used for validation purposes in the present work. The CO and H₂ conversions were analyzed using Equation (14).

The comparative data between the results reported by Rafiq et al. [35] and those calculated using the Gibbs energy minimization model are presented in

Table 4. Comparison between the minG thermodynamic model for CO and H₂ conversion and 2D kinetic model results and experimental data for the FTS process from Rafiq et al. [35].

		GHSV (Nml.gcat−1.h−1)
		37		74		111		148
	minG Model	Exp.	2D Model	Exp.	2D Model	Exp.	2D Model	Exp.	2D Model
CO conversion (%)	99.999	90	91	68	68	40	50	36	39
H2 Conversion (%)	99.948	92	93	71	70	43	52	40	40

. The reaction was simulated using the proposed minG thermodynamic model under the same operating conditions described by Rafiq et al. [35]: a reactor temperature of 473 K, pressure of 20 bar, and a feed composition of 33% H₂, 17% CO, and 50% N₂ on a molar basis.

An analysis of the results presented in Table 4 reveals that the thermodynamic model overestimated the CO and H₂ conversion values compared to both the experimental data and the predictions of the 2D kinetic model proposed by Rafiq et al. [35]. This outcome is expected, as the thermodynamic model assumes that the system reaches equilibrium by the end of the process, whereas in the experimental setup, equilibrium is often not fully achieved under the tested experimental conditions.

This discrepancy becomes more pronounced when comparing the results across different GHSV values shown in Table 4. As the GHSV increases, the deviation between the experimental data and the thermodynamic predictions grows. This is due to the shorter residence time at a higher GHSV, which limits the extent to which the system can approach equilibrium.

Another important aspect to highlight is that the minG model assumes perfect isothermal operation, whereas the experimental prototype used by Rafiq et al. [35] does not guarantee strict isothermal conditions. Temperature fluctuations along the reactor can lead to localized variations in reaction behavior, further contributing to discrepancies between the experimental results and the thermodynamic minG model predictions.

The closest agreement between the thermodynamic model and the experimental data for both CO and H₂ conversion was observed at a GHSV of 37 NmL·gcat⁻¹·h⁻¹. Considering only the data at this GHSV, the absolute mean relative deviation between the experimental results and the thermodynamic model was 4.99% for CO conversion and 5.85% for H₂ conversion. In contrast to the 2D kinetic model, the average relative deviation was 6.66% for CO conversion and 5.09% for H₂ conversion.

Another aspect evaluated by Rafiq et al. [35] was the application of the recursive ASF (Anderson–Schulz–Flory) theory, originally proposed by Anderson et al. [36], using the chain growth probability parameter (α). This value was calculated based on the C₃ to C₃₀ region of the ASF plot in experimental results and C₃ to C₂₀ in the minG thermodynamic model. The α values reported by Rafiq et al. [35] were 0.77, 0.779, 0.784, and 0.80 for GHSV values of 37, 74, 111, and 148 NmL·gcat⁻¹·h⁻¹, respectively.

Under the same conditions, the α value estimated by the minG thermodynamic model was 0.625. This result suggests that the thermodynamic model predicts a lower formation of long-chain hydrocarbons compared to the experimental data. This outcome is consistent with expectations, as thermodynamic modeling tends to favor the formation of products with lower standard Gibbs free energy, which typically corresponds to shorter-chain hydrocarbons. Unlike kinetic models, thermodynamic models do not account for mass transfer limitations or surface kinetics, which are critical factors in promoting chain growth during the FTS process.

The comparison between the proposed minG model and the experimental dataset highlights the inherent characteristics of thermodynamic modeling when applied to a kinetically controlled, mass-transfer-dependent process like Fischer–Tropsch synthesis. Nevertheless, it is important to emphasize that thermodynamic modeling remains a valuable tool for understanding reaction behavior and assessing the influence of operating variables (pressure, temperature, and feed composition) without the high costs and complexity associated with experimental campaigns. Additionally, thermodynamic models offer significantly reduced computational time compared to detailed kinetic simulations.

3.2. Temperature Effects

Figure 4 presents the conversion of CO and H₂ for the HC synthesis process carried out at a pressure of 10 bar and a constant molar ratio of H₂/CO in the feed (2.0/1.0). Analyzing the results presented in Figure 4, it is possible to see that the conversion of both reagents decreased due to the increase in reaction temperature. The conversion of the reagents was elevated under conditions where liquid–vapor equilibrium (L + V) was observed in the system, specifically at temperatures of 350 and 400 K; at temperatures above 400 K, only the vapor phase (V) was present, and conversions of the reactants were significantly reduced. It is important to emphasize that, in all figures where liquid-phase formation occurs, the equilibrium predicted by the thermodynamic minG model corresponds to a liquid–liquid–vapor (LLV) equilibrium. However, the data related to the aqueous liquid phase were omitted from the graphs. Only the results for the hydrocarbon-rich liquid phase (represented as L when present in simulations) and the vapor phase (V) are shown in the figures.

Notably, no solid carbon (C_(s)) formation was detected in the system under any of the conditions examined. This is a significant observation, as coke deposition is directly linked to a reduction in catalyst lifespan [37]. This finding also highlights a correlation: the formation of an organic liquid phase is directly associated with increased CO consumption and greater production of hydrocarbons with carbon chains longer than five (C₅₊). Similar outcomes were reported by Rane et al. [38] in their study on the behavior of different alumina phases (δ, θ, and α-Al₂O₃) during the FTS reaction using cobalt catalysts. The absence of coke formation in the FTS reactor offers significant advantages from an engineering perspective. Without coke deposition, routine reactor maintenance is greatly simplified, as there is minimal risk of fouling, plugging, or pressure drop increases caused by solid buildup [39].

Figure 5 details the effect of temperature on both the number of moles of C₅₊ hydrocarbons produced and the H₂ yield from these hydrocarbons (calculated by Equation (15)). Analyzing the H₂ yield from HCs is particularly relevant because hydrogen is a high-cost reactant, and this metric serves as a key indicator of process efficiency by quantifying how effectively it is converted to the hydrocarbon products of interest. It is important to highlight in this figure that the H₂ yield from HCs increased with increasing reaction temperature throughout the analyzed range, although the increase was less pronounced for temperatures above 450 K. However, this trend reveals a critical operational trade-off. While a higher H₂ yield from HCs suggests a more complete consumption of the reactants, the increase in temperature was mainly reflected in the formation of low-molecular-weight hydrocarbons (C₂–C₄). This behavior, which can be confirmed by analyzing the results in Figure 5 and Figure 6, demonstrates that the conditions that maximize the H₂ yield are not the same as those that optimize the production of valuable long-chain hydrocarbons (C₅₊). It is evident, therefore, that the highest yields of C₅₊ hydrocarbons are associated with lower-temperature regions, a result consistent with experiments conducted by Horáček [40].

Figure 6 presents the behavior of the produced HCs distributed between the liquid and gas phases. The formation of HCs with carbon chain lengths of C₅₊ was observed only at reaction temperatures below 400 K, and it is also possible to see that most of the higher-molecular-weight HC were produced in the liquid phase. In all cases, the main HC formed in the gas phase was ethane (C₂). At a temperature of 350 K, it was observed that the main product obtained in the liquid phase was HCs with carbon chain lengths greater than 5, with a significant production of butane (C₄).

At reaction temperatures above 400 K, the formation of a liquid phase was not observed, and the main product formed in this phase was water, representing 84.23% of the products formed at 350 K and 95.27% of the products formed at 400 K. This indicates that high reaction temperatures combined with the tested reaction conditions are associated with an operational range of low HC production.

The behavior of the mole fraction of the obtained HCs as a function of reaction temperature and the number of carbon atoms in the formed molecular chain is presented in Figure 7. It can be observed from this figure that the production of high-carbon-chain hydrocarbons is inversely related to the reaction temperature; that is, the higher the reaction temperature, the lower the production of long-chain HCs. The production of C₂₀ HC, which was the highest-carbon-chain HC formed during the simulations, was observed only when the reaction was carried out at 350 K. This result emphasizes the close relationship between reaction temperature and the formation of HCs throughout the LTFT FTS process.

The size of the largest hydrocarbon depended on the temperature. It was C₁₄ at 400 K, C₉ at 450 K, C₈ at 500 K, and C₆ at 550 K. This behavior was expected since the increase in reaction temperature is directly related to the production of short-chain HCs and the consequent decrease in the production of longer-chain HCs. This behavior is supported by some experimental studies found in the literature [41,42,43,44]. The results obtained by the thermodynamic model proposed by Freitas and Guirardello [21] also emphasize similar behavior for the formation of HCs throughout the FTS process.

By analyzing the results, it was possible to verify that temperature exerts a significant influence on the products obtained during HC synthesis by FTS reaction. The production of long-chain HCs, which are the main products of interest in this type of process, was favored by conducting the reaction at lower temperatures.

The results presented in Figure 7 emphasize that the products formed in a state of thermodynamic equilibrium follow the Anderson−Schulz−Flory (ASF) general polymerization distribution at carbon numbers greater than about three. These results are similar to those reported by Tavakoli et al. [45], who studied the application of the ASF equation on the product distribution of FT synthesis using nanosized iron catalysts.

Lowering the reaction temperature improves selectivity toward long-chain hydrocarbons while reducing the formation of unwanted light gases like methane and ethane. The downside is that lower temperatures slow down reaction kinetics, necessitating larger reactors or longer residence times to maintain conversion rates in experimental and industrial systems. Catalyst performance may also suffer at lower temperatures due to potential deactivation mechanisms, as reported in [39].

3.3. Pressure Effects

Pressure is reported as one of the process variables that most significantly interferes with the composition of the products obtained in the FTS processes. Figure 8 shows the conversion of CO and H₂ as a function of the process operating pressure. Within the analyzed pressure range and at a constant operating temperature of 373.15 K and a constant H₂/CO molar ratio in the feed stream of 2.0/1.0, the formation of the liquid phase was observed at all pressures up to 30 bar. For pressures of 5, 10, 15, and 20 bar, liquid–vapor equilibrium conditions (L + V) were observed in the system. At 30 bar, the system presented only the formation of the liquid phase (L), which was mathematically constituted by the presence of the organic liquid phase (l₁) and the aqueous phase (l₂) in the proposed minG thermodynamic model. Coke formation was not observed in any of the analyzed conditions, similar to what was noted during the tests related to the effects of temperature.

The H₂ conversion was total throughout the pressure range analyzed, while the CO conversion was favored by increasing the operating pressure, reaching its highest value at the highest pressure analyzed (90.76% at 30 bar). This behavior is consistent with fundamental thermodynamic principles. According to Le Chatelier’s principle, an increase in pressure shifts the reaction equilibrium in the direction that results in a decrease in the number of moles of gas. Since the Fischer–Tropsch reaction (Equation (11)) involves a net molar contraction, higher pressures thermodynamically favor the formation of products, thus boosting the conversion of the reactants. This robust theoretical basis, together with the observation that similar behavior was verified in the experiments conducted by Bochniak and Subramaniam [46], reinforces the validity and predictive capacity of the developed model.

Figure 9 presents the behaviors of H₂ yield in HCs and the total number of moles of C₅₊ HCs produced as a function of operating pressure. It is possible to verify that the H₂ yield in HCs showed a significant reduction as the operating pressure increased. The yield decreased by approximately 5% between the lowest pressure studied (5 bar) and the highest (30 bar).

The main reason associated with the observed decrease in H₂ yield in HCs is the increase in the number of moles of C₅₊ HCs produced, which are formed in higher concentrations under high-pressure conditions. As shown in Figure 9, the number of moles of hydrocarbons with a molecular chain longer than C₅ showed a significant increase with the rise in operating pressure. An increase of approximately 30 fold was observed for hydrocarbons with carbon chains longer than C₅ when the pressure was increased from 5 bar to 30 bar. Similar effects were reported by Horáček [40].

Thus, the decrease in HC yield is interesting because the production of higher-molecular-weight HCs is favored under these conditions, which results in greater H₂ consumption, thereby reducing the yield of this compound in HCs in terms of total moles produced. However, the quality of these hydrocarbons is superior due to the long chains formed under these process conditions.

The distribution of the molar fraction of the produced HCs as a function of the number of carbon atoms in the HC chain and the operating pressure is presented in Figure 10. The behaviors of the curves indicate that the increase in pressure is directly associated with the production of hydrocarbons with a higher number of carbon atoms in the chain. For example, when the reaction was conducted at 5 bar, the HC with the longest carbon chain formed was C₁₅. Starting from 10 bar, the formation of C₂₀ was observed at all evaluated pressures, with greater quantities being formed as the operating pressure increased.

The behavior observed for the effect of pressure was similar to that reported in the literature, including experiments conducted by Das et al. [47] and simulations performed by Fernandes [34]. The increase in pressure resulted in a proportional increase in the production of higher-carbon-chain HCs and greater conversions of CO. Furthermore, it was possible to observe that most of the produced HC were in the liquid phase; similar behaviors are reported in the literature [11].

From the results, it was possible to see that the influence of operating pressure was extremely significant on the production of HCs from syngas during the FTS reaction. The production of higher-carbon-chain HCs is favored by the increase in operating pressure during LTFT FTS. At 30 bar and 373.15 K, the gas phase was eliminated, and only the liquid phase was observed in the products. Under these conditions, significant productions of HCs were achieved. Once again, analyzing the results presented in Figure 10, it is possible to verify that the products formed in a state of thermodynamic equilibrium follow the Anderson−Schulz−Flory general polymerization distribution. Similar results for the effects of temperature and pressure throughout the FTS process are described by Chen and Yang [48].

The observation of a liquid-only phase at 30 bar in the low-temperature Fischer–Tropsch (LTFT) process is highly significant, as it indicates that the heavy hydrocarbons produced condense entirely within the reactor under operating conditions. This greatly influences mass and heat transfer characteristics, as the absence of a gas–liquid interface minimizes issues such as bubble formation or phase separation inside the reaction zone.

3.4. Inlet Composition Effects

The H₂/CO molar ratio is another factor reported to have a significant influence on the behavior of the obtained HC products during FTS processes [49]. In the cases evaluated in this study, the effects of suppression of H₂ and CO under HC production were analyzed during LTFT FTS.

For all factors evaluated, the effects of H₂ suppression were studied at the first two molar ratios (mol/mol), while the effects of CO suppression were evaluated at the last two studied molar ratios (mol/mol). Additionally, the H₂/CO molar ratio of 2.0/1.0 was used as a reference guideline, as it is the molar ratio typically employed in LTFT FTS processes from syngas.

These modifications were made to simulate situations where H₂ acts as a limiting reagent and situations where CO acts as the limiting reagent in the process. In all simulations, pressure and temperature were kept constant at 15 bar and 373.15 K, respectively. In all cases, conditions of liquid–vapor equilibrium (L + V) were observed, and in none of the analyzed conditions was coke formation predicted. Thus, it can be concluded that thermodynamically, under the tested conditions, coke formation is of lesser relevance in an LTFT FTS process using pure syngas as feed for the process.

Figure 11 shows the effect of the H₂/CO feed molar ratio on CO and H₂ conversion. Comparing the two molar ratios in which H₂ was suppressed, it is possible to see a significant reduction in CO conversion, which occurs due to the lack of H₂ as a reagent in the system. Similar results for CO conversion were observed in the experiments conducted by Pirola et al. [44] when studying and simulating a fixed-bed reactor with iron catalysts conducting an FTS process.

CO conversion decreased from 88.45% to 41.54% when the H₂/CO molar ratio varied from 2.0/1.0 to 1.0/1.0. The conversion of H₂, in turn, decreased due to the suppression of CO. When the H₂/CO molar ratio was reduced from 2.0/1.0 to 2.0/0.5, the conversion of H₂ dropped from 100% to 62.50%. In cases where H₂ was suppressed, its conversion was 100%, and the same behavior was observed for CO when it was the suppressed reagent. Similar behaviors for CO conversion are reported by Lu and Lee [49].

Figure 12 presents the yield of H₂ in HCs and the number of moles of C₃, C₄, and C₅₊ produced as a function of the H₂/CO molar ratio used in the feed stream. The yield of H₂ in HCs showed an inverse relationship with the decrease in the molar ratio of H₂ in the feed, meaning that the yield of H₂ in HCs increased as the molar ratio of H₂ in the feed decreased. The suppression of CO also increased selectivity, such that at the H₂/CO molar ratio of 2.0/1.0, the yield of H₂ in HC was 55.95%, while with the H₂/CO molar ratio of 2.0/0.5, the yield increased to 60.00%.

Figure 12 also presents the number of moles of C₅₊ HCs produced as a function of the H₂/CO molar ratio in the feed. In all compositions, the main hydrocarbon formed was C₂; however, these results were suppressed, and only HCs with carbon chains longer than five are shown in Figure 12.

The production of these HCs was negatively affected by the decrease in the molar ratio of H₂ used in the feed, a result that was expected due to the chain growth characteristics presented in the FTS process reaction (Equation (11)). This behavior can be explained by the fact that the suppression of H₂ is directly related to the growth of the carbon chain during LTFT FTS and represents an important reaction characteristic predicted by the proposed thermodynamic model.

At the H₂/CO molar ratio of 2.0/1.0, a significant production of HCs with carbon chains longer than five was observed. With the suppression of H₂, HCs with shorter carbon chains began to be produced in greater proportions. The suppression of CO proved to be even more detrimental to HC productivity; starting from the H₂/CO molar ratio of 2.0/0.75, the production of hydrocarbons with carbon chains longer than four ceased to occur.

This behavior can be more clearly visualized by analyzing Figure 13, which presents the production of hydrocarbons as a function of the H₂/CO molar ratio used in the feed and the number of carbon atoms in the produced hydrocarbon chain. It is possible to see that the suppression of H₂ had a negative effect on the production of HC with higher molecular chains. This behavior occurs because H₂ is a necessary compound in the step of constructing the carbon chains of the formed hydrocarbons, as emphasized before.

At the H₂/CO molar ratios of 1.5/1.0 and 2.0/1.0, the formation of C₂₀ was not observed, while at the molar ratio of 1.0/1.0, the largest hydrocarbon produced was C₁₉. It is worth noting that in all cases where H₂ was suppressed, these hydrocarbons were produced in very low concentrations (less than 10⁻⁶ moles, which represents around 1 to 2% of the total hydrocarbons formed in molar fraction).

Another factor to emphasize is the effect observed due to the suppression of CO in the feed. The suppression of CO proved to be more influential and more detrimental to the quality of the formed HC products than the suppression of H₂. In the case of using a H₂/CO molar ratio of 2.0/0.75, the largest hydrocarbon formed was C₄, while at a H₂/CO molar ratio of 2.0/0.5, the largest hydrocarbon formed was C₃. This behavior is relevant and indicates that the consequence of limiting CO to the reactive system is a reduction in the growth capacity of the carbon chains of HCs produced through the LTFT FTS process. This result is an important finding that thermodynamic modeling provided: CO suppression is very harmful to C₅₊ production. CO is the fundamental carbon source for FTS; its deficiency directly limits hydrocarbon formation capacity. H₂ plays a supportive role in hydrogenation and chain termination but can be adjusted more flexibly. Thus, gas conditioning strategies prioritize maintaining sufficient CO levels to sustain high productivity of higher-chain HCs.

The H₂/CO molar ratio was found to have one of the most significant influences on the composition of the obtained HC products. Analyzing the results, it was possible to observe that the production of HCs with longer carbon chains showed significant dependence on the presence of CO in the system; in systems where this compound was suppressed, the production of hydrocarbons was highly disadvantaged. The suppression of H₂ also had a negative impact on HC production, although its effect was less significant than that of CO suppression.

The results presented in Figure 13 highlight the linear behavior of HC formation as a function of the number of carbon atoms in the HC chain. Thus, once again, we emphasize that the products formed in a state of thermodynamic equilibrium during the FTS reaction follow the Anderson−Schulz−Flory general polymerization distribution. It is important to emphasize that the observation that FTS product distributions follow the Anderson-Schulz-Flory (ASF) model highlights the dominant role of surface kinetics in hydrocarbon chain growth, which is characterized by a probability of constant chain growth. While this indicates that product formation is governed by polymerization-like kinetics rather than thermodynamic equilibrium, thermodynamic modeling remains essential to comprehensively understand the system. Gibbs free energy minimization helps define the thermodynamic limits of the reactions, predicting feasible equilibrium compositions, secondary reactions such as the water–gas shift, and potential formation of undesired byproducts. These insights establish the boundary conditions within which the kinetic mechanisms operate, ensuring that process conditions favor desired pathways while minimizing side reactions.

4. Conclusions

The influence of reaction conditions on the synthesis of hydrocarbons (HCs) from syngas through the low–temperature Fischer–Tropsch (LTFT) process was analyzed. The impacts of temperature (350–550 K), pressure (5–30 bar), and the H₂/CO molar ratio in the feed (1.0–2.0/0.5–1.0) were thermodynamically evaluated using a Gibbs free energy minimization (minG) model with nonlinear programming and a non-stoichiometric formulation.

The synthesis demonstrated a marked dependence on operating conditions, with temperature, pressure, and feed composition significantly influencing HC production. It was found that most of the higher-carbon-chain HCs (C₅₊) were generated in the liquid phase under the conditions where this phase manifested, signaling the optimal operating range for HC production in the LTFT process.

The effects of temperature and pressure were prominent in HC formation. Temperature acted to suppress carbon chain growth during the FTS process, while pressure played an important role in improving the quality of the HCs formed, with the highest C₅₊ hydrocarbon production observed at the highest tested pressure (30 bar) under all investigated operational conditions.

The data obtained highlight the fundamental role of the H₂ molecule in the development of HC chains. CO also proved indispensable, and its restriction was shown to be even more detrimental to HC quality. Such behavior is significant, indicating that limiting CO in the reaction system results in a decreased capacity for carbon chain growth of HCs produced via LTFT.

It was verified that the products formed at thermodynamic equilibrium follow the Anderson-Schulz-Flory (ASF) general polymerization distribution, a consistent trend for all operational variables studied. The optimal operational conditions identified by the simulations were low temperatures (below 450 K), high pressures (above 25 bar), and a H₂/CO molar ratio of 2.0/1.0.

The developed thermodynamic model and the premises adopted in its formulation confirmed its technical applicability for predicting the thermodynamic behavior of LTFT systems. The model exhibited robustness and reliability, with simulation times of less than 5 s, successfully harmonizing complexity and predictive capability, which qualifies it as a valuable tool for the study and characterization of FTS processes.