Determination of the Diffusion Coefficients of Binary CH₄ and C₂H₆ in a Supercritical CO₂ Environment (500–2000 K and 100–1000 atm) by Molecular Dynamics Simulations

Abstract

The self-diffusion coefficients of carbonaceous fuels in a supercritical CO₂ environment provide transport information that can help us understand the Allam Cycle mechanism at a high pressure of 300 atm. The diffusion coefficients of pure CO₂ and binary CO₂/CH₄ and CO₂/C₂H₆ at high temperatures (500 K~2000 K) and high pressures (100 atm~1000 atm) are determined by molecular dynamics simulations in this study. Increasing the temperature leads to an increase in the diffusion coefficient, and increasing the pressure leads to a decrease in the diffusion coefficients for both methane and ethane. The diffusion coefficient of methane at 300 atm is approximately 0.012 cm²/s at 1000 K and 0.032 cm²/s at 1500 K. The diffusion coefficient of ethane at 300 atm is approximately 0.016 cm²/s at 1000 K and 0.045 cm²/s at 1500 K. The understanding of diffusion coefficients potentially leads to the reduction in fuel consumption and minimization of greenhouse gas emissions in the Allam Cycle.

1. Introduction

Diffusion is a fundamental process driven by the random thermal motion of molecules, which significantly impacts chemical and biological reaction rates [1]. It manifests differently across gas, liquid, and solid phases, and governs the overall rate of these processes. This study explores diffusion within supercritical fluid (SCF), which is a unique state where liquid and gas properties coexist. Such unique properties pose experimental and computational challenges [2]. SCF mass transfer processes hold immense potential in various fields: (i) the selective extraction of valuable components from food and environmental samples; and (ii) design and optimization of SCF reactors with precise control of temperature and pressure parameters to ensure proper mixing for efficient reactions. Supercritical carbon dioxide (sCO₂) is a promising tool in materials science and biological applications. Notably, the Allam Power Cycle, a recent breakthrough technology, utilizes sCO₂ for thermal energy conversion from carbon fuels [3]. The cycle transforms greenhouse gas CO₂ in traditional power plants into a reusable working fluid for spin turboexpanders and then generates electricity. The power generation cycle was developed in the last 10 years and has turned the air pollution problem into a solution [3]. However, further research is necessary to fully unlock its potential. Understanding diffusion is crucial for optimizing several key aspects of the technology of the Allam Cycle. First, diffusion dictates the mixing of fuels, like natural gas, with CO₂ at the molecular level. Improved mixing enhances combustion completeness. Second, efficient heat transfer from hot combustion products to CO₂ working fluid relies on diffusion within the gas stream. Third, diffusion influences the separation mechanism of CO₂ from flue gas after combustion. This study investigates the diffusion of hydrocarbon in sCO₂ due to its unique liquid-like and gaseous-like properties, which influences the diffusion coefficients of various hydrocarbons. Understanding the behavior of hydrocarbons in high-temperature and -pressure supercritical environments is vital [4].

The direct-fired sCO₂ cycle that utilizes CO₂ as the working fluid directly is an alternative to current electricity production. The Allam Cycle is a novel power generation technology and it converts carbon fuels into thermal energy and electricity while capturing the generated CO₂ and O₂. The cycle involves four stages. First, the cycle begins with the burning of fuels and oxygen. Fuels and pure O₂ combust within high-pressure and high-temperature sCO₂ in a combustor. The purpose of using pure O₂ is to replace air for cleaner combustion and to avoid nitrogen dilution. Second, the hot combustion products, CO₂ and water vapor, transfer heat to sCO₂ in a heat exchanger. Third, the heated CO₂ expands through a turbine and generates electricity. Lastly, cooled-down sCO₂ is collected in a compressor to complete the cycle. The relatively low critical temperature (37 °C) and critical pressure (8.4 MPa) of sCO₂ make it an attractive choice as an SCF solvent in the Allam Cycle. By elucidating the diffusion coefficients, we aim to gain a deeper understanding of the molecular transport phenomena within this cycle, ultimately paving the way for further power cycle design improvements. The Allam Cycle operates at high pressures (30–300 atm) and high temperatures (500–1150 °C) [3,4].

SCFs offer significant advantages due to their low viscosity and high solute diffusivity, leading to a multitude of industrial applications. The combination of CO₂ with hydrocarbons, like methane and ethane, in supercritical environments holds particular interest in various fields. For example, efficient carbon capture and storage are crucial for mitigating climate change. sCO₂ can effectively capture CO₂ emissions from industrial processes. sCO₂ can also be used to selectively separate and purify desired components from mixtures to avoid the usage of hazardous organic solvents. Furthermore, sCO₂ is a valuable solvent for extracting materials from various sources. These characteristics show the importance of understanding diffusion properties in the sCO₂ environment to optimize these industrial procedures. With its near-perfect CO₂ capture capability, the Allam Cycle requires stable diffusion in the combustor. This is the main reason for investigating the diffusion coefficients of methane and ethane in an sCO₂ environment to see how stable the diffusion of hydrocarbon is in such gaseous-like and liquid-like conditions. The Allam Cycle operates at a pressure range of 30–300 atm. At 300 atm, CO₂ exhibits optimal thermodynamic properties and translates to high thermal efficiency and reduced material selection challenge. In addition, the turbine inlet temperature can reach as high as 1150 °C, which is comparable to modern natural gas combined cycle plants. The broad pressure and temperature windows play significant roles in the Allam Cycle’s performance and design [3,4], and this is the goal of our study.

This paper describes the results of the determination of the binary diffusion coefficients CO₂/CH₄ and CO₂/C₂H₆ by classical molecular dynamics (MD) simulations. CH₄ and C₂H₆ are prevalent hydrocarbon fuels employed in power cycles. In the past, Stubbs comprehensively reviewed MD and Monte Carlo (MC) simulations of supercritical H₂O as well as sCO₂ systems in detail, encompassing crucial aspects of the selection of force fields, the size of the simulation boxes, and the pH at different temperatures, pressures, and densities [5]. The thermophysical properties in the sCO₂ environment have been summarized in his review article. Several studied have employed various force fields to evaluate diffusion coefficients in systems relevant to this work. Aimoli et al. [6] and Moultos et al. [7] utilized the transferable potentials for phase equilibria (TraPPE) force field to assess the diffusion coefficients of pure CO₂, CH₄, and CO₂-H₂O mixtures across a broad range of temperatures (273–623 K) and pressures (0.1–100 MPa). Zhu et al. [8] employed the elementary physical model (EPM2) force field for a Gibbs ensemble MC simulation to determine diffusion coefficients at 304 K. The radial distribution function of CO₂ was analyzed to evaluate the validity of their simulations that the first-neighbor C-C distance is around 4.0 Å. Abbaspour and Nameni used a two-body Hartree–Fock dispersion-like potential to determine the self-diffusion coefficients of CO₂ and CO₂-CH₄ mixtures at approximately 300 K [9].

Several studies have investigated diffusion in an sCO₂ environment relevant to our research. Guevara-Carrion et al. performed CH₄ diffusion experiments with the Taylor dispersion technique and MD simulations [10] at temperatures in the range of 293–333 K and pressures in the range of 9.0–14.7 MPa, which is typical of Allam Cycle conditions. This group determined the self-diffusion, Fick, and Maxwell–Stefan coefficients of CH₄ diluted in an sCO₂ environment. Feng et al. collected a series of diffusion coefficients of n-hydrocarbon (C1–C14) in near-critical and sCO₂ environments (308 K and 323 K, respectively, at 10.5 MPa) from experiments and simulations at an infinite dilution. The ratios of carbon dioxide to hydrocarbon in their simulations are CO₂:CH₄ = 4000:110 and CO₂:C₂H₆ = 4000:58 to mimic the infinite dilution of hydrocarbon molecules [11]. Furthermore, in recent years, Asadov et al. experimentally studied the diffusion coefficients of CO₂-C₂H₆-heavy oil and CO₂-C₃H₈-heavy oil in an sCO₂ environment at temperatures in the range of 320–355 K and pressures in the range of 2–15 MPa [12]. It is important to note that these examples of previous studies focused on conditions below 623 K and 100 MPa.

More MD and MC simulations were applied to study various physical and chemical properties of hydrocarbons in an sCO₂ environment. Other MD studies included of the following: (1) the thermodynamic properties of CH₄ in an sCO₂ environment (CO₂:CH₄ = 400:100, 323 K at 9.94 MPa), such as potential energy and pressure, mean square force, and torque, were studied by Skarmoutsos et al. [13]; and (2) Gong et al. studied the evaporation mode transition of hydrocarbon fuels in subcritical and supercritical fluids (750–3600 K and 4–36 MPa) to gain insights into air–fuel mixing and combustion processes [14]. Other MC studies included the following: (1) the chemical potential of non-polar hydrocarbon in an sCO₂ environment (300–350 K at 10–500 bar) was examined by Chang [15]; and (2) the free energy of the solvation and structural properties of CH₄ in an sCO₂ environment (304 K at 80 and 200 atm) were studied by Tafazzoli et al. [16] These studies show the importance of providing thermodynamic parameters to understand hydrocarbon reactions in an sCO₂ environment.

Here, we focus on the self-diffusion coefficients of CO₂/CH₄ and CO₂/C₂H₆ mixtures in extreme temperatures (500–2000 K) and pressures (100–1000 atm) that were not easily studied experimentally and/or computationally in the past to mimic the condition of the Allam Cycle. Studying diffusion under such extreme conditions presents significant challenges for both experimental and theoretical approaches. From the perspective of the experimental challenges, specialized equipment is needed to handle such conditions safely and accurately. The high mobility of molecules and potential for chemical reactions at these extremes introduce noise to the measurements. Some materials used for the equipment might be unstable and potentially lead to contamination [3,4,17]. From the perspective of computer simulation challenges, the accuracy of MD simulations relies heavily on the selection of force fields for high-temperature and -pressure systems. In addition, diffusion is a relatively slow process. Simulating realistic timescales at high temperatures and pressures can be computationally prohibitive [18]. To the best of our knowledge, it is the first time extremely high temperatures (~2000 K) and pressures (~1000 atm) have been achieved by computer simulations, and this is systematically discussed under such conditions. Our simulations explore diffusion behavior at high temperatures and pressures relevant to the Allam Cycle. While these conditions might be challenging to achieve experimentally, our simulations provide valuable data for the understanding the system’s behavior at its operating limits.

2. Computational Methods

The initial pure CO₂ diffusion coefficients at various temperatures and pressures were determined by ChemKin II Fortran version for comparison [19]. All the molecular dynamics simulations were performed using the LAMMPS program [20]. The force fields of CO₂ [21,22], CH₄ [23], and C₂H₆ [23,24] we selected were united-atom TraPPE because of its broad applicability and transferability to carbon dioxide and hydrocarbons. TraPPE has been successfully validated across various diffusion coefficient evaluations involving sCO₂ by comparison with available computational CO₂ self-diffusion coefficient results [7]. The cutoff radius of Coulomb and Lennard–Jones potentials was set to 14.0 Å. A time step of 1.0 fs was adopted. Simulations were performed under periodic boundary conditions in a cubic box measuring 35.0 × 35.0 × 35.0 ų and 40.0 × 40.0 × 40.0 ų, with a particle–particle particle–mesh calculation in the k-space to better estimate long-range interactions. The initial geometry of all the simulations was generated by Packmol v20.15.0 [25]. A total of 77 CO₂ molecules and binary CO₂:CH₄ = 16:16, 32:32, or 42:42, and CO₂:C₂H₆ = 16:16 or 32:32 molecules were placed in the simulation box. Different numbers of molecules in simulation boxes of different sizes will lead to different pressures. The reason for using a one-to-one ratio of carbon dioxide versus hydrocarbon is to achieve high pressure in the simulations. The selected simulation temperatures are 750, 1000, 1250, 1500, 1750, and 2000 K for pure CO₂, and 300, 500, 1000, 1500, and 2000 K for both CO₂/CH₄ and CO₂/C₂H₆ mixtures. Energy minimization was the first step to stabilize the simulation box, which was generated originally using Packmol software. The criteria for stopping minimization were: a tolerance for energy of 10⁻⁵, a tolerance for force of 10⁻⁷ kcal/mol/Å, max iterations of the minimizer of 5 million steps, and a max number of force/energy evaluations of 10 million steps. After energy minimization, a constant volume and temperature ensemble (NVT) were applied for the first 5 million steps (5 ns), followed by a constant volume and energy ensemble (NVE) for 5 million steps (5 ns) as the production run. A Nosé–Hoover thermostat was applied for the NVT [26,27]. The damping parameter was 200.0, meaning we relaxed the temperature at a timespan of 200 femtoseconds for the NVT. At least 5 to 10 independent test runs were averaged (with different initial geometries) to reduce fluctuations and then receive statistically reasonable results.

The calculations of the self-diffusion coefficients of the Einstein relation are based upon the mean squared displacement (MSD):

$$MSD\equiv 〈|r\left(t\right)-r(0){|}^2〉.$$

(1)

The equation describes the rate at which individual molecules move around due to random thermal motion within a medium. The slope of the MSD versus time is proportional to the diffusion coefficient of the diffusing atoms. The displacement of an atom is from its reference position, which is the original position at the time the simulation was started. MSD reflects the average squared distance a molecule travels over time, indicating how far it explores its surroundings. The MSD was collected in the NVE production runs. The unit of diffusion is cm²/s.

3. Results and Discussion

Pure CO_2. In order to validate our simulation methods, we compared our 16, 49, 148, 288, 411, and 538 atm MD simulation results for diffusion coefficient D_ij to the ideal gas kinetic theory (IGKT, the selected pressures are 1, 49, 99, 147, and 296 atm) predictions by ChemKin II in Figure 1. In the MD simulation, the NVT and NVE are not able to assign the same pressure as the IGKT results. Therefore, the selection of pressures in the MD simulation should be as close to the IGKT pressure as possible. This comparison serves a two-fold purpose. Firstly, it ensures consistency with established principles of the IGKT. Secondly, it provides a reference point for interpreting the behavior observed in the MD simulations. D_ij of pure CO₂ from the IGKT creates a gas where particles are point masses with negligible interactions. The equation for calculating D_ij by the ideal gas kinetic theory is presented as:

$$D_{ij}={\displaystyle \frac{3}{16}}{\displaystyle \frac{\sqrt{\frac{2\pi k_B^3{T}^3}{m_{ij}}}}{P\pi {\sigma }_{ij}^2{\mathsf{\Omega }}^{\left(\mathrm{1,1}\right)*}}},$$

(2)

where k_B is the Boltzmann constant and m_ij is the reduced molecular mass for the (i,j) species pair:

$$m_{ij}={\displaystyle \frac{m_i{m}_j}{m_i+m_j}}$$

(3)

Σ_ij is the reduced collision diameter, and Ω^(1,1)* is the collision integral based on the Stockmayer potential [28]. Figure 1 shows the similarities between our MD simulation results and fitted ideal gas theory results, except at a low pressure of 16 atm at 750 K. The increase in temperature leads to the increase in D_ij. A higher temperature leads to faster molecular motion and a higher diffusion coefficient. The increased density in SCFs due to the pressure increase results in more frequent collisions and more efficient shuffling between molecules. At a high pressure (above 100 atm), the MD simulation results are closer to those of the ideal gas kinetic theory (within a factor of 10 of each other). At a very high pressure, potentially, sCO₂ can start to resemble a solid and diffusion slows down. This causes the slope of D_ij to decline as the temperature increases. An underestimate of D_ij at a lower temperature (750 K) aligns with our previous study of the chemical kinetics of combustion reactions in an sCO₂ environment well [29,30].

Table 1. Diffusion coefficients (D_ij) of CO₂ under various temperatures (748–2040 K) and pressures (16–538 atm).

T (K)	P (atm)	Dij (10−4 cm2/s)
748	16 ± 3	2 ± 1
1012	49 ± 32	58 ± 24
1243	148 ± 7	128 ± 11
1530	288 ± 14	171 ± 9
1783	411 ± 23	215 ± 13
2040	538 ± 21	245 ± 16

shows the average diffusion coefficients of pure CO₂ under various temperatures and pressures by MD simulations and how spread out the data are. The results support the validity of using the chosen simulation conditions, such as force field TraPPE, to account for intermolecular interactions for the binary systems containing carbon dioxide and hydrocarbon, particularly at higher pressures above 50 atm, for accurate predictions compared to ideal gas assumptions.

Binary CO₂ and CH₄ mixtures. To understand the combustion of methane in an sCO₂ environment, it is necessary to study the diffusion of methane in CO₂/CH₄ mixtures. Figure 2 presents the diffusion coefficient of binary mixtures at various temperatures and pressures. The general trend of diffusion coefficient D_ij is that, as the temperature increases, the molecules in the mixture gain kinetic energy and result in the increase in D_ij. Both CO₂ and CH₄ move faster and collide with each other more frequently. A higher temperature provides enough thermal energy for molecules to diffuse. As the pressure increases at a constant temperature, D_ij decreases. A higher pressure depresses the diffusion of molecules as the mean free path of the fluid decreases and starts to resemble a solid so that a more frequent collision between molecules occurs and then hinders diffusion. Our simulation results reach as high as 930 atm at 2000 K. D_ij at near 300 atm is 0.012 at 1000 K and 0.032 cm²/s at 1500 K. Interestingly, compared with the results of D_ij at 300 K from different groups, our results are ten-times higher than those of Guevara-Carrion et al.’s result of ~0.0002 cm²/s at 300 K, 9 MPa [10], and Feng et al.’s result of 0.000272 cm²/s at 299 K, 10.5 MPa [11]. It is important to remember that D_ij at low temperatures (300 and 500 K) in our simulation results is systematically tenfold higher than those the other groups reported. However, when we compare the diffusion coefficients in terms of absolute values, they are quite similar. These differences may be from the relatively low mole fraction of CH₄ in their study, while our study has a 1:1 carbon dioxide and hydrocarbon ratio. Such a discrepancy at lower temperatures aligns with our previous study of the chemical kinetics of combustion reactions in an sCO₂ environment well [29,30]. We are currently looking into the additional reasons behind this discrepancy to better understand the implications for our research.

Binary CO₂ and C₂H₆ mixtures. To understand the combustion of ethane in an sCO₂ environment, it is necessary to study the diffusion of ethane in CO₂/C₂H₆ mixtures. Figure 3 presents the trend of the diffusion coefficient (D_ij) of CO₂/C₂H₆ that is similar to that of CO₂/CH₄ mixtures. We successfully obtained D_ij for all studied temperatures (500, 1000, 1500, and 2000 K), except 300 K and pressures below 600 atm. As the temperature increases, D_ij increases as a higher thermal energy for molecules. As the pressure increases at a constant temperature, D_ij decreases as more frequent collisions hinder diffusion. Our simulation can reach as high as 560 atm at 2000 K. D_ij at almost 300 atm is 0.016 cm²/s at 1000 K and 0.045 cm²/s at 1500 K. Our results can be compared with Feng et al.’s results, 0.000597 cm²/s at 323 K, 10.5 MPa as well [11], though we are not able to achieve the simulation condition of 300 K at a mole fraction of 0.50 for C₂H₆.

Overall, compared with the CO₂/CH₄ and CO₂/C₂H₆ results in Figure 2 and Figure 3, respectively, the diffusion coefficient is inversely proportional to the molar volume of the solute and decreases quickly with the increasing carbon chain for short-chain n-alkanes. A smaller molar volume indicates a smaller molecule that experiences less friction as it moves through a solvent molecule. In general, both methane and ethane will experience similar increases in D_ij with the increasing temperature. The pressure influences the D_ij values of methane and ethane differently due to their subtle size difference.

Table 2. Diffusion coefficients (D_ij) of hydrocarbon in CO₂/CH₄ and CO₂/C₂H₆ mixtures under various temperatures (300–2000 K) and pressures (10–1000 atm).

	CO2/CH4		CO2/C2H6
T (K)	P (atm)	Dij (10−4 cm2/s)	P (atm)	Dij (10−4 cm2/s)
300	12 ± 4	185 ± 29
	20 ± 11	111 ± 24
	31 ± 11	52 ± 8
	104 ± 38	32 ± 3
500	13 ± 7	283 ± 58	41 ± 3	404 ± 54
	42 ± 11	113 ± 27	74 ± 5	192 ± 34
	94 ± 4	69 ± 9	95 ± 14	124 ± 18
	121 ± 21	56 ± 17
1000	91 ± 7	356 ± 40	73 ± 5	653 ± 88
	140 ± 9	229 ± 20	179 ± 14	266 ± 20
	238 ± 18	129 ± 21	266 ± 16	163 ± 15
	274 ± 20	125 ± 16	490 ± 31	105 ± 11
	341 ± 31	119 ± 15
1500	232 ± 17	484 ± 69	144 ± 12	884 ± 146
	351 ± 27	330 ± 51	304 ± 22	444 ± 43
	567 ± 54	202 ± 16	384 ± 29	376 ± 37
	726 ± 74	167 ± 20	557 ± 37	242 ± 19
2000	371 ± 32	624 ± 68	275 ± 14	928 ± 109
	510 ± 54	397 ± 53	564 ± 25	490 ± 31
	729 ± 51	353 ± 34
	936 ± 104	276 ± 45

shows the value of the diffusion coefficients of CO₂/CH₄ and CO₂/C₂H₆ under various temperatures and pressures. It is expected that the diffusion coefficient decreases as the mass and size of the molecule increase, while the difference is limited. This table also shows, in general, a greater fluctuation in D_ij as the pressure reduces. Feng et al. showed the limited difference of D_ij by experiments and MD simulations from CH₄ to C₁₄H₃₀ at 10.5 MPa [11]. For example, D_ij ranges from 0.000691 to 0.000220 cm²/s at 323 K. Our simulations show the importance of the diffusion coefficients of binary CO₂/CH₄ and CO₂/C₂H₆ in an sCO₂ environment at high temperatures and pressures, which are challenging to achieve experimentally [2].

Table 3. Fitted equations of D_ij (y) of binary CO₂/CH₄ and CO₂/C₂H₆ mixtures under various temperatures (300–2000 K) and pressures (x).

	CO2/CH4		CO2/C2H6
T (K)	Fitted Equation	R2 Value	Fitted Equation	R2 Value
300	y = 0.1200x−0.813	0.9487
500	y = 0.1786x−0.723	0.9979	y = 7.3009x−1.392	0.9979
1000	y = 1.8117x−0.882	0.9860	y = 4.1986x−0.977	0.9981
1500	y = 8.5061x−0.949	0.9993	y = 9.5610x−0.940	0.9978
2000	y = 7.3284x−0.817	0.9339	y = 13.7450x−0.890	1.0000

shows the fitted equations under various temperatures in the range of 300–2000 K. The temperature is around 1100 °C (1423 K) and the pressure is around 300 bar (296 atm) of the Allam Power Cycle. The fitted D_ij for methane at 1500 K is 0.038 cm²/s and for ethane at 1500 K, it is 0.045 cm²/s.

4. Conclusions

We determined the self-diffusion coefficients of pure CO₂ and simplest hydrocarbons in binary CO₂/CH₄ and CO₂/C₂H₆ mixtures at temperatures in the range of 300–2000 K and pressures in the range of 10–1000 atm. For the comparison to other experimental and computational results, we tested lower temperatures (as low as ~300 K) and pressures (as low as ~10 atm) as well. Our simulations at higher temperatures (as high as ~2000 K) and pressures (as high as ~900 atm) provide the data to better understand the Allam Power Cycle at 300 atm and make improvements in the future. The diffusion coefficients of methane at 300 atm are approximately 0.012 cm²/s at 1000 K and 0.032 cm²/s at 1500 K. The diffusion coefficients of ethane at 300 atm are approximately 0.016 cm²/s at 1000 K and 0.045 cm²/s at 1500 K. Our simulation results are far from a perfect fit and still need improvements, especially at lower temperatures and pressures. Nonetheless, this study provides valuable information for the improvement of the Allam Cycle in the future. The technology of the Allam Cycle is still under development. Besides the understanding of diffusion coefficients, the Allam Cycle requires further research and experiments to improve its overall performance and effectiveness to achieve commercial viability.