Effect of Pressure on the Selectivity of Supercritical CO₂ Extraction During the Fractionation of a Fatty Acid Ethyl Ester Mixture: Numerical Simulation and Experiment

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

In the manuscript entitled “Effect of Pressure on the Selectivity of Supercritical CO2 Extraction during the Fractionation of a Fatty Acid Ethyl Ester Mixture: Numerical
Simulation and Experiment”, authors, Sergey V. Mazanov, Almaz U. Aetov, Alexander S. Zakharov, use the computational fluid dynamics (CFD) modeling to study the fractionation process for a model ethyl oleate/ethyl palmitate mixture (25.28:74.72
wt.%) in supercritical CO2 at pressures of 11 and 14 MPa and a temperature of 40 °C. Authors developed a three-dimensional model of a laboratory-scale extractor using the ANSYS Fluent software environment. Standard material database, a custom property library and compiled User-Defined Function (UDF) routines were developed to describe the temperature dependence of density, viscosity, heat capacity, and thermal conductivity for both the individual components and their mixture using established mixing rules. Results indicate pronounced selectivity: under the chosen thermodynamic conditions, ethyl palmitate is extracted preferentially over ethyl oleate, with this difference becoming more pronounced as pressure increases.

 

Before the manuscript is accepted for publication, authors must make several corrections and modifications, such as:

 

1) The authors need to detail how the experimental equipment (Figure 1) works and what data was obtained. Has this equipment been shown in any other publication?

2) The authors should present the uncertainties of each device, piece of equipment, as well as those related to the experimental measurements.

3) Why did the authors choose pressures of 11 and 14 MPa and a temperature of 40°C?

4) How can the authors validate the results obtained in the CFD modeling?

5) The manuscript could better contextualize the environmental or economic impact of viscosity reduction on an industrial scale, beyond the laboratory.

6) A limitation of the manuscript is that the study only considers primarily laminar flow; but turbulence effects on an industrial scale can be more complex.

7) Under P and T conditions, solid-liquid phase transitions may appear, especially with palmitates at reduced temperatures. The authors did not take this criterion into account.

8) Authors should reformulate the discussions regarding the scalability of the CFD model for larger industrial units.

9) In the case of biodiesel fuels, it would be interesting to include an analysis of the cost or energy efficiency of the process at different pressures to assess industrial viability.

Author Response

All changes made in the revised manuscript are highlighted in green.

Comments 1: The authors need to detail how the experimental equipment (Figure 1) works and what data was obtained. Has this equipment been shown in any other publication?

Response 1: The experimental setup allows for the extraction of substances in supercritical fluid media at temperatures up to 100 °C and pressures up to 25 MPa. The setup includes a pressure generation and maintenance system, a temperature control and maintenance system, a measuring cell, and a system for measuring and recording the extract mass. The pressure generation system consists of a CO₂ cylinder (1) with a volume of 10000 ml and a high-pressure pump (3) model “Supercritical 24” (Teledyne SSI, Newark, United States), which provides a liquid CO₂ flow rate in the range of 0.01–24 ml/min (uncertainty up to ±2%) and in the pressure range of 0–69 MPa (uncertainty up to ±0.5%). The temperature control and maintenance system consists of an electric heater (6), a thermocouple, and a meter-controller (7) (accuracy class 0.25). The system for measuring and recording the extract mass includes an electronic laboratory balance with automatic recording, model LLC OKB “Vesta” (Saint-Petersburg, Russia) (measurement accuracy 0.001 g), and a personal computer (13). The internal volume of the extractor is 92 ml. At the bottom of the extractor, there are glass balls with a diameter of 3.0 mm, designed to ensure uniform treatment of the liquid-phase charge by the gas-phase extractant across the extractor cross-section and to increase the phase contact area, thereby improving mass transfer.

Experimental Procedure. The pre-washed extractor cell (5) is “dried” using a vacuum pump to remove solvent vapors and atmospheric air. A pre-calculated volume of the test liquid is filled into the measuring cell through the outlet control valve (8) using a dispenser, with valve (4) closed. The amount of the liquid-phase component in the cell is determined by the gravimetric method. The solvent, carbon dioxide, from the cylinder (1) enters the high-pressure pump (3), where it is pre-liquefied, and is then supplied in the SCF state to the solubility cell at a flow rate of 0.8–1.0 ml/min under a pressure below the experimental pressure. Next, the temperature control system, including the sensor, temperature controller (7), and electric heater (6), is activated. Consequently, the cell is heated to the set temperature.

To minimize heat loss, the outer surface of the cell is thermally insulated. After reaching the set temperature regime, the supercritical fluid solvent is pumped up to the pressure planned for the experiment. Gas supply is accompanied by intensive mixing of the cell contents by rotating the cell itself at ±45° around its axis. After reaching the state of saturation and the set pressure, the system is held for 30 minutes. The fact of reaching an equilibrium state in the thermodynamic system is established by the absence of pressure change in the cell, after which sample mixing is stopped. Then, after opening valve (4) and the outlet control valve (8), the SCF solution enters the separator (10), where the gas and liquid phases are separated. Separator (10) is located on the platform of the electronic laboratory balance with automatic recording of readings (12) to a personal computer (13). Based on the information on the kinetics of mass change, the parameters of the time period during which the equilibrium dissolution regime is maintained are established.

This description of the setup's operation was previously described in [Gumerov, F.M.; Zaripov, Z.I.; Nakipov, R.R.; Mazanov, S.V.; Sagdeev, A.A. Highly efficient supercritical fluid extraction process: solubility and pseudosolubility. Tech. Phys. 2025. 70, 1881–1895, https://journals.ioffe.ru/articles/viewPDF/62090].

Comments 2: The authors should present the uncertainties of each device, piece of equipment, as well as those related to the experimental measurements.

Response 2: The uncertainties of the devices are provided above in the description of the experimental setup and have been added to the article. The uncertainty of the experimental measurements of extraction yield does not exceed ±5.6%.

Comments 3: Why did the authors choose pressures of 11 and 14 MPa and a temperature of 40°C?

Response 3: The selection of pressures of 11 and 14 MPa is based on the phase diagrams of the binary mixtures CO₂-ethyl palmitate and CO₂-ethyl oleate (Fig. 3 in the article), where at t=40 °C and at these pressures, type I–II phase behavior is formed, according to the classification proposed by Williams [Williams, D.F. Extraction with supercritical gases. Chem. Eng. Sci. 1981, 36, 1769–1788] with continuous critical curves and the presence of critical points. Regarding the choice of temperature, studies show that the region of maximum solvent power only begins to form just beyond the critical point. At 40 °C, the fluid properties are already sufficiently stable, while maintaining high sensitivity to pressure changes. This allows for an effective study of how pressure affects fluid density and, consequently, the solubility of various substances. Many experimental solubility data in the literature are reported precisely for t=40 °C, which enables comparison of results. If the temperature were set near the critical point, any slight cooling would return the CO₂ to a two-phase state (gas-liquid), rendering the solubility measurements invalid. Operating at 40 °C provides a temperature "safety margin", guaranteeing that the fluid remains in a single-phase supercritical state throughout the experiment.

Comments 4:  How can the authors validate the results obtained in the CFD modeling?

Response 4: The results of the CFD modeling in the present work were validated by comparison with experimental data obtained from a laboratory-scale supercritical CO₂ extraction setup under the same operating conditions used in the numerical setup: t = 40 °C; P = 11 and 14 MPa, scCO₂ flow rate of 2 ml/min, initial mixture composition of ethyl oleate/ethyl palmitate 25.28/74.72 wt.%. Statistical processing showed that increasing the pressure from 11 to 14 MPa leads to a significant increase in the ethyl palmitate content in the extract (p < 0.05), confirming the reproducibility of the observed effect. The numerical modeling used the geometry of the same laboratory extractor and the boundary conditions corresponding to the experiment. The simulation reproduced the same key trend established experimentally: ethyl palmitate is extracted more efficiently than ethyl oleate, and increasing pressure enhances this selectivity. Specifically, according to the CFD data, the mass fraction of ethyl palmitate at the extractor outlet increases from 0.69 at 11 MPa to 0.76 at 14 MPa, while experimentally, the ethyl palmitate content in the extract increases from 88.28 wt.% to 94.34 wt.% at a process time of 30 min. Thus, the CFD model correctly reproduces the direction of change in the extract composition and the influence of pressure on process selectivity.

Comments 5: The manuscript could better contextualize the environmental or economic impact of viscosity reduction on an industrial scale, beyond the laboratory.

Response 5: Regarding the environmental impact, it was stated in the introduction that "biodiesel offers several significant environmental advantages: its combustion leads to a substantial reduction in CO emissions, unburnt hydrocarbons, soot, and almost completely eliminates SOx emissions." This pertains to fuel with a low viscosity value. Additionally, the following sentence was added to the article concerning ecology: "Furthermore, biodiesel exhibits excellent lubricating properties, is non-toxic, and biodegradable, which reduces risks to soil and groundwater in the event of potential spills."

The economic benefits of using low-viscosity biodiesel manifest at all stages – from production to end-use. The reduction in viscosity allows for a decrease or complete elimination of the energy costs associated with heating the fuel in the fuel system. Improved atomization and combustion of low-viscosity fuel directly translate into an increase in effective thermal power by 2–4% [Ramalingam, K.; Kandasamy, A.; Subramani, L.; Balasubramanian, D.; Thadhani, J.P.J. Challenges and Opportunities of Low Viscous Biofuel—A Prospective Review. ACS Omega 2023, 8, 16545–16560], meaning that the fuel's energy is used more productively. All of this extends the service life of expensive fuel equipment and the engine itself.

This text has been inserted into the article.

Comments 6: A limitation of the manuscript is that the study only considers primarily laminar flow; but turbulence effects on an industrial scale can be more complex.

Response 6: The authors fully agree with the reviewer's comment. Turbulent effects in industrial-scale apparatus can be significantly more complex than in the laboratory-scale setup under consideration, which limits the direct transferability of the obtained results. In the present work, the study was performed for a laboratory extractor with a length of 408 mm, an internal diameter of 16 mm, a volume of 92 ml, an scCO₂ flow rate of 2 ml/min, at 40 °C and pressures of 11 and 14 MPa. For these conditions, a characteristic calculation for the packing yields Re ≈ 4.57, which corresponds to a low-Reynolds number flow regime and confirms that the primary physical interpretation of the results pertains specifically to laminar flow at the laboratory scale. Thus, the developed model is primarily intended for analyzing the interrelationship between hydrodynamics, mass transfer, and selectivity during laboratory-scale supercritical fractionation. When scaling up to larger industrial apparatus, inertial and turbulent effects may become more pronounced, along with changes in the flow distribution within the packing. This would require separate re-tuning of the model, the selection of appropriate closure models for the flow, and new experimental validation. We believe that further experimental research should be directed towards flow turbulization, based on which a model accounting for the turbulent effects present at industrial scales can be developed.

The text provided below has been added to the article.

For the conditions considered in the present work, the estimated Reynolds number in the packed bed is on the order of Re ≈ 4.57, which corresponds to a laminar flow regime. Therefore, from a physical point of view, the flow under investigation is not considered to be fully turbulent. The use of the "k–ε Realizable" model with "Scalable Wall Functions" in this setup is primarily related to the necessity of numerical stabilization for the multiphase calculation involving interphase mass transfer and the complex structure of the packed bed. In this sense, it was used as a means of numerical regularization and to account for local sub-grid mixing, rather than as an assertion of developed turbulence throughout the entire apparatus volume. This setup pertains specifically to the laboratory scale. When transitioning to industrial-scale apparatus, the hydrodynamic flow structure and the role of turbulent effects may change substantially; therefore, the direct transfer of the chosen closure to industrial-scale problems requires separate justification.  

Comments 7: Under P and T conditions, solid-liquid phase transitions may appear, especially with palmitates at reduced temperatures. The authors did not take this criterion into account.

Response 7: The authors fully agree with the reviewer's comment and observation. Indeed, for systems containing saturated fatty acid esters, including ethyl palmitate, solid-liquid phase transitions are possible at reduced temperatures, and this factor can significantly influence both hydrodynamics and mass transfer. However, in the present work, this effect was not specifically addressed, as the study was limited to a fixed isothermal regime of 40 °C and pressures of 11 and 14 MPa. These conditions were precisely selected based on the visual observation of the phase diagrams of the CO₂-ethyl palmitate and CO₂-ethyl oleate binary systems at 40 °C, which were used to set up both the experiment and the CFD modeling.

Within the framework of this article, the model was applied only to the parameter region corresponding to a stable liquid organic phase and the supercritical state of CO₂. We agree that when moving to lower temperatures, close to the crystallization region of saturated esters, the single-fluid approach becomes insufficient, and an explicit consideration of the liquid-solid phase transition becomes necessary.

To eliminate ambiguity, a clarification has been added to the article stating that the developed property model and CFD setup are intended for conditions where the organic mixture remains in a liquid state. For calculations at lower temperatures, where solidification of ethyl palmitate or another part of the mixture may occur, an extension of the model using the Solidification & Melting (enthalpy-porosity) approach, or a separate restriction of the operating temperature range, is required.

The text provided below has been added to the article.

It should be noted that the developed CFD model in the present work was applied for a fixed isothermal regime of 40 °C and pressures of 11–14 MPa, i.e., for the parameter region in which the organic phase was considered as a stable liquid mixture of ethyl oleate and ethyl palmitate. At reduced temperatures, for systems containing saturated fatty acid esters, the appearance of a solid phase due to crystallization is possible, which can significantly alter flow hydrodynamics, interphase contact, and the intensity of mass transfer. Therefore, the proposed setup is applicable only within the region of a stable liquid phase, whereas for describing regimes with possible solidification, a separate consideration of the liquid-solid phase transition using the Solidification & Melting model in ANSYS Fluent is required.

Comments 8: Authors should reformulate the discussions regarding the scalability of the CFD model for larger industrial units.

Response 8: The main objective of the present model is to identify and quantitatively analyze the fundamental relationships between hydrodynamics, mass transfer, and selectivity under conditions of laboratory-scale supercritical fractionation of an ethyl oleate and ethyl palmitate mixture. In this sense, the model should be regarded not as a ready-to-use tool for direct industrial application, but as a methodological and physical basis for subsequent scale-up to larger apparatus. When transitioning to an industrial scale, an additional consideration of factors that were not determining in this work will be required: changes in the flow structure, possible intensification of inertial and turbulent effects, the influence of scale on flow distribution across the apparatus cross-section, the specifics of real packings and distributors, and repeated experimental validation of the model for apparatus of different sizes.

Consequently, the proposed model was built and verified for a laboratory extractor with fixed geometry and operating parameters (Fig. 2, Table 1). Therefore, it should primarily be considered as a tool for the physical interpretation and analysis of the interrelationship between hydrodynamics, mass transfer, and selectivity under laboratory conditions. The direct transfer of the obtained quantitative results to larger industrial apparatus requires separate consideration of scale effects, possible changes in the flow structure, phase redistribution across the apparatus cross-section, and repeated experimental validation for new geometric and operating conditions. In this sense, the present model serves as a methodological foundation for the subsequent transition to industrial systems.

The reformulated discussion text has been added to the article.

Comments 9: In the case of biodiesel fuels, it would be interesting to include an analysis of the cost or energy efficiency of the process at different pressures to assess industrial viability.

Response 9: The authors agree with the reviewer that the analysis of the cost and energy efficiency of the process at different pressures is of significant interest from the perspective of industrial applicability of the technology. At the same time, the present work was primarily focused on the experimental and CFD investigation of the influence of pressure on hydrodynamics, mass transfer, and separation selectivity of a binary ethyl oleate and ethyl palmitate mixture in a laboratory-scale extractor. From a practical point of view, the result of the manuscript shows that a higher pressure can ensure more efficient removal of the saturated high-viscosity component from the initial mixture, i.e., enhance the degree of target fractionation. However, such an improvement in selectivity is inevitably accompanied by an increase in energy costs for CO₂ compression and maintaining the high-pressure regime. Therefore, the industrially optimal pressure should be determined as a compromise between separation quality and additional energy expenditure. In the present work, this compromise is discussed qualitatively, whereas its quantitative assessment requires a separate techno-economic analysis considering productivity, the CO₂ recycling scheme, heat exchange, and the energy consumption of compressor equipment. This presents a promising direction for future work and the preparation of a separate publication on this topic.

The text provided below has been added to the article.

From the perspective of industrial applicability, the obtained results indicate a compromise between separation selectivity and the energy costs of the process. It was experimentally established that increasing the pressure from 11 to 14 MPa at 40 °C leads to an increase in the ethyl palmitate content in the extract (Table 2), indicating more efficient fractionation of the mixture. However, an increase in pressure in supercritical CO₂ processes is inevitably associated with higher energy consumption for compression and maintaining the operating regime. Therefore, from a practical standpoint, a pressure of 14 MPa can be considered more effective in terms of separation selectivity, but its ultimate industrial feasibility must be determined based on a separate techno-economic analysis considering energy consumption, plant capacity, and the CO₂ recycling scheme.

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

Flow Model and Turbulent Regime

The flow regime is stated to be laminar, but the k–ε Realizable model is used for numerical stability reasons. This creates a conceptual inconsistency:

If the flow is truly laminar (low Reynolds number), the use of a turbulent model must be quantitatively justified.

It is recommended to include an estimate of the Reynolds number in the packing. Add an approximate calculation of Reynolds number. Explicitly justify the use of the model as a numerical regularization technique.

 

The following reference should be included:

Dávila, P., Bourouis, M., Nicolalde, J. F., & Martínez-Gómez, J. (2023). Modelling and analysis of a compression/resorption heat pump system with a zeotropic mixture of acetone/CO2. Applied Thermal Engineering, 227, 120388.

This reference should be included in the Introduction section, specifically in the subsection that discusses:

Thermodynamic modeling with CO₂
Multiphase systems with binary mixtures
Use of equations of state for mixtures with CO₂
Influence of thermodynamic conditions on system performance

Justification for inclusion:

The work by Dávila et al. is relevant because it:
Analyzes systems involving binary mixtures with CO₂.
Applies advanced thermodynamic modeling.
Evaluates the influence of thermodynamic properties on system performance.

This represents a recent precedent in CO₂ modeling under pressurized conditions.

Its inclusion will strengthen:
The thermodynamic context of the work.
Its positioning within the state of the art in CO₂ modeling.
Its connection to broader energy applications.

It is suggested that it be included alongside the references [39–44] or in the paragraph where CFD models and equations of state are discussed.

 

LDF Model and Mass Transfer Coefficient

km​=4⋅10⁻³–7⋅10⁻²s⁻¹: adopted

Its origin is not explained.

It is not indicated whether it comes from literature or experimental fitting.

Recommended:
Justify the adopted range.
Indicate the model's sensitivity 

 

Isothermal Assumption

The process is modeled as isothermal.

However:
The possible effect of the heat of solution is not discussed.

The possible Joule-Thomson effect of CO₂ is not evaluated.

Recommendations:
Include a brief thermodynamic discussion justifying the assumption.

Clarify whether or not the energy equation was solved.

 

Correct minor typographical errors:

"intel" → "inlet" (Table 1)

"in-terest" → "interest"

Double ":" in equation (6)

Standardize unit notation (MPa, °C, etc.)

Check consistency between wt.% and mass fraction.

 

 

Author Response

All changes made in the revised manuscript are highlighted in green.

Comments 1: 

Flow Model and Turbulent Regime

The flow regime is stated to be laminar, but the k–ε Realizable model is used for numerical stability reasons. This creates a conceptual inconsistency:

If the flow is truly laminar (low Reynolds number), the use of a turbulent model must be quantitatively justified.

It is recommended to include an estimate of the Reynolds number in the packing. Add an approximate calculation of Reynolds number. Explicitly justify the use of the model as a numerical regularization technique.

Response 1: 

The authors agree with the reviewer that in the original version of the manuscript, the formulation describing a laminar flow regime while simultaneously using the k–ε Realizable model could be perceived as ambiguous. In the revised version of the article, this point will be clarified more precisely. For the characteristic conditions of the laboratory extractor considered in this work, an estimate of the Reynolds number in the packed bed was performed. The calculation yields a value of Re≈4.57 [Re=(ρ⋅u⋅d_p)/μ=(780⋅1.66∙10^-4⋅0.003)/(8.5∙10^-5)≈4.57], which corresponds to a low-Reynolds number laminar flow regime. Consequently, from a physical point of view, the flow in the laboratory setup is not considered by us to be fully turbulent.

The use of the k–ε Realizable model in this work was not intended to assert the presence of developed turbulence in the apparatus, but rather served primarily as a numerical means of regularization and stabilization of the solution. In a strictly laminar setup for this multiphase Eulerian model with interphase mass transfer, complex packed bed geometry, and sharp local gradients, the solution exhibited deteriorated stability and convergence. Therefore, the k–ε Realizable model was applied here as a closure to account for local sub-grid mixing and to ensure a stable computational process, rather than as a direct physical statement about the turbulent nature of the flow throughout the entire apparatus volume.

Simultaneously, we agree with the reviewer that when transitioning to larger industrial apparatus, the situation may change substantially. At an industrial scale, increased characteristic dimensions, flow rates, non-uniformity of flow distribution across the apparatus cross-section, the influence of distribution devices, and intensified inertial effects can lead to more complex hydrodynamics and to a truly significant role of turbulent mixing. Therefore, the setup chosen in the present work and the applied numerical closure should not be interpreted as a universal model for direct industrial scale-up. On the contrary, the low-Reynolds number laboratory model in this article is considered as a tool for the physical interpretation of the process, whereas for the industrial scale, a separate hydrodynamic description, the selection of an appropriate flow model, and new experimental validation would be required—an effort comparable in scope to a separate publication.

The text provided below has been added to the article.

For the conditions considered in the present work, the estimated Reynolds number in the packed bed is on the order of Re≈4.57, which corresponds to a laminar flow regime. Therefore, from a physical point of view, the flow under investigation is not considered to be fully turbulent. The use of the k–ε Realizable model in this setup is primarily related to the necessity of numerical stabilization for the multiphase calculation involving interphase mass transfer and the complex structure of the packed bed. In this sense, it was used as a means of numerical regularization and to account for local sub-grid mixing, rather than as an assertion of developed turbulence throughout the entire apparatus volume. This setup pertains specifically to the laboratory scale. When transitioning to industrial-scale apparatus, the hydrodynamic flow structure and the role of turbulent effects may change substantially; therefore, the direct transfer of the chosen closure to industrial-scale problems requires separate justification.

Comments 2: 

The following reference should be included:

Dávila, P., Bourouis, M., Nicolalde, J. F., & Martínez-Gómez, J. (2023). Modelling and analysis of a compression/resorption heat pump system with a zeotropic mixture of acetone/CO2. Applied Thermal Engineering, 227, 120388.

This reference should be included in the Introduction section, specifically in the subsection that discusses:

Thermodynamic modeling with CO₂
Multiphase systems with binary mixtures
Use of equations of state for mixtures with CO₂
Influence of thermodynamic conditions on system performance

Justification for inclusion:

The work by Dávila et al. is relevant because it:
Analyzes systems involving binary mixtures with CO₂.
Applies advanced thermodynamic modeling.
Evaluates the influence of thermodynamic properties on system performance.

This represents a recent precedent in CO₂ modeling under pressurized conditions.

Its inclusion will strengthen:
The thermodynamic context of the work.
Its positioning within the state of the art in CO₂ modeling.
Its connection to broader energy applications.

It is suggested that it be included alongside the references [39–44] or in the paragraph where CFD models and equations of state are discussed.

Response 2: The authors are grateful to the reviewer for a careful reading of our manuscript and for the recommendation to include the work of Davila et al. (2023). The authors agree that this reference will strengthen the thermodynamic context of the article and reinforce its connection with modern research in the field of modeling CO₂-based systems. In accordance with the recommendation, the authors have included a reference to the work of Davila et al. (2023) in the introduction [60].

Comments 3: 

LDF Model and Mass Transfer Coefficient

km​=4⋅10⁻³–7⋅10⁻²s⁻¹: adopted

Its origin is not explained.

It is not indicated whether it comes from literature or experimental fitting.

Recommended:
Justify the adopted range.
Indicate the model's sensitivity 

Response 3: The range for k_m was not introduced as an arbitrary fitting parameter, but was considered as a physically constrained parametric interval based on an order-of-magnitude estimate of external mass transfer in the packed bed under the conditions of the present work. The calculation is presented below.

1. Theoretical basis for calculating the mass transfer coefficient.

In "supercritical CO₂ – organic liquid" systems, the mass transfer coefficient in the LDF (Linear Driving Force) model is related to the external mass transfer coefficient k_f (m/s) and the specific interfacial area (a; m²/m³) by the relationship:

k_m=k_f ⋅ a

For the calculation of k_f in packed columns, dimensionless correlations of the form are used:

Sh = (k_f ⋅ d_p)/D = C ⋅ Re^m ⋅ Scⁿ

where Sh is the Sherwood number; Re = (ρ ⋅ u ⋅ d_p)/μ is the Reynolds number; Sc = μ / (ρ ⋅ D)  is the Schmidt number; d_p​ is the characteristic size (packing particle diameter), m; D is the diffusion coefficient of ethyl palmitate/ethyl oleate in scCO₂, m²/s; ρ and μ  are the density and dynamic viscosity of scCO₂ under operating conditions; u is the superficial velocity, m/s.

2. Initial data for the calculation (for the regime P = 14 MPa, t = 40 °C).

scCO₂ parameters: ρ = 780 kg/m³; dynamic viscosity μ = 8.5 ∙ 10^-5 Pa·s; kinematic viscosity ν = μ/ρ = 1.09 ∙10⁻⁷ m²/s.

Geometric and regime parameters: scCO₂ flow rate:  = 2 ml/min = 3.33 ∙ 10^-8 m³/s; extractor cross-sectional area: S = π ⋅ (0.008)² = 2.01 ∙ 10⁻⁴ m²; superficial velocity: u = Q/S = 1.66 ∙ 10⁻⁴ m/s; packing sphere diameter d_p​ = 3 mm = 0.003 m; specific surface area of the packing a  (for spherical particles with porosity ε ≈ 0.4:

a = 6(1−ε)/d_p = (6 ∙ 0.6)/0.003 = 1200 m²/m³

The diffusion coefficient for "fatty acid ester – scCO₂" systems at 14 MPa and 40 °C is approximately D ≈ 2 ∙ 10^-9 m²/s [López-Padilla, A.; Ruiz-Rodriguez, A.; Reglero, G.; Fornari, T. Study of the diffusion coefficient of solute-type extracts in supercritical carbon dioxide: Volatile oils, fatty acids and fixed oils. J. Supercrit. Fluids 2016, 109, 148–156. DOI: 10.1016/j.supflu.2015.11.017; Funazukuri, T.; Kong, C.Y.; Kagei, S. Effects of molecular weight and degree of unsaturation on binary diffusion coefficients for lipids in supercritical carbon dioxide. Fluid Phase Equilib. 2004, 219, 67–73. DOI: 10.1016/j.fluid.2004.01.017; Liong, K.K.; Wells, P.A.; Foster, N.R. Diffusion coefficients of long-chain esters in supercritical carbon dioxide. Ind. Eng. Chem. Res. 1991, 30, 1329–1335. DOI: 10.1021/ie00054a039].

3. Calculation of similarity criteria.

Reynolds number:

Re = (ρ ⋅ u ⋅ d_p)/μ = (780 ⋅ 1.66 ∙ 10⁻⁴ ⋅ 0.003) / (8.5 ∙ 10⁻⁵) ≈ 4.57

The obtained value Re < 10 confirms the laminar flow regime in the packing.

Schmidt number:

Sc = μ / (ρ ⋅ D) = (8.5 ∙ 10⁻⁵) / (780 ⋅ 2 ∙ 10⁻⁹) ≈ 54.5

4. Calculation of the Sherwood number and the coefficient k_f.

For laminar flow in packed columns, a widely used correlation for external mass transfer in packed beds can be employed as an engineering estimate:

Sh = 0.38 ⋅ Re^0.83 ⋅ Sc^0.33= 0.38 ⋅ (4.57)^0.83 ⋅ (54.5)^0.33≈ 5.02

This relationship was not used as a strictly validated correlation specifically for the supercritical system under consideration, but as an engineering estimate of the order of magnitude of the external mass transfer coefficient in the packed bed. This approach is consistent with current literature, where mass transfer coefficients in packed beds are described through dimensionless correlations of the form Sh = f(Re, Sc) [Versteeg, F.A.; Picchioni, F.; Versteeg, G.F. On the mass transfer of supercritical fluids, specifically supercritical CO₂: An overview. Chem. Eng. J. 2024, 493, 152521. DOI: 10.1016/j.cej.2024.152521]. And since Re≈4.57 (the flow regime is above the region of very low Reynolds numbers where classical data for particle beds can be noticeably distorted by the influence of natural convection), this correlation is used in the work as a calculation relationship for estimating the range of k_m.

We find k_f​:

k_f = (Sh ⋅ D) / d_p = (5.02 ⋅ 2 ∙ 10⁻⁹) / 0.003 = 3.35 ∙ 10⁻⁶ m/s

5. Calculation of (k_m):

k_m = k_f ⋅ a = 3.35 ∙ 10⁻⁶ ⋅ 1200 = 4.02 ∙ 10⁻³ s⁻¹.

6. Upper boundary of the range.

The upper boundary of the range k_m = 7 ⋅ 10⁻² s⁻¹ was considered in this work as an upper estimate for locally intensified interfacial mass transfer in the constrictions of the pore space of the packed bed. In contrast to the lower boundary, based on the superficial velocity and particle scale, the pore-geometric scale m = ε / a was used for the upper boundary.

With ε = 0.4 and a = 1200 m²/m³, we obtain m = 3.33 ⋅ 10⁻⁴  m. For the molecular diffusion coefficient, a value of D = 5.0 ⋅ 10⁻⁹ m²/s was adopted, corresponding to the lower part of the experimentally measured range for long-chain ethyl esters in scCO₂. The average interstitial velocity was determined as u_i = u_s / ε, where u_s=1.66 ⋅ 10⁻⁴ m/s; additionally, for the upper boundary, a local acceleration coefficient α ≈ 4 was introduced to account for the increased velocity in the inter-sphere regions, so that:

u_loc = α ∙ u_i ≈ 1.66⋅10⁻³ m/s

Then Sc ≈ 21.79, Re_loc ≈ 5.08, Sh ≈ 4.05, k_f ≈ 6.07 ⋅ 10⁻⁵ m/s, and consequently, k_m ≈ 7.28 ⋅ 10⁻² s^-1. Thus, the upper boundary of the accepted interval corresponds to a scenario where mass transfer is controlled by the local hydrodynamics of interparticle constrictions and the literature-confirmed range of the diffusion coefficient.

For a pressure of 11 MPa, the deviation from the obtained k_m value is ~0.5%, considering changes in Re and Sh within 0.5–1% (considering the mixture properties and packing geometry).

Therefore, the range 10^-3…10^-2 s^-1 is physically justified for our conditions, while the rounded value of the upper boundary 7 ∙ 10^-2 s⁻¹ was considered in this work as an upper estimate, accounting for possible local intensification of mass transfer and the uncertainty of the effective interfacial area.

A justification for the selection of the mass transfer coefficient range in the LDF model and references to the used literature [65-68] were added to the article.

Comments 4: 

Isothermal Assumption

The process is modeled as isothermal.

However:
The possible effect of the heat of solution is not discussed.

The possible Joule-Thomson effect of CO₂ is not evaluated.

Recommendations:
Include a brief thermodynamic discussion justifying the assumption.

Clarify whether or not the energy equation was solved.

Response 4: In the present work, the CFD model was initially formulated as isothermal for a fixed experimental regime of 40 °C and pressures of 11 and 14 MPa. This corresponds to the assumption adopted in the manuscript of constant temperature throughout the apparatus volume, as well as to the experimental setup scheme, which includes thermal insulation, a heater, and a CO₂ supply preheating unit, i.e., elements designed to maintain the specified temperature regime.

The energy equation was solved within the model; however, the heat of dissolution of the components and the Joule–Thomson effect for CO₂ were not separately evaluated or considered as an independent subject of study, but were attributed to the limitations/assumptions of the adopted isothermal formulation.

The adoption of the isothermal approximation in this work is related to the fact that the study was primarily focused on the influence of pressure on hydrodynamics, mass transfer, and separation selectivity at a fixed temperature of 40 °C, chosen based on the phase diagrams of the CO₂–ethyl oleate and CO₂–ethyl palmitate binary systems. Therefore, the current model is intended for analyzing the process within the specified thermostated regime, rather than for a detailed investigation of coupled heat and mass transfer effects.

The text provided below has been added to the article.

In the present work, the process was modeled using an isothermal approximation at a fixed temperature of 40 °C, corresponding to the experimental conditions. This assumption is consistent with the laboratory setup scheme, which includes thermal insulation, a heater, and preheating of the scCO₂ supply, and was adopted to focus the analysis on the influence of pressure on hydrodynamics, interphase mass transfer, and separation selectivity. Possible local thermal effects associated with the heat of dissolution of the components, as well as CO₂ compression/expansion and the Joule–Thomson effect, were not analyzed separately and were not the subject of an independent study. Therefore, these effects are considered within the limitations of the model in the present work.

Comments 5: 

Correct minor typographical errors:

"intel" → "inlet" (Table 1)

"in-terest" → "interest"

Double ":" in equation (6)

Standardize unit notation (MPa, °C, etc.)

Check consistency between wt.% and mass fraction.

Response 5: All corrections have been incorporated into the text of the article, and the units of measurement have been standardized.

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

Good morning

Authors made the corrections and responded to the suggestions made, and the article can be accepted.

Reviewer 2 Report

Comments and Suggestions for Authors

Dear

 

I accept they article.