Parametric Optimization of Sensible Thermocline Packed Bed Thermal Energy Storage Systems: A Computation Fluid Dynamics Study

Comments 1: The manuscript should more clearly distinguish its contributions from prior CFD-based TES optimization studies. What differentiates this work from existing literature in terms of modeling innovation, parameter coupling, scale effects, or analytical insight?

Response 1: We sincerely thank the reviewer for this insightful comment. To address this, we have thoroughly revised the manuscript to clearly position our contribution within the existing literature. Specifically, the novelty of this work lies in the coupling of non-Darcian flow dynamics (utilizing the Darcy-Forchheimer law) with a transient Local Thermal Non-Equilibrium (LTNE) framework under mixed (forced and natural) convection regimes. While many prior CFD-based TES studies assume Local Thermal Equilibrium (LTE) or neglect turbulent inertial effects in the porous matrix, our approach provides a more rigorous physical representation of the heat transfer bottleneck at the fluid-solid interface during dynamic charging phases.

Furthermore, we have deepened our analytical interpretation in the Results and Discussion section. Rather than simply reporting temperature and velocity profiles, we have expanded our analysis to explain how variations in porosity and particle size directly manipulate the viscous momentum transport and ultimately govern the thickness and stability of the thermocline layer.

The modification is highlighted in red for your convenience.

Comments 2: Certain physical explanations appear simplified. For example, the statement that increased porosity results in faster temperature rise due to larger interfacial area requires more careful justification, as porosity changes also affect solid fraction, heat capacity, and flow resistance.

Response 2: Agree. We have substantially rewritten the discussion surrounding Figure 11 to provide a rigorous physical justification. Specifically, we now clarify that an increase in porosity (from 0.35 to 0.55) drastically reduces the solid fraction, which inherently lowers the effective thermal mass and heat capacity of the storage bed. Consequently, with less solid material to heat, the bed reaches thermal equilibrium much faster. Furthermore, we expanded the explanation to note that higher porosity increases the permeability of the medium, thereby reducing flow resistance and altering the interstitial velocity, which collectively govern the transient thermal response.

This change can be found at the second paragraph of the section “ 5.2. Transient Temperatures, Thermocline and Efficiency” in the revised manuscript:

“[This effect seems to be due to decreased the solid fraction. A higher porosity fundamentally decreases the total mass, and consequently the effective thermal capacity, of the solid medium, yielding to faster temperature rise in both phases and enhancing heat transfer efficiency (likely due to enhanced convection effects) and reducing the time required to charge the tank. Also, interesting to note that higher void fraction enhances the permeability, which reduces flow resistance and alters the interstitial fluid velocity, thereby modifying the transient convective heat transfer dynamics. In addition, a comparative analysis of the decoupled temperature profiles showed the significant impact of the porosity on LTNE dynamics. It’s observed that compacted bed (ε = 0.35) resulted in a maximum temperature difference of ≈ 40 °C compared to the loose bed (ε = 0.55) where it decreased to ≈ 30 °C at 10 h of the entire charging phase. This demonstrates the important heat capacity of the solid fraction, which resists rapid heat saturation. ]”

Comments 3: In Equations (14)–(15), thermocline thickness is defined using a 20–80% temperature threshold. Why was this range selected instead of a 10–90% definition? A brief sensitivity discussion would strengthen methodological rigor.

Response 3: We agree that providing a justification for the chosen thermocline threshold strengthens the rigor of the study. The 20–80% threshold was selected to exclude the asymptotic temperature tails, which are highly susceptible to residual axial conduction and numerical diffusion. This ensures a more robust metric that strictly isolates the active heat exchange zone without artificially inflating the thermocline thickness.

This change can be found at the second paragraph of the section “ 3.3.1. Thermocline thickness” in the revised manuscript:

“[This range was selected to exclude the asymptotic temperature tails, thereby avoiding artificial inflation of the thermocline thickness caused by slow residual axial conduction and numerical diffusion]”

Comments 4: The neglect of radiative heat transfer should be further justified. At an inlet temperature of 390°C, radiation effects may not be negligible, and a short quantitative or literature-based justification would improve credibility.

Response 4: We agree that quantitative or literature-based justification would enhance the credibility of ignoring radiation effects in our work. It has been established in the current literature that radiative heat transfer can be safely neglected in practical applications below 600 °C, due to the fluid's high optical opacity and the overwhelming dominance of convection.

We have mentioned this in the revised draft in section “3. Mathematical Model" with the corresponding reference:

“[Radiation exchanges within the bed are neglected (operating temperature below 600°C) [33]]”

Comments 5: The formulation of storage efficiency (Equation 18) requires clarification. As currently written, it appears to integrate temperature over both time and volume, which raises dimensional consistency concerns. The expression should be carefully reviewed and clarified.

Response 5: We agree that calculating the stored energy at a specific time t requires only a spatial integration of the solid temperature field over the bed volume. The extraneous time integral has been removed, and Equation 18 has been corrected in the revised manuscript to ensure strict dimensional accuracy (yielding Joules).

Comments 6: Although mesh independence is stated to be achieved at 51,256 elements, no quantitative comparison is provided. Including a table showing variation in key outputs (e.g., temperature, velocity, efficiency) across mesh densities would strengthen the numerical validation.

Response 6: We agree, to rigorously validate our numerical resolution, we would like to kindly correct a typographical error from our original manuscript: the actual number of elements for our chosen mesh is 51,276, rather than 51,256. We have incorporated a quantitative mesh sensitivity analysis into the revised manuscript. A new table has been added to compare key thermodynamic outputs (specifically, solid temperature and interstitial velocity at the bed center) across three different mesh densities. The results demonstrate that velocity converged at very low mesh densities (0%) due to the strong porous flow resistance. Our grid independence was entirely dictated by the thermal field (Temperature, energy stored and charge efficiency), refining the grid beyond 51,276 elements yields a relative deviation of less than 1%, confirming grid independence while optimizing computational cost.

We have mentioned this in the revised draft in section “4. Numerical Modeling and Gride Size Independence":

 “[To ensure the reliability of the simulation results, grid size independence verification was performed. Three different mesh densities were evaluated. The temperatures (of both fluid and solid) at the center of the bed (r = 0, z = H/2) at 10h, the total heat stored and the charge efficiency were recorded and compared across the different mesh densities. As shown in

Table 1, increasing the number of elements beyond 51,276 results in a relative variation of less than 1% for all metrics. Therefore, the mesh with 51,276 elements was selected for all subsequent simulations to provide an optimal balance between numerical accuracy and computational time]”

Table 1. Mesh Independence Study

Comments 1: What are the Reynolds numbers corresponding to 3.8–9.4 kg/s?

Response 1: We sincerely thank the reviewer for this insightful comment. We completely agree that providing the dimensionless Particle Reynolds number (Re_p) is crucial for characterizing the flow regime and ensuring the results can be accurately scaled. In the original manuscript, the mass flow rates of 3.8 to 9.4 kg/s corresponded to a Re_p range of 17.2 to 42.

However, to strengthen the scientific contribution of the paper during this major revision, we critically re-evaluated our parametric flow study. We have updated the evaluated mass flow rates to 8 kg/s (reference case), corresponding to a Re_p of ≈ 36, and 15 kg/s (representing partial-load operation) corresponding to a Re_p of ≈ 68.25.

To enhance methodological rigor of the revised manuscript, we have explicitly incorporated these dimensionless numbers into the first paragraph of the section “5.2. Transient Temperatures, Thermocline and Efficiency” 

“[mass flow rate is changed from 8 kg/s (Re_p ≈ 36) to 15 kg/s (Re_p ≈ 68.25]”

Comments 2: Are you within the validity range of the hi correlation (Eq. 7)? You quote 10⁴ < Redp/ε < 2×10⁷. Please compute actual values.

Response 2: Agree; computing the actual values for our system reveals that we were operating well outside the validity range of the previously cited high-turbulence correlation. With mass flow rates of 8 kg/s and 15 kg/s, the computed values are approximately 73.47 and 139.28, respectively. This confirms that applying the previous correlation was physically inappropriate for our Forchheimer-regime flow. To correct this and ensure strict thermodynamic accuracy, we have replaced the heat transfer correlation in our numerical model with the widely validated Wakao and Kaguei spherical particle correlation  which is valid for

Comments 3: Did you perform time-step independence analysis?

Response 3: Agree; to ensure the complete reliability of our computational model, we rigorously addressed both spatial and temporal discretization. As detailed in the revised manuscript (Table 2), a quantitative spatial grid independence study was explicitly performed to determine the optimal mesh density (51,276 elements). For the temporal discretization, rather than conducting a manual fixed time-step analysis, we utilized the robust adaptive time-stepping algorithm inherent to the numerical solver to intrinsically guarantee temporal convergence. The transient simulations were resolved using an implicit Backward Differentiation Formula (BDF) scheme. The solver dynamically regulates the time step Δt at each iteration to ensure the local truncation error remains strictly below a predefined relative tolerance of 10^-3 and an absolute tolerance of 10^-4.

we have explicitly incorporated that into the revised manuscript at the second paragraph of the section’’4. Numerical Modeling and Gride Size Independence’’

“[In addition, temporal accuracy was intrinsically enforced utilizing an implicit Backward Differentiation Formula (BDF) adaptive time-stepping algorithm.]”

Comments 4: How sensitive are results to interfacial heat transfer coefficient correlation?

Response 4: Because the specific surface area of the packed bed is high (As = 6(1-ε)/dp), the volumetric heat transfer between the fluid and solid phases is exceptionally efficient. Consequently, the solid and fluid phases operate relatively close to Local Thermal Equilibrium (LTE) during the charging process. Because the interfacial heat transfer is not the primary limiting thermal resistance in this system, the macroscopic results, specifically the thermocline thickness, temperature distribution maps, and efficiency, are highly insensitive to the exact choice of the interfacial heat transfer coefficient (hi) correlation.

Although the system is so efficient that it essentially operates in local thermal equilibrium (LTE), LTNE approach was strictly required because there is a distinct temperature difference between the HTF and storage medium inside the active thermocline zone. However, because the fluid flow (advection) is so strong, the macroscopic behavior of the tank is "advection-dominated," making it insensitive to minor math variations in the film coefficient.

Comments 5: Why is radiation neglected at 390°C without radiative Biot number estimation?

Response 5: Thermal radiation is safely neglected at 390°C because the HTF (liquid thermal oil) acts as an optically thick medium that effectively suppresses long-wave radiative exchange between the solid particles. Consequently, the calculation of a radiative Biot number is unnecessary, as the high optical opacity of the fluid absorbs emitted radiation over microscopic distances and prevents macroscopic bed-scale radiative transport. It has been established in the current literature that radiative heat transfer can be safely neglected in practical applications below 600 °C, due to the fluid's high optical opacity and the overwhelming dominance of convection.

We have mentioned this in the revised draft in section “3. Mathematical Model" with the corresponding reference:

“[Radiation exchanges within the bed are neglected (operating temperature below 600°C) [34]]”

Comments 1: Figure 1: The description of the workflow is rather brief; please provide more detailed explanation of each step in the flowchart in the text

Response 1: We agree that a more comprehensive explanation of the numerical workflow enhances the clarity and reproducibility of the study. In the last paragraph of the introduction of the revised manuscript, we have significantly expanded the text accompanying Figure 1 to explicitly detail each sequential step of the computational procedure.

“[The approach adopted in this work is presented systematically and concisely in Figure 1, which schematizes the sequential progression from the initial conceptual frame-work and methodology design to data acquisition, analysis, and final validation of results. The model incorporates key parameters such as transient temperature distribution, flow dynamics (velocity and pressure filed distributions), and efficiency under varying parameters: bed porosity, solid particles size, and mass flow rate. Numerical resolution of the governing equations was based on the finite element method (FEM). The 2D-Axisymmetric assumption was used to design the cylindrical tank geometry in a two-dimensional r-z plane. The Darcy-Forchheimer flow model was employed to account for the fluid inertia effects, especially regarding the clear/porous interface. In addition, The Local Thermal Non-Equilibrium (LTNE) approach was used to solve the energy conservation equations and the transient interstitial thermal lag between temperatures of the solid and fluid phases (Tf and Ts) critical in systems using large-diameter particles (5 – 20 cm). In addition, this study explicitly characterizes the spatiotemporal degradation of the thermocline under varying Reynolds numbers (mass flow rates of 9.4 – 15 kg/s). The effect of bed porosity (0.35–0.7) was studied to determine the optimal effect that maximize energy storage density without compromising the heat transfer rates required for fast charging cycles.]”

Comments 2: Only the total number of grid elements is reported. Please also provide key mesh quality indicators (e.g., maximum/minimum element size, growth rate, equiangle skewness, aspect ratio).

Response 2: We agree that reporting comprehensive mesh quality indicators is essential for ensuring the computational reproducibility and reliability of the numerical model. To address this, we have extracted the detailed mesh statistics from our spatial discretization. The optimized 51,276-element grid utilized an excellent average element quality of 0.8221 (evaluated via equiangle skewness) and a minimum element quality of 0.2288. Additionally, the grid maintains a highly refined element area ratio of 0.001011.

Comments 1: The similarity percentage based on iThenticate report should be more reduced.

Response 1: We sincerely thank the reviewer for this insightful comment.  We have carefully reviewed the manuscript and comprehensively revised the text to ensure a high degree of originality and strict adherence to the journal's publication standards. Specifically, we have substantially paraphrased the Introduction, completely reworded the descriptive text in the Methodology, and restructured our literature review to significantly reduce the overall similarity percentage.

Comments 2: The literature review doesn’t lead the readers understand the reason what the research gap in this field is and how they are fulfilling this objective in their paper. Should be improved.

Response 2: Agree. To rectify this, we have completely restructured the final paragraphs of the Introduction in the revised manuscript. We now explicitly articulate the primary research gap: while small-scale and simplified mathematical models are prevalent, there is a critical scarcity of comprehensive, multidimensional numerical studies evaluating transient thermocline degradation in true industrial-scale (e.g., 4-meter diameter) packed beds using rigorously coupled LTNE and Darcy-Forchheimer physics under dynamic, high-mass-flow conditions. Immediately following the identification of this gap, we clearly define our study's primary objectives and state exactly how our numerical methodology and scale fulfill this specific void in the existing literature.

Comments 3: The number of keywords is too high. “Heat transfer in fluid and solid” is not a keyword.

Response 3: Agree. We have condensed the keyword list to strictly adhere to standard publication limits. The keyword list was reduced and refined to the following five terms: Thermal Energy Storage; Packed Bed; Local Thermal Non-Equilibrium; CFD; Finite Element Method.

Comments 4: Some important quantitate results should be added to the Abstract.

Response 4: We agree. we have comprehensively revised the Abstract to include specific, quantitative outcomes derived from model validation error and our parametric studies.

We have mentioned this in the abstract of the revised draft:

“[The model was in close agreement with the experiment, with an overall mean absolute percentage error (MAPE) of 5.03%.]”

“[The simulation results demonstrate that the system achieves a high charge efficiency of 92.3 % at a nominal charging rate of 15 kg/s.]”

“[Furthermore, increasing the porosity from 0.35 to 0.55 reduced charging time, decreased the temperature difference between the HTF and the storage medium by 10°C, and in-creased the final heat charging efficiency by 8%. On the contrary, an increase in particle size from 5 to 20cm leads to a slower rise in temperature within the solid phase, creating an important LTNE lag of ≈ 34 °C, thereby reducing the final heat charge efficiency by 16%, and prolonging the time required to charge the tank.]”

Comments 5: The description of Flowchart in Figure 1 should be more elaborated.

Response 5: We agree that a more comprehensive explanation of the numerical workflow enhances the clarity and reproducibility of the study. In the last paragraph of the introduction of the revised manuscript, we have significantly expanded the text accompanying Figure 1 to explicitly detail each sequential step of the computational procedure.

“[The approach adopted in this work is presented systematically and concisely in Figure 1, which schematizes the sequential progression from the initial conceptual frame-work and methodology design to data acquisition, analysis, and final validation of results. The model incorporates key parameters such as transient temperature distribution, flow dynamics (velocity and pressure filed distributions), and efficiency under varying parameters: bed porosity, solid particles size, and mass flow rate. Numerical resolution of the governing equations was based on the finite element method (FEM). The 2D-Axisymmetric assumption was used to design the cylindrical tank geometry in a two-dimensional r-z plane. The Darcy-Forchheimer flow model was employed to account for the fluid inertia effects, especially regarding the clear/porous interface. In addition, The Local Thermal Non-Equilibrium (LTNE) approach was used to solve the energy conservation equations and the transient interstitial thermal lag between temperatures of the solid and fluid phases (Tf and Ts) critical in systems using large-diameter particles (5 – 20 cm). In addition, this study explicitly characterizes the spatiotemporal degradation of the thermocline under varying Reynolds numbers (mass flow rates of 8 – 15 kg/s). The effect of bed porosity (0.35–0.55) was studied to determine the optimal effect that maximize energy storage density without compromising the heat transfer rates required for fast charging cycles.]”

Comments 6: Don’t put degree sign for Kelvin. It should be just “K”

Response 6: We agree. We completely agree, and we have carefully reviewed the entire manuscript to ensure strict adherence to SI unit conventions.

Comments 7: In nomenclature how you considered “ext” for “Ambient environment” ? It is not consistent.

Response 7: We agree. we have updated the nomenclature table in the revised manuscript. The subscript "ext" is now correctly defined as "External"

Comments 8: CFD stands for “computational fluid dynamics” not computational fluid dynamic.

Response 8: We agree. we have updated the abbreviation ‘CFD’ in the entire revised manuscript.  The abbreviation "CFD" is now correctly defined as "computational fluid dynamics’’.

Comments 9: In Table 1, the Units are not correct when you utilized /.

Response 9: We agree. we have updated Units in the entire revised manuscript. 

Comments 10: The Quality of figure 4 should be increased.

Response 10: We agree. in the revised manuscript, we have regenerated graphical plots, with specific attention directed to Figure 4.

Comments 11: Provide grid size independence results thorough a table or diagram

Response 11: We agree, to rigorously validate our numerical resolution, we would like to kindly correct a typographical error from our original manuscript. We have incorporated a quantitative mesh sensitivity analysis into the revised manuscript. A new table has been added to compare key thermodynamic outputs (specifically, solid and fluid temperature, heat stored and charge efficiency) across three different mesh densities.

We have mentioned this in the revised draft in section “4. Numerical Modeling and Gride Size Independence":

Table 2. Mesh Independence Study

Comments 12: In the legends of contours what is the meaning of “degC”? Should be modified.

Response 12: We agree. In the legends of contours, “degC” means (°C) unit. We have updated this in the revised manuscript.

Comments 13: All sections and of sub-sections should be numbered and well organized. This format is messed up. Please check the template.

Response 13: We agree. We have carefully reviewed the template and have completely reorganized the manuscript. All main sections and sub-sections have been sequentially re-numbered and strictly formatted to comply with the journal's guidelines.

“[5.3. Thermocline Behavior]”

“[5.4. Transient Temperatures and Efficiency]”

“[5.4.1. Effect of Porosity]”

“[5.4.2. Effect of Particle Diameter]”

Comments 14: Provide proper references for all equations.

Response 14: We agree. We have comprehensively reviewed the "Mathematical Modeling" section and explicitly added peer-reviewed references for every equation presented.

Comments 15: What is the application of Eq.(7) in your study?

Response 15: Equation (7) defines the volumetric interphase convective heat transfer coefficient (hi), which serves as the fundamental coupling mechanism between the fluid and solid phases in our Local Thermal Non-Equilibrium (LTNE) model.

We would like to inform you that due to its inappropriateness for our Forchheimer-regime flow, we have replaced the heat transfer correlation (Equation (7)) in our numerical model with the widely validated Wakao and Kaguei spherical particle correlation:

 which is valid for

Comments 16: The figure captions should be self-explanatory.

Response 16: We agree. We have systematically revised all figure captions throughout the manuscript to ensure they are fully self-explanatory. Each caption now explicitly details the physical quantities plotted, the specific operational conditions (e.g., mass flow rate, inlet temperature, porosity), and the definitions of all lines, colors, and variables shown.

Comments 17: Did you investigate the porosity impact? Where is its interpretation?

Response 17: We agree that the physical interpretation of the impact of varying three different porosities (0.35, 0.49 and 0.55), on the temperature’s evolution of fluid and solid (presented in figure 11) and on the charging efficiency (showed in figure 12), was previously underdeveloped. To rectify this, we have significantly expanded the discussion in Section (5.2. Transient Temperatures, Thermocline and Efficiency) at the second and third paragraphs. The revised text now provides a rigorous physical interpretation of the porosity graphs, explicitly detailing how an increase in void fraction simultaneously reduces the effective thermal capacity of the solid matrix and alters the interstitial fluid velocity.

We have mentioned this in the revised draft in section “5.2. Transient Temperatures, Thermocline and Efficiency ":

“[ To illustrate the transient variation in temperatures for different porosities, Figure 11 shows the solid and fluid transient temperatures at the mid-point of the tank throughout the heat charging cycle for varying values of ε (with a fixed other parameters). From Figure 11, it can be seen that varying the porosity has a significant effect on the temperatures, with higher values of ε (0.55) yielding higher temperatures, especially at the intermediate stage of the cycle. This effect seems to be due to decreased the solid fraction. A higher porosity fundamentally decreases the total mass, and consequently the effective thermal capacity, of the solid medium, yielding to faster temperature rise in both phases and enhancing heat transfer efficiency (likely due to enhanced convection effects) and reducing the time required to charge the tank. Also, interesting to note that higher void fraction enhances the permeability, which reduces flow resistance and alters the interstitial fluid velocity, thereby modifying the transient convective heat transfer dynamics. In addition, a comparative analysis of the decoupled temperature profiles showed the significant impact of the porosity on LTNE dynamics. It’s observed that compacted bed (ε = 0.35) resulted in a maximum temperature difference of ≈ 40 °C compared to the loose bed (ε = 0.55) where it decreased to ≈ 30 °C at 10 h of the entire charging phase. This demonstrates the important heat capacity of the solid fraction, which resists rapid heat saturation.]”.

“[To evaluate the effect of changing the porosity on the efficiency, Figure 12 presents the heat charging efficiency for three different porosity values ε = 0.35, ε = 0.49 and ε = 0.55. From the figure, increasing the value of ε results in a discernible increase in final ηch by ≈ 8%]”

Comments 18: Put subscripts in a right position in the axis of diagrams.

Response 18: We have carefully reviewed all graphical results in the manuscript and meticulously updated the axis labels, legends, and titles. All variables now feature properly formatted subscripts (e.g., T_f, T_s, d_p) and superscripts

Comments 19: Result should be more improved from the physical point of view.

Response 19: we have comprehensively rewritten the "5. Results and Parametric Analysis" section. The discussion of each parametric study (mass flow rate, porosity, particle diameter…) has been expanded to explicitly link the graphical data to the governing physical phenomena.

Comments 20: Conclusion section should be enhanced and all optimum outcomes should be presented and compared.

Response 20: we Agree. We have substantially rewritten the Conclusion section to explicitly highlight the optimum design parameters identified in our research. The revised text now systematically presents the quantitative outcomes for the optimal mass flow rate, bed porosity, particle diameter. Furthermore, we have included direct numerical comparisons against the baseline configuration to explicitly demonstrate the improvements in thermal performance and charging efficiency.

Comments 1: What is the clear scientific novelty of this study compared to existing LTNE-based packed-bed TES models?

The modeling approach and numerical tools are well established. The authors should clearly define the research gap and explicitly state what differentiates this work from recent CFD thermocline studies (e.g., scale, optimization method, efficiency definition, boundary conditions). A short comparative positioning within recent literature is required.

Response 1: We agree. To explicitly define the scientific novelty of our work, we have substantially revised the final paragraphs of the Introduction section. We now provide a concise comparative positioning against recent CFD thermocline studies, explicitly highlighting that while most recent literature focuses on lab-scale systems or idealized micro-particles and assume Local Thermal Equilibrium (LTE) or neglect turbulent inertial effects in the porous matrix, our study uniquely quantifies the spatiotemporal LTNE dynamics (with the coupling of non-Darcian flow dynamics (utilizing the Darcy-Forchheimer law)) induced by extreme macroscopic particles (up to dp = 25 cm) and bed porosity (0.35-0.55) under high mass flow rates (8 kg/s and 15 kg/s). Our approach provides a more rigorous physical representation of the heat transfer bottleneck at the fluid-solid interface during dynamic charging phases. This work provides a novel, quantitative techno-economic optimization framework for industrial-scale sensible storage design.

Furthermore, we have deepened our analytical interpretation in the Results and Discussion section. Rather than simply reporting temperature and velocity profiles, we have expanded our analysis to explain how variations in porosity and particle size directly manipulate the viscous momentum transport and ultimately govern the thickness and stability of the thermocline layer.

The modification is highlighted in red for your convenience.

Comments 2: Is the model formulation fully consistent with the numerical implementation?

The manuscript describes axial one-dimensional flow, while the model appears to be 2D axisymmetric. The authors should clarify whether the governing equations are solved in full 2D and revise the description to ensure conceptual consistency between theory and implementation.

Response 2: Agree. To rectify this and ensure complete conceptual consistency, we have comprehensively revised the "3.1. Governing Equations" section. The text has been updated to explicitly state that the computational domain is 2D axisymmetric. Furthermore, all governing equations (continuity, momentum, and LTNE energy balances) have been rewritten using generalized vector notation (∇ ) to correctly represent the multi-dimensional nature of the solved physics.

3.1.1. Continuity Equation:

3.1.2. Momentum Equation:

The pressure drop :

3.1.3. Conservation Energy Equations

Comments 3: Is the neglect of radiative heat transfer at 390°C quantitatively justified?

The justification for neglecting radiation is too general. The authors should provide either an order-of-magnitude estimation comparing radiative and convective heat fluxes or a literature-based justification specific to packed-bed systems in this temperature range.

Response 3: Thermal radiation is safely neglected at 390°C because the HTF (liquid thermal oil) acts as an optically thick medium that effectively suppresses long-wave radiative exchange between the solid particles. Consequently, the calculation of a radiative Biot number is unnecessary, as the high optical opacity of the fluid absorbs emitted radiation over microscopic distances and prevents macroscopic bed-scale radiative transport. It has been established in the current literature that radiative heat transfer can be safely neglected in practical applications below 600 °C, due to the fluid's high optical opacity and the overwhelming dominance of convection.

We have mentioned this in the revised draft in section “3. Mathematical Model" with the corresponding reference:

“[Radiation exchanges within the bed are neglected (operating temperature below 600°C) [34]]”

Comments 4: Is the small-scale validation sufficient to support industrial-scale extrapolation?

The model is validated on a small-scale setup but applied to a 9.17 m industrial system. The authors should discuss scale effects and include relevant dimensionless analysis (e.g., Reynolds, Peclet, Biot numbers) to justify the applicability of the model at larger scale.

Response 4: Agree. The extrapolation is quantitatively justified by the dimensionless nature of the governing macroscopic equations. The LTNE and Darcy-Forchheimer formulations rely on generalized empirical closure relations (e.g., Ergun, Wakao) whose validity is strictly governed by dimensionless numbers (specifically the particle Reynolds number, Re_p, and Prandtl number, Pr), rather than absolute physical dimensions. Because our industrial-scale mass flow rates maintain the flow within the valid turbulent regimes of these correlations, the physics remain fully applicable. Furthermore, moving to a 9 m industrial scale significantly increases the tank-to-particle diameter ratio (D/d_p). This actively suppresses the localized wall-channeling effects that typically introduce errors in small-scale setups, rendering the continuum porous media assumption substantially more robust at the industrial scale.

We have added a concise paragraph to the ‘’5.1. Numerical Model Validation’’ section to explicitly state this dimensionless justification.

“[While the experimental setup consisted of a small TES tank (0.3 m), the extrapolation to our industrial scale (9.17m) is justified because the governing empirical correlations rely strictly on dimensionless flow regimes (Rep, Pe) rather than absolute dimensions, while the larger geometry actively suppresses the boundary wall-channeling errors prevalent in small-scale setups. ]”

Level

Relative Error (%)

Global Average

Error (%)

A

3.92

1.06

3.01

4.07

3.62

C

8.11

1.44

1.08

0.66

E

10.8

2.4

4.41

2.84

G

3.85

0.96

5.47

5.37

I

5

2.45

2.22

3.29